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Interdendritic fluid flow 1974

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INTERDENDRITIC FLUID FLOW by NORMAN STREAT B.Sc.(Eng.), University of London, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 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 ŵ . 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<z<1, the logarithm may be expanded 2 3 1 fi / _. * \ \ / • r \ (z + 6z) (z + 6z) l n i l - (z + 6z) } = (z + 6z) - ^ — - 3 — z 2 6z 2 z 3 = z + 6z - — - z6z - — - — S i m i l a r l y l n ( l - yl) = l n { l - z} 2 3 and l n { l - z } = z - ~ y But e ± = l n { l - (z + <5z)} - l n { l - z} 2 <5z i s small, therefore one may neglect 6z and higher order terms. Also, 2 since -1<z<1, z &z and higher order terms w i l l be small, therefore as a f i r s t approximation e. - &z - z6z x i . e . e = (1 - z)6z When z = 0 , l n { l - (z + 6z)} - l n { l - z} = l n ( l - 6z) = 6z „ j . . . e r r o r at z Defxnxng w. = w i error at z = 0 (1 - z)6z i 6z i . e . w i = (1 ~ z) The standard e r r o r of Y, which estimates the "goodness of f i t " of the logarithmic p l o t i s ZCw^..)2 n-2 48 The value of has been calculated for the constant velocity data from the test in Section 3.7. It represents the "goodness of f i t " one would expect i f one were to do a least squares f i t on this data, and the only errors were random experimental errors. This value may now be compared with the value of Oy for a given number of points from a flow c e l l experiment. Using a standard test of significance, the F test, one can decide whether the observed scatter for the flow c e l l data is larger than one would expect i f i t were due to experimental errors alone. The method used to calculate the best straight line i s to start with the f i r s t six points from the flow c e l l data and f i t a straight line on the ln(h t/h Q) versus time graph using the method of least squares, and calculate Oy and the variance (equal to a ^ ) . Using the F test with a significance level of 0.05, one can say whether the scatter i s greater than one would expect from random experimental errors alone. If the scatter is less, seven points would be taken, and a new straight line fitted and a new Oy calculated. This procedure would be repeated with eight points etc. until the scatter is greater than expected for this significance level. In other words, this approach finds the largest number of points which contribute to the straight line portion of the graph within the experimental error associated with the technique. The slope of this line i s considered to give the best estimate of the i n i t i a l permeability. One can also say with certainty that the deviations seen (such as those shown in Figure 17) are due to effects other than random errors, or weighted errors associated with a logarithmic plot. Because of the large amount of data associated with each flow test, the calculations were done on a d i g i t a l computer which also provided 49 plots showing the best f i t line according to the above method. The FORTRAN program for processing the results is given in Appendix II. 4.2.2 Results The results of the permeability calculations are given in Table V. Deviations from Darcy's Law are considered to be positive when the values of ln(h t/h Q) are larger than would be expected, as in Figure 17(a), and negative for the reverse (Figure 17(b)). This means that for a positive deviation the flow rate becomes slower than predicted by Darcy's Law, indicating that the flow is becoming impeded, and for a negative deviation the flow rate is more rapid, indicating that the flow channels are possibly becoming larger. Those experiments where the deviation is listed as zero in Table V are either early experiments which, because they were less accurate, were not used to study deviations, or the resulting plot did not show a clear trend in either direction. From the table, one can see that positive deviations only occur when the dendrite spacing i s greater than 71 microns, and negative deviations only when the spacing i s less than 51 microns. The height of liquid in the riser pipe when deviations begin is given in the table, and i t can be seen that negative deviations begin in a range 4.5 to 12.7 mm, whereas positive deviations begin in a range 9.1 to 22.8 mm. In general, one can conclude that positive deviations start later than negative deviations (also seen in the two examples in Figure 17). The drop in the liquid level in the reservoir when deviations begin has also been calculated and is given in Table V. The calculations TABLE V RESULTS OF FLOW MEASUREMENTS Average Dendrite Structure I n i t i a l Tracer Deviations Height of Distance f a l - Total Spacing (ym) Permea- Used? from liquid in len in reser- time at Primary Secon- b i l i t y q Darcy's Lav riser when voir when temp. dary (K x 10 (+ or -) deviations deviations (hrs) 2. cm ) begin (mm) begin (mm) 28 23 COLUMNAR 0.199 (R) 0 1.07 0.239 (R) 0 1.08 " 0.105 - 7.3 2.4 2.52 " " 0.152 - 7.3 2.9 2.13 " 0.156 - 8.3 2.7 1.67 0.136 - 7.6 3.2 2.07 48 21 EQUIAXED 0.0569 (R) - 8.1 1.9 3.81 51 33 COLUMNAR 0.367 _ 12.7 3.4 1.18 II II II 0.346 0 0.72 II II II 0.244 - 4.5 1.9 1.32 II II II 0.407 - 8.5 2.7 1.22 II II II 0.436 (R) 0 0.74 71 49 COLUMNAR 0.499 (R) 0 0.67 " 1.47 * (R) + 17.3 3.9 0.44 0.821 (R) + 12.2 3.4 0.63 II 1.27 * (R) + 17.8 4.0 0.53 " 1.17 * (R) + 15.7 3.8 0.54 1.49 * (R) + 13.2 3.7 0.44 77 - EQUIAXED 0.300 (R) 0 1.86 80 54 EQUIAXED 0.618 (R) + 9.1 2.9 1.19 TABLE CONTINUED TABLE V CONTINUED Average Dendrite Structure I n i t i a l Tracer Deviations Height of ;Distance f a l - Total Spacing (pm) Permea- Used? from liquid in len in reser- time at Primary Secon- b i l i t y Darcy's Law riser when ; voir when temp. dary (K x 10 (+ or -) deviations deviations (hrs) 2 cm ) begin (mm) begin (mm) 83 57 COLUMNAR 0.546 (R) 0 0.62 it II II 1.06 (R) 0 0.52 n II ti 0.780 (R) 0 0.43 ii II it 0.543 0 0.57 ii II II 1.21 * (R) 0 0.41 103 51 EQUIAXED 0.820 (R) + 11.7 3.1 0.60 116 57 COLUMNAR 2.15 (R) + 19.9 4.5 0.29 130 61 EQUIAXED 1.95 (R) + 20.0 4.4 0.32 175 83 COLUMNAR 6.27 (R) + 22.8 4.4 0.15 n II ti 8.20 (R) + 19.1 3.8 0.12 Autoradiography showed evidence of flow channelling, therefore these results are considered less reliable. (R) Radioactive tracer was added to the reservoir (casting B) in these tests. 52 were based on the f i n a l height of the Pb-Sn alloy after testing, there- fore they do not necessarily correspond to the height of liquid in the riser pipe. This i s because despite careful machining to f i t the Pb-Sn castings to the flow c e l l , small spaces inevitably remained, and these were f i l l e d by the liquid before any flow was detected in the riser pipe. For this reason the curves in Figure 16 and 17 do not pass through t = 0. Negative deviations begin when the liquid has fallen between 1.9 and 3.4 mm, and positive deviations between 2.9 and 4.5 mm. The i n i t i a l height of liquid in the reservoir was calculated as 8.4 mm. 4.3 Dendrite Spacings and Structure The relationship between primary dendrite spacing and distance from the c h i l l was determined for columnar castings produced by the four different quenches described in Table II. The results are given in Figure 18, and they show that the dependence on distance is essentially linear, in agreement with previously published work on unidirectionally cast copper alloys j_ n the latter work a linear dependence was also found for secondary arm spacings versus distance from the c h i l l . Consequent- ly, the average spacings could be determined non-destructively by taking the mean value for the top and bottom surfaces of the casting. Throughout this work, average primary and secondary spacings have been used as the parameters which characterize the structure of the casting with respect to interdendritic f l u i d flow behaviour. In these experiments, the bulk flow is one dimensional, and in the case of columnar castings, the flow is in the same direction as the primary dendrites. The columnar castings are placed in the flow c e l l so that the interdendritic channels 5 3 FIGURE 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 . 54 vary in size only in the direction of flow, parallel to the axis of the casting. Therefore, visualizing the columnar dendritic structure as a stack of resistances in series, the flow rate w i l l be a function of the sum of these resistances, or alternatively, a function of the average resistance. This reasoning would not necessarily hold for the equiaxed castings, nevertheless, the average spacings have been used because the spacing measured in different locations on the top and bottom surfaces was fai r l y uniform, and the difference between measurements on the two surfaces was relatively small. 4.3.1 Autoradiography Autoradiography made i t possible to examine the Pb-Sn alloy samples after testing to determine whether the measurements which had been made were truly representative of uniform flow through the casting A. Previous experiments on interdendritic f l u i d flow by Kaempffer^^ showed that there is a strong tendency for superheated liquid to form preferential channels by dissolving dendrite branches, and i f this occurred to the same extent in the present work i t would invalidate the use of Darcy's Law. Those samples which contained radiocative tracer were sectioned at the mid-point of the cylinder A perpendicular to the axis, and the upper half was then sectioned parallel to the axis. Autoradiographs were made from the cross sections and longitudinal sections (examples are shown in Figure 19). In the majority of cases there was either uniform (Figure 19(a)) or no dark- ening of the film (Figure 19(b)) for the cross sections^except for five of the tast3 (marked with an asterisk in Table V). The cross section autoradio- FIGURE 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. M a g n i f i c a t i o n 2x. 56 graphs of these tests showed a dark spot (Figure 19(c)), and longitudinal sections showed a non-uniform penetration of tracer which indicated that flow had taken place preferentially in this region. Longitudinal sections for the majority of samples showed a relatively uniform penetration of radioactive material, and for large dendrite spacings the structure was revealed. For the smaller spacings the structure was not clearly revealed, probably because i t was too fine to be resolved. Figures 20 and 21 show autoradiographs from cross sections taken at various levels for two of the samples. Figure 20 shows uniform penetra- tion of tracer, and microexamination showed no evidence of pipes of the type seen by Kaempffer. For the sample shown in Figure 21 the tracer penetration is non-uniform, and i t is also evident that radioactive material has pene- trated much further down casting A. Microexamination revealed several small pipes close to the top which extended approximately 0.7 cm down. Enlarged views of one of the pipes are also shown in Figure 21 for the levels on which they were seen. The position of the dark patches on the autoradio- graphs from lower sections showed that flow had taken place preferentially down the pipes, even though the pipes themselves were no longer visible on the lower sections. From this evidence i t was fel t that a single cross section and longitudinal section would provide sufficient information to determine whether the flow was uniform or not. Although pipes were seen close to the top of the cylinder A in the tests which showed non-uniform flow, channels which penetrated right through the dendritic region and caused catastrophic flow, as reported by Kaempffer, were never observed. Although i t is f e l t that the permeabilities for the five tests marked with asterisks in Table V are less reliable than the others, 7 FIGURE 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 . Tracer has penetrated uniformly as f a r as the t h i r d s e c t i o n , 4.6 mm from the top. ~ = 28 um. M a g n i f i c a t i o n 1.5x. FIGURE 21: An example of an unre l i a b l e flow t e s t , showing uneven pene- t r a t i o n of t r a c e r . \ = 83 ym. M a g n i f i c a t i o n of autoradiographs 1.5x. M i c r o s t r u c t u r e s from the bottom r i g h t hand corner of the top two s e c t i o n s , showing a pipe. Lower s e c t i o n s d i d not show t h i s d e f e c t . M a g n i f i c a t i o n 33x. 59 the r e s u l t s have nevertheless been i n c l u d e d because i t i s of i n t e r e s t that they are l a r g e r than the more r e l i a b l e values by a f a c t o r of between only one and two, f o r the same d e n d r i t i c spacing. I t i s not completely c l e a r why pipes should form i n a few of the t e s t s when the same procedure was used throughout. One can suggest that the flow c e l l was not p o s i t i o n e d c o r r e c t l y i n the furnace f o r these t e s t s , which r e s u l t e d i n higher temperatures at the top. Since the samples were 2.46 cm i n diameter, quenching r a t e s were not as r a p i d as i n some previous experiments where t r a c e r was used to observe (21 22) convective flow patterns ' , ther e f o r e one might question whether quenching the flow c e l l a f f e c t e d the observed flow behaviour. In view of the f o l l o w i n g reasons, the autoradiographs are b e l i e v e d t o be t r u l y r e p r e - s e n t a t i v e of the nature of the flow behaviour (uniform or v i a p r e f e r e n t i a l channels): 1) Both water quenching and a i r c o o l i n g were used, and despi t e the d i f f e r e n t c o o l i n g r a t e s , there were no obvious d i f f e r e n c e s i n flow p a t t e r n s between s i m i l a r samples cooled i n d i f f e r e n t ways. 2) The p o s i t i o n of the c a s t i n g A w i t h i n the flow c e l l i s o f f - c e n t r e (Figure 9 ) , and would cause i t to c o o l more r a p i d l y on one si d e than the other, yet no patterns were observed which could be a t t r i b u t e d to t h i s e f f e c t . 4.4 Microexamination Microexamination of the Pb-Sn castings was d i r e c t e d towards f i n d i n g the reasons f o r the d e v i a t i o n s from Darcy's Law which emerged from the flow c a l c u l a t i o n s . L o n g i t u d i n a l and cross s e c t i o n s from the centre 60 region of casting A before and after testing are shown in Figures 22-25. 4.4.1 Negative deviations from Darcy's Law For the largest dendrite spacing, 175 ym (Figure 22), there is no apparent difference between the structures before and after testing, however, for the smaller spacings, 71, 51 and 28 ym, the differences become more obvious as the spacing decreases. The interdendritic regions appear to coalesce to some extent to form a continuous network, and there is l i t t l e difference in appearance between the cross sections and the longitudinal sections. The microstructures taken after testing also show an overall background of white dots which resemble spheroidized precipitates in other systems. These white dots exist on a much finer scale in the microstructures taken before testing, and are attributed to the formation of a dendritic (23) substructure, described in the literature The microstructures therefore indicate that a ripening mechanism is taking place, especially for the smallest dendrite spacings (which were held at temperature for the longest times), which increases the effective diameter of the flow channels with time. In Figure 25 the dendritic structure has changed to a type of cellular structure, and some flow channels appear to have grown at the expense of others. The flow paths also appear less tortuous, which would cause the flow velocity to increase. Negative deviations from Darcy's Law, which were observed to occur only when the spacing was less than 51 ym, are therefore attributed to this ripening mechanism. 61 L o n g i t u d i n a l Sections before flow a f t e r flow Cross Sections before flow a f t e r flow FIGURE 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 , t = 175 um, time at temperature 0.12 hours. M a g n i f i c a t i o n 60x. 62 Cross Sections before flow a f t e r flow FIGURE 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 . }, = 71 um, time at temperature 0.44 hours. M a g n i f i c a t i o n 60x. 63 L o n g i t u d i n a l Sections before flow a f t e r flow Cross Sections before flow a f t e r flow FIGURE 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, time at temperature 1.18 hours, M a g n i f i c a t i o n 60x. 64 L o n g i t u d i n a l Sections before flow a f t e r flow Cross Sections before flow a f t e r flow FIGURE 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 . A = 28 um, time at temperature 1.67 hours, M a g n i f i c a t i o n 60x. 65 4,4.2 Positive deviations from Darcy's Law To account for positive deviations from Darcy's Law the micro- structures were examined for evidence that the flow channels for the larger dendrite spacings were becoming more constricted with time. However, no evidence of this type was found. As in Figure 22, these castings were held at temperature for a relatively short time, and showed l i t t l e change after- wards. Microexamination of the top reservoir, which was originally casting B, revealed a structure which probably accounts for the positive deviations. Three examples of this structure are shown in Figure 26. The microstructures are from longitudinal sections through the reservoir, with the casting A at the bottom. It can be seen in Figures 26(a) and (b) that the material i n the reservoir has separated into two layers, the lower layer containing spherical precipitates of high lead content, and the layer above in one case has a fine dendritic structure (Figure 26(b), water quenched) and in the other a coarser dendritic structure (Figure 26(a), air cooled). The structure of the upper layer is therefore related to the cooling conditions, but the spherical precipitates are not. The latter material probably consists of dendrites of the primary phase which, owing to their high lead content, f e l l to the bottom of the reservoir and spheroidized as the alloy was held at temperature. Although the testing temperature was set equal to the liquidus temperature of casting B, i t appears that the dissolution of dendrites i s time dependent at this temperature. It also appears as though an Ostwald ripening mechanism is causing the average particle size to increase with time. Other workers have recently reported seeing the same type of precipitate i n a Pb-Sn alloy heated above i t s equilibrium liquidus 66 (a) A i r cooled, showing coarse d e n d r i t i c s t r u c t u r e i n the upper l a y e r . T o t a l time at temperature 1 .86 hours. (b) Water quenched, showing f i n e d e n d r i t i c s t r u c t u r e i n the upper l a y e r . T o t a l time at temperature 0.62 hours. (c) L i q u i d l e v e l i s equal to the top of the lower l a y e r . T o t a l time at temperature 0.15 hours. This corresponds to a drop i n l i q u i d l e v e l of 5.4 mm ( i n i t i a l height of r e s e r v o i r = 8.4 mm). FIGURE 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 . M a g n i f i c a t i o n 23x. 67 (24) temperature . I t does not appear, from the m i c r o s t r u c t u r e s , as though the l a y e r of s p h e r o i d i z e d dendrites would act as an appreciable b a r r i e r t o f l u i d f l o w , since the l i q u i d channels appear to be much l a r g e r than i n the c a s t i n g below. However, the presence of the l a y e r means that the l i q u i d l e v e l can only f a l l as f a r as the top of the l a y e r . I t i s t h e r e f o r e suggested that the flow r a t e becomes lower than p r e d i c t e d by Darcy's Law, because the l i q u i d l e v e l i n the r e s e r v o i r i s constrained by c a p i l l a r y e f f e c t s to f a l l at the same v e l o c i t y as the s p h e r i c a l p a r t i c l e s . This c a p i l l a r y e f f e c t would not be important when the concentration of p a r t i c l e s i s low, but would become i n c r e a s i n g l y important as the concentration i n c r e a s e s . Since i t was shown e a r l i e r that p o s i t i v e d e v i a t i o n s begin when the l i q u i d l e v e l i n the r e s e r v o i r has f a l l e n between 2.9 and 4.5 mm, t h i s would mean that c a p i l l a r y e f f e c t s s t a r t to i n f l u e n c e the flow r a t e when the r e s e r v o i r has f a l l e n to approxi- mately h a l f i t s o r i g i n a l h e i g ht. Using Stoke's Law f o r t e r m i n a l v e l o c i t y of a s p h e r i c a l p a r t i c l e , one can make a rough estimate of whether t h i s mechanism i s p o s s i b l e : v = 2 g r 2 ( p ' - P ) where v = t e r m i n a l v e l o c i t y of p a r t i c l e , radius r g = a c c e l e r a t i o n due to g r a v i t y u = v i s c o s i t y of l i q u i d p ' = d e n s i t y of p a r t i c l e p = d e n s i t y of l i q u i d . An estimated mean radius of s p h e r i c a l p a r t i c l e s i n Figure 26 i s 68 3 0.002 cm, and the density would be approximately 10.0 g/cm . Liquid 3 density and viscosity would be 8.33 g/cm and 0.03 poise. This gives a terminal velocity of 0.048 cm/sec, which is of the same order of magnitude as the flow velocity calculated in section 4.1.1. Ideally, i t should be possible to check this mechanism by quench- ing a sample when positive deviations begin, and then examining the structure of the reservoir. This is unfortunately impractical, since the point of deviation is not known unti l the data have been processed by computer. Therefore the sample which came closest to f u l f i l l i n g these conditions was examined, and i s shown in Figure 26(c). The drop in liquid level for this sample was 4.4 mm, and the microstructure represents a drop of 5.4 mm. The liquid level i s equal to the top of the precipitate layer, which supports the proposed mechanism. The density of precipitate particles appears to increase downwards, therefore the rate at which the liquid level f a l l s beyond the point where positive deviations begin may be related to a change in packing of the spheres, and dissolution effects, in addition to the terminal velocity for a single particle. 4.5 Permeability and Dendrite Spacing The permeabilities calculated in section 4.2 were plotted against both secondary and primary dendrite spacings (Figures 27 and 28). Vertical bars were used to show the scatter between repeated experiments on columnar castings produced under the same quenching conditions, and the data known to be less reliable (marked with asterisks in Table V) was not included. Since -9 the scatter ranges from 1.93 x 10 for a primary spacing of 175 ym, to _9 0.134 x 10 for a primary spacing of 28 ym, i.e., the scatter is a function 69 «r io'9- r- < LU or bJ Q_ IO-'°-i • columnar A equiaxed 10 T t — r T I I I I I I 100 SECONDARY DENDRITE SPACING (microns ± 10%) FIGURE 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 I0"8' ^ 10 I- < LiJ or LvJ Q_ I0-'°H slope = 2 • columnar A equiaxed T T 1—I I I I I I 1 10 100 PRIMARY DENDRITE SPACING (microns ± 1 0 % ) FIGURE 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 71 of the magnitude of K, log-log plots have been chosen as the best method of representing this data. Figure 27 shows that at the higher secondary arm spacings the permeability increases very rapidly for a small increase in spacing. In view of the stated accuracy with which spacings can be measured, i t is f e l t that primary dendrite spacing is therefore the more useful parameter for characterizing the structure in terms of the interdendritic fluid flow behaviour. Plotting permeability as a function of primary dendrite spacing, Figure 28 shows that values of permeability measured for columnar castings were slightly higher than those for equiaxed castings. This difference is probably not significant in view of the size of the error bars for the columnar castings. The permeability is clearly a sensitive function of the cast structure, and theoretical relationships can be calculated, based on models of the structure. The simplest model considers the porous medium (the casting) to be a bundle of straight, parallel, capillary tubes aligned in the direction of flow. 4.5.1 Straight Capillary Model Flow through a single, straight capillary tube of radius r can be described by the well known Hagen-Poiseulle equation 72 where q i s the flow r a t e along a tube of length L. For n c a p i l l a r i e s per u n i t area, the t o t a l flow r a t e per u n i t area ( i . e . v e l o c i t y ) i s 4 n i r r AP , . v = " r 4 - 4 Comparing t h i s equation w i t h Darcy's Law: v = - V 4.1 tiL Thus, by analogy K - ^ 4.5 I t i s common to add a " t o r t u o s i t y f a c t o r " t to t h i s expression to account f o r the f a c t that the flow paths are n e i t h e r s t r a i g h t nor symmetrical, thus: The length TL then represents the " e f f e c t i v e l e n g t h " of the flow channels. The l i q u i d f r a c t i o n g i s given by; , _ l i q u i d volume 'L t o t a l volume 2 m r r TLA 8 L ~ L A i . e . g L = nrrr x 4.7 2 4 g L From t h i s expression r = 2 2 2 n i T 73 4 Therefore, r e p l a c i n g r i n Equation (4.6) K = n 7 T 2 8T \ 2 2 2 n IT T 2 i . e . K = ^ 4- 8 8nir x (1 2) Piwonka ' demonstrated that p e r m e a b i l i t y was p r o p o r t i o n a l to the square of the f r a c t i o n l i q u i d f o r A l - 4.5%Cu which was c o n s i s t e n t w i t h the c a p i l l a r y model. Therefore, t a k i n g the model one step f u r t h e r , one may i n t u i t i v e l y set the number of c a p i l l a r i e s equal to the number of channels between primary dendrite s t a l k s . The spacing between channels equals the primary dendrite spacing A, th e r e f o r e 1 s L 2 * 2 and K = 4.9 8TTT Since the p e r m e a b i l i t y measurements i n Figure 28 were made at constant temperature, the volume f r a c t i o n of l i q u i d would be constant. Assuming the t o r t u o s i t y f a c t o r remains constant f o r the range of den d r i t e spacings s t u d i e d , the t h e o r e t i c a l r e l a t i o n s h i p would be the s t r a i g h t l i n e of slope 2 which has been drawn. In a d d i t i o n , t h i s l i n e goes through the point K = 0, A = 0; which cannot be represented i n Figure 28. This 2 can be checked by showing that the r a t i o K/A , i s constant f o r a l l p o i n t s along the l i n e . ^2 = 1.46 x 10~ 5 A 74 2 2 A Since x~ = • ̂ R from Equation (4.9) 3 8 L A Therefore, taking g L = 0.19 at 193°C T 3 = 99.0 i . e . x = 4.6 A t o r t u o s i t y factor of 4.6 implies that the " e f f e c t i v e length" of the channels i s 4.6 times longer than i f . t h e y were considered to be s t r a i g h t and p a r a l l e l . In most other p r a c t i c a l a p p lications of flow through f i l t e r beds etc., t o r t u o s i t y factors are usually considered to l i e between 1 and 2, the argument being that the average i n c l i n a t i o n of c a p i l l a r i e s around random i r r e g u l a r p a r t i c l e s i s about 45°, therefore the mean c a p i l l a r y length would be approximately /2L. However, since dendrite branches mesh together i n a highly regular manner, a t o r t u o s i t y value of 4.6 i s not unreasonable. Tortuosity has been introduced as a property equal to the average length of the flow path of a f l u i d p a r t i c l e , and attempts have been made i n the l i t e r a t u r e to measure t o r t u o s i t y by e l e c t r i c a l r e s i s t i v i t y measurements based on the concept that current would flow along the same paths as the f l u i d p a r t i c l e s . This work has been reviewed by S c h e i d e g g e r b u t no d i r e c t c o r r e l a t i o n between the e l e c t r i c a l and geometrical properties appears to have been shown. Scheidegger points out that the concept of t o r t u o s i t y i s therefore somewhat doubtful, yet i t i s c l e a r that the c a p i l l a r y model w i l l f i t any porous medium i f one adjusts the value of x appropriately. 75 4.5.2 Hydraulic Radius Theory: Other Theories (25) A more elaborate model of porous media developed by Kozeny attempts to relate the actual shape of the particles to the flow behaviour in a more systematic manner than by assigning a tortuosity factor. The theory is based on the observation that permeability, in absolute units, has the dimensions of a length squared, therefore, i t is argued, there should be a characteristic length which describes the permeability. This length is called the "hydraulic radius" (r ) and is defined as: n pore volume H wetted surface area Alternatively, the characteristic length may be expressed as the specific surface of particles (S ) where P g _ surface area of particle p volume of particle liquid volume since g T = _ \ \ z L total volume gL rH S (1 - g T) 4.10 p L. Using this approach, the flow velocity through the bed i s described by the Kozeny equation: 3 1 8 L 1 AP ... K" s / d - g T T u L P Li Therefore, from Equation 4.1 3 1 g L K = — —= - 5- 4.12 K- s^ a - g T r p " 76 K" i s g e n e r a l l y known as the Kozeny c o n s t a n t , and has the commonly ac c e p t e d v a l u e o f 5 f o r most porous media. T h i s approach i s f a m i l i a r i n m e t a l l u r g y , where i t i s used to d e s c r i b e the f l o w of gases and l i q u i d m e t a l i n the b l a s t f u r n a c e . To a p p l y the approach to i n t e r d e n d r i t i c f l u i d f l o w r e q u i r e s an e x p r e s s i o n f o r the s p e c i f i c s u r f a c e o f a d e n d r i t e , p r e f e r a b l y i n terms of the d e n d r i t e s p a c i n g . From E q u a t i o n (4.12): l K « S 2 P and from F i g u r e 28 2 K = A ( a p p r o x i m a t e l y ) t h e r e f o r e 1 P * There i s no a c c e p t e d method o f measuring the s p e c i f i c s u r f a c e of d e n d r i t e s t h a t the a u t h o r i s aware o f , and a l t h o u g h one c o u l d p o s t u l a t e a model which would s a t i s f y the above c o n d i t i o n , f o r example, by c o n s i d e r i n g p l a t e l i k e d e n d r i t e s , t h i s would not r e a l l y p r o v i d e more i n f o r m a t i o n t h a n the use of a t o r t u o s i t y f a c t o r i n the c a p i l l a r y model. In a d d i t i o n , the Kozeny t h e o r y i s g e n e r a l l y recommended o n l y f o r beds o f s m a l l p a r t i c l e s which are n e a r l y s p h e r i c a l i n shape. In p a r t i c u l a r , (25) d e v i a t i o n s o c c u r when the t h e o r y i s used f o r beds o f f i b r e s , which, i f a n y t h i n g , the d e n d r i t e s most c l o s e l y resemble, and a l s o v e r y h i g h and v e r y low v o i d f r a c t i o n s s h o u l d be a v o i d e d . F o r beds of spheres the densest p o s s i b l e p a c k i n g can g i v e a v o i d f r a c t i o n i n the range 0.2 to 0.3, t h e r e f o r e one would 77 expect poor agreement with the present work where the liquid fraction was held at 0.19. The only experimental evidence which is available to test the applicability of the Kozeny theory is Piwonka's data, discussed earlier in 3 section (3.1). His results have been plotted in the form of K versus g T /(1-j following Equation 4.12, in Figure 29, and i t can be seen that this plot does not correspond well to a straight line, compared to the previous plot of 2 K versus g L (Figure 7). Scheidegger^^ has given a comprehensive criticism of the Kozeny theory, drawing attention in particular to i t s ina b i l i t y to describe aniso- tropic permeability, and he reviews other theories which are more applicable to beds pf fibres, which come under the general heading of drag theories of permeability. Unfortunately, a basic assumption in the drag theory is.that the spacing between individual fibres i s large compared to the fibre diameters (high void fractions) and that flow disturbance due to adjacent fibres i s negligible. This would clearly not be applicable to the present studies on interdendritic f l u i d flow. Consequently, i t was fe l t that the simple capillary model provided a useful empirical approach for relating permeability to structure despite the limitation that i t does not give much information regarding the nature of flow on a microscopic scale. 78 79 4.6 Dendrite Coarsening Negative d e v i a t i o n s from Darcy's Law, which occurred when the primary dendrite spacings were l e s s than 51 ym, have been a t t r i b u t e d to m i c r o s t r u c t u r a l changes such as those shown i n Figures 24 and 25. S i m i l a r e f f e c t s have been reported by Kattamis et a l . on Al-Cu a l l o y s h e l d i n the s o l i d - l i q u i d region f o r various times. The authors observed an apparent increase i n the secondary dendrite arm spacing w i t h time, and they proposed two mathematical models f o r the d i s s o l u t i o n of dendrite arms, based on the d i f f e r e n c e i n s o l u b i l i t y between two surfaces of d i f f e r e n t curvature. The d r i v i n g f o r c e f o r the process was the reduction of surface area, and the rate of d i s s o l u t i o n was c o n t r o l l e d by d i f f u s i o n i n the l i q u i d . Both mathe- m a t i c a l models gave q u a l i t a t i v e agreement w i t h the observations, though i t was not p o s s i b l e to choose one model over the others. A s i m i l a r coarsening process i s b e l i e v e d to take place i n the Pb-Sn castings 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 , although Figures 23-25 appear to show s p h e r o i d i z a t i o n of the d e n d r i t i c s t r u c t u r e , r a t h e r than the d i s s o l u t i o n of secondary branches. This process i s r e l a t e d to the work done by Ostwald i n 1900, who observed an increase i n s o l u b i l i t y w i t h a decrease i n p a r t i c l e s i z e f o r various s a l t s o l u t i o n s i n water. La t e r workers who observed that large p a r t i c l e s tended to grow at the expense of smaller p a r t i c l e s named the process Ostwald r i p e n i n g . D i f f u s i o n c o n t r o l l e d growth of s p h e r i c a l p r e c i p - (27 28) i t a t e s was analyzed by Greenwood ' , and he derived the f o l l o w i n g expression f o r the growth rate w i t h respect to the radius of the p a r t i c l e ( r ) : 80 dr dt 2DS Va 0 0 kTr 4.13 where d i f f u s i o n c o e f f i c i e n t of s o l u t e Sro = s o l u b i l i t y of a p a r t i c l e of i n f i n i t e radius r = mean radius of the system of p a r t i c l e s V = molar volume of p a r t i c l e a = i n t e r f a c i a l energy k = Boltzmann's constant T = absolute temperature. From t h i s equation Greenwood noted that the maximum growth r a t e corresponds to the p a r t i c l e w i t h twice the mean r a d i u s . Extending h i s a n a l y s i s to a c o n s i d e r a t i o n of p a r t i c l e s i z e d i s t r i b u t i o n , he a r r i v e d at the f o l l o w i n g expression f o r the v a r i a t i o n of the mean p a r t i c l e radius w i t h time (t) Experimental evidence f o r a dependence of r on time has been found i n s t u d i e s of the growth of p a r t i c l e s i n a s t r a i n f r e e medium, u s u a l l y a l i q u i d , but good agreement has a l s o been found f o r c o b a l t p a r t i c l e s i n a copper ma t r i x , copper i n i r o n , manganese i n magnesium, and others reviewed ( 2 8} by Greenwood The growth rate derived by Kattamis et a l . i s e s s e n t i a l l y the same as Equation (4.13), except that i t i s based on the growth and d i s s o l u t i o n of c y l i n d r i c a l p a r t i c l e s r a t h e r than s p h e r i c a l ones. The model cannot r e a d i l y be extended to a c o n s i d e r a t i o n of p a r t i c l e s i z e d i s t r i b u t i o n , t h e r e f o r e no r e l a t i o n corresponding to Equation (4.14) has been der i v e d . As a f i r s t -3 8DS oVt r 9kT 4.14 8 1 approximation, t h e r e f o r e , the r e s u l t s of the Greenwood theory have been ap p l i e d to the present work. The me t a l l o g r a p h i c evidence, e s p e c i a l l y Figure 25, shows that the i n t e r d e n d r i t i c channels (pores) become more s p h e r i c a l , yet they must of course remain connected or flow would stop. Assuming the mean radius of i n t e r d e n d r i t i c pores i s a f u n c t i o n of time as derived i n Equation (4.14) r 3 « t From the c a p i l l a r y model of p e r m e a b i l i t y (Equation 4.6) K « r 4 i t f o l l o w s that K « t 4 / 3 4 ' 1 5 A d i f f e r e n t view of the coarsening of i n t e r d e n d r i t i c channels can be obtained by comparing the process to s i n t e r i n g i n ceramics. Considering two spheres of radius P i n contact (Figure 30), where the area of contact i s (29) a c i r c l e of radius x, i t can be shown that x 1/5 - 1 t P f o r d i f f u s i o n c o n t r o l l e d growth. The radius of curvature at the neck r , increases w i t h x: 2 x r = 4p" 2/5 Therefore the radius of curvature grows as a f u n c t i o n of t I f one p i c t u r e s the i n t e r d e n d r i t i c channels as sharp c r e v i c e s FIGURE 30: Growth of a neck during s i n t e r i n g . TABLE VI RESULTS OF DENDRITE COARSENING CALCULATIONS Primary Dendrite I n i t i a l Power Constant Spacing (pm) P e r m e a b i l i t y (K x 10 9cm 2) o (b) (m) 28 0.105 2.6 3.7 X i o - 1 7 ti 0.152 2.3 4.8 X i o - 1 6 ii 0.156 3.3 4.1 X i o " 1 9 II 0.136 2.6 6.0 X i o " 1 7 48 0.0569 2.7 4.4 X i o " 1 8 51 0.367 1.9 3.9 X i o " 1 4 it 0.244 1.8 1.1 X i o " 1 3 II 0.407 2.2 8.1 X i o " 1 5 From l e a s t squares f i t t i n g : e r r o r i n b, approximately 30% e r r o r i n m, approximately 65% 83 between secondary arms, one can assume that the e f f e c t i v e radius of the channels changes i n the same manner as the radius of curvature of the necked r e g i o n . Following the same argument as bef o r e : K oc r °c t 4.16 Using e i t h e r the Ostwald r i p e n i n g or the s i n t e r i n g approach, the r e l a t i o n between p e r m e a b i l i t y and time may be w r i t t e n as a power f u n c t i o n : K K + a t 1 o 4.17 where K q i s the p e r m e a b i l i t y at t = 0, and 'a' i s a constant. Consequently K q i s equal to the value of p e r m e a b i l i t y c a l c u l a t e d from the i n i t i a l slope of a curve such as Figure 17(b), and t = 0 i s defined as the time when l n ( h /h ) = 0. The d i f f e r e n t i a l form of Darcy's Law (Equation 8, t o Appendix I) may now be r e w r i t t e n f o r a v a r i a b l e K dh. dt = (K + a t p ) d t o -c (K + at*) t dh. -c 4.18 o (p+1) t P + 1 = -c In o j i . e . p+1 ,P+1 •c In oJ - K t o 4.19 P l o t t i n g t against the r i g h t hand s i d e of Equation (4.19) on a l o g - l o g s c a l e should give a s t r a i g h t l i n e of slope (p+1), and an example i s 84 shown i n Figure 31. A l t e r n a t i v e l y , one may use the method of l e a s t squares to f i t an expression of the form Y = mX*3 to the data where: X = t m = a/(P + 1 ) and b = P + 1. This has been done f o r the e i g h t t e s t s i n Table V which showed negative d e v i a t i o n s , and the r e s u l t s are l i s t e d i n Table VI. Following the Ostwald r i p e n i n g argument the power b should equal 2.33, and f o l l o w i n g the s i n t e r i n g argument b should equal 2.6. Within the accuracy of the e x p e r i - mental technique, the r e s u l t s show f a i r l y good agreement w i t h e i t h e r of the proposed mechanisms, though i t i s not p o s s i b l e to choose one over the other. The constant 'm' which has a l s o been c a l c u l a t e d i s a f u n c t i o n of a l a r g e number of parameters i n both mechanisms, i n c l u d i n g the d i f f u s i o n c o e f f i - c i e n t and i n t e r f a c i a l t e n s i o n , f o r which accurate data are not a v a i l a b l e . I t would a l s o probably vary w i t h d e n d r i t e spacing, but i n view of the range of values (6 orders of magnitude of a very small number), i t i s not p o s s i b l e to a t t a c h too much s i g n i f i c a n c e to t h i s constant. On the b a s i s of these r e s u l t s , i t appears that the change i n p e r m e a b i l i t y of the c a s t i n g when i t i s heated above the s o l i d u s temperature can be compared e i t h e r to Ostwald r i p e n i n g , or to the changes which take place during s i n t e r i n g . In e i t h e r case, d i f f u s i o n has been chosen as the mechanism by which m a t e r i a l i s removed from convex s o l i d regions and i s deposited on concave regions. This process would take place i n a l l f l u i d flow experiments, Y K t o FIGURE 3 1 : Dendrite coarsening p l o t f o r a sample w i t h K Q = 0.152 cm , average primary d e n d r i t e spacing 28 pm. oo 86 but i t would have the greatest effect in those tests where the casting was held in the solid-liquid region for the longest times. Conversely, positive deviations, which depend on the velocity with which particles f a l l in the reservoir, would probably have the greatest effect in those castings which had the highest flow rates. Microexamination of the reservoir from castings with the smallest dendrite spacings revealed that the liquid level had not fallen to the level of the particle layer, therefore positive deviation effects were considered negligible, and ignored in the calculation of the data in Table VI. 4.7 The Scatter of Permeability Results The results have shown that permeability measurements are affected by the dendrite spacing, dendrite coarsening effects, preferential flow due to the formation of pipes, and external effects i n the li q u i d reservoir above the casting. Other factors which could influence the results were variations in composition and temperature. To check whether composition changes had taken place within casting A during the f l u i d flow experiments, density measurements were made before and after testing on 10 of the samples. The machined cylinder of casting A was used for the f i r s t measurement (weighing in air and in water), and a cylindrical sample was machined after the test for the second measurement. Densities of standard specimens were measured and plotted to give the c a l i - bration curve in Figure 32. The densities could be measured to an accuracy of about ± 0.5% by this method, and within this error, no difference in the composition before and after the test could be detected. The effect of temperature variations was tested by making flow WEIGHT FRACTION Sn Ol 0.2 0.3 ATOMIC FRACTION Sn FIGURE 32: 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 3) at 25°C as a f u n c t i o n of composition. I 1 • — l 1 1 r Eutectic Temperature 183 °C 51 i , I i I 180 190 200 210 2 2 0 TEMPERATURE °C FIGURE 33: The r e l a t i o n s h i p between p e r m e a b i l i t y and temperature. 88 measurements on one sample at d i f f e r e n t temperatures. A f t e r s u f f i c i e n t data p o i n t s were recorded at one temperature the sample was r a p i d l y heated to a new temperature and another s e r i e s of p o i n t s was recorded. This technique was l i m i t e d by the tendency towards p r e f e r e n t i a l flow c h a n n e l l i n g (pipe formation) i f the i n t e r d e n d r i t i c l i q u i d became superheated, t h e r e f o r e i t was only p o s s i b l e to make measurements clo s e to the e u t e c t i c temperature. Subsequent m e t a l l o g r a p h i c examination of t h i s sample showed that changes i n the s t r u c t u r e of the c a s t i n g A were minimal and that the l i q u i d l e v e l i n the r e s e r v o i r had not f a l l e n to the top of the p a r t i c l e l a y e r , t h e r e f o r e p o s i t i v e and negative d e v i a t i o n s from Darcy's Law were considered to be n e g l i g i b l e . The p e r m e a b i l i t y was c a l c u l a t e d f o r each temperature, and the r e s u l t s , p l o t t e d i n Figure 33, show that the p e r m e a b i l i t y measurements are r e l a t i v e l y i n s e n s i t i v e to s m a l l changes i n temperature close to the e u t e c t i c . I t was f o r t h i s reason that the t o l e r a n c e of ± 3°C on the t e s t i n g temperature was considered acceptable. Using the c a p i l l a r y model described i n s e c t i o n 4.5 a t h e o r e t i c a l r e l a t i o n s h i p between p e r m e a b i l i t y and temperature may be c a l c u l a t e d , s i n c e : Using x = 4.6, A = 67 um, and g as a f u n c t i o n of temperature from Li the t a b l e i n Appendix I I I , the t h e o r e t i c a l curve has been p l o t t e d i n Figure 33 and a l s o shows that the p e r m e a b i l i t y would not be very s e n s i t i v e to temper- ature close to the e u t e c t i c . From the t h e o r e t i c a l l i n e , a v a r i a t i o n of 89 ± 3 C at 193 C would produce an e r r o r of about ± 5% i n the value of p e r m e a b i l i t y , which i s considerably s m a l l e r than the e r r o r bars i n Figure 28. Consequently composition and temperature v a r i a t i o n s are not thought to be the major f a c t o r s r e s p o n s i b l e f o r the s c a t t e r i n the r e s u l t s . Since the p e r m e a b i l i t y has been shown to be very s e n s i t i v e to s t r u c t u r e , the major f a c t o r i n f l u e n c i n g the s c a t t e r i n Figure 28 i s probably the u n c e r t a i n t y i n the measured value of the dendrite spacing. In a d d i t i o n , there would be e r r o r s i n v o l v e d i n d e s c r i b i n g the s t r u c t u r e of the equiaxed castings i n terms of an average value. However, i t i s f e l t t h a t , d e s p i t e the s c a t t e r , the r e s u l t s cover a s u f f i c i e n t l y wide range (approximately one order of magnitude i n primary dendrite spacing, and two orders of magnitude i n p e r m e a b i l i t y ) f o r the e m p i r i c a l r e l a t i o n s h i p to be v a l i d . 90 CHAPTER 5 THE EFFECT OF DENSITY DIFFERENCES ON THE FORMATION OF CHANNELS 5.1 I n t r o d u c t i o n and Review of Previous Work To apply the r e s u l t s of the i n t e r d e n d r i t i c f l u i d flow experiments to a p r a c t i c a l c a s t i n g problem, a study was conducted on the formation of channel-type d e f e c t s , namely, f r e c k l e s and A segregates. The term f r e c k l e s has been used f o r a number of d i f f e r e n t types of defects which are probably not a l l caused by the same mechanism. F r e c k l e s i n i r o n and n i c k e l base s u p e r a l l o y s can appear as d i s t i n c t t r a i l s of equiaxed grains on the surface of d i r e c t i o n a l l y s o l i d i f i e d c a s t i n g s . These f r e c k l e t r a i l s are observed to begin at some distance from the c h i l l f a c e , and the t o t a l number decreases w i t h distance from the c h i l l ^ 3 ^ . The t r a i l s are c a l l e d f r e c k l e s because of the speckled appearance of the equiaxed g r a i n s . Figure 34 shows examples of f r e c k l e l i n e s i n ingots of Mar-M200 (compositions of various s u p e r a l l o y s are given i n Table V I I ) . The composition of the m a t e r i a l i n the t r a i l s has been shown to be r i c h i n those s o l u t e elements which segregate normally, and t h i s , coupled w i t h the evidence of feeding shrinkage w i t h i n the t r a i l s (Figure 34(b)), leads to the con c l u s i o n that the f r e c k l e s represent the l a s t l i q u i d i n the system to s o l i d i f y . The photographs i n Figure 34 have been taken from the published work of Giamei and K e a r ^ 3 ^ . A second type, of f r e c k l e defect i s i l l u s t r a t e d i n Figure 35. In t h i s case, m a t e r i a l containing higher concentrations of the s o l u t e elements has accumulated i n patches or spots i n the i n t e r i o r of the c a s t i n g . These 9 1 FIGURE 34: F r e c k l e t r a i l s i n d i r e c t i o n a l l y s o l i d i f i e d Mar-M200 ; (a) 10 cm diameter ingot showing t r a i l s of equiaxed g r a i n s ; (b) 3.8 cm diameter - s i n g l e c r y s t a l showing feeding shrinkage along f r e c k l e l i n e . FIGURE 35: Fr e c k l e s i n as-cast Inconel 718, perpendicular to the ax i s 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 i n g ot. M a g n i f i c a t i o n lOx. TABLE VII COMPOSITION OF SUPERALLOYS^3°^ (wt. pet.) Cr A l T i W Mo Nb Ta V Mn S i B C Fe Co N i Mar-M200 (nominal) 9.5 5.0 2.0 12.5 - 1.0 - - - <0.2 0.015 0.15 - 10.0 b a l . I n c onel 718 19.0 0.6 0.9 - 3.0 (5.0 sum) - - - - 0.10 18.0 - b a l . A286 14.7 - 2.1 - 1.25 - - 0.30 1.5 0.70 0.005 - b a l . - 25.5 93 patches can be seen on a macroetched s u r f a c e , but they do not n e c e s s a r i l y form t r a i l s . Areas of l o c a l segregation of t h i s type appear to be a r e l a t i v e - l y common feature of some i r o n and n i c k e l base consumable arc melted i n g o t s . I t has been pointed out that defects of t h i s type can be d e t r i m e n t a l even (31) i n i n gots that are to be hot worked (DeVries and Mumau reported that they could not be removed even a f t e r 90% hot r e d u c t i o n ) . In a d d i t i o n to these two types, the term f r e c k l e s has been used f o r many c a s t i n g defects that have a "s p o t t y " appearance, e i t h e r on the surface or on a p o l i s h e d s e c t i o n , whether or not the spots are accumulations of s o l u t e or p o r o s i t y , and whether or not they a r i s e due to the s o l i d i f i c a - t i o n c h a r a c t e r i s t i c s of the a l l o y , or mould w a l l e f f e c t s . The present work was t h e r e f o r e r e s t r i c t e d to a study of the s o l i d i f i c a t i o n c o n d i t i o n s which could cause the formation of the f i r s t (channel) type of f r e c k l e . Two main explanations f o r the o r i g i n of channel-type f r e c k l e s (32) have appeared i n the l i t e r a t u r e . Gould , n o t i n g that f r e c k l e t r a i l s were r e l a t e d to the d i r e c t i o n of g r a v i t y , suggested that they might be formed by gas bubbles i n the l i q u i d . I f a bubble were nucleated at the s o l i d - l i q u i d i n t e r f a c e , i t would r i s e v e r t i c a l l y and i t s t r a c e would be f i l l e d by lower melting point l i q u i d . Gould rep o r t e d , however, that n e i t h e r varying the nit r o g e n and oxygen content, nor e l i m i n a t i n g hydrogen i n experiments on A-286 or Inconel 718, had any di s c e r n a b l e e f f e c t on the incidence of f r e c k l i n g . A more comprehensive study of the r e l a t i o n s h i p between blowholes and "spot segregates" ( i . e . , the second type of f r e c k l e s ) was made by (33) Mukherjee i n e l e c t r o s l a g i n g o t s . He found that the occurrence of spot 94 segregates could be i n f l u e n c e d by va r y i n g the oxygen content and by v i b r a t i o n , supporting the concept that f r e c k l e s may be caused by gas bubbles. He suggested a mechanism s i m i l a r to that of Gould, to account f o r f r e c k l e t r a i l s caused by r i s i n g bubbles. (34) Copley, Giamei e t a l . o f f e r e d an e x p l a n a t i o n based on the formation of lower d e n s i t y l i q u i d c l o s e to the bottom of the s o l i d - l i q u i d r egion during s o l i d i f i c a t i o n . I f the lowest m e l t i n g p o i n t i n t e r d e n d r i t i c l i q u i d i s l e s s dense than the bulk l i q u i d above, t h i s would cause upward flow of the l i g h t e r l i q u i d . As the l i q u i d r i s e s , i t would move towards h o t t e r regions of the c a s t i n g , becoming superheated. This would lead t o d i s s o l u t i o n of den d r i t e branches i n i t s path, and the formation of a channel. They examined the s o l i d i f i c a t i o n of a transparent ammonium chloride-water model i n which a " d e n s i t y i n v e r s i o n " of t h i s type occurred, and observed upward flo w i n g pipes through the s o l i d - l i q u i d r e g i o n , which supported t h e i r hypothesis. They a l s o presented a s e m i q u a n t i t a t i v e mathematical a n a l y s i s which considered the maximum d r i v i n g f o r c e f o r f r e c k l i n g to be the p o t e n t i a l energy d i f f e r e n c e between the unstable l i q u i d c o n f i g u r a t i o n , w i t h the most dense l i q u i d l a y e r on top, and a s t a b l e c o n f i g u r a t i o n , w i t h the most dense l a y e r at the bottom. The model d i d not consider 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 r e g i o n , but i t was p o s s i b l e to make q u a l i t a t i v e p r e d i c t i o n s on the e f f e c t of changing the c o o l i n g c o n d i t i o n s and changing the a l l o y composition. They explained the l o c a t i o n of f r e c k l e s on the surface of the ingot i n terms of the shape of the s o l i d - l i q u i d i n t e r f a c e , and by var y i n g t h i s shape i n the ammonium chloride-water model they produced e i t h e r i n t e r n a l or surface pipes. 95 The concept that d e n s i t y d i f f e r e n c e s cause f r e c k l e s i s c o n s i s t e n t (35) w i t h the observations of Smeltzer who noted that the defect could be (31) e l i m i n a t e d by changing the a l l o y composition, and DeVries and Mumau who reported that the accepted i n d u s t r i a l p r a c t i c e f o r reducing f r e c k l e s was to lower the power input to the consumable furnace. This would have the e f f e c t of changing the c o o l i n g c o n d i t i o n s . However, i t i s not p o s s i b l e to draw any a p r i o r i conclusions regarding the e f f e c t on the temperature gradient o r the f r e e z i n g r a t e . I t i s i n t e r e s t i n g to note that they a l s o reported that f r e c k l e s could be reduced by decreasing the dendrite spacing. Part of the extensive theory of macrosegregation pu b l i s h e d by (9) Mehrabian et a l . , which i s discussed i n more d e t a i l i n s e c t i o n 7.1, deals w i t h the formation of channel-type d e f e c t s . B a s i c a l l y , the theory i n v o l v e s the c a l c u l a t i o n of the i n t e r d e n d r i t i c f l u i d v e l o c i t y at every po i n t i n the s o l i d - l i q u i d region when the forces a c t i n g on the f l u i d are s o l i d i f i c a t i o n c o n t r a c t i o n s and g r a v i t y . They propose that the c r i t i c a l c o n d i t i o n f o r the formation of channel-type defects i s when the d i r e c t i o n of the i n t e r d e n d r i t i c f l u i d flow vector goes from the colder to the h o t t e r regions of the c a s t i n g . (34) This hypothesis i s s i m i l a r to that of Copley et a l . , s i n c e density, i n v e r s i o n s can lead to i n t e r d e n d r i t i c f l u i d flow i n the d i r e c t i o n of i n c r e a s i n g temperature, but i t d i f f e r s i n that the c r i t i c a l c o n d i t i o n i s not the s i g n of the density change, but the magnitude and d i r e c t i o n of the flow v e l o c i t y v e c t o r . Thus f r e c k l e s need not merely be upward t r a i l s , but they can go i n any d i r e c t i o n of i n c r e a s i n g temperature, depending on the 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 zone. The model permits c a l c u l a t i o n of the flow patterns i n an i d e a l i z e d 96 ingot which s o l i d i f i e s u n i d i r e c t i o n a l l y from one s i d e - w a l l , and they provide (36) experimental evidence of a channel-type defect i n a degassed Al-20%Cu ingot of t h i s type. The authors extend t h e i r theory to commercial ingots by suggesting a number of assumed flow patterns which would lead to channel- type d e f e c t s . (34) The experiments done by Copley et a l . were s i m i l a r to others (37) done by McDonald and Hunt, who examined an ammonium chl o r i d e - w a t e r model of a conventional c a s t i n g , and observed that f l u i d flow occurred through the s o l i d - l i q u i d regions i n the form of r i s i n g pipes. Using d e n s i t y data f o r ammonium chlo r i d e - w a t e r , they suggested that the pipes were formed by lower d e n s i t y l i q u i d i n the i n t e r d e n d r i t i c r e g i o n s , and they considered these pipes to be analogous to A segregates i n large s t e e l c a s t i n g s . There are a l t e r n a t i v e explanations f o r the o r i g i n of A segregates, (3 8) f o r example the suggestion by Blank and P i c k e r i n g t h a t a s o l u t e enriched l a y e r i s formed between the columnar grains at the s i d e s , and the equiaxed grains i n the centre of the in g o t . This enriched l a y e r i s subsequently drawn back i n t o the columnar regions to form A segregates by volume shrinkage during s o l i d i f i c a t i o n . Explanations of t h i s type are l e s s s a t i s f a c t o r y than the mechanism suggested by McDonald and Hunt since they do not account f o r e i t h e r the c h a r a c t e r i s t i c c h a n n e l - l i k e shape of the segregates, or t h e i r c h a r a c t e r i s t i c i n c l i n a t i o n . The ammonium chloride-water experiments e f f e c t i v e l y proved that channels c l o s e l y resembling f r e c k l e s and A segregates could be formed by the upward flow of l e s s dense l i q u i d i n t h i s system. However, the use of t r a n s - parent models to simulate s o l i d i f i c a t i o n i n metal c a s t i n g s has been questioned 97 by a number of w o r k e r s v ~ " ' " 1 . I t has been shown by S t e w a r t } using r a d i o a c t i v e t r a c e r techniques, that the convective flow v e l o c i t y and flow path i n water based systems can be markedly d i f f e r e n t to that which occurs i n l i q u i d metals. This i s due to the large d i f f e r e n c e i n the P r a n d t l (21) number (0.013 f o r l i q u i d t i n , 10.0 f o r water ) , a dimensionless parameter which c h a r a c t e r i z e s convective flow. The extent to which thermal convection would i n f l u e n c e the mechanism of f r e c k l e and A segregate formation observed by Copley, McDonald and others i s u n c e r t a i n when a p p l i e d to a metal system. Therefore i t was considered important to i n v e s t i g a t e the i n t e r d e n d r i t i c f l u i d flow behaviour i n an a l l o y , when l e s s dense l i q u i d e x i s t s at the bottom of the s o l i d - l i q u i d zone. The f o l l o w i n g experiments were t h e r e f o r e d i r e c t e d towards e s t a b l i s h i n g whether the proposed mechanisms, based on the obser- v a t i o n of transparent models, were p o s s i b l e during s o l i d i f i c a t i o n of an a l l o y w i t h a s i m i l a r d ensity c o n f i g u r a t i o n i n the l i q u i d . Consequently, they would not imply that t h i s i s the only mechanism of f r e c k l e formation. In s p i t e of the advantages of t r a c e r techniques, the f a c t that metals are opaque makes i t extremely d i f f i c u l t to reproduce the ammonium chloride-water experiments i n an a l l metal system. For example, to e s t a b l i s h that pipes flow upwards during s o l i d i f i c a t i o n , i t would be necessary to introduce t r a c e r close to the bottom of the advancing s o l i d - l i q u i d r e g i o n . Experimentally t h i s would be very d i f f i c u l t , because one would f i r s t have to l o c a t e the p o s i t i o n of the i n t e r f a c e between the s o l i d and the s o l i d - l i q u i d r e g i o n , and then f o l l o w the motion of the t r a c e r as s o l i d i f i c a t i o n progresses. 98 For t h i s reason experiments described i n t h i s chapter were designed to e s t a b l i s h the f o l l o w i n g : a) whether a density i n v e r s i o n can cause l i q u i d t o flow upward through the mushy zone; b) whether l i q u i d can flow upward through t h i s zone even i f there i s no density i n v e r s i o n , due to s o l u b i l i t y e f f e c t s ; c) whether the r i s i n g l i q u i d tends to advance along a smooth f r o n t , or breaks down i n t o pipe flow. While these experiments do not simulate the s o l i d i f i c a t i o n of r e a l c a s t i n g s , the r e s u l t s could e s t a b l i s h the p r i n c i p l e that density d i f f e r e n c e s cause upward flowing pipes to be formed through the s o l i d - l i q u i d r e g i o n . This would provide a d d i t i o n a l support to the mechanisms f o r f r e c k l i n g and A segregate formation p r e v i o u s l y proposed. The experimental work i n t h i s chapter was published i n M e t a l l u r g - i c a l T ransactions inDecember 1972^*^. Recently, Hebditch and H u n t ^ 4 ^ have reported experiments where they i n j e c t e d r a d i o a c t i v e m a t e r i a l i n t o the s o l i d - l i q u i d region of a growing Sn-Zn a l l o y using a s y r i n g e . They observed upward flow of the l e s s dense r a d i o a c t i v e l i q u i d , supporting the concept that l e s s dense l i q u i d can r i s e through the s o l i d - l i q u i d r e g i o n , but they d i d not i d e n t i f y a c t u a l channels resembling f r e c k l e s or A segregates associated w i t h t h i s upward flow. 5.2 Experimental Procedure The t e s t assembly used i s shown sc h e m a t i c a l l y i n Figure 36. Casting A i s a d i r e c t i o n a l l y s o l i d i f i e d l e a d - t i n a l l o y 2.3 cm i n diameter 99 and 4.5 cm long w i t h a columnar d e n d r i t i c s t r u c t u r e having a primary dendrite spacing of approximately 50 microns. M a t e r i a l from both ends of the c a s t i n g was removed by c a r e f u l machining and replaced by c y l i n d r i c a l i n s e r t s B and C of l e a d - t i n a l l o y having d i f f e r e n t compositions to the ca s t i n g A. The c a s t i n g and two i n s e r t s were t i g h t l y enclosed i n a copper block which had been coated w i t h c o l l o i d a l g r a p h i t e , and placed i n s i d e the tube furnace i l l u s t r a t e d i n Figure 1(b). The columnar c a s t i n g s and i n s e r t s were made by techniques described i n se c t i o n s 3.3 and 3.4, and the dimensions are shown to s c a l e i n Figure 36. The assembly was heated at about 5°C/min to the d e s i r e d tempera- ture T, ta k i n g care not to overshoot t h i s v a l u e , held at t h i s temperature (± 0.5°C) f o r 45 minutes, and then quenched. The r e s u l t a n t c a s t i n g was then sectioned l o n g i t u d i n a l l y , p o l i s h e d and etched. In each experiment 500 ppm of r a d i o a c t i v e t h a l l i u m was uniformly d i s t r i b u t e d throughout i n s e r t B. The flow of l i q u i d from i n s e r t B i n t o the c a s t i n g A could then be determined d i r e c t l y from autoradiographs of p o l i s h e d l o n g i t u d i n a l s e c t i o n s . The a l l o y compositions used f o r the castings A, and the i n s e r t s B and C are l i s t e d i n Table V I I I . The temperature of t e s t i n g , and the d e n s i t i e s of B and G at t h i s temperature, together w i t h the density d i f f e r e n c e , are included i n the t a b l e . The holding temperature T was made equal to the l i q u i d u s tempera- ture at the top i n s e r t B. At t h i s temperature, from the phase diagram, the bottom i n s e r t C would be e n t i r e l y l i q u i d and the c a s t i n g A would have l i q u i d i n t e r d e n d r i t i c channels. THERMO- C O U P L E C O P P E R BLOCK TOP INSERT C O L U M N A R CAST ING B O T T O M INSERT C O V E R FIGURE 36: The t e s t assembly f o r isothermal experiments. M a g n i f i c a t i o n 1 . 5x . 1 0 1 TABLE V I I I TEST CONDITIONS FOR ISOTHERMAL EXPERIMENTS Test Composition Pb + wt % Sn Temp. T°C L i q u i d D e n s i t y ^ 4 2 ^ z 3 g/cm Density D i f f e r e n c e , 3 g/cm Series I Casting A 20 224 -0.65 Upper i n s e r t B 44 8.70 Lower i n s e r t C 62 8.05 Series I I Casting A 20 239 -0.93 Upper i n s e r t B 37 8.97 Lower i n s e r t C 62 8.04 Series I I I Casting A 20 254 -1.21 Upper i n s e r t B 30 9.24 Lower i n s e r t C 62 8.03 Series IV Casting A 96 206 +0.66 Upper i n s e r t B 84 7.41 Lower i n s e r t C 62 8.07 Series V Casting A 15 254 Lower i n s e r t C 62 8.03 102 The r e s u l t s of these experiments cannot be q u a n t i f i e d , and are presented i n the form of autoradiographs and m i c r o s t r u c t u r e s . Consequently, the r e s u l t s from only f i v e sets of experiments ( s e r i e s I-V) are presented. These were part of a more extensive s e r i e s of observations on d i f f e r e n t a l l o y compositions and h o l d i n g temperatures, a l l of which gave r e s u l t s compatible w i t h those reported below. 5.3 Results In s e r i e s I , I I and I I I the top i n s e r t had a higher d e n s i t y than the bottom i n s e r t (Table V I I I ) , which i s a c o n d i t i o n of d e n s i t y i n v e r s i o n as (34) defined by Copley et a l . . In s e r i e s IV, the top i n s e r t had a lower d e n s i t y , i . e . , there was no density i n v e r s i o n . The composition of the bottom i n s e r t was f i x e d at 62% Sn (the e u t e c t i c composition) i n a l l the t e s t s . In each of these four cases, the autoradiographs show that l i q u i d from i n s e r t B flowed uniformly down through i n t e r d e n d r i t i c channels near the top of the c a s t i n g A, without the formation of p i p e s . These r e s u l t s d i f f e r from those of K a e m p f f e r ^ ^ who observed downward flow through pipes because, i n h i s experiment, l i q u i d above the mushy zone was heated above the l i q u i d u s temperature. One can reasonably conclude that i f the t r a c e r showed uniform flow near the top, then the downward flcr-.T of l i q u i d through the whole of c a s t i n g A was v i a i n t e r d e n d r i t i c channels, s i n c e a uniform temperature was maintained. Consequently, any pipes revealed by p o l i s h i n g and etching are due t o l i q u i d f lowing upward. A) Series I The r e s u l t s f o r s e r i e s I are shown i n Figures 37(a) and (b). In (a) FIGURE 38: (a) Macrostructure of columnar c a s t i n g ( s e r i e s I I ) . (b) Corresponding autoradiograph. M a g n i f i c a t i o n 2x. 104 Figure 37(a) l i q u i d from the sides of the bottom i n s e r t has r i s e n i n the i n t e r i o r of the casting. The autoradiograph i n Figure 37(b) shows that l i q u i d from the top i n s e r t B flowed e s s e n t i a l l y uniformly down through i n t e r d e n d r i t i c channels. S l i g h t l y more l i q u i d flowed down the righ t hand side, but there i s no evidence that t h i s i s due to pipes. The enhanced flow probably r e s u l t s from s l i g h t differences i n s i z e or o r i e n t a t i o n of the i n t e r d e n d r i t i c channels on the r i g h t side of the casting. The etched surface of i n s e r t B shows two layers which are s i m i l a r to the separation which occurred i n the r e s e r v o i r of the flow c e l l , discussed e a r l i e r i n section 4 .4.2. Although the bottom l i q u i d has r i s e n , i t does not appear to show a marked breakdown in t o pipe flow, i n the sense that t h i s term i s used by previous workers (in the ammonium chloride-water model studied by Copley et a l . , a t y p i c a l upward flowing pipe occupied an area of 4 to 9 ( 3 4 ) dendrites ). Apart from the region i n the centre of i n s e r t C, where the lower i n s e r t does not make contact with the casting, the l i q u i d has r i s e n f a i r l y uniformly by d i s s o l v i n g part of the d e n d r i t i c structure of casting A. B) Series II In series II the temperature T and the density d i f f e r e n c e were increased, and the results are shown i n Figures 38(a) and (b). The uneven interface between the bottom i n s e r t C and the casting A shows the beginning of breakdown into pipe flow. The l i q u i d on both sides of i n s e r t C has r i s e n i n a less uniform manner than series I. The autoradiograph i n Figure 38(b) 105 shows u n i f o r m downward f l o w a l o n g i n t e r d e n d r i t i c c h a n n e l s . The l a c k of c o n t a c t i n the c e n t r e o f i n s e r t C i n F i g u r e s 37(a) and 38(a), and the p o r o s i t y i n some o f the upward f l o w i n g c h a n n e l s , was p r o b a b l y caused e i t h e r by volume s h r i n k a g e of the l i q u i d on f r e e z i n g , o r by i m p e r f e c t f i t t i n g o f the i n s e r t s i n t o the c a s t i n g . C) S e r i e s I I I In s e r i e s I I I the temperature and d e n s i t y d i f f e r e n c e were a g a i n i n c r e a s e d , and the c a s t i n g s were s e c t i o n e d t r a n s v e r s e l y and then l o n g i t u d - i n a l l y . The r e s u l t s o f two samples are shown i n F i g u r e s 39-42. F i g u r e 39 shows the two s u r f a c e s r e v e a l e d a f t e r s e c t i o n i n g one sample normal t o the a x i s . S u r f a c e B-B shows a number o f channels i n s e c t i o n , and some p o r o s i t y . Two channels can be seen on s u r f a c e C-C, the 3 s m a l l e r of which c o v e r s an e s t i m a t e d a r e a of a p p r o x i m a t e l y 10 d e n d r i t e s . The c a s t i n g was reassembled and s e c t i o n e d l o n g i t u d i n a l l y a l o n g A-A and the s t r u c t u r e and c o r r e s p o n d i n g a u t o r a d i o g r a p h a r e shown i n F i g u r e s 40(a) and ( b ) . The a u t o r a d i o g r a p h shows u n i f o r m downward f l o w , which c o n f i r m s t h a t upward f l o w i n g l i q u i d produced the wide channels i n F i g u r e 40(a). F i g u r e 41 shows the two s u r f a c e s r e v e a l e d a f t e r t r a n s v e r s e s e c t i o n i n g of the second sample. One c h a n n e l i s seen on each s u r f a c e . The l o n g i t u d i n a l s e c t i o n i n F i g u r e 42(a) shows a s i m i l a r s t r u c t u r e t o F i g u r e 40(a) and the a u t o r a d i o g r a p h a g a i n shows u n i f o r m downward f l o w . F i g u r e 42(a) shows a v e r y uneven s u r f a c e between the lower l i q u i d and the c a s t i n g . One FIGURE 3 9 : 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 . M a g n i f i c a t i o n 2x. FIGURE 41: Surfaces revealed a f t e r sample from s e r i e s I I I . transverse s e c t i o n i n g M a g n i f i c a t i o n 2x. of a FIGURE 42: (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 (b) Corresponding autoradiograph. M a g n i f i c a t i o n 2x. 1 0 8 wide channel (which contains some p o r o s i t y ) has r i s e n up one si d e and then s p l i t i n t o s m aller channels higher up the c a s t i n g . The columnar s t r u c t u r e near the top of c a s t i n g A i n Figures 40(a) and 42(a) appears to have changed during the t e s t . Such changes were r e g u l a r l y observed i n cas t i n g s h e l d f o r some time at temperatures where the f r a c t i o n of l i q u i d was r e l a t i v e l y h i g h. This e f f e c t i s a t t r i b u t e d to the beginnings of a general c o l l a p s e of the d e n d r i t i c s t r u c t u r e , a s s o c i a t e d w i t h coarsening e f f e c t s discussed i n s e c t i o n 4.5. D) Se r i e s IV In s e r i e s IV the two i n s e r t s had the same density d i f f e r e n c e as s e r i e s I , w i t h the heavier i n s e r t at the bottom of the c a s t i n g . The r e s u l t - ant s t r u c t u r e i s shown i n Figure 43(a) and the corresponding autoradiograph i n Figure 4 3 ( b ) . There i s no evidence of pipes i n the c a s t i n g from e i t h e r the bottom or the top i n s e r t . The f l a t i n t e r f a c e between the columnar s t r u c t u r e and the bottom i n s e r t i n Figure 43(a) c l e a r l y shows that l i q u i d d i d not flow upwards i n the centre of the c a s t i n g . Figure 43(b) shows that the t r a c e r has o u t l i n e d the d e n d r i t i c s t r u c t u r e to a grea t e r extent than the other t e s t s , yet comparing the o r i g i n a l autoradiographs, there appears to have been l e s s o v e r a l l p e n e t r a t i o n of the c a s t i n g . The long t r a i l s are not due to pip e s , but i n d i c a t e that the t r a c e r has advanced f u r t h e r down g r a i n bound- a r i e s . Since there i s no evidence of s o l u t e convection, t h i s t r a c e r pene- t r a t i o n must be due to l i q u i d d i f f u s i o n or p o s s i b l y shrinkage e f f e c t s during quenching. (a) (b) FIGURE 43: (a) Macrostructure of columnar c a s t i n g ( s e r i e s I V ) . (b) Corresponding autoradiograph. M a g n i f i c a t i o n 2x. FIGURE 44: Macrostructure from s e r i e s V. The m a t e r i a l which etches darker i s Pb-15%Sn homogenized to remove the d e n d r i t i c s t r u c t u r e . Lower i n s e r t i s e u t e c t i c Pb-62%Sn. M a g n i f i c a t i o n 1.5x. 110 Figure 43(a) shows that some l i q u i d from the bottom i n s e r t has r i s e n along the outside of the c a s t i n g . This probably occurred when the supporting m a t e r i a l between the lower i n s e r t and the copper block melted and the c a s t i n g s h i f t e d s l i g h t l y . P o s s i b l y some r a d i o a c t i v e m a t e r i a l from i n s e r t B emerged at the sides and d i s s o l v e d i n t h i s o u t side l i q u i d . T h i s e f f e c t i s not considered to be r e l a t e d to f r e c k l i n g . E) Series V In Figure 44, a Pb-15%Sn a l l o y , homogenized f o r 46 hours at 177°C was used i n place of c a s t i n g A. The lower i n s e r t was e u t e c t i c m a t e r i a l , and the specimen was subjected to the same is o t h e r m a l treatment as previous samples from s e r i e s I I I , which showed long upward flowing p i p e s . The dimensions of the Pb-15%Sn a l l o y and the lower i n s e r t were the same as i n the other samples, however, the top i n s e r t was omitted. A f t e r c o o l i n g the sample was sectioned t r a n s v e r s e l y and l o n g i t u d i n a l l y as before. The transverse s e c t i o n revealed no p i p e s , and the l o n g i t u d i n a l s e c t i o n (Figure 44) showed that d i s s o l u t i o n took place at the i n t e r f a c e e s s e n t i a l l y along a smooth, planar f r o n t . 5.4 D i s c u s s i o n (34) In the mechanism proposed f o r f r e c k l i n g ( a l s o a p p l i c a b l e to (37) A segregate formation ) the l i q u i d at the root of i n t e r d e n d r i t i c channels was considered to be close to e u t e c t i c composition. I f t h i s l i q u i d was of lower d e n s i t y than the bulk l i q u i d above the mushy zone, s o l u t e convection could occur, and the l i g h t e r l i q u i d would r i s e up i n t e r d e n d r i t i c channels. I l l The r i s i n g l i q u i d would move from c o o l e r t o h o t t e r regions of the c a s t i n g , becoming superheated, and would e v e n t u a l l y merge i n t o d i s c r e t e pipes by d i s s o l v i n g away den d r i t e branches. This i s a steady s t a t e process; as the mushy zone advances during s o l i d i f i c a t i o n , the pipe a l s o advances, and the composition of l i q u i d e n t e r i n g the bottom of the pipe would be expected t o remain the same. The main d i f f e r e n c e between the arrangement used i n the present work and a true s o l i d i f i c a t i o n process i s that these t e s t s were done at uniform temperature, whereas i n a r e a l c a s t i n g there would be a temperature gra d i e n t . The r i s i n g l i q u i d would be superheated i n t h i s case, because of the d i f f e r e n c e i n composition, and would s i m i l a r l y be able to d i s s o l v e dendrite branches. However, no steady s t a t e process could be e s t a b l i s h e d s i n c e the mushy zone i n these t e s t s d i d not advance. Consequently, the upward f l o w i n g channels shown L̂n Figures 40(a) and 42(a) are not i d e n t i c a l to the pipes formed i n the ammonium chloride-water experiments. However, they do demonstrate that a d e n s i t y i n v e r s i o n can cause upward flo w i n g pipes through the mushy zone. In a d d i t i o n , the r e s u l t s of s e r i e s IV demonstrate that pipes do not form i f there i s no d e n s i t y i n v e r s i o n . Table IX l i s t s the maximum s o l u b i l i t y of the dendrites i n e u t e c t i c l i q u i d heated t o the h o l d i n g temperature, estimated from the phase diagram. I t can be seen that the s o l u b i l i t y of dendrites i n s e r i e s IV, where pipes do not form, i s about the same as s e r i e s I I I . The a b i l i t y of the bottom l i q u i d to d i s s o l v e dendrites i s therefore not a s u f f i c i e n t c o n d i t i o n f o r the formation of upward flowing p i p e s , unless there i s a l s o a d e n s i t y i n v e r s i o n . 1 1 2 TABLE IX SOLUBILITY DATA FOR ISOTHERMAL EXPERIMENTS Test D e n d r i t i c Phase T °C S o l u b i l i t y of Dendrites i n E u t e c t i c L i q u i d at T °C, grams per gram e u t e c t i c . S e r i e s I a 224 0.66 Series I I a 239 0.95 Series I I I a 254 1.63 Series IV B 206 1.58 Series V a 254 1.25 113 F o l l o w i n g the p u b l i c a t i o n of the r e s u l t s shown i n Figures 37-43 ( s e r i e s I-IV), Hebditch and Hunt^ suggested t h a t the s t r u c t u r e s observed were the r e s u l t of convection w i t h i n the lower i n s e r t C, and had l i t t l e to do w i t h i n t e r d e n d r i t i c f l u i d flow through the c a s t i n g A. This suggestion was t e s t e d by s u b s t i t u t i n g a s i n g l e phase Pb-Sn a l l o y f o r the o r i g i n a l columnar c a s t i n g A ( s e r i e s V, Figure 44). In t h i s case there would be no l i q u i d i n t e r d e n d r i t i c channels, yet the a l l o y would s t i l l be s o l u b l e i n the superheated l i q u i d of the lower i n s e r t (Table I X ) . I f Hebditch and Hunt's contention were c o r r e c t , t h i s experimental arrangement should produce the same type of s t r u c t u r e s as seen i n Figures 37-42. The r e s u l t i n Figure 44 shows that pipes were not formed i n the absence of i n t e r d e n d r i t i c channels i n the c a s t i n g A. Convection w i t h i n the lower i n s e r t probably plays a pa r t i n the development of the s t r u c t u r e s seen, but the experimental evidence leads to the conc l u s i o n that the flow p a t t e r n caused by m a t e r i a l d i s s o l v i n g at the i n t e r f a c e leads to a smooth f r o n t , and would not account f o r the long channels. Although p o r o s i t y due to trapped a i r has been observed i n some of these samples, i t i s not b e l i e v e d that these experiments provide any support- ing evidence f o r the proposed mechanism of f r e c k l i n g by r i s i n g gas bubbles. Trapped a i r would c e r t a i n l y form a bubble above the lower i n s e r t C, but only superheated l i q u i d would be able to d i s s o l v e dendrite branches and r i s e . The bubbles thought to be r e s p o n s i b l e f o r producing f r e c k l e t r a i l s are considered to be on top of the advancing mushy zone, r a t h e r than on the bottom. Bubbles which become trapped by the advancing s o l i d are thought to be responsible f o r "spot segregates" and Figure 42(a) appears to show some 114 i s o l a t e d spots of t h i s type, a s s o c i a t e d w i t h p o r o s i t y . However, one cannot r u l e out the p o s s i b i l i t y that these "spots" were o r i g i n a l l y connected w i t h the main channel, but appear i s o l a t e d because the plane of s e c t i o n i n g has separated them. A l t e r n a t i v e l y , they may be r e l a t e d to the general c o l l a p s e of the d e n d r i t i c s t r u c t u r e , mentioned e a r l i e r . Therefore these experiments do not provide any c o n c l u s i v e evidence i n support of t h i s mechanism. They do, however, e s t a b l i s h the p r i n c i p l e that upward flow i n g pipes can be formed by superheated l i q u i d , when d e n s i t y gradients i n the l i q u i d cause i n t e r d e n d r i t i c f l u i d flow. 115 CHAPTER 6 SOLUTE CONVECTION AND FRECKLE FORMATION DURING SOLIDIFICATION 6.1 I n t r o d u c t i o n The model experiments described i n Chaper 5 demonstrated that pipes can form through a s o l i d - l i q u i d r e g i o n due to s o l u t e convection, and that the d i r e c t i o n of flow w i t h i n the pipes was upwards. A c c o r d i n g l y , the l o g i c a l extension of t h i s work was to study the a c t u a l s o l i d i f i c a t i o n of an a l l o y system where the l i q u i d at 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. I f the proposed mechanism f o r the formation of channel-type defects i s c o r r e c t , i t should be p o s s i b l e to demonstrate s o l u t e convection i n such a system, and produce the defects under appropriate c o o l i n g c o n d i t i o n s . Pb-Sn a l l o y s c o n t a i n i n g l e s s than 62%Sn were th e r e f o r e used because they produce the required " d e n s i t y i n v e r s i o n " when s o l i d i f i e d from the bottom. The purpose of the f o l l o w i n g experiments was to experimentally determine the extent of macrosegregation and f r e c k l e formation i n v e r t i c a l l y s o l i d i f i e d samples as a f u n c t i o n 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 . 6.2 Experimental Procedure 6.2.1 Apparatus C y l i n d r i c a l ingots 14 cm long and 1.27 cm i n diameter were s o l i d i f i e d by lowering the mould through a furnace w i t h two heating zones. The graphite mould i s shown dismantled i n Figure 45. A f t e r assembly, the 116 1 2 3 4 5 6 7 8 9 JO 1 2 3 4 5 6 7 8 III FIGURE 4 5 : S p l i t graphite mould f o r making long c y l i n d r i c a l i n g o t s . The mould was assembled by s l i d i n g the two sleeves over the ends and then w i r i n g the pa r t s together. 117 mould was suspended v e r t i c a l l y i n the furnace, and lowered at a c o n t r o l l e d r a t e by a low speed synchronous motor. The two-zone furnace was constructed from two tube furnaces, s i m i l a r to those shown i n Figure 1, placed v e r t i - c a l l y end to end, and connected t o a two-zone temperature c o n t r o l l e r which maintained a constant gradient. By a d j u s t i n g the temperature of the furnace zones and the speed of descent, the temperature gradient and growth rate could be v a r i e d independently. S o l i d i f i c a t i o n rates between 0.0033 and 0.24 cm/sec were used i n t h i s work, w i t h temperature gradients between 1.0 and 2.3°C/cm. In some t e s t s , the a l l o y was quenched during s o l i d i f i c a t i o n by surrounding the mould w i t h water. For quenching, a t h i n w a l l e d g r a p h i t e mould was used, having the same i n t e r n a l dimensions as the standard mould. Water was introduced i n t o the quartz furnace tube from the bottom, quenching the ingot i n approximately 20 sec. The graphite mould used f o r quenching was enclosed i n a t h i n w a l l e d quartz tube to prevent the mould cracking and water coming i n contact w i t h the molten metal. A p r o t e c t i v e atmosphere was not used. 6.2.2 Macrosegregation st u d i e s For each set of s o l i d i f i c a t i o n c o n d i t i o n s , four c a s t i n g s were made. The temperature gradient and f r e e z i n g r a t e were e s t a b l i s h e d i n one c a s t i n g using three thermocouples p o s i t i o n e d along the ingot a x i s . The cast s t r u c t u r e was determined from l o n g i t u d i n a l and transverse s e c t i o n s of two of the castings which were p o l i s h e d and etched a f t e r s e c t i o n i n g . The f o u r t h c a s t i n g was used f o r measurements of macrosegregation. Approximately 500 ppm of r a d i o a c t i v e t r a c e r was w e l l mixed i n t o the l i q u i d p r i o r to s o l i d - i f i c a t i o n . A f t e r s o l i d i f i c a t i o n , the ingot was placed i n a l a t h e , the 118 outside surface was machined t o remove the t a r n i s h e d surface l a y e r , and cu t t i n g s were taken i n a plane perpendicular to the ingot a x i s . S t a r t i n g at one end of the i n g o t , the c u t t i n g s were c o l l e c t e d at 0.37 cm i n t e r v a l s , weighed (approximately 4 g ) , placed i n t o t e s t tubes, and the a c t i v i t y of each tube measured i n a P i c k e r Nuclear Twinscaler I I automatic s c i n t i l l a - t i o n counter. 113 The isotopes used were e i t h e r Sn ( h a l f l i f e 112 days, p r i - m a r i l y a low energy y e m i t t e r , i r r a d i a t e d to a s p e c i f i c a c t i v i t y of 0.5 204 m i l l i c u r i e s / g ) or T l ( h a l f l i f e 4.1 years, p r i m a r i l y a 3 e m i t t e r , but also some low energy y, i r r a d i a t e d to a s p e c i f i c a c t i v i t y of 5 m i l l i - 113 20A c u r i e s / g ) . The Sn was used i n the lead r i c h a l l o y s , and T l i n the t i n r i c h a l l o y s . The y emission spectrum was measured f o r both these isotopes using the s c i n t i l l a t i o n counter (Figures 46 and 47), and s i n c e the a c t i v i t y l e v e l s T^ere f a i r l y low, an appropriate window was set to reduce the r a t i o between sample a c t i v i t y and background. No c o r r e c t i o n s f o r decay of the isotopes were made, because the time taken f o r each set of measure- 113 2 OA ments (2-3 hours f o r Sn , or 12 hours f o r T l ) was very short compared to the h a l f l i v e s . A c t i v i t y p r o f i l e s , as a f u n c t i o n of distance from the bottom of the i n g o t , f o r castings quenched d i r e c t l y from the l i q u i d were compared to the d i r e c t i o n a l l y s o l i d i f i e d c a stings to confirm that the e f f e c t s observed were due to the s o l i d i f i c a t i o n c o n d i t i o n s , and not due to poor mixing of the o r i g i n a l a l l o y charge, or other extraneous f a c t o r s . 2 2 i 20\ 119 ' o 181 o o 14 12 10 CL Window d 2 7 0 - 4 8 0 Kev 100 2 0 0 3 0 0 4 0 0 ENERGY (Kev) 5 0 0 FIGURE 46: Spectnrai of y emission f o r S n 1 1 3 . 2 9 2 7 2 5 2 3 o x in 2! z 3 O " 19 17 15 5 0 T 1- ,--o. Window 5 0 - 1 0 0 K e v 7 0 _ l _ 9 0 E N E R G Y (Kev) 110 130 FT.GURE 47: Spectrum of y emission f o r TI 204 120 6.2.3 Determination of composition from a c t i v i t y measurements I t i s u s u a l l y assumed that the measured a c t i v i t y of the r a d i o - a c t i v e isotope i s d i r e c t l y p r o p o r t i o n a l to the s o l u t e c o n c e n t r a t i o n . The major f a c t o r s c o n t r i b u t i n g to e r r o r s would be counting s c a t t e r , which i s a f u n c t i o n of the number of counts, and geometrical s c a t t e r , s i n c e the lat h e turnings would not always present the same geometry towards the counter. The number of counts per sample was always between 300,000 and 1 m i l l i o n i n t h i s work, which would r e s u l t i n an o v e r a l l s c a t t e r of (43) b e t t e r than ± 0.5% due to counting s c a t t e r I t was b e l i e v e d that the e f f e c t of geometrical s c a t t e r would be magnified i n the Pb-Sn system due to the h i g h absorption of the r a d i a t i o n -1 (44) by lead (absorption c o e f f i c i e n t 1.127 cm f o r 0.5MeV y rays ). To reduce t h i s e f f e c t , and to o b t a i n a more constant geometry, some of the 113 samples containing Sn were t r e a t e d w i t h a s o l u t i o n of hot 50% HNO^. This formed s o l u b l e lead n i t r a t e , and a white p r e c i p i t a t e of t i n oxide which s e t t l e d i n the bottom of the t e s t tube. Each tube contained 40 ml of s o l u t i o n , and from the published s o l u b i l i t y of PbtNO.^ (37.7 g/100 g s o l u t i o n ) i t was c a l c u l a t e d that a l l the l e a d would be i n s o l u t i o n . The a c t i v i t y measurements would t h e r e f o r e only be a f f e c t e d by absorption i n the p r e c i p i t a t e l a y e r . i To convert the measured a c t i v i t i e s to composition, the f o l l o w i n g s e r i e s of c a l i b r a t i o n t e s t s were done: 113 i ) a c t i v i t y versus sample weight, f o r a constant Sn c o n c e n t r a t i o n ; 113 i i ) a c t i v i t y versus Sn c o n c e n t r a t i o n , f o r a constant sample weight; 121 113 i i i ) a c t i v i t y versus a l l o y composition, f o r a constant Sn concen- t r a t i o n ; 113 i v ) a c t i v i t y versus a l l o y composition, when the Sn concentration was d i r e c t l y p r o p o r t i o n a l to the a l l o y composition. The c a l i b r a t i o n samples were prepared by c a r e f u l weighing of the co n s t i t u e n t s and subsequent treatment w i t h n i t r i c a c i d . The r e s u l t s are shown i n Figures 48-51. The measured a c t i v i t i e s as a f u n c t i o n of both 113 sample weight and Sn concentration (Figures 48 and 49) both deviated from l i n e a r i t y when l a r g e r amounts of r a d i o a c t i v e m a t e r i a l were present, which i n d i c a t e d that the s c i n t i l l a t i o n counter began to satu r a t e at these l e v e l s . I t was f o r t h i s reason that r e l a t i v e l y low concentrations of the r a d i o a c t i v e isotopes were used. Figure 50 shows that the s p e c i f i c a c t i v i t y measurements were r e l a t i v e l y independent of the a l l o y composition f o r a 113 constant Sn concentration. Therefore from Figures 48-50 one can conclude that i f the samples do not vary over a wide range of weights or compositions the s o l u t e content of the sample can be assumed to be d i r e c t l y p r o p o r t i o n a l tp the s p e c i f i c a c t i v i t y . This was t e s t e d and shown to be true over a wide composition range i n Figure 51. The method used f o r c a l c u l a t i n g s o l u t e content was to determine the s p e c i f i c a c t i v i t y of the whole ingot (a. ) , which was taken as the sum m g ' of the a c t i v i t i e s of each sample (a ) from the i n g o t , d i v i d e d by the sum of the sample weights (w^) . The s p e c i f i c a c t i v i t y was then set equal to the prepared a l l o y composition ( C Q ) . Za. ^ _ i _ ing Ew^ ~ o  > < 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 (PL̂ 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 10 5 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.

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