@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Materials Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Kostić, Miodrag Miloš"@en ; dcterms:issued "2010-02-22T20:41:23Z"@en, "1977"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """The morphology of bainite plates and widmanstätten needles formed in ordered bcc β* phase of a Ag-45 at. pct Cd alloy at temperatures 160-320° C was studied by optical and scanning electron microscopy. Both precipitate forms were similar in appearance to precipitates reported for Cu-Zn alloys. The structure of the bainite plates in the various stages of their growth was studied by X-ray diffraction and by transmission electron microscopy. Initially, the plates formed with a 3R stacking fault modulation of the fee structure and contained a high density of random stacking faults. The stacking faults annealed out during a prolonged isothermal treatment and the structure gradually changed to a regular fee. The orientation relationship between the bcc matrix and the fcc bainite was as follows: [111]b 0.7° from [011]f, [110]b 1.1° from [100]f and [011]b 4.3° from the stacking fault plane pole [111]f. The habit plane of the bainite plates, determined by two surface trace analysis, was close to (144)b. The surface relief of the plates was observed by the interference microscopy. It was in the form of a simple tilt indicating an invariant plane strain transformation. The features of the transformation agreed with the predictions of the Bowles-Mackenzie theory of martensite formation. The growth kinetics of both bainite plates and widmanstätten needles were measured by interrupted annealing and scanning electron microscopy. Using the bainite thickening kinetics measured at 160, 200 and 240°C, the Frank-Zener model for growth of planar precipitates, and supersaturation data obtained from the Ag-Cd metastable phase diagram enabled the effective chemical diffusivities, Deff , to be calculated for the three transformation temperatures. The results were in good agreement with the expected diffusivities. The lengthening kinetics of bainite plates at 160°C and of widmanstätten needles at 240°C were analyzed using Trivedi's model for diffusion-controlled growth. Deff obtained from the lengthening kinetics of the needles was in good agreement with the D value obtained from the thickening kinetics of the plates, indicating that widmanstätten needles lengthened and bainite plates thickened at rates controlled by volume diffusion. Bainite plates lengthened only in the early stage of growth and at a rate approximately 180 times larger than that permitted by volume diffusion. It was concluded that the morphology, structure and other characteristics of the freshly formed bainite plates were consistent with their formation by a thermally activated martensitic process."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/20730?expand=metadata"@en ; skos:note "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 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, 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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"@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0078961"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Materials Engineering"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Morphology, structure and growth kinetics of bainite plates in the β' phase of A Ag-45 AT. PCT Cd Alloy"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/20730"@en .