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Interdendritic fluid flow Streat, Norman 1974

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INTERDENDRITIC FLUID FLOW by NORMAN STREAT B.Sc.(Eng.), University of London, 1966  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department of METALLURGY  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December 1973  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  further  for  agree  scholarly  by  his  of  this  thesis at  partial  the U n i v e r s i t y  make  it  freely  that permission  for  It  financial  of  Columbia,  is  J a n u a r y 14,  1974  British for  for extensive  gain  Columbia  shall  the  requirements  reference copying of  I  agree  and this  not  copying or  be a l l o w e d  for  that  study. thesis  t h e Head o f my D e p a r t m e n t  understood that  Metallurgy  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Date  of  available  written permission.  Department o f  fulfilment  p u r p o s e s may be g r a n t e d by  representatives. thesis  in  or  publication  without  my  i  ABSTRACT  F l u i d flow through l i q u i d i n t e r d e n d r i t i c channels of a p a r t i a l l y remelted l e a d - t i n casting has been measured d i r e c t l y , with gravity as the driving force. with Darcy's Law.  The results were shown to be consistent  The permeability of the d e n d r i t i c array was found  to be a function of the square of the primary dendrite spacing, and was observed to increase with time due to coarsening of the d e n d r i t i c structure.  The formation of casting defects i n lead-tin alloys was studied with isothermal and u n i d i r e c t i o n a l s o l i d i f i c a t i o n  experiments.  Solute convection was observed when the l i q u i d close to the bottom of the s o l i d - l i q u i d region was less dense than the l i q u i d above, using radioactive tracer techniques.  Macrosegregation was shown to be related  to the s o l i d i f i c a t i o n conditions, and channel-type defects, resembling freckles and A segregates, were formed when the r i s i n g i n t e r d e n d r i t i c l i q u i d dissolved dendrite branches i n i t s path.  A simple mathematical model i s proposed, which predicts the composition p r o f i l e s i n v e r t i c a l , d i r e c t i o n a l l y s o l i d i f i e d lead-tin castings, as a function of the structure, growth rate, and temperature gradient.  The model i s shown to agree q u a l i t a t i v e l y with the experi-  ments, and can be used to recommend s p e c i f i c changes i n casting practice to reduce gravity segregation e f f e c t s .  ii  ACKNOWLEDGEMENTS  I would l i k e t o e x p r e s s s i n c e r e thanks t o my  research  s u p e r v i s o r , Dr. F. Weinberg, f o r h i s a d v i c e , s u p p o r t and encouragement throughout t h i s work.  Thanks a r e a l s o extended t o o t h e r  f a c u l t y members and f e l l o w g r a d u a t e s t u d e n t s f o r many h e l p f u l discussions.  In addition the assistance of the technical s t a f f  of t h e M e t a l l u r g y Department has been g r e a t l y  appreciated.  F i n a n c i a l a i d from t h e K i l l a m F o u n d a t i on i n t h e form of a P r e d o c t o r a l F e l l o w s h i p , and from t h e N a t i o n a l Research C o u n c i l of Canada (Grant Number A-4609) i s g r a t e f u l l y acknowledged. Thanks a r e a l s o due t o t h e programming s t a f f o f t h e U.B.C. Computing C e n t r e , i n p a r t i c u l a r t o M r s . J a n e t S t r e a t f o r her i n v a l u a b l e  assistance.  iii  TABLE OF CONTENTS Page CHAPTER 1  Introduction  1  1.1  I n t e r d e n d r i t i c F l u i d Flow i n C a s t i n g s  .  .  .  1  1.2  Purpose  .  .  .  4  1.3  O r g a n i z a t i o n o f the T h e s i s  CHAPTER 2  of the Present I n v e s t i g a t i o n .  .  .  G e n e r a l E x p e r i m e n t a l Apparatus  .  7  2.3  Metallography  2.4  Measurement o f D e n d r i t e S p a c i n g  .  .  .  . 1 1  2.4.1  Primary dendrite spacings  .  .  .  . 1 1  2.4.2  Secondary  .  . .  .  .  .  .  .  .  .  .  .  .  3.1  Review o f P r e v i o u s Work  .  3.2  G e n e r a l D e s c r i p t i o n o f t h e Technique P r e s e n t Work  .  .  .  .  .  .  .  .  .  .  3.5  Flow Measurement Equipment  3.6  Flow T e s t i n g P r o c e d u r e  3.7  P r e c i s i o n o f t h e Flow Measurement Technique  .  . 1 7 .  17  .  .  . .  . .  .  22 .  .  4.1.1  Laminar f l o w  .  30  .  .  .  31  .  .  .  34  4.1.2  Interaction effects  .  .  .  .  .  27  .  .  R e s u l t s and D i s c u s s i o n o f Flow Measurements  I n t e r p r e t a t i o n U s i n g Darcy's Law  .  .  35 38  .  38  .  .  .  40  .  .  .  42  A p p l i c a t i o n t o t h e Flow C e l l Experiments 4.2.1  . 1 6 Flow  .  P r e p a r a t i o n o f C a s t i n g s B and C  .  16  Used i n t h e  3.4  .  9  .  .  P r e p a r a t i o n o f t h e A l l o y under T e s t (A)  4.2  .  .  .  9  .  .  3.3  4.1  .  The Measurement o f I n t e r d e n d r i t i c F l u i d Rates  CHAPTER 4  . .  .  dendrite spacings  Autoradiography  CHAPTER 3  .  7  P r e p a r a t i o n of Lead-Tin A l l o y s .  .  and P r o c e d u r e s  2.2  .  .  5  Apparatus  .  .  .  2.1  2.5  .  .  .  .  43  The method f o r f i n d i n g t h e i n i t i a l permeability  .  .  .  .  .  .  .  45  iv  TABLE OF CONTENTS  (Continued) P a  4.2.2 4.3  Results  4.5  .  .  .  .  .  D e n d r i t e Spacings and S t r u c t u r e 4.3.1  4.4  .  Autoradiography  .  .  Microexamination  .  .  .  .  .  . .  . .  .  52  .  .  .  54  .  .  .  59  N e g a t i v e d e v i a t i o n s from Darcy's Law  .  .  60  4.4.2  P o s i t i v e d e v i a t i o n s from Darcy's Law  .  .  65  P e r m e a b i l i t y and D e n d r i t e S p a c i n g  .  4.5.1  S t r a i g h t C a p i l l a r y Model  .  4.5.2  H y d r a u l i c Radius Theory:  4.7  The S c a t t e r o f P e r m e a b i l i t y R e s u l t s  CHAPTER 5  .  .  .  .  .  . 6 8  .  .  Other T h e o r i e s  Dendrite Coarsening  .  .  .  71 .  .  .  .  . .  75 79  .  86  .  90  The E f f e c t o f D e n s i t y D i f f e r e n c e s on t h e F o r m a t i o n o f Channels  .  .  5.1  I n t r o d u c t i o n and Review o f P r e v i o u s Work  5.2  Experimental Procedure  5.3  Results  .  .  5.4  Discussion  .  .  CHAPTER 6  . . .  .  . .  .  .  .  .  .  .  . .  .  .  .  .  .  .  .  90  .  98  .  102  .  .110  S o l u t e C o n v e c t i o n and F r e c k l e F o r m a t i o n During S o l i d i f i c a t i o n  6.1  Introduction  .  6.2  Experimental Procedure  .  .  .  .  .  .  .115  .  .  .  .  .  .115  .  .  .  .  .  .115  .  .  .  .  .  .115  6.2.1  Apparatus  6.2.2  Macrosegregation studies  6.2.3  D e t e r m i n a t i o n o f c o m p o s i t i o n from measurements  6.2.4 Results  .  Solute convection .  .  . .  .  .  6.3.1  Composition p r o f i l e s  6.3.2  C o n v e c t i o n i n the l i q u i d  .  . .  .  .  .  117  activity  .  .  .  .  120  .  .  .  .  126  .  .  .  .  .127  .  .  .  .  .127  .  e  49  4.4.1  4.6  6.3  .  g  .  .  .  136  V  TABLE OF CONTENTS  (Continued) Page  6.3.3 6.4  Freckles  139  Discussion of Results  CHAPTER 7  .  .  .  .  .  .  139  A N u m e r i c a l Model f o r M a c r o s e g r e g a t i o n i n Pb-Sn A l l o y s  .  144  7.1  I n t r o d u c t i o n and Review o f P r e v i o u s Work  .  .  .  7.2  Model o f t h e S o l i d i f i c a t i o n P r o c e s s  .  .  .  .  147  7.3  I n t e r d e n d r i t i c F l u i d Flow Model  .  .  .  .  150  7.4  U n i d i r e c t i o n a l S o l i d i f i c a t i o n of a V e r t i c a l  7.5  Results of Calculations f o r S o l i d i f i c a t i o n of a  Casting  .  Pb-Sn A l l o y 7.6  Conclusions .  .  8.1  Summary  8.2  Conclusions  8.3  S u g g e s t i o n s f o r F u t u r e Work  . .  . .  .  . .  .  .  .161  .  .  .  164  .  .  .  .  .  .  .  164  .  .  .  .  .  .  .  165  .  166  .  .  .  .  REFERENCES  APPENDIX I  APPENDIX I I  APPENDIX I I I  168  I n t e g r a t i o n o f Darcy's Law f o r a F a l l i n g Head  The S o l i d i f i c a t i o n o f Pb-20%Sn  175  .  .  179  .  .  181  FORTRAN Program f o r C a l c u l a t i n g Macrosegregation i n Lead-Tin Castings  APPENDIX V  172  FORTRAN Program f o r P r o c e s s i n g F l u i d Flow Data  A Table of S o l i d i f i c a t i o n V a r i a b l e s APPENDIX IV  151 155  Comparison w i t h Experiment  CHAPTER 8  144  .  D i r e c t Observation of S o l i d i f i c a t i o n E l e c t r o n Microscopy  .  .  Using .  .  .188  vi  TABLE OF CONTENTS  (Continued) Page  V.l  Introduction  V.2  E x p e r i m e n t a l Method  V.3  Results  .•  .  .  .  .  .  .  .  .  .  .  V.3.1  Pure b i s m u t h  V.3.2  Other pure m e t a l s ( t i n , aluminum and indium)  V.3.3  .  188  .188 189  .  Lamellar eutectics  .  . .  . .  . .  . .  .  .  194 194  vii  LIST OF ILLUSTRATIONS Figure Number  Page  (a) Tube f u r n a c e and quenching apparatus f o r p r o d u c i n g columnar c a s t i n g s . (b) Tube f u r n a c e f o r ± 0.5°C temperature c o n t r o l . . . . . . Cross s e c t i o n o f a group o f p r i m a r y d e n d r i t e s (schematic) . . . . . . (a) L o n g i t u d i n a l s e c t i o n o f a d i r e c t i o n a l l y casting. (b) Corresponding c r o s s s e c t i o n  .  solidified  4  E n l a r g e d views o f r e g i o n s A' and B' i n F i g u r e 3(b)  5  Three d i m e n s i o n a l composite, from w h i c h one can e s t i m a t e t h a t the d e n d r i t e s i n the top c o r n e r a r e t i l t e d a p p r o x i m a t e l y 20° . . . . .  6  Schematic v i e w o f t h e s t r u c t u r e i n F i g u r e 4(b)  7  P e r m e a b i l i t y as a f u n c t i o n o f the square o f t h e volume f r a c t i o n l i q u i d , u s i n g e x p e r i m e n t a l d a t a o b t a i n e d by Piwonka'^' . . . .  .  Schematic diagram showing t h e p r i n c i p l e o f a F a l l i n g Head Permeameter . . . .  .  S e c t i o n a l v i e w s o f t h e f l o w c e l l and t h e l e a d - t i n alloy inserts . . . . . . . 10  Three p i e c e s o f Pb-Sn a l l o y used f o r f l o w measurement  11  P a r t i a l l y assembled f l o w c e l l  12  Pb-Sn a l l o y b e f o r e and a f t e r f l o w t e s t  13  Flow measurement apparatus  14  C i r c u i t used f o r r e c o r d i n g the p o s i t i o n o f t h e probe on t h e temperature t r a c e  15  Apparatus f o r t e s t i n g t h e p r e c i s i o n o f t h e f l o w measurement t e c h n i q u e . . . . .  16  .  .  .  .  .  .  .  .  .  .  (a) Flow measurement r e s u l t s ; d i s t a n c e o f f l o w up the r i s e r p i p e v e r s u s time f o r X = 116 ym. (b) S i m i l a r p l o t f o r A = 28 pin . . . . . .  viii LIST OF ILLUSTRATIONS  (Continued)  Figure Number  17  18  p g a  (a) D a t a from F i g u r e 1 6 ( a ) , r e p l o t t e d a c c o r d i n g t o Darcy's Law, showing a p o s i t i v e d e v i a t i o n . (b) S i m i l a r p l o t f o r d a t a from F i g u r e 1 6 ( b ) , showing a negative deviation . . . . . . .  44  P r i m a r y d e n d r i t e s p a c i n g as a f u n c t i o n of d i s t a n c e from the c h i l l , f o r the quenching c o n d i t i o n s i n Table I I  53  19  A u t o r a d i o g r a p h s from c r o s s s e c t i o n s and l o n g i t u d i n a l s e c t i o n s o f Pb-Sn samples used f o r i n t e r d e n d r i t i c f l u i d f l o w s t u d i e s ; (a) and (b) show u n i f o r m f l o w , (c) shows f l o w down a p r e f e r e n t i a l c h a n n e l . . . .  20  Cross s e c t i o n a u t o r a d i o g r a p h s the c a s t i n g A, a f t e r t e s t i n g  21  An example o f an u n r e l i a b l e f l o w t e s t , showing uneven p e n e t r a t i o n of t r a c e r . . .  at v a r i o u s l e v e l s down . . . . .  22  M i c r o s t r u c t u r e s o f c a s t i n g A b e f o r e and a f t e r f l o w t e s t i n g X = 175 urn . .  23  M i c r o s t r u c t u r e s o f c a s t i n g A b e f o r e and a f t e r t e s t i n g I = 71 ym .  flow  24  M i c r o s t r u c t u r e s o f c a s t i n g A b e f o r e and a f t e r t e s t i n g X = 51 ym  flow  25  M i c r o s t r u c t u r e s o f c a s t i n g A b e f o r e and a f t e r f l o w t e s t i n g X = 28 ym  26  M i c r o s t r u c t u r e s o f the r e s e r v o i r ( c a s t i n g B) testing . . . . . . .  27  R e l a t i o n s h i p between the i n i t i a l p e r m e a b i l i t y and the secondary d e n d r i t e arm s p a c i n g f o r Pb-Sn at 193  28  after . .  R e l a t i o n s h i p between the i n i t i a l p e r m e a b i l i t y and the p r i m a r y d e n d r i t e s p a c i n g f o r Pb-20%Sn a t 193°C  29  3 2 P e r m e a b i l i t y as a f u n c t i o n of g^ /(1-g^) u s i n g o b t a i n e d by P i w o n k a ' ^ . . . . .  30  Growth o f a neck d u r i n g  sintering  data .  C  e  ix  LIST OF ILLUSTRATIONS  (Continued)  Figure Number 31  'age D e n d r i t e c o a r s e n i n g p l o t f o r a sample w i t h K = 0.152 cm , average p r i m a r y d e n d r i t e s p a c i n g 28 urn  85  3 C a l i b r a t i o n c u r v e ; d e n s i t y o f Pb-Sn a l l o y s (g/cm ) a t 25°C as a f u n c t i o n o f c o m p o s i t i o n  87  33  The r e l a t i o n s h i p between p e r m e a b i l i t y and temperature  87  34  F r e c k l e t r a i l s i n d i r e c t i o n a l l y s o l i d i f i e d Mar-M200  91  35  F r e c k l e s i n a s - c a s t I n c o n e l 718  91  36  The  37  (a) M a c r o s t r u c t u r e o f columnar c a s t i n g ( s e r i e s I ) (b) C o r r e s p o n d i n g a u t o r a d i o g r a p h  103  (a) M a c r o s t r u c t u r e o f columnar c a s t i n g ( s e r i e s I I (b) C o r r e s p o n d i n g a u t o r a d i o g r a p h  103  Surfaces revealed a f t e r transverse s e c t i o n i n g of a sample from s e r i e s I I I . . . .  106  (a) L o n g i t u d i n a l s e c t i o n o f same sample as i n F i g u r e 39. (b) C o r r e s p o n d i n g a u t o r a d i o g r a p h  106  Surfaces revealed a f t e r transverse s e c t i o n i n g of a sample from s e r i e s I I I . . . .  107  (a) L o n g i t u d i n a l s e c t i o n o f same sample as i n F i g u r e 41. (b) C o r r e s p o n d i n g a u t o r a d i o g r a p h  107  (a) M a c r o s t r u c t u r e o f columnar c a s t i n g ( s e r i e s I V ) (b) Corresponding a u t o r a d i o g r a p h  109  44  M a c r o s t r u c t u r e from s e r i e s V  109  45  S p l i t g r a p h i t e mould f o r making l o n g c y l i n d r i c a l ingots . . . . . . .  116  113 Spectrum o f y e m i s s i o n f o r Sn  119  q  32  38 39 40  41 42  43  46  t e s t assembly f o r i s o t h e r m a l experiments  .  .  204  47 48  .  Spectrum o f y e m i s s i o n f o r T l C a l i b r a t i o n curve;  100  119 .  .  .  a c t i v i t y v e r s u s sample weight  122  X  LIST OF ILLUSTRATIONS  (Continued)  Figure Number  Page  49  Calibration curve;  a c t i v i t y versus S n  50  Calibration curve; s p e c i f i c a c t i v i t y versus composition for constant S n concentration  1 1 3  concentration  1 1 3  51  alloy . .  122 .  123  Calibration curve; s p e c i f i c a c t i v i t y versus a l l o y composition, when S n concentration i s proportional to the solute content . . . . . . .  123  Composition p r o f i l e for one ingot using lathe turning treated with n i t r i c acid (open c i r c l e s ) , and untreated samples (closed c i r c l e s ) . . . . . .  125  1 1 3  52  53  (a) Solute d i s t r i b u t i o n ,  (b) Cooling conditions  .  128  54  (a) Solute d i s t r i b u t i o n ,  (b) Cooling conditions  .  129  55  (a) Solute d i s t r i b u t i o n .  (b) Cooling conditions  .  130  56  (a) Solute d i s t r i b u t i o n .  (b) Cooling conditions  .  131  57  (a) Solute d i s t r i b u t i o n .  (b) Cooling conditions  .  132  58  Autoradiographs showing the extent of tracer movement one hour a f t e r tracer was added; (a) d i r e c t i o n a l l y s o l i d i f i e d , (b) quenched from the l i q u i d . .  59  60  61 62  63  Shrinkage t r a i l , approximately outside of an ingot s o l i d i f i e d i n Table XI. (b) Longitudinal showing a f r e c k l e t r a i l on the  7 cm long, along the under conditions given and transverse sections right hand side . .  .137  140  (a) Transverse section of the freckle t r a i l i n Figure 59(b). (b) Longitudinal section showing that the t r a i l originates from widening i n t e r d e n d r i t i c channels i n the i n t e r i o r of the ingot . . . . . .  141  Schematic representation of u n i d i r e c t i o n a l s o l i d i f i c a t i o n assumed i n the model .  .  148  Equilibrium diagram for a binary a l l o y . The nonequilibrium solidus i s shown by the dashed l i n e .  .148  .  .  D i r e c t i o n a l l y s o l i d i f y i n g ingot divided into layers. Temperature, composition and density p r o f i l e s given by the s o l i d i f i c a t i o n model . . . . . .  153  xi  LIST OF ILLUSTRATIONS  (Continued)  Figure Number  64  65  Page (a) Assumed flow pattern showing two main flow cells. (b) Resistances Ri_5» and flow rates q^_5 for flow between s i x layers . . . . Solute d i s t r i b u t i o n as a function of the number of layers  66  .153  .  .  .  .  .  .  .  .  Solute d i s t r i b u t i o n as a function of structure (effective number of channels) . . . .  .  .157  .  157  67  Solute d i s t r i b u t i o n as a function of ingot height  .  159  68  Solute d i s t r i b u t i o n as a function of growth rate .  .  159  69  Solute d i s t r i b u t i o n as a function of temperature gradient . . . . . . . . .  .  70  Pb-Sn alloy i n the flow c e l l , a f t e r a time t  71  (a-c) Alternate freezing, melting and freezing i n pure bismuth, showing evidence of faceted growth. (d) Enlarged view of the s o l i d - l i q u i d interface showing high angle grain boundaries emerging at the interface . . . . . . . . .  191  Alternate melting, freezing, melting and freezing in pwre bismuth . . . . . . . .  193  72 73  The melting of pure aluminum, photographed from the fluorescent screen using a 35 mm camera  .  160  .  .172  .195  xii  LIST OF TABLES Table I  Page Dimensions and Composition of Castings Used for Interdendritic F l u i d Flow Studies . . . .  .  .  .  Quench Data  III  Thermal Conditions for Pb-20%Sn Columnar Castings  .  29  IV  Precision of the Flow Measurement Technique  .  36  V  Results of Flow Measurements  VI  Results of Dendrite Coarsening  VII  Composition  VIII  Test Conditions f o r Isothermal Experiments  .  .  101  IX  S o l u b i l i t y Data for Isothermal Experiments  .  .  112  X  S o l i d i f i c a t i o n Variables and Macrosegregation  .  .  133  XI  Cooling Conditions  .  XII  S o l i d i f i c a t i o n Variables Used for Theoretical Plots  .  .  .  .  .  .  .  .  .  29  . .  .  .  .  .  Calculations .  .  .  26  II  of Superalloys  .  .  50 .  .  82 92  .138  .  162  1  CHAPTER 1 INTRODUCTION 1.1  I n t e r d e n d r i t i c F l u i d Flow i n C a s t i n g s  N e a r l y a l l m e t a l p r o d u c t s are made by c a s t i n g and f a b r i c a t i o n of the c a s t m a t e r i a l .  subsequent  S i n c e the v a s t m a j o r i t y of m e t a l p r o d u c t s  are a l l o y s o f two o r more c o n s t i t u e n t s , as the a l l o y s o l i d i f i e s the compos i t i o n of the s o l i d must be d i f f e r e n t to the c o m p o s i t i o n of the a d j a c e n t l i q u i d from which i t grows (except f o r the s p e c i a l case of solidification).  congruent  Assuming the s o l i d i s of a s i n g l e phase and s o l u t e i s  c o n s e r v e d , the c o m p o s i t i o n o f b o t h s o l i d and l i q u i d must v a r y d u r i n g solidification.  I n the s o l i d s t a t e , c o m p o s i t i o n changes can o n l y occur by means o f s o l i d d i f f u s i o n , which i s r e l a t i v e l y slow even a t h i g h  temperatures.  A c c o r d i n g l y , f o r a l l p r a c t i c a l purposes, i t i s i m p o s s i b l e to c a s t homogeneous alloys.  I n g e n e r a l , a l l o y c a s t i n g s have c o m p o s i t i o n d i f f e r e n c e s on a m i c r o s c o p i c s c a l e , w h i c h can a f f e c t the m e c h a n i c a l , c o r r o s i o n and s u r f a c e p r o p e r t i e s , depending stituents.  on the e x t e n t and d i s t r i b u t i o n o f the s e g r e g a t e d  con-  The e f f e c t s o f c o m p o s i t i o n v a r i a t i o n s on a m i c r o s c o p i c s c a l e  may  n o t n e c e s s a r i l y be d e t r i m e n t a l , p a r t i c u l a r l y i f the c a s t i n g i s to be p r o c e s s e d further.  However, when m i c r o s c o p i c v a r i a t i o n s are c o n c e n t r a t e d i n l o c a l  r e g i o n s , they can s e r i o u s l y reduce the s t r e n g t h and d u c t i l i t y of the c a s t i n g . Large s c a l e c o m p o s i t i o n v a r i a t i o n s are termed " m a c r o s e g r e g a t i o n " and a number o f d i f f e r e n t types are r e c o g n i z e d .  These i n c l u d e c e n t r e l i n e s e g r e g a t i o n , A and  2  V segregates i n l a r g e s t e e l i n g o t s , inverse segregation,  f r e c k l e s and s o l u t e  banding: 1)  Centreline segregation  i s a l i n e of s o l u t e r i c h m a t e r i a l along the a x i s  of an i n g o t w h i c h has c o o l e d from t h e s i d e w a l l s . 2)  A s e g r e g a t e s have been d e s c r i b e d  as r o p e - l i k e c o n c e n t r a t i o n s  of solute  w h i c h form i n the upper r e g i o n s o f t h e columnar zone o f l a r g e s t e e l ingots.  They a r e c a l l e d A s e g r e g a t e s because they a r e i n c l i n e d a few  degrees from t h e v e r t i c a l on each s i d e o f t h e a x i s , g i v i n g t h e appearance of an A o r Greek A when seen on a s u r f a c e s e c t i o n e d p a r a l l e l t o the a x i s . 3)  S i m i l a r l y , V s e g r e g a t e s a r e cones o f h i g h s o l u t e content w h i c h form i n the lower equiaxed r e g i o n s on the s e c t i o n e d  4)  Inverse  o f s t e e l c a s t i n g s , and a r e V-shaped when seen  surface.  segregation  i s a r e g i o n o f h i g h s o l u t e content c l o s e t o t h e c h i l l  f a c e o f a c a s t i n g w h i c h can, i n some c a s e s , be observed as e x u d a t i o n s o r beads o f s o l u t e r i c h m a t e r i a l on the s u r f a c e . 5)  F r e c k l e s a r e patches o r v e r t i c a l l i n e s o f s o l u t e r i c h m a t e r i a l w h i c h o c c u r i n a number o f d i f f e r e n t types o f c a s t i n g s , i n p a r t i c u l a r , consumable arc melted ingots.  They a r e so named because o f t h e i r s p o t t y o r s p e c k l e d  appearance when seen on t h e o u t e r s u r f a c e , o r on p o l i s h e d s e c t i o n s o f the 6)  ingot.  S o l u t e banding i s the term a p p l i e d when a l t e r n a t i n g r e g i o n s o f s o l u t e r i c h and s o l u t e d e p l e t e d m a t e r i a l a r e o b s e r v e d t o o c c u r i n t h e columnar r e g i o n s o f an i n g o t .  I t i s d e s i r a b l e t o m i n i m i z e these d e f e c t s i n c a s t i n g s t h a t a r e used i n c r i t i c a l a p p l i c a t i o n s , p a r t i c u l a r l y when t h e c a s t i n g i s used w i t h o u t subsequent w o r k i n g .  F o r example, d i r e c t i o n a l l y s o l i d i f i e d c a s t i n g s o f n i c k e l - b a s e  3  superalloys  (designed  f o r h i g h t e m p e r a t u r e s e r v i c e ) a r e used f o r t u r b i n e  blades i n j e t a i r c r a f t engines. formation  These c a s t i n g s a r e s u s c e p t i b l e t o t h e  o f f r e c k l e s , w h i c h can s e r i o u s l y a f f e c t the s t r e n g t h and creep  p r o p e r t i e s when the b l a d e s a r e i n s e r v i c e . produced i n the U n i t e d  The c o s t o f s u p e r a l l o y  castings  S t a t e s , w h i c h a r e used i n many c r i t i c a l a p p l i c a t i o n s  of t h i s t y p e , and a r e p o t e n t i a l l y s u s c e p t i b l e t o f r e c k l i n g , i s p r e s e n t l y about $500 m i l l i o n a n n u a l l y .  Considering  r a t e o f 20%, and t h e cost o f r e m e l t i n g  that there i s a p o s s i b l e r e j e c t i o n  the r e j e c t e d c a s t i n g s i s about 40% o f  the c o s t o f the p r o d u c t , the p o t e n t i a l cost o f t h i s one type o f c a s t i n g d e f e c t i s about $40 m i l l i o n p e r y e a r .  In general, a l l o y s s o l i d i f y w i t h a d e n d r i t i c s t r u c t u r e , that i s , the s o l i d grows i n the form o f c l u s t e r s o f t r e e - l i k e s p i k e s w i t h s i d e b r a n c h e s . The  d e n d r i t i c s t r u c t u r e i n a l l o y c a s t i n g s can n o r m a l l y  etching a polished surface. r i c h or s o l u t e d e p l e t e d  The e t c h a n t i s s e l e c t e d t o r e a c t w i t h  solute  regions producing p r e f e r e n t i a l attack of the i n t e r -  d e n d r i t i c regions, or dendrite  The  be examined by s u i t a b l y  centres.  e x t e n t and d i s t r i b u t i o n o f t h e s e g r e g a t e d s o l u t e i n t h e c a s t i n g  i s r e l a t e d to the d e n d r i t i c s t r u c t u r e , w h i c h i n t u r n i s a f u n c t i o n o f t h e a l l o y c o m p o s i t i o n and c a s t i n g c o n d i t i o n s .  These would i n c l u d e t h e c r y s t a l l o -  g r a p h i c p r o p e r t i e s o f t h e a l l o y c o n s t i t u e n t s , t h e t h e r m a l environment, and i n p a r t i c u l a r , l i q u i d t r a n s p o r t , e i t h e r by f o r c e d o r n a t u r a l c o n v e c t i o n .  The  d r i v i n g f o r c e s f o r l i q u i d t r a n s p o r t i n a c a s t i n g w i l l be r e l a t e d t o the temperature d i f f e r e n c e s w h i c h cause n a t u r a l c o n v e c t i o n , w h i c h can l e a d t o s o l u t e c o n v e c t i o n , and gas e v o l u t i o n .  composition differences  and o t h e r f a c t o r s such as volume  shrinkage  4  The s p a c i n g between s i d e branches o f d e n d r i t e s i n t h e c e n t r e o f l a r g e , s l o w l y c o o l e d c a s t i n g s can be o f the o r d e r o f m i l l i m e t r e s .  For s m a l l  c a s t i n g s which c o o l r a p i d l y the s p a c i n g can be o f the o r d e r o f t e n m i c r o n s , c o n s e q u e n t l y , l i q u i d t r a n s p o r t through the growing d e n d r i t i c network w i l l be r e s t r i c t e d by the narrow, t o r t u o u s channels through w h i c h the l i q u i d must move.  S i n c e m a c r o s e g r e g a t i o n i s s t r o n g l y i n f l u e n c e d by t h e e x t e n t o f f l u i d  f l o w d u r i n g s o l i d i f i c a t i o n , an u n d e r s t a n d i n g of the f o r c e s a c t i n g on the l i q u i d , and the e x t e n t o f i n t e r d e n d r i t i c f l u i d f l o w r e s t r i c t i o n s i s e s s e n t i a l to account f o r , and m o d i f y , c e r t a i n types o f m a c r o s e g r e g a t i o n i n c a s t i n g s . 1.2  Purpose of t h e P r e s e n t I n v e s t i g a t i o n  The purpose o f the p r e s e n t work was  t o measure i n t e r d e n d r i t i c f l u i d  f l o w i n a m e t a l l i c system under a known d r i v i n g f o r c e , and determine  the  r e l a t i o n s h i p between the r e s i s t a n c e t o f l o w and the s t r u c t u r e o f the c a s t i n g . W i t h s u i t a b l e measurements, t h e r e s u l t s would be c o n s i d e r e d i n terms o f e s t a b l i s h e d e m p i r i c a l r e l a t i o n s h i p s f o r f l o w through porous media. most m e t a l s have s i m i l a r t h e r m a l and v i s c o u s c h a r a c t e r i s t i c s , and  Since solidify  i n a s i m i l a r manner, as compared t o non m e t a l s , i t i s c o n s i d e r e d t h a t a d e t a i l e d e x a m i n a t i o n o f one m e t a l system can g i v e r e s u l t s a p p l i c a b l e t o most o t h e r systems.  In c o n j u n c t i o n w i t h the f l u i d f l o w c o n s i d e r a t i o n s , o t h e r a s p e c t s o f m a c r o s e g r e g a t i o n would be examined, s p e c i f i c a l l y , d e f e c t s w h i c h resemble f r e c k l e s and A s e g r e g a t e s .  t h e f o r m a t i o n of c h a n n e l - t y p e To combine the r e s u l t s of  b o t h the f l u i d f l o w and t h e m a c r o s e g r e g a t i o n s t u d i e s , a s i m p l e m a t h e m a t i c a l model has been d e r i v e d .  The model c o n s i d e r s the s o l i d i f i c a t i o n  o f an i n g o t  where the d r i v i n g f o r c e f o r m a c r o s e g r e g a t i o n i s d e n s i t y d i f f e r e n c e s i n the  5  l i q u i d , and t h e i n t e r d e n d r i t i c f l u i d f l o w i s a f u n c t i o n o f t h e c a s t s t r u c t u r e as e s t a b l i s h e d e x p e r i m e n t a l l y .  The model i s compared w i t h t h e  experimental r e s u l t s . 1.3  Organizat ion of the Thesis  The  t h e s i s i s d i v i d e d i n t o f o u r main s e c t i o n s .  The f i r s t  section  (Chapter 2) g i v e s a d e s c r i p t i o n o f t h e apparatus and p r o c e d u r e s common t o a l l the f o l l o w i n g s e c t i o n s .  The second s e c t i o n (Chapters 3 and 4) d e a l s w i t h the  development o f t h e i n t e r d e n d r i t i c f l o w measurement t e c h n i q u e , and t h e i n t e r p r e t a t i o n o f t h e r e s u l t s i n terms o f t h e t h e o r y o f f l o w t h r o u g h porous media.  The  t h i r d s e c t i o n c o n s i s t s o f t h e e x p e r i m e n t s on t h e e f f e c t o f  d e n s i t y d i f f e r e n c e s i n t h e l i q u i d on a c a s t i n g h e l d a t u n i f o r m t e m p e r a t u r e i n the s o l i d - l i q u i d r e g i o n defect formation The  (Chapter 5) and t h e s t u d y o f m a c r o s e g r e g a t i o n and  i n i n g o t s s o l i d i f i e d under known c o o l i n g c o n d i t i o n s  (Chapter 6 ) .  e x p e r i m e n t a l work on m a c r o s e g r e g a t i o n i s t i e d t o g e t h e r w i t h t h e r e s u l t s o f  i n t e r d e n d r i t i c f l u i d f l o w measurements i n t h e m a t h e m a t i c a l model, p r e s e n t e d i n the f o u r t h s e c t i o n (Chapter 7 ) .  A r e v i e w o f p r e v i o u s work r e l e v a n t t o t h e p a r t i c u l a r s e c t i o n i s p r e s e n t e d a t t h e b e g i n n i n g o f Chapters 3, 5 and 7.  D u r i n g t h e course o f t h e s e e x p e r i m e n t s an attempt was made t o d i r e c t l y observe s o l i d i f i c a t i o n i n a t h i n f i l m o f m e t a l u s i n g an e l e c t r o n m i c r o s c o p e . The  aim o f t h i s work was t o study s o l i d - l i q u i d i n t e r f a c i a l e n e r g i e s ,  which  would have been r e l e v a n t t o t h e work on i n t e r d e n d r i t i c f l u i d f l o w , i n r e l a t i o n t o t h e i n t e r p r e t a t i o n o f changes w h i c h take p l a c e i n a c a s t i n g h e l d i n t h e  s o l i d - l i q u i d region for long periods of time. microscope study were inconclusive. i n Appendix V.  The results of the elect  A b r i e f summary of the work i s giv<  7  CHAPTER 2  GENERAL EXPERIMENTAL APPARATUS AND PROCEDURES 2.1  Apparatus  Two tube f u r n a c e s were c o n s t r u c t e d and used t o produce c a s t i n g s and t o heat t h e a l l o y samples f o r f l u i d f l o w s t u d i e s .  columnar Columnar  c a s t i n g s were produced i n the f u r n a c e shown i n F i g u r e 1(a) w h i c h had a c e n t r a l copper tube t o w i t h s t a n d t h e r m a l shock when w a t e r quenches were used the  furnace.  inside  The copper tube was wrapped w i t h a s b e s t o s t a p e , and chromel  w i n d i n g s were wrapped o v e r t h e tape t o p r e v e n t s h o r t c i r c u i t i n g on t h e m e t a l tube.  A c o n t r o l thermocouple (Chromel/Alumel) was p l a c e d next t o t h e w i n d i n g s .  Temperatures measured i n a m o l t e n m e t a l charge i n t h i s f u r n a c e c o u l d be h e l d c o n s t a n t t o ± 1°C.  F o r b e t t e r c o n t r o l a second f u r n a c e was b u i l t o f s i m i l a r  d e s i g n ( F i g u r e 1 ( b ) ) w i t h a c e n t r a l c e r a m i c tube.  I n t h i s case the w i n d i n g s  were d i r e c t l y i n c o n t a c t w i t h t h e ceramic tube, w h i c h was a b e t t e r c o n d u c t o r than t h e a s b e s t o s t a p e , and ± 0.5°C c o n t r o l was p o s s i b l e .  T h i s f u r n a c e was  used f o r f l u i d f l o w s t u d i e s where i t was n e c e s s a r y t o heat t h e samples r a p i d l y t o a p r e d e t e r m i n e d temperature w i t h o u t o v e r s h o o t i n g , and then h o l d them c o n s t a n t .  U n l e s s o t h e r w i s e s t a t e d , a l l temperature measurements i n t h i s work were made u s i n g i r o n - c o n s t a n t a n thermocouples c a l i b r a t e d a g a i n s t t h e m e l t i n g p o i n t o f pure t i n .  Bare thermocouple j u n c t i o n s were used f o r r a p i d r e s p o n s e .  The thermocouple w i r e s were i n s e r t e d i n s m a l l d i a m e t e r c e r a m i c t u b i n g (approxi m a t e l y 1.6 mm) t o ensure t h a t when t h e thermocouple was immersed i n m o l t e n m e t a l t h e r e a d i n g was r e p r e s e n t a t i v e o f c o n d i t i o n s a t t h e t i p .  T h i s was  *— Quench Medium  (a)  (b)  (a)  Tube furnace and quenching apparatus f o r producing columnar  (b)  Tube furnace for ± 0.5°C temperature control.  9  confirmed by breaking apart the ceramic tubes after the test, where i t could be seen that molten metal did not r i s e up the bore.  In a l l tests the  thermocouples were connected v i a a cold junction i n i c e water to a Honeywell Electronik 194 m i l l i v o l t recorder. 2.2  Preparation of Lead-Tin Alloys  A l l lead-tin alloys were prepared using high purity Cominco Pb (99.999%) and high purity Vulcan Sn (99.999%).  The required compositions  were f i r s t melted i n lots of approximately 1500 g i n a s t a i n l e s s s t e e l beaker over a bunsen burner, s t i r r e d thoroughly, and cast into graphite moulds to produce starting ingots about 2.2 cm diameter and 5 cm long.  Both lead and t i n have a low vapour pressure and no composition changes were expected due to evaporation.  This was confirmed by p l o t t i n g  the cooling curves of two samples of Pb-20% Sn. the melting point, i n a i r , respectively.  The samples were held above  for approximately 15 minutes and 17% hours  There was no s i g n i f i c a n t difference between the liquidus arrest  points seen on the cooling curves.  In  those cases where radioactive T l  lead-tin a l l o y s , the tracer was  or Sn  was added to the  f i r s t dissolved i n 100 g of pure Sn i n a pyrex  test tube with an argon flow to prevent oxidation.  This 'master' a l l o y was  then used i n the preparation of the required lead-tin a l l o y . 2.3  Metallography  Since Pb-Sn alloys are extremely s o f t , s p e c i a l techniques for metallographic preparation were used.  Samples were cut with a coarse toothed  10  hacksaw to prevent clogging, and care was  taken to prevent  overheating.  Normally the cut samples were mounted i n 'Quickmount', a cold setting p l a s t i c that did not heat up more than about 30°C during curing.  In addition,  though 'Quickmount' i s harder than the Pb-Sn a l l o y s , i t i s s o f t e r than other mounting materials and can r e a d i l y be polished. The mounted specimens were machined on a lathe to give a f l a t surface using a sharp angled tool which cut the surface but did not tear. Properly machined specimens could be taken d i r e c t l y to a 5 micron alumina lapping wheel where s u f f i c i e n t material was marks and any flowed layer.  polished away to remove machine  Grinding papers were not used since i t was  found that they would immediately clog and tear at the surface, and p a r t i c l e s of abrasive from coarser papers would become embedded i n the soft metal surface and be impossible to remove l a t e r .  F i n a l polishing was  speed with a thick s l u r r y of 1 micron alumina. cold water or alcohol, since i t was  done at slow  The specimens were washed i n  found that hot water etched the structure.  The etchant used for high Pb content alloys was acid, 15 parts hydrogen peroxide, and 100 parts water.  50 parts a c e t i c  This acted r a p i d l y ,  both as an etch and as a mild chemical p o l i s h to remove remaining polishing marks.  High Sn alloys were etched i n a f e r r i c chloride base etch.  As an a l t e r n a t i v e , e l e c t r o p o l i s h i n g was frequently necessary  attempted, but since i t was  to polish r e l a t i v e l y large areas, the currents required  proved to be impractical.  Since some degree of preparation was  necessary  because sawn surfaces could not be electropolished, the mechanical method above was  found to be the most s a t i s f a c t o r y .  11  2.4  Measurement of Dendrite  Spacing  Dendrites i n lead (fee) and t i n (bet) have orthogonal branches, and primary dendrites are defined as those growing i n the general freezing direction, starting near the c h i l l .  Secondary branches grow from primary  dendrite stalks, and are therefore perpendicular to the freezing d i r e c t i o n . Tertiary branches growing perpendicular to secondary branches and p a r a l l e l to primary stalks can form at large primary spacings In lead r i c h Pb-Sn a l l o y s , the dendrites may  (over approximately  200  ym).  contain up to 19% Sn, and the  i n t e r d e n d r i t i c regions 62% Sn, the eutectic composition, which forms a fine lamellar structure.  2.4.1  Primary dendrite spacings When sectioned perpendicular to the growth d i r e c t i o n , primary  dendrites are seen to form the close packed arrangement, shown schematically i n Figure 2.  The distance between the centres of nearest neighbours i s the  primary spacing X. One method of measuring this spacing from a section perpendicular to the growth d i r e c t i o n i s to mark a l l the dendrite centres on a photograph, and then count the number of centres (n) i n a given area (A).  The spacing i s  2 then given by  A  A/n.  A second method i s to measure the distance between the centres of primary stalks on a longitudinal section ( p a r a l l e l to the growth d i r e c t i o n ) . The stalks can be recognized when secondary branches are v i s i b l e on both sides. The spacing measured i n this case w i l l be A' i n Figure 2, from which the nearest neighbour spacing can be calculated (A  = A'//2).  12  FIGURE 3: (a) L o n g i t u d i n a l s e c t i o n of a d i r e c t i o n a l l y dendrites  i n r e g i o n A are v e r t i c a l ,  (b) C o r r e s p o n d i n g c r o s s s e c t i o n .  solidified  and i n r e g i o n B are  Magnification  18x.  casting, tilted.  13  I t was dendrite  spacings  d i r e c t i o n * so tests,  and  ively.  considered be  that  the  measured  the  ends o f  the  difficulties  of the  primary  the  i t difficult  not  always  in  however,  phase;  would work r e l i a b l y  dicular  to  the  primary  stalks  on  Pb-Sn  corresponding  the  r e g i o n marked A  seen  area  longitudinal periodic  plane  The  also  other  centre of  cross section stalks  are  the  An  (in this  show a s i m i l a r  Although  the  checked  the  of  and  of marking  centres  c l o s e to the  present  in  completely,  branches.  t y p e was  In  solubility  This  reveals composition  this  flow  non-destruct-  dendrite branches  not  can  be  gradients  found  on  i s m a r k e d A'.  angle  area the  longitudinal  tilted  that  s i n c e they  on  lines  the are  are  perpenVertical  s e c t i o n marked  On  away f r o m  such  and  o f Pb-20% S n .  either  the  side  vertical,  also  tilted  cross section  shows  45°  B'  left  correspond  t o make m e a s u r e m e n t s  a  of to  tilted  lines.  logical  with  the  about  A.  of  a r e g i o n i s B on  cross s e c t i o n which  pattern of  i t w o u l d be most  the  example o f  case  the  sections parallel  cast cylinder  corresponding  areas  growth  s m a l l amounts o f e u t e c t i c  etchant  to determine  pattern of l i n e s the  i n the the  be  compositions  c a s t s t r u c t u r e on  of s e c t i o n .  section.  v e r t i c a l ) , and dendrites  on  the  primary  polished before  the method  etchant which  an  the  alloys.  the primary  though i t i s d i f f i c u l t t o the  the  a x i s of a d i r e c t i o n a l l y can be  work t h a t  revealed i n cast structures.  alloy  outline  u s i n g an  3 shows t h e  The  respect  using  t o d i s t i n g u i s h between s t a l k s by  Figure  c o u l d then  clearly  arise with  o v e r c o m e i n some c a s e s the p r i m a r y  c a s t i n g s c o u l d be  of each  always  present  sections perpendicular to  phase, because  c a s t s t r u c t u r e do  making  i n the  were e n c o u n t e r e d  because d e n d r i t e s are not  limit  from  dendrite spacing  Difficulties  particular,  important  from  the  14  region A', the enlarged view of this area i n Figure 4(a) shows the d i f f i culty i n distinguishing between stalks and branches. individual dendrites, i t i s frequently d i f f i c u l t  Although one can see  to i d e n t i f y the centres of  the neighbouring primary s t a l k s . It i s easy, however, to measure the l i n e spacing i n region B', shown enlarged i n Figure 4(b), f o r a comparatively large number of l i n e s . A composite of the microstructures was constructed from three orthogonal sections through region B', and when the model was turned the angle of the primary stalks became apparent.  A photograph of the composite  (Figure 5)  i s shown oriented such that the primary stalks i n the corner of the cube are normal to the plane of the paper.  For t h i s example, i t was  primary stalks were t i l t e d approximately 20  found that the  from the v e r t i c a l , i n a plane  perpendicular to the d i r e c t i o n of the l i n e s .  It was found that the pattern of l i n e s i s only seen when the dendrites are t i l t e d between approximately 10° and 30° from the v e r t i c a l . The lines probably appear because the plane of section passes through secondary branches at an appropriate angle.  Figure 6 shows a schematic view  of the t i l t e d dendrites (secondary branches are s l i g h t l y elongated i n the plane of t i l t ) .  The l i n e spacing L can be used to calculate the primary  dendrite spacing since X = /2Lcos6, where 0 i s the angle of t i l t . i s between 10° and 30°, cos8 i s between 0.98 and 0.87,  Since 6  therefore the use of  an average value of 0 = 20° introduces an error of less than 8%, which i s less than the error involved i n measuring L, which i s no better than ± 10%. Thus the formula used i n this work was X = 1.3L.  15  FIGURE 5:  Three d i m e n s i o n a l composite,  from which one can e s t i m a t e t h a t the d e n d r i t e s i n t h e top c o r n e r a r e t i l t e d approximately  20  FIGURE 6:  Schematic view of  the s t r u c t u r e i n F i g u r e 4(b)  16  Since primary spacings were measured from sections perpendicular to the growth d i r e c t i o n , no d i s t i n c t i o n could be made between primary and t e r t i a r y dendrite arms.  2.4.2  Secondary dendrite spacings Secondary  spacings were determined by measuring  the spacing  between a large number of c l e a r l y delineated secondary arms on polished sections p a r a l l e l to the growth d i r e c t i o n .  This could not be done non-  destructively for the columnar castings, and therefore the measurements were made on samples produced under the same cooling conditions as those used i n the flow tests. 2.5  Autoradiography Polished specimens (down to 5 micron alumina) which contained  radioactive tracer were placed f l a t against X-ray or orthochromatic f i l m to 204 make autoradiographs.  For a concentration of 500 ppm T l  (irradiated to  a s p e c i f i c a c t i v i t y of 5 millicuries/gm) s a t i s f a c t o r y exposures were obtained i n 16 hours on X-ray f i l m , or approximately 14 days on orthochromatic f i l m . Although exposures were long f o r orthochromatic f i l m , the resolution was appreciably better.  A l l the autoradiographs i n this thesis are printed so  that dark areas indicate the presence of radioactive material.  17  CHAPTER 3 THE 3.1  MEASUREMENT OF INTERDENDRITIC FLUID FLOW RATES  Review of Previous Work  The f i r s t direct measurements of i n t e r d e n d r i t i c f l u i d flow rates (1 2) were reported by T.S. Piwonka  '  from experiments  on Al-4.5%Cu a l l o y s .  Samples of the molten a l l o y were poured into a U-tube and allowed to s o l i d i f y . They were then reheated to the testing temperature between the solidus and liquidus, where the i n t e r d e n d r i t i c l i q u i d i n the a l l o y sample was expelled by applying pressure to one branch of the U-tube.  The f l u i d flow rates  were calculated from the time taken f o r the displaced l i q u i d to make contact with a probe i n the other branch of the U-tube. Two methods of applying pressure were used.  In one case the  inderdendritic l i q u i d was displaced using an inert gas (nitrogen), and i n the other case l i q u i d lead was used i n addition to gas pressure.  Piwonka  acknowledged that surface tension e f f e c t s at the liquid-gas interface may have caused the gas displacement results to be unreliable, since the pressure required to force gas into the i n t e r d e n d r i t i c regions might have been a s i g n i f i c a n t proportion of the t o t a l pressure required to expel the i n t e r dendritic l i q u i d .  An approximate estimate of the magnitude of this effect  can be made as follows: The surface tension (a) of l i q u i d aluminum i s 520 dynes/cm at 750°C (surface tensions for Al-Cu alloys are not readily a v a i l a b l e ) . Assuming an i n t e r d e n d r i t i c channel size of 20ym diameter, the pressure required to force gas into such a channel would be equal to the pressure  18  r e q u i r e d t o blow a h e m i s p h e r i c a l b u b b l e o f r a d i u s r = 10 urn.  P  =  |2. = 1.04 x 1 0  According  6  dynes/cm . 2  t o Piwonka's t h e s i s , t h e a p p l i e d p r e s s u r e s  i n the  4 n i t r o g e n gas e x p e r i m e n t s were i n t h e range 3 x 1 0 I f t h e assumed c h a n n e l s i z e i s r e a s o n a b l e ,  6 t o 1.8 x 10  the surface tension  2 dynes/cm . could  p o s s i b l y account f o r a l l t h e r e s i s t a n c e t o f l o w t h a t was o b s e r v e d . the i n c r e a s i n g r e s i s t a n c e w i t h d e c r e a s i n g increase i n the pressure  Indeed,  temperature might be due t o the  (P) w i t h decrease i n t h e r a d i u s ( r ) .  Since the  aim o f t h e experiments was t o determine t h e r e s i s t a n c e t o f l o w caused by f l u i d drag w i t h i n t h e i n t e r d e n d r i t i c c h a n n e l s , r e s u l t s must be c o n s i d e r e d The  t h e gas d i s p l a c e m e n t  completely u n r e l i a b l e .  use o f l i q u i d l e a d i n s t e a d o f gas would reduce t h e s u r f a c e  t e n s i o n e f f e c t and g i v e a b e t t e r measure o f f l o w r a t e .  However, i n a r e a l  s o l i d i f i c a t i o n s i t u a t i o n , the flowing l i q u i d w i l l react w i t h the d e n d r i t i c s o l i d , and t h i s i n t e r a c t i o n would p r o b a b l y the measurements.  have an i m p o r t a n t  e f f e c t on  T h i s would n o t be t h e case when l e a d i s used i n t h e  A l - C u system, s i n c e aluminum i s i n s o l u b l e i n l e a d . Piwonka examined some o f t h e a l l o y samples a f t e r t e s t i n g t o ensure u n i f o r m p e n e t r a t i o n o f l e a d i n t o the i n t e r d e n d r i t i c r e g i o n s ; measurements r e l a t i n g t o the s t r u c t u r e were g i v e n .  T h i s i s an  o m i s s i o n , s i n c e the s t r u c t u r e was assumed t o be c o n s t a n t tions.  however, no unfortunate  i n a l l the c a l c u l a -  He acknowledged t h a t p r e f e r e n t i a l channels were formed by the i n t e r -  19  dendritic l i q u i d as i t was being displaced i n some of the higher temperature t e s t s , which would suggest that uniform flow d i d not always occur.  Piwonka believed that his results were consistent with a model of  the s o l i d - l i q u i d region which considered the region to be a bundle of straight c a p i l l a r y tubes.  This could be demonstrated by p l o t t i n g the  logarithm of the permeability of the a l l o y , which i s the r e c i p r o c a l of the resistance to flow (defined i n more d e t a i l i n subsection 4.1  of the present  work) against the logarithm of the l i q u i d f r a c t i o n , calculated from the Al-Cu phase diagram.  A straight l i n e of slope 2 should be obtained i f the model i s applicable (the t h e o r e t i c a l r e l a t i o n s h i p i s derived i n subsection 4.5.1  of  (2) the present work).  Piwonka's results  of l i q u i d lead displacement  i n Al-Cu  are replotted i n Figure 7 as permeability versus the square of volume f r a c t i o n l i q u i d (from the thesis, i t appears that Piwonka i n c o r r e c t l y used weight fractions instead of volume f r a c t i o n s ) .  The results agree f a i r l y well  with a straight l i n e , for l i q u i d fractions less than about 0.3,  and the con-  clusion can be drawn that the s o l i d - l i q u i d region can be treated i n this s i m p l i f i e d manner, i . e . , as a bundle of straight c a p i l l a r y tubes, within this range, provided the assumption of constant structure i s correct. The results of this i n v e s t i g a t i o n have led a number of workers to explain certain effects i n s o l i d i f i c a t i o n i n a semiquantitative manner, based on the standard equations  for flow through porous media.  The explanations  are  semiquantitative i n the sense that numerical values for the parameters which describe the structure of the porous medium must be assumed, since they cannot be obtained from Piwonka's experiments.  20  FIGURE 7:  Permeability as a function of the square of the volume f r a c t i o n l i q u i d , using experimental data obtained by Piwonka^^ .  21  Thus Piwonka' ', C a m p b e l l  w  and Tien*'''' used these r e s u l t s i n  u y  theoretical predictions of hydrostatic tensions which could lead to pore formation during s o l i d i f i c a t i o n . A segregate formation  Standish  used the results to argue that  i s not caused by i n t e r d e n d r i t i c f l u i d flow, i n contrast  (9) to Mehrabian et a l .  who  have formulated  a comprehensive macrosegregation  theory which can be used to semiquantitatively explain the formation  of A  segregates, based on Piwonka's findings. Attempts to measure i n t e r d e n d r i t i c f l u i d flow i n the Pb-Sn system were made by K a e m p f f e r ^ ^ ' ^ \ who  observed the formation  of droplets on the  bottom surface of an ingot with a thin layer of eutectic material placed top, when i t was was  heated above the eutectic temperature.  Radioactive  added to the eutectic layer, and the aim of the experiment was  the flow rate by measuring the a c t i v i t y  Kaempffer found that i t was dendritic flow with this experiment.  on  tracer  to measure  of the droplets.  not possible to produce uniform i n t e r At the eutectic temperature no flow  was  observed, but as the temperature increased drops began appearing on the bottom surface of the ingot. constant  These would coalesce before f a l l i n g , and, with a  rate of heating, the f i r s t drop would f a l l at about 230°C, and  remainder would follow rapidly.  the  Examination of polished and etched sections  of the ingot showed that one or two wide channels had  formed, and  autoradio-  graphs showed that f l u i d flow was mostly confined to these channels.  Kaempffer interpreted h i s results as follows; of the ingot was  as the temperature  increased above the eutectic temperature, the lead-rich  dendrites became soluble i n the superheated eutectic l i q u i d ,  therefore, as  22  material from the top layer began to flow down, i t was  able to dissolve  dendrite branches i n i t s path, forming the wide channels. therefore not representative of uniform  This work was  flow through porous media, and i t  remained to be shown whether one could produce uniform flow of i n t e r d e n d r i t i c l i q u i d which was not superheated. 3.2  General Description of the Technique Used i n the Present Work  Measurement of the permeability of a packed bed i s often done using a F a l l i n g Head Permeameter (Figure 8).  This consists of two concentric tubes  with the porous material packed i n the inner tube.  A s t a t i c head of f l u i d  i n this tube w i l l cause flow through the porous bed and up the space between the two tubes.  The permeability K can be calculated from the time required  for the f l u i d head (h) to f a l l a given amount, using an integrated form of Darcy's Law  t  =  - f  ln(h /h ) t  o  where c and K are constants, h i s the head at time t, and h i s the head ' t ' o at t = 0. This equation i s derived i n d e t a i l i n Appendix I.  The same p r i n c i p l e i s used i n the design of the flow c e l l for measuring i n t e r d e n d r i t i c f l u i d flow (Figure 9). are side by side instead of concentric. of brass and resembles a s p l i t mould.  In this case the two tubes  The flow c e l l was made from four pieces This design was  chosen so that the  various pieces of the alloy under test could be assembled before the test and removed afterwards without damage.  The brass was  completely  covered with a  thin coating of graphite (Aquadag) which prevented contact with the molten alloy and also prevented  leaks.  Steel screws were used to hold the pieces  23  Outflow  Porous  FIGURE 8:  "TV  Bed  Inner  Tube  Outer  Tube  Schematic diagram showing the p r i n c i p l e of a F a l l i n g Head Permeameter.  argon  Pb-55Sn  uE  Pb-20Sn  , 2 - 3 5c m  FIGURE 9 :  Pb-55Sn d i o  Sectional views of the flow c e l l and the lead-tin alloy inserts (to s c a l e ) .  24  together.  Brass was  chosen for i t s machinability, strength and  conductivity (0.27 cal/cm sec°C).  thermal  High conductivity was desirable to ensure  isothermal conditions within the flow c e l l , and brass was  found to be s a t i s -  factory by making temperature measurements at various locations inside the c e l l while i t was being heated. tivity  Copper would have provided higher conduc-  (0.88 cal/cm sec°C), but i t was  difficult  to machine to the complex  shape of the flow c e l l , and from previous experience i t was  found that  threads tapped i n copper did not hold a f t e r repeated heating and cooling.  To carry out a flow test three pieces of l e a d - t i n alloy were inserted into the c e l l  (Figure 10).  The p a r t i a l l y assembled flow c e l l with  the lead-tin a l l o y i s shown i n Figure 11.  A c y l i n d r i c a l casting A of the  a l l o y under test was placed i n the appropriate cavity, and two  other  castings, B and C, of d i f f e r e n t composition, were placed above and below i t . The compositions  of these three pieces of lead-tin alloy were chosen such  that, at the testing temperature, both B and C would be l i q u i d , and the casting A would be p a r t i a l l y l i q u i d .  Thus there would be a hydrostatic  pressure i n the s o l i d - l i q u i d region through A, and the l i q u i d l e v e l would tend to f a l l on the l e f t , and r i s e i n the smaller diameter ' r i s e r ' pipe on the r i g h t .  The l e v e l of the l i q u i d metal i n the r i s e r could be measured at  any time using a copper wire probe, which closed a c i r c u i t on contact. Therefore, under known conditions of pressure, temperature, l i q u i d  composition  and dendrite spacing, graphs of height of l i q u i d i n the r i s e r versus time were plotted, from which the permeability could be calculated.  At the end of the test the flow c e l l was removed.  c h i l l e d and the a l l o y  Figure 12 shows the a l l o y before and a f t e r testing.  was  The alloy could  FIGURE 12:  Pb-Sn a l l o y b e f o r e and a f t e r f l o w t e s t .  TABLE I DIMENSIONS AND COMPOSITION OF CASTINGS USED FOR INTERDENDRITIC FLUID FLOW STUDIES  Diameter  Length  cm  cm  Composition  Casting  A  2.46  3.37  Pb-20%Sn  Casting  B  1.91  0.76  Pb-55%Sn  Casting  C  1.91  0.64  Pb-55%Sn  Comments  204 Approx. 500 ppm T l added i n c e r t a i n t e s t s as t r a c e r T h i s c a s t i n g was made u s i n g t h e flow c e l l as a mould  27  then be examined by sectioning and p o l i s h i n g . Autoradiography  of these  sections was used to observe the f l u i d flow d i r e c t l y i n some of the castings 204 where radioactive T l was added as a tracer. The compositions  and dimensions of A, B and C are given i n  Table I.  3.3  Preparation of the Alloy under Test (A)  In most of the tests the cylinders of a l l o y (A) were columnar castings.  For the range of primary dendrite spacings from 28 to 83 microns  these were produced by remelting the required weight (255 g) of s t a r t i n g ingots i n a v e r t i c a l graphite mould and c h i l l i n g from the bottom. The furnace, mould and c h i l l i n g arrangement are shown i n Figure 1(a).  When the a l l o y was molten, an iron-constantan thermocouple was placed i n the melt and the temperature was adjusted to the required l e v e l by adjusting the furnace c o n t r o l l e r .  When the alloy reached a constant  temperature (checked by moving the thermocouple around i n the l i q u i d ) the thermocouple was withdrawn and the alloy s o l i d i f i e d u n i d i r e c t i o n a l l y from the bottom.  Different cooling rates were produced by using either a blast  of nitrogen or a constant pressure of water against the c h i l l , and also by changing the size of the nozzle and the thickness of the c h i l l .  Careful  control of the temperature of the melt and the cooling conditions made i t possible to reproduce d i r e c t i o n a l castings with a given dendrite spacing within the precision with which the spacing could be measured 10%).  (approximately  Four d i f f e r e n t quenches were used, and the d e t a i l s are given i n  Table I I .  28  Cooling curves at three positions i n the casting were plotted by inserting iron-constantan thermocouples i n the melt and s o l i d i f y i n g .  These  curves were taken to be representative of the actual castings held at the same temperature and quenched i n the same manner.  Table III l i s t s the  thermal conditions f o r the four d i f f e r e n t quenches used.  The c h i l l face  cooling rate was calculated from the cooling curve of a bare i n the l i q u i d , placed i n contact with the c h i l l .  thermocouple  The cooling rate i s taken  from the slope of the cooling curve at the liquidus temperature of the a l l o y . The freezing rate i s given at two points and i s calculated from the established relationship ^^'^"^  x  = A/t  where x i s the distance from the c h i l l , t i s the time elapsed from the start of freezing, and A i s a constant. equal to x/2t.  The freezing rate i s therefore  The mean primary dendrite spacing i s also l i s t e d i n  Table I I I . Columnar castings with dendrite spacings larger than 83 microns were prepared by cooling the alloy very slowly under a shallow temperature gradient using the apparatus described i n section 6.2.  Ingots 2.5 cm i n  diameter and approximately 12 cm long were produced i n graphite moulds that were lowered through the two zone furnace.  Each of these ingots was machined  to produce c y l i n d r i c a l samples with the dimensions of casting A.  In'addition,  equiaxed castings of d i f f e r e n t dendrite spacings were produced by pouring molten alloy into a simple graphite mould, 2.5 cm i n diameter and approximately 5 cm long, using different mould preheats and alloy superheats.  29 TABLE II QUENCH DATA  Quench  Coolant*  Pressure of Coolant  Thickness of C h i l l (cm)  Nozzle Dia.(cm)  Al  N gas  35 l b / i n  2  1.98  0.31  A2  N gas  35 l b / i n  2  0.88  0.57  Wl  water  23.5 i n . head  1.98  0.31  W2  water  23.5 i n . head  2.86**  0.31  2  2  * Coolants were at room temperature.  ** Two copper discs with t o t a l thickness of 2.86 cm were used.  TABLE I I I THERMAL CONDITIONS FOR Pb-20%Sn COLUMNAR CASTINGS Quench  Temp. of melt(°C)  C h i l l face Cooling Rate* (°C/sec.)  Freezing Rate A B (cm/sec) (cm/sec)  Av. primary dendrite spacing (microns)  Al  310  0.29  0.015  0.014  71  A2  310  0.18  0.015  0.016  83  Wl  310  6.00  0.042  0.037  28  W2  310  0.68  0.015  0.019  51  A = 1.31 cm from c h i l l B = 2.77 cm from c h i l l  * Measured at the liquidus temperature.  30  After cooling to room temperature the dimensions given i n Table I.  a l l castings were machined to  The ends were polished, etched and  examined microscopically to determine the dendrite spacing, and before testing i n the flow c e l l the ends were again polished and cleaned to remove oxide or other extraneous material that might i n t e r f e r e with f l u i d flow. 3.4  Preparation of Castings B and C  The upper cylinder B was machined from starting ingots of the required composition.  In the majority of cases approximately 500 ppm  of  204 radioactive T l  was  dissolved i n the a l l o y .  After t e s t i n g , f l u i d flow  patterns were obtained from autoradiographs of sections taken from samples containing radioactive tracer. The lower casting C was made by using the lower part of the flow c e l l as a mould which was preheated before pouring alloy of the required composition (radioactive tracer was not added to the lower castings). After cooling, the flow c e l l was  dismantled and the castings were removed  and machined to the required length. The .composition of castings B and C used throughout was Pb-55%Sn. ience^^  this work  The choice of composition was based on Kaempffer's exper-  using eutectic material (62%Sn) as the l i q u i d reservoir above  the casting.  Since the main requirement i n this work was  should not become superheated, an o f f - e u t e c t i c alloy was to be more suitable.  found by experience  The p a r t i c u l a r composition chosen was  best for permeability measurements approximately temperature.  that casting B  found to work  10°C above the eutectic  Problems of p r e f e r e n t i a l channelling, s i m i l a r to those  31  encountered by Kaempffer, arose when attempts were made to use d i f f e r e n t alloy compositions to measure permeabilities at higher temperatures.  3.5  Flow Measurement Equipment Flow measurements were made as l i q u i d metal rose up the ' r i s e r '  pipe of the flow c e l l , shown i n Figure 9.  In most experiments the actual  distance involved was only 3 cm or l e s s , therefore i t was essential to hold the furnace, flow c e l l and the measuring probe firmly i n p o s i t i o n , so that accurate measurements could be made. apparatus i s shown i n Figure 13.  A schematic diagram of the  The flow c e l l was held i n position  inside the furnace from the metal tube which was also connected to the argon supply.  The measuring probe consisted of a copper wire i n a ceramic  tube which was inserted i n the flow c e l l down the r i s e r pipe.  When the  probe touched the surface of the l i q u i d metal an e l e c t r i c c i r c u i t was giving the position of the interface.  closed  The probe was attached to a long  feed screw and crank handle so that i t could be accurately positioned at predetermined intervals (usually 0.5 mm)  and the time required f o r the  l i q u i d to make contact could be measured.  The position of the probe was  given on the d i a l gauge, which was accurate to ± 0.0125 mm.  Thus up to  50 data points of distance versus time could be obtained as the flow took place.  The temperature of the flow c e l l was continuously monitored on a chart recorder during the test by means of an iron-constantan thermocouple i n a 3 mm diameter glass sheath.  A simple c i r c u i t (shown i n  Figure 14) was used to connect the measuring probe to the thermocouple  32  Lamp  O  —'I  100 K  1|  FIGURE 14:  I K  -  >  A  M  M  o r r eAc o r dN ing ^ used \ fA the p o s i t i o n o f t h e  .022/zF  probe on t h e tempera-  T/C  ture  Probe  M Flowcell  Circuit  Recorder  trace.  |  33  wires, on  so that  at the instant  the temperature  were used  trace.  Chart  i n t h i s work, so t h a t  between s u c c e s s i v e  positions  o f ± 0.1  sec ( i f necessary).  produced  a blip  signal  the t i p , i n this  The  equipment  total  o f 30 t e s t s  early  tests  the  I t was  understood  measuring graphing contact  an e l e c t r o n i c  The  thermocouple  required could  produced  forliquid  be measured that  sec/in  to rise t o an a c c u r a c y  the c i r c u i t  merely  the thermocouple  was n e c e s s a r y t o e l e c t r i c a l l y  above,  which  were done, i n f a c t deviations felt  that  the probe  was u s e d  f o r 22 o f t h e  evolved gradually  from Darcy's  since the  Law o c c u r r e d i n some o f  the nature o f these d e v i a t i o n s would  touched  was u s e d .  the l i q u i d  described  previously,  of the experimenter over a pointer  involved  a lamp  just  photo-  showed  as a c c u r a t e and  however i t r e q u i r e d t h e  long  periods.  and s c a l e w h i c h  The  position  was l e s s  precise  gauge.  results  from  a l l t h e t e s t s were used  subsequently  found  scatter.  that  and d e n d r i t e  the results  However, o n l y  i n the study o f d e v i a t i o n s  from  to determine the  spacing, since  the early  the results  from  from Darcy's  Law.  be  E a r l y methods o f  metal  that  T h i s m e t h o d was p o t e n t i a l l y  between p e r m e a b i l i t y  used  was  case).  relationship  observed  a blip  10 m i n / i n a n d 10  I t s h o u l d be n o t e d  described  as t h e method  attention  the d i a l  t h e time  o f the probe  t h e p r o b e was m e a s u r e d u s i n g  than  between  t i m e r a t t h e same i n s t a n t  h a d b e e n made.  undivided  speeds  i f more a c c u r a t e equipment  t h e t i m e when  reproducible  of  which  showed t h a t  flow t e s t s .  better  c o n t a c t was made  on c o n t a c t , b u t d i d n o t o t h e r w i s e a l t e r  (however a s h e a t h e d  insulate  when  tests  i t was  l a y within the  the l a t e r  tests  were  34  3.6  Flow Testing Procedure Before assembly, the faces of the castings which were to be  placed i n contact were painted with a thin layer of a soldering-type flux paste, to ensure that when castings B and C became l i q u i d they would completely wet the end surfaces of casting A. The flow c e l l was  assembled with the Pb-Sn i n s e r t s , and placed  inside the v e r t i c a l tube furnace.  The central tube was  connected to an  argon supply (1000 ml/min) to provide an inert atmosphere inside the c e l l which would prevent oxidation that might i n h i b i t f l u i d flow.  The  measuring probe was inserted down the r i s e r pipe u n t i l contact was made with the surface of the branched portion of casting C.  This established  the datum l e v e l f o r measuring f l u i d pressures, and the d i a l gauge was set to zero f o r this point.  The probe was  in preparation f o r flow measurements.  then moved up the required amount The sheathed thermocouple was  inserted i n the appropriate hole i n the flow c e l l .  The power supply to the furnace was adjusted to heat the flow c e l l rapidly (approximately 5°C/min) to the required temperature, without overshooting, and once this temperature was attained the automatic c o n t r o l l e r held the c e l l at constant temperature while flow measurements were made. supply was  It was  found that manual control of the power  the most e f f e c t i v e method of heating the c e l l i n the i n i t i a l  stages, and a reproducible procedure could be developed a f t e r two or three 'dummy' t r i a l s .  The zero point for timing measurements was  instant when the temperature  reached 183°C, the eutectic  taken as the  temperature,  35  i.e.,  the instant melting  would be expected to begin.  Flow took place  r e l a t i v e l y slowly, and i n the majority of cases the temperature of the c e l l had s t a b i l i z e d at 193°C by the time the f i r s t measurements were made. In those cases where the temperature had not s t a b i l i z e d within ± 3°C of the required temperature, the points were not used u n t i l the temperature had stabilized.  When the required number of data points had been measured, the flow c e l l was c h i l l e d , either by lowering into a water bath or by using a 50 P i s  air blast.  In the l a t t e r case s o l i d i f i c a t i o n was complete 45 sees a f t e r the  a i r blast was turned on. After cooling to room temperature the flow c e l l was dismantled and the  t o t a l height of the Pb-Sn sample was measured.  Comparing  the height of  the  column i n the r i s e r pipe after cooling, to the measured height when  l i q u i d , i t was found that thermal and s o l i d i f i c a t i o n contraction caused a reduction i n length of 9% between the testing temperature and room temperature for this composition.  This information was therefore used when calculating  the  height of l i q u i d from room temperature measurements.  The majority of  the  samples were subsequently sectioned both at right angles and p a r a l l e l to  the  axis of the cylinder A. Microexamination and autoradiography were used  to determine the f l u i d flow paths and the effect of flow on the microstructure.  3.7  Precision of the Flow Measurement Technique  The method of using a copper wire probe to locate the position of the  surface of a r i s i n g column of l i q u i d metal was f i r s t tested by using the  36  Motor Probe Contact Glass Guide  Tube  (7.9 mm  Tube  Mercury  Flexible  Tube  FIGURE 15: Apparatus f o r testing the precision of the flow measurement technique.  TABLE IV PRECISION OF THE FLOW MEASUREMENT TECHNIQUE  No. of observations  flow rate (cm/sec)  std. error (cm/sec)  95% conf. interval (cm/sec)  95% conf. interval (pet)  20  0.004397  2.06 x 10"  5  ± 4.33 x 10"  5  ± 1.0%  45  0.004430  0.73 x 10"  5  ± 1.47 x 10~  5  ± 0.3%  48  0.004417  0.66 x 10~  5  ± 1.33 x 10"  5  ± 0.3%  50  0.004420  0.68 x 10"  5  ±1.37 x 10"  5  ± 0.3%  ID.)  37  the apparatus to measure the flow rate of a r i s i n g column of mercury.  A  constant flow rate was imposed by r a i s i n g one branch of a f l e x i b l e U-tube using a low speed synchronous motor (12 rph).  The U-tube contained mercury,  and the fixed branch was made the same diameter as the r i s e r tube i n the flow cell.  A schematic diagram of the equipment i s shown i n Figure 15. Four experimental runs were done ranging from 20-50 observations.  Points were taken approximately method of least squares was values of distance and time.  every 0.75 mm  as the mercury rose.  The  used to f i t a straight l i n e to the measured The flow rate, standard error of the flow rate,  and the 95% confidence intervals were calculated and are l i s t e d i n Table IV.  From these tests one may measurement technique was ± 0.3% rate.  conclude that the error i n the flow  over 50 observations for a constant flow  38  CHAPTER 4 RESULTS AND DISCUSSION OF FLOW  4.1  Interpretation  The of  fluid  Darcy's  experimental  f l o w up t h e r i s e r  shown i n F i g u r e of  Using  the curves,  16.  versus  The flow v e l o c i t y  and i t i s c l e a r  magnitude d i f f e r e n c e  Law  technique pipe  that  supplied data time  there  velocities  The curves  the  d e n d r i t e spacing, y e t i t remains  smaller  the flow v e l o c i t y  r e s u l t s were i n t e r p r e t e d  a p o r o u s medium w h i c h o b e y s D a r c y ' s Darcy  through tional  plots are  f o r these  decreases  fairly  an o r d e r o f  constant  two d e n d r i t e  with  time f o r  f o r the  spacing.  The  by  ( t ) , a n d two t y p i c a l  i s approximately  spacings. larger  o n t h e d i s t a n c e (&)  a t any p o i n t i s g i v e n by t h e s l o p e s  between t h e i n i t i a l show t h a t  MEASUREMENTS  i n 1856 a sand  by c o n s i d e r i n g t h e c a s t i n g  Law.  The c l a s s i c a l  c o n s i s t e d o f measurements filter  bed.  to the pressure  and i n v e r s e l y  experiment  performed  of the q u a n t i t y o f water  T h e q u a n t i t y was f o u n d  drop,  A t o be  t o be d i r e c t l y  flowing  propor-  proportional to the length of  (14) the bed.  From d i m e n s i o n a l  arguments  one c a n d e d u c e  the following  relationship:  4.1  v where  v =  bulk v e l o c i t y  K  permeability  =  viscosity L  =  length  AP  =  pressure  of the f l u i d o f t h e porous  of the l i q u i d  o f the porous drop  medium  (measured medium  over  t h e whole  area)  39  A V E R A G E PRIMARY DENDRITE SPACING 116 microns  o  COLUMNAR  m CNJ  E  o o — cvi uJ O  <  -  0 0 5  °  o  460  200 FIGURE 16(a):  TIME  6 0 0 8 (seconds)  Flow measurement r e s u l t s ;  r i s e r pipe versus time f o r A = 116 ym.  0  1000  distance of flow up the I n i t i a l slope = 0.0055 cm/sec.  AVERAGE PRIMARY DENDRITE SPACING  o.  0  28 microns  COLUMNAR  in (VI  So —  CVl  UJ  <  Is m 6  1 0 0 0  2 0 0 0  3 0 0 0  4 0 0 0  5 0 0 0  T I M E (seconds) FIGURE 16(b):  Similar plot f o r X = 28 ym. I n i t i a l slope =0.00039 cm/sec.  6 0 0 0  7 0 0 0  40  The permeability K i s a property of the porous medium and has 2 the dimensions of area (cm ).  The minus sign i n the expression indicates  that flow i s i n the opposite d i r e c t i o n of increasing A P . Darcy's Law has been v e r i f i e d experimentally for flow through many types of porous media, and Carman^"^ has stated that there i s good reason to believe that i t can always be applied under the following conditions: i) ii)  the flow must be laminar the f l u i d must be inert to the porous medium, i . e . , chemical, adsorptive, e l e c t r i c a l ,  electrochemical and c a p i l l a r y effects are  absent. 4.1.1  Laminar flow  Laminar flow i s related to the Reynold's number, a dimensionless group defined as Re  =  i£i M  where V i s the (scalar) v e l o c i t y measured over the whole area of the bed, p i s the density of the f l u i d , u i s the v i s c o s i t y of the f l u i d , and 6 i s a diameter associated with the porous medium, i . e . , the average p a r t i c l e or pore diameter, or some length corresponding to the hydraulic radius theory. The representation of the flow by means of the Reynold's number i s therefore dependent on the choice of the length 6, which i n turn i s dependent on the model chosen to describe the porous medium. Many investigations have been directed towards finding the c r i t i c a l Reynold's number where flow through the bed ceases to be laminar.  These have  41  been reviewed by S c h e i d e g g e r , and the range of values reported f o r the c r i t i c a l :Reynold's number l i e s between 0.1 and 75. i  Scheidegger has  commented that the uncertainty of a factor of 750 i s probably related to the fact that 6 i s not c l e a r l y defined, and he points out that the difference between these values and the Reynold's number of ZOOO^which i s normally  taken as the c r i t i c a l value for turbulent flow i n straight tubes,  makes the Reynold's number concept somewhat doubtful when applied to porous media. Nevertheless, experiments have shown that the c r i t i c a l  range  e x i s t s , therefore, to check f o r laminar flow i n the present work, the following simple approach was adopted.  The maximum observed v e l o c i t y was  calculated from Equation 4.1:  v where  =  K - —  , pgh —8  K  =  8.2x10  y  =  0.03 poise  L  =  3.37 cm  P  =  8.33 g/cm  2 cm when X  =  175 ym and g  Li  - 0.2  3 2  i.e.  g  =  981 cm/sec  h(max)  =  -4.51 cm  v  =  0.03 cm/sec  As a f i r s t approximation, <5 i s taken, as equal to the primary dendrite spacing X , then  42  R  =  i.e.  ^ u  e X  =  175 x 10~  u  =  0.03  R  =  4  cm  poise  0.14  e This value i s approximately  equal to the lowest  estimate of the c r i t i c a l Reynold's number.  published  However, i t i s reasonable  to  assume that the value of 6 chosen i s a conservative estimate, since the e f f e c t i v e diameter of flow channels or p a r t i c l e s (depending on the model chosen) i s l i k e l y to be much less than X.  Flow i s therefore considered to  be laminar i n a l l the tests done i n the present work, and the f i r s t condition i s s a t i s f i e d . 4.1.2  Interaction effects  As well as an upper l i m i t to Darcy's Law, references i n the l i t e r a t u r e to a lower l i m i t .  there are a number of  Carman reviewed the early  observations of this behaviour i n 1 9 3 7 ^ ^ \ and drew the conclusion that the deviations were related to surface forces between the s o l i d and This has remained the general consensus  liquid.  since then, and the l i s t of  possible surface effects reviewed by S c h e i d e g g e r i n c l u d e s tension, adsorption and molecular d i f f u s i o n .  surface  Electrochemical effects have (18  been of interest recently, and work has been published the effects i n terms of an e l e c t r i c a l double layer.  19) '  explaining  Since l i q u i d metals are  not i o n i c , this would not be relevant to the present work. There i s no doubt that the i n t e r d e n d r i t i c l i q u i d w i l l interact chemically with the dendrites, consequently  deviations from Darcy's Law  will  43  be interpreted i n terms of the i n t e r a c t i o n e f f e c t s .  It i s assumed that  because the dendrites form a r i g i d network, i n t e r a c t i o n effects over short time periods w i l l not cause a general collapse of the structure. i t should be possible to use the data from these experiments  Therefore  to draw some  conclusions regarding the nature of the effects which cause the deviations. 4.2  Application to the Flow C e l l For the experiments  Experiments  done i n the flow c e l l AP varies with time,  therefore Equation 4.1 can be expressed i n the following form t  where h  -  - |  ln(h /h ) t  i s the head of l i q u i d at time t, h  t  a constant.  4.2  o  o  i s the i n i t i a l head and c i s  The complete derivation of Equation 4.2 i s given i n Appendix I.  The distance versus time data were therefore replotted as In  (h /h ) versus time. t  o  When the permeability K i s a constant, these  plots should be l i n e a r , with a slope equal to -c/K.  The data from Figure 16 have been replotted i n this manner i n Figure 17, and i n both cases the plots deviate from l i n e a r i t y .  The castings  were examined metallographically to investigate the reasons f o r these deviations and the results w i l l be described i n d e t a i l i n a l a t e r section (4.5).  In both cases the plots show that the mechanisms which caused the  deviation are time dependent, therefore the permeability has been estimated from the i n i t i a l slopes.  It i s clearly a problem to decide how many points contribute to the i n i t i a l slope, and at which point the data begin to deviate from l i n e a r -  FIGURE 17(b);  Similar plot f o r data from Figure 16(b), showing a negative deviation.  45  ity.  A s t a t i s t i c a l argument was therefore developed, and the straight  lines i n Figure 17 are the best estimates of the i n i t i a l slope by this method.  4.2.1  The method f o r finding the i n i t i a l permeability The tests described i n section 3.7 to establish the precision of  the flow measurement technique can be used to separate the random experimental errors inherent i n the technique from the systematic effects which cause the deviations from l i n e a r i t y .  The method of least squares can be used to estimate the rate of flow of the mercury from the data i n section 3.7.  Since the flow v e l o c i t y  was constant, the observed scatter was only due to experimental errors i n using the copper wire probe to locate the mercury surface.  Therefore the  data was f i t t e d to a l i n e y  =  mx  where y i s the dependent variable (position of the mercury surface) and x i s the independent variable (time).  The slope of the regression l i n e  (m)  would therefore be the v e l o c i t y .  The scatter of points about the best f i t l i n e i s described by the standard error of Y (°y) where  °Y  =  V  n-2  (y.-y.) i s the difference between the value of the i t h point y_. and the  46  value y_^;  estimated by the regression l i n e , i n other words, the error;  and (n-2) i s the number of degrees of freedom.  Therefore a  i s a measure  of the "goodness of f i t " of the data.  However, the data from the flow c e l l experiments  are plotted i n  the form of ln(h /h ) versus time, therefore one must consider a plot of t o •. •» . l n ( l - y ^ ) versus x_^ instead of the simple y^ versus x^, where  h , a Q  2  and a^ are constants defined i n Appendix I.  (From the numerical  values of the constants, i t follows that y! i s normally less than  1.)  S i m i l a r l y , the f i t t e d value i s y!^ where  The error i n the dependent variable, e^, plotted i n this manner i s :  e  i  -  l n ( l - yj)  -  ln(l-  v[)  It i s well known that when data i s plotted on a logarithmic graph the points become weighted, of position along the l i n e .  One can take this effect into account by  including a weighting factor w^. calculated as follows: Let y! J  1  =  z, and y' - y! . i ^l J  i n other words, the error e i s a function  =  6z  A suitable weighting factor can be  47  then l n ( l - y\)  =  l n { l - (z + 6z)}  S i n c e -1<z<1, the l o g a r i t h m may be expanded  1  / _. * \ \  fi  lnil  Similarly  - (z + 6z) }  l n ( l - yl)  =  /  (z + 6z) -  =  z + 6z  But  e  z  z  3  (z + 6z) 3—  -  6z — -  z  2  2  —  -  z6z -  3  —  3  ~  -  y  l n { l - (z + <5z)} -  =  ±  =  -  (z + 6z) ^—  l n { l - z}  2 and l n { l - z }  2  • r \  =  l n { l - z}  2 <5z i s s m a l l , t h e r e f o r e one may n e g l e c t  6z  and h i g h e r  order  terms.  Also,  2 s i n c e -1<z<1, z &z and h i g h e r first  terms w i l l be s m a l l , t h e r e f o r e  as a  approximation -  e.  x  &z  e =  i.e.  When z = 0 , „ j.. . Defxnxng  w. i  w  -  z6z  (1 - z)6z  l n { l - (z + 6z)}  w  i.e.  order  =  -  l n { l - z}  =  l n ( l - 6z) =  6z  e r r o r at z e r r o r at z = 0 (1 - z)6z  6z  i  i  =  (1 ~  z  The s t a n d a r d  )  e r r o r o f Y, which e s t i m a t e s the "goodness o f f i t "  of the l o g a r i t h m i c p l o t i s ZCw^..)  n-2  2  48  The value of  has been calculated for the constant v e l o c i t y  data from the test i n Section 3.7.  It represents the "goodness of f i t " one  would expect i f one were to do a least squares f i t on this data, and the only errors were random experimental errors.  This value may now  be  compared with the value of Oy for a given number of points from a flow c e l l experiment.  Using a standard test of s i g n i f i c a n c e , the F t e s t , one  can decide whether the observed scatter for the flow c e l l data i s larger than one would expect i f i t were due to experimental errors alone.  The method used to calculate the best straight l i n e i s to start with the f i r s t s i x points from the flow c e l l data and f i t a straight l i n e on the l n ( h / h ) versus time graph using the method of least squares, and t  Q  calculate Oy and the variance (equal to a ^ ) . significance l e v e l of 0.05,  Using the F test with a  one can say whether the scatter i s greater than  one would expect from random experimental errors alone.  I f the scatter i s  less, seven points would be taken, and a new straight l i n e f i t t e d and a new Oy calculated.  This procedure would be repeated with eight points etc. u n t i l  the scatter i s greater than expected for this significance l e v e l .  In other  words, this approach finds the largest number of points which contribute to the straight l i n e portion of the graph within the experimental error associated with the technique.  The slope of this l i n e i s considered to  give the best estimate of the i n i t i a l permeability.  One can also say with  certainty that the deviations seen (such as those shown i n Figure 17) are due to effects other than random errors, or weighted errors associated with a logarithmic p l o t .  Because of the large amount of data associated with each flow test, the calculations were done on a d i g i t a l computer which also provided  49  plots showing the best f i t l i n e according to the above method. The FORTRAN program for processing the results i s given i n Appendix I I .  4.2.2  Results  The results of the permeability calculations are given i n Table V. Deviations from Darcy's Law are considered to be positive when the values of l n ( h / h ) are larger than would be expected, as i n Figure 17(a), and t  Q  negative for the reverse (Figure 17(b)).  This means that for a positive  deviation the flow rate becomes slower than predicted by Darcy's Law, indicating that the flow i s becoming impeded, and for a negative deviation the flow rate i s more rapid, indicating that the flow channels are possibly becoming larger.  Those experiments where the deviation i s l i s t e d as zero  in Table V are either early experiments which, because they were less accurate, were not used to study deviations, or the resulting plot did not show a clear trend i n either d i r e c t i o n .  From the table, one can see that positive deviations only occur when the dendrite spacing i s greater than 71 microns, and negative deviations only when the spacing i s less than 51 microns. The height of l i q u i d i n the r i s e r pipe when deviations begin i s given i n the table, and i t can be seen that negative deviations begin i n a range 4.5 to 12.7 mm, whereas positive deviations begin i n a range 9.1 to 22.8 mm.  In general, one can conclude  that positive deviations start l a t e r  than negative deviations (also seen i n the two examples i n Figure 17).  The drop i n the l i q u i d l e v e l i n the reservoir when deviations begin has also been calculated and i s given i n Table V.  The calculations  TABLE V RESULTS OF FLOW MEASUREMENTS Average Dendrite Spacing (ym) Primary Secondary  Structure  Initial Permeability (K x 10 2. cm )  Tracer Used?  " "  0.199 0.239 0.105 0.152 0.156 0.136  (R) (R)  21  EQUIAXED  0.0569  (R)  51  33  COLUMNAR  II  II  II  0.367 0.346 0.244 0.407 0.436  28  23  " " 48  q  COLUMNAR  II  II  II  II  II  II  II  II  II  71  49  COLUMNAR  " II  "  0.499 1.47 0.821 1.27 1.17 1.49  Deviations from Darcy's Lav (+ or -)  -  0  -  * * *  Distance f a l len i n reservoir when deviations begin (mm)  0 0  _  *  Height of liquid in r i s e r when deviations begin (mm)  (R)  -0  (R) (R) (R) (R) (R) (R)  + + + + +  7.3 7.3 8.3 7.6  2.4 2.9 2.7 3.2  1.07 1.08 2.52 2.13 1.67 2.07  8.1  1.9  3.81  12.7  3.4  1.18 0.72 1.32 1.22 0.74  4.5 8.5  1.9 2.7  17.3 12.2 17.8 15.7 13.2  3.9 3.4 4.0 3.8 3.7  0  77  -  EQUIAXED  0.300  (R)  0  80  54  EQUIAXED  0.618  (R)  +  Total time at temp. (hrs)  0.67 0.44 0.63 0.53 0.54 0.44 1.86  9.1  2.9  1.19  TABLE CONTINUED  TABLE V Average Dendrite Spacing (pm) Primary Secondary  57  83 it n ii ii  Structure  COLUMNAR  II  II  II  ti it  II II  II  Initial Permeability (K x 10 2 cm )  CONTINUED  Tracer Used?  Deviations from Darcy's Law (+ or -)  0.546 1.06 0.780 0.543 1.21 *  (R) (R) (R) (R)  0 0 0 0 0  Height of liquid i n r i s e r when deviations begin (mm)  ;Distance f a l len i n reser; voir when deviations begin (mm)  Total time at temp. (hrs)  0.62 0.52 0.43 0.57 0.41  103  51  EQUIAXED  0.820  (R)  +  11.7  3.1  0.60  116  57  COLUMNAR  2.15  (R)  +  19.9  4.5  0.29  130  61  EQUIAXED  1.95  (R)  +  20.0  4.4  0.32  83  COLUMNAR ti  6.27 8.20  (R) (R)  + +  22.8 19.1  4.4 3.8  0.15 0.12  175 n  II  Autoradiography showed evidence of flow channelling, therefore these results are considered less r e l i a b l e . (R)  Radioactive tracer was added to the reservoir (casting B) i n these tests.  52  were based on the f i n a l height of the Pb-Sn a l l o y a f t e r testing, therefore they do not necessarily correspond r i s e r pipe.  to the height of l i q u i d i n the  This i s because despite c a r e f u l machining to f i t the Pb-Sn  castings to the flow c e l l , small spaces inevitably remained, and were f i l l e d by the l i q u i d before any flow was  these  detected i n the r i s e r pipe.  For this reason the curves i n Figure 16 and 17 do not pass through t = 0. Negative deviations begin when the l i q u i d has f a l l e n between 1.9 and 3.4 and p o s i t i v e deviations between 2.9 and 4.5 mm. l i q u i d i n the reservoir was 4.3  calculated as 8.4  mm,  The i n i t i a l height of mm.  Dendrite Spacings and Structure  The relationship between primary dendrite spacing and distance from the c h i l l was  determined for columnar castings produced by the four  d i f f e r e n t quenches described i n Table I I .  The results are given i n  Figure 18, and they show that the dependence on distance i s e s s e n t i a l l y linear, i n agreement with previously published work on u n i d i r e c t i o n a l l y cast copper alloys  j _ the l a t t e r work a l i n e a r dependence was n  found f o r secondary arm spacings versus distance from the c h i l l .  also  Consequent-  l y , the average spacings could be determined non-destructively by taking the mean value for the top and bottom surfaces of the casting.  Throughout this work, average primary and secondary spacings have been used as the parameters which characterize the structure of the casting with respect to i n t e r d e n d r i t i c f l u i d flow behaviour.  In these experiments,  the bulk flow i s one dimensional, and i n the case of columnar castings, the flow i s i n the same d i r e c t i o n as the primary dendrites.  The columnar  castings a r e placed i n the flow c e l l so that the i n t e r d e n d r i t i c channels  53  FIGURE 18:  Primary  d e n d r i t e s p a c i n g as a f u n c t i o n o f d i s t a n c e  from the c h i l l , f o r the quenching c o n d i t i o n s i n Table I I .  54  vary i n size only i n the d i r e c t i o n of flow, p a r a l l e l to the axis of the casting.  Therefore, v i s u a l i z i n g the columnar dendritic structure as a stack  of resistances i n s e r i e s , the flow rate w i l l be a function of the sum of these resistances, or a l t e r n a t i v e l y , a function of the average resistance. This reasoning would not necessarily hold for the equiaxed castings, nevertheless, the average spacings have been used because the spacing measured i n d i f f e r e n t locations on the top and bottom surfaces was f a i r l y uniform, and the difference between measurements on the two surfaces was r e l a t i v e l y small. 4.3.1  Autoradiography Autoradiography  made i t possible to examine the Pb-Sn a l l o y  samples after testing to determine whether the measurements which had been made were t r u l y representative of uniform flow through the casting A. Previous experiments on i n t e r d e n d r i t i c f l u i d flow by K a e m p f f e r ^ ^ showed that there i s a strong tendency for superheated  l i q u i d to form p r e f e r e n t i a l  channels by dissolving dendrite branches, and i f this occurred to the same extent i n the present work i t would invalidate the use of Darcy's Law.  Those samples which contained radiocative tracer were sectioned at the mid-point was  of the cylinder A perpendicular to the axis, and the upper half  then sectioned p a r a l l e l to the axis.  Autoradiographs  were made from the  cross sections and longitudinal sections (examples are shown i n Figure 19). In the majority of cases there was either uniform (Figure 19(a)) or no darkening of the f i l m (Figure 19(b)) for the cross sections^except for f i v e of the tast3 (marked with an asterisk i n Table V).  The cross section autoradio-  FIGURE 19:  A u t o r a d i o g r a p h s from c r o s s s e c t i o n s and l o n g i t u d i n a l s e c t i o n s of Pb-Sn samples used f o r i n t e r d e n d r i t i c f l u i d flow s t u d i e s ; p r e f e r e n t i a l channel.  (a) and (b) show uniform  M a g n i f i c a t i o n 2x.  f l o w , (c) shows flow down a  56  graphs of these tests showed a dark spot (Figure 19(c)), and l o n g i t u d i n a l sections showed a non-uniform penetration of tracer which indicated that flow had taken place p r e f e r e n t i a l l y i n this region.  Longitudinal sections  for the majority of samples showed a r e l a t i v e l y uniform penetration of radioactive material, and f o r large dendrite spacings the structure was revealed.  For the smaller spacings the structure was not c l e a r l y revealed,  probably because i t was too fine to be resolved. Figures 20 and 21 show autoradiographs at various levels f o r two of the samples.  from cross sections taken  Figure 20 shows uniform  penetra-  tion of tracer, and microexamination showed no evidence of pipes of the type seen by Kaempffer.  For the sample shown i n Figure 21 the tracer penetration  i s non-uniform, and i t i s also evident that radioactive material has penetrated much further down casting A.  Microexamination revealed several small  pipes close to the top which extended approximately  0.7 cm down.  Enlarged  views of one of the pipes are also shown i n Figure 21 f o r the levels on which they were seen.  The p o s i t i o n of the dark patches on the autoradio-  graphs from lower sections showed that flow had taken place p r e f e r e n t i a l l y down the pipes, even though the pipes themselves were no longer v i s i b l e on the lower sections.  From this evidence i t was f e l t that a single cross  section and longitudinal section would provide s u f f i c i e n t information to determine whether the flow was uniform or not.  Although pipes were seen close to the top of the cylinder A i n the tests which showed non-uniform flow, channels which penetrated right through the dendritic region and caused catastrophic flow, as reported by Kaempffer, were never observed.  Although i t i s f e l t that the permeabilities for the  five tests marked with asterisks i n Table V are less r e l i a b l e than the others,  7  FIGURE 20: Cross s e c t i o n  autoradiographs  at v a r i o u s l e v e l s down the c a s t i n g A, a f t e r  testing.  T r a c e r has p e n e t r a t e d u n i f o r m l y as f a r as the t h i r d 4.6 mm ~ = 28  from the top. um.  Magnification  1.5x.  section,  FIGURE 21: An example o f an unre l i a b l e flow  test,  showing uneven penet r a t i o n of t r a c e r . \ = 83 ym. M a g n i f i c a t i o n of a u t o r a d i o g r a p h s 1.5x.  Microstructures  from the  bottom r i g h t hand  corner  o f the top two s e c t i o n s , showing a p i p e .  Lower  s e c t i o n s d i d not show t h i s defect. M a g n i f i c a t i o n 33x.  59  the r e s u l t s have n e v e r t h e l e s s been i n c l u d e d because i t i s o f i n t e r e s t t h a t they a r e l a r g e r t h a n t h e more r e l i a b l e v a l u e s by a f a c t o r o f between o n l y one and two, f o r t h e same d e n d r i t i c s p a c i n g .  I t i s not completely  c l e a r why p i p e s s h o u l d form i n a few o f the  t e s t s when t h e same procedure was used t h r o u g h o u t .  One can suggest t h a t  the f l o w c e l l was n o t p o s i t i o n e d c o r r e c t l y i n t h e f u r n a c e f o r these  tests,  w h i c h r e s u l t e d i n h i g h e r temperatures a t t h e t o p . S i n c e t h e samples were 2.46 cm i n d i a m e t e r ,  quenching r a t e s were  not as r a p i d as i n some p r e v i o u s experiments where t r a c e r was used t o observe (21 22) convective flow patterns  '  , t h e r e f o r e one might q u e s t i o n whether  quenching t h e f l o w c e l l a f f e c t e d t h e observed f l o w b e h a v i o u r . the f o l l o w i n g r e a s o n s ,  the autoradiographs  I n view o f  a r e b e l i e v e d t o be t r u l y  s e n t a t i v e o f the nature of the flow behaviour  repre-  ( u n i f o r m or v i a p r e f e r e n t i a l  channels): 1)  Both w a t e r quenching and a i r c o o l i n g were used, and d e s p i t e  the  d i f f e r e n t c o o l i n g r a t e s , t h e r e were no o b v i o u s d i f f e r e n c e s i n f l o w p a t t e r n s between s i m i l a r samples c o o l e d i n d i f f e r e n t ways. 2)  The p o s i t i o n o f the c a s t i n g A w i t h i n t h e f l o w c e l l i s o f f - c e n t r e ( F i g u r e 9 ) , and would cause i t t o c o o l more r a p i d l y on one s i d e than t h e o t h e r , y e t no p a t t e r n s were observed w h i c h c o u l d be a t t r i b u t e d t o t h i s effect.  4.4  Microexamination  Microexamination  o f t h e Pb-Sn c a s t i n g s was d i r e c t e d towards  f i n d i n g t h e reasons f o r t h e d e v i a t i o n s from Darcy's Law w h i c h emerged from the f l o w c a l c u l a t i o n s . L o n g i t u d i n a l and c r o s s s e c t i o n s from t h e c e n t r e  60  region of casting A before and after testing are shown i n Figures 22-25. 4.4.1  Negative deviations from Darcy's Law  For the largest dendrite spacing, 175 ym (Figure 22), there i s no apparent difference between the structures before and after testing, however, for the smaller spacings, 71, 51 and 28 ym, obvious as the spacing decreases.  the differences become more  The i n t e r d e n d r i t i c regions appear to  coalesce to some extent to form a continuous  network, and there i s l i t t l e  difference i n appearance between the cross sections and the longitudinal sections.  The microstructures taken after testing also show an o v e r a l l  background of white dots which resemble spheroidized p r e c i p i t a t e s i n other systems.  These white dots exist on a much f i n e r scale i n the microstructures  taken before testing, and are attributed to the formation of a d e n d r i t i c (23) substructure, described i n the l i t e r a t u r e The microstructures therefore indicate that a ripening mechanism i s taking place, especially for the smallest dendrite spacings  (which were  held at temperature for the longest times), which increases the e f f e c t i v e diameter of the flow channels with time.  In Figure 25 the d e n d r i t i c  structure has changed to a type of c e l l u l a r structure, and some flow channels appear to have grown at the expense of others.  The flow paths also  appear less tortuous, which would cause the flow v e l o c i t y to increase. Negative deviations from Darcy's Law,  which were observed to occur only  when the spacing was  are therefore attributed to this  ripening mechanism.  less than 51 ym,  61 Longitudinal Sections  before flow  a f t e r flow Cross S e c t i o n s  before  FIGURE 22:  after  flow  flow  M i c r o s t r u c t u r e s of c a s t i n g A b e f o r e and a f t e r testing,  t = 175 um,  Magnification  60x.  time at temperature 0.12  flow hours.  62  Cross S e c t i o n s  before flow FIGURE 23:  a f t e r flow  M i c r o s t r u c t u r e s of c a s t i n g A b e f o r e and a f t e r f l o w testing.  }, = 71 um,  Magnification  60x.  time at temperature 0.44  hours.  63  Longitudinal Sections  before  after  flow  flow  Cross S e c t i o n s  after  flow  M i c r o s t r u c t u r e s of c a s t i n g A b e f o r e and a f t e r  flow  testing.  hours,  before  FIGURE 24:  flow  X = 51 ym,  Magnification  60x.  time at temperature  1.18  64 Longitudinal Sections  before  a f t e r flow  flow Cross S e c t i o n s  before  FIGURE 25:  after  flow  flow  M i c r o s t r u c t u r e s of c a s t i n g A b e f o r e and a f t e r f l o w testing.  A = 28 um,  Magnification  60x.  time at temperature  1.67  hours,  65 4,4.2  Positive deviations from Darcy's  Law  To account for p o s i t i v e deviations from Darcy's Law  the micro-  structures were examined for evidence that the flow channels for the larger dendrite spacings were becoming more constricted with time. evidence of this type was  found.  However, no  As i n Figure 22, these castings were held  at temperature for a r e l a t i v e l y short time, and showed l i t t l e change a f t e r wards.  Microexamination of the top r e s e r v o i r , which was  originally  casting B, revealed a structure which probably accounts for the p o s i t i v e deviations.  Three examples of t h i s structure are shown i n Figure 26.  microstructures  The  are from longitudinal sections through the r e s e r v o i r , with  the casting A at the bottom.  It can be seen i n Figures 26(a)  and  (b) that  the material i n the reservoir has separated into two layers, the lower layer containing spherical precipitates of high lead content, and the layer above in one case has a fine dendritic structure (Figure 26(b), water quenched) and in the other a coarser dendritic structure (Figure 26(a), a i r cooled).  The  structure of the upper layer i s therefore related to the cooling conditions, but the spherical precipitates are not. consists of dendrites  The l a t t e r material probably  of the primary phase which, owing to t h e i r high lead  content, f e l l to the bottom of the reservoir and spheroidized was  held at temperature.  Although the testing temperature was  as the a l l o y set equal to  the liquidus temperature of casting B, i t appears that the d i s s o l u t i o n of dendrites i s time dependent at t h i s temperature. an Ostwald ripening mechanism i s causing with time.  It also appears as though  the average p a r t i c l e size to increase  Other workers have recently reported seeing the same type of  p r e c i p i t a t e i n a Pb-Sn a l l o y heated above i t s equilibrium liquidus  66 (a) A i r c o o l e d , showing  coarse  dendritic structure  i n the  upper l a y e r . temperature  T o t a l time at 1.86  hours.  (b) Water quenched, showing  fine  d e n d r i t i c s t r u c t u r e i n the upper l a y e r .  T o t a l time at  temperature 0.62  hours.  (c) L i q u i d l e v e l i s e q u a l t o the top of the lower l a y e r .  Total  time at temperature 0.15  hours.  T h i s corresponds t o a drop i n l i q u i d l e v e l of 5.4 mm  (initial  h e i g h t of r e s e r v o i r = 8.4  FIGURE 26:  Microstructures testing.  of the r e s e r v o i r  Magnification  23x.  (casting B), after  mm).  67 (24) temperature It  . does n o t appear, from t h e m i c r o s t r u c t u r e s , as though t h e l a y e r  o f s p h e r o i d i z e d d e n d r i t e s would a c t as an a p p r e c i a b l e b a r r i e r t o f l u i d  flow,  s i n c e the l i q u i d channels appear t o be much l a r g e r t h a n i n the c a s t i n g below.  However, the presence o f t h e l a y e r means t h a t t h e l i q u i d l e v e l can  o n l y f a l l as f a r as t h e t o p o f the l a y e r .  I t i s t h e r e f o r e suggested t h a t  the f l o w r a t e becomes l o w e r than p r e d i c t e d by Darcy's Law, because t h e l i q u i d l e v e l i n t h e r e s e r v o i r i s c o n s t r a i n e d by c a p i l l a r y e f f e c t s t o f a l l a t the same v e l o c i t y as t h e s p h e r i c a l p a r t i c l e s .  T h i s c a p i l l a r y e f f e c t would n o t  be i m p o r t a n t when t h e c o n c e n t r a t i o n o f p a r t i c l e s i s l o w , b u t would become i n c r e a s i n g l y important  as t h e c o n c e n t r a t i o n i n c r e a s e s .  S i n c e i t was shown  e a r l i e r t h a t p o s i t i v e d e v i a t i o n s b e g i n when t h e l i q u i d l e v e l i n t h e r e s e r v o i r has  f a l l e n between 2.9 and 4.5 mm, t h i s would mean t h a t c a p i l l a r y e f f e c t s  s t a r t t o i n f l u e n c e t h e f l o w r a t e when t h e r e s e r v o i r has f a l l e n t o a p p r o x i mately h a l f i t s o r i g i n a l  height.  U s i n g Stoke's Law f o r t e r m i n a l v e l o c i t y o f a s p h e r i c a l p a r t i c l e , one can make a rough e s t i m a t e o f whether t h i s mechanism i s p o s s i b l e :  v  where  =  2  gr (p' 2  -  P)  v  = terminal v e l o c i t y of p a r t i c l e , radius r  g  = a c c e l e r a t i o n due t o g r a v i t y  u  = v i s c o s i t y of l i q u i d  p' p  = density of p a r t i c l e = density of l i q u i d . An e s t i m a t e d mean r a d i u s o f s p h e r i c a l p a r t i c l e s i n F i g u r e 26 i s  68  0.002 cm,  and the density would be approximately 10.0 g/cm  3  .  Liquid  3 density and v i s c o s i t y would be 8.33 g/cm  and 0.03 poise.  This gives a  terminal v e l o c i t y of 0.048 cm/sec, which i s of the same order of magnitude as the flow v e l o c i t y calculated i n section 4.1.1. Ideally, i t should be possible to check t h i s mechanism by quenching  a sample when positive deviations begin, and then examining the structure  of the reservoir.  This i s unfortunately impractical, since the point of  deviation i s not known u n t i l the data have been processed by computer. Therefore  the sample which came closest to f u l f i l l i n g these conditions  examined, and i s shown i n Figure 26(c). sample was  4.4 mm,  and the microstructure  was  The drop i n l i q u i d l e v e l for t h i s represents  a drop of 5.4 mm.  The  l i q u i d l e v e l i s equal to the top of the p r e c i p i t a t e layer, which supports the proposed mechanism.  The density of p r e c i p i t a t e p a r t i c l e s appears to  increase downwards, therefore the rate at which the l i q u i d l e v e l f a l l s beyond the point where p o s i t i v e deviations begin may  be r e l a t e d to a change  in packing of the spheres, and d i s s o l u t i o n effects, i n addition to the terminal v e l o c i t y for a single p a r t i c l e .  4.5  Permeability and Dendrite  Spacing  The permeabilities calculated i n section 4.2 were p l o t t e d against both secondary and primary dendrite spacings (Figures 27 and 28).  Vertical  bars were used to show the scatter between repeated experiments on columnar castings produced under the same quenching conditions, and the data known to be less r e l i a b l e (marked with asterisks i n Table V) was  not included.  Since  -9  the scatter ranges from 1.93 x 10  for a primary spacing of 175 ym,  to  _9  0.134 x 10  for a primary spacing of 28 ym,  i . e . , the scatter i s a function  69  «r  io' 9  r-  < LU  or  bJ Q_  IO-'°-i  10  • columnar A equiaxed  T  t — r  I  I  I I I I T 100 SECONDARY DENDRITE SPACING (microns ± 10%)  FIGURE 27: R e l a t i o n s h i p between t h e i n i t i a l p e r m e a b i l i t y and t h e secondary d e n d r i t e arm s p a c i n g  f o r Pb-Sn a t 193°C  I0" ' 8  slope = 2  ^  10  I-  <  LiJ  or LvJ Q_  • columnar A equiaxed  I0-'°H  10  T  T  1—I  I  I I I I  100  1  PRIMARY DENDRITE SPACING (microns ± 1 0 % )  FIGURE 28: R e l a t i o n s h i p between t h e i n i t i a l p e r m e a b i l i t y and t h e primary dendrite spacing  f o r Pb-20%Sn a t 193°C  71  of the magnitude of K, log-log plots have been chosen as the best method of representing this data.  Figure 27 shows that at the higher secondary arm spacings the permeability increases very rapidly f o r a small increase i n spacing.  In view  of the stated accuracy with which spacings can be measured, i t i s f e l t that primary dendrite spacing i s therefore the more useful parameter f o r characterizing the structure i n terms of the i n t e r d e n d r i t i c f l u i d flow behaviour.  Plotting permeability as a function of primary dendrite spacing, Figure 28 shows that values of permeability measured f o r columnar castings were s l i g h t l y higher than those for equiaxed castings.  This difference i s  probably not s i g n i f i c a n t i n view of the size of the error bars for the columnar castings.  The permeability i s clearly a sensitive function of the cast structure, and t h e o r e t i c a l relationships can be calculated, based on models of the structure.  The simplest model considers the porous medium (the  casting) to be a bundle of straight, p a r a l l e l , c a p i l l a r y tubes aligned i n the direction of flow.  4.5.1  Straight Capillary Model  Flow through a single, straight c a p i l l a r y tube of radius r can be described by the well known Hagen-Poiseulle  equation  72  where q i s t h e f l o w r a t e a l o n g a tube o f l e n g t h L.  For n c a p i l l a r i e s per  u n i t area, the t o t a l flow r a t e per u n i t area ( i . e . v e l o c i t y ) i s 4 nirr v  AP  "  =  ,  r  4  -  . 4  Comparing t h i s e q u a t i o n w i t h Darcy's Law:  v  -V  4.1  ^  4.5  =  tiL  Thus, by analogy  K  -  I t i s common t o add a " t o r t u o s i t y f a c t o r " t  to t h i s  expression  t o account f o r t h e f a c t t h a t the f l o w p a t h s a r e n e i t h e r s t r a i g h t n o r symmetrical,  thus:  The l e n g t h T L then r e p r e s e n t s t h e " e f f e c t i v e l e n g t h " o f t h e f l o w channels.  The l i q u i d f r a c t i o n g , 'L  _  i s g i v e n by;  l i q u i d volume t o t a l volume 2 mrr  8  i.e.  ~  L  g  L  =  TLA LA  nrrr x  From t h i s e x p r e s s i o n r  4  4.7 2 L 2 2 2 n i T g  =  73 Therefore, r e p l a c i n g r  4  i n E q u a t i o n (4.6) 2  K  =  n 7 T  2 2 2  8T \  n  IT  T  2 i.e.  K  =  8nir x  ^  4-  8  (1 2) Piwonka  '  demonstrated t h a t p e r m e a b i l i t y was p r o p o r t i o n a l t o  the square o f t h e f r a c t i o n l i q u i d f o r A l - 4.5%Cu w h i c h was c o n s i s t e n t w i t h the c a p i l l a r y model.  T h e r e f o r e , t a k i n g t h e model one s t e p f u r t h e r , one may  i n t u i t i v e l y s e t t h e number o f c a p i l l a r i e s e q u a l t o t h e number o f channels between p r i m a r y d e n d r i t e s t a l k s .  The s p a c i n g between channels e q u a l s t h e  p r i m a r y d e n d r i t e s p a c i n g A, t h e r e f o r e 1  s * 2  2  L  and  K  =  4.9 8TTT  S i n c e t h e p e r m e a b i l i t y measurements i n F i g u r e 28 were made a t c o n s t a n t temperature,  t h e volume f r a c t i o n o f l i q u i d would be c o n s t a n t .  Assuming t h e t o r t u o s i t y f a c t o r remains c o n s t a n t f o r t h e range o f d e n d r i t e s p a c i n g s s t u d i e d , t h e t h e o r e t i c a l r e l a t i o n s h i p would be t h e s t r a i g h t l i n e o f s l o p e 2 w h i c h has been drawn. p o i n t K = 0,  A = 0;  I n a d d i t i o n , t h i s l i n e goes through t h e  w h i c h cannot be r e p r e s e n t e d i n F i g u r e 28.  This  2 can be checked by showing t h a t t h e r a t i o K/A , i s c o n s t a n t f o r a l l p o i n t s along the l i n e . ^2 A  =  1.46 x 1 0 ~  5  74  3 S i n c e x~  8  =  Therefore,  2 2 L A •^ A  from E q u a t i o n  R  taking g  T  i.e.  =  L  0.19  3  =  99.0  x  =  4.6  at 193°C  A t o r t u o s i t y f a c t o r o f 4.6 of the channels i s 4.6  (4.9)  i m p l i e s t h a t the " e f f e c t i v e  times l o n g e r than i f . t h e y were c o n s i d e r e d  s t r a i g h t and p a r a l l e l .  length" to  be  In most o t h e r p r a c t i c a l a p p l i c a t i o n s of flow  through  f i l t e r beds e t c . , t o r t u o s i t y f a c t o r s are u s u a l l y c o n s i d e r e d  t o l i e between  1 and  capillaries  2, the argument b e i n g  t h a t the average i n c l i n a t i o n of  around random i r r e g u l a r p a r t i c l e s i s about 45°, l e n g t h would be a p p r o x i m a t e l y /2L. together  i n a highly  t h e r e f o r e the mean c a p i l l a r y  However, s i n c e d e n d r i t e branches mesh  r e g u l a r manner, a t o r t u o s i t y v a l u e  o f 4.6  i s not  unreasonable.  T o r t u o s i t y has  been i n t r o d u c e d  as a p r o p e r t y  l e n g t h o f the flow path o f a f l u i d p a r t i c l e , and  equal  t o the  average  attempts have been made i n  the l i t e r a t u r e to measure t o r t u o s i t y by e l e c t r i c a l r e s i s t i v i t y measurements based on the concept that c u r r e n t would flow along fluid particles. direct  T h i s work has been reviewed by  the same paths as  Scheideggerbut  c o r r e l a t i o n between the e l e c t r i c a l and g e o m e t r i c a l  to have been shown.  Scheidegger p o i n t s out  that the  the no  p r o p e r t i e s appears  concept o f t o r t u o s i t y  i s t h e r e f o r e somewhat d o u b t f u l , y e t i t i s c l e a r t h a t the c a p i l l a r y model w i l l f i t any  porous medium i f one  a d j u s t s the v a l u e  of x a p p r o p r i a t e l y .  75  4.5.2  Hydraulic Radius Theory:  Other Theories  A more elaborate model of porous media developed by Kozeny  (25)  attempts to r e l a t e the actual shape of the p a r t i c l e s to the flow behaviour in a more systematic manner than by assigning a tortuosity factor. theory i s based on the observation that permeability, i n absolute  The units,  has the dimensions of a length squared, therefore, i t i s argued, there should be a c h a r a c t e r i s t i c length which describes the permeability.  This  length i s called the "hydraulic radius" (r ) and i s defined as: n pore volume wetted surface area  H  A l t e r n a t i v e l y , the c h a r a c t e r i s t i c length may  be expressed  as the s p e c i f i c  surface of p a r t i c l e s (S ) where P g  _  surface area of p a r t i c l e volume of p a r t i c l e  p  since  g  l i q u i d volume _\ \ z t o t a l volume  =  L  T  L S (1 - g ) p L. g  r  H  4.10  T  Using t h i s approach, the flow v e l o c i t y through the bed i s described by the Kozeny equation: 3 1 K"  8  L  s/d P  - g T T  1  AP  u  L  ...  Li  Therefore, from Equation  4.1 3  K  =  1 — K-  g  —=  s^a p  L -  g r  5-  T  "  4.12  76  K" accepted  value  metallurgy, in  the  i s g e n e r a l l y known a s  to d e s c r i b e  constant,  This  the  and  approach  flow  has  the  commonly  is familiar  of gases  and  in  liquid  metal  furnace.  To  the  Kozeny  f o r most p o r o u s m e d i a .  where i t i s u s e d  blast  expression  of 5  the  apply  the  f o r the  dendrite  approach  specific  spacing.  K  to  interdendritic  surface  of  From E q u a t i o n  fluid  a dendrite,  flow  requires  an  p r e f e r a b l y i n terms  of  (4.12):  l  « S  2  P and  from Figure  28  2 K  =  A  (approximately)  therefore  1 P  *  T h e r e i s no dendrites  that  the  model which would platelike use  of  author satisfy  dendrites,  this  method o f m e a s u r i n g  i s aware o f , the  would not  a d d i t i o n , the Kozeny  small  particles  which  and  the  although  specific  one  could  are  really  capillary  provide  surface  of  postulate  above c o n d i t i o n , f o r e x a m p l e , by  a t o r t u o s i t y f a c t o r i n the  In beds o f  accepted  a  considering  more i n f o r m a t i o n  than  the  model.  theory  i s generally  recommended  nearly  s p h e r i c a l i n shape.  In  only  for  particular,  (25) deviations anything, void  occur the  when t h e  dendrites  f r a c t i o n s should  packing  can  give  be  a void  theory  most  i s used  closely  avoided. fraction  f o r beds  resemble,  For i n the  beds  of  and  of  fibres  also very  spheres  r a n g e 0.2  , which, i f  to  the  0.3,  high  and  densest therefore  very  low  possible one  would  77  expect poor agreement with the present work where the l i q u i d f r a c t i o n held at  was  0.19.  The only experimental evidence which i s available to test the a p p l i c a b i l i t y of the Kozeny theory i s Piwonka's data, discussed e a r l i e r i n  3 section (3.1).  His results have been plotted i n the form of K versus g /(1-j T  following Equation 4.12,  i n Figure 29, and i t can be seen that this plot does  not correspond w e l l to a straight l i n e , compared to the previous plot of 2 K versus g  (Figure 7).  L  Scheidegger^^  has given a comprehensive c r i t i c i s m of the Kozeny  theory, drawing attention i n p a r t i c u l a r to i t s i n a b i l i t y to describe anisotropic permeability, and he reviews other theories which are more applicable to beds pf f i b r e s , which come under the general heading of drag theories of permeability.  Unfortunately, a basic assumption i n the drag theory is.that  the spacing between i n d i v i d u a l fibres i s large compared to the f i b r e diameters  (high void fractions) and that flow disturbance due to adjacent  fibres i s n e g l i g i b l e .  This would clearly not be applicable to the present  studies on i n t e r d e n d r i t i c f l u i d flow.  Consequently, i t was  f e l t that the simple c a p i l l a r y model provided  a useful empirical approach for r e l a t i n g permeability to structure despite the l i m i t a t i o n that i t does not give much information regarding the nature of flow on a microscopic scale.  78  79  4.6  Dendrite  Coarsening  Negative  d e v i a t i o n s from Darcy's Law, w h i c h o c c u r r e d when the  p r i m a r y d e n d r i t e s p a c i n g s were l e s s than 51 ym, have been a t t r i b u t e d t o m i c r o s t r u c t u r a l changes such as those shown i n F i g u r e s 24 and 25. e f f e c t s have been r e p o r t e d by K a t t a m i s  et a l .  the s o l i d - l i q u i d r e g i o n f o r v a r i o u s t i m e s .  Similar  on A l - C u a l l o y s h e l d i n  The authors  observed an apparent  i n c r e a s e i n the secondary d e n d r i t e arm s p a c i n g w i t h t i m e , and they proposed two m a t h e m a t i c a l models f o r t h e d i s s o l u t i o n o f d e n d r i t e arms, based on the d i f f e r e n c e i n s o l u b i l i t y between two s u r f a c e s of d i f f e r e n t c u r v a t u r e .  The  d r i v i n g f o r c e f o r t h e p r o c e s s was the r e d u c t i o n o f s u r f a c e a r e a , and the r a t e o f d i s s o l u t i o n was c o n t r o l l e d by d i f f u s i o n i n the l i q u i d .  Both mathe-  m a t i c a l models gave q u a l i t a t i v e agreement w i t h the o b s e r v a t i o n s , though i t was not p o s s i b l e t o choose one model over t h e o t h e r s .  A s i m i l a r coarsening process  i s b e l i e v e d t o t a k e p l a c e i n the Pb-Sn  c a s t i n g s used f o r i n t e r d e n d r i t i c f l u i d f l o w s t u d i e s , a l t h o u g h F i g u r e s  23-25  appear t o show s p h e r o i d i z a t i o n of t h e d e n d r i t i c s t r u c t u r e , r a t h e r than the d i s s o l u t i o n o f secondary b r a n c h e s .  T h i s process  i s r e l a t e d t o the work done by Ostwald i n 1900, who  observed an i n c r e a s e i n s o l u b i l i t y w i t h a decrease i n p a r t i c l e s i z e f o r various s a l t s o l u t i o n s i n water.  L a t e r workers who observed t h a t l a r g e  p a r t i c l e s tended t o grow at the expense o f s m a l l e r p a r t i c l e s named the process Ostwald r i p e n i n g .  D i f f u s i o n c o n t r o l l e d growth o f s p h e r i c a l p r e c i p (27 28)  i t a t e s was a n a l y z e d by Greenwood  '  , and he d e r i v e d t h e f o l l o w i n g  e x p r e s s i o n f o r the growth r a t e w i t h r e s p e c t t o the r a d i u s o f the p a r t i c l e ( r ) :  80 2DS  dr dt  00  Va 4.13  kTr  where  diffusion S  coefficient  of s o l u t e  =  s o l u b i l i t y of a p a r t i c l e of i n f i n i t e  r  =  mean r a d i u s of the system of  V  =  m o l a r volume of p a r t i c l e  a  =  i n t e r f a c i a l energy  k  =  Boltzmann's  T  =  absolute  ro  radius  particles  constant  temperature.  From t h i s e q u a t i o n Greenwood n o t e d t h a t the maximum growth r a t e corresponds t o the p a r t i c l e w i t h t w i c e the mean r a d i u s .  Extending  his  a n a l y s i s to a c o n s i d e r a t i o n of p a r t i c l e s i z e d i s t r i b u t i o n , he a r r i v e d following  e x p r e s s i o n f o r the v a r i a t i o n  at  the  of the mean p a r t i c l e r a d i u s w i t h  time ( t ) 8DS  -3 r  oVt 4.14  9kT  Experimental  e v i d e n c e f o r a dependence of r  found i n s t u d i e s of the growth of p a r t i c l e s a l i q u i d , but good agreement has  on time has  been  i n a s t r a i n f r e e medium, u s u a l l y  a l s o been found f o r c o b a l t p a r t i c l e s  copper m a t r i x , copper i n i r o n , manganese i n magnesium, and o t h e r s  in a  reviewed  ( 2 8} by Greenwood The as E q u a t i o n  growth r a t e d e r i v e d by K a t t a m i s e t a l . i s e s s e n t i a l l y  the same  (4.13), except t h a t i t i s based on the growth and d i s s o l u t i o n  c y l i n d r i c a l p a r t i c l e s r a t h e r than s p h e r i c a l  ones.  The model cannot  readily  be extended t o a c o n s i d e r a t i o n of p a r t i c l e s i z e d i s t r i b u t i o n , t h e r e f o r e r e l a t i o n corresponding  to E q u a t i o n  (4.14) has been d e r i v e d .  As a  of  first  no  81  approximation,  t h e r e f o r e , t h e r e s u l t s o f t h e Greenwood t h e o r y have been  a p p l i e d t o the p r e s e n t work.  The m e t a l l o g r a p h i c e v i d e n c e ,  especially  F i g u r e 25, shows t h a t t h e i n t e r d e n d r i t i c channels (pores) become more s p h e r i c a l , y e t they must o f course  remain connected o r f l o w would s t o p .  Assuming t h e mean r a d i u s o f i n t e r d e n d r i t i c pores i s a f u n c t i o n o f time as d e r i v e d i n E q u a t i o n r  3  «  (4.14)  t  From t h e c a p i l l a r y model o f p e r m e a b i l i t y ( E q u a t i o n 4.6) K « r  4  i t follows that K « t / 4  3  4  A d i f f e r e n t view o f t h e c o a r s e n i n g be o b t a i n e d by comparing t h e p r o c e s s two spheres o f r a d i u s P i n c o n t a c t  '  1 5  o f i n t e r d e n d r i t i c channels can  t o s i n t e r i n g i n ceramics.  Considering  ( F i g u r e 3 0 ) , where t h e a r e a o f c o n t a c t i s (29)  a c i r c l e o f r a d i u s x, i t can be shown x -  1  P  t  that  1/5  for diffusion controlled  growth.  The r a d i u s o f c u r v a t u r e a t the neck r , i n c r e a s e s w i t h x:  r  =  2 x 4p" 2/5  Therefore  t h e r a d i u s o f c u r v a t u r e grows as a f u n c t i o n o f t I f one p i c t u r e s t h e i n t e r d e n d r i t i c channels as sharp c r e v i c e s  FIGURE 30:  Growth o f a neck d u r i n g s i n t e r i n g .  TABLE V I RESULTS OF DENDRITE COARSENING CALCULATIONS  Initial Permeability (K x 10 cm ) o  Power (b)  Constant (m)  28 ti  0.105  2.6  3.7 X i o -  1 7  0.152  2.3  4.8 X i o -  1 6  ii  0.156  3.3  4.1 X i o "  1 9  II  0.136  2.6  6.0 X i o "  1 7  48  0.0569  2.7  4.4 X i o "  1 8  51 it  0.367  1.9  3.9 X i o "  1 4  0.244  1.8  1.1 X i o "  1 3  II  0.407  2.2  8.1 X i o "  1 5  Primary Dendrite Spacing (pm)  9  From l e a s t squares f i t t i n g :  2  e r r o r i n b, a p p r o x i m a t e l y 30% e r r o r i n m, a p p r o x i m a t e l y 65%  83  between secondary arms, one can assume t h a t the e f f e c t i v e r a d i u s o f the channels changes i n the same manner as the r a d i u s o f c u r v a t u r e o f the necked r e g i o n . F o l l o w i n g the same argument as b e f o r e :  K  oc  r  °c  t  4.16  Using e i t h e r the Ostwald r i p e n i n g o r the s i n t e r i n g a p p r o a c h , the r e l a t i o n between p e r m e a b i l i t y and time may be w r i t t e n as a power f u n c t i o n :  K  where K K  q  K  +  o  at  1  4.17  i s the p e r m e a b i l i t y a t t = 0, and 'a' i s a c o n s t a n t .  q  Consequently  i s e q u a l t o t h e v a l u e o f p e r m e a b i l i t y c a l c u l a t e d from t h e i n i t i a l  slope  of a curve such as F i g u r e 1 7 ( b ) , and t = 0 i s d e f i n e d as t h e time when l n ( h /h ) = 0. t o  The d i f f e r e n t i a l form o f Darcy's Law ( E q u a t i o n  8,  Appendix I ) may now be r e w r i t t e n f o r a v a r i a b l e K dt  (K  (K  o  i.e.  p+1  (p+1)  ,P+1  t  P  + 1  4.18  + at*) t  + at )dt p  o  dh.  -c  =  dh.  -c  =  -c In oj  •c In  -  oJ  K  o  t  P l o t t i n g t a g a i n s t the r i g h t hand s i d e o f E q u a t i o n  4.19 (4.19) on a  l o g - l o g s c a l e s h o u l d g i v e a s t r a i g h t l i n e o f s l o p e (p+1), and an example i s  84  shown i n F i g u r e 31.  A l t e r n a t i v e l y , one may  use the method o f l e a s t  squares  to f i t an e x p r e s s i o n of the form Y = mX* t o the d a t a where: 3  Y  K t o  X  =  t  m  =  a/(P  and b  =  P +  +1) 1.  T h i s has been done f o r the e i g h t t e s t s i n Table V w h i c h showed n e g a t i v e d e v i a t i o n s , and the r e s u l t s a r e l i s t e d i n T a b l e V I . Ostwald  r i p e n i n g argument the power b s h o u l d e q u a l 2.33,  s i n t e r i n g argument b s h o u l d e q u a l 2.6.  F o l l o w i n g the  and f o l l o w i n g  the  W i t h i n the a c c u r a c y o f the e x p e r i -  m e n t a l t e c h n i q u e , the r e s u l t s show f a i r l y good agreement w i t h e i t h e r o f the proposed mechanisms, though i t i s not p o s s i b l e t o choose one over t h e o t h e r . The  c o n s t a n t 'm'  w h i c h has a l s o been c a l c u l a t e d i s a f u n c t i o n of a  l a r g e number of parameters i n b o t h mechanisms, i n c l u d i n g the d i f f u s i o n  coeffi-  c i e n t and i n t e r f a c i a l t e n s i o n , f o r w h i c h a c c u r a t e d a t a a r e not a v a i l a b l e .  It  would a l s o p r o b a b l y v a r y w i t h d e n d r i t e s p a c i n g , but i n v i e w o f the range of v a l u e s (6 o r d e r s of magnitude o f a v e r y s m a l l number), i t i s not p o s s i b l e t o a t t a c h too much s i g n i f i c a n c e t o t h i s  constant.  On the b a s i s of these r e s u l t s , i t appears t h a t the change i n p e r m e a b i l i t y of the c a s t i n g when i t i s heated above t h e s o l i d u s can be compared e i t h e r t o Ostwald during s i n t e r i n g .  temperature  r i p e n i n g , o r t o the changes w h i c h take p l a c e  I n e i t h e r c a s e , d i f f u s i o n has been chosen as the mechanism  by w h i c h m a t e r i a l i s removed from convex s o l i d r e g i o n s and i s d e p o s i t e d concave r e g i o n s .  T h i s p r o c e s s would take p l a c e i n a l l f l u i d f l o w  on  experiments,  FIGURE  31:  D e n d r i t e c o a r s e n i n g p l o t f o r a sample w i t h  K  Q  =  0.152  cm  , average p r i m a r y d e n d r i t e  s p a c i n g 28 pm. oo  86  but i t would have the greatest e f f e c t i n those tests where the casting held i n the s o l i d - l i q u i d region f o r the longest times.  was  Conversely, p o s i t i v e  deviations, which depend on the v e l o c i t y with which p a r t i c l e s f a l l i n the reservoir, would probably have the greatest e f f e c t i n those castings which had the highest flow rates.  Microexamination of the reservoir from  castings with the smallest dendrite spacings revealed that the l i q u i d l e v e l had not f a l l e n to the l e v e l of the p a r t i c l e layer, therefore positive deviation effects were considered n e g l i g i b l e , and ignored i n the calculation of the data i n Table VI.  4.7  The Scatter of Permeability Results The results have shown that permeability measurements are affected  by the dendrite spacing, dendrite coarsening e f f e c t s , p r e f e r e n t i a l flow due to the formation of pipes, and external e f f e c t s i n the l i q u i d r e s e r v o i r above the casting.  Other factors which could influence the results were  variations i n composition and  temperature.  To check whether composition changes had taken place within casting A during the f l u i d flow experiments, density measurements were made before and after testing on 10 of the samples.  The machined cylinder of casting A  was used f o r the f i r s t measurement (weighing i n a i r and i n water), and a c y l i n d r i c a l sample was machined a f t e r the test for the second measurement. Densities of standard specimens were measured and plotted to give the c a l i bration curve i n Figure 32.  The densities could be measured to an accuracy  of about ± 0.5% by this method, and within this error, no difference i n the composition before and after the test could be detected.  The effect of temperature  variations was  tested by making flow  WEIGHT  FRACTION  Ol  ATOMIC  FRACTION  FIGURE 32: C a l i b r a t i o n c u r v e ; d e n s i t y at 25°C as a f u n c t i o n I  51  180  1  r Eutectic i  ,  Sn  0.2  •  Temperature I 190  —  0.3  Sn  o f Pb-Sn a l l o y s  (g/cm ) 3  of composition. l  1  183 ° C i 200 TEMPERATURE  I 210  1  220  °C  FIGURE 33: The r e l a t i o n s h i p between p e r m e a b i l i t y  and temperature.  88  measurements on one  sample at d i f f e r e n t t e m p e r a t u r e s .  d a t a p o i n t s were r e c o r d e d t o a new  temperature the sample was  temperature and a n o t h e r s e r i e s of p o i n t s was  t e c h n i q u e was  r a p i d l y heated  recorded.  This  l i m i t e d by the tendency towards p r e f e r e n t i a l f l o w c h a n n e l l i n g  (pipe formation) i t was  at one  After sufficient  i f the i n t e r d e n d r i t i c l i q u i d became s u p e r h e a t e d , t h e r e f o r e  o n l y p o s s i b l e t o make measurements c l o s e t o the e u t e c t i c t e m p e r a t u r e .  Subsequent m e t a l l o g r a p h i c e x a m i n a t i o n o f t h i s sample showed t h a t changes i n the s t r u c t u r e of the c a s t i n g A were m i n i m a l and t h a t the l e v e l i n the r e s e r v o i r had not  f a l l e n t o the top of the p a r t i c l e l a y e r ,  t h e r e f o r e p o s i t i v e and n e g a t i v e t o be  liquid  d e v i a t i o n s from Darcy's Law  were  considered  negligible. The  p e r m e a b i l i t y was  r e s u l t s , p l o t t e d i n F i g u r e 33,  c a l c u l a t e d f o r each t e m p e r a t u r e , and  the  show t h a t the p e r m e a b i l i t y measurements are  r e l a t i v e l y i n s e n s i t i v e t o s m a l l changes i n temperature c l o s e t o the e u t e c t i c . I t was was  f o r t h i s r e a s o n t h a t the t o l e r a n c e of ± 3°C  considered  Using  on the t e s t i n g temperature  acceptable.  the c a p i l l a r y model d e s c r i b e d i n s e c t i o n 4.5  r e l a t i o n s h i p between p e r m e a b i l i t y and temperature may  Using  x = 4.6,  A = 67 um,  and g  a theoretical  be c a l c u l a t e d , s i n c e :  as a f u n c t i o n of temperature from Li  the t a b l e i n Appendix I I I , the t h e o r e t i c a l curve has been p l o t t e d i n F i g u r e and a l s o shows t h a t the p e r m e a b i l i t y would not be v e r y s e n s i t i v e t o tempera t u r e c l o s e t o the e u t e c t i c .  From the t h e o r e t i c a l l i n e , a v a r i a t i o n of  33  89  ± 3 C at 193 C would produce an e r r o r of about ± 5% i n the v a l u e  of  p e r m e a b i l i t y , w h i c h i s c o n s i d e r a b l y s m a l l e r than the e r r o r bars i n F i g u r e 28.  Consequently c o m p o s i t i o n  and t e m p e r a t u r e v a r i a t i o n s a r e  thought t o be the major f a c t o r s r e s p o n s i b l e f o r the s c a t t e r i n the  not results.  S i n c e the p e r m e a b i l i t y has been shown t o be v e r y s e n s i t i v e t o s t r u c t u r e , the m a j o r f a c t o r i n f l u e n c i n g the s c a t t e r i n F i g u r e 28 i s p r o b a b l y the u n c e r t a i n t y i n the measured v a l u e o f the d e n d r i t e s p a c i n g .  In a d d i t i o n ,  t h e r e would be e r r o r s i n v o l v e d i n d e s c r i b i n g the s t r u c t u r e of the e q u i a x e d c a s t i n g s i n terms o f an average v a l u e .  However, i t i s f e l t t h a t , d e s p i t e  the s c a t t e r , the r e s u l t s cover a s u f f i c i e n t l y wide range one o r d e r o f magnitude i n p r i m a r y  (approximately  d e n d r i t e s p a c i n g , and two o r d e r s  magnitude i n p e r m e a b i l i t y ) f o r the e m p i r i c a l r e l a t i o n s h i p to be  of  valid.  90  CHAPTER 5 THE EFFECT OF DENSITY DIFFERENCES ON THE FORMATION OF CHANNELS 5.1  I n t r o d u c t i o n and Review o f P r e v i o u s Work To apply t h e r e s u l t s o f t h e i n t e r d e n d r i t i c f l u i d f l o w e x p e r i m e n t s  to a p r a c t i c a l c a s t i n g problem, a s t u d y was conducted on t h e f o r m a t i o n o f channel-type  d e f e c t s , namely, f r e c k l e s and A  segregates.  The term f r e c k l e s has been used f o r a number o f d i f f e r e n t types o f defects which are probably not  a l l caused by t h e same mechanism.  Freckles i n  i r o n and n i c k e l base s u p e r a l l o y s can appear as d i s t i n c t t r a i l s o f e q u i a x e d g r a i n s on t h e s u r f a c e o f d i r e c t i o n a l l y s o l i d i f i e d c a s t i n g s .  These f r e c k l e  t r a i l s a r e o b s e r v e d t o b e g i n a t some d i s t a n c e from t h e c h i l l f a c e , and t h e t o t a l number decreases w i t h d i s t a n c e from the c h i l l ^ ^ . 3  The t r a i l s a r e  c a l l e d f r e c k l e s because of the s p e c k l e d appearance o f t h e e q u i a x e d g r a i n s . F i g u r e 34 shows examples o f f r e c k l e l i n e s i n i n g o t s o f Mar-M200  (compositions  of v a r i o u s s u p e r a l l o y s a r e g i v e n i n Table V I I ) .  of the  The c o m p o s i t i o n  m a t e r i a l i n the t r a i l s has been shown t o be r i c h i n those s o l u t e elements w h i c h segregate  n o r m a l l y , and t h i s , c o u p l e d w i t h the e v i d e n c e o f f e e d i n g  s h r i n k a g e w i t h i n the t r a i l s  (Figure 3 4 ( b ) ) , leads to the c o n c l u s i o n that the  f r e c k l e s r e p r e s e n t t h e l a s t l i q u i d i n t h e system t o s o l i d i f y .  The photographs  i n F i g u r e 34 have been taken from the p u b l i s h e d work o f Giamei and K e a r ^ ^ . 3  A second type, o f f r e c k l e d e f e c t i s i l l u s t r a t e d i n F i g u r e 35.  In  t h i s c a s e , m a t e r i a l c o n t a i n i n g h i g h e r c o n c e n t r a t i o n s o f the s o l u t e elements has accumulated i n patches o r s p o t s i n t h e i n t e r i o r o f t h e c a s t i n g .  These  91  FIGURE 34:  Freckle t r a i l s i n directionally  s o l i d i f i e d Mar-M200  (a) 10 cm d i a m e t e r i n g o t showing t r a i l s of e q u i a x e d g r a i n s ; diameter - s i n g l e c r y s t a l showing f e e d i n g s h r i n k a g e  FIGURE 35:  axis of a d i r e c t i o n a l l y  s o l i d i f i e d ingot.  (b) 3.8 cm  along f r e c k l e  F r e c k l e s i n a s - c a s t I n c o n e l 718, p e r p e n d i c u l a r  ;  t o the  M a g n i f i c a t i o n lOx.  line.  TABLE V I I COMPOSITION OF SUPERALLOYS^ °^ (wt. p e t . ) 3  Mar-M200 (nominal) I n c o n e l 718  A286  Ti  W  Mo  Nb  Ta  V  Mn  -  1.0  -  -  -  -  -  Cr  Al  9.5  5.0  2.0 12.5  19.0  0.6  0.9  -  3.0  14.7  -  2.1  -  1.25  (5.0 sum)  -  -  0.30  1.5  Si  <0.2  -  B  C  0.015 0.15  -  0.70 0.005  Fe  -  Co  Ni  10.0  bal.  0.10  18.0  -  bal.  -  bal.  -  25.5  93  patches can be seen on a macroetched s u r f a c e , b u t they do n o t n e c e s s a r i l y form t r a i l s .  Areas o f l o c a l s e g r e g a t i o n o f t h i s type appear t o be a r e l a t i v e -  l y common f e a t u r e o f some i r o n and n i c k e l base consumable a r c m e l t e d i n g o t s . I t has been p o i n t e d out t h a t d e f e c t s o f t h i s type can be d e t r i m e n t a l even (31) i n i n g o t s t h a t a r e t o be h o t worked (DeVries  and Mumau  reported  that  they c o u l d n o t be removed even a f t e r 90% h o t r e d u c t i o n ) . I n a d d i t i o n t o t h e s e two t y p e s , t h e term f r e c k l e s has been used f o r many c a s t i n g d e f e c t s t h a t have a " s p o t t y " appearance, e i t h e r on t h e s u r f a c e o r on a p o l i s h e d s e c t i o n , whether o r n o t t h e spots a r e a c c u m u l a t i o n s of s o l u t e o r p o r o s i t y , and whether o r n o t they a r i s e due t o t h e s o l i d i f i c a t i o n c h a r a c t e r i s t i c s o f t h e a l l o y , o r mould w a l l e f f e c t s . was  The p r e s e n t  work  t h e r e f o r e r e s t r i c t e d t o a study o f t h e s o l i d i f i c a t i o n c o n d i t i o n s w h i c h  c o u l d cause the f o r m a t i o n  of the f i r s t  Two main e x p l a n a t i o n s  (channel) type o f f r e c k l e .  f o r the o r i g i n o f channel-type f r e c k l e s (32)  have appeared i n t h e l i t e r a t u r e .  Gould  , n o t i n g t h a t f r e c k l e t r a i l s were  r e l a t e d t o t h e d i r e c t i o n o f g r a v i t y , suggested t h a t they might be formed by gas b u b b l e s i n t h e l i q u i d .  I f a bubble were n u c l e a t e d  a t the s o l i d - l i q u i d  i n t e r f a c e , i t would r i s e v e r t i c a l l y and i t s t r a c e would be f i l l e d by lower melting point l i q u i d .  Gould r e p o r t e d , however, t h a t n e i t h e r v a r y i n g t h e  n i t r o g e n and oxygen c o n t e n t , n o r e l i m i n a t i n g hydrogen i n e x p e r i m e n t s on A-286 o r I n c o n e l 718, had any d i s c e r n a b l e e f f e c t on t h e i n c i d e n c e o f f r e c k l i n g . A more comprehensive study o f t h e r e l a t i o n s h i p between b l o w h o l e s and "spot s e g r e g a t e s "  ( i . e . , t h e second type o f f r e c k l e s ) was made by  (33) Mukherjee  i n electroslag ingots.  He found t h a t t h e o c c u r r e n c e o f spot  94  segregates  c o u l d be i n f l u e n c e d by v a r y i n g the oxygen c o n t e n t  and by  v i b r a t i o n , s u p p o r t i n g t h e concept t h a t f r e c k l e s may be caused by gas bubbles.  He suggested a mechanism s i m i l a r t o t h a t o f G o u l d , t o account  f o r f r e c k l e t r a i l s caused by r i s i n g  bubbles.  (34) Copley, Giamei e t a l .  o f f e r e d an e x p l a n a t i o n b a s e d on t h e  f o r m a t i o n o f lower d e n s i t y l i q u i d c l o s e t o the bottom o f the s o l i d - l i q u i d region during s o l i d i f i c a t i o n .  I f the l o w e s t m e l t i n g p o i n t  interdendritic  l i q u i d i s l e s s dense than t h e b u l k l i q u i d above, t h i s would cause upward flow of the l i g h t e r l i q u i d .  As the l i q u i d r i s e s , i t would move towards  h o t t e r r e g i o n s o f the c a s t i n g , becoming s u p e r h e a t e d .  T h i s would l e a d t o  d i s s o l u t i o n o f d e n d r i t e branches i n i t s p a t h , and the f o r m a t i o n o f a channel.  They examined t h e s o l i d i f i c a t i o n o f a t r a n s p a r e n t  ammonium  c h l o r i d e - w a t e r model i n which a " d e n s i t y i n v e r s i o n " o f t h i s type  occurred,  and observed upward f l o w i n g p i p e s t h r o u g h t h e s o l i d - l i q u i d r e g i o n , w h i c h supported  their  hypothesis.  They a l s o p r e s e n t e d which considered  a semiquantitative mathematical a n a l y s i s  the maximum d r i v i n g f o r c e f o r f r e c k l i n g t o be the p o t e n t i a l  energy d i f f e r e n c e between the u n s t a b l e l i q u i d c o n f i g u r a t i o n , w i t h t h e most dense l i q u i d l a y e r on t o p , and a s t a b l e c o n f i g u r a t i o n , w i t h t h e most dense l a y e r a t t h e bottom.  The model d i d n o t c o n s i d e r r e s i s t a n c e t o f l o w t h r o u g h  the s o l i d - l i q u i d r e g i o n , b u t i t was p o s s i b l e t o make q u a l i t a t i v e p r e d i c t i o n s on the e f f e c t o f changing t h e c o o l i n g c o n d i t i o n s and changing t h e a l l o y composition.  They e x p l a i n e d t h e l o c a t i o n o f f r e c k l e s on t h e s u r f a c e o f t h e  i n g o t i n terms o f t h e shape o f t h e s o l i d - l i q u i d i n t e r f a c e , and by v a r y i n g t h i s shape i n t h e ammonium c h l o r i d e - w a t e r model they produced e i t h e r i n t e r n a l or surface pipes.  95  The  concept t h a t d e n s i t y d i f f e r e n c e s cause f r e c k l e s i s c o n s i s t e n t (35)  with the observations  of Smeltzer  who n o t e d t h a t the d e f e c t c o u l d be (31)  e l i m i n a t e d by changing t h e a l l o y c o m p o s i t i o n ,  and D e V r i e s and Mumau  reported that the accepted i n d u s t r i a l p r a c t i c e f o r reducing t o l o w e r t h e power i n p u t t o t h e consumable f u r n a c e . e f f e c t o f changing t h e c o o l i n g c o n d i t i o n s . draw any a p r i o r i c o n c l u s i o n s  regarding  gradient o r the f r e e z i n g r a t e . reported  f r e c k l e s was  T h i s would have t h e  However, i t i s n o t p o s s i b l e t o  t h e e f f e c t on t h e temperature  I t i s i n t e r e s t i n g t o n o t e t h a t they a l s o  t h a t f r e c k l e s c o u l d be reduced by d e c r e a s i n g Part of the extensive  who  theory  the d e n d r i t e  spacing.  o f m a c r o s e g r e g a t i o n p u b l i s h e d by  (9) Mehrabian e t a l . ,  which i s discussed  i n more d e t a i l i n s e c t i o n 7.1, d e a l s  with the formation  of channel-type defects.  B a s i c a l l y , the theory  involves  the c a l c u l a t i o n o f t h e i n t e r d e n d r i t i c f l u i d v e l o c i t y a t e v e r y p o i n t i n t h e s o l i d - l i q u i d r e g i o n when t h e f o r c e s a c t i n g on t h e f l u i d a r e s o l i d i f i c a t i o n c o n t r a c t i o n s and g r a v i t y . formation  They propose t h a t t h e c r i t i c a l c o n d i t i o n f o r t h e  o f c h a n n e l - t y p e d e f e c t s i s when t h e d i r e c t i o n o f t h e i n t e r d e n d r i t i c  f l u i d f l o w v e c t o r goes from t h e c o l d e r t o t h e h o t t e r r e g i o n s (34) This hypothesis  i s s i m i l a r t o t h a t of Copley e t a l . ,  since  of the c a s t i n g . density,  i n v e r s i o n s can l e a d t o i n t e r d e n d r i t i c f l u i d f l o w i n t h e d i r e c t i o n o f i n c r e a s i n g temperature, but i t d i f f e r s i n that the c r i t i c a l c o n d i t i o n i s not the s i g n o f t h e d e n s i t y change, b u t t h e magnitude and d i r e c t i o n o f t h e f l o w velocity vector.  Thus f r e c k l e s need n o t m e r e l y be upward t r a i l s , b u t they  can go i n any d i r e c t i o n o f i n c r e a s i n g t e m p e r a t u r e , depending on the f l o w p a t t e r n w i t h i n t h e s o l i d - l i q u i d zone. The model p e r m i t s c a l c u l a t i o n o f t h e f l o w p a t t e r n s  i n an i d e a l i z e d  96  i n g o t w h i c h s o l i d i f i e s u n i d i r e c t i o n a l l y from one s i d e - w a l l , and they p r o v i d e (36) experimental evidence ingot of t h i s type.  o f a c h a n n e l - t y p e d e f e c t i n a degassed  Al-20%Cu  The authors e x t e n d t h e i r t h e o r y t o commercial i n g o t s by  s u g g e s t i n g a number o f assumed f l o w p a t t e r n s w h i c h would l e a d t o c h a n n e l type d e f e c t s . (34) The experiments done by Copley e t a l .  were s i m i l a r t o o t h e r s  (37) done by McDonald and Hunt,  who examined an ammonium c h l o r i d e - w a t e r model  of a c o n v e n t i o n a l c a s t i n g , and observed t h a t f l u i d f l o w o c c u r r e d through the s o l i d - l i q u i d r e g i o n s i n the form o f r i s i n g p i p e s .  Using d e n s i t y data  f o r ammonium c h l o r i d e - w a t e r , they suggested t h a t t h e p i p e s were formed by lower d e n s i t y l i q u i d i n t h e i n t e r d e n d r i t i c r e g i o n s , and they c o n s i d e r e d these p i p e s t o be analogous t o A s e g r e g a t e s i n l a r g e s t e e l c a s t i n g s . There a r e a l t e r n a t i v e e x p l a n a t i o n s f o r t h e o r i g i n o f A s e g r e g a t e s , (3 8) f o r example the s u g g e s t i o n by Blank and P i c k e r i n g  that a solute enriched  l a y e r i s formed between the columnar g r a i n s a t t h e s i d e s , and t h e e q u i a x e d g r a i n s i n t h e c e n t r e o f the i n g o t .  T h i s e n r i c h e d l a y e r i s s u b s e q u e n t l y drawn  back i n t o t h e columnar r e g i o n s t o form A s e g r e g a t e s by volume s h r i n k a g e during s o l i d i f i c a t i o n .  E x p l a n a t i o n s o f t h i s type a r e l e s s s a t i s f a c t o r y  than  the mechanism suggested by McDonald and Hunt s i n c e they do n o t account f o r e i t h e r t h e c h a r a c t e r i s t i c c h a n n e l - l i k e shape o f t h e s e g r e g a t e s , o r t h e i r characteristic inclination. The ammonium c h l o r i d e - w a t e r experiments e f f e c t i v e l y proved t h a t channels c l o s e l y r e s e m b l i n g f r e c k l e s and A s e g r e g a t e s c o u l d be formed by t h e upward f l o w of l e s s dense l i q u i d i n t h i s system.  However, t h e use o f t r a n s -  parent models t o s i m u l a t e s o l i d i f i c a t i o n i n m e t a l c a s t i n g s has been q u e s t i o n e d  97  by a number of w o r k e r s ~ " ' " . v  1  I t has been shown by S t e w a r t  using  }  r a d i o a c t i v e t r a c e r t e c h n i q u e s , t h a t the c o n v e c t i v e f l o w v e l o c i t y and  flow  p a t h i n w a t e r based systems can be markedly d i f f e r e n t t o t h a t w h i c h o c c u r s i n l i q u i d metals.  T h i s i s due t o the l a r g e d i f f e r e n c e i n the P r a n d t l (21)  number (0.013 f o r l i q u i d t i n , 10.0  f o r water  ), a dimensionless  parameter w h i c h c h a r a c t e r i z e s c o n v e c t i v e f l o w . The e x t e n t to w h i c h t h e r m a l c o n v e c t i o n would i n f l u e n c e the mechanism o f f r e c k l e and A s e g r e g a t e f o r m a t i o n observed by  Copley,  McDonald and o t h e r s i s u n c e r t a i n when a p p l i e d t o a m e t a l system. i t was  considered important  behaviour  Therefore  to i n v e s t i g a t e the i n t e r d e n d r i t i c f l u i d  flow  i n an a l l o y , when l e s s dense l i q u i d e x i s t s a t the bottom o f the  s o l i d - l i q u i d zone.  The  f o l l o w i n g experiments  were t h e r e f o r e d i r e c t e d  towards e s t a b l i s h i n g whether the proposed mechanisms, based on the  obser-  v a t i o n o f t r a n s p a r e n t models, were p o s s i b l e d u r i n g s o l i d i f i c a t i o n o f an a l l o y w i t h a s i m i l a r d e n s i t y c o n f i g u r a t i o n i n the l i q u i d .  Consequently,  they would not i m p l y t h a t t h i s i s the o n l y mechanism of f r e c k l e f o r m a t i o n .  In s p i t e of the advantages of t r a c e r t e c h n i q u e s , the f a c t t h a t m e t a l s are opaque makes i t e x t r e m e l y d i f f i c u l t t o reproduce c h l o r i d e - w a t e r experiments  i n an a l l m e t a l system.  the ammonium  For example, t o e s t a b l i s h  t h a t p i p e s f l o w upwards d u r i n g s o l i d i f i c a t i o n , i t would be n e c e s s a r y i n t r o d u c e t r a c e r c l o s e t o the bottom of the advancing  to  s o l i d - l i q u i d region.  E x p e r i m e n t a l l y t h i s would be v e r y d i f f i c u l t , because one w o u l d f i r s t have to l o c a t e t h e p o s i t i o n of the i n t e r f a c e between the s o l i d and t h e  solid-liquid  r e g i o n , and then f o l l o w the m o t i o n of the t r a c e r as s o l i d i f i c a t i o n  progresses.  98  For t h i s reason experiments d e s c r i b e d i n t h i s c h a p t e r were d e s i g n e d t o e s t a b l i s h the f o l l o w i n g :  a)  whether a d e n s i t y i n v e r s i o n can cause l i q u i d t o f l o w upward through  the  mushy zone; b)  whether l i q u i d can f l o w upward through t h i s zone even i f t h e r e i s no d e n s i t y i n v e r s i o n , due to s o l u b i l i t y  c)  effects;  whether the r i s i n g l i q u i d tends t o advance a l o n g a smooth f r o n t , o r breaks down i n t o p i p e f l o w . While these experiments do not s i m u l a t e the s o l i d i f i c a t i o n o f  r e a l c a s t i n g s , the r e s u l t s c o u l d e s t a b l i s h the p r i n c i p l e t h a t d e n s i t y d i f f e r e n c e s cause upward f l o w i n g p i p e s t o be formed through the s o l i d - l i q u i d region.  T h i s would p r o v i d e a d d i t i o n a l s u p p o r t to the mechanisms f o r  f r e c k l i n g and A s e g r e g a t e f o r m a t i o n p r e v i o u s l y  proposed.  The e x p e r i m e n t a l work i n t h i s c h a p t e r was i c a l T r a n s a c t i o n s inDecember 1 9 7 2 ^ * ^ .  published i n Metallurg-  R e c e n t l y , H e b d i t c h and H u n t ^ ^ have 4  r e p o r t e d experiments where they i n j e c t e d r a d i o a c t i v e m a t e r i a l i n t o the s o l i d - l i q u i d r e g i o n o f a growing Sn-Zn a l l o y u s i n g a s y r i n g e .  They observed  upward f l o w of the l e s s dense r a d i o a c t i v e l i q u i d , s u p p o r t i n g the  concept  t h a t l e s s dense l i q u i d can r i s e through the s o l i d - l i q u i d r e g i o n , but they d i d not i d e n t i f y a c t u a l channels r e s e m b l i n g f r e c k l e s o r A s e g r e g a t e s a s s o c i a t e d w i t h t h i s upward f l o w .  5.2  E x p e r i m e n t a l Procedure The t e s t assembly  used i s shown s c h e m a t i c a l l y i n F i g u r e 36.  C a s t i n g A i s a d i r e c t i o n a l l y s o l i d i f i e d l e a d - t i n a l l o y 2.3  cm i n diameter  99  and 4.5 cm l o n g w i t h a columnar d e n d r i t i c s t r u c t u r e h a v i n g a p r i m a r y d e n d r i t e s p a c i n g o f a p p r o x i m a t e l y 50 m i c r o n s .  M a t e r i a l from b o t h ends o f  the c a s t i n g was removed by c a r e f u l m a c h i n i n g and r e p l a c e d by c y l i n d r i c a l i n s e r t s B and C o f l e a d - t i n a l l o y h a v i n g d i f f e r e n t c o m p o s i t i o n s t o t h e c a s t i n g A.  The c a s t i n g and two i n s e r t s were t i g h t l y e n c l o s e d i n a copper  b l o c k w h i c h had been c o a t e d w i t h c o l l o i d a l g r a p h i t e , and p l a c e d i n s i d e t h e tube f u r n a c e i l l u s t r a t e d i n F i g u r e 1 ( b ) .  The columnar c a s t i n g s and i n s e r t s  were made by t e c h n i q u e s d e s c r i b e d i n s e c t i o n s 3.3 and 3.4, and t h e dimensions a r e shown t o s c a l e i n F i g u r e 36.  The assembly was heated a t about 5°C/min t o t h e d e s i r e d t u r e T, t a k i n g care n o t t o o v e r s h o o t t h i s v a l u e , h e l d a t t h i s (± 0.5°C) f o r 45 m i n u t e s , and then quenched. s e c t i o n e d l o n g i t u d i n a l l y , p o l i s h e d and e t c h e d .  tempera-  temperature  The r e s u l t a n t c a s t i n g was then I n each experiment 500 ppm  o f r a d i o a c t i v e t h a l l i u m was u n i f o r m l y d i s t r i b u t e d t h r o u g h o u t i n s e r t B.  The  f l o w of l i q u i d from i n s e r t B i n t o t h e c a s t i n g A c o u l d then be determined d i r e c t l y from a u t o r a d i o g r a p h s o f p o l i s h e d l o n g i t u d i n a l s e c t i o n s .  The a l l o y c o m p o s i t i o n s used f o r t h e c a s t i n g s A, and t h e i n s e r t s B and C a r e l i s t e d i n T a b l e V I I I .  The temperature o f t e s t i n g , and t h e  d e n s i t i e s o f B and G a t t h i s t e m p e r a t u r e , t o g e t h e r w i t h t h e d e n s i t y d i f f e r e n c e , are i n c l u d e d i n the t a b l e .  The h o l d i n g temperature T was made e q u a l t o t h e l i q u i d u s t u r e a t t h e top i n s e r t B.  tempera-  At t h i s t e m p e r a t u r e , from t h e phase diagram, t h e  bottom i n s e r t C would be e n t i r e l y l i q u i d and t h e c a s t i n g A would have l i q u i d i n t e r d e n d r i t i c channels.  THERMOCOUPLE  COPPER BLOCK  TOP  INSERT  COLUMNAR CASTING  BOTTOM INSERT  COVER  FIGURE 36: The t e s t assembly  f o r i s o t h e r m a l experiments.  Magnification 1.5x.  101  TABLE V I I I TEST CONDITIONS FOR ISOTHERMAL EXPERIMENTS  Test  Composition Pb + wt % Sn  Temp. T°C  Liquid Density^ ^ z 3 g/cm 4 2  Density Difference , 3 g/cm  Series I Casting A Upper i n s e r t B Lower i n s e r t C  20 44 62  224  20 37 62  239  20 30 62  254  96 84 62  206  15 62  254  -0.65 8.70 8.05  Series I I Casting A Upper i n s e r t B Lower i n s e r t C  -0.93 8.97 8.04  Series I I I Casting A Upper i n s e r t B Lower i n s e r t C  -1.21 9.24 8.03  S e r i e s IV Casting A Upper i n s e r t B Lower i n s e r t C  +0.66 7.41 8.07  Series V Casting A Lower i n s e r t C  8.03  102  The r e s u l t s o f t h e s e e x p e r i m e n t s cannot be q u a n t i f i e d , and are p r e s e n t e d i n the form of a u t o r a d i o g r a p h s and m i c r o s t r u c t u r e s . the r e s u l t s from o n l y f i v e s e t s o f experiments  Consequently,  ( s e r i e s I-V) are p r e s e n t e d .  These were p a r t o f a more e x t e n s i v e s e r i e s o f o b s e r v a t i o n s on d i f f e r e n t a l l o y c o m p o s i t i o n s and h o l d i n g t e m p e r a t u r e s , a l l o f w h i c h gave r e s u l t s c o m p a t i b l e w i t h those r e p o r t e d below. 5.3  Results In s e r i e s I , I I and I I I the top i n s e r t had a h i g h e r d e n s i t y than  the bottom i n s e r t  (Table V I I I ) , which i s a c o n d i t i o n o f d e n s i t y i n v e r s i o n as (34)  d e f i n e d by Copley e t a l .  .  I n s e r i e s IV, the top i n s e r t had a l o w e r  d e n s i t y , i . e . , t h e r e was no d e n s i t y i n v e r s i o n . bottom i n s e r t was  The c o m p o s i t i o n o f the  f i x e d a t 62% Sn (the e u t e c t i c c o m p o s i t i o n ) i n a l l  the  tests. I n each of t h e s e f o u r c a s e s , the a u t o r a d i o g r a p h s show t h a t  liquid  from i n s e r t B f l o w e d u n i f o r m l y down through i n t e r d e n d r i t i c channels near the top of t h e c a s t i n g A, w i t h o u t the f o r m a t i o n o f p i p e s . from those o f K a e m p f f e r ^ ^  who  These r e s u l t s  differ  observed downward f l o w through p i p e s because,  i n h i s e x p e r i m e n t , l i q u i d above the mushy zone was heated above the l i q u i d u s temperature.  One  can r e a s o n a b l y conclude t h a t i f the t r a c e r showed u n i f o r m  f l o w near the t o p , then the downward flcr-. of l i q u i d through the whole o f T  c a s t i n g A was v i a i n t e r d e n d r i t i c c h a n n e l s , s i n c e a u n i f o r m temperature maintained.  was  C o n s e q u e n t l y , any p i p e s r e v e a l e d by p o l i s h i n g and e t c h i n g are  due t o l i q u i d f l o w i n g upward. A)  Series I The r e s u l t s f o r s e r i e s I are shown i n F i g u r e s 37(a) and ( b ) .  In  (a) FIGURE 38:  (a) M a c r o s t r u c t u r e of columnar c a s t i n g (b) C o r r e s p o n d i n g  autoradiograph.  (series I I ) .  M a g n i f i c a t i o n 2x.  104  F i g u r e 3 7 ( a ) l i q u i d from the s i d e s of the bottom i n s e r t has i n t e r i o r of the c a s t i n g . liquid  The  autoradiograph  from the top i n s e r t B flowed  i n t e r d e n d r i t i c channels. s i d e , but  t h e r e i s no  flow p r o b a b l y  i n F i g u r e 3 7 ( b ) shows t h a t  e s s e n t i a l l y u n i f o r m l y down  S l i g h t l y more l i q u i d  evidence  r i s e n i n the  flowed  t h a t t h i s i s due  through  down the r i g h t hand  to p i p e s .  The  enhanced  r e s u l t s from s l i g h t d i f f e r e n c e s i n s i z e or o r i e n t a t i o n o f  the i n t e r d e n d r i t i c channels  The  on the r i g h t  s i d e of the c a s t i n g .  etched s u r f a c e o f i n s e r t B shows two  l a y e r s which are  s i m i l a r t o the s e p a r a t i o n which o c c u r r e d i n the r e s e r v o i r o f the flow  cell,  discussed e a r l i e r i n s e c t i o n 4.4.2.  Although  the bottom l i q u i d has  r i s e n , i t does not appear to show  a marked breakdown i n t o p i p e flow, i n the sense t h a t t h i s term i s used by p r e v i o u s workers et  ( i n the ammonium c h l o r i d e - w a t e r model s t u d i e d by  Copley  a l . , a t y p i c a l upward f l o w i n g p i p e o c c u p i e d an area o f 4 t o 9 (34)  dendrites  ).  Apart  from the r e g i o n i n the c e n t r e of i n s e r t  C, where  the lower i n s e r t does not make c o n t a c t w i t h the c a s t i n g , the l i q u i d risen f a i r l y casting B)  has  u n i f o r m l y by d i s s o l v i n g p a r t o f the d e n d r i t i c s t r u c t u r e of  A.  Series II  In s e r i e s I I the temperature T and i n c r e a s e d , and  the d e n s i t y d i f f e r e n c e were  the r e s u l t s are shown i n F i g u r e s 3 8 ( a ) and  i n t e r f a c e between the bottom i n s e r t C and of breakdown i n t o p i p e flow.  The  liquid  i n a l e s s uniform manner than s e r i e s I .  (b).  The  the c a s t i n g A shows the on both The  sides of i n s e r t  autoradiograph  uneven  beginning C has  i n Figure  risen 38(b)  105  shows u n i f o r m  downward f l o w  The and  38(a),  probably  C)  l a c k of contact  and  interdendritic  i n the  centre of  t h e p o r o s i t y i n some o f t h e  caused  imperfect  along  e i t h e r by  fitting  of  the  volume inserts  shrinkage into  the  channels.  insert  C i n Figures  upward  flowing  of the  liquid  37(a)  channels, on  was  freezing,  or  by  casting.  Series III  In increased, inally.  series  and  The  the  I I I the  39  of  and  two  samples  shows t h e  sample normal t o the section,  and  d e n s i t y d i f f e r e n c e were  c a s t i n g s w e r e s e c t i o n e d t r a n s v e r s e l y and  results  Figure  temperature  axis.  two  are  Two  B-B  sectioning  shows a number o f  channels  can be  seen  then l o n g i t u d -  39-42.  surfaces revealed after  Surface  some p o r o s i t y .  shown i n F i g u r e s  again  on  channels  one  in  s u r f a c e C-C,  the  3 smaller  of which The  A-A 40(a)  and  the  and  confirms  covers  casting  an  was  s t r u c t u r e and  (b).  The  t h a t upward  estimated  area  reassembled  and  corresponding  autoradiograph  of  approximately  dendrites.  sectioned longitudinally  autoradiograph  shows u n i f o r m  flowing l i q u i d  10  produced  are  downward  the wide  along  shown i n F i g u r e s flow,  channels  which i n Figure  40(a). Figure sectioning  of the  longitudinal 40(a)  and  the  shows a v e r y  41  shows t h e  second  section  two  sample.  i n Figure  autoradiograph  surfaces revealed after One  42(a)  again  uneven s u r f a c e between  channel  i s seen  shows a s i m i l a r shows u n i f o r m the  lower  on  each  surface.  The  s t r u c t u r e to Figure  downward  liquid  transverse  and  flow. the  Figure  casting.  42(a) One  FIGURE 3 9 :  S u r f a c e s r e v e a l e d a f t e r t r a n s v e r s e s e c t i o n i n g of a sample from s e r i e s I I I . M a g n i f i c a t i o n 2x.  FIGURE 41:  Surfaces  revealed  a f t e r transverse  sample from s e r i e s I I I .  FIGURE 42:  s e c t i o n i n g of a  M a g n i f i c a t i o n 2x.  (a) L o n g i t u d i n a l s e c t i o n of same sample as i n F i g u r e (b) C o r r e s p o n d i n g a u t o r a d i o g r a p h .  M a g n i f i c a t i o n 2x.  108  wide c h a n n e l  (which c o n t a i n s some p o r o s i t y ) has r i s e n up one s i d e and  then s p l i t i n t o s m a l l e r channels  The  h i g h e r up t h e c a s t i n g .  columnar s t r u c t u r e n e a r t h e top o f c a s t i n g A i n F i g u r e s 4 0 ( a )  and 4 2 ( a ) appears t o have changed d u r i n g t h e t e s t .  Such changes were  r e g u l a r l y o b s e r v e d i n c a s t i n g s h e l d f o r some time a t t e m p e r a t u r e s where the f r a c t i o n o f l i q u i d was r e l a t i v e l y h i g h . the b e g i n n i n g s  This e f f e c t i s a t t r i b u t e d t o  of a general c o l l a p s e of the d e n d r i t i c s t r u c t u r e , associated  w i t h coarsening e f f e c t s discussed i n s e c t i o n 4 . 5 .  D)  S e r i e s IV  I n s e r i e s IV t h e two i n s e r t s had the same d e n s i t y d i f f e r e n c e as s e r i e s I , w i t h t h e h e a v i e r i n s e r t a t t h e bottom o f t h e c a s t i n g . ant s t r u c t u r e i s shown i n F i g u r e 4 3 ( a ) and t h e c o r r e s p o n d i n g  The r e s u l t -  autoradiograph  i n Figure 4 3 ( b ) .  There i s no evidence bottom o r t h e top i n s e r t .  o f p i p e s i n t h e c a s t i n g from e i t h e r the  The f l a t i n t e r f a c e between the columnar s t r u c t u r e  and t h e bottom i n s e r t i n F i g u r e 4 3 ( a ) c l e a r l y shows t h a t l i q u i d d i d n o t f l o w upwards i n the c e n t r e o f the c a s t i n g .  F i g u r e 4 3 ( b ) shows t h a t t h e t r a c e r  has o u t l i n e d the d e n d r i t i c s t r u c t u r e t o a g r e a t e r e x t e n t than t h e o t h e r t e s t s , y e t comparing t h e o r i g i n a l a u t o r a d i o g r a p h s , less o v e r a l l penetration of the c a s t i n g .  t h e r e appears t o have been  The long t r a i l s a r e n o t due t o  p i p e s , b u t i n d i c a t e t h a t t h e t r a c e r has advanced f u r t h e r down g r a i n boundaries.  S i n c e t h e r e i s no e v i d e n c e  o f s o l u t e c o n v e c t i o n , t h i s t r a c e r pene-  t r a t i o n must be due t o l i q u i d d i f f u s i o n o r p o s s i b l y s h r i n k a g e e f f e c t s quenching.  during  (a) FIGURE 43:  (b)  (a) M a c r o s t r u c t u r e of columnar c a s t i n g (b) Corresponding  autoradiograph.  (series IV).  Magnification  2x.  FIGURE 44: M a c r o s t r u c t u r e from s e r i e s V.  The  m a t e r i a l which etches darker i s Pb-15%Sn homogenized t o remove the dendritic structure. i s e u t e c t i c Pb-62%Sn. Magnification  1.5x.  Lower i n s e r t  110  F i g u r e 43(a)  shows t h a t some l i q u i d from t h e bottom i n s e r t has  r i s e n along the o u t s i d e o f t h e c a s t i n g .  This probably  o c c u r r e d when t h e  s u p p o r t i n g m a t e r i a l between t h e lower i n s e r t and t h e copper b l o c k m e l t e d and t h e c a s t i n g s h i f t e d s l i g h t l y .  P o s s i b l y some r a d i o a c t i v e m a t e r i a l from  i n s e r t B emerged a t t h e s i d e s and d i s s o l v e d i n t h i s o u t s i d e l i q u i d . e f f e c t i s not considered E)  This  t o be r e l a t e d t o f r e c k l i n g .  Series V  I n F i g u r e 44, a Pb-15%Sn a l l o y , homogenized f o r 46 hours a t 177°C was used i n p l a c e o f c a s t i n g A.  The lower i n s e r t was e u t e c t i c m a t e r i a l , and  the specimen was s u b j e c t e d t o t h e same i s o t h e r m a l treatment  as p r e v i o u s  samples from s e r i e s I I I , which showed l o n g upward f l o w i n g p i p e s .  The  dimensions o f t h e Pb-15%Sn a l l o y and t h e lower i n s e r t were the same as i n the o t h e r samples, however, the top i n s e r t was o m i t t e d .  A f t e r c o o l i n g the  sample was s e c t i o n e d t r a n s v e r s e l y and l o n g i t u d i n a l l y as b e f o r e .  The  t r a n s v e r s e s e c t i o n r e v e a l e d no p i p e s , and t h e l o n g i t u d i n a l  s e c t i o n ( F i g u r e 44) showed t h a t d i s s o l u t i o n took p l a c e a t t h e i n t e r f a c e e s s e n t i a l l y a l o n g a smooth, p l a n a r 5.4  front.  Discussion (34) I n t h e mechanism proposed f o r f r e c k l i n g  (also applicable to  (37) A segregate  formation  ) the l i q u i d a t t h e r o o t o f i n t e r d e n d r i t i c channels  was c o n s i d e r e d t o be c l o s e t o e u t e c t i c c o m p o s i t i o n .  I f t h i s l i q u i d was o f  lower d e n s i t y than t h e b u l k l i q u i d above t h e mushy zone, s o l u t e c o u l d o c c u r , and the l i g h t e r l i q u i d would r i s e up i n t e r d e n d r i t i c  convection channels.  Ill  The  rising  l i q u i d would move from c o o l e r t o h o t t e r r e g i o n s  of the c a s t i n g ,  becoming s u p e r h e a t e d , and would e v e n t u a l l y merge i n t o d i s c r e t e p i p e s  by  d i s s o l v i n g away d e n d r i t e b r a n c h e s .  as  This i s a steady s t a t e process;  mushy zone advances d u r i n g s o l i d i f i c a t i o n ,  the p i p e a l s o advances, and  the the  c o m p o s i t i o n of l i q u i d e n t e r i n g the bottom o f the p i p e would be e x p e c t e d t o remain the same.  The main d i f f e r e n c e between the arrangement used i n the p r e s e n t work and  a true s o l i d i f i c a t i o n  p r o c e s s i s t h a t t h e s e t e s t s were done at  u n i f o r m t e m p e r a t u r e , whereas i n a r e a l gradient.  The  c a s t i n g t h e r e would be a temperature  r i s i n g l i q u i d would be s u p e r h e a t e d i n t h i s  the d i f f e r e n c e i n c o m p o s i t i o n , dendrite branches.  and would s i m i l a r l y  c a s e , because of  be a b l e t o d i s s o l v e  However, no s t e a d y s t a t e p r o c e s s c o u l d be  s i n c e the mushy zone i n these t e s t s d i d not advance. upward f l o w i n g channels shown ^Ln F i g u r e s 40(a)  and  t o the p i p e s formed i n the ammonium c h l o r i d e - w a t e r  established  Consequently,  42(a)  a r e not  experiments.  the  identical However,  they do demonstrate t h a t a d e n s i t y i n v e r s i o n can cause upward f l o w i n g through the mushy zone.  pipes  In a d d i t i o n , the r e s u l t s of s e r i e s IV demonstrate  t h a t p i p e s do not form i f t h e r e i s no d e n s i t y i n v e r s i o n .  T a b l e IX l i s t s the maximum s o l u b i l i t y  of the d e n d r i t e s  l i q u i d h e a t e d t o the h o l d i n g t e m p e r a t u r e , e s t i m a t e d I t can be seen t h a t the s o l u b i l i t y  formation  from the phase diagram.  of d e n d r i t e s i n s e r i e s IV, where p i p e s  not form, i s about the same as s e r i e s I I I . The to d i s s o l v e dendrites  in eutectic  a b i l i t y of the bottom l i q u i d  i s t h e r e f o r e not a s u f f i c i e n t  of upward f l o w i n g p i p e s , u n l e s s  do  c o n d i t i o n f o r the  there i s also a density i n v e r s i o n .  112  TABLE IX SOLUBILITY DATA FOR ISOTHERMAL EXPERIMENTS  Test  D e n d r i t i c Phase  T °C  S o l u b i l i t y of Dendrites i n E u t e c t i c L i q u i d at T °C, grams p e r gram eutectic.  Series I  a  224  0.66  Series I I  a  239  0.95  Series I I I  a  254  1.63  S e r i e s IV  B  206  1.58  Series V  a  254  1.25  113  F o l l o w i n g t h e p u b l i c a t i o n o f t h e r e s u l t s shown i n F i g u r e s 37-43 ( s e r i e s I - I V ) , H e b d i t c h and Hunt^  suggested t h a t t h e s t r u c t u r e s observed  were t h e r e s u l t o f c o n v e c t i o n w i t h i n t h e l o w e r i n s e r t C, and had l i t t l e t o do w i t h i n t e r d e n d r i t i c f l u i d f l o w through the c a s t i n g A. was  This suggestion  t e s t e d by s u b s t i t u t i n g a s i n g l e phase Pb-Sn a l l o y f o r t h e o r i g i n a l  columnar  c a s t i n g A ( s e r i e s V, F i g u r e 44). I n t h i s case t h e r e would be no  l i q u i d i n t e r d e n d r i t i c c h a n n e l s , y e t t h e a l l o y would s t i l l be s o l u b l e i n the superheated l i q u i d o f t h e lower i n s e r t  (Table I X ) .  I f H e b d i t c h and Hunt's  c o n t e n t i o n were c o r r e c t , t h i s e x p e r i m e n t a l arrangement s h o u l d produce t h e same type o f s t r u c t u r e s as seen i n F i g u r e s 37-42.  The r e s u l t i n F i g u r e 44 shows t h a t p i p e s were n o t formed i n t h e absence o f i n t e r d e n d r i t i c channels i n t h e c a s t i n g A.  Convection w i t h i n the  l o w e r i n s e r t p r o b a b l y p l a y s a p a r t i n t h e development o f t h e s t r u c t u r e s  seen,  but t h e e x p e r i m e n t a l e v i d e n c e l e a d s t o t h e c o n c l u s i o n t h a t t h e f l o w p a t t e r n caused by m a t e r i a l d i s s o l v i n g a t t h e i n t e r f a c e l e a d s t o a smooth f r o n t , and would n o t account f o r t h e l o n g c h a n n e l s .  A l t h o u g h p o r o s i t y due t o t r a p p e d a i r has been observed i n some o f these samples, i t i s n o t b e l i e v e d t h a t t h e s e experiments p r o v i d e any s u p p o r t i n g e v i d e n c e f o r t h e proposed mechanism o f f r e c k l i n g by r i s i n g gas b u b b l e s . Trapped a i r would c e r t a i n l y form a bubble above t h e l o w e r i n s e r t C, b u t o n l y superheated l i q u i d would be a b l e t o d i s s o l v e d e n d r i t e branches and r i s e . The b u b b l e s thought t o be r e s p o n s i b l e f o r p r o d u c i n g f r e c k l e t r a i l s a r e c o n s i d e r e d t o be on top o f t h e advancing mushy zone, r a t h e r than on t h e bottom.  Bubbles which become t r a p p e d by t h e advancing s o l i d a r e thought t o  be r e s p o n s i b l e f o r " s p o t s e g r e g a t e s " and F i g u r e 42(a) appears t o show some  114  i s o l a t e d spots of t h i s type, a s s o c i a t e d w i t h p o r o s i t y .  However, one  cannot r u l e out the p o s s i b i l i t y t h a t t h e s e " s p o t s " were o r i g i n a l l y connected w i t h t h e main c h a n n e l , but appear i s o l a t e d because the p l a n e o f s e c t i o n i n g has s e p a r a t e d them.  A l t e r n a t i v e l y , they may  the g e n e r a l c o l l a p s e o f the d e n d r i t i c s t r u c t u r e , mentioned  be r e l a t e d t o earlier.  T h e r e f o r e t h e s e experiments do not p r o v i d e any c o n c l u s i v e e v i d e n c e i n s u p p o r t o f t h i s mechanism.  They do, however, e s t a b l i s h the p r i n c i p l e t h a t upward  f l o w i n g p i p e s can be formed by superheated l i q u i d , when d e n s i t y g r a d i e n t s i n the l i q u i d cause i n t e r d e n d r i t i c f l u i d f l o w .  115  CHAPTER 6 SOLUTE CONVECTION AND FRECKLE FORMATION DURING SOLIDIFICATION  6.1  Introduction The model e x p e r i m e n t s d e s c r i b e d i n Chaper 5 demonstrated t h a t  p i p e s can form through a s o l i d - l i q u i d r e g i o n due t o s o l u t e c o n v e c t i o n , and t h a t the d i r e c t i o n o f f l o w w i t h i n the p i p e s was upwards.  Accordingly, the  l o g i c a l e x t e n s i o n o f t h i s work was t o study the a c t u a l s o l i d i f i c a t i o n o f an alloy  system where the l i q u i d a t the bottom o f the s o l i d - l i q u i d  l e s s dense than t h e l i q u i d above. f o r m a t i o n o f channel-type  zone was  I f t h e proposed mechanism f o r t h e  d e f e c t s i s c o r r e c t , i t s h o u l d be p o s s i b l e t o  demonstrate s o l u t e c o n v e c t i o n i n such a system, and produce t h e d e f e c t s under a p p r o p r i a t e c o o l i n g c o n d i t i o n s . Pb-Sn a l l o y s c o n t a i n i n g l e s s than 62%Sn were t h e r e f o r e used because they produce t h e r e q u i r e d " d e n s i t y i n v e r s i o n " when s o l i d i f i e d from the bottom.  The purpose o f t h e f o l l o w i n g e x p e r i m e n t s was t o e x p e r i m e n t a l l y  determine the e x t e n t o f m a c r o s e g r e g a t i o n and f r e c k l e f o r m a t i o n i n v e r t i c a l l y s o l i d i f i e d samples as a f u n c t i o n o f t h e s o l i d i f i c a t i o n v a r i a b l e s .  6.2  6.2.1  Experimental  Procedure  Apparatus Cylindrical  i n g o t s 14 cm l o n g and 1.27 cm i n d i a m e t e r were  s o l i d i f i e d by l o w e r i n g t h e mould through a f u r n a c e w i t h two h e a t i n g The g r a p h i t e mould i s shown d i s m a n t l e d  i n F i g u r e 45.  zones.  A f t e r assembly, the  116  1  2  3  4  5  6  7  8  9  JO  1  2  3  4  5  6  7  8  III FIGURE 4 5 :  S p l i t g r a p h i t e mould f o r making long c y l i n d r i c a l i n g o t s . The mould was assembled by s l i d i n g the two s l e e v e s o v e r the ends and then w i r i n g the p a r t s  together.  117  mould was suspended v e r t i c a l l y  i n the f u r n a c e , and l o w e r e d at a c o n t r o l l e d  r a t e by a low speed synchronous motor.  The two-zone f u r n a c e was  constructed  from two tube f u r n a c e s , s i m i l a r t o those shown i n F i g u r e 1, p l a c e d v e r t i c a l l y end t o end, and connected t o a two-zone temperature c o n t r o l l e r w h i c h maintained a constant gradient.  By a d j u s t i n g the temperature o f the  f u r n a c e zones and the speed of d e s c e n t , the temperature g r a d i e n t and growth r a t e c o u l d be v a r i e d i n d e p e n d e n t l y .  S o l i d i f i c a t i o n r a t e s between 0.0033 and 0.24  cm/sec were used i n  t h i s work, w i t h temperature g r a d i e n t s between 1.0 and 2.3°C/cm. t e s t s , the a l l o y was quenched w i t h water. the  I n some  d u r i n g s o l i d i f i c a t i o n by s u r r o u n d i n g the mould  F o r quenching, a t h i n w a l l e d g r a p h i t e mould was used, h a v i n g  same i n t e r n a l dimensions as the s t a n d a r d mould.  Water was  introduced  i n t o the q u a r t z f u r n a c e tube from the bottom, quenching the i n g o t i n a p p r o x i m a t e l y 20 s e c .  The g r a p h i t e mould used f o r quenching was e n c l o s e d  i n a t h i n w a l l e d q u a r t z tube t o p r e v e n t the mould c r a c k i n g and w a t e r i n c o n t a c t w i t h the m o l t e n m e t a l .  6.2.2  A p r o t e c t i v e atmosphere was not used.  Macrosegregation studies  For made.  coming  each s e t of s o l i d i f i c a t i o n c o n d i t i o n s , f o u r c a s t i n g s were  The temperature g r a d i e n t and f r e e z i n g r a t e were e s t a b l i s h e d i n one  c a s t i n g u s i n g t h r e e thermocouples p o s i t i o n e d a l o n g the i n g o t a x i s .  The  c a s t s t r u c t u r e was determined from l o n g i t u d i n a l and t r a n s v e r s e s e c t i o n s of two of the c a s t i n g s w h i c h were p o l i s h e d and e t c h e d a f t e r s e c t i o n i n g . f o u r t h c a s t i n g was used f o r measurements of m a c r o s e g r e g a t i o n . 500 ppm  The  Approximately  of r a d i o a c t i v e t r a c e r was w e l l mixed i n t o the l i q u i d p r i o r t o s o l i d -  ification.  A f t e r s o l i d i f i c a t i o n , the i n g o t was p l a c e d i n a l a t h e , the  118  o u t s i d e s u r f a c e was machined t o remove t h e t a r n i s h e d s u r f a c e l a y e r , and c u t t i n g s were t a k e n i n a p l a n e p e r p e n d i c u l a r t o t h e i n g o t a x i s .  Starting  a t one end o f t h e i n g o t , t h e c u t t i n g s were c o l l e c t e d a t 0.37 cm i n t e r v a l s , weighed ( a p p r o x i m a t e l y  4 g ) , p l a c e d i n t o t e s t t u b e s , and t h e a c t i v i t y o f  each tube measured i n a P i c k e r N u c l e a r T w i n s c a l e r I I a u t o m a t i c tion  scintilla-  counter. 113 The  i s o t o p e s used were e i t h e r Sn  (half l i f e  112 d a y s , p r i -  m a r i l y a low energy y e m i t t e r , i r r a d i a t e d t o a s p e c i f i c a c t i v i t y o f 0.5 204 m i l l i c u r i e s / g ) or T l ( h a l f l i f e 4.1 y e a r s , p r i m a r i l y a 3 e m i t t e r , b u t a l s o some low energy y, i r r a d i a t e d t o a s p e c i f i c a c t i v i t y o f 5 m i l l i 113 20A c u r i e s / g ) . The Sn was used i n t h e l e a d r i c h a l l o y s , and T l i n the tin  rich alloys.  The y e m i s s i o n spectrum was measured f o r b o t h  isotopes using the s c i n t i l l a t i o n counter  these  ( F i g u r e s 46 and 4 7 ) , and s i n c e  the a c t i v i t y l e v e l s T^ere f a i r l y l o w , an a p p r o p r i a t e window was s e t t o reduce the r a t i o between sample a c t i v i t y and background.  No c o r r e c t i o n s f o r decay  of t h e i s o t o p e s were made, because t h e time taken f o r each s e t o f measurements (2-3 hours f o r Sn to  the h a l f  113  2 OA , o r 12 hours f o r T l ) was v e r y s h o r t compared  lives.  A c t i v i t y p r o f i l e s , as a f u n c t i o n o f d i s t a n c e from t h e bottom o f the i n g o t , f o r c a s t i n g s quenched d i r e c t l y from t h e l i q u i d were compared t o the d i r e c t i o n a l l y s o l i d i f i e d c a s t i n g s t o c o n f i r m t h a t t h e e f f e c t s o b s e r v e d were due t o t h e s o l i d i f i c a t i o n c o n d i t i o n s , and n o t due t o poor m i x i n g o f the o r i g i n a l a l l o y c h a r g e , o r o t h e r extraneous f a c t o r s .  22i  119  20\  ' o 181  o  o  14  12  10  CL  100  Window 2 7 0 - 4 8 0 Kev  d  200  300 ENERGY  FIGURE 46:  400  Spectnrai o f y e m i s s i o n f o r S n T  29  500  (Kev) 1 1 3  .  1,--o.  27  25  23  o  x  in  2!  z  3 O  "  19  17  Window 50-100  Kev  15  50  70  _l_  90 ENERGY  (Kev)  110  FT.GURE 47: Spectrum o f y e m i s s i o n f o r T I204  130  120  6.2.3  Determination  of composition  from a c t i v i t y measurements  I t i s u s u a l l y assumed t h a t the measured a c t i v i t y of the r a d i o a c t i v e i s o t o p e i s d i r e c t l y p r o p o r t i o n a l t o the s o l u t e c o n c e n t r a t i o n .  The  major f a c t o r s c o n t r i b u t i n g t o e r r o r s would be c o u n t i n g s c a t t e r , w h i c h i s a f u n c t i o n of the number o f c o u n t s ,  and g e o m e t r i c a l s c a t t e r , s i n c e  l a t h e t u r n i n g s would not always p r e s e n t counter.  the  the same geometry towards the  The number of counts per sample was  always between 300,000 and  1 m i l l i o n i n t h i s work, w h i c h would r e s u l t i n an o v e r a l l s c a t t e r of (43) b e t t e r than ± 0.5% I t was magnified  due  to c o u n t i n g s c a t t e r  b e l i e v e d t h a t the e f f e c t o f g e o m e t r i c a l s c a t t e r would be  i n the Pb-Sn system due  t o the h i g h a b s o r p t i o n of the r a d i a t i o n  by l e a d ( a b s o r p t i o n c o e f f i c i e n t 1.127 reduce t h i s e f f e c t , and  cm  -1  f o r 0.5MeV y r a y s  t o o b t a i n a more c o n s t a n t  (44)  ).  To  geometry, some of  the  113 samples c o n t a i n i n g Sn  were t r e a t e d w i t h a s o l u t i o n of hot 50%  HNO^.  T h i s formed s o l u b l e l e a d n i t r a t e , and a w h i t e p r e c i p i t a t e of t i n o x i d e w h i c h s e t t l e d i n the bottom o f t h e t e s t tube.  Each tube c o n t a i n e d  of s o l u t i o n , and from the p u b l i s h e d s o l u b i l i t y of P b t N O . ^ (37.7 s o l u t i o n ) i t was  40  ml  g/100  g  c a l c u l a t e d t h a t a l l the l e a d would be i n s o l u t i o n .  The  a c t i v i t y measurements would t h e r e f o r e o n l y be a f f e c t e d by a b s o r p t i o n i n the precipitate layer. i To convert  the measured a c t i v i t i e s to c o m p o s i t i o n ,  the f o l l o w i n g  s e r i e s of c a l i b r a t i o n t e s t s were done: i)  a c t i v i t y versus  ii)  a c t i v i t y versus  113 sample w e i g h t , f o r a c o n s t a n t Sn concentration; 113 Sn c o n c e n t r a t i o n , f o r a c o n s t a n t sample w e i g h t ;  121  iii)  a c t i v i t y v e r s u s a l l o y c o m p o s i t i o n , f o r a c o n s t a n t Sn  113  concen-  tration; 113 iv)  a c t i v i t y v e r s u s a l l o y c o m p o s i t i o n , when t h e Sn was d i r e c t l y p r o p o r t i o n a l t o t h e a l l o y The  concentration  composition.  c a l i b r a t i o n samples were p r e p a r e d by c a r e f u l w e i g h i n g o f t h e  c o n s t i t u e n t s and subsequent treatment w i t h n i t r i c a c i d . shown i n F i g u r e s 48-51.  The r e s u l t s a r e  The measured a c t i v i t i e s as a f u n c t i o n o f b o t h  113 sample w e i g h t and Sn  c o n c e n t r a t i o n ( F i g u r e s 48 and 49) b o t h d e v i a t e d  from l i n e a r i t y when l a r g e r amounts o f r a d i o a c t i v e m a t e r i a l were p r e s e n t , which i n d i c a t e d t h a t t h e s c i n t i l l a t i o n c o u n t e r began t o s a t u r a t e a t t h e s e levels.  I t was f o r t h i s reason t h a t r e l a t i v e l y low c o n c e n t r a t i o n s o f the  r a d i o a c t i v e i s o t o p e s were used.  F i g u r e 50 shows t h a t t h e s p e c i f i c a c t i v i t y  measurements were r e l a t i v e l y independent  o f the a l l o y c o m p o s i t i o n f o r a  113 constant Sn  concentration.  T h e r e f o r e from F i g u r e s 48-50 one can conclude  t h a t i f t h e samples do not v a r y over a wide range o f w e i g h t s o r c o m p o s i t i o n s the s o l u t e content o f t h e sample can be assumed t o be d i r e c t l y p r o p o r t i o n a l tp t h e s p e c i f i c a c t i v i t y .  T h i s was t e s t e d and shown t o be t r u e over a wide  c o m p o s i t i o n range i n F i g u r e 51. The method used f o r c a l c u l a t i n g  s o l u t e c o n t e n t was t o determine  the s p e c i f i c a c t i v i t y o f t h e whole i n g o t ( a . ) , which was taken as t h e sum mg  '  of t h e a c t i v i t i e s o f each sample (a ) from t h e i n g o t , d i v i d e d by t h e sum o f the sample w e i g h t s  (w^) .  The s p e c i f i c a c t i v i t y was then s e t e q u a l t o the  prepared a l l o y composition ( C ) . Q  Za. ^  _  ing  i  _  Ew^  ~  o  > < 71  1  1  1  1  10  20  30  40  ALLOY  COMPOSITION  (%  Sn)  FIGURE 50: C a l i b r a t i o n c u r v e ;  s p e c i f i c a c t i v i t y versus 113 a l l o y c o m p o s i t i o n f o r c o n s t a n t Sn concentration.  ALLOY  COMPOSITION  FIGURE 51: C a l i b r a t i o n c u r v e ;  (wt  %  Sn)  s p e c i f i c a c t i v i t y versus 113  a l l o y c o m p o s i t i o n , when Sn .  concentration i s  p r o p o r t i o n a l t o the solute content.  124 The sample c o m p o s i t i o n  i  was then g i v e n by:  a.  C  W.  a.  1  mg  A l t h o u g h t h e method o f t r e a t i n g samples w i t h n i t r i c a c i d reduced  probably  some o f t h e s c a t t e r a s s o c i a t e d w i t h t h e a n a l y s i s t e c h n i q u e , i t  introduced  o t h e r problems.  I n p a r t i c u l a r , the d i s p o s a l o f r a d i o a c t i v e  waste i n t h e form o f a p r e c i p i t a t e i n a s t r o n g l y a c i d s o l u t i o n , i n a l a r g e number o f i n d i v i d u a l t e s t t u b e s , p r o v e d t o be v e r y time consuming.  I t was  found t h a t t h e time r e q u i r e d t o s a f e l y t r a n s f e r t h e samples t o one, nonc o r r o s i v e c o n t a i n e r , and then reduce t h e b u l k o f l i q u i d waste by e v a p o r a t i o n was much l o n g e r than a n t i c i p a t e d .  S i n c e t h e main advantage o f u s i n g t h e  i s o t o p e a n a l y s i s t e c h n i q u e was i t s speed compared t o o t h e r methods, t h i s reduced  i t s value appreciably.  Consequently,  c o m p o s i t i o n p r o f i l e s f o r one i n g o t were compared  u s i n g t h e r e s u l t s f o r samples o f t u r n i n g s w h i c h had not been t r e a t e d , and for  t h e same samples w h i c h had been t r e a t e d w i t h n i t r i c a c i d .  are p l o t t e d t o g e t h e r i n F i g u r e 52.  Both s e t s  A l t h o u g h t h e r e appears t o be l e s s  scatter  a s s o c i a t e d w i t h t h e t r e a t e d samples, n e i t h e r p l o t i s smooth, because o f t h e e f f e c t of microsegregation.  T h e r e f o r e i t was f e l t t h a t t h e improved  accuracy d i d n o t m e r i t the time i n v o l v e d i n t r e a t i n g a l l t h e samples w i t h n i t r i c a c i d , and t h e c o m p o s i t i o n p r o f i l e s f o r o t h e r i n g o t s were c a l c u l a t e d u s i n g c u t t i n g s from t h e i n g o t s , and t h e d i r e c t p r o p o r t i o n a l i t y between c o m p o s i t i o n and s p e c i f i c a c t i v i t y was t a k e n t o be c o r r e c t o v e r the range considered.  The  r e p r o d u c i b i l i t y o f the a c t i v i t y measurements from u n t r e a t e d  FIGURE 52:  C o m p o s i t i o n p r o f i l e f o r one i n g o t u s i n g l a t h e t u r n i n g s t r e a t e d w i t h n i t r i c a c i d (open c i r c l e s ) , and u n t r e a t e d samples ( c l o s e d c i r c l e s ) .  ho  126  l a t h e t u r n i n g s was  t e s t e d by c o u n t i n g a s i n g l e sample s e v e r a l t i m e s , emptying  and r e f i l l i n g the t e s t t u b e , t o v a r y the geometry o f the p a c k i n g . bars used i n the f o l l o w i n g c o m p o s i t i o n p r o f i l e s l i m i t s based on these t e s t s .  ( F i g u r e s 53-57) a r e ± 2s  For c a s t i n g s w i t h a mean c o m p o s i t i o n  Pb-20%Sn, they r e p r e s e n t a s c a t t e r o f ± 0.38%Sn (percentage i n the a n a l y s i s t e c h n i q u e .  The e r r o r  T h i s was  of  error +  1.9%)  c o n s i d e r e d a c c e p t a b l e s i n c e the compo-  s i t i o n d i f f e r e n c e between the ends o f the i n g o t s was,  i n general, s i g n i f i -  c a n t l y l a r g e r than the s c a t t e r .  For Sn-4%Pb c a s t i n g s , the e r r o r bars r e p r e s e n t a s c a t t e r of ± 0.22%Pb, w h i c h i s a l a r g e r percentage  e r r o r ( 4 . 3 % ) , p r o b a b l y due t o g r e a t e r 204  a b s o r p t i o n of the low energy e m i s s i o n from T l 6.2.4  Solute convection  To observe  c o n v e c t i o n through the b u l k l i q u i d and  solid-rliquid  113 r e g i o n s , 0.1  g p e l l e t s of the c a s t i n g a l l o y , c o n t a i n i n g Sn  , were p l a c e d  i n the l i q u i d a t the top of the mould d u r i n g s o l i d i f i c a t i o n . was  c o n t i n u e d f o r one h o u r ;  Solidification  f o l l o w i n g w h i c h t h e c a s t i n g s were quenched i n  the f u r n a c e , then s e c t i o n e d and p o l i s h e d .  Autoradiographs  of transverse  and l o n g i t u d i n a l s e c t i o n s showed the e x t e n t o f t r a c e r movement. experiment  was  A similar  performed on a c a s t i n g of the same c o m p o s i t i o n h e l d  l i q u i d under the same temperature  g r a d i e n t f o r one hour.  In t h i s  completely case,  s p r e a d i n g of the t r a c e r c o u l d be a t t r i b u t e d t o the c u m u l a t i v e e f f e c t s of d i s t u r b a n c e s a s s o c i a t e ^ w i t h adding the p e l l e t s and quenching. f l o w r e s u l t i n g from the s o l i d i f i c a t i o n the r e s u l t s of these  tests.  The  p r o c e s s c o u l d be e v a l u a t e d by  fluid comparing  127 6.3  Results  6.3.1  Composition  profiles  The c o m p o s i t i o n p r o f i l e s o f t h e i n g o t s , determined by t h e r a d i o a c t i v e t r a c e r a n a l y s i s , a r e shown i n p a r t (a) o f F i g u r e s 53-57.  The s o l i d  l i n e s a r e t h e o r e t i c a l curves c a l c u l a t e d from t h e m a t h e m a t i c a l model w h i c h i s developed i n Chapter 7.  Correspondence  between t h e o r y and experiment i s  d i s c u s s e d i n s e c t i o n 7 6. F i g u r e 54(a) i s ^ i n f a c t , t h e same c o m p o s i t i o n 4  p r o f i l e as F i g u r e 52, and the d a t a p o i n t s used were those f o r samples t r e a t e d with n i t r i c acid.  S i n c e the e r r o r b a r s were o b t a i n e d by t e s t i n g the geo-  m e t r i c a l s c a t t e r a s s o c i a t e d w i t h l a t h e t u r n i n g s , they p r o b a b l y r e p r e s e n t a more c o n s e r v a t i v e e s t i m a t e o f t h e e r r o r s on t h i s p a r t i c u l a r  graph.  The r e s u l t s from the c o o l i n g curves f o r each i n g o t a r e r e p r e s e n t e d g r a p h i c a l l y i n p a r t (b) o f F i g u r e s 53-57, a c c o r d i n g t o t h e method proposed (45) by Flemings and Nereo  . Measurements o f the time r e q u i r e d f o r the  l i q u i d u s and s o l i d u s i s o t h e r m s t o pass each thermocouple p l o t t e d on a d i s t a n c e - t i m e graph.  p o s i t i o n are  I f t h e l i q u i d u s and s o l i d u s l i n e s a r e  s t r a i g h t and p a r a l l e l , t h i s i n d i c a t e s t h a t b o t h the growth r a t e and temperat u r e g r a d i e n t remained  constant during s o l i d i f i c a t i o n .  T h i s i s shown t o be  e s s e n t i a l l y t r u e under t h e slow f r e e z i n g c o n d i t i o n s imposed i n these e x p e r i ments . The s o l i d i f i c a t i o n v a r i a b l e s and m a c r o s e g r e g a t i o n i n T a b l e X. AC where C  a r e summarized  M a c r o s e g r e g a t i o n i s n o r m a l l y d e f i n e d as =  C - C x o  6.1  i s t h e c o m p o s i t i o n a t a p a r t i c u l a r l o c a t i o n (x) and C  i s t h e mean  128 o AVERAGE  TEMPERATURE  GRADIENT  AVERAGE  GROWTH R A T E  00047  l-5°C/cm cm/sec  o  C\J • C\J  "00  2 0  40  DISTANCE  FROM  FIGURE 5 3 ( a ) :  6 0  8 0  B O T T O M OF CASTING  Solute  100  (cm)  distribution.  12 0  14 0  129 AVERAGE  TEMPERATURE  AVERAGE  GROWTH R A T E  GRADIENT  l-5°C/cm  0-013 cm/sec  o  CM CM  o  I"D0  0  20  4 0  DISTANCE  FROM  FIGURE 54(a):  6 0  8 0  B O T T O M OF CASTING  Solute  100  (cm)  12 0  14 0  distribution.  FIGURE 54(b): Cooling  100 TIME  (min)  140  conditions.  130 o AVERAGE  TEMPERATURE  AVERAGE  GROWTH R A T E  GRADIENT  2-3°C/cm  0011 c m / s e c  o  CM CM  (_> O  rr b  1  UJ CM CL  O  CO'  O "00  —I  20  1  1  1  4 0  DISTANCE  1  6 0  1  1  8 0  1  FR0IV1 B O T T O M O F C A S T I N G  FIGURE 5 5 ( a ) :  Solute  1  100  12 0  14  0  (cm)  distribution.  FIGURE 5 5 ( b ) : Cooling conditions,  TIME  140 (min)  180  131  C\J  AVERAGE  TEMPERATURE  AVERAGE  GROWTH R A T E  GRADIENT  I 0°C/cm  0 2 4 cm/sec  o  c\l • |2  N  CE O  LlJ  CM  Q .  5 co-  o to • "  0  0  2  0  4 0 DISTANCE  FROM  FIGURE 5 6 ( a ) :  6 0 S O ' B O T T O M OF CASTING  Solute  lOO (cm)  '  I2"0  •'  | '0 4  distribution.  FIGURE 56(b) : Cooling conditions.  132 ob  AVERAGE  TEMPERATURE  AVERAGE  GROWTH RATE  GRADIENT  l-9°C/cm  0 0 0 3 3 cm/sec  or oX  — |5 o  C\J"  '  00  ™  '  ^  DISTANCE  '  iS  '  8^  ^ o "  FROM BOTTOM OF CASTING  FIGURE 57(a):  '  [£o  (cm)  Solute d i s t r i b u t i o n .  FIGURE 57(b): Cooling conditions.  TIME  (min)  TABLE X SOLIDIFICATION VARIABLES AND MACROSEGREGATION  Figure Number:  Alloy Composition (wt-pct)  Average Temperature Gradient (°C/cm)  Average Growth Rate (cm/sec)  Calculated d i s tance between l i q u i d u s and s o l i d u s isotherms (cm)  Primary Dendrite Spacing (microns)  Structure  Macrosegregation (AC p e t )  53(a)  Pb-20Sn  1.5  0.0047  62.0  206  colmnar  1.07  54(a)  Pb-20Sn  1.5  0.013  62.0  172  h a l f columnar h a l f equiaxed  0.73  55(a)  Pb-20Sn  2.3  0.011  40.4  119  equiaxed  0.13  56(a)  Pb-20Sn  1.0  0.240  93.0  58  equiaxed  0.27  Pb-20Sn  - -  Quenched  - -  equiaxed  -0.04  57(a)  Sn-4Pb  1.9  0.0033  24.7  - -  h a l f columnar h a l f equiaxed  -0.35  - - -  Sn-4Pb  - -  Quenched  - -  - - ' equiaxed  0.11  134  composition.  In this case, however, the amount of macrosegregation over  the whole casting has been re-defined as the difference i n mean  composition  between the upper and lower halves of the ingot  An A  C  1 1 1 =  ~rw.  1 i  2 1 1 o r 2 i  , 6  -  „ 2  where £^ and Z^ are the sums from x = H (the t o t a l length of the casting) to x = H/2, and x = H/2 to x = 0 respectively, and composition  and weight of the i t h sample.  and  are the  Macrosegregation i s considered  positive when the solute concentration increases i n the d i r e c t i o n of s o l i d i f i c a t i o n , and negative for the reverse. Comparing the macrosegregation f o r d i r e c t i o n a l l y s o l i d i f i e d and quenched castings, l i s t e d i n Table X, i t can be seen that the solute d i s t r i b u t i o n i s a function of the s o l i d i f i c a t i o n conditions. the amount of macrosegregation for Pb-20%Sn alloys decreased dendrite spacing decreased.  In general, as the primary  Figures 53(a) and 54(a) show that macrosegre-  gation increases for the slower growth rate at the same temperature gradient, and Figures 54(a) and 55(a) show the same effect for the shallower gradient when the growth rates are almost the same.  Figure 56(a), s o l i d i f i e d under  the lowest temperature gradient and highest growth rate, showed s l i g h t l y more macrosegregation than Figure 55(a).  For the Sn-4%Pb a l l o y (Figure 57(a))  there was more solute at the bottom of the casting and less at the top, resulting i n some negative macrosegregation. An estimate of the s i g n i f i c a n c e of the macrosegregation values i n Table X can be obtained by using the "Student's t - t e s t " .  The composition  135  measurements f o r each sample a r e s u b j e c t t o m i c r o s e g r e g a t i o n and g e o m e t r i c a l s c a t t e r , however, i t i s r e a s o n a b l e t o assume t h a t t h e mean o f t h e samples over h a l f t h e i n g o t w i l l o n l y be s u b j e c t t o g e o m e t r i c a l s c a t t e r , s i n c e m i c r o s e g r e g a t i o n o n l y extends over r e l a t i v e l y s h o r t d i s t a n c e s . i t i s a l s o reasonable  Therefore  t o assume t h a t t h e s t a n d a r d d e v i a t i o n o f t h e mean i s  e q u a l t o the s t a n d a r d d e v i a t i o n o f each sample.  Thus, knowing the means  and s t a n d a r d d e v i a t i o n s f o r t h e top and bottom h a l v e s o f t h e i n g o t , one may use t h e t - t e s t t o check whether t h e c o m p o s i t i o n d i f f e r e n c e s a r e s i g n i f i c a n t .  In g e n e r a l  x^ t  = s /J n  l  .1  + n  2  where x^ and x^ a r e t h e two means w i t h s t a n d a r d d e v i a t i o n s assumed e q u a l t o s , and n^ and n^ a r e the number of samples used t o e s t i m a t e t h e means.  The  number o f degrees o f freedom i s n^ + n^ - 2.  The number o f samples used t o c a l c u l a t e t h e c o m p o s i t i o n o f each h a l f of the c a s t i n g i s approximately degrees o f freedom as 28.  15, t h e r e f o r e one may t a k e t h e number o f  F o r a s i g n i f i c a n c e l e v e l o f 0.01, t h e v a l u e o f t  i s 2.763, t h e r e f o r e i t i s u n l i k e l y t h a t the mean c o m p o s i t i o n s  o f the two  h a l v e s o f t h e i n g o t come from t h e same p o p u l a t i o n when:  2.763  since  <  AC  A  =  C  x^ -  One can t h e r e f o r e conclude t h a t t h e r e i s no s i g n i f i c a n t macrosegreg a t i o n when AC < 0.19%Sn f o r Pb-20%Sn i n g o t s ;  o r AC < 0.11%Pb f o r Sn-4%Pb  136  ingots.  Thus, t h e v a l u e s f o r b o t h q u e n c h e d - i n g o t s show no s i g n i f i c a n t  macrosegregation,  n o r can the v a l u e o f 0.13%Sn f o r t h e i n g o t i n F i g u r e  55(a) be regarded  as a s i g n i f i c a n t d i f f e r e n c e .  the i n g o t i n F i g u r e 56(a) i s p r o b a b l y  The v a l u e o f 0.27%Sn f o r  only marginally s i g n i f i c a n t since  the s c a t t e r i n t h e upper h a l f i s l a r g e r than i n the o t h e r i n g o t s , making the assumption r e g a r d i n g m i c r o s e g r e g a t i o n  less valid.  however, show a s i g n i f i c a n t amount o f m a c r o s e g r e g a t i o n  The r e m a i n i n g i n g o t s , compared t o t h e  quenched i n g o t s , and s i g n i f i c a n t d i f f e r e n c e s when compared t o one a n o t h e r . 6.3.2  Convection  i n the l i q u i d  The r e s u l t s d e m o n s t r a t i n g  convection i n the l i q u i d  s o l i d i f i c a t i o n a r e g i v e n i n F i g u r e 58.  The a u t o r a d i o g r a p h s  during shown a r e of  s e c t i o n s p a r a l l e l and p e r p e n d i c u l a r t o t h e f r e e z i n g d i r e c t i o n o f a Pb-20%Sn a l l o y under t h e c o n d i t i o n s l i s t e d i n T a b l e X I .  In Figure 58(a), the regions  w h i c h a r e u n i f o r m l y d a r k ( s e c t i o n s i - i v ) i n d i c a t e t h a t the l i q u i d u s passed through liquid.  t h i s r e g i o n a f t e r t r a c e r had become mixed through  isotherm  the b u l k  In s e c t i o n ( v ) , only the i n t e r d e n d r i t i c regions are dark,  i n d i c a t i n g t r a c e r p e n e t r a t i o n i n t o t h e s o l i d - l i q u i d zone.  F i g u r e 58(b) shows a u t o r a d i o g r a p h s the l i q u i d .  f o r t h e i n g o t quenched from  T r a c e r has moved l e s s than 3 cm down t h e i n g o t as compared t o  6.5 cm f o r F i g u r e 5 8 ( a ) .  The d i f f e r e n c e i n p e n e t r a t i o n i s a t t r i b u t e d t o  s o l u t e c o n v e c t i o n a s s o c i a t e d w i t h the s o l i d i f i c a t i o n p r o c e s s . i n d i c a t i o n , however, o f t h e f l o w p a t t e r n w h i c h caused  mixing.  There i s no  137  FIGURE 58:  A u t o r a d i o g r a p h s showing the e x t e n t of t r a c e r movement one hour a f t e r t r a c e r was added; (b) quenched from the l i q u i d .  (a) d i r e c t i o n a l l y s o l i d i f i e d , Magnification  2.2x.  TABLE XI COOLING  Figure Number  Alloy Composition (wt p e t )  Average Temperature G r a d i e n t (oc/cm)  CONDITIONS  Average Growth Rate (cm/sec)  Calculated Distance Between L i q u i d u s and S o l i d u s Isotherms (cm)  58(a)  Pb-20Sn  1.9  0.0033  48.9  58(b)  Pb-20Sn  - -  Quenched  -- -  59(a)  Pb-20Sn  1.5  0.0047  62.0  59(b) 60  Pb-20Sn  1.9  0.0033  48.9  139  6.3.3  Freckles E v i d e n c e of s t r u c t u r e s r e s e m b l i n g  s o l i d i f i e d a t the s l o w e s t growth r a t e s .  f r e c k l e s was  seen i n i n g o t s  I n one i n g o t s o l i d i f i e d  at  0.0047 cm/sec, the o u t e r s u r f a c e showed a s h r i n k a g e  trail  7 cm l o n g n e a r the top ( F i g u r e 5 9 ( a ) ) .  defect of t h i s  A shrinkage  i n d i c a t e s t h a t a l o n g channel o f e u t e c t i c l i q u i d was the i n g o t became c o m p l e t e l y  solid.  approximately  present  just  type before  T h i s b e a r s a c l o s e resemblance t o  photographs o f f r e c k l e s i n n i c k e l - b a s e s u p e r a l l o y s shown i n F i g u r e One  c a s t i n g grown at 0.0033 cm/sec r e v e a l e d an i n t e r n a l  w h i c h c o u l d be c l a s s e d as a f r e c k l e .  F i g u r e 59(b)  34(b). trail  shows t r a n s v e r s e  l o n g i t u d i n a l s e c t i o n s through the f r e c k l e t r a i l w h i c h was  4.5  cm l o n g .  e n l a r g e d v i e w of a t r a n s v e r s e s e c t i o n ( F i g u r e 6 0 ( a ) ) shows t h a t the has a f i n e r d e n d r i t i c s t r u c t u r e .  F i g u r e 60(b)  and An  trail  i s an e n l a r g e d view o f  the  lowest p o r t i o n o f the t r a i l , showing t h a t i t o r i g i n a t e d i n the i n t e r i o r  of  the i n g o t as an i n t e r d e n d r i t i c channel w h i c h widened and moved towards the mould w a l l , i n the same d i r e c t i o n as the primary  6.4  dendrite  stalks.  D i s c u s s i o n of R e s u l t s The  curves i n F i g u r e s 53-56 showing p o s i t i v e m a c r o s e g r e g a t i o n  resemble curves f o r normal s e g r e g a t i o n w i t h d i f f u s i o n c o n t r o l l e d m i x i n g ahead of a p l a n a r s o l i d - l i q u i d i n t e r f a c e . There i s , however, c o n s i d e r a b l e (2347) experimental  e v i d e n c e i n the l i t e r a t u r e  '  t o show t h a t o n l y a  n e g l i g i b l e amount o f s o l u t e i s r e j e c t e d ahead of d e n d r i t e t i p s when growth i s not p l a n a r .  Normal s e g r e g a t i o n takes p l a c e over a d i s t a n c e o f  the  o r d e r of microns i n the l i q u i d between d e n d r i t e b r a n c h e s , l e a d i n g t o m i c r o -  (b)  (a) FIGURE 59:  (a) S h r i n k a g e t r a i l , a p p r o x i m a t e l y 7 em  l o n g , along  s o l i d i f i e d under c o n d i t i o n s g i v e n i n Table X I . (b) L o n g i t u d i n a l and Magnification  3x.  transverse  the o u t s i d e s u r f a c e of an  ingot  M a g n i f i c a t i o n 1.7x.  s e c t i o n s showing a f r e c k l e t r a i l on the r i g h t hand s i d e .  §  141  FIGURE 6 0 ( a ) :  Transverse  s e c t i o n of the  freckle  in Figure  5 9 ( b ) , showing  fine dendritic w i t h i n the  trail  structure  trail.  Magnification  25x.  142  segregation.  However, one would not expect a net movement o f s o l u t e i n  the d i r e c t i o n o f growth u n l e s s t h e r e was scale.  l i q u i d m i x i n g on a m a c r o s c o p i c  T h i s m i x i n g c o u l d t a k e p l a c e e i t h e r w i t h i n the s o l i d - l i q u i d  or between t h i s zone and the b u l k l i q u i d ahead o f the d e n d r i t e  The  experiment where t r a c e r was  of the c a s t i n g ( F i g u r e 58)  zone,  tips.  added t o the l i q u i d a t the  c o n f i r m s t h a t s o l u t e c o n v e c t i o n took p l a c e ,  w h i c h i s a t t r i b u t e d t o the f o r m a t i o n o f lower d e n s i t y l i q u i d i n the l i q u i d region.  top  solid-  I n the case o f Pb-20%Sn, the i n t e r d e n d r i t i c l i q u i d becomes  e n r i c h e d i n t i n , up t o the e u t e c t i c c o m p o s i t i o n  (62% Sn).  The  d e n s i t y of  3 the b u l k l i q u i d a t the i n t e r f a c e would be 9.7 3 be 8.2  g/cm  g/cm  , and the e u t e c t i c would  (42) ,  solid-liquid One  thus t h e r e would be a d e n s i t y i n v e r s i o n through r e g i o n w h i c h causes the l e s s dense l i q u i d t o r i s e . can t h e r e f o r e conclude  t h a t the s o l u t e p r o f i l e s i n F i g u r e s  53-56 are a f u n c t i o n o f the growth r a t e , temperature g r a d i e n t and spacing.  S i n c e i t was  cannot draw any f i r m c o n c l u s i o n s r e g a r d i n g  the e f f e c t of a v a r i a t i o n i n any one.  through  dendrite  o n l y p o s s i b l e to h o l d two of the t h r e e v a r i a b l e s  c o n s t a n t f o r any two i n g o t s , one  model was  the  For t h i s r e a s o n , a s i m p l e  mathematical  d e r i v e d ( c h a p t e r 7 ) , w h i c h i n c l u d e d the concept of mass t r a n s f e r  the s o l i d - l i q u i d The  r e g i o n caused by d e n s i t y d i f f e r e n c e s i n the  Sn-4%Pb a l l o y was  liquid.  chosen as an example o f a c o m p o s i t i o n where  the i n t e r d e n d r i t i c l i q u i d would be more dense than the b u l k l i q u i d above. The  comparable d e n s i t i e s would be 7.1  g/cm  3  i n the b u l k l i q u i d , and 8.2  g/cm  (42) i n the e u t e c t i c  .  T h i s d e n s i t y c o n f i g u r a t i o n would be s t a b l e , and  would not expect any c o n v e c t i o n .  The  r e s u l t i n g composition p r o f i l e  one  (Figure  57(a)) shows a s m a l l i n c r e a s e i n l e a d content c l o s e t o the bottom of the  3  143  i n g o t , w h i c h i s p o s s i b l y due  t o the e f f e c t of i n v e r s e  segregation.  A s m a l l sample (150 g) o f the m o l t e n a l l o y was  placed i n a  g r a p h i t e c r u c i b l e and observed d u r i n g s o l i d i f i c a t i o n under vacuum. b u b b l e s were seen.  This evidence,  No  t o g e t h e r w i t h the i n f o r m a t i o n i n the  l i t e r a t u r e t h a t o n l y oxygen i s v e r y s l i g h t l y s o l u b l e i n m o l t e n Pb-Sn alloys^  4 8  \  confirmed  t h a t gas e v o l u t i o n c o u l d be i g n o r e d as a p o s s i b l e  f r e c k l i n g mechanism i n t h i s system. C o n s e q u e n t l y , the f r e c k l e t r a i l s observed i n i n g o t s a t the s l o w e s t convection.  solidified  f r e e z i n g r a t e s are a t t r i b u t e d t o the e f f e c t of s o l u t e  As the l e s s dense l i q u i d towards the bottom of the mushy  zone b e g i n s t o r i s e , i t becomes s u p e r h e a t e d and branches i n i t s p a t h .  can d i s s o l v e d e n d r i t e  I f s u f f i c i e n t i n t e r d e n d r i t i c c h a n n e l s widen i n t h i s  f a s h i o n , they can converge and r e s u l t i n the f o r m a t i o n o f a l a r g e v e r t i c a l pipe.  D i r e c t e v i d e n c e of t h i s mechanism i s shown i n F i g u r e 6 0 ( b ) .  experimental  The  e v i d e n c e s u g g e s t s t h a t f r e c k l e s do not always form when  solute convection  takes p l a c e , but  they appear when the v e l o c i t y of  r i s i n g i n t e r d e n d r i t i c l i q u i d reaches a c r i t i c a l  value.  the  144  CHAPTER 7 A NUMERICAL MODEL FOR  MACROSEGREGATION  IN Pb-Sn ALLOYS 7.1  I n t r o d u c t i o n and Review o f P r e v i o u s Work Macrosegregation  caused by i n t e r d e n d r i t i c  f l u i d f l o w has been  t r e a t e d a n a l y t i c a l l y i n a number o f p u b l i s h e d p a p e r s .  Most o f these models  are f o r i n v e r s e s e g r e g a t i o n , where an a n a l y t i c a l s o l u t i o n can be o b t a i n e d by c o n s i d e r i n g b a c k f l o w  through a volume element as the s o l i d and l i q u i d  c o n t r a c t d u r i n g s o l i d i f i c a t i o n . C h i l l f a c e s e g r e g a t i o n under these con(49) d i t i o n s was f i r s t p r e d i c t e d by S c h e i l  , and h i s model was l a t e r extended  by K i r k a l d y and Y o u d e l i s t o p r e d i c t t h e s o l u t e d i s t r i b u t i o n a l o n g t h e whole «.. (50,51) casting A more g e n e r a l s o l u t i o n f o r m a c r o s e g r e g a t i o n ,  considering f l u i d (45)  f l o w i n t h r e e dimensions,  was f i r s t p u b l i s h e d by Nereo and F l e m i n g s  T h e i r model i s based on t h e use o f t h e P f a n n E q u a t i o n t o p r e d i c t m i c r o s e g r e gation i n binary a l l o y s : C  = S  where  C  kC O  (1 - f )  k  _  ' " 7.1  1  S  g  =  s o l i d composition at the s o l i d - l i q u i d i n t e r f a c e  k  =  equilibrium distribution  =  weight f r a c t i o n s o l i d  =  i n i t i a l alloy  f C  q  In a d d i t i o n : and  composition.  C s  =  kC  f  =  (1 - f ) L  s  coefficient  L T  T  145  where C  T  Li  and f a r e the c o m p o s i t i o n Lt  respectively. shrinkage  and w e i g h t f r a c t i o n of  C o n s i d e r i n g a c o n s t a n t k, and a c o n s t a n t  ( 3 ) , they s o l v e h e a t and mass b a l a n c e s  liquid,  solidification  i n a volume element to  o b t a i n the f o l l o w i n g g e n e r a l e x p r e s s i o n :  Ih.  ! S  (Lz-A\ I" +5 yil 1  1  9C L  \ l - k^  T  where P  s ~  P  p  L  e J  72  3t  L  s  V  =  i n t e r d e n d r i t i c flow v e l o c i t y vector  VT  =  temperature g r a d i e n t i n t h r e e dimensions  p  =  solid density  =  l i q u i d density  s  p^  J  and e i s d e f i n e d as the r a t e o f t e m p e r a t u r e change ( 9 T / 3 t ) .  Equation  i s r e f e r r e d t o as the " l o c a l s o l u t e r e d i s t r i b u t i o n e q u a t i o n " .  7.2  It i s written  f o r the g e n e r a l case o f t h r e e - d i m e n s i o n a l heat and f l u i d f l o w , assuming c o n s t a n t s o l i d d e n s i t y d u r i n g s o l i d i f i c a t i o n , n e g l i g i b l e d i f f u s i o n and pore f o r m a t i o n .  I f f u r t h e r assumptions are now  no  made t o c a l c u l a t e the f l o w  v e l o c i t y v e c t o r ( v ) , the e q u a t i o n can be used t o e s t i m a t e  macrosegregation  in castings. Flemings and Nereo used the model t o p r e d i c t s o l u t e d i s t r i b u t i o n s for inverse segregation i n Al-Cu a l l o y s . (52 r e f i n e d and extended  The model has s u b s e q u e n t l y  been  53) '  , and has been a p p l i e d by M e h r a b i a n , Keane and  (9) Flemings  to p r e d i c t macrosegregation  caused by a c o m b i n a t i o n  c a t i o n c o n t r a c t i o n and s o l u t e c o n v e c t i o n .  of  solidifi-  They c o n s i d e r the f l u i d dynamics  through a volume element where the f o r c e s a c t i n g are s o l i d c o n t r a c t i o n ,  146  l i q u i d c o n t r a c t i o n and g r a v i t y .  The l i q u i d i s o f v a r i a b l e d e n s i t y , and t h e  s o l i d - l i q u i d r e g i o n i s t r e a t e d as a porous medium o f v a r i a b l e p o r o s i t y . Equations a r e derived r e l a t i n g i n t e r d e n d r i t i c f l u i d pressure, f l o w v e l o c i t y , f r a c t i o n l i q u i d and l i q u i d c o m p o s i t i o n  interdendritic  w h i c h can, i n t h e o r y ,  be s o l v e d t o g i v e t h e s e v a r i a b l e s as a f u n c t i o n o f p o s i t i o n .  In practice,  s o l u t i o n s f o r t h e g e n e r a l case a r e d i f f i c u l t t o o b t a i n , s i n c e t h i s would i n v o l v e the s o l u t i o n o f s i m u l t a n e o u s p a r t i a l d i f f e r e n t i a l e q u a t i o n s .  They  have t h e r e f o r e made t h e f o l l o w i n g s i m p l i f y i n g a s s u m p t i o n s : 1)  The f r a c t i o n l i q u i d v a r i e s w i t h p o s i t i o n i n t h e mushy zone o n l y as a f u n c t i o n o f t e m p e r a t u r e , and i s c a l c u l a t e d by assuming u n i d i r e c t i o n a l heat and f l u i d  2)  Planar isotherms  flow.  a r e assumed, so t h a t f o r a c o n s t a n t  the l i q u i d c o m p o s i t i o n  steady-state,  liquidus slope,  v a r i e s l i n e a r l y w i t h p o s i t i o n i n one d i r e c t i o n .  3)  The d e n s i t y o f l i q u i d v a r i e s l i n e a r l y w i t h  4)  The d e n s i t y o f s o l i d i s  composition.  constant.  The model has been a p p l i e d t o t h e s p e c i a l - case o f h o r i z o n t a l , u n i d i r e c t i o n a l heat f l o w and s t e a d y - s t a t e s o l i d i f i c a t i o n , where a v a l u e f o r t h e parameter c h a r a c t e r i z i n g t h e s t r u c t u r e and t h e r m a l  c o n d i t i o n s has been assumed.  I n Chapter 6, m a c r o s e g r e g a t i o n e x p e r i m e n t s were done on v e r t i c a l l y s o l i d i f i e d Pb-Sn a l l o y s .  S i n c e i t i s d i f f i c u l t t o apply t h e model d e r i v e d  by Mehrabian e t a l . t o t h i s type o f i n g o t and a l l o y system, a s i m p l e mathemati c a l model was d e v e l o p e d , w h i c h would take i n t o account t h e e f f e c t o f growth r a t e , temperature g r a d i e n t and s t r u c t u r e .  Some o f t h e assumptions f o r t h i s  model d i f f e r from those used by M e h r a b i a n e t a l . , t h e major d i f f e r e n c e s being: 1)  The p a r t i t i o n r a t i o v a r i e s as a f u n c t i o n o f t e m p e r a t u r e .  147  2)  The d e n s i t y o f the l i q u i d i s a f u n c t i o n of temperature  and  composition,  o b t a i n e d from e x p e r i m e n t a l d a t a . 3)  The s t r u c t u r e of the mushy zone i s c h a r a c t e r i z e d by a parameter o b t a i n e d from the r e s u l t s o f the experiments  on i n t e r d e n d r i t i c f l u i d  flow  (Chapter 4 ) . 4)  S i n c e t h e model i s a p p l i e d to the s o l i d i f i c a t i o n o f Pb-Sn a l l o y s , b a c k f l o w due t o volume s h r i n k a g e i s n e g l e c t e d .  7.2  Model o f the S o l i d i f i c a t i o n The  Process  s o l i d - l i q u i d c o n f i g u r a t i o n d u r i n g the p r o g r e s s i v e v e r t i c a l  s o l i d i f i c a t i o n o f a s m a l l i n g o t i s assumed t o be t h a t shown s c h e m a t i c a l l y i n F i g u r e 61.  The a c t u a l s t r u c t u r e r e p r e s e n t e d by the s p i k e s may  d e n d r i t i c or equiaxed.  The  be  columnar  f o l l o w i n g assumptions a r e made:  1)  The  l i q u i d i s c o m p l e t e l y mixed i n the h o r i z o n t a l p l a n e s .  2)  There i s no s i g n i f i c a n t d i f f u s i o n i n the s o l i d  3)  L o c a l e q u i l i b r i u m e x i s t s a t the i n t e r f a c e between the l i q u i d and the a d j a c e n t  state. interdendritic  solid.  Under these c o n d i t i o n s , the c o m p o s i t i o n of the s o l i d a t the l i q u i d i n t e r f a c e i s g i v e n by the Pfann E q u a t i o n a constant.  ( E q u a t i o n 7.1), p r o v i d e d k i s  I t i s a l s o p o s s i b l e t o use E q u a t i o n 7.1  incrementally to describe  the s o l i d i f i c a t i o n of an a l l o y where k v a r i e s w i t h t e m p e r a t u r e , t h a t k remains c o n s t a n t over a s m a l l temperature  solid-  i n t e r v a l AT.  g e n e r a l case o f s o l i d i f i c a t i o n between temperatures  T^ and T^,  by assuming For the as shown i n  F i g u r e 62, l i q u i d c o m p o s i t i o n i s g i v e n by the l i q u i d u s l i n e , k i s e q u a l t o the average d i s t r i b u t i o n c o e f f i c i e n t between these two  temperature:  148  FIGURE 62:  E q u i l i b r i u m diagram f o r a b i n a r y  alloy.  The non-  e q u i l i b r i u m s o l i d u s i s shown by t h e dashed  line.  149  C s  where  and  7.3  r e f e r t o c o m p o s i t i o n s at t h e e q u i l i b r i u m s o l i d u s  and  liquidus.  The w e i g h t f r a c t i o n o f l i q u i d w h i c h s o l i d i f i e s as the a l l o y c o o l s between T^ and  is: -l/(l-k) 7.4 J  1  J  The s o l i d which f r e e z e s i n t h i s increment i s o f c o m p o s i t i o n  C' 2 , but a S  and weight f r a c t i o n (1 - f ) . b e t t e r e s t i m a t e of C'  For s m a l l increments  AT, C' s  = kC L  2  can be o b t a i n e d f o r l a r g e r i n c r e m e n t s by  the t o t a l w e i g h t of s o l u t e i n b o t h s o l i d and l i q u i d a t T^, c o n s e r v a t i o n o f s o l u t e mass a t T^'  T n e  2  calculating  then a p p l y i n g  c o m p o s i t i o n o f the e n t i r e s o l i d  any temperature w i l l be l e s s than the e q u i l i b r i u m v a l u e , because no i n the s o l i d has been assumed.  at  diffusion  The n o n - e q u i l i b r i u m s o l i d u s (shown dashed i n  F i g u r e 62) can be c a l c u l a t e d by summing the t o t a l amount o f s o l u t e i n the solid.  The  c o m p o s i t i o n o f the m a t e r i a l which s o l i d i f i e s a t T  i s C' , y e t 1 s  1  x  the c o m p o s i t i o n o f the e n t i r e s o l i d i s C_ . I f the weight o f l i q u i d at T c o m p o s i t i o n of s o l i d a r e W  and C S  %  (1  "V  \  +  C  L  f 2  l  L \  S  +  is W  , and the w e i g h t and average  , solute conservation gives: l  %  W  C  o  (W_ L l  + w  s  ) x  7.5  150  and t h e average c o m p o s i t i o n C  W S  s  of s o l i d at  l  2. 0  l s :  + C (1 - f ) W 2 l W + (1 - f _ ) W Sj L S  T  l  S  L  L  7.6  T  Consequently t h e l i q u i d c o m p o s i t i o n ,  average s o l i d  composition  and the w e i g h t f r a c t i o n s o f s o l i d and l i q u i d can be d e f i n e d a t any temperat u r e between t h e l i q u i d u s and s o l i d u s by u s i n g d a t a from t h e phase diagram.  I n a d d i t i o n , i f d e n s i t y d a t a a r e a v a i l a b l e as a f u n c t i o n o f  temperature and c o m p o s i t i o n , volume f r a c t i o n s and d e n s i t i e s a r e a l s o d e f i n e d a t any t e m p e r a t u r e .  7.3  I n t e r d e n d r i t i c F l u i d Flow Model The model o f t h e s o l i d i f i c a t i o n p r o c e s s  r e d i s t r i b u t e d perpendicular t o the dendrite s t a l k s .  d e s c r i b e s how s o l u t e i s I f no i n t e r d e n d r i t i c  f l u i d f l o w o c c u r s , t h e i n g o t would show m i c r o s e g r e g a t i o n  on a s c a l e  e q u i v a l e n t t o t h e d e n d r i t e s p a c i n g , b u t t h e r e would be no n e t movement o f s o l u t e over g r e a t e r d i s t a n c e s .  The assumption i s now made t h a t s o l u t e can  be moved o v e r much g r e a t e r d i s t a n c e s by i n t e r d e n d r i t i c f l u i d f l o w caused by d e n s i t y d i f f e r e n c e s i n the l i q u i d .  I t i s f u r t h e r assumed t h a t t h e network  of d e n d r i t e s i n the s o l i d - l i q u i d r e g i o n produces a r e s i s t a n c e t o f l o w , and t h a t t h i s r e s i s t a n c e i s a f u n c t i o n o f the volume f r a c t i o n , and d e n d r i t e s p a c i n g o f the s o l i d .  T h e r e f o r e , u s i n g the c a p i l l a r y model d e s c r i b e d i n  s e c t i o n (4.5.1), t h e p e r m e a b i l i t y o f t h e d e n d r i t i c network i s g i v e n b y : 2 K  =  4-8 8niTT  T h i s e q u a t i o n was d e r i v e d f o r one d i m e n s i o n a l  flow;  however, the  151  r e d i s t r i b u t i o n o f s o l u t e i n a c a s t i n g o f the type shown i n F i g u r e 61 involves three dimensional flow. down one t h i r d o f t h e channels  In t h i s case, s i n c e flow only takes place  i n any one d i r e c t i o n , E q u a t i o n 4.8 becomes:  2 K  =  ^  7.7  24mTT  P r e v i o u s e x p e r i m e n t a l work  (1 2) ' has shown t h a t K i s p r o p o r t i o n a l  2 to g  when g J-j  i s l e s s t h a n 0.3.  I n the absence o f a b e t t e r m o d e l , i t has  Li  been assumed t h a t E q u a t i o n  7.7 h o l d s f o r a l l v a l u e s o f g .  Values  o f the  Li  3 f a c t o r nx model.  have been t a k e n from F i g u r e 28, and used i n c o n j u n c t i o n w i t h t h i s  D e n d r i t e c o a r s e n i n g has been n e g l e c t e d . Macrosegregation  i s t h e r e f o r e determined  u s i n g Darcy's Law t o  c a l c u l a t e the f l o w r a t e of i n t e r d e n d r i t i c l i q u i d when the p e r m e a b i l i t y v a r i e s w i t h temperature,  and t h e d r i v i n g f o r c e (AP) i s g i v e n by the d e n s i t y  d i f f e r e n c e s i n the l i q u i d . 7.4  U n i d i r e c t i o n a l S o l i d i f i c a t i o n of a V e r t i c a l Casting As an example o f t h e a p p l i c a t i o n o f t h e model, c o n s i d e r the  s o l i d i f i c a t i o n of a v e r t i c a l , c y l i n d r i c a l c a s t i n g of constant cross s e c t i o n . To s i m p l i f y t h e c a l c u l a t i o n s , c o n s t a n t growth r a t e (R) and t e m p e r a t u r e gradient  (G) a r e assumed.  I n a d d i t i o n , s i n c e t h i s model w i l l be a p p l i e d t o the  s o l i d i f i c a t i o n o f Pb-Sn a l l o y s where t h e volume change on f r e e z i n g i s s m a l l (of the o r d e r o f 2 % ) , i t has been assumed t h a t b a c k f l o w  due t o volume  shrinkage i s n e g l i g i b l e .  The c a s t i n g i s d i v i d e d i n t o a number o f h o r i z o n t a l l a y e r s o f length i  y  and the temperature w i t h i n each l a y e r i s assumed t o be  uniform.  152  The G£.  temperature d i f f e r e n c e between a d j o i n i n g l a y e r s i s t h e r e f o r e e q u a l t o Increments o f time ( A t ) a r e chosen such t h a t the c o o l i n g r a t e i s e q u a l  to G£/At.  A f t e r each time i n c r e m e n t , t h e temperature o f each l a y e r i s then  e q u a l t o t h e temperature o f t h e l a y e r below and t h e growth r a t e (R) i s e q u a l t o £/At.  The l e n g t h I c a n , t h e r e f o r e , be e l i m i n a t e d  as a v a r i a b l e ,  S i n c e i t can be d e f i n e d i n terms o f R and A t . Solidification  i s considered  t o b e g i n when t h e temperature o f t h e  bottom l a y e r i s e q u a l t o t h e l i q u i d u s temperature T . At,  W i t h i n each time  s o l i d i f i c a t i o n and f l u i d f l o w a r e t r e a t e d s e p a r a t e l y .  step  Conservation of  s o l u t e w i t h i n each l a y e r a p p l i e s d u r i n g each s o l i d i f i c a t i o n s t e p , and c o n s e r v a t i o n o f s o l u t e through t h e whole c a s t i n g a p p l i e s i n each f l u i d f l o w  step.  F l u i d f l o w through a l a y e r s t o p s when the temperature f a l l s below t h e e u t e c t i c temperature T^, and t h e f i n a l c o m p o s i t i o n p r o f i l e i s obtained  when t h e t o p l a y e r r e a c h e s T . h  F i g u r e 63 g i v e s a s c h e m a t i c r e p r e s e n t a t i o n intermediate  o f t h e c a s t i n g a t an  t i m e , and shows t h e t e m p e r a t u r e , c o m p o s i t i o n and d e n s i t y  i n the l i q u i d obtained  The  profiles  u s i n g t h e s o l i d i f i c a t i o n model f o r an a l l o y where t h e  s o l u t e r i c h l i q u i d has a lower d e n s i t y than the i n i t i a l  liquid.  d r i v i n g f o r c e f o r f l o w t h r o u g h the s o l i d - l i q u i d  g i v e n by t h e d e n s i t y d i f f e r e n c e between T where Ap  o f the c a s t i n g  and T  region i s  and i s e q u a l t o Ap g h ,  i s the d e n s i t y d i f f e r e n c e i n t h e l i q u i d , g i s g r a v i t y , and h i s t h e  Li  d i s t a n c e between T L  and T . hi  S i n c e the l i q u i d i s a continuum, one would  expect t h e d r i v i n g f o r c e a t e v e r y p o i n t through t h e s o l i d - l i q u i d be the same.  region to  However, s i n c e t h e l i q u i d f r a c t i o n d e c r e a s e s downwards, t h e  r e s i s t a n c e t o f l o w would i n c r e a s e  towards the bottom o f the s o l i d - l i q u i d  zone.  153  Q" - -  --->-- _  T  L  _  -/-  J E _  J I Ki I jRAt Jon t \^ N — \  TEMPERATURE  LIQUID DENSITY  LIQUID COMPOSITION  FIGURE 63: D i r e c t i o n a l l y s o l i d i f y i n g i n g o t d i v i d e d i n t o  layers.  Temperature, c o m p o s i t i o n and d e n s i t y p r o f i l e s g i v e n by t h e s o l i d i f i c a t i o n model.  6 5  •  « •  R4.Q4  4  R3  3  1  3  R. ,qi  I  (b)  (a)  FIGURE 64:  ^  R2 .<*2  2 |  .  (a) Assumed f l o w p a t t e r n showing two main f l o w c e l l s . (b) R e s i s t a n c e s R  1 - 5  between s i x l a y e r s .  » and f l o w r a t e s  f o r flow  154  A f l o w p a t t e r n w i t h i n the s o l i d - l i q u i d been assumed where f l o w can take p l a c e v e r t i c a l l y n e x t , and h o r i z o n t a l l y t h r o u g h the l a y e r .  r e g i o n has, from one  For v e r t i c a l  therefore,  l a y e r t o the  f l o w , h a l f the  c r o s s s e c t i o n a l a r e a c o n t r i b u t e s t o downward f l o w , and h a l f t o upward f l o w . F i g u r e 64(a)  shows the assumed f l o w p a t t e r n w i t h two main f l o w  cells.  However, p r o v i d e d downward and upward f l o w each occupy h a l f the  cross  s e c t i o n a l a r e a , the a c t u a l number of f l o w c e l l s i s u n i m p o r t a n t .  The  r e s i s t a n c e o f the d e n d r i t i c network t o f l u i d f l o w i s r e p r e -  sented s c h e m a t i c a l l y i n F i g u r e 6 4 ( b ) .  The  r e s i s t a n c e symbols  represent  porous media of a r e a e q u a l t o h a l f the c r o s s s e c t i o n a l a r e a o f the c a s t i n g , and l e n g t h e q u a l t o the l e n g t h o f the l a y e r s .  I t i s assumed t h a t t h e r e i s  no r e s i s t a n c e t o h o r i z o n t a l f l o w through the l a y e r s , s i n c e the w i l l be s h o r t , e s p e c i a l l y f o r a l a r g e number of f l o w  distances  cells.  Porous l a y e r s s t a c k e d i n t h i s manner obey the laws o f s e r i e s resistances;  t h e r e f o r e , s i n c e the magnitude o f the r e s i s t a n c e can be c a l c u -  l a t e d i n terms o f the l i q u i d f r a c t i o n and s t r u c t u r e , and the p r e s s u r e known, the v e l o c i t y of the i n t e r d e n d r i t i c l i q u i d u s i n g Darcy's Law. to 2v/Ag  The  drop i s  ( v / g ) can be c a l c u l a t e d L  flow rate of i n t e r d e n d r i t i c l i q u i d  where A i s the c r o s s s e c t i o n a l a r e a o f the i n g o t .  (q) i s then e q u a l For b r i e f t i m e  Li i n t e r v a l s , A t , the q u a n t i t y o f l i q u i d w h i c h f l o w s between l a y e r s w i l l s m a l l , t h e r e f o r e the d r i v i n g f o r c e f o r f l o w i s assumed t o remain  be  constant.  I n F i g u r e 6 4 ( b ) , s i x l a y e r s are shown, the r e s i s t a n c e s between the l a y e r s a r e numbered -^_5» and the f l o w r a t e s are qj_5« R  o f each l a y e r , expressed  The  liquid  compositions  as w e i g h t p e r u n i t volume, are e q u a l t o (PL^T.,^1-6*  The volumes o f the l a y e r s , V,  ,, remain  constant.  155  Each l a y e r exchanges l i q u i d w i t h the a d j o i n i n g l a y e r s , f o l l o w i n g t h i s n o t a t i o n , the volume f l o w r a t e a c r o s s the top" and surfaces  of the i t h l a y e r are q^ and  b a l a n c e can,  v  i i t  (P  q^ ^, r e s p e c t i v e l y .  and bottom  A s o l u t e mass  t h e r e f o r e , be w r i t t e n f o r the i t h l a y e r :  LVI  V i W i - i  =  +  ^ i W i + i -  (  q  i  +  V i  )  (  p  L V i  7  -  8  T h i s mass b a l a n c e can be w r i t t e n f o r each l a y e r , g i v i n g a s e r i e s of s i m u l t a n e o u s o r d i n a r y  d i f f e r e n t i a l e q u a t i o n s w h i c h can be s o l v e d  c o m p o s i t i o n of each l a y e r , a f t e r a time i n t e r v a l A t , u s i n g  for  the  standard  n u m e r i c a l methods. Thus, the net e f f e c t of f l u i d f l o w i s t h a t the average c o m p o s i t i o n of each l a y e r i s no l o n g e r e q u a l t o C , q  y e t on the n e x t s o l i d i f i c a t i o n  step,  the l i q u i d c o m p o s i t i o n w i l l be e q u a l to the v a l u e g i v e n by the l i q u i d u s on the e q u i l i b r i u m diagram. differences 7.4  and  7.5  For each l a y e r , t h i s w i l l r e s u l t i n s l i g h t  i n the average c o m p o s i t i o n of the p r i m a r y s o l i d from E q u a t i o n s  7.5 and  Results The  i n the f i n a l f r a c t i o n of e u t e c t i c .  of C a l c u l a t i o n s model was  f o r S o l i d i f i c a t i o n of a Pb-Sn A l l o y  used to c a l c u l a t e the f i n a l s o l u t e d i s t r i b u t i o n i n  a Pb-20%Sn a l l o y , as an example of a system w h i c h shows a d e n s i t y during k and  line  solidification.  inversion  D a t a f o r the e q u i l i b r i u m d i s t r i b u t i o n c o e f f i c i e n t  the e q u i l i b r i u m l i q u i d u s l i n e , as a f u n c t i o n of t e m p e r a t u r e , were  o b t a i n e d from the phase diagram and  converted to polynomial expressions  s t a n d a r d curve f i t t i n g t e c h n i q u e s .  D a t a f o r the d e n s i t y of l i q u i d Pb-Sn  a l l o y s as a f u n c t i o n of temperature and  c o m p o s i t i o n were a v a i l a b l e i n  using  the  156  form o f a t a b l e  y  and i n t e r m e d i a t e v a l u e s were o b t a i n e d by l i n e a r (54)  interpolations.  The v a l u e o f v i s c o s i t y  (y) was t a k e n as 0.03 p o i s e  The f i n a l s o l u t e p r o f i l e was o b t a i n e d by r e c a l c u l a t i n g t h e v a l u e s o f each s o l i d i f i c a t i o n parameter f o r a s o l i d i f i c a t i o n s t e p f o l l o w e d by a f l u i d f l o w s t e p a f t e r every time increment  A t as t h e c a s t i n g c o o l e d between T J_i  and Tg, u s i n g a d i g i t a l computer.  The FORTRAN program i s g i v e n i n  Appendix I V . F i g u r e 65 shows t h e s o l u t e d i s t r i b u t i o n f o r v a r i o u s v a l u e s of t h e time i n t e r v a l , i . e . , d i f f e r e n t numbers o f l a y e r s , when H, G, R, y and NC have t h e v a l u e s shown.  H i s t h e l e n g t h o f t h e c a s t i n g , and NC i s the e f f e c 3  t i v e number o f channels  (NC = nx A ) .  The curves show t h a t , except f o r t h e  ends o f t h e i n g o t , as t h e number o f l a y e r s i n c r e a s e s ( A t d e c r e a s e s ) t h e s o l u t e d i s t r i b u t i o n converges t o a s i n g l e s o l u t i o n .  The c o m p o s i t i o n s a t  the extreme ends d i v e r g e a t A t d e c r e a s e s , because t h e assumption t h a t t h e f l o w r a t e (q) i s s m a l l compared t o t h e amount of l i q u i d i n each l a y e r no l o n g e r h o l d s when t h e s i z e o f l a y e r s becomes v e r y s m a l l . The h i g h e s t v a l u e o f q would be a t t h e t o p , t h e r e f o r e one would expect t h i s assumption t o break down f i r s t i n t h i s r e g i o n o f the i n g o t .  The  c o m p o s i t i o n s w h i c h a r e c a l c u l a t e d a t t h e extreme ends o f the i n g o t a r e t h e r e f o r e n o t c o n s i d e r e d m e a n i n g f u l , b u t t h e shape o f t h e curves and the i n t e g r a t e d amount o f s o l u t e which has moved from t h e bottom o f t h e c a s t i n g t o the top a r e a measure o f t h e r e l a t i v e amount o f  macrosegregation.  The s t r u c t u r e o f t h e s o l i d - l i q u i d r e g i o n i s e x p r e s s e d i n terms o f the e f f e c t i v e number o f channels  (NC) , and the s o l u t e p r o f i l e s f o r d i f f e r e n t  v a l u e s o f NC a r e shown i n F i g u r e 66.  I t can be seen t h a t t h e amount o f  157  CVI  ro O ob CVJ  o CVJ  L= 37  o o  I  o  UJ  —  length of casting (H) = 14 cm temperature gradient (G) = l-5°C/cm growth rate(R) = 0 005 cm/sec number of channels (NC) =3-3 x I0 viscosity of the liquid = 0 0 3 poise 5  o  CVJ  CO,  00  20  40 60 80 DISTANCE FROM BOTTOM OF CASTING  100 (cm)  120  FIGURE 65: S o l u t e d i s t r i b u t i o n as a f u n c t i o n o f t h e number o f layers.  FIGURE 66:  S o l u t e d i s t r i b u t i o n as a f u n c t i o n o f s t r u c t u r e ( e f f e c t i v e number o f c h a n n e l s ) .  14 0  158 macrosegregation,  c o n s i d e r e d i n terms o f t h e amount o f s o l u t e w h i c h moves  from t h e bottom h a l f o f t h e c a s t i n g t o t h e t o p , i n c r e a s e s as NC  decreases.  S i n c e NC i s r e l a t e d t o t h e d e n d r i t e s p a c i n g , t h i s means t h a t f o r l a r g e r spacings  the r e s i s t a n c e t o flow through the s o l i d - l i q u i d region  decreases,  t h e r e f o r e , f o r t h e same p r e s s u r e drop t h e r e i s more f l o w . F i g u r e 67 shows t h e s o l u t e d i s t r i b u t i o n as a f u n c t i o n o f i n g o t height.  As t h e h e i g h t i n c r e a s e s , so t h e f l u i d head w i l l i n c r e a s e , c a u s i n g  more f l o w t h r o u g h the mushy zone.  However, t h i s o n l y a p p l i e s when t h e  l e n g t h o f t h e mushy zone i s g r e a t e r than o r e q u a l t o t h e i n g o t h e i g h t . t h e o r e t i c a l l e n g t h o f t h e mushy zone i s (T  The  - T )/G, w h i c h f o r t h e c o n d i t i o n s  used i n F i g u r e 67 i s 62 cm.  F i g u r e 68 shows t h a t t h e amount o f m a c r o s e g r e g a t i o n i n c r e a s e s as the growth r a t e d e c r e a s e s .  T h i s would be e x p e c t e d ,  a v a i l a b l e f o r f l o w i n c r e a s e s , as R d e c r e a s e s .  s i n c e t h e amount o f time  F i g u r e 69 shows t h a t t h e  amount o f m a c r o s e g r e g a t i o n i n c r e a s e s as t h e temperature g r a d i e n t i n c r e a s e s , (34 c o n t r a r y t o t h e s e m i q u a n t i t a t i v e t h e o r y proposed by C o p l e y , G i a m e i , e t a l . T h i s can be v i s u a l i z e d when one c o n s i d e r s t h a t t h e c o m p o s i t i o n  gradient  through t h e mushy zone w i l l be s t e e p e r f o r the h i g h e r temperature g r a d i e n t . T h i s w i l l l e a d t o a h i g h e r d e n s i t y d i f f e r e n c e and c o n s e q u e n t l y when a l l o t h e r v a r i a b l e s a r e h e l d c o n s t a n t . c o n t r a d i c t experience  The reason why t h i s appears t o  i s t h a t h i g h temperature g r a d i e n t s a r e u s u a l l y a s s o c -  i a t e d w i t h h i g h growth r a t e s , and i t i s n o t n o r m a l l y two parameters  more f l o w ,  independently.  f e a s i b l e t o vary  these  159  FIGURE 67:  FIGURE 68:  S o l u t e d i s t r i b u t i o n as a f u n c t i o n o f i n g o t h e i g h t .  S o l u t e d i s t r i b u t i o n as a f u n c t i o n o f growth r a t e .  160  o I m H  00  1  1  1  2 0  i  4 0  1  1  6 0  1  1  1  8 0  DISTANCE FROM BOTTOM OF CASTING  1  100  (cm)  1  1  120  FIGURE 69: S o l u t e d i s t r i b u t i o n as a f u n c t i o n o f temperature gradient.  1  1  14  0  161  7.6  Comparison w i t h  Experiment  The d a t a used t o g e n e r a t e the t h e o r e t i c a l curves i n F i g u r e s 56,  t o g e t h e r w i t h t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s of  a c c o r d i n g t o the p r e s e n t d e f i n i t i o n ( E q u a t i o n 6.2),  53-  macrosegregation  are g i v e n i n T a b l e X I I .  I n g e n e r a l , the model p r e d i c t s p r o f i l e s o f the same shape as the e x p e r i m e n t a l p l o t s , but the c o m p o s i t i o n s at the ends o f t h e i n g o t do not always w e l l w i t h those p r e d i c t e d . the model.  T h i s i s due to the assumptions  used i n d e r i v i n g  I n a d d i t i o n t o those a l r e a d y d i s c u s s e d , E q u a t i o n 7.1  take i n t o account  agree  does not  t h a t the l i q u i d c o m p o s i t i o n cannot r i s e above the e u t e c t i c  composition.  The aim o f the experiments macrosegregation  was  i n Chapter 6 was  t o demonstrate t h a t  r e l a t e d t o the s o l i d i f i c a t i o n v a r i a b l e s .  They were  not s p e c i f i c a l l y d e s i g n e d t o t e s t the model, c o n s e q u e n t l y o n l y q u a l i t a t i v e comparisons  have been made.  When attempts were made t o use the computer  program t o c a l c u l a t e the s o l u t e p r o f i l e s f o r h y p o t h e t i c a l i n g o t s w i t h v e r y l a r g e d e n d r i t e s p a c i n g s , i t was r i s e to a very large value. assumption  found t h a t the c o m p o s i t i o n a t the top would  T h i s was  p r o b a b l y due t o the breakdown i n the  t h a t the f l o w r a t e between l a y e r s i s s m a l l compared t o the amount  of l i q u i d i n each l a y e r .  However, i n the case o f t h e d a t a i n T a b l e X I I , the  maximum t h e o r e t i c a l c o m p o s i t i o n f o r each i n g o t was w e l l below the e u t e c t i c , which i s an i n d i c a t i o n t h a t t h i s assumption was d e n d r i t e s p a c i n g s i n v o l v e d i n these  For F i g u r e s 53(a) temperature  r e a s o n a b l y v a l i d f o r the  experiments.  and 54(a), which were s o l i d i f i e d under the same  g r a d i e n t , but w i t h d i f f e r e n t growth r a t e s and d e n d r i t e s p a c i n g s ,  b o t h the t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s show more m a c r o s e g r e g a t i o n  at  TABLE X I I  S O L I D I F I C A T I O N VARIABLES USED FOR THEORETICAL PLOTS  Figure  Number  Temperature Gradient Growth Rate  of Casting  Number  o f Channels  Number AC AC  of  Layers  (theoretical) (experimental)  55(a)  56(a)  1.5  1.5  2.3  1.0  0.013  0.011  14  14  14  (H) cm  3  (NC) = n r A  of the Liquid  54(a)  0.0047  (R) c m / s e c  Length  Viscosity  (G) ° C / c m  53(a)  (u)  poise  2.96 x  10  5  4.24 x  10  5  8.86 x  0.24 14 10  5  3.73 x  10  0.03  0.03  0.03  0.03  28  28  28  29  1.95  0.39  0.31  0.002  1.07  0.73  0.13  0.27  6  163  the l o w e r growth r a t e . at a p p r o x i m a t e l y  For F i g u r e s 54(a)  and 5 5 ( a ) , w h i c h were s o l i d i f i e d  the same growth r a t e , but w i t h d i f f e r e n t temperature  grad-  i e n t s and d e n d r i t e s p a c i n g s , the t h e o r y p r e d i c t s s l i g h t l y more macrosegregat i o n at the l o w e r temperature g r a d i e n t . w i t h experiment.  T h i s i s q u a l i t a t i v e l y i n agreement  I t s h o u l d be n o t e d t h a t a l t h o u g h  m a c r o s e g r e g a t i o n a t lower temperature g r a d i e n t s  F i g u r e 69 p r e d i c t s l e s s  (when a l l o t h e r v a r i a b l e s  are h e l d c o n s t a n t ) , t h i s e f f e c t has been outweighed by the d i f f e r e n c e i n (31) dendrite spacing.  T h i s a l s o c o r r e s p o n d s w i t h the e v i d e n c e c i t e d  earlier  t h a t a r e d u c t i o n i n d e n d r i t e s p a c i n g e l i m i n a t e d f r e c k l e s i n consumable a r c melted i n g o t s . The  t h e o r e t i c a l r e s u l t f o r F i g u r e 56(a)  shows h a r d l y any macroseg-  r e g a t i o n , w h i c h would c o r r e s p o n d w i t h the e a r l i e r s u g g e s t i o n p o s s i b l e t o c l a i m any e x p e r i m e n t , due  s i g n i f i c a n t macrosegregation i n t h i s  t h a t i t i s not  particular  t o the l a r g e amount o f s c a t t e r .  U s i n g the model p r e d i c t i o n s , i t i s p o s s i b l e t o recommend a number o f changes i n c a s t i n g p r a c t i c e t h a t would reduce g r a v i t y s e g r e g a t i o n e f f e c t s in v e r t i c a l directional castings: 1)  Refinement of the d e n d r i t i c s t r u c t u r e w i l l i n c r e a s e the r e s i s t a n c e t o f l o w through the mushy zone.  2)  R e d u c t i o n of i n g o t h e i g h t  f o r a l l o y s w i t h a wide f r e e z i n g range w i l l  reduce the d r i v i n g f o r c e f o r f l o w . 3)  I n c r e a s i n g the growth r a t e w i l l reduce the time a v a i l a b l e f o r f l o w .  4)  Decreasing flow.  the temperature g r a d i e n t w i l l reduce the d r i v i n g f o r c e f o r  164  CHAPTER 8 CONCLUSIONS 8.1  Summary  I n t e r d e n d r i t i c f l u i d f l o w r a t e s have been measured i n the l e a d t i n a l l o y system w i t h g r a v i t y as the d r i v i n g f o r c e .  The r e s u l t s have been  used t o c a l c u l a t e the p e r m e a b i l i t y o f the d e n d r i t i c s t r u c t u r e , as d e f i n e d by Darcy's Law - t h e s t a n d a r d e m p i r i c a l r e l a t i o n s h i p w h i c h d e s c r i b e s t h r o u g h porous media.  flow  I t was found t h a t the p e r m e a b i l i t y o f a d e n d r i t i c  a r r a y i s a s e n s i t i v e f u n c t i o n o f the p r i m a r y  dendrite spacing.  The permea-  b i l i t y r e s u l t s were shown t o be c o n s i s t e n t w i t h a s i m p l e model o f the porous medium, w h i c h c o n s i d e r s t h e i n t e r d e n d r i t i c channels t o be e q u i v a l e n t to a bundle of c a p i l l a r y  tubes.  I t was shown t h a t t h e i n t e r d e n d r i t i c l i q u i d f l o w e d t h r o u g h the d e n d r i t i c a r r a y , w i t h o u t by d i r e c t e x a m i n a t i o n techniques.  uniformly  the formation of p r e f e r e n t i a l  channels,  o f the e t c h e d s t r u c t u r e and w i t h r a d i o a c t i v e t r a c e r  D e v i a t i o n s from Darcy's Law, w h i c h o c c u r r e d when t h e samples  were h e l d above the e u t e c t i c temperature f o r l o n g p e r i o d s of t i m e , were discussed i n r e l a t i o n t o d e n d r i t e coarsening e f f e c t s , s i m i l a r t o Ostwald r i p e n i n g , or s i n t e r i n g i n ceramics.  L e a d - t i n a l l o y s were used t o i n v e s t i g a t e the f o r m a t i o n o f c h a n n e l type c a s t i n g d e f e c t s ( f r e c k l e s and A s e g r e g a t e s ) .  I s o t h e r m a l and u n i d i r e c t -  i o n a l s o l i d i f i c a t i o n e x p e r i m e n t s were used t o study p i p e f o r m a t i o n and s o l u t e c o n v e c t i o n , caused by d e n s i t y d i f f e r e n c e s i n the i n t e r d e n d r i t i c liquid.  Macrosegregation  was observed i n i n g o t s where the l i q u i d c l o s e  165  t o the bottom of the s o l i d - l i q u i d zone was  l e s s dense than the  liquid  above, and the r e s u l t i n g p r o f i l e s were shown t o be r e l a t e d t o the  growth  r a t e , temperature g r a d i e n t , d e n d r i t e s p a c i n g , and a l l o y c o m p o s i t i o n . S h r i n k a g e t r a i l s and p i p e s were produced i n some o f t h e s e experiments when the growth r a t e s were v e r y low.  These f i n d i n g s s u p p o r t the p r e v i o u s l y  proposed mechanism f o r the f o r m a t i o n of c h a n n e l - t y p e d e f e c t s , based  on  d e n s i t y d i f f e r e n c e s i n the l i q u i d c a u s i n g i n t e r d e n d r i t i c f l u i d f l o w . A n u m e r i c a l model i s p r o p o s e d , which p r e d i c t s the c o m p o s i t i o n p r o f i l e s i n v e r t i c a l , d i r e c t i o n a l l y s o l i d i f i e d c a s t i n g s , as a f u n c t i o n o f the s o l i d i f i c a t i o n v a r i a b l e s .  D e n s i t y d i f f e r e n c e s i n the l i q u i d are taken  t o be the d r i v i n g f o r c e f o r m a c r o s e g r e g a t i o n , and the d e n d r i t i c s t r u c t u r e i s c o n s i d e r e d t o be a porous medium o f v a r i a b l e p o r o s i t y .  8.2 i)  Conclusions U s i n g the s i m p l e c a p i l l a r y model t o d e s c r i b e the d e n d r i t i c a r r a y , the p e r m e a b i l i t y i s p r o p o r t i o n a l t o the square o f the p r i m a r y d e n d r i t e spacing.  ii)  The " t o r t u o s i t y f a c t o r " , w h i c h a l l o w s f o r the f a c t t h a t the i n t e r d e n d r i t i c channels are n e i t h e r s t r a i g h t nor s y m m e t r i c a l i s e q u a l t o  iii)  C a s t i n g s h e l d f o r l o n g p e r i o d s of time above the e u t e c t i c  4.6  temperature  show d e n d r i t e c o a r s e n i n g e f f e c t s w h i c h can modify the s t r u c t u r e .  The  p e r m e a b i l i t y of c a s t i n g s h e l d a few degrees above the e u t e c t i c temperature was iv)  observed t o i n c r e a s e due t o t h i s  effect.  M a c r o s e g r e g a t i o n and c h a n n e l - t y p e d e f e c t s can be produced  i n the  l e a d - t i n system by s o l u t e c o n v e c t i o n c a u s i n g upward f l o w of l e s s dense l i q u i d .  The r i s i n g l i q u i d becomes superheated and can form a  166 pipe or channel by dissolving dendrite branches i n i t s path, v)  Using the relationship between permeability and structure determined from the i n t e r d e n d r i t i c f l u i d flow measurements, the numerical macrosegregation model i s q u a l i t a t i v e l y i n agreement with the d i r e c t i o n a l s o l i d i f i c a t i o n experiments.  The model can therefore be used to  recommend changes i n casting practice to reduce gravity segregation effects.  8.3 i)  Suggestions  for Future Work  Using the i n t e r d e n d r i t i c flow measurement technique developed i n this work, permeabilities could be measured i n other alloy systems, i n p a r t i c u l a r , those systems where the dendrites do not have orthogonal branches.  ii)  With suitable permeability data for high l i q u i d f r a c t i o n s , the fundamental nature of i n t e r d e n d r i t i c f l u i d flow could be investigated further, leading to a better theory than the simple c a p i l l a r y model used i n this work.  iii)  The effect of dendrite coarsening could be used as a method of modifying the cast structure.  iv)  Since density differences i n the l i q u i d have been shown to produce freckles i n ammonium chloride-water models, and i n lead-tin a l l o y s , the next step would be to add radioactive tracers to commercial ingots which are prone to this defect.  This method would rapidly provide  information on solute convection i n these castings, since density data i s often not available for many l i q u i d alloys used v)  commercially.  The most important improvement i n the mathematical model would be to  167  have a b e t t e r d e s c r i p t i o n o f the s o l i d - l i q u i d zone than g i v e n by the s i m p l e c a p i l l a r y model.  S i n c e the model shows t h a t macro-  s e g r e g a t i o n i s v e r y s e n s i t i v e t o the s t r u c t u r e , t h i s w o u l d p r o b a b l y have the most pronounced e f f e c t . improved  I n a d d i t i o n , the model c o u l d be  by c o n s i d e r i n g b a c k f l o w due t o volume s h r i n k a g e , and by  u s i n g a v a i l a b l e d a t a on l i q u i d v i s c o s i t y as a f u n c t i o n o f and c o m p o s i t i o n , r a t h e r than a c o n s t a n t .  temperature  168  REFERENCES  1.  T.S. Piwonka: " I n t e r d e n d r i t i c Flow d u r i n g S o l i d i f i c a t i o n o f Aluminum A l l o y s " , D.Sc. T h e s i s , 1963, Mass. I n s t , o f Technology.  2.  T.S. Piwonka and M.C. F l e m i n g s :  T r a n s TMS-AIME, 1966^ v o l . 236,  pp. 1157-65. 3.  J . Campbell:  Trans TMS-AIME, 1968, v o l . 242, p. 264.  4.  J . Campbell:  Trans TMS-AIME, 1968, v o l . 242, p. 268.  5.  J . Campbell:  Trans TMS-AIME, 1968, v o l . 242, p. 1464.  6. 7.  J . Campbell: Trans TMS-AIME, 1969, v o l . 245, p. 2325. R.H. T i e n : J . A p p l . Mech., 1972, v o l . 3, p. 333.  8.  N. S t a n d i s h :  9.  R. M e h r a b i a n , M. Keane and M.C. F l e m i n g s : pp. 1209-20.  10.  Met. T r a n s . , 1970, v o l . 1, p. 2026.  F.L. Kaempffer:  Met. T r a n s . , 1970, v o l . 1,  " i n t e r d e n d r i t i c F l u i d Flow", M.A.Sc. T h e s i s , 1970,  U n i v . o f B r i t i s h Columbia. 11.  F.L. Kaempffer and F. Weinberg:  12.  F. Weinberg and R.K. Buhr:  13.  M.C. F l e m i n g s : R.V. Barone, S.Z. Uram and H.F. T a y l o r : Trans TMS-AIME, 1961, v o l . 69, p. 422. M. Muskat: "The Flow o f Homogeneous F l u i d s t h r o u g h Porous M e d i a " , 1937, M c G r a w - H i l l .  14.  Met. T r a n s . , 1971, v o l . 2, pp. 3051-54.  I . S . I . P u b l i c a t i o n 110, 1968, p. 295.  15.  P.C. Carman: "Flow o f Gases t h r o u g h Porous M e d i a " , B u t t e r w o r t h s S c i e n t i f i c P u b l i c a t i o n s , London, 1956.  16.  A.E. S c h e i d e g g e r : "The P h y s i c s o f Flow through Porous M e d i a " , o f T o r o n t o P r e s s , 1957.  17.  P.C. Carman: T r a n s . I n s t . Chem. Eng. ( B r i t i s h ) , 1937, v o l . 15, pp. 150-166.  18.  C.L. R i c e and R. Whitehead:  19.  N.F. Bondarenko and V.G. Karmanov: v o l . 13, No. 8, p. 791.  Univ.  J . Phys. Chem., 1965, v o l . 69, pp. 4017-24. S o v i e t P h y s i c s - D o k l a d y , 1969,  169  20.  F. Weinberg and E. T e g h t s o o r i i a n : Met. T r a n s . , 1972, v o l . 3, pp. 93-111.  21.  M.J. S t e w a r t : " N a t u r a l C o n v e c t i o n i n L i q u i d M e t a l s " , Ph.D. T h e s i s , 1970, U n i v e r s i t y o f B r i t i s h Columbia.  22.  L.C. MacAulay:  " L i q u i d M e t a l Flow i n H o r i z o n t a l Rods", Ph.D. T h e s i s ,  1972, U n i v e r s i t y  of B r i t i s h  Columbia.  23.  F. Weinberg:  24.  T.E. Strangman and T.Z. K a t t a m i s :  25.  J.M. Coulson and J.F. R i c h a r d s o n : "Chemical E n g i n e e r i n g " , 1968, v o l . 2, Pergammon P r e s s . T.Z. K a t t a m i s , J.C. C o u g h l i n and M.C. F l e m i n g s : T r a n s . TMS-AIME, 1967,  26.  T r a n s . TMS-AIME, 1961, v o l . 221, pp. 844-850. Met. T r a n s . , 1973, v o l . 4, p. 2219.  v o l . 239, pp. 1504-1511. 27.  G.W.  28.  G.W.  29.  Greenwood:  A c t a Met.,  1956, v o l . 4, pp. 243-248.  Greenwood: Monograph and Report S e r i e s No. 33, 1969, I n s t , o f M e t a l s , London, p. 103. W.D. K i n g e r y : " i n t r o d u c t i o n t o Ceramics", 1960, John W i l e y & Sons I n c . , p. 375.  30.  A.F. Giamei and B.H. K e a r :  Met. T r a n s . , 1970, v o l . 1, pp. 2185-91.  31.  R.P. D e V r i e s and G.P. Mumau:  32. 33.  G.C. G o u l d : T r a n s . TMS-AIME, 1965, v o l . 233, p. 1345. T. Mukherjee: 3 r d I n t . Symp. on E l e c t r o s l a g and o t h e r s p e c i a l m e l t i n g t e c h n o l o g y , A.S.M. and M e l l o n I n s t . , June 1971, Symposium P r o c e e d i n g s P a r t I I , p. 215.  34.  S.M. C o p l e y , A.F. G i a m e i , S.M. Johnson and F. Hornbecker:  J . M e t a l s , Nov. 1968, v o l . 20, p. 33.  Met. T r a n s . ,  1970, v o l . 1, pp. 2193-2204. 35.  C.E. S m e l t z e r :  36.  R. M e h r a b i a n , M. Keane and M.C. F l e m i n g s : Met. T r a n s . , 1970, v o l . 1, pp. 3238-41. R . J . McDonald and J.D. Hunt: T r a n s . TMS-AIME, 1969, v o l . 245, pp. 1993-97.  37.  38.  I r o n Age, 1959, v o l . 184, No. 11, p. 188.  J.R. B l a n k and F.B. P i c k e r i n g : "The S o l i d i f i c a t i o n P u b l i c a t i o n 110, pp. 370-376.  of Metals", I.S.I.  170  39.  H.P. U t e c h , W.S. Bower and J.G. E a r l y : " C r y s t a l Growth", P r o c e e d i n g s of an I n t e r n a t i o n a l Conference on C r y s t a l Growth, B o s t o n , June 1966, p. 201.  40.  N . S t r e a t and F. Weinberg:  41.  D.J. H e b d i t c h and J.D. Hunt:  42.  H.R. T h r e s h , A.F. Crawley and D.W.G. White: v o l . 242, pp. 819-22.  43.  J . K o h l , R.D. Z e n t n e r and H.R. Lukens:  Met. T r a n s . , 1972, v o l . 3, pp. 3181-84. Met. T r a n s . , 1973, v o l . 4, pp. 2008-10. T r a n s . TMS-AIME, 1968,  "Radioisotope Applications  E n g i n e e r i n g " , 1961, Van N o s t r a n d and Co. 44.  B. P r a b h a k a r :  45.  M.C. Flemings and G.E. Nereo: pp.  M.A.Sc. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 1973. T r a n s . TMS-AIME, 1967, v o l . 239,  1449-1461.  46.  D.J. H e b d i t c h and J.D. Hunt: Met. T r a n s . , 1973, v o l . 4 , p. 2474.  47.  T.F. Bower, H.D. Brody and M.C. F l e m i n g s : T r a n s . TMS-AIME, 1966, v o l . 236, pp. 624-34. C . J . S m i t h e l l s : " M e t a l s R e f e r e n c e Book", v o l . 2, 4 t h E d i t i o n ,  48.  B u t t e r w o r t h s , London, 1967. 49.  E. S c h e i l :  50.  J . S . K i r k a l d y and W.V. Y o u d e l i s : T r a n s . TMS-AIME, 1958, v o l . 58, p. 212. W.V. Y o u d e l i s : "The S o l i d i f i c a t i o n o f M e t a l s " , I . S . I . P u b l i c a t i o n 110, December 1967, p. 112. M.C. F l e m i n g s , R. Mehrabian and G.E. Nereo: T r a n s . TMS-AIME, 1968, v o l . 242, pp. 41-49.  51. 52.  M e t a l l f o r s c h u n g , 1942, v o l . 20, p. 69.  53.  M.C. Flemings and G.E. Nereo:  T r a n s . TMS-AIME, 1968, v o l . 242, pp. 50-55.  54.  H.R. Thresh and A.F. Crawley:  Met. T r a n s . , 1970, pp. 1531-35.  55.  M.E. G l i c k s m a n and C.L. V o i d :  A c t a Met.,  56.  M.E. G l i c k s m a n and C.L. V o i d : "The S o l i d i f i c a t i o n o f M e t a l s " , I . S . I . P u b l i c a t i o n 110, December 1967, pp. 37-42.  57.  M.E. G l i c k s m a n and C.L. V o i d : pp. 73-77.  1967, v o l .15, pp. 1409-12.  J o u r n a l o f C r y s t a l Growth, 1972, v o l . 13,  171  58.  M.E. G l i c k s m a n  and C.L. V o i d :  A c t a Met., 1969, v o l . 17, pp. 1-11.  59.  M.E. G l i c k s m a n  and C.L. V o i d :  S c r i p t a Met., 1971, v o l . 5, pp. 493-498.  60.  M. Hansen: " C o n s t i t u t i o n o f B i n a r y A l l o y s " , M c G r a w - H i l l , p. 302.  Second E d i t i o n ,  1958,  172  APPENDIX I INTEGRATION OF DARCY'S LAW FOR A FALLING HEAD area  a.  FIGURE 70: Pb-Sn a l l o y i n t h e f l o w c e l l , a f t e r a time t .  Darcy's Law s t a t e s : where  v  =  PL  AP  v  =  bulk velocity  K  =  permeability  AP  =  p r e s s u r e drop a c r o s s t h e porous medium  u  =  v i s c o s i t y of the l i q u i d  .L  =  l e n g t h o f t h e porous medium.  A.l  D u r i n g a b r i e f time i n t e r v a l d t , t h e q u a n t i t y w h i c h f l o w s t h r o u g h t h e porous medium w i l l be dq, t h e r e f o r e : A  dt  A.2  173  where  A  Thus  =  a r e a o f t h e porous medium.  dq  =  -KA ~ pgh dt  p  =  density of the l i q u i d  g  =  gravity  =  head a t time t .  where  h  A. 3  t  I f t h e volume o f l i q u i d w h i c h has r i s e n up t h e r i s e r p i p e i s a2& (where a  2  i s t h e a r e a o f t h e r i s e r , and £ t h e l e n g t h r i s e n ) ,  w i l l have f a l l e n i n t h e r e s e r v o i r above t h e bed.  The d i s t a n c e f a l l e n  t h e r e f o r e be a2^/a^, where a^ i s t h e a r e a o f t h e r e s e r v o i r . h  fc  an e q u a l volume  The d i s t a n c e  i s t h e r e f o r e g i v e n by:  i.e.  where h  Q  h. t  =  h  - (a-A/a.) - I  h  = h o  - £(1 + a./a.) 2 1  t  o  i  l  A.4  i s the o r i g i n a l head a t t = 0.  Thus f o r a s m a l l change i n t h e head:  dh  =  fc  - (1 + a / ) d £ 2  ai  A.5  The volume w h i c h f l o w s up t h e r i s e r p i p e d u r i n g time d t i s dq, where dq = - a d£ 0  2  A. 6  S u b s t i t u t i n g i n E q u a t i o n A.5  dh  t  =  (1 + a /a.) / 1 9  will  S i n c e the q u a n t i t y f l o w i n g i n the r i s e r e q u a l s the q u a n t i t y f l o w i n g through the porous medium, E q u a t i o n A.7 Equation  A.3:  -KApgh UL  i.e.  can be combined w i t h  t  A  dt  d h  t  (1 + a.^/a.^)  dt  =  - (c/K)dh /h  c  =  a^L/(1  where  2  A.8  + a /a )Apg 2  1  Integrating: h t dt  =  - (c/K) |  dh /h t  h t  =  o - (c/K) I n (h /h ). t o  t  175  APPENDIX I I FORTRAN PROGRAM FOR PROCESSING  INTERDENDRITIC  FLUID FLOW DATA  THIS PROGRAM CALCULATES THE PERMEABILITY OF A CASTING FROM THE FLUID PLOH MEASUREMENTS. IT READS L AND T DATA (DISTANCE PLOWED UP THE RISER PIPE, AND TIME) WHICH IT CONVERTS TO THE FORM OF EQUATION 4.2 IN THE TEXT. THE INITIAL PERMEABILITY IS THEN FOUND USING THE METHOD OF LEAST SQUARES ITERATIVELY, AS DESCRIBED IN SECTION 4.2.1. THE TIME DEPENDENCE OF THE PERMEABILITY IS SUBSEQUENTLY CALCULATED BY FITTING THE DATA TO AN EQUATION OF THE FORM GIVEN IN EQUATION 4.19 OF THE TEXT. EXTERNAL LINE DIMENSION F (4 5) . WW (60) ,YF(60) ,E1 (2) ,E2 (2) ,P (2) ,T (60) , ALH (60) ,TT (60 • ) ,A(4) ,TU(60) ,W(60) REAL L (60),LN(60),H,K1,LU(60) DATA TS/'SEC. V.TH/'MIN. V . L I / ' I I ' - ' / . f / ' M H . V » I " F / ' / 2 6 V C F-TEST TABLE FOR A SIGNIFICANCE LEVEL OF 0.05 DATA F/2.43,2.27,2.16,2.07,2.00,1.94,1.89,1.85,1.82,1.79,1.76, 11.74,1.71,1.69,1.68,1.66,1.64,1.63,1.6 2,1.61,1.60,1.59,1.58,1.57,1 1.56, 1.55, 1.54, 1.53, 1.53, 1.52,1.51,1.51 ,1.50, 1.49,1.49,1.48,1.47, 1. 147, 1.46, 1.46, 1.46, 1.46, 1.45, 1. 45, 1.45/ V=.8740711E-05 C V-WBIGHTED VARIANCE OBTAINED FROM THE CALIBRATION TEST DESCRIBED IN C SECTION 3.7. IT IS USED IN THE F-TEST COMPARISON CA=0.315*0.315 CB=0.75*0.75 C1=CA»1.33/ (0.93*0.93*(1.0+CA/CB) ) *2.54 C C1=LOWER CASE C IN THE TEXT VISC=.03 C VISC=GREEK MU IN THE TEXT C1=C1*VISC/(981.*8.33) WRITE(7,131) 900 HEAD (5,1,END=901)A 1 FORMAT(4A4) READ (5,2)H,HO,TA,TC,LC,IF C H=NUHBER OF DATA POINTS PER TEST C H0=INITIAL HEAD OF LIQUID-LOWER CASE H,SUBSCRIPT 0 IN THE TEXT C TA=TIHE BETWEEN MELTING AND THE ZERO POINT OF FLOW MEASUREMENTS C TC=UHITS OF TIME C LC=0BXTS OF LENGTH C IF=1 OR 0, DEPENDING ON PORMAT OF L AND T DATA 2 F0RMAT(I3,F6.3,F7.2,A4,A3,I1) IP (IF.EQ.O)GO TO 4 C READ L DATA AND T DATA IN DIFFERENT FORMATS READ(5,3) (L(I),T(I),I=1,N) 3 FORHAT(F5.3,P10.3) GO TO 65 4 READ (5,5) (T(I) , 1=1,N) 5 FORMAT (11F7.2) RSAD(5,6) (L (I) ,1=1,N) 6 FORMAT(13F6. 3) 65 IF (LC.EQ.LI) PL=1. IP (LC.EQ.LP)PL*1./26. IP (LC.EQ.LH)PL-1./25.4  176  DO 7 1*1,H T ( I ) = T ( I ) *Tk L O ( I ) * L (I) TO (I)«T(I) 7 L(I)»PL*L(I) 8 IP (TC.BQ.TS) GO TO 9 DO 85 1*1,1 C COBVEBT T TO SECONDS PROH THE INSTANT OF BELTING 85 T(I)=60.»T(I) 9 HO=(1.0+CA/CB)/HO DO 10 I*1,8 HM(I) = 1 • 0-H0*L (I) C L I ( I ) * L N (HT/HO) IN EQUATION Q.2 OF THE TEXT 10 LI(I)=ALOG(Hli(I)) HBITB (6 , 10 1) A 101 FORMAT(////IX,20A4) WHITE(6,103)LC.TC 103 FOBNAT(/1X,' NO. L(»,*3,') T(«,AU,«) L (IB. ) T(SEC.) 1L) •) HBITE(6,102) (I,LU(I) ,T0(I) ,L(I) ,T(I) ,LN(I) ,1=1,«) 102 FOBHAT (1X,I<» F8.3,2F9.3,F8.0,21,E1U.7) C LEAST SQUARES FITTING ROUTINE (LN(I) VEBSUS T ( I ) ) J=6 JJ=1 111 SH=0. SXYH=0. SXB>0. SYW=0. SWXS=0. DO 12 1=1, J H(I)=HH(I) • WH(I) C » (I) WEIGHTING FACTOR SH=SH + H(I) SXYH=SXYH*T(I)*LN (I)*H (I) SXH=SXN*T(I)*W(I) SYH=SYH*LH(I)*H (I) 12 SBXS=S»XS*B(I)*T ( I ) * T (I) DBH*SHXS*Sll-SXW*SXi1 IF (DEB.EQ.0.)GO TO 121 NUH=(SXYW*SW-SXH*SYH) R=NUR/DEN GO TO 220 121 H=0. 220 AN*FLOAT(J) SX*0. SI=»0„ DO 13 1=1,J SX=SX*T(I) 13 SI»SY*LN(I) TBAB=SX/AN C BEST FIT VALUES OF LB(I) ARE ALBAR,AND FOB T ( I ) ARE TBAR AL8AH SY/AN C=ALBAR-H*TBAR SBES=0„ DO 1»4 1=1,J BES=H(I)*(LN(I)-C-H*T(I))*(LN (I)-C-H*T (I)) 1«» SRES=SRES*RES C CALCULATE VARIANCE VAB=SBES/(AN-2.) IP (VAB.LE.V)GO TO 112 #  Z  LN(1-HO*  177  C C  DO P-TEST, AND ITERATE TO FIHD THE MAXIMUM NUMBER OF POINTS WHICH CAN BE USED FOR THE INITIAL SLOPE PP-VAR/V IP (PP.GE.P(JJ))GO TO 161 IP (J.EQ.SO)GO TO 161 112 J«J*1 JJ-JJM GO TO 111 C CALCULATE THE STANDARD ERROR OP Y PROS THE VARIANCE 161 SEY*SQRT (VAB) SWYS*0. DO 15 I»1,J 15 SWYS<*SWYS«LM (I) *LN (I) *W (I) R«NUH/SQRT (DEN* (SWIS*SW-SYW*SYW) ) WRITE(6,104)H,C,R,SBY 104 PORHAT(//1X,»SLOPE=',E16.7, • INTERCEPT *',E16.7//IX,'CORRELATION C 10EPPICIBNT='.E16.7//1X,»STD. ERROR OF Y=',E16.7) WRITE (6,105)VAR 105 FORMAT(//IX,'WEIGHTED VARIANCE OF Y=',E16.7) WRITE (6,106) J 106 FORMAT(//IX,'NO. OF DATA POINTS USED TO ESTIMATE SLOPE=',I3) C CALCULATE THE INITIAL PERMEABILITY ( K 1=K IN THE TEXT) K1=-H*C1 TO*-C/H DO 18 I»1,N C FIT DATA TO THE PORM OF EQUATION 4.19 OF THE TEXT TT(I)=T(I)-TO 18 ALH (I)=-C1*LN (I)-K1*TT (I) C ALH»RIGHT HAND SIDE OF EQUATION 4.19 WRITE(6,1061)TO,K1 1061 FORMAT(////1X,'TIME BETWEEN MELTING AND ZERO POINT OF FLOW MEASURE 1 HENTS (TO)=',P7.2,'SEC. '//1X,'PERMEABILITY(K1) AT TIME TO=',E16.7,' • (SQ.CM.) •) WRITE (6,140) 140 FORMAT(/1X,'RESULTS FROM LQF'/) P(1)=0.0 P(2) =0.0 C USE THE LIBRARY LEAST SQUARES FITTING ROUTINE TO CHECK THE LEAST C SQUARES ROUTINE THAT WAS WRITTEN FOR THIS PROGRAM. CALL LQF(T,LN,YF,W,E1,E2,P,1.0,J,2,1,ND,1.E-4,LINE) WRITE(8) A,N,P (1) ,P(2) EM=E2 (1)/P(1) BC=B2(2)/P(2) P(2)=-P(2)/P(1) P(1)=-P(1) *C1 WRITE (8) P (1) ,P (2) WRITE (8) (L(I) ,T(I) ,LN(I) ,I=1,N) WRITE(6,141)El (1),E2 (1) 141 FOR9AT(1X, STATISTICAL ERROR IN SLOPE=•,E16.7,3X,•TOTAL ERROR IB S •LOPE=«,E16.7) WRITE (6, 142) E1 (2) , E2 (2) 142 FORMAT (IX,'STATISTICAL ERROR IN INTERCEPT=•,E16.7,3X,'TOTAL ERROR • IN INTERCEPT ' , E16. 7) ETO*SQRT((EM*EM*EC*EC) *P(2)*P(2) ) ETO=2.0*ETO EK1= (C1*E2(1))*2.0 WRITE (6, 143) P (1) ,EK1,P(2) , ETO 143 FORMAT(/1X,'PERMEABILITY (K1) AT TIME TO=•,E16.7,• (SQ.CH.)•,2X,•95* • CONF. INTERVAL ',E16.7//1X,'TIME BETWEEN MELTING AND ZERO POINT 0 •F PLOW MEASUREMENTS(TO)=•,F7.2,'SEC.•,2X,'951 CONF. INTERVAL=•,P7. 3  0  31  3  178  107 108 109 131 132 901  •3) BR ITE (8) (ALH(I) ,TT(I) ,1=1,N) WRITE (6,107) FORMAT{////IX,'DATA FOR HON LINEAR LEAST SQUARES FITTING'//1X,6X, • •T-TO',10X,•ALH') WRITE(6,108) ( I , T T ( I ) ,ALH (I) ,1=1,N) FORHAT (1I,I3,F7.0,2X,E16.7) WRITE(6,109) FORMAT (//1X,120(**')) WRITE(7,132)A,N,J,R,P(2) ,ETO,P(1) ,EK1 GO TO 9 0 0 FORMAT(<*5X,'SUMMARY OF TEST RESULTS'/20X,* N',6 X,' J•,7X,• R',81,•TO* •,«X,'ERROR',8X,»K1 ',8X,'ERROR') FORMAT («AU,2X,I3 <*X,I3,UX,F6.ft,ftX,P5.0,2X,P7.3 E13. <*, E12. U) STOP END FUNCTION LINE (P.D.T.LQ) DIMENSION P (2),D (2) D(1)=T D(2)=1.0 LINE=P (1) *T*P (2) RETURN END #  f  179  APPENDIX I I I  THE SOLIDIFICATION OF Pb-20%Sn - A TABLE OF SOLIDIFICATION  VARIABLES  V a l u e s o f t h e p a r t i t i o n r a t i o k , and t h e l i q u i d c o m p o s i t i o n C^, were Q  o b t a i n e d as a f u n c t i o n o f t e m p e r a t u r e from t h e phase diagram. composition  C  and the w e i g h t f r a c t i o n l i q u i d f  S  The s o l i d  were c a l c u l a t e d u s i n g t h e  Li  Pfann e q u a t i o n , as d e s c r i b e d i n s e c t i o n 7.2. The volume f r a c t i o n o f (42) l i q u i d was c a l c u l a t e d u s i n g t h e d e n s i t y d a t a f o r Pb-Sn a l l o y s  T°C  k  276.0 275.0 274.0 273.0 272.0 271.0 270.0 269.0 268.0 267.0 266.0 265.0 264.0 263.0 262.0 261.0 260.0 259.0 258.0 257.0 256.0 255.0 254.0 253.0 252.0 251.0 250.0 249.0 248.0 247.0 246.0 245.0 244.0 243.0 242.0 241.0 240.0 239.0 238.0  0.501 0.497 0.493 0.490 0.486 0.482 0.478 0.474 0.471 0.467 0.463 0.459 0.456 0.452 0.449 0.445 0.442 0.438 0.435 0.432 0.428 0.425 0.422 0.419 0.416 0.413 0.410 0.407 0.404 0.402 0.399 0.396 0.394 0.391 0.389 0.387 0.384 0.382 0.380  0  C  L  20.005 20.488 20.974 21.461 21.951 22.442 22.935 23.430 23.926 24.424 24.923 25.423 25.925 26.427 26.931 27.436 27.941 28.447 28.954 29.462 29.970 30.478 30.986 31.495 32.004 32.513 33.022 33.531 34.039 34.548 35.055 35.563 36.070 36.576 37.081 37.586 38.089 38.592 39.093  C  s  0.000 10.107 10.184 10.259 10.332 10.402 10.470 10.535 10.598 10.659 10.718 10.776 10.831 10.884 10.936 10.987 11.035 11.082 11.128 11.173 11.216 11.258 11.298 11.338 11.376 11.414 11.450 11.486 11.520 11.554 11.586 11.618 11.650 11.680 11.710 11.739 11.767 11.795 11.822  f  L  1.000 0.953 0.910 0.870 0.833 0.798 0.765 0.734 0.706 0.679 0.654 0.630 0.608 0.587 0.567 0.548 0.531 0.514 0.498 0.483 0.469 0.455 0.442 0.430 0.418 0.407 0.397 0.386 0.377 0.368 0.359 0.350 0.342 0.334 0.327 0.320 0.313 0.306 0.300  g  L  1.000 0.956 0.915 0.877 0.841 0.808 0.776 0.747 0.720 0.694 0.670 0.647 0.626 0.606 0.587 0.569 0.552 0.535 0.520 0.506 0.492 0.488 0.475 0.464 0.452 0.436 0.426 0.416 0.407 0.398 0.389 0.381 0.373 0.365 0.358 0.351 0.344 0.338 0.332  T°C 237.0 236.0 235.0 234.0 233.0 232.0 231.0 230.0 229.0 228.0 227.0 226.0 225.0 224.0 223.0 222.0 221.0 220.0 219.0 218o0 217.0 216.0 215.0 214.0 213.0 212.0 211.0 210.0 209.0 208.0 207.0 206.0 205.0 204.0 203.0 202.0 201.0 200.0 199.0 198.0 197.0 196.0 195.0 194.0 193.0 192.0 191.0 190.0 189.0 188.0 187.0 186.0 185.0 184.0  "0 0.378 0.376 0.374 0.372 0.370 0. 368 0.367 0. 365 0. 363 0. 362 0.360 0. 359 0. 358 0.356 0.355 0. 354 0.352 0. 351 0.350 0.349 0.348 0.347 0.346 0. 345 0.344 0.343 0.342 0.341 0. 340 0.339 0.338 0.337 0. 336 0. 335 0.334 0.333 0.332 0.331 0. 330 0.329 0.327 0. 326 0.325 0o 324 0.322 0. 321 0. 319 0.318 0.316 0.314 0. 312 0. 310 0.308 0. 306  C.'L 39.594 40.092 40.590 41.086 41.581 42.074 42.565 43.054 43.542 44.027 44.511 44.992 45.471 45.947 46.421 46.893 47.362 47.828 48.292 48.752 49.210 49.664 50.1 16 50.564 51.009 51.450 51.888 52.322 52.753 53.180 53.603 54.022 54.437 54.847 55.254 55.656 56.054 56.447 56.836 57.221 57.600 57.975 58.344 58.709 59.068 59.423 59.772 60.1 15 60.453 60.786 61.113 61.434 61.750 62.059  C  s  11.849 11.875 11.901 11.926 11.950 11.974 11.998 12.021 12.043 12.066 12.088 12.109 12.130 12. 151 12.171 12.191 12.211 12.230 12.249 12.267 12.286 12.304 12.321 12.339 12.356 12.372 12.389 12.405 12.421 12.436 12.452 12.467 12.481 12.496 12.510 12.523 12.537 12.550 12.563 12.575 12.588 12.599 12.611 12.622 12.633 12.644 12.654 12.664 12.674 12.683 12.692 12.700 12.709 12.716  f  L  0.294 0.288 0.282 0.277 0. 272 0.267 0.262 0.257 0. 253 0.248 0.244 0.240 0.236 0.232 0.229 0.225 0.222 0.218 0.215 0.212 0.209 0.206 0.203 0.201 0.198 0. 195 0.193 0. 190 0. 188 0. 186 0. 184 0.181 0.179 0. 177 0.175 0. 173 0.172 0. 170 0.168 0. 166 0. 165 0. 163 0.162 0. 160 0. 159 0. 157 0.156 0. 155 0.153 0. 152 0. 151 0. 150 0.149 0. 148  9L  0.326 0.319 0.313 0.308 0.303 0.298 0.293 0.288 0. 284 0.280 0.275 0.271 0.268 0.264 0.260 0.257 0.253 0.250 0.247 0.251 0.248 0.245 0.235 0.232 0.230 0.227 0.225 0.222 0.220 0.218 0.215 0.213 0.211 0.209 0.207 0. 205 0.203 0.202 0.200 0. 198 0.197 0. 195 0. 193 0. 192 0. 191 0. 189 0.188 0. 191 0.190 0. 189 0.187 0. 186 0.185 0. 184  180  181  APPENDIX IV FORTRAN PROGRAM FOR CALCULATING MACROSEGREGATION IN LEAD-TIN CASTINGS THIS PROGRAM CALCULATES MACROSEGREGATION ACCORDING TO THE MODEL DESCRIBED IN CHAPTER 7. THE METHOD BASICALLY INVOLVES THE FOLLOWING STEPS: 1) THE CASTING IS DIVIDED INTO A NUMBER OF HORIZONTAL LAYERS 2) THE TEMPERATURE OF THE BOTTOM LAYER IS SET EQUAL TO THE LIQUIDUS TEMPERATURE (SEE SECTION 7.4) 3) KNOWING THE TEMPERATURE GRADIENT, THE TEMPERATURE OF ALL THE OTHER LAYERS ARE CALCULATED. 4) THEREFORE KNOWING THE GROWTH RATE, THE TEMPERATURE OF EACH LAYER AT ANY POINT IN TINE IS DEFINED. THEREFORE, DURING SOLIDIFICATION, ALL OTHER VARIABLES CAN BE DETERMINED AS A FUNCTION OF TEMPERATURE. THUS, LIQUID COMPOSITION AND THE PARTITION RATIO ARE A FUNCTION OF TEMPERATURE FROM THE PHASE DIAGRAM. USING THE PFANN EQUATION (SECTION 7.2) THE FRACTION LIQUID AND COMPOSITION OF SOLID CAN BE CALCULATED FOR EACH LAYER AT EVERY POINT IN TIME. SINCE THE COMPOSITION AND TEMPERATURE OF EACH LAYER IS NOW DEFINED, ITS DENSITY IS GIVEN USING THE SUBROUTINE DENS. DENSITY DIFFERENCES THROUGH THE MUSHY ZONE PROVIDE THE DRIVING FORCE FOR FLUID FLOW, AND THE RESISTANCE OF THE DENDRITIC STRUCTURE IS CALCULATED USING DARCY'S LAW (SECTION 7.3). THE MAIN ROUTINE READS IN THE DATA, AND THEN SETS THE INITIAL TEMPERATURE OP EACH LAYER OF THE CASTING. IT USES THE SOLIDIFICATION MODEL TO CALCULATE SOLID AND LIQUID COMPOSITIONS AND FRACTIONS. THE PHASE DIAGRAM DATA IS CONTAINED IN SUBROUTINE PBSN. EACH SOLIDIFICATION STEP IS FOLLOWED BY A FLUID FLOW STEP: THE LATTER IS CONTAINED IN SUBROUTINE FLOW. TWO DIMENSIONAL ARRAYS ARE USED FOR SOME OF THE VARIABLES. IN THIS CASE THE COLUMNS (1ST DIMENSION) REPRESENT THE POSITION - LAYER NUMBER - IN THE CASTING, AND THE ROWS (2ND DIMENSION) REPRESENT THE POIBT IN TIME. HEAHIBG OF SYMBOLS IN MAIN PROGRAM AK=EQUILIBBIUM DISTRIBUTION COEFFICIENT AKO=AVERAGE DISTRIBUTION COEFF. BETWEEN TWO TEMPERATURES AL,ALAYER=NUMBER OF LAYERS (REAL NUMBER) ALEN=LENGTH OF INGOT AH=NUMBER OF INTERDENDRITIC CHANNELS C=TORTUOSITY FACTOR (EQUALS GREEK TAU CUBED) CL=COHPOSITION OF LIQUID COHP=TOTAL COMPOSITION OF LAYER CS=COHPOSITIOH OF SOLID DIST=DISTANCE FROH BOTTOM OF INGOT DHSTY=DENSITY OF LIQUID DT'TIHE INTERVAL (INCREMENT) FL=1-PS PRLIQ=WT. FRACTION OF LIQUID PS=PBOPORTION OF FRLIQ THAT IS FREEZING  182  G=TEHPERATURE GRADIENT GL=VOLUME FRACTION OF LIQOID L=NUHBER OF LAYERS (INTEGER) P=A POWER PERH=PERHEABILITY OF LAYER R=GROWTH RATE SD=SOLID DENSITY SSOL=WT. OF SOLID SSOLUT=TOTAL WT. OF SOLUTE IN SOLID STATE= STATE OF LAYER (SOLID,MUSHY OR LIQUID) T=TOAL SOLIDIFICATION TIME TB,TB1,TB2=DATA TABLES GENERATED BY THE PROGRAM TE=SOLIDUS TEMPERATURE TIHB=TOTAL TIME AFTER THE START OF FREEZING TEMP=TEHPERATORE OF LAYER TL=LIQUIDUS TEMPERATURE TSOLOT=TOTAL WT. OF SOLUTE IN SOLID AND LIQUIC TWT=TOTAL WT. OF SOLID AND LIQUID VISC=VISCOSITY OF THE LIQUID VL=VOLUHB OF LIQUID VS=VOLUHE OF SOLID WSLIQ=WT. OF SOLUTE IN FRLIQ WSOLID=WT. OF SOLID THAT IS FREEZING WSOLUT=WT. OF SOLUTE IN WSOLID DIMENSION TEMP(100,2),CL(100,2) ,AK (100,2) ,DNSTY (100),FRLIQ (100,2) DIMENSION COMP (100) ,CS (100,2) ,WSLIQ (100) DIMENSION SSOL(100),SSOLUT (100),GL (100) DIMENSION DIST (100) ,PERM (100) INTEGER TB,TB1,TB2 DATA ALIQ/« LIQ•/,*HUSH/«HUSH•/,SOL/ SLD'/ READ (5,203)DT READ(5,201) ALEN,G,R READ (5,202)TL,TE RBAD(5,204) AN,C,VISC READ (5,205)TB1,TB2 FORMAT(F6.1,F6.2,F7.tt) FORMAT (2F10.U) FORMAT(F10. 1) FORMAT(E9.3,F6.2,F6.3) FORMAT(215) ALAYER=ALEN/ (R*DT) L=INT (ALAYER) AL=FLOAT (L) T=((TL*AL*G*R*DT)-TE)/(G*R) TEMP(1,1)=TL DO 1 1=2,L TEMP(I,1)=TEHP(1-1,1)•G*H*DT CONTINUE DO 11 1=1,L CALL PBSN(TEMP (1,1) ,AK ( I , 1) ,CL (I, 1)) CL(I,1)=CL(1,1) COMP ( I ) = C L ( I , 1) PRLIQ(I,1) = 1. SSOL (I)=0. SSOLUT(I)=0. WSLIQ(I)=CL(1,1) GL (I)=1. CS(I,1)=0. ,  201 202 203 20U 205  1  183  11  101 102  103 2 7  501  302 301  303  307  308  CONTINUE TIHE-0. PL-1. WSOLID=0. WSOLUT-0. TSOLUT=CL(1,1) TWT=1. STATE=ALIQ TB=1 WRITE (6,101) TIRE FORMAT (//IX,• TIME=',F7. 1, 'SECONDS -BEFORE FLUID FLOW•/) WRITE(6,102) FORMAT (IX,' L TEMP STATE CL AK PRLIQ DNSTI FL • WSOLID WSLIQ WSOLUT SSOL SSOLUT CS GL TNT • COMP'/) DO 2 1=1,L CALL DENS(TEMP(1,1),CL(1,1),DNSTX (I)) WRITE(6,103)I,TEMP(1,1) ,STATE,CL (1,1),AK(1,1),FRLIQ (1,1),DNSTT ( I ) , • FL,WSOLID.WSLIQ(I),WSOL0T,SSOL(I),SSOLUT(I),CS (I,1),GL (I),TWT,COHP •d) FORMAT (1X,12,F6.1,5X,AO,1X.F8.5,IX,F6.3,12 (2X,F6.3)) CONTINUE TIME=TIHE»DT IF(TIME.GT.T) GO TO 999 IP (TB.GT.TB1.AND.TB.LT.TB2)GO TO 501 WRITE{6,101)TIME WRITE (6,102) CONTINUE DO 3 1=1,L TEMP(1,2)=TEMP(I,1)-G*R*DT AK (I,2)=AK (I,1) CL(I,2)=CL(I, 1) PRLIQ (I,2)=FRLIQ (I,1) CS(I,2)=CS(I,1) STATE=ALIQ CALL DENS(TEMP(I,2),CL(I,2),DNST¥(I)) IP (TEHP(I,2).GE.TL)GO TO 304 IF (TEMP ( I , 2) . LT. TE) GO TO 303 STATE=AHUSH CALL PBSN(TEMP (1,2) ,AK(1,2),CL (1,2)) AKO= (AK (1,2) • AK (I, 1) )*0.5 P=-1./(1.-AKO) FL= (CL (I,2)/CL (I, 1) )*»P FRLIQ(I,2)=FRLIQ(I,1) *FL GO TO 307 STATE=SOL AK(I,2)=1. FL=1„ WSOLID=0. WSOLUT=0. GO TO 308 WSOLID=(1.-PL)*FBLIQ(1,1) BSLIQ (I)=FRLIQ(1,2)*CL(1,2) WSOLUT=COHP(I)-WSLIQ ( I ) - (CS (1,1)* (1.-FRLIQ (1,1))) IP (HSOLUT.LE.O.)WSOLUT=0. SSOLUT(I)=SSOLUT(I)*WSOLUT SSOL (I)=SSOL (I)*HSOLID IP(SSOLUT(I).LT.0.00001)GO TO 333 CS (I,2)=SSOLUT (IJ/SSOL (I) GO TO 33H  184  333  CONTINUE CS(1,2)=0. 330 CONTINUE TWT-SSOL(I) *FRLIQ (1,2) TSOLUT=SSOLUT(I)•HSLIQ(I) COMP(I)=TSOLUT/TWT IP (CS (1,2) . LB. 0. )CS (1,2) =0. CALL DENS(TBBP(1,2) ,CS (1,2) ,SD) VS=SSOL(I)/SD VL=FBLIQ (1,2) /DNSTY (I) GL (I)=VL/(VS^VL) IP(TEMP(I,2) .LT.TE)GL(I)=0. GO TO 309 304 CONTINOE STATE=ALIQ FL»1. WSOLID=0. WSOLUT*0. TSOLUT=CL(1,1) TNT* 1. 309 IP(TB.GT.TB1.AND.TB.LT.TB2)GO TO 502 WRITE(6,103) I,TEMP(1,2) ,STATE,CL (I,2),AK(1,2),FRLIQ (1,2),DNSTY ( I ) , •FL,WSOLID,WSLIQ(I),WSOLOT,SSOL(I),SSOLDT(I),CS(1,2),GL(I),TWT,COUP •(I) 502 CONTINUE AI=FLOAT(I) DIST (I)=(AI-0.5)*R*DT PERN (I)=0. 3 CONTINUE CALL FLOW (L , TBBP (1,2) ,DNSTY,GL,R,DT,TL,TE,VISC,AN,C,PERM , CL (1,2) ,C • S (1,2),FRLIQ (1,2),WSLIQ,SSOLDT,SSOL,COHP,TB,TB1,TB2) IF (TB.GT.TB1.AND.TB.LT.TB2)GO TO 503 WRITE(6,1011)TIME 1011 FORMAT (/1X,' TIHE=',F7.1,'SECONDS -AFTER FLUID FLOW'/) WRITE(6,106) 106 FORMAT(1X, * L TEMP PERM CL FRLIQ DIST COM •P') WRITE (6, 107) (I,TEMP (I, 2) ,PERM (I) ,CL (I, 2) ,PRLIQ (1,2) ,DIST (I) ,COMP (I •) #1=1. M 107 FORMAT (1X,I3,F6.1,1X,E14.6,F9.4,F7.3,F7.2,F10.4) 503 CONTINUE DO 6 1=1,L TEMP(1,1)=TEHP(1,2) CL(I,1)=CL(I,2) AK(I,1)=AK(1,2) FRLIQ(I,1)=FRLIQ (1,2) CS(I,1)=CS(I,2) 6 CONTINUE TIB B-TIM E*DT T3=TB*2 GO TO 7 999 WRITE (6,990)ALEN,G,R,DT,AH,C, VISC WRITE(6,992) WRITE(6,991)(DIST(I),COMP(I),1=1,L) 990 PORRAT(/1X,' LENGTH OF CASTING=•,P6.2,'CM. '/1X, • TEMPERATURE GRA • DIENT=',F6.3,'OEG.C/CM.«/1X,« GROWTH RATE=',F7.4,'CM./SEC */IX,« •TIME INTERVAL=',F6.1,'SEC.'/1X,• NUMBER OF CHANNELS= • ,E12.4,/1X,• • TORTUOSITY FACTOR=',F6.2,/1X,• VISCOSITY OP THE LIQUID*',P6.3,• •POISE') 992 FORBAT(/1X,' DIST COMP')  185  991  FORMAT (1X , F6. 2.F9.4) STOP BID ;  C C C C C  100 101  SOBROUTINB DENS (TEMP,C,DNSTY) THIS SUBROUTINE CALCULATES THE DENSITY OP LIQUID LEAC-TIN ALLOYS AS A PUNCTION OP COMPOSITION AND TEMPERATURE. THE DATA ARE TAKEN FROM A TABLE OF VALUES PUBLISHED BY THRESH ET AL, TRANS. TMS-AIME 1968,PAGE 819. INTERMEDIATE VALUES ARE OBTAINED BY A LINEAR INTERPOLATION DIMENSION CP(14) ,A(14) ,B(14) DATA CP/0.,10.,20.,30.,32.5,40.,48.75,50.,60.,62.5,70.,83.,85.,100 •./ DATA A/11.06,10.49, 9.956,9.497,9.383,9.079,8.697,8.671,8.321.8.22 • 9,7.995,7.603,7.543,7. 139/ DATA B/12.22, 1 1.582,10.481,10. 109,9.762,9.708,8. 688,8.761,8. 69,8.6 •52,8.443,7.76,7.775,7.125/ DO 100 1=1,14 E=C-CP (I) IP(E.LT.O.)GO TO 101 CONTINUE 1=1-1 CC= (CP (1+ 1) -C) / (CP (1*1) -CP (I) ) AA=A ( I O ) • (A (I) - A (1*1) ) *CC BB=B (I)-B (1 + 1) BB=ABS(BB) BB=B (1*1)•BB*CC DNSTY=AA-BB*0.0001*TEHP RETURN END SUBROUTINE PBSN (T,AK,CL) T2=T*T T3=T2*T T4=T2*T2 P1=-13.86709 P2=.2432528 P3=-. 1552426E-2 P4=.435223E-5 P5=-.448334E-8 Q1=-75.10938 02=2.297987 Q3=-. 1115845E-1 Q4 = . 147874E-4 AK=PUP2*T»P3*T2*P4*T3 + P5*T4 CL=Q1*Q2*T+Q3*T2*Q4*T3 RETURN BHD  C C C C  SUBROUTINE PLOW(L,TEMP,DNSTY,GL,R,DT,TL,TE,VISC,AN,C,PERM,CL,CS,PR •LIQ,HSLIQ,SSOLUT,SSOL,COMP,TB,TB1,TB2) THIS SUBROUTINE CALCULATES THE NEW COMPOSITION PROFILE APTER EVERY TIME INCREMENT. PERMEABILITY IS CALCULATED FROM EQUATION 7.7 IN THE TEXT, AND THE SERIES OP SIMULTANEOUS DIFFERENTIAL EQUATIONS (7.8) ABB SOLVED USING A RONGE-KUTTA TECHNIQUE - LIBRARY ROUTINE DRKC. DIMENSION TEMP (L) ,DNSTY (L) ,GL (L) ,PERM (L) ,CL (L) ,CS (L) ,FRLIQ (L) DIMENSION HSLIQ(L),SSOLUT(L).SSOL (L),COMP (L)  186  550 560  570 580  1 21 100 200  U  10 99  DIMENSION BY (99),B(99) REAL*8 X.Z,Y(99) ,P(99) ,H,HflIN,E,G (99) ,S (99) ,T (99) , VLIQ (99) ,QT (99) INTEGER TB,TB1,TB2 EXTERNAL FUNC COHHON /ZVQ/ ?LIQ,QY,LL,K,J J=1 DO 550 1=1,L TH»TL-0.01 IF(TEHP(I).GT.TB)GO TO 560 J«I CONTINUE CONTINUE DO 570 1=1,L K=I IP (TEflP(I).GT.TE)GO TO 580 CONTINUE IF(K.EQ.J)GO TO 99 GR=981. PI = 3. 142 YA=.6334 YD=R*DT XA=R*DT*0.5*2.54 DO 1 I=K,J PERM (I)=GL(I) *2.*YA/(24.*AN*PI*C) BY (I)=VISC*YD/(PERN(I)*YA) PBRH (I)=PERH (I) *GL (I) N=J-1 DO 21 I=K,N HY (I) = (HY (I) + HY ( I * 1) ) *0.5 DO 100 I=K,N AJK=PLOAT(J-K) B(I)=(DNSTY(J)-DNSTY(K))*GR*AJK*R*DT DO 200 I=R,N QY(I)=B(I)/(2.*RY(I)) X=0.D0 Z=DT H=Z/64.D0 HHIN=H*1.D-3 B=1.D-5 PB=R*DT*9.667*1.27*1.27*PI/«. H=1 DO U I=K,J X(0) =DHSTY(I)*CL (I) VLIQ (I)=PRLIQ(I)*FR/DNSTY(I) »=H*1 N=J-K*1 LL=N CALL DRKC(N,X,Z,Y, F, H,HHIN,E,PUNC,G,S,T) 11=1 DO 10 I=K,.J CL (I)=Y (H) /DNSTY (I) HSLIQ (I) =FRLIQ (I) *CL (I) TSOLUT=SSOLUT(I)•HSLIQ(I) THT=SSOL(I)•FRLIQ(I) COHP (I)=TSOLUT/TWT !!=!!• 1 CONTINUE RETURN END  187  C C  1 2  SUBROUTINE FUNC(X,Y,F) THIS SUBROUTINE SETS OP THE DIFFERENTIAL EQUATIONS TEXT) POR THE RUNGE-KUTTA TECHNIQUE DRKC. IMPLICIT REAL*8(A-H,0-Z) DIMENSION 1(1) ,F (1) COMMON /ZVQ/ VLIQ (99),QY(99) ,LL,K,J A1-QY (K)/VLIQ (K) F(1)=A1*Y(2)-A1*Y(1) IF (LL.BQ.2) GO TO 2 JJ=LL- 1 KK=K»1 H-2 DO 1 I=KK,JJ A1»QY(I)/VLIQ(I) A2=»QY (I-1)/VLIQ(I) A3= (QY(I) •QY(I-1))/VLIQ(I) F(H)=A1*Y (M*1)*A2*Y (M-1) -A3*Y (M) H=H*1 A2=QY (J-1)/VLIQ (J) F(LL) =»A2*Y (LL-1) -A2*Y(LL) RETURN END  (7.8 IN THE  188  APPENDIX V DIRECT OBSERVATION OF SOLIDIFICATION USING ELECTRON MICROSCOPY  V.l  Introduction The aim o f t h i s work was t o d i r e c t l y observe m e l t i n g and s o l i d i f i -  c a t i o n i n t h i n f i l m s o f pure m e t a l s and a l l o y s , u s i n g e l e c t r o n m i c r o s c o p y . The method was e s s e n t i a l l y t h e same as t h a t developed by G l i c k s m a n and (55-59) Void  , who observed m e l t i n g and s o l i d i f i c a t i o n i n t h i n f i l m s o f pure  b i s m u t l i / " ^ ' " ' ^ , and a number o f d i l u t e a l l o y s a n d  used t h e i r o b s e r v a -  tions to obtain the absolute value of the s o l i d - l i q u i d i n t e r f a c i a l .. ,,(58,59) f o r pure bismuth  energy  The p r e s e n t work was f i r s t d i r e c t e d towards r e p r o d u c i n g G l i c k s m a n ' s experiments on pure b i s m u t h , and t h e n u s i n g the t e c h n i q u e , t o observe s o l i d i f i c a t i o n i n o t h e r pure m e t a l s , and t h e r e b y c a l c u l a t e t h e s o l i d - l i q u i d i n t e r f a c i a l energies.  I t was hoped t h a t s u f f i c i e n t e x p e r t i s e would be  g a i n e d t o observe t h e growth o f a l a m e l l a r e u t e c t i c from t h e l i q u i d . V.2  E x p e r i m e n t a l Method The t h i n f i l m s , produced by v a r i o u s methods d i s c u s s e d below, were  examined u s i n g a H i t a c h i H U - l l A e l e c t r o n m i c r o s c o p e ( t h e m i c r o s c o p e was t h e same as t h a t used by G l i c k s m a n ) .  Both a s i m p l e h e a t i n g s t a g e , and a h e a t i n g -  t i l t i n g s t a g e were used, b u t i t was found t h a t temperatures c o u l d n o t be c o n t r o l l e d w i t h s u f f i c i e n t p r e c i s i o n u s i n g the l a t t e r s t a g e , t h e r e f o r e the r e s u l t s o n l y a p p l y t o work done w i t h the s i m p l e h e a t i n g s t a g e .  189  The specimen was h e a t e d s l o w l y u s i n g t h e h e a t i n g s t a g e , u n t i l a s m a l l m o l t e n zone was produced u s i n g t h e a d d i t i o n a l heat i n d u c e d by f o c u s i n g the  lOOkV e l e c t r o n beam.  G l i c k s m a n e s t i m a t e d t h a t t h e optimum temperature  of the specimen was about 10°C below t h e m e l t i n g t e m p e r a t u r e , however, t h i s c o u l d n o t be a c c u r a t e l y determined i n t h e p r e s e n t work u s i n g t h e a v a i l a b l e equipment.  The e l e c t r o n beam s i m u l t a n e o u s l y p r o v i d e d image i l l u m i n a t i o n and  l o c a l h e a t i n g t o produce the m o l t e n zone.  I t was found t h a t t h e m o l t e n zone  c o u l d be made t o expand and c o n t r a c t by a d j u s t i n g the c u r r e n t t o the second condenser l e n s .  In a l l e x p e r i m e n t s , one o f t h e major problems was s t a b i l i t y o f the m o l t e n zone.  The a v a i l a b l e power s u p p l y d i d n o t p r o v i d e s u f f i c i e n t l y  s e n s i t i v e c o n t r o l t o h o l d t h e specimen a t t h e r e q u i r e d temperature f o r l o n g periods.  F o r t h i s r e a s o n , i t was found t h a t t h e b e s t r e s u l t s were o b t a i n e d  by s e t t i n g t h e power s u p p l y t o heat t h e specimen v e r y s l o w l y .  This usually  a l l o w e d about t e n minutes f o r o b s e r v a t i o n o f s o l i d i f i c a t i o n and m e l t i n g w h i l e t h e specimen was i n a s u i t a b l e temperature range.  I n c o n t r a s t to  Glicksman's f i n d i n g s , i t was e x t r e m e l y d i f f i c u l t t o h o l d t h e s o l i d - l i q u i d i n t e r f a c e s t a b l e enough f o r photography d u r i n g t h i s p e r i o d .  Exposure t i m e s  of about 1-5 seconds were r e q u i r e d , and the image f r e q u e n t l y s h i f t e d d u r i n g the  course o f t h e exposure.  V.3  V.3.1  Results  Pure bismuth T h i n f i l m s were p r e p a r e d by vacuum e v a p o r a t i o n onto carbon s u p p o r t  f i l m s u s i n g s t a n d a r d methods.  A l l m e t a l s used i n t h i s work were 6-9's p u r i t y ,  190  and the vacuum system was  f l u s h e d s e v e r a l times w i t h o x y g e n - f r e e n i t r o g e n  b e f o r e pumping, and a t i t a n i u m g e t t e r was vacuum b e f o r e e v a p o r a t i o n was f i l m s was  about  2.0  used b e f o r e e v a p o r a t i o n .  x 10 ^ T o r r .  The  The  t h i c k n e s s of the carbon  and the b i s m u t h t h i c k n e s s was  i n the range 1000-2000A  1  (measured u s i n g an i n t e r f e r e n c e m i c r o s c o p e ) . T y p i c a l r e s u l t s a r e shown i n F i g u r e s 71 and  72.  Figure  71(a-c)  shows f r e e z i n g , f o l l o w e d by m e l t i n g , f o l l o w e d by f r e e z i n g i n the same r e g i o n . F i g u r e 71(d) Faceted  shows the s o l i d - l i q u i d i n t e r f a c e a t h i g h e r m a g n i f i c a t i o n .  growth o f the s o l i d i s seen i n the lower r i g h t - h a n d c o r n e r  F i g u r e s 71(a) and  (c), similar  t o t h a t seen by G l i c k s m a n .  o f the l a r g e g r a i n s i n F i g u r e s 71(b) the m o l t e n zone.  The  and  (d) i s due  The  of  double image  t o the i n s t a b i l i t y of  s m a l l g r a i n s i n the c o r n e r s show the o r i g i n a l s t r u c t u r e  of the e v a p o r a t e d f i l m .  The  l i q u i d r e g i o n s appear u n i f o r m l y dark because the l i q u i d phase  s c a t t e r s the e l e c t r o n beam randomly.  The  l i g h t p a t c h , w h i c h appears to be  growing i n the c e n t r e o f the l i q u i d zone, i s caused by t h i n n i n g of the i n this region.  E v e n t u a l l y t h i s would l e a d t o d e - w e t t i n g  draw back i n t o g l o b u l e s around a c e n t r a l h o l e .  film  and the l i q u i d would  When t h i s o c c u r r e d ,  l i q u i d became too t h i c k f o r the e l e c t r o n beam t o p e n e t r a t e , and the  the inter-  f a c e c o u l d no l o n g e r be observed.  The  advantage o f u s i n g b i s m u t h f o r i n t e r f a c i a l energy measurements  i s t h a t i t tends t o d e p o s i t from the vapour phase w i t h the b a s a l p a r a l l e l t o the p l a n e o f the specimen.  plane  Thus the boundary between n e i g h b o u r -  i n g g r a i n ? i n the t h i n f i l m i s u s u a l l y a s i m p l e t i l t boundary, and the method  FIGURE 71:  (a-c)  A l t e r n a t e f r e e z i n g , m e l t i n g , and f r e e z i n g i n pure  b i s m u t h , showing e v i d e n c e o f f a c e t e d growth.  Magnification  5000x. (d)  E n l a r g e d view of the s o l i d - l i q u i d  i n t e r f a c e , showing h i g h  a n g l e g r a i n b o u n d a r i e s emerging a t the i n t e r f a c e . M a g n i f i c a t i o n lOOOOx.  192  used by G l i c k s m a n was t o s e a r c h f o r l o w - a n g l e t i l t b o u n d a r i e s w h i c h emerged a t t h e s o l i d - l i q u i d i n t e r f a c e .  The a n g l e o f t i l t  c o u l d be measured  by t a k i n g a s e l e c t e d a r e a d i f f r a c t i o n p a t t e r n a c r o s s the boundary, o r by c o u n t i n g t h e d i s l o c a t i o n s p a c i n g a l o n g b o u n d a r i e s w i t h v e r y low t i l t a n g l e s . The i n t e r f a c i a l energy was c a l c u l a t e d from measurements o f t h e cusp a n g l e , where t h e g r a i n boundary  emerged a t t h e s o l i d - l i q u i d i n t e r f a c e ^ ^ ' " ^ .  No  attempt was made i n the p r e s e n t work t o r e p e a t these measurements on pure bismuth.  All  the bismuth samples observed i n t h e p r e s e n t work showed dark  s p e c k l e s o v e r the f i e l d o f view.  These s p e c k l e s were n o t seen b e f o r e h e a t i n g ,  b u t seemed t o form when t h e specimens are  came c l o s e t o t h e m e l t i n g p o i n t .  seen i n F i g u r e 71, and a r e even more pronounced  i n F i g u r e 72.  They  They  a l s o appear i n photographs p u b l i s h e d by G l i c k s m a n , b u t a r e n o t as common as in  the p r e s e n t work.  A l t h o u g h o x i d a t i o n was s u s p e c t e d i n b o t h the p r e v i o u s  and p r e s e n t work, no o x i d e r i n g s were o b s e r v e d i n the d i f f r a c t i o n p a t t e r n s . The n a t u r e o f t h e s p e c k l e s t h e r e f o r e remains unknown.  One can s p e c u l a t e  t h a t they might be due t o some i n t e r a c t i o n between t h e b i s m u t h and t h e carbon substrate,, s i n c e l i q u i d b i s m u t h can d i s s o l v e minute amounts o f carbon (0.0028 atomic p e r c e n t a t 300°C), w h i c h i t r e j e c t s as g r a p h i t e c r y s t a l s on solidification^^ .  T h e r e f o r e , t h e s p e c k l e s might be c r y s t a l l i t e s o f  g r a p h i t e , which one c o u l d n o t d i s t i n g u i s h from t h e s u b s t r a t e by e l e c t r o n diffraction.  The s p e c k l e s i n F i g u r e 72(c) appear t o p i n the i n t e r f a c e , which has a rougher c o n t o u r than i n F i g u r e 71.  I t i s a l s o p o s s i b l e that the  s p e c k l e s may be r e l a t e d t o some k i n d o f c o n t a m i n a t i o n .  The " i s l a n d s " o f  FIGURE 72:  A l t e r n a t e m e l t i n g , f r e e z i n g , m e l t i n g and f r e e z i n g i n pure bismuth.  Note the h i g h c o n c e n t r a t i o n of " s p e c k l e s " and the  i n t e r f a c e p i n n i n g e f f e c t i n (b) and ( d ) . The " i s l a n d s " o f s o l i d i n (a) and (c) resemble photographs of m e l t i n g publ i s h e d by G l i c k s m a n , and are p r o b a b l y caused by i n a t i o n o f the m e t a l f i l m .  M a g n i f i c a t i o n 5000x.  contam-  194  s o l i d w h i c h remain a f t e r m e l t i n g ( F i g u r e s 72(a) and ( c ) ) , a r e s i m i l a r t o the photographs o f B i - S n a l l o y m e l t i n g p u b l i s h e d by G l i c k s m a n , and t h e y can be c o n s i d e r e d p a r t o f a "mushy" zone.  T h i s would i n d i c a t e t h a t c o n t a m i n a t i o n  w h i c h causes some a l l o y i n g o c c u r r e d i n the specimen shown i n F i g u r e 72. V.3.2  Other pure m e t a l s ( t i n , aluminum and indium) S a t i s f a c t o r y t h i n f i l m s o f t i n and aluminum were produced by vacuum  e v a p o r a t i o n onto carbon s u p p o r t f i l m s , but i t was n o t p o s s i b l e t o produce o x i d e - f r e e f i l m s of i n d i u m . Both t i n and aluminum behaved i n the manner shown i n the sequence pf photographs shown i n F i g u r e 73.  D u r i n g h e a t i n g , g r a i n growth would be  observed ( F i g u r e s 73(b) and ( c ) ) , f o l l o w e d by m e l t i n g ( F i g u r e 73(d)) and immediate d e - w e t t i n g ( F i g u r e s 73(e) and ( f ) ) . u s i n g a 35 mm  These photographs were t a k e n  camera t o photograph the f l u o r e s c e n t s c r e e n , because the u s u a l  t e c h n i q u e s were too slow t o r e c o r d the r a p i d e v e n t s w h i c h o c c u r r e d .  The  grain  s i z e o f the s c r e e n , p l u s the use o f a f a s t f i l m account f o r the poor q u a l i t y . The h i g h t h e r m a l c o n d u c t i v i t y of aluminum may be r e s p o n s i b l e f o r the d i f f i c u l t y i n k e e p i n g the m o l t e n p o o l l o c a l i z e d .  S i n c e s t a b l e molten zones c o u l d not be produced i n pure m e t a l s o t h e r t h a n b i s m u t h , no i n t e r f a c i a l energy c a l c u l a t i o n s were attempted. V.3.3  Lamellar eutectics Three t e c h n i q u e s f o r p r o d u c i n g a l l o y f i l m s were a t t e m p t e d .  These  were e v a p o r a t i o n o f the two c o n s t i t u e n t pure m e t a l s , m i c r o t o m i n g , and e l e c t r o l y t i c thinning.  The work was c o n c e n t r a t e d on aluminum-copper, but the problem  of; de-rwetting p e r s i s t e d , and l a m e l l a r growth was n o t o b s e r v e d .  195  FIGURE 73:  The m e l t i n g of pure aluminum, photographed from the f l u o r e s c e n t  screen u s i n g a 35 mm (b) and  camera;  (a) o r i g i n a l vapour d e p o s i t e d s t r u c t u r e ;  (c) g r a i n growth (approx.  and d e - w e t t i n g ;  (f)  450°C);  t o t a l de-wetting.  (d) and  (e) b e g i n n i n g of m e l t i n g  M a g n i f i c a t i o n approx. 5000x.  

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