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 in the Department of METALLURGY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1973 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r equ i r emen t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l owed w i thout my w r i t t e n p e r m i s s i o n . Department o f M e t a l l u r g y The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date January 14, 1974 i ABSTRACT Fluid flow through liquid interdendritic channels of a partially remelted lead-tin casting has been measured directly, with gravity as the driving force. The results were shown to be consistent with Darcy's Law. The permeability of the dendritic 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 dendritic structure. The formation of casting defects in lead-tin alloys was studied with isothermal and unidirectional s o l i d i f i c a t i o n experiments. Solute convection was observed when the liquid close to the bottom of the solid-liquid region was less dense than the liquid above, using radioactive tracer techniques. Macrosegregation was shown to be related to the solidification conditions, and channel-type defects, resembling freckles and A segregates, were formed when the rising interdendritic liquid dissolved dendrite branches in i t s path. A simple mathematical model i s proposed, which predicts the composition profiles in vertical, directionally 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 qualitatively with the experi-ments, and can be used to recommend specific changes in casting practice to reduce gravity segregation effects. i i ACKNOWLEDGEMENTS I would l i k e t o express s i n c e r e thanks to my research s u p e r v i s o r , Dr. F. Weinberg, f o r h i s advice, support and encourage-ment throughout t h i s work. Thanks are als o extended to other f a c u l t y members and f e l l o w graduate students f o r many h e l p f u l d i s c u s s i o n s . In a d d i t i o n the a s s i s t a n c e of the t e c h n i c a l s t a f f of the 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 the K i l l a m Foundati on i n the 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 the 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 are als o due to the programming s t a f f of the U.B.C. Computing Centre, i n p a r t i c u l a r t o Mrs. Janet S t r e a t f o r her i n v a l u a b l e a s s i s t a n c e . i i i TABLE OF CONTENTS Page CHAPTER 1 I n t r o d u c t i o n 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 Castings . . . 1 1.2 Purpose of the Present I n v e s t i g a t i o n . . . 4 1.3 Organization of the Thesis . . . . . 5 CHAPTER 2 General Experimental Apparatus and Procedures 7 2.1 Apparatus . . . . . . . . 7 2.2 Pr e p a r a t i o n of Lead-Tin A l l o y s . . . . 9 2.3 Metallography . . . . . . . . 9 2.4 Measurement of Dendrite Spacing . . . . 1 1 2.4.1 Primary dendrite spacings . . . . 1 1 2.4.2 Secondary dendrite spacings . . . 16 2.5 Autoradiography . . . . . . . 1 6 CHAPTER 3 The Measurement of I n t e r d e n d r i t i c F l u i d Flow Rates . . . . . . . 1 7 3.1 Review of Previous Work . . . . . . 17 3.2 General D e s c r i p t i o n of the Technique Used i n the Present Work . . . . . . . 22 3.3 P r e p a r a t i o n of the A l l o y under Test (A) . . 27 3.4 Pr e p a r a t i o n of Castings B and C . . . . 30 3.5 Flow Measurement Equipment . . . . . 31 3.6 Flow T e s t i n g Procedure . . . . . . 34 3.7 P r e c i s i o n of the Flow Measurement Technique . 35 CHAPTER 4 Results and D i s c u s s i o n of Flow Measurements 38 4.1 I n t e r p r e t a t i o n Using Darcy's Law . . . . 38 4.1.1 Laminar flow . . . . . . 40 4.1.2 I n t e r a c t i o n e f f e c t s . . . . . 42 4.2 A p p l i c a t i o n to the Flow C e l l Experiments . . 43 4.2.1 The method f o r f i n d i n g the i n i t i a l permea-b i l i t y . . . . . . . 45 i v TABLE OF CONTENTS (Continued) P a g e 4.2.2 Results . . . . . . . . 49 4.3 Dendrite Spacings and S t r u c t u r e . . . . 52 4.3.1 Autoradiography . . . . . . 54 4.4 Microexamination . . . . . . . 59 4.4.1 Negative 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 4.5 P e r m e a b i l i t y and Dendrite Spacing . . . . 6 8 4.5.1 S t r a i g h t C a p i l l a r y Model . . . . 71 4.5.2 H y d r a u l i c Radius Theory: Other Theories . 75 4.6 Dendrite Coarsening . . . . . . . 79 4.7 The S c a t t e r of P e r m e a b i l i t y Results . . . . 86 CHAPTER 5 The E f f e c t of Density D i f f e r e n c e s on the Formation of Channels . . . . . 90 5.1 I n t r o d u c t i o n and Review of Previous Work . . . 90 5.2 Experimental Procedure . . . . . . 98 5.3 Results . . . . . . . . . 102 5.4 D i s c u s s i o n . . . . . . . . . 1 1 0 CHAPTER 6 Solute Convection and F r e c k l e Formation During S o l i d i f i c a t i o n . . . . . 1 1 5 6.1 I n t r o d u c t i o n . . . . . . . . 1 1 5 6.2 Experimental Procedure . . . . . . 1 1 5 6.2.1 Apparatus . . . . . . . 1 1 5 6.2.2 Macrosegregation s t u d i e s . . . . 117 6.2.3 Determination of composition from a c t i v i t y measurements . . . . . . . 120 6.2.4 Solute convection . . . . . . 126 6.3 Results . . . . . . . . . 1 2 7 6.3.1 Composition p r o f i l e s . . . . . 1 2 7 6.3.2 Convection i n the l i q u i d . . . . 136 V TABLE OF CONTENTS (Continued) Page 6.3.3 Fr e c k l e s 139 6.4 Di s c u s s i o n of Results . . . . . . 139 CHAPTER 7 A Numerical Model f o r Macrosegregation 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 of Previous Work . . . 144 7.2 Model of the S o l i d i f i c a t i o n Process . . . . 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 Casting . 151 7.5 Re s u l t s of C a l c u l a t i o n s 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 155 7.6 Comparison w i t h Experiment . . . . . 1 6 1 CHAPTER 8 Conclusions . . . . . . . 164 8.1 Summary . . . . . . . . . 164 8.2 Conclusions . . . . . . . . 165 8.3 Suggestions f o r Future Work . . . . . 166 REFERENCES 168 APPENDIX I I n t e g r a t i o n of Darcy's Law f o r a F a l l i n g Head 172 APPENDIX I I FORTRAN Program f o r Processing F l u i d Flow Data 175 APPENDIX I I I The S o l i d i f i c a t i o n of Pb-20%Sn -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 . . 179 APPENDIX IV FORTRAN Program f o r C a l c u l a t i n g Macro-segregation i n Lead-Tin Castings . . . 181 APPENDIX V D i r e c t Observation of S o l i d i f i c a t i o n Using E l e c t r o n Microscopy . . . . . 1 8 8 v i TABLE OF CONTENTS (Continued) Page V . l I n t r o d u c t i o n . • . . . . . . 188 V.2 Experimental Method . . . . . . 1 8 8 V.3 R e s u l t s V.3.1 Pure bismuth 189 V.3.2 Other pure metals ( t i n , aluminum and indium) . . . . . . . 194 V.3.3 Lamellar e u t e c t i c s . . . . . 194 v i i LIST OF ILLUSTRATIONS Figure Number Page 4 5 6 7 10 11 12 13 14 15 16 (a) Tube furnace and quenching apparatus f o r producing columnar c a s t i n g s . (b) Tube furnace f o r ± 0.5°C temperature c o n t r o l . . . . . . Cross s e c t i o n of a group of primary dendrites (schematic) . . . . . . . (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 s o l i d i f i e d c a s t i n g . (b) Corresponding cross s e c t i o n Enlarged views of regions A' and B' i n Figure 3(b) Three dimensional composite, from which one can estimate t h a t the dendrites i n the top corner are t i l t e d approximately 20° . . . . . Schematic view of the s t r u c t u r e i n Figure 4(b) Pe r m e a b i l i t y as a f u n c t i o n 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'^' . . . . . Schematic diagram showing the p r i n c i p l e of a F a l l i n g Head Permeameter . . . . . S e c t i o n a l views of the flow c e l l and the l e a d - t i n a l l o y i n s e r t s . . . . . . . Three pieces of Pb-Sn a l l o y used f o r flow measurement P a r t i a l l y assembled flow c e l l . . . . Pb-Sn a l l o y before and a f t e r flow t e s t Flow measurement apparatus . . . . . C i r c u i t used f o r recording the p o s i t i o n of the probe on the temperature t r a c e Apparatus f o r t e s t i n g the p r e c i s i o n of the flow measurement technique . . . . . . (a) Flow measurement r e s u l t s ; d i s t ance of flow up the r i s e r pipe versus 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 . . . . . . v i i i LIST OF ILLUSTRATIONS (Continued) Figure Number p a g e 17 (a) Data from Figure 16(a), r e p l o t t e d according to 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 data from Figure 16(b), showing a negative d e v i a t i o n . . . . . . . 44 18 Primary dendrite spacing as a f u n c t i o n of distance 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 Autoradiographs from cross 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 ; (a) and (b) show uniform flow, (c) shows flow down a p r e f e r e n t i a l channel . . . . 20 Cross s e c t i o n autoradiographs at various l e v e l s down the c a s t i n g A, a f t e r t e s t i n g . . . . 21 An example of an u n r e l i a b l e flow t e s t , showing uneven p e n e t r a t i o n of t r a c e r . . . . 22 M i c r o s t r u c t u r e s of c a s t i n g A before and a f t e r flow 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 of c a s t i n g A before and a f t e r flow t e s t i n g I = 71 ym . 24 M i c r o s t r u c t u r e s of c a s t i n g A before and a f t e r flow t e s t i n g X = 51 ym 25 M i c r o s t r u c t u r e s of c a s t i n g A before and a f t e r flow 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 of the r e s e r v o i r ( c a s t i n g B) a f t e r t e s t i n g . . . . . . . . . 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 dendrite arm spacing f o r Pb-Sn at 193 C 28 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 primary dendrite spacing f o r Pb-20%Sn at 193°C 3 2 29 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^) using data obtained by Piwonka'^ . . . . . . 30 Growth of a neck during s i n t e r i n g i x LIST OF ILLUSTRATIONS (Continued) Figure Number 'age 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Dendrite coarsening p l o t f o r a sample w i t h K q = 0.152 cm , average primary dendrite spacing 28 urn 3 C a l i b r a t i o n curve; d e n s i t y of Pb-Sn a l l o y s (g/cm ) at 25°C as a f u n c t i o n of composition 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 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 Fre c k l e s i n as-cast Inconel 718 The t e s t assembly f o r i s o t h e r m a l experiments (a) Macrostructure of columnar c a s t i n g ( s e r i e s I) (b) Corresponding autoradiograph (a) Macrostructure of columnar c a s t i n g ( s e r i e s I I (b) Corresponding autoradiograph 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 . . . . (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 Figure 39. (b) Corresponding autoradiograph 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 . . . . (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 Figure 41. (b) Corresponding autoradiograph (a) Macrostructure of columnar c a s t i n g ( s e r i e s IV) (b) Corresponding autoradiograph Macrostructure from s e r i e s V . . . S p l i t g r aphite mould f o r making long c y l i n d r i c a l ingots . . . . . . . Spectrum of y emission f o r Sn 113 204 Spectrum of y emission f o r T l . . . C a l i b r a t i o n curve; a c t i v i t y versus sample weight 85 87 87 91 91 100 103 103 106 106 107 107 109 109 116 119 119 122 X LIST OF ILLUSTRATIONS (Continued) Figure Number 51 62 63 49 Calibration curve; activity versus S n 1 1 3 concentration 122 50 Page Calibration curve; specific activity versus alloy composition for constant S n 1 1 3 concentration . . . 123 Calibration curve; specific activity versus alloy composition, when S n 1 1 3 concentration is proportional to the solute content . . . . . . . 123 (b) Cooling conditions . 128 (b) Cooling conditions . 129 (b) Cooling conditions . 130 (b) Cooling conditions . 131 (b) Cooling conditions . 132 52 Composition profile for one ingot using lathe turning treated with n i t r i c acid (open cir c l e s ) , and untreated samples (closed circles) . . . . . . 125 53 (a) Solute distribution, 54 (a) Solute distribution, 55 (a) Solute distribution. 56 (a) Solute distribution. 57 (a) Solute distribution. 58 Autoradiographs showing the extent of tracer movement one hour after tracer was added; (a) directionally s o l i d i f i e d , (b) quenched from the liquid . . . 1 3 7 59 Shrinkage t r a i l , approximately 7 cm long, along the outside of an ingot s o l i d i f i e d under conditions given in Table XI. (b) Longitudinal and transverse sections showing a freckle t r a i l on the right hand side . . 140 60 (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 interdendritic channels in the interior of the ingot . . . . . . 141 61 Schematic representation of unidirectional solidification assumed in the model . . . . 148 Equilibrium diagram for a binary alloy. The non-equilibrium solidus is shown by the dashed line . . 1 4 8 Directionally solidifying ingot divided into layers. Temperature, composition and density profiles given by the solidification model . . . . . . 153 xi LIST OF ILLUSTRATIONS (Continued) Figure Number Page 64 (a) Assumed flow pattern showing two main flow c e l l s . (b) Resistances Ri_5» and flow rates q^_5 for flow between six layers . . . . . 1 5 3 65 Solute distribution as a function of the number of layers . . . . . . . . . . 1 5 7 66 Solute distribution as a function of structure (effective number of channels) . . . . . 157 67 Solute distribution as a function of ingot height . 159 68 Solute distribution as a function of growth rate . . 159 69 Solute distribution as a function of temperature gradient . . . . . . . . . . 160 70 Pb-Sn alloy in the flow c e l l , after a time t . . 1 7 2 71 (a-c) Alternate freezing, melting and freezing in pure bismuth, showing evidence of faceted growth. (d) Enlarged view of the solid-liquid interface showing high angle grain boundaries emerging at the interface . . . . . . . . . 191 72 Alternate melting, freezing, melting and freezing in pwre bismuth . . . . . . . . 193 73 The melting of pure aluminum, photographed from the fluorescent screen using a 35 mm camera . . 1 9 5 x i i LIST OF TABLES Table Page I Dimensions and Composition of Castings Used for Interdendritic Fluid Flow Studies . . . . 26 II Quench Data . . . . . . . . 29 III Thermal Conditions for Pb-20%Sn Columnar Castings . 29 IV Precision of the Flow Measurement Technique . . 36 V Results of Flow Measurements . . . . . 50 VI Results of Dendrite Coarsening Calculations . . 82 VII Composition of Superalloys . . . . . 92 VIII Test Conditions for Isothermal Experiments . . 101 IX Solubility Data for Isothermal Experiments . . 112 X Solidification Variables and Macrosegregation . . 133 XI Cooling Conditions . . . . . . 1 3 8 XII Solidification Variables Used for Theoretical Plots . 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 Castings Nearly a l l metal products are made by c a s t i n g and subsequent f a b r i c a t i o n of the cast m a t e r i a l . Since the vast m a j o r i t y of metal products are a l l o y s of two or 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 compo-s 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 composition of the adjacent 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 congruent s o l i d i f i c a t i o n ) . 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 conserved, the composition of both s o l i d and l i q u i d must vary during s o l i d i f i c a t i o n . In the s o l i d s t a t e , composition changes can only occur by means of 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 at high 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 impossible to cast homogeneous a l l o y s . In general, a l l o y castings have composition d i f f e r e n c e s on a microscopic s c a l e , which can a f f e c t the mechanical, c o r r o s i o n and surface p r o p e r t i e s , depending on the extent and d i s t r i b u t i o n of the segregated con-s t i t u e n t s . The e f f e c t s of composition v a r i a t i o n s on a microscopic s c a l e may not 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 processed f u r t h e r . However, when microscopic v a r i a t i o n s are concentrated 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 composition v a r i a t i o n s are termed "macrosegregation" and a number of d i f f e r e n t types are recognized. These i n c l u d e c e n t r e l i n e segregation, A and 2 V segregates i n la r g e s t e e l i n g o t s , i n v e r s e segregation, f r e c k l e s and s o l u t e banding: 1) C e n t r e l i n e 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 ingot which has cooled from the si d e w a l l s . 2) A segregates have been described as r o p e - l i k e concentrations of s o l u t e which form i n the upper regions of the columnar zone of la r g e s t e e l i n g o t s . They are c a l l e d A segregates because they are i n c l i n e d a few degrees from the v e r t i c a l on each side of the a x i s , g i v i n g the appearance of an A or Greek A when seen on a surface sectioned p a r a l l e l to the a x i s . 3) S i m i l a r l y , V segregates are cones of high s o l u t e content which form i n the lower equiaxed regions of s t e e l c a s t i n g s , and are V-shaped when seen on the sectioned s u r f a c e . 4) Inverse segregation i s a region of high s o l u t e content c l o s e to the c h i l l face of a c a s t i n g which can, i n some cases, be observed as exudations or beads of s o l u t e r i c h m a t e r i a l on the surface. 5) F r e c k l e s are patches or v e r t i c a l l i n e s of s o l u t e r i c h m a t e r i a l which occur i n a number of d i f f e r e n t types of c a s t i n g s , i n p a r t i c u l a r , consumable arc melted i n g o t s . They are so named because of t h e i r s p o t t y or speckled appearance when seen on the outer s u r f a c e , or on p o l i s h e d s e c t i o n s of the i n g o t . 6) Solute banding i s the term a p p l i e d when a l t e r n a t i n g regions of s o l u t e r i c h and s o l u t e depleted m a t e r i a l are observed t o occur i n the columnar regions of an in g o t . I t i s d e s i r a b l e to minimize these defects i n c a s t i n g s that are 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 the c a s t i n g i s used without sub-sequent working. For 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 of n i c k e l - b a s e 3 su p e r a l l o y s (designed f o r h i g h temperature s e r v i c e ) are 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. These c a s t i n g s are s u s c e p t i b l e to the formation of f r e c k l e s , which can s e r i o u s l y a f f e c t the st r e n g t h and creep p r o p e r t i e s when the blades are i n s e r v i c e . The cost of s u p e r a l l o y c a s t i n g s produced i n the United S t a t e s , which are 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 type, and are p o t e n t i a l l y s u s c e p t i b l e to 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 annually. Considering that there i s a p o s s i b l e r e j e c t i o n rate of 20%, and the cost of remelting the r e j e c t e d c a s t i n g s i s about 40% of the cost of the product, the p o t e n t i a l cost of t h i s one type of c a s t i n g defect i s about $40 m i l l i o n per year. In g e n e r a l , 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 of c l u s t e r s of t r e e - l i k e spikes w i t h s i d e branches. 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 astings can normally be examined by s u i t a b l y etching a p o l i s h e d surface. The etchant i s s e l e c t e d to react w i t h s o l u t e r i c h or s o l u t e depleted regions producing p r e f e r e n t i a l a t t a c k of the i n t e r -d e n d r i t i c r e g i o n s , or dendrite centres. The extent and d i s t r i b u t i o n of the segregated s o l u t e i n the 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 , which i n turn i s a f u n c t i o n of the a l l o y composition 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 the c r y s t a l l o -graphic p r o p e r t i e s of the a l l o y c o n s t i t u e n t s , the thermal 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 forced or n a t u r a l convection. The d r i v i n g forces f o r l i q u i d transport i n a c a s t i n g w i l l be r e l a t e d to the temperature d i f f e r e n c e s which cause n a t u r a l convection, composition d i f f e r e n c e s which can lead to s o l u t e convection, and other f a c t o r s such as volume shrinkage and gas e v o l u t i o n . 4 The spacing between s i d e branches of d e n d r i t e s i n the centre of l a r g e , s l o w l y cooled c a s t i n g s can be of the order of m i l l i m e t r e s . For small castings which c o o l r a p i d l y the spacing can be of the order of ten microns, consequently, l i q u i d transport 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, tortuous channels through which the l i q u i d must move. Since macrosegregation i s s t r o n g l y i n f l u e n c e d by the extent of f l u i d flow during s o l i d i f i c a t i o n , an understanding of the forces a c t i n g on the l i q u i d , and the extent of i n t e r d e n d r i t i c f l u i d flow 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 modify, c e r t a i n types of macrosegregation i n c a s t i n g s . 1.2 Purpose of the Present I n v e s t i g a t i o n The purpose of the present work was to measure i n t e r d e n d r i t i c f l u i d flow 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 to flow and the s t r u c t u r e of the c a s t i n g . With s u i t a b l e measurements, the r e s u l t s would be considered i n terms of 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 flow through porous media. Since most metals have s i m i l a r thermal and viscous c h a r a c t e r i s t i c s , and s o l i d i f y i n a s i m i l a r manner, as compared to non metals, i t i s considered that a d e t a i l e d examination of one metal system can give r e s u l t s a p p l i c a b l e to most other systems. In conjunction w i t h the f l u i d flow c o n s i d e r a t i o n s , other aspects of macrosegregation would be examined, s p e c i f i c a l l y , the formation of channel-type defects which resemble f r e c k l e s and A segregates. To combine the r e s u l t s of both the f l u i d flow and the macrosegregation s t u d i e s , a simple mathematical model has been de r i v e d . The model considers the s o l i d i f i c a t i o n of an ingot where the d r i v i n g f o r c e f o r macrosegregation 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 the i n t e r d e n d r i t i c f l u i d flow i s a f u n c t i o n of the cast 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 the experimental r e s u l t s . 1.3 Organizat i o n of the Thesis The t h e s i s i s d i v i d e d i n t o four main s e c t i o n s . The f i r s t s e c t i o n (Chapter 2) gives a d e s c r i p t i o n of the apparatus and procedures common to 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) deals w i t h the development of the i n t e r d e n d r i t i c flow measurement technique, and the i n t e r -p r e t a t i o n of the r e s u l t s i n terms of the theory of flow through porous media. The t h i r d s e c t i o n c o n s i s t s of the experiments on the e f f e c t of den s i t y d i f f e r e n c e s i n the l i q u i d on a c a s t i n g h e l d at uniform temperature i n the s o l i d - l i q u i d r e gion (Chapter 5) and the study of macrosegregation and defect formation i n ingots 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 ). The experimental work on macrosegregation i s t i e d together w i t h the r e s u l t s of i n t e r d e n d r i t i c f l u i d flow measurements i n the mathematical model, presented i n the f o u r t h s e c t i o n (Chapter 7 ). A review of previous work re l e v a n t to the p a r t i c u l a r s e c t i o n i s presented at the beginning of Chapters 3, 5 and 7. During the course of these experiments an attempt was made to 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 of metal using an e l e c t r o n microscope. The aim of t h i s work was to 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 to the work on i n t e r d e n d r i t i c f l u i d flow, i n r e l a t i o n to the i n t e r p r e t a t i o n of changes which take place i n a c a s t i n g h e l d i n the solid-liquid region for long periods of time. The results of the elect microscope study were inconclusive. A brief summary of the work is giv< in Appendix V. 7 CHAPTER 2 GENERAL EXPERIMENTAL APPARATUS AND PROCEDURES 2.1 Apparatus Two tube furnaces were constructed and used to produce columnar castings and to heat the a l l o y samples f o r f l u i d flow s t u d i e s . Columnar castings were produced i n the furnace shown i n Figure 1(a) which had a c e n t r a l copper tube to withstand thermal shock when water quenches were used i n s i d e the furnace. The copper tube was wrapped w i t h asbestos tape, and chromel windings were wrapped over the tape to prevent short c i r c u i t i n g on the metal tube. A c o n t r o l thermocouple (Chromel/Alumel) was placed next to the windings. Temperatures measured i n a molten metal charge i n t h i s furnace could be hel d constant to ± 1°C. For b e t t e r c o n t r o l a second furnace was b u i l t o f s i m i l a r design (Figure 1(b)) w i t h a c e n t r a l ceramic tube. In t h i s case the windings were d i r e c t l y i n contact w i t h the ceramic tube, which was a b e t t e r conductor than the asbestos tape, and ± 0.5°C c o n t r o l was p o s s i b l e . T h i s furnace was used f o r f l u i d flow s t u d i e s where i t was necessary to heat the samples r a p i d l y to a predetermined temperature without overshooting, and then h o l d them constant. Unless otherwise s t a t e d , a l l temperature measurements i n t h i s work were made using iron-constantan thermocouples c a l i b r a t e d against the melti n g p o i n t of 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 response. The thermocouple wires were i n s e r t e d i n s m a l l diameter ceramic tubing (approx-imately 1.6 mm) to ensure that when the thermocouple was immersed i n molten metal the reading was r e p r e s e n t a t i v e of co n d i t i o n s at the t i p . This was *— Quench Medium (a) (b) (a) (b) Tube furnace and quenching apparatus for producing columnar 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 rise up the bore. In a l l tests the thermocouples were connected via a cold junction i n ice 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 in lots of approximately 1500 g in a stainless steel beaker over a bunsen burner, stirred thoroughly, and cast into graphite moulds to produce starting ingots about 2.2 cm diameter and 5 cm long. Both lead and tin have a low vapour pressure and no composition changes were expected due to evaporation. This was confirmed by plotting the cooling curves of two samples of Pb-20% Sn. The samples were held above the melting point, in air, for approximately 15 minutes and 17% hours respectively. There was no significant difference between the liquidus arrest points seen on the cooling curves. In those cases where radioactive T l or Sn was added to the lead-tin alloys, the tracer was f i r s t dissolved in 100 g of pure Sn in a pyrex test tube with an argon flow to prevent oxidation. This 'master' alloy was then used in the preparation of the required lead-tin alloy. 2.3 Metallography Since Pb-Sn alloys are extremely soft, special 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 in 'Quickmount', a cold setting plastic that did not heat up more than about 30°C during curing. In addition, though 'Quickmount' is harder than the Pb-Sn alloys, i t i s softer than other mounting materials and can readily 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 directly to a 5 micron alumina lapping wheel where sufficient material was polished away to remove machine marks and any flowed layer. Grinding papers were not used since i t was found that they would immediately clog and tear at the surface, and particles of abrasive from coarser papers would become embedded in the soft metal surface and be impossible to remove later. Final polishing was done at slow speed with a thick slurry of 1 micron alumina. The specimens were washed in cold water or alcohol, since i t was found that hot water etched the structure. The etchant used for high Pb content alloys was 50 parts acetic acid, 15 parts hydrogen peroxide, and 100 parts water. This acted rapidly, both as an etch and as a mild chemical polish to remove remaining polishing marks. High Sn alloys were etched in a f e r r i c chloride base etch. As an alternative, electropolishing was attempted, but since i t was frequently necessary to polish relatively 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 satisfactory. 1 1 2.4 Measurement of Dendrite Spacing Dendrites in lead (fee) and t i n (bet) have orthogonal branches, and primary dendrites are defined as those growing in 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 direction. Tertiary branches growing perpendicular to secondary branches and parallel to primary stalks can form at large primary spacings (over approximately 200 ym). In lead rich Pb-Sn alloys, the dendrites may contain up to 19% Sn, and the interdendritic regions 62% Sn, the eutectic composition, which forms a fine lamellar structure. 2.4.1 Primary dendrite spacings When sectioned perpendicular to the growth direction, primary dendrites are seen to form the close packed arrangement, shown schematically in Figure 2. The distance between the centres of nearest neighbours is the primary spacing X. One method of measuring this spacing from a section perpendicular to the growth direction is to mark a l l the dendrite centres on a photograph, and then count the number of centres (n) in a given area (A). The spacing is 2 then given by A A/n. A second method is to measure the distance between the centres of primary stalks on a longitudinal section (parallel to the growth direction). The stalks can be recognized when secondary branches are visible on both sides. The spacing measured in this case w i l l be A' in 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 s o l i d i f i e d c a s t i n g , dendrites i n region A are v e r t i c a l , and i n region B are t i l t e d . (b) Corresponding cross s e c t i o n . M a g n i f i c a t i o n 18x. 13 I t was c o n s i d e r e d i m p o r t a n t i n the p r e s e n t work t h a t the p r i m a r y d e n d r i t e s p a c i n g s be measured from s e c t i o n s p e r p e n d i c u l a r t o the growth d i r e c t i o n * so t h a t the ends of the c a s t i n g s c o u l d be p o l i s h e d b e f o r e the f l o w t e s t s , and the d e n d r i t e s p a c i n g of each c o u l d then be checked n o n - d e s t r u c t -i v e l y . D i f f i c u l t i e s were encountered u s i n g the method of marking c e n t r e s because d e n d r i t e s a r e not always c l e a r l y r e v e a l e d i n c a s t s t r u c t u r e s . In p a r t i c u l a r , d i f f i c u l t i e s a r i s e w i t h a l l o y c o m p o s i t i o n s c l o s e t o the s o l u b i l i t y l i m i t o f t h e p r i m a r y phase, because the s m a l l amounts of e u t e c t i c p r e s e n t i n the c a s t s t r u c t u r e do not always o u t l i n e the d e n d r i t e branches c o m p l e t e l y , making i t d i f f i c u l t t o d i s t i n g u i s h between s t a l k s and b r a n c h e s . T h i s can be overcome i n some cases by u s i n g an e t c h a n t which r e v e a l s c o m p o s i t i o n g r a d i e n t s i n the p r i m a r y phase; however, an e t c h a n t of t h i s t y p e was n o t found t h a t would work r e l i a b l y on Pb-Sn a l l o y s . F i g u r e 3 shows the c a s t s t r u c t u r e on 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 the a x i s of a d i r e c t i o n a l l y c a s t c y l i n d e r o f Pb-20% Sn. V e r t i c a l p r i m a r y s t a l k s can be seen i n the c e n t r e o f the l o n g i t u d i n a l s e c t i o n marked A. The c o r r e s p o n d i n g a r e a on the c r o s s s e c t i o n i s marked A'. On e i t h e r s i d e of the r e g i o n marked A the p r i m a r y s t a l k s are t i l t e d away from th e v e r t i c a l , though i t i s d i f f i c u l t t o determine the a n g l e s i n c e they a r e a l s o t i l t e d w i t h r e s p e c t t o the p l a n e of s e c t i o n . An example o f such a r e g i o n i s B on the l o n g i t u d i n a l s e c t i o n . The c o r r e s p o n d i n g a r e a on the c r o s s s e c t i o n B' shows a p e r i o d i c p a t t e r n o f l i n e s ( i n t h i s case the l i n e s a r e about 45° l e f t o f v e r t i c a l ) , and the o t h e r a r e a s on the c r o s s s e c t i o n which c o r r e s p o n d t o t i l t e d d e n d r i t e s a l s o show a s i m i l a r p a t t e r n o f l i n e s . A l t h o u g h i t would be most l o g i c a l to make measurements from the 14 region A', the enlarged view of this area in Figure 4(a) shows the d i f f i -culty in distinguishing between stalks and branches. Although one can see individual dendrites, i t is frequently d i f f i c u l t to identify the centres of the neighbouring primary stalks. It is easy, however, to measure the line spacing in region B', shown enlarged in Figure 4(b), for a comparatively large number of lines. 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) is shown oriented such that the primary stalks in the corner of the cube are normal to the plane of the paper. For this example, i t was found that the primary stalks were t i l t e d approximately 20 from the vertical, in a plane perpendicular to the direction of the lines. It was found that the pattern of lines i s only seen when the dendrites are t i l t e d between approximately 10° and 30° from the ver 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 slightly elongated in the plane of t i l t ) . The line spacing L can be used to calculate the primary dendrite spacing since X = /2Lcos6, where 0 is the angle of t i l t . Since 6 is between 10° and 30°, cos8 is between 0.98 and 0.87, therefore the use of an average value of 0 = 20° introduces an error of less than 8%, which is less than the error involved in measuring L, which is no better than ± 10%. Thus the formula used in this work was X = 1.3L. 15 FIGURE 5: Three dimensional composite, from which one can estimate that the dendrites i n the top corner are t i l t e d FIGURE 6: Schematic view of the s t r u c t u r e i n Figure 4(b) approximately 20 16 Since primary spacings were measured from sections perpendicular to the growth direction, no distinction could be made between primary and tertiary dendrite arms. 2.4.2 Secondary dendrite spacings Secondary spacings were determined by measuring the spacing between a large number of clearly delineated secondary arms on polished sections parallel to the growth direction. 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 in 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 film to 204 make autoradiographs. For a concentration of 500 ppm T l (irradiated to a specific activity of 5 millicuries/gm) satisfactory exposures were obtained in 16 hours on X-ray film, or approximately 14 days on orthochromatic film. Although exposures were long for orthochromatic film, the resolution was appreciably better. A l l the autoradiographs in this thesis are printed so that dark areas indicate the presence of radioactive material. 17 CHAPTER 3 THE MEASUREMENT OF INTERDENDRITIC FLUID FLOW RATES 3.1 Review of Previous Work The f i r s t direct measurements of interdendritic f l u i d flow rates (1 2) were reported by T.S. Piwonka ' from experiments on Al-4.5%Cu alloys. Samples of the molten alloy 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 interdendritic liquid in the alloy 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 for the displaced liquid to make contact with a probe in the other branch of the U-tube. Two methods of applying pressure were used. In one case the inderdendritic liquid was displaced using an inert gas (nitrogen), and in the other case liquid lead was used in addition to gas pressure. Piwonka acknowledged that surface tension effects at the liquid-gas interface may have caused the gas displacement results to be unreliable, since the pressure required to force gas into the interdendritic regions might have been a significant proportion of the total pressure required to expel the inter-dendritic liquid. An approximate estimate of the magnitude of this effect can be made as follows: The surface tension (a) of liquid aluminum is 520 dynes/cm at 750°C (surface tensions for Al-Cu alloys are not readily available). Assuming an interdendritic channel size of 20ym diameter, the pressure required to force gas into such a channel would be equal to the pressure 18 required to blow a hemispherical bubble of radius r = 10 urn. P = |2. = 1.04 x 10 6 dynes/cm 2. According to Piwonka's t h e s i s , the a p p l i e d pressures i n the 4 6 2 n i t r o g e n gas experiments were i n the range 3 x 1 0 to 1.8 x 10 dynes/cm . I f the assumed channel s i z e i s reasonable, the surface t e n s i o n could p o s s i b l y account f o r a l l the r e s i s t a n c e to flow that was observed. Indeed, 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 decreasing temperature might be due to the increase i n the pressure (P) w i t h decrease i n the radius ( r ) . Since the aim of the experiments was to determine the r e s i s t a n c e to flow caused by f l u i d drag w i t h i n the i n t e r d e n d r i t i c channels, the gas displacement r e s u l t s must be considered completely u n r e l i a b l e . The use of l i q u i d l e a d i n s t e a d of gas would reduce the surface tension e f f e c t and give a b e t t e r measure of flow 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 flo w i n g 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 probably have an important e f f e c t on the measurements. This would not be the case when lead i s used i n the Al-Cu system, s i n c e aluminum i s i n s o l u b l e i n lea d . Piwonka examined some of the a l l o y samples a f t e r t e s t i n g to ensure uniform p e n e t r a t i o n of lead 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 ; however, no measurements r e l a t i n g to the s t r u c t u r e were given. This i s an unfortunate omission, s i n c e the s t r u c t u r e was assumed to be constant i n a l l the c a l c u l a -t i o n s . He acknowledged that p r e f e r e n t i a l channels were formed by the i n t e r -19 dendritic liquid as i t was being displaced in some of the higher temperature tests, which would suggest that uniform flow did not always occur. Piwonka believed that his results were consistent with a model of the solid-liquid region which considered the region to be a bundle of straight capillary tubes. This could be demonstrated by plotting the logarithm of the permeability of the alloy, which is the reciprocal of the resistance to flow (defined in more detail in subsection 4.1 of the present work) against the logarithm of the liquid fraction, calculated from the Al-Cu phase diagram. A straight line of slope 2 should be obtained i f the model is applicable (the theoretical relationship is derived in subsection 4.5.1 of (2) the present work). Piwonka's results of liquid lead displacement i n Al-Cu are replotted in Figure 7 as permeability versus the square of volume fraction liquid (from the thesis, i t appears that Piwonka incorrectly used weight fractions instead of volume fractions). The results agree f a i r l y well with a straight line, for liquid fractions less than about 0.3, and the con-clusion can be drawn that the solid-liquid region can be treated in this simplified manner, i.e., as a bundle of straight capillary tubes, within this range, provided the assumption of constant structure is correct. The results of this investigation have led a number of workers to explain certain effects in sol i d i f i c a t i o n in a semiquantitative manner, based on the standard equations for flow through porous media. The explanations are semiquantitative in 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 fraction liquid, using experimental data obtained by Piwonka^^ . 21 Thus Piwonka' ', Campbell w u y and Tien*'''' used these results in theoretical predictions of hydrostatic tensions which could lead to pore formation during so l i d i f i c a t i o n . Standish used the results to argue that A segregate formation is not caused by interdendritic fl u i d flow, in 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 interdendritic f l u i d flow in the Pb-Sn system were made by Kaempffer^^'^\ who observed the formation of droplets on the bottom surface of an ingot with a thin layer of eutectic material placed on top, when i t was heated above the eutectic temperature. Radioactive tracer was added to the eutectic layer, and the aim of the experiment was to measure the flow rate by measuring the activity of the droplets. Kaempffer found that i t was not possible to produce uniform inter-dendritic flow with this experiment. At the eutectic temperature no flow was observed, but as the temperature increased drops began appearing on the bottom surface of the ingot. These would coalesce before f a l l i n g , and, with a constant rate of heating, the f i r s t drop would f a l l at about 230°C, and the remainder would follow rapidly. 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 his results as follows; as the temperature of the ingot was increased above the eutectic temperature, the lead-rich dendrites became soluble in the superheated eutectic liquid, therefore, as 22 material from the top layer began to flow down, i t was able to dissolve dendrite branches in i t s path, forming the wide channels. This work was therefore not representative of uniform flow through porous media, and i t remained to be shown whether one could produce uniform flow of interdendritic liquid which was not superheated. 3.2 General Description of the Technique Used in the Present Work Measurement of the permeability of a packed bed is often done using a Falling Head Permeameter (Figure 8). This consists of two concentric tubes with the porous material packed in the inner tube. A static head of f l u i d in 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 t/h o) where c and K are constants, h is the head at time t, and h is the head ' t ' o at t = 0. This equation is derived in detail in Appendix I. The same principle is used in the design of the flow c e l l for measuring interdendritic f l u i d flow (Figure 9). In this case the two tubes are side by side instead of concentric. The flow c e l l was made from four pieces of brass and resembles a s p l i t mould. 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 Outflow Porous Bed Inner Tube Outer Tube 23 "TV FIGURE 8: Schematic diagram showing the principle of a Falling Head Permeameter. argon E u Pb-55Sn Pb-20Sn Pb-55Sn , 2 - 3 5 c m d i o FIGURE 9 : Sectional views of the flow c e l l and the lead-tin alloy inserts (to scale). 24 together. Brass was chosen for i t s machinability, strength and thermal conductivity (0.27 cal/cm sec°C). High conductivity was desirable to ensure isothermal conditions within the flow c e l l , and brass was found to be satis-factory by making temperature measurements at various locations inside the c e l l while i t was being heated. Copper would have provided higher conduc-t i v i t y (0.88 cal/cm sec°C), but i t was d i f f i c u l t to machine to the complex shape of the flow c e l l , and from previous experience i t was found that threads tapped in copper did not hold after repeated heating and cooling. To carry out a flow test three pieces of lead-tin alloy were inserted into the c e l l (Figure 10). The partially assembled flow c e l l with the lead-tin alloy i s shown in Figure 11. A cylindrical casting A of the alloy under test was placed in the appropriate cavity, and two other castings, B and C, of different 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 liquid, and the casting A would be partially liquid. Thus there would be a hydrostatic pressure in the solid-liquid region through A, and the liquid level would tend to f a l l on the l e f t , and rise in the smaller diameter 'riser' pipe on the right. The level of the liquid metal in the riser could be measured at any time using a copper wire probe, which closed a circuit on contact. Therefore, under known conditions of pressure, temperature, liquid composition and dendrite spacing, graphs of height of liquid in the riser versus time were plotted, from which the permeability could be calculated. At the end of the test the flow c e l l was chilled and the alloy was removed. Figure 12 shows the alloy before and after testing. The alloy could FIGURE 12: Pb-Sn a l l o y before and a f t e r flow t e s t . TABLE I DIMENSIONS AND COMPOSITION OF CASTINGS USED FOR INTERDENDRITIC FLUID FLOW STUDIES Diameter cm Length cm Composition Comments Casting A 2.46 3.37 Pb-20%Sn Casting B 1.91 0.76 Pb-55%Sn 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 Casting C 1.91 0.64 Pb-55%Sn This c a s t i n g was made using the flow c e l l as a mould 27 then be examined by sectioning and polishing. Autoradiography of these sections was used to observe the fl u i d flow directly in 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 in Table I. 3.3 Preparation of the Alloy under Test (A) In most of the tests the cylinders of alloy (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 starting ingots in a vertical graphite mould and chi l l i n g from the bottom. The furnace, mould and ch i l l i n g arrangement are shown in Figure 1(a). When the alloy was molten, an iron-constantan thermocouple was placed in the melt and the temperature was adjusted to the required level by adjusting the furnace controller. When the alloy reached a constant temperature (checked by moving the thermocouple around in the liquid) the thermocouple was withdrawn and the alloy s o l i d i f i e d unidirectionally 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 directional castings with a given dendrite spacing within the precision with which the spacing could be measured (approximately 10%). Four different quenches were used, and the details are given in Table II. 28 Cooling curves at three positions in the casting were plotted by inserting iron-constantan thermocouples in the melt and solidifying. 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 for the four different quenches used. The c h i l l face cooling rate was calculated from the cooling curve of a bare thermocouple in the liquid, placed in contact with the c h i l l . The cooling rate is taken from the slope of the cooling curve at the liquidus temperature of the alloy. The freezing rate is given at two points and is calculated from the estab-lished relationship ^ ^'^"^ x = A/t where x is the distance from the c h i l l , t is the time elapsed from the start of freezing, and A is a constant. The freezing rate is therefore equal to x/2t. The mean primary dendrite spacing is also listed in Table III. 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 in section 6.2. Ingots 2.5 cm in 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 cylindrical samples with the dimensions of casting A. In'addition, equiaxed castings of different dendrite spacings were produced by pouring molten alloy into a simple graphite mould, 2.5 cm in 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 Ch i l l (cm) Nozzle Dia.(cm) Al N2gas 35 l b / i n 2 1.98 0.31 A2 N2gas 35 l b / i n 2 0.88 0.57 Wl water 23.5 in. head 1.98 0.31 W2 water 23.5 in. head 2.86** 0.31 * Coolants were at room ** Two copper discs with total temperature. thickness of 2.86 cm were used. TABLE III 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 = B = 1.31 cm from c h i l l 2.77 cm from c h i l l * Measured at the liquidus temperature. 30 After cooling to room temperature a l l castings were machined to the dimensions given in Table I. The ends were polished, etched and examined microscopically to determine the dendrite spacing, and before testing in the flow c e l l the ends were again polished and cleaned to remove oxide or other extraneous material that might interfere 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 in the alloy. After testing, 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 this work was Pb-55%Sn. The choice of composition was based on Kaempffer's exper-i e n c e ^ ^ using eutectic material (62%Sn) as the liquid reservoir above the casting. Since the main requirement in this work was that casting B should not become superheated, an off-eutectic alloy was found by experience to be more suitable. The particular composition chosen was found to work best for permeability measurements approximately 10°C above the eutectic temperature. Problems of preferential channelling, similar to those 31 encountered by Kaempffer, arose when attempts were made to use different alloy compositions to measure permeabilities at higher temperatures. 3.5 Flow Measurement Equipment Flow measurements were made as liquid metal rose up the 'riser' pipe of the flow c e l l , shown in Figure 9. In most experiments the actual distance involved was only 3 cm or less, therefore i t was essential to hold the furnace, flow c e l l and the measuring probe firmly in position, so that accurate measurements could be made. A schematic diagram of the apparatus is shown in Figure 13. The flow c e l l was held in position inside the furnace from the metal tube which was also connected to the argon supply. The measuring probe consisted of a copper wire in a ceramic tube which was inserted in the flow c e l l down the riser pipe. When the probe touched the surface of the liquid metal an electric circuit was closed giving the position of the interface. 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 for the liquid to make contact could be measured. The position of the probe was given on the dial 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 thermo-couple in a 3 mm diameter glass sheath. A simple circuit (shown in Figure 14) was used to connect the measuring probe to the thermocouple 32 Probe M Flowcell Lamp O 100 K I K —'I 1 | - > A M M ^ \ A A N - | .022/zF T/C FIGURE 14: C i r c u i t used f o r recording the p o s i t i o n of the probe on the tempera-ture t r a c e . Recorder 33 w i r e s , so t h a t a t the i n s t a n t when c o n t a c t was made a b l i p was produced on t h e temperature t r a c e . Chart speeds between 10 m i n / i n and 10 s e c / i n were used i n t h i s work, so t h a t the time r e q u i r e d f o r l i q u i d t o r i s e between s u c c e s s i v e p o s i t i o n s o f the probe c o u l d be measured to an a c c u r a c y of ± 0.1 s e c ( i f n e c e s s a r y ) . I t s h o u l d be n o t e d t h a t the c i r c u i t m e r e l y produced a b l i p on c o n t a c t , but d i d not o t h e r w i s e a l t e r the thermocouple s i g n a l (however a sheathed 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 i n s u l a t e the t i p , i n t h i s c a s e ) . The equipment d e s c r i b e d above, which was used f o r 22 o f the t o t a l o f 30 t e s t s which were done, i n f a c t e v o l v e d g r a d u a l l y s i n c e the e a r l y t e s t s showed t h a t d e v i a t i o n s from Darcy's Law o c c u r r e d i n some o f the f l o w t e s t s . I t was f e l t t h a t the n a t u r e o f these d e v i a t i o n s would be b e t t e r u n d e r s t o o d i f more a c c u r a t e equipment was used. E a r l y methods o f measuring the time when the probe touched the l i q u i d m e t a l i n v o l v e d photo-g r a p h i n g an e l e c t r o n i c t i m e r a t the same i n s t a n t t h a t a lamp showed c o n t a c t had been made. T h i s method was p o t e n t i a l l y j u s t as a c c u r a t e and r e p r o d u c i b l e as the method d e s c r i b e d p r e v i o u s l y , however i t r e q u i r e d the u n d i v i d e d a t t e n t i o n o f the e x p e r i m e n t e r over l o n g p e r i o d s . The p o s i t i o n o f the probe was measured u s i n g a p o i n t e r and s c a l e which was l e s s p r e c i s e than the d i a l gauge. The r e s u l t s from a l l the t e s t s were used t o determ i n e 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 d e n d r i t e s p a c i n g , s i n c e i t was s u b s e q u e n t l y found t h a t the r e s u l t s from the e a r l y t e s t s l a y w i t h i n t h e observed s c a t t e r . However, o n l y t h e r e s u l t s from the l a t e r t e s t s were used i n the study o f d e v i a t i o n s from Darcy's Law. 34 3.6 Flow Testing Procedure Before assembly, the faces of the castings which were to be placed in contact were painted with a thin layer of a soldering-type flux paste, to ensure that when castings B and C became liquid they would completely wet the end surfaces of casting A. The flow c e l l was assembled with the Pb-Sn inserts, and placed inside the vertical 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 inhibit f l u i d flow. The measuring probe was inserted down the riser pipe u n t i l contact was made with the surface of the branched portion of casting C. This established the datum level for measuring f l u i d pressures, and the dial gauge was set to zero for this point. The probe was then moved up the required amount in preparation for flow measurements. The sheathed thermocouple was inserted in the appropriate hole in 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 auto-matic controller held the c e l l at constant temperature while flow measurements were made. It was found that manual control of the power supply was the most effective method of heating the c e l l in the i n i t i a l stages, and a reproducible procedure could be developed after two or three 'dummy' t r i a l s . The zero point for timing measurements was taken as the instant when the temperature reached 183°C, the eutectic temperature, 35 i.e., the instant melting would be expected to begin. Flow took place relatively slowly, and in the majority of cases the temperature of the c e l l had stabilized at 193°C by the time the f i r s t measurements were made. In those cases where the temperature had not stabilized within ± 3°C of the required temperature, the points were not used unt i l the temperature had stabilized. When the required number of data points had been measured, the flow c e l l was chilled, either by lowering into a water bath or by using a 50 P s i air blast. In the latter case sol i d i f i c a t i o n was complete 45 sees after the air blast was turned on. After cooling to room temperature the flow c e l l was dismantled and the total height of the Pb-Sn sample was measured. Comparing the height of the column in the riser pipe after cooling, to the measured height when liquid, i t was found that thermal and sol i d i f i c a t i o n contraction caused a reduction in length of 9% between the testing temperature and room temperature for this composition. This information was therefore used when calculating the height of liquid from room temperature measurements. The majority of the samples were subsequently sectioned both at right angles and parallel to the axis of the cylinder A. Microexamination and autoradiography were used to determine the flu i d flow paths and the effect of flow on the micro-structure. 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 rising column of liquid metal was f i r s t tested by using the 36 Motor Contact Guide Tube Mercury Probe Glass Tube (7.9 mm ID.) Flexible Tube FIGURE 15: Apparatus for 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% 37 the apparatus to measure the flow rate of a rising column of mercury. A constant flow rate was imposed by raising one branch of a flexible 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 riser tube in the flow c e l l . A schematic diagram of the equipment is shown in Figure 15. Four experimental runs were done ranging from 20-50 observations. Points were taken approximately every 0.75 mm as the mercury rose. The method of least squares was used to f i t a straight line to the measured values of distance and time. The flow rate, standard error of the flow rate, and the 95% confidence intervals were calculated and are listed in Table IV. From these tests one may conclude that the error in the flow measurement technique was ± 0.3% over 50 observations for a constant flow rate. 38 CHAPTER 4 RESULTS AND DISCUSSION OF FLOW MEASUREMENTS 4.1 I n t e r p r e t a t i o n U s i n g Darcy's Law The e x p e r i m e n t a l t e c h n i q u e s u p p l i e d d a t a on the d i s t a n c e (&) of f l u i d f l o w up the r i s e r p i p e v e r s u s time ( t ) , and two t y p i c a l p l o t s a r e shown i n F i g u r e 16. The f l o w v e l o c i t y a t any p o i n t i s g i v e n by the s l o p e s of t h e c u r v e s , and i t i s c l e a r t h a t t h e r e i s a p p r o x i m a t e l y an o r d e r o f magnitude d i f f e r e n c e between the i n i t i a l v e l o c i t i e s f o r t h e s e two d e n d r i t e s p a c i n g s . The cur v e s show t h a t the f l o w v e l o c i t y d e c r e a s e s w i t h time f o r the l a r g e r d e n d r i t e s p a c i n g , y e t i t remains f a i r l y c o n s t a n t f o r t h e s m a l l e r s p a c i n g . The r e s u l t s were i n t e r p r e t e d by c o n s i d e r i n g the c a s t i n g A t o be a porous medium w h i c h obeys Darcy's Law. The c l a s s i c a l experiment p e r f o r m e d by Darcy i n 1856 c o n s i s t e d o f measurements o f the q u a n t i t y o f water f l o w i n g through a sand f i l t e r bed. The q u a n t i t y was found to be d i r e c t l y p r o p o r -t i o n a l t o the p r e s s u r e drop, and i n v e r s e l y p r o p o r t i o n a l t o the l e n g t h of (14) the bed. From d i m e n s i o n a l arguments one can deduce the f o l l o w i n g r e l a t i o n s h i p : v 4.1 where v = b u l k v e l o c i t y o f the f l u i d (measured o v e r the whole area) K = p e r m e a b i l i t y o f the porous medium v i s c o s i t y o f the l i q u i d L = l e n g t h o f the porous medium A P = p r e s s u r e drop 39 o m CNJ E o o — cvi uJ O < -0 0 5 ° o AVERAGE PRIMARY DENDRITE SPACING 116 microns COLUMNAR 460 6 0 0 8 0 0 T IME (seconds) 200 1000 FIGURE 16(a): Flow measurement results; distance of flow up the riser pipe versus time for A = 116 ym. I n i t i a l slope = 0.0055 cm/sec. o . in (VI So — CVl UJ < Is m 6 AVERAGE PRIMARY DENDRITE SPACING 28 microns COLUMNAR 3 0 0 0 4 0 0 0 5 0 0 0 TIME (seconds) Similar plot for X = 28 ym. I n i t i a l slope =0.00039 cm/sec. 1 0 0 0 2 0 0 0 6 0 0 0 7 0 0 0 FIGURE 16(b): 40 The permeability K is a property of the porous medium and has 2 the dimensions of area (cm ). The minus sign in the expression indicates that flow is in the opposite direction of increasing AP. Darcy's Law has been verified experimentally for flow through many types of porous media, and Carman^"^ has stated that there is good reason to believe that i t can always be applied under the following conditions: i) the flow must be laminar i i ) 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 capillary effects are absent. 4.1.1 Laminar flow Laminar flow is related to the Reynold's number, a dimensionless group defined as Re = i£i M where V is the (scalar) velocity measured over the whole area of the bed, p is the density of the f l u i d , u is the viscosity of the f l u i d , and 6 is a diameter associated with the porous medium, i.e., the average particle or pore diameter, or some length corresponding to the hydraulic radius theory. The representation of the flow by means of the Reynold's number is therefore dependent on the choice of the length 6, which in 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 4 1 been reviewed by S c h e i d e g g e r , and the range of values reported for the c r i t i c a l :Reynold's number li e s between 0.1 and 75. Scheidegger has i commented that the uncertainty of a factor of 750 is probably related to the fact that 6 is not clearly defined, and he points out that the difference between these values and the Reynold's number of ZOOO^which is normally taken as the c r i t i c a l value for turbulent flow in 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 exists, therefore, to check for laminar flow in the present work, the following simple approach was adopted. The maximum observed velocity was calculated from Equation 4.1: K , v = - — pgh —8 2 where K = 8.2x10 cm when X = 175 ym and g - 0.2 Li y = 0.03 poise L = 3.37 cm 3 P = 8.33 g/cm 2 g = 981 cm/sec h(max) = -4.51 cm i.e. v = 0.03 cm/sec As a f i r s t approximation, <5 is taken, as equal to the primary dendrite spacing X, then 42 R = ^ e u X = 175 x 10~4 cm u = 0.03 poise i.e. R = 0.14 e This value is approximately equal to the lowest published estimate of the c r i t i c a l Reynold's number. However, i t is reasonable to assume that the value of 6 chosen is a conservative estimate, since the effective diameter of flow channels or particles (depending on the model chosen) is likely to be much less than X. Flow is therefore considered to be laminar in a l l the tests done in the present work, and the f i r s t condition is satisfied. 4.1.2 Interaction effects As well as an upper limit to Darcy's Law, there are a number of references in the literature to a lower limit. Carman reviewed the early observations of this behaviour in 1937^^\ and drew the conclusion that the deviations were related to surface forces between the solid and liquid. This has remained the general consensus 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 surface tension, adsorption and molecular diffusion. Electrochemical effects have (18 19) been of interest recently, and work has been published ' explaining the effects in terms of an e l e c t r i c a l double layer. Since liquid metals are not ionic, this would not be relevant to the present work. There is no doubt that the interdendritic liquid w i l l interact chemically with the dendrites, consequently deviations from Darcy's Law w i l l 43 be interpreted in terms of the interaction effects. It i s assumed that because the dendrites form a r i g i d network, interaction effects over short time periods w i l l not cause a general collapse of the structure. Therefore i t should be possible to use the data from these experiments to draw some conclusions regarding the nature of the effects which cause the deviations. 4.2 Application to the Flow Cell Experiments For the experiments done in the flow c e l l AP varies with time, therefore Equation 4.1 can be expressed in the following form t - - | ln(h t/h o) 4.2 where h is the head of liquid at time t, h is the i n i t i a l head and c is t o a constant. The complete derivation of Equation 4.2 is given in Appendix I. The distance versus time data were therefore replotted as In (h t/h o) versus time. When the permeability K is a constant, these plots should be linear, with a slope equal to -c/K. The data from Figure 16 have been replotted in this manner in Figure 17, and in both cases the plots deviate from linearity. The castings were examined metallographically to investigate the reasons for these deviations and the results w i l l be described in detail in a later 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 linear-FIGURE 17(b); Similar plot for 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 in Figure 17 are the best estimates of the i n i t i a l slope by this method. 4.2.1 The method for finding the i n i t i a l permeability The tests described in section 3.7 to establish the precision of the flow measurement technique can be used to separate the random experi-mental errors inherent in the technique from the systematic effects which cause the deviations from linearity. 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 velocity was constant, the observed scatter was only due to experimental errors in using the copper wire probe to locate the mercury surface. Therefore the data was fi t t e d to a line y = mx where y is the dependent variable (position of the mercury surface) and x is the independent variable (time). The slope of the regression line (m) would therefore be the velocity. The scatter of points about the best f i t line 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 ith point y_. and the 46 value y_^ ; estimated by the regression line, in other words, the error; and (n-2) is the number of degrees of freedom. Therefore a is 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 in the form of ln(h /h ) versus time, therefore one must consider a plot of t o •. •» . ln(l-y^) versus x_^ instead of the simple y^ versus x^, where h Q, a 2 and a^ are constants defined in Appendix I. (From the numerical values of the constants, i t follows that y! is normally less than 1.) Similarly, the f i t t e d value is y!^ where The error in the dependent variable, e^, plotted in this manner is: e i - l n ( l - yj) - l n ( l - v[) It i s well known that when data is plotted on a logarithmic graph the points become weighted, in other words, the error e is a function of position along the line. One can take this effect into account by including a weighting factor w^ . A suitable weighting factor can be calculated as follows: Let y! = z, and y' - y! = 6z J 1 . J i ^ l 47 then l n ( l - y\) = l n { l - (z + 6z)} Since -1 < 71 1 1 1 1 10 2 0 3 0 4 0 A L L O Y COMPOSITION (% Sn) FIGURE 50: C a l i b r a t i o n curve; s p e c i f i c a c t i v i t y versus 113 a l l o y composition f o r constant Sn concentration. A L L O Y COMPOS IT ION (wt % Sn) FIGURE 51: C a l i b r a t i o n curve; s p e c i f i c a c t i v i t y versus 113 a l l o y composition, when Sn . concentration i s p r o p o r t i o n a l to the so l u t e content. 124 The sample composition was then given by: a. C i W. a. 1 m g Although the method of t r e a t i n g samples w i t h n i t r i c a c i d probably reduced some of the s c a t t e r a s s o c i a t e d w i t h the a n a l y s i s technique, i t introduced other problems. In p a r t i c u l a r , the d i s p o s a l of r a d i o a c t i v e waste i n the form of 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 of i n d i v i d u a l t e s t tubes, proved to be very time consuming. I t was found that the time required to s a f e l y t r a n s f e r the samples to one, non-c o r r o s i v e container, and then reduce the bulk of l i q u i d waste by evaporation was much longer than a n t i c i p a t e d . Since the main advantage of using the isotope a n a l y s i s technique was i t s speed compared to other methods, t h i s reduced i t s value appreciably. Consequently, composition p r o f i l e s f o r one ingot were compared using the r e s u l t s f o r samples of turnings which had not been t r e a t e d , and f o r the same samples which had been t r e a t e d w i t h n i t r i c a c i d . Both s e t s are p l o t t e d together i n Figure 52. Although there appears to be l e s s s c a t t e r a s s o c i a t e d w i t h the t r e a t e d samples, n e i t h e r p l o t i s smooth, because of the e f f e c t of microsegregation. Therefore i t was f e l t that the improved accuracy d i d not merit the time i n v o l v e d i n t r e a t i n g a l l the samples w i t h n i t r i c a c i d , and the composition p r o f i l e s f o r other ingots were c a l c u l a t e d using c u t t i n g s from the i n g o t s , and the d i r e c t p r o p o r t i o n a l i t y between composition and s p e c i f i c a c t i v i t y was taken to be c o r r e c t over the range considered. The r e p r o d u c i b i l i t y of the a c t i v i t y measurements from untreated FIGURE 52: Composition p r o f i l e f o r one ingot using l a t h e turnings t r e a t e d with n i t r i c a c i d (open c i r c l e s ) , and untreated samples (cl o s e d c i r c l e s ) . ho 126 lathe turnings was t e s t e d by counting a s i n g l e sample s e v e r a l times, emptying and r e f i l l i n g the t e s t tube, to vary the geometry of the packing. The e r r o r bars used i n the f o l l o w i n g composition p r o f i l e s (Figures 53-57) are ± 2s l i m i t s based on these t e s t s . For castings w i t h a mean composition of Pb-20%Sn, they represent a s c a t t e r of ± 0.38%Sn (percentage e r r o r + 1.9%) i n the a n a l y s i s technique. This was considered acceptable 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 of the ingots was, i n g e n e r a l , 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 represent a s c a t t e r of ± 0.22%Pb, which i s a l a r g e r percentage e r r o r (4.3%), probably due to g r e a t e r 204 absorption of the low energy emission from T l 6.2.4 Solute convection To observe convection through the b u l k l i q u i d and s o l i d - r l i q u i d 113 regions, 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 placed i n the l i q u i d at the top of the mould during s o l i d i f i c a t i o n . S o l i d i f i c a t i o n was continued f o r one hour; f o l l o w i n g which the castings were quenched i n the furnace, then sectioned 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 extent of t r a c e r movement. A s i m i l a r experiment was performed on a c a s t i n g of the same composition h e l d completely l i q u i d under the same temperature gradient f o r one hour. In t h i s case, spreading of the t r a c e r could be a t t r i b u t e d to the cumulative e f f e c t s of disturbances a s s o c i a t e ^ w i t h adding the p e l l e t s and quenching. The f l u i d flow 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 process could be evaluated by comparing the r e s u l t s of these t e s t s . 127 6.3 Results 6.3.1 Composition p r o f i l e s The composition p r o f i l e s of the i n g o t s , determined by the 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 , are shown i n pa r t (a) of Figures 53-57. The s o l i d l i n e s are t h e o r e t i c a l curves c a l c u l a t e d from the mathematical model which i s developed i n Chapter 7. Correspondence between theory and experiment i s discussed i n s e c t i o n 746. Figure 54(a) i s ^ i n f a c t , the same composition p r o f i l e as Figure 52, and the data p o i n t s used were those f o r samples t r e a t e d w i t h n i t r i c a c i d . Since the e r r o r bars were obtained 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 probably represent a more conservative estimate of the 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 ingot are represented g r a p h i c a l l y i n part (b) of Figures 53-57, according to the method proposed (45) by Flemings and Nereo . Measurements of the time required f o r the l i q u i d u s and s o l i d u s isotherms to pass each thermocouple p o s i t i o n are p l o t t e d on a distance-time graph. I f the l i q u i d u s and s o l i d u s l i n e s are 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 that both the growth ra t e and tempera-ture gradient remained constant during s o l i d i f i c a t i o n . This i s shown to be e s s e n t i a l l y true under the 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 macrosegregation are summarized i n Table X. Macrosegregation i s normally defined as AC = C - C 6.1 x o where C i s the composition at a p a r t i c u l a r l o c a t i o n (x) and C i s the mean 128 o o C\J • C\J AVERAGE TEMPERATURE GRADIENT l-5°C/cm AVERAGE GROWTH RATE 0 0 0 4 7 cm/sec " 0 0 2 0 4 0 6 0 8 0 1 0 0 DISTANCE F R O M B O T T O M OF CAST ING (cm) 12 0 14 0 FIGURE 53(a): Solute d i s t r i b u t i o n . 129 o CM CM o ID AVERAGE TEMPERATURE GRADIENT l-5 °C/cm AVERAGE GROWTH RATE 0-013 cm/sec " 0 0 2 0 4 0 6 0 8 0 1 0 0 DISTANCE F R O M B O T T O M OF CAST ING (cm) 12 0 14 0 FIGURE 54(a): Solute d i s t r i b u t i o n . FIGURE 54(b): Cooling c o n d i t i o n s . 100 TIME (min) 140 130 o o CM CM ( _> O rr b 1 UJ CM CL O CO' O AVERAGE TEMPERATURE GRADIENT 2-3°C/cm AVERAGE GROWTH RATE 0011 cm/sec " 0 0 —I 1 1 1 1 1 1 1 1 2 0 4 0 6 0 8 0 1 0 0 DISTANCE FR0IV1 B O T T O M OF CAST ING (cm) 12 0 14 0 FIGURE 55(a): Solute d i s t r i b u t i o n . FIGURE 55(b): Cooling c o n d i t i o n s , 140 TIME (min) 180 131 C\J o c\l • | 2 N C E O L l J C M Q . 5 co-o to • AVERAGE TEMPERATURE GRADIENT I 0 ° C / c m AVERAGE GROWTH RATE 0 24 cm/sec " 0 0 2 0 4 0 6 0 S O ' l O O ' I2"0 • ' | 4 ' 0 D ISTANCE F R O M B O T T O M OF CAST ING (cm) FIGURE 56(a): Solute d i s t r i b u t i o n . FIGURE 56(b) : Cooling c o n d i t i o n s . 132 ob AVERAGE TEMPERATURE GRADIENT l-9°C/cm AVERAGE GROWTH RATE 0 0 0 3 3 cm/sec or o-X — |5 o C\J" ' 0 0 ™ ' ^ ' iS ' 8^ ^ o " ' [£o DISTANCE F R O M B O T T O M OF CAST ING (cm) FIGURE 57(a): Solute distribution. FIGURE 57(b): Cooling conditions. TIME (min) TABLE X SOLIDIFICATION VARIABLES AND MACROSEGREGATION Figure Number: A l l o y Composition (wt-pct) Average Temperature Gradient (°C/cm) Average Growth Rate (cm/sec) Ca l c u l a t e d 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) S t r u c t u r e Macro-segre-g a t i o n (AC pet) 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 in mean composition between the upper and lower halves of the ingot An 1 1 1 2 1 1 , „ A C = ~rw. o r 6 - 2 1 i 2 i where £^ and Z^ are the sums from x = H (the total length of the casting) to x = H/2, and x = H/2 to x = 0 respectively, and and are the composition and weight of the it h sample. Macrosegregation is considered positive when the solute concentration increases in the direction of solidification, and negative for the reverse. Comparing the macrosegregation for directionally s o l i d i f i e d and quenched castings, listed in Table X, i t can be seen that the solute distribution i s a function of the solidification conditions. In general, the amount of macrosegregation for Pb-20%Sn alloys decreased as the primary dendrite spacing decreased. 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 slightly more macrosegregation than Figure 55(a). For the Sn-4%Pb alloy (Figure 57(a)) there was more solute at the bottom of the casting and less at the top, resulting in some negative macrosegregation. An estimate of the significance of the macrosegregation values in Table X can be obtained by using the "Student's t-test". The composition 135 measurements f o r each sample are subject to microsegregation and geometrical s c a t t e r , however, i t i s reasonable to assume that the mean of the samples over h a l f the ingot w i l l only be subject to geometrical s c a t t e r , s i n c e microsegregation only extends over r e l a t i v e l y short d i s t a n c e s . Therefore i t i s a l s o reasonable to assume that the standard d e v i a t i o n of the mean i s equal to the standard d e v i a t i o n of each sample. Thus, knowing the means and standard d e v i a t i o n s f o r the top and bottom halves of the i n g o t , one may use the t - t e s t to check whether the composition d i f f e r e n c e s are s i g n i f i c a n t . In general x^ -t = s / J . 1 + -n l n 2 where x^ and x^ are the two means w i t h standard d e v i a t i o n s assumed equal to s, and n^ and n^ are the number of samples used to estimate the means. The number of degrees of freedom i s n^ + n^ - 2. The number of samples used to c a l c u l a t e the composition of each h a l f of the c a s t i n g i s approximately 15, th e r e f o r e one may take the number of degrees of freedom as 28. For a s i g n i f i c a n c e l e v e l of 0.01, the value of t i s 2.763, therefore i t i s u n l i k e l y that the mean compositions of the two halves of the ingot come from the same population when: 2.763 < A C since AC = x^ -One can th e r e f o r e conclude that there i s no s i g n i f i c a n t macrosegre-gat i o n when AC < 0.19%Sn f o r Pb-20%Sn i n g o t s ; or AC < 0.11%Pb f o r Sn-4%Pb 136 i n g o t s . Thus, the values f o r both quenched-ingots show no s i g n i f i c a n t macrosegregation, nor can the value of 0.13%Sn f o r the ingot i n Figure 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 value of 0.27%Sn f o r the ingot i n Figure 56(a) i s probably only m a r g i n a l l y s i g n i f i c a n t s i n c e the s c a t t e r i n the upper h a l f i s l a r g e r than i n the other i n g o t s , making the assumption regarding microsegregation l e s s v a l i d . The remaining i n g o t s , however, show a s i g n i f i c a n t amount of macrosegregation compared to the 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 to one another. 6.3.2 Convection i n the l i q u i d The r e s u l t s demonstrating convection i n the l i q u i d during s o l i d i f i c a t i o n are given i n Figure 58. The autoradiographs shown are of sec t i o n s p a r a l l e l and perpe n d i c u l a r to the f r e e z i n g d i r e c t i o n of a Pb-20%Sn a l l o y under the c o n d i t i o n s l i s t e d i n Table X I . In Figure 58(a), the regions which are uniformly dark (sections i - i v ) i n d i c a t e that the l i q u i d u s isotherm passed through t h i s r e gion a f t e r t r a c e r had become mixed through the bulk l i q u i d . 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 the s o l i d - l i q u i d zone. Figure 58(b) shows autoradiographs f o r the ingot quenched from the l i q u i d . Tracer has moved l e s s than 3 cm down the ing o t as compared to 6.5 cm f o r Figure 58(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 to s o l u t e convection 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 process. There i s no i n d i c a t i o n , however, of the flow p a t t e r n which caused mixing. 137 FIGURE 58: Autoradiographs showing the extent of t r a c e r movement one hour a f t e r t r a c e r 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 . M a g n i f i c a t i o n 2.2x. TABLE XI COOLING CONDITIONS Figure Number A l l o y Composition (wt pet) Average Temperature Gradient (oc/cm) Average Growth Rate (cm/sec) C a l c u l a t e d Distance Between Liquidus and Solidus 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 F r e c k l e s Evidence of s t r u c t u r e s resembling f r e c k l e s was seen i n ingots s o l i d i f i e d at the slowest growth r a t e s . In one ingot s o l i d i f i e d at 0.0047 cm/sec, the outer surface showed a shrinkage t r a i l approximately 7 cm long near the top (Figure 5 9 ( a ) ) . A shrinkage defect of t h i s type i n d i c a t e s that a long channel of e u t e c t i c l i q u i d was present j u s t before the ingot became completely s o l i d . This bears a c l o s e resemblance to photographs of 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 Figure 34(b). One c a s t i n g grown at 0.0033 cm/sec revealed an i n t e r n a l t r a i l which could be classed as a f r e c k l e . F i gure 59(b) shows transverse and 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 which was 4.5 cm long. An enlarged view of a transverse s e c t i o n (Figure 60(a)) shows that the t r a i l 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 . Figure 60(b) i s an enlarged view of the lowest p o r t i o n of the t r a i l , showing that 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 ingot as an i n t e r d e n d r i t i c channel which 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 dendrite s t a l k s . 6.4 D i s c u s s i o n of Results The curves i n Figures 53-56 showing p o s i t i v e macrosegregation resemble curves f o r normal segregation w i t h d i f f u s i o n c o n t r o l l e d mixing ahead of a planar s o l i d - l i q u i d i n t e r f a c e . There i s , however, considerable ( 2347) experimental evidence i n the l i t e r a t u r e ' to show that only a n e g l i g i b l e amount of 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 planar. Normal segregation takes place over a d i s t a n c e of the order of microns i n the l i q u i d between dendrite branches, le a d i n g to micro-(a) (b) FIGURE 59: (a) Shrinkage t r a i l , approximately 7 em long, along the outside surface of an ingot s o l i d i f i e d under conditions given i n Table XI. M a g n i f i c a t i o n 1.7x. § (b) L o n g i t u d i n a l and transverse 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 . M a g n i f i c a t i o n 3x. 141 FIGURE 60(a): Transverse s e c t i o n of the f r e c k l e t r a i l i n Figure 59(b), showing f i n e d e n d r i t i c s t r u c t u r e w i t h i n the t r a i l . M a g n i f i c a t i o n 25x. 142 segregation. However, one would not expect a net movement of s o l u t e i n the d i r e c t i o n of growth unless there was l i q u i d mixing on a macroscopic s c a l e . This mixing could take 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 zone, or between t h i s zone and the bulk l i q u i d ahead of the dendrite t i p s . The experiment where t r a c e r was added to the l i q u i d at the top of the c a s t i n g (Figure 58) confirms that s o l u t e convection took p l a c e , which i s a t t r i b u t e d to the formation of lower d e n s i t y l i q u i d i n the s o l i d -l i q u i d r e g i o n . In the case of 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 enriched i n t i n , up to the e u t e c t i c composition (62% Sn). The d e n s i t y of 3 the bulk l i q u i d at the i n t e r f a c e would be 9.7 g/cm , and the e u t e c t i c would 3 (42) be 8.2 g/cm , thus there would be a d e n s i t y i n v e r s i o n through the s o l i d - l i q u i d region which causes the l e s s dense l i q u i d to r i s e . One can t h e r e f o r e conclude that the s o l u t e p r o f i l e s i n Figures 53-56 are a f u n c t i o n of the growth r a t e , temperature gradient and d e n d r i t e spacing. Since i t was only p o s s i b l e to hold two of the three v a r i a b l e s constant f o r any two i n g o t s , one cannot draw any f i r m conclusions regarding the e f f e c t of a v a r i a t i o n i n any one. For t h i s reason, a simple mathematical model was derived (chapter 7 ) , which i n c l u d e d the concept of mass t r a n s f e r through the s o l i d - l i q u i d region 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 . The Sn-4%Pb a l l o y was chosen as an example of a composition 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 bulk l i q u i d above. 3 3 The comparable d e n s i t i e s would be 7.1 g/cm i n the bulk l i q u i d , and 8.2 g/cm (42) i n the e u t e c t i c . This 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 one would not expect any convection. The r e s u l t i n g composition p r o f i l e (Figure 57(a)) shows a s m a l l i n c r e a s e i n lead content close to the bottom of the 143 i n g o t , which i s p o s s i b l y due to the e f f e c t of i n v e r s e segregation. A small sample (150 g) of the molten a l l o y was placed i n a graphite c r u c i b l e and observed during s o l i d i f i c a t i o n under vacuum. No bubbles were seen. This evidence, together 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 that only oxygen i s very s l i g h t l y s o l u b l e i n molten Pb-Sn a l l o y s ^ 4 8 \ confirmed that gas e v o l u t i o n could be ignored 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. Consequently, the f r e c k l e t r a i l s observed i n i n g o t s s o l i d i f i e d at the slowest f r e e z i n g r a t e s are a t t r i b u t e d to the e f f e c t of s o l u t e convection. As the l e s s dense l i q u i d towards the bottom of the mushy zone begins to r i s e , i t becomes superheated and can d i s s o l v e dendrite branches i n i t s path. 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 channels 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 formation of a large v e r t i c a l pipe. D i r e c t evidence of t h i s mechanism i s shown i n Figure 60(b). The experimental evidence suggests that f r e c k l e s do not always form when s o l u t e convection takes p l a c e , but they appear when the v e l o c i t y of 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 reaches a c r i t i c a l value. 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 of Previous Work Macrosegregation caused by i n t e r d e n d r i t i c f l u i d flow 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 of published papers. Most of these models are f o r inverse segregation, where an a n a l y t i c a l s o l u t i o n can be obtained by considering backflow through a volume element as the s o l i d and l i q u i d c o ntract during s o l i d i f i c a t i o n . C h i l l face segregation 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 Youdelis to p r e d i c t the s o l u t e d i s t r i b u t i o n along the whole «.. (50,51) c a s t i n g A more general s o l u t i o n f o r macrosegregation, c o n s i d e r i n g f l u i d (45) flow i n three dimensions, was f i r s t p u b lished by Nereo and Flemings Their model i s based on the use of the Pfann Equation to p r e d i c t microsegre-g a t i o n i n binary a l l o y s : C = kC (1 - f ) k _ 1 ' " 7.1 S O S where C 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 = 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 f = weight f r a c t i o n s o l i d C q = i n i t i a l a l l o y composition. In a d d i t i o n : C = kC T s L and f = (1 - f T ) s L 145 where CT and f are the composition and weight f r a c t i o n of l i q u i d , Li Lt r e s p e c t i v e l y . Considering a constant k, and a constant s o l i d i f i c a t i o n shrinkage ( 3 ) , they solve heat and mass balances i n a volume element to o b t a i n the f o l l o w i n g general expression: L I h . (Lz-A\ I"1 +51yil ! S 7 2 9C T \ l - k ^ L e J 3t where P s ~ P L p 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 v e c t o r VT = temperature gradient i n three dimensions p = s o l i d d e n s i t y s J p^ = l i q u i d d e n s i t y and e i s defined as the r a t e of temperature change (9T/3t). Equation 7.2 i s r e f e r r e d to 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 equation". I t i s w r i t t e n f o r the general case of three-dimensional heat and f l u i d f l o w , assuming constant s o l i d d e n s i t y during 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 no pore formation. I f f u r t h e r assumptions are now made to c a l c u l a t e the flow v e l o c i t y v e c t o r ( v ) , the equation can be used to estimate macrosegregation i n c a s t i n g s . Flemings and Nereo used the model to 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 f o r i nverse segregation i n Al-Cu a l l o y s . The model has subsequently been (52 53) r e f i n e d and extended ' , and has been a p p l i e d by Mehrabian, Keane and (9) Flemings to p r e d i c t macrosegregation caused by a combination of 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 and s o l u t e convection. They consider the f l u i d dynamics through a volume element where the forces 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 of v a r i a b l e d e n s i t y , and the s o l i d - l i q u i d region i s t r e a t e d as a porous medium of v a r i a b l e p o r o s i t y . Equations are 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, i n t e r d e n d r i t i c flow 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 composition which can, i n theory, be solved t o give these v a r i a b l e s as a f u n c t i o n of p o s i t i o n . In p r a c t i c e , s o l u t i o n s f o r the general case are d i f f i c u l t to 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 of simultaneous p a r t i a l d i f f e r e n t i a l equations. They have th e r e f o r e made the f o l l o w i n g s i m p l i f y i n g assumptions: 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 the mushy zone only as a f u n c t i o n of temperature, and i s c a l c u l a t e d by assuming s t e a d y - s t a t e , u n i d i r e c t i o n a l heat and f l u i d flow. 2) Planar isotherms are assumed, so that f o r a constant l i q u i d u s s l o p e , the l i q u i d composition 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 density of l i q u i d v a r i e s l i n e a r l y w i t h composition. 4) The den s i t y of s o l i d i s constant. The model has been a p p l i e d to the s p e c i a l - case of 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 flow and steady-state s o l i d i f i c a t i o n , where a value f o r the parameter c h a r a c t e r i z i n g the s t r u c t u r e and thermal c o n d i t i o n s has been assumed. In Chapter 6, macrosegregation experiments 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 . Since i t i s d i f f i c u l t to apply the model derived by Mehrabian et a l . to t h i s type of ingot and a l l o y system, a simple mathemat-i c a l model was developed, which would take i n t o account the e f f e c t of growth r a t e , temperature gradient and s t r u c t u r e . Some of the assumptions f o r t h i s model d i f f e r from those used by Mehrabian et a l . , the 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 of temperature. 147 2) The d e n s i t y of the l i q u i d i s a f u n c t i o n of temperature and composition, obtained from experimental data. 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 obtained from the r e s u l t s of 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) Since the 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 of Pb-Sn a l l o y s , backflow due to volume shrinkage i s neglected. 7.2 Model of the S o l i d i f i c a t i o n Process The s o l i d - l i q u i d c o n f i g u r a t i o n during 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 of a small ingot i s assumed to be that shown s c h e m a t i c a l l y i n Figure 61. The a c t u a l s t r u c t u r e represented by the spikes may be columnar d e n d r i t i c or equiaxed. The f o l l o w i n g assumptions are made: 1) The l i q u i d i s completely mixed i n the h o r i z o n t a l planes. 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 s t a t e . 3) L o c a l e q u i l i b r i u m e x i s t s at the i n t e r f a c e between the i n t e r d e n d r i t i c l i q u i d and the adjacent s o l i d . Under these c o n d i t i o n s , the composition of the s o l i d at the s o l i d -l i q u i d i n t e r f a c e i s given by the Pfann Equation (Equation 7.1), provided k i s a constant. I t i s a l s o p o s s i b l e to use Equation 7.1 i n c r e m e n t a l l y 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 temperature, by assuming that k remains constant over a s m a l l temperature i n t e r v a l AT. For the general case of s o l i d i f i c a t i o n between temperatures T^ and T^, as shown i n Figure 62, l i q u i d composition i s given by the l i q u i d u s l i n e , k i s equal to 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 bin a r y a l l o y . The non-e q u i l i b r i u m s o l i d u s i s shown by the dashed l i n e . 149 C s 7.3 where and r e f e r to compositions at the e q u i l i b r i u m s o l i d u s and l i q u i d u s . The weight f r a c t i o n of l i q u i d which s o l i d i f i e s as the a l l o y cools between T^ and i s : J 1 J - l / ( l - k ) 7.4 The s o l i d which freezes i n t h i s increment i s of composition C' S 2 and weight f r a c t i o n (1 - f ). For s m a l l increments AT, C' = kC , but a s 2 L 2 b e t t e r estimate of C' can be obtained f o r l a r g e r increments by c a l c u l a t i n g the t o t a l weight of s o l u t e i n both s o l i d and l i q u i d at T^, then applying conservation of s o l u t e mass at T^' T n e composition of the e n t i r e s o l i d at 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 d i f f u s i o n i n the s o l i d has been assumed. The non-equilibrium s o l i d u s (shown dashed i n Figure 62) can be c a l c u l a t e d by summing the t o t a l amount of s o l u t e i n the s o l i d . The composition of the m a t e r i a l which s o l i d i f i e s at T 1 i s C' , yet 1 s x the composition of the e n t i r e s o l i d i s C_ . I f the weight of l i q u i d at T i s W , and the weight and average composition of s o l i d are W and C , s o l u t e conservation g i v e s : S l S l % (1 " V \ + C L 2 f L \ + % W C (W_ + w ) o L l s x 7.5 150 and the average composition of s o l i d at l s : C W + C (1 - f T ) W S l S l S2 L L l s 0 W + (1 - f_) WT 7.6 2. Sj L Consequently the l i q u i d composition, average s o l i d composition and the weight f r a c t i o n s of s o l i d and l i q u i d can be defined at any tempera-ture between the l i q u i d u s and s o l i d u s by using data from the phase diagram. In a d d i t i o n , i f d e n s i t y data are a v a i l a b l e as a f u n c t i o n of temperature and composition, volume f r a c t i o n s and d e n s i t i e s are a l s o defined at any temperature. 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 of the s o l i d i f i c a t i o n process describes how s o l u t e i s r e d i s t r i b u t e d perpendicular to the dendrite s t a l k s . I f no i n t e r d e n d r i t i c f l u i d flow occurs, the ingot would show microsegregation on a s c a l e equivalent to the dendrite spacing, but there would be no net movement of so l u t e over greater d i s t a n c e s . The assumption i s now made that s o l u t e can be moved over much greater distances by i n t e r d e n d r i t i c f l u i d flow caused by density 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 that the network of dendrites i n the s o l i d - l i q u i d region produces a r e s i s t a n c e to flow, and that 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 of the volume f r a c t i o n , and dendrite spacing of the s o l i d . Therefore, using the c a p i l l a r y model described i n s e c t i o n (4.5.1), the p e r m e a b i l i t y of the d e n d r i t i c network i s given by: 2 K = 4 - 8 8niTT This equation was der i v e d f o r one dimensional flow; however, the 151 r e d i s t r i b u t i o n of s o l u t e i n a c a s t i n g of the type shown i n Figure 61 i n v o l v e s three dimensional flow. In t h i s case, s i n c e flow only takes place down one t h i r d of the channels i n any one d i r e c t i o n , Equation 4.8 becomes: 2 K = ^ 7.7 24mTT (1 2) Previous experimental work ' has shown that K i s p r o p o r t i o n a l 2 to g when g i s l e s s than 0.3. In the absence of a b e t t e r model, i t has J-j Li been assumed that Equation 7.7 holds f o r a l l values of g . Values of the Li 3 f a c t o r nx have been taken from Figure 28, and used i n conjunction w i t h t h i s model. Dendrite coarsening has been neglected. Macrosegregation i s t h e r e f o r e determined using Darcy's Law to c a l c u l a t e the flow 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 the d r i v i n g f o r c e (AP) i s given by the de 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 of the a p p l i c a t i o n of the model, consider 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 the c a l c u l a t i o n s , constant growth ra t e (R) and temperature grad-i e n t (G) are assumed. In 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 to the s o l i d i f i c a t i o n of Pb-Sn a l l o y s where the volume change on f r e e z i n g i s s m a l l (of the order of 2%), i t has been assumed that backflow due to 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 of h o r i z o n t a l l a y e r s of length iy and the temperature w i t h i n each l a y e r i s assumed to be uniform. 152 The 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 equal to G£. Increments of time (At ) are chosen such that the c o o l i n g r a t e i s equal to G£/At . A f t e r each time increment, the temperature of each l a y e r i s then equal to the temperature of the l a y e r below and the growth r a t e (R) i s equal to £/At. The length I can, 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 , Since i t can be defined i n terms of R and A t . S o l i d i f i c a t i o n i s considered to begin when the temperature o f the bottom l a y e r i s equal to the l i q u i d u s temperature T . Wit h i n each time step A t , s o l i d i f i c a t i o n and f l u i d flow are t r e a t e d s e p a r a t e l y . Conservation of sol u t e w i t h i n each l a y e r a p p l i e s during each s o l i d i f i c a t i o n s t e p , and conser-v a t i o n of s o l u t e through the whole c a s t i n g a p p l i e s i n each f l u i d flow step. F l u i d flow through a l a y e r stops when the temperature f a l l s below the e u t e c t i c temperature T^, and the f i n a l composition p r o f i l e of the c a s t i n g i s obtained when the top l a y e r reaches T . h Figure 63 gives a schematic r e p r e s e n t a t i o n of the c a s t i n g at an intermediate time, and shows the temperature, composition and den s i t y p r o f i l e s i n the l i q u i d obtained using the 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 the so 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 l i q u i d . The d r i v i n g f o r c e f o r flow through the s o l i d - l i q u i d region i s given by the den s i t y d i f f e r e n c e between T and T and i s equal to Ap gh, where Ap i s the density d i f f e r e n c e i n the l i q u i d , g i s g r a v i t y , and h i s the Li distance between T and T . Since the l i q u i d i s a continuum, one would L hi expect the d r i v i n g f o r c e at every point through the s o l i d - l i q u i d r e gion to be the same. However, sin c e the l i q u i d f r a c t i o n decreases downwards, the r e s i s t a n c e to flow would increase towards the bottom of the s o l i d - l i q u i d zone. 153 Q " - ---->-- _ T L _ -/-J E _ J I Ki I J o n t \ ^ N — \ j RA t TEMPERATURE LIQUID COMPOSITION LIQUID DENSITY 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 ingot d i v i d e d i n t o l a y e r s . Temperature, composition and den s i t y 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. 6 5 « • • R4.Q4 4 R3 . ^ 3 3 R2 .<*2 2 R. ,qi | I 1 (a) (b) FIGURE 64: (a) Assumed flow p a t t e r n showing two main flow c e l l s . (b) Resistances R 1 - 5 » and flow r a t e s f o r flow between s i x l a y e r s . 154 A flow 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 region has, t h e r e f o r e , been assumed where flow can take place v e r t i c a l l y from one l a y e r to the next, and h o r i z o n t a l l y through the l a y e r . For v e r t i c a l f l o w , h a l f the cross s e c t i o n a l area c o n t r i b u t e s to downward flow, and h a l f to upward flow. Figure 64(a) shows the assumed flow p a t t e r n w i t h two main flow c e l l s . However, provided downward and upward flow each occupy h a l f the cross s e c t i o n a l area, the a c t u a l number of flow c e l l s i s unimportant. The r e s i s t a n c e of the d e n d r i t i c network to f l u i d flow i s repre-sented s c h e m a t i c a l l y i n Figure 64(b). The r e s i s t a n c e symbols represent porous media of area equal to h a l f the cross s e c t i o n a l area of the c a s t i n g , and length equal to the length of the l a y e r s . I t i s assumed that there i s no r e s i s t a n c e to h o r i z o n t a l flow through the l a y e r s , s i n c e the distances 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 flow c e l l s . Porous l a y e r s stacked i n t h i s manner obey the laws of s e r i e s r e s i s t a n c e s ; t h e r e f o r e , s i n c e the magnitude of 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 of 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 pressure drop i s 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 (v/g L) can be c a l c u l a t e d using Darcy's Law. The flow 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 (q) i s then equal to 2v/Ag where A i s the cross s e c t i o n a l area of the i n g o t . For b r i e f time Li i n t e r v a l s , At, the q u a n t i t y of l i q u i d which flows between l a y e r s w i l l be s m a l l , t h e r e f o r e the d r i v i n g force f o r flow i s assumed to remain constant. In Figure 64(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 are numbered R-^_5» and the flow r a t e s are qj_5« The l i q u i d compositions of each l a y e r , expressed as weight per u n i t volume, are equal to (PL^ T.,^ 1-6* The volumes of 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 , and f o l l o w i n g t h i s n o t a t i o n , the volume flow r a t e across the top" and bottom surfaces of the i t h l a y e r are q^ and q^ ^, r e s p e c t i v e l y . A s o l u t e mass balance can, 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 : v i i t ( PLVI = V i W i - i + ^ i W i + i - ( q i + V i ) ( p L V i 7 - 8 This mass balance 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 simultaneous ordinary d i f f e r e n t i a l equations which can be solved f o r the composition of each l a y e r , a f t e r a time i n t e r v a l At, using standard numerical methods. Thus, the net e f f e c t of f l u i d flow i s that the average composition of each l a y e r i s no longer equal to C q, yet on the next s o l i d i f i c a t i o n s t e p , the l i q u i d composition w i l l be equal to the value given by the l i q u i d u s l i n e on the e q u i l i b r i u m diagram. 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 d i f f e r e n c e s i n the average composition of the primary s o l i d from Equations 7 . 4 and 7.5 and 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 . 7.5 Results of C a l c u l a t i o n s 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 The model was 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 which shows a d e n s i t y i n v e r s i o n during s o l i d i f i c a t i o n . Data 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 k and 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 temperature, were obtained from the phase diagram and converted to polynomial expressions using standard curve f i t t i n g techniques. Data 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 composition were a v a i l a b l e i n the 156 form of a t a b l e y and intermediate values were obtained by l i n e a r (54) i n t e r p o l a t i o n s . The value of v i s c o s i t y (y) was taken as 0.03 poise The f i n a l s o l u t e p r o f i l e was obtained by r e c a l c u l a t i n g the values of 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 step followed by a f l u i d flow step a f t e r every time increment A t as the c a s t i n g cooled between T J_i and Tg, using a d i g i t a l computer. The FORTRAN program i s given i n Appendix IV. Figure 65 shows the s o l u t e d i s t r i b u t i o n f o r various values of the time i n t e r v a l , i . e . , d i f f e r e n t numbers of l a y e r s , when H, G, R, y and NC have the values shown. H i s the length of the c a s t i n g , and NC i s the e f f e c -3 t i v e number of channels (NC = nx A). The curves show t h a t , except f o r the ends of the i n g o t , as the number of l a y e r s increases (At decreases) the so l u t e d i s t r i b u t i o n converges to a s i n g l e s o l u t i o n . The compositions at the extreme ends diverge at At decreases, because the assumption that the flow r a t e (q) i s small compared to the amount of l i q u i d i n each l a y e r no longer holds when the s i z e of l a y e r s becomes very s m a l l . The highest value of q would be at the top, t h e r e f o r e one would expect t h i s assumption to break down f i r s t i n t h i s r e g ion of the i n g o t . The compositions which are c a l c u l a t e d at the extreme ends of the ingot are there-fore not considered meaningful, but the shape of the curves and the i n t e -grated amount of s o l u t e which has moved from the bottom of the c a s t i n g to the top are a measure of the r e l a t i v e amount of macrosegregation. The s t r u c t u r e of the s o l i d - l i q u i d region i s expressed i n terms of the e f f e c t i v e number of 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 values of NC are shown i n Figure 66. I t can be seen that the amount of 157 CVI ro O ob CVJ o CVJ o o I o UJ — o CVJ CO, L= 37 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 I05 viscosity of the liquid = 0 0 3 poise 0 0 2 0 4 0 6 0 8 0 100 DISTANCE FROM BOTTOM OF CASTING (cm) 120 14 0 FIGURE 65: Solute d i s t r i b u t i o n as a f u n c t i o n of the number of l a y e r s . FIGURE 66: Solute d i s t r i b u t i o n as a f u n c t i o n of s t r u c t u r e ( e f f e c t i v e number of channels). 158 macrosegregation, considered i n terms of the amount of s o l u t e which moves from the bottom h a l f of the c a s t i n g to the top, increases as NC decreases. Since NC i s r e l a t e d to the dendrite spacing, t h i s means that f o r l a r g e r spacings the r e s i s t a n c e to 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 the same pressure drop there i s more flow. Figure 67 shows the 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 of ingot height. As the height i n c r e a s e s , so the f l u i d head w i l l i n c r e a s e , causing more flow through the mushy zone. However, t h i s only a p p l i e s when the length of the mushy zone i s greater than or equal to the ingot height. The t h e o r e t i c a l length of the mushy zone i s (T - T )/G, which f o r the c o n d i t i o n s used i n Figure 67 i s 62 cm. Figure 68 shows that the amount of macrosegregation increases as the growth r a t e decreases. This would be expected, since the amount of time a v a i l a b l e f o r flow i n c r e a s e s , as R decreases. Figure 69 shows that the amount of macrosegregation increases as the temperature g r a d i e n t i n c r e a s e s , (34 contrary to the s e m i q u a n t i t a t i v e theory proposed by Copley, Giamei, et a l . This can be v i s u a l i z e d when one considers that the composition g r a d i e n t through the mushy zone w i l l be steeper f o r the higher temperature g r a d i e n t . This w i l l lead to a higher d e n s i t y d i f f e r e n c e and consequently more fl o w , when a l l other v a r i a b l e s are h e l d constant. The reason why t h i s appears to c o n t r a d i c t experience i s that h i g h temperature gradients are u s u a l l y assoc-i a t e d w i t h high growth r a t e s , and i t i s not normally f e a s i b l e to vary these two parameters independently. 159 FIGURE 67: Solute d i s t r i b u t i o n as a f u n c t i o n of ingot height. FIGURE 68: Solute d i s t r i b u t i o n as a f u n c t i o n of growth r a t e . 160 o I m H 1 1 1 i 1 1 1 1 1 1 1 1 1 1 00 2 0 4 0 6 0 8 0 1 0 0 1 2 0 14 0 DISTANCE FROM BOTTOM OF CASTING (cm) FIGURE 69: Solute d i s t r i b u t i o n as a f u n c t i o n of temperature g r a d i e n t . 161 7.6 Comparison w i t h Experiment The data used to generate the t h e o r e t i c a l curves i n Figures 53-56, together w i t h t h e o r e t i c a l and experimental values of macrosegregation according to the present d e f i n i t i o n (Equation 6.2), are given i n Table X I I . In 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 of the same shape as the experimental p l o t s , but the compositions at the ends of the ingot do not always agree w e l l w i t h those p r e d i c t e d . This i s due to the assumptions used i n d e r i v i n g the model. In a d d i t i o n to those already d i s c u s s e d , Equation 7.1 does not take i n t o account that the l i q u i d composition cannot r i s e above the e u t e c t i c composition. The aim of the experiments i n Chapter 6 was to demonstrate that macrosegregation was r e l a t e d to 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 designed to t e s t the model, consequently only q u a l i t a t i v e comparisons have been made. When attempts were made to use the computer program to 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 ingots w i t h very l a r g e dendrite spacings, i t was found that the composition at the top would r i s e to a very l a r g e value. This was probably due to the breakdown i n the assumption that the flow r a t e between l a y e r s i s small compared to the amount of l i q u i d i n each l a y e r . However, i n the case of the data i n Table X I I , the maximum t h e o r e t i c a l composition f o r each ingot 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 that t h i s assumption was reasonably v a l i d f o r the dendrite spacings i n v o l v e d i n these experiments. For Figures 53(a) and 54(a), which were s o l i d i f i e d under the same temperature 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 dendrite spacings, both the t h e o r e t i c a l and experimental r e s u l t s show more macrosegregation at TABLE X I I SOLIDIFICATION VARIABLES USED FOR THEORETICAL PLOTS F i g u r e Number 53(a) 54(a) 55(a) 56(a) Temperature G r a d i e n t (G) °C/cm 1.5 1.5 2.3 1.0 Growth Rate (R) cm/sec 0.0047 0.013 0.011 0.24 Lengt h o f C a s t i n g (H) cm 14 14 14 14 3 Number o f Channels (NC) = nr A 2.96 x 105 4.24 x 10 5 8.86 x 1 0 5 3.73 x 1 0 6 V i s c o s i t y o f the L i q u i d ( u ) p o i s e 0.03 0.03 0.03 0.03 Number of L a y e r s 28 28 28 29 AC ( t h e o r e t i c a l ) 1.95 0.39 0.31 0.002 AC ( e x p e r i m e n t a l ) 1.07 0.73 0.13 0.27 163 the lower growth r a t e . For Figures 54(a) and 55(a), which were s o l i d i f i e d at approximately 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 dendrite spacings, the theory p r e d i c t s s l i g h t l y more macrosegrega-t i o n at the lower temperature g r a d i e n t . This i s q u a l i t a t i v e l y i n agreement w i t h experiment. I t should be noted that although Figure 69 p r e d i c t s l e s s macrosegregation at lower temperature gradients (when a l l other 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. This a l s o corresponds w i t h the evidence c i t e d e a r l i e r that a r e d u c t i o n i n dendrite spacing e l i m i n a t e d f r e c k l e s i n consumable arc 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 Figure 56(a) shows h a r d l y any macroseg-r e g a t i o n , which would correspond w i t h the e a r l i e r suggestion that i t i s not p o s s i b l e to claim any s i g n i f i c a n t macrosegregation i n t h i s p a r t i c u l a r experiment, due to the large amount of s c a t t e r . Using 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 to recommend a number of changes i n c a s t i n g p r a c t i c e that would reduce g r a v i t y segregation e f f e c t s i n v e r t i c a l d i r e c t i o n a l c a s t i n g s : 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 to flow through the mushy zone. 2) Reduction of ingot height 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 force f o r flow. 3) Increasing 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 flow. 4) Decreasing the temperature gradient w i l l reduce the d r i v i n g force f o r flow. 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 flow rates 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 to c a l c u l a t e the p e r m e a b i l i t y of the d e n d r i t i c s t r u c t u r e , as defined by Darcy's Law - the standard e m p i r i c a l r e l a t i o n s h i p which describes flow through porous media. I t was found that the p e r m e a b i l i t y of a d e n d r i t i c array i s a s e n s i t i v e f u n c t i o n of the primary dendrite spacing. The permea-b i l i t y r e s u l t s were shown to be c o n s i s t e n t w i t h a simple model of the porous medium, which considers the i n t e r d e n d r i t i c channels to be equivalent to a bundle of c a p i l l a r y tubes. I t was shown that the i n t e r d e n d r i t i c l i q u i d flowed uniformly through the d e n d r i t i c a r r a y , without the formation of p r e f e r e n t i a l channels, by d i r e c t examination of the etched 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 techniques. Deviations from Darcy's Law, which occurred when the samples were h e l d above the e u t e c t i c temperature f o r long periods of time, were discussed i n r e l a t i o n to dendrite coarsening e f f e c t s , s i m i l a r to Ostwald r i p e n i n g , or s i n t e r i n g i n ceramics. Lead-tin a l l o y s were used to i n v e s t i g a t e the formation of channel-type c a s t i n g defects ( f r e c k l e s and A segregates). 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 were used to study pipe formation and so l u t e convection, caused by density 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 l i q u i d . Macrosegregation was observed i n ingots where the l i q u i d c l o s e 165 to the bottom of the s o l i d - l i q u i d zone was l e s s dense than the l i q u i d above, and the r e s u l t i n g p r o f i l e s were shown to be r e l a t e d to the growth r a t e , temperature g r a d i e n t , dendrite spacing, and a l l o y composition. Shrinkage t r a i l s and pipes were produced i n some of these experiments when the growth rates were very low. These f i n d i n g s support the p r e v i o u s l y proposed mechanism f o r the formation of channel-type d e f e c t s , based on density d i f f e r e n c e s i n the l i q u i d causing i n t e r d e n d r i t i c f l u i d flow. A numerical model i s proposed, which p r e d i c t s 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 c a s t i n g s , as a f u n c t i o n of the s o l i d i f i c a t i o n v a r i a b l e s . Density d i f f e r e n c e s i n the l i q u i d are taken to be the d r i v i n g f o r c e f o r macrosegregation, and the d e n d r i t i c s t r u c t u r e i s considered t o be a porous medium of v a r i a b l e p o r o s i t y . 8.2 Conclusions i ) Using the simple c a p i l l a r y model to describe 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 to the square of the primary d e n d r i t e spacing. i i ) The " t o r t u o s i t y f a c t o r " , which allows f o r the f a c t that 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 symmetrical i s equal to 4.6 i i i ) Castings h e l d f o r long periods of time above the e u t e c t i c temperature show dendrite coarsening e f f e c t s which 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 castings h e l d a few degrees above the e u t e c t i c temperature was observed t o increase due to t h i s e f f e c t . i v ) Macrosegregation and channel-type defects can be produced i n the l e a d - t i n system by s o l u t e convection causing upward flow 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 in i t s path, v) Using the relationship between permeability and structure determined from the interdendritic f l u i d flow measurements, the numerical macro-segregation model is qualitatively in agreement with the directional solidification experiments. The model can therefore be used to recommend changes in casting practice to reduce gravity segregation effects. 8.3 Suggestions for Future Work i) Using the interdendritic flow measurement technique developed in this work, permeabilities could be measured in other alloy systems, in particular, those systems where the dendrites do not have orthogonal branches. i i ) With suitable permeability data for high liquid fractions, the funda-mental nature of interdendritic f l u i d flow could be investigated further, leading to a better theory than the simple capillary model used in this work. i i i ) The effect of dendrite coarsening could be used as a method of modifying the cast structure. iv) Since density differences in the liquid have been shown to produce freckles in ammonium chloride-water models, and in lead-tin alloys, the next step would be to add radioactive tracers to commercial ingots which are prone to this defect. 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Symp. on E l e c t r o s l a g and other s p e c i a l m e l t i n g technology, A.S.M. and M e l l o n I n s t . , June 1971, Symposium Proceedings Part I I , p. 215. 34. S.M. Copley, A.F. Giamei, S.M. Johnson and F. Hornbecker: Met. Trans., 1970, v o l . 1, pp. 2193-2204. 35. C.E. Smeltzer: Iron Age, 1959, v o l . 184, No. 11, p. 188. 36. R. Mehrabian, M. Keane and M.C. Flemings: Met. Trans., 1970, v o l . 1, pp. 3238-41. 37. R.J. McDonald and J.D. Hunt: Trans. TMS-AIME, 1969, v o l . 245, pp. 1993-97. 38. J.R. Blank 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 of Me t a l s " , I . S . I . P u b l i c a t i o n 110, pp. 370-376. 170 39. H.P. Utech, W.S. Bower and J.G. E a r l y : " C r y s t a l Growth", Proceedings 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, Boston, June 1966, p. 201. 40. N.Streat and F. Weinberg: Met. Trans., 1972, v o l . 3, pp. 3181-84. 41. D.J. Hebditch and J.D. Hunt: Met. Trans., 1973, v o l . 4, pp. 2008-10. 42. H.R. Thresh, A.F. Crawley and D.W.G. White: Trans. TMS-AIME, 1968, v o l . 242, pp. 819-22. 43. J . K o h l , R.D. Zentner and H.R. Lukens: "Radioisotope A p p l i c a t i o n s Engineering", 1961, Van Nostrand and Co. 44. B. Prabhakar: M.A.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, 1973. 45. M.C. Flemings and G.E. Nereo: Trans. TMS-AIME, 1967, v o l . 239, pp. 1449-1461. 46. D.J. Hebditch and J.D. Hunt: Met. Trans., 1973, v o l . 4 , p. 2474. 47. T.F. Bower, H.D. Brody and M.C. Flemings: Trans. TMS-AIME, 1966, v o l . 236, pp. 624-34. 48. C.J. S m i t h e l l s : "Metals Reference Book", v o l . 2, 4th E d i t i o n , Butterworths, London, 1967. 49. E. S c h e i l : M e t a l l f o r s c h u n g , 1942, v o l . 20, p. 69. 50. J.S. K i r k a l d y and W.V. Y o u d e l i s : Trans. TMS-AIME, 1958, v o l . 58, p. 212. 51. 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 of Metals" , I . S . I . P u b l i c a t i o n 110, December 1967, p. 112. 52. M.C. Flemings, R. Mehrabian and G.E. Nereo: Trans. TMS-AIME, 1968, v o l . 242, pp. 41-49. 53. M.C. Flemings and G.E. Nereo: Trans. TMS-AIME, 1968, v o l . 242, pp. 50-55. 54. H.R. Thresh and A.F. Crawley: Met. Trans., 1970, pp. 1531-35. 55. M.E. Glicksman and C.L. Void: Acta Met., 1967, v o l . 15, pp. 1409-12. 56. M.E. Glicksman and C.L. Void: "The S o l i d i f i c a t i o n of 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. Glicksman and C.L. Void: J o u r n a l of C r y s t a l Growth, 1972, v o l . 13, pp. 73-77. 171 58. M.E. Glicksman and C.L. Void: Acta Met., 1969, v o l . 17, pp. 1-11. 59. M.E. Glicksman and C.L. Void: 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 of Binary A l l o y s " , Second E d i t i o n , 1958, McGraw-Hill, p. 302. 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 the flow c e l l , a f t e r a time t . Darcy's Law st a t e s : v = P L AP A . l where v = bulk v e l o c i t y K = p e r m e a b i l i t y AP = pressure drop across the porous medium u = v i s c o s i t y of the l i q u i d . L = length of the porous medium. During a b r i e f time i n t e r v a l d t , the q u a n t i t y which flows through the porous medium w i l l be dq, t h e r e f o r e : A dt A.2 173 where A = area of the porous medium. -KA Thus dq = ~ pgh tdt A. 3 where p = density of the l i q u i d g = g r a v i t y h = head at time t . I f the volume of l i q u i d which has r i s e n up the r i s e r pipe i s a2& (where a 2 i s the area of the r i s e r , and £ the len g t h r i s e n ) , an equal volume w i l l have f a l l e n i n the r e s e r v o i r above the bed. The distance f a l l e n w i l l t h e refore be a2^/a^, where a^ i s the area of the r e s e r v o i r . The distance h f c i s t h e r e f o r e given by: h. = h - (a-A/a.) - I t o i l i . e . h = h - £(1 + a./a.) A.4 t o 2 1 where h Q i s the o r i g i n a l head at t = 0. Thus f o r a s m a l l change i n the head: dh f c = - (1 + a 2/ a i)d£ A.5 The volume which flows up the r i s e r pipe during time dt i s dq, where dq = - a 0d£ 2 A. 6 S u b s t i t u t i n g i n Equation A.5 dh = (1 + a9/a.) t / 1 Since the q u a n t i t y f l o w i n g i n the r i s e r equals the q u a n t i t y fl o w i n g through the porous medium, Equation A.7 can be combined w i t h Equation A.3: -KApgh t A 2 d h t dt UL (1 + a.^/a.^) i . e . dt = - (c/K)dh /h A.8 where c = a ^ L / ( 1 + a 2/a 1)Apg I n t e g r a t i n g : h t dt = - (c/K) | d h t / h t h o t = - (c/K) In (h /h ). t o 175 APPENDIX II 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.LI/'II'-'/.f/'MH. 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 LO(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 LI(I)*LN (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.) LN(1-HO* 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 AL8AHZSY/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 112 177 C DO P-TEST, AND ITERATE TO FIHD THE MAXIMUM NUMBER OF POINTS WHICH C 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 J J - J J M 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, • INTERCEPT3*',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,0 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 INTERCEPT31' , 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. INTERVAL3',E16.7//1X,'TIME BETWEEN MELTING AND ZERO POINT 0 •F PLOW MEASUREMENTS(TO)=•,F7.2,'SEC.•,2X,'951 CONF. INTERVAL=•,P7. 178 •3) BR ITE (8) (ALH(I) ,TT(I) ,1=1,N) WRITE (6,107) 107 FORMAT{////IX,'DATA FOR HON LINEAR LEAST SQUARES FITTING'//1X,6X, • •T-TO',10X,•ALH') WRITE(6,108) (I,TT(I) ,ALH (I) ,1=1,N) 108 FORHAT (1I,I3,F7.0,2X,E16.7) WRITE(6,109) 109 FORMAT (//1X,120(**')) WRITE(7,132)A,N,J,R,P(2) ,ETO,P(1) ,EK1 GO TO 9 0 0 131 FORMAT(<*5X,'SUMMARY OF TEST RESULTS'/20X,* N',6 X,' J•,7X,• R',81,•TO* •,«X,'ERROR',8X,»K1 ',8X,'ERROR') 132 FORMAT («AU,2X,I3#<*X,I3,UX,F6.ft,ftX,P5.0,2X,P7.3f E13. <*, E12. U) 901 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 APPENDIX I I I 179 THE SOLIDIFICATION OF Pb-20%Sn - A TABLE OF SOLIDIFICATION VARIABLES Values of the p a r t i t i o n r a t i o k Q , and the l i q u i d composition C^, were obtained as a f u n c t i o n of temperature from the phase diagram. The s o l i d composition C and the weight f r a c t i o n l i q u i d f were c a l c u l a t e d using the S Li Pfann equation, as described i n s e c t i o n 7.2. The volume f r a c t i o n of (42) l i q u i d was c a l c u l a t e d using the den s i t y data f o r Pb-Sn a l l o y s T°C k0 C L C s f L g L 276.0 0.501 20.005 0.000 1.000 1.000 275.0 0.497 20.488 10.107 0.953 0.956 274.0 0.493 20.974 10.184 0.910 0.915 273.0 0.490 21.461 10.259 0.870 0.877 272.0 0.486 21.951 10.332 0.833 0.841 271.0 0.482 22.442 10.402 0.798 0.808 270.0 0.478 22.935 10.470 0.765 0.776 269.0 0.474 23.430 10.535 0.734 0.747 268.0 0.471 23.926 10.598 0.706 0.720 267.0 0.467 24.424 10.659 0.679 0.694 266.0 0.463 24.923 10.718 0.654 0.670 265.0 0.459 25.423 10.776 0.630 0.647 264.0 0.456 25.925 10.831 0.608 0.626 263.0 0.452 26.427 10.884 0.587 0.606 262.0 0.449 26.931 10.936 0.567 0.587 261.0 0.445 27.436 10.987 0.548 0.569 260.0 0.442 27.941 11.035 0.531 0.552 259.0 0.438 28.447 11.082 0.514 0.535 258.0 0.435 28.954 11.128 0.498 0.520 257.0 0.432 29.462 11.173 0.483 0.506 256.0 0.428 29.970 11.216 0.469 0.492 255.0 0.425 30.478 11.258 0.455 0.488 254.0 0.422 30.986 11.298 0.442 0.475 253.0 0.419 31.495 11.338 0.430 0.464 252.0 0.416 32.004 11.376 0.418 0.452 251.0 0.413 32.513 11.414 0.407 0.436 250.0 0.410 33.022 11.450 0.397 0.426 249.0 0.407 33.531 11.486 0.386 0.416 248.0 0.404 34.039 11.520 0.377 0.407 247.0 0.402 34.548 11.554 0.368 0.398 246.0 0.399 35.055 11.586 0.359 0.389 245.0 0.396 35.563 11.618 0.350 0.381 244.0 0.394 36.070 11.650 0.342 0.373 243.0 0.391 36.576 11.680 0.334 0.365 242.0 0.389 37.081 11.710 0.327 0.358 241.0 0.387 37.586 11.739 0.320 0.351 240.0 0.384 38.089 11.767 0.313 0.344 239.0 0.382 38.592 11.795 0.306 0.338 238.0 0.380 39.093 11.822 0.300 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. C s f L 9L 180 '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 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 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 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. 184 181 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 DETER-MINED 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 201 FORMAT(F6.1,F6.2,F7.tt) 202 FORMAT (2F10.U) 203 FORMAT(F10. 1) 20U FORMAT(E9.3,F6.2,F6.3) 205 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 1 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 )=CL(I, 1) PRLIQ(I,1) = 1. SSOL (I)=0. SSOLUT(I)=0. WSLIQ(I)=CL(1,1) GL (I)=1. CS(I,1)=0. 183 11 CONTINUE TIHE-0. PL-1. WSOLID=0. WSOLUT-0. TSOLUT=CL(1,1) TWT=1. STATE=ALIQ TB=1 WRITE (6,101) TIRE 101 FORMAT (//IX,• TIME=',F7. 1, 'SECONDS -BEFORE FLUID FLOW•/) WRITE(6,102) 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) 103 FORMAT (1X,12,F6.1,5X,AO,1X.F8.5,IX,F6.3,12 (2X,F6.3)) 2 CONTINUE TIME=TIHE»DT 7 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) 501 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 304 302 IF (TEMP (I, 2) . LT. TE) GO TO 303 301 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 307 303 STATE=SOL AK(I,2)=1. FL=1„ WSOLID=0. WSOLUT=0. GO TO 308 307 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. 308 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; SOBROUTINB DENS (TEMP,C,DNSTY) C THIS SUBROUTINE CALCULATES THE DENSITY OP LIQUID LEAC-TIN ALLOYS C AS A PUNCTION OP COMPOSITION AND TEMPERATURE. THE DATA ARE TAKEN C FROM A TABLE OF VALUES PUBLISHED BY THRESH ET AL, TRANS. TMS-AIME C 1968,PAGE 819. INTERMEDIATE VALUES ARE OBTAINED BY A LINEAR C 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 101 100 CONTINUE 101 1=1-1 CC= (CP (1+ 1) -C) / (CP (1*1) -CP (I) ) AA=A (IO) • (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 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) C THIS SUBROUTINE CALCULATES THE NEW COMPOSITION PROFILE APTER EVERY C TIME INCREMENT. PERMEABILITY IS CALCULATED FROM EQUATION 7.7 IN THE C TEXT, AND THE SERIES OP SIMULTANEOUS DIFFERENTIAL EQUATIONS (7.8) C 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 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 550 CONTINUE 560 CONTINUE DO 570 1=1,L K=I IP (TEflP(I).GT.TE)GO TO 580 570 CONTINUE 580 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) 1 PBRH (I)=PERH (I) *GL (I) N=J-1 DO 21 I=K,N 21 HY (I) = (HY (I) + HY (I* 1) ) *0.5 DO 100 I=K,N AJK=PLOAT(J-K) 100 B(I)=(DNSTY(J)-DNSTY(K))*GR*AJK*R*DT DO 200 I=R,N 200 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) U »=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 10 CONTINUE 99 RETURN END 187 SUBROUTINE FUNC(X,Y,F) C THIS SUBROUTINE SETS OP THE DIFFERENTIAL EQUATIONS (7.8 IN THE C 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) 1 H=H*1 2 A2=QY (J-1)/VLIQ (J) F(LL) =»A2*Y (LL-1) -A2*Y(LL) RETURN END 188 APPENDIX V DIRECT OBSERVATION OF SOLIDIFICATION USING ELECTRON MICROSCOPY V . l I n t r o d u c t i o n The aim of t h i s work was to 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 of pure metals and a l l o y s , using e l e c t r o n microscopy. The method was e s s e n t i a l l y the same as that developed by Glicksman 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 of pure bi s m u t l i / " ^ ' " ' ^ , and a number of d i l u t e a l l o y s a n d used t h e i r observa-t i o n s to obt a i n 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 energy .. ,,(58,59) f o r pure bismuth The present work was f i r s t d i r e c t e d towards reproducing Glicksman's experiments on pure bismuth, and then using the technique, to observe s o l i d i f i c a t i o n i n other pure metals, and thereby c a l c u l a t e the 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 that s u f f i c i e n t e x p e r t i s e would be gained to observe the growth of a l a m e l l a r e u t e c t i c from the l i q u i d . V.2 Experimental Method The t h i n f i l m s , produced by various methods discussed below, were examined using a H i t a c h i HU-llA e l e c t r o n microscope (the microscope was the same as that used by Glicksman). Both a simple heating stage, and a heating-t i l t i n g stage were used, but i t was found that temperatures could not 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 using the l a t t e r stage, the r e f o r e the r e s u l t s only apply to work done w i t h the simple heating stage. 189 The specimen was heated sl o w l y using the heating stage, u n t i l a sm a l l molten zone was produced using the a d d i t i o n a l heat induced by focusing the lOOkV e l e c t r o n beam. Glicksman estimated that the optimum temperature of the specimen was about 10°C below the melti n g temperature, however, t h i s could not be a c c u r a t e l y determined i n the present work using the a v a i l a b l e equipment. The e l e c t r o n beam simultaneously provided image i l l u m i n a t i o n and l o c a l heating to produce the molten zone. I t was found that the molten zone could be made to expand and contract by a d j u s t i n g the current to the second condenser l e n s . In a l l experiments, one of the major problems was s t a b i l i t y of the molten zone. The a v a i l a b l e power supply d i d not provide 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 to ho l d the specimen at the required temperature f o r long p e r i o d s . For t h i s reason, i t was found that the best r e s u l t s were obtained by s e t t i n g the power supply to heat the specimen very s l o w l y . This u s u a l l y allowed about ten minutes f o r observation of s o l i d i f i c a t i o n and me l t i n g w h i l e the specimen was i n a s u i t a b l e temperature range. In con t r a s t to Glicksman's f i n d i n g s , i t was extremely d i f f i c u l t to ho l d the 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 during t h i s p e r i o d . Exposure times 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 during the course of the exposure. V.3 Results V.3.1 Pure bismuth Thin f i l m s were prepared by vacuum evaporation onto carbon support f i l m s using standard methods. A l l metals 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 oxygen-free n i t r o g e n before pumping, and a t i t a n i u m g e t t e r was used before evaporation. The vacuum before evaporation was 2.0 x 10 ^ Torr. The thickness of the carbon f i l m s was about and the bismuth thickness was i n the range 1000-2000A1 (measured using an i n t e r f e r e n c e microscope). T y p i c a l r e s u l t s are shown i n Figures 71 and 72. Figure 71(a-c) shows f r e e z i n g , followed by m e l t i n g , followed by f r e e z i n g i n the same region. Figure 71(d) shows the s o l i d - l i q u i d i n t e r f a c e at higher m a g n i f i c a t i o n . Faceted growth of the s o l i d i s seen i n the lower right-hand corner of Figures 71(a) and ( c ) , s i m i l a r to that seen by Glicksman. The double image of the l a r g e g r a i n s i n Figures 71(b) and (d) i s due to the i n s t a b i l i t y of the molten zone. The s m a l l g r a i n s i n the corners show the o r i g i n a l s t r u c t u r e of the evaporated f i l m . The l i q u i d regions appear uniformly 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 patch, which appears to be growing i n the centre of the l i q u i d zone, i s caused by t h i n n i n g of the f i l m i n t h i s region. E v e n t u a l l y t h i s would lea d to de-wetting and the l i q u i d would draw back i n t o globules around a c e n t r a l hole. When t h i s occurred, the 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 to penetrate, and the i n t e r -face could no longer be observed. The advantage of using bismuth f o r i n t e r f a c i a l energy measurements i s that i t tends to deposit from the vapour phase w i t h the b a s a l plane p a r a l l e l to the plane of the specimen. Thus the boundary between neighbour-i n g grain? i n the t h i n f i l m i s u s u a l l y a simple 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 bismuth, showing evidence of faceted growth. M a g n i f i c a t i o n 5000x. (d) Enlarged view of the s o l i d - l i q u i d i n t e r f a c e , showing high angle g r a i n boundaries emerging at 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 Glicksman was to search f o r low-angle t i l t boundaries which emerged at the s o l i d - l i q u i d i n t e r f a c e . The angle of t i l t could be measured by t a k i n g a s e l e c t e d area d i f f r a c t i o n p a t t e r n across the boundary, or by counting the d i s l o c a t i o n spacing along boundaries w i t h very low t i l t angles. 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 of the cusp angle, where the g r a i n boundary emerged at the 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 present work to repeat these measurements on pure bismuth. A l l the bismuth samples observed i n the present work showed dark speckles over the f i e l d of view. These speckles were not seen before h e a t i n g , but seemed to form when the specimens came close to the m e l t i n g p o i n t . They are seen i n Figure 71, and are even more pronounced i n Figure 72. They a l s o appear i n photographs p u b l i s h e d by Glicksman, but are not as common as i n the present work. Although o x i d a t i o n was suspected i n both the previous and present work, no oxide rings were observed i n the d i f f r a c t i o n p a t t e r n s . The nature of the speckles t h e r e f o r e remains unknown. One can speculate that they might be due to some i n t e r a c t i o n between the bismuth and the carbon substrate,, s i n c e l i q u i d bismuth can d i s s o l v e minute amounts of carbon (0.0028 atomic percent at 300°C), which i t r e j e c t s as gra p h i t e c r y s t a l s on s o l i d i f i c a t i o n ^ ^ . Therefore, the speckles might be c r y s t a l l i t e s of gr a p h i t e , which one could not d i s t i n g u i s h from the substrate by e l e c t r o n d i f f r a c t i o n . The speckles i n Figure 72(c) appear to p i n the i n t e r f a c e , which has a rougher contour than i n Figure 71. I t i s a l s o p o s s i b l e that the speckles may be r e l a t e d to some k i n d of contamination. The " i s l a n d s " of 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 high concentration of "speckles" and the i n t e r f a c e pinning e f f e c t i n (b) and (d). The " i s l a n d s " of s o l i d i n (a) and (c) resemble photographs of mel t i n g pub-l i s h e d by G l i c k s m a n , and are probably caused by contam-i n a t i o n of the metal f i l m . M a g n i f i c a t i o n 5000x. 194 s o l i d which remain a f t e r m e l t i n g (Figures 72(a) and ( c ) ) , are s i m i l a r to the photographs of Bi-Sn a l l o y m e l t i n g published by Glicksman, and they can be considered p a r t of a "mushy" zone. This would i n d i c a t e that contamination which causes some a l l o y i n g occurred i n the specimen shown i n Figure 72. V.3.2 Other pure metals ( 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 of t i n and aluminum were produced by vacuum evaporation onto carbon support f i l m s , but i t was not p o s s i b l e to produce oxide - f r e e f i l m s of indium. Both t i n and aluminum behaved i n the manner shown i n the sequence pf photographs shown i n Figure 73. During h e a t i n g , g r a i n growth would be observed (Figures 73(b) and ( c ) ) , followed by m e l t i n g (Figure 73(d)) and immediate de-wetting (Figures 73(e) and ( f ) ) . These photographs were taken using a 35 mm camera to photograph the f l u o r e s c e n t screen, because the usual techniques were too slow to record the r a p i d events which occurred. The g r a i n s i z e of the screen, plus the use of a f a s t f i l m account f o r the poor q u a l i t y . The high thermal 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 keeping the molten pool l o c a l i z e d . Since s t a b l e molten zones could not be produced i n pure metals other than bismuth, 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 e u t e c t i c s Three techniques f o r producing a l l o y f i l m s were attempted. These were evaporation of the two c o n s t i t u e n t pure metals, microtoming, and e l e c t r o -l y t i c t h i n n i n g . The work was concentrated 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 not observed. 195 FIGURE 73: The melting of pure aluminum, photographed from the f l u o r e s c e n t screen using a 35 mm camera; (a) o r i g i n a l vapour deposited s t r u c t u r e ; (b) and (c) g r a i n growth (approx. 450°C); (d) and (e) beginning of m e l t i n g and de-wetting; (f) t o t a l de-wetting. M a g n i f i c a t i o n approx. 5000x.