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Dissolution of grain boundary allotriomorphs in the Al-Cu and Al-Ag systems 1972

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I l l b O DISSOLUTION OF GRAIN BOUNDARY ALLOTRIOMORPHS IN THE Al-Cu AND Al-Ag SYSTEMS BY ARISTEDES PASPARAKIS B.S c , U n i v e r s i t y of P r e t o r i a , 1964 M.Sc. (Tech.), University of S h e f f i e l d , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY . i n the Department of METALLURGY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1972 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 requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree 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 reference and study . I f u r t h e r agree t h a t pe rmiss ion fo r e x t e n s i v e copying 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 granted by the Head of my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or 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 ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department of M e t a l l u r g y The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 4, 1972 - i i - ABSTRACT The d i s s o l u t i o n behaviour of grain boundary allotriomorphs has been studied i n the systems Al-Cu and Al-Ag. Using e l e c t r o n probe microanalysis, isoconcentration contours around d i s s o l v i n g allotriomorphs i n an e f f e c t i v e l y i n f i n i t e matrix have been determined. The influences of allotriomorph shape, grain boundary misorientation, solute super- saturation i n the matrix, volume and grain boundary d i f f u s i o n , and homologous temperature on the shape of these contours have been examined. It was established that the shape of the isoconcentration contours i s dependent only on homologous temperature (T R) for both systems studied. Above T„ = 0.92, no grain boundary d i f f u s i o n contribution hi to the d i s s o l u t i o n process i s observed. The grain boundary d i f f u s i o n contribution was- found to increase with decreasing T and to completely H dominate the d i s s o l u t i o n process below T = 0.72. H Using the back scattered electron image on the e l e c t r o n probe microanalyzer, the d i s s o l u t i o n rate of indivTatial grain boundary allotriomorphs was determined f o r various values of T , under conditions i n which there was impingement of d i f f u s i o n f i e l d s from adjacent p r e c i p i t a t e s . An exponential r e l a t i o n s h i p between a x i a l length or width and d i s s o l u t i o n time was found to adequately describe the observed d i s s o l u t i o n k i n e t i c s . A change i n a x i a l r a t i o accompanied the d i s s o l u t i o n of the grain boundary allotriomorphs. At high T an increase i n a x i a l r a t i o with d i s s o l u t i o n time was observed, whereas at low T„, a decrease i n a x i a l r a t i o ( i . e . , spheroidization) was observed, rl A model has been proposed to account f o r t h i s behaviour. - i i i - TABLE OF CONTENTS Page 1. INTRODUCTION 1 1.1 Grain Boundary P r e c i p i t a t e s 1 1.2 Growth of P r e c i p i t a t e s 4 1.2.1 Thickening of Grain Boundary Allotriomorphs. 5 1.2.2 Lengthening of Grain Boundary Allotriomorphs 5 1.2.3 The "Grain Boundary C o l l e c t o r P l a t e " Model . 6 1.3 D i s s o l u t i o n Studies of P r e c i p i t a t e s 7 1.3.1 Observations of the D i s s o l u t i o n of P r e c i p i t a t e s by Electron Microscopy 12 1.3.2 D i s s o l u t i o n Studies Using the Electron Probe Mlcroanalyzer 1.4 D i s s o l u t i o n Mechanisms Derived from P r e c i p i t a t e Growth i g 1.5 Objective of Present Investigation 17 2. EXPERIMENTAL PROCEDURE 20 2.1 Reasons f o r Systems Chosen 20 2.2 Preparation of Master A l l o y s 21 2.3 Procedure Used to Grow Large Grain Boundary Allotriomorphs 24 2.4 D i s s o l u t i o n Heat Treatment Procedure 25 2.5 D i f f u s i o n Couple Work 28 2.6 Ele c t r o n Probe Microanalysis 29 2.6.1 Isoconcentration Contours 29 2.6.1.1 Resolution of Experimental Technique 32 2.6.2 K i n e t i c Studies 36 - i v - Page 2.6.3 D i f f u s i o n Couple Measurements 38 2.7 V a l i d i t y of Experimental Technique 3 ^ 2.7.1 Isoconcentration Contour Measurements 38 2.7.2 K i n e t i c Measurements 41 3. ISOCONCENTRATION CONTOUR RESULTS AND DISCUSSION 44 3.1 Introduction 44 3.1.1 D i f f u s i o n Couple Results 45 3.1.2 The Microstructure of E q u i l i b r a t e d A l l o y s .. 49 3.1.3 D i s s o l u t i o n Isoconcentration Contours 50 3.2 Rate C o n t r o l l i n g Mechanism 54 3.3 The E f f e c t of P r e c i p i t a t e Shape T.S /R 1 56 3.4 Homologous Temperature [T^] 60 3.4.1 Calculations of D y and Cj 71 3.5 The E f f e c t of Supersaturation (k) and Volume . D i f f u s i o n C o e f f i c i e n t (Dy) 77 3.6 The E f f e c t of Grain Boundary Miso r i e n t a t i o n (R) ... 81 3.6.1 Evaluation of the Grain Boundary D i f f u s i o n C o e f f i c i e n t ( D , ) 84 g.b. 3.7 Semi-Quantitative Analysis of Isoconcentration Contours 89 3.8 The Al-15.75 wt.% Ag System 94 3.9 A p p l i c a b i l i t y of Contours 99 - v - Page 4. KINETIC RESULTS AND DISCUSSION 101 4.1 Introduction 101 4.2 A p p l i c a t i o n of the E x i s t i n g Models f o r D i s s o l u t i o n K i n e t i c s 102 4.3 An Exponential Model for Planar D i s s o l u t i o n K i n e t i c s 111 4.4 P r e c i p i t a t e Shape Change During D i s s o l u t i o n 122 4.4.1 High Homologous Temperatures 131 4.4.2 Low Homologous Temperatures 137 4.4.3 Intermediate Homologous Temperatures 138 5. CONCLUSIONS: 142 BIBLIOGRAPHY 144 APPENDIX I Correction Procedures f o r Quantitative Electron Probe Microanalysis 148 1.1 The Al-Cu System 148 1.2 The Al-Ag System 153 APPENDIX II Electron Spot Size Determination 156 APPENDIX III C a l c u l a t i o n of the Flux Line Divergence f o r D i f f e r e n t Sections of a Two Dimensional E l l i p t i c a l geometry 159 APPENDIX IV D v C a l c u l a t i o n from the D i f f u s i o n Couple Results 162 APPENDIX V C a l c u l a t i o n of Standard Deviation f o r the Isoconcentration Contours 164 - v i - Page APPENDIX VI Ca l c u l a t i o n of ij> 166 APPENDIX VII Calculations of the Impingement Resulting from the Interaction of the Confocal Spheroids About an Allotriomorph and a Surrounding Spherical F i e l d 170 - v i i - LIST OF FIGURES Figure Page 1 A schematic diagram of the "Grain Boundary C o l l e c t o r P l a t e " model showing the three steps that take place i n the growth of grain boundary allotriomorphs 8 2 (a) Section of an equilibrium phase diagram showing Tg and T^ for an a l l o y of composition C n. (b) D i s s o l u t i o n composition p r o f i l e for an a l l o y of composition C n 10 3 A comparison of composition contours for (a) growth, (b) d i s s o l u t i o n 18 4 (a) The aluminum-copper equilibrium phase diagram (b) The aluminum r i c h end of the aluminum-copper equilibrium phase, diagram 22 5 The aluminum-silver equilibrium phase diagram 23 6 O p t i c a l micrograph of a t y p i c a l a l l o y microstructure for the Al-2.81 wt.% Cu a l l o y 26 7 Schematic representation of the d i r e c t i o n s of movement of a specimen r e l a t i v e to the el e c t r o n beam, i n the electron probe microanalyzer 30 8 O p t i c a l micrograph of a t y p i c a l grain boundary allotriomorph, with contamination marks showing the f u l l extent of scans across the p r e c i p i t a t e and grain boundary 31 - v i i i - Figure Page 9 The (a) specimen current, (b) background concentration, and (c) Cu^Kct^ concentration, f o r the blank p r o f i l e (1) Through the centre of the p r e c i p i t a t e , (2) at the allotriomorph t i p and (3) 5 u away from the t i p across the grain boundary 34 10 Schematic diagram depicting the e f f e c t of spot s i z e on the evaluation of concentration at a point x^ away from the p r e c i p i t a t e i n t e r f a c e 35 11 A schematic depiction of the d i s s o l u t i o n f l u x l i n e s f o r ; (a) an allotriomorph sectioned above the minor axis b, (b) an allotriomorph sectioned at the minor axis and (c) an allotriomorph sectioned below the minor axis 35 12 O p t i c a l micrograph showing (a) el e c t r o n probe traces across the bonded i n t e r f a c e of an Al-CuA^ d i f f u s i o n couple, (b) microstructure of large CuA^ grain i n an a + 6 matrix 39 13 D i s s o l u t i o n isoconcentration contours of a grain boundary allotriomorph on the surface and 3u below the surface (A and B) and a second allotriomorph i n the bulk material (C). At T„ = 0.97 for a l l o y A ... 40 H 14 Population d i s t r i b u t i o n curve f o r allotriomorphs of varying a x i a l r a t i o S Q / R o for a l l o y A 42 15 Composition contour f o r Al-CuA^ d i f f u s i o n couple at 500°C 46 - i x - Figure Page 16 P r o b a b i l i t y p l o t of atomic % Cu versus d i f f u s i o n distance (x) over the s o l i d s o l u b i l i t y range of the a phase f o r the 500°C d i f f u s i o n couple 47 17 Arrhenius p l o t of versus 1/T°K ''"for the d i f f u s i o n couple r e s u l t s 48 18 Size d i s t r i b u t i o n (S o) of p r e c i p i t a t e s as a function of S /R f o r a l l o y A 51 o o J 19 Isoconcentration contours at T = 0.91 for a l l o y B.. 52 rl 20 Concentration contour along the minor axis of a grain boundary allotriomorph at 545°C for a l l o y A .. 55 21 Isoconcentration contours f o r three allotriomorphs having a x i a l r a t i o s S Q/R o of 1) 18, 2) 3.5, 3) 1.7 for a l l o y A at T • = 0.93 57 22 Isoconcentration contours of two Widmanstatten plates f o r a l l o y A at T„ = 0.93 59 rl 23 Isoconcentration contours f o r a l l o y A at T = 0.97.. 62 H 24 Isoconcentration contours f o r a l l o y B at T = 0.94.. 63 H 25 Isoconcentration contours for a l l o y B a t T = 0.92.. 64 ri 26 Isoconcentration contours f o r a l l o y B at T = 0.905. 65 ti 27 Isoconcentration contours f o r a l l o y B at T = 0.86.. 66 H 28 Isoconcentration contours for a l l o y C at T = 0.80.. 67 ti 29 Isoconcentration contours f o r a l l o y C at T = 0.77.. 68 30 Isoconcentration contour f o r a l l o y D at T = 0.72... 70 rl 31 Interface concentration Ĉ . versus vTTt" f or a l l o y B at T R = 0.92 73 - X - Figure Page 32 C versus e r f ( — — ) for a l l o y B at T„ = 0.92 74 2vfit~ H C ( x t) ~ CM 33 P r o b a b i l i t y p l o t of [ — ~ 2 — ] versus d i f f u s i o n 4 ~ LM distance (x) for a l l o y B at T„ = 0.92 75 n 34 Isoconcentration contours for a l l o y A at T = 0.92.. 79 H 35 Isoconcentration contours of two allotriomorph shaped p r e c i p i t a t e s s i t u a t e d on a low angle grain bounary f o r the Al-Ag a l l o y at T„ = 0.84 82 36 Isoconcentration contours f o r the Al-Ag a l l o y at T R = 0.84 83 37 Schematic diagram for Shewmon surface d i f f u s i o n model (af t e r P.G. Shewmon) 85 38 Evaluation of D , fo r a l l o y B at T, = 0.86 86 g.b. H m, " ^. f l u x v i a grain boundary d i f f u s i o n ,,s 39 The function — : — TTF^—. W f l u x v i a volume d i f f u s i o n versus temperature T 91 40 A comparison of t h e o r e t i c a l l y derived versus homologous temperature T with the experimental rl , _. _ solute v i a grain boundary d i f f u s i o n observation of — ; — ; =—: ,. — .. 92 solute v i a volume d i f f u s i o n 41 Schematic diagram of method used to cal c u l a t e 1) solute v i a grain boundary d i f f u s i o n , and 2) solute v i a volume d i f f u s i o n 94 42 Isoconcentration contours f o r the Al-Ag a l l o y at T u = 0.885 95 rl 43 Isoconcentration contours f o r the Al-Ag a l l o y at T R = 0.93 96 - x i - Figure Page 44 Isoconcentration contour f o r a l l o y B at T = 0.83... 97 rl 45 Isoconcentration contours for a l l o y B at T = 0.885. 98 rl 46 Arrhenius p l o t of D y versus l/T°K - 1for the Al-15.75 wt.% Ag a l l o y 100 2 47 A p l o t of R versus t and R versus /t f o r a l l o y B at T„ = 0.92 104 ri 48 Log [ S / S q ] versus t and log [ R / R q ] versus t for a l l o y A at T = 0.97 106 rl 49 Log [ S / S q ] versus t and log [ R / R q ] versus t f o r a l l o y D at T„ = 0.86 107 rl 50 Log [ S / S q ] versus t and log [ R / R q ] versus t f o r a l l o y B at T„ = 0.86 108 rl 51 Log [ S / S q ] versus-t and log [ R / R q ] versus t for a l l o y D at T • = 0.84 109 n 52 Log [ S / S O ] versus t and log [ R / R q ] versus t f o r a l l o y D at T u = 0.76 110 rl •53 (a) Geometrical model of a p a r t i c l e with h a l f - thickness R q i n a matrix with i n t e r p a r t i c l e spacing 111 I. (b) An appropriate phase diagram with growth of p r e c i p i t a t e s taking place at and d i s s o l u t i o n at T^* (c) D i s s o l u t i o n p r o f i l e p r i o r to impingement. 112 54 (a) The concentration d i s t r i b u t i o n s during d i s s o l u - t i o n a f t e r impingement takes place, (b) Schematic diagram of model used with a constant concentration C„, at x = 1 113 - x i i - Figure Page 55 R versus t f o r equations 11 and 12 117 56 Log [R / R q] versus t f o r d i s s o l u t i o n of p r e c i p i t a t e s i n an Al-4 wt.% Cu system at 520°C 119 57 Log [ R / R Q ] versus t f o r the d i s s o l u t i o n of planar p r e c i p i t a t e i n a f i n i t e system w i t h k = 0.22 120 58 Arrhenius p l o t of versus 1/T°K "'"for the k i n e t i c r e s u l t s 121 59 Back s c a t t e r e d e l e c t r o n image micrographs,used to observe the d i s s o l u t i o n k i n e t i c s of an a l l o t r i o m o r p h at T R = 0.97 f o r a l l o y A 123 60 Back s c a t t e r e d e l e c t r o n image micrographs,used to observe the d i s s o l u t i o n k i n e t i c s of an a l l o t r i o m o r p h at T R = 0.97 f o r a l l o y A 124 61 Back s c a t t e r e d e l e c t r o n image micrographs,used to observe the d i s s o l u t i o n k i n e t i c s of an a l l o t r i o m o r p h at T„ = 0.88 f o r the Al-Ag a l l o y 125 ri 62 Back s c a t t e r e d e l e c t r o n image micrographs, used to observe the d i s s o l u t i o n k i n e t i c s of an a l l o t r i o m o r p h at T H » 0.88 f o r a l l o y B 126 63 Back s c a t t e r e d e l e c t r o n image micrographs,used to observe the d i s s o l u t i o n k i n e t i c s of an a l l o t r i o m o r p h at T D = 0.84 f o r the Al-Ag a l l o y 127 n 64 F r a c t i o n of p r e c i p i t a t e s showing an increase i n a x i a l r a t i o g r e a t e r than 10% (F) versus super- s a t u r a t i o n (k) 129 - x i i i - Figure Page 65 Frac t i o n of p r e c i p i t a t e s showing an increase i n a x i a l r a t i o greater than 10% (F) versus T 130 n 66 (a) Symmetrical d i s t r i b u t i o n of p r e c i p i t a t e s with [S/R ] = 4 131 o o (b) Schematic representation of concentration f i e l d s around a symmetrical array of p r e c i p i t a t e s . . . 132 (c) Diagram of s p h e r i c a l l y symmetrical f i e l d surrounding a grain boundary allotriomorph 132 67 The r e s u l t i n g impingement between the confocal spheroid.; isoconcentration contours about an allotriomorph and the contours of the surrounding s p h e r i c a l f i e l d aftex t = 500 seconds f o r various [C/C^.] values 134 6 8 (a) [C/Cj] versus x for the minor axis Ob ^35 (b) [C/C ] versus x for the eq u a t o r i a l plane Oa 136 S ,R 69 Percentage increase i n [_, ] versus [S /R ] 139 S /R o o o o 70 (a) Log [R/R Q] versus t (b) [S/R] versus t 140 1-1 Cn versus K_ 152 Cu Cu I- 2 C A versus K. 155 Ag Ag I I - l Schematic diagram of the e l e c t r o n spot s i z e 157 I I I - l Two dimensional sections of an oblate spheroid 159 IV- 1 Composition p r o f i l e of d i f f u s i o n couple at 500°C .... 162 VII-1 Schematic diagram comparing actual contour to confocal spheroid 170 - x i v - Figure Page VII-2 P o s i t i o n of error function p r o f i l e 171 VII-3 Pertaining to Equation 4 172 VII-4 Schematic diagram of the summing up of two i n - dependent p r o f i l e s 174 - XV - LIST OF TABLES Table Page I Morphology C l a s s i f i c a t i o n and Relationship to Boundary Mis o r i e n t a t i o n 3 II (a) P r e c i p i t a t e Growth Equations 13 (b) P r e c i p i t a t e D i s s o l u t i o n Equations 14 III Average S q for Allotriomorphs with S Q / R 0 = 3.5 and Average Width of P r e c i p i t a t e Free Zone f o r the D i f f e r e n t Alloys 50 IV Contour Experiments 61 V Interface Concentration (Ĉ .) and Volume D i f f u s i o n C o e f f i c i e n t (Dy) f o r Contour Experiments 7 5 VI Comparison of the E f f e c t s of k, T and T on the rl Shape of Isoconcentration Contours 80 VII Grain Boundary D i f f u s i o n C o e f f i c i e n t s (D^ ^ ) Determined from Contours 88 VIII K i n e t i c Experiments 103 I I - l E , A, Z and p Values for Ag Lot, , A l Ka and Cu Ka^ Radiation 156 I I I - l A9° for D i f f e r e n t Values of x/b 160 IV- 1 D v Values of the D i f f u s i o n Couple Results 163 V- l D i f f u s i o n Distances, x, a and — 164 2 2 ^ VII-1 Values of (r^ - B ) 2 and n^ of Confocal Spheroids for D i f f e r e n t C/Cj. Ratios 172 VII-2 Values of r for Concentric Spheres for D i f f e r e n t C/C T Ratios 173 - x v i - ACKNOWLEDGEMENTS The author would l i k e to thank Dr. L.C. Brown for invaluable help and encouragement throughout the duration of t h i s p r o j ect. Thanks are also due to Dr. D.E. Coates f o r numerous h e l p f u l discussions and guidance during the preparation of t h i s t h e s i s . The c r i t i c i s m s and suggestions of Dr. E.B. Hawbolt, Dr. F. Weinb and R.V. Krishnan of the f i n a l d r a f t of the thesis are s i n c e r e l y appreciated. The assistance of the t e c h n i c a l s t a f f and i n p a r t i c u l a r that of Mr. A. Lacis i s gr e a t l y appreciated. Thanks are extended to Mr. A. Bakas for h i s help with the figures and drawings f o r the t h e s i s . F i n a n c i a l assitance provided by the National Research Council under grant number A 2459, and the graduate fellowship awarded by the University of B r i t i s h Columbia are g r a t e f u l l y appreciated. 1. INTRODUCTION The growth of precipitates i n s o l i d - s t a t e systems has been examined extensively and the r e s u l t s have been analysed t h e o r e t i c a l l y on the basis of d i f f u s i o n theory. Reasonably good c o r r e l a t i o n between experiment and theory has been obtained f o r grain boundary a l l o t r i o - morph growth, by incorporating volume and grain boundary d i f f u s i o n i n the a n a l y s i s . Much les s experimental e f f o r t has been expended on the d i s s o l u t i o n of p r e c i p i t a t e s and, as yet, no conclusive experimental observations have been made on the d i s s o l u t i o n of grain boundary allotriomorphs. I t has been suggested that the same analysis as used f o r allotriomorph growth i s applicable to allotriomorph d i s s o l u t i o n . For a number of reasons, which w i l l be discussed l a t e r , t h i s analysis i s considered to be at best only a f i r s t approximation when applied to d i s s o l u t i o n . The theory pertinent to the growth and the d i s s o l u t i o n of p r e c i p i t a t e s w i l l follow a short discussion of the morphology of grain boundary p r e c i p i t a t e s . 1.1 Grain Boundary P r e c i p i t a t e s The morphology of grain boundary p r e c i p i t a t e s as a function of misorientation across the boundary on which they are s i t u a t e d was - 2 - 1 o r i g i n a l l y studied and c l a s s i f i e d by Dube et a l . This c l a s s i f i c a t i o n 2 3 was amended s l i g h t l y by Aaronson. In subsequent work Aaronson has shown that grain boundary misorientation can be characterized by the parameter R where: and X, Y, and Z represent the minimum rotations about the three ortho- gonal axes required to bring the two adjacent l a t t i c e s i n t o coincidence. This c l a s s i f i c a t i o n of p r e c i p i t a t e morphology as a function of R i s given i n Table I. The fact that p r e c i p i t a t e shape i s a function of L 4 only R has been v e r i f i e d by the work of Toney and Aaronson i n the Fe-Si system, Clark"* i n the Al-18 wt.% Ag system and Hawbolt and Brown 6 i n the Ag-5.64 wt.% A l system. Grain boundary allotriomorphs, the most common type of grain boundary p r e c i p i t a t e s , are confined to grain boundary misorientations of R > 15°. These grain boundaries are considered to be high energy disordered boundaries and comprise a large proportion of the boundaries i n a random p o l y c r y s t a l . ^ The high energy of the boundaries i s 'instrumental i n : ( i ) grain boundary allotriomorphs being the f i r s t p r e c i p i t a t e s . t o nucleate and hence appear during growth,and ( i i ) a l i o - triomorphs forming i n preference to Widmanstatten plates at low degrees of supersaturation. In general 9allotriomorphs have an oblate spheroidal shape, being c i r c u l a r i n the plane of the grain boundary and having a lens shaped R Morphology Classification and Relationship to Boundary Misorientation Grain Boundary Allotriomorphs Primary Sideplate Widmanstatten Secondary Sideplates Primary and Secondary Sawteeth Idiomorphs Morphology ill A Grain Boundary Misorientation Associated with l a r g e angle or disordered— type grain boundaries i . e . , >15° misorientation » Associated with small angle or dislocation type boundaries and grain interiors <15°. Generally associated with intermediate D i s o r i e n t a t i o n s 15"- 25°. Generally associated with intermediate misorientations 15*- 25°. Usually an intragranular morphology although were observed for low angle grain boundaries in the Pe-l.55 Si system. (Clark , Dube and Aaronson ) - 4 - (e l l i p t i c a l ) cross section. In the Al-18 wt.% Ag and the Al -4 wt.% g Cu systems, the allotriomorph-matrix interface is generally found to be incoherent and the allotriomorphs develop equally in both 4 8 grains. It has been noted, ' however, that nucleation often takes place in only one grain. This results in a planar, semi-coherent interface with a definite crystallographic relationship between the allotriomorph and the matrix grain. I n i t i a l growth of the a l l o t r i o - morph takes place in the adjacent matrix grain. At a later stage of growth the semi-coherent interface becomes mobile followed by growth into both matrix grains. In an examination of grain boundary ferrite precipitation from 9 austenite in a Co-20% Fe alloy, Ryder and Pitsch and P.L. Ryder et a l . ^ found that grain boundary precipitates on high angle grain boundaries always have an orientation relationship with at least one and sometimes both of the austenite grains. However, they did not make sufficient measurements to determine whether the austenite- austenite relationships are completely random. Thus some doubt is cast on the observations of an orientation relationship between the precipitate and both grains. 1.2 Growth of Precipitates Theoretical growth rate calculations for the diffusion-controlled growth of planar, cylindrical and spherical precipitates were originally 11 12 made by Zener and Frank. These solutions were extended to cover 13 14 the growth of spheroids and ellipsoids by Ham. ' The rate of growth of the major and minor axes of oblate spheroids, for various • - 5 - a x i a l r a t i o s , was numerically evaluated by Horvay and Cahn.^ These solutions do not include impingement e f f e c t s and are therefore only v a l i d during the i n i t i a l period of growth p r i o r to the i n t e r a c t i o n of adjacent d i f f u s i o n f i e l d s . The experimental observations and analysis of the growth and d i s s o l u t i o n of allotriomorphs reported i n the l i t e r a t u r e i s summarized below. F i r s t , growth i s considered i n terms of thickening and lengthening k i n e t i c s . 1.2.1 Thickening of Grain Boundary Allotriomorphs The f i r s t measurements of growth k i n e t i c s of grain boundary 16 allotriomorphs were made i n commercial s t e e l s by Mazanec and Cadek 17 and Hickley and Woodhead. Both reported a parabolic thickening rate f o r the growth of f e r r i t e p r e c i p i t a t e s i n bulk s t e e l samples, which i s consistent with d i f f u s i o n c o n t r o l l e d growth. Hawbolt and Brown, using a s t a t i s t i c a l approach, found that at high homologous temperatures (T > 0.90) the thickness of g phase rl grain boundary allotriomorphs i n the Ag-5.64 wt.% A l system increases 18 p a r a b o l i c a l l y with time. Aaron and Aaronson examined the thickening of eCCuA^) grain boundary allotriomorphs i n an Al-4 wt.% Cu a l l o y for homologous temperatures i n the range T = 0.54-0.69. They assumed the true thickness to be that of the th i c k e s t allotriomorphs i n a ser i e s of f o i l s reacted f o r successively increasing times at each temperature. Their d i f f u s i o n c o e f f i c i e n t s , c a l c u l a t e d using Z e n e r s ^ planar model, were found to be several orders of magnitude higher than the l i t e r a t u r e volume d i f f u s i o n c o e f f i c i e n t s . They conclude - 6 - that at lower homologous temperatures grain boundary d i f f u s i o n i s operative and contributes to the enhanced growth rate. 1.2.2 Lengthening of Grain Boundary Allotriomorphs Less data on the lengthening rate has been reported than on the thickening rate of grain boundary allotriomorphs. Dube et al.''" and 3 Aaronson measured the largest f e r r i t e p r e c i p i t a t e r e s u l t i n g from the Fe-C pro-eutectoid reaction and reported a l i n e a r lengthening rate up to impingement. Aaronson's r e s u l t s f o r the Fe-C system showed reasonable agreement with the lengthening rate equation proposed by 19 6 H i l l e r t . Hawbolt and Brown found that the lengthening of grain boundary allotriomorphs increases p a r a b o l i c a l l y with time i n the Ag-5.64 wt.% A l system at high homologous temperatures T u > 0.90. 18 At low homologous temperatures, Aaron and Aaronson found that the lengthening rate of grain boundary allotriomorphs i n the Al-4% Cu 19 system, cannot be adequately accounted for by the H i l l e r t analysis This analysis gave values several orders of magnitude higher than the l i t e r a t u r e value. 1.2.3 The "Grain Boundary C o l l e c t o r P l a t e " Model The "grain boundary c o l l e c t o r p l a t e " model was formulated to account for the greatly enhanced growth k i n e t i c s observed by Aaron 18 and Aaronson. This model makes use of the high d i f f u s i v i t y paths provided by the grain boundary and the disordered allotriomorph-matrix i n t e r f a c e . The model assumes D , /DTI -»- °° i . e . the volume d i f f u s i o n g.b.' V contribution i s i n s i g n i f i c a n t ^ a n d grain boundary d i f f u s i o n completely dominates the growth process. Growth of grain boundary allotriomorphs - 7 - according to t h i s model occurs by: 1. volume d i f f u s i o n of solute to the grain boundary; 2. solute transport along the grain boundary c o l l e c t o r plate to the allotriomorph (a f i n i t e grain boundary area of h a l f length (R^-R) i s associated with each allotriomorph); 3. interphase boundary d i f f u s i o n and deposition of solute over the surface of the growing allotriomorph as shown i n Figures 1(a) and 1(b). 20 18 B r a i l s f o r d and Aaron modified the Aaron and Aaronson C o l l e c t o r Plate Model to accommodate s i t u a t i o n s i n which the volume d i f f u s i o n contribution i s s i g n i f i c a n t . The "Grain Boundary C o l l e c t o r P l a t e " model predicts a thickening rate proportional to t^*"* and a lengthening rate proportional to t^" 2"' f o r the experimental 18 conditions used i n the work of Aaron and Aaronson. These predicted rates are i n reasonable agreement with the l a t t e r ' s experimental 0 34 observations of thickening proportional to t " and lengthening 0.27 21 proportional to t . Goldman et a l . , using the B r a i l s f o r d and , 20 Aaron analysis f o r allotriomorph growth rates i n the temperature range T = 0.69-0.78, found that volume d i f f u s i o n d i r e c t l y to the rl allotriomorph contributes s i g n i f i c a n t l y to growth only at the highest homologous temperature ( T R = 0.78). Below t h i s temperature the major contribution to growth i s v i a grain boundary d i f f u s i o n . 1.3 D i s s o l u t i o n Studies of P r e c i p i t a t e s D i s s o l u t i o n of p r e c i p i t a t e s occurs when a two-phase material i s heated or cooled to a temperature where the p r e c i p i t a t e phase becomes unstable. This r e s u l t s i n mass transport from the unstable second phase Figure 1. A schematic diagram of the "Grain Boundary Collector P l a t e " model showing the three steps that take place i n the growth of grain boundary allotriomorphs. - 9 - t o t h e s t a b l e m a t r i x p h a s e , as shown i n F i g u r e 2 ( a ) , where a l l o y C n may be p r e s e n t as e i t h e r (a+6) o r a , depending on t h e t e m p e r a t u r e . The d i s s o l u t i o n c o m p o s i t i o n p r o f i l e f o r an a l l o y o f c o m p o s i t i o n C n , a f t e r i n s t a n t a n e o u s h e a t i n g from T^ t o T^, i s d e p i c t e d i n F i g u r e 2 ( b ) . The c o m e n c l a t u r e i s : C n = e q u i l i b r i u m a l l o y c o n c e n t r a t i o n ; = c o n c e n t r a t i o n i n t h e m a t r i x a t t h e p r e c i p i t a t e - m a t r i x i n t e r f a c e ; C^ = m a t r i x c o n c e n t r a t i o n ; Cp = c o m p o s i t i o n o f p r e c i p i t a t e ; Tg = p r e c i p i t a t e g r o w t h t e m p e r a t u r e ; Tp = p r e c i p i t a t e d i s s o l u t i o n t e m p e r a t u r e . 22 N o l f i e t a l . d e v e l o p e d a n a l y t i c a l e x p r e s s i o n s f o r t h e growth and d i s s o l u t i o n o f s p h e r i c a l p r e c i p i t a t e s f o r d i f f u s i o n c o n t r o l l e d , i n t e r f a c e r e a c t i o n c o n t r o l l e d , a n d m i x e d c o n t r o l mass t r a n s p o r t . I n subsequent e x p e r i m e n t a l work on t h e d i s s o l u t i o n o f Fe^C i n f e r r i t e , 23 24 N o l f i e t a l . ' found t h a t l o c a l e q u i l i b r i u m w i t h r e s p e c t t o t h e i r o n i s ; n o t m a i n t a i n e d a t t h e F e ^ C - f e r r i t e i n t e r f a c e . They c o n c l u d e t h e d i s s o l u t i o n o f F e ^ C - f e r r i t e i s c o n t r o l l e d e n t i r e l y by an i n t e r f a c i a l r e - a c t i o n . 25 26 , U s i n g a n u m e r i c a l method, T a n z i l i and H e c k e l ' t r e a t e d : t h e d i f f u s i o n c o n t r o l l e d d i s s o l u t i o n , from an u n s t a b l e s p h e r i c a l , c y l i n d r i c a l o r p l a n a r second<phase t o a s t a b l e f i n i t e m a t r i x phase,as a f u n c t i o n o f , C^, Dg, C n and R f o r complete p r e c i p i t a t e s o l u t i o n and subsequent o v e r a l l h o m o g e n i s a t i o n o f t h e m a t r i x . Because t h e p r e s e n t i n v e s t i g a t i o n i s p r i m a r i l y c o n c e r n e d w i t h d i f f u s i o n c o n t r o l l e d d i s s o l u t i o n , i t i s a p p r o p r i a t e t h a t t h i s t o p i c be d i s c u s s e d i n g r e a t e r d e t a i l . F o r a s p h e r i c a l p r e c i p i t a t e t h e - 10 - Z o I — < \— z uu U z o U J ? T, D < ? T, c c c C O N C E N T R A T I O N C, - 'O -M •R(t)- R, DISTANCE <r> Figure 2. (a) Section of an equilibrium phase diagram showing T and T for an a l l o y of composition C^. (b) D i s s o l u t i o n composition p r o f i l e f o r an a l l o y of composition C 0' -11- d i f f u s i o n f i e l d f o r d i f f u s i o n c o n t r o l l e d d i s s o l u t i o n must s a t i s f y the following f i e l d equation: DV2C = £ 1 dt where C = C(r,t) i s the concentration of solute i n the matrix surrounding the p r e c i p i t a t e , r (defined as i n Figure 2 ( b ) ) i s the distance from the centre of tie precipitate,and i t i s assumed D f D(C) . The following boundary conditions apply: C(r = R,t) = C. 0 < t < C(r, t=0) = C. M r > R C(r = o o }t) = C. M 0 < t « oo where R i s the p r e c i p i t a t e radius. Ĉ . i s the concentration i n the matrix at the precipitate-matrix i n t e r f a c e for t > 0 and i s the i n i t i a l matrix composition. A further boundary condition i s the interface f l u x balance: f C - C ) ^ = D l L P V dt D 9r r=R where i t i s assumed f ( r , t ) . This assumption i s j u s t i f i e d , as the change i n composition of CuAl^ over a range of 548°C i s less than 0.5 wt.% A l (see Figure 4a). - 12 - Several approximations have been used to tre a t the d i f f u s i o n c o n t r o l l e d d i s s o l u t i o n of an i s o l a t e d p r e c i p i t a t e i n an i n f i n i t e matrix. These models include the Reversed Growth model, the Laplace 2 dR model (V C= 0), the Stationary Interface model (-j-̂ - = 0), and the Linear- i z e d Gradients model. Of these,the Stationary Interface approximation appears to be the most accurate. In a t h e o r e t i c a l comparison of the d i f f e r e n t mathematical models for d i f f u s i o n l i m i t e d growth and 27 d i s s o l u t i o n Aaron et a l . showed that the stationary i n t e r f a c e approximation i s the only one v a l i d f o r both growth and d i s s o l u t i o n of s p h e r i c a l and planar p r e c i p i t a t e s (see Table I I ) . 1.3.1 Observations of the D i s s o l u t i o n of P r e c i p i t a t e s by Electron Microscopy Using electron microscopy at high homologous temperatures, 2 8 Thomas and Whelan (1960) observed p r e c i p i t a t e d i s s o l u t i o n i n an Al-A wt.% Cu a l l o y . They found that i n the f i n a l stages of d i s s o l u t i o n three out of twelve p r e c i p i t a t e s s a t i s f y the following equation: = - kD 4 dt They suggest that equation (4) should adequately describe s o l u t i o n k i n e t i c s for small k values where: 2 < c i - y s i s the supersaturation. Equation (5) i s derived by an approximation c r ° M using the steady state d i f f u s i o n theory, f or small (7;—pr -) r a t i o s . T> I - 13 - T A B L E n o PRECIPITATE GROWTH SPHERE R = \.(Dt)"s" P L A N E S = X.(Dt)1 j Exact Solution Invariant Size Approx. Invariant F i e l d Approx. Linearized Gradients Approx. X, = 2X ^ e X 2 [ e ~ x 2 - \ erfc \] = - 7 4 lim X1 - Xz k -> 0 lim \ 1 = x 3 k - 0 k < 0: X x > X2 > X 3 lim X x = XA', k ̂  -2j Xx > XA k - 0 x 2 = - k lim X2 - X 3 k - 0 /4E f(k) = -4k L 1 + 4k Y l_r„X(k). f(k) X4 - /-k/r X1 = 2X X2 k /TT X e erfc X = - - lim X x = X2 k - 0 k < 0: \ x > X 2 |k| > 1 .98: \ < XA X2 = -k//Tf N O T DEFINED (cannot satisfy the far field condition) 2(1+ § )* lim \A = -k/2 k - 0 Note: k, C , C„, and C T are defined i n Section 1.3 p M I The X , j = 1, 2, 3, and 4,correspond to the rate constants for the exact s o l u t i o n , i n v a r i a n t s i z e , i n v a r i a n t f i e l d and l i n e a r i z e d gradients approximations re s p e c t i v e l y . - 14 - Exact Solution Invariant Size Approx. Invariant F i e l d Approx. TABLE l i b PRECIPITATE DISSOLUTION SPHERE Linearized Gradients Approx. (y + 2p ./> y + t) arctan / T _Y_ + p where' R/R , _ a f t " R 2 2 _ kD, p 2 - kAn and reduces to y 2 = i - T i n the limit as p (or k) goes to zero 2 - Tj2 or y 2 = 1 - kDt o Sufficiently complex to negate i t s usefulness. 27 PLANE S = S Q - X.(Dt)2 >.x = 2X X eXZer£c(-\) = k/2 lim X1 = A.2 > X 4 k - 0 k > 0 : \ 2 > 0 ̂  k ^ 1.5: ^ s ^ i > X* X 2 = k / / f f NOT DEFINED (cp - cMF ( C p 2(1 + | ) S lim XA - ~ k - 0 ^ - 15 - The f a c t that only a small f r a c t i o n of the p r e c i p i t a t e s observed obey equation (4) i s a t t r i b u t e d to surface d i f f u s i o n enhancement of d i s s o l u t i o n . In most a l l o y systems of i n t e r e s t |k| < 0.3,and values of |k| < 0.1 are quite common. For t h i s small k range, R i s a slowly varying function of time,and the s o l u t i o n to equation (1) reduces to that 29 obtained by assuming a stationary i n t e r f a c e . Whelan used the stationary i n t e r f a c e approximation to obtain the following equation for the d i s s o l u t i o n rate: dR - k , f D , dt" = 2 [ D / R + 1 6 The D/R term a r i s e s from the steady state part of the d i f f u s i o n f i e l d (Equation 4) yand the term arises from the transient part of the f i e l d . In the case of planar p r e c i p i t a t e s (the one dimensional case), there i s only a transient term i n the expression for d i s s o l u t i o n rate^ and one obtains: R = R - k ,F^~ 7 . O y 7T For s p h e r i c a l p r e c i p i t a t e s (the three dimensional case) 9 both terms i n equation (6) contribute to the d i s s o l u t i o n rate. In the f i n a l stages of d i s s o l u t i o n of s p h e r i c a l p r e c i p i t a t e s , the transient term becomes n e g l i g i b l e and equation (6) reduces to equation (4). A good p h y s i c a l d e s c r i p t i o n of the d i f f u s i o n f i e l d s around d i s s o l v i n g spheres i s 30 given by Readey and Cooper. - 16 - 1.3.2 D i s s o l u t i o n Studies Using the Ele c t r o n Probe Microanalyzer. 31 H a l l and Hayworth used e l e c t r o n probe microanalysis to determine the concentration p r o f i l e s around d i s s o l v i n g Widmanstatten plates and needles i n an Al-5 wt.% Cu a l l o y . They found that at high homologous temperatures (T = 0.91-0.95), the plates dissolve v i a a volume n d i f f u s i o n mechanism and i n t e r f a c e equilibrium i s maintained. E i f e r t 32 et a l . used the probe to observe the g (B.C.C.) -* a(F.C.C) transformation i n the Cu-12.5 wt.% A l system. They reported a d i f f u s i o n - c o n t r o l l e d l a t t i c e transformation i n which i n t e r f a c e equilibrium i s not maintained. 1.4 D i s s o l u t i o n Mechanisms Derived from P r e c i p i t a t e Growth 33 34 Both T a n z i l i and Heckel and Aaron have proposed an experimental procedure, based on t h e o r e t i c a l considerations, whereby d i s s o l u t i o n experiments may be used to determine the growth mode of p r e c i p i t a t e s . That the observed growth mode cannot be used to a r r i v e at d i s s o l u t i o n 27 k i n e t i c s has been shown by Aaron et a l . In t h e i r treatment of d i s s o l u t i o n at the reverse of growth, they show that to treat d i s s o l u - t i o n as the mathematical inverse of growth i s at best a crude approximation, as the boundary conditions are i n t r i n s i c a l l y d i f f e r e n t 27 i n the two cases. Table II summarizes the r e l a t i o n s h i p between supersaturation (k) and transformation rate constant (^j)> f ° r t n e d i f f e r e n t mathematical models used to determine growth and d i s s o l u t i o n k i n e t i c s . 22 I t has further been shown by N o l f i et a l . that curvature and int e r f a c e reaction k i n e t i c s play an important part i n the early stages of growth and that both tend to decrease the growth v e l o c i t y . - 17 - Curvature e f f e c t s on the other hand, are only important i n the l a t e r stages of d i s s o l u t i o n and tend to increase the d i s s o l u t i o n v e l o c i t y . The inference that research on p r e c i p i t a t e growth precludes the need for extensive d i r e c t i n v e s t i g a t i o n of p r e c i p i t a t e d i s s o l u t i o n , may 35 be shown to be i n c o r r e c t by the following reasoning. In the case of growth,the formation of a c r i t i c a l nucleus, a f t e r some incubation time, i s followed by the growth of the p r e c i p i t a t e , causing depletion of solute i n the matrix immediately ahead of the advancing i n t e r f a c e (as shown i n Figure 3(a)). At any p o s i t i o n r i n the matrix phase (r > R ( t ) ) , the solute concentration i s a monotonically decreasing function of time. During d i s s o l u t i o n , on the other hand, there i s no incubation or nucleation period. The p r e c i p i t a t e has a f i n i t e s i z e ( R Q ) and decreases i n s i z e by mass transf e r of solute i n t o the matrix behind the receding i n t e r f a c e (as shown i n Figure 3(b)). The v a r i a t i o n of solute concentration with time f o r d i s s o l u t i o n i s d i f f e r e n t and more complex than i n growth. At a p o s i t i o n w e l l away from the p r e c i p i t a t e (» > r >> R ( t ) ) , the solute concentration increases with time; close to the p r e c i p i t a t e (r > R ( t ) ) , the concentration decreases with time; and at intermediate p o s i t i o n s , the solute concentration w i l l increase, decrease or remain unchanged. Furthermore, the extent of the regions corresponding to 'near', 'intermediate' and 'far' vary with time. 1.5 Objective of Present Investigation The purpose of the present i n v e s t i g a t i o n i s to provide experimental data on the d i s s o l u t i o n of grain boundary allotriomorphs and thereby  - 1 9 n e l l u c i d a t e the d i s s o l u t i o n process. An attempt to a r r i v e at a s a t i s f a c t o r y t h e o r e t i c a l analysis to account for the r e s u l t s w i l l be made. The atomic processes involved i n d i s s o l u t i o n of p r e c i p i t a t e s have thus f a r been i n d i r e c t l y deduced from s t a t i s t i c a l and other re- lat e d k i n e t i c measurements. The objective of t h i s i n v e s t i g a t i o n i s to determine the mechanisms of the d i s s o l u t i o n phenomenon i n a more d i r e c t and fundamental way by the use of electron probe microanalysis. - 20 - 2. EXPERIMENTAL PROCEDURE The electron probe microanalyzer was used as an instrument f o r both quantitative and q u a l i t a t i v e a n a l y s i s . Quantitative e l e c t r o n probe microanalysis was used to determine the concentration p r o f i l e s around d i s s o l v i n g grain boundary allotriomorphs. A technique was established whereby the concentration p r o f i l e s around a si n g l e p r e c i p i t a t e were measured as a function of d i s s o l u t i o n time. K i n e t i c studies of the d i s s o l u t i o n rate of grain boundary allotriomorphs were made using the back-scattered electron image on the ele c t r o n probe to determine the s i z e of a p r e c i p i t a t e as a function of d i s s o l u t i o n time. Here once again, the d i s s o l u t i o n of i n d i v i d u a l p r e c i p i a t a t e s was observed. 2.1 Reasons f o r Systems Chosen The aluminium-copper system was chosen f o r the following reasons: 1. the solidus composition changes r a p i d l y with temperature, gi v i n g r i s e to a s i g n i f i c a n t v a r i a t i o n i n concentration around a 36 37 di s s o l v i n g p r e c i p i t a t e , see the phase diagram (Figure 4); ' 2. the low vapour pressure of the a l l o y at the experimental temperatures; 3. the existence of r e l i a b l e phase diagram data; - 21 - 4. the well-established r e l a t i o n s h i p between p r e c i p i t a t e shape and grain boundary misorientations; and another consideration i s 5. the high degree of accuracy that can be achieved i n the quantitative analysis of these a l l o y s using the electron probe microanalyzer. This accuracy i s due to the small correction required to convert measured Cu X-ray i n t e n s i t y to weight percent of Cu present i n an a l l o y . 36 The aluminium-silver system ' (Figure 5), was chosen for supplementary experiments as i t also f u l f i l l s a l l the above c r i t e r i a . 2.2 Preparation of Master A l l o y s Four Al-Cu master alloys(A,B,C,and D) of composition Al-4.83wt.% Cu, Al-2.81wt.% Cu,Al-1.57wt.% Cu, and Al-1.22wt.% Cu were prepared by melting 99.99% pure copper and 99.99% pure aluminium i n a graphite c r u c i b l e at 720°C. This was followed by casting i n a s p l i t graphite mold to produce a slab of dimensions 6"x 2 3/8" x 1/2". An aluminium-15.75 wt.% s i l v e r a l l o y was produced i n the same way using s i l v e r of 99.95% p u r i t y . Following casting, the slabs were homogenized for seven days at 550°C, then c o l d - r o l l e d i n several passes to give a reduction i n thickness of 50%. This was followed by a two hour anneal at 550°C and further cold r o l l i n g to give a f i n a l sheet thickness of 0.05". The sheet was then given a f i n a l anneal at 550°C for f i v e days. The resultant sheet had an equiaxed grain structure, the grain s i z e varying from 2.5 to 4 mm. The grain boundaries were s t r a i g h t and extended from the top to the bottom surface of the sheet. The majority of the grain boundaries were found to be stable on subsequent heat treatment. Figure 4 . °C 800 (a) The aluminum-copper equilibrium phase diagram °F 36 Atomic Percentage Copper / 2 3 700 600 500 400 300 200 100 Al 'DC 2 B 3 4 A 5 6 7 8 Weight Percentage Copper i i | L i 1 - 660° is i t | K K + L 1 548" i 1 - y. — I i l! 1 1 1 I 1 l | 1 1 - -I- 1 1 1 1 ' 1 1 1 j I 1 i i - 1400 1200 1000 800 600 400 10 Figure 4 . (b) The aluminum rich end of the aluminum-copper equilibrium phase diagram. - 23 - Figure 5. The aluminum-silver equilibrium phase diagram. - 24 - The copper content of the aluminium-copper a l l o y s , as w e l l as the s i l v e r content of the aluminium-silver a l l o y , were determined using two d i f f e r e n t techniques: (the analyses were performed by Warnock-Hersey International Ltd. and Cantest L t d . ) . (a) The Atomic Absorption technique, which has an accuracy of + 0.02% up to 2% Cu and + 0.05% f o r up to 5% Cu. (b) The e l e c t r o p l a t i n g of copper on a platinum electrode,which has an accuracy of 0.02% for the range of i n t e r e s t . The analyses were i n a l l cases within + 0.08% of the cast a l l o y compositions based on the weight of components used. Microscopic sections of specimens were scanned i n the e l e c t r o n probe microanalyzer to evaluate the homogeneity of the material. No s i g n i f i c a n t microsegregation was detected i n a s e r i e s of samples taken across the width of the sheet. 2.3 Procedure Used to Grow Large Grain Boundary Allotriomorphs To study the concentration p r o f i l e s associated with i n d i v i d u a l p r e c i p i t a t e s , a heat t r e a t i n g procedure had to be established which would produce: (1) a minimum number of p r e c i p i t a t e s , (2) p r e c i p i t a t e s large enough i n s i z e f o r accurate probe analysis (thickness of p r e c i p i t a t e s > 4y), and (3) large p r e c i p i t a t e free zones adjacent to the grain boundaries, as w e l l as large i n t e r p a r t i c l e spacings on the grain boundaries to avoid the impingement of neighbouring d i f f u s i o n f i e l d s at short d i s s o l u t i o n times. The desired microstructure was produced i n the following way. Ten discs of 1 1/4" diameter were cut from the f i n a l sheet of a p a r t i c u l a r a l l o y . These discs were t i g h t l y stacked together, t i e d with - 25 - chromel wire>and a hole 1/16" i n diameter was made through h a l f of the discs to accommodate a temperature c o n t r o l l i n g chromel-alumel thermocouple. The assembly was inser t e d i n t o a 2" diameter v e r t i c a l tube furnace with a 4" long maximum temperature zone at 550°C. The temperature was con t r o l l e d to within + 1°C f o r prolonged periods using a deviation a m p l i f i e r . A f t e r three days the temperature of the furnace was reduced slowly to a temperature approximately 10°C above, the phase diagram solvus temperature. Further cooling at 3°C per day was used to nucleate and grow p r e c i p i t a t e s . The slow cooling was continued u n t i l the p r e c i p i t a t e s had grown to a s u i t a b l e s i z e f o r a n a l y s i s . The sample was then allowed to e q u i l i b r a t e f o r seven days to remove any concentration gradients introduced i n t o the matrix by the p r e c i p i t a t e growth. The samples were f i n a l l y quenched i n i c e water from the e q u i l i b r a t i o n temperatures (T ) (as depicted i n Figures 4 and 5). A t y p i c a l e q u i l i b r a t e d microstructure i s shown i n Figure 6. Small t e s t samples were quenched and o p t i c a l l y inspected at various stages of the heat treatment. It was established that nucleation occurred at temperatures within + 3°C of the l i t e r a t u r e phase diagram solvus temperature. The number of grain boundary allotriomorphs present j u s t below the solvus was comparable to that observed i n the e q u i l i b r a t e d structure. 2.4 D i s s o l u t i o n Heat Treatment Procedure Specimens approximately 3/8" x 1/4" were cut from the discs ( i n some cases spark machined), cold mounted and polished to 1 u diamond. Figure 6. Optical micrograph of a typical alloy microstructure for the Al-2.81 wt.% Cu al loy. - 27 - Mechanical polishing,was used i n preference to el e c t r o - p o l i s h i n g , a s t h i s resulted i n a surface f i n i s h more s u i t a b l e f or microprobe a n a l y s i s . P o l i s h i n g produced a s l i g h t etching of the surface i n the case of the Al-4.83 wt.% Cu, the Al-2.83 wt.% Cu and the Al-Ag a l l o y , f a c i l i t a t i n g the l o c a t i o n of appropriate grain boundary a l l o t r i o - morphs. For the other a l l o y s s u i t a b l e allotriomorphs were selected using the absorbed electron image on the microprobe, thus eliminating the need for etching the specimen to o u t l i n e the structure. The same p r e c i p i t a t e s could be relocated a f t e r several heat treatments by the probe contamination marks and by micro-hardness indentations made at s t r a t e g i c points on the surface. For d i s s o l u t i o n heat treatment,the specimens were immersed i n a 60% KNO.j-40% NaNO^ s a l t bath c o n t r o l l e d to + 1°C by a steady state power input unit. Surface contamination by the s a l t was avoided by wrapping the specimens i n 0.0005" thick, high p u r i t y tantalum f o i l . The s a l t did not penetrate the wrapping f o i l because of the high v i s c o s i t y of the s a l t at the temperatures used. The immersion of the small mass of the sample produced a n e g l i g i b l e temperature change i n the s a l t (< 1°C). Temperature s t a b i l i z a t i o n occurred i n less than 30 seconds a f t e r immersion of the specimen i n the s a l t . I t i s estimated that the specimen reached bath temperature within two seconds of immersion i n the s a l t . A f t e r heat treatment the specimens were quenched i n water, the transfer time from the s a l t to the beaker being less than 1.5 seconds. Using t h i s procedure, as many as twenty successive heat treatments - 28 - were c a r r i e d out on some specimens without any observable trace of surface contamination by the s a l t . 2.5 D i f f u s i o n Couple Work D i f f u s i o n couples were prepared by clamping an aluminium b i - c r y s t a l , with a grain boundary angle R = 17°, to a specimen with a structure containing large grains of CuA^ surrounded by a fine e u t e c t i c of a + 0 (see Figure 12(b)). The aluminium b i - c r y s t a l was used to permit the measurement of both grain boundary and volume d i f f u s i o n c o e f f i c i e n t s . The specimen containing the CuA^ grains was produced by melting stoichiometric amounts of copper (53 wt.%) and aluminium i n a vacuum sealed quartz tube at 700°C and quenching i n brine. Several attempts were made to produce pure CuA^ without success. The bonding surfaces of both the aluminium and the CuA^ were polished to 6 y diamond and u l t r a s o n i c a l l y cleaned p r i o r to bonding. The couples were d i f f u s e d at a f i x e d temperature i n a cracked ammonia atmosphere. The temperature-controlling thermocouple was placed j u s t above the couple and the maximum temperature f l u c t u a t i o n r egistered was < 3°C. Unfortunately, bonding was achieved only i n . four couples at temperatures of 480°C, 500°C, 520°C, and 535°C. Repeated attempts at lower temperatures were unsuccessful, probably due to the presence of the oxide f i l m on the aluminium. - 29 - 2.6 Electron Probe Microanalysis 2.6.1 Isoconcentration Contours A JEOLCO (JXA-3A) electron probe microanalyzer with a step-scan attachment was used to determine the composition contours around d i s s o l v i n g grain boundary allotriomorphs. Figure 7 gives a schematic representation of the specimen i n the microprobe specimen holder and the micrometer screws, as w e l l as the possible d i r e c t i o n s of movement of the specimen. The step-scan attachment i s an e l e c t r o n i c device that may be attached to e i t h e r of the micrometer screws to automatically displace the specimen r e l a t i v e to the electron beam by steps of 1.25 u, 5 u or 20 u. Most of the measurements i n t h i s i n v e s t i g a t i o n used 1.25 u step scanning. Each displacement i s followed by a stationary spot count of 10 seconds and the resultant counts are registered automatically. Grain boundary allotriomorphs s i t u a t e d on s t r a i g h t grain boundaries were selected f o r i n v e s t i g a t i o n and were rotated i n t o a p o s i t i o n per- v pendicular to one micrometer. Step scanning was then c a r r i e d out across the width of the p r e c i p i t a t e i n t o the matrix. The scan was continued u n t i l the equilibrium matrix concentration was reached. This procedure was repeated along the length of the p r e c i p i t a t e and the adjacent grain boundary. The scanning procedure i s depicted i n Figure 8. The dark contamination streaks mark the trace of each scan. An i n i t i a l trace was taken on each p r e c i p i t a t e before any d i s s o l u t i o n heat treatment was c a r r i e d out to give a blank p r o f i l e . In the analysis of the Al-Cu a l l o y s , one spectrometer was set to coincide with the CuK^, peak and the second was set at 30' above the 0 1 Specimen 2 Rotating Specimen Holder 3 Electron Beam 4 Spectrometers A&B Micrometer Screws X-Rays Figure 7. Schematic representation of the directions of movement of a specimen relative to the electron beam, in the electron probe microanalyzer. - 31 - Figure 8. Optical micrograph of a t y p i c a l grain boundary allotriomorph, with contamination marks showing the f u l l extent of scans across the p r e c i p i t a t e and grain boundary. CuKa^ peak to measure the background r a d i a t i o n i n t e n s i t y . These settings were chosen i n preference to the A l La^ r a d i a t i o n because the maximum v a r i a t i o n of the aluminium concentration i n these a l l o y s i s between 95 wt.% and 98.5 wt.% Al,and therefore cannot be picked up with any degree of accuracy. In the Al-Ag a l l o y , the Ag Lct-̂  peak and a background at 30' above the peak were reg i s t e r e d . Appendix I gives the d e t a i l s of the correction procedure used to convert measured Cu Ka^ and Ag Lcc^ i n t e n s i t i e s to weight percent coppi and weight percent s i l v e r , r e s p e c t i v e l y . 2.6.1.1 Resolution of Experimental Technique The ac c e l e r a t i n g p o t e n t i a l used i n the evaluation of the concen- 38 t r a t i o n p r o f i l e s was 25 K.V. This s a t i s f i e s the c r i t e r i o n which has been established for optimum accuracy i n quantitative analysis; K where: V = accelerating p o t e n t i a l V„ = e x c i t a t i o n p o t e n t i a l f o r any p a r t i c u l a r r a d i a t i o n . The p r i n c i p a l r e s o l u t i o n . l i m i t a t i o n i n quantitative e l e c t r o n probe microanalysis , i s the f i n i t e spot s i z e ( i . e . , the volume from which X-rays are generated f o r any one spot count). The experimental conditions i n t h i s work resulted i n a t h e o r e t i c a l l y - c a l c u l a t e d spot s i z e of about 6 u i n diameter i n the aluminium-rich matrix,and 3 u i n diameter i n the copper-rich CuA^. In the Al-Ag a l l o y , the spot s i z e i s approximately 5 u i n the aluminium-rich matrix and 2.5 u i n the s i l v e r - r i c h phase (see Appendix II f o r c a l c u l a t i o n s ) . The spot - 33 - s i z e was also determined experimentally from the measured Cu concen- t r a t i o n i n the matrix adjacent to a p r e c i p i t a t e i n t e r f a c e . A copper concentration above the equilibrium matrix concentration was found to extend over a distance of approximately 5 u i n the blank p r o f i l e (Figure 9). This increase at a distance 5 y away from the determined i n t e r f a c e was never greater than 1-2% above the matrix composition. Since 1.25 y step scanning was used,the e f f e c t i v e spot s i z e w i l l be > 3.75 y i n radius,which i s i n good agreement with the t h e o r e t i c a l l y c a l c ulated value. That step-scanning i n 1.25 y steps gives meaningful values of concentration, even though the spot s i z e i s - 3.75 y i n radius, may be appreciated i f one considers that an error function p r o f i l e (erf x) of C versus x i s very close to a l i n e a r p r o f i l e over the range x = 0 t o x = 0 . 8 . Accordingly f o r a l i n e a r composition p r o f i l e , adjacent to the allotriomorph i n t e r f a c e , the mean composition of the c i r c u l a r area with centre x^ as i n Figure 10, i s equal to the C^ the composition at the point x^. In the present work the spot s i z e l i m i t s the X-ray r e s o l u t i o n at the precipitate-matrix i n t e r f a c e and the grain boundary. The p o s i t i o n of the precipitate-matrix i n t e r f a c e was determined from the blank p r o f i l e by a large decrease i n specimen current and a sudden increase i n background count, associated with going from the aluminium-rich matrix to the copper-rich p r e c i p i t a t e (see Figure 9). The grain boundary p o s i t i o n was roughly established from the p o s i t i o n of adjacent p r e c i p i t a t e s , t o the p r e c i p i t a t e being observed,along the grain boundary (see Figure 6 ) . i t was found that the maximum solute concentration on traversing a grain boundary - 34 - E X c 0 k_ u c a) E U 'u cu 10 a OS o "* o u O 00 150 h 100h 501- 0 ^ D u 2000 |r 1000 C M 0 Figure 9 c p I 4000 30001- The (a) specimen current, (b) background concentration, and (c) Cu-Kc^ concentration,for the blank profile (1) through the centre of the precipitate, (2) at the allotriomorph tip and (3) 5u away from the tip across the grain boundary. - 35 - Figure 10. Schematic diagram depicting the e f f e c t of spot s i z e on the evaluation of concentration at a point x^ away from the p r e c i p i t a t e i n t e r f a c e . Figure 11. A schematic depiction of the d i s s o l u t i o n f l u x l i n e s f o r ; (a) an allotriomorph sectioned above the minor axis b, (b) an a l l o - triomorph sectioned at the minor axis,and (c) an allotriomorph sectioned below the minor axis. - 36 - i n v a r i a b l y coincided with the p o s i t i o n of the grain boundary as established from the back scattered e l e c t r o n image. Concentration gradients on e i t h e r side of the maximum concentration were also very s i m i l a r , providing further v e r i f i c a t i o n of the grain boundary p o s i t i o n . 2.6.2 K i n e t i c Studies Attempts were made to study p r e c i p i t a t e d i s s o l u t i o n by heat- tre a t i n g the specimen, etching and examining the micro-structure using an o p t i c a l microscope. As i t was necessary to p o l i s h the specimen each time p r i o r to etching, a d i f f e r e n t p o s i t i o n on the p r e c i p i t a t e was observed a f t e r each heat treatment. Thus the r e s u l t i n g data i s not useful f o r a k i n e t i c a n a l y s i s . The back-scattered electron image (b.s.e.i.) of the microprobe . proved much more s a t i s f a c t o r y since i t did not require etching of the specimen. The experimental procedure developed approximates very w e l l the d i r e c t observation of p r e c i p i t a t e d i s s o l u t i o n . The procedure involves photographing the back-scattered e l e c t r o n image of i n d i v i d u a l grain boundary allotriomorphs a f t e r successive heat treatments i n the s a l t . A po l a r o i d camera,attached to a monitor on the microprobe console, was used with p o l a r o i d N52 type f i l m . The largest allotriomorphs found i n any one specimen were used for k i n e t i c studies. This was done to ensure that the allotriomorphs studied were sectioned as close to the maximum diameter as pos s i b l e . As shown i n Figure 11, p r e c i p i t a t e s sectioned above the maximum h a l f length w i l l dissolve more slowly than p r e c i p i t a t e s sectioned through the - 37 - center, whereas p r e c i p i t a t e s sectioned below the maximum h a l f length w i l l dissolve f a s t e r . For the two-dimensional e l l i p t i c a l geometry depicted i n Figure 11, the divergence of the f l u x l i n e s at x^,(Figure 11(b)),may be r e l a t e d to the change i n the angle of the tangent at x^, r e l a t i v e to the tangent at the maximum h a l f thickness b. For an e l l i p s e with an a x i a l r a t i o of 3.5 to 1,the change i n angle A0 i s 4° and 10° f o r x^ equal to 0.25 a and 0.5 a re s p e c t i v e l y (see Appendix III f o r c a l c u l a t i o n s ) . As the precaution was taken to study the d i s s o l u t i o n of only the largest grain boundary allotriomorphs i n any one sample, tha e f f e c t of the f l u x l i n e divergence, which as calculated i s quite small, i s considered n e g l i g i b l e i n t h i s work. Photographic observations were made at magnifications of x500, xlOOO, x2084, and x4168; at higher magnifications the r e s o l u t i o n was found to be impaired. f Measurements of the length of p r e c i p i t a t e s were made d i r e c t l y from the photograph to within 0.001" using vernier c a l l i p e r s . The thickness measurement was made at a p o s i t i o n halfway between the t i p s of the p r e c i p i t a t e . The lengthy heat treatments i n t h i s part of the work increased the p o s s i b i l i t y of contamination by the s a l t , of grain boundary migration and of extensive thermal grooving of the grain boundaries. I f there was any detectable evidence of contamination of the specimen surface, the specimen was rejected. - 38 - 2.6.3 D i f f u s i o n Couple Measurements An etchant of composition 47% HCl, 6% HI and 47% H 20, when f r e s h l y prepared, delineated the copper composition i n d i f f u s i o n couples. The etchant was used to o u t l i n e the d i f f u s i o n f i e l d and thus to v e r i f y the extent of bonding which had taken place. For each s u c c e s s f u l l y prepared d i f f u s i o n couple, at l e a s t two composition p r o f i l e s ( i n some cases three) were taken across the centre of the two larges t CuA^ grains at the bonded i n t e r f a c e (see Figure 12(a)). The step scan device was used, and i n most cases 1.25 u steps were taken. 2.7 V a l i d i t y of Experimental Technique 2.7.1 Isoconcentration Contour Measurements The v a l i d i t y of the.experimental technique depends on whether the dissolution contours measured on the sample surface are representative of those present about p r e c i p i t a t e s i n the i n t e r i o r of the sample. Experiments were c a r r i e d out to e s t a b l i s h that the surface measurements i n t h i s i n v e s t i g a t i o n are representative of the bulk material. This was done by measuring dissolution contours as follows: (A) around a given p r e c i p i t a t e at the surface; (B) around the same p r e c i p i t a t e a f t e r a layer of material 2-3 u thick was c a r e f u l l y removed from the specimen surface; and (C) around a second p r e c i p i t a t e i n the bulk material. The resultant dissolution contours are shown i n Figure 13. Within reasonable s c a t t e r results(A),(B)and (C) coincide, r u l i n g out any - 39 - Figure 1 2(b). Microstructure of large C u A l 2 grain i n an a + 9 matrix. TH=0.97 A l l o y - A Figure 1 3 . D i s s o l u t i o n isoconcentration contours of a grain boundary allotriomorph on the surface and 3u below the surface (A and B) and a second allotriomorph i n the bulk material (C). At T = 0.97 f o r H a l l o y A. - 41 - surface d i f f u s i o n e f f e c t . This also indicates that the e f f e c t of f l u x divergence i s small r e l a t i v e to the experimental e r r o r , an observa- t i o n that i s important for the k i n e t i c studies. A further important question was whether impingement from adjacent d i f f u s i o n f i e l d s j u s t below the surface was a f f e c t i n g the measured r e s u l t s . To detect t h i s p o s s i b i l i t y : (1) A scan was obtained along the d i r e c t i o n of the p r e c i p i t a t e minor axis from the bulk matrix on one side to the bulk matrix on the other sid e . (2) A p a r a l l e l scan was also obtained j u s t beyond the t i p s on e i t h e r side of the p r e c i p i t a t e continuing to a distance, on both sides the grain boundary, associated with an equilibrium matrix composition. The t o t a l d i f f u s i o n f i e l d around the p r e c i p i t a t e could then be examined, and any anomalies i n the d i f f u s i o n f i e l d became apparent. 2.7.2 K i n e t i c Measurements Indi v i d u a l grain boundary p r e c i p i t a t e s were examined during d i s s o l u t i o n . No s t a t i s t i c a l analysis of d i s s o l u t i o n k i n e t i c s was ca r r i e d out i n the present work, as i s generally done, because there were a r e l a t i v e l y small number of p r e c i p i t a t e s on the grain boundaries, with a large range of p r e c i p i t a t e shapes (s /R varying o o from > 10/1 to 2/1) (see Figure 14), With the small number of p r e c i p i t a t e s i n any one specimen, the population d i s t r i b u t i o n curves are inadequate f o r s t a t i s t i c a l a n a l y s i s . 20 18 16 z «/> 14 | 0 12 ' a <D 10 Q_ M— 0 8 m O Z 6 - - • - >10 i 3.5 4 [So/Ro] 8 10 Figure 14. Population d i s t r i b u t i o n curve for allotriomorphsof varying a x i a l r a t i o S^/R^ for a l l o y A . - 43 - To determine i f surface d i f f u s i o n increased the d i s s o l u t i o n rate i n the k i n e t i c studies, four grain boundary allotriomorphs were examined a f t e r a 12 minute d i s s o l u t i o n treatment at 527°C. This d i s s o l u t i o n heat treatment res u l t e d i n a decrease i n the i n i t i a l h alf-thickness R of about 40%. The p r e c i p i t a t e s i z e was observed o using the back-scattered e l e c t r o n image. A very t h i n surface layer was removed, the specimen was etched with a 5% NaOH sol u t i o n 5 a n d the p r e c i p i t a t e measurements were taken again o p t i c a l l y . For a l l four p r e c i p i t a t e s the o p t i c a l measurements were i n good agreement with measurements obtained using the b . s . e . i . , thus eliminating the p o s s i b i l i t y of surface d i f f u s i o n contribution to the d i s s o l u t i o n . The use of the back-scattered e l e c t r o n image to measure the 'true' p r e c i p i t a t e s i z e was tested by measuring the p r e c i p i t a t e s i z e o p t i c a l l y and comparing the r e s u l t s with those obtained by the b . s . e . i . and the absorbed electron image ( a . e . i . ) . The absorbed electron image was found to be unreliable, as a change i n contrast could give a d r a s t i c a l l y d i f f e r e n t p r e c i p i t a t e s i z e . The b . s . e . i . , however, could be picked up at a l l settings without d i s t o r t i n g the image. A l l the settings gave measurements of p r e c i p i t a t e s i z e equal to those obtained o p t i c a l l y . - 44 - 3. ISOCONCENTRATION CONTOUR RESULTS AND DISCUSSION 3.1 Introduction Experimental determinations of the isoconcentration contours around d i s s o l v i n g grain boundary allotriomorphs, i n an e f f e c t i v e l y i n f i n i t e matrix, w i l l be presented and analyzed i n t h i s chapter. One anticipates that isoconcentration contours associated with d i s s o l v i n g grain boundary p r e c i p i t a t e s would be influenced by the following parameters: (1) D i s s o l u t i o n rate c o n t r o l l i n g mechanism, i . e . , whether the d i s s o l u t i o n process i s d i f f u s i o n c o n t r o l l e d or i n t e r f a c e reaction c o n t r o l l e d . (2) The p r e c i p i t a t e shape ( S 0 / R Q ) . (3) Homologous temperature T . (4) The supersaturation (k) of the a l l o y . (5) The volume d i f f u s i o n c o e f f i c i e n t (D^) of the a l l o y . (6) The grain boundary misorientation (R) and the grain boundary d i f f u s i o n c o e f f i c i e n t D , g.b. These parameters have each been considered experimentally. The r e s u l t s are presented, following a discussion of: (a) The d i f f u s i o n couple r e s u l t s , t o e s t a b l i s h whether D^ i s independent of composition i n the a phase; - 45 - (b) The microstructure of the e q u i l i b r a t e d alloys,and (c) The nature of isoconcentration contours to e s t a b l i s h the r e p r o d u c i b i l i t y of the technique used. 3.1.1 D i f f u s i o n Couple Results The concentration p r o f i l e s of d i s s o l v i n g p r e c i p i t a t e s are strongly dependent on the value of the volume d i f f u s i o n c o e f f i c i e n t of a given material. The analysis of the concentration p r o f i l e s i s greatly s i m p l i f i e d i f D^ can be considered as independent of a l l o y composition. To e s t a b l i s h to what extent D^ i s r e l a t e d to composition, composition measurements were made on a s e r i e s of Al-CuA^ d i f f u s i o n couples annealed at 480°C, 500°C, 520°C, and 535°C for 163 hours. The r e s u l t s for the 500°C d i f f u s i o n couple are shown i n Figures 15 and 16. In Figure 15 three runs are shown which are e s s e n t i a l l y coincident, i n d i c a t i n g reproducible measurements. The anomalous r i s e i n concentration near the i n t e r f a c e i s believed due to a two hour d i s s o l u t i o n heat treatment at 535°C. This was done to dissolve CuA^ p r e c i p i t a t e s formed i n the d i f f u s i o n zone upon furnace cooling of the couple. Representative values of concentration from Figure 15 are p l o t t e d on p r o b a b i l i t y paper, versus distance from the couple i n t e r f a c e . The r e s u l t shows a l i n e a r dependence, c l e a r l y i n d i c a t i n g that D^ i s independent of composition over the complete range of s o l i d s o l u t i o n i n the a phase,( see Figure 16). (The complete d i f f u s i o n couple data i s l i s t e d i n Appendix IV). The temperature dependence of D^ from the present r e s u l t s i s shown i n an Arrhenius p l o t i n Figure 17, along 39 with values of D for an Al-0.5 wt.% Cu a l l o y published by Murphy. 3OO0f- 400(- 100C[- J 30* COMPOSITION PROFILES - 500 C •O Run-I • • Run-2 Run-3 550 600 tso Pittance, (H) Figure 15. Composition contour for Al-CuAl„ d i f f u s i o n couple at 500°C. - 47 - Diffusion Couple-500C X x 107 Cm. Figure 16. P r o b a b i l i t y plot of atomic % Cu versus d i f f u s i o n distance (x) over the s o l i d s o l u b i l i t y range of the a phase f or the 500°C d i f f u s i o n couple. - 48 - V T x l O 4 , ^ " 1 Figure 17. Arrhenius p l o t of D versus l / T ° K _ 1 f o r the d i f f u s i o n couple r e s u l t s . - 49 - The present r e s u l t s are observed to follow an Arrhenius p l o t and agree c l o s e l y with the values of Murphy. Since the r e s u l t s reported by Murphy were obtained for a wider temperature range than i n the present case, his values of w i l l be used i n the analysis of the present r e s u l t s . Grain boundary d i f f u s i o n c o e f f i c i e n t s could not be determined, as the pressure applied to achieve bonding caused a certain:.amount of r e c r y s t a l l i z a t i o n i n the b i - c r y s t a l . 3.1.2 The Microstructure of E q u i l i b r a t e d A l l o y s The shape of t y p i c a l grain boundary allotriomorphs i n t h i s i n v e s t i g a t i o n can be observed i n Figure 6 (on page 26 ) which i s a representative micrograph of a l l o y B. The p r e c i p i t a t e shape i s taken as oblate spheroidal and i s defined by the length to width r a t i o SQ/RQ. The grain boundary allotriomorphs are widely spaced with 3 < SQ/RQ < 6. A number of "snakes" of high a x i a l r a t i o are also observed i n the micrographs, which are believed to r e s u l t from the impingement of two or more adjacent allotriomorphs. Widmanstatten plates can be observed, however, they are not present i n the areas immediately adjacent to the grain boundaries. The presence of a p r e c i p i t a t e - f r e e zone adjacent to the grain boundaries i s a r e s u l t of the depletion of vacancies i n t h i s area by the grain boundary p r e c i t a t e s which form f i r s t . The vacancy concentration i s 40 41 as a r e s u l t too low for the nucleation of a second phase. ' - 50 - The s i z e d i s t r i b u t i o n ( S q ) of the grain boundary p r e c i p i t a t e s obtained i n A l l o y A i s p l o t t e d i n Figure 18 as a function of the a x i a l r a t i o ( S /R ) f o r the precipitate d i s t r i b u t i o n p l o t t e d i n o o Figure 14. The average allotriomorph s i z e and the width of the depleted zone i s shown i n Table I I I , for the d i f f e r e n t a l l o y compositions studied. Table I I I . Average S q for Allotriomorphs with S /R =3.5 and Average o o 6 Width of P r e c i p i t a t e Free Zone for the D i f f e r e n t A l l o y s . A l l o y Half length of Width of P r e c i p i t a t e a l l o t r i o m o r p h s ( S Q ) y free zone y A 25 90 B 15 70 C 10 60 D 10 30 3.1.3 D i s s o l u t i o n Isoconcentration Contours Five separate grain boundary allotriomorphs were examined by electron probe microanalysis and concentration contours determined for both sides of each p r e c i p i t a t e . The r e s u l t s are shown i n Figure 19. (One side of one p r e c i p i t a t e was not used because of early impingement from an adjacent allotriomorph.) The top h a l f of the f i g u r e shows d i s s o l u t i o n contours a f t e r 90 seconds from the s t a r t of d i s s o l u t i o n , the bottom a f t e r 360 seconds. The f i v e allotriomorphs were i n an a l l o y B specimen (Al-2.81 wt.% Cu) which was homogenized at 400°C (T ) and then upquenched to 520°C (T ). The maximum s o l u b i l i t y of Cu 60 - 50 - >10 3.5 4 [SQ/RQ] 8 10 Figure 18. Size d i s t r i b u t i o n (S ) of p r e c i p i t a t e s as a function of S /R for a l l o y A. ° o o o Figure 19. Isoconcentration contours at T = 0.91 f o r a l l o y B. - 53 - at 400°C i n a i s 1.9 wt.%; at 520°C i t i s 5.2 wt.%. Accordingly the p r e c i p i t a t e s w i l l dissolve on upquenching u n t i l they u l t i m a t e l y disappear. The concentration measurements were confined to the ea r l y part of the d i s s o l u t i o n process to avoid i n t e r a c t i o n e f f e c t s due to solute f i e l d s from adjacent allotriomorphs. A l l composition measure- ments were made on scans perpendicular to the allotriomorph eq u a t o r i a l axis and grain boundary. In general, concentration measure- ments could not be made at distances less than 6 u from the CuA^ p r e c i p i t a t e , since within t h i s distance Cu i n the p r e c i p i t a t e could contribute to the measured concentration. In Figure 19 two values of concentration are p l o t t e d , 2.1 and 2.5 wt.% Cu. Higher values of concentration occurred within 6 u of the p r e c i p i t a t e surface and were therefore considered u n r e l i a b l e . Lower values were approaching the matrix composition (1.9 wt.% Cu), thus making composition differences comparable to the experimental uncertainty of the measurements. Examining the values of concentra- t i o n p l o t t e d i n Figure 19, i t i s observed that the p o s i t i o n of points from sides I and II of a given p r e c i p i t a t e are i n close proximity i n d i c a t i n g a high degree of symmetry of the isoconcentration contours about a given p r e c i p i t a t e . However, points of equal concentration f o r the f i v e p r e c i p i t a t e s are not i n exact coincidence, as might be expected. The points for 2.5 wt.% Cu can e s s e n t i a l l y be enclosed i n an envelope, shown by the two s o l i d l i n e s c l o s e s t to the p r e c i p i t a t e . The 2.1 wt.% Cu points are contained i n the envelope furthest from tfie p r e c i p i t a t e . To a f i r s t approximation, l i n e s drawn through the centre of each zone can be taken as the isoconcentration contours f o r - 54 - the composition being considered. The standard deviation (a) for these p r o f i l e s varies from 0.68 u at a mean distance (x) of 6.11 u from the p r e c i p i t a t e i n t e r f a c e , to a standard deviation (a) of 1.17 u at a mean distance (x) of 17.59 u from the p r e c i p i t a t e i n t e r f a c e . (See Appendix V for c a l c u l a t i o n s ) . In the f i r s t case a i s 11.12% of the mean distance x whereas i n the second case a i s only 6.6% of the mean distance (x)(calculations i n Appendix V). For the evaluation of and Ĉ . values, the longer time d i s s o l u t i o n experiments with d i f f u s i o n distance greater than 20 u from the i n t e r f a c e were used, wherever p o s s i b l e , to minimize the e f f e c t of sc a t t e r j u s t established as inherent i n this experimental procedure. 3.2 Rate C o n t r o l l i n g Mechanism To e s t a b l i s h that the d i s s o l u t i o n process i s d i f f u s i o n c o n t r o l l e d , values of concentrations at various time i n t e r v a l s were p l o t t e d as a function of x//t , where x i s the distance along the minor axis of the p r e c i p i t a t e s t a r t i n g at the p r e c i p i t a t e surface,and t i s the time. The average r e s u l t f o r two p r e c i p i t a t e s i n the Al-4.83 wt.% Cu a l l o y at a d i s s o l u t i o n temperature of 545°C (T = 0.97) i s shown i n Figure 20. The points are observed to be e f f e c t i v e l y coincident, i n d i c a t i n g that the composition C(x,t) = f ( )• The p r o f i l e vTTt obtained extrapolates at =0, to an i n t e r f a c e concentration C T = /t 1 5.5. This i s i n good agreement with the phase diagram i n d i c a t i n g l o c a l equilibrium at the i n t e r f a c e and therefore no i n t e r f a c e reaction control. A s i m i l a r analysis of data f o r the range of homologous temperatures between T = 0.72 and 0.97 gave s i m i l a r r e s u l t s for d i f f u s i o n distances 5.5 5-0 D 4.5 V 4.0 3.5 3-0 h • A A O TH= 0.97 Alloy -A. T = 545C Ttme(t) Seconds • 100 • 225 • 400 A 900 • 1600 ± 0.2 0-4 0-6 0.8 1.0 x x 1 0 7 Cm. t 1.2 1.4 1.6 Figure 20. Concentration p r o f i l e along the minor axis of a grain broundary allotriomorph at 545°C fo a l l o y A. - 5 6 - along the minor axis of the allotriomorphs. 3 . 3 The E f f e c t of the P r e c i p i t a t e Shape (S /R ) , . , £ . E 0 0 To determine the e f f e c t of p r e c i p i t a t e shape on the concentration contours,three p r e c i p i t a t e s were examined with d i f f e r e n t a x i a l r a t i o s : ( 1 ) a t y p i c a l allotriomorph with S^/R^ = 3 . 5 , ( 2 ) a snake with S^/R^ = 1 8 and ( 3 ) a nearly s p h e r i c a l p r e c i p i t a t e having S Q / R Q = 1 . 7 The resultant concentration contours are shown i n Figure 2 1 f o r concentrations of 3 . 2 5 and 3 . 5 0 wt.% Cu and d i s s o l u t i o n times of 30 and 1 2 0 seconds. The d i s s o l u t i o n temperature was the same f o r a l l three p r e c i p i t a t e s . The r e s u l t s shown i n Figure 2 1 c l e a r l y i n d i c a t e that the composition contours f o r the three p r e c i p i t a t e s are e s s e n t i a l l y coincident,and therefore that under these experimental conditions the distance from the p r e c i p i t a t e surface of the isoconcentration contours i s independent of the a x i a l r a t i o . This i s not meant to imply that the o v e r a l l shape of isoconcentration contours about a sphere and a snake are the same, as may be misconstrued from Figure 2 1 . I t should be pointed out that i n Figure 2 1 as i n a l l the figures of isoconcentra- t i o n contours the points at the extreme l e f t hand side denote average d i f f u s i o n distances from the i n t e r f a c e at the p r e c i p i t a t e centre. The isoconcentration contours attempt to give an accurate representation of the isoconcentration l i n e s , about the t i p of an allotriomorph, and along the adjacent grain boundary, with only an average d i f f u s i o n a l distance away from the broad allotriomorph faces. In general, two e f f e c t s have to be considered f o r d i s s o l u t i o n of precipitates, of d i f f e r e n t a x i a l r a t i o s : (a) the Gibbs-Thomson e f f e c t Figure 21. Isoconcentration contours for three allotriomorphs having a x i a l r a t i o s S /R of 1) 3 , 5 , 2) 18 3) 1.7 for a l l o y A at T = 0.93. - 58 - and (b) the point e f f e c t of d i f f u s i o n . (a) The Gibbs-Thomson e f f e c t i s the v a r i a t i o n with curvature of the equilibrium solute concentration at the precipitate-matrix 42 i n t e r f a c e . Aaron and K o t l e r have ca l c u l a t e d the influence of this e f f e c t on the d i s s o l u t i o n k i n e t i c s of Al-Cu a l l o y s and conclude that i t i s n e g l i g i b l e f o r the t o t a l d i s s o l u t i o n time of a p r e c i p i t a t e . Therefore the influence of i n t e r f a c e curvature on the isoconcentration con- tours i s expected to be minimal. (b) Because of the point e f f e c t of d i f f u s i o n , the f l u x from the t i p of an allotriomorph i s d i s t r i b u t e d over a greater s o l i d angle than the f l u x perpendicular to the side of an allotriomorph. Thus the point e f f e c t tends to round o f f the isoconcentration contours at the t i p of an allotriomorph. As a further v e r i f i c a t i o n that the Gibbs-Thomson e f f e c t does not influence the p o s i t i o n of the isoconcentration curves around a d i s s o l v i n g p r e c i p i t a t e for short d i s s o l u t i o n times, concentration measurements were made around two Widmanstatten plates located i n the centre of a grain under the same conditions as those f o r Figure 21. The average r e s u l t s are shown i n Figure 22 and show comparable d i f f u s i o n distances, along the minor axis, to those for the grain boundary allotriomorphs i n Figure 21. Hence, the shape of the p r e c i p i t a t e does not influence the p o s i t i o n of the isoconcentration curve r e l a t i v e to the p r e c i p i t a t e surface. In addition, Widmanstatten p l a t e s , l i k e grain boundary allotriomorphs, are found to dis s o l v e by a d i f f u s i o n mechanism i n 31 agreement with the r e s u l t s of H a l l and Hayworth i n an Al-5 wt.% Cu a l l o y . Figure 22. Isoconcentration contours of two Widmanstatten plates f o r a l l o y A at T = 0.93. - 60 - 3.4 Homologous Temperature To determine the e f f e c t of T„ on the p o s i t i o n and shape of the n isoconcentration l i n e s around an allotriomorph, concentration measurements were made on a se r i e s of a l l o y s , over a range of d i s s o l u t i o n times, f o r 0.72 < T < 0.97. The s p e c i f i c experiments H are tabulated i n Table IV. Some of the corresponding concentration contours are given i n Figures 23-29 f o r the times and concentrations indicated. No sc a t t e r bars have been placed on the contour p l o t s ; the s c a t t e r as established i n Section 3.1.3 i s expected to be the same for these r e s u l t s . Comparing the isoconcentration contours for T = 0.97 (Figure 23) H with T = 0.77 (Figure 29), i t i s observed that the isoconcentration rl l i n e s i n the former are i n c l i n e d at approximately 90° to the grain boundary plane, whereas i n the l a t t e r the corresponding angle i s about 20°. The decreasing of the angle of i n c l i n a t i o n i s clear evidence that enhanced grain boundary d i f f u s i o n has occurred at the lower homologous temperature. Figures 23-29 show a progressive decrease i n the angle of i n c l i n a t i o n with decreasing T and therefore a rl progressively greater contribution from grain boundary d i f f u s i o n to the d i s s o l u t i o n process. The contours f o r the highest homologous temperature T = 0.97 (Figure 23) e x h i b i t e s s e n t i a l l y no grain rl boundary contribution to d i s s o l u t i o n . The s l i g h t rounding-off of the contours at the p r e c i p i t a t e t i p i s thought to be due to the point e f f e c t of d i f f u s i o n . At T = 0.92 there i s a very small amount of grain boundary ri d i f f u s i o n and the d i f f u s i o n distance along the grain boundary at the t i p of the p r e c i p i t a t e i s s l i g h t l y greater than the d i f f u s i o n distance - 61 - Table IV. Contour Experiments T H a l l o y No. of allotriomorphs D i s s o l u t i o n times (seconds) Figure No. .97 A 2 25,100,225,400,900,1600 23 .94 B 1 400,900,1600 24 .93 A * 5 30,120 21 and 22 .93 Al-Ag 3 40,160 43 .92 A ** 3 100,400,1600 34 .92 B 2 400,900,1600,3200 25 .91 B 9 90,360 19 .905 B 1 400,900,1600 26 .885 B 2 500,1600 45 .885 Al-Ag 2 50,100,200 42 .86 B 1 400,900,1600 27 .84 Al-Ag *** 3 70,150,225,400 35 and 36 .83 B 1 100,400,1600,6400 44 .80 C 1 1600,6400 28 .77 C 2 1600,3600,6400 29 .72 C 2 1600,3600 30 2 Widmanstatten p l a t e s , plus 3 d i f f e r e n t shape grain boundary p r e c i p i t a t e s . ** 1 p r e c i p i t a t e only at 1600 seconds *** 2 precpitates on low angle grain boundary Figure 23. Isoconcentration contours for a l l o y A at T = 0.97.  Figure 25. Isoconcentration contours for a l l o y B at T = 0.92. Figure 26. Isoconcentration contours for a l l o y B at T = 0.905. Figure 27. Isoconcentration contours for a l l o y B at T = 0.86. Figure 28. Isoconcentration contours for a l l o y C at T = 0.80. TH=Q.77 A l b y - C wt./£Cu oUO • UO A 1.60 Figure 29. Isoconcentration contours for a l l o y C at T = 0.77. - 69 - perpendicular to i t s f l a t surface. As the homologous temperature decreases, t h i s e f f e c t becomes more pronounced u n t i l at the lowest temperature studied T = 0.72 (Figure 30), the d i f f u s i o n f i e l d s along the grain boundaries impinge with those of adjacent allotriomorphs before there i s enough volume d i f f u s i o n from the p r e c i p i t a t e to be resolved by the electron probe. The grain boundary d i f f u s i o n contribution to growth was estimated 21 experimentally using s t a t i s t i c a l techniques, by Goldman et a l . Their r e s u l t s i n d i c a t e that below T = 0.78 solute transport should be due s o l e l y to grain boundary d i f f u s i o n . No experiments were done above T„ = 0.78. However, between T„ = 0.79 and 0.90, they postulate volume d i f f u s i o n should become progressively more dominant, and above T = 0.91 they p r e d i c t transport should be e n t i r e l y by rl volume d i f f u s i o n . Hawbolt and Brown,^ i n t h e i r studies of the growth of grain boundary allotriomorphs i n Ag-5.64 wt.% A l , f i n d that at T R = 0.92, growth i s e n t i r e l y volume d i f f u s i o n c o n t r o l l e d . At T = 0.90, however, rl there i s some evidence i n t h e i r r e s u l t s for grain boundary d i f f u s i o n enhancing the growth rate of p r e c i p i t a t e s . The present r e s u l t s show quite conclusively that volume d i f f u s i o n completely dominates the d i s s o l u t i o n process above a homologous temperature of between 0.92 and 0.93. I t i s further shown that at T R = 0.77, the e f f e c t of volume d i f f u s i o n on d i s s o l u t i o n i s n e g l i g i b l e , and that there i s e s s e n t i a l l y t o t a l grain boundary d i f f u s i o n c o n t r o l at T = 0.72. Thus the present allotriomorph d i s s o l u t i o n r e s u l t s are i n excellent 21 6 agreement with the work of both Goldman et a l . and Hawbolt and Brown TH^0.72 Alloy-D p 0.45wt.% Cu • 0-45 wt. L Cu Precipitate 1 Precipitate 2 o o • o • o • o a o 0 O ° O D O o o o • a • o o 10u f o = 3600 Seconds Q O o • o n O OD o o o a • a • • o o o o o p o Figure 30. Isoconcentration contour f o r a l l o y D at T = 0.72. - 71 - regarding the t r a n s i t i o n from complete volume d i f f u s i o n c o n t r o l to complete grain boundary d i f f u s i o n c o n t r o l f o r the growth of a l l o t r i o - morphs. The r e l a t i v e importance of volume d i f f u s i o n to gr a i n boundary d i f f u s i o n , i n the t r a n s i t i o n region between complete volume d i f f u s i o n c o n t r o l and complete grain boundary d i f f u s i o n c o n t r o l , i s also w e l l i l l u s t r a t e d f o r the f i r s t time experimentally i n these r e s u l t s . In the following section the volume d i f f u s i o n c o e f f i c i e n t Dy i s calculated using the concentration p r o f i l e s p a r a l l e l to the minor axis of the grain boundary allotriomorph,where enhancement by grain boundary d i f f u s i o n can be neglected. Values of Ĉ ., the allotriomorph i n t e r f a c e composition, are also determined. 3.4.1 Calculations of. and C^ Evaluations of and C^ are made to test the accuracy of the concentration contour experiments. The following assumptions were made i n determining and Ĉ .. (1) i s independent of concentration. This i s based on the d i f f u s i o n couple r e s u l t s (Section 3.1.1) which in d i c a t e that the volume d i f f u s i o n c o e f f i c i e n t i s independent of concentration over the complete s o l i d s o l u b i l i t y range of the a-phase. (2) The allotriomorph-matrix i n t e r f a c e i s e s s e n t i a l l y f l a t at the minor axis,so that a one-dimensional s o l u t i o n to the d i f f u s i o n equation can be used with distances measured perpendicular to the centre of the f l a t surface of the allotriomorph. (Figure 20 indicates an error function p r o f i l e i n t h i s d i r e c t i o n ) . (3) There i s very l i t t l e movement of the i n t e r f a c e r e l a t i v e to - 72 - t h e d i f f u s i o n a l d i s t a n c e . The p r e c i p i t a t e i n t e r f a c e i s c o n s i d e r e d s t a t i o n a r y r e l a t i v e t o t h e d i f f u s i o n d i s t a n c e s c o n s i d e r e d d u r i n g d i s s o l u t i o n . A c a l c u l a t i o n f o r t h e p r e v a i l i n g e x p e r i m e n t a l c o n d i t i o n s , 31 by H a l l and Hayworth, i n d i c a t e s a d i f f u s i o n d i s t a n c e w h i c h i s a p p r o x i m a t e l y two o r d e r s o f magnitude g r e a t e r t h a n t h e d i s p l a c e m e n t o f t h e i n t e r f a c e . S u b j e c t t o t h e s e a s s u m p t i o n s , t h e s o l u t i o n t o F i c k l s law r e d u c e s t o : c o u l d n o t be measured e x p e r i m e n t a l l y due t o t h e f i n i t e s i z e o f t h e e l e c t r o n beam i n t h e e l e c t r o n p r o b e , t h e r e f o r e , b o t h C^ and have unknown v a l u e s i n e q u a t i o n 1. V a l u e s o f and Ĉ . were o b t a i n e d from e q u a t i o n 1 u s i n g t h e f o l l o w i n g g r a p h i c a l a n a l y s i s . (1) F o r t h r e e v a l u e s o f x (x = 10 y, 15 y, 20 y), t h e c o r r e s - p o n d i n g e x p e r i m e n t a l v a l u e s o f C, . were o b t a i n e d f r o m t h e i s o - ( x , t ) c o n c e n t r a t i o n l i n e s i n F i g u r e 25 and were u s e d i n c o n j u n c t i o n w i t h 1/2 a range o f (D^t) v a l u e s t o s o l v e e q u a t i o n 1. The gra p h o f Ĉ . v s . 1/2 (D^t) ( F i g u r e 31) c o n s i s t s o f t h r e e c u r v e s w h i c h c r o s s a t a p o i n t 1/2 c o r r e s p o n d i n g t o a p p r o x i m a t e v a l u e s o f Ĉ . and (D^t) (2) A p l o t o f C, •> v e r s u s e r f . [ — T77] w a s made u s i n g 1/2 *KUyt) t h e v a l u e of. (Dyt) o b t a i n e d f r o m F i g u r e 31. T h i s i s a l i n e a r p l o t w h i c h i s c o n s i s t e n t w i t h e q u a t i o n 1. When e x t r a p o l a t e d t o e r f . [ r - T T T - ] = 0 a more a c c u r a t e v a l u e o f C T i s o b t a i n e d as shown 2 ( D y t ) i / Z - 73 - - 74 - TH = 0.92 Al loy- B O S i d e • S i d e 0.2 0.4 0.6 0.8 Erf ( ^ ) Figure 32. C versus e r f ( ) f o r a l l o y B at T R = 0.92 1.0 2>/Dt  - 76 - in Figure 32. C ( x t ) _ C M (3) Accurate values of D were obtained by plotting •' V V S l versus x on probability paper. can be calculated from the i slope of the line in Figure 33, which is equal to 2 ( D t ) 1 / 2 ' (4) Iteration of (2) and (3) could be used to determine even more accurate values of and C^. This was, however, considered unnecessary considering the good agreement between the and values calculated and the literature values. Table V gives a comparison of the experimentally determined values with the 36 37 equilibrium phase diagram values. ' There i s , in most cases, excellent agreement between the experimental and the literature values, which i s consistent with the assumption t h a t the dissolution process is diffusion controlled. Table V. Interface Concentration (C^) and Volume Diffusion Coefficients (Dy) for Contour Experiments. Alloy Experimental values Cj wt.% Cu Equilibrium diagram values 5.60 4.76 5.62 4.97 4.57 4.27 3.60 2.85 2.40 2.15 5.6 4.6 5.7 5.1 4.6 4.3 3.4 2.8 2.5 1.9 T H .97 .92 .940 .92 .90 .885 .86 .83 .80 .77 Experimental Dy xlO cm se 12.0 12.6 13.3 11.6 11.6 8.2 5.1 4.8 4.5 2.3 Murphy's values of D^ 13.2 9.5 13.2 10.5 9.5 6.1 3.0 1.9 1.5 0.6 Temperature °C 545 517 545 532 517 500 480 448 442 418 - 77 - The probability plots for the evaluation of were a l l found to be linear and thus indicated a concentration-independent diffusion coefficient, confirming the diffusion couple results. Table V also gives the calculated values of D̂ . The volume diffusion coefficients determined from the profile studies are in good agreement with the literature values at the high experimental temperatures. The experimental conditions at the low temperatures, resulted in: (1) smaller concentration gradients in the matrix, (2) shorter diffusion distances,and, (3) a greater possibility of a surface diffusion contribution to dissolution. This effect would enhance the dissolution rate but would leave the shape of the profiles unaffected. A l l these factors may be instrumental in the less accurate Dy values obtained at low temperatures. The activation energy for volume diffusion has not been evaluated from the results, as the good correlation between individual volume diffusion coefficients and literature values, as well as the excellent agreement between calculated Ĉ . values and equilibrium phase diagram values, is adequate demonstration of the consistency and accuracy of the concentration contour experiments. 3.5 The Effect of Supersaturation (k) and Volume Diffusion Coefficient In any one alloy, changing the temperature changes both the volume diffusion coefficient (D.r) and the supersaturation (k). This makes - 78 - i t d i f f i c u l t to estimate the i n d i v i d u a l contributions of these two v a r i a b l e s . They w i l l therefore be considered i n one section,and an attempt w i l l be made to show the e f f e c t of each of these factors on the shape of the isoconcentration contours. In order to determine the e f f e c t of k on the shape of the i s o - concentration l i n e s , measurements were c a r r i e d out at T = 0.92 f o r ri a l l o y A with k = 0.035. The r e s u l t s are shown i n Figure 34,which can be compared to Figure 25,in which the homologous temperature i s the same but k i s appreciably higher (k = 0.094). The d i s s o l u t i o n temperatures were d i f f e r e n t i n the two cases (517°C for Figure 34 and 532°C for Figure 25). Comparing the two f i g u r e s , i t i s evident that although the actual concentration values and d i f f u s i o n distances are quite d i f f e r e n t the o v e r a l l shape of the isoconcentration contours i s the same. In the above experiment both the temperature and the super- saturation were d i f f e r e n t . An evaluation of the influence of these two variables w i l l be attempted by making a comparison between pai r s of experiments i n which e i t h e r T, k or T are kept constant. Five sets H of such experiments are l i s t e d i n Table (VI), three sets have T constant, one set has T constant and one has k approximately constant. A l l three sets of isoconcentration l i n e s having a constant T have ri the same shape. In both the cases of a constant volume d i f f u s i o n c o e f f i c i e n t and constant k, but d i f f e r e n t T , there i s no s i m i l a r i t y ri i n the shape of the isoconcentration contour p a i r s . One may s a f e l y conclude that the influence of homologous temperature on the shape of the isoconcentration contours i s independent of both the volume  - 80 - Table VI. Comparison of the E f f e c t s of k, T and T on the Shape of n Isoconcentration Contours. A l l o y Temp. °K H D x l O 1 0 2 -1 cm sec S/sat. k Shape of Contours F i g . No. A B 790 805 .92 .92 7.3. 10.7 .035 .094 same 34 25 A B 790 790 .92 .905 7.3 7.3 .035 .078 d i f f e r e n t 34 26 Al-Ag A 823 800 .93 .93 50 9.4 .597 .045 same 43 21 Al-Ag B 786 777 .885 .885 22 5.4 .341 .0653 same 42 45 A B 818 753 .97 .86 7.3 2.8 0.515 0.511 d i f f e r e n t 23 27 - 81 - d i f f u s i o n c o e f f i c i e n t and the supersaturation. 3.6 The E f f e c t of Grain Boundary Mi s o r i e n t a t i o n (R) The Al-15.75 wt.% Ag a l l o y had a micro-structure with a s i g n i f i c a n t number of low angle grain boundaries. These were i d e n t i f i e d using the 4 Clark c l a s s i f i c a t i o n of p r e c i p i t a t e morphology i n t h i s system. On one grain boundary i t was found that nearly a l l the p r e c i p i t a t e s were i n the form of primary side plates and sawteeth (R ^ 9.5°). However, two p r e c i p i t a t e s on t h i s boundary resembled allotriomorphs., Composition contours were measured around these and were compared with composition contours obtained from an allotriomorph on a high angle grain boundary at the same temperature. In Figure 35 the p r o f i l e s c l e a r l y show that grain boundary d i f f u s i o n e f f e c t s are not important for low angle grain boundaries, whereas i n Figure 36 there i s a s i g n i f i c a n t grain boundary contribution to d i s s o l u t i o n for a high angle grain boundary. Investigations of the e f f e c t of grain boundary misorientation 43 44 on the grain boundary d i f f u s i o n c o e f f i c i e n t ' have shown that beyond a misorientation of R 'v 20° there i s very l i t t l e s i g n i f i c a n t v a r i a t i o n of.D , . Since grain boundary, allotriomorphs generally g.b. form on grain boundaries with a misorientation R > 17°, the grain boundary contribution would not be expected to vary s i g n i f i c a n t l y from allotriomorph to allotriomorph. The experimental r e s u l t s appear to confirm t h i s because the d i f f u s i o n distances along the grain boundary vary by no more than 15% from p r e c i p i t a t e to p r e c i p i t a t e i n any one system, f o r any one set of experimental conditions. Figure 35. Isoconcentration contours of two allotriomorph shaped precipitates s i t u a t e d on a low angle grain boundary for the Al-Ag a l l o y at T = 0.84. Figure 36. Isoconcentration contours for the Al-Ag a l l o y at T = 0.84. - 84 - 3.6.1 Evaluation of the Grain Boundary D i f f u s i o n C o e f f i c i e n t (D ^ ) 45 A model developed by Shewmon f o r surface d i f f u s i o n was used to evaluate D , i n the present work. In Shewmon's model, solute i s g.b. considered to quickly d i f f u s e out along a surface from a c y l i n d r i c a l source. From the surface the solute then d i f f u s e s into the bulk of the material by a volume d i f f u s i o n process. Shewmon's model i s the 46 c y l i n d r i c a l analogue of Fisher's model f o r one-dimensional surface d i f f u s i o n (see Figure 37). I t su f f e r s from the same approximations as used by Fisher; viSi: that d i f f u s i o n occurs i n the bulk only i n the d i r e c t i o n perpendicular to the surface and that steady state conditions on the surface are established e a r l y i n the d i f f u s i o n process. However Turnbull et a l . ^ 7 i n a c r i t i c a l evaluation of the Fisher Model and 48 the more soph i s t i c a t e d Whipple Model, found that D , values calculated from the d i f f e r e n t models d i f f e r e d only by a few percent. The Shewmon model i s immediately applicable to d i s s o l u t i o n from c i r c u l a r p r e c i p i t a t e s by d i f f u s i o n along the grain boundaries and has 20 been used i n t h i s connection by B r a i l s f o r d and Aaron. In t h i s case the s o l u t i o n takes the form: C ( p , z , x ) = C ( p , x ) erfc(z/2v^~F) g. D . V where C g . b . ( p ' T ) = S A W [K 0(a-P)/K 0(a.b)] a 2 = 2D . ./D 6 (T ID . . T ) 1 / 2 V g.b. V - 85 - T i s the annealing time, and b i s the source radius. and the boundary conditions on C , (O.T) are g • • C g . b . ( b ' T ) = C i g.b. M - H 2 b h - High Diffusivity Surface Layer Z •Source Low Diffusivity Material Figure 37. Schematic diagram f o r Shewmon surface d i f f u s i o n model (aft e r P.G. Shewmon). The concentration contours at low temperatures i n the present work were used to determine D , from equations(2) and (3). In g.b. these contours there was l i t t l e d i r e c t volume d i f f u s i o n c ontribution to the gradient along the grain boundary. Figure 38 represents the c a l c u l a t i o n s of D , . The t h e o r e t i c a l p r o f i l e s along the grain 1.0 0.8 3 u % 0,6 u 3 - 0-41 0.2 r TH=0.86 A l l o y - B W \ \ W \ x X X \ \ \. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ . " 3 ^ 2 Dg.b.x lQ^cmPsecT^ 1 = 1 2 = 5 3 = 7.5 4 = 10 • Exptl. Results(t = 400sec.) 10 2 0 3 0 4 0 5 0 px lof Cm 6 0 7 0 Figure 38. Evaluation of D , for a l l o y B at T = 0.86. g-b. ^ H - 87 - boundary were calculated f o r various values of D , and the best f i t g.b. with the experimental data of the d i s s o l u t i o n of a l l o y B at a homologous temperature T = 0.86 (see Figure 27) was taken as the experimental value of ^ (see Figure 38). Grain boundary d i f f u s i o n c o e f f i c i e n t s were determined for several temperatures, the r e s u l t s being shown i n Table VII. An approximate grain boundary d i f f u s i o n coefficient has also been calculated assuming: ^g.b. 2^volume n g . b . r « V O l . r> o n 2. D° = D n = 0.29 cm /sec The a c t i v a t i o n energy for volume d i f f u s i o n i s taken as Q , = 31.13 O J ^volume 39 kcal/g.mole. The r e s u l t i s included i n Table VII,for T = 480°C. The experimental value of ^ i s i n reasonable agreement with the value of D , calculated at 480°c. However, the inconsistency g.b. of the values of D , calculated at the other temperatures i s not g.b. s u r p r i s i n g considering:(1) the approximations made i n a r r i v i n g at values of Q , and D^*^'(2) that the experimental observations show g.b. 0 a d i f f u s i o n a l distance along the grain boundary which obeys an almost parabolic r e l a t i o n s h i p with time, whereas the sbewmon model 0 25 predicts a t " r e l a t i o n s h i p , and (3) the f a c t that the experimental technique used i s not one from which accurate evaluations of ^ are to be expected. These D , c a l c u l a t i o n s are intended as p e r i p h e r a l g.b. information based on the experimental r e s u l t s . - 88 - Table VII. Grain Boundary D i f f u s i o n C o e f f i c i e n t s (D b ) Determined from Contours 8* " T°C D , x 10 cm sec g.b. * 520 7.5 ** 504 9.0 480 0.75 448 7.5 420 5.0 480 0.83 (calculated) ' The grain boundary d i f f u s i o n c o e f f i c i e n t has been determined at these temperatures,even though the r e l a t i v e amount of volume to grain boundary d i f f u s i o n i s very large r e s u l t i n g i n a high degree of error i n the c a l c u l a t i o n of D - 89 - 3.7 Semi-Quantitative Analysis of Isoconcentration Contours I t has generally been found i n d i f f u s i o n studies that no e f f e c t due to grain boundary d i f f u s i o n can be seen at temperatures above 0.6 T^. I t i s rather s u r p r i s i n g therefore that i n the present work a grain boundary d i f f u s i o n contribution p e r s i s t s to a homologous temperature greater than 0.92. One of the e s s e n t i a l differences between the two phenomena i s s t a t i s t i c a l i n nature. For grain boundary d i f f u s i o n i n a s i n g l e phase p o l y c r y s t a l l i n e material, one must consider the .mean time that an atom spends i n the grain boundary as w e l l as the p r o b a b i l i t y of i t s getting onto the boundary. In dealing with the d i s s o l u t i o n of grain boundary allotriomorphs, t h i s s t a t i s t i c a l aspect i s eliminated since the p r e c i p i t a t e I s , i n f a c t , s i tuated on the grain boundary. A t h e o r e t i c a l analysis of the r e l a t i v e contributions of grain boundary and volume d i f f u s i o n to allotriomorph d i s s o l u t i o n has been developed by D.E. Coates (see Appendix VI f o r d e t a i l s ) . The main features of t h i s analysis are: 2 (1) The use of Laplace's equation (V C=0) to c a l c u l a t e the volume d i f f u s i o n f i e l d about the allotriomorph. This approximation i s good for low supersaturations and f o r longer d i f f u s i o n times. (2) The use of two extreme conditions: ( i ) where volume d i f f u s i o n i s the sole contributor to d i s s o l u t i o n , as i n Figure 22, and ( i i ) where grain boundary d i f f u s i o n i s the overwhelming contributor to d i s s o l u t i o n as i n Figure 30. (3) A function ty i s evaluated }where ty i s the r e l a t i v e contribu- tion to the t o t a l f l u x out of an allotriomorph due to grain boundary - 90 - d i f f u s i o n and volume d i f f u s i o n , v i z : ( f J . dA) ^ = — £ L _ J ( J l \ • d l ) v o l . I f ^ >> 1 grain boundary d i f f u s i o n dominates. I f ty « 1 volume d i f f u s i o n dominates. As shown i n Appendix VI, ty i s given by K l [ a V -Q/2RT 2e i 2 ( T T E r c ) 1 / 2 f (a/b) ( T T D T ) 1 / 2 where and K q are modified Bessel functions of f i r s t and zero orde res p e c t i v e l y and i t i s assumed -Q/2RT D / D = e g.b. V where Q i s the a c t i v a t i o n energy for volume d i f f u s i o n . 1/2 For a given p r e c i p i t a t e s i z e a and d i f f u s i o n distance (TTDT) equation (2) i s e s s e n t i a l l y an equation of the form ty = I J J( T ) . ty i s plot t e d i n Figure 39 f o r the experimental conditions corresponding t three of the a l l o y s used. The progressive transformation from grain boundary d i f f u s i o n c o n t r o l to volume d i f f u s i o n c o n t r o l with increasing temperature can b - 91 - - 92 - Figure 40. A comparison of t h e o r e t i c a l l y derived i|; versus homologous tempera- ture T R with the experimental observation of solute v i a grain boundary d i f f u s i o n solute v i a volume d i f f u s i o n - 93 - c l e a r l y seen. The three p l o t s are drawn as a function of homologous temperature i n Figure 40, and are found to coincide giving one curve. This i s a f o r t u i t o u s e f f e c t , as the increase i n the p r e c i p i t a t e s i z e ( p r e c i p i t a t e h a l f length a) with increasing a l l o y content compensates 1/2 f o r the longer experimental d i s s o l u t i o n times (TTDT) (see equation 2). I t i s thus c l e a r why for the present experimental conditions iji i s only a function of T , rather than a function of the actual temperature. The d i s s o l u t i o n of grain boundary allotriomorphs s o l e l y by grain boundary d i f f u s i o n occurs below T = 0.72, which i s i n excellent ri agreement with the t h e o r e t i c a l l y predicted value at which grain boundary d i f f u s i o n dominates. The t h e o r e t i c a l c a l c u l a t i o n s confirm the s i g n i f i c a n c e of the contribution of grain boundary d i f f u s i o n to the t o t a l f l u x even at high homologous temperatures. From the experimental data, i t i s p o s s i b l e to make an approximate c a l c u l a t i o n of the t o t a l f l u x due to volume d i f f u s i o n and that due to grain boundary d i f f u s i o n . The composition contour corresponding to volume d i f f u s i o n i s e s s e n t i a l l y an oblate spheroid. The extra contribution i s assumed to be due to grain boundary d i f f u s i o n as i n Figure 41. By measuring the two areas and i n t e g r a t i n g over the s o l i d angle corresponding to an oblate spheroid, the r e l a t i v e grain boundary and volume contributions to d i s s o l u t i o n can be calculated The experimental points i n Figure 40 represent the r a t i o : area due to g.b. d i f f u s i o n .. , ^ , , . ^ •, . ... -,-, 5 ° .. ,. r r : , calculated as depicted schematically i n area due to volume d i f f u s i o n Figure 41 for a l l the p r o f i l e s measured. The experimental r e s u l t s are i n good agreement with the t h e o r e t i c a l values. The t h e o r e t i c a l analysis (equation 2) predicts no e f f e c t of supersaturation and l i t t l e - 94 - 1 2 3 4 G.B. Allotriomorph Solute via Volume Diffusion Solute via G.B. Diffusion Grain Boundary (G.B.) Figure 41. Schematic diagram of method used to calculate 1) solute v i a grain boundary d i f f u s i o n , and 2) solute v i a volume d i f f u s i o n . e f f e c t of a x i a l r a t i o , both of which are i n agreement with the experimental observations (see Appendix VI). 3.8 The Al-15.75 wt.% Ag System Figures 36, 42, and 43 show the concentration contours at three d i f f e r e n t homologous temperatures i n the Al-Ag system. As can be seen by comparing these r e s u l t s with those i n the Al-Cu system ( v i z : Figures 36 and 44, Figure 42 and Figure 23, Figure 43 and 45) there i s excellent agreement i n the shape of these contours at equivalent homologous temperatures. In both the Al-Cu and the Al-Ag systems, allotriomorphs having a r e l a t i v e l y complex structure are d i s s o l v i n g i n t o an f . c . c . matrix. The d i f f u s i o n c o e f f i c i e n t s of these a l l o y s d i f f e r by a f a c t o r of 39 49 f i v e ' at equivalent homologous temperatures. The supersaturation Figure 42. Isoconcentration contours for the Al-Ag a l l o y at T = 0.885. Figure 43. Isoconcentration contours for the Al-Ag a l l o y at T = 0.93. TH=i0.83 Alloy-B Figure 44. Isoconcentration contour for a l l o y B at T = 0.83.  - 99 - however may vary by more than an order of magnitude (see Table V). The observation that the isoconcentration contour shapes are comparable f o r comparable homologous temperatures confirms the singular importance of the homologous temperature; changing the supersaturation only a l t e r s the values for the concentration contours, not t h e i r shape. Calculations of C i n t h i s system are i n good agreement with phase diagram data(as i n Figure 5). i s found to be independent of concentration over the t o t a l s o l i d s o l u b i l i t y range, as was found by 49 Heuman and Bohmer. Figure 46 compares values of calculated from the contours with values of Dy found i n the l i t e r a t u r e . The agreement i s good f o r both i n d i v i d u a l values of as w e l l as f o r the c a l c u l a t i o n of the a c t i v a t i o n energy. 3.9 A p p l i c a b i l i t y of Contours I t has been found that the d i s s o l u t i o n contours i n both the Al-Cu and Al-Ag systems are very s i m i l a r at comparable homologous tempera- tures. I t thus seems probable that the present r e s u l t s are applicable to any d i s s o l v i n g grain boundary allotriomorph. The d i f f u s i o n c o e f f i c i e n t s i n Al-Cu and Al-Ag are s i m i l a r at s i m i l a r homologous temperatures and so the transformation from grain boundary d i f f u s i o n to volume d i f f u s i o n c o n t r o l occurs at the same T i n both cases. The H r e l a t i o n s h i p of volume d i f f u s i o n c o e f f i c i e n t to homologous temperature i n other systems, w i l l determine the T at which the transformation between grain boundary and volume d i f f u s i o n c o n t r o l occurs. The basic shapes of the contours w i l l however, be the same and can simply be translated to d i f f e r e n t T 1 T values.  - 101 - 4. KINETIC RESULTS AND DISCUSSION 4.1 Introduction In t h i s chapter, observed d i s s o l u t i o n k i n e t i c s of grain boundary allotriomorphs are analysed. The r e s u l t s are discussed i n terms of the most appropriate e x i s t i n g a n a l y t i c a l models f o r allotriomorph d i s s o l u t i o n . The l a t t e r include Whelan's s p h e r i c a l and planar models 28 29 50 f o r p r e c i p i t a t e d i s s o l u t i o n ' and Aaron's planar model. I t i s important to note that under the present experimental conditions, impingement of the d i f f u s i o n f i e l d s from adjacent p r e c i p i t a t e s takes place at a very early stage i n the d i s s o l u t i o n process (see Table I I I ) . On the other hand, the three models j u s t c i t e d are based on the assumption that the d i f f u s i o n f i e l d s of adjacent p r e c i p i t a t e s do not impinge at a l l ( i . e . , these models r e l a t e to the d i s s o l u t i o n of a si n g l e p r e c i p i t a t e i n an i n f i n i t e matrix). Accordingly i t i s ant i c i p a t e d that the c o r r e l a t i o n between these models and the present experimental results w i l l be, at best, semi-quantitative. Thus an attempt i s made to provide a firmer basis for i n t e r p r e t a t i o n of the r e s u l t s by formulating a d i s s o l u t i o n model which accounts f o r impingement of the d i f f u s i o n f i e l d s . - 102 - 4.2 Analysis of D i s s o l u t i o n K i n e t i c s Using E x i s t i n g Planar and Spherical Models Table VIII gives a resume of a l l the k i n e t i c experiments and includes the number of p r e c i p i t a t e s studied, the d i s s o l u t i o n times and temperatures, and the supersaturation values (k). I t should be noted that the time f o r complete d i s s o l u t i o n of p r e c i p i t a t e s v a r i e d from 5 minutes to 116 hours, depending on the experimental conditions. 28 29 Whelan's s p h e r i c a l model ' was formulated f o r a p p l i c a t i o n to the l a t e r stages of the dissolution of s p h e r i c a l p r e c i p i t a t e s . D i f f u s i o n f i e l d impingement i s completely ignored i n the d e r i v a t i o n . For the l a t e r stages one can approximate that only the steady state d i f f u s i o n f i e l d contributes to the d i s s o l u t i o n rate (see Section 1.3.1, equation (4 ) ) . 2 To apply the Whelan s p h e r i c a l model a p l o t of R versus t was made for each p r e c i p i t a t e studied. I f the model i s v a l i d , such p l o t s should be l i n e a r . In f a c t , these p l o t s were not l i n e a r , i n d i c a t i n g the model i s not consistent with the present data (see Figure 47). 2 I t should be noted however that whereas the p l o t s of R versus t are not l i n e a r f o r the f u l l extent of the d i s s o l u t i o n heat treatment. There i s a tendency at the l a t e r stages of the d i s s o l u t i o n process f o r 2 the r e l a t i o n s h i p R versus t to become l i n e a r , i n d i c a t i n g at l e a s t a 29 q u a l i t a t i v e agreement with Whelan's experimental observations. 29 50 Whelan's planar and Aaron's planar models are formulated f o r planar p r e c i p i t a t e s and for the e a r l y stages of d i s s o l u t i o n before impingement takes place,and are thus not applicable to the present set _ 103 - Table VIII. K i n e t i c Experiments. A l l o y Number of P r e c i p i t a t e s T H Super- saturation (k) T°C Time D 2 0.76 0.014 424 0,10,20,37,65,125,180,125, 307,442,560,686,816,1003 min Al-Ag 6 0.84 0.123 460 116,220,320,457,678,1123 min C 6 0.79 0.036 450 0,55,110,220,440,700 min B 4 0.83 0.032 460 0,1/3,1,3,4,6 1/2,9,12 1/2, 16,20,25,34,44,51,62,89, 116 hrs. D 6 0.84 0.039 489 0,5,8,11,15,24,36,50,69, 82 min D 5 0.86 0.092 500 0,1/2,1 1/2,3,6,12 min B 8 0.88 0.065 500 0,1 1/2,6,24,48,72,96,144 mir B 6 0.86 0.059 489 15,30,45,60,80,100,156,212, 297 min D 5 0.89 0.116 534 0,1/2,1,2,3,5 min Al-Ag 4 0.88 0.328 500 5,17,38,60,90 min B 5 0.90 0.078 513 1 1/2,6,12,18,24,48 min Al-Ag 6 0.92 0.597 534 1/2,1 1/2,3,5,7 1/2 min A 5 0.95 0.048 520 1,4,16,66 min A 9 0.97 0.052 534 5,10,20,30,50 min B 5 0.92 0.09 534 3,6,10,15,20 min TH = 0.92 Alloy-B - 105 - of experimental data. 29 Whelan's planar model (see Section 1.3.1) d i f f e r s from Aaron's planar model"^ by a fa c t o r of /n"/2 i n the constant k of equation (7) i n Section 1.3.1. A p l o t of R versus /t f o r the experimental r e s u l t s should give a l i n e a r r e l a t i o n s h i p i f t h i s model i s v a l i d . The experimental r e s u l t s for some p r e c i p i t a t e s give a l i n e a r r e l a t i o n s h i p f o r the t o t a l amount of d i s s o l u t i o n observed^with a s i g n i f i c a n t amount of sc a t t e r . A l i n e a r f i t i s observed i n the case of a few p r e c i p i t a t e s f o r only the e a r l i e r d i s s o l u t i o n times (see Figure 47). However, no conclusion can be drawn as to the a p p l i c a b i l i t y of t h i s model to the ea r l y stages (that i s before impingement),as the majority of the observations were made a f t e r a c e r t a i n amount of impingement had occurred. In an attempt to f i n d the best c o r r e l a t i o n between R and t, the data was p l o t t e d In the form log R/RQ versus t and, to a good approximation, a l i n e a r r e l a t i o n s h i p was obtained. Figures 48 to 52 are t y p i c a l examples of log R/RQ versus t and log S / S q versus t p l o t s f o r the homologous temperature range from T„ = 0.97 to T„ = 0.76. The observed l i n e a r r e l a t i o n s h i p indicates H H that a r e l a t i o n s h i p of the form -k t R = R e 1 o i s applicable to the change i n h a l f thickness and - k 2 t S • » S e o 2 - 106 - 0 10 20 30 40 50 Time(min) Figure 48. Log [S/S ] versus t and log [R/R ] versus t for a l l o y A at T = 0.97. u O H - 107 - - 108 - - 109 - - 110 - 0 200 400 600 800 1000 Time(min) Figure 52. Log [S/S ] versus t and log [R/R ] versus t for a l l o y D at 1 = 0.76. - I l l - i s v a l i d f o r the change i n h a l f length. In the following section, the observed exponential nature of the d i s s o l u t i o n k i n e t i c s i s r a t i o n a l i z e d on the basis of an a n a l y t i c a l model which accounts for d i f f u s i o n f i e l d impingement. The model assumes planar symmetry. 4.3 An Exponential Model for Planar D i s s o l u t i o n K i n e t i c s It was shown i n section 3.4.1 that f or the present experimental conditions, impingement of the d i f f u s i o n f i e l d s occurs early i n the d i s s o l u t i o n process, before there i s any s i g n i f i c a n t movement of the interphase boundary. In view of t h i s f a c t , i t was decided to use an approximate s o l u t i o n to the d i f f u s i o n equation f o r a f i n i t e system, assuming that the movement of the phase boundary can be neglected ( i . e . , the stationary i n t e r f a c e approach i s adopted). The i n i t i a l s i t u a t i o n f o r t h i s geometrical model i s shown i n Figure 53(a) i Figure 53(a) Geometrical model of a p a r t i c l e with half-thickness R i n a matrix with i n t e r p a r t i c l e spacing £. where R^ i s the i n i t i a l p a r t i c l e h a l f thickness and H i s the i n t e r p a r t i c l e distance. I t i s assumed throughout t h i s d e r i v a t i o n that R Q << i/2. The approrpriate phase diagram i s shown i n Figure 53(b). - 112 - Cq CQ C i Figure 53(b) An appropriate phase diagram with growth of p r e c i p i t a t e s taking place a t . l ^ and d i s s o l u t i o n at 1^. i s the annealing temperature f o r growth and i s t n e d i s s o l u t i o n temperature. From the lever r u l e i t follows that R 1/2 C -C, o 3 C -C P o since the sample i s assumed to be completely e q u i l i b r a t e d at temperature ( i . e . , p r i o r to upquenching to the d i s s o l u t i o n temperature T^). Assume that the d i s s o l u t i o n process has commenced at 1^. P r i o r to impingement^ the concentration d i s t r i b u t i o n s are as shown i n Figure 53(c). Figure 53(c) D i s s o l u t i o n p r o f i l e p r i o r to impingement. - 113 - During the impingement period, the d i s t r i b u t i o n s are as shown i n Figure 54(a). Ultimately the a l l o y w i l l have a uniform composition C . Figure 54(a) The concentration d i s t r i b u t i o n s during d i s s o l u t i o n s a f t e r impingement takes place. Hence the concentration at the centre of the impingement region (&/2) w i l l gradually approach C^, but w i l l a t t a i n i t only a f t e r the second phase has completely disappeared and the concentration gradients i n the matrix have been eliminated. There ex i s t s no a n a l y t i c a l s o l u t i o n to the d i f f u s i o n equation for the impingement s i t u a t i o n . Therefore an approximate s o l u t i o n i s used i n conjunction with the assumption that there i s a constant concentration Figure 54(b) Schematic diagram of model used with a constant concentration C-*3 £L fc X —~ & • - 114 - at x = H, which over a reasonably long time period y i e l d s concentra- t i o n contours near the i n t e r f a c e which approximate those due to impingement as shown i n Figure 54(b). A reasonable value f o r t h i s concentration i s C 0 = 2C - C 2 o I as t h i s y i e l d s a concentration of C q (the bulk composition) at i/2 f o r i n f i n i t e time. The present model i s d e f i c i e n t i n that i t p r e d i c t s a f i n i t e gradient at ft/2 f o r a l l times, whereas there should be no f l u x across t h i s point. However, i t i s emphasized that we are concerned with the gradients at x = 0 (the interface) and, therefore, the approximation should involve l i t t l e e r r o r . The s o l u t i o n to the d i f f u s i o n equation for a p l a t e of thickness £, faces of constant concentration C _ and C_ = 2C -C_, and an i n i t i a l concentration C„ i s 1 2 o I 3 given i n Crank"^ (page 47 ). This s o l u t i o n involves a s e r i e s expansion. For the present problem i t i s assumed that the leading term i n t h i s s e r i e s dominates. This assumption requires that 2 2 _ n DTT t 2 e % « 1 (n = 2,3 ) 2 Under these conditions the s o l u t i o n i s „ 2 _ DTT t C = C T + 2 ( C -CT)x/£, - - (C - C j s i n ^ e ft2 3 I O I TT O 3 I On d i f f e r e n t i a t i n g and s e t t i n g x = 0, one obtains - 115 - 9£ 9x = 2 < C o - V ( C -C„) -/ ° 3 4 e D T T 2t x=0 This gradient i s substituted i n t o the i n t e r f a c e mass balance (C -C ) ^ % V dt 3x x=0 to give dR dt pr2 -<w * (cQ-c3) £ L (C -C T) + (c-c T) J e £ p " I ' p - I ' The r a t i o (C -C )/(C -C_) i s f ,, the supersaturation f o r d i s s o l u t i o n I o p I d' (Figure 53(b)). In the second term the r a t i o C -C- C -C_ 0 3 ^ o 3 C -C T P I C -C p o since >> C ^ , C q ( i . e . , for low supersaturation) In view of Equation (1), i t follows that C -CL R o 3 ^ o V c i 1 , 2 Using Equation (8) and the d e f i n i t i o n of f ^ , Equation 6 becomes - 116 - This i s integrated from t = 0 (R = R ) to t(R ) , to give 2Df,t 8R RQ-R = -f ~f (e ll - 1) 10 TT 2 Assuming 8/IT =1, Equation 10 becomes R = R e ft - 2 — — 11 o ft If one substitutes values of the parameters D, t , ft and f ^ which are t y p i c a l of the present experimental conditions, i t i s found that the 2Df d t l i n e a r term — - — only becomes important i n the l a t e r stages of d i s s o l u t i o n (see Figure 55). Thus f o r a considerable (intermediate) portion of the d i s s o l u t i o n period the k i n e t i c s should follow the equation R = R e ft2 o 2 i . e . ftn R/R = - ] L ~ 12 ft2 which i s i n agreement with the experimental r e s u l t s (Figures 48-52). An attempt was made to generalize the above treatment to s p h e r i c a l symmetry. Unfortunately, i n order to obtain an equation of the same form as Equation (12),so many assumptions were required that the deri v a t i o n was lacking i n a n a l y t i c a l r i g o r and was therefore deemed unacceptable. I t was f e l t that i t would be worthwhile to check the v a l i d i t y of the exponential r e l a t i o n s h i p with d i s s o l u t i o n k i n e t i c s obtained by r Time(sec) Figure 55. R versus t for equations 11 and 12. - 118 - 52 Baty et a l . They used a s t a t i s t i c a l method to follow the d i s s o l u t i o n of C11AI2 p r e c i p i t a t e s i n an Al-4wt.% Cu system at 520°C. Using t h e i r data, a p l o t of log R/RQ versus t i s shown i n Figure 56 and a reasonably l i n e a r f i t i s obtained. Further support for the exponential model i s shown i n Figure 57 where the numerically calculated s o l u t i o n 25 by T a n z i l l i and Heckel for d i s s o l u t i o n of planar p r e c i p i t a t e s i n a f i n i t e system i s p l o t t e d i n logarithmic form. The supersaturation (k = 0.22) i s much greater i n t h i s instance than f o r most of the present experimental work. Equation 12 was derived using a stationary i n t e r f a c e assumption, the r e l i a b i l i t y of which increases with decreasing supersaturation. That a reasonably l i n e a r f i t was obtained from the numerically derived s o l u t i o n which involves a r e l a t i v e l y high supersaturation i s very encouraging. The experimental r e s u l t s obey the exponential r e l a t i o n s h i p as established i n Equation 12, for the whole range of homologous temperatures observed. The greatest deviation from l i n e a r i t y i n the pl o t s of £n R/RQ versus t i s i n most cases found during the early stages of d i s s o l u t i o n , before impingement. This i s to be expected as the model does not account for d i s s o l u t i o n k i n e t i c s p r i o r to impingement. The values of calculated using equation 12 are co n s i s t e n t l y higher than the l i t e r a t u r e values as may be seen i n Figure 58. A l l other things being equal, the d i s s o l u t i o n rate of a curved surface i s fa s t e r than that of a planar surface. The surfaces of the allotriomorphs are i n fa c t curved. Thus the a p p l i c a t i o n of a planar d i s s o l u t i o n model to the d i s s o l u t i o n of a curved surface w i l l of necessity y i e l d values which are higher than the a c t u a l values. Furthermore the ommision of the Time(sec) Figure 56. Log [R/RQ] versus t for d i s s o l u t i o n of p r e c i p i t a t e s i n an Al-4 wt.% Cu system at 520°C. 0 0.1 0 . 2 0 . 3 0.4 0.5 0 . 6 0 . 7 Const. X t Figure 57. Log [R/R0] versus t for the dissolution of planar precipitate in a f in i te system with k = 0.22. - 121 - - 122 - D f d t l i n e a r term (2 ——) i n the f i n a l form of the exponential equation I (equation 12) w i l l also r e s u l t i n higher calculated D v values. Finally,any surface d i f f u s i o n e f f e c t present w i l l increase the observed Dy value. I t should be noted, however, that a surface d i f f u s i o n e f f e c t i f present would become more important at lower temperatures tending , . , r_, c a l c u l a t e d lT. l i t e r a t u r e , ^. ^ •> , to give higher [D^ /D^ J r a t i o s at low homologous temperatures. I f anything the r e s u l t s show the opposite trend. 4.4 P r e c i p i t a t e Shape Change During D i s s o l u t i o n In general i t was observed that a constant a x i a l r a t i o S/R was not maintained during d i s s o l u t i o n . Indeed, i n c e r t a i n cases the f i n a l a x i a l r a t i o , S /R , was s i g n i f i c a n t l y d i f f e r e n t from the i n i t i a l a x i a l r r r a t i o , S /R . The present section deals with t h i s observed change i n o o r & a x i a l r a t i o . Figures 59-63 are examples of s i t u a t i o n s i n which V R o - V V V R o > V R F A N D V R o < V R ! I t was shown i n Section 3.4 that the shape of the concentration p r o f i l e s during d i s s o l u t i o n and hence the mode of d i s s o l u t i o n , f or a c e r t a i n homologous temperature i s independent of supersaturation. Figure 64 i s a p l o t of supersaturation versus the f r a c t i o n (F) of p r e c i p i t a t e s i n a given experiment showing an increase i n a x i a l r a t i o of greater than 10%, i . e . , the f r a c t i o n of p r e c i p i t a t e s f o r which ( V V > 1.1 o o This procedure i s crude for several reasons. - 123 - - 124 - Figure 60. Back scattered electron image micrographs,used to observe the d i s s o l u t i o n k i n e t i c s of an allotriomorph at = 0.97 for a l l o y A. - 125 - Figure 61. Back scattered electron image microgranhs,used to observe the d i s s o l u t i o n k i n e t i c s of an allotriomorph at T = 0.88 for ri the Al-Ag a l l o y . - 1 2 6 - Figure 62. Back scattered electron image micrographs, used to o the dissolution kinetics of an allotriomorph at T}{ = for alloy B. - 127 - Figure 63. Back scattered electron image micrographs,used to observe the d i s s o l u t i o n k i n e t i c s of an allotriomorph at = 0 . 8 4 for the Al-Ag a l l o y . - 128 - 1. The use of 10% as the f i d u c i a l a x i a l r a t i o increase i s admittedly a r b i t r a r y . 2. The comparison i s made on the basis of the l a s t a x i a l r a t i o measurement taken. However, there may have been further changes i n a x i a l r a t i o before the p r e c i p i t a t e completely d i s s o l v e s . This however could not be observed due to the r e s o l u t i o n of the technique used. 3. The number of p r e c i p i t a t e s studied i n any experiment was t y p i - c a l l y 3-10, which i s i n s u f f i c i e n t f o r unquestionably meaningful s t a t i s t i c a l studies. In s p i t e of these deficiences Figure 64 indicates that at a given T , the change i n a x i a l r a t i o i s independent n of supersaturation which i s i n agreement with the concentration contour studies. In Figure 65 the same function (F), as i n Figure 64, i s p l o t t e d versus the homologous temperature. This p l o t shows a d e f i n i t e increase i n F with increasing homologous temperature. At high homologous temperatures, T R > 0.95, a l l the p r e c i p i t a t e s studied show an increase i n t h e i r a x i a l r a t i o during d i s s o l u t i o n (see Figures 59 and 60). About h a l f of the p r e c i p i t a t e s examined at 0.95 > T > 0.80 show an ri increase i n a x i a l r a t i o and at low homologous temperatures T < 0.80 ri the p r e c i p i t a t e s e i t h e r show no change or else a s l i g h t decrease i n a x i a l r a t i o (see Figure 63). These observations can be explained i n the following manner. 1.00 0.75 aso a25 Alloy TH Alloy TH Alloy TH Alloy TH • A 0.9/ • B 0.90 • D 0.86' 0 C 0-79 O A 0.95 A • D 0.89 B V B A Al-Ag 0.86 0 D 0-76. 0 B 0.92 A • B 0-88 0.84 C © Al-Ag 0.92. E Al-Ag 0.88. * A B 0.84 A D 0.84. 0 • • Q L 0.02 0.2 D a 0.3 0.4 0.5 0.03 0.04 0.05 0.1 Supersaturation(k) F i g u r e 64. F r a c t i o n o f p r e c i p i t a t e s showing an i n c r e a s e i n a x i a l r a t i o g r e a t e r t h a n 10% (F) v e r s u s s u p e r s a t u r a t i o n ( k ) . * D i s s o l u t i o n e x p e r i m e n t b e l o w t h e s o l v u s , r e s u l t e d i n s p h e r i o d i - s a t i o n (see F i g u r e 70). A A l - C u Alloys O A l - A g Alloys - 131 - A.4.1 High Homologous Temperatures At high homologous temperatures (T > 0.92), the con t r i b u t i o n of grain boundary d i f f u s i o n to the d i s s o l u t i o n process may be neglected and the p r e c i p i t a t e may be considered to be an oblate spheroid d i s s o l v i n g by volume d i f f u s i o n into the matrix. Figure 66(a) i s a schematic diagram of the d i s t r i b u t i o n of p r e c i p i t a t e s with an a x i a l r a t i o of approximately 4 to 1 and an i n t e r p a r t i c l e spacing of L = 10 x R . These values are consistent with most of the experimental work. Figure 66(b) gives a schematic representation of the concentration f i e l d s around these p r e c i p i t a t e s during the i n i t i a l stages of d i s s o l u t i o n . Because only bulk d i f f u s i o n contributes, these contours are confocal spheroids. In view of the fa c t that a p a r t i c u l a r p r e c i p i t a t e p a r t i c l e has a symmetrical d i s t r i b u t i o n of other p a r t i c l e s about i t , i t i s reasonable to assume that the d i f f u s i o n f i e l d from surrounding p r e c i p i t a t e s has approximately s p h e r i c a l symmetry as shown i n Figure 66(c). o 28 p 0 0 0 0 0 Figure 66(a) Symmetrical d i s t r i b u t i o n of p r e c i p i t a t e s with [S /R ] = 4. - 132 - I <§ c° (in / ® \ n t Figure 66(b) Schematic representation of concentration f i e l d s around a symmetrical array, of p r e c i p i t a t e s . Figure 66(c) Diagram of s p h e r i c a l l y symmetrical f i e l d surrounding a grain boundary allotriomorph. However, even i f one assumes a random d i s t r i b u t i o n of p r e c i p i t a t e s about any one p r e c i p i t a t e , the d i f f u s i o n f i e l d from these l a t t e r p r e c i p i t a t e s i s on the average r a d i a l l y symmetrical with respect to the p r e c i p i t a t e i n question. Accordingly, as a crude approximation the concentration contours a r i s i n g from neighbouring p r e c i p i t a t e s can be "smeared" into confocal s p h e r i c a l contours with respect to the p r e c i p i t a t e i n question. Figure 66(c) shows c l e a r l y that impingement f i r s t - 133 - takes place at the t i p rather than the faces of the allotriomorph. The t i p s are therefore expected to dissolve more slowly r e s u l t i n g i n an increase i n S/R. Because the p r e c i p i t a t e s i n question are grain boundary a l l o t r i o - morphs, the assumption of a random d i s t r i b u t i o n of p r e c i p i t a t e s about a p a r t i c u l a r p r e c i p i t a t e i s open to question. However, any asymmetry would be due to the presence of a number of p r e c i p i t a t e s on the same grain boundary being i n closer proximity to the p r e c i p i t a t e under observation than the Widmansta'tten plates i n the adjacent matrix. Such a d i s t r i b u t i o n w i l l further enhance the l i k e l i h o o d of impingement f i r s t taking place i n the e q u a t o r i a l plane ( i . e . , the plane of the grain boundary). The impingement p r o f i l e i s assumed to be a superposition of the confocal spheroids of the c e n t r a l p r e c i p i t a t e and the surrounding s p h e r i c a l f i e l d (see Appendix VII for d e t a i l s ) . Figure 67 gives the c a l c u l a t e d p r o f i l e s a f t e r 500 seconds. S i g n i f i c a n t i n t e r a c t i o n of the d i f f u s i o n f i e l d s i n the equatorial plane i s evident. In Figure 68(a) i t can be seen that on section Ob (from Figure 67), which i s along the minor axis, the d i f f u s i o n f i e l d adjacent to the p r e c i p i t a t e i s hardly affected by d i f f u s i o n inward from the s p h e r i c a l f i e l d ( i . e . , from the adjacent p r e c i p i t a t e s ) . In Figure 68(b), however, the magnitude of the gradients i n s e c t i o n Oa, which i s the e q u a t o r i a l plane, are reduced s i g n i f i c a n t l y by d i f f u s i o n inward from the s p h e r i c a l surface. The concentration contours w i l l not a c t u a l l y be confocal spheroids but w i l l have somewhat higher a x i a l r a t i o . This w i l l r e s u l t i n impingement along the equator at e a r l i e r times and w i l l tend to accentuate the above e f f e c t . - 134 - t= 500 Seconds D=SxlO"10cm2sec1 ®-CA Figure 67. The r e s u l t i n g impingement between the confocal spheroid isoconcentra- t i o n contours about an allotriomorph and the contours of the surrounding s p h e r i c a l f i e l d a f t e r t = 500 seconds f o r various [C/Cj].   - 137 - Thus i f during the ea r l y stages of impingement, the low curvature surfaces are d i s s o l v i n g at a rate such as to maintain the a x i a l r a t i o , then because of impingement,the high curvature e q u a t o r i a l region i s d i s s o l v i n g at a rate slower than that required to maintain the a x i a l r a t i o . The net r e s u l t i s therefore an increase i n a x i a l r a t i o . This e f f e c t w i l l continue even a f t e r s i g n i f i c a n t impingement. 4.4.2 Low Homologous Temperatures At low homologous temperatures,the grain boundary d i f f u s i o n c o e f f i c i e n t i s so much higher than the volume d i f f u s i o n c o e f f i c i e n t , that the grain boundary i s e f f e c t i v e l y supersaturated with solute before any s i g n i f i c a n t d i s s o l u t i o n takes place, (see Figure 30, for concentration p r o f i l e s at low T ). This r e s u l t s i n a solute concentration along the grain boundary which i s equal to the i n t e r f a c e solute concentration Ĉ .. Thus the t i p of the allotriomorph tends to dissolve p r e f e r e n t i a l l y , with the corresponding solute moving along the grain boundary and dispersing from here into the matrix. This model f o r d i s s o l u t i o n suggests a spheroidizing process with the t i p d i s s o l v i n g much f a s t e r than the low curvature faces of the p r e c i p i t a t e . However th i s 53 tendency i s probably reduced by i n t e r f a c i a l d i f f u s i o n along the precipitate-matrix i n t e r f a c e which would tend to feed solute to the grain boundary from other areas over the p r e c i p i t a t e surface. A s l i g h t spheroidizing tendency i s observed at low homologous i n the present work. - 138 - 4 . 4 . 3 Intermediate Homologous Temperatures At intermediate homologous temperatures the tendencies applicable to both high and low homologous temperatures are probably operative. The r e l a t i v e importance of these tendencies i s determined, among other things, by: (1) The i n t e r p a r t i c l e spacing on the grain boundary: For instance, i t was q u a l i t a t i v e l y observed that i n cases where a grain boundary p r e c i p i t a t e was s i t u a t e d between two other grain boundary p r e c i p i t a t e s , the rate of change i n h a l f length S was reduced s i g n i f i c a n t l y . (2) The i n i t i a l a x i a l r a t i o ( S Q / R ) : This r a t i o i s important i n determining the degree to which e a r l i e r impingement of d i f f u s i o n f i e l d s takes place at the t i p as opposed to the sides of the allotriomorph. The model for high homologous temperatures indicates that the higher the i n i t i a l a x i a l r a t i o , ( S / R ) , i s , t h e e a r l i e r impingement of ' o o d i f f u s i o n f i e l d s at the t i p takes place, and therefore the greater i s the expected increase i n a x i a l r a t i o . i s plotted against ( S O / R Q ) . There i s a d e f i n i t e trend towards a larger change i n ( S / R ) with higher i n i t i a l a x i a l r a t i o s , a r e s u l t which i s consistent with the preceding discussion. experiment at a temperature below the solvus l i n e f o r an intermediate homologous temperature (T = 0.83 f o r a l l o y B)) i n Figure 70(a), shows a good f i t f o r the early stages of d i s s o l u t i o n . A p l o t of [ S / R ] versus t i n Figure 70(b) shows a s l i g h t o v e r a l l decrease i n [ S / R ] f o r the In Figure 69 the percentage change i n a x i a l r a t i o ( F F / S /R ) o o A p l o t of log R/R versus t, ( f o r a d i s s o l u t i o n o corresponding e a r l y stages of d i s s o l u t i o n . A l l four p r e c i p i t a t e s 7o Increase in SEZEEI So/R< KJ O o - 6CT " - 140 - 9 • 7 5 • • # • 5° 3 - 2 25 50 t . h r s . 7 5 , 0 ° 1 2 5 3.0 1 • • • •• • 2.5 - LOO£ • • 2.0 • • • 1.6 • Figure 70. (a) Log [R/R ] versus t. (b) [S/R] versus t. - 141 - i n t h i s experiment show a small change i n [S/R] for the i n i t i a l stages of d i s s o l u t i o n . This i s i n good agreement with the predicted change i n [S/R] with homologous temperature shown i n Figure 65. I t should be noted that the point of departure of the r e s u l t s from the exponential r e l a t i o n s h i p i n Figure 70(a) coincides reasonably w e l l with the point i n Figure 70(b) where rapid spheroidization begins to take place. - 142 - 5. CONCLUSIONS The r e s u l t s of the present study of the d i s s o l u t i o n behaviour of grain boundary allotriomorphs i n the Al-Cu and Al-Ag systems y i e l d the following conclusions. 1. Isoconcentration contours around d i s s o l v i n g grain boundary allotriomorphs can be accurately determined by the use of el e c t r o n probe microanalysis. 2. For allotriomorphs i n an e f f e c t i v e l y i n f i n i t e matrix, the shape of the isoconcentration contours i s dependent only on the homologous temperature, T . H 3. At a p a r t i c u l a r T , the r e l a t i v e contribution of grain boundary d i f f u s i o n to volume d i f f u s i o n can be obtained d i r e c t l y from the isoconcentration contours. 4. The d i s s o l u t i o n of grain boundary allotriomorphs i s dominated by volume d i f f u s i o n above T u = 0.92 and by grain boundary d i f f u s i o n rl below T = 0.72. For the intermediate range T u = 0.92-0.72, the grain n H boundary d i f f u s i o n contribution increases continuously as T decreases.. H 5. Enhanced d i s s o l u t i o n due to grain boundary d i f f u s i o n i s l i m i t e d to p r e c i p i t a t e s on high angle grain boundaries. Grain boundary a l l o t r i c - morphs are therefore the only p r e c i p i t a t e morphology that w i l l e x h i b i t t h i s phenomenon. 6. The d i s s o l u t i o n k i n e t i c s of an i n d i v i d u a l grain boundary a l l o t r i o - morph can be established from i t s back scattered e l e c t r o n image on the ele c t r o n probe microanalyzer. 7. The d i s s o l u t i o n k i n e t i c s of grain boundary allotriomorphs, under conditions i n which impingement of d i f f u s i o n f i e l d s from adjacent - 143 - precipitates takes place early in the dissolution process, are found to satisfy the following equations: _ TT2Dt R - R e £ 2 o for the change in half thickness and _ v D t S = S e I1 o for the change in half length. 8. At high homologous temperatures, where volume diffusion dominates, an increase in the axial ratio of the allotriomorphs is observed as dissolution proceeds. On the other hand, at low homologous temperatures where grain boundary diffusion dominates, a slight decrease in axial ratio (i.e., slight spheroidization) is observed. A continuous change in behaviour from one extreme to the other is observed over the range of intermediate homologous temperatures. - 144 - BIBLIOGRAPHY 1. Dube, C.A., Aaronson, H.I. and Mehl, R.F., Rev. Met., 55_, 201 (1958). 2. 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Castaing, R., Advan. Elec t r o n . E l e c t r o n Phys. , 13, 317 (1960). 64. Murdoch, D.C, A n a l y t i c Geometry with an Introduction -to Vectors and Matrices, John Wiley and Sons, Inc., N.Y. (1966) 65. Coates, D.E., Dept. of Met., U.B.C, Private Communication. 66. Hawbolt, E.B., Ph.D. Thesis, Dept. of Met., U.B.C. (1967). 67. Brown, L . C , Dept. of Met., U.B.C, unpublished work (1964). 68. Crank, J . , Mathematics of D i f f u s i o n , The Univ e r s i t y Press, Oxford, p. 86 (1957). 69. Ham, F.S., Quart. Appl. Math. 17, 137 (1959). - 148 - APPENDIX I. Correction Procedures for Quantitative E l e c t r o n Probe Microanalysis The measured X-ray i n t e n s i t y of element A i n a l l o y AB may be converted to the true mass concentration (C^) of element A i n a l l o y AB, by applying the following corrections i n the order i n d i c a t e d : 1. dead time; 2. background; 3. secondary fluorescent enhancement by c h a r a c t e r i s t i c r a d i a t i o n and white r a d i a t i o n ; 4. X-ray mass absorption i n the sample; 5. the atomic number e f f e c t , which i s made up of two separate components: (a) the e l e c t r o n stopping power (S); and (b) the e l e c t r o n backscattering e f f e c t (R). 1.1 The Al-Cu System Pure Cu was used as a reference standard for CuKa^ r a d i a t i o n . The CuKa^ peak was found at a Bragg angle of 26°34* + 3' using a quartz analyzing c r y s t a l to r e f l e c t the X-rays to the gas flow proportional counter. The background count was picked up on the second spectrometer at an angle of 27°4'. To convert the measured i n t e n s i t y of CuKa^ r a d i a t i o n to the concentration of Cu i n the Al-Cu a l l o y only the deadtime, background, and atomic number corrections need be applied to the experimentally determined r a t i o of i n t e n s i t i e s (K ) where: - 149 - Kc^ Cu Ka 1 I n = the Ka., r a d i a t i o n i n t e n s i t y of Cu i n an Al-Cu a l l o y , Ka^ and I = the Ka^ r a d i a t i o n i n t e n s i t y of Cu i n pure Cu. A n e g l i g i b l e secondary fluorescent enhancement i s expected i n t h i s 54 system and t h i s was v e r i f i e d by applying the Reed and Long c o r r e c t i o n for fluorescent e x c i t a t i o n . The absorption f o r t h i s system i s also n e g l i g i b l e as"'"' Cu Cu y A l = ^Cu where u^j* = 49.6 = (mass absorption c o e f f i c i e n t of CuKa^ r a d i a t i o n by Al) and = 53.7 = (mass absorption c o e f f i c i e n t of CuKa^ r a d i a t i o n by Cu). (1) The dead time,d, may be ca l c u l a t e d from the following 56 r e l a t i o n s h i p ^ ( b b s . ) '(true) 1 - C , , x x d (obs.) C. =true counts a f t e r the dead time, c o r r e c t i o n i s taken i n t o (true) consideration, C, , . = observed counts, d = dead time = .4 ysec. (obs.) ' for the electron probe used. This simple c o r r e c t i o n need only be - 150 - applied for high count rates (eg.>2,000 c.p.s.). (2) The background count per ten seconds i s r e g i s t e r e d f o r each ten second CuKa^ count. The background count i s subtracted d i r e c t l y from the CuKa^ count, thereby eli m i n a t i n g the e f f e c t of background r a d i a t i o n . (3) The Duncumb and Read"'7 atomic number cor r e c t i o n was used to convert the i n t e n s i t y r a t i o of CuKa^ X-rays produced ( K Q ) T O the true mass concentration of Cu (C„ ). This c o r r e c t i o n has the form: Cu K = C * R a 1 ~ C u S c u Al-Cu where R = backscatter c o e f f i c i e n t ; S = el e c t r o n stopping power; Al-Cu refers to the a l l o y specimen and Cu r e f e r s to the standard. 58 The stopping power i s evaluated using the following equation: _ " _ . Z 1 „. (1.66 • E) S = constant * — • — • m — A E where Z = atomic number; A = atomic weight; J = mean i o n i z a t i o n p o t e n t i a l (tabulated values i n reference 57); v E = electron energy ; and the backscatter c o e f f i c i e n t , R, i s given by: 1 n(W).Q/S.dW FW K 1 - R = E K - 151 - where: E q = i n c i d e n t e l e c t r o n energy; J E = c r i t i c a l K e x c i t a t i o n p o t e n t i a l ; Q = i o n i z a t i o n c r o s s - s e c t i o n ; • W = E / E ; S as i n equation 2. n (W) .=. backscattered e l e c t r o n energy d i s t r i b u t i o n expressed i n i n t e g r a l form. Values of R are ta b u l a t e d by Duncumb and Reid"* 7 as a f u n c t i o n of 1/U (U = V/Vv) and Z where: V = a c c e l e r a t i n g p o t e n t i a l ; V^ = e x c i t a t i o n p o t e n t i a l of K a , c h a r a c t e r i s t i c K i r a d i a t i o n . F i g ure 1-1 i s a p l o t of versus . f o r the Al-Cu system, a f t e r making the c o r r e c t i o n s described above. 59 The Belk atomic number c o r r e c t i o n was a p p l i e d to the Al-Cu system as a check on the v a l i d i t y of the Duncumb and Reed c o r r e c t i o n used and was found to be i n good agreement w i t h i t . This c o r r e c t i o n procedure i s very simple to apply and has the form: K - "A A Z -Z 1 + <mr> „ -7 Z A C A + Z B C B where Z = C A + C B and Z = atomic number. fin The above has been found by D. M. Poole to be as accurate as any of the more s o p h i s t i c a t e d c o r r e c t i o n procedures. - 152 - - 153 - 1.2 The Al-Ag System For t h i s system the background, the dead time, the absorption and the atomic number co r r e c t i o n , S, were used to convert measured La^ X-ray i n t e n s i t y to the concentration of Ag i n the a l l o y (C. ). The A g dead time, background and atomic number c o r r e c t i o n were applied i n the same manner as f o r the Al-Cu system. 61 The following P h i l i b e r t absorption c o r r e c t i o n was used F(X) " f ( x ) - ?(0) = ( 1 +' x / a ) t l + h ( l + x-/a>] where f (x) = the f r a c t i o n a l transmission a f t e r absorption and represents the absorption e f f e c t only. F(0) = the atomic number e f f e c t only, h = 1.2 A/Z 2; A =.atomic weight; X = p/p ( s e c e); u/p = mass absorption c o e f f i c i e n t ; 9 = take of angle on the probe = 20°; 0 = modified Leonard's c o e f f i c i e n t as calculated by Duncumb 62 and Shields and i s given by 2.39 x 10 5 E 1 - 5 - E J ' 5 o K Ag 5 5 v l l = 779.2 p f S = 521.9 Ag - 154 - Figure 1-2 gives a c a l i b r a t i o n curve of C. versus K. f o r the Al-Ag Ag Ag 6 system, a f t e r applying the corrections described above. - 155 - - 156 - APPENDIX I I . E l e c t r o n Spot Size Determination. The e l e c t r o n spot s i z e for the absorbed, backscattered and topo- graphic e l e c t r o n image on the JEOLCO JXA 300 i s approximately 0.7 u. This corresponds to the r e s o l u t i o n f o r these images. The X-ray spot 6 3 s i z e however va r i e s according to the r e l a t i o n s h i p : spot s i z e S = 0.033 ( E 1 , 7 - E ^ , 7 ) A / P Z o K K where E q = operating voltage = 25 K.V. E = e x c i t a t i o n p o t e n t i a l f o r the r a d i a t i o n considered; and P = specimen density. The relevent data i s given i n Table I I - l . Table I I - l . E , A, Z and p Values for Ag La.. , A l Ka, and Cu Ka.. Radiation. E__ (K.V.) A Z p g cm Ag La^ 4.0 107.868 47 10.49 A l Ka-̂  1.6 26.98 13 2.699 Cu Ka-L 9.0 63.546 29 8.96 A schematic diagram of the electron d i s t r i b u t i o n i n the sample i s shown i n Figure I I - l . The t o t a l spot s i z e i s the sum of the e l e c t r o n beam diameter and the calculated spot s i z e (S). On the basis of the above, the t o t a l spot s i z e f o r the aluminium-rich matrix i s - 5.5 u, - 157 - Figure I I - l . Schematic diagram of the e l e c t r o n spot s i z e . - 158 - taking i n t o account the fa c t that the density of the ct-solid s o l u t i o n i s s l i g h t l y l a r g e r than that for pure aluminum. The spot s i z e f o r C11AI2 i s approximately 3 u , the spot s i z e i n the Ag r i c h matrix of the Ag-Al system i s approximately 5 y , and the spot s i z e i n Ag^Al - 2.25 u . - 159 - APPENDIX I I I . C a l c u l a t i o n of the Flux Line Divergence f o r D i f f e r e n t Sections of a Two Dimensional E l l i p t i c a l Geometry. The general equation f o r an e l l i p s o i d as i n Figure I l l - l ( a ) i s given by: 2 2 2 (a) (b) (c) Figure I I I - l . Two dimensional sections of an oblate spheroid. If a = d > b one obtains an oblate spheroid, whereas i s a = b < d one obtains a prolate spheroid. Assume a grain boundary allotriomorph can be approximated by an oblate spheroid and assume the grain boundary lies on the y = 0 plane ( i . e . x-z plane),(Figure I l l - l ( b ) ) . A cross s e c t i o n perpendicular to the grain boundary, the z=0 plane ( i . e . x-y pla n e ) , i s shown i n Figure I I I - l ( c ) which i s an e l l i p s e of equation 2 2 1 I f c i s the f o c a l length of the e l l i p s e , the following r e l a t i o n s h i p s , 64 ex i s t - 160 - e = c/a = e c c e n t r i c i t y , and I f c -+ a, e = 1, and b -* 0, therefore the spheroid reduces to a disc. I f c + 0, e = 0, and a = b, therefore the spheroid reduces to a sphere. In the Al-Cu system the most common allotriomorphs present have an a x i a l r a t i o of 3.5. With an a x i a l r a t i o , a/b = 3.5, Eq (1) i s used to determine the angle between the tangent to the e l l i p s e at x = 0, y = b, and at a r b i t r a r y values of x and y. Results are given i n Table I I I - l . Table I I I - l A0° for D i f f e r e n t Values of x/b. x/b y/b A 6 ° 0 1 0 0.875 0.97 4 1.75 0.867 10 2.625 0.658 20 3.20 0.407 39 - 161 - APPENDIX IV. D v C a l c u l a t i o n from the D i f f u s i o n Couple Results As shown i n Figure 16, a p r o b a b i l i t y p l o t of C ^ ^ versus x w i t h i n the supersaturated a phase gives a s t r a i g h t l i n e for the d i f f u s i o n couple annealed at 500°C. Consequently a volume d i f f u s i o n c o e f f i c i e n t Dy, which i s independent of concentration, i s i n d i c a t e d . A l l the d i f f u s i o n couple r e s u l t s when p l o t t e d on p r o b a b i l i t y paper gave d i f f u s i o n c o e f f i c i e n t s independent of concentration. In Figure IV-1, the concentration p r o f i l e at 500°C i s used to evaluate D^ from the equation c v - IF where Z i s the point at which the tangent to the concentration gradient at the i n t e r f a c e cuts the concentration l i n e C=0. This equation gives accurate values of D^ when D^ £ f (concentration), and the i n t e r f a c e i s e s s e n t i a l l y s t a tionary. From Figure IV-1 _ _ (2.80 x 1 0 " 2 ) 2 -10 2, D - - r - r = 4.26 x 10 cm /sec V TT x 163 x 3,600 Dy values were also determined i n an alternate manner using the p r o b a b i l i t y p l o t s for each of the four d i f f u s i o n couples. Using a standard d i f f u s i o n c o e f f i c i e n t , D = 5 x 10 ^" cm 2sec \ and the appropriate value of time, a standard l i n e was calculated and drawn on each p r o b a b i l i t y p l o t . Then D^ could be calculated from the r e l a t i o n s h i p : Distance in (fJ) Figure IV-1. Composition p r o f i l e of d i f f u s i o n couple at 500°C. (?'2 x v where X g and x̂ . are the d i f f u s i o n distances i n the aluminum corresponding to the same copper concentration f o r the standard and experimental l i n e s r e s p e c t i v e l y . The value of D̂ . calculated i n t h i s way at 500°C d i f f e r s only i n the second decimal place from the D^ value calculated using equation 1. Table IV-1. D v Values of the D i f f u s i o n Couple Results. x_(50-l%) = 560 y x 2 = 31.36 x 10 4 y 2 - 163 - D.. T°C T°K 1/T°K - Ixl0 4 v y (50-1%) 2 i n " 4 2 x v xlO y D vxl0 480 753 13.28 368 13.54 2.16 500 773 12.94 514 26.40 4.21 520 793 12.60 700 49.00 7.80 535 808 12.38 745 55.50 8.84 - 164 - APPENDIX V. Calcu l a t i o n of Standard Deviation f o r the Isoconcentration Contours. The mean (X) of a set of data X^ i s given by: Mean = X n = E X. i = l 1 n The standard deviation (a) i s given by: Standard deviation = a = - 2 E (X.-X) Z 1=1 1 n-1 Table V - l gives the mean (X) and standard deviation (a) f o r the d i s s o l u t i o n r e s u l t s of 5 p r e c i p i t a t e s i n a l l o y B at T = 0.91 shown H i n Figure 19. Table V - l . D i f f u s i o n Distance, X, a and "=f P r e c i p i t a t e number At centre 90 sec At centre 360 sec 2.1% 2.5% 2.1% 2.5% 1 10.62 6.87 20 13.2 2 9.37 6.87 17.5 15.0 1 10.00 5.63 17.5 10.0 2 8.75 6.25 13.5 1 10.00 6.87 17.5 11.25 2 8.75 5.0 16.25 11.25 1 8.75 5.63 17.5 11.25 1 10.00 6.25 11.25 8.75 5.63 16.87 11.25 Table continued Table V - l Continued - 165 - 2.1% 2.5% 2.1% _2-5% X 9.44 6.11 17.59 11.88 o 0.73 0.73 1.17 1.47 a/X 0.08 0.11 0.066 0.11 - 166 - APPENDIX VI. Ca l c u l a t i o n of \p Coates^^ has attempted to c a l c u l a t e , as a function of temperature, the r e l a t i v e contributions of volume and grain boundary d i f f u s i o n to d i s s o l u t i o n of grain boundary allotriomorphs. The following i s an ou t l i n e of h i s calcu- t i o n . Consider the grain boundary to l i e on the x-y plane. The allotriomorph i s approximated by an oblate spheroid whose major axis i s i n the x-y plane ( i . e . , an e l l i p s e i n the x-z plane i s rotated about i t s minor a x i s , the z a x i s ) . The f o c i are at x = + c and the semi-major and semi-minor axes are a and b re s p e c t i v e l y ( i . e . , the spheroid i n t e r s e c t s the x, y and z axes at + a, + a and + b r e s p e c t i v e l y ) . The f i r s t step i n the c a l c u l a t i o n i s to determine the d i s t r i b u t i o n of solute (the d i f f u s i o n f i e l d ) about the pr e c i p i t a t e assuming only volume d i f f u s i o n . Provided the supersaturation i s low, the d i f f u s i o n 2 f i e l d can be approximated by a s o l u t i o n of Laplace's equation V C=0. P h y s i c a l l y t h i s i s equivalent to assuming a steady state d i s t r i b u t i o n of solute. Using t h i s approximation the d i f f u s i o n f i e l d about the allotriomorph i s determined. Then the i n t e r f a c e gradient of t h i s f i e l d V^C i s computed to give i n turn the i n t e r f a c e f l u x J^. = -D^V^C. F i n a l l y the t o t a l i n t e r f a c e outflow of material from the allotriomorph, J « dA; i s cal c u l a t e d to give 1 where C T and C., are the i n t e r f a c e and bulk matrix concentrations I M respectively,and a/b i s the a x i a l r a t i o of the spheroid. - 167 - The next p r i n c i p a l step i s the c a l c u l a t i o n of the d i f f u s i o n f i e l d about the allotriomorph assuming the material leaves v i a the grain boundary only. Thus one i s e f f e c t i v e l y considering d i f f u s i o n from a c y l i n d r i c a l source (the p r e c i p i t a t e ) of radius a and height 6 46 47 (the grain boundary thickness). Shewmon has modified the Fisher analysis of grain boundary d i f f u s i o n so that i t i s applicable to the same c y l i n d r i c a l symmetry as i s involved here. Using Shewmon's so l u t i o n f o r the d i s t r i b u t i o n of solute i n the grain boundary, Coates cal c u l a t e s the i n t e r f a c e f l u x J* = -D C V TC. The t o t a l i n t e r f a c e outflow of material from the allotriomorph i s simply Jj -oA = 27ra6J , i . e . , cf 3yaA)_ u = 2Tra6D 0a(C T-Cj K^Caa) g.b. -M-V^g^j. R (33) where a = , .. 6(TTD V T) I D S / D V x i s time and K Q and are modified Bessel Functions of zeroth and f i r s t order r e s p e c t i v e l y . Now a function \\i i s defined such that (f 3 V 2 A ) O K Jj_ I g.b. (f J - 3 A ) . I v o l . which i s to be used as an index of the r e l a t i v e contributions of volume and grain boundary d i f f u s i o n . - 168 - I f ij; >> 1,grain boundary d i f f u s i o n dominates. I f 1/ip » 1,volume d i f f u s i o n dominates. A f t e r s u b s t i t u t i n g equations (1), (2) and (3) i n t o (4), Coates obtains, a f t e r considerable rearrangement: D Q / D„6 K 1 ( a a ) * " / ' TT^a) f ( a / b ) 9 , _ .1/2 V a a ; 2(T T D V T ) where f (a/b) - a ^ 2 ~ ' ^ [ { U / b ) 2 - ! } ^ 2 ] /(a/b) 2-] Notice that the supersaturation (C^-C^) has cancelled out i n Equation (6). For a sphere (a/b = 1) and a f l a t disk (a/b = °°)} the corresponding values of f (a/b) are 1 and TT/2 r e s p e c t i v e l y . C l e a r l y therefore, a x i a l r a t i o has l i t t l e influence since f(a/b) i s the only te i n equation (5) i n which a x i a l r a t i o appears. The f a c t o r /TTDT i s approximately the c h a r a c t e r i s t i c d i f f u s i o n distance f o r the matrix. Notice that D appears only i n the r a t i o D / D (cf. equations o o V —0 /RT (3) and (5)). Assume that = D^e , where Q i s the a c t i v a t i o n energy for volume d i f f u s i o n . I f the a c t i v a t i o n energy for grain boundary d i f f u s i o n i s % Q/2 and the pre-exponential remains unchanged, then V D V > eQ/2RT which can be substituted i n t o equations (3) and (5) to give Coates' f i n a l result: - 169 - ^ 2 e - Q / 2 R T 6 0 T D V T ) 1 / 2 e Q / 2 R T 6 * _ _ _ _ _ f(a/b) 2(,D x ) 1 / 2 v . / 2 e - Q / 2 R T . V K [a / JJ] For a given p r e c i p i t a t e s i z e a and d i f f u s i o n distance (irD T ) ^ 2 equation (8) i s e s s e n t i a l l y an equation of the form ty =. KT) i . e . a function describing the r e l a t i v e contributions of volume and grain boundary d i f f u s i o n as a function of temperature. - 170 - APPENDIX VII. Calculations of the Impingement Resulting from the Inter a c t i o n of the Confocal Spheroids about an A l l o - triomorph and a Surrounding Concentric Spherical F i e l d . For the f i e l d around the e l l i p s o i d A i n Figure 66(c) assume that the d i f f u s i o n distances are shorter than the sphere radius so that we can consider the p r e c i p i t a t e i s d i s s o l v i n g i n an i n f i n i t e medium. Consider that the concentration contours are a se r i e s of confocal spheroids; i n the case of a d i s s o l v i n g p r e c i p i t a t e the p r o f i l e s w i l l be l e s s s p h e r i c a l than t h i s at short times as may be seen i n Figure VII-1. As discussed i n the text t h i s w i l l make the a x i a l r a t i o increase f a s t e r on d i s s o l u t i o n . The equation f o r confocal oblate Actual Point effect negligable Confocal Spheroid Point effect important Figure VII-1. Schematic diagram comparing actual contour to confocal spheroid. spheroids i s given by 69 2 2 + = 1 In two dimensions the contour shapes are defined by: - 171 - The value of g (the f o c a l length) may be obtained d i r e c t l y f o r the 2 2 case of growth from a p l o t of the square of the minor a x i s ^ . ^ -0 )versus the n l a x i a l r a t i o ^ — 2 1/2— ^ o r v a r-"- o u s v a l u e s of the supersaturation ( n l "r?7 ) (f) by L.C. Brown.' From analogy with growth at f = 0.1 and an a x i a l r a t i o of 4, g = 1.095 and Is found to vary from 1.13 to 2.82. The actual concentrations are c a l c u l a t e d assuming that the p r o f i l e at the centre of the f l a t face of the oblate spheroid i s an error function as i n Figure VTI-2. c = e r f c m Figure VII-2. P o s i t i o n of error function p r o f i l e . Distances are — — = [ (n., 2 - g 2 ) 1 / 2 - 0.283] where: B = 1.095, n, = 1.13; . = 4; and (n, '-0 ) = 0.283. The ( n i 6 } -10 -1 confocal spheroids are calculated f o r D - 5 x 10 cm sec and t = 500 seconds. 2/67 = 2 /25 x 10 = 10 y 2 2 1/2 The length of the semi-minor axis w i l l therefore be 10 x (n^ -g ) and the semi-major axis w i l l be 1 0 n ^ . These values are l i s t e d i n Table V I I - 1 . - 172 - Table VII-1. Values of ( n 1 - B ) 2 and n 1 of Confocal Spheroids f o r D i f f e r e n t C/C Ratios. [C/CT] - x 10(n. 2-B 2) 1 / 2 10n. 1 2/57 1 1 -1 0 0 2.83 11.3 0.75 0.23 2.3 5.1 12.1 0.5 0.48 4.8 7.6 13.5 0.25 0.81 • 8.1 10.9 15.5 0.1 1.16 11.6 14.4 18.1 0.05 1.39 13.9 16.7 20.0 For d i f f u s i o n inward from the sphere, again consider that d i f f u s i o n distances are shorter than the sphere radius so that the presence of the oblate spheroid can be neglected. The p r o f i l e i s of the form oo C/Cj. = a/r I [erfc C 2 n + 1 ) a ~ r - e r f c ( 2n+l)a+r ] 4 n=0 2/nt 2/Dt Figure VII-3. P e r t a i n i n g t o Equation 4 - 173 - Calculations f o r the concentric spheres are made using a p l o t of C 2 68 versus r/a for d i f f e r e n t values of Dt/a i n Crank (page 86, Figure 6.1). In this' case jDtj = 5 x 1 0 - 1 0 x 500 m 0 Q 3 1 9 a (28 x 10 ) Table VII-2 gives the locations of the concentric spheres f o r various [C/C T] r a t i o s . Table VII-2. Values of r f o r Concentric Spheres f o r D i f f e r e n t C/C T Ratios. [C/Cj] r/a 1 1 28.3 0.75 0.89 24.9 0.50 0.79 22.2 0.25 0.66 18.5 0.10 0.51 14.3 0.05 0.42 11.8 The a c t u a l p r o f i l e can be found by summing these two independent p r o f i l e s as i n Figure VII-4. The f i n a l p r o f i l e s are given i n Figures 67 and 68 on pages 134-136. - 174 - Figure VII-4. Schematic diagram of the summing up of two independent p r o f i l e s .

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