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Periodic precipitation in Cu-Ni-Co Vandermousen, Roland F. 1967

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PERIODIC PRECIPITATION IN Cu-Ni-Co  by  ROLAND F. VANDERMOUSEN  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n the Department of METALLURGY  We a c c e p t t h i s  t h e s i s as conforming t o the  standard required  from c a n d i d a t e s f o r the  degree o f MASTER OF APPLIED SCIENCE  Members o f the Department of M e t a l l u r g y The  University  of B r i t i s h  J u l y , 1967  Columbia  In p r e s e n t i n g requirements Columbia, for  by  the  the  of  this  gain  Department  I further thesis  that shall  of  copying not  or  August  or  for  in p a r t i a l at  agree  publication  8 , 1967  Columbia  University  that  scholarly  by h i s  Metallurgy  the  fulfilment  s h a l l make i t  be a l l o w e d w i t h o u t  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date  Library  Head o f my D e p a r t m e n t  understood cial  that  and s t u d y .  copying  thesis  an a d v a n c e d d e g r e e  I agree  reference  tensive  for  this  of  freely  the  British available  permission  for  ex-  p u r p o s e s may be  representatives. of  of  this  thesis  my w r i t t e n  It for  gran is  finan-  permission.  -  11  -  ABSTRACT An i n v e s t i g a t i o n of p r e c i p i t a t i o n was  c a r r i e d out on the system  Cu-Ni-Co a t temperatures v a r y i n g from 600°C to 800°C over a wide range of c o m p o s i t i o n .  The p r e c i p i t a t i o n p r o c e s s was  studied primarily  by  use of X-ray d i f f r a c t i o n t e c h n i q u e s .  The p r e c i p i t a t i o n was composition process.  observed to procede by a s p i n o d a l  The i n i t i a l wavelength  o f the modulated  de-  structure  o  was  i n the range 40 to 55A  s p i n o d a l decomposition. a l l o y s aged  , i n good agreement w i t h the theory f o r  On a g e i n g , a c o a r s e n i n g was  i n s i d e the s p i n o d a l .  observed f o r a l l  T h i s c o a r s e n i n g appeared  to f o l l o w  a law of the form Q where Q  = wavelength  Qo = wavelength  m  - Qo  m  = k  (t-to)  a t time t b e f o r e the c o a r s e n i n g b e g i n s at time to  k and m = c o n s t a n t s The a c t i v a t i o n energy o f the c o a r s e n i n g p r o c e s s was + 10 k c a l s u g g e s t i n g t h a t volume d i f f u s i o n was  found to be 65 kcal/mole  the c o n t r o l l i n g  factor.  RESUME Le pre'sent t r a v a i l e t u d i e l a f o r m a t i o n de p r e c i p i t e ' s dans d i v e r s a l l i a g e s du systeme Cu-Ni-Co au cours de v i e i l l i s s e m e n t s a des de 600°C a 800°C.  i / e v o l u t i o n de l a p r e c i p i t a t i o n a 6te  p r i n c i p a l e m e n t par d i f f r a c t i o n des rayons  temperatures  suivie  X.  On a observe que l a p r e c i p i t a t i o n procede par decomposition spinodale.  La l o n g e u r d'onde i n i t i a l e de l a s t r u c t u r e modulee e s t comprise o  e n t r e 40 e t 55A;  en bon a c c o r d avec l a . t h e o r i e de l a decomposition s p i n o d a l e  de Cahn.  Au cours du v i e i l l i s s e m e n t  longeur d'onde pourtous  on a mesure un accroissement de l a  l e s a l l i a g e s t r a i t e s a l ' i n t e r i e u r du s p i n o d a l .  Cet a c c r o i s s e m e n t de longeur d'onde ob£it a une l o i de l a forme Cf _ ou  Q  m Q  o  =  k  (t-to)  = longeur d'onde au temps  tjf  Qo = l o n g e u r d onde avant que l e grossisement  commence, a to  k . et m = c o n s t a n t e s if e n e r g i e d ' a c t i v a t i o n determinee; est  du mecanisme de c r o i s s a n c e des p r e c i p i t e s  a ete  sa v a l e u r , 65 kcal/mole suggere que l a d i f f u s i o n en volume  l e facteur  preponderant.  -  iv  -  TABLE OF CONTENTS  Page 1.  2.  INTRODUCTION The S p i n o d a l Concept  1  X-Ray D i f f r a c t i o n O b s e r v a t i o n s  3  Theory o f S p i n o d a l Decomposition  4  Wavelength I n c r e a s e  7  Systems I n v e s t i g a t e d  8  Present I n v e s t i g a t i o n  12  EXPERIMENTAL Specimen P r e p a r a t i o n X-Ray D i f f r a c t i o n  3.  4.  13 •.  14  Metallography  15  E l e c t r o n Microscopy  15  RESULTS P r e l i m i n a r y experiments f o r the d e t e r m i n a t i o n o f t i e l i n e s i n the t e r n a r y Cu-Ni-Co phase diagram  15  Sidebands  22  Metallography  27  E l e c t r o n Microscopy  28  DISCUSSION L a t t i c e Parameters S p i n o d a l Decomposition  37 ..  D e t e r m i n a t i o n o f the Chemical S p i n o d a l E f f e c t o f S t r a i n Energy on the L i m i t o f S p i n o d a l Decomposition  40 41  . .  43  T a b l e o f Contents  (Cont)  -  v Page  I n i t i a l Wavelength  44  Coarsening  47  A c t i v a t i o n Energy Metallography  5.  . .. .  58 60  CONCLUSIONS  61  REFERENCES  62  APPENDICES I. Ila.  R e s u l t s o f X-Ray Photographs  64  D e t e r m i n a t i o n o f the S p i n o d a l  70  - Cook and H i l l i a r d  70  Method  - Regular S o l u t i o n Model  72  lib.  D e t e r m i n a t i o n o f the I n i t i a l Wavelength ....  73  III.  D e t e r m i n a t i o n o f the Change of the S t a b i l i t y L i m i t due to S t r a i n Energy  75  E v a l u a t i o n of the S t r a i n Energy  77  E f f e c t of the C o n f i g u r a t i o n a l Entropy Change  78  IV. V.  - vi LIST OF FIGURES Page 1. 2.  Au-Pt Phase Diagram  2  Schematic F r e e Energy Curves of a B i n a r y System w i t h a M i s c i b i l i t y Gap  2  3.  Cu-Ni-Co Phase Diagram  16  4.  X-Ray D i f f r a c t i o n P a t t e r n s  23  5.  V a r i a t i o n of 69 w i t h Ageing Time  25  6. t o l 4 .  Wavelength I n c r e a s e w i t h Ageing Time  26,29-32  15.to20.  O p t i c a l m i c r o s t r u c t u r e s ( A l l o y C a t 700°C)  33-35  21.to22,  E l e c t r o n Microscope P i c t u r e s Hours a t 700°C)  23.  ( A l l o y C Aged  V a r i a t i o n of the L a t t i c e Parameter Lines  10 36  Along the T i e 38  24.  L i n e s of Constant L a t t i c e Parameter  25.  Pseudo-Binary  26.  T h e o r e t i c a l I n i t i a l Wavelength as a F u n c t i o n o f Temperature, Composition and T i e L i n e  46  Composition P r o f i l e s f o r D i f f e r e n t A l l o y s Same T i e L i n e  48  27.  i n Cu-Ni-Co..  Phase Diagram and S p i n o d a l  28. to30.  Average Ageing Curves  31.  Change i n S p e c i f i c I n t e r f a c i a l Energy w i t h p o s i t i o n i n Cu-Ni-Co  39 42  on the  49-51 Com^52  32.  Semi-log P l o t of the Average Ageing Curves  54  33.  Log-Log  55  34.  Theoretical Coarsening  35.  P l o t of the Average Ageing Curves Composition P r o f i l e s During  Arrhenius Plot  57 59  - v i i LIST OF TABLES Page I.  Summary of Systems p r e v i o u s l y I n v e s t i g a t e d  8  II.  L a t t i c e Parameters of A l l o y s 1^5  17  III.  Heat-Treatments and L a t t i c e Parameters of A l l o y s A and B .  18  IV.  L a t t i c e Parameters a f t e r Long A n n e a l i n g Times  19  V.  A l l o y Compositions  20  VI.  Probe A n a l y s i s of Tungsten i n A l l o y C  21  VII.  C a l c u l a t i o n s of Wavelength of Modulated S t r u c t u r e  24  VIII.  A l l o y s forming Sidebands a t V a r i o u s Temperatures  27  IX.  V a l u e s of k and m f o r the Four Ageing Curves  56  X.  C a l c u l a t i o n s f o r the D e t e r m i n a t i o n of the Spinodal^ Cook and H i l l i a r d Method  71  C a l c u l a t i o n s f o r the D e t e r m i n a t i o n of the S p i n o d a l , Regular S o l u t i o n Model  72  C a l c u l a t i o n s f o r the D e t e r m i n a t i o n of the I n i t i a l Wavelength  74  C a l c u l a t i o n s f o r the D e t e r m i n a t i o n of the E f f e c t of the C o n f i g u r a t i o n a l Entropy Change....  79  XI.  XII.  XIII.  a f t e r Quench  ACKNOWLEDGEMENT The author wishes to express h i s s i n c e r e g r a t i t u d e to Dr. L. C. Brown f o r h i s a d v i c e and a s s i s t a n c e throughout t h i s work.  He  a l s o wishes t o express h i s thanks t o Dr. E. P e t e r s f o r h i s h e l p i n interpreting  the r e s u l t s . S p e c i a l thanks a r e extended to Mr. E. Armstrong  for carrying  out t h e c h e m i c a l a n a l y s i s on t h e a l l o y s and to Mr. W. I r v i n e of the B. C. I n s t i t u t e o f Technology f o r t h e use o f a microphotometer. F i n a n c i a l a s s i s t a n c e p r o v i d e d by the N a t i o n a l Research c o u n c i l under g r a n t No. A2459 and by t h e Canada C o u n c i l as a r e s e a r c h s c h o l a r s h i p t o t h e author i s g r a t e f u l l y  acknowledged.  - 1 1.  INTRODUCTION  T h i s t h e s i s i s concerned w i t h a study o f a l l o y systems which a r e completely  m i s c i b l e a t h i g h temperatures but which show a m i s c i b i l i t y  gap.at low temperatures.  A t y p i c a l system  Such systems a r e o f c o n s i d e r a b l e cooled  (Au-Pt) i s shown i n F i g . ( 1 ) .  i n t e r e s t s i n c e a s i n g l e phase a l l o y  from h i g h temperatures i n t o . t h e 2-phase r e g i o n p r e c i p i t a t e s the  new phases without the d i f f i c u l t y o f n u c l e a t i n g a new c r y s t a l l o g r a p h i c structure.  Systems o f t h i s type i n c l u d e Cu-Ni-Fe, Cu-Ni-Co,  Al-Zn,  N i - A l and Au-Pt. The  Spinodal  Concept The  By c o n s i d e r i n g composition fig.  s p i n o d a l c u r v e w i l l be o f g r e a t  the schematic Helmoltz. f r e e energy " f " as a f u n c t i o n of  f o r the temperature T^ f i g ( 1 ) , a curve o f the shape shown i n  (2) i s o b t a i n e d .  By d e f i n i t i o n the s p i n o d a l curve i s the l o c u s of  ^2 p o i n t s l i k e A and B, where 0 f \2 0 f  importance i n t h i s work.  = 0~fig.  ( 2 ) . O u t s i d e the s p i n o d a l  ^ \2 i s p o s i t i v e and i n s i d e the s p i n o d a l 0 f i s n e g a t i v e . c  Gibbs has shown t h a t l i q u i d f l u c t u a t i o n s l a r g e i n extent  to  composition  but s m a l l i n amplitude i n s i d e the s p i n o d a l .  T h i s concept can be extended to s o l i d complexities  systems are unstable  s o l u t i o n s provided- the e x t r a  o f i n t e r f a c i a l and e l a s t i c energy a r e taken i n t o account.  T h i s w i l l be d i s c u s s e d  i n more d e t a i l s h o r t l y .  A  u  composition Au-Pt Phase Diagram Fig.  (1)  ^  pt  -  3  -  X-Ray D i f f r a c t i o n O b s e r v a t i o n s Bradley  (1) w h i l e i n v e s t i g a t i n g the system  Cu-Ni-Fe  o b t a i n e d unusual X-ray i n t e r f e r e n c e p i c t u r e s f o r the a l l o y Cu^ Fe N i ^ which had been homogenised a t 1000°C i n a s i n g l e phase r e g i o n of the phase diagram and then annealed a t 600°C i n s i d e the m i s c i b i l i t y The De.bye-Sherrer remained  l i n e s of the o r i g i n a l l y homogeneous f . c . c .  unchanged, but each l i n e was  gap.  lattice  f l a n k e d by d i f f u s e though  quite  s t r o n g sidebaq'ds. D a n i e l and L i p s o n (2) gave the f o l l o w i n g e x p l a n a t i o n f o r these i n t e r f e r e n c e p i c t u r e s : a n n e a l i n g the coherency was edges  a  d u r i n g the r e l a t i v e l y  s h o r t time of  conserved, but i n the d i r e c t i o n of the cube  p e r i o d i c v a r i a t i o n of the l a t t i c e c o n s t a n t had appeared.  By  s i m u l a t i n g a one d i m e n s i o n a l model and c a l c u l a t i n g the expected d i f f r a c t i o n e f f e c t s they found t h a t t h e i r i d e a was t h e i r o b s e r v a t i o n s , the wavelength Q  =  h N  where — i n the [100]  of the v a r i a t i o n b e i n g g i v e n by:  tan 9 6 0  C; = wavelength  (1.1)  expressed i n a number of u n i t - c e l l s  direction -  9  -  69=  =  a n g l e of the Bragg r e f l e c t i o n ( h k l ) d i f f e r e n c e i n a n g l e between the c e n t e r o f the sideband and i t s Bragg 2  -  N  =  h  2 + k  a  reflection  2 + 1  However t h i s model has a d i f f i c u l t y increases, n  i n agreement w i t h  i n t h a t when  i n c r e a s e o f the wavelength  the a n n e a l i n g time  of the f l u c t u a t i o n can be  A c c o r d i n g to t h i s model, t h i s would mean a d r a s t i c rearrangement a process very d i f f i c u l t  to v i s u a l i z e p h y s i c a l l y .  seen.  of atoms,  T h i s shortcoming l e d  -  d i f f e r e n t authors  to propose o t h e r models to e x p l a i n the  Hargreaves ( 3 ) l a m e l l a e of two  e n  the t r a n s f o r m a t i o n p r o c e e d i n g  growth p r o c e s s .  the theory of f o r m a t i o n of one  proposed a r e g u l a r arrangement of  t l a t t i c e between each p a i r of l a m e l l a e ,  form b e i n g r e c t a n g u l a r and  composition  proceeding  indeed  by a normal  by normal d i f f u s i o n without  the  requiring  However, d i r e c t  (See "Systems i n v e s t i g a t e d " ) i n d i c a t e t h a t t h e r e i s  Cohen and Averbach  (5)  theoretical considerations  show t h a t t h e . G u i n i e r model can  be a p p l i c a b l e o u t s i d e the s p i n o d a l and  Theory of S p i n o d a l The  by  only  even here i t n e i t h e r p r e d i c t s the  o r i g i n a l w i d t h of the zones nor accounts f o r t h e i r i n i t i a l  formation.  Decomposition  t h e o r e t i c a l approach to p r e c i p i t a t i o n i n systems w i t h  a m i s c i b i l i t y gap was formation  t h e wave  l a m e l l a e of another c o m p o s i t i o n ,  a p e r i o d i c a r r a y of l a m e l l a e , and  Hillert,  intermediate  a central lamella  the d r a s t i c rearrangement of atoms mentioned above. observations  coherent  on the o t h e r hand p r e f e r r e d  of i s o l a t e d zones c o m p r i s i n g  between two  growth of which was  (4)  Guinier  -  sidebands.  m e t a s t a b l e t e t r a g o n a l phases, w i t h and without  l a y e r s of the p a r  n u c l e a t i o n and  4  first  developed by H i l l e r t ( 6 )  of modulated s t r u c t u r e s .  to e x p l a i n the  He developed a s o l i d - s o l u t i o n model  allowing compositional  v a r i a t i o n i n one  c o n s i d e r a t i o n s l e d him  to the c o n c l u s i o n t h a t a p e r i o d i c v a r i a t i o n of  composition  m o d u l a t i n g the parent  r e d u c t i o n of the f r e e energy. p r e c i p i t a t i o n may  occur  dimension.  l a t t i c e was  Thermodynamic  consistent with  a  H i s model a l s o predicted t h a t p e r i o d i c  o u t s i d e the s p i n o d a l where the energy a c t i v a t i o n  b a r r i e r f o r n u c l e a t i o n i s low.  This  led  to the i d e a that the  might be of no r e a l s i g n i f i c a n c e to the decomposition  process.  spinodal  -  T h i s mathematical treatment was by C  a  n  and H i l l i a r d  n  (7).  They d e r i v e d an e x p r e s s i o n  f o r the Helmoltz  ^'(c) V  J  =  ^' * e  ^  o r  (f*  v  + K  ( c )  (v-c) ) dv  and  i s the c o e f f i c i e n t of the  term of an expansion r e p r e s e n t i n g f r e e energy due  a composition  f l u c t u a t i o n of the  2  t h a t i n s i d e the s p i n o d a l  x  dV  Cj£"2~ ^  °)  t n e  s o l u t i o n was  stable  wavelengths,  •  1  (~^7"~ \ 0) the s o l u t i o n was  unstable  to i n f i n i t e s i m a l f l u c t u a t i o n s of wavelengths g r e a t e r  with  than  where 2  2TT/3C =  T h i s equation  [-8ir  2  K/(d  1/2  2  f/dc  )]  that 2n/g  (1.3)  shows r i g h t away that at the s p i n o d a l 2n/g  I and  composition^  form:  to i n f i n i t e s i m a l s i n u s o i d a l f l u c t u a t i o n s o f , a l l d f  2TT 6c  3  showed that o u t s i d e the s p i n o d a l  respect  the i n c r e a s e i n  to i n t r o d u c i n g a g r a d i e n t . i n  C - Co = A Cos  but  first  c  Cahn (8) c o n s i d e r e d  with respect  c  gradient  = i s a constant  V  (1.2)  2  homogeneous m a t e r i a l of composition  = composition  c  K  and  molar volume  imperfections.  F= where:  -  extended to t h r e e dimensions  f r e e energy f o r an i s o t r o p i c b i n a r y s o l u t i o n of constant f r e e from  5  c  decreases where  dV I  I i^ 2 c  )  spinodal.  =  i n c r e a s e s , i . e . the  wavelength becomes s m a l l e r w i t h p r o g r e s s i v e l y g r e a t e r i n s i d e the  c  °° ,  critical  supersaturation  -  He f u r t h e r developed  6  -  the theory to take i n t o account  the  e f f e c t of s t r a i n energy accompanying c o m p o s i t i o n f l u c t u a t i o n s and showed the new  l i m i t of s t a b i l i t y  to correspond to the l o c u s of  i£where:  -»  n  =  l i n e a r expansion per u n i t c o m p o s i t i o n change  E  =  Young modulus f o r the average  p  =  Poisson  composition  ratio  T h i s m o d i f i e d s p i n o d a l becomes the o r d i n a r y one f o r n=0. t h a t the c l a s s i c a l  T h i s shows  thermodynamic concept of a r e g i o n i n the phase diagram  in.which the s o l u t i o n i s u n s t a b l e to i n f i n i t e s i m a l f l u c t u a t i o n s i s u n a l t e r e d by i n t r o d u c t i o n o f s u r f a c e t e n s i o n and  e l a s t i c energy.  the d e t a i l s a r e a l t e r e d ^ s u r f a c e t e n s i o n p r e v e n t s the on too f i n e a s c a l e w i t h o u t a l t e r i n g  However  decomposition  the c r i t e r i o n f o r s t a b i l i t y ,  elastic  energy a l t e r s the c r i t e r i o n i t s e l f by l o w e r i n g the e f f e c t i v e s p i n o d a l and n e c e s s i t a t i n g a l a r g e degree of s u p e r c o o l i n g to i n i t i a t e  decomposition.  A l t h o u g h a l l c o m p o s i t i o n f l u c t u a t i o n s w i t h wavelength than Q  = — C  can form, some f l u c t u a t i o n s develop more r a p i d l y  c  than  P£  o t h e r s because than Q  greater  of k i n e t i c c o n s i d e r a t i o n s .  w i l l not tend to develop because  for t h e i r formation i s quite small.  Wavelengths  l i t t l e greater  the thermodynamic d r i v i n g  Very l o n g wavelengths,  force  on the o t h e r  hand, w i l l not tend to develop e i t h e r s i n c e l o n g range d i f f u s i o n i s required.  I n t e r m e d i a t e wavelengths  that the amplitude o f the  grow most r a p i d l y and Cahn (8)  wavelength  grows most q u i c k l y o f  all.  The magnitude of the c o m p o s i t i o n f l u c t u a t i o n i n c r e a s e s w i t h time. as the f l u c t u a t i o n  approaches  showed  However  the e q u i l i b r i u m c o m p o s i t i o n v a l u e s , the  r a t e o f growth slows down.and the f l u c t u a t i o n begins to develop a square  -7 wave form w i t h compositions g i v e n i n the phase diagram  corresponding (Cahn 9 ) .  the e q u i l i b r i u m compositions quench i t s e l f  to the e q u i l i b r i u m v a l u e s  In f a c t , a c c o r d i n g to Cahn(9)  are e s t a b l i s h e d very q u i c k l y during  the  s i n c e o n l y a veryr s m a l l amount of d i f f u s i o n i s r e q u i r e d  owing to the f i n e s c a l e of the  fluctuation.  Cahn (10) extended h i s study of s p i n o d a l to  an a e o l o t r o p i c s o l i d of c u b i c symmetry.  in  the c r y s t a l i s of g r e a t importance.  seek out the plane w i t h the lowest  decomposition  In t h i s case the o r i e n t a t i o n  The: s p i n o d a l mechanism w i l l  e l a s t i c modulus.  T h i s may  be  c o n s i d e r a b l y l e s s than i n d i c a t e d by the average e l a s t i c c o n s t a n t . H i s calculations,showed  t h a t for most metals  and a l l o y s , s p i n o d a l  should g i v e p l a n e waves p r i m a r i l y on 3 (100) the f i r s t  stage of the decomposition  The  s t r u c t u r e at  would a c c o r d i n g l y be composed of  a p e r i o d i c a r r a y of rods l i n e d up i n the Wavelength  planes.  [l00]  direction.  Increase In s e v e r a l e x p e r i m e n t a l  i n v e s t i g a t i o n s , an i n c r e a s e of  the wavelength of the f l u c t u a t i o n w i t h ageing  time was  noticed.  d i f f e r e n t e x p l a n a t i o n s have been g i v e n f o r t h i s phenomenon. u s i n g a k i n e t i c treatment  growth of the average one, A l l other authors  depending on the decrease  stage of the t r a n s f o r m a t i o n and  Hillert  a gradual  seem to agree to a p a r t i c l e c o a r s e n i n g of s u r f a c e energy  (13,15,16).  Both  process  processes  to f o l l o w a k i n e t i c  equation  the form: m Q  - Q  m 0  (6)  because the l o n g e r ones have a g r e a t e r growth  b e i n g d i f f u s i o n c o n t r o l l e d might be expected of  Two  p r e d i c t e d the f o r m a t i o n of a wide spectrum of  wavelengths d u r i n g the f i r s t  rate.  decomposition  -.k(t-to)  (1.5)  - 8 where:  Qo = wavelength of the f l u c t u a t i o n b e f o r e the c o a r s e n i n g process  starts  to = time of f o r m a t i o n of the modulation  with  Qo  and m and k a r e c o n s t a n t s . T h i s  formula was  d e r i v e d by Greenwood (11) f o r the c o a l e s c e n c e of  spherical  p a r t i c l e s , where m takes the v a l u e 3, and k i s g i v e n by 3DS where  -gS  (1.6)  D = Diffusion.coefficient S = Solute concentration M = M o l e c u l a r weight a = specific  interfacial  energy  p = density The p a r t i c l e s formed by s p i n o d a l decomposition r a t h e r than s p h e r i c a l .  are p l a t e  However i t has been suggested  t h a t t h i s should make l i t t l e  d i f f e r e n c e to the growth  (or rod)  by Wagner  shaped  (12)  law.  Systems I n v e s t i g a t e d We decomposition.  have now  reviewed  the main t h e o r i e s r e l a t i v e to s p i n o d a l  S e v e r a l a l l o y s have been i n v e s t i g a t e d  work c a r r i e d out on the system Cu-Ni-Fe. important  A b r i e f summary of the more  i n v e s t i g a t i o n s w i l l be g i v e n .  Table I:  (Table I) w i t h most  Summary of Systems I n v e s t i g a t e d  System  Reference  Cu-Ni-Fe Al-Ni Cu-Ni-Co Al-Zn Au-Pt  13-14-15 16 14-15-17 18 19  D a n i e l and L i p s o n wavelength w i t h ageing  (13) s t u d i e d the i n c r e a s e of  time and  opted  the  f o r an e x p o n e n t i a l law of the form  Q = A log t +  0  (T)  (1.7)  but admitted  t h a t t h i s e q u a t i o n c o u l d not h o l d f o r v e r y s h o r t and  l o n g times.  In f a c t t h e i r r e s u l t s can be p l o t t e d e q u a l l y w e l l to g i v e  the c o a r s e n i n g law, (Eq.  1.5)  i n accordance  t h a t the e q u i l i b r i u m compositions times of a n n e a l i n g and  a r e reached  observations  even w i t h the s h o r t e s t  t h a t the wavelength i n c r e a s e i s s i m i l a r to a  phenomenon of g r a i n growth. . T h i s agrees of  with t h e i r  s p i n o d a l decomposition  completely w i t h the  s u b s e q u e n t l y developed  by X-ray d i f f r a c t i o n and  e l e c t r o n microscopy.  (Eq. 1.7)  was  the s h o r t ones.  compositions  They d e c i d e d  t h a t the  f o l l o w e d but w i t h a d i s c o n t i n u i t y ,  the c o n s t a n t A b e i n g 5 times g r e a t e r f o r the l o n g ageing for  theory  by Cahn.  Biedermann and K n e l l e r (14) s t u d i e d d i f f e r e n t  e x p o n e n t i a l law  very  times  They a l s o n o t i c e d a range i n the v a l u e of  wavelength l e s s than 15%,  i n o p p o s i t i o n to H i l l e r t ' s  than the  theory f o r  wavelength i n c r e a s e which r e q u i r e s a wide range. Hillert, corresponding  Cohen and Averbach (5) found  a growth  law  to Q  m  - Qo  = k (t-to)  m  (1.5)  w i t h the q u a n t i t y to-Qo /k s m a l l enough to be n e g l e c t e d . m  t h a t the k i n e t i c model of H i l l e r t  (6) was  They  the a c t i v e p r o c e s s  concluded  in this  precipitation. Tufton found  (15) i n a p u r e l y e l e c t r o n m i c r o s c o p i c study of t h i s  the same growth law and  growth, i n accordance  observed  syst  the f o l l o w i n g sequence f o r p a r t i c l e  w i t h Cahn's l a t e s t  s t u d i e s about s p i n o d a l  - l O ci ecompo s i t I o n : f  1)  n u c l e a t i o n of a c u b i c a l array  2)  growth of rods l y i n g i n p l a t e l e t s c o n f i n e d  Q  n e  ar  spherical particles to  the  cube f a c e s 3)  a f t e r l o n g e r ageing, form l a m e l l a e  The  l a t e r a l spread  i n agreement w i t h  the x - r a y  x-ray measurements were always made i n . s t a g e s  a l s o c l e a r l y showed t h a t the growth r a t e was ageing  of the rods to  2 and  evidence.  3.  He  v e r y dependant on  the  temperature.  System N i - A l T h i s system has Nicholson was  (16) who  been i n v e s t i g a t e d r e c e n t l y by A r d e l l  concluded t h a t  spinodal  and  decomposition  not. r e s p o n s i b l e f o r the modulated s t r u c t u r e which appeared at  supersaturations.  T h e i r c o n c l u s i o n was  based on the o b s e r v a t i o n  the f o r m a t i o n of randomly d i s t r i b u t e d n u c l e i d u r i n g subsequent p a r t i c l e alignment due  to e l a s t i c  the quench and  small of of  i n t e r a c t i o n among them.  They c o u l d not d e c i d e whether s p i n o d a l decomposition or n u c l e a t i o n r e s p o n s i b l e f o r the p r e c i p i t a t i o n at h i g h e r the g r e a t e r number of p a r t i c l e s i n c r e a s e d at the f i r s t  stage of the r e a c t i o n .  agreement w i t h increased  the law  from zero at v e r y  d e c o m p o s i t i o n was  not  supersaturations  because  (Eq. 1.5).  observed to be i n However the wavelength  s h o r t ageing times s u g g e s t i n g  occurring.  was  the p r o b a b i l i t y of alignment  Coarsening was  g i v e n by H i l l e r t  the  that  spinodal  11  -  -  System Cu-Ni-Co T h i s system i s p h y s i c a l l y q u i t e s i m i l a r to Cu-Ni-Fe. has  It  not been much i n v e s t i g a t e d presumably because i t s m e c h a n i c a l p r o p e r t i e s  do not a l l o w i t to be used so e a s i l y f o r permanent magnet a l l o y s . A first in  1949,  but  i n v e s t i g a t i o n was  they were not  conducted by G e i s l e r and  They e x p l a i n e d  quench of two  t h i s phenomenon by  they always o b t a i n e d  the f o r m a t i o n  mixed t e t r a g o n a l l a t t i c e s .  during  They a l s o found  p r e c i p i t a t i o n t a k i n g over, a f t e r the f i r s t growing out  from the g r a i n b o u n d a r i e s and  stages  of the  side-  the  discontinuous transformation,  consuming the g r a i n s .  They  a l s o r e l a t e d a p h y s i c a l hardening to the coherent p r e c i p i t a t e and subsequent s o f t e n i n g to the l o s s of coherency and the m a t r i x  grains.  miscibility  (17)  a b l e to quench s u f f i c i e n t l y f a s t to r e t a i n  an u n a l t e r e d m a t e r i a l at room temperature and bands.  Newkirk  They o n l y s t u d i e d one  the  recrystallisation  a l l o y i n . t h e c e n t r e of  of  the  gap. L a t e r Biedermann and  Kneller  (14) p o i n t e d  out,that  such  a d i f f e r e n c e i n p r e c i p i t a t i o n behaviour between the s i m i l a r Cu-Ni-Fe and  Cu-Ni-Co systems seemed improbable.  the same a l l o y as G e i s l e r and decomposition process F i n a l l y , Tufton Cu-Ni-Co.like  (15)  was  They subsequently i n v e s t i g a t e d  Newkirk and  concluded t h a t the  b a s i c a l l y the same as f o r the Cu-Ni-Fe system.  s t u d i e d the same a l l o y composition  Cu-Ni-Fe.is  alloy  and  confirmed  an a l l o y which decomposes S p i n o d a l l y , but  k i n e t i c s of the r e a c t i o n are f a s t e r . He was  undergoing s e v e r e s t r a i n i n g d u r i n g  and  Newkirk n o t i c e d the d i s c o n t i n u o u s  a l s o i n d i c a t e d t h a t the the p r e c i p i t a t i o n and  that  the  lattice  like Geisler  p r e c i p i t a t i o n at g r a i n b o u n d a r i e s  - 12 and  r e c r y s t a l l i z a t i o n at the l a t e r stages  none of these  i n v e s t i g a t i o n s was  wavelength d u r i n g Present  any  of the p r e c i p i t a t i o n .  In  study made of the i n c r e a s e i n  ageing.  Investigation As can be seen from the p r o c e e d i n g  composition  has  text, only  one.alloy  been i n v e s t i g a t e d i n the Cu-Ni-Co. system.  I t seemed  t h a t i t would be i n t e r e s t i n g to i n v e s t i g a t e t h i s system i n more d e t a i l , e s p e c i a l l y by v a r y i n g specimen composition This.work was  t h e r e f o r e undertaken w i t h  and.temperature of  ageing.  the f o l l o w i n g o b j e c t i v e s ' i n  mind: (a)  -- to o b t a i n x-rray i n t e r f e r e n c e . p i c t u r e s c o n f i r m i n g  modulated s t r u c t u r e s occur d u r i n g the (b)  —  to t r y to d e c i d e  that  precipitation.  to which k i n d of p r o c e s s  t h i s modulated  s t r u c t u r e c o u l d be a t t r i b u t e d : - Spinodal  decomposition  - Alignment of p a r t i c l e s because of e l a s t i c i n t e r a c t i o n s between .them (c)  —to  check i f t h e r e i s an i n c r e a s e i n the wavelength of  f l u c t u a t i o n i n Cu-Ni-Co s i m i l a r to Cu-Ni-Fe. if a  t h e r e i s a wavelength i n c r e a s e and  To f i n d  the growth  the  law  to see i f t h i s can be r e l a t e d to  theory. (d)  —  to see i f any  i n f o r m a t i o n can be p r o v i d e d  by  consideration  of the i n i t i a l wavelength of the modulated s t r u c t u r e ^ r e l a t i n g i t to comp o s i t i o n and (e)  —  on the growth  annealing  temperature.  to observe the law.  effect  of the degree of  supersaturation  - 13 (f)  —  to r e l a t e the s p i n o d a l w i t h the l i m i t s of o c c u r r e n c e of the  decomposition.  2. Specimen  EXPERIMENTAL  Preparation A l l o y s were prepared from e l e c t r o l y t i c copper and  sintered  c o b a l t and n i c k e l powders manufactured by S h e r r i t t Gordon L i m i t e d .  All  m a t e r i a l s used were of 99.9 + % p u r i t y not c o n s i d e r i n g the presence of N i i n the Co and Co i n the N i . Charges weighing a p p r o x i m a t e l y 50 gms were melted i n an a r c f u r n a c e u s i n g a non-rconsumable atmosphere.  t u n g s t e n e l e c t r o d e and an Argon  Each specimen was melted t h r e e times and was  between m e l t i n g s to ensure homogeneity.  turned over  The a l l o y s were then annealed  f o r 50 hours i n a tube f u r n a c e u s i n g c r a c k e d ammonia as the p r o t e c t i v e atmosphere.  The temperature used was  chosen to be a t l e a s t 50°C  into  the s i n g l e - p h a s e r e g i o n o f the phase diagram, and v a r i e d from 900°C to 1150°C depending on the a l l o y c o m p o s i t i o n . specimen was  r e c r y s t a l l i z e d by c o l d r o l l i n g  A f t e r quenching, the and a n n e a l i n g f o r 30 .minutes  the s i n g l e - p h a s e r e g i o n o f the phase diagram and.then quenched. quenches were made i n t o  All  brine.  At t h i s s t a g e the specimens used f o r m e t a l l o g r a p h y ^ e l e c t r o n microscopy, and d i f f r a c t o m e t e r a n a l y s i s were ready f o r the subsequent heat-treatments.  Those used f o r powder x - r a y d i f f r a c t i o n had to undergo  t h i s a d d i t i o n a l p r e p a r a t i o n : . a f i n e powder was j e w e l l e r ' s saw.  T h i s was  diameter q u a r t z tube.  o b t a i n e d by use of a  then s e a l e d under vacuum i n a 0.5 mm  inside  To remove the e f f e c t of c o l d work caused by  - 14 sawing,  the powder was heated f o r 30 minutes  of the phase diagram. slightly  i n the s i n g l e phase  I n the c o u r s e of t h i s a n n e a l t h e powder  t o form f i n e r o d s .  region  sintered  A f t e r quenching, the tube was opened and the  s i n t e r e d r o d c u t i n t o l i t t l e p i e c e s , a p p r o x i m a t e l y 1/2 cm l o n g , were s e a l e d s e p a r a t e l y under vacuum i n 1 mm  which  i n s i d e diameter q u a r t z tubes.  These i n d i v i d u a l tubes were used f o r the d i f f e r e n t heat t r e a t m e n t s . The specimens,  e i t h e r the s i n t e r e d powder i n a s e a l e d  o r . t h e b u l k specimen^were annealed a t the d e s i r e d desired  time i n a hydrogen  into.brine.  temperature and f o r the  f u r n a c e o r i n a s a l t b a t h and then quenched  C o r r e c t i o n s were made when n e c e s s a r y f o r the time r e q u i r e d  to r e a c h a g e i n g X-Ray  tube  temperature.  Diffraction Attempts were made t o study sidebands u s i n g the x - r a y d i f f r a c t o -  meter.  However they were never observed u s i n g t h i s  technique, possibly '  because o f t h e i r low i n t e n s i t y r e l a t i v e to background. observed u s i n g the Debye-Sherrer t h i n specimens  Sidebands were  Camera p r o v i d e d c a r e was taken to use  (^0.4 mm diameter) and t o a l i g n them c a r e f u l l y . A l l  photographs were taken i n a camera 360/TT mm i n diameter u s i n g Cu. fa  o radiation  (X= 1.542 A) w i t h a normal The powder photographs  visible  exposure time of 4 hours.  gave sidebands which were q u i t e  to the unaided eye and c o u l d be c o n v e n i e n t l y measured u s i n g the  s t a n d a r d l i g h t b o x f o r powder p a t t e r n s . photometer  Attempts  t r a c e s o f the sidebands were c o m p l e t e l y u n s u c e s s f u l , however,  because o f t h e i r low i n t e n s i t y above background. to r e s o l v e s m a l l d i f f e r e n c e s i n i n t e n s i t y , unexcelled  to obtain micro-  f o r locating  Because  of i t s a b i l i t y  the human eye.appeared  the exact p o s i t i o n o f the s i d e b a n d s .  t o be.  - 15 Metallography  The b u l k specimens were m e c h a n i c a l l y p o l i s h e d and etched w i t h c o n c e n t r a t e d n i t r i c a c i d b e f o r e Electron  then  examination.  Microscopy The sample was  n i t r i c acid.  The  p o l i s h e d m e c h a n i c a l l y and etched w i t h  s u r f a c e was  diluted  examined u s i n g a 2-step r e p l i c a t i o n  technique,  This consisted of: - making a c e l l u l o s e a c e t a t e r e p l i c a and  shadowing i t w i t h Cr.  - making a carbon r e p l i c a , evaporated n o r m a l l y to the s u r f a c e of the f i r s t  one. - d i s s o l v i n g the o r i g i n a l c e l l u l o s e a c e t a t e r e p l i c a i n acetone. The HU  v o l t a g e was  used  11A  e l e c t r o n microscope,  to take the  micrographs.  3.  RESULTS  o p e r a t i n g a t 50 KV  accelerating  B r e l i m i n a r y Experiments f o r the D e t e r m i n a t i o n of the T i e - L i n e s i n the Ternary Cu-Ni-Co Phase Diagram.  .« The Neumann (20) temperatures  t e r n a r y phase diagram.for Cu-Ni-Co as g i v e n by Dannohl and [ f i g . (3)3 shows the l i m i t s o f . s o l u b i l i t y a t v a r i o u s  but does not i n c l u d e any The method chosen  tie-line5.  f o r t h e i r d e t e r m i n a t i o n was  the l a t t i c e parameters ,with.xomposition.  o Co  S i n c e the l a t t i c e parameters  of  o  (3.537A) and Ni(3.517A) a r e almost  find  the v a r i a t i o n of  the same i t would not be p r a c t i c a l to  the p o s i t i o n of the t i e - l i n e on the Co r i c h s i d e of the m i s c i b i l i t y  gap.  o  Cu and Ni.have  s i g n i f i c a n t l y d i f f e r e n t parameters  end of the t i e - l i n e should be d e t e c t a b l e .  (Cu 3.6153A) and  so  this  - 17 -  In o r d e r to f i n d v a l u e s f o r the l a t t i c e parameters as a f u n c t i o n of c o m p o s i t i o n a t the Cu c o r n e r of the phase diagram, a s e r i e s of Cu r i c h a l l o y s  (1-5) were prepared  and annealed  i n the 1-phase r e g i o n of the phase diagram and quenched. parameters of these a l l o y s  f o r 73 hours The  lattice  i n the quenched s t a t e are g i v e n i n T a b l e I I  below. Table I I :  L a t t i c e parameters of a l l o y s 1 to 5 a f t e r  quench  o  L a t t i c e parameter  Alloy  1  3 610  +  0 001  2  3 602  ±  0 001  3  3 595  +  0 001  4  3 589  ±  0 001  5  3 582  +  0 001  These v a l u e s p l u s the known v a l u e s f o r the  (A)  binary.system  Cu-Ni (21) enable the f i r s t p a r t of the i s o - l a t t i c e parameter to be.drawn f i g . ( 3 ) . pared and annealed  To determine  curves  t i e - l i n e s , a l l o y s A and B were p r e -  f o r a l o n g time i n the 2-phase r e g i o n of the phase  diagram so as to r e a c h the e q u i l i b r i u m compositions and  then quenched.  The l a t t i c e parameters of the 2 - e q u i l i b r i u m phases a r e g i v e n i n Table I I I , along with t h e i r  heat-treatments.  - 18 Table I I I  Heat Treatments  and L a t t i c e Parameters  o f A l l o y s A and B  0  Heat  Alloy  Treatment  L a t t i c e Parameter  (A)  A  117 hours a t 700°C  3.596±0.003  3.556+0.002  B  90 hours a t 700°C  3.603*0.003  3.548*0.001  The v a l u e s g i v e n by the a l l o y s 1-5 l e d to knowledge o f the v a r i a t i o n o f the l a t t a i c e parameter solubility limit. parameters  w i t h c o m p o s i t i o n a l o n g the  The measurement o f the l a r g e r o f the 2 l a t t i c e  i n t h e e q u i l i b r a t e d a l l o y then enabled the C u - r i c h end o f  the t i e - l i n e t o be.determined  a t the temperature  of i n t e r e s t .  This  c o m p o s i t i o n t o g e t h e r w i t h the c o m p o s i t i o n o f the o r i g i n a l a l l o y  enabled  the t r u e t i e - l i n e t o be c o m p l e t e l y d e f i n e d . With  the 2 t i e - l i n e s known i t was p o s s i b l e to prepare the  a l l o y s l y i n g on these t i e - l i n e s ,  f o r the study of t h e p r e c i p i t a t i o n  (Alloys C to J ) . In the next p a r t o f t h i s work, r e s u l t s w i l l be p r e s e n t e d f o r a l l o y s heat t r e a t e d a t v a r i o u s temperatures.  I t i s therefore  n e c e s s a r y to check i f the t i e - l i n e changes w i t h temperature From the r e s u l t s of t h e x - r a y i n v e s t i g a t i o n s  (Appendix  or not.  I ) , i t can be  seen t h a t a l l a l l o y s l y i n g on,one t i e - l i n e g i v e the same l a t t i c e f o r the two e q u i l i b r i u m phases when annealed (Table I V ) .  temperature parameter one  f o r a l o n g time a t 700°C  T h i s c o n f i r m s t h a t a t t h i s temperature  on a t i e - l i n e .  parameters  they r e a l l y do l i e  When t h e same a l l o y s . a r e e q u i l i b r a t e d a t another  (e.g. 800°C o r 600°C) they a g a i n a l l have the same l a t t i c e  values i n d i c a t i n g  tie-line.  t h a t a t each temperature  they a r e a l l l y i n g on  - 19 T a b l e IV  Temperature °C  L a t t i c e parameters a f t e r l o n g a n n e a l i n g ( T i e l i n e I)  Alloy  E q u i l i b r i u m l a t t i c e parameters Cu r i c h  600  700  800  times  corner  Cu poor s i d e  F B G H  3.601±0.002 3.600*0.002 3.599±0.002 3.600+0.002  3.547+0.002 3.547+0.002 3.545±0.002 3.546+0.002  Average  3.600+0.002  3.546+0.002  B E F G H J  3.603±0.003 3.601*0.002 3.600+0.001 3.600+0.001 3.601+0.001 3.599+0.001  3.548±0.001 3.547±0.002 3.548±0.001 3.548+0.001 3.549±0.001 3.550+0.001  Average  3.601db0.002  3.548±0.002  F B G H  3.597±0.002 3.598+0.002 3.597±0.002 3.600+0.002  3.551±0.002 3.550±0.002 3.549+0.002 3.552+0.0pl  Average  3.598+0.002  3.550+0.002  - 20 The a c t u a l v a l u e s of l a t t i c e parameter agreement w i t h the known with  a r e i n complete  s l i g h t change o f the s o l u b i l i t y  limit  temperature. The  f a c t t h a t the t i e - l i n e s do n o t change w i t h  temperature  i s extremely c o n v e n i e n t s i n c e a l l o y s l y i n g on one t i e - l i n e can be t r e a t e d as b e l o n g i n g t o a q u ^ s i - b i n a r y system w i t h much in  analysis. The compositions o f the a l l o y s as g i v e n by  are given i n f i g .  ( 3 ) . T a b l e V shows the comparison  Table V  N°  Cu  1 2 3 4 5 C A D B E F  83 78 73 68 63 51.5 41.5 31.5 40 13 25 55 66 81  analysis  alloys .  Compositions  Chemical A n a l y s i s Composition At %  Prepared Composition At %  Alloy  G  Alloy  chemical  between the  prepared c o m p o s i t i o n s and the a n a l y s i s of the completed  H J  simplification  Ni  Co  Cu  10 15 20 25 30 30.5 33.5 36.5 25 34 30 20 16 11  7 7 7 7 7 18 25 32 35 53 45 25 18 8  50.2 38.9 30.7 41.6 12.9 25.0 53.5 68.5 80.6  Ni  Co  W  30.1 34.4 36.4 24.4 34.0 29.9 22.5 15.1 11.0  18.2 26.0 32.3 33.2 53.0 44.9 23.8 16.3 8.3  1.5 0.6 0.7 0.8 0.03 .0.1 0.2 0.2 0.1  - 21 The a n a l y s i s  showed  a l l o y s .  This  arc  furnace.  detectable It  i s  haye  sets  had  As  i n  disturbed  considered  of  composition  considerable  shown  tungsten,  segregation  of  tungsten  p o s s i b l e  a l l o y ,  two  that the  chemical did  i s  to  of  the  a l l o y  W  a  or  not very  VI  L&  2  electrode  low  i t s  the  r e s u l t s .  Probe  a n a l y s i s  r a d i a t i o n  at  30  cps  A l l o y  C  11  cps  Cu  10  cps  N i  10  cps  Co  6  cps  at  most  2 cps l e s s  a n a l y s i s  a  above than  of  W  probe  At.%  i n  W  i n  Chemical the  i n  showed  contamination  acc.  background 0.25  of  containing  presence.  KV  265  has  probe  contamination  W  C  the  supposedly  Pure  A l l o y  from  some  3%.  come  although  reveal  w i t h i n i n  l o c a l i z e d  a n a l y s i s  a f f e c t  Table  I n t e n s i t y  i n  agree  contamination  Vljhoweverj  surface  there  i n s u f f i c i e n t  presumably Table  even  tungsten  that  tungsten  values  1.5  the no At.%.  might  search  for  It  therefore  the  a l l o y  voltage  i s bulk  C  of  such  the  a  - 22 Sidebands T y p i c a l Patterns at D i f f e r e n t  Stages of the P r e c i p i t a t i o n  F i g . (4) shows t y p i c a l x-ray d i f f r a c t i o n patterns obtained for a l l o y C a f t e r d i f f e r e n t ageing times. - (4a):  a f t e r quench:  the t y p i c a l l i n e s of the s i n g l e phase f.c.c.  s t r u c t u r e are r e t a i n e d . - (4b):  a f t e r a short annealing time (30min at 700°C)marked sidebands  have appeared f l a n k i n g - (4c):  the o r i g i n a l f.c.c.  lines.  a f t e r a longer annealing time (60min at 700°C) the sidebands  p r o g r e s s i v e l y move closer to the main l i n e s and are almost superposed w i t h them. - (4d):  a f t e r a very long annealing time (703 0 min at 700°C)f.c.c. l i n e s  the t o e q u i l i b r i u m phases are obtained. W  D e t a i l e d Examination of One X-Ray Photograph We have examined i n d e t a i l photograph number 3 where sidebands were c l e a r l y v i s i b l e f o r several main l i n e s .  The  following  procedure was used to f i n d the value of the wavelength of the modulation: a)  Determination of the l a t t i c e parameter by use of the  Bragg formula: A =  where  CuKa d =  so that  =  o  2d s i n 9  1.542A (Cu f i l t e r e d with Ni)  F i g . 4a  A f t e r Quench  F i g . 4b  30 Minutes  F i g . 4c  60 Minutes A n n e a l i n g  F i g . 4d  7080 Minutes  F i g . 4:  X-Ray D i f f r a c t i o n P a t t e r n s .  Annealing  Annealing  Alloy  C  - 24 The v a l u e o f a was c a l c u l a t e d f o r s e v e r a l l i n e s and the average v a l u e used i n the second p a r t o f the c a l c u l a t i o n . b)  Determination  o f the Wavelength o f the M o d u l a t i o n  The D a n i e l and L i p s o n formula was used: _ ah .tan^9 Q  (1.1)  " 7h^+k2+17) 6 e  where 69 i s the d i s t a n c e between the main l i n e and t h e s i d e b a n d . T a b l e V I I g i v e s the complete c a l c u l a t i o n s f o r one photograph . Table VII:  Line (hkl)  9 (degree)  C a l c u l a t i o n s o f Wavelength o f Modulated S t r u c t u r e  tanG  sin9  a  a  o  (A)  111  22.06  0.4052  200  25.68  0.4808  220  37.645  0.6107 0.7714  average  69  69  degree  radian  Q o  (A)  0.56+0. 020 1.97+0.03 50+2  3.571  04 O O  0.88+0. 021 ,53±0.03 56±1  O*  0.74+0. 021 ..30±0.03 53±2  +1  311  45.71  0.7158 1.0250  3.572  CN  222  48.39  0.7477  3.572  ro  400  59.60  0.8625  3.575  1.00+0. 021 ..74±0.03 57±1  Q average=54 I t i s to be n o t i c e d t h a t the v a l u e s g i v e n f o r 69 a r e the r e s u l t of s e v e r a l readings. mainly  The e r r o r a s c r i b e d to each r e a d i n g depends  on the q u a l i t y o f the photograph .  T a b l e V I I shows t h a t t h e  v a l u e s o f 0 g i v e n by the d i f f e r e n t l i n e s a r e a p p r o x i m a t e l y c o n f i r m i n g the D a n i e l and L i p s o n e q u a t i o n photographs, the measurement line,  (1.1).  For a l l other  o f .69^wasronly c a r r i e d out on the (200)  t h e one g e n e r a l l y g i v i n g the b e s t sidebands  sharpness).  the same, thus  ( l a r g e 69 and b e s t  QCA)  I I _  I  to ON  I  time i n hours  _  0  _  j  I  I  I  L  2  3  4  - 27 Fig.  (5) shows, the v a r i a t i o n o f (SO w i t h a g e i n g time f o r 1  a l l o y A aged a t 700°C.  The l i m i t s on each p o i n t g i v e the u n c e r t a i n t y  i n l o c a t i n g t h e p o s i t i o n o f the sideband i n the x - r a y Fig.  photograph.  (6) shows the r e s u l t o f the c a l c u l a t i o n f o r the v a r i a t i o n o f  wavelength  Q w i t h ageing time.  A d e f i n i t e i n c r e a s e i n wavelength  w i t h time can be seen. R e s u l t s f o r a l l the a l l o y s i n v e s t i g a t e d a r e t a b u l a t e d i n Appendix  I.  T a b l e V I I I l i s t s a l l o y s g i v i n g sidebands a t the v a r i o u s  temperatures  studied. Table VIII  T  A l l o y s g i v i n g sidebands a t v a r i o u s T  S.B. observed  No. S.B. observed  600°C  B-G-H  E-F-J  700°C  B-G-H-A-C-D  E-F-J  800°C  B-G-H  E-F-J  Graphs o f wavelength v e r s u s time f o r those a l l o y s showing plotted  in fig.  sidebands a r e  7 to 14.  Metallography F i g s . 15 to 20 show o p t i c a l m i c r o s t r u c t u r e s f o r a l l o y C aged  f o r v a r i o u s p e r i o d s a t 700°C.  A l l specimens  showed many twin bands  caused by the r e c r y s t a l l i s a t i o n anneal a t 1000°C.  F o r a g e i n g p e r i o d s up  to 10 hours no t r a c e o f any p r e c i p i t a t e c o u l d be seen except f o r a v e r y s m a l l amount quenched 4 at the g 11  r a  i n boundaries.  At l o n g e r times a s i g n i f i c a n t  amount o f p r e c i p i t a t i o n took p l a c e a t g r a i n b o u n d a r i e s and a l o n g d e f i n i t e  - 28 p l a n e s i n the m a t r i x . E l e c t r o n Microscopy Fig.  (21) and (22) show  the s t r u c t u r e o f a l l o y C a f t e r  10  hours o f a g e i n g a t 700°C. Groups o f about 5 p l a t e l e t s  are observable.  The average  o wavelength  i s about 1000A.  T h i s v a l u e i s a p p r o x i m a t e l y 8 times  l a r g e r than the one expected from e x t r a p o l a t i o n o f the x - r a y d i f f r a c t i o n results  t o t h i s a g e i n g time (see f i g . 33). I t i s p o s s i b l e t h a t the c o o l i n g r a t e o b t a i n e d i n the b u l k  specimen used f o r e l e c t r o n microscopy was c o n s i d e r a b l y s m a l l e r than the one f o r x - r a y d i f f r a c t i o n .  The s t r u c t u r e observed might  then be due  to the s m a l l c o o l i n g r a t e d u r i n g the quench, c a u s i n g t h e f o r m a t i o n of a modulated  s t r u c t u r e w i t h an a p p r e c i a b l e wavelength  d u r i n g the  quench i t s e l f .  ( T h i s has been observed f o r example by G e i s l e r and  Newkirk ( 1 7 ) ) .  The subsequent  wavelength  10 hours o f a n n e a l i n g would i n c r e a s e the  to the observed v a l u e .  0  1 Fig. 8  2  3 Alloy D  700°C  4  5  F i g . 10  Alloy H  600°C  140 120  Q(A)  -  -  100  80  -  60 l 40  time i n hours  20  0  5  10  F i g . 11  Alloy B  15  20  25  600°C  14u Q'  (A)  120  •  800°C  700°C  100  80  i I  I  /  60  40  time i n min.  20  0  i  i  2  4  6 F i g . 12.  8  1  i  i  10  12  14  Alloy H  700°C 800°C  i 16  i 18  20  - 32 -  140 Q(A)  120  . A  800°C  100  ^  -J  700°C  80  60  i x  1  /  »  i  40  1 20  time i n min. .J  1  i  '  0  1 10  F i g . 14  A l l o y B.  '  '  •  i  20 700°C 800°C  - 33 -  F i g . 16:  A l l o y C - Aged 1 hour a t 700°C - X650  F i g . 18:  A l l o y C - Aged 27 Hours a t 700°C - X650  - 35 -  F i g . 20:  A l l o y C - Aged 120 Hours a t 700°C - X230  -  F i g . 22:  A l l o y C - Aged 10 Hours a t 700°C - X 10,000  36  - 37 4.  DISCUSSION  L a t t i c e Parameters Fig.  (23) shows the v a r i a t i o n of the l a t t i c e parameter  along the two t i e - l i n e s investigated using the r e s u l t s given i n Appendix I .  These curves w i l l be used l a t e r i n discussing the e f f e c t  of s t r a i n energy on precipitation.From these curves, the compositions o  at which l a t t i c e parameters of 3.600, 3.590.-.3.520A occur can be found.  The curve showing the v a r i a t i o n of the l a t t i c e parameter along  a l i n e of constant Co content (7%) can also be drawn from the r e s u l t s obtained f o r a l l o y s 1 to 5.  F i n a l l y , the v a r i a t i o n of the l a t t i c e  parameter with composition i s known f o r the two binary systems Cu-Ni (21) and Ni-Co (22). Enough information i s provided by these r e s u l t s to enable l i n e s of constant l a t t i c e parameter to be drawn over most of the ternary diagram^fig. (24). I t i s to be noticed that the values of the l a t t i c e parameter along the Cu-Co axis are not a v a i l a b l e i n the l i t e r a t u r e , presumably because of the very low s o l i d s o l u b i l i t y between Cu and Co.  However,  i t can be seen, i n f i g . (24), that f o r any reasonable v a r i a t i o n of the l a t t i c e parameter w i t h composition, the isoparameter curves already drawn can be extended smoothly to the Cu-Co a x i s . The v a r i a t i o n of the l a t t i c e parameter when  adding Co  to a Cu-Ni a l l o y i s rather unexpected since the l a t t i c e parameter i n i t i a l l y increases with increasing Co, but r i s e s to a maximum and then decreases. Taylor (22) found a s i m i l a r phenomenon i n the binary Ni-Co system and explained i t by considering that there was a tendency towards ordering at about 25 wt.% Co.  I t may be possible that t h i s Co-Ni  a (A) 3.610  3.600  t  3.590  L  ?~  38 -  0.0168  3.580  I 3.570  1  3.560  3.550 1 J  i  H  1  G  1  B  i  3.540 Cu 0.91 Co 001 N i 0.08  V a r i a t i o n o f the l a t t i c e parameter a l o n g t h e tie lines  a (A)  3.560 h  3.550 Co 0.035 Cu 0.82 N i 0.145  Tie Line II  E  :  i  Cu 0.085 Co 0.56_> N i 0.355  Tie Line I  F i g . 23  !  Co  0.37 Cu 0.2 N i 0.43  - 40 i n t e r a c t i o n extends into the ternary system as suggested by the ridge of the l a t t i c e parameter surface pointing approximately towards the composition mentioned above.  Another p o s s i b i l i t y i s that there  might exist some kind of magnetic i n t e r a c t i o n between the three species.  However no detailed consideration has been given to this  possiblility. Spinodal Decomposition As already stated i n the introduction, one of the purposes of this work was process was  to find out whether spinodal decomposition or another  responsible for the p r e c i p i t a t i o n . The f i r s t general observation which can be made i s that  x-ray interference pictures showed that modulated structures were obtained  from the very early stages of p r e c i p i t a t i o n over a wide  range of a l l o y compositions and annealing  temperatures.  The electron micrographs gave remarkable confirmation of the modulated structure.  Lamellae were observed i n groups of 5 or 6  and had a very regular wavelength.  They were v i s i b l e i n a l l parts of  the micrograph, with no tendency for p r e f e r e n t i a l p r e c i p i t a t i o n on grain boundaries, showing the homogeneous nature of the p r e c i p i t a t i o n process. These r e s u l t s are exactly what would be expected from a spinodal decomposition process. process i s occurring i s provided  Further confirmation that this i n subsequent sections of this  chapter.  - 41 As d i s c u s s e d i n the next s e c t i o n , the f a c t t h a t  modulated  s t r u c t u r e s a r e observed o n l y i n s i d e the s p i n o d a l i s a v e r y s t r o n g i n d i c a t i o n t h a t s p i n o d a l decomposition i s o c c u r r i n g . the i n i t i a l wavelength  Furthermore,  the v a l u e of  i s i n good agreement w i t h t h e o r e t i c a l v a l u e s  f o r s p i n o d a l decomposition.  The o b s e r v a t i o n t h a t the curves do not  e x t r a p o l a t e to zero wavelength  a t s h o r t ageing times s t r o n g l y  indicates  t h a t the mechanism o f n u c l e a t i o n f o l l o w e d by p a r t i c l e alignment, suggested  i n Ni-Al  as  ( 1 6 ) i s not o c c u r r i n g i s Cu-Ni-Co. ;  These o b s e r v a t i o n s ^ a l o n g w i t h p r e v i o u s work c a r r i e d on the same Cu-Ni-Co system by T u f t o n , seem evidence  out  enough.that  p r e c i p i t a t i o n i n t h i s sytem takes p l a c e by a s p i n o d a l decomposition process. D e t e r m i n a t i o n of the Chemical S p i n o d a l  temperature  I t has been,found  t h a t t i e - l i n e I does not change i n the  range 600-800°C.  A l l o y s l y i n g on t h i s t i e - l i n e  t h e r e f o r e be c o n s i d e r e d as b e l o n g i n g to a pseudo-binary  can  system^and the  l i m i t s of s o l u b i l i t y on the t i e - l i n e drawn as a f u n c t i o n of T ( f i g . g i v e a phase diagram  s i m i l a r to t h a t i n a b i n a r y system.  o f the s p i n o d a l a t any temperature procedures a v a i l a b l e f o r b i n a r y Two  position  can then be found by the v a r i o u s  systems.  d i f f e r e n t procedures were used  to l o c a t e the p o s i t i o n  of the c h e m i c a l s p i n o d a l . Regular s o l u t i o n model -  The  25)  Cook and H i l l i a r d Model (Ref. 23).  - 43 A f u l l d i s c u s s i o n o f these t e c h n i q u e s i s presented i n Appendix I I a .  The Cook and H i l l i a r d method gave the most r e a s o n a b l e  r e s u l t s f o r the p o s i t i o n of the s p i n o d a l .  This i s plotted  E f f e c t o f S t r a i n Energy on the L i m i t o f S p i n o d a l  in fig.  25.  Decomposition  As p r e v i o u s l y d i s c u s s e d i n the i n t r o d u c t i o n , s t r a i n  energy  causes the l i m i t s f o r s p i n o d a l decomposition to be c o r r e l a t e d to the curve locus o f : d f-'  .  2  Where n  2n  2  E  ,..  = l i n e a r e x p a n s i o n per u n i t c o m p o s i t i o n change, and  can  be e v a l u a t e d from the curves g i v i n g the v a r i a t i o n o f the l a t t i c e a l o n g the  parameter  tie-lines.  E  = Young modulus  v  - Poisson r a t i o  r a t h e r than to the c h e m i c a l s p i n o d a l . The maximum temperature o c c u r s i s a l s o reduced and  a t which s p i n o d a l decomposition  the d e c r e a s e of t h i s maximum  temperature  i s g i v e n by n E 2  T  - T  -  c where  T k  £  = critical  , .  (4 .2)  2(1-v) k Nv temperature  = Boltzmann c o n s t a n t  Nv = number of atoms per u n i t volume T h i s formula i s much e a s i e r to e v a l u a t e than e q u a t i o n 4.1  above.  -44D e t a i l s of the e v a l u a t i o n of the terms of e q u a t i o n a r e g i v e n i n Appendix I I I . 3.13°K, and  For t i e - l i n e I , T  £  - T was  to be  1.83°K f o r t i e - l i n e I I . Although  these v a l u e s a r e v e r y s m a l l , they a r e c o n s i d e r a b l e  o v e r e s t i m a t e s s i n c e the room temperature  e l a s t i c modulus used  them i s much g r e a t e r than t h a t a t the temperature e f f e c t of s t r a i n energy c h e m i c a l s p i n o d a l may spinodal  found  (4.2)  to c a l c u l a t e  of i n t e r e s t .  The  i s t h e r e f o r e c o m p l e t e l y n e g l i g i b l e , and  the  be c o n s i d e r e d to be the t h e o r e t i c a l l i m i t f o r  decomposition. E x p e r i m e n t a l l y , o n l y the a l l o y s aged i n s i d e the s p i n o d a l  showed sidebands spinodal  (Table V I I ) .  decomposition  This i s a strong i n d i c a t i o n that  i s o c c u r i n g and' not a n u c l e a t i o n and  p r o c e s s which would o c c u r f o r any c o m p o s i t i o n and  growth  temperature  inside  the m i s c i b i l i t y gap and not o n l y i n s i d e the s p i n o d a l . I n i t i a l Wavelength Sidebands were not o b t a i n e d i n the quenched s t a t e and would p o s s i b l y suggest  t h a t the decomposition  this  does not occur d u r i n g  the quench but v e r y q u i c k l y as soon as the ageing i s s t a r t e d .  However  a more l i k e l y p o s s i b i l i t y i s t h a t d u r i n g the quench a spectrum  of wave-  l e n g t h s i s formed as found  i n the A l - Z n system  would g i v e r i s e to v e r y d i f f u s e sidebands x - r a y photograph.  (18).  Such a  spectrum  i m p o s s i b l e to d e t e c t on  the  A f t e r a v e r y s h o r t time the optimum wavelength  would develop w h i l e the o t h e r ones decrease w i t h the subsequent appearance of a sharp sideband. a g e i n g curves to zero time cannot it  appears  Although  the e x t r a p o l a t i o n of  the  be c a r r i e d out w i t h g r e a t a c c u r a c y ,  t h a t the i n i t i a l wavelength l i e s between 40 and  o  55A  i n a l l cases.  - 45 These r e s u l t s a r e i n agreement w i t h those of H i l l e r t on Ni-Fe.  W i t h i n the s p i n o d a l he found o n l y a s l i g h t v a r i a t i o n of  wavelength w i t h c o m p o s i t i o n and  f i n d a d e f i n i t e i n c r e a s e i n i n i t i a l wavelength w i t h  temperature.  However the temperature  He  increasing  range i n v e s t i g a t e d i n the  0.64  0.83),  p r e s e n t work was  q u i t e s m a l l ( T / x v a r y i n g from  t h i s temperature  range H i l l e r t ' s e x p e r i m e n t a l curves g i v e o n l y a  15%  initial  the v a r i a t i o n which he observed d i d not  change i n a r e g u l a r f a s h i o n w i t h i n c r e a s i n g s u p e r s a t u r a t i o n . did  Cu-  c  to  and i n  i n c r e a s e i n i n i t i a l wavelength, a d i f f e r e n c e which i s s c a r c e l y  detectable. The r e s u l t s a r e a l s o i n good agreement w i t h v a l u e s of  Q c a l c u l a t e d from the theory of s p i n o d a l decomposition  A c c o r d i n g to Calm's theory i s going to develop f i r s t  ( I n t r o d u c t i o n p.6)the wavelength which i s g i v e n by  Q opt where Qc  =  VT  i s the c r i t i c a l wavelength.  d i f f e r e n t temperatures,  .  Qc The v a l u e s of Qopt f o r  compositions and  t i e - l i n e s have been  worked out i n Appendix l i b . The r e s u l t i s shown i n f i g .  26.  Although  the v a l u e s o b t a i n e d i n the p r e s e n t work are not a c c u r a t e enough to v e r i f y the change i n Qopt w i t h temperature  composition and t i e l i n e ,  they a r e i n good agreement w i t h the t h e o r e t i c a l v a l u e s . For comparison Hillert  (5)  the v a l u e s of i n i t i a l  wavelength c a l c u l a t e d by the theory  have been p l o t t e d on f i g 26 as w e l l .  Agreement w i t h  the  r e s u l t s of Cahn's theory and w i t h the experimental r e s u l t s i s good.  - 46 -  - 47 Coarsening An increase of the wavelength of the composition f l u c t u a t i o n with ageing time has d e f i n i t e l y been observed i n Cu-Ni-Co. Sidebands were found i n the x-ray photographs at an early stage i n the ageing process.  This implies that there i s only a very  small wavelength range i n the m a t e r i a l even at these short times. H i l l e r t ' s theory f o r increase i n wavelength i s not a p p l i c a b l e i n t h i s case.  Hence  (see i n t r o d u c t i o n p.7)  Furthermore, the very large increase i n o  wavelength observed during ageing (40 to 150A) i s much more than could be explained by t h i s theory. I t i s believed that the increase i n coarsening process.  must be due to a  There can be no volume free energy d r i v i n g the  process since the e q u i l i b r i u m compositions must be established very e a r l y i n the ageing ( r e f . 13 and 24).  Two suggestions f o r the d r i v i n g  force f o r the process can be made, one that i t i s a decrease i n i n t e r f a c i a l energy, the other that there i s a decrease i n s t r a i n energy.  C a l c u l a t i o n s i n Appendix IV show that the s t r a i n energy i s  n e g l i g i b l e i n comparison with the i n t e r f a c i a l energy. Comparison of the d i f f e r e n t coarsening curves gives the f o l l o w i n g q u a l i t a t i v e information: (1)  For a set of a l l o y s l y i n g at d i f f e r e n t points on the  same t i e - l i n e and aged at the same temperature, the curves are very similar.  The composition p r o f i l e s which would be observed at d i f f e r e n t  points on the t i e l i n e are shown i n f i g . (27).  - 48 Ce,  Ce,  Near one end o f the t i e l i n e  F i g . 27:  Symmetric composition  Near the other end o f the t i e line.  Composition p r o f i l e s f o r d i f f e r e n t a l l o y s on the same t i e - l i n e .  T h i s r e s u l t can be e x p l a i n e d because i n each case: - D i s the same (same temperature  and same c o m p o s i t i o n  range) - The s u r f a c e energy  i s the same (same e q u i l i b r i u m phases)  i . e . the d r i v i n g f o r c e f o r the t r a n s f o r m a t i o n i s the same. In o r d e r . t o improve accuracy f o r subsequent  calculations,  the ageing r e s u l t s on one t i e l i n e have been combined to g i v e  average  ageing curves, F i g . ( 2 8 ) , (29) and ( 3 0 ) . (2)  A comparison o f the average  l y i n g on d i f f e r e n t  ageing c u r v e s f o r two s e t s o f a l l o y s  t i e - l i n e s but aged a t the same temperature  t h a t t h e growth r a t e i s d i f f e r e n t ( f i g  (28) and (29)).  shows  In t h i s case D  might be d i f f e r e n t b e i n g a f u n c t i o n o f c o m p o s i t i o n but t h i s seems unlikely.  I t i s more p r o b a b l e t h a t the d i f f e r e n c e i n growth r a t e i s due  to a d i f f e r e n c e i n s p e c i f i c i n t e r f a c i a l  energy.  o  Q(A)  O  100  50  •  A  •  D  O  c time i n hours _l_  2  0 F i g . 28,  Average A g e i n g Curve f o r Ar-C^D a t 700°C  F i g . 30  Average A g e i n g Curves f o r B-G-H  a t 600°C  - 52 For the by  the c o a r s e n i n g p r o c e s s , the r e l a t i o n s h i p between  change i n wavelength AQ AF  • KAQ  and  the change i n f r e e energy AF  a, where a i s the s p e c i f i c  d i f f e r e n t f o r d i f f e r e n t t i e - l i n e s and o f the r e a c t i o n .  possible  i n t e r f a c i a l energy.  energy can be understood i n the  The  when  is force  to compute  change i n  f o l l o w i n g manner:  from p o s i t i o n I to p o s i t i o n I I  a  so a f f e c t s the d r i v i n g  U n f o r t u n a t e l y i t i s not  a b s o l u t e v a l u e of the  shifts  i n t e r f a c i a l energy,  i s given  the  interfacial the  tie-line  ( f i g . 31)  Ni  Fig.  the  31:  Change i n s p e c i f i c  equilibrium  c o m p o s i t i o n on  interfacial  energy  the Cu-poor s i d e of the  changes from a p o s i t i o n where the Cu and  Co  tie-line,  atoms r e p e l one  E,  another  ( h i g h - s p e c i f i c i n t e r f a c i a l energy) towards a p o s i t i o n where the  strongly  repulsion  i s weaker because E becomes c l o s e r to a r e g i o n where complete m u s c i b i l i t y i s achieved  (low  specific  i n t e r f a c i a l energy).  This  reduction  i n o from  - 53 I t o I I l e a d s t o a s m a l l e r d r i v i n g f o r c e and growth r a t e .  The  a smaller  e f f e c t of change i n c o n f i g u r a t i o n a l entropy would  be t o i n c r e a s e the c o a r s e n i n g r a t e and (Appendix  consequently  thus i t s e f f e c t  i s negligible  V) (3)  A comparison of the average  ageing curves f o r  one  s e t of a l l o y s l y i n g on the same t i e - l i n e but aged a t d i f f e r e n t  fig  (29) and  (30), shows a g a i n a d i f f e r e n c e i n growth r a t e .  Two  temperatures  effects  have to be considered h e r e : -  when T i n c r e a s e s , D i n c r e a s e s and  t h i s tends to  i n c r e a s e the c o a r s e n i n g r a t e . -  when T i n c r e a s e s , the s p e c i f i c i n t e r f a c i a l  d e c r e a s e s , t h i s tends to d e c r e a s e the c o a r s e n i n g r a t e . Becker  (25), the s p e c i f i c i n t e r f a c i a l energy  energy  A c c o r d i n g to  i s p r o p o r t i o n a l : , to  2 (Ax)  , where  Ax i s the width of the m i s c i b i l i t y gap (Ax)  2  a t 600°C  =  1.00  (Ax)  2  a t 700°C  =  0.85  (Ax)  2  a t 800°C  =  0.69  and  The change i n c o n f i g u r a t i o n a l entropy i s shown to be n e g l i g i b l e i n Appendix The growth r a t e i n c r e a s e s v e r y markedly w i t h i n agreement w i t h the i n c r e a s e of D b e i n g the predominant  temperature,  factor.  With a view to d i s c u s s i n g the c o a r s e n i n g r e s u l t s i n a quantitative f a s h i o n , a growth law should be found. the average  ageing curves were p l o t t e d i n two -  law suggested  d i f f e r e n t ways.  Q v e r s u s l o g t to check i f the e x p o n e n t i a l growth  by D a n i e l and L i p s o n i s v e r i f i e d -  For t h i s purpose  (fig.  32).  l o g Q v e r s u s l o g t to check the c o a r s e n i n g law  by Greenwood ( f i g .  33).  In t h i s case i t i s n e c e s s a r y  proposed  to c o n s i d e r t h a t  V.  F i g . 32  Semi-log p l o t of the average ageing  curves  I  0.1  time i n minute  100  10 F i g . 33  Log-Log P l o t o f the Average Ageing Curves  1000  I  u i  - 56 -  Qo  and k t  a r e of s i m i l a r m a g n i t u d e  Q  as suggested  by H i l l e r t  so .that--Qo - k t  i s negligible  f o r Cu-Ni-Fe.  A b e t t e r agreement w i t h the Greenwood law seems to be o b t a i n e d a l t h o u g h i t i s f u l l y r e a l i z e d t h a t f i n d i n g a s t r a i g h t . l i n e on a l o g - l o g equation;.  p l o t i s not a , p a r t i c u l a r l y s t r o n g t e s t of  T h i s suggests  an  t h a t the growth law i s a p p r o x i m a t e l y  of  the  form Q  m  -kt  T a b l e IX g i v e s the v a l u e s of k and m f o r the f o u r ageing T a b l e IX  Tie Line  I  II  curves.  Values of k and m f o r the 4 ageing  Ageing Temp. °C  curves.  k  m  A /min m  700  5.18  2.1  10  600  6.6  2.5  x  10  700  4.91  5.1  x  10  800  4.0  7.6  x  x  10  8  7  8  1 0  m i s markedly d i f f e r e n t . f r o m 3 and v a r y i n g w i t h temperature.  Q u a n t i t a t i v e agreement w i t h the theory i s t h e r e f o r e  not too good.  The v a r i a t i o n of k would be expected,  to Greenwood's theory energy  (eg.  i t i s proportional  to D and  specific  interfacial  1.6). The  s m a l l d i f f e r e n c e i n k between the t i e l i n e s I and  c o u l d be due  to the d i f f e r e n c e i n s p e c i f i c  previously.  The  be due  since according  i n t e r f a c i a l energy as e x p l a i n e d  l a r g e i n c r e a s e i n k w i t h i n c r e a s i n g temperature would  to the v a r i a t i o n of D.  because of the v a r i a t i o n of m, t h i s i n t e r p r e t a t i o n of the I t i s important wavelength d e f i n i t e l y  However k should not be c o n s i d / e r e d  alone  and not much weight should be g i v e n to  results. to r e a l i z e t h a t a l t h o u g h an i n c r e a s e i n ,  takes p l a c e the mechanism by which t h i s  occurs i s very d i f f i c u l t the c o m p o s i t i o n  II  to v i s u a l i z e p h y s i c a l l y .  p r o f i l e a t two  different  F i g . (34)  times of the  coarsening represents  coarsening.  I t can be seen t h a t the c o a r s e n i n g p r o c e s s r e q u i r e s the t r a n s p o r t of one.part  of a p l a t e l e t  to another  one.  T h i s would i n v o l v e d o w n - h i l l  Ce F i g . 34:  Composition p r o f i l e s a t two times, of the c o a r s e n i n g .  different  2  and  - 58 u p - h i l l d i f f u s i o n i n a d j a c e n t r e g i o n s of the specimen.  The  for  and moreover  it  t h i s k i n d of d i f f u s i o n i s v e r y d i f f i c u l t is difficult  to e x p l a i n why  to imagine  process  one p l a t e l e t r a t h e r than another  one  i s going t o d i s s o l v e or to grow. A c t i v a t i o n Energy Measurements of the c o a r s e n i n g r a t e s a t 600°C, 700°C and enable an a c t i v a t i o n energy The  f o r the p r o c e s s to be  800°C  determined.  change i n a c t i v a t i o n energy w i t h temperature  due  to  a c o n f i g u r a t i o n a l entropy c o n t r i b u t i o n i s shown to be n e g l i g i b l e i n Appendix V.  The v a r i a t i o n of the i n t e r f a c i a l energy w i t h  cannot  be taken i n t o account  energy  and has  i n the d e t e r m i n a t i o n of the  temperature activation  to be n e g l e c t e d .  Assuming the c o a r s e n i n g p r o c e s s i s s i m i l a r to t h a t of many o t h e r r e a c t i o n s the f o l l o w i n g r e l a t i o n s h i p can be assumed between t i m e of r e a c t i o n t , temperature d e f i n e d by the wavelength t(Q,T) where E i s the a c t i v a t i o n or:  T and  the extent of the  reaction  0. -  f(Q) exp  (E/RT)  energy  l o g t (Q,T)  = l o g f (Q) +  J-J- I R  the v a l u e of E can be o b t a i n e d by p l o t t i n g l o g t v e r s u s JL f o r a g i v e n T Q at d i f f e r e n t No  t e m p e r a t u r e s ^ f i g . (35).  E i s found e q u a l ' t o 65  i n f o r m a t i o n i s a v a i l a b l e f o r d i f f u s i o n i n Cu-Ni-Co.  system appears  s i m i l a r to Cu-Ni-Fe i n which D a n i e l s has  a c t i v a t i o n energy  f o r volume d i f f u s i o n of 65  kcal/mole.  kcal/mole,-  However the found  +  an  -  I t t h e r e f o r e appears  60  -  t h a t the growth of the wavelength i n  Cu-Ni-Co i s c o n t r o l l e d by volume d i f f u s i o n as suggested  by the c o a r s e n i n g  model. Metallography The o b s e r v a t i o n o f the m i c r o s t r u c t u r e o f a l l o y C, aged a t 700°C, shows t h a t the sidebands be observed.  a r e gone b e f o r e any m i c r o s t r u c t u r e can  A v e r y s m a l l amount o f g r a i n b o u n d a r y . p r e c i p i t a t i o n o c c u r s  from the quenched s t a t e .  A f t e r l o n g ageing times p r e c i p i t a t e s appear  and develop a t g r a i n and twin boundaries observed  and a l o n g d e f i n i t e p l a n e s as  by G e i s l e r and Newkirk, who suggest  this i s localized  p r e c i p i t a t i o n on s t r a i n l i n e s p r o b a b l y caused by thermal s t r e s s e s induced d u r i n g the quench. Even a f t e r v e r y l o n g ageing times, where x - r a y d i f f r a c t i o n shows two e q u i l i b r i u m phases, no t r a c e o f the homogeneous is visible,  precipitation  s u g g e s t i n g t h a t t h e heterogeneous p r e c i p i t a t i o n  c o m p l e t e l y d u r i n g the l a t e r stages o f a g e i n g .  took  over  - 61 5. (1)  CONCLUSIONS  S p i n o d a l decomposition was observed  i n Cu-Ni-Co over a  s i g n i f i c a n t range o f c o m p o s i t i o n s and temperatures, but o n l y i n s i d e the spinodal.  o (2)  The i n i t i a l wavelength  over a l l compositions and temperatures  l a y i n the range 40 to 55A s t u d i e d , i n good agreement w i t h  the t h e o r y f o r s p i n o d a l decomposition. (3) aged i n s i d e  A c o a r s e n i n g was observed on ageing f o r a l l a l l o y s  the spinodal.  The growth r a t e changes were e x p l a i n e d i n  terms o f d i f f u s i o n c o e f f i c i e n t dependance and e f f e c t s o f change o f s p e c i f i c i n t e r f a c i a l energy.  A c o a r s e n i n g law o f the form Q  m  - Qo  m  = k(t-to),  w i t h m i n the range 4 to 6.6 was found. (4)  The a c t i v a t i o n  energy f o r the c o a r s e n i n g p r o c e s s was  found to be 65 kcal/mole + 10 k c a l was t h e c o n t r o l l i n g  factor.  s u g g e s t i n g t h a t volume d i f f u s i o n  - 62 REFERENCES  1.  A. J .  2.  V. D a n i e l and H.Lipson,  3.  M. E. Hargreaves,  4.  A. G u i n i e r , A c t a . Met. _3_ (1955) , p. 510  B r a d l e y , P r o c . Phys. Soc. (London) 52_ (1940) p. 80. P r o c . Roy. Soc. (London) A 181 (1943) p. 368.  A c t a C r y s . _4_(1951) p. 301.  S o l i d S t a t e Phys._9_(1959) p. 293. 5.  M. H i l l e r t , M. Cohen and B. L. Averbach, A c t a . Met. _9_ (1961) p. 536.  6.  M. H i l l e r t , A c t a Met.J9_(1961)  p. 525.  7.  J.  J.  W.  Cahn and J .  E. H i l l i a r d ,  Chem. Phys. _28_ (1958) p. 258 and  31 (1959) p. 668. 8.  J.  W. Cahn, A c t a Met. _9_ (1961) p. 795.  9.  J.  W.  10.  J.  W. Cahn, A c t a . Met. 10_ (1962) p. 179.  11.  G. W. Greenwood, A c t a . Met._4_ (1956) p. 243.  12.  C. Wagner, Z. Elektrochem.  13.  V. D a n i e l and H. L i p s o n , P r o c . Roy. Soc. (London) A182 p. 378.  14.  E. Biedermann and E. K n e l l e r , Z. Met. 47 (1956) p. 289.  15.  P. J .  T u f t o n , Ph.D. D i s s e r t a t i o n , Cambridge  16.  A. J .  A r d e l l and R. B. N i c h o l s o n , A c t a . Met. JL4_ (1966),  p.  Cahn, A c t a . Met. 14_ (1966) p. 1685 and 1053.  (1944)  (1963).  1295.  17.  A. H. G e i s l e r and J .  18.  K. B. Rundmann and J .  19.  T. J . p.  65_ (1961) p. 581.  Tiedema, J .  B. Newkirk, TAIME 180 (1949) p. 101. E. H i l l i a r d , A c t a Met. 15_ (1967) p. 1024.  Bouman and W. G. Burgers, A c t a . Met. 5  310.  20.  W. Dannohl and H. Neumann, Z. Met._7_ (1938) p. 217.  21.  E. A. Owen and L. P i c k u p , Z. K r i s t . 88_ (1934) p. 116.  22.  A. T a y l o r , J .  23.  H. E. Cook and J .  I n s t . M e t a l s . _7_7 (1950) p. 585. E. H i l l i a r d ,  TAIME 233 (1965) p. 142.  (1957),  - 63 - . References 24.  (Cont)  R. E. V i l l a g r a n a and G. Thomas, Paper p r e s e n t e d a t AIME S p r i n g Meeting, Los Angeles  (Feb. 1967).  25.  R. Becker, Z. Met. _29 (1937) p.  245.  26.  " C o b a l t Monograph' Centre d i n f o r m a t i o n du C o b a l t ^ B r u x e l l e s - B e l g i u m .  27.  "Metals R e f e r e n c e Book", S m i t h e l l s _2_ (1962) p.  -  614.  APPENDIX I  - 64 -  R e s u l t s o f X-ray Photographs Photo  Alloy  Annealing Temp. °C  Annealing time min.  69 (°) (on l i n e 200)  a(A°)  Q (A ) 0  Min.  Av. Max.  2  B  700  0  -  -  3  B  700  1  0.88+.0.02  54.4 55.7  4  B  700  5  0.55+0.01  87.5 89.1 91.0  5  B  700  3  0.665±0.01 72.5 73.7 75.0  6  B  700  90  7  B  700  10  0.50±0.01 , 96.0 98.0 100.0 3.57L+0.004  8  B  700  2  0.79±0.001 61.2 62.0 63.0  10  B  700  5400  -  -  22  A  700  0  -  -  23  A  700  5  0.96±0.01  50.5 51.0 51.6  24  A  700  10  0.82±0.01  59.0 59.8 60.0  25  A  700  30  0.645±0.02 73.7 76 78.4  26  A  700  20  0.72t0.01  67.1 68.0 69.0  27  A  700  45  0.60±0.01  80.3 81.7 83.0  28  A  700  60  0.57±0.01  84.3 81.7 83.0  29  A  700  120  0.48±0.01  100.0 102.1 104  29&is  A  700  300  0.4p±_0.01  119.5 122.5 125  30  A  700  7020  -  -  32  D  700  0  -  -  33  D  700  10  0.79+0.02  58.5 62 65  34  D  700  5  0.89±0.02  53.5 55 56.5  3.573±0.002 7.0  II II ti  Decomposition has started  ii  3.603+.0.003 3.548+.0.001 3.575±0.001 it  II  3.574±0.03 II 1!  II II II  3.596+.0.002 3.558±0.001 3.573±0.002 II II  65Photo  Alloy  Annealing Temp.  °c  Annealing time min  69;(°) (on l i n e 200)  Q(A°)  35  . D  .700  30  0.63S±0.03  73.7 77.2 81.0  36  D  700  20  6.72+0.02  66.2 68.0 70.0  37  D  700  45  0.55+0.03  84.5 89.1 94.2  D  700  120  0.36+0.01  132 136 140  D  700  60  0.46+0.02  102.1 106.5 111,3  D  700  180  0.34+0.01  140 144 149  39  D  700  300  N S B *  40  D  700  7020  -  42  C  700  0  -  43  C  700  30  0.58+0.01  83.0 84.5 86.0  44  C  700  10  0.73±0.02  71.0 73.1 75.4  45  C  700  20  0.67±0.02  71.0 73.1 75.4  46  C  700  7  0.87+0.02  55.0 56.3 57.6  47  C  700  40  0.5?±0.02  90.7 94.2 98  48  C  700  60  0.47+0.02  100.0 104.3 108.9  49  C  700  120  0.41±0.02  114.0 119.5 125.6  C  700  300  0.35+0.02  133.0 140.0 148.0  50  C  700  7080  -  -  82  E  700  0  -  -  83  E  70p  10  N . S . B .  84  E  700  30  N . S . B .  85  E  700  66  N . S . B .  86  E  700  300  N . S . B .  3 7  bis  38 3 8  bis  * N S B  i  = no sidebands i  a (A)  Min. Av. Max.  »  »  -  3 .596±0.002 3 .558*0.001 3 •583±0.001  -  •  ••  3 •596±0.002 3 .-558±0.001 3 .549*0:001 fi  It  -  IT  -  1!  Photo  Alloy  Annealing Temp. C°  Annealing Time min.  60(°) fcn l i n e 200)  Q(A°)  - 66 a (A ) 0  Min. Av. Max.  90  E  700 «,  7200  -  92  F  700  0  -  93  F  700  5  N.S.B.  94  F  700  10  N.S.B.  95  F  700  20  N.S.B.  96  F  700  60  N.S.B.  100  F  700  7200  -  -  102'  G  700  0  -  -  103  G  700  5  0.68±0.01  71.0 72.1 73.1  104  G  700  25  0.83±0.02  57.6 59.0 60.0  105  G  700  10  0.60±0.02  79.0 81.7 84.5  106  G  700  7.5  0.62+/). 02  76.6 79.0 81.7  107  G  700  1  0.90±0.06  53.3 54.4 55.7  »  108  G  700  15  0.40+0,02  116.7 122.5 128  »  109  G  700  21.5  0.35±0.02  132 140 148  110  G  700  7200  -  -  112  H  700  0  -  -  113  H  700  5  0.67±0.03  70.0 73.1 76.6  114  H  700  10  0.48±0.01  100.0 102.1 104  115  H  700  15  0.54±0.01  89.1 90.7 92.4  116  H  700  20  0.40±0.01  119.5 122.5 125  117  H  700  40  N.M.*  *  N.M.  -  3. 601*0.002 3. 547+.0.002 3. 558+0.001  -  it  -  ii II  II  3. 600+.0.001 3. 548±0.001 3. 576±0.001  3.' 600±0.001 3. 548±0.001 3. 592+0.002  -  Sidebands-too c l o s e t o main r e f l e c t i o n t o be measured.  11  - 67 Photo,  Alloy  Annealing Temp. C°  Annealing Time min  66 0°) (on l i n e 200)  Q(A°) Min.  a(A°) Av. Max.  -  ll8  H  700  60  N.M.  119  H  700  2.5  0.59+0.01  81.7 83.0 84.5  H  700  1  0.65+0.01  74.2 75.4 76.6  120  H  700  7200  -  -  3.601*0.001 3.549*0.1001  122  J  700  0  -  -  3.601*0.001  123  J  700  30  N.S.B.  124  J  700  60  N.S.B.  -.  125"  J  700  10  N.S.B.  -  126  J  700  300  N.S.B.  -  130  J  700  7200  -  -  3.599*0.001 3.550*0.001  132  F  800  0  -  -  3.558*0.001  133  F  800  5  N.S.B.  -  ti  134  F  800  1  N.S.B.  -  II  135  F  800  3  N.S.B.  -  II  140  F  800  1560  -  -  3.597*0.002 3.551*0.002  142  B  800  0  -  -  3.571*0.004  143  B  800  5  N.M.  -  144  B  800  1  0.52+0.02  90.7 94.2 98.0  145  B  800  3  0.41+0.01  116.5 119. 5 122.5  146  B  800  2  0.45+0.01  106.5 108. 7 111.4  147  B  800  0.5  0.64+0.02  74.2 76.5 79.0  150  G  800  0  U  9  b i s  —  •An  -  3.592*0.002  it  II II  it II  3.598*0.002 3.550*0.002  - 68 Photo  Alloy  Annealing T&mp. C°  Annealing Time min  66 (°) Q(A°) (on l i n e Min. Av . Max. 200)  -  -  a (A ) 0  152  G  800  1560  153  €  800  5  N.M.  154  G  800  3  0.35±0.03  130 140 153  II  156  G  800  2  0.67+0.02  71.0 73.1 75.4  II  157  G  800  0,5  0.67+0.02  71.0 73.1 75.4  II  160  G  800  1560  -  -  162  H  800  0  -  -  163  H  800  5  0.40 (£=:)  122.5(*very)  164  H  800  1  0.52±0.01  93 95 97 * ...  »  165  H  800  3  0.4;5±0.2  104 108.9 114  »  166  H  800  2  0.65±0.01  74 75.4 76.5  167  H  800  0.5  0.56±0.02  84.5 87.5 90.7  170  H  800  1560  -  -  172  F  600  0  -  -  173  F  600  10  N.S.B.  174  F  600  30  N.S.B.  175  F  600  60 ,  N.S.B.  176  F  600  300  N.S.B.  177  F  600  1500  N.S.B.  180  F  600  14.400  -  -  182  B  6,00  0  -  -  -  3.576+.0.001  it  3.597+0.002 3.549+0.002 3.592+0.002  3.600+0.002 3.552±0.001 3.558+0.001 »  3.601±0.002 3.547±0.002 ' 3.573±0.002  Photo  A l l o y  Annea-  Annea-  69  l i n g  l i n g  (on  Temp.  Time  200)  C°  min  (°)  a  Q(A°)  69  -  (A ) 0  l i n e Min.  Av.  Max.  -  II  183  B  600  10  184  B  600  30  0.80±0.01  60  185  B  600  60  0.74+0.01  65.3  66.2  6 7 . 1  186  B  600  300  0.63+0.02  75.4  77.8  80.3  187  B  600  1500  0.50+0.02  94.2  98  188  B  600  600  0.59±0.01  81.7  83.0  190  B  600  14.400  -  61.2  ti  62.0  II  it  it  1 0 2 . 1  II  84.5  -  3.600*0.002 3.547*0.002  -  3.576*0.001  192  G  600  0  193  G  600  10  0.86±0.01  56.3  57.0  57.6  194  G  600  30  0.75±0.01  64.0  65.3  66.2  195  G  600  60  0.70+0.03  67.1  70.0  7 3 . 1  196  G  600  300  0.58+0.01  83.0  84.4  86.0  197  G  600  300  0.40±0.01  119.5  198  G  600  600  0.50±0.01  96.0  200  G  600  14.400  -  122.5 98.0  ii  II  ii  it  II  125  II  100  -  3.599*0.002 3.545±0.002  -  3.592±0.002  202  H  600  0  203  H  600  10  0.83±0.02  57.6  59.0  60.0  204  H  600  30  0.90±0.01  54.0  54.4  55.0  205  H  600  60  0.6Q±0.01  80.3  81.7  83.0  206  H  600  300  0.48+0.01  100.0  102.1  104  207  H  600  x$po  0.40+0.01  119.5  122.5  125  208  H  600  600  0.46+0.01  104.3  106.5  108  2 1 0  H  600  14.400  -  -  II II II II II II 3.600*0.002 -i  3 . 5 4 5 * 0 . 0 0 2 —  - 70 APPENDIX  Ha  D e t e r m i n a t i o n o f the S p i n o d a l I  Cook and H i l l i a r d  Method  The method d e s c r i b e d by Cook and H i l l i a r d a T a y l o r expansion o f (dF/dc) about a formula where no assumption  the c r i t i c a l p o i n t , and l e a d s t o  has t o be made f o r the f r e e energy  The s p i n o d a l composittons a t v a r i o u s temperatures the f o l l o w i n g  curves.  a r e o b t a i n e d by use o f  formula  Cs-Ce= where  (Ce-Cc)  [1-0.422 (T/Tc) ]  AII.l  Cs • s p i n o d a l c o m p o s i t i o n Ce = e q u i l i b r i u m Cc = c r i t i c a l T  composition  composition  = Temperature °K  Tc = c r i t i c a l The peudo-binary  temperature  °K  s o l u b i l i t y curve makes i t p o s s i b l e t o e s t i m a t e  Ce, Cc and Tc, and Cs can be c a l c u l a t e d . i n T a b l e X.  (23) i s based on  The c a l c u l a t i o n s a r e grouped  Table X  Tie Line  Ji I  T °K  T °K  T/Tc  873  1365  0,639  0.730  1073  1365  0.785  1273  1365  873  1073  [•]  C  c  C a l c u l a t i o n s f o r the d e t e r m i n a t i o n o f the Spinodal; Cook and H i l l i a r d Method  Ce  Ce-Cc  Cs-Cc  L  R  0.438  0  1  -0,438  0.562  0.669  0.438  0.565  0.887  -0.382  0.932  0.606  0.438  0.215  0.688  1168  0.747  0.685  0.437  0  1168  0.918  0.614  0.437  0.11  L  R  L  Cs R  L  R  -0.340  0,410  0.118  0.848  0.449  -0.255  0.300  0.183  0.738  -0.223  0.250  -0.135  0.152  0.303  0.590  1  -0.437  0.563  -0.299  -0.385  0.138  0.822  0.805  -0.327  0.368  -0.201  0.226  0.236  0.673  11  II.  Regular S o l u t i o n Model For a r e g u l a r s o l u t i o n AF mix -  where  fi=  2  R  ^  ttc(l-c)  > * ^ anc  C  s a  + RT ( c i i  n  c + (1-c) I n (1-c)  n  i°n  n t e r a c t  (AII-2)  energy  Tie l i n e I  T = 1.365°K c  il  =5.42 kcal/mole  Tie line II  T  £2  = 4.65 kcal/mole  = 1168°K  c  D i f f e r e n t i a t i o n o f (AII-2) g i v e s  < f  d  =  m i x )  dc  2 and d (AFmix) _ dc2 ~ for  spinodal  n(l-2c) + RT ( l n c - l n ( l - c ) ) , 1 c(l-c) f  2  "  +  R  T  d (A Fmix) dc^  (  ,  at G = Cs  _ =  U  0 = - 2 8 + RT [  giving  1  Cs = 2Q  Cs(l-Cs)  ±  ]  0  V 4  - 8 Q RT  40 The f o l l o w i n g t a b l e g i v e s the complete c a l c u l a t i o n s : T a b l e XI  C a l c u l a t i o n s f o r the determination Regular s o l u t i o n model  of the Spinodal:  Tie Line I  4n  8ftRT  873  117.5  75.2  6.50  10.84  21.68  0.20  0.80  1073  5.42 117.5  92.5  5.00  10.84  21.68  0.27  0.73  1273  5.42 117.5  109.5  2.83  10.84  21.68  0.37  0.63  T  ft  2  Cs  29,  1  Cs  2  Tie Line I I  873  4.65 86.5  64.6  4.68  9.3  18.6  0.25  0.75  1073  4.65 86.5  79.3  2.661  9.3  18.6  0.36  0.64  - 73 Cook and  Hilliard  showed t h a t f o r r e g u l a r  would g i v e r e s u l t s w i t h i n 4% i s greater  proving  of the  that t h i s  exact v a l u e .  The  -  s o l u t i o n t h e i r model d i f f e r e n c e here  peudo-binary does not behave l i k e a  regular  solution. The  Cook and  H i l l i a r d model seems to f i t the pseudo-binary better^  p r o b a b l y because i t does not c u r v e s , and  assume the symmetry of the  enables s e p a r a t e c a l c u l a t i o n s of the s p i n o d a l  solubility compositions  to be made on each s i d e of the phase diagram.  APPENDIX l i b C a l c u l a t i o n of the I n i t i a l Wavelength A c c o r d i n g to Cahn, the f i r s t wavelength to develop i s 0  = VT •  opt  Oc 1  where Qc For  a regular  =  [-:8  2 lir  /  2  K/(d f/dc ) ] 2  2  s o l u t i o n model:  K - RTc  11  a  x y-  - j-  2  (Cahn and  Hilliard  (7))  2 3-4-  *  '  n  =  dc-'  ^ opt "  3  -4R  Tc  r  +  -rr^-r  RT  11  hn-2_T a 4 Tc  The  (l-c)c J  i  ci(l-c)  f o l l o w i n g t a b l e shows the c a l c u l a t i o n s f o r 2 d i f f e r e n t Tc,  T and  3 d i f f e r e n t C.  3 different  - 74 Table X I I :  Tc  °K  T  C  °K  Calculations Wavelength.  1  f o r the D e t e r m i n a t i o n o f the i n i t i a l  JS  C ( l - C )  4Tc  T 1 -  1  oOpt  TTC C(1-C) (  A  873  0.5 0.4 0.3  4.00 4.17 4.765  0.16 0.16 0.16  0.360 0.334 0.238  2.09 2.17 2.565  23.5 24.4 28.8  973  0.5 0.4 0.3  4.00 4.17 4.765  0.178 0.178 0.178  0.288 0.258 0.152  2.34 2.47 3.22  26.3 27.8 36.2  1073  0.5 0.4 0.3  4.00 4.17 4.765  0.1965 0.1965 0.1965  0.215 0.180 0.064  2.70 2.955 4.95  30.5 33.2 55.5  973  0.5 0.4 0.3  4.00 4.17 4.765  0.2085 0.2085 0.2085  0.166 0.13 0.006  3.015 3.48 16.2  34.6 34.1 182  1365  1168  - 75 APPENDIX I I I  D e t e r m i n a t i o n of the change of the S t a b i l i t y L i m i t Due  to S t r a i n Energy  As d i s c u s s e d i n p . A 3 the d i f f e r e n c e i n the maximum at which s p i n o d a l decomposition T  "  C  =  T  o c c u r s i s g i v e n by  ? ( l - v ) kNv  ( 4  • where.:^^n .-'-'. ?.---.linear--.expansion'-per/unit :  temperature  ,  composition  '  2 )  change  E = e l a s t i c modulus v = Poisson  ratio  k - Boltzman c o n s t a n t Nv = number of atoms per u n i t volume E v a l u a t i o n of n E v e r y t h i n g happening i n s i d e the m i s c i b i l i t y  gap,  and  any  e x t r a p o l a t i o n of the t i e - l i n e o u t s i d e i t having no s i g n i f i c a n c e , i t seemed l o g i c a l to take f o r u n i t c o m p o s i t i o n change the d i s t a n c e between the two  e q u i l i b r i u m phases a t the average  temperature  of 700°C.  Aa(700°C) a (Average)  n  From f i g . 23 TI ( T i e - l i n e  I) = 0.0168  E v a l u a t i o n of E and  n (Tie-line  not a v a i l a b l e .  = 0.0128  v.  The E v a l u e s f o r Cu|,': N i , and c o n s i d e r e d -(26)  II)  (27)  —  Co a t room temperature  will  s i n c e v a l u e s a t e l e v a t e d temperatures  The E ( A l l o y ) w i l l be taken as the average  of the t h r e e  values. . E  N ±  = 30 x 1 0 .\  6  psi^E  E(Alloy)i^  C u  = 17 x l O ^ s i ^ E ^  26X10  6  psi  are  = 30 x  10 psi 6  be  - 76 Similarly  forv  A c c o r d i n g to (26) (27)  v... = 0 . 3 6 Ni  v_ = 0.27 Co vflloy)^  v„ = 0.33 Cu  0.32  CU  Evalution of N  Taking t h e average l a t t i c e N v =  ^fTs)  3  x 10  parameter as 3.58A g i v e s = 8.7 1 0  2 4  2 2  at/cm  3  The d i f f e r e n t f a c t o r s o f eq. (4.2) i n a c o n s i s t e n t system o f u n i t s are  E = 26 1 0  6  p s i = 1.82 x 1 0  k = 1.38 x 1 0 " N v- 8.710  00  /cm  1 6  "\  kg/m  1 0  2  erg/°K = 1.38 x 1 0 "  = 8 . 7 10  2 3  joule/°K  2R 3 /m  For T i e - L i n e I  Tc - T = (0.0168)  2  1 32 x 10^^  2(  i_ .32) 0  1.38 10"23 x . 7 x 1 0 ^  x  8  = 3.13°K For T i e L i n e I I „• _ Tc-T . -  ,„ ...„,2 1.82 x 1 0 ^ (0.0128) x _ . 2 ( 1  = 1.83°K  0  3 2 )  x  1  >  3  8  1 0  -23'  x  8  ,. . 7  1 0  28  APPENDIX IV Evaluation  o f ,the S t r a i n Energy Two  s t a t e s may  State  be c o n s i d e r e d  f o r the modulated  I  structure.  State I I  Stress free High s t r a i n Low s t r a i n energy  High s t r e s s Strain free „. , A„. Highstraxnt energy (Due to d i s p l a c e m e n t of the atoms from I to I I )  e , J  £*£ 1-v where  e =>  strain  No sidebands would be observed i f the a l l o y s were i n s t a t e I I and t h e r e f o r e  state I i s a closer representation  of the s t r u c t u r e .  Furthermore i n s t a t e I I the s t r a i n energy i s independent of the number o f i n t e r f a c e s and so t h e r e would be no d r i v i n g f o r c e f o r c o a r s e n i n g . However, the e v a l u a t i o n  of the s t r a i n energy i s f a i r l y s i m p l e i n s t a t e  ,  I I and w i l l p r o v i d e a maximum.possible v a l u e of the s t r a i n energy i n the system. If phases i s 0.3%,  the d i f f e r e n c e i n l a t t i c e e i s 0.15%  c • / S t r a i n energy/cc =  parameter between the 2 e q u i l i b r i u m  f o r the complete a l l o y .  -  3  (1.5 x 10 -Q  2  E  - 78 From the v a l u e s computed i n appendix I I I _E_ 1-v  S t r a i n energy this  26 x  =  7.0-1 x 1 0 0.68  p e r c - 2.25 C  x IO"  1 5  x 2.7  x 10  erg/cc  6  r 6  x 10  i s a very considerable overestimate  11  =  =0.6  1 1  10  6  , erg/cc  erg/cc  of the a c t u a l s t r a i n  energy  i n the system Interfacial  Energy  o  I f the l a m e l l a e have a 100A interface i n a  cm cube of a l l o y .  s p a c i n g , t h e r e i s 10  Taking the s p e c i f i c  6  cm  2  of  interfacial  energy  2 of  a coherent boundary 20 erg/cm , g i v e s the t o t a l  = 20 x 10^ e r g / c c  .  T h e r e f o r e the s t r a i n  comparison w i t h i n t e r f a c i a l  energy  interfacial  energy  is.negligible  in  energy. APPENDIX V  Effect  of the C o n f i g u r a t i o n a l Entropy  Change  To a l l o w the c o a r s e n i n g p r o c e s s , i t seems n e c e s s a r y that r e g i o n s of the m a t e r i a l a t one to become d e p l e t e d i n t h i s  time r i c h  i n one component have  component d u r i n g c o a t s e i a g and v i c e - v e r s a .  T h i s phenomenon r e q u i r e s a mixing  and unmixing\ of the phases w i t h  i n t e r m e d i a t e s t a t e comparable to the i n i t i a l  solid  a s s o c i a t e d c o n f i g u r a t i o n a l entropy of m i x i n g . w i l l be a f u n c t i o n of t i e - l i n e and The the two  s o l u t i o n and  a  n  an  T h i s entropy of mixing  temperature.  c o n f i g u r a t i o n a l entropy . ( A s )  to mix  and unmix atoms of  e q u i l i b r i u m phases i s of the form AS =R' E x ^ l n x ^ where x£  c o m p o s i t i o n of the e q u i l i b r i u m phases.  The  i s the  f r e e energy a s s o c i a t e d  ( A F = - T A S ) which i s to be c o n s i d e r e d l i k e an a c t i v a t i o n energy  due  to the  -  configurational and  so has  entropy has t o v a n i s h a t t h e c r i t i c a l  79  -  temperature  AS.  The  f r e e energy  a t any temperature  AF(T) = -RT [2Zxi Tfie c a l u l a t i o n s a r e p r e s e n t e d  lnxj.  w i l l be g i v e n by  -E xi ..lnxiT] T  i n Table XII  T a b l e X I I : C a l c u l a t i o n f o r the d e t e r m i n a t i o n o f the e f f e c t o f t h e c o n f i g u r a t i o n a l entropy change i  T Tie Line °K  x  Co  equilibrium at % Cu  Ni  873 0.560 0.085 0.355  973 0.530 0.130 0.340  1073 0.500 0.175 0.325  x  Co  equilibrium at.% Cu  0.010 0.910  0.020 0.895  0.040 0.865  AS kcal °Kmole  AS Real kcal °Kmole  AF kcal/mole  -2.45 10-3.  -1.50 IO"  1.31  -2.81 10-3  -1.14 IO  1.11  -2.95 IO"  -1.00 10-3  1*07  0  0  Ni  0.080  0.085  0.095  3  I  3  - 3  1365 0.250 0.500 0.200  0.250 0.550  0.200  -3.95 10-3  973 0 3 7 0 0.200 0.430  0.035 0.820  0-.145  -0.92 -3.16 10-3 10" -4x10-3 0  0.8  J  II  1168 -.190 0.535 0.275  0.190 0.535  0.275  0  - 80 Conclusions - The change l  n  AF w i t h temperature f o r a g i v e n t i e l i n e i s  v e r y s m a l l (0.24 kcal/mole)  and i s n e g l i g i b l e w i t h r e s p e c t to the a c t i v a t i o n  energy. - The change i n v e r y s m a l l (0.31 kcal/mole) activation  energy.  AF from one t i e - l i n e to the o t h e r one i s and i s a g a i n n e g l i g i b l e w i t h r e s p e c t to the  

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