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Phase transformations in the silver-aluminum system Hawbolt, Edward Bruce 1967

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The U n i v e r s i t y o f B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  :  of EDWARD BRUCE HAWBOLT B.A.Sc., The U n i v e r s i t y of B r i t i s h  Columbia, 1960  M.A.Sc, The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1964 TUESDAY, OCTOBER 31, 1967, AT 3:30 P.M. . IN ROOM 201, METALLURGY BUILDING COMMITTEE IN CHARGE Chairman: E. T e g h t s o o n i a n A. M i t c h e l l L. Young  G.M. Tener D. Tromans L.C. Brown H.J. Greenwood  E x t e r n a l Bxaminer: J.B. C l a r k , P r o f e s s d r Department o f M e t a l l u r g i c a l E n g i n e e r i n g The U n i v e r s i t y o f M i s s o u r i a t R o l l a  Research Supervisor:  L.C. Brown  PHASE TRANSFORMATIONS  IN THE  SILVER-ALUMINUM SYSTEM  ABSTRACT  The f o r m a t i o n o f g r a i n boundary p r e c i p i t a t e s o f the h i g h temperature ^ phase from the s u p e r s a t u r a t e d cx' phase has been examined i n Ag-5.64 wt.%, aluminum a l l o y s a t 688 C. Large g r a i n e d samples were used and the boundary m i s o r i e n t a t i o n s were determined by X-ray d i f f r a c t i o n . A t low angle b o u n d a r i e s o n l y p r i m a r y s i d e p l a t e s formed w h i l e above a m i s o r i e n t a t i o n o f 17 l e n t i c u l a r p r e c i p i t a t e s were dominant. P r e c i p i t a t e growth was s t u d i e d on i n d i v i d u a l g r a i n b o u n d a r i e s u s i n g a s t a t i s t i c a l t e c h n i q u e . The l e n g t h e n i n g and t h i c k e n i n g r a t e s were independent o f the g r a i n boundary m i s o r i e n t a t i o n i n d i c a t i n g t h a t g r a i n boundary d i f f u s i o n was not s i g n i f i c a n t under these c o n d i t i o n s . The p r e c i p i t a t e s grew w i t h c o n s t a n t shape, w i t h b o t h the l e n g t h and t h i c k n e s s i n c r e a s i n g p a r a b o l i c a l l y w i t h time. By a p p r o x i m a t i n g the shape o f the p r e c i p i t a t e t o t h a t of an o b l a t e spheroid growing w i t h c o n s t a n t shape, an e q u i v a l e n t d i f f u s i o n c o e f f i c i e n t was c a l c u l a t e d . The v a l u e o b t a i n e d was i n good agreement w i t h measurements o b t a i n e d from d i f f u s i o n c o u p l e s . The n a t u r e o f the quenched f£ phase was a l s o examined u s i n g o p t i c a l and e l e c t r o n m i c r o s c o p y . The (?> phase t r a n s f o r m e d r a p i d l y on c o o l i n g , f o r m i n g a massive ^ p r o d u c t or an a c i c u l a r m a r t e n s i t e a t h i g h e r quenching r a t e s . The s t r u c t u r e s were v e r y s i m i l a r t o those r e p o r t e d f o r the Cu-Ga and Cu-Al systems. Many g r a i n boundary p r e c i p i t a t e s showed unequal growth i n t o the two m a t r i x g r a i n s . Measurements o f the m a t r i x h a b i t plane suggested t h a t a p o s s i b l e o r i e n t a t i o n r e l a t i o n s h i p e x i s t e d between the p r e c i p i t a t e and t h a t g r a i n i n t o w h i c h no development o c c u r r e d . I n t h i s case the p r e c i p i t a t e n u c l e a t e d i n one g r a i n b u t grew  i n t o the o p p o s i t e g r a i n . P r e c i p i t a t e s w h i c h developed e q u a l l y i n t o b o t h g r a i n s e x h i b i t e d no apparent h a b i t relationship with either grain.  GRADUATE -STUDIES  F i e l d of Study: M e t a l l u r g y M e t a l l u r g i c a l Thermodynamics Metallurgical Kinetics S t r u c t u r e of M e t a l s Diffusion Nuclear M e t a l l u r g y  C . S . Samis E, P e t e r s E. T e g h t s o o n i a n L o C . •Brown W.M-. Armstrong  R e l a t e d F i e l d s of Study: S t a t i s t i c a l Mechanics Nuclear Physics Thin Films Mathematics  R.F, S n i d e r J.C. G i l e s L. Young E. Macskasy  PUBLICATIONS  G r a i n Boundary P r e c i p i t a t i o n i n Ag - 5.64 A l a l l o y s T r a n s . AIME., i n p r e s s  wt.%  PHASE TRANSFORMATIONS IN THE SILVER-ALUMINUM SYSTEM  by  EDWARD BRUCE HAWBOLT B.A.Sc., University of British Columbia, i960 M.A.Sc, University of British Columbia, 196^  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in th§ Bapartmtnt of METALLURGY  Wt aeeipt t h i s t h e i i s as eenfGriming to th© required standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1 9 6 7  In  presenting  for  an  that  advanced  the  I  thesis  for  Department  of  Date  8,  the  it  that  of  freely  of  November  thesis  for  British  Columbia  1967  It  is  financial  of  British  available  permission for  permission.  1,  of  p u r p o s e s may be g r a n t e d  this  Canada  fulfilment  University  Metallurgy  The U n i v e r s i t y  Vancouver  partial  b y h.i.'s r e p r e s e n t a t i v e s .  my w r i t t e n  Department  in  make  agree  scholarly  or  at  shall  further  publication  without  thesis  degree  Library  Study.  or  this  for  the  Columbia,  I  reference  and  extensive  by  the  requirements  copying of  this  Head o f my  understood that  gain  agree  shall  not  be  copying  allowed  i ABSTRACT The formation of grain boundary precipitates of the high temperature  phase from the supersaturated o< phase has been examined  in Ag-5.6^ vt.fi) aluminum alloys at 688°C . Large grained samples were used and the boundary misorientations were determined by X-ray diffraction. At low angle boundaries only primary sideplates formed while above a misorientation  of 17° lenticular precipitates were dominant.  Precipitate  growth was studied on individual grain boundaries using a statistical technique.  The lengthening and thickening rates were independent of the  grain boundary misorientation  indicating that grain boundary diffusion  was not significant under these conditions.  The precipitates grew with  constant shape, with both the length and thickness increasing parabolically with time.  By approximating the shape of the precipitate to that of an  oblate spheroid growing with constant shape, an equivalent diffusion coefficient was calculated.  The value obtained was in good agreement  with measurements obtained from diffusion couples. The nature of the quenched ^> phase was also examined using optical and electron microscopy. cooling, forming a massive ^ quenching rates.  The ^ phase transformed rapidly on  product or an acicular martensite at higher  The structures were very similar to those reported for  the Cu-Ga and Cu-Al systems. Many grain boundary precipitates showed unequal growth into the two matrix grains.  Measurements of the matrix habit plane suggested that  a possible orientation relationship existed between the precipitate and that grain into which no development occurred.  In this case the preci-  pitate nucleated in one grain but grew into the opposite grain.  Precipitates  which developed equally into both grains exhibited no apparent habit relationship with either grain.  ii  ACKNOWLEDGEMENT The author wishes, to express his sincere appreciation to Dr. L. C. Brown for his direction and encouragement given throughout the period of study and during the preparation of this thesis.  Thanks are  also extended to Dr. D. Tromans for his assistance with the electron microscopy and to the faculty, staff and graduate students of the Metallurgy Department for many.helpful discussions. Particular thanks are extended to Mr. R. Ziccarelli and Mr. R. Olsen of The Boeing Company, for supplying microprobe composition traverses for the diffusion couples. Financial assistance was received in the form of a Steel Company of Canada'Fellowship, 196^ to 1965, and a National Research Council Studentship, .1965 to 1967.  iii TABLE OF CONTENTS Page I  GENERAL INTRODUCTION -A. Significance of Project and Choice of System B. The Organization of the Pertinent Theory  II  THEORY OF PHASE TRANSFORMATIONS  3  Mh  B. Nucleation Theory  6  (i)  Homogeneous Nucleation  (ii)  Heterogeneous Nucleation  7 13 .15  (i)  Diffusion Controlled Growth  17  (ii)  Stability of a Growing Shape  23  PRECIPITATE MORPHOLOGY A. Widmanstatten Precipitates  25 25  (i)  Thickening of Widmanstatten Plates  27  (ii)  Lengthening of W-idmanstatten Plates  28  (iii)  Shear Theories Applied to Widmanstatten Growth.  •B. Grain Boundary. Allotriomorphs (i)  .30 30  .Kinetics of Nucleation of Grain Boundary . Allotriomorphs  IV  1  A. Driving Force for a Precipitation Reaction  C. Growth in Nucleation and Growth Reactions  III  1  32  (ii)  Thickening of Grain Boundary Allotriomorphs ...  33  (iii)  Lengthening of Grain Boundary Allotriomorphs ..  3^  (iv)  Shear Theories as Applied to the Nucleation of Grain Boundary Precipitates  36"  MASSIVE AND MARTENSITTC REACTIONS A. Massive  or SRD Reactions  38 38  iv Table of Contents (Continued)  Page  B. Massive (Lath) Martensitic and Acicular Martensitic Reactions ........................................... V  A5  EXPERIMENTAL A. Details of Ag-Al Binary System  .k^  B. Choice of Experimental Procedure ....................... C.  Preparation of Samples for Kinetic Measurements  D. Details of Kinetic Experiments E.  VI  Preparation of Diffusion Couples to Determine D as a Function of the Solute Concentration  ^7 ^7 .1+9  F.  Preparation of Samples for Examination of the Quenched $  G.  Preparation of Samples for'Matrix-Precipitate Orientation •Relationship and Habit Plane Analyses 51  . RESULTS A.  a B 6 B O B S B « i » B 0 » . » a a a a u < » a a a B B u a u s B U u u 6 < i B a a B B B B a » B a » a 0 e a » 9 B < >  Precipitate Morphology  53 53  B. Nucleation Kinetics  56  C . Growth Kinetics  60  (i)  (ii)  Statistical Treatment a) Lengthening ............................ b) Thickening ............................  60 68  Experimental Lengthening and Thickening Kinetics ..................................  71  D. Diffusion Couple Data ................ ..................  80  E.  8^  Structure of the Quenched ^> Phase .....................  F. -Matrix-Precipitate Orientation and Habit Analyses ...... .91 VII  DISCUSSION ....................................................  .97  A. Morphology  97  B. Nucleation  97  V  Table of Contents (Continued) C.  Growth  D. Massive and Martensitic Transformations of the ^ Phase.. E. . Matrix-Precipitate Orientation and Habit Relationships.. VIII  CONCLUSIONS  IX  APPENDICES I  Composition Analyses  II  Theoretical Precipitate Size Distributions  Page 99 106 108 110  112 llA  III Compilation of Kinetic Data  116  IV  Diffusion Couple Data  125  V  Order of Magnitude Calculations on Grain Boundary C r i t i c a l Nucleus Formation  VI  Calculation of Composition Contours for:  131 136  A) Bipolar Coordinate System X  B) Oblate Spheroid Shape Approximation BIBLIOGRAPHY  ihO  vi LIST OF FIGURES No. 1.  Page General binary diagrams exhibiting precipitation reactions  .1  2.  Binary diagram exhibiting a proeutectoid reaction  k  3.  Schematic diagrams of free energy vs composition for the and *]f phases of Figure 2. Free energy of a spherical nucleus as a function of i t s radius  8  C r i t i c a l nucleus size as a function of the degree of supersaturation, where T -^ T -^' Tequilibrium  8  Schematic drawing of the volume strain energy as a function of the axial ratio  9  Schematic representation of three types of interphase boundaries  11  8.' a) b)  Lenticular precipitate developing at a two-grain interface. .Interfacial energy balance at t i p of precipitate  1^  9.  Precipitation reactions and resulting composition profiles normal to the developing interphase boundaries  17  10.  Growth.rates for planar, cylindrical and spherical precipitates  21  11.  Relationship between the dimensionless supersaturation, f, and the squared reduced radius, X I , of spheroids having various aspect ratios  22  Schematic diagram of the ledge-type thickening mechanism for Widmanstatten plates  27  13.  Relationship between curvature and supersaturation  29  Ik.  Allotriomorph t i p shape as determined by the balance of interfacial energies and showing Hillert's postulated growth direction .. ;  35  Structure of massive transformation product in Cu-23 .87 at. # Ga . . .  39  Internal structure of massive product in Cu-23.87 at.•$ Ga, showing stacking faults . . .?  k0  h. 5.  ,  1  6. 7.  12.  15. 16. 17.  2  Massive martensite in brine-quenched Fe-7-5 "t.$> Cu w  5  kl  vii List of Figures (Continued) No.  Page  18.  Acicular martensite in  19.  Ag-Al binary diagram  20.  Variation of precipitate shape with grain boundary misorientation, R  21.  Cu-23.87  at. $ Ga  k-2  k-6  Results of two-surface trace analysis for the Widmanstatten habit plane  22. 25.  2k, 25.  5^  55  .5-Dimensional shape of a typical grain boundary allotriomorph 5 7 Effect of aging time at 688°C on the number of nuclei per unit boundary length Effect of aging temperature on the final number of nuclei formed per unit boundary length  58 59  The effect of boundary misorientation on the final number of nuclei formed per unit length of boundary  6 l  26.  Micrographs illustrating precipitate growth  27.  (Maximum precipitate length)  28. 29.  Typical population distributions of the precipitate length . 6 5 Effect of different nucleation times on the resulting precipitate size after k and 8 minutes aging 66  30.  Theoretical population distribution for the length o f a single-sized precipitate randomly sectioned  2  62  vs the precipitation time .... 6 5  67  2  31.  (Maximum precipitate width) vs the precipitation time  69  32.  Typical population distributions for the precipitate width..  70  33.  Theoretical population distribution for the width o f a single-sized lenticular precipitate (axial ratio 1 . 8 ) randomly sectioned  72  Typical growth data obtained at reproducibility of the results  73  jk.  35. 36.  688°C,  Growth data plotted as (parameter size) precipitation time  showing the vs  Single immersion precipitate growth compared to interrupted aging data  7^ 76  viii List of Figures (Continued) No.  Page  37- ' 38. 39'  Rate constant K (x = K J~t") vs the misorientation parameter, R  77  Axial ratio of the precipitate/length\vs the boundary misorientation, R ^width '  78  Effect of temperature on the growth kinetics of an 18° misorientation boundary  79  hO.  Composition profile from the diffusion couple prepared at 688°C. 8l  kl.  Diffusion coefficient vs the solute composition  82  k2.  Log. D vs 1 for 5.56 wt.$ aluminum T(°A) Structure of the ^> phase, water-quenched from 688°C  83  . kk.  -Lattice parameter of.the j£ phase extrapolated to the composition of the ^ phase, compared with the experimentally obtained values  Qk  86  k^>.  Structure of the brine-quenched (2> phase  87  k6.  Fine structure of the water-quenched product  89  kj.  Fine structure of the brine-quenched product  $0  k8.  Development of the massive original ^ grain boundaries  91  k°,.  product with respect to the  Typical grain boundary precipitates and details of diffraction patterns obtained  50. a)  •b)  c)  Results of trace analyses showing the poles of a l l possible planes in the cx matrix grain adjacent to the flat precipitatematrix boundary which could give rise to the flat interface ... .9^ Results of trace analyses showing the poles of a l l possible planes in that o< matrix grain adjacent to the curved surface of the precipitate which could give rise to the flat, undeveloped matrix-precipitate interface  95  Results of trace analyses showing the poles of a l l possible planes in the massive ^ product which could give rise to the flat matrix-precipitate interface  96  m  51.  93  -Composition profiles associated with a) a lenticular precipitate, and b) an oblate spheroid  100  ix  LIST OF TABLES Page Table I  Nucleation models and c r i t i c a l parameters .........  .12  Table II  Thermally activated growth reactions  16  Table III  Morphology classification and relationship to boundary misorientation  26  Table IV  Diffusion coefficients calculated for specific growth shapes as compared to the measured value ..  10k  Table V  Diffusion coefficients calculated from the data obtained at 700, 688, 682 and 67^°C, assuming a single surface oblate spheroid shape approximation and including the effect of inaccuracies of 'tl°C and'±0.01 wt.# aluminum  105  1 GENERAL INTRODUCTION A.  Significance of Project'and Choice of System Precipitation reactions can occur when a single phase material i s  heated or cooled to a temperature at which a second phase becomes stable. The temperature changes lead to a supersaturation of the matrix which is i  removed by precipitation of the second phase.  The general binary  diagrams shown in Fig..1 illustrate these conditions, a)  -p  b)  Composition  Composition Figure 1. General binary diagrams exhibiting precipitation reactions.  2 Type a] and Type b] are of technological importance, being comparable to the proeutectoid  steel reaction and the age hardening reaction  respectively. Precipitation produces a phase having a different composition, requiring atom transport, i.e., diffusion.  If the temperature is rela-  tively high and diffusion rapid, the equilibrium products may form, the proeutectoid  steel reaction being an example.  are low and supersaturation  However, i f the temperatures  is large, a randomly distributed, very fine,  non-equilibrium precipitate, characteristic of a transitional development process may result.  This phase may not be stable and may approach equi-  librium composition i f further diffusion occurs. the Al-k.^  The aging products in  Cu system are typical of this type of reaction.  This thesis  w i l l deal primarily with equilibrium precipitation. In slowly cooled steels the nature and distribution of the proeutectoid phase is known to affect the final mechanical properties. Hence an examination of the nucleation and growth mechanisms controlling this reaction product would be of considerable importance.  This would  involve 1]  establishing nucleation conditions, requiring an examination  of nucleation sites  and parent-product orientation relationships with  respect to the precipitate shape, and  2]  establishing the growth kinetics of the second phase.  However, in the Fe-C binary alloy i t is not possible to retain the parent ^ .austenite, to room temperature and the transformation is irreversible. Thus no room temperature examination of the matrix phase with respect to the precipitate is possible.  ,  3 Several non-ferrous systems exhibit  temperature-composition  relationships in which the matrix becomes supersaturated on heating, Type c] reactions.  In such systems the transformation temperatures are sufficiently  high to ensure an equilibrium transformation product and there is no problem in retaining, the matrix to low temperatures.  Examination of precipitation  in such a system should produce results which are generally applicable to equilibrium phase precipitation and in particular to the steel proeutectoid reaction. The Ag-rich end of the Ag-Al system, was chosen for this investigation for the following reasons: 1]  The binary exhibited an f.cc. to bee. transformation comparable to  the Fe-C system and thus involving very simple crystal structures. .2]  Low vapour pressures could be expected from both components at  the relatively high transformation temperatures and thus surface composition would be maintained during precipitation. • B.  The Organization of the Pertinent Theory The necessary background theories are organized such that nuclea-  tion and growth relevant to solid state transformations are treated f i r s t . Precipitate morphology is then considered and precipitates are classified with respect to shape and nucleation site.  The nucleation and growth of  grain boundary precipitates follows, applying the general theories as they relate to this specific morphology. A f i n a l section is included outlining the characteristics of massive and martensitic reactions as they apply to the properties of the quenched precipitating phase.  .k  II  THEORY OF PHASE TRANSFORMATIONS A. Driving Force for a Precipitation Reaction'*' The driving force for a l l phase transformations is the difference  in volume free energy between the i n i t i a l and the final state of the system.  The isobaric proeutectoid reaction shown in Fig. 2 can be used  as a specific example.  General precipitation from supersaturated solid  solutions would have a comparable form.  -l  -  To  -  X  X' Composition- X.  Figure 2. Binary diagram exhibiting a proeutectoid reaction.  At T^ the free energy of the ]f phase w i l l l i e below that of the phase, as shown in Fig..3a, and hence the 9 phase w i l l be stable over this composition range. At Tg however, the common tangent to the «=< and ^ free energy curves corresponds to a minimum free energy as shown in Fig. 3b. Thus a mixture of ^  of compositionX  and Q of composition X w i l l have the  Composition X.  G  Composition X^  Figure  Schematic diagrams of free energy vs composition for the e< and }f phases of Figure 2. a) b)  At Tx At T  2  lowest free energy and ©< w i l l begin to precipitate with the resulting phase composition changing from X"*" to X . The total volume free energy change associated with the reaction is given by the free energy difference between the i n i t i a l and the final state, i.e., the distance BC.  As the  supersaturation increases or the precipitation temperature decreases^it can be seen that the magnitude of this free energy change increases. :  The volume free energy change per atom of the precipitating phase is greater than that for the matrix phase^and is given by the distance AE . For the f i r s t particles of precipitate which form this value is s t i l l greater, AF, being obtained by constructing a tangent to the surface at X . 1  The value w i l l tend towards zero in the f i n a l stages of precipitation, AE being an average value. • Although the thermodynamic treatment is effective in explaining experimental observationsjthe actual energy changes involved in solid state reactions are not easily obtained and in most systems are unknown. B.  Nucleation Theory The transformation path of any activated process is dependent on  the type of fluctuation that initiates the reaction.  Gibbs"'" has emphasized  that two types of fluctuations exist: 1]  Fluctuations small in extent but large in magnitude; this normally  includes a i l reactions which involve a nucleation process. and  2]  Fluctuations small in magnitude but large in extent; this  includes reactions such as spinodal decomposition where the initiation of the transformation involves a developing wave-like composition variation and as such no classical nucleation.  7 Only transformations of type 1], as applicable to equilibrium precipitation from a supersaturated solid solution or proeutectoid reaction are  examined in this thesis. 1]  Homogeneous Nucleation Nucleation theory begins with the concepts developed by Volmer  and Weber  •3  k  and extended by Becker and Doring.  The nucleus is considered  a very small, spherical volume of material, having similar properties to the equilibrium phase.  The formation of the nucleus creates an interface  between the matrix and the precipitate the surface energy of which hinders the transformation. For very small precipitates the increase in surface energy i s greater than the volume free energy decrease and the embryo is unstable and w i l l go back into solution.  However, i f the fluctuation  producing the nucleus is large enough to produce the minimum c r i t i c a l size, r*, shown in Fig..k, the volume free energy term w i l l dominate and the particle w i l l grow spontaneously. The energy required to form the nucleus i s supplied by thermal fluctuations.  Assuming that the c r i t i c a l nuclei form by a reversible  process the number of nuclei at any instant having the c r i t i c a l radius r * is given by  where N  Q  o i s the number of atoms in the system and AG  is the c r i t i c a l  free energy for nucleus formation. The interfacial energy i s relatively insensitive to temperature and the volume free energy has been shown to be a function of the degree of supersaturation.  The c r i t i c a l radius, r , is inversely proportional to the  volume free energy change and consequently i t s size w i l l increase as the degree of supersaturation decreases, as shown in Fig. 5-  8 Interfacial energy terra  Volume free energy Figure k. Free energy of a spherical nucleus as a function of i t s radius.  Figure 5. C r i t i c a l nucleus size as a function of the degree of supersaturation, where T - * - T p - » - T^> —»-T -i.-u • ' equilibriumn  1  2  3  Cohen and Turnbull et a l have generalized nucleation to solid 5  6  state transformations by including a strain energy term to account for the difference in the specific volume between the parent and the product phase. Nabarro  has shown that this term is markedly dependent on the precipitate  shape as shown in Figure 6.  Figure 6.  Schematic drawing of the volume strain energy as a function of the axial ratio, where f (b/a) is a function of the axial ratio (b/a), of a precipitating oblate spheroid and is proportional to the strain energy. (Nabarro )  The strain energy is a maximum for a spherical precipitate and a minimum for a disc shaped product.  Therefore, there i s a preference for  plate shaped precipitates when there is a large difference in specific volume between the matrix and the precipitate. When both phases have  10 comparable volumes spherical precipitates w i l l be favored as the specific surface area i s much smaller than for the disc shaped form. A precipitate forming from a supersaturated solid solution generally has a crystal structure different from that of the matrix. In such cases i t i s commonly observed that the phase tends to form with a crystallographic plane parallel to a closely matching plane of the matrix^. The surface energy, is reduced by.maintaining a coherency between the phases. In the early stages of nucleus development where the interfacial energy term predominates, coherency effectively lowers the c r i t i c a l nucleus size thereby aiding the nucleation p r o c e s s ' ^ ' ^ The atomic spacings in the matching planes are not generally perfect and some elastic strain may be required to accommodate the coherency. Where the atomic f i t i s less perfect some elastic strain and/or dislocation structure may be present.  Fig.. 7 illustrates the planes of closest f i t  for three common interphase boundaries. The coherent precipitates have a high strain energy which i n creases as the precipitate size increases. .As a result, coherency i s normally associated only with the early stages of nucleation where the precipitate is very small. The classical model for nucleation considers the c r i t i c a l nucleus to have the equilibrium composition and to simply grow.in size."^  Another  approach by Borelius"'""'" assumes that the nuclei start at some final size Ik  and concentrate themselves.to reach the stable composition. Hobstetter combined these approaches and allowed both composition and size to vary. A specific value for the volume free energy change for each size and composition is obtained, .that combination which yields a minimum free  11  matrix nucleus matrix nucleus matrix nucleus  Figure 7-  Schematic representation of three types.of interphase boundaries. a)  Represents an incoherent interface containing a maximum of dislocations and no gross elastic strain.  .b)  Represents a semi^coherent interface containing some dislocations and some strain. ;  c) .Represents a coherent interface maintained by elastic strain and no dislocation structure. ("Hobstetter )  energy increase being preferred.  Near equilibrium temperatures, .i.e.,  small supersaturations, the nucleus :is specialized, approaching the shape and equilibrium composition, of the classical nucleus.  At high super-  saturations the nucleus becomes very diffuse, exhibits a composition not too different from the matrix phase, and thus is similar to a spinodal form. The mathematical formalism and the c r i t i c a l values derived from each of the theories are compared in Table I.  TABLE I Nucleation.Models and C r i t i c a l Parameters  Free Energy Change for Transformation  Model 1) .Classical Theory -surface and volume energy considered  AG=4TT  r cr- + krfr — 2  3  AG*= 16CT3 AG  AG v  3  3 2) Cohen -surface, volume and strain energy considered 7  3) Fisher, Hollomon and Turnbull -surface, volume and strain energy considered 8  Critical Nucleus Size  r* = -2Cr-  2  V  AG=^-TT ac A + 2"rTa a-+ 2  C r i t i c a l Free Energy Required for Spontaneou 3 Growth  ITa c AG  2  3  2  3  V  AG*= 32TTA cr 3 AG* 2  3  c* = -2CT AG a* = UACT AG  v  V  2  AG=2Tfrfo>+ Tfrft A G + CTTr t y  0  2  AG* =8192 T f c c r 2  27  2  A G *  3  t* = -16 <T  3AG  v  t* = -AG At low supersaturatioiis the nucleus r* Uci s equivalent to the clas5sical nucleus, approaching Becker's 1todel 10  Y  k) Hobstetter -size,composition change as related to the volume free energy term and surface enerj included 9  AG=  - n i l [fiG (x) v  - G (Y) V  (Y-X)|G (x)l > A G v  +  s u r f  •  AG*= 4k a?  a* = 2k C T  1  2  1  V  27.Q AG r i s the radius of a lenticular disc t i s the thickness of a lenticular disc c i s proportional to the shear modulus of the n i s the nucleus size parent phas X i s parent phase composition Y i s the variable embryo composition -Q i s the atomic volume 2  is the surface free energy AGy is vol. free energy change A. .is a strain constant related to the axial ratio c/a k's are shape constants  3-QAG  D  2  !5 ii]  Heterogeneous Nucleation Homogeneous nucleation does not commonly occur in phase trans-  formations and in fact i t is usually necessary to evoke elaborate practices 13 Ik  to ensure i t s occurrence. ^'  In solid state transformations structural  singularities such as external surfaces,,impurities, grain boundaries and dislocations act as preferential nucleation sites.  The removal of the  high energy regions associated with these sites provides an extra term tending to drive the transformation.  The c r i t i c a l free energy.is therefore  less for heterogeneous nucleation and thus i t is the more likely process to occur. The role of grain boundaries in assisting n u c l e a t i o n " ' " ^ i s of direct importance to this thesis. Clemm and Fisher"'"^ have examined the nucleation rate of a second phase on 2,.3 and A grain intersections of the parent phase.  They assumed the precipitating particle to have  symmetrical, lenticular surfaces and dihedral angles controlled by an interfacial energy balance, as shown in Fig. 8.  The surface energy term  included a summation of the surfaces produced minus the grain boundary area consumed. Thus the nucleation energy required .is reduced by a term corresponding to the grain boundary energy removed. This approach establishes that the nucleation energy decreases as the number of grain boundaries defining the intersection increases. Hence the effectiveness of a site increases in going from a grain interior to a grain, boundary to a three grain intersection. However, as the complexity of the site increases the number of such sites per unit volume l8  decreases..  In a transformation where a large volume of the parent phase  is transforming, .extensive nucleation is normally observed along 2-grain  Ik  a)  V  if.  Figure 8. a) Lenticular precipitate developing at a two-grain interface, b)  Interfacial energy balance at tip of precipitate. (Clemm and Fisher  )  where Q i s the interfacial energy.  intersections as well as the more complex sites. Grain boundaries may.also act as preferential sites due to accelerated diffusion along these paths.  The more rapid diffusion could  permit greater transformation per unit time. Grain boundaries are also sinks for,impurity and vacancy segregation, both of which may enhance nucleation. As. grain interfaces are considered regions of high misfit, transformation stresses may.be reduced by'localized plastic flow or accommodation of strain^thereby assisting the. transformation reaction. In summary, the nucleation theory describes a fluctuation process whereby a nucleus having, the c r i t i c a l size may be formed. As such, i t i s  15  most effective in determining time zero, that time after which growth •19  occurs.  For. this reason.it is usually more useful to focus attention  on the growth aspects of precipitation as this is a parameter more amenable to measurement. - C.  Growth in Nucleation and Growth Reactions In precipitation from a supersaturated solid solution, growth  of the second phase occurs by an atom transfer across the interphase boundary.  It has been useful to classify growth processes by the nature of  the atomic transfer to the growing phase.• Table II illustrates the broad range of thermally activated growth reactions. The discussion .is restricted to continuous reactions associated with long-range atom transport. A continuous reaction.is one that proceeds simultaneously .in a l l parts of 20 the system although the rate 'may vary from one region to another. There are two types of continuous transformation depending on the mechanism of the interface movement. Reactions which are interface controlled have a slowly moving interphase boundary such that the interface advances into a matrix of essentially constant composition.  Thus the rate  of the process controlling growth is independent of the interface position and hence the time.  o° o  x (the boundary ..movement)  t (time)  Of greater importance are those reactions in which the boundary mobility, i s high compared ,to the diffusion rate and the boundary can move as rapidly.as the required segregation can occur.  From nucleation theory  i t i s probable that the early stages of nucleation may involve coherency with the matrix and an interface controlled growth. However, with increase  TABLE I I C l a s s i f i c a t i o n o f Thermally. A c t i v a t e d Growth P r o c e s s e s Thermally. A c t i v a t e d Growth Long-Range  No Long-Kange T r a n s p o r t (Interface Controlled) Polymorphic T r a n s i t i o n s  Recrystallization, G r a i n Growth  Continuous Reactions  Interface  Controlled  •Precipitation, .Dissolution  Transport  Order-Disorder  D i s c o n t i n u o u s Rxs ( D i f f u s i o n and I n t e r f a c e C o n t r o l )  Diffusion Controlled Precipitation, .Dissolution  Eutectoid Reactions  (Christian ) 2 0  Discontinuous Precipitation  in s i z e to precipitates of Measurable dimensions coherency would of neces }  be destroyed and diffusion controlled growth would be expected. 1  1  Diffusion Controlled Growth When a ^) particle develops in a supersaturated  - matrix the  following composition profiles normal to the interface ere present.  Coo a)  b)  Figure 9 .  Precipitation reactions and resulting composition profiles normal to the developing interphase' boundaries.  The growth of the precipitate i s in both cases dependent or. diffusion down the composition gradient. The kinetics thus depend cn diffusion in the supersaturated solid solution and the mass balance at th interface. Theoretical grov/th rate calculations have been made for several  i  18  p r e c i p i t a t e shapes c o n s i s t e n t w i t h t h e f o l l o w i n g assumptions: •1]  The d i f f u s i o n c o e f f i c i e n t D i s c o n s t a n t .  Since D i s normally  dependent on c o m p o s i t i o n t h i s r e s t r i c t s t h e t r e a t m e n t t o low degrees o f s u p e r s a t u r a t i o n where t h e c o m p o s i t i o n v a r i a t i o n and hence t h e v a r i a t i o n i n D i s small. 2] in  The p r e c i p i t a t e grows from a z e r o i n i t i a l s i z e w i t h no change  shape. .3]  diffusion .k]  'Each p r e c i p i t a t e i s i s o l a t e d and a f f e c t e d o n l y by i t s own field. The i n t e r f a c e c o m p o s i t i o n s  compositions.  a r e c o n s i d e r e d t o be t h e e q u i l i b r i u m  The d e v i a t i o n s r e l a t e d t o t h e i n t e r f a c e c u r v a t u r e would  o n l y be s i g n i f i c a n t when t h e p r e c i p i t a t e i s v e r y s m a l l .  5]  Changes i n volume per. atom as t h e phase grows a r e i g n o r e d .  6]  A l l i n t e r f a c e s are considered  incoherent.  S o l u t i o n s have been o b t a i n e d f o r growth o f p l a n a r , c y l i n d r i c a l and s p h e r i c a l s u r f a c e s by Zener  21  and Frank  22  2^5 and f o r e l l i p s o i d s by Ham.  As t h e s e s o l u t i o n s a r e o f c o n s i d e r a b l e importance i n t h e p r e s e n t work, one s o l u t i o n , t h e growth o f a p l a n a r s u r f a c e , w i l l be c o n s i d e r e d i n d e t a i l . In t h e example chosen, t h e s a t u r a t i o n c o n d i t i o n s and r e s u l t i n g c o n c e n t r a t i o n p r o f i l e have a form c o n s i s t e n t w i t h F i g u r e 9b.  The d i f f u s i o n  e q u a t i o n must be s a t i s f i e d f o r atom t r a n s f e r down t h e c o n c e n t r a t i o n g r a d i e n t i n t h e s u p e r s a t u r a t e d cK m a t r i x . i.e.,  "c^cfx.t)  =  r-v " ^ c 2  where c = £(x,1  19  Combining the x and t variables using ^\ =  X  Jot"  gives  - _A dc 2  = cTc  ^  2>A  2  and permits solution of.the equation by double integration.  The f i r s t  integration is carried out without limits giving  c) c • = -Ae  ax  i)  4  The second i s integrated between C = C«o i.e.,X  =  e  and *\  i.e.,  0  C = C =A  yielding (C . C „ ) = A j Substituting  ^  =  A  produces 2  fC  -  i.e., (C -  = 2 A j ^ e  dTj,  ) = JTTA (1 - erf 2.) 2  •This defines the diffusion gradient in the supersaturated phase. •A i s found using the fact that the concentration at the interface A = L i s C  S  (C - Coc) = <[rfA (1 - erf L)  2)  s  2  The mass balance across the phase interface equates the diffusion flux down the concentration gradient with the movement of the phase boundary, i.e.,  .Flux /  unit area  = -D  Flux permitting precipitate growth = (C  - C-, ) *b f. J t  where ^ i s the interface position, L =  ^-  and W = L at the interface  20 .Lfb~  .  (A£.)  i  = (c  -  s  C l  )  Substituting 1) and 2) into 3) gives L .. = L Jar ( i W e  F  where f = C c  s  s  - Cpo C  }  r  f  the fractional supersaturation.  l  From this equation a graph of L as a function of the supersaturation, f, can be constructed (Fig.10) and the interface growth would have the form x = L^Dt.  Solutions for the growth of a cylinder and a sphere can be  obtained in a similar manner and are included in the diagram. It can be seen that the growth rate w i l l increase in going .from the planar.to cylindrical to spherical case; this is expected since a spherically growing particle obtains i t s atomic flux from a larger solid angle.  The  growth rate w i l l also increase with increasing supersaturation since the composition gradient in the matrix is increased.  23 Ham  extended the solutions of Zener and Frank to cover growth  of ellipsoids including the important cases of oblate (rotation of an ellipse about i t s minor axis) and prolate spheroid (rotation about the major axis).  For oblate spheroids the growth of the major axis is given by  and the minor axis  Ct  = Tj^  JDt  Numerical evaluation  ofY/i and; Y[±  - Q  24  Horvay and Cahn  }  2  for varying axial ratios was made by  the results being shown in Figure'"11.' By knowing the rate  21  22 IO  I """I 1  2  ,I  1  1  IMH|  1  I  I  1  I I I  I I I Mil  1  IO " 1  10"2  CM  i CM H II  i i mil 10  F i g u r e 11.  10  10  ''  '  I  l  l 111 i l l  i  i  10  I 11  Ml  1.0  R e l a t i o n s h i p between t h e d i m e n s i o n l e s s s u p e r s a t u r a t i o n , f , and squared reduced radius,XI , o f s p h e r o i d s h a v i n g v a r i o u s / w i d t h \ ratios, length where ag_•= 1 r e p r e s e n t s the sphere, ai ag_ = 0 r e p r e s e n t s t h e p l a n e . oo r e p r e s e n t s the c i r c u l a r c y l i n d e r ai and  ..= L , where t h e growth r a t e o f t h e minor a x i s i s k 2  g i v e n by'X  Dt  24  (Horvay and Cahn  )  o f growth o f the minor a x i s and t h e a x i a l r a t i o , t h e r a t e o f growth of the major a x i s can be d e t e r m i n e d .  The p l o t a l s o i n c l u d e s the growth o f p l a n a r  and s p h e r i c a l s u r f a c e s . The o n l y o t h e r f o r m a l s o l u t i o n o f t h e d i f f u s i o n e q u a t i o n has been o b t a i n e d f o r growth o f a p a r a b o l i c t i p o f c o n s t a n t s i z e and c o n s t a n t growth 24  T h i s i s an i m p o r t a n t s o l u t i o n s i n c e i t may be used t o d e s c r i b e  velocity.  Widmanstatten and d e n d r i t i c growth. f =J TT* P where  Q,  '2D  erfc J p  The f i n a l e q u a t i o n has the form  = f r a c t i o n a l supersaturation  i s the r a d i u s o f c u r v a t u r e o f the t i p , and ~\l i s the t i p growth v e l o c i t y .  23  Hence the solution for a t i p growing with constant velocity yields a value for the product i f , but does not give a unique growth velocity. The smaller the radius of tip curvature the faster w i l l be the growth velocity. This effect i s known as the point effect of diffusion and can be rationalized in that as the tip gets finer the solute diffusing to the growing area is supplied from a larger solid angle of surrounding matrix. 1  This approach  w i l l be considered in slightly more detail in a later section on Widmanstatten growth. When a solution to the diffusion equation,  M p & ) = he dx  dt  does not exist i t i s common practice to use solutions to the steady state equation,ai .e^aLaplacees equation, as an approximation.  l(p'£) = o a* This latter form implies that there i s no change in composition with time, i.e.,  = 0.  ii]  Stability of a Growing Shape Mullins and Sekerka  26  examined the problem of shape stability  under diffusion controlled growth.  Their treatment included the effect of  surface energy which tends to maintain a smooth surface by decreasing the interface composition  (supersaturation) and hence the driving force.. This  effect is balanced by the diffusion field which tends to increase the instability of a surface to permit more rapid growth.  The treatment involved  applying a perturbation to the surface and determining whether i t would develop or decay.  The results, only applicable to the very early devia-  tions from a spherical shape and to small degrees of supersaturation,  2k  indicate that the shape preserving solutions of Frank  22  and Zener  21  are correct. 27  Shewmon  also examined this problem and concluded that i f inter-  facial equilibrium is maintained at an interface growing under diffusional control, the precipitate should tend towards a dendritic shape.  This  morphology i s seldom seen in high-temperature solid-state transformations and consequently he argues that an interfacial reaction must be rate determining and that growth should be linear with time until the precipitate is approximately one micron thick.  25  III  PRECIPITATE MORPHOLOGY A large range of precipitate shapes are seen in precipitation /28r29  from supersaturated solid solutions.. Dube  developed a classification 30  for the various forms which was amended slightly by Aaronson illustrated in Table III.  and I s  The system was developed for the proeutectoid  ferrite reaction but later work has shown these morphologies to be more general and applicable to a wide range of alloy systems and to precipita3 1 > 3 2  tion both from solid solution and proeutectoid  > 3 3 ) 3 4  reactions.  Only Widmanstatten and grain boundary allotriomorphs  w i l l be considered  in this text. •A. Widmanstatten Precipitates Widmanstatten precipitates are plate-shaped having a needle-like appearance when sectioned and viewed on a single plane of polish. The orientation of the precipitate i s related to the orientation of the matrix; in the f.c.c. austenite to b.c.c. ferrite proeutectoid 35  Kurdjumow and Sachs  reaction^the common  relationship (llDfcc/AllO^  )CC  ^ W / ^ c c is observed.  These are also the habit relationships in this transformation.  A comparable relationship has been reported for the Al-20$ Ag f.c.c. to h.c.p. transformation and has the form (iii)  // ( o o o n  fee''  [HO].  v  ;  hcp  // [1120].  fee''  hep 3T  This is.also the habit plane for the transformation.  Both are common  orientation relationships for the crystal structures involved.  It i s also  common for the plates to form on matrix planes of low index as in the cited  TABLE III Morphology Classification and Relationship to Boundary Misorientation Widmanstatten Grain Boundary Allotriomorphs Morphology  Grain Boundary Misorientation  Associated with large angle or disordered— type grain boundaries i.e., >.15° misorientation »  Primary Sideplate  Primary and Secondary Sawteeth  Secondary Sideplates  i, I  Associated with small angle or dislocation type boundaries and grain interiors  Idiomorphs  /I ^  Generally associated with intermediate misorientations 1 5 ° - 25°  Generally associated with intermediate misorientations 1 5 ° - 25 . 0  <15°.  /Clark > vDube and Aardnson 33  29  Usually an int ragranula r morphology although were observed for low angle grain boundaries in the Fe-l.55 Si system.  27' reactions but irrational habit planes have been observed  38j38  Widmanstatten precipitates are usually associated with a large grain size and a fair degree of supersaturation.  The shape does not 41  spheroidize with prolonged annealing at high temperatures.  In the case  of Widmanstatten Al^Ag in A l , the precipitates go back into solution in a sequence opposite to precipitation indicative of the inherent stability 34  of the morphology. plates in  .Transmission electron microscopy, of Widmanstatten  Cu -39-6$> Zn  revealed the presence of stepped boundaries  at both flat interphase boundaries and plate tips.  The facets corresponded  to variants of the irrational habit plane observed. i ] -Thickening of Widmanstatten Plates'4 0 Parabolic thickening is observed indicative of diffusion controlled growth, but the measured rates are much slower (X100) than predicted 41  by the planar growth model. Aaronson has adopted Smith's  concept that a  semi-coherent boundary has low mobility, to explain the Tower growth rate. He considers the .-boundary to have the structure shown in Fig. .12. Thickening Lengthening  disordered  coherent or semicoherent Figure 12.  Schematic diagram of the Ledge-type thickening mechanism for Widmanstatten plates. 40  (Aaronson  )  28  The disordered boundaries move freely and the lateral growth rate i s controlled by uncoupled atomic migration, i.e., volume diffusion in the matrix.  The coherent or semi-coherent areas are considered regions of  relatively high dislocation density and thus thickening would be dependent on a synchronized movement or climb of these defects.  This is not a likely  mechanism to occur and hence Aaronson proposes that the thickening rate depends upon: 1)  The rate of ledge formation.  2)  The height of the disordered edges.  .3)  The rate of lateral migration of the edges.  A parabolic thickening rate smaller than that representative of a disordered interface would thus be expected. ii]  Lengthening of Widmanstatten Precipitates Widmanstatten plates lengthen much more rapidly than they thicken.  Mehl and Barrett  proposed that the lengthening is related to the point  effect of diffusion at the edge of the growing plate, the crystallographic aspects only being significant during nucleation.  The shape of the nucleated  precipitate i s simply reproduced by the geometry of the diffusion f i e l d . Howeverjthis would not explain the inherent stability of the interface. The theory governing growth of a parabolic tip (Section IIC, Page 22) has been applied to Widmanstatten plates. This mathematical treatment predicts that the growth rate w i l l tend to ^ 6 as the tip radius 42  approaches zero, this being an unrealistic concept.  Boiling and T i l l e r  introduced the concept of surface energy, reducing the composition gradient in the matrix, which would thus limit the effect of curvature.  Although  44  this has been criticized by Paxton and Pound because i t includes only the  29  variation in composition of one of the binary constituents, i t is probably a fair approximation at low degrees of supersaturation.  The variation  43  of equilibrium matrix composition with curvature is given by: /composition at the\ C  I interface having  J = C  \radius r  /  ^quilibriumN •g  _  Composition/ 1 + J  J  p  where  P =t l RT  and  V  is the surface energy  V is the partial molar volume T the abs. temp. ^  is the total radius of curvature .  This would result in a decrease of the equilibrium concentration at the tip as shown in Fig. .13.  sr  Figure 13.  decreasing radius of curvature  Relationship between curvature and supersaturation for small degrees of supersaturation. (Boiling and T i l l e r ) 4 2  With a decrease in the composition gradient the diffusion rate w i l l also decrease.  The point effect of diffusion w i l l thus be balanced  by the surface energy effect resulting in a growing tip which has a constant radius of curvature. .A linear growth rate would be expected under these conditions, the equation being consistent rfith the qualitatively developed  30 Zener-Hillert treatment.  43  Measurements reported for Widmanstatten growth in Fe-.3$C and 40  Ti-Cr  are in reasonable agreement with the calculated values and linear  growth rates were observed. i i i ] Shear Theories Applied to Widmanstatten Growth Widmanstatten plates grow relatively rapidly and have an appearance similar to martensite.  Consequently, shear theories have been suggested 45 } 46  to explain this structure.  The fact that relief effects have also  been observed for both Widmanstatten ferrite and for Bainite lends support 4*7  to this approach.  Otte and Massalski  associated with Widmanstatten  observed that the flat interfaces  formation in the ^)*°^ brass transforma-  tion formed parallel to martensitic boundaries.  The habit plane calculated  using the phenomenological martensitic theory agreed with the irrational plane observed.  The calculated habit planes for the Fe N and Fe P forma4  3  tion in <^< iron were also close to the observed values. B. Grain Boundary Allotriomorphs 40  The type of precipitate formed at grain boundaries depends on the misorientation across the boundary. Aaronson  40  and C l a r k  34  have shown  that grain boundary misorientation can be characterized by the parameter where  =J  R = I f X+ f+ Y+ Z 2  2  2  and X, Y, Z represent the minimum rotations about three orthogonal axes required to bring the two adjacent lattices into coincidence.  Although  this parameter does not include the orientation of the grain boundary 33  i t s e l f , in both the Fe-Si only on the parameter R.  34  and Al-Ag  the precipitate shape was dependent  31 Allotriomorphs, the most common form of grain boundary precipio tate, occur at grain boundary misorientations  R } 15".  These boundaries  are considered disordered and comprise a large proportion of the boundaries in a random polycrystalline material.  48  The precipitate is lenticular in  shape being circular in the plane of the grain boundary and having a lensshaped cross section.  They form under low degrees of  supersaturation,  lengthen rapidly along the grain boundary and thicken at a slower rate. This normally results in impingement of separate allotriomorphs relatively short times.  after  This is usually the f i r s t precipitate form to  appear, indicative of the heterogeneous nature of the nucleation site. Qualitative observations by Dube  28  and Heckel and Paxton ' 3  1  indicate that growth rates increase with increasing supersaturation.  How-  ever, l i t t l e quantitative rate data has been obtained. Available information is inconclusive as to whether a crystallographic relationship exists between the precipitating phase and either of the matrix grains.  Hultgren and Ohlin  found that portions of a grain  boundary network of ferrite developed from separate allotriomorphs, were single crystalline, indicative of a parent-product orientation relationship. Ryder and Pitsch ° examined grain boundary precipitates in a Co-20$ Fe 5  alloy to determine i f a parent-product orientation relationship existed. Across the flat interfaces of particles similar in appearance to a sawteeth form, a Kurdjumow and Sachs orientation relationship was observed.  Pre-  cipitates exhibiting a curved interface^typical of allotriomorphs, bore the K. and S. orientation relationship with that grain into which no growth had occurred. >  Clark s  3 4  observation of precipitation of allotriomorphs  in  A1-2C$ Ag alloys indicated that under small degrees of supersaturation, nucleation took place in one of the two grains forming the boundary with a definite crystallographic relationship.  This effect was not constant over  long distances as the grain boundaries changed direction bringing other equivalent habit planes into effectiveness. The habit relationship can only be indicative of a semi-coherency in an observable precipitate.  However, this degree of coherency may affect  the resulting growth characteristics.  Interface migration would involve  movement of a semi-coherent, low mobility boundary fthat boundary at which the precipitate nucleated)and the disordered boundary (the boundary between the precipitate and adjacent matrix grain).  A precipitate developing under  these conditions would be expected to develop in an asymmetrical manner as growth would occur mainly at the disordered interface.  Clark observed  this effect but also noted other allotriomorphs which developed equally into both grains.  At later stages of growth both interfaces became mobile,  indicating a loss of coherency. At intermediate boundaries (R = Ik to 20°) allotriomorphs were observed which grew along specific crystallographic orientations initiated by a kinking in the grain boundary.  Growth was dependent on both a diffu-  sional atom transport and a continuous relocation of the boundary. i]  Kinetics of Nucleation of Grain Boundary Allotriomorphs The number of nuclei formed per second per unit area of unreacted  grain boundary, N , has been examined by Mazanec and Cadek  51  for a steel  of composition 0.35$C, .1.2$ Cr and 0.31$ Mo . They observed that N initially.increases with time, and then decreases.  g  Turnbull and F i s h e r  52  explained the maximum in that upon quenching the steel to a specific degree  33 of supersaturation a certain number of nuclei are potentially stable.  How-  everjdiffusion and hence time is required to produce the necessary atomic concentrations.  If an unlimited number of nuclei could form, a constant  rate of nucleation would be expected.  If only a limited number can form,  consistent with carbon enrichment and the heterogeneous nature of specific grain boundary nucleation sites, then a maximum would be expected.  Initially  the number of nuclei would increase with time due to the diffusion required and as the potential sites were consumed, fewer and fewer nuclei could form. The effect of temperature on N  g  w a s  also examined by comparing  i t at times when 2$ of the austenite had! transformed to ferrite, ignoring the time dependence of N . A marked increase in N  with increasing super-  saturation was observed. 53  Hickley and Woodhead  also examined the effect of temperature  and carbon content on the nucleation rate and noted that the rate of nucleation at a given reaction time increased rapidly with increasing supersaturation . These effects are consistent with nucleation theory in that the larger the supersaturation, the larger w i l l be the volume free energy available for the transformation and the smaller will.be the stable nucleus size, resulting in higher nucleation rates. i i ] -Thickening of Grain Boundary Allotriomorphs Thickening can be treated as a diffusion controlled reaction 21  using the solutions of Zener described in Section TIB.  , Frank , Ham , and Horvay and Cahn 22  23  24  For a meaningful thickness evaluation of a grain  boundary precipitate in bulk samples, i t is necessary to know the direction  3i+  of the boundary with respect to the surface of examination. . The kinetics of ferrite precipitation in bulk steel samples was examined by Mazanec and Cadek and Hickley and Woodhead , the direction 51  53  of the grain boundaries being neglected.  A statistical technique was  employed and a thickness cX ^time relationship established which was consistent with diffusion controlled growth.  The calculated D value agreed  within.a factor of 2 with the volume diffusion coefficient reported for the system. Aaron and Aaronson  54  examined the thickening of CuAl grain 2  boundary precipitates in the Al-k$> Cu system.  The maximum allotriomorph  width for a given reaction time was assumed to be the true thickness. The precipitates were found to thicken at a rate approximately proportional i to t . 3  The D values were calculated using Zener's planar model, and were  found to be several orders of magnitude greater than volume diffusion. Short circuit interfacial diffusion was invoked to explain the large diffusion coefficients observed. , i i i ] Lengthening of Grain Boundary Allotriomorphs Very.little experimental work has been reported concerning the lengthening rate of grain boundary allotriomorphs.  It is generally  accepted that the growth rate along the boundary is greater than that perpendicular to the boundary.  Dube  28  and Aaronson  40  measured the longest  ferrite precipitate resulting from the Fe-C proeutectoid reaction and found a linear relationship with time up to impingement. Aaron and Aaronson  54  Recent work by  using the maximum length criteria for lengthening of 1.  CuAl  2  in Cu-k'fc Al indicated the length varied as t. . Lengthening was  considered to proceed by volume diffusion of solute to the matrix grain  35 boundaries, followed by grain boundary diffusion to the advancing edge of the allotriomorph. The theory of growth of a parabolic shaped t i p , developed in Section IIC, has been applied to the lengthening of allotriomorphs.  Grain  boundary precipitates exhibit an infinitely sharp t i p related to the surface energy balance as described in Fig. 8, page Ik. Since the growth rate is inversely proportional to the radius of curvature (page 22) an infinite 43  growth rate at the t i p would be predicted.  Hillert  assumed that growth  was controlled by the curvature adjacent to the t i p . The growth rate in the direction AB of Figure Ik was given by the standard equation for t i p growth.  The lengthening rate along the grain boundary was then the growth  velocity.in direction AB x 1 . No attempt was made to obtain a formal sin 0  Figure Ik. Allotriomorph t i p shape as determined by the balance of interfacial energies and showing Hillert's postulated growth direction. 43  (Hillert  )  solution to the diffusion equation for this particular precipitate shape and no explanation was given as to why the growth should be controlled by the curvature adjacent to the t i p . Aaronson  40  made a preliminary test of  this approach using data for the Fe-0.6$C alloy and a radius of curvature measured by electron microscopy.  A f a i r agreement between the calculated  and observed growth rates was noted.  However, insufficient data is  available to truly test the validity of this approach. The work of Toney and Aaronson  33  on 1) precipitation in  Fe-1.5$ Si alloys indicates that grain boundary misorientation may also affect the lengthening kinetics. .An increase in the precipitate length width ratio was observed for increasing grain boundary misorientation.  However,  the morphology changed from grain boundary idiomorphs to allotriomorphs over the misorientation range examined and as a result the overall effect was not clearly defined. iv]  Shear Theories as.Applied to the Nucleation of Grain Boundary Precipitates Similar dislocation mechanisms and therefore similar crystallo-  graphic lattice relationships have been proposed for the i n i t i a l stages of certain martensitic and nucleation-and-growth transformations.  Otte  47  and Massalski  examined the habit plane of flat interfaces formed on  precipitating < = > < from  in a Cu -25.31$ Zn -7.23$ Ga alloy.  A common  habit plane for planar-boundaries in the equilibrium °< precipitate and martensite plates in the quenched Ryder and P i t s c h  50  phase was observed.  combined the martensitic nucleation approach  with the appearance of bee. intragranular and grain boundary precipitates in the Co-20$Fe f.c.c.alloy. As no evidence of internal structure was  37  observed in the precipitates the "second shear" requirement of the martensite theory was relaxed and hence no habit plane was expected.  A "region of  coherency" was calculated, applying the modified martensitic approach, and yielded a Kurdjumow and Sachs orientation relationship consistent with that observed.  38 IV  M A S S I V E ;AND M A R T E N S I T T C  If librium  a  grain  single  time  phase  retained  may  be  reaction.  composition  but  w i l l  Massive  dissociate  to  and  the  There diffusionless such  the  movement  grain but  is  the a  ,  or  an  at  the  to  prevent  show  no  evidence  of  s h e a r .  The is  shown  in  typical  Pig.  the  15.  processes  parent  7  the  ,  5  8  with  interface,  appearance  of  and a  the  not  and  of  may  of  a l l  different.  appearance) develops  impeded by stage  usually  and  relatively  the 5  9  free  short-range  ¥  equilibrium  rate  quite  generally  Massalski  massive  same  classification  is  or by  parent  martensitie  bears  sufficiently  diffusion  i.e.,  a  A  the  the  Reactions  of  structure  K i t t l  have  the or  metastable  are  (3RD)  is  occur,  massive  w i l l  the  nucleation  grain.  long-range 3  which  resulting  associated ¥  that  a  transformation  and  equi-  permitted.  concerning  boundaries  interface  undergo  is  that  diffusion,  considered  the  growth  i.e.,  approximating  diffusion  Although  thought  required  parent-massive  are  confusion  grain  is  the  structure  such  cannot  product  descriptive  rate  at  the  being  is  was  may  term  incoherent;  process  or  Diffusion  relationship  growth  case  i f  actual  incoherent  with  segregation,  products  form  rapidly  precipitation  Range  tion  the  latter  very  Short  (the  It  for  crystal  the  nucleates of  is  In  rapid  cooled  room t e m p e r a t u r e  considerable  boundaries.  growth  to  have  massive'  product  3RD  available  equilibrium  is  is  Widmanstatten  martensitie  Massive  A  alloy  transformations.  reactions  A.  or  being  martensitie  phase.  phase  boundary  insufficient  REACTIONS  fast  no  cooling  resulting concluded atomic  diffusion  transformation  orienta-  surfaces that  movement (SRD).  product  39  The crystal structure of massive £> i s equivalent to the nearest equilibrium structure, the lattice parameters being consistent with an extension of the equilibrium phase to the required composition.  56  When the alloy composition i s such that the quenched product is intermediate between two equilibrium phases, the massive form of either phase or a mixture of both may be obtained.  i n the hypoeutectoid  Cu-Ga alloys the microstructure consists of grains having many straight boundaries and surface striations.  Diffraction patterns of the striated  regions exhibited streaking and partial spot splitting which has been interpreted as being due to strain relief or diffraction effects associated with the presence of a duplex massive structure. Thin film electron microscopy ' 60  61  has shown that the typical  single phase massive £ product exhibits faulting as shown in Fig. 16.  This has been related to deformation resulting from the transformation volume strain, i.e., the difference between the specific volumes of the parent and the product phase.  The deformation is usually unevenly  distributed whereas the lattice invariant shear of a martensitie reaction produces a regular array of deformation markings. B.  Massive Martensite (Lath Martensite ) and Acicular Martensite Reactions 55  64  Martensitie reactions occur at faster cooling rates than required to form massive £  and involve a true shear process resulting in the  appearance of surface shear markings. gated the massive  In many of the alloy systems investi-  and martensitie reactions are competing processes.  By increasing the cooling rate the massive martensite or acicular martensite may r e s u l t .  5 7 , 5 9 , 6 2  This is dependent on the grain size and  composition  of the parent structure.  The increased probability that a massive  reaction w i l l occur at small grain sizes has been related to the grain boundary nucleation of this phase, whereas martensitic reactions are much less affected by grain size.  Figure 17.  Massive martensite in brine-quenched Fe-7.5 wt.# Cu.  x 160 63  (Lund and Lawson  Figure 17 shows a typical massive martensitic structure. phase boundaries are very jagged and irregular. Owen  64  )  The  examined the  internal structure of Fe-Ni massive martensites, finding only an appreciable density of tangled dislocations, indicating that the inhomogeneous shear is accommodated by slip. The more familiar acicular martensite, illustrated in Fig. l 8 a , generally consists of single or connected martensitic plates.  The internal  Figure  18.  A c i c u l a r martensite i n a) b)  Cu-23-87  a t . $ Ga.  O p t i c a l micrograph Transmission e l e c t r o n micrograph ( S a b u r i and Wayman ) 60  ^3 structure exhibits twinning for Fe-29 to jM> N i Cu-12# Al ° and Cu-2^ Ga 6  62  6 5  and stacking faults for  (Fig. l 8 b ) .  The crystallography of martensites has received much attention but because of i t s very specialized nature w i l l not be considered in this 66  text.  The theoretical approaches of Wechsler, Lieberman and Read  Bowles and Mackenzie  67  and  have been successful in predicting habit planes and  orientation relationships between parent and product phases in internally twinned martensites and have been very useful in massive martensite reactions where i t i s usually not possible to retain the parent phase. The calculation is based on the assumption that the interface between the parent and the product phase i s an invariant strain plane i.e., a plane that is undistorted (a small uniform dilation ~2# i s permitted in the Bowles and Mackenzie approach) and unrotated by the transformation. The following lattice changes are required to produce such a condition:  :  1)  A homogeneous lattice deformation which transforms the parent to the product phase.  The Bain distortion is normally used as i t represents  the smallest atomic displacements for a fee. to bjec. (hc.t.) transformation. 2)  .A simple shear e.g., slip or twinning, commonly termed the second shear which produces an undistorted or nearly undistorted "habit plane".  The shear i s required to change the vector lengths,.previously  decreased by the lattice deformation, to their original length. 3)  ..A rigid body rotation which returns the "habit plane" to i t s original position thereby f u l f i l l i n g the conditions of an invariant strain plane.  The requirements have been reduced to mathematical analogues permitting comparison of predicted and experimental data.  1^5  X  EXPERIMENTAL A.  Details of the Ag-Al Binary The phase diagram for the Ag-Al binary system is shown in Fig..19-  The  wt. % A l alloy has a single phase, f.c.c. structure over the  temperature range ^35 to 671°C ., On cooling below 1+35°C the o< phase becomes supersaturated but the precipitation of the ^-manganese p. phase i s very sluggish and the ©< phase i s normally retained at room temperature. On heating above 671°C the b.c.c. ^> phase precipitates in the °< matrix. Experiments were designed ,to determine the kinetics of precipitation of this phase. B.  Choice of Experimental Procedure Preliminary experiments were conducted in an effort to directly  observe precipitate growth.  Hot stage metallography was used and a highly  polished surface was heated to the precipitation temperature under an inert' gas atmosphere.  Although thermal grooving of the boundaries was  observed, the shape of the developing precipitates could not be followed. Several gaseous etchants - Br , HC1, were intermittently introduced to the 2  system to reveal the interphase boundary positions but this was unsuccessful. Specimens were immersed into high temperature salt baths and periodically withdrawn for surface examination. Although many salt mixtures were used the specimen surface was always stained and required some polishing prior to etching for precipitate examination. A statistical technique was finally adopted which involved the measurement of the dimensions of many precipitates lying on one grain boundary.  he  WEIGHT 1 1000  2  3  4  5  1 1 1 1i  6  7 8 9 10  :  i  i  i  12  i  CENT  PER  14 16  i  I  ALUMINUM  18 20 I  25  30  1  I  35  40  111  50  60  70 80 90  1  1 1  1  960.5°  900  '780°  800  17.84%* (5.15) 1  S727°  \ 0 /  700  £  20.34  600  16.0)  610°  r  <x Ag or (Ag)  \ \  660°  56°  X  \  .—  5 1  62.5 (29.5)  76.2\ (44.4)  a  500 19.92 (5.86).  (72)  448°  \  A  400  97.9\  (92.0)T"  >  \  300  1  M 200  100  0 Ag  Al or(Al)  N  8 . 7 5 / (2.34) /  1 1  10  1 1 >  20  1 1 1  99.2  • 1 1  30  40 A T O M I C PER  99.82 (99.3)  50 CENT  60 ALUMINUM  Ag-Al  Figure 19. Ag-Al binary diagram. (Hansen  )  1  (96.8)1  70  80  90  100 Al  k  7  C.  Preparation of Samples for Kinetic Measurements A master alloy of nominal composition 5-7 wt  Al was prepared  by melting fine silver of 99.95$ purity and aluminum of 99-999$ purity under argon and c h i l l casting.  Following homogenization at 600°C for 10  days, sections weighing approximately 50 gms. were remelted under.hydrogen and furnace cooled to 600°C at a rate of approximately 5°C/min. This procedure produced ©< grains up to l/2 cm. in diameter.  The alloy was  then annealed for 15 days at 600°C under a hydrogen atmosphere to ensure homogeneity and to straighten the grain boundaries. . The resulting alloy composition as determined by chemical analysis and comparison of the experimentally determined (©<  ) boundary with  the published phase diagram, was 5.6^ ± .01 wt.$ aluminum. The details are contained in Appendix I. Long, relatively straight grain boundaries were located by metallographic examination of the surfaces of the coarse grained material. .Test coupons l/k x l/k x l/8 inch containing at most four grain boundaries were spark machined from the bulk sample. Sections at right angles to the sample surface were polished to determine the trace of the grain boundaries and only those boundaries which were flat and essentially.perpendicular to the surface were selected for growth measurements. The boundaries usually exhibited a slight curvature which was not eliminated even after very extended aging times. Back reflection Laue photographs were obtained for each grain to establish the grain boundary misorientation. D.  Details of Kinetic Experiments The test samples were heat treated at the precipitation tempera-  h8  ture for a measured time^then quenched to room temperature.  After measure-  ment of the size of the resulting precipitates the procedure was repeated. Rather than observing the growth of an individual precipitate, the size of a l l precipitates on a specific boundary was examined after progressively increasing aging times. The samples were heat treated in a neutral salt (Houghton 300) controlled to t ±°Q by a steady state power input.  This salt was especially  suitable as i t has a very rapid heat transfer rate and causes minimal pitting or surface corrosion.  Specimens took from 3 to 5 seconds to reach  temperature and because of their small thermal mass had a negligible effect on the salt temperature  (^.0.1°C).  The specimen surface became stained during aging in the salt. A layer approximately 20 ^i thick was removed with 3/0  emery paper prior  to final polishing and etching for precipitate examination. .The following 6 8  etchant was used:  1 part  5 parts  6 20 12 .300  gm cc cc cc  h0 gms  380 cc  KC r 0 2  H S0 2  7  4  NaCl (sat.)  H0 2  Chromic acid  H0 2  Individual precipitates were measured at 250 or 500 x magnification.  Measurements were made directly on the viewing screen of the  Reichert metallograph, the screen width being used to determine the number of precipitates per unit length of grain boundary. Approximately 100 precipitates were observed for each boundary, this being considered sufficient for a statistical evaluation.  Specimens were normally aged  for a total of 10 minutes with five observations being made during this period.  Most of the kinetic measurements were made at a precipitation temperature of 688°C, representing a 7 $ supersaturation.  However, three  short term tests, conducted primarily to determine nucleation rates, were carried out at 7 0 0 , 6 8 2 and 67k C. 0  E.  Preparation of Diffusion Couples to Determine D as a Function of Solute Composition In order to check whether the precipitates were growing by a  diffusion controlled reaction, i t was necessary to determine values of the diffusion coefficient in the supersaturated C K phase. Diffusion couples were prepared using fine silver and a twophase A g - 6 . 6 5 wt.$> Al alloy and aged at temperatures of 5 0 0 , 55O, 6 0 0 , 65O and 688°C.  The 6 . 6 5 wt.$ Al alloy was prepared by melting under argon  c h i l l casting and homogenizing at 600°C for 1 0 days.  The sample surfaces  were polished to k/0- emery paper and were ultrasonically cleaned prior to bonding.  The couples were aged at temperature in an H  2  atmosphere in a  tube furnace. The controlling thermocouple was placed adjacent.to the couple and the maximum temperature fluctuation was found to be ~5°C.  One  sample was heat treated in a salt medium at 688°C as the temperature control in this system was -2°C over several days of annealing.  After  agingjthe couples were quenched, sectioned and polished to a 1 p. diamond finish.  The f i n a l polish was parallel to the isoconcentration lines, i.e.  parallel to the bonded surface, to minimize the effect of smearing on surface concentration measurements.  Composition profiles were measured  +  on an Applied Research  +  The Boeing Co. kindly consented to measure the composition profiles and corrected the Al Ko< line intensities to an equivalent wt.$> aluminum.  50  Laboratories (ARL) electron microprobe and the Aluminum Ke* line intensity was converted to an equivalent wt.$ using the Ziebold-Ogilvie correction factor.  69  Preparation of Samples for Examination, of the Quenched S Phase  F.  The kinetic measurements were obtained using an interrupted annealing technique in which the sample was repeatedly immersed in salt at the desired temperature and after a measured length of time water— quenched to room temperature.  I n i t i a l experiments indicated that the  matrix phase was retained to room temperature but i t was necessary to establish the nature of the quenched ^  phase.  Samples of the 6.65 wt;$ Al alloy (preparation described in Section_V, E) were aged at 688°C for 1 hour.  At this temperature the  alloy is 95$ ^? having a composition equivalent to the grain boundary precipitate examined in the kinetic analysis. After aging^the samples were water-quenched to room temperature and the structure was examined using metallographic techniques and wide angle X-ray diffractometry. •Sheet specimens of the 6.65 wt.$ aluminum alloy, 0.010" thick, were prepared by hot rolling at 550°C.  After quenching from the  phase  and electropolishing the resulting surface, the sample was placed in the Norelco Back Focusing Camera to obtain the lattice parameters of the quenched structure. Electropolishing was carried out in a solution of:'° 70$ Ethyl alcohol 20$ perchloric acid 10$ glycerine maintained at -20°C in an alcohol-dry ice bath.  51 The 0.010" sheet was also used as starting material for the preparation of thin films to determine the fine structure of the quenched ^> phase. A l l transmission electron microscopy was carried out using a Hitachi HU-11A Electron microscope at 100 KV. 7 1  using the Bollman technique,  Specimens were prepared  J 7 2  and the electropolishing solution previously  mentioned. G.  Preparation of Material for Matrix-Precipitate Relationship and Habit Plane Analyses  Orientation  It was necessary to prepare thin film specimens of the ^.Sh wt.$ aluminum alloy in order to examine the possibility that an orientation relationship or habit plane existed between the matrix and the precipitating phase.  The alloy preparation i s described in Section V^,C.  Using the  electron microscope for this purpose permitted the precipitate shape and crystal orientations to be obtained simultaneously.  As only a small area  can be examined in the electron microscope i t was necessary to decrease the grain size of the o< phase to increase the probability of observing a grain boundary allotriomorph  in a thin film section.  For this reason  sections from the original alloy were cold rolled with intermediate 15 minute annealing at 550°C to produce a starting sheet 0.010 inches thick having a grain size of approximately 50 p. Samples were aged at 688°C for 1 to h minutes, water quenched and the resulting^) precipitates examined. The shape was similar in appearance, being lenticular and exhibiting both single and double surface development. No intragranular Widmanstatten precipitate had formed. Thin film specimens, suitable for transmission electron microscopy, were prepared from samples aged 30 sec. at 688°C and water-quenched.  52  The Bollman technique was employed with the alcohol-perchloric acid electropolishing solution being used.  53  VI  RESULTS A.  Precipitate Morphology Figure 20 shows the effect of grain misorientation on the result-  ing precipitate shape. At low angle boundaries a relatively small number of primary Widmanstatten sideplates were observed.  No attempt was made  to measure the growth kinetics of this form as their number was insufficient for a statistical treatment. nature of the low angle boundary. in straightening these interfaces.  Figure 20 also exhibits the irregular Long annealing times were not effective This i s a common feature believed to  be related to the low energy and low mobility of the boundary. The primary sideplates formed parallel to the intragranular 40  Widmanstatten precipitates in agreement with Aaronson's observations for the Fe-C proeutectoid reaction. A two surface trace analysis was conducted to determine the Widmanstatten habit plane. .The results are shown in Figure 21. The data indicates a habit plane very close to the (111)^ plane. Boundaries having a misorientation of 18° to 60° exhibited predominantly allotriomorphic growth, although a small number of secondary sideplates were observed for 20° and 23° misorientations.  The secondary  sideplates were also parallel to the intragranular plates as shown in Figure 20. Some sawteeth development was apparent over the complete misorientation range.  However these were very few in number and did not  appear related to either a specific boundary or any particular section on a boundary. As a large number of allotriomorphs were present on boundaries  CO  X500 (10 min.)  UJ  CL <  X250 (16 min.)  to  1 UJ  CL  o UJ or  J  R=5°  CL  u.  R=20°  o  UJ  o z < or  k-1 Secondary Sideplates, Sawteeth, Allotriomorphs  Primory Sideplates  10  20  j " * - Mainly Allotriomorphs  30 R  Figure 20.  R=60°  =  40 y 2 X  +  Y  2  *  Z  50  +\  60  2  Variation of precipitate shape with grain boundary misorientation, R.  55  100  100  F i g u r e 21.  R e s u l t s o f a t w o - s u r f a c e t r a c e a n a l y s i s o f the f . c . c . m a t r i x h a b i t p l a n e o f t h e i n t r a g r a n u l a r Widmanstatten p r e c i p i t a t e , showing-the p o l e s o f t h e e x p e r i m e n t a l l y d e t e r m i n e d h a b i t p l a n e s , A , p l o t t e d on a s t a n d a r d (001) projection. ( a c c u r a c y ^* -5°)  56 between 18 and 60° misorientation, a statistical growth analysis was applicable.  It was f i r s t necessary to determine the general 5-dimensional  shape of the allotriomorphs to interpret the growth statistics.  The shape  of several precipitates was examined by progressive removal of a known depth of surface and recording the change in precipitate dimensions. Microhardness indentations were used as fiducial marks and the change in length of the diamond diagonal permitted calculation of the depth removed. The 3-dimensional shape of a typical allotriomorph is shown in Figure 22. The precipitate i s essentially circular in the plane of the grain boundary and is bounded by two spherical surfaces of different radii.  The  asymmetry of growth of the precipitate into the two adjacent matrix grains was a commonly.observed feature. B.  Nucleation Kinetics Figure 23 shows the number of nuclei per unit length of grain  boundary as a function of the precipitation time at 688°C. The number i s essentially constant after 2 minutes indicating that nucleation is completed within this time. The effect of supersaturation on the nucleation kinetics was also examined. Figure 2^ illustrates the number of precipitates per unit length of an 18° misorientation boundary as a function of the precipitation temperature.  Prior to each test the precipitate from the previous  test was dissolved by aging the sample at 6lO°C for 50 hours.  The data  also indicated that nucleation was completed within 2 minutes precipitation time, the maximum values being reported in Figure 2k. The number of precipitates increased rapidly with temperature. The high temperature produced too many precipitates and consequently early  57  PRECIPITATE  HEIGHT  35 JJ i-H  10  Ll  PRECIPITATE  LENGTH  PRECIPITATE WIDTH Figure 22,  3-Dimensional shape of a typical grain boundary allotriomorph. (R=23°, 8 minutes at 688°c;  -O-  r  2  •  Q  E=6o°  o  6 /  /  •  /  /  /  /  / /  •  R=l8°  • _  •  •  / /  >, 11  •  ,  o  o  o  o  R=23°  ' i[  o  o  °  o  //9>  n  i  11/  f h  6 8 Precipitation Time (minutes)  10  Figure 23. Effect of aging time at 688°C on the number of nuclei per unit boundary length.  12  Ik  16  800  _ L _ ^  670  1  680  Precipitation Temperature (°C)  1  690  L_  700  ,Figure 2h. Effect of aging temperature on the final number of nuclei formed per unit boundary length.  (R=l8°)  VJl  60  impingement whereas the low temperatures produced too few for a statistical evaluation. 688°C.  The optimum temperature was considered to be between 682 and  The bulk of the kinetic data was obtained at 688°C. The number of nuclei was constant along the length of any one  boundary and was reproducible upon reprecipitation; ,the original precipitate was dissolved by aging the sample 50 hours at 6l0°C. The number of nuclei per unit length of grain boundary varied from boundary to boundary.  No obvious relationship with  misorientation  rfas observed, although in general the lower misorientation boundaries contained slightly fewer precipitates. •C.  The data is illustrated in Figure 25.  Growth Kinetics A typical series of micrographs showing the development of  precipitates with progressive annealing time is shown in Figure .26. i]  Statistical Treatment a]  Lengthening  In the literature i t has been common practice to measure the precipitate having the maximum length and to assume that this corresponded to one which had nucleated at the start of precipitation and had been sectioned through i t s true diameter.  Test results for 18° and 60° mis2  orientation boundaries are shown, plotted as the (max length)  vs time,  in Figure 27. Up to 10 minutes, the maximum length of precipitate on both boundaries shows a parabolic relationship with the precipitation time. Two typical distributions for the precipitate length are shown in Figure 28.  The curves are generally slightly asymmetric and the average  6oo  O  500 _  o  o  koo  300 _  o 200  20 Figure 25-  30  i+o Boundary Misorientation, R  50  The effect of boundary misorientation on the final number of nuclei formed per unit length of boundary. (688°C)  60  ON  I MIN.  Figure 26.  4 MIN.  9 MIN.  Micrographs illustrating precipitate growth.  (R*l8% 688°C, x 250)  18 MIN.  20  Precipitation Time (minutes) Figure 27.  (Maximum precipitate length)  2 v s  the precipitation time. (688°C)  6k  size differs at most by 10$ from the most common size, represented by the peak of the distribution curve.  Precipitates are observed which are much  larger than the most common size and the maximum length is seen to be representative of very few precipitates. -Since not a l l precipitates are nucleated at t , this would tend 0  to spread the population data.  Using the nucleation curve for the 18°  boundary shown in Figure 23, a corresponding histogram of the true size distribution at the end of k and 8 minutes can be constructed.  The results  are shown in Figure 29- -At the end of 8 minutes precipitation time, precipitates w i l l be present which have growth times of from 6.8 to 8 minutes.  As shown in Figure 29 this would produce only.an 8$ variation  in the final size and could not account for the wider size range observed. As the plane of polish is a random section, even precipitates of identical size would produce a size distribution varying from extremely small to a maximum corresponding to the true diameter.  For a single sized  circular precipitate, sectioned at random, a theoretical population d i s t r i bution can be calculated as shown in'Figure 30.  The mathematical develop-  ment is contained in Appendix II. . The curve is markedly asymmetric and bears l i t t l e resemblance to the experimental size distribution.  It i s  apparent that the experimentally observed precipitates are of various true sizes.  The distribution curve obtained is then a superposition of individual  distribution curves related to each specific size and the random nature of the  section. If the peak of the distribution curve is considered representative  of the true size of the most commonly occurring precipitate shape, the contribution made by the random sectioning of larger precipitates i s neglected.  However, as shown in Figure 30, a larger number of precipitates  a) Average 50_  65  ^ N  CO CD  •p ctJ •P  •H  p-  20  •H O CU *H  (R=23, 8 minutes at 688°C)  I  P4 <H  o cu  -O  10 /  _L 16 2k 32 Precipitate Length (cms. x 10 )  ZL  1*8  4  U0 \ Average : \ 30 (R=60, k minutes at 688°C) CO CD •P  \  cd -P •H  p-  20  \  •H O CU PL, CH O  |  / 10  \  / /  \  /  8 12 16 Precipitate Length (cms. x 10 Figure 28.  20  Typical population distributions of the precipitate length.  tn cu -P cd -p  h minutes  •H  o  P.  •H  8 minutes 0  o—o  O  cu  u  PH CH O  U  <u  ^3  CU  > •r-i  -P cd r-l  <u  IK  Figure 29.  2K Precipitate Size (x = K ft;)  Effect of different nucleation times on the resulting precipitate size after k and 8 minutes aging.  (R=l8°)  3K  ON ON  CO 0)  -p aJ  -P •H  f-u •H  o cuu  o CU  a; >  •H -P  ai H  «  0.2  0.6  'Relative Precipitate Length  Figure 30- - Theoretical population distribution for the length of a single-sized precipitate randomly sectioned.  68 having nearly the true size are observed and the number having less than the true size f a l l s off rapidly. Hence the contribution of a random section of a larger precipitate must always be less than the number of precipitates associated with that true size. -The use of the average value as being representative of the most commonly occurring precipitate shape includes the effect of the asymmetry and is thus more justified.  It is clearly not  meaningful to use the maximum length as i t is representative of only a very few precipitates. b]  Thickening The measurement of the thickness of a precipitate from a random  section is also dependent upon the angle the grain boundary.makes with the examination surface.  The maximum thickness is not a meaningful value in  this case as i t could represent a precipitate sectioned at a glancing angle.  In this.investigation the grain boundaries examined were approxi-  mately perpendicular to the surface of polish.  Although the boundaries  were never completely straight they always appeared to be within 20° of being perpendicular.  No correction was necessary since even 20° causes  an error of only. +6% in the thickness measurement. For a single sized precipitate sectioned at random, the maximum thickness would represent the true thickness.  Figure 3 1 shows the  2  (maximum width) plotted against the precipitation time, a parabolic relationship being observed. Two typical experimental thickness distributions are shown in Figure 32. The curves are generally fairly symmetric.  The maximum width  is seen to be representative of a very few precipitates.  30  70  Average I  \  /  (R=23,8 minutes at 688°C)  20 _ to  -<u p d -cp •H  p<  g  u  10  /  CH  \ \  O  u  \r  6 8 Precipitate Width (cms. x i 0 )  10  4  50  \ \ Average  1+0  .1  (R=60, .1+ minutes at 688°C)  \  \  30  I  2C  \ 1C-  I  I  2  I  3 ^ Precipitate Width (cms. x 10 )  I  L  5  4  Figure 52.  Typical population distributions for the precipitate width.  12  71  The effect of precipitate nucleation time can only account for a small range of thickness values. The thickness distribution for a single sized lenticular precipitate, sectioned at random, is shown in Figure 33-  The mathematical  development is contained in Appendix II. Comparison with the experimental distribution again indicates that the precipitates:,'areohot of one single size. This conclusion is consistent with that made for the length measurements. It is apparent that the average value would be a close approximation to the most common true size and is more representative of typical precipitate growth. .The range of precipitate sizes is believed to be related-to the effect of precipitate shape on the growth kinetics.  Precipitates with 73  different axial ratios w i l l grow at different rates,  consistent with  the concept of the point effect of diffusion;.those precipitates having surfaces with a small radius of curvature w i l l grow faster than surfaces having a large curvature. ii]  Experimental-Lengthening and Thickening Kinetics A l l data reported are average values  ^ representative of the  true size of the most common precipitate shape. Figure  shows a typical growth curve obtained at 688°C and  demonstrates the reproducibility of the results.  In Figure 35 the  average of the two tests is plotted as (parameter size) .vs precipitation time.  Parabolic growth is observed for both lengthening and thickening  for a l l the grain boundary misorientations examined, the data being recorded in Appendix III.  72  CO u -ccp d -p •H  p-  •ri  O <U H PH CH O SH 0) ,0  CU >  •ri -P  cd  H  <U  K  0.2  0..^  0.6  0.'8  Relative Precipitate Length  Figure 35•  Theoretical population distribution for the width of a single-sized lenticular precipitate (axial ratio 1.8) randomly sectioned.  1.0  V Legend: 0-,0 ^ •  0.2k  i n i t i a l test. precipitation after resolution.  o Lengthening 0.20  h  4 o. v  0.16  V  h  v  /  /  v  /  0.12  7o v  / o  / 0.08  "O  / /  o>.ok  O  Thickening  12  16  o  •  l /  8  Precipitation Time (minutes)  Figure Jk.  Typical growth data obtained at 688°C reproducibility of the results.  >  showing the (R=l8°;  20  Jk  Figure 35.' Growth data plotted as (parameter size) s . precipitation time. (R=l8°, 688°C) V  75  The thickening data remains parabolic up to 16 minutes aging time whereas the length results begin to deviate after 10 minutes. is also shown in Figures 27 and 31 precipitates are plotted.  i n  This  which the dimensions of the largest  The deviation from parabolic behaviour is  probably related to precipitate impingement.  Total precipitation times  of 10 minutes were normally employed to minimize this effect. Following kinetic measurements, several specimens were returned to the single phase condition by heating for 50 hours at 6l0°C.  Precipi-  tation was repeated using a single anneal to check i f the repeated annealingquenching technique affected the growth process.  The results are shown  in Figure 36, no significant difference being observed. The rate constant, K, for lengthening and thickening was calculated assuming the precipitate growth rate to be of the form x = K {T . Figure 37 shows that the grain boundary misorientation has l i t t l e effect on the growth rates. 8 x 10  5  cm/ £ se  The average rate constant for lengthening i s  and for thickening i s 2.3 x 10  5  c m  /  s e  c  ^  e  P  rec  ipitate  axial ratio is also independent of the grain boundary misorientation as illustrated in Figure 38.  The most common precipitate shape is therefore  similar over the complete misorientation range. The i n i t i a l growth of precipitates on an 18° misorientation boundary was measured at 700, 682 and 67^°C.  The data was obtained primarily  for nucleation kinetics but growth-results were also tabulated. The values are compared with the data obtained at 688°C for the same boundary and the maximum and minimum values for a l l the boundaries examined at. 688°C. results are shown in Figure 39  a n a  The  the data in Appendix III. The axial  76  Legend:  Precipitation Time (minutes) Legend:  Precipitation Time (minutes) Figure 36.  Single immersion precipitate growth compared to interrupted aging data. (688°C)  20  10 _  Lengthening  2 "  Thickening  io  _•  •  •  •  _°-o- -  1  •  ^mean  •  G  .20  O  30  mean  O  Uo  50  Misorientation Parameter, R.  Figure 57- Rate constant K(x = K^T) vs the misorientation parameter, R.  6o  _!  10  F i g u r e 38.  I  20  I  I  30 k-0 M i s o r i e n t a t i o n Parameter, R  P r e c i p i t a t e a x i a l r a t i o ^ l e n g t h \ v s t h e g r a i n boundary parameter, R. width  !  50  misorientation  L_  60  79 a) Lengthening  1  2  Precipitation Time (minutes)  -.Figure 39*  Effect of temperature on the growth kinetics on an 18° misorientation boundary, including the maximum and minimum 688°C growth kinetics.  3  80 ratio of the precipitates remained essentially constant over the temperature range examined.  It should be remembered that the nucleation kinetics  limited the temperature range over which truly meaningful growth results could be obtained (Section VI, B). D.  Diffusion Couple Data The diffusion coefficients at the compositions and temperatures  of interest were determined using diffusion couples. Calculated diffusion coefficient vs. composition plots are recorded in Appendix IV.  Figure 1+0  shows the composition profile for the couple prepared at 688°C, the most important precipitation temperature.  A graph of the diffusion coefficient 74  vs. the solute composition, obtained using the Boltzmann-Matano method, is illustrated in Figure kl. The diffusion coefficient corresponding to the boundary composition, 5-56" wt.$> Al at 688°C, was calculated in two ways. The graph for D as a function of the composition, obtained at 688°C,.was extrapolated to the boundary composition and also an activation energy plot was constructed using the 65O, 600, 55O, and 500°C results and was extrapolat.ft4.-to 688°C. As shown in Figure 42 the resulting value is similar 5.56  wt.fo  in both cases. .At  Aluminum and 688°C, D = 7-5 x 1 0 " cm / 9  2  sec  with  D = 2.7 cm /  and  E = 37.4 kcal/mole (1.62 e.v.)  2  0  sec  A  The diffusion coefficient was calculated using the phase boundary composition (5-56 wt.$> aluminum), whereas for growth of the sion in the  phase, diffu-  5»64  matrix occurs over the composition range  This is equivalent to a variation in the D value from 9 1 0 X  9  to 5.56 wt.$ aluminum.  to 7.5x10 c m / 9  2  sec  O  O  o  Q-rO O Q Q ° -o O O  I  !  Couple Composition  o Matano Interface  I Final Interface Position  2h  1 U  300  100  Distance (u) Figure kO. Composition profile from the diffusion couple prepared at 688°C.  10  composition range of supersaturated phase  -wt.$ Aluminum Figure kl,  Diffusion coefficient vs the solute composition (688°C)  -11  _J 1.04  !  1.08  ! 1.12  I 1.16 1  T(°A) Figure 42.  Log. D vs 1 T(°A) for 5.56 wt.# aluminum.  X  I  I  1.20  1.24  103  I 1.28  I 1.32  Qk  respectively.  Since a l l formulae for the growth rates of precipitates  assume D to be a constant, and since the value of D at the phase interface was taken to be representative of diffusion in the supersaturated  phase,  there is an uncertainty of +20$ in any comparison of diffusion coefficients and growth rates. E.  Structure of the Quenched ^> Phase A 6.65 wt.$ Aluminum alloy was heated at 688°C for 1 hour and  water-quenched to room temperature. At 688°C this alloy is 95$ P> the ^  with  having a composition equivalent to the grain boundary precipitate  examined in the kinetic analysis. The quenched structure is shown in Figure kj> - the material was photographed using polarized light to illustrate the anisotropy of the quenched phase.  Figure kj>.  Structure of the ^) phase, water-quenched from 688°C. (polarized light, x250)  8  5  The crystallographic nature of the product was established using wide-angle X-ray diffractometry.  The structure was found to be h.c.p. and  the following lattice parameter values were obtained using a Norelco Back Re-flection Focusing Camera:  2.868 t .002  4.680 "± .002  Calculations were based on Cohen's method of least squares as applied to ' 75  the hexagonal system.  Using a fine grained, strain free powder, the •+  °  back focusing camera should yield lattice parameters accurate to -.0001 A. However, sheet specimens were employed for this analysis as i t was not possible to anneal a powder sample after preparation, as a transformation could result.  The relatively large uncertainty quoted for the lattice  parameters is related to the diffuse nature of the diffraction rings analyzed; this was probably due to transformation strains in the quenched product although no specimen warping was observed. The transformation occurred rapidly on quenching, the product exhibiting equiaxed grains, suggestive of a massive £ transformation. The lattice parameter expected for such a transformation would correspond to an extension of the stability of the equilibrium ^* phase to the composition of i n t e r e s t .  56  Figure kk illustrates the effect of atomic $ Al on  the lattice parameters of the h.c.p. ^  phase  76  and shows that the predicted  lattice parameter for no composition change is in reasonable agreement with the observed value. Samples of 6.65 wt.$ aluminum were quenched into 16$ NaOH after 1 hour aging at 688°C.  The higher quenching rate produced rumpling and  surface shears as shown in Figure-k^a).  The resulting microstructure was  a 2.88_ ^ . 7 0  u CU  -p cu S cd u  $  a; o  •H •P -P  2.87  U.60  cd  i-J  2.86L  4.50 20  t l  Equilibrium Composition Range (550ocV" 1*0  30  50  Atomic $ Aluminum Figure .Mi. Lattice parameter of the ^ phase extrapolated to the composition of the Q phase, compared with the experimentally obtained values. (Hansen ) 76  co  87 a)  Figure k^>.  Structure of the brine-quenched ^> phase. a) Shear markings on quenched surface ( x l 2 5 ) b) Acicular martensitie microstructure fx250) c) Mixture of massive 6 and acicular martensite  (xl25)  88 typical of an acicular martensite and i s illustrated in Figure 45b. Regions which contained a mixture of the massive and the martensite phases were also noted (Figure 45c).  Diffraction patterns of the martensitie  structure were obtained using the back focusing equipment. The structure was similar to the water-quenched material but the lines were extremely broad, probably due to the transformation strains, and accurate lattice parameter measurements could not be made. Thin film specimens were prepared to permit an electron microscope examination of the fine structure in both quenched products.  Figure  46 contains two examples of the internal structure of the water-quenched phase. The parallel striations are consistent with the basal plane, indicative of stacking faults, and the streaking directionj,as shown in Figure 46 c and 46d,is seen to be consistent with the projection of the basal plane on the f o i l surface. Although this was the commonly observed structure one region was examined which showed striations consistent only with internal twinning. Thin films of the brine-quenched material showed an acicular structure having a uniform density of. parallel^ internal striations. The structure i s illustrated in Figures 47a and 47b. Diffraction patterns from single plates were analyzed according to the h.c.p. structure and the striations were found to be consistent with the basal plane, as shown in Figures 47c and 47d. Some spot multiplicity was observed, however no satisfactory explanation is available as the internal markings were not consistent with the twin plane.  89  Standard (0001) h.c.p. d) stereographic projection.  giving rise to striations in b). Figure kS.  Fine structure of the water-quenched P) phase. a) b) c) d)  General striated structure (x 8p00). Striated structure in single grain (x 30,000). Standard area diffraction pattern from b). Interpretation of striations with respect to the diffraction pattern.  90  a)  b)  c)  d)  a) b) c) d)  Standard (0001) h.c.p. stereographic projection  General acicular appearance (x 3 0 , 0 0 0 ) . Striations in individual needles (x 3 0 , 0 0 0 ) , Standard area diffraction pattern from needle 1, b) . Interpretation of striations with respect to the diffraction pattern.  91 F.  Matrix - Precipitate Orientation and Habit Analyses The data indicates that the Q) phase transforms to an h.c.p.  massive  structure on water-quenching to room temperature.  It is neces-  sary therefore to determine whether an orientation relationship exists between the parent ^ -  and the massive product before pursuing the possibility of any  relationship. Samples of 6.65  wt.$ aluminum were held at 688°C for 1 hour in  H2 and were water-quenched.  Thermal grooving of the (j*> boundaries fixed  the ^> structure relative to the massivej product.  Examination of a  lightly electropolished sample surface under polarized light revealed that the massive boundaries crossed the original shown in Figure k8.  Figure ^-8.  boundaries at random, as  This indicates that no orientation relationship exists  Development of the massive product with respect to original (2, grain boundary. (polarized light, x  100)  92 between the parent and the product phase.  If this i s indeed the case then  i t i s only possible to examine the precipitate shape with respect to any habit relationship in the matrix phase. To this end sheet samples of 5»64 wt.$ aluminum were annealed at 688°C for JO seconds to produce grain boundary precipitates equivalent to those from which the kinetic data was determined.  The precipitate and  bounding matrix grains were examined using transmission electron microscopy and electron diffraction patterns were obtained for the allotriomorph and both adjacent matrix grains.  A typical set of results i s shown in Figure  h°.; the interpretation of the results i s equivalent to a single surface trace analysis. -As shown in'Figures ^9a and -49b, precipitates exhibiting both single and double surface development were examined. The possibility that the flat o(  interface was related to a specific matrix habit plane  was investigated.  Figure 50a shows the poles of a l l possible planes in  the <X grain adjacent to the flat interface, which could give rise to the flat interface.  The data indicates that the undeveloped interfaces are  consistent with an irrational habit plane in the is approximately 5  0  from the (111)^ .  matrix; the plane  Those precipitates which developed  into both matrix grains exhibited no preferred relationship with respect to the original grain boundary position. The same analysis applied to the matrix grain adjacent to the curved surface of the allotriomorph produced the results shown in Figure 50b. No apparent relationship exists. The structure of the massive  product was also analyzed with  respect to the undeveloped, flat interphase boundary, the results being  93 illustrated in Figure 50c.  No obvious habit relationship existed.  In  addition, no simple orientation relationship was observed between the matrix grain adjacent to the flat interface and the resulting massive precipitate. a)  Figure U9.  b)  Typical grain boundary precipitates a) Showing single surface development (x 8p00), b) Showing double surface development (x 8p00), c) Schematic diagram showing diffraction patterns obtained for habit analysis with respect to the precipitate shape.  9k  100  100  Figure ^0a. Results of trace analyses showing the poles of a l l possible planes in that cK. matrix grain adjacent to a flat precipitate-matrix boundary which could give rise to the flat interface.  95  100  Figure 50b.  Results of trace analyses showing the pole of a l l possible planes in that «K matrix grain adjacent to the curved surface of the precipitate which could give rise to the f l a t , undeveloped matrixprecipitate interface.  96  1010  'Figure ^Oc.  Results of trace analyses showing the pole of a l l possible planes in the massive product which could give rise to the flat matrix-precipitate interface.  97 VII  DISCUSSION A.  Morphology The range of precipitate morphologies observed is in good agree-  ment with Aaronson's studies on the Fe-C proeutectoid reactions Clark's observations for precipitation of the h.c.p. ^  40  and  phase from the  34  f.c.c. o< phase in Al-rich-Al-Ag alloys.  This is additional evidence  that the shape controlling mechanisms are generally operative and not related to the peculiarities of any one system. B.  Nucleation Nucleation was completed within the f i r s t 2 minutes of precipi-  tation and thus the system was not ideal for examining nucleation kinetics. The relatively high homologous temperature would account for the rapid nucleation. The decrease in the nucleation rate to zero after 2 minutes is not due to a shortage of nucleation sites since, as Figure 2h indicates, increasing the temperature produces a large number of additional sites; the effectiveness of any one sitejhowever, may be dependent on the precipitation temperature.  Furthermore, the number of sites would be expected  to increase with misorientation and yet the number of nuclei bore no obvious relationship with misorientation.  The reason for the marked decrease in  nucleation rate would appear to be due rather to a lack of aluminum atoms able to segregate to form nuclei. Those aluminum atoms on and adjacent to the grain boundary w i l l diffuse along the boundary to preferred nucleation sites.  Once this supply of atoms is exhausted nucleation w i l l  cease as further atomic movement must occur by volume diffusion, a much  98 slower process. The aluminum atoms diffusing along the boundary w i l l form clusters having the ^ composition. When a cluster attains the c r i t i c a l nucleus size growth w i l l ensue. At low degrees of supersaturation, the c r i t i c a l nucleus size i s large and therefore only that number of nuclei consistent with the available supply of aluminum atoms, can form.  At  higher degrees of supersaturation the stable nucleus size is much smaller, each nuclei requiring fewer of the available Al atoms, and therefore a larger number of nuclei can form.  Order of magnitude calculations of the  size of a c r i t i c a l nucleus and the aluminum atoms required, indicate that a sufficient number of aluminum atoms are present on and immediately adjacent to the grain boundary.  The calculations, contained in Appendix V,  also show that the grain boundary and adjacent regions are denuded of aluminum atoms and-that growth after nucleus formation must occur by volume diffusion.  More atoms w i l l be supplied to the growing t i p by volume  diffusion to the grain boundary and diffusion along the boundary. This effect combined with the increased curvature associated with a t i p is thought to give rise to the more rapid lengthening kinetics. . The temperature range over which nucleation was studied is small (674-700°C) and thus the grain boundary diffusion coefficient would be expected to change very l i t t l e over this range.  Consequently, i f the  suggested mechanism i s correct, the time required for formation of the c r i t i c a l nucleus"would not change significantly.  This is consistent with  the experimental observation that nucleation was completed within two minutes over the temperature range examined.  99 C.  Growth It has been shown that grain boundary allotriomorphs lengthen  and thicken parabolically with time. diction with the Hillert theory Aaronson  40  44  Parabolic lengthening i s in contra,  and the experimental work of Dube and  and Aaron and Aaronson.  54  As both the length and the thickness obey a parabolic growth law the precipitate grows with constant shape.  This is in agreement with  T3  the theoretical work of Ham  who showed that for a precipitate growing  with constant shape, a l l points develop according to a parabolic growth law.  Hence once the precipitate has nucleated and lost coherency i t main-  tains i t s shape during subsequent growth. It i s possible that the lengthening results may have been affected by interactions of the diffusion fields of adjacent allotriomorphs. As shown in Figures 20 and .26 the spacing between adjacent precipitates at later stages of growth i s sometimes less than the precipitate size i t s e l f . Impingement would decrease the lengthening rate and so give a false picture of the growth law. Two approximate models can be used to represent the composition gradients around developing precipitates.  The closest shape representation TT  for a lenticular allotriomorph is given by a bipolar coordinate system. However, no solution to the diffusion equation is available in this coordinate system, and solutions to Laplace's equation (i.e., steady state diffusion, Section II C) have to be used to obtain a qualitative picture. The mathematical evaluation i s contained in Appendix VI.  The composition  profile illustrated in Figure 51a shows that there is no composition gradient and consequently no diffusion head of the growing t i p .  Solutions to the  (a) Figure ^1.  Composition profiles associated with a)  and b)  a lenticular precipitate an oblate spheroid.  101  diffusion equation are available for the growth of oblate spheroids having 24 ) 7 3  a constant shape,  (Fig.11).  Such precipitates do not have a sharp t i p  characteristic of lenticular precipitates but otherwise are similar in appearance.  Composition contours adjacent to an oblate spheroid having  an axial ratio approximately equal to the experimental lenticular precipitates are shown in Figure 5 1 ° ; the calculation is included in Appendix VI. The restricted development of the diffusion field adjacent to the t i p remains, but is less pronounced as compared to that of the bipolar case. It would appear that there should be no significant impingement of the diffusion fields until the precipitates are very close to one another. The experimental results support this observation. A marked change in shape in the lengthening data would be expected after impingement and was not observed except possibly after 12 minutes growth (Figure 35)-  Further-  more, the basic shape distribution did not change with time, while i f impingement were taking place a progressively sharper distribution of precipitate shapes would be expected. Figure 37 shows that the lengthening rate is independent of the grain boundary misorientation, although i t is known that the grain boundary 7*8  diffusion coefficient increases with increasing misorientation.  The  sharp t i p of the lenticular precipitate, having essentially no composition gradient ahead of i t (Figure 51a), would not be expected to grow by grain boundary diffusion.  Furthermore, i f , as suggested in the nucleation  discussion, the grain boundary i s depleted of aluminum atoms during nucleation, atoms can only transfer to the boundary by volume diffusion.  Therefore  growth at the t i p would be volume diffusion controlled with some enhancement expected, due to the larger area supplying aluminum atoms to*the grain boundary.  102 These observations are different from the iron-silicon results 33  of Toney and Aaronson  who found an increase in the average axial ratio  with increasing misorientation.  However their precipitate shapes changed  from predominantly idiomorphs at lower angle boundaries to allotriomorphs at the high angle boundaries.  In the present investigation no idiomorphs  were observed and measurements were made exclusively on allotriomorphs. From the growth data an estimate can be made of the diffusion coefficient in the supersaturated o< phase, assuming diffusion control. The D value obtained depends on the model adopted for the shape of the grain boundary precipitate.  In previous work the thickening kinetics have  been evaluated using planar development.  This means that the curvature  of the lenticular precipitate, which enables solute to diffuse to the growing front from a wider solid angle, has been ignored. An overestimate of the diffusion coefficient is thus obtained, just as assuming the lenticular precipitate to be spherical would produce an underestimate of i t s true value. There is no solution to the diffusion equation for lenticular precipitates.  An oblate spheroid, growing with constant shape and having  the same radius of curvature on i t s minor axis as the experimentally observed precipitates, is the closest shape approximation for which a solution does exist.  Because growth often occurs  asymmetrically into  the two matrix grains i t is also necessary to include the extreme cases of growth into one grain and equal growth into both grains; the true value is likely an intermediate value. The comprehensive kinetic data was obtained at 688°C. .. Equivalent D values were calculated using a:  and  a)  planar  b)  spherical  c)  oblate spheroid shape approximation.  The results are recorded in Table IV  and the true D value obtained from  diffusion couple measurements i s included for comparison. As can be seen, the thickening results are best interpreted when the curvature of the precipitate i s included. The oblate spheroid shape approximation produces D values which are in good agreement with the diffusion couple measurements.  Both the lengthening and thickening  kinetics are used in the calculation as the axial ratio describes the shape of the oblate spheroid. The comparison of D values is quite good and thus volume diffusion in the supersaturated cx» phase is considered the rate determining process.  Aaron and-Aaronson  54  recently reported that the lengthening of  grain boundary precipitates in Al-4# Cu alloys was controlled by Cu diffusion along the matrix grain boundaries, while precipitate thickening was controlled by diffusion along the matrix-precipitate interphase boundaries.  However, their data was obtained at temperatures where T  varied from O.55 to 0.70 (1^ being the absolute liquidus temperature). In this range grain boundary diffusion may predominate, whereas our data was obtained at T_ = 0.91^where volume diffusion should be dominant. Equivalent D values were also calculated comparing the kinetics of precipitation at 700, 688, 682 and 674°C . A single surface, oblate spheroid shape approximation was used, the data being recorded in Table V.'. The effect of inaccuracies of ±i°C and -0.0lwt.$ aluminum is included as are D values obtained from the diffusion couple data.  TABLE IV Diffusion Coefficients Calculated for Specific Growth Shapes as Compared to the Measured Value  AVERAGE TEST DATA ( D  D  PLANAR SINGLE DEVELOPMENT  -rf \  /sec) D  SPHERICAL  ELLIPSOIDAL  M  rh 76  DOUBLE DEVELOPMENT  C m  X  10  -9 2.5  x  "9 10  5-5  x  -9 10 7  h -9 19 x 10  O.63  x  10"  9  D FROM DIFFUSION COUPLE MEASUREMENTS = 7-5 x 10"9 cm /sec.  -9 3-^ x 10  TABLE  V  Equivalent D values obtained from kinetic data using a single surface oblate spheroid shape approximation and including the effect of the experimental errors -1°C and ±.01 wt A A l .  Test Temperature  Effect of Experimental Errors on D Calculation  Equivalent cm / 2  Max.fo-l, - 0 J ) ^  sec  "10 "  C I  / s e c  M  l  n  Ifru, . j )cm2/ /sec cng  +  0  c  .D Value Obtained byDiffusion Couples /sec cm2  700°C  5.3X  9  5.6 x 10"  4.0 x 10"  8.5 x 10"  688°C  6.6 x 10"  10.4 x 10"  6.x x 10"  7.5 x 10"  682°C  10.0 x 10"  15.0. x 10 "  9  8.6 x 10"  7.1 x 10"  67k C  37.6 x 10"  x 10"  13.5 x 10"  6.8 x 10"  0  9  9  9  9  9  1050  9  9  9  9  9  9  9  9  9  taken as 5-64 to permit calculation  H  o  106 The accuracy of the 700°C, 682 and 6lk°C data i s poor as i t was  /  obtained after very short growth times (—J> min.).  The 688°C and 682°C  data provided sufficient precipitates for a statistical evaluation, and the experimental errors were a minimum for the 688°C data. At 700°C too many precipitates were formed, leading to early impingement, while at 67^-°C too few were produced for statistical evaluation and the effect of experimental composition and temperature errors was very pronounced. For these reasons the data i s not useful for an activation analysis.  Rather,  i t confirms the choice of 688°C for the precipitation temperature.  To  obtain data for an activation analysis i t would be necessary to prepare several compositions such that the precipitation temperature for each would duplicate the supersaturation over, the range of temperatures to be examined. D.  Massive and Martensitic Transformations in the Quenched ^) Phase On water-quenching the  phase from 688°C a product exhibiting  the following characteristics was obtained: l]  The structure of the-quenched'phase was h.c.p., the lattice  parameters representing an extension of the equilibrium high temperature 2]  phase to the  composition. -..  ,The boundaries of the product phase crossed the original  boundaries at random indicating that no parent-product orientation relationship existed. 3]  The grain structure of the quenched alloy was equiaxed and no  surface rumpling or shears were observed on quenching. k]  .The internal fine structure was consistent with stacking faults;  a range of stacking fault densities was observed in any one sample. The structure characteristics are identical to those reported  io>7  by Massalski et a l .  5 6 , 5 9  j Saburi and Wayman  60  a n (  for a massive <p  obtained by quenching the ^> phase in Cu-23$ Ga alloys. have also been observed for the quenched  in Cu-38$ Z n  product  Similar structures 57  that the equilibrium concentration of vacancies in the CsCl  i t Is believed structure  at high temperatures i s approximately an order of magnitude greater than 79  that for pure metals.  This high quenched-in vacancy concentration  would enhance short-range diffusion, explaining the rapidity of the transformation. The  phase when subjected to a more rapid quench produced a  structure exhibiting: 1]  Sample rumpling and surface shears.  2]  Either a typical acicular microstructure or a mixture of the  massive and the acicular product. 3]  Internal striations of the acicular product which were uniformly  distributed and were consistent with stacking faults. k]  A slightly distorted h.c.p. structure. These observations are also similar to those obtained by Saburi  and Wayman  60  for the Cu-23$ Ga acicular martensites.  The structures are  also comparable to those reported by Swann and Warlimont  62  for Cu-12$  aluminum martensites. Although the fine structure in both the massive f and the acicular martensites exhibited stacking faults the origin of the faults is believed to be different in each case.  The faults in the martensite  are believed to be due to the lattice invarient or "second" shear characteristic of a martensitic reaction, while those in the massive product are considered to be growth and deformation faults arising from 60  the transformation volume change.  108  E.  Matrix - Precipitate Orientation and Habit Relationships The experiments established that the  on water-quenching by a massive  precipitate transformed  reaction to the h.c.p. "P phase. A 3d  comparison of the thermally grooved ^ boundaries with the boundaries of the massive £ product indicated that no relationship existed between the parent b.c.c and the product h.c.p. phase.  In addition, no orientation  relationship was present between the massive  (quenched precipitates) On  and the adjacent matrix grains. Examination of the precipitate shape with respect to the parent grain boundaries.indicated that those interphase boundaries which exhibited l i t t l e or no growth were consistent with a habit plane within 5° of the ( i l l ) plane of the associated matrix grain.  No distinct habit plane was  observed with that grain into which the precipitate was growing.  This is  in agreement with the concept that i n i t i a l l y a precipitate nucleates in one grain producing a low energy interface and grows into the adjacent grain by movement of the high energy, mobile boundary.  The matrix habit  plane could imply a Kurdjumow and Sachs orientation relationship  which  has been observed for flat interfaces between f.c.c.and b.c.c. phases. ' 36  50  Any coherency would necessarily be restricted to the early stages of nucleation and hence, the observation of non-symmetric development at later stages must be related to the growth behaviour.  Furthermore, with the  interrupted annealing technique used in this work any lattice coherency would be lost after the f i r s t quench due to the characteristics.  -massive transformation  If growth of the precipitate is diffusion controlled  then the i n i t i a l shape would be retained during later development due to the nature of the established diffusion gradients; rounded interfaces would develop more rapidly.than flat interfaces, thereby maintaining the  109  original shape. Not a l l precipitates exhibited a single surface development. Those which developed symmetrically  into both matrix grains displayed no  habit relationship with either grain.  It is known that precipitates w i l l  preferentially nucleate at sites on a grain boundary but apparently these sites need not be associated with a coherency with one of the parent grains. Those which do nucleate on one of the parent grains would require less free energy to develop which may explain the predominance of this form. .This interpretation is also consistent with the observation that the habit plane of the Widmanstatten intragranular precipitate is very similar to that determined for the f l a t , undeveloped allotriomorph surfaces.  A coherency effect is normally associated with the flat Widmanstatten  faces.  However, the probability of coherency controlling growth at later  development stages i s suspect in this system. Well developed Widmanstatten plates were observed after repeated quenching and precipitation even though the ^) -massive transformation should have effectively removed any lattice coherency.  This would indicate that the coherency is mainly  effective at the nucleation stage; the shape is determined by diffusional growth. It is interesting to note that the habit plane observed for the matrix grain is consistent with the Wechsler, Lieberman and Read martensitie prediction for an inhomogeneous shear (111) [112].in the parent phase which also produces an orientation relationship very close to the K-S.  66  This would then agree with Otte and Massalski^s observations on several 47  nucleation and growth reactions.  However, i t is d i f f i c u l t to envisage  a martensitie reaction nucleating a nucleation and growth process and i t is felt that the coherent nucleation model can explain the results in a more reasonable fashion.  110  VIII CONCLUSIONS In the precipitation of the ^) phase from supersaturated  o\  of composition Ag-5<6h wt. $ aluminum l]  Primary sideplates formed at low misorientation boundaries,  while allotriomorphs were dominant on boundaries having R ^ 18°. Secondary sideplates were observed at intermediate misorientations and a small number of sawteeth forms were present over the complete allotriomorph range. 2]  The number-of grain boundary precipitates increased rapidly  with increasing supersaturation. No obvious relationship with grain boundary misorientation was observed although fewer precipitates were associated with the low misorientation boundaries. 3l  Histograms of allotriomorph length and thickness distributions  indicated that a range of precipitate shapes were present on any one boundary. k]  .Allotriomorphs grew with constant shape, the length and thickness  increasing parabolically with time. 5]  The growth kinetics were independent of the grain boundary  misorientation. 6]  Growth rates were in good agreement with independently measured  values of the volume diffusion coefficient. On quenching the l]  phase -  An internally faulted, massive G> , transformation product On  developed.  With a higher quench rate an acicular martensite resulted.  Ill  2]  The resulting massive phase exhibited no orientation relation-  ship with the parent (2> phase and thus parent-product orientation relationships and habit planes i n the precipitate could not be examined. Examination for habit planes in the matrix grains as related to the shape of the associated grain boundary precipitate revealed l]  Flat interfaces i.e., surfaces where l i t t l e or no precipitate  development had occurred, were consistent with an irrational plane (V5° from (111) in that matrix grain. 2]  Curved surfaces, indicative of growth into both matrix grains  exhibited no habit relationship with either of the matrix grains.  112  APPENDIX I Composition Analyses Chemical analysis was carried out using a technique described in R. Belcher and C.L.Wilson, "New Methods of Analytical Chemistry", Chapman and Hall Ltd., London, (l964) p.343. ?  In this procedure the aluminum concentration of the Ag-Al alloys i s complexed with an excess of EDTA solution.  The consumption of the EDTA i s  then determined by back-titrating with a lead nitrate solution in the presence of Xylenol Orange. The composition of the alloy used for the kinetic evaluation was found to be 5-64 - 0 . 0 1 wt.$ aluminum. The average of at least three analyses was used for each sample with the individual results being within ±0.01 wt.$ aluminum of the quoted average value. The composition with respect to the published phase diagram was also determined by conducting a series of precipitation experiments. The coarse grained, ©< phase material was cold rolled and recrystallized to produce a grain size of approximately ^0 p.. Samples were heat treated over a temperature range which traversed the single °^ phase and the two phase (o(. + Q ) equilibrium regions.  The boundary between the single  and the double phase was established by determining the maximum temperature at which the sample could be annealed for 15 minutes without the appearance of a precipitate.  This temperature was found to be 671 - 1°C which is in  excellent agreement with the published phase diagram for a 5-64 wt.#> aluminum alloy. The Ag-Al alloy used both for the preparation of diffusion  113  couples and for determining the nature of the quenched \p phase was also chemically analyzed and found to be 6.65 wt.$ aluminum.  Ilk APPENDIX II Distribution of Precipitate Shapes on a J Random Plane of Polish Figure II-A shows two views of a grain boundary allotriomorph of length 2 \\ and thickness 2 Tr. A typical section through the precipitate, perpendicular to the grain boundary,is shown with length 2I and thickness 2t.  Since the precipitate can be sectioned with equal probability  at any point along line AB, the distribution of precipitate shapes on a random plane of polish can be calculated.  Fig. II-A Two views of a grain boundary Length Distribution  allotriomorph.  115 The number of precipitates observed is proportional to the length p. Hence, the relative number of precipitates with length between  X and (X, •+ dl,) i s given by equation 2). at  = 0 to infinity at X, = ^  This function varies from zero  and is markedly asymmetric.  In any real  length distribution where the number of precipitates i s limited, no problem arises from the distribution function tending to infinity since finite ranges of length are being measured. Thickness Distribution From Fig. II-A  p  i.e.,  p  Using;  2  p  ^p t  = 2X R  +  =, t  =i  t  ?  d  + (R - +  £  1  and  2  1  t) = R 2  2 7 j t - 2tR - T T a n  |  x  2  _  2  k)  2  - t ; x  0 .2 5)  =  1  J (0  -t  X  = \(f+.tja = 1 - 2t  2  d 0 = _2i_ (the axial ratio)  X  1  3)  2  + t )(1 - t )  The relative number of precipitates with thickness between t and (t .+ dt ) i s given by equation 5)- The function varies from i  (1 - 0) s  20  for t = 0 = t ,to infinity for t = f 1  i.e., t =1. 1  The  thickness curve is thus less asymmetric than the curve for the length distribution. Calculated length and thickness histograms are plotted in Figures 30 and 33 respectively. ratio of 3.6 i s used.  In Figure 33 "the observed average axial  APPENDIX III  Compilation of Kinetic Data  117  Figure I I I - l .  Kinetic data for R=l8°, 688°C.  118  •h  Figure III-2.  8 12 Precipitation Time (minutes)  Kinetic data for R=23°, 688°C.  16  119  .Figure I I I - 3 .  Kinetic data for R=20° a n d 3 6 % 688°C . ;  120  6 U  Precipitation Time (minutes)  Figure I l l - k .  Kinetic data for R=il2° and ky,  688°C .  121  Figure  Kinetic data for R=45° and 5O , 688°C. 0  122  2  Figure III-6.  k. 6 8 Precipitation Time (minutes)  Kinetic data for R=60°, 6'88°C.  10  123  Precipitation Time (minutes)  Figure III-7.  Effect of ..temperature on precipitate thickening kinetics. (R=l8°)  12k  Figure III-8.  Effect of temperature on precipitate lengthening kinetics. (R=l8°)  APPENDIX IV Diffusion Couple Data The Boltzmann-Matano method  74  D(c) .=  was employed to calculate D =  (composition) from the composition-  distance data obtained using the electron probe microanalyzer. calculated D vs composition profiles are included for couples prepared at 500, 55O, 600, 6 5 O and 688°C.  The  2.dx  Figure IV-1.  D(c) vs solute concentration (500°C).  ro OA  Figure IV-2.  D(c) vs solute concentration (550°C)  ro  •Figure IV-3.  D(c) vs. solute concentration (600°C).  H  ro  Figure TV-4.  D(c) vs. solute concentration (650°C).  10 I  I  l  2  3  Wt.$ Aluminum  Figure TV-5.  D(c) vs. solute concentration (688°C).  k  5  131  APPENDIX V Order of Magnitude Calculation to Check the C r i t i c a l Nucleus Model Adopted To calculate  Af ; y  Assume c r i t i c a l nucleus forms when the gradient in the super42  saturated solid solution i s removed. Figure V - l . -Composition gradient in the supersaturated matrix and the resultant spherical nucleus.  Coo=  5-64  Spherical nucleus of radius E . c  Coo  =C (1+iP)  R  R„  P  c  -  ^spl" (C.  1  C  s ) P  where P assuming  = fl^V RT  ^ = 500 e r g s / ^ AB  = 1.38 x 1 0 "  k  T(°K)  16  erg/°K  = 688+273 = 96l°K  V for Ag a = 4.086 A at 20°C Coeff. of expansion = l 8 . 8 x 10 /°c 6  at 688°C  a  = 4.086[l + (18.8x10~ )(688j 6  t  = 4.i4 A  4 atoms/ ' = (4.14 x 1 0 ) c m unit fee. c e l l _8  o  °  o  1 atom = 17.7 x 10  • cm  3  3  3  = 70.7 x 10" cm 4 24  3  1J2  |  — ,  = (500.)(17.7 x 10~ ) = 6.66 x 10~ cm. 8  24  (1..38)( 10-is j 961) (  C^, = 5.6U- wt.# = 19.30 at pet. = 5.56 wt.$ = 19.08 at pet. .(5  c  2 sp  (C 06  P  -2  sp  ,°Af =  "til AS  > e)  V  'sp> 'sp  P  = -(500)(0..22) (19.08)(6.66 x 10-  = -8.65 x 10 ergs enr  This compares favorably with the A f value of -4.35x£0 ' ergs used by Clemm , cmand Fisher for the austenite -to ferrite transformation! ) 7  v  17  Using Clemm and Fisher's Model for a lenticular precipitate on a grain boundary, the shape i s determined by the interfacial energy balance shown in Figure V-2.  Figure V-2.  Lenticular precipitate formed at a grain boundary, showing the balance of interfacial energies and the resulting precipitate shape.  133  The work done in forming the nucleus, neglecting strain energy, is W =  ^ b r  .  2  / ar  + Af cr  2  w  1)  3  y  b = 4 T T (1 - cos 6 )  where  a = Ti sm  9  c = 2 T f ( 2 - 3 cos 9 + cos 6 ) 3  3  The c r i t i c a l nucleus size is obtained by differentiating equation 1) with respect to r and equating the result to zero. 0 =2 ^  B  br - 2 ^  ar + 3 A f c r r  o o  = r ( 2 tf^b-2 ^  2  y  b +  = -2  3  a + 3 Af cr) v  a  2  -Af c  2  v  =. 1 .'42 adopted by Clemm and Fisher^the balance  Using the ratio of  of the interfacial energies yields cos 0  =  ^A*  =  .707  or 9 = 45° 00  r = -2  41Y*(1 - cos 6 ) + 2 3 Af  Using  ^  = 500 ergs/ a.and  sin  2  21< (2 - 3 cos © + cos 9 ) 3  v  3  tf  cra  and 0 = 45°, A f  AB  = .707  ^  = -8.65 x 1 0 ergs/. cnr 7  = l.lk  I.98  Af  x 10 cm. 5  v  Precipitate l / 2 length = r sin 8 = .81 x 10 cm. 5  Precipitate l / 2 width Volume of ppt. =  )  2~\C (2  -  = r - r cos © = .33 x 10  3cos  O.73 x 10" cm 15  3  © + cos 0) 3  r  3  •5  cm.  13+  From the experimental data there are approximately 370 nuclei per cm of boundary.  For equally spaced sites there i s 1  = 2.7 x 10  "3  cm between each nuclei.  370  At the instant of c r i t i c a l nucleus formation, assuming a l l nuclei formed at the  Figure V-3.  same time, the following distribution would be present:  Distribution of nuclei at instant of c r i t i c a l nucleus formation.  The ^ precipitate of composition 23 at.$ aluminum is formed from a matrix of 19-30 atomic $> leaving a depleted zone of 19.08 at.$. Thus to produce the  phase the matrix Al composition must be increased  by 3-7 at, $ at the expense of material depleted by 0.22 at.$. o 0  0  The volume of depleted matrix = 3-7  volume of  x  0.22  ^  formed .  Considering one quarter volume of the ^> precipitate as being produced from the shaded area of Figure V-3 , the volume of matrix required = 3/7_ 0.22  = 30.7  7.3 x k x  10"  10" cm 16  16  3  Assuming the depth of boundary is equivalent to the width of the  precipitate, (i.e., 2 x 0.81 x 10 m) 5  C  the area required to supply the  necessary A l atoms i s 30.7 x 10 1.62  X  = 19  1 6  10 cm . _11  x  2  10-5  If this atomic flux i s obtained from the complete length of grain boundary then A l atoms must come from 19 x 10  = 1.1U x 10~ cm 7  1 1  I3U.I9 x 10"  5  or 11.k angstroms from the boundary. This i s consistent with the concept that i n i t i a l nucleation is essentially controlled by a grain boundary reaction thereby depleting the grain boundaries^requiring that further growth be a volume diffusion controlled mechanism.  156  APPENDIX VI Calculation of Composition Contours Associated with a Specific Precipitate Shape A.  Two-dimensional solution to Laplace s equation for a lenticular 77  allotriomorph using bipolar coordinates. a - a cos fyx = a ( l - cos  sin^-i  .Figure VI-1.  sin ^  s i n ^  Allotriomorph shape,  Using a precipitate axial ratio = sin £ i =5-6 1 - cos £ ! f  --31.25 or. 180 --31.25 = 148.75°  x  Laplace s equation in 2^D i s  c) c +^ c 2  2  Taking C to be a function of ^ only i.e., uniform composition around the precipitate.  c2 = 0 and C = A £ + B Applying the boundary conditions i.e., at  I = J>  i  = s  C  C  C  =  Coc  137  o ° o  f  G: - C»0 = S - cO C  £  C  Example Calculation: Contour  C - C oo  0.2  Radius of Contour •f 0  (.  a  C - Coo  - Ji C g  Coo  29-75  (&=!")  sin 89.2°  2.02  Center of Contour  a cot 89.2°  1.75  1"—^1  B.  2 4 > 7 3  Solution to the Diffusion Equation for an Oblate Spheroid.  Using as an approximation an axial ratio of k, and the experimental supersaturation, f = O.O7O5, values for  • = axial ratio  and (Tj  2  ~^ )  can be obtained .from Figure VI-2a.  2  Figure VI-2b  yields a value for  ( n_2-e>2) 2  From these values solutions can be obtained for •Solving the equation . 2  s '•rti  dt by numerical integration  t t  2  (  _(^ ) 2  using Simpson's rule for a range of values of t, yields the function  The growth rate of the oblate spheroid i s defined by |2  « A  2«  22 2  f c_^=7ii ri;.& ) F 7? 8)e^ =  C s  Coo  2  (  (  _t where F f ^ ! (J) =  / ^  2  ±  1  2  e , t ( t -0, )2  and thus values of 7^ and composition contour.  2  dt  2  can be obtained as a function of the  For the oblate spheroid the contour shapes  are defined by x  2  + jf  = 1  and thus the diagram as shown in Figure 51b can be constructed.  Figure VI-2.  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