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Morphology, structure and growth kinetics of bainite plates in the β' phase of A Ag-45 AT. PCT Cd Alloy Kostić, Miodrag Miloš 1977

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MORPHOLOGY, STRUCTURE AND GROWTH KINETICS OF BAINITE PLATES IN THE 3' PHASE OF A Ag-45 AT. PCT Cd ALLOY by MIODRAG MILOS KOSTIC D i p l . Ing./ University of Belgrade, 1963 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of METALLURGY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA JULY, 1977 © Miodrag Miloa Kosti& 1977 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 e q u i r e m e n 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 Co lumb ia , I a g 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 purposes 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 o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f Metallurgy  The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e July 8/77 ABSTRACT The morphology of ba i n i t e plates and widmanstatten needles formed i n ordered bcc 6* phase of a Ag-45 at. pet Cd a l l o y at temperatures 160-320° C was studied by o p t i c a l and scanning electron microscopy. Both p r e c i p i t a t e forms were s i m i l a r i n appearance to p r e c i p i t a t e s reported for Cu-Zn a l l o y s . The structure of the b a i n i t e plates i n the various stages of t h e i r growth was studied by X-ray d i f f r a c t i o n and by transmission electron microscopy. I n i t i a l l y , the plates formed with a 3R stacking f a u l t modulation of the fee structure and contained a high density of random stacking f a u l t s . The stacking f a u l t s annealed out during a prolonged isothermal treatment and the structure gradually changed to a regular fee. The orientation r e l a t i o n s h i p between the bcc matrix and the fee bainite was as follows: [ l l l ] b 0.7° from [ 0 l l ] f , [110] b 1.1° from [100]f and [011] b 4.3° from the stacking f a u l t plane pole [ l l l ] f . The habit plane of the b a i n i t e plates, determined by two surface trace analysis, was close to (144) b. The surface r e l i e f of the plates was observed by the interference microscopy. I t was i n the form of a simple t i l t i n d i c a t i n g an invariant plane s t r a i n transformation. The features of the transformation agreed with the predictions of the Bowles-Mackenzie theory of martensite formation. The growth k i n e t i c s of both bain i t e plates and widmanstatten needles were measured by interrupted annealing and scanning electron microscopy. Using the b a i n i t e thickening k i n e t i c s measured at 160, 200 and 240°C, the Frank-Zener model for growth of planar p r e c i p i t a t e s , and supersaturation data obtained from the Ag-Cd metastable phase diagram enabled the e f f e c t i v e chemical d i f f u s i v i t i e s , De f f # to be calculated for the three transformation temper-atures. The results were i n good agreement with the expected d i f f u s i v i t i e s . The lengthening k i n e t i c s of bain i t e plates at 160°C and of widmanstatten needles at 240°C were analyzed using Trivedi's model for d i f f u s i o n - c o n t r o l l e d growth. D e f f obtained from the lengthening k i n e t i c s of the needles was i n good agreement with the D value obtained from the thick-er f ening k i n e t i c s of the plates, i n d i c a t i n g that widmanstatten needles lengthened and bain i t e plates thickened at rates controlled by volume d i f f u s i o n . Bainite plates lengthened only i n the early stage of growth and at a rate approximately 180 times larger than that permitted by volume d i f f u s i o n . I t was concluded that the morphology, structure and other c h a r a c t e r i s t i c s of the freshly formed b a i n i t e plates were consistent with t h e i r formation by a thermally activated martensitic process. i i i TABLE OF CONTENTS CHAPTER PAGE 1 THE INTRODUCTION . . . . . ...... . 1 1.1. Nucleation and Growth Transformations and Martensitic Transformations 1 1.2. B a i n i t i c Transformations 5 1.3. Martensite and Bainite i n Cu- and Ag-Based Alloys 8 1.4. Aim of the Present Work 11 2 EXPERIMENTAL . . . 13 2.1. Preparation of Alloys 13 2.2. Quenching 15 2.3. E l e c t r o l y t i c Polishing and Thinning.... I 8 2.4. X-Ray Analysis 1 8 2.5. Surface R e l i e f » 1 9 2.6. Habit Plane Measurements 19 2.7. Electron Microscopy 21 2.8. Growth Kinetics Measurements 22 3 . RESULTS AND DISCUSSION 25 3.1. Morphology of Precipitates Formed during Quenching 25 3.2. Morphology of Precipitates Formed during Isothermal Annealing 27 3.3. X-Ray Structure Analysis 35 3.4. Surface R e l i e f 4 1 3.5. Bainite Habit Plane Measurements ...... 4 4 i v CHAPTER PAGE 3.6. Transmission Electron Microscopy Results.. 44 3.6.1. Morphology and Structure of Bainite 46 3.6.2. Orientation Relationship 61 3.7. Application of the Phenomenological Martensite Theory to the Formation of Bainite 67 3.8. Comparison with the Martensitic Products Observed i n Ag-Cd, Ag-Zn and Cu-Zn Alloys 73 3.9. Origin and S t a b i l i t y of the 3R Structure of Bainite 75 3.10. Growth Kinetics 78 3.10.1. Analysis of Bainite Thickening Data 79 3.10.2. Analysis of Bainite Lengthening Data 96 3.10.3. Analysis of Widmanstatten Lengthening Data 102 3.10.4. Discussion of the Growth Kinetics Results 105 3.10.5. General Discussion 108 4 CONCLUSIONS 110 SUGGESTIONS FOR FUTURE WORK 112 APPENDIX A - Structure Analysis 113 A . l . D i s t o r t i o n of the FCC Reciprocal L a t t i c e due to a High Density of Random Stacking Faults 117 A. 2. Long Period Stacking Order Modulation of the FCC L a t t i c e .... 118 v CHAPTER PAGE APPENDIX B <* A n a l y t i c a l Treatment of Martensitic Transformations 125 APPENDIX C - Theory of the Volume Di f f u s i o n Controlled P r e c i p i t a t e Growth 139 C l . Thickening of Plates 139 C.2. Lengthening of Plates and Needles 140 APPENDIX D - An estimate of the Chemical D i f f u s i v i t y i n the g ' Phase of Ag-Cd Al l o y s on the Basis of a Comparison between the Cu-Zn and Ag-Cd Systems 144 APPENDIX E - The Equilibrium and the Metastable Ag-Cd Phase Diagram 148 REFERENCES 151 v i LIST OF TABLES v i i TABLE PAGE . • • 2 A-1 Calculated Relative I n t e n s i t i e s , |F| , for the 3R Modulation of the CuAu I-type Structure Based on Equations (A-3) and (A-4) • for k - 0, 1, -1. .. 123 B-I Application for the Bowles-Mackenzie Martensite Theory to the 3' to a Transformation i n the Ag-45 At. Pet Cd A l l o y - Summary of the Used Data and Results I 3 6 v i i i LIST OF FIGURES FIGURE PAGE 1. The Relevant Portion of the Ag-Cd Equilibrium • Phase Diagram. Dotted Lines Indicate Compo-s i t i o n of the Alloys Used 14 2. Schematic Diagram of the Induction Heating and Quenching Apparatus. 16 3. Bainite Plates and Massive otm i n the B' matrix of a Ag-46 at. Pet Cd A l l o y Quenched from 600°C. The Quenching Rate was I n s u f f i c i e n t to Retain the Untransformed 0' Phase, Resulting i n Formation of Bainite (a). Upon Decreasing the Quenching Rate, F i r s t Formed on Grain Boundaries (b), and Then i n the In t e r i o r of the Grains (c) . 26 4. A Scanning Electron Micrograph of the Edge (Included Angle of Approximately 90°) of a severely Etched Specimen of Ag-45 at. Pet Cd A l l o y Annealed for 1,225 Seconds at 200°C. Both Widmanstatten Needles and Bainite Plates are V i s i b l e . 28 5. Bainite Plates (a) and a Mixture of Bainite Plates and Widmanstatten Needles (b)formed i n a Ag-45 at. Pet Cd A l l o y During Annealing at 160°C for 57,600 Seconds (a) and at 200°C for 1,225 Seconds (b). 3 0 i x A Mixture of Bainite Plates and Widmanstatten Needles Formed i n a Ag-46 at. Pet Cd A l l o y During Annealing at 200°C for 25,600 Seconds. Most Plates Degenerated to Needles Isomor-phous with the Widmanstatten Needles.The Broad Faces of the Pair of Plates i n the Centre are Approximately P a r a l l e l to the , Plane of Po l i s h Bainite Plates i n Ag-45 at. Pet Cd A l l o y Formed After Approximately 2 s at 280°C. .. The Vari a t i o n of Bainite Plate Morphology i n Different Matrix Grains Annealing Temperature 160°C; Annealing Time 12,600 s(a), 25,600 s (b), and 57,600 s (c).-Annealing Temperature 200°C; Annealing Time 529 s (a), 900 a (b), and 2,116 s (c) Annealing Temperature 240°C; Annealing Time 25 s (a), 64 s (b), and 144 s (c). Interference Micrographs of the Surface R e l i e f Caused by. Formation of Bainite Plates Interference Micrographs of Surface R e l i e f Caused by Formation of Widmanstatten Needles x FIGURE PAGE 14. Portion of the Standard [001]^ Stereographic Projection of the Matrix Showing the Measured Habit Plane Poles of Bainite Plates Formed During Annealing at 240°C. The C i r c l e Below the [011]^ Pole (radius approximately 3.5°) Encompasses Two Thirds of A l l Measurements. The C i r c l e Above the [011]j3 Pole Has the Same Size and i s Centered i n the Cry s t a l l o g r a p h i c a l l y Equivalent Position With Respect to the [011] b Pole. The Poles Marked With Numbers Belong to Individual Plates Joined i n Pairs, e.g., 31 and 31a. The Open Triangle Represents the Theoretical Habit Plane Pole [0.180747; 0.667566; 0.722279] b (See Section 3.7). 45 15. Bainite Plates i n a Ag-45 at. Pet Cd A l l o y A f t e r 15,900 s at 160°C (a,b) and 36 s at 240°C (Dark Field) (c) T . 47 16. Selected Area D i f f r a c t i o n Pattern of a Bainite Plate A f t e r 15,900 s at 160°C. The structure i s 3R 4 8 17. Bainite i n a Ag-45 at. Pet Cd Alloy A f t e r 19,800 s (a) and 25,600 s (b) at 160°C 50 x i FIGURE PAGE 18. Selected Area D i f f r a c t i o n Patterns of Bainite i n a Ag-45 at. Pet Cd Al l o y A f t e r 19,800 s (a) and 25,600 s (b) at 160°C. Note the Appearance of fee Spots and Dissapearance of 3R Spots 52 19. Changes i n the D i f f r a c t i o n Patterns Due to the 3R to fee Structure Transformation 54 20. Micrographs of a Bainite Plate Af t e r 900 s at 240°C. Note That the Stacking Fault i n the Upper Righthand Corner i n (a) Disappeared i n (b) Leaving a Dislocation Resolved into Two P a r t i a l s (A) 56 21. Micrograph of a Bainite Plate Af t e r 900 s at 240°C 57 22. Micrograph of a Bainite Plate Af t e r 900 s at 240°C 5 8 23. Micrographs of Bainite Plates A f t e r 900 s at 240°C. The Portions with Zero Stacking Fault Density Thickened Faster Than the Rest of the Plates 6 0 24. Selected Area D i f f r a c t i o n Pattern Composed of (111)^ and (011) f Reciprocal L a t t i c e Planes 6 2 x i i Schematic Stereographic Projection Diagram of the Orientation Relationship Between the 3' Parent and B a i n i t e . The Bainite L a t t i c e i s Indexed i n Cubic Notation; Although the [ l l l J b and [ Q l l ] f P-oles are Here Shown to Coincide, They Are Actually Approximately 0.7° Appart The Composite Matrix - Bainite D i f f r a c t i o n Patterns (a,b) Obtained from the Branches of the Chevron Shown i n (c). Orientation Relationship Between the Two Bainite Plates (I and II) and the Matrix. The Normal to the Projection i s P a r a l l e l to the Optical Axis i n F i g . 26. Poles Marked p 1 I and P ^ 1 are the Theoretical Habit Plane Boles of the Plates I and I I . Their Indices are p^ 1 = [-0.667566; -0.180774; 0.722279] b and P! 1 1 = [0.722279; -0.180747; -0.667566] b. Optical Micrographs of L i g h t l y Etched Surface of a Ag-45 at. Pet Cd Specimen Annealed at 200°C. (a) A f t e r 625 s: a Number of Chevron Shaped Bainite Traces Appeared with an Occasional Widmanstatten Needle (A). (b) A f t e r 900 s: : Bainite Traces Present i n (a) Have Either Maintained t h e i r O r i g i n a l Length x i i i or Have Lengthened S l i g h t l y , but A l l Have Increased Their Thickness. The Tips of Some of the Plates Apparently Acted as Nucleation Sites for Widmanstatten Needles(B).Widmanstatten Needles Continued to Lengthen. A Number of New Bainite Traces Appeared (C) . (c) AFter l,225:;s: The Same Behavior i s Continued; Old Bainite Traces Thicken and New Ones Keep Appearing, While Widmanstatten Needles Which Have Not Impinged Upon other Precipitates Continue to Lengthen Scanning Electron Micrographs of the Unetched Surface of a Ag-45 at. pet Cd Specimen Annealed at 240°C. (a) A f t e r 16 s: A Bainite Chevron Appeared. (b) Afer 36 s: The Lower Arm of the Chevron Erom (a) Dad Not Lengthen Although i t Thickened Appreciably, While Traces of New Plates Appeared from the L e f t , the Lower One Stopping Before Impinging Upon the O r i g i n a l Plate, (c) A f t e r 49 s: Thickening Continued Without Lengthening Scanning Electron Micrographs of a Pair of Bainite Plates i n a Ag-45 at.Pet Cd A l l o y Showing Their Early Growth at 160°C. Both Lengthening and Thickening are V i s i b l e . xiv FIGURE 31. Scanning Electron Micrographs Showing Thickening of the Trace of a Bainite Plate at 240°C i n a Ag-45 at. pet Cd A l l o y . .................................... 32. Thickening Kinetics of a Bainite Plate Trace at 200°C i n a Ag-45 at. pet Cd A l l o y . . 2 33. The X U 6 v - . t a p l o t f o r the Bainite Plate Trace From F i g . 3 2 * 34. Schematic Representation of the Dependence of the Plate Trace Width, 2Xfc, on the Angle Between the Plate and the Specimen Surface, a, and on the Plate Thickness, T^ (sin a = V2xt>-- • 35. Thickness of Plate Traces at a Given Growth Time Plotted As a Function of the Angle Between the Plate and the Specimen Surface, a. § Angle a Calculated Assuming That the Growth Rate was the Same for A l l Plates and That Plate No.6 was Perpendicular to the Surface of the Specimen. • Angle a Measured by S e r i a l Dissolution. The Numbers Refer to the 240°C Bainite Plates i n Table V 36. LogD e££ V6. 1/T f o r a Ag-45 at. pet Cd A l l o y . . xv FIGURE PAGE 37. Schematic Diagram of a Pair of Bainite Plates Which Nucleated i n the In t e r i o r of the Specimen at Point N, Emerged on the Surface of Observation at Point E and Formed the Trace ABC at Time t ( a ) , and Lengthening Kinetics of the Trace EC (b) 9 7 99 38. Lengthening Kinetics at 160°C of Bainite Plate Traces i n a Ag-45 at. pet Cd A l l o y . . . 39. Scanning Electron Micrographs Showing the Growth of a Widmanstatten Needle (a) and lengthening Kinetics of Widmanstatten Needles (b) i n a Ag-45 at. pet Gd All o y at 240°C. 103 A-1 Stacking Sequence of Close Packed [111]^ Layers i n the fee Lattice.Atoms A are i n the Plane of the Drawing; the Layer Beneath Has Atoms i n C Positions, the Layer Above i n B Positions. The Shear Vectors R of a Stacking Fault are Indicated i n the Diagram. H 4 A-2 (101) f Reciprocal L a t t i c e Plane with Twinned La t t i c e Spots. The Plane Consists of Rows of Reflections with Successive Phase S h i f t s 0, 2ir/3 and - 2 T T / 3 , Every Third Layer Having the Same Phase S h i f t . Stacking Faults on (111)^ Plane Cause Broadening and Displace-ment or S p l i t t i n g of Spots with $=±2u^3 i n the Direction P a r a l l e l to [ l l l ] f H 6 FIGURE PAGE A-3. Intensity D i s t r i b u t i o n i n the 3R Reciprocal L a t t i c e Plane (110) i n the Orthorhombic o Notation or (101)£ i n the Cubic Notation.„. 120 A-4. (a) The L a t t i c e Correspondence Between the FCC (CuAu I-Type) and Orthorhombic L a t t i c e , (b) The Unit C e l l of the Basal Plane of the Orthorhombic L a t t i c e . The Orthorhombic Coordinates of Atoms i n the Plane are: Ag - 0, 0; Cd - h, h- (c) The D i s t r i b u t i o n of Atoms i n the \Basal Plane i n the A,B and C Layers. The Orthorhombic Coordinates of the Ag Atoms i n the Layers are; A - 0, 0, 0; B - 0, 1/3, 1/9; C - 0, 2/3; 2/9 121 B-1. Schematic Representation of the Correspondence Between the Parent CsCl-Type L a t t i c e (b Basis) and the Product CuAu I-Type L a t t i c e (f Basis). 126 B-2. Stereographic Projection Showing Some of the Operations i n the Determination of Invariant Line Strains Compatible with the Shear System (011) [011] b. 130 D-l. Comparison of the D i f f u s i v i t y Data for a-Cu-Zn and a-Ag-Cd Phase. I 4 5 D-2. Comparison of the, D i f f u s i v i t y Data f o r g'-Cu-Zn and B'-Ag-Cd Phase 147 x v i i FIGURE PAGE E - l . The Relevant Portion of the Ag-Cd Equlibrium Phase Diagram (Thin Lines) and the Ag-Cd Metastable Phase Diagram (Thick Lines). In the Metastable Phase Diagram the Formation of the r, Phase i s Suppressed by Rapid Quenching from the B Phase to the B1 phase 149 x v i i i ACKNOWLEDGMENTS I am very g r a t e f u l to Dr. E.B. Hawbolt, Dr. L.C. Brown and Dr. D. Tromans, as well as to my colleagues, for t h e i r help. My work on t h i s thesis has been made possible by the NRC research assistantship. xix 1. INTRODUCTION When a system i n e q u i l i b r i u m c o n s i s t s of d i f f e r e n t phases a t d i f f e r e n t temperatures, c o o l i n g through a temperature i n t e r v a l may g i v e r i s e t o a phase t r a n s f o r m a t i o n . The d r i v i n g f o r c e f o r the t r a n s f o r m a t i o n i s the d i f f e r e n c e between the f r e e e n e r g i e s of the i n i t i a l and f i n a l s t a t e s . The f i n a l s t a t e does not n e c e s s a r i l y have t o be an e q u i l i b r i u m s t a t e , but the requirement i s t h a t i t s t o t a l f r e e energy be lower than the t o t a l f r e e energy of the i n i t i a l s t a t e . 1.1. N u c l e a t i o n and Growth Transformations and M a r t e n s i t i c T r a n s f o r m a t i o n s . The phase t r a n s f o r m a t i o n s are u s u a l l y d i v i d e d i n t o two main groups: nactzation and growth t r a n s f o r m a t i o n s and mah.t<Ln&4.t<Lc. t r a n s f o r m a t i o n s . A t y p i c a l t r a n s f o r m a t i o n o f the f i r s t group i s p r e c i p i t a t i o n from a s u p e r s a t u r a t e d s o l i d s o l u t i o n . The p r e c i p i t a t e phase grows a t the expense of the ma t r i x phase by the r e l a t i v e l y slow m i g r a t i o n of the h i g h energy i n t e r p h a s e boundary, r e s u l t i n g from atom by at6m t r a n s f e r across the boundary. Compositions o f the mat r i x and p r e c i p i t a t e phases are d i f f e r e n t and the motion of the i n t e r f a c e r e q u i r e s d i f f u s i o n o f atoms of d i f f e r e n t s p e c i e s towards or away from the i n t e r f a c e . The t r a n s p o r t o f atoms i s a t h e r m a l l y 1 2 activated process and transformation proceeds isothermally at a rate which i s a function of temperature. Transformations of the second group, martensitic transformations, occur by the co-operative movements of many atoms over small distances of the order of the atom s i z e . A unit c e l l of the parent phase i s homogeneously deformed into unit c e l l of the product phase. Such a transformation mechanism causes the transformed regions to change t h e i r shape, giving r i s e to a surface r e l i e f on an o r i g i n a l l y f l a t surface. The boundary between the parent and the product i s coherent and g l i s s i l e and the compo-s i t i o n s of the phases are i d e n t i c a l . Thus, discrete regions of the parent can transform with a high v e l o c i t y almost independent of temperature. In most martensitic transfor-mations, the amount of transformation i s c h a r a c t e r i s t i c of the temperature and does not increase with time; at any temperature, the product i s i n a thermoelastic equilibrium with the parent, the number and the size of the i n d i v i d u a l martensite plates r e f l e c t i n g the equilibrium between the d r i v i n g force £or the transformation and the e l a s t i c stresses created by the transformation. However, some martensitic reactions are thermally activated and hence occur isothermally. In the case of athermal martensite, the transformation on cooling begins spontaneously at a fixed temperature (M_ temperature), but deformation and external e l a s t i c stresses can play an important role i n a s s i s t i n g or i n h i b i t i n g the 3 transformation. The martensitic product i s usually i n the form of plates l y i n g p a r a l l e l to a uniquely defined parent plane the habit pt&ne. Formation of the plates i s accompanied by a shape deformation, which, as mentioned, causes a surface r e l i e f , i.e., a t i l t i n g of the specimen surface about i t s l i n e of i n t e r s e c t i o n with the plate-matrix i n t e r f a c e . The e f f e c t of the transformation on surface scratches and the nature of the surface r e l i e f indicate that the martensitic shape deformation i s , apart from a possible small and uniform d i s t o r t i o n , an invariant plane s t r a i n , with the habit plane being the invariant plane. The r e l a t i v e orientation of the parent and product l a t t i c e s , or orientation, relationship, i s also uniquely defined. Another c h a r a c t e r i s t i c of martensitic transformations, which can sometimes be d i r e c t l y observed using a microscope, i s the inhomogeneous shear ( s l i p or twinning) of the product, or tattixie invariant shear. In most cases, the habit plane, amount of shape deformation, amount of shear and o r i e n t a t i o n r e l a t i o n s h i p for a given martensitic transformation can be experimentally determined. These c h a r a c t e r i s t i c s of martensitic transformations have been described mathematically i n terms of the pheno-menological theories of martensite formation. The best known are the theories of Wechsler, Lieberman and Read (1) and Bowles and Mackenzie (2-4). The theories enable a prediction 4 of the habit plane, shape deformation and orientation r e l a t i o n s h i p by assuming a shear system i n the product and a l a t t i c e correspondence between the parent and product, providing that the l a t t i c e parameters and c r y s t a l structures of the parent and product phases are known.+ A description of the Bowles and Mackenzie theory i s given i n Appendix B. Since t h e i r formulation, the theories have demons-trated a remarkable agreement with the experimental obser-vations (5) and thus provide a plausible rationale for the rather complex crystallographic features of d i f f e r e n t martensitic transformations. During rapid cooling of the high temperature phase, massive, t/utrU) formation may occur i n the absence of competitive nucleation and growth or martensitic transformations. Massive transformation i s a s p e c i a l kind of composition invariant phase transformation which becomes possible as soon as s u f f i c i e n t free energy d r i v i n g force i s generated below the temperature at which the free energy of the high temperature, matrix phase becomes equal to the free energy of the low temperature, product phase. The transformation occurs by a short range d i f f u s i o n a l process that involves a rapid non-cooperative transfer of atoms accross a r e l a t i v e l y 5 high energy i n t e r f a c e , but does not involve any change of the o v e r a l l composition. Also, unless the changes of c r y s t a l structure involve considerable changes i n the volume, the product phase of the massive transformation i s expected to develop without accompanying d i s t o r t i o n of the free surface of the specimen. 1 . 2 . B a i n i t i c Transformations Martensitic transformations generally require a larger dri v i n g force and hence occur at lower temperatures than do nucleation and growth transformations. The t r a n s i t i o n from nucleation and growth transformations to martensitic transformations as the reaction temperature i s lowered i s not sharply defined; baZnltic transformations occur before the onset of the martensitic transformation. The b a i n i t i c transformation i s generally regarded as being intermediate between a true nucleation and growth transformation and a martensitic transformation (6). Unlike many nucleation and growth transformations, b a i n i t i c reactions are accompanied by changes of shape of the transformed regions. In an early interpretation, Ko and C o t t r e l l (7) suggested that the structure change during a b a i n i t i c transformation i s e s s e n t i a l l y martensitic, but that due to the reduced dr i v i n g force available at temperatures above the M s temperature, growth i s only possible i f the free energy i s further decreased by d i f f u s i o n a l composition change, d i f f u s i o n 6 rate l i m i t i n g the rate of the martensitic structure change. The b a i n i t e i n iron-carbon allows consists of a non-lamellar aggregate of f e r r i t e and carbides. Inlowoji balnito,, which forms at temperatures below approximately 350°C, f e r r i t e i s plate-shaped with carbides p r e c i p i t a t e d within the plates, the o v e r a l l structure resembling that of tempered martensite. The plates form on a d e f i n i t e habit plane and produce an invariant plane s t r a i n type of surface r e l i e f . The growing t i p s of the plates have sharp r a d i i of curvature and are free of carbide (8). The bai n i t e forms isothermally, i t s TTT curves being C-shaped, as expected for a nucleation and growth transfor-mation. However, the transformation starts only below a well defined temperature (B temperature), with the f r a c t i o n of austenite transforming to b a i n i t e being a function of the temperature of isothermal annealing. I t was stated that the b a i n i t i c transformations were o r i g i n a l l y interpreted as being a martensitic formation of supersaturated f e r r i t e with a secondary p r e c i p i t a t i o n of carbides within the f e r r i t e , the l a t t e r p r e c i p i t a t i o n being necessary to s t a b i l i z e f e r r i t e by decreasing i t s free energy (Ko and C o t t r e l l ) . Nevertheless, some re s u l t s (9,10) indicated that a c e r t a i n amount of r e d i s t r i b u t i o n of carbon occurs at the f e r r i t e - a u s t e n i t e i n t e r f a c e , and thus the 7 lengthening of plates could be controlled by d i f f u s i o n of carbon into austenite. Supported by the experimental evidence of a l i n e a r dependence of lengthening on time, the Zener-Hillerb model (for d i f f u s i o n controlled lengthening was applied to the baini t e growth k i n e t i c s (11,12). The agreement with the theory was d i f f i c u l t to ascertain due to uncertainties of concentration factors and the dependence of carbon d i f f u s i v i t y on concentration. The theory of Ko and C o t t r e l l has the advantage of explaining the s i m i l a r i t y between ba i n i t e and martensite crystallographies (13). I t does not necessarily c o n f l i c t with the assumption that a cert a i n amount of d i f f u s i v e r e d i s t r i b u t i o n of components occurs i n front of the moving martensitic boundary, as long as the a l l o y contains components of widely d i f f e r e n t m o b i l i t i e s . This i s the case for ferrous alloys containing i n t e r s t i t i a l carbon, where the martensitic correspondence of l a t t i c e s can be preserved by the slowly moving iron component. But i n the case of alloys i n which the composition change occurs by d i f f u s i o n of a su b s t i t u t i o n a l component (e.g., Cu-Zn), i t seems necessary to assume that bain i t e i n h e r i t s the unchanged composition of the parent. In spite of the considerable attention which has been given to the b a i n i t i c transformation over the l a s t f i f t e e n years, controversy concerning the growth mechanism s t i l l e x ists and widely d i f f e r i n g views are encountered i n 8 the l i t e r a t u r e (14) . 1.3. Martensite and Bainite i n Cu- and Ag-Based Alloys Martensites forming from the B-phase of Cu-based (15-24) and Ag-based (25-28) al l o y s have been investigated extensively. The B-phase i n these alloys i s an electron compound with the electroh-to-atom-ratio of approximately 1.5. In the Cu-Zn, Ag-Zn and Ag-Cd a l l o y s , i t occurs i n the 50 at. pet range, but i t can generally be retained to room temperature over a wider composition range by rapid quenching. On cooling to room temperature, i t undergoes an ordering transformation to a B" CsCl- type structure. At lower zinc or cadmium contents, a massive type transfor-mation of B to a m takes place on cooling(29). In the absence of external stresses, the metastable B' phase transforms to thermoelastic martensite only on cooling below the Mg temperature. However, an i n t e r e s t i n g mode of martensitic transformation has been observed by Ayers (27) i n a Ag-37.8 at.pet Zn a l l o y ; large plates of martensite formed i n quenched B' phase when i t was reheated to approximately 280°C for a period of one second or l e s s . The transformation was apparently thermally activated. Longer times at the temperature caused the formation of much smaller, chevron shaped p r e c i p i t a t e s , which could probably be c l a s s i f i e d as b a i n i t e (see below). 9 The structure of the thermoelastic martensite can generally be described as being e i t h e r fee or a close packed stacking variant of the fee structure, such as 3R (stacking sequence ABCBCACAB), 2H (AB), 11H (ABCBCACABAB), or a lamellar mixture of these. In burst type (bulk)thermoelestic marten-s i t e i n Cu-Zn al l o y s (24) and i n thermally activated martensite i n the Ag-Zn a l l o y (27), the martensite plates consisted of fine twin lamellae having a s l i g h t l y d i s t o r t e d fee structure. Application of the phenomenological theory of martensite resulted i n a good agreement between/the experimental and t h e o r e t i c a l features of the martensites (24, 27, 28). Plates of the a phase exhibiting chevron-shaped surface traces were found to form isothermally above room temperature i n the 3 V phase of some Cu- and Ag-based alloys (2 8-35). The plates have been considered to be b a i n i t e by analogy to the b a i n i t e i n ferrous a l l o y s . Garwood (35) found that the b a i n i t e plates i n 3 1 Cu-Zn phase formed at temperatures up to 350°C; at higher temperatures only a needle shaped (widmanstatten) p r e c i p i t a t e formed. The plates gave r i s e to an invariant plane s t r a i n type of surface r e l i e f . Their habit plane was i d e n t i c a l to that of low temperature martensite i n the same a l l o y . Garwood con-cluded that the plates formed by a shear transformation, but that t h e i r growth was controlled by d i f f u s i o n . 10 Later research (31-33,36) confirmed Garwood's observations and supplied new data supporting his conclusion that the bainit e plates formed by shear. Hornbogen and Warlimont (31) studied the structure of bainit e plates by electron d i f f r a c t i o n and found that the stacking order of the close packed planes i n fr e s h l y formed plates was 3R, sim i l a r to that found i n &1 martensite i n copper a l l o y s , where i t was explained as o r i g i n a t i n g from a l a t t i c e invariant shear (17). Prolonged annealing gradually destroyed that structure and transformed the plates to almost equilibrium a phase with a random d i s t r i b u t i o n of stacking f a u l t s . Srinavasan and Hepworth (33) found that the crystallographic c h a r a c t e r i s t i c s of bainit e plates were consistent with the phenomenological theory of marten-s i t e formation. Cornells and Wayman (24,36) performed a rigorous crystallographic study on sub-zero martensite and isothermally formed a plates. They confirmed the res u l t s of Srinavasan and Hepworth and established that the l a t t i c e o r ientation r e l a t i o n s h i p , habit plane and magnitude of shape deformation for martensite plates were i d e n t i c a l to the respective c h a r a c t e r i s t i c s for the a plates. However, they found that the martensite was i n t e r n a l l y twinned, not faulted. Plewitt and Towner (32) and Cornells and Wayman (37) observed that the bainit e plates formed i n i t i a l l y without change i n composition, and that p a r t i t i o n i n g of copper and zinc atoms occurred only a f t e r the plates had formed. 11 Lorimer Z,t a.l.:. (38) disputed the v a l i d i t y of the results of Cornells and Wayman and, i m p l i c i t l y , those of Flewitt and Towner. Lorimer ctaZ. repeated the experiment of Cornells and Wayman and found that the composition of the a plates (bainite) d i f f e r e d from the bulk composition even i n the i n i t i a l stages of the transformation. Kostic and Hawbolt (39) c r i t i c i z e d the method of analysis used by Lorimer zt ft£.and suggested that t h e i r conclusions were not convincing since they reported an abnormally low zinc concentration for the a plates. In t h e i r studies of massive (29) and martensitic (28) transformations i n Ag-Cd and Ag-Zn a l l o y s , Ayers (29) and Krishnan (28) observed a plate shaped a pr e c i p i t a t e which formed during quenching from the high temperature 3-phase region. In both instances the pr e c i p i t a t e was designated as bain i t e due to i t s morphological s i m i l a r i t i e s to bainit e i n the 3' phase i n Cu-Zn a l l o y s . The nature of the pre-c i p i t a t e was not investigated i n more d e t a i l . 1.4. Aim of the Present Work Results i n the l i t e r a t u r e indicate that martensitic shear may play a prominent, role i n the formation of bainit e i n Cu-Zn- a l l o y s . The b a i n i t i c transformation i n Ag-based alloys has been examined much less extensively. In the present work, the morphology, crystallographic features and growth k i n e t i c s of b a i n i t e i n Ag-Cd al l o y s are examined i n order to ascertain which mechanisms control b a i n i t e formation. 2. EXPERIMENTAL 2.1. Preparation of Alloys The a l l o y s were prepared by melting measured amounts of s i l v e r and cadmium of 99.999 pet purity i n evacuated s i l i c a capsules. The capsules were heated to 850°C for one hour, and the l i q u i d metal was shaken to ensure adequate mixing. The a l l o y s were then s o l i d i f i e d by dipping one end of the capsules i n t o cold water i n order to prevent piping. The s o l i d i f i e d ingots were hot r o l l e d at 650°C to approxi-mately 50 pet reduction i n cross-sectional area, sealed again i n s i l i c a capsules and homogenized for 48 hours at approximatly 15°C below the solidus temperature. Homogenized ingots were then hot r o l l e d (450°C) and f i n a l l y cold r o l l e d to sheets of the desired thickness (0.38 mm). Preliminary experiments were ca r r i e d out using Ag-Cd alloys of three nominal compositions: 44,45 and 46 at. pet Cd. These are marked i n the equilibrium Ag-Cd phase diagram i n F i g . 1. The alloys were analyzed for s i l v e r using a s i l v e r chloride gravimetric method (43) i n order to determine the t o t a l losses of cadmium associated with the sample preparation procedures. I t was found that the melting, subsequent heat tr e a t i n g and hot forming resulted i n a cadmium loss of approximately 0.1 at. pet Cd. 13 14 700 b SILVER, AT. PCT 55 50 CADMIUM, AT. PCT FIGURE 1 The relevant portion of the Ag~Cd equilibrium phase diagram. Dotted l i n e s indicate composition of the alloys used. 15 2.2. Quenching • The growth k i n e t i c s experiments required that specimens be quenched from the high-temperature 3-phase region i n order to r e t a i n the untransformed 3 * phased. _ _______ . . _ _______ I t i s assumed that quenching; cannot preserve the high-temperature disordered 3 phase (26, 31, 40). Some preliminary tests were c a r r i e d out by quenching evacuated pyrex capsules containing the specimen into iced brine. In general, the quenching rate was not s u f f i c i e n t l y f a s t to prevent the formation of b a i n i t e or other trans-formation' products. Rapidly t r a n s f e r r i n g the bare specimen from the s a l t pot to iced brine was also unsatisfactory; large cadmium losses resulted from the necessity of using high quenching temperatures to ensure that the specimen remained untransformed during transfer to the quenching bath. The composition change due to the cadmium loss induced formation of a phase which consumed substantial portions m of the parent 3-phase grains. Therefore, i t was necessary to use a method which would enable rapid heating to the quenching temperature,< a v i r t u a l l y instantaneous transfer to the quenching medium and a vigorous s t i r r i n g i n the quenching medium. Such a method was devised u t i l i z i n g high frequency induction heating. The apparatus consisted of a s i l i c a glass tube connected to an iced brine reservoir (Fig.2). The tube .was placed i n a water-cooled induction c o i l . Specimens ARGONj T H E R M O -COUPLE SILICA GRAPHITE FIGURE 2 Schematic diagram of the induction heating and quenching apparatus. 17 (15 x 3 x 0.38 mm) were placed i n a groove i n a small graphite core inside the tube. The tube was f i l l e d with argon before heating. The temperature was monitored using a chrome1-alumel thermocouple having i t s hot junction on the surface supporting the specimens. The specimens were heated to the desired temperature i n approximately 3 seconds, held for another 17 seconds, and then quenched by flushing the brine through the s i l i c a tube. This method usually ensured the retention of the B * phase, although some a m phase developed i n the t h i n cadmium-depleted surface layer. This layer was removed by e l e c t r o l y t i c thinning immediately afte r quenching. The average B' grain diameter produced by the heating and quenching was approximately 0.25-0.5 mm. For the habit plane measurements i t was necessary to have larger grains of at least 2 mm i n diameter. For that purpose, quenched s t r i p s consisting e n t i r e l y of the B' phase were strained approximately 5 pet and then lowered into a 630°C s a l t bath at a rate of approximately 10 mm/min. They were then quenched manually into iced brine. Large B 1 grains 2-4 mm wide and over 10 mm long were produced. Although most of the grains contained large areas of a m and ba i n i t e , some were completely free from any transformation product and thus were useful for habit plane analysis. 18 2.3 E l e c t r o l y t i c Polishing and Thinning The specimens for the o p t i c a l and scanning electron microscopy had to be polished e l e c t r o l y t i c a l l y i n order to remove the cadmium depleted layer and obtain a clean, smooth surface. Mechanical pol i s h i n g could not be used since the B1 phase transforms m a r t e n s i t i c a l l y when deformed at room temperature (28). E l e c t r o l y t i c p o l i s h i n g of specimens was accomplished using a 10 pet KCN-water solution with a l t e r -nating current. A st a i n l e s s s t e e l water-cooled beaker was used as the other electrode, a voltage of 13.5 V and a speci-men current density of 0.06 A/mm2 being maintained. Smoothly polished B' grains, and l i g h t l y etched grain boundaries and transformation products were revealed by t h i s procedure. The specimens for the o p t i c a l microscope were also etched using a solution of lOg CrOg and 1 g Na 2S0 4 i n 100 ml of water. This was necessary to enhance the contrast of the grain and p r e c i p i t a t e boundaries. Specimens for transmission electron microscopy, i n the form of 3 mm diameter discs, were cut out of the s t r i p s by spark machining. The centre of the disk was thinned using a j e t e l e c t r o l y t i c procedure at a voltage of 15 V a.c. and using the same e l e c t r o l y t e as for e l e c t r o l y t i c p o l i s h i n g . 2.4. X-Ray Analysis A 57.4 mm diameter Straumanis-type loading Debye-Scherrer camera with a revolving rectangular (2 x 0.4 s l i t diaphragm and CuK„ radiation was used for.the c r y s t a l structure analysis. The specimen, i n the form of a thin p o l y c r y s t a l l i n e wire, was cut from the sheet by spark machining and then e l e c t r o l y t i c a l l y thinned to approximately 0.3 mm diameter. A voltage of 26kV and long exposures of 15-100 hours were used to minimize the background i n t e n s i t y and to enable detection of weak p r e c i p i t a t e l i n e s i n the high angle region of the pattern. The specimen was also moved a x i a l l y i n increments of approximately 1 mm during the exposure to incorporate as many matrix grains as possible. The background i n t e n s i t y was further reduced by placing a 0.05 mm aluminum f i l t e r i n front of the f i l m . 2.5. Surface Relief The surface r e l i e f e f f e c t s associated with p r e c i p i t a t i o n were observed on the prepolished specimen surfaces following the p r e c i p i t a t i o n treatment. They were examined using the Zeiss interference microscope and a monochromatic thallium-vapor l i g h t . The fringe patterns c h a r a c t e r i s t i c of the surface r e l i e f were photographed and enlarged to allow measurements of the fringe displace-ments . 2.6. Habit Plane Measurements Habit planes of bainite plates were determined using a modification of the two-surface trace analysis. 20 Quenched and polished large-grain specimens were f i r s t cut by spark machining to obtain a sharp edge. The samples were then annealed at 240°C u n t i l a f a i r l y large number o f ! i s o l a t e d bainite-plate traces appeared on the surface. Before measurement, the specimens were l i g h t l y repolished i n order to remove the thin oxidized layer, and were etched. They were then fastened onto a goniometer which allowed rotation of the specimen around the axis of i t s sharp edge. The gonio-meter was f i r s t attached to the stage of the X-ray machine and a back-reflection Laue photograph was taken to determine the orie n t a t i o n of the matrix grain. The goniometer was then attached to the microscope stage and photographs of the two surfaces of the edge were taken. The edge of the specimen was always s l i g h t l y rounded and the traces quite short, thus i t rarely occurred that a trace extended far enough beyond the edge on both sides to allow a r e l i a b l e measurement of i t s p o s i t i o n to be made. In most cases, i t was necessary to attempt to i d e n t i f y other traces which seemed to belong to the plates of the same habit plane variant but which did not extend a l l the way to the edge. This proved possible, although, as the r e s u l t s w i l l show, a large scatter resulted, probably due to errors i n the matching of traces belonging to d i f f e r e n t plate variants. 2.7. Electron Microscopy The morphology and structure of pr e c i p i t a t e s as well as the orientation r e l a t i o n s h i p between the matrix and p r e c i p i t a t e l a t t i c e s was examined using the Hitachi HU-11A 100 kV electron microscope. A wide-angle (±30°) t i l t i n g stage was employed for the orientation r e l a t i o n -ship measurements and the structure analysis of the b a i n i t e . The orientation r e l a t i o n s h i p between the ba i n i t e and the matrix was determined from composite electron d i f f r a c t i o n patterns containing the {011} f + r e c i p r o c a l _ In t h i s work, the subscripts f and b are assigned to symbols r e f e r r i n g to the fee and bee structures respectively. l a t t i c e planes of the bainit e and the {111}^ planes of the matrix. These patterns were obtained using selected area d i f f r a c t i o n from an area straddling portions of both the bainite plate and the surrounding matrix. By making t h i s area s u f f i c i e n t l y small (maximum 0.5 um square) i t was possible to avoid any v a r i a t i o n of the in t e n s i t y of the spots due to buckling of the f o i l . Under these conditions the r e l a t i v e i n t e n s i t y d i s t r i b u t i o n of the equivalent spots was s o l e l y and d i r e c t l y related to the angle between the zone axis of a pattern and the o p t i c a l axis of the microscope. This angle was determined to an accuracy of better than ±0.5° by using an experimental procedure developed by Ryder and P i t s c h (41, 42). The (Oil) f-zone b a i n i t e p a t t e r n s were s u i t a b l e f o r the o r i e n t a t i o n r e l a t i o n s h i p d e t e r m i n a t i o n f o r the f o l l o w i n g reason; B a i n i t e p l a t e s c o n t a i n e d random s t a c k i n g f a u l t s which caused s t r e a k i n g i n the r e c i p r o c a l l a t t i c e a l o n g a ^111^ ^ d i r e c t i o n . When the d i r e c t i o n o f s t r e a k i n g d i d not c o i n c i d e w i t h the Ewald sphere, i . e . , when i t d i d not show i n the d i f f r a c t i o n p a t t e r n , the i n t e r p r e t a t i o n o f the p a t t e r n was very d i f f i c u l t . I t was, t h e r e f o r e , necessary to r o t a t e the f o i l u n t i l the ^ O l l ^ ^ zone c o n t a i n i n g the s t r e a k i n g d i r e c t i o n was i n the r e f l e c t i n g p o s i t i o n , i . e . , u n t i l the s t a c k i n g f a u l t s were n e a r l y p a r a l l e l to the o p t i c a l a x i s . F o r t u n a t e l y , t h i s brought the m a t r i x to an o r i e n t a t i o n having a ^ l l l ^ k zone i n the r e f l e c t i n g p o s i t i o n , e n a b l i n g the d i r e c t d e t e r m i n a t i o n o f i t s o r i e n t a t i o n r e l a t i o n s h i p w i t h the b a i n i t e from a s i n g l e photograph. 2.8. Growth K i n e t i c s Measurements For the p r e c i p i t a t e growth k i n e t i c s measurements, a p r e l i m i n a r y r e s e a r c h p l a n proposed u s i n g a hot stage o p t i c a l microscope. However, the m a g n i f i c a t i o n and r e s o l u t i o n c a p a b i l i t i e s o f an o p t i c a l microscope d i d not make p o s s i b l e even the c r u d e s t k i n e t i c measurements o f the p r e c i p i t a t e s i n a B' Ag-Cd a l l o y . T h e r e f o r e , the p l a n to d i r e c t l y observe i s o t h e r m a l p r e c i p i t a t e growth had to be r e p l a c e d by one i n c o r p o r a t i n g an a n n e a l i n g method t h a t separated growth from the observation and measurement of growth and so allowed the use of a scanning electron microscope. Specimens were i n i t i a l l y annealed i n a s i l i c o n o i l bath, maintained to within ±1°C of the desired temperature, u n t i l traces of pre c i p i t a t e s , v i s i b l e due to t h e i r surface r e l i e f , appeared on the electropolished surface. The traces were immediately photographed using an ETEC scanning electron microscope operated i n the secondary electron image mode. Specimens were then annealed for an additional increment of time, and the traces of the same pre c i p i t a t e s were photographed again. The procedure was repeated to monitor the entire growth of the p r e c i p i t a t e . S t a b i l i t y of the magnification was frequently tested by comparing the distance between unchanging d e t a i l s on the.surface of the specimen. The thickness of the b a i n i t e plate traces was measured d i r e c t l y on the negatives using a microdensitometer. Measurements of each trace were repeated ten times to minimize any measurement error.- A l l magnifications were corrected for t i l t i n g of the specimen r e l a t i v e to the electron beam. An attempt was made to determine the angle a between the ba i n i t e plates and the specimen surface. Thin layers of the specimen were e l e c t r o l y t i c a l l y removed and a measurement of the displacements of the plate traces r e l a t i v e to several a r t i f i c i a l reference marks was made. The error of these measurements was large due to d i f f i c u l t i e s i n estimating 24 the t h i c k n e s s of the e l e c t r o l y t i c a l l y removed l a y e r s and due to the s m a l l s i z e of the p l a t e s . The t h i c k n e s s of the removed l a y e r s was estimated by weighing the specimen bef o r e and a f t e r d i s s o l u t i o n and assuming t h a t the d i s s o l u -t i o n was uniform over the whole s u r f a c e . However, the specimen edges and the m a t r i x i n the v i c i n i t y o f the p l a t e s d i s s o l v e d f a s t e r thereby i n t r o d u c i n g an e r r o r i n the c a l c u l a t e d t h i c k n e s s o f the removed l a y e r . The average l e n g t h o f the p l a t e s being approximately 5-15 um r e q u i r e d t h a t l a y e r s of o n l y a few microns i n t h i c k n e s s be removed. The r e s u l t i n g s m a l l t r a c e displacement was d i f f i c u l t t o measure. The p r e c i s i o n o f the measurement was f u r t h e r reduced by e l e c t r o l y t i c e t c h i n g o f the m a t r i x - p l a t e boundaries and consequent rounding o f the p l a t e p r o t r u d i n g from the s u r f a c e . D e s p i t e t h i s l a r g e e r r o r , the data from these measurements showed t h a t the v a r i a t i o n of the measured t h i c k e n i n g r a t e s was predominantly due t o the v a r i a t i o n o f the angle a . A d e t a i l e d d i s c u s s i o n o f the r e s u l t s i s g i v e n i n S e c t i o n 3.10.1. The width of the s u r f a c e t r a c e s o f s e v e r a l l a r g e p l a t e s were c a r e f u l l y measured a f t e r the e l e c t r o l y t i c removal o f the s u r f a c e l a y e r and a l i g h t mechanical p o l i s h i n g t o l e v e l the p r o t r u d i n g p o r t i o n s of the p l a t e s . The widths were found t o be equal t o the widths o f t r a c e s measured on the o r i g i n a l s u r f a c e s a f t e r a n n e a l i n g . T h i s showed t h a t the t r a c e width o f the p l a t e s used f o r k i n e t i c measurements c o r r e s -ponded to the a c t u a l t h i c k n e s o f the p l a t e s below the s u r f a c e . 3. RESULTS AND DISCUSSION 3.1. Morphology of P r e c i p i t a t e s Formed during Quenching In some specimens which were encapsulated and quenched manually from the s a l t bath, a spectrum of cooling rates was achieved due to uneven contact with the quenching medium aft e r the capsule was broken. When the cooling rates were not f a s t enough to completely r e t a i n the 0 1 phase, p a r t i a l l y transformed structures resulted. The d e t a i l s of the trans-formation products were noted. Large, regular traces of b a i n i t e plates appeared f i r s t With decreasing cooling rate (Fig. 3a), then patches of am on the grain boundaries with b a i n i t e i n the i n t e r i o r of the grains (Fig. 3b). With a further decrease i n the cooling rate, a m spread into the i n t e r i o r of the grains (Fig. 3c). Slower cooling also produced thicker b a i n i t e plates with i r r e g u l a r sides which deviated l o c a l l y from the habit plane defined by the thin regular plates. F i n a l l y , at very slow quenching rates, i t was observed that the a m phase spread over the entire surface. In the micrograph i n F i g . 3c, there i s evidence of a m adapting to the shape of b a i n i t e . This indicates that the b a i n i t e plates were present p r i o r to the development of the massive product and acted as a boundary to the growth of the am-26 FIGURE 3 B a i n i t e p l a t e s and massive a m i n the B' m a t r i x of a Ag-46 at . pet Cd a l l o y quenched from 600°C. The quenching r a t e was i n s u f f i c i e n t t o r e t a i n the untransformed 3• phase, r e s u l t i n g i n formation o f b a i n i t e (a). Upon d e c r e a s i n g the q u e n c h i n g , r a t e , a m f i r s t formed on g r a i n boundaries (b), and then i n the i n t e r i o r o f the g r a i n s ( c ) . 27 Similar observations of pre c i p i t a t e s formed during quenching i n Ag-Cd, Ag-Zn and Cu-Zn alloys were made by other researchers (28,29). The rapid quenching device produced specimens with regular polygonal 6' grains, i n d i c a t i n g that quenching was successful. Occasionally, the grain boundaries showed traces of a , and, less frequently, portions of the specimen contained several long, narrow bainite plates formed during quenching. The average & 1 - g r a i n diameter was approximately 0.25-0.5mm. 3.2. Morphology of Prec i p i t a t e s Formed during Isothermal Annealing The morphology of the pre c i p i t a t e s developing during isothermal treatment at 160-320°C i n alloys with 44,45 and 46 at.pet Cd was examined using the o p t i c a l microscope, the scanning and the transmission electron microscope. The transmission electron microscope studies w i l l be presented i n Section 3.6. A two-surface analysis of specimens with varied amounts of pr e c i p i t a t e showed that two kinds of pr e c i p i t a t e s formed; a needle-like widmanstatten p r e c i p i t a t e and a plate-l i k e b a i n i t e . The ba i n i t e plates usually formed i n p a i r s , joined at an obtuse angle, giving r i s e to a chevron-shaped trace on the specimen surface. F i g . 4 shows a scanning-electron micrograph of the edge of a severely etched specimen containing both kinds of p r e c i p i t a t e s . 2 8 FIGURE 4 A scanning e l e c t r o n micrograph of the edge ( i n c l u d e d angle of approximately 90°) of' a s e v e r e l y etched specimen of Ag-45 a t . pet _Cd a l l o y annealed f o r 1,225 seconds a t 200°C. Both widmanstatten needles and b a i n i t e p l a t e s are v i s i b l e . 29 A examination of the e f f e c t of p r e c i p i t a t i o n temperature on the type and amount of p r e c i p i t a t e formed produced the following r e s u l t s . Widmanstatten needles formed both at the grain boundaries and i n the i n t e r i o r of the 3 ' grains, with grain boundary needles nucleating at an e a r l i e r stage. Bainite plates usually nucleated i n the i n t e r i o r of the grains. Judging by the r e l a t i v e area f r a c t i o n of each type of p r e c i p i t a t e , the ba i n i t e plates were predominant at the lower temperatures. Needles and plates both formed at the; intermediate temperatures. Bainite plates i n these mixed structures always nucleated before intragranular widmanstatten needles. Fig.5 shows o p t i c a l photomicrographs of two polished and etched Ag-45 at. pet Cd specimens. F i g . 5a shows only the p l a t e - l i k e b a i n i t e product formed at 160°C, while F i g . 5b contains a mixture of baini t e plates and widman-statten needles formed at 200°C. The r e l a t i v e area fractions of the two kinds of precip i t a t e s i n the mixed structures also depended on the amount of cadmium i n the a l l o y . In the 44 at.pet Cd a l l o y , intragranular needles did not form below 300°C, while i n the 45 at. pet Cd a l l o y they did not form below 200°C. Increasing the cadmium content of the a l l o y (44 to 46 at. pet) increased the thickness-to-length r a t i o of the plates. The thicker, shorter p r e c i p i t a t e s exhibited 30 FIGURE 5 Bainite plates (a) and a mixture of b a i n i t e plates and widmanstatten needles (b) formed i n a Ag-45 at. pet Cd alloy during annealing at 160°C for 57,600 seconds (a) and at 200°C f o r 1,225 seconds (b). 31 h-30um-H FIGURE 6 A mixture of b a i n i t e p l a t e s and widmanstatten needles formed i n a Ag-46 a t . p e t Cd a l l o y d u r i n g a n n e a l i n g a t 200°C f o r 25,600 seconds. Most p l a t e s degenerated t o needles isomorphous w i t h the widmanstatten n e e d l e s . The broad faces o f the p a i r of p l a t e s i n the c e n t r e are approximately p a r a l l e l to the plane of p o l i s h . 32 i r r e g u l a r boundaries and needle-like protuberances seemingly isomorphous with the widmanstatten needles (Fig. 6). An increased cadmium concentration and lower transformation temperatures increased the incubation time for formation of plates and needles. Lower transformation temperatures also decreased the average plate s i z e . It was observed that the size and appearance of plates formed during quenching was s i m i l a r to plates formed isothermally a f t e r 1-2 seconds at temperature of 280-320°C (Fig. 7). Thus, i t was assumed that the plates formed during quenching nucleate and grow at or above temperatures equivalent to those employed for isothermal p r e c i p i t a t i o n . The observed c h a r a c t e r i s t i c s of the p r e c i p i t a t e s i n the 3 ' phase of Ag-Cd alloys are very s i m i l a r to the c h a r a c t e r i s t i c s of the analogous pr e c i p i t a t e s i n Cu-Zn alloys (31-36). The phenomenological s i m i l a r i t y between these systems, which also have p a r a l l e l chemical and physical properties, w i l l be made use of i n further discussion. The incubation time for nucleation of b a i n i t e plates was d i f f e r e n t i n d i f f e r e n t matrix grains. Often, during the early annealing times, one grain was free of any v i s i b l e p r e c i p i t a t e s , while a neighboring grain had several hundred bainite chevrons. The average size of the plates and the number of b a i n i t e habit plane variants also changed from one grain to another grain. In most grains, the plates grew on several variants, r e s u l t i n g i n chevrons 33 / i-—200 um—H FIGURE 7 Bainite plates i n Ag-45 at. pet Cd a l l o y formed afte r approximately 2 s at 2 80°C. j—100 um—i FIGURE 8 The v a r i a t i o n of b a i n i t e p l a t e morphology i n d i f f e r e n t - m a t r i x g r a i n s . 35 oriented i n many d i f f e r e n t d i r e c t i o n s . However, i n some grains, chevrons tended to belong to the same variants, as i l l u s t r a t e d i n F i g . 8. This suggests that the formation of b a i n i t e depends on the orientation of the matrix grain, most probably due to the e f f e c t of d i f f e r e n t quenching stresses i n d i f f e r e n t grains. Thus, stress may have an important influence on the nucleation of b a i n i t e . 3.3. X-Ray Structure Analysis Debye-Scherrer patterns were obtained from the Ag-45 at. pet Cd a l l o y at d i f f e r e n t stages of transformation. The patterns contained two sets of l i n e s ; one set of r e l a t i v e l y sharp and intense but spotty l i n e s corresponding to the bcc structure of the coarse-grained matrix and a set of much weaker, dif f u s e l i n e s from the p r e c i p i t a t e . The major pr e c i p i t a t e l i n e s appeared at 20 values corresponding to an fee structure. However, some additional, extremely weak l i n e s , for which the 28 values could not be accurately determined, were observed. No superlattice l i n e s due to ordering could be observed since the atomic scattering factors for Ag and Cd are nearly equal. The values of the l a t t i c e parameters calculated from the patterns of the matrix and the p r e c i p i -tate are given i n Table I. The microstructures of the analyzed specimens are shown i n Figs .9-11. It was d i f f i c u l t to obtain well defined p r e c i p i t a t e 36 TABLE I La t t i c e Parameters of the bcc Matrix, a b , and the fee Pr e c i p i t a t e , af, i n the Ag-45 at.pet Cd Alloy A f t e r Annealing at 160, 200 and 240°C.+ Treatment 0 a b , A a f , A Quenched 3.324 -Annealing time at 160°C, s 12,600 3.325 4.178 25,600 3.326 4.177 57,600 3.329 4.175 Annealing time at 200°C, s 529 3.324 4.186 900 3.326 4.180 2,116 3.327 4.177 Annealing time at 240°C, s 25 3.324 4.185 64 3.326 4.183 144 3.326 4.180 + Each figure i s a mean of 2-4 measurements. The error for a b i s approximately ±0.001 A, for a f approximately ±0.002 A. 37 FIGURE 9 Annealing temperature 160°C; annealing time 12,600 s (a), 25,600 s(b), and 57,600 s (c) . 38 FIGURE 10 Annealing temperature 200°C; annealing time 529 s (a), 900 s (b), and 2,116 s (c). 39 F I G U R E 11 Annealing temperature 240°C; annealing time 25 s (a), 64 s (b), and 144 s (c). 40 l i n e s at high 2© angles for the specimens annealed for very short periods of time. This was due i n part to the small volume f r a c t i o n of the p r e c i p i t a t e which tended to decrease the l i n e i n t e n s i t y with respect to the background i n t e n s i t y , but also due to the small size of the p r e c i p i t a t e s which caused broadening of the l i n e s . For these reasons, the measurement error of the 26 values for i n i t i a l p r e c i p i t a t e growth was quite large, making i t impossible to investigate any tetragonal d i s t o r t i o n of the face-centred unit c e l l . The l a t t i c e parameter values l i s t e d i n Table I show that the density of the bcc parent and the fee product — 7 was almost i d e n t i c a l (for a b= 3.324 x 10 mm, the bcc 19 3 —7 l a t t i c e density i s 5.446 x 10 atoms/mm ; for af=4.186 x 10 19 ? mm, the fee l a t t i c e density i s 5.453 x 10 atoms/mmJ),meaning that the volume change during the transformation was n e g l i g i b l e . The v a r i a t i o n of the l a t t i c e parameters with annealing temp-erature and time was consistent with depletion of the p r e c i -p i t a t e and enrichment of the matrix i n cadmium. When the p r e c i p i t a t e i n i t i a l l y develops with a composition close to that of the matrix, i t w i l l be r i c h i n cadmium (see Table VII and the metastable phase diagram i n Appendix E). I f composition of the p r e c i p i t a t e as i t grows changes towards the lower, i . e . , equilibrium cadmium content, then the loss of the larger cadmium atoms should be r e f l e c t e d i n a decrease i n the l a t t i c e parameter of the p r e c i p i t a t e and a corresponding 41 increase i n the l a t t i c e parameter of the matrix. The change was also more pronounced at lower annealing temperatures, which i s consistent with the shape of the metastable phase diagram. However, the error of measurement was r e l a t i v e l y large, of the same order of magnitude as the l a t t i c e parameter changes. This prevented quantitative i n t e r p r e t a t i o n of the data. 3.4. Surface Relief The bain i t e plates formed a s i m p l e - t i l t type of surface r e l i e f , which i s t y p i c a l of an invariant plane s t r a i n (Fig. 12). In most cases, the t i l t was accompanied by some accomodation s t r a i n i n the matrix. The t i l t angle 0 was calculated from the following expression: fx tane = 2TF -where f i s the fringe displacement across the trace measured i n the d i r e c t i o n normal to the fringes i n the matrix, X(=0.54ym) i s the wavelength of the thallium-vapor l i g h t , T i s the width of the plate trace measured on the scanning electron micrograph, and F i s the fringe distance i n the matrix. The maximum observed t i l t angle i n the Ag-45 at.pet Cd a l l o y was 12+1°. Accuracy of the measurement was l i m i t e d by the small size of the plates. The surface r e l i e f created by the widmanstatten needles growing p a r a l l e l to the specimen surface was 42 HlO/tmn FIGURE 12 Interference micrographs of the surface r e l i e f caused by formation of b a i n i t e plates. HO/xmH FIGURE 13 Interference micrographs of surface r e l i e f caused by formation of widmanstatten needles. 44 d i f f e r e n t . I t was symmetrical about the centre of the needle, forming the so c a l l e d tent-type r e l i e f (Fig. 13). 3.5. Bainite Habit Plane Measurements Bainite plate habit planes were measured i n eight matrix grains within four d i f f e r e n t specimens, a l l annealed at 240°C. A l l of the measured habit plane poles are plotted in Fig.14 but rotated to the unit t r i a n g l e [001]-[Oil]-[111] b of the matrix. In several cases i t was possible to determine the habit plane poles of both plates i n a chevron. In such cases, one of the poles was rotated to the standard unit t r i a n g l e , and the other was plotted i n i t s o r i g i n a l p o s i t i o n r e l a t i v e to the f i r s t . The large scatter of results i n Fig.14 may be due to wrong matching of traces on two surfaces. However, two thirds of the poles l i e within the marked c i r c l e of approximately 3.5° radius. The plates forming pairs are oriented i n such a way as to have t h e i r poles symmetric to each other with respect to the nearest [011]^ pole of the matrix. 3.6. Transmission Electron Microscopy Results Transmission electron microscopy was used to study the morphology and structure of the b a i n i t i c p r e c i p i t a t e s , t h e i r orientation r e l a t i o n s h i p with the matrix and the 45 22o o FIGURE 14 Portion of the standard [001]^ stereographic projection of the matrix showing the measured habit plane poles of bai n i t e plates formed during annealing at 240°C. The c i r c l e below the [011] b pole (radius approximately 3.5) encompasses two thirds of a l l measurements. The c i r c l e above the [011]^ pole has the same size and i s centered i n the c r y s t a l l o g r a p h i c a l l y equivalent p o s i t i o n with respect to the [011] b pole. The poles marked with numbers belong to in d i v i d u a l plates joined i n pairs, e.g., 31 and 31a. The open t r i a n g l e represents the t h e o r e t i c a l habit plane pole [0.180747; 0.667566; 0.722279] b (see Section 3.7). 46 changes that occurred during a prolonged isothermal anneal. 3.6.1. Morphology and Structure of Bainite Freshly formed b a i n i t e plates were long and narrow, had p a r a l l e l sides and a very f i n e t i p . Fig.15a shows a micrograph of a ba i n i t e plate i n a Ag-45 at. pet Cd a l l o y a f t e r 15,900 s at 160°C. A micrograph of a plate i n the same specimen but at a higher magnification i s shown i n Fig.15b. The plates contain a high density of s t r i a t i o n s . A range of s t r i a t i o n densities were v i s i b l e i n a l l plates except those annealed for long periods of time. Fig.15c shows that the s t r i a t i o n s are i n fact two-dimensional planar f a u l t s l y i n g within the plates. These f a u l t s were always p a r a l l e l to the { l l l } f planes of the plates. A selected area d i f f r a c t i o n pattern from a baini t e plate a f t e r 15,900 s representing a {110} f r e c i p r o c a l l a t t i c e plane i s shown i n Fig.16. The structure i s recognized as the 3R structure, a stacking modulation of the fee structure, which i s derived from the fee l a t t i c e by i n t r o -ducing a stacking f a u l t at each t h i r d close packed layer. (A d e t a i l e d structure analysis i s given i n Appendix A.) The {110}f r e c i p r o c a l l a t t i c e plane of the 3R structure i s characterized by s p l i t t i n g of the o r i g i n a l fee spots into a series of three spots each i n one of the ( i l l ) f d i r e c t i o n s . In each series, the spots are arranged i n the order strong-weak-medium. The s p l i t t i n g should not occur i n the (c) FIGURE 15 Bainite plates i n a Ag-45 at. pet Cd a l l o y a f t e r 15,900 s at 160°C (a,b) and 36s at 240°C (dark f i e l d ) (c) . FIGURE 16 S e l e c t e d are d i f f r a c t i o n p a t t e r n of a b a i n i t e p l a t e a f t e r 15,900 s at 160°C. The s t r u c t u r e i s 3R. 4 9 r e c i p r o c a l l a t t i c e l i n e s of the zeroth kind. The fact that i n F i g . 16 the extra spots do appear along the l i n e through the o r i g i n , as well as along the l i n e s which are t h i r d from the o r i g i n (both kinds of l i n e s are of the zeroth kind) can be rea d i l y traced to double d i f f r a c t i o n - predominantly the double d i f f r a c t i o n from the strong 3R r e f l e c t i o n s , for example r e f l e c t i o n 114. This was confirmed by the disappearance of the extra spots when the specimen was rotated around the ( l l l ) f d i r e c t i o n of s p l i t t i n g . The s p l i t spots i n Fig.16 are accompanied by streaks i n the d i r e c t i o n of s p l i t t i n g . The streaks are caused by the random stacking f a u l t s i n the 3R l a t t i c e , which are apparent as the s t r i a t i o n s shown i n Fig.15. Prolonged annealing of b a i n i t e plates causes them to thicken and to change t h e i r structure. Figs. 17a, b show micrographs of t y p i c a l b a i n i t e plates aft e r successively longer periods of time at 160°C. I t can be seen that the density of s t r i a t i o n s decreases with increasing time, i n d i c a t i n g that the random stacking f a u l t s are annealing out. In Figs. 18a, b, selected area d i f f r a c t i o n patterns are shown which correspond to the above micrographs. The most s i g n i f i c a n t e f f e c t of increasing time i s the appearance of d i f f u s e fee spots of increasing i n t e n s i t y and sharpness. Simultaneously, the 3R structure spots are becoming weaker and more d i f f u s e . The change can best be seen i n Fig.19, i n FIGURE 17 B a i n i t e i n a Ag-45 a t . pet Cd a l l o y a f t e r 19,800 s (b) and 25,600 s (b) at 160°C. FIGURE 17 - continued 52 (a) FIGURE 18 Selected area d i f f r a c t i o n patterns of ba i n i t e i n a Ag-45 at. pet Cd alloy a f t e r 19,800 s (a) and 25,600 s (b) at 160°C. Note the appearance of fee spots and dissappearance of 3R spots. 5 3 (b) FIGURE 18 - continued FIGURE 19 Changes i n the d i f f r a c t i o n patterns due to the 3R to fee structure transformation. which are reproduced only single magnified rows of spots. The change i n the appearance of the 3R spots i s accompanied by t h e i r displacement from the o r i g i n a l equidistant positions. The strong-weak distance remains approximately the same, while the medium-strong becomes larger and the weak-medium shorter. Aft e r a s u f f i c i e n t l y long annealing time, whole plates or large areas within plates become free of the stacking f a u l t s . The micrograph i n F i g . 20a shows a plate with only a few stacking,faults, e a s i l y recognizable by th e i r c h a r a c t e r i s t i c e x t i n c t i o n fringes. Fig.20b shows the same f i e l d a few moments l a t e r , a f t e r one of the stacking f a u l t s has disappeared leaving only what appears to be a d i s l o c a t i o n resolved into two p a r t i a l s . A number of these resolved:dislocations are v i s i b l e i n the lower part of the micrograph i n F i g . 21. F i g . 22 shows a plate completely free of large stacking f a u l t s . Dislocation debris l e f t by the stacking f a u l t s can be observed. This behaviour can be explained i n terms of a 3 R to fee structure transformation which occurs by a random disappearance of the regularly d i s t r i b u t e d stacking f a u l t s . The 3 R structure with some random stacking fau l t s i s characterized by strong, sharp and regularly d i s t r i b u t e d s p l i t spots accompanied by streaks. The random disappearance of the regularly d i s t r i b u t e d stacking f a u l t s introduces disorder, causing the 3 R spots to become dif f u s e and displaced 56 FIGURE 20 Micrographs of a bainite plate a f t e r 900 s at 240°C. Note that the stacking f a u l t i n the upper righthand corner i n (a) disappeared i n (b) leaving a d i s l o c a t i o n resolved into two p a r t i a l s (A). FIGURE 21 Micrograph of a b a i n i t e p l a t e a f t e r 900 s a t 240°C. FIGURE 22 Micrograph of a bainite plate a f t e r 900 s at 240°C 59 from t h e i r positions. Simultaneously, fee stacking order i s introduced l o c a l l y , but due to the large number of random stacking f a u l t s , fee spots are also broadened. With a decrease i n the number of 3R stacking f a u l t s , the 3R spots become weaker, more di f f u s e and more displaced. Simul-taneously, the fee spots become stronger and sharper, u n t i l eventually the structure becomes fee with random stacking f a u l t s . An additional factor causing broadening of the 3R and fee spots i s the extremely small thickness of the discrete bands of these two structures which e x i s t during the tran-s i t i o n a l period of the structure transformation. Stacking f a u l t s probably disappear by generating a p a r t i a l d i s l o c a t i o n on the interface between the matrix and the b a i n i t e plate. This i s supported by the observation that the stacking f a u l t s always s t a r t disappearing from the plate-matrix:interface (Figs. 2 0 a , b). The p a r t i a l d i s l o c a t i o n then sweeps across the plane of the stacking f a u l t , canceling i t . I f the p a r t i a l d i s l o c a t i o n i s prevented from t r a v e l l i n g to the other interface by meeting another p a r t i a l d i s l o c a t i o n coming from the opposite d i r e c t i o n , a pair of p a r t i a l dislocations remains i n the plate, t h e i r separation distance r e f l e c t i n g the stacking f a u l t energy of the a l l o y and magnitude of the l o c a l l a t t i c e stress. I t was observed that the thickening of b a i n i t e plates was influenced by the random stacking f a u l t s within the plates. FIGURE 23 Micrographs of bainite plates a f t e r 900 s at 240°C. The portions with zero stacking f a u l t density thickened faster than the rest of the plates. 61 Figs. 23 a, b show micrographs of plates a f t e r 900 s at 240°C. The plates contain portions with zero stacking f a u l t density which have thickened faster.The p r e c i p i t a t e -matrix interface has bulged out between the portions that were seemingly slowed down by the remaining stacking f a u l t s . It appears that the disappearance of stacking f a u l t s changed the nature of the interface i n such a way to allow i t to migrate more e a s i l y . 3.6.2. Orientation Relationship The o r i e n t a t i o n r e l a t i o n s h i p between the b a i n i t e and the matrix was determined from the ( l l l ) b - (OlD^ composite electron d i f f r a c t i o n patterns, one of which i s shown i n F i g . 24. The r e s u l t s , expressed as angles between the four d i f f e r e n t sets of directions i n the matrix and i n the b a i n i t e l a t t i c e (the b a i n i t e l a t t i c e was indexed i n the cubic notation), are given i n Table I I . The same res u l t s are presented schematically i n F i g . 25. The orientation r e l a t i o n s h i p established i s within several degrees of the Bain orientation r e l a t i o n s h i p . The mutual orientation of plates joined i n a p a i r arid t h e i r orientation r e l a t i o n s h i p with the matrix were determined using the same technique. In Figs. 26a, b are shown d i f f r a c t i o n patterns which were taken from the branches of the b a i n i t e chevron shown i n F i g . 26c. The branches FIGURE 24 Selected area d i f f r a c t i o n pattern composed of ( l l l ) b and ( O l l ) f r e c i p r o c a l l a t t i c e planes. TABLE II Experimental Orientation Relationship Between the g» Parent Phase and the Bainite* Direction L a t t i c e i n the of Angle between poles, degrees Parent Bainite , D i f f r a c t i o n Pattern Number Mean value ±0 1 2 3 4 5 6 7 8 9 10 [ U l ] b [011] f 0.9 0.9 0.8 0.6 0.6 1.0 0.3 0.7 0.6 0.5 0.7±0.2 [100] f - 0.9 1.6 1.2 1.3 1.1 0.7 0.6 1 .1±0.4 [ o i i ] b [ l l l ] f + + .3.9 3.8 4 .0 4.3 4.7 4.5 4.8 4.6 3.8 4 .1 4.3±0.4 U 1 2 ] b [011] f — 1.3 1.3 1.6 0.9 0.9 1.0 0.9 1 .1±0.3 + A l l measurements were performed on electron d i f f r a c t i o n patterns composed of [ l l l ] b and [ 0 l l ] f zones. The angle between the zone axes was determined by the method developed by Ryder and Pitsch (41,42). ++ The pole of the stacking f a u l t plane (Tl LO 64 FIGURE 25 Schematic stereographic projection diagram of the orientation relationship between the B 1 parent and ba i n i t e . The ba i n i t e l a t t i c e i s indexed i n cubic notation; although the [111]^ and [011]f poles are here shown to coincide, they are actually approximately 0.7° appart. 65 The composite matrix - bainite d i f f r a c t i o n patterns (a,b) obtained from the branches of the chevron shown i n (c). 66 FIGURE 27 Orientation relationship between the two b a i n i t e plates (I and II) and the matrix. The normal to the projection i s p a r a l l e l to the o p t i c a l axis i n F i g . 26. Poles marked p, 1 and P-^11 are the th e o r e t i c a l habit plane poles of the plates I and I I . Their indices are px 1 -[-0.667566; -0.180774; 0.722279] b and p - _ H = [0.722279; -0.180747; -0.667566] b. 67 were oriented i n such a way that a rotation of approximately 51.5° about the o p t i c a l axis, followed by a rotation of approximately 180° about the [111]^ d i r e c t i o n , the d i r e c t i o n along which the streaking occurred, brought the two d i f f r a c t i o n patterns, a and b, into coincidence. This i s shown schematically i n F i g . 27. 3.7. Application of the Phenomenological Martensite Theory to the Formation of Bainite In the previous sections, i t was shown that the bai n i t e i n i t s early stages of formation displayed a highly regular p l a t e - l i k e morphology. The presence of an i n t e r n a l inhomogeneity, stacking f a u l t s , was established i n the plates and i t was found that the surface r e l i e f caused by the formation of the plates was of an invariant plane s t r a i n type. I t was also shown that the orientation r e l a t i o n s h i p between the matrix and the b a i n i t e was close to the Bain correspondence. These c h a r a c t e r i s t i c s , usually associated with a martensitic product, naturally led to the application of the phenomenological theory of martensite formation to the formation of the b a i n i t e plates i n the bcc B' matrix of the Ag-Cd a l l o y . The r e s u l t s predicted by the theory were compared with the experimental observations. The p r i n c i p l e s of the a n a l y t i c a l treatment of the martensite phenomenology, as applied to the present system, > are given i n Appendix B. The Bain l a t t i c e correspondence 68 was assumed, as shown i n F i g . B-1. The l a t t i c e parameters of the parent and the product at the transformation temper-ature (160-240°C) were assumed to be those values measured at room temperature i n the Ag-45 at. pet Cd a l l o y annealed at 20Q°C for 529 s (Table I ) . The e f f e c t of temperature on the l a t t i c e parameter values was neglected since the data on the thermal expansion of the ordered 8 1 phase were not available. A possible tetragonal d i s t o r t i o n of the product, r e s u l t i n g from the ordering (CuAu I type) inherited from the parent, was also neglected, since i t was not observed i n the d i f f r a c t i o n patterns. I t was further assumed that the shear system operating i n the product was (111) [ l l 2 ] f . The choice of the ( l l l ) f shear plane was consistent with the observation of stacking f a u l t s on {111} f planes. The shear d i r e c t i o n [112] f was chosen because t h i s i s the only one of the three (112) f directions i n the (111) f plane which does not v i o l a t e the atomic order. Two solutions of the invariant l i n e s t r a i n , S, corresponding to two c r y s t a l l o g r a p h i c a l l y d i s t i n c t variants of the invariant l i n e - invariant plane normal combinations, were obtained. The two variants are referred to as the (x , n 1) and the (x-j_, n 2) variants. The results for both variants derived from the theory are given i n Table I I I . The application of the invariant l i n e s t r a i n matrix (bSb) on the l a t t i c e of the parent re s u l t s i n the orientation relationship given i n Table IV. The mean values of the 69 TABLE III Strains and Habit Planes Predicted by the Martensitic Theory Assuming the Bain L a t t i c e Correspondence with L a t t i c e Parameters a b = 3.324 A and a f = 4.186 & and the shear system (111) [112] f. Variant Invariant Line Strain Matrix, (bSb) Habit Plane Pole, p± ( x l ' n l > 0.884593 -0.091389 0.064720 0.099149 0.864098 -0.270022 •0.024814 0.194768 1.228335 •0.667566 •0.722279 0.180747 (X!,n 2) 0.884593 -0.012685 0.143425 0.024814 0.883812 -0.149754 •0.099149 0.108019 1.242139 0.667566 0.180747 •0.722279 Direction of Magnitude of Magnitude of Angle of the Variant the Shape the Shape the L a t t i c e L a t t i c e Deformation, Deformation, Invariant Invariant m. Shear, m2 Shear, <x2 ( x ^ n ^ 0.748615 -0.643169 0.160966 0.230924 0.428838 24.20v (x 1,n 2) -0.748615 -0.160966 -0.643169 0.230924 0.237831 13.56' 70 TABLE IV Theoretical and Experimental Orientation Relationship Between the Parent and Product L a t t i c e s Direction i n the La t t i c e of Angle Between Poles, Degrees Theoretical Parent Bainite Variant (x 1,n 1) Variant (x 1,n 2) Experimental [ H l ] b [ i i o ] b [011] b [112]. [011] f [100] f [ H I ] / [011], 0.78 9.51 4.30 9.54 0.78 1.25 4.30 1.05 0.7±0.2 1.1±0.4 4.3±0.4 1.1±0.3 + Pole of the plane of l a t t i c e invariant shear. 71 experimental orientation r e l a t i o n s h i p from Table II are also included i n Table IV for comparison. Rotation of the habit plane poles of the two variants to the standard unit stereographic t r i a n g l e r e s u l t s i n the pole with indices [0.180747; 0.667566; 0.722279] b, which i s 2.25° from the pole [144] b, as shown i n F i g . 14. I t can be seen that the t h e o r e t i c a l habit plane pole i s approxi-m a t e l y 1° from the centre of the c i r c l e encompassing two thirds of the experimental habit plane measurements. The angle between the habit plane normal and the d i r e c t i o n of the shape deformation for both variants i s close to 90° (90.35° for variant ( x 1 # and 93.69° for variant (x-p n 2) ), which means that the maximum t i l t angle should occur when the habit plane i s approximately perpendi-cular to the surface of the specimen. For the t h e o r e t i c a l magnitude of the shape deformation m-^  = 0.230924, the maximum t i l t angle i s 13.3°. This value agrees well with the maximum measured value of 12°, considering the large error of the • measurement. The two variants predict an i d e n t i c a l magnitude for the shape deformation. However, the magnitude of the l a t t i c e invariant shear for variant (x-^, n 2) i s approximately a factor of two smaller than the magnitude for the variant (x^, n^), i n d i c a t i n g that the transformation i s more l i k e l y 72 to occur by the mechanism associated with the (x^, n 2) variant. The magnitude of the l a t t i c e invariant shear for the variant (x^, n 2) corresponds to a shear angle of 13.56°, which i s very close to the shear angle of 13.26° r e s u l t i n g from the creation of an i d e n t i c a l stacking f a u l t on each t h i r d ( l l l ) f plane, as i s present i n the 3R structure. The experimentally determined mutual orientation of chevron pairs (Figs. 26, 27) was described i n Section 3.6.2. If each of the two plates i s assigned a t h e o r e t i c a l habit plane of the variant (x^, n 2 ) , as shown i n F i g . 27, the poles of t h e i r habit planes are symmetrical with respect to the nearest (110)^ pole of the matrix, i n t h i s case the [101]^ pole. The same r e s u l t was obtained by the two-surface trace analysis (Section 3.5.). I t can be shown that when the t h e o r e t i c a l habit planes are assigned to two plates having the described orientation r e l a t i o n s h i p , the angle between the traces of the plates ( i . e . , the branches of the chevron) i n the f o i l has to be 165.5° providing that the specimen i s oriented so that the directions [ O i l ] f 1 , [ O i l ] f 1 1 and [ l l l ] b are approximately p a r a l l e l to the o p t i c a l axis . This i s i n excellent agreement with the measured angle of 164° between the branches of the chevron contained i n Fig.26c. This shows that the t h e o r e t i c a l habit plane i s not only c r y s t a l l o g r a p h i c a l l y s i m i l a r to the experimental habit plane, but i d e n t i c a l to i t , i . e . , the habit plane i s predicted 73 uniquely. A comparison of the t h e o r e t i c a l and experimental or i e n t a t i o n relationships, Table IV, shows that the t h e o r e t i c a l orientation r e l a t i o n s h i p of the variant (x^, n^) i s very close to the experimental one, the differences between the two being less than 0.5°. 3.8. Comparison with the Martensitic Products Observed i n Ag-Cd, Ag-Zn and Cu-Zn Alloys The crystallographic c h a r a c t e r i s t i c s of bain i t e i n the Ag-Cd a l l o y are s i m i l a r to the crystallographic charac-t e r i s t i c s of martensitic products i n Ag-Cd, Ag-Zn and Cu-Zn a l l o y s . Krishnan (28) reported a good agreement between the experimental and t h e o r e t i c a l results for the thermal martensite with the 2H structure found i n the Ag-45 at.pet Cd a l l o y . Ayers (27) found even better agreement for the plates of the isothermal twinned fee martensite formed at 280°C i n the Ag-37.8 at. pet Zn a l l o y ; the experimental habit plane poles were clustered around the t h e o r e t i c a l habit plane poles and the difference between the experimental and t h e o r e t i c a l orientation relationship was less than 0.5°. Cornells and Wayman (24, 36) also obtained excellent agreement between the experimental and t h e o r e t i c a l values for both the martensite and bainite i n Cu-Zu a l l o y s . They found that the characteris-t i c s of martensite and b a i n i t e i n t h i s a l l o y were i d e n t i c a l , 74 except that the b a i n i t e plates had a 3R structure and were much smaller than the martensite plates which had a twinned fee structure - 1" with a s l i g h t orthorhombic d i s t o r t i o n . + Such differences r e s u l t from the d i f f e r e n t modes of the l a t t i c e invariant shear and do not necessarily influence the macroscopic c h a r a c t e r i s t i c s of the transformation. In a l l cases, the t h e o r e t i c a l predictions were based on the assumption of a Bain correspondence and {111}(112^ l a t t i c e invariant shear. The experimental habit plane poles were clustered close to the l i n e connecting the [133]^ and [1443^ poles, s i m i l a r to that obtained i n the present r e s u l t s . The l a t t i c e orientation r e l a t i o n s h i p was e s s e n t i a l l y i d e n t i c a l to that found for b a i n i t e i n the Ag-Cd a l l o y . The Ag-Cd data also agreed well with the orientation r e l a t i o n s h i p obtained for the isothermal martensite i n the Ag^-Zn a l l o y , the difference being less than 0.5°. In l i g h t of the excellent agreement between the observed crystallography of b a i n i t e i n the Ag-45 at. pet Cd a l l o y and the results predicted by the martensitic theory and considering the s i m i l a r i t y with the crystallographies of ba i n i t e and martensite formed from the g 1 phase of Cu-Zn type a l l o y s , i t can be concluded that the nucleation and early growth of ba i n i t e occurs by a thermally activated martensitic process. 75 3.9 Origin and S t a b i l i t y of the 3R Structure of Bainite In a systematic analysis of equilibrium structures i n Au-Mn alloys with 20-28 at. pet Mn (the composition near to AugMn) , Sato e-t al. (44, 45) established the existence of close-packed structures having a long period stacking f a u l t modulation i n which the type of modulation was quite sensi-t i v e to composition. The s t a b i l i t y of these structures was thought to be due to the lowering of the energy of the conducting electrons by the creation of the B i r i l l o u i n zone boundaries at the Fermi surface. The opposing energy term due to the introduction of stacking fau l t s was assumed to be n e g l i g i b l e , since the stacking f a u l t boundaries are low energy boundaries accross which the number of nearest neighbor atoms does not change. Similar structures were found i n other noble and t r a n s i t i o n metal alloys and i n martensites of nonferrous alloys (46) including some martensites found i n the 3-phase alloys of Au-Cd (46), Ag-Cd (28) and Cu-Zn (24) , and bainite i n Cu-Zn (36). In a l l cases the electron-to-atom r a t i o i n the 3 phase i s close to 1.5. The modulation of the fee structure i n these alloys occurred because the fee structure i s not stable i n the composition range of the 3-phase. I t should be pointed out that i n the case of a martensitic transformation the possible r e s u l t i n g structures are l i m i t e d by the s t a r t i n g structures. In the 76 present work i t has been shown that the observed 3R structure i s compatible with a martensitic transformation. The order of appearance of modulated structures i n 8 phase al l o y s with increasing electron-to-atom r a t i o i s always 3R to 2H (24, 28, 46), with the 3R appearing at approximately 45 at. pet of the divalent component. An important question i s whether the b a i n i t e i n the Ag-Cd a l l o y can i n i t i a l l y form with the equilibrium composition of the fee a phase stable at that annealing temperature. An assumption that the composition of the b a i n i t e adjusts to i t s equilibrium value by long range d i f f u s i o n during the i n i t i a l stages of the transformation would not explain the 3R modulation i n the structure. The observed 3R modulation could occur eit h e r as a d i r e c t r e s u l t of a martensitic transformation, or as a way of s t a b i l i z i n g a close packed structure having a higher electron-to-atom r a t i o than allowed by equilibrium. In either case, the following conclusion i s reached. The bain i t e forms with a composition either i d e n t i c a l or very close to the composition of the parent. In the view of the dependence on composition of the stacking modulation of the fee structure, i t should be expected that the stacking f a u l t energy decreases with increasing electron-to-atom r a t i o . Indeed, Howie and Swann (41) found that the stacking f a u l t energies of copper 77 and s i l v e r continually decreased with increasing amounts of added aluminum or zinc, reaching a value less than one tenth of t h e i r values for pure copper and s i l v e r at the electron-to-atom r a t i o of approximately 1.35 (17.5 at.pet A l or 35 at.pet Zn). Howie and Swann also established that the stacking f a u l t energy i n Ni-Co alloys varied with an increasing amount of cobalt i n a s i m i l a r fashion, extra-polating to zero at approximately 75 wt. pet Co, which i s the composition where the fee to hep t r a n s i t i o n occurred. If the stacking f a u l t energy changes with composition i n a s i m i l a r manner throughout the a-phase region i n the Ag-Cd system, the bainite plates with t h e i r high density of stacking f a u l t s w i l l tend to form with a composition which has a lower stacking f a u l t energy, i . e . , a higher concen-t r a t i o n of cadmium, than the equilibrium a phase. This has already been deduced from the structure analysis. Only the prolonged annealing allows the r e a l i z a t i o n of the lower energy equilibrium a phase composition by long range d i f f u s i o n . This, i n turn, increases the energy of the stacking f a u l t s and t h e i r disappearance further reduces the energy of the system. To summarize, i t appears that the long period stacking f a u l t modulation of the fee structure i s an important energy factor i n s t a b i l i z i n g the bain i t e i n Ag-45 at. pet Cd a l l o y . However, t h i s s t a b i l i z a t i o n e f f e c t i s necessary and possible only i n the i n i t i a l stages of bain i t e formation. The 78 prolonged annealing at elevated temperatures allows p a r t i t i o n i n g of s i l v e r and cadmium between the bai n i t e and the matrix, thereby decreasing the electron-to-atom r a t i o of the bainite and d e s t a b i l i z i n g the 3R structure with i t s i n t r i n s i c high density of stacking f a u l t s . This simultaneously s t a b i l i z e s the fee structure, which has the lowest possible volume free energy at that temperature. 3.10 Growth Kinetics Growth k i n e t i c s measurements were made on the Ag-45 at. pet Cd a l l o y . This a l l o y was considered to be the most suitable as the 3 1 phase could be completely retained on quenching and p r e c i p i t a t i o n at low temperatures produced the desired p r e c i p i t a t e morphology. Bainite plates and widmanstatten needles exhibited d i f f e r e n t modes of nucleation and growth. The plates nucleated, grew rapidly to a given length and maintained that length for extended transformation times, although they continued to thicken. The termination of the rapid lengthening of the f i r s t i s o l a t e d plates i n most cases appeared not to be due to impingement with other p r e c i p i t a t e s . Some observations indicated that plates continued to lengthen very slowly a f t e r the termination of the rapid lengthening. However, the additional length deviated from the o r i g i n a l plate d i r e c t i o n or branched o f f into two new plates or needles. At l a t e r trnasformation times, a f t e r plates had ceased to 79 lengthen, new plates continued to nucleate and grow i n the same fashion. This behavior i s i l l u s t r a t e d by the series of micrographs shown i n Figs. 28 and 29. In contrast to the plates, the widmanstatten needles continued to both lengthen and thicken for extended growth times u n t i l the impingement with other p r e c i p i t a t e s . The growth k i n e t i c s of i s o l a t e d plates were measured at 160, 200 and 240°C. Lengthening of the plates o was so rapid at 200 and 240 C that i t was not possible to measure the i n i t i a l lengthening using the interrupted annealing method. However, the lengthening rate was measured at 160°C; a series of micrographs i l l u s t r a t i n g the lengthening of a pair of plates i s shown i n F i g . 30. Thickening of the plates was much slower and could be successfully monitored using the interrupted annealing procedure at a l l three temperatures. Micrographs shown i n Fig.31 i l l u s t r a t e thickening of a bainite plate; the plate thickened on both sides. 3.10.1 Analysis of Bainite Thickening Data A plo t of the half-width of the plate trace, X, as a function of the annealing time for a t y p i c a l plate i s shown i n F i g . 32. The curve has a parabolic shape, suggesting that b a i n i t e thickening obeys the parabolic rate equation for d i f f u s i o n controlled thickening. 80 3 0 ^ . m FIGURE 28 Optical micrographs of l i g h t l y etched surface of a Ag-45 at. pet Cd specimen annealed at 200°C. (a) After 625 s: A number of chevron shaped bainite traces appeared with an occasional widmanstatten needle (A). (b) After 900s: Bainite traces present i n (a) have either maintained t h e i r o r i g i n a l length or have lengthened s l i g h t l y , but a l l have increased t h e i r thickness. The t i p s of some of the plates apparently acted as nucleation s i t e s for widmanstatten needles (B). Widmanstatten needles continued to lengthen. A number of new bainite traces appeared (C). (c) After 1,225 s: The same behaviour i s continued; old bainite traces thicken and new ones keep appearing, while widmanstatten needles which have not impinged upon other precipitates continue to lengthen. 81 FIGURE 29 Scanning electron micrographs of the unetched surface of a Ag-45 at. pet Cd specimen annealed at 240°C. (a) Af t e r 16 s: A ba i n i t e chevron appeared, (b) After 36 s: The lower arm of the chevron from (a) did not lengthen although i t thickened appreciably, while traces of new plates appeared from the l e f t , the lower one stopping before impinging upon the o r i g i n a l plate, (c) After 49 s: Thickening continued without lengthening. 82 9,025 sec 11,025 sec 15,625 sec 24,025 sec FIGURE 30 Scanning e l e c t r o n micrographs o f a p a i r o f b a i n i t e p l a t e s i n a Ag-45 a t . pet Cd a l l o y showing t h e i r e a r l y growth a t 160°C. Both length e n i n g and t h i c k e n i n g are v i s i b l e . FIGURE 31 Scanning e l e c t r o n micrographs showing t h i c k e n i n g o f the t r a c e o f a b a i n i t e p l a t e at 240°C i n a Ag-45 a t . p e t Cd a l l o y 84 0 2000 4000 6000 8000 10000 ANNEALING TIME, S FIGURE 32 Thickening k i n e t i c s of a bainite plate trace at 200°C i n a Ag-45 at. pet Cd a l l o y 85 FIGURE 33 The X 2 t a p l o t f o r the b a i n i t e p l a t e t r a c e from F i g . 3 2 . 8 The parabolic rate equation can be written i n the + An o u t l i n e of the theory of volume d i f f u s i o n controlled p r e c i p i t a t e growth i s given i n Appendix C. following form: X = L[D ( t a - x ) ] h , t a > T (1) or t_ = — . x 2 + T , (la) L 2 D where t a i s the t o t a l annealing time and T i s the incubation time for nucleation of the plate. Eq. (la) shows that t_ i s ci l i n e a r l y dependent on X 2. Square of the half-width of the 2 plate traces, X , was plotted as a function of the annealing time, t a , as shown i n F i g . 33 for the plate from F i g . 32. Linear relationships were obtained i n a l l cases. The slopes d ( X 2 ) / d t a are l i s t e d i n Table V. The X 2 vs. t plots obtained a f t e r long growth times deviated from a straight l i n e behaviour. I t was assumed that the deviation occurred when the d i f f u s i o n f i e l d of the precip tate overlapped with the d i f f u s i o n f i e l d s of neighboring p r e c i p i t a t e s . The width of the plate traces and therefore the measured growth rates obviously have to depend on the angle a between the plates and the specimen surface. An attempt was made to determine that angle. The plates gorwn at 240°C ( l i s t e d i n Table V) were chosen because of t h e i r r e l a t i v e l y 87 TABLE V 2 2 Bainite Plate Thickening Slope d(X ) / d t a i n nr/s for Traces of Bainite Plates Grown i n a Ag-45 at. pet Cd A l l o y at 160, 200 and 240°C Plate No. 160°C [d(X 2)/dt a]x 1 0 1 8 200°C [d(X 2)/dt a]x 1 0 1 6 240°C [d(X 2)/dt a]x 1 2.71 5.43 1.82 2 2.41 2.37 2.04 3 1.72 5.43 1.88 4 1.92 17.31 2.23 5 1.13+ 2.02+ 2.75 6 2.38 2.34 1.73+ 7 2.01 3.10 4.74 8 1.24 2.82 4.00 9 1.95 2.22 1.84 10 1.31 2.34 3.56 11 1.15 3.20 3.29 12 1.17 3.28 2.31 13 2.31 5.20 5.00 14 2.41 2.46 15 2.10 + Minimum observed growth rate 88 large s i z e . I n i t i a l l y , i t was assumed that the growth rate i s equal for a l l plates growing without interference with the neighboring p r e c i p i t a t e s , so that a l l the plates grow to the same thickness T at time t a f t e r nucleation. Then the v a r i a t i o n of the measured trace thickness, 2X^, at time t can be assumed to be only due to the v a r i a t i o n of the angle a for d i f f e r e n t plates (Fig.34). I t was also assumed that the plate with the slowest measured growth rate, or plate No.6 i n Table V, i s normal to the surface, i . e . , that the angle a for t h i s plate i s 90°. Using these assumptions, i t i s possible to calculate the angle a for a l l plates using the following r e l a t i o n : T t s i n a c a l c " — ' ( 2 ) with the parameter T equal to the measured trace width of plate No.6, for which s i n a = 1. In F i g . 35, the measured trace thickness, 2X t at t=49 s i s plotted as a function of a c a l c * In F i g . 35 the same measured trace width was plotted as a function of a e X p / the angle determined experimentally by the method described i n Section 2.8. The experimental curve shows that the v a r i a t i o n of the trace width i s predominantly due to the v a r i a t i o n i n « e Xp/ although there i s a substantial scatter of the data, e s p e c i a l l y for angles 89 FIGURE 34 Schematic representation of the dependence of the plate trace width, 2X t, on the angle between the plate and the specimen surface, q, and on the plate thickness, T t (sin a = T+./2X+.) . 90 FIGURE 35 Thickness of plate traces at a given growth time plotted as a function of the angle between the plate and the specimen surface, a. ©-Angle a calculated assuming that the growth rate was the same f o r a l l plates and that plate No.6 was perpendicular to the surface of the specimen. • — Angle a measured by s e r i a l d i s s o l u t i o n . The number re f e r to the 240°C b a i n i t e plates i n Table V. * less than 70 . There i s no reason to believe that the trace width depends on a exp i n any other way than as predicted by the Eq.(2). Therefore, the discrepancy of the two curves i n F i g . 35 i s explained by a large systematic error i n measuring a e Xp. This error increases with increasing a e X p , approaching 100 pet for the thickest traces. The sources of the error are probably the technique f o r measuring the thickness of the e l e c t r o l y t i c a l l y removed specimen layers (see Section 2.8) and the p r e f e r e n t i a l e l e c t r o l y t i c attack of the matrix i n the v i c i n i t y of the plates. These may also cause the scatter of the data around the experimental curve, although the scatter might to a certain degree r e f l e c t a true v a r i a t i o n of the plate thickness. In the l i g h t of the above discussion i t 2 was concluded that the observed scatter i n the d(X )/dta values i n Table V for a l l three temperatures was predomi-nantly due to the v a r i a t i o n of the angle between the i n d i v i d u a l plates and the specimen surface. Also, i t was assumed that those plates exhibiting a minimum growth rate (marked i n Table V) were perpendicular to the surface of the specimen and therefore exhibited the true thickening rate. D i f f e r e n t i a t i n g Eq. (la) and solving for D, the following expression was obtained: 1 d(x2)  d t a (3) 92 The e f f e c t i v e d i f f u s i v i t i e s f o r the thickening of plates at 160, 200 and 240°C were calculated using Eq. (3) and the minimum values of d(X )/dt from Table V. The r e s u l t s a are given i n Table VI. The values of L, calculated using Eq. (C-2), and the values of other physcial parameters used i n the c a l c u l a t i o n are summarized i n Table VII. The ac t i v a t i o n energy and the frequency factor for d i f f u s i o n were found from the Arrhenius plo t shown i n Fig.36. The calculated value f o r the a c t i v a t i o n energy, EA=1.89 x 10^ J/mole, agreed very well with the estimated value of 1.658 x 10 J/mole (see Appendix D). The calculated 4 2 value for the frequency factor D Q = 3.74 x 10 m /s was several orders of magnitude higher than the expected value. One possible reason for t h i s could be the large error introduced due to the small temperature range, 80°C, over which the Arrhenius p l o t was drawn. A rough estimate of t h i s error can be ca r r i e d out assuming that the error made 2 by choosing the minimum measured values of d(X ) / d t a as representative of the true growth rates at the given temperatures i s not larger than the scatter of the data i n Table V. Thus, standard deviation about the mean value of the data i n Table V was used to calculate the error bars i n F i g . 36. Drawing a l i n e with minimum slope through the 5 2 error bars, E A = 1.61 x 10 J/mole and D Q = 31 m /s were obtained; for a l i n e with maximum slope, EA=2.09 x 10 J/mole 93 TABLE VI Calculated E f f e c t i v e D i f f u s i v i t i e s and Estimated D i f f u s i v i t i e s 2 Calculated E f f e c t i v e D i f f u s i v i t y , m /sec Temperature ' Estimated o Bainite Bainite Widmanstatten D i f f u s i v i t y Thickening Lengthening Lengthening m 2/sec 160 6.1 x 10" 1 9 1.1 x 10" 1 6 - 2 x 10" 1 9 200 5.0 x IO" 1 7 - - 1 x 10" 1 7 240 2.1 x 10" 1 5 - 1.9 x 10" 1 5 3 x 10~ 1 6 94 TABLE VII Physical Parameters Used i n the Calculations i Parameter Value Obtained From Temperature, °C 160 200 240 L 1.36 2.01 2.84 Eq. (C-2) c , a t . p e t Cd 42.5 43.6 44.2 Metastable Ag-Cd phase diagram c Q , at. pet Cd 49.5 49.4 49.3 (Fig.E-l,App.E) ftQ 0.643 0.759 0.843 QQ=(c„-c0)/(cop-c0) V . m3/mole 1.12 x 10"5 - 1.13 x 10~ 5 Ref. (88) a 0.5 J/m2 (=o a/gi for Cu-Zn) Ref. (71) e c d g i 13 (=e Z ngi for Cu-Zn) Ref. (71) 95 TEMPERATURE, °C 240 200 160 14 -15 h -16 h 0J o -17 K -18 h -19 1-9 20 21 2-2 2-3 2 1/T, l / °Kx l0 3 FIGURE 36 Log D e f f V-6 .1/T for a Ag-45 at. pet Cd alloy. 96 and DQ=7.6 x 10 m /s. Cl e a r l y , the calculated value of D Q i s very sens i t i v e to small changes i n slope of the Arrhenius pl o t . The e f f e c t i v e d i f f u s i v i t i e s obtained from the bainite thickening k i n e t i c s could not be compared with independently measured d i f f u s i v i t i e s i n the Ag-Cd B 1 phase. Such information was not available i n the l i t e r a t u r e . Experimental measurement of d i f f u s i v i t y i n the metastable ordered B ' phase e x i s t i n g only i n a narrow temperature-composition region of the phase diagram i s very d i f f i c u l t . However, the s i m i l a r i t y between the Ag-Cd and the Cu-Zn systems and the extensive d i f f u s i o n data available for both the a and B phases of the Cu-Zn system and for the a phase of the Ag-Cd system allowed an estimate of the d i f f u s i v i t i e s i n the B'Ag-Cd to be made (see Appendix D). The results are shown i n Table VI. It i s evident that the calculated e f f e c t i v e d i f f u s i v i t i e s for thickening of bainit e plates agree with the estimated d i f f u s i v i t i e s for thickening of ba i n i t e plates within one order of magnitude. This agreement i s sat i s f a c t o r y considering the uncertainty of the exact p o s i t i o n of the a / ( a + B ' ) and the ( a + B ' J / B 1 phase boundaries i n the Ag-Cd binary phase diagram at low temperatures (see Appendix E). 3.10.2 Analysis of Bainite Lengthening Data Consider the t i p of a bainit e plate (Fig.37a) which T ( t o - T ) TIME, t FIGURE 37 Schematic diagram of a p a i r of bain i t e plates which nucleated i n the i n t e r i o r of the specimen at point N, emerged on the surface of observation at point E and formed the trace ABC at time t ( a ) , and lengthening k i n e t i c s of the trace EC(b). 98 nucleated i n the i n t e r i o r of the specimen and which continued to grow at a steady-state rate v. At time t f i t forms a semi-circular plate of radius r ( t ) = v ( t - T ) , where T i s the incubation time. The length of the trace of the plate on the surface of observation measured from the point of emergency to the t i p i s then given by the following equation: l(t) = v [ ( t - T ) 2 - ( t Q - T)2]H, t * t Q f (4) where t i s the time when the plate f i r s t emerges on the surface of observation. Equation (4) describes a part of a hyperbola, as shown i n F i g . 37b. When t Q = T, i . e . , when the plate i s nucleated on the observation surface, the hyperbola becomes the straight l i n e I = v(t - x). D i f f e r e n t i a t i n g Eq. (4) with respect to t and rearranging gives [(t - t ) 2 - ( t D - x)2]h v = v (t) (5) obs t — x where v o b s ( t ) = (d£/dt) t_ t, the observed growth rate of the trace at time t. When t = x, v = v . , as expected. o ob s The length of the traces of ba i n i t e plates grown at 160°C was plotted as a function of the annealing time, as shown i n F i g . 38 for three t y p i c a l plates covering the range of observed lengthening rates. The curves conformed 99 Lengthening k i n e t i c s at 160°C of ba i n i t e plate traces i n a Ag-45 at.pet Cd alloy. 100 to the hyperbolic shape, although at longer growth times a negative deviation was observed. I t was thought that t h i s r e f l e c t e d the gradual decrease i n growth rate associated with the t i p of the plate entering the d i f f u s i o n f i e l d s of other p r e c i p i t a t e s . In order to e s t a b l i s h the true growth rate, v, from an observed growth rate, v i t was necessary to know the incubation time, x, and the emergence time, t Q . In p r i n c i p l e , T for a p a r t i c u l a r plate could be found by constructing the asymptote of i t s hyperbola. However, th i s was d i f f i c u l t because the curves i n Fig.38 deviated from the hyperbolic r e l a t i o n s h i p at longer times. The fact that the measured plates were the f i r s t ones to be observed indicated that they had a l l nucleated close to the surface. I t was therefore assumed that the plate exhibiting the minimum growth rate (plate 1 i n Fig.38) had nucleated on the surface. Its intercept with the time axis (200 seconds) was then equal to the incubation time, x. I t was further assumed that the fastest growing plate (plate 3 i n Fig.38) had also nucleated at the same time, x = 200 s. Extrapolating t h i s l i n e to I = 0 gave -9 t o~7.500 s. From the same l i n e , v Q b s (10,000 s) = 1.2 x 10 m/s. Substituting these values into Eq.(4) yielded v = 6.76 x 10~^° m/s. The parameter values necessary for c a l c u l a t i n g d i f f u s i v i t y from the growth rate are l i s t e d i n Table VIII. These were obtained i n the following way. The 101 TABLE VIII Values of the Parameters for Calculation of D e f f from the Lengthening Kinetics of Bainite Plates at 160°C and of Widmanstatten Needles at 240°C i n the Ag-45 At. Pet Cd Alloy C r i t i c a l Plate Dimensionless Plate Tip Tip Radius, m Growth Rate Radius, m Bainite Plate _ R Lengthening Pc=9.50 x 10~ 9 p=0.276 p=8.83 x 10 0 at 160°C Widmanstatten 8 Needle Lengthe- p'=2.33 x 10" 8 p=2.480 p=6.87 x 10 ning at 240°C 102 c r i t i c a l plate t i p radius p = 9.50 x 10 *" m was obtained c using the modified Gibbs-Thornson Equation (Eq. (C-6)) with the data from Table VII. The dimensionless growth rate — 8 p=0.276 and plate t i p radius p=8.83 x 10 m, both corres-ponding to the maximum growth rate for the given dimensionless supersaturation fiQ=0.643 for the Ag-45 at. pet Cd a l l o y at 160°C, were obtained from Eq. (C-3) by Trivedi's method described i n Appendix C. Knowing v, p and p, the e f f e c t i v e d i f f u s i v i t y for lengthening of b a i n i t e plates at 160°C was then calculated from the expression D=vp/2p; a value of D e f f = 1.1 x 10*"-^ m2/s was obtained. The modified Zener-H i l l e r t Equation (Eq. (C-5)) yielded the same value. This was more than two orders of magnitude larger than the e f f e c t i v e d i f f u s i v i t y obtained from the 160°C baini t e thickening k i n e t i c s . This means that the bain i t e lengthening rate was two orders of magnitude larger than that allowed by a volume d i f f u s i o n controlled growth process. 3.10.3. Analysis of Widmanstatten Lengthening Data The lengthening.rate of widmanstatten needles growing on and along the specimen surface was measured at 240°C. A series of micrographs i l l u s t r a t i n g the growth i s shown i n F i g . 39a. The p l o t of the needle length as a function of the growth time i s shown i n Fig.39b f o r three t y p i c a l needles. A l i n e a r r e l a t i o n s h i p was obtained. The fastest and the slowest observed growth rates were approximately factor of three d i f f e r e n t , the average growth rate being equal to FIGURE 39 Scanning electron micrographs showing the growth of a widmanstatten needle (a) and lengthening k i n e t i c s of widmanstatten needles (b) i n a Ag-45 at.pet Cd al l o y at 240°C. 104 FIGURE 39 - continued 105 v=1.38 x 10~ 7 m/s. _ Q The c r i t i c a l needle t i p radius, p '=2.33 x 10 m, c was calculated using the modified Gibbs-Thomson Equation (Eq. (C-6a)) with the data from Table VII. The calculated parameter values are l i s t e d i n Table VIII. The dimensionless _ g growth rate p=2.48 and needle t i p radius p=6.87 x 10 m for the dimensionless supersaturation nQ=0.84.3 for the Ag-45 at. pet Cd a l l o y at 240°C were obtained by the method described i n Appendix C. The e f f e c t i v e d i f f u s i v i t y for lengthening of widmanstatten needles was then calculated using the average measured needle lengthening rate and the calculated values for p and p. The obtained value of —15 2 D e f f = 1.9 x 10 m/s agreed very well with the value of the ef f e c t i v e d i f f u s i v i t y of 2.1 x 10~5m2/s obtained from the 240°C baini t e thickening k i n e t i c s (Table VI). 3.10.4 Discussion of the Growth Kinetics Results The i n t e r p r e t a t i o n of the thickening k i n e t i c s for bainit e plates at 160, 200 and 240° C i s straightforward when i t i s assumed that the estimated values for d i f f u -s i v i t i e s are correct within one order of magnitude; t h i s i s supported by the comparable value for the d i f f u s i v i t y obtained by measurements of the lengthening k i n e t i c s of the widmanstatten needle at 240°C. The res u l t s of the thickening k i n e t i c s are then i n good agreement with the 106 Zener-Frank model of volume d i f f u s i o n controlled growth of a p r e c i p i t a t e plate. This means that the interface between the matrix and the broad sides of the bai n i t e plates i s disordered and advances uniformly over i t s entire area. This conclusion does not agree with the studies which found that thickening of p r e c i p i t a t e plates occurred by a ledge growth mechanism (48, 49), i n agreement with Aaronson's general theory of p r e c i p i t a t e morphology (50, 51). The int e r p r e t a t i o n of the lengthening k i n e t i c s measurements of bain i t e plates during t h e i r early stage of growth at 160°C requires a reappraisal i n l i g h t of the estimated values for the d i f f u s i v i t i e s (Table VI). If the estimated d i f f u s i v i t i e s are correct, the larger than expected d i f f u s i v i t y obtained from the lengthening k i n e t i c s shows that the lengthening i s accelerated beyond the rate permitted by volume d i f f u s i o n . Similar observations have been reported for bainite plates i n Cu-Zn alloys (31, 52). Repas (52) found that the plates lengthened at rates up to two orders of magnitude higher than that predicted by the Zener-Hillert volume d i f f u s i o n model. Hornbogen and Warlimont (31) observed s i m i l a r lengthening rates and found that lengthening ceased very soon, while thickening continued for an extended time. (Both studies were performed by measuring the size of the largest plate i n the f i e l d of observation as a function of the specimen annealing time.) I t should be emphasized, though, that i t i s d i f f i c u l t to ascertain the si g n i f i c a n c e 107 of these observations since i t i s not clear whether they referred to the same, early stage of growth during which the present observations on Ag-Cd alloys were made. Onthe other hand, i f i t i s allowed that the actual d i f f u s i v i t i e s are approximately two orders of magnitude higher than was estimated, the measured baini t e lengthening rate would agree with that of a volume d i f f u s i o n controlled growth process. This p o s s i b i l i t y i s less l i k e l y since i t requires anomalously high d i f f u s i v i t e s . Nevertheless, i t i s a t t r a c t i v e because i t then allows the p o s s i b i l i t y that the broad faces of the plates are p a r t i a l l y coherent and growing at a slower rate, controlled by the l a t e r a l movement of incoherent ledges, i n agreement with Aaronson's theory of p r e c i p i t a t e morphology. However, the present measurements have c l e a r l y shown that the thickening k i n e t i c s i s parabolic, which i s c h a r a c t e r i s t i c of a planar disordered interface. The k i n e t i c s of thickening r e s u l t i n g from the movement of regularly d i s t r i b u t e d ledges w i l l be the same as the k i n e t i c s of i n d i v i d u a l ledges, which i s l i n e a r for widely spaced ledges having i s o l a t e d d i f f u s i o n f i e l d s (53).At the other extreme, i f the ledges were closer to each other, causing t h e i r d i f f u s i o n f i e l d s to overlap, t h e i r k i n e t i c s would deviate from l i n e a r towards the parabolic d i f f u s i o n rate of a competely disordered i n t e r f a c e . Thickening measurements on various systems (48, 49, 54) have confirmed t h i s conclusion q u a l i t a t i v e l y . Ultimately, i f thickening by ledges occurred at a volume-diffusion controlled rate, the distance between 108 the ledges would have to be zero and the interface would become indistinguishable from a completely disordered interface. Thus, on balance, i t i s more l i k e l y that the b a i n i t e plates lengthened at a rate faster than permitted by the volume d i f f u s i o n controlled model. This reinforces the conclusion derived from the crystallographic studies that the nucleation and early growth of b a i n i t e plates occurs by a martensitic process. Later thickening then occurs by a d i f f u s i o n controlled growth process. 3.10.5 General Discussion The choice of the b a i n i t e growth mechanism may depend on competitive growth k i n e t i c s . The martensitic process i s able to lower the free energy of the system by producing a quantity of b a i n i t e faster than i s possible by a d i f f u -sional mechanism. The nucleation and growth of b a i n i t e by the thermally activated martensitic process may be assisted by the residual stresses i n the quenched 8' phase. As these stresses are accommodated by a martensitic growth process and adverse stresses accumulate, the martensitic growth ceases. In the meantime the p a r t i t i o n i n g of s i l v e r and cadmium through long range d i f f u s i o n becomes s i g n i f i c a n t . The r e s u l t i n g composition changes a f f e c t the structure of the b a i n i t e by an n i h i l a t i n g the stacking f a u l t s , transforming i t into 109 equilibrium a phase. Thus, the l a t t e r stage of growth i s controlled by long range d i f f u s i o n giving the parabolic growth rate. 4. CONCLUSIONS (1) There are many s i m i l a r i t i e s i n the morphology, structure and growth of pre c i p i t a t e s formed at low temperatures i n the g' phase of Cu-Zn and Ag-Cd a l l o y s . (2) The b a i n i t i c transformation i n 44-46 at. pet Cd g' Ag-Cd alloys can be suppressed by rapid quenching. When formed during quenching, the bainite nucleates and grows before or during the competitive trans-formation to the massive a phase. (3) P l a t e - l i k e b a i n i t i c p r e c i p i t a t e s form isothermally i n 44-46 at.pet Cd alloys i n the temperature range 160-320°C. At higher temperatures the plates form competitively with the needle-like widmanstatten p r e c i p i t a t e . At lower temperatures the plates are the only intergranular transformation product. Some grain boundary side needles are always present. (4) The 3R structure of the freshly formed bainite plates, t h e i r surface r e l i e f , habit plane and orientation r e l a t i o n s h i p with the matrix were consistent with the phenomenological theory of martensite formation. (5) Prolonged annealing of the ba i n i t e plates at t h e i r temperature of formation causes t h e i r structure to change to fee. 110 I l l (6) The i n i t i a l lengthening of the b a i n i t e plates appears to be faster than permitted by a volume d i f f u s i o n controlled process. (7) Volume d i f f u s i o n probably controls the thickening of the b a i n i t e plates during the l a t e r growth stages. (8) Needle-like widmanstatten pr e c i p i t a t e s lengthen at a rate controlled by volume d i f f u s i o n . (9) The morphology, structure and other c h a r a c t e r i s t i c s of the freshly formed b a i n i t e plates are consistent with t h e i r formation by a thermally activated martensitic process. SUGGESTIONS FOR FUTURE WORK A deeper insight into the processes of baini t e formation i n the Ag-45 at. pet Cd a l l o y would be achieved by investigating the ef f e c t s of stress on the nucleation and growth of bainite plates. The res u l t s of such a study, i f used with the predictions of the martensitic theory, could help explain the mechanism of nucleation of plates and determine the factors r e s t r i c t i n g t h e i r lengthening to the i n i t i a l stage of growth. Another important object of study i s the ba i n i t e -matrix in t e r f a c e . The information about the degree of the coherency of the interface and i t s mobility, as well as about the e f f e c t of the 3R to fee structure transformation would contribute towards the understanding of the mechanism of the ba i n i t e growth i n the various stages of i t s formation. 112 APPENDIX A Structure Analysis The e f f e c t of a high density of stacking f a u l t s i n the fee l a t t i c e was studied by Paterson (55), Whelan and Hirsh (56) and Sato et al. (44,45,57). By using the kinematical theory of X-ray d i f f r a c t i o n , Paterson showed that a high density of random stacking f a u l t s d i s t o r t e d the fee r e c i p r o c a l l a t t i c e , while by using the dynamical theory df electron d i f f r a c t i o n , Whelan and Hirsch found that the d i s t o r t i o n of the l a t t i c e was accompanied by streaking along the (ill)£ axis perpendicular to the stacking f a u l t s . Sato et al' studied the modulation of the fee structure due to a regular d i s t r i b u t i o n of stacking f a u l t s using the kinematical theory of electron d i f f r a c t i o n . They found that the r e c i -procal l a t t i c e of the modulated structure was characterized by s p l i t t i n g of c e r t a i n fee r e c i p r o c a l l a t t i c e points i n the ( i l l ) d i r e c t i o n . They also analyzed the e f f e c t of random stacking f a u l t s on the r e c i p r o c a l l a t t i c e of the modulated structures and found that they could cause broadening and displacement of the s p l i t spots. A close packed structure i s s p e c i f i e d by a stacking order of the close packed hexagonal layers which occupy one of the three possible positions, A,B and C, i n the projection of the plane of layers (Fig. A - l ) . I t can be assumed that 113 114 FIGURE A-1 Stacking sequence of close packed [ l l l ] f layers i n the fee l a t t i c e . Atoms A are i n the plane of the drawing; the layer beneath has atoms i n C positions, the layer above i n B positions. The shear vectors R of a stacking f a u l t are indicated i n the diagram. 115 the fundamental close packed structure i s the fee structure with the simple stacking order ABCABCABC, as shown i n Fig.A-1. Shearing a B layer r e l a t i v e to the A layer by any of the vectors R, e.g., R-^  = [112]^, w i l l change the pattern to ABCA/CABC introducing a stacking f a u l t i n the middle of the sequence. The phase change for a r e f l e c t i o n corresponding to the r e c i p r o c a l l a t t i c e vector g = (hk£) produced by the shear R^ i s $ = 2 TT g. R x = i (-h-k+2£) . Since h,k,£ are eit h e r a l l odd or a l l even, $ w i l l assume the following values: For h+k+£ = 3N, $ = 0, for h+k+l = 3N±1,$= ± 2TT/3, where N i s any integer, choosing the p r i n c i p a l values of $ l y i n g between - I T and T T . Therefore, only those r e f l e c t i o n s for which h+k+£ i s equal to 3N w i l l be unaffected by the stacking f a u l t s ; those for which h+k+-£ i s equal to 3N+1 ($=2TT/3) and 3N-1 (0=-2TT/3) w i l l be affected i n a way depending on the d i s t r i b u t i o n of the stacking f a u l t s , as w i l l be described shortly. However, the affected r e f l e c t i o n s w i l l remain positioned along the ( i l l ) f d i r e c t i o n perpendicular to the stacking f a u l t plane, and consequently a l l features of the modulated r e c i p r o c a l l a t t i c e can be v i s u a l i z e d i n one 116 h + k + l 3 N 3N-I 3N + I 3 N 3 N - I 151 2 4 2 131 I I I - i — 0 3 3 3 2 2 2 I I 0 0 0 I I 27T ' 3 27T 3 III 0 2 0 27T 3 • REGULAR FCC REFLECTIONS x TWINNED FCC REFLECTIONS FIGURE A-2 (101)f r e c i p r o c a l l a t t i c e plane with twinned l a t t i c e spots. The plane consists of rows of r e f l e c t i o n s with successive phase s h i f t s 0, 2TT/3 and -2ir/3, every t h i r d layer having the same phase s h i f t . Stacking f a u l t s on (111)f plane cause broadening and displacement or s p l i t t i n g of spots with $=±2T\/3 i n the d i r e c t i o n p a r a l l e l to [ l l l ] f . 117 of the {110}^ re c i p r o c a l l a t t i c e planes which contains the <111>£ d i r e c t i o n of stacking, for example, the (110)^ plane i n F i g . A-2. The r e c i p r o c a l l a t t i c e l i n e s p a r a l l e l to the [ l l l ] f d i r e c t i o n i n F i g . A-2 can be c l a s s i f i e d into three kinds, the zeroth, f i r s t and second kind, for which the phase changes due to a stacking f a u l t are 0, -2TT/3 and 27r/3 respectively. A . l . D i s t o r t i o n of the FCC Reciprocal L a t t i c e due to a High Density of Random Stacking Faults (55,56) A high density of randomly d i s t r i b u t e d stacking f a u l t s causes broadening of the r e f l e c t i o n s and t h e i r d i s -placement. The e f f e c t s occur p a r a l l e l to the (111) f d i r e c t i o n , with the r e f l e c t i o n s being displaced towards the nearest twin spots. In the electron d i f f r a c t i o n patterns of the ( l l O ) ^ -zone, streaks are observed running along the r e c i p r o c a l l a t t i c e l i n e s of the f i r s t and second kind, although due to the multiple d i f f r a c t i o n they may be also observed along the r e c i p r o c a l l a t t i c e l i n e s of the zeroth kind. Paterson has established the following relationship between the amount of displacement of the affected r e f -lections and the density of stacking f a u l t s : h 3 = 3N - I + | arc tan [/T ( l - 2 a ) ] , (A-1) where h^ i s the co-ordinate of a r e f l e c t i o n measured i n the 118 d i r e c t i o n of displacement (for undisplaced r e f l e c t i o n s hg i s equal to h+k+Z = 3N±1; for displaced r e f l e c t i o n s to 3N±1 plus the f r a c t i o n of the distance to the twin spot, which i s equal to l/3d^^^). N i s any integer, a i s the p r o b a b i l i t y of a f a u l t occurring at any layer and the sign (+)corres-ponds to the phase change sign $ = ±2TT/3 respectively. Inspecting Eq.(A-l) i t i s seen that the r e f l e c t i o n s which for a = 0 are at hg, = 3N±1 become displaced towards the nearest twin spots for a>0. For a = 0.5, the peaks of the re f l e c t i o n s occur at values of hg = 3N-(3/2), where, according to Paterson, the i n t e g r a l breadth of r e f l e c t i o n s has i t s largest value. As a becomes smaller or larger than 0.5, the r e f l e c t i o n s gradually decrease i n breadth and approach the sharp r e f l e c t i o n s c h a r a c t e r i s t i c of the perfect c r y s t a l : or i t s twinned couterpart respectively. A.2. Long Period Stacking Order Modulation of the FCC L a t t i c e (44,45,57) If i t i s assumed that the fundamental close packed structure i s the fee structure with the simple stacking order ABCABCABC, a l l close packed structures can be derived from i t by in s e r t i n g stacking f a u l t s i n an appropriate way. When the stacking fau l t s are introduced at every t h i r d layer, the res u l t i n g stacking order i s ABC/BCA/CAB and the period which brings a close packed layer A of the modulated l a t t i c e into coincidence with a close packed layer A of the fee l a t t i c e i s 9. This modulation of the fee structure i s s p e c i f i e d by the 1 1 9 symbol 3 R . The number 3 s i g n i f i e s that the period i s divided into series of three layers each, with series related to each other by a unit stacking s h i f t , ( 1 / 6 ) a [ 1 1 2 ] £ , and the l e t t e r R s i g n i f i e s that the modulation has a rhombohedral symmetry. Modulation of the fee structure i s characterized by a s p l i t t i n g of ,the o r i g i n a l r e c i p r o c a l l a t t i c e points l y i n g i n the f i r s t and second kind of l i n e s into series of spots. The number of the s p l i t spots i n the series i s equal to the number of layers i n the fundamental series of modulation. Therefore, as shown i n F i g . A -3 , the number of s p l i t spots i n a 3 R l a t t i c e i s three. The rhombohedral symmetry i s manifested i n the manner of s p l i t t i n g ; i.e., the s p l i t spots are s h i f t e d by 1 / 3 of the unit of s p l i t t i n g (or 1 / 9 of the I / C ^ - Q ) away from the positions of the fee spots, the spots i n the adjacent layers being s h i f t e d i n the opposite d i r e c t i o n s . The r e s u l t i s that the spots e x i s t at positions which are a multiple of l / ^ d - ^ ^ . In the rec i p r o c a l l a t t i c e l i n e s of the zeroth kind, d i f f r a c t i o n spots appear at each unit distance, which i s the re c i p r o c a l i n t e r l a y e r spacing (1/d^-^) . The modulated structure can be best described i n terms of an orthorhombic unit c e l l with the close packed ( l l l ) f plane as i t s basal plane (Fig. A-4), and the d i r e c t i o n of modulation, [ l l l ] f , as i t s o 3 axis. The structure factor 120 orth 18 6 020, Pri-ll I o - ® -009 I orth " 3 0 •I 112 II 0 0 0 11 172 114 117 020 27T 3 111 H § H - 0 009 8 27T 3 O FCC REFLECTIONS • 3R REFLECTIONS FIGURE A-3 Intensity d i s t r i b u t i o n i n the 3R rec i p r o c a l l a t t i c e plane (110)o i n the orthorhombic notation or (101)£ i n the cubic notation. 121 R o A g • Cd (a) - 9 Eid, © L Ag Cd A B C O • A ( c ) FIGURE A-4 (a) The l a t t i c e correspondence between the fee (CuAu I-type) and orthorhombic l a t t i c e , (b) The unit c e l l of the basal plane of the orthorhombic l a t t i c e . The orthorhombic coordinates of atoms i n the plane are: Ag - 0, 0; Cd - h,h- (c) The d i s t r i b u t i o n of atoms i n the basal plane i n the A, B and C layers. The orthorhombic coordinates of the Ag atoms i n the layers are; A - 0, 0, 0; B - 0, 1/3, 1/9; C - 0, 2/3; 2/9. 122 for the orthorhombic l a t t i c e can be written as the product of three terms: F = where (A-2) F 1 The product F^.F^ i s usually designated as F^. The factor F A i s the structure factor for the basal plane. F-^  indicates that the stacking order i n each series of the unit c e l l period i n the Or> d i r e c t i o n i s ABC, and F indicates that the series of the unit c e l l period are related to each other by a unit stacking s h i f t . The calculated r e l a t i v e i n t e n s i t y d i s t r i b u t i o n , neglecting the modulation of i n t e n s i t y by the specimen thickness, for the Ag-Cd a l l o y of the stoichiometric composition i s given i n Table A-I. The i n t e n s i t y d i s t r i b u t i o n i n the orthorhombic r e c i p r o c a l l a t t i c e plane (110)o (which i s equivalent to the cubic r e c i p r o c a l l a t t i c e plane (101) f) i s shown schematically i n F i g . A-3. 123 TABLE A-1 Calculated Relative I n t e n s i t i e s , |F| , for the 3R Modulation of the CuAu I-Type Structure Based on Equations (A-3) and (A-4) for k = 0, 1, -1. Reciprocal L a t t i c e Reciprocal L a t t i c e Relative Intensity, Point i n the Ortho- Point i n the Cubic |F| 2 rhombic Coordinates Coordinates 110 0 111 0 112 905 113 0 114 0 115 6217 116 020 0 117 0 118 1217 119 0 001 0 002 0 003 0 004 0 005 0 006 0 007 0 008 0 009 I I I 10034 110 0 111 2178 112 0 113 111 0 114 6631 115 0 116 0 117 598 118 0 119 0 124 Sato zt at. found that a random d i s t r i b u t i o n of stacking f a u l t s i n the 3R structure can cause broadening and displacement of c e r t a i n 3R r e f l e c t i o n s as well. Their analysis followed e s s e n t i a l l y that of Paterson and Whelan and Hirsch, who analyzed the e f f e c t of random stacking faul t s on the r e c i p r o c a l l a t t i c e of the fee structure. As i t was shown above, there are three r e f l e c t i o n s i n a unit distance, and t h e i r i n t e n s i t y can be s p e c i f i e d as weak (W), medium (M) and strong (S). The arrangement of the r e f l e c t i o n s i n the pattern i s i n the order S-W-M. The regular 3R structure can be s p e c i f i e d as having the stacking f a u l t density parameter a = 0. Since the structure has the R symmetry (similar to fee structure, or IR), the other d i s t i n c t state to be taken into account i s that of a twin, which can be s p e c i f i e d as having a = 1. Thus, ; the broadening and displacement of the r e f l e c t i o n s of the a = 0 state occurs i n the d i r e c t i o n of the nearest r e f l e c t -ions of the a= 1 state, weighted by t h e i r r e l a t i v e i n t e n s i -t i e s . The r e s u l t i s that W r e f l e c t i o n s become most di f f u s e and S r e f l e c t i o n s l e a s t d i f f u s e , and that a l l are accompanied by streaks. The distances between the spots are no longer equal; the S-W distance i s about one t h i r d of the unit distance, while M-S i s longer and W-M i s shorter. APPENDIX B A n a l y t i c a l Treatment of Martensitic Transformations The phenomenological theory of martensite formation allows pred i c t i o n of the habit plane, d i r e c t i o n and magnitude of shape deformation, magnitude of l a t t i c e invariant shear and orientation r e l a t i o n s h i p between the parent and product for an assumed l a t t i c e correspondence between the parent and product and known l a t t i c e parameters, as well as a known or assumed shear system i n the product. The matrix analysis of the bcc (CsCl) to fee (CuAu I) martensitic transformation, as applied to the transformation of g'-AgCd to a-AgCd i s outlined here. The theory follows e s s e n t i a l l y the formulation of Bowles and Mackenzie (2-4,58) with the d e t a i l s of the mathematical development and notation borrowed from Wayman (5). The l a t t i c e correspondence i s defined by the correspondence matrix, (fCb), so that [ x ] f = ( f C b ) [ x ] b , where [x]^ and [ x ] b symbolize a column matrix x (vector x) r e l a t i v e to the f and b bases respectively. The b basis defines the i n i t i a l bcc unit c e l l with l a t t i c e parameter a b , while the basis f defines the face centred unit c e l l with l a t t i c e parameters /2" a b , /2 a b and a b (Fig. B-1). 125 o Ag • Cd FIGURE B-1 Schematic representation of the correspondence between the parent CsCl-type l a t t i c e (b basis) and the product CuAu I-type l a t t i c e (f bas i s ) . 127 A l t e r n a t i v e l y , ( n ) f = ( n ) b (bCf) , where (n) f and (n)]-, symbolize a row matrix n (plane normal n) r e l a t i v e to the bases f and b respectively. The notation of the correspondence matrix (fCb) symbolizes the transformation of coordinates from the b to f basis. Naturally (bCf) = (f C b ) " 1 . In the present case, the following correspondence (Fig.B-1) was assumed: (fCb) = h h 0 -h h 0 0 0 1 which i s a variant of the Bain correspondence. Therefore, the shear system (111)[112] f, which was assumed to operate i n the product, corresponds to the shear system (Oil)[011]^ i n the parent. The p r i n c i p a l axes of s t r a i n associated with t h i s correspondence can be taken as p a r a l l e l to the vectors defining the basis b. Referred to these p r i n c i p a l axes, the diagonal matrix (bBb) representing the s t r a i n which compresses (or extends) the base vectors b to t h e i r f i n a l lengths without rotation i s the following: (bBb) = diag ( n ^ r ^ r " ^ ) r 128 where the p r i n c i p a l d i s t o r t i o n s are as follows: n 1 2 = 0.890478 af 1.259326. n 3 According to the theory, the t o t a l s t r a i n due to the transformation (i.e the shape deformation), P 1' i s composed of a simple shear (the l a t t i c e invariant shear),P, a l a t t i c e deformation (the Bain s t r a i n ) , B, and a r i g i d body rotation, R, i . e . , P 1P = RB. Since both P^ and P are invariant plane s t r a i n s , RB must be an invariant l i n e s t r a i n , S, the invariant l i n e of which must l i e i n the shear plane. However, since the invariant l i n e of S becomes Bx^ afte r the s t r a i n B, the rotation R must be such that i t restores Bx^ to x^. Also, R must simul-taneously restore n|B to n| (see Footnote 4 -) , where n| i s the P_L = RBP, or -1 + The prime, as i n n!, symbolizes the transposition operation. 129 invariant plane normal of S. This rotation w i l l now be determined. The f i r s t step i s to i d e n t i f y which w i l l be represented as unit vectors. The cone which s a t i s f i e s the equation x' (bBb)" x = x'x gives the i n i t i a l p o s i t i o n of a l l l i n e s that are not changed i n length by the s t r a i n S. In the present case, the cone i s c i r c u l a r with [°01] b as i t s axis and the semiapex angle $ = arc tan 1 - n 2s 2 i = 59.27' The in t e r s e c t i o n of the cone with the unit sphere i s shown i n Fig.B-2. S i m i l a r l y , the c i r c u l a r cone C 2• also shown i n F i g . B-2, which s a t i s f i e s the equation -2 x' (bBb) x = x'x gives the f i n a l positions a f t e r the s t r a i n B of a l l l i n e s that are not changed i n length by the s t r a i n S. Its semiapex angle i s < 2 $' = arc tan 1 -n- "1 49.95°. The same cone, C 9, gives the i n i t i a l p o s i t i o n of plane normals 130 FIGURE B-2 Stereographic projection showing some of the operations i n determination of invariant l i n e strains compatible with the shear system (Oil) [ O i l ] , . 131 that are not changed i n length by s t r a i n S, i . e . , -2 n' (bBb) n = n'n. Now, since the invariant l i n e x. of the s t r a i n S x must l i e i n the shear plane with normal P 2', the following relationship holds: P 2 ' * ± = 0, and the two possible values of x.,x, and x~, are the l 1 2' intersections of with the great c i r c l e (011)^ (Fig.B-2). The algebraic solutions for x^ are as follows (the unit vectors with p o s i t i v e b^ components were chosen): x1 = [0.691214; -0.510991; 0.510991] b x„ = [-0.691214; -0.510991; 0.510991] . b Also, since the plane with an invariant normal must contain the shear d i r e c t i o n d 2 , the following holds: V d 2 = 0, and the two possible values of n. 1, n ' and n„', are the intersections of C 2 with the great c i r c l e (011)j-,. The algebraic solutions for n-^ ' are the following row vectors: n 1 = ( 0.414493; 0.643504; 0.643504), n 2« = (-0.414493; 0.643504; 0.643504)b . To f a c i l i t a t e the determination of the rotation which ca r r i e s Bx. into x. , the rotation R i s resolved into a 1 i rotation which makes the shear plane p 2 1 an unrotated plane and a generalized rotation about x^. F i r s t , two subsidiary orthonormal bases are introduced. The f i r s t basis, basis i , i s defined by the unit vectors x^, p 2 (where p 2' i s the i n i t i a l p o s i t i o n of the normal to the plane of the l a t t i c e invariant shear) and u = x^*p 2. Thus, the matrix (bR 1i) = (x i 7p 2,u) represents the rotation of the basis i into the basis b. The second subsidiary basis, basis j , i s defined by the unit vectors x.^  , p 2 and v = x^ x p 2 , where x^ = (bBb) x i i s the position of the invariant l i n e due to the l a t t i c e deformation and p 2' i s the unit vector p a r a l l e l to the f i n a l p o s i t i o n of the normal p 2 1 due to the l a t t i c e deformation. The l a t t e r i s given by the following r e l a t i o n : p ' (bBb) - 1 p2 ' = ~ =F — ' _ 2 [p 2' (bBb) 2 P2V2 The s t r a i n SQ defined so that (fS f) = R R ' (bBb) O 1 2 leaves the invariant l i n e x^ invariant and the shear plane with normal p_* unrotated. Since the desired invariant l i n e 133 s t r a i n (fSf) does i n fac t rotate the plane P 2 1 / a n c * necessarily only about the invariant l i n e i t s e l f , the generalized rotation about x^ has to be introduced. When the s t r a i n (fS f) i s referred to the basis i , i t follows o that (iS i ) = R ' (bBb)'R , O 2 1 and the desired invariant: l i n e s t r a i n becomes (iSi) = The rotation angle B i s determined using the following r e l a t i o n s h i p : n i' (iSi) = n i' F i n a l l y , the invariant l i n e s t r a i n can be referred to the o r i g i n a l b basis, i . e . , (bSb) = R (iSi) R-_ ' . As mentioned e a r l i e r , the s t r a i n S leaves invariant a l i n e x^ and a normal n / . Since i n the present case the st r a i n B leaves two l i n e s (x and x_) and two normals 1 -> (n^' and n 2 1 ) unchanged i n length, there are four possible rotations R corresponding to four possible combi-nations of x ^ » n i " ( i . e . , 1. x^,n^'; 2. x 2,n 2'; 3. x^,n 2'; 4. x 0,n.'), which w i l l make these l i n e s and normals invariant 1 0 0 cos 6 0 s i n 0 0 -sinB COSB R 0 (bBb) R, 134 as well. This leads to four solutions for S. However, the positions of x_^  and n^ 1 are symmetrical to the positions of and n 2' with respect to the plane (100) b, which means that the combinations 1 and 3 are e r y s t a l l o g r a p h i c a l l y equivalent to the combinations 2 and 4. This reduces the number of rotations to two, and consequently leads to only two e r y s t a l l o g r a p h i c a l l y d i s t i n c t solutions for S. Thus, only the solutions corresponding to combinations 1 and 3 need be considered, and they w i l l be referred to as variants (x 1,n 1) and (x-^n ). The invariant l i n e s t r a i n S i s now resolved into i t s two component s t r a i n s : the invariant plane s t r a i n on the habit plane (the shape deformation) and the simple shear (the l a t t i c e invariant shear). In the matrix represen-tati o n t h i s has the following form: (bSb) = (I + d 1p 1»)(l + d 2p 2') , (B-1) where d^ i s the d i r e c t i o n of the shape deformation, p-^ " i s the habit plane normal, and d 2 and p 2 1 , as s p e c i f i e d •earlier, are the d i r e c t i o n and the normal to the plane of the l a t t i c e invariant shear respectively. The habit plane, shape deformation and magnitude of the simple shear can be determined from the following expressions that are derived from Eq. (B-1): 135 P l « || p ' (bSb)" 1 - P 2 (bSb) d 2 - A d = P l d2 The normalization factor mi of the vector d determines the 1 1 magnitude of the shape deformation. Also, the normalization factor m_ of the shear vector d given by the following Z 2 expression: , 2 . y - ( b S b ) " l y ! P 2' (bSb) _ 1y determines the magnitude of the simple shear. The vector y i n Eq. (B-2) i s any unit vector d i s t i n c t from x^ and l y i n g i n the plane P j ' • The orientation r e l a t i o n s h i p results from the application of the invariant l i n e strains (bSb) and (bSb)"' to vectors and normals of the parent l a t t i c e respectively. The data used i n the application of the Bowles-Mackenzie martensite theory to the present case of the B' to a transformation i n the Ag-45 at. pet Cd a l l o y and the results of the theory are summarized i n Table B-I. TABLE B-I Application of the Bowles-Mackenzie Martensite Theory to The g' to a Transformation i n the Ag-45 At. Pet Cd A l l o y Summary of the Used Data and Results Structure and L a t t i c e Parameters: Parent - Ordered bcc (CsCl); a b = 3.324 A Product - Ordered fee (CuAl I ) ; a f = 4.186 A Parent - Product L a t t i c e Correspondence: h h 0 (fCb) = h 0 _ 0 0 l Shear System: Parent - ( O i l ) [ 0 1 1 ] b ; Product - (111) [112] Homogeneous Strain (Bain S t r a i n ) : (bBb)=diag(0.890478; 0.890478; 1.259326) Semiapex Angle of the I n i t i a l Cone of Unextended Lines: $ = 59.27° Semiapex Angle of the F i n a l Cone of Unextended Lines: $' = 49.95° Invariant Lines: x1 = [0.691214; -0.510991; 0.510991] b x 2 = [-0.691214; -0.510991; 0.150991] b -Continued-137 TABLE B-I - Cont. Invariant Plane Normals: n ' = ( 0.414493; 0.643504; 0.643504)b n 2' = (-0.414493; 0.643504; 0.643504)b Invariant Line Strain: Variant ( x ^ n ^ — (bSb) = Variant (x^,n 2) — (bSb) = 0.884593 0.099149 •0.024814 0.884593 0.024814 -0.099149 •0.091389 0.864098 0.194768 •0.012685 0.883812 0.108019 0.064720 •0.270022 1.228335_ 0.143425" •0.14.9754 1.242139 Habit Plane Pole: Variant (x^,n^) — •0.667566 •0.722279 0.180747 0.667566 0.180747 -0.722279 Variant ( X ^ n ^ — P i = Direction and Magnitude of the Shape Deformation: Variant (x^,n^) — d-^  -0.748615 -0.643169 0.160966 ; m = 0.230924 Variant (x-^,n2) — d^ -0.748615 -0.160966 -0.643169 ; m1 = 0.230924 Magnitude and Angle of the L a t t i c e Invariant Shear: Variant ( x ^ n ) — m2 =0.428838; a 2 = 24.20° Variant ( x l f n 2 ) — ™ 2 = .0.237831; a 2 = 13.56 -Continued-138 TABLE B-I - Cont. Orientation Relationship between the Parent and Product Lat t i c e s Direction i n the L a t t i c e of Angle between Poles, degrees Parent Product Variant (x 1 jn-^ Variant (x^ ,n^) [ l l l ] b [ o i i ] b [112] b [Oil] 0.78 0.78 [100] f 9.51 1.25 [ l l l ] f 4.30 4.30 [011] f 9.54 1.05 APPENDIX C Theory of the Volume Dif f u s i o n Controlled P r e c i p i t a t e Growth C l . Thickening of Plates Zener (59) and Frank (6) showed that the h a l f -thickness, X, of the p r e c i p i t a t e plate Whose growth i s controlled by d i f f u s i o n of solute through the matrix i s related to the d i f f u s i v i t y , D, and the growth time, t, i n the following way: X = L(Dt) J s, (C-l) where L i s a dimensionless growth c o e f f i c i e n t that depends only on the dimensionless supersaturation, fiQ =(0^ - c Q ) / '(c„:-- c„) . c m i s the concentration of solute i n the op o matrix far away from the p r e c i p i t a t e , and c o p and C q are respectively the concentrations i n the p r e c i p i t a t e and i n the matrix at the p r e c i p i t a t e - matrix interface. Assuming that the interface remains planar, that the p r e c i p i t a t e s are i s o l a t e d during growth, and that the d i f f u s i v i t y does not depend on the concentration, the following r e l a t i o n exists between SlQ and L: 2 n = : £ L Lexp (—) erf c h. . (C-2) 0 2 4 2 139 140 C.2. Lengthening of Plates and Needles The most advanced treatment of the volume d i f f u s i o n controlled lengthening of p r e c i p i t a t e plates and needles was presented by T r i v e d i (61,62) . His solutions, based on the o r i g i n a l Ivantsov treatment of the problem (63,64), included the e f f e c t of the non-isoconcentrate nature of the interface at the p r e c i p i t a t e t i p . The concentration at the t i p can vary due to the c a p i l l a r i t y e f f e c t and due to the interface k i n e t i c s e f f e c t . Other authors who previously studied the problem either disregarded t h i s e f f e c t , or gave only an approximate mathematical treatment (65-69). The p r i n c i p a l approximations used by T r i v e d i were: (1) the steady state shape of the interface near the growing t i p i s a parabolic cylinder f o r the case of a p l a t e - l i k e p r e c i p i t a t e , (2) the e l a s t i c s t r a i n energy and anisotropy of surface properties can be neglected, (3) the concentration of solute i n the matrix i s such that the theory of c a p i l l a r i t y applicable to d i l u t e solutions can be used, and (4) the d i f f u s i v i t y i s independent of concentration. Trivedi's r e s u l t s , r e l a t i n g the dimensionless supersaturation, -fl ,-to the dimensionless growth rate, p a vp/2D, of the t i p of the p r e c i p i t a t e , when the interface k i n e t i c s e f f e c t i s neglected, are: o = (ifp^ePerfctp 3 5) [1+ — S (p) ] O D 0 —' (C-3) 141 for plates, and P ' 1 fi = pePEi(p)[±+-± fiQ R (p)] (C-4) ° P for needles, where v i s the growth rate of the t i p of the pre c i p i t a t e , p i s the radius of curvature at the advancing t i p , p and p ' are the c r i t i c a l r a d i i for growth (the C • c r a d i i at which the concentration gradient i n the matrix vanishes), E i i s the exponential i n t e g r a l function, and S 2 and R 2 are complicated,functions defined i n the o r i g i n a l papers by T r i v e d i . > The f i r s t term on the righthand side of each equation i s the r e s u l t obtained by Ivantsov for the case of the isoconcentrate i n t e r f a c e . The second term i s a correction due to. the c a p i l l a r i t y e f f e c t . The value of fiQ can.be obtained from the phase diagram knowing the average composition of the al l o y and the transformation temperature. However, many exact solutions of Eqns. (C-3) and (C-4) are possible for a given value of fi , depending upon the value of the radius of curvature. In accord with the experimental observations, T r i v e d i assumed that only one of these solutions i s stable with respect to a small perturbation i n the radius of curvature of the t i p of the plate. In agreement with Zener (68), Tri v e d i stated that t h i s corresponds to the radius of * * curvature, p , which gives the maximum growth rate, v . The 142 maximum growth r a t e can be o b t a i n e d by d i f f e r e n t i a t i n g Eq. ( 0 3 ) or (C-4) w i t h r e s p e c t t o p, and s e t t i n g 9v/8p=0, g i v i n g another r e l a t i o n s h i p between fi , p and p . The simultaneous s o l u t i o n o f t h i s e q u a t i o n w i t h Eq. (C-3) or (C-4) then g i v e s unique v a l u e s f o r p (=p*) and p ( = p * ) f o r a g i v e n v a l u e o f fiQ+. F i n a l l y , the expected maximum + An e q u i v a l e n t , g r a p h i c a l procedure can be a p p l i e d u s i n g T r i v e d i ' s (62) diagrams f o r the v a r i a t i o n o f P * / P c and p* with fiQ. growth r a t e , v*, i s c a l c u l a t e d from p* ( = v *p*/2D) when p c and D are known. Rece n t l y , H i l l e r t (70) r e p o r t e d the f o l l o w i n g new m o d i f i c a t i o n o f the w e l l known Z e n e r - H i l l e r t Equation f o r t h e growth o f p r e c i p i t a t e p l a t e s : v * p c 1 ao = i _ _ _ exp[-5.756 (1-fi ) ] . (C-5) D 4 ( i - j j o ) 0 I t was r e p o r t e d t h a t i t agrees w i t h T r i v e d i ' s a n a l y s i s when used f o r medium and hig h v a l u e s o f fiQ. The c r i t i c a l r a d i u s f o r n u c l e a t i o n , p , can be c c a l c u l a t e d from the Gibbs-Thompson Equation; i n i t s o r i g i n a l form t h i s equation i s a p p l i c a b l e o n l y to i d e a l o r d i l u t e 143 solutions. In the case of a p r e c i p i t a t e growing i n a r i c h nonideal solution, the following modified form of the Gibbs-Thomson Equation has to be used (71): p - C ° ( 1 - C ° ' V « ( C-6) or p' = 2p , (C-6a) c c where a . , i s the i n t e r f a c i a l free energy, V i s the molar a/0 a volume of the a phase, and the thermodynamic factor S18' = 1 + 9 1 n Y 1 8 ' / 3 1 n X l 8 - ' APPENDIX D An Estimate of the Chemical D i f f u s i v i t y i n the 0' Phase of Ag-Cd Alloys on the Basis of a Comparison Between the Cu-Zn and Ag-Cd Systems Dif f u s i o n data for the ordered 0 1 phase of Ag-Cd alloys are not available i n the l i t e r a t u r e . Therefore, the s i m i l a r i t y between the Ag-Cd system and the Cu-Zn system was used to examine the d i f f u s i v i t y data obtained from the growth k i n e t i c s . A comparison of Home and Mehl's data (72) for chemical d i f f u s i v i t i e s i n the a phase of a Cu-25 at.pet Zn a l l o y i n the i n t e r v a l 724-915°C (Fig.D-1) and a set of analogous data f o r the a phase of Ag-Cd alloys i n the temperature i n t e r v a l 600-780°C (73-75) showed that the d i f f u s i v i t i e s i n the a phase of Ag-Cd alloys were approximately three times larger and that the act i v a t i o n energy was 10 pet larger. The early work of Petrenko and Rubinstein (76) provides the only available d i f f u s i o n data for the 0 phase of a Ag-Cd a l l o y . They reported an a c t i v a t i o n energy of 3.767 x 10 4 J/mole, which was s i g n i f i c a n t l y less than the analogous a c t i v a t i o n energies for zinc i n the 0 phase of Cu-Zn alloys reported by Kuper tt al, (77) (7.86 x 10 4 J/mole) and Camagni (78) (9.22 x 10 4 J/mole). However, the data 144 TEMPERATURE, °C 1000 800 700 600 ^ i i i 7 8 9 10 II 1/T, IO"4 l/°K FIGURE D - l Comparison of the d i f f u s i v i t y data for a-Cu-Zn and a-Ag-Cd phase. 146 of Petrenko and Rubinstein are of dubious q u a l i t y ; they 4 also reported and a c t i v a t i o n energy of 1.109 x 10 J/mole for the d i f f u s i v i t y of zinc i n the B phase of a Cu-Zn a l l o y (500-800°C). This i s almost an order of magnitude less than the more r e l i a b l e r e s u l t s of Kuper and Camagni. Thus the only basis for comparison i s the previously described relationship between the d i f f u s i v i t i e s i n the a phases of the Cu-Zn and the Ag-Cd a l l o y s . Ugaste and Pimenov (79) reported an a c t i v a t i o n energy of 1.507 x 10 J/mole and a frequency factor 0.144 mz/s for the chemical d i f f u s i v i t y i n the B' phase of a Cu-48 at. pet Zn a l l o y i n the temperature i n t e r v a l 318-447°C (Fig.D-2). Increasing the ac t i v a t i o n energy by 10 pet. to 1.658 x 10^ J/mole, and d i f f u s i v i t i e s approximately three times, the re s u l t i n g d i f f u s i v i t i e s (Table VI) are within one order of magnitude agreement with the d i f f u s i v i t i e s obtained from the b a i n i t e thickening k i n e t i c s . However, i t should be stressed that the d i f f u s i v i t y values obtained from the b a i n i t e thickening k i n e t i c s also depend on the uncertain p o s i t i o n of the metastable a / ( a + B ' ) and ( a + B ' ) / B ' phase boundaries. 147 TEMPERATURE, °C 240 200 160 19 20 21 22 23 24 1/T, IO'4 l/°K FIGURE D-2 Comparison of the d i f f u s i v i t y data for B'-Cu-Zn and B'-Ag-Cd phase. APPENDIX E The E q u i l i b r i u m and the Metastable Ag-Cd Phase Diagram The g e n e r a l l y accepted (80) ex t e n t o f cadmium s o l u b i l i t y i n the a phase of the Ag-Cd system i s based on the m e t a l l o g r a p h i c work of Hume-Rothery at al. (81) and the l a t t i c e parameter work of Owen t t a l . (82,83). T h e i r v a l u e s , which agree on l y a t temperatures near 700°C, are p l o t t e d i n F i g . E - l . In the same f i g u r e are a l s o p l o t t e d the p o i n t s which d e f i n e the upper l i m i t s f o r the formation o f massive a duri n g p u l s e h e a t i n g of the quenched g' phase, as measured by Ayers (29). These p o i n t s match the curve o b t a i n e d by the e x t r a p o l a t i o n o f the high-temperature p o r t i o n of the s o l u b i l i t y l i m i t . The p o i n t s show t h a t a m formed i n a l l o y s c o n t a i n i n g up t o 1.6 a t . pet Cd i n excess of the upper l i m i t of the a-phase f i e l d . The e x t e n s i o n o f the formation of a m i n t o the e q u i l i b r i u m two-phase f i e l d was l e s s i n o t h e r systems s i m i l a r t o the Ag-Cd system: 0.3-0.45 a t . pet i n . Cu-Zn (29,34) and 1.2 a t . pet i n Cu-Al (34). Ayers a l s o found t h a t i n the Ag-Zn system a m formed on l y i n the a l l o y s c o n t a i n i n g up to 39.1 a t . pet Zn, i . e . , 1.1 a t . pet l e s s . than the s o l u b i l i t y l i m i t . These r e s u l t s i n d i c a t e d t h a t the a-phase f i e l d i n the metastable Ag-Cd diagram may extend i n t o the e q u i l i b r i u m two-phase f i e l d 1-1.5 a t . pet beyond the 148 149 SILVER, AT. PCT CADMIUM, AT. PCT FIGURE E - l The relevant portion of the Ag-Cd equilibrium phase diagram (thin lines) and the Ag-Cd metastable phase diagram (thick l i n e s ) . In the metastable phase diagram,the formation of the c phase is suppressed by rapid quenching from theBphase to the 6 1 phase. 150 previously assumed l i m i t s . A sharp change i n the d i r e c t i o n of the s o l u b i l i t y l i n e at 440°C i n the equilibrium Ag-Cd diagram i s due to the appearance of the eutectoid hep ? phase. If the formation of the r, phase i s suppressed by quenching, the s o l i d s o l u b i l i t y of cadmium i n s i l v e r should be larger than the equilibrium s o l u b i l i t y , since i t was observed that i n a system of t h i s kind the:primary solution can reach higher concentrations when i t i s followed by cubic 3 phase, rather than by hep 5 phase (84). The s o l u b i l i t y l i n e should therefore extend uniformly to the 3 phase ordering temperature (240°C), as indicated i n F i g . E - l , and then bend towards the s i l v e r side, s i m i l a r to the s o l u b i l i t y l i n e i n the Cu-Zn diagram. In F i g . 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