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UBC Theses and Dissertations

Sintering and grain growth of nonstoichiometric rutile. Thiriar, Jacques Pierre Jean 1964

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SINTERING AND GRAIN GROWTH OF NONSTOICHIOMETRIC RUTILE  BY  JACQUES PIERRE JEAN THIRIAR  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 c o n f o r m i n g . t o t h e  s t a n d a r d r e q u i r e d from c a n d i d a t e s degree of MASTER OF APPLIED  for  the  SCIENCE.  Members of the Department of M e t a l l u r g y THE UNIVERSITY OF BRITISH COLUMBIA February  196k  In  presenting t h i s thesis  the requirements B r i t i s h Columbia,  f o r an advanced degree a t t h e U n i v e r s i t y o f I agree t h a t the L i b r a r y s h a l l make i t  a v a i l a b l e f o r reference for  extensive  i n p a r t i a l fulfilment of  and s t u d y .  I f u r t h e r agree t h a t p e r m i s s i o n  copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be  g r a n t e d by the Head o f my Department o r by h i s It  freely  representatives.  i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s  for  f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n .  Department o f  Metallurgy,  The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver 8, Canada. D a t e  February 28th. 1964  ABSTRACT  R u t i l e powders i n f l a k e d form were p r e s s e d and heated a t temperatures  (100G°C t o 1300°C) under r e d u c i n g ( H / H 0 ) atmospheres 2  study the r a t e of weight l o s s ,  The  to  2  the g r a i n growth and the d e n s i f i c a t i o n .  weight l o s s measurements  stoichiometric  f o r r e d u c t i o n o f r u t i l e t o two n o n -  c o m p o s i t i o n s o f TiOi.92 and T i O i . g s y i e l d e d an a c t i v a t i o n  energy f o r weight l o s s o f 82 ± 2 k c a l / m o l e . the r a t e - d e t e r m i n i n g in  different  step.  No attempt was made t o i d e n t i f y  P r e v i o u s weight loss.measurements  carried  out  e q u i l i b r i u m c o n d i t i o n s produced an e n t h a l p y o f 83 * 10 k c a l / m o l e f o r  f o r m a t i o n of an oxygen i o n v a c a n c y .  T h i s c o u l d suggest  t h a t the  the  rate-  d e t e r m i n i n g step might be the f o r m a t i o n of an oxygen i o n vacancy.  The  g r a i n growth study r e v e a l e d t h a t the  composition of T i 0 i . g The  results  d i d not obey the t h e o r e t i c a l  2  can be expressed by the  D  2  - D  0  2  = K t°-  6  non-stoichiometric r e l a t i o n of B u r k e .  following  exp (-  2%^)  T h i s a c t i v a t i o n energy f o r g r a i n growth i s e q u a l t o t h e a c t i v a t i o n for  oxygen i o n d i f f u s i o n i n T i 0 . 2  energy  T h i s suggests t h a t the oxygen i o n  d i f f u s i o n may be t h e r a t e - c o n t r o l l i n g s t e p f o r g r a i n growth.  The measurements tried,  d e n s i f i c a t i o n on s i n t e r i n g was e v a l u a t e d from l i n e a r o f the compacts  during reduction to T i 0 i . g .  t o f i n d the b e s t f i t f o r the p r e s e n t  graphs suggest  2  data.  t h e C o b l e model f o r b u l k d i f f u s i o n ,  diffusion coefficients  are  shrinkage  A few. models were  While the p h o t o m i c r o and the v a l u e s f o r  o f the r i g h t o r d e r of magnitude, the  activation  energy f o r t h e r a t e d e t e r m i n i n g s t e p i s about 118 k c a l / m o l e , which i s in  agreement  w i t h t h e p r e v i o u s s i n t e r i n g study on r u t i l e .  the  not  iii.  • From g r a i n growth d a t a f o r those compacts at 1200°C and those s i n t e r e d i n open a i r ,  reduced t o  i t was seen t h a t  TlOi.gs  the  d i f f u s i o n c o e f f i c i e n t was not s i g n i f i c a n t l y a f f e c t e d by v a r i a t i o n of. the  oxygen p a r t i a l p r e s s u r e .  - T h i s d i s c r e p a n c y i n the a c t i v a t i o n  v a l u e may be e x p l a i n e d by a . p o s s i b l e e r r o r  i n measurement  energy  and o t h e r  unknown v a r i a b l e s which may c o n t r o l t h e d e n s i f i c a t i o n p r o c e s s .  1V-.  • ACKNOWLEDGEMENT  The a u t h o r wishes t o g r a t e f u l l y acknowledge,the g i v e n by members o f the Department of M e t a l l u r g y . to'Dr. A. C Mrs. the  D. C h a k l a d e r f o r h i s a d v i c e ,  assistance  He i s e s p e c i a l l y  grateful  guidance and a s s i s t a n c e and t o  A . M . - A r m s t r o n g f o r her c r i t i c a l d i s c u s s i o n s i n the p r e p a r a t i o n  of  thesis.  The work was f i n a n c e d by a g r a n t p r o v i d e d by the Defence Research .Board o f Canada,  D.R.B.  7501-02.  V.  • TABLE.OF CONTENTS Page 1  INTRODUCTION E a r l i e r T h e o r i e s o f S i n t e r i n g ( S p h e r i c a l Models) D e n s i f i c a t i o n o f Powder Compacts  2  . . . . . . . . .  . . . . . . . . . .  5  The E n d - P o i n t D e n s i t y  6  Pore S t r u c t u r e  7  Model f o r Complete D e n s i f i c a t i o n  8  ......  9  S i n t e r i n g o f Oxides . . . . . . . . . . . . . . . .  . . . .... . . . 9  a) Neck Growth E x p e r i m e n t s . . . b) D e n s i f i c a t i o n o f Powder Compacts Crystal Structure of Titania  11  . . . . . . . . . . . . . .  11  . . . . . . . . . . . .  Defect Reaction i n R u t i l e . . . . . . . . .  12  . . . . . . .  13  Aim o f t h e P r e s e n t Work EXPERIMENTAL  lk 1^  Titania-Powder Compacting o f t h e Powder  16,  . . . . . ... ... . .  The Furnace  16  C o n t r o l o f t h e Furnace Atmosphere  18  Description  19  o f a Run w i t h t h e Furnace  19  Measurements f o r Weight Loss and D i m e n s i o n a l Change . . . . . . . . • Measurements f o r G r a i n Growth  . . . . . . . . . . . . . . . . ... . . .  .20  a) P o l i s h i n g a n d E t c h i n g .  20  b) G r a i n S i z e Measurements  21 .22  RESULTS X-Ray I n v e s t i g a t i o n  .  ...  .  .22 .22  Weight Loss S t u d y  30  G r a i n Growth S t u d y S i n t e r i n g Study . . . . . . . .  ....  .  32  vi. Table o f Contents Continued... Page DISCUSSION  kk  .  Defect S t r u c t u r e of R u t i l e  . . . . . . . . . . . . . . . . . . . . .  -kf>  Weight L o s s G r a i n Growth  k6  "  Temperature Dependence o f G r a i n Growth Sintering  kk  k&  .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  D e n s i t y - T i m e Curve Diffusion Coefficients  50  . . . . . . .  51  . . . . . . . . . . . . . . . . . . . . . .  52  .  CONCLUSIONS  56  RECOMMENDATIONS FOR FUTURE INVESTIGATION . . . . . . . . . . . . . . . .  57  BIBLIOGRAPHY . . . . . . . . . . . . . . . . .  ...  58  APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  61  . . . . . . . . . . . . . . . . .  6l  . . . . . . . . . . . . . . . .  65  Appendix I I I . S i n t e r i n g Study. B u l k d i f f u s i o n Model ( C o b l e ) . . .  66  Appendix I V . D e f e c t E q u i l i b r i a and Oxygen I o n D i f f u s i o n f o r Non-stoichiometric R u t i l e ...  78  Appendix V.  79  Appendix I .  '/.Weight Loss' S t u d y  Appendix I I . G r a i n Growth S t u d y  Boundary D i f f u s i o n Model ( C o b l e )  .  Vll .  LIST OF FIGURES Figure 1.  2.  3«  k.  Page Schematic R e p r e s e n t a t i o n o f t h e C o n t a c t A r e a Between Two P a r t i a l l y S i n t e r e d S p h e r e s , (a). C e n t e r - t o - c e n t e r d i s t a n c e c o n s t a n t , (b) C e n t e r - t o - c e n t e r d i s t a n c e s h r i n k s ( a f t e r Kuczynski-'-) . .  • 2  (a). I n i t i a l Stage o f S i n t e r i n g , (b).Near End o f I n i t i a l S t a g e . Spheres have bjegun t o c o a l e s c e , (c). I n t e r mediate S t a g e . Dark g r a i n s have adopted shape o f tetrakaidecahedron, e n c l o s i n g white pore.channels a t g r a i n edges. ( d ) , F i n a l S t a g e . P o r e s a r e t e t r a h e d r a l i n c l u s i o n s a t c o r n e r s where f o u r t e t r a k a i d e c a h e d r a meet, ( a f t e r C o b l e ^ ) .  10  (a) T y p i c a l I n t e r m e d i a t e Stage S t r u c t u r e . (b) T y p i c a l F i n a l Stage S t r u c t u r e , ( c ) End Stag.e a t T h e o r e t i c a l D e n s i t y , (d) F i n a l Stage A f t e r D i s c o n t i n u o u s G r a i n Growth, ( a f t e r C o b l e ^ )  10  E l e c t r o n Micrograph of T i 0  2  Powders Showing t h e S i z e and  Shape o f t h e P a r t i c l e s , X 3000 5.  E l e c t r o n Micrograph of T i 0  2  15  Powders Showing t h e S i z e and  Shape o f t h e P a r t i c l e s , X U000  15  6.  Schematic Diagram o f t h e Furnace  17  7.  P e r Cent Weight Loss as a F u n c t i o n o f Time f o r T i 0 . g  8.  P e r Cent Weight Loss as a F u n c t i o n o f Time f o r TiOx.gs  9.  P e r Cent Weight L o s s V e r s u s Square Root o f Time f o r T i 0 i . g  . 1  . . .  2  2  10.  P e r Cent Weight Loss V e r s u s Square Root o f Time f o r T i 0 ! . g  11.  A Log-Log P l o t f o r A ° '  5  8  .  26  .  27  as a F u n c t i o n o f Oxygen P a r t i a l  P r e s s u r e s a t D i f f e r e n t Temperatures 12. 13-  25  ...  ....  Temperature Dependence o f Weight L o s s T y p i c a l M i c r o s t r u c t u r e s o f S i n t e r e d T i 0 ! . g . (a) I n i t i a l S t a g e . F i r e d a t 1150°C f o r 190 m i n u t e s , X 1500. (b) . I n t e r m e d i a t e S t a g e . Showing Channel; and S p h e r i c a l B p r e s . F i r e d a t 1150°C f o r 280, m i n u t e s , X I5OO. ( c ) F i n a l  29 29  2  S t a g e . . F i r e d a t 1200°C f o r 2800 minutes, X 1500. (d) B e g i n n i n g o f D i s c o n t i n u o u s G r a i n Growth.  1300°C f o r 1200 minutes, X 600  F i r e d at  lk.  Log-Log P l o t f o r Average G r a i n D i a m e t e r V e r s u s Time  15.  G r a i n Growth i n TIOX.QZ  Compacts w i t h Temperature  31 33 J>h  viii.  L i s t of Figures Continued. • Figure  Page  16.  I s o t h e r m a l G r a i n Growth  17.  L o g a r i t h m o f G r a i n Growth Rate V e r s u s R e c i p r o c a l ture  18.  \ Tempera-  36  D e n s i f i c a t i o n o f Compacts f o r the F i n a l C o m p o s i t i o n of.,. TiOx.92 (a) a t T = 1000°C (b) at'T = 1050°C (c) a t T = 1100°C (d) a t T = 1150°C I . . . (e) a t T = 1200°C  38 38 39 39 *f0  D i f f u s i o n C o e f f i c i e n t V e r s u s t h e R e c i p r o c a l o f the A b s o l u t e Temperature. D i r e c t l y , measured oxygen d i f f u s i o n c o e f f i c i e n t s measured i n s i n g l e c r y s t a l s are compared w i t h v a l u e s c a l c u l a t e d f r o m s i n t e r i n g e x p e r i m e n t s and models  k^>  D e n s i f i c a t i o n of TiOx.gs (a) (b) (c) (d) (e)  73 74 7k 75 75  :  ;  .19.  A-.III-l.  35  :  Compacts f o r the F i n a l C o m p o s i t i o n o f a t T ="1000°C a t T = 1050°C a t T = 1100°C a t T = 1150°C a t T = 1200°C  A.III-2.  D e n s i f i c a t i o n of T i 0  A.III-3.  V a r i a t i o n o f G r a i n S i z e w i t h Time and Temperature. i s t o d e t e r m i n e A of e q u a t i o n (6)  2  a t D i f f e r e n t Temperatures . . . . . . .  76  This 77  ix. . LIST OF TABLES Page lk  Table I .  A n a l y s e s o f t h e Two R u t i l e Powder Samples  Table I I .  V a l u e s o f Exponent n f o r D i f f e r e n t Ceramic Oxides . . . . . .  k"J  Table I I I .  A c t i v a t i o n E n e r g y Data f o r D i f f e r e n t Ceramic Oxides . . . .  k&  T a b l e IV.  C a l c u l a t e d D i f f u s i o n C o e f f i c i e n t s a t D i f f e r e n t Oxygen ^>k  . P a r t i a l Pressure  i  INTRODUCTION  Because o f t h e v e r y h i g h m e l t i n g temperature  o f most o x i d e c e r a m i c s ,  d e n s i f i c a t i o n b y s i n t e r i n g o f powder compacts i s a common f a b r i c a t i o n t e c h nique.  A l t h o u g h t h i s t e c h n o l o g y may be a s o l d as t h e a r t s o f c e r a m i c s , an  u n d e r s t a n d i n g of. t h e mechanisms i n v o l v e d d u r i n g t h e s i n t e r i n g p r o c e s s e s i s o n l y a r e c e n t development.  The t e r m " s i n t e r i n g " as u s e d by t h e powder m e t a l l u r g i s t s ,  means  an o p e r a t i o n b y w h i c h a mass o f compacted powder i s t r a n s f o r m e d i n t o a more dense p r o d u c t b y t h e a p p l i c a t i o n  o f heat a l o n e .  Observation o f the increased  c o h e s i o n between t h e p a r t i c l e s suggests t h a t t h e p r o c e s s may be d i v i d e d stages.  into  I n t h e f i r s t s t a g e , t h e growth o f b r i d g e s between a d j a c e n t p a r t i c l e s  occurs, but with very l i t t l e d e n s i f i c a t i o n .  I n t h e subsequent s t a g e s , t h e  i n t e r p a r t i c l e necks grow b i g g e r p r o d u c i n g a n o t i c e a b l e s h r i n k a g e .  Further  neck g r o w t h would r e s u l t i n t h e f o r m a t i o n o f i s o l a t e d .pores.  The most s u c c e s s f u l d e t e r m i n a t i o n s o f t h e k i n e t i c s o f s i n t e r i n g have been made u t i l i z i n g  systems o f s i m p l i f i e d geometry, where t h e s t u d y o f  t h e neck growth i s q u i t e f e a s i b l e b y s i m p l e e x p e r i m e n t a l t e c h n i q u e s . . F o r t h i s reason, experiments  i n t h i s f i e l d have been m o s t l y c o n f i n e d t o t h e  measurement o f t h e r a t e o f neck growth between a sphere and a p l a n e , between two s p h e r e s , between a w i r e and a p l a n e o r between two w i r e s a s a f u n c t i o n of time and temperature.  Such systems have t h e d i s t i n c t advantage t h a t t h e i r  geometry i s w e l l known, b u t t h e y have l i m i t e d a p p l i c a b i l i t y t o t h e d e n s i f i c a v t i o n o f powder compacts.  - 2  -  E a r l i e r T h e o r i e s o f S i n t e r i n g ( S p h e r i c a l Models) 1  Kuczynski for  was t h e f i r s t t o attempt t o d e r i v e t h e r a t e e x p r e s s i o n s  t h e growth o f t h e neck a t t h e p o i n t o f c o n t a c t between two s p h e r i c a l  p a r t i c l e s . • A c c o r d i n g t o him, t h e f i r s t stage o f s i n t e r i n g may be c h a r a c t e r i z e d by t h e f o r m a t i o n o f a neck between, two p a r t i c l e s as shown i n - F i g u r e . i .  -Figure 1.  Schematic R e p r e s e n t a t i o n o f t h e C o n t a c t A r e a Between Two P a r t i a l l y S i n t e r e d Spheres. (a) C e n t e r - t o - c e n t e r d i s t a n c e c o n s t a n t , (b) C e n t e r - t o - c e n t e r d i s t a n c e s h r i n k s , ( a f t e r Kuszynski" ") . 1  T h i s can be b r o u g h t about by one o r more o f t h e f o l l o w i n g p r o c e s s e s : t h e v i s c o u s o r p l a s t i c f l o w ; e v a p o r a t i o n and c o n d e n s a t i o n ; and volume or s u r f a c e diffusion.  The well-known r e l a t i o n s h i p between t h e r a d i i x o f t h e neck and  a, o f t h e s p h e r e s , t i m e t and temperature ;  equation  T c a n be d e s c r i b e d by one g e n e r a l  n  x_ .=  a  m  F (T)t  where F (T) i s a f u n c t i o n o f temperature  .....(1)  only.  -.5  The  -  d i f f e r e n t mechanisms i n v o l v e d i n t h e s i n t e r i n g p r o c e s s a r e r e l a t e d  by t h e f o l l o w i n g n u m e r i c a l n =  values:  2  m = 1  for  viscous or p l a s t i c  n- = •3  . m = 1  for  evaporation.and  n = 5  m = 2  for  volume d i f f u s i o n  n = 7  •m = 3  for  surface d i f f u s i o n .  Although  flow  condensation  i t has been r e c o g n i z e d t h a t t h e d e c r e a s e o f t o t a l  s u r f a c e energy o f t h e compacts i s a m o t i v a t i n g f o r c e f o r s i n t e r i n g ,  this  e x c e s s s u r f a c e energy i n a c t u a l compacts has n e v e r been v e r y g r e a t .  For  i n s t a n c e , t h e net d e c r e a s e i n f r e e energy o c c u r i n g on s i n t e r i n g a l p . p a r t i c l e s i z e m a t e r i a l c o r r e s p o n d s t o an energy d e c r e a s e o f about 1 cal/gm. However, i n t h e neck a r e a t h e r e e x i s t K may  s t r e s s e s due  t o the curvature which  produce mass f l o w o r a l o w e r i n g o f t h e v a p o u r p r e s s u r e  region.  The v a p o u r p r e s s u r e  over a f l a t s u r f a c e p  Q  and  i n . the neck  ^p,  the  lowering  o f t h e v a p o u r p r e s s u r e due t o t h e r a d i u s o f c u r v a t u r e p^ a r e r e l a t e d , by 2  t h e f o l l o w i n g e x p r e s s i o n , f i r s t d e r i v e d by K e l v i n  AS. = P where 2f solid.  The  0  -  ^ Vo  .(2)  RTp  i s t h e s u r f a c e energy and V© t h e m o l a r volume o f t h e  evaporation-condensation  mechanism f o r m a t e r i a l t r a n s p o r t d u r i n g  s i n t e r i n g can be e a s i l y u n d e r s t o o d from t h i s r e l a t i o n s h i p .  I f t h e vapour  p r e s s u r e o v e r t h e f l a t s u r f a c e ( o r o v e r t h e convex s u r f a c e i n t h e s p h e r i c a l model, F i g u r e 1) i s h i g h e r t h a n t h a t o f t h e neck r e g i o n ( r a d i u s p ) , p a r t i c u l a r l y at h i g h temperatures, sequent c o n d e n s a t i o n  e v a p o r a t i o n f r o m t h e convex r e g i o n and  i n t h e c a v i t y o f t h e neck can be e x p e c t e d  r e s u l t i n g i n neck growth.  to  sub-  occur,  - k -  A n o t h e r consequence o f t h e e x i s t e n c e o f the s t r e s s e s i n t h e neck can.be d e t e r m i n e d by c o n s i d e r i n g t h e vacuum s u r r o u n d i n g t h e neck as a composed o f v a c a n c i e s w h i c h evaporate  i n t o the s o l i d .  For these  t h e c a v i t y o f the neck w i l l be a c o n v e x i t y and c o n s e q u e n t l y  the  fluid  vacancies, pressure  o f v a c a n c i e s i n t h e neck area, w i l l be g r e a t e r t h a n under the s u r f a c e s o f the o t h e r p a r t s .  Assuming t h a t t h e vacancy p r e s s u r e i s p r o p o r t i o n a l t o  t h e i r c o n c e n t r a t i o n i n t h e s o l i d , e q u a t i o n (2) may  Ac Cb where C surface.  Due  Q  =  be r e w r i t t e n as  ,(3)  Vp  RTP  .  i s the e q u i l i b r i u m vacancy c o n c e n t r a t i o n under'a  t o t h e excess c o n c e n t r a t i o n o f v a c a n c i e s  flat  A C i n t h e neck  a r e a , t h e r e e x i s t s a g r a d i e n t o f v a c a n c i e s between t h i s a r e a and the  interior  o f t h e system. . T h i s w i l l r e s u l t i n t h e v a c a n c y m i g r a t i o n a l o n g t h e i r g r a d i e n t accompanied by t h e volume or s u r f a c e d i f f u s i o n o f atoms i n t h e direction.  opposite  The e x c e s s o f v a c a n c i e s have t o be removed f r o m the system by  d e p o s i t i n g them a t t h e p o s s i b l e s i n k s , w h i c h may the g r a i n boundaries.  be the nearby s u r f a c e o r  The v a c a n c i e s d e p o s i t e d - i n the g r a i n b o u n d a r i e s  can  e i t h e r be d i f f u s e d r a p i d l y t o t h e s o l i d v a p o u r i n t e r f a c e , because t h e r a t e o f d i f f u s i o n i s much l a r g e r i n t h e g r a i n boundary t h a n i n the b u l k o f t h e b o d y j o r i f t h e f l u i d i t y ' o f the m a t e r i a l i n t h e g r a i n boundary i s h i g h (as i n t h e case o f the v i s c o u s flow)  the vacancies w i l l  w i l l . i n a l i q u i d o f low v i s c o s i t y . vacancies v i a g r a i n boundaries  c o l l a p s e , as any v o i d  I n any c a s e , t h e e l i m i n a t i o n o f  w i l l . r e s u l t i n t h e s h r i n k a g e accompanied  by c e n t e r - t o - c e n t e r approach o f t h e p a r t i c l e s .  On t h e o t h e r hand, i f t h e  e l i m i n a t i o n o f vacancies, o c c u r s a t t h e f r e e s u r f a c e , , the c e n t e r - t o - c e n t e r • d i s t a n c e o f t h e spheres w i l l n o t change as would.be t h e case i n d e n s i f i c a t i o n  -.5 -  by t h e e v a p o r a t i o n and c o n d e n s a t i o n i  mechanism.  . From t h e d e r i v e d e x p r e s s i o n f o r AC,  e q u a t i o n (3)  and b y u s i n g  t h e f i r s t F i c k ' s d i f f u s i o n e q u a t i o n , t h e e q u a t i o n f o r volume d i f f u s i o n mechanism can.be o b t a i n e d .  . .xf. = a  K  y  A Q  rv t  v  (k)  RT  2  • where D  I t s f i n a l form i s  i s t h e volume s e l f - d i f f u s i o n c o e f f i c i e n t ;  K: i s a  n u m e r i c a l c o n s t a n t and has a v a l u e o f about 1 0 0 .  D e n s i f i c a t i o n o f Powder Compacts The i n c r e a s e i n d e n s i t y o f a powder compact d u r i n g s i n t e r i n g i s o f t h e g r e a t e s t p r a c t i c a l importance,, b u t due t o t h e number o f unknown v a r i a b l e s o r p o o r l y d e f i n e d parameters i n v o l v e d i n t h e s i n t e r i n g p r o c e s s , i t i s d i f f i c u l t t o e v a l u a t e t h e b a s i c mechanisms o f m a t e r i a l t r a n s p o r t . The e a r l i e r measurements o f powder compacts c o n s i s t e d m o s t l y o f d e t e r m i n i n g t h e i r d e n s i t y as a f u n c t i o n o f t e m p e r a t u r e ,  a l t h o u g h t h e time v a r i a b l e i s  of great s i g n i f i c a n c e from the p o i n t o f view o f the k i n e t i c s o f t h e process.  K i n g e r y and Berg-^ t r i e d t o a p p l y t h e volume d i f f u s i o n e q u a t i o n f o r neck g r o w t h between two p a r t i c l e s t o powder compacts o f o x i d e s . • They assumed t h a t t h e volume o f t h e neck a t any time i s e q u a l t o t h e volume o f t h e pore space removed.from t h e system, and o b t a i n e d t h e f o l l o w i n g r e l a t i o n ship by u s i n g equation  AV V  =  Q  where  AV  (k) kO g V  - 3n 8 and V  a Q  3  n  Py  i V5  V5 ••(5)  k T.  a r e t h e change o f volume w i t h time and i n i t i a l  - 6 -  volume r e s p e c t i v e l y ,  n i s t h e number of p o i n t s o f c o n t a c t determined, by  the c o o r d i n a t i o n s t a t e of the p a r t i c l e s .  From t h e e x p e r i m e n t a l o b s e r v a t i o n s  t h e y have f o u n d t h a t t h i s e q u a t i o n has.very, l i m i t e d a p p l i c a b i l i t y . i t can be a p p l i e d f o r a volume s h r i n k a g e o f up t o 2$>,  i t failed  Although  completely  k  f o r a s h r i n k a g e g r e a t e r t h a n Qfo.  Kuczynski  approached t h e problems o f  d e n s i f i c a t i o n of compacts from t h e p o i n t o f view o f pore s h r i n k a g e  and  c o n s i d e r e d two p o s s i b l e mechanisms f o r m a t e r i a l t r a n s p o r t i n t o t h e ' p o r e s : v i s c o u s o r p l a s t i c f l o w and volume d i f f u s i o n . The E n d - P o i n t  Density  E a r l i e r s t u d i e s on d e n s i f i c a t i o n o f powdered m e t a l s and  oxides  i n d i c a t e d t h a t i t was n o t p o s s i b l e t o r e a c h t h e t h e o r e t i c a l d e n s i t y by s i n t e r i n g a l o n e and t h i s l e d t h e powder m e t a l l u r g i s t s t o b e l i e v e t h a t t h e r e i s an ,"end-point d e n s i t y " f o r a l l m a t e r i a l s . 5 I n a recent study, Coble  showed t h a t t h e e n d - p o i n t d e n s i t y i s  t h e r e s u l t o f g r a i n growth, w h i c h t a k e s p l a c e s i m u l t a n e o u s l y d u r i n g s i n t e r i n g . He p o s t u l a t e d a model and d e r i v e d an e q u a t i o n t o s u p p o r t t h i s h y p o t h e s i s t h a t as l o n g as t h e g r a i n growth i s c o n t i n u o u s and t h e pores a r e connected by t h e g r a i n b o u n d a r i e s , complete  e l i m i n a t i o n o f pores i n t h e b o u n d a r i e s can be  a c h i e v e d , and t h i s w i l l r e s u l t i n complete  densification.  Only when d i s -  c o n t i n u o u s g r a i n growth, t o o k p l a c e i n a system would t h e pores be i n s i d e the g r a i n s .  trapped  These t r a p p e d p o r e s would.not s h r i n k any f u r t h e r .  I n the  l a t t e r c a s e , t h e powder compact would r e a c h an e n d - p o i n t d e n s i t y which would be below t h e t h e o r e t i c a l l i m i t of" d e n s i f i c a t i o n .  C o b l e , u s i n g doped a l u m i n a  ( t o c o n t r o l g r a i n growth) showed e x p e r i m e n t a l l y t h a t i t i s p o s s i b l e t o a c h i e v e t h e o r e t i c a l d e n s i t y o n l y by s i n t e r i n g . . On t h e o t h e r hand, undoped  AI2O3 w i t h i t s u n r e s t r i c t e d g r a i n growth r e a c h e d an e n d - p o i n t d e n s i t y a f t e r a certain period of s i n t e r i n g .  T h i s phenomenon o f e x a g g e r a t e d g r a i n growth  and appearance o f t r a p p e d p o r e s a f t e r a few hours o f s i n t e r i n g occurs o n l y 6 at  o r above a c e r t a i n t e m p e r a t u r e c a l l e d t h e Sauerwald t e m p e r a t u r e , Tg.  T h i s i s g e n e r a l l y 2/3  to 3A  of the m e l t i n g p o i n t of the m a t e r i a l .  Pore • S t r u c t u r e The change i n shape and s i z e of t h e p o r e s i s v e r y d i f f i c u l t t o s t u d y i n r e a l compacts as t h e p o r e s a r e o f d i f f e r e n t s i z e s and shapes. the  In  f i r s t s t a g e , t h e necks grow between a d j a c e n t p a r t i c l e s and t h e g r a i n  boundary does n o t move, because any d i s p l a c e m e n t towards t h e c e n t e r o f the  p a r t i c l e would mean an i n c r e a s e i n boundary a r e a and t h u s , an i n c r e a s e  i n g r a i n boundary energy.  A f t e r w a r d s , as s i n t e r i n g proceeds and as t h e  v a c a n c y f l u x i s i n v e r s e l y p r o p o r t i o n a l t o the p o r e r a d i u s , t h e l a r g e r p o r e s may  i n c r e a s e i n s i z e as a r e s u l t o f t h e c o n d e n s a t i o n o f v a c a n c i e s o r i g i n a t i n g  from t h e s m a l l e r ones.  Thus, t h e p o r e s may be c l a s s i f i e d as s m a l l ones,  w h i c h s h r i n k , and l a r g e ones, w h i c h i n c r e a s e i n s i z e , d u r i n g s i n t e r i n g . s m a l l e r p o r e s a r e much more numerous and may i n t h e compact.  The  i n c l u d e t h e b u l k of t h e p o r o s i t y  When t h e y d i s a p p e a r g r a d u a l l y t h e o v e r a l l d e n s i t y of t h e  compact i n c r e a s e s .  On t h e o t h e r hand, t h e p o r e s o f t h e second group  may  c o n t r o l t h e g r a i n growth by a n c h o r i n g . t h e g r a i n b o u n d a r i e s . The c r i t i c a l d i a m e t e r ('J  ) o f t h e p o r e s o f r a d i u s r , w h i c h i s most  e f f e c t i v e i n i n h i b i t i n g g r a i n growth, and t h e volume f r a c t i o n p o r o s i t y f 7 i n t h e m a t e r i a l a r e r e l a t e d , a c c o r d i n g t o Zener  by t h e f o l l o w i n g r e l a t i o n s h i p  - 8  As soon as t h e l a r g e r pores-  r e a c h the c r i t i c a l d i a m e t e r ,  -  exaggerated  g r a i n g r o w t h w i l l o c c u r and t h e t r a p p e d pores w i l l not s h r i n k any more.  Model f o r Complete D e n s i f i c a t i o n Coble  was.,-the f i r s t t o f o r m u l a t e b u l k d i f f u s i o n models f o r t h e  t o t a l c o u r s e o f s h r i n k a g e i n powder compacts, l e a d i n g , t o t h e o r e t i c a l l y dense products.  He assumed t h a t t h e r e a r e t h r e e s t a g e s o f d e n s i f i c a t i o n .  In  the i n i t i a l or f i r s t stage, i n t e r p a r t i c l e contact area i n c r e a s e d from zero to  0.2  of the c r o s s - s e c t i o n a l area of the p a r t i c l e .  This stage,also r e -  f e r e d t o as t h e neck g r o w t h s t a g e , u s u a l l y i s accompanied by an i n c r e a s e i n r e l a t i v e d e n s i t y o f powder compacts f r o m 0.5 F i g u r e s 2a and 2b.  During.the  t o 0.6.  T h i s i s shown i n  i n i t i a l ' s t a g e o f s i n t e r i n g , g r a i n growth can  not o c c u r , as i t w o u l d . r e q u i r e m i g r a t i o n o f t h e g r a i n boundary f r o m t h e minimum a r e a p o s i t i o n w h i c h i n t u r n would r e s u l t i n an i n c r e a s e i n a r e a .and energy.  In t h e second o r i n t e r m e d i a t e s t a g e , g r a i n growth b e g i n s  and  pore shape changes t o produce a m a t r i x o f p o r e s and g r a i n boundary.  The  e q u i l i b r i u m a n g l e s formed between thgm a r e d i c t a t e d by s u r f a c e t e n s i o n such t h a t t h e . t h r e e i n t e r a c t i n g s u r f a c e s f o r m a s p a t i a l f o r c e b a l a n c e . . T h i s stage can be r e p r e s e n t e d by F i g u r e 2c.  The pore phase i s v e r y s i m i l a r t o a con- •  t i n o u s c h a n n e l and i s assumed t o be c y l i n d r i c a l i n shape. The f i n a l s t a g e b e g i n s when t h e pore becomes d i s c o n t i n u o u s and c h a n n e l s a r e r e p l a c e d by t h e g r a i n b o u n d a r i e s .  the  The pores o n l y occupy t h e ,  f o u r g r a i n c o r n e r s and a r e n e a r l y s p h e r i c a l i n shape as shown i n F i g u r e These p o r e s a t t h e f o u r g r a i n c o r n e r s w i l l g r a d u a l l y s h r i n k t o z e r o s i z e t h e s i n t e r i n g w i l l p r o c e e d t o t h e o r e t i c a l d e n s i t y o f the compact.  An  2d. and  - 9  -  a l t e r n a t i v e f i n a l s t a g e would be when d i s c o n t i n u o u s g r a i n growth o c c u r s b e f o r e a l l t h e p o r o s i t y i s removed. of p o r e s w o u l d be i m p o s s i b l e . his  I n t h i s case complete e l i m i n a t i o n  C o b l e s u p p o r t e d h i s arguments by  comparing  h y p o t h e t i c a l model w i t h t h e m i c r e s t r u c t u r e s o f t h e specimens a t  d i f f e r e n t stages of s i n t e r i n g .  These s t a g e s a r e shown i n F i g u r e . 3 .  . The f i n a l e q u a t i o n r e l a t i n g t h e r a t e . o f pore s h r i n k a g e w i t h o t h e r parameters has t h e f o l l o w i n g f o r m  dp dt where  =  3  '.  'my{&  n  .(6)  r k T  N = n u m e r i c a l c o n s t a n t : f o r c y l i n d r i c a l pore case N = 10 f o r c l o s e d pore case N = 6 TT Dv= b u l k d i f f u s i o n c o e f f i c i e n t ,1f = s u r f a c e energy aQ = vacancy; volume 1 = average g r a i n d i a m e t e r k =; Boltzmann's constaux T; = a b s o l u t e t e m p e r a t u r e  S i n t e r i n g o f Oxides a) Neck Growth E x p e r i m e n t s To'test the k i n e t i c s of s i n t e r i n g ^ ( p a r t i c u l a r l y the i n i t i a l stage) spheres o f A l 0 3 , T i 0 2 and ZnO have been u s e d by s e v e r a l w o r k e r s . 2  Kiczynski'  p a r t i c u l a r l y u s e d spheres of s a p p h i r e t o t e s t h i s volume d i f f u s i o n model o f neck growth. . P a r r a v a n o and N o r r i s ^ used spheres o f ZnO t o s t u d y t h e r a t e o f neck growth as a f u n c t i o n o f t e m p e r a t u r e .  T h e i r r e s u l t s u p p o r t e d t h e model  of e v a p o r a t i o n and c o n d e n s a t i o n f o r m a t e r i a l t r a n s p o r t i n t h a t system. o'Bryan a n d P a r r a v a n o " ^ s t u d i e d t h e s i n t e r i n g o f s i n g l e c r y s t a l s o f r u t i l e i n a i r and i n r e d u c i n g atmosphere u s i n g a s p h e r e - t o - s p h e r e model.  i n t h e t e m p e r a t u r e range o f 900-1350°C, T h e i r work i n d i c a t e d t h a t t h e predominant  mechanism o f m a t e r i a l t r a n s p o r t f o r s i n t e r i n g was volume d i f f u s i o n and t h e y  - 10 F i g u r e 2. (a) I n i t i a l Stage o f S i n t e r i n g , (c) (b) Wear End o f t h e I n i t i a l Stage; spheres have begun t o c o a l e s c e , ( c ) I n t e r m e d i a t e Stage; dark g r a i n s have adopted shape o f t e t r a k a i d e c a h e d r o n , e n c l o s i n g w h i t e pore channels a t g r a i n edges, (d) F i n a l Stage; pores a r e t e t r a h e d r a l i n c l u s i o n s a t c o r n e r s where f o u r t e t r a k a i d e c a h e d r a meet, ( a f t e r Coble5). (b)  F i g u r e 3•  ( ) (b) (c) (d) a  T y p i c a l I n t e r m e d i a t e Stage S t r u c t u r e T y p i c a l F i n a l Stage S t r u c t u r e End Stage a t T h e o r e t i c a l D e n s i t y F i n a l Stage A f t e r D i s c o n t i n u o u s G r a i n Growth (after Coble ). 5  - 11  an a c t i v a t i o n energy o f 70 * h k c a l / m o l e f o r t h e p r o c e s s .  obtained  and Whitmore''""'" a l s o i n d e p e n d e n t l y s t u d i e d t h e s p h e r e - t o - p l a t e of vacuum-reduced m o n o c r y s t a l l i n e 1200-1275°C.  -  Kawai  bonding  r u t i l e over t h e t e m p e r a t u r e range o f  The r a t e law g o v e r n i n g t h e i n t e r f a c i a l growth i n d i c a t e d  a l s o t h a t t h e volume d i f f u s i o n was t h e predominant mechanism o f m a t e r i a l transport i n the s i n t e r i n g process.  b), Dens i f i c a t i o n o f Powder Compacts  v  The  d e n s i f i c a t i o n o f powder compacts o f oxide was s t u d i e d by  12 Coble  as d i s c u s s e d p r e v i o u s l y .  ;He used a l u m i n a as t h e s t a n d a r d  material  13 to.test h i s hypothesis.  V e r y r e c e n t l y , Johnson and C u t l e r  out i n v e s t i g a t i o n s on t h e l i n e a r s h r i n k a g e  also carried  r a t e o f a l u m i n a powder compacts.  B o t h o f t h e s e i n v e s t i g a t i o n s i n d i c a t e d t h a t b u l k d i f f u s i o n , and n o t t h e g r a i n boundary d i f f u s i o n was t h e basic- mechanism, o f d e n s i f i c a t i o n .  Clark  lh and W h i t e  used magnesia powder compacts t o s t u d y t h e r a t e o f d e n s i f i c a t i o n  and e x p l a i n e d t h e i r r e s u l t s on.the b a s i s o f a p l a s t i c f l o w model.  The  e f f e c t o f n o n - s t o i c h i o m e t r y on t h e r a t e o f d e n s i f i c a t i o n has been s t u d i e d 15 by s e v e r a l w o r k e r s with U0 + . 2  x  No r e s u l t s have been r e p o r t e d compacts, a l t h o u g h s i n t e r e d T i 0 wave g u i d e t u b e s a t the, p r e s e n t  2  on t h e d e n s i f i c a t i o n o f T i 0  discs are b e i n g w i d e l y  2  powder  used i n t h e micro-  time.  16 Crystal Structure of Titania Titanium  d i o x i d e c a n c r y s t a l l i z e i n t h r e e forms.  t h e y a r e known as r u t i l e , a n a t a s e and b r o o k i t e .  Mineralogically  R u t i l e i s t h e s t a b l e form .1  above 820°C b u t m e t a s t a b l e a t room t e m p e r a t u r e and e x i s t s i n a l l commercial titania  products.  - 12  -  R u t i l e has a t e t r a g o n a l s t r u c t u r e w i t h a.= k.k-923 A and c = 2 . 8 9 3 0 A.  From a . c o n s i d e r a t i o n o f p u r e l y i o n i c s t r u c t u r e , t h e  r a d i u s r a t i o o f t i t a n i u m t o oxygen i o n p r e d i c t s a . s i x - f o l d of t i t a n i u m . w i t h , oxygen.  coordination  The r u t i l e s t r u c t u r e may be d e s c r i b e d as  b u i l t up f r o m d i s t o r t e d T i 0 . o c t a h e d r a 6  >  t h e octahedra forming  chains  i n t h e c - d i r e c t i o n and each o c t a h e d r o n s h a r i n g an edge w i t h t h e a d j a c e n t members o f t h e c h a i n s .  The c r y s t a l d e n s i t y ls-k.26  gm/cm , as d e t e r m i n e d 3  from t h e X - r a y measurements. are p r e s e n t  I n t h e r u t i l e s t r u c t u r e t i t a n i u m and oxygen d i n t h e i r h i g h e s t v a l e n c e s t a t e +k and - 2 . T i t a n i u m i s a  t r a n s i t i o n m e t a l ©f t h e i r o n group and i t s normal e l e c t r o n i c c o n f i g u r a t i o n 2  2  i s (^s) (3d)  outside t h e argon core.  Defect Reaction i n R u t i l e 17  Straumanis  e t a l . r e c e n t l y measured t h e d e n s i t y and l a t t i c e  parameter o f r u t i l e powders f o r t h e oxygen  d e f i c i e n c i e s from 0.5 t o 0.8  a t o m i c p e r c e n t . . T h e i r r e s u l t s i n d i c a t e t h a t t h e s i z e and shape o f t h e u n i t c e l l i n t h i s range d© n o t change a p p r e c i a b l y so t h a t t h e change i n d e n s i t y c a n be a t t r i b u t e d t o oxygen v a c a n c i e s  alone.  K i n e t i c s t u d i e s o f t h e o x i d a t i o n o f t i t a n i u m , under c o n d i t i o n s such t h a t r u t i l e i s the only oxide i n t h e t a r n i s h l a y e r , d i d not provide unambiguous i n f o r m a t i o n c o v e r i n g e i t h e r t h e predominant p o i n t d e f e c t o r the s l o w e r moving i o n i c s p e c i e s i n r u t i l e . 18  and- Andrew- , and K i n n a a n d K n o r r  19  The i n v e s t i g a t i o n s , o f G u l b r a n s e n  s u p p o r t t h e case f o r i n t e r s t i t i a l c a t i o n 20  d i f f u s i o n being r a t e - c o n t r o l l i n g , while those o f B i r c h e n a l l  21  , Hauffe  and  22  others  show oxygen i o n d i f f u s i o n c o n t r o l l i n g t h e o x i d a t i o n o f t i t a n i u m .  -.13  -  From t h e s e m i - c o n d u c t i n g b e h a v i o u r o f r u t i l e i t i s w e l l e s t a b l i s h e d t h a t r u t i l e becomes a m e t a l excess n-type s e m i - c o n d u c t o r upon r e d u c t i o n .  The predominant p o i n t d e f e c t s a r e oxygen i o n v a c a n c i e s  w h i c h a r e c a p a b l e o f t r a p p i n g e l e c t r o n s and t h e r e b y a c t i n g as a donor center.  Aim.of t h e P r e s e n t I n v e s t i g a t i o n T h i s p r e s e n t i n v e s t i g a t i o n has been m a i n l y concerned w i t h s i n t e r i n g and. g r a i n growth o f n o n - s t o i c h i o m e t r i c t i t a n i a powder compact's. A l l t h e e x p e r i m e n t s were c a r r i e d . o u t i n a. c o n t r o l l e d oxygen p a r t i a l p r e s s u r e over a t e m p e r a t u r e range o f .1000 t o 125G°C. - T h i s was t o m a i n t a i n a c o n s t a n t v a c a n c y c o n c e n t r a t i o n o f oxygen i n the system, i . e . a c o n s t a n t r a t i o o f t i t a n i u m t o oxygen i n t h e n o n - s t o i c h i o m e t r i c t i t a n i a .  I n a d d i t i o n , t h e r a t e o f w e i g h t l o s s was a l s o d e t e r m i n e d i n t h e specimens used f o r t h e i n v e s t i g a t i o n o f s i n t e r i n g a n d g r a i n g r o w t h . :  - 14 -  - EXPERIMENTAL  . T i t a n i a Powder A l l experiments were performed u s i n g . r u t i l e powders s u p p l i e d by t h e J . J . Baker Chemical Company, P h i 1 1 i p s b u r g , N.-J.  • Two one-pound :samples were used, t h e c h e m i c a l c o m p o s i t i o n o f which i s . g i v e n i n t h e f o l l o w i n g  table. Table I .  A n a l y s e s o f t h e Two R u t i l e Powder Samples  L o t 21303 Water S o l u b l e S a l t s Arsenic Iron Lead . Zinc  L o t 28363  0.05 $ 0.0001 0.002 0.008 0.004  0.02 % 0.00005 0.002 0.004 0.005  To determine t h e g r a i n s i z e and shape o f t h e powders, t h r e e methods were tried:  f i r s t , , t h e s t a n d a r d T y l e r s i e v e s ; second, a s e d i m e n t a t i o n t e c h n i q u e  u s i n g Andreasen's p i p e t t e ; and, t h i r d s ; measurement o f tl^.i.^rain.^sSgjegrdiBi'ived 5  from p i c t u r e s t a k e n b y t h e e l e c t r o n m i c r o s c o p e .  F i g u r e s 4 and 5 r e p r e s e n t t h e p i c t u r e s taken by t h e e l e c t r o n microscope w i t h t h e m a g n i f i c a t i o n o f 3000 and 4000. g r a i n s were f l a k e d , having.two  These show t h a t t h e  l a r g e dimensions b u t l i t t l e  thickness.  The p a r t i c l e s were a l l i n s u b - s i e v e range. • A c c o r d i n g . t o t h e measurements c a r r i e d out by t h e Andreasen's p i p e t t e t e c h n i q u e , • o v e r 80 $ o f t h e p a r t i c l e s are l e s s . t h a n 4 m i c r o n s .  The p a r t i c l e s i z e was a l s o determined by t a k i n g  t h e average o f the p a r t i c l e dimensions i n F i g u r e s 4 and 5  a  n  <  i  i s 1.5 J * 1  The l a r g e s t and s m a l l e s t dimensions a r e 4.25 M ^ d 0 . 2 5 p. r e s p e c t i v e l y .  - 15 -  F i g u r e k.  E l e c t r o n M i c r o g r a p h o f T i 0 Powders Showing t h e S i z e and Shape o f t h e P a r t i c l e s . 2  X 3000  Figure 5.  E l e c t r o n M i c r o g r a p h o f T i 0 Powders Showing the S i z e and Shape o f the P a r t i c l e s . X 4000 2  -16  -  However, t h e average g r a i n s i z e o f s m a l l e r p a r t i c l e s , w h i c h a r e t h e l a r g e r f r a c t i o n , h a v e , an average d i a m e t e r of about 0,8 u.  Compaction o f t h e Powder One  c y l i n d r i c a l d i e h a v i n g a n ~ i n t e r n a l d i a m e t e r o f 0.5 i n c h e s  and a n o t h e r one o f r e c t a n g u l a r shape, h a v i n g the i n t e r n a l  cross-sectional  dimensions o f 0.5 X k i n c h e s were used f o r compacting t h e powder. specimen t h i c k n e s s was k e p t a p p r o x i m a t e l y  O.25  specimens tended t o break down d u r i n g . f i r i n g .  i n c h e s because  The  thicker  E a r l i e r i n v e s t i g a t i o n s made  on compacting powders o f b r i t t l e ' m a t e r i a l s , such as o x i d e s , used compacting p r e s s u r e s r a n g i n g f r o m ^,000 i t was  t o 40,000 p s i .  In the present i n v e s t i g a t i o n  found t h a t an i n c r e a s e o f p r e s s u r e from 766O t o 40,000 p s i i n c r e a s e d  the i n i t i a l g r e e n d e n s i t y o f t h e compacts from 1^3 as t h e compacting p r e s s u r e was specimens.  t o 2.30 gm/cm . 3  However,  i n c r e a s e d , l a m i n a r c r a c k s appeared on t h e  These caused t h e specimens'to b r e a k i n t o p i e c e s on  sintering.  As t h e specimens were a l s o used f o r weight l o s s s t u d i e s , no l u b r i c a n t o f any k i n d c o u l d be used. . T h e r e f o r e , a compacting p r e s s u r e o f 12,000 p s i was  adopted f o r a l l the weight l o s s , g r a i n . g r o w t h and s i n t e r i n g measurements.  The  Furnace The f u r n a c e u s e d f o r a l l experiments was  essentially a horizontal  t y p e tube f u r n a c e h e a t e d by f o u r globar;, h e a t i n g elements. F i g u r e 6,  As shown i n  t h e main tube c o n s i s t e d o f a l o n g Z i r c o t u b e o f 1 l / 8 i n c h e s i n  d i a m e t e r p a s s i n g t h r o u g h t h e h e a t i n g chamber, which was b u i l t w i t h i n s u l a t i n g bricks.  Both ends o f t h e tube were c o o l e d by c i r c u l a t i n g water t h r o u g h  copper j a c k e t s .  Thus the specimen, w h i l e s t i l l i n s i d e t h e f u r n a c e was  q u i c k l y a f t e r i t was  removed from t h e h e a t i n g zone.  A second Z i r c o t u b e  cooled  - 18  -  o f 0 . 9 i n c h e s i n t e r n a l d i a m e t e r was f i t t e d i n s i d e t h e main one t o a v o i d any t h e r m a l shock on t h e l a t t e r when t h e h o t boat was p u l l e d i n t o t h e c o o l zone.  The i n l e t o f t h e tube was c o n n e c t e d t o a h e a t e d copper tube  w h i c h was j o i n e d t o a b u b b l e r and w h i c h c a r r i e d r e d u c i n g atmosphere. The o u t l e t was d i r e c t l y c o n n e c t e d t o a l o n g g l a s s tube bent a t r i g h t a n g l e s , t h e end o f w h i c h was d i p p e d i n t o a w a t e r b a t h .  By t h i s  arrangement  t h e f u r n a c e was e s s e n t i a l l y a c l o s e d system.  W i t h t h e h e l p o f a s t r i n g system a s shown i n F i g u r e 6 i t was p o s s i b l e t o move t h e b o a t f r o m t h e c o o l zone t o t h e h o t zone and v i c e - v e r s a . The t e m p e r a t u r e was measured b y t h e u s e o f a P t - P t 10$ Rh thermoc o u p l e i n s e r t e d i n t o a p r o t e c t i o n tube and p l a c e d j u s t above t h e main Zircotube.  T h i s thermocouple was c o n n e c t e d t o a Wheelco t e m p e r a t u r e  r e g u l a t o r w h i c h c o n t r o l l e d t h e t e m p e r a t u r e w i t h i n * 10°C.  There was a t e m p e r a t u r e g r a d i e n t a l o n g t h e l e n g t h o f t h e main Z i r c o t u b e and a l s o a t e m p e r a t u r e d i f f e r e n c e between t h e thermocouple and the specimen.  F o r t h i s r e a s o n t h e p o s i t i o n o f t h e boat was m a i n t a i n e d i n  t h e t h r e e i n c h e s o f h o t zone where t h e temperature was r e l a t i v e l y c o n s t a n t 1 ; and a l m o s t e q u a l t o t h e r e c o r d e d t e m p e r a t u r e i n t h e Wheelco r e g u l a t o r .  The  t e m p e r a t u r e o f t h e specimens was o c c a s i o n a l l y measured w i t h a s e p a r a t e s t a n d a r d i z e d P t - P t 10$. Rh t h e r m o c o u p l e . C o n t r o l o f t h e Furnace Atmosphere The r a t i o o f the p r e s s u r e s o f H  2  and H 0 i n t h e i n l e t gas o f 2  t h e f u r n a c e was d e t e r m i n e d by p a s s i n g hydrogen s l o w l y i n t o two b u b b l e r s , submerged i n a b a t h o f w a t e r w h i c h was m a i n t a i n e d a t a f i x e d t e m p e r a t u r e . The f l o w o f hydrogen was k e p t as low as p o s s i b l e t o have an e q u i l i b r i u m '  -19  atmosphere  -  i n s i d e , t h e b u t j b l e r s . . From t h e e a r l y r e s u l t s o f r e d u c t i o n i t  appeared t h a t f l o w s •. o f l e s s t h a n 7Q b u b b l e s a minute gave r e p r o d u c i b l e • • results. The b u b b l e r s were connected w i t h t h e f u r n a c e b y a copper p i p e w h i c h was h e a t e d b y a r e s i s t a n c e element a t a t e m p e r a t u r e o f 70°C, s l i g h t l y h i g h e r t h a n t h e bath, t e m p e r a t u r e t o a v o i d any c o n d e n s a t i o n .  D e s c r i p t i o n o f a .Rim i n t h e Furnace The a l u m i n a specimen h o l d e r was l o a d e d w i t h f i v e t o f i f t e e n specimens, and was p u t i n s i d e the. t u b e i n t h e c o o l zone. .. The f u r n a c e atmosphere was g r a d u a l l y changed t o t h e d e s i r e d atmosphere  f i r s t by  f l a s h i n g i t w i t h a f l o w , of" a r g o n f o r 10 minutes and s u b s e q u e n t l y b y a f l o w , o f p r e d e t e r m i n e d H / H 0 m i x t u r e f o r 15 m i n u t e s . 2  2  Afterwards the  specimens h o l d e r was pushed i n t o t h e h o t e s t zone o f t h e t u b e .  The r e c o r d e d  time f o r a l l runs began when t h e b o a t s were f i r s t i n t h e h o t zone. a . d e f i n i t e t i m e o f f i r i n g . t h e r e v e r s e p r o c e d u r e was f o l l o w e d .  After  The specimen  h o l d e r was p u l l e d back i n t o t h e c o o l i n g zone b u t t h e same flow-was m a i n t a i n e d for  s e v e r a l m i n u t e s t o a v o i d any change i n t h e s t o i c h i o m e t r y o f r u t i l e .  When t h e specimen t e m p e r a t u r e was s u f f i c i e n t l y l o w t h e tube was f l a s h e d w i t h a r g o n f o r 10 m i n u t e s and f i n a l l y t h e specimens were t a k e n out f o r measurements.  - A f t e r t h e a p p r o p r i a t e measurements were t a k e n t h e specimens were r e p l a c e d i n t h e b o a t and t h e p r o c e s s was r e p e a t e d .  Measurements f o r Weight L o s s and D i m e n s i o n a l Change The w e i g h t l o s s and s h r i n k a g e measurements.were c a r r i e d o u t on the  same specimens, as b o t h o f t h e s e measurements were needed t o c a l c u l a t e  the  relative density.  - 20 -  From the weight:  r a t i o o f t i t a n i u m t o t i t a n i a which i s  ^7-9/79S9J t h e weight o f t i t a n i u m i n t h e specimen b e f o r e r e d u c t i o n was  deduced.  The weight o f t h e oxygen was determined by s u b t r a c t i n g  the  weight o f t h e t i t a n i u m from t h e weight o f t h e i n i t i a l r u t i l e h a v i n g  a s t o i c h i o m e t r i c composition of T i 0 . 2  The l o s s . o f oxygen a f t e r  reduction  was  o b t a i n e d by s u b t r a c t i n g t h e i n i t i a l f i x e d weight o f t i t a n i u m from  the  weight o f t h e specimen.  These measurements were t a k e n on a C h r i s t i a n  Beckers manual b a l a n c e .  The s h r i n k a g e was c a l c u l a t e d by measuring t h e l e n g t h and w i d t h of  t h e specimen w i t h a micrometer h a v i n g an a c c u r a c y o f * 0.001  The average o f t h e s e two l i n e a r s h r i n k a g e s was used t o deduce  inches.  t h e volume  s h r i n k a g e from t h e f o l l o w i n g . r e l a t i o n  .X  100  From t h e weight measurements and from t h e dimensions o f t h e specimens a f t e r d i f f e r e n t p e r i o d s o f heat treatment, t h e volume and t h e b u l k d e n s i t i e s were c a l c u l a t e d and r e c o r d e d .  Measurements f o r G r a i n Growth a) P o l i s h i n g and E t c h i n g For  t h e s t u d y o f g r a i n growth, t h e specimens were p o l i s h e d  first  u s i n g sandpaper o f v a r i o u s s i z e s and t h e n u s i n g wheels h a v i n g suspensions of  f i n e alumina powder. The sandpapers used were o f t h e types 1,  alumina powder had an average g r a i n s i z e o f 0.05 the  0,; 00 and 000; t h e  u.  When the s u r f a c e s o f  p o l i s h e d specimens were f r e e from any v i s i b l e s c r a t c h e s , t h e y were  e t c h e d by immersing them i n a b a t h o f c o n c e n t r a t e d b o i l i n g H S 0 2  4  for 2 to  -.21  3 minutes.  Some specimens were s u b j e c t e d t o a. t h e r m a l  etch f o r three  -  y  m i n u t e s a t 6^>0°C b e f o r e a c i d e t c h i n g . .This i s t o make t h e g r a i n b o u n d a r i e s more a p p a r e n t .  I n t h i s case however, t h e t i m e f o r t h e H S 0 2  4  immersion was  r e d u c e d t o 30 seconds.  b)  G r a i n S i z e Measurements S e v e r a l p i c t u r e s o f known m a g n i f i c a t i o n o f these  specimens  were t a k e n w i t h a R e i c h e r t m e t a l l o g r a p h i c m i c r o s c o p e u s i n g  reflected  light.  A m a g n i f i c a t i o n o f I5OO was used f o r t h e specimens  having  s m a l l e r g r a i n s whereas a m a g n i f i c a t i o n o f 600 was used f o r t h o s e larger grains.  having  The average g r a i n s i z e measurements were made u s i n g 23  t h e i n t e r c e p t ( o r Heyn) p r o c e d u r e  .  - 22 -  RESULTS  X-Ray I n v e s t i g a t i o n To i n v e s t i g a t e t h e s t a b i l i t y range o f r u t i l e f o l l o w i n g p r e l i m i n a r y experiment was c a r r i e d o u t .  on r e d u c t i o n t h e  Compacts o f r u t i l e were  reduced, t o a f i x e d . r a t i o o f t i t a n i u m t o oxygen b y h e a t i n g i n t h e f u r n a c e a l r e a d y d e s c r i b e d and b y u s i n g d i f f e r e n t r a t i o s o f H /H 0 p a r t i a l p r e s s u r e s 2  over t h e temperature  2  range o f 1 0 0 0 . t o 1300°C.  . The specimens were scanned i n a N o r e l c o X - r a y d i f f r a c t o m e t e r f o r t h e Bragg a n g l e s o f 10° t o 80°.  The "d" v a l u e s were c a l c u l a t e d  from.the  c o r r e s p o n d i n g d i f f r a c t o m e t r i c peaks and. compared w i t h t h e s t a n d a r d A.S.T.M. cards f o r i d e n t i f i c a t i o n - o f , t h e o x i d e s p r e s e n t .  I t was e s t a b l i s h e d t h a t t h e  r u t i l e was s t a b l e up t o a n o n - s t o i c h i o m e t r i c c o m p o s i t i o n o f compound T i 3 p of  5  The  T±0 .Q . 1  2  was d e t e c t e d o n l y i n reduced r u t i l e h a v i n g a c o m p o s i t i o n  TiOi.gi.  I n t h e same i n v e s t i g a t i o n , t h e r e l a t i o n s h i p between t h e b u b b l i n g rate of H  i n t h e w a t e r b a t h and t h e c o r r e s p o n d i n g e q u i l i b r i u m r a t i o o f  2  t i t a n i u m t o oxygen o b t a i n e d a t t h e d i f f e r e n t t e m p e r a t u r e s was d e t e r m i n e d .  These r a t i o s were s u b s e q u e n t l y used d u r i n g s i n t e r i n g , weight  loss  and- g r a i n growth measurements. Weight Loss The  Study rates of r e d u c t i o n of T i 0  compositions having the r u t i l e were determined.by  2  t o two w e l l d e f i n e d n o n - s t o i c h i o m e t r i c  structure (Ti0i.g  weight l o s s measurements.  out a t 1000, 1050, 1100,.  2  ± o-oi  a  n  (  i T i ^ . g a , * o-oi)  These e x p e r i m e n t s were c a r r i e d  II50 and 1200°C under t h e a p p r o p r i a t e H / H 0 2  2  - 23  -  atmospheres t o y i e l d t h e s e c o m p o s i t i o n s a t e q u i l i b r i u m .  ^r^-  The f r a c t i o n a l weight l o s s  o f t h e compacts can be r e l a t e d  w t o t h e t i m e o f r e d u c t i o n by t h e f o l l o w i n g t y p e o f g e n e r a l i z e d e q u a t i o n . AW  = ' ,(  A  t)  .....(7)  w Z\W = w e i g h t l o s s a t t i m e t W = o r i g i n a l weight A = a f u n c t i o n o f t e m p e r a t u r e , p r e s s u r e and geometry o f t h e sample.  I n o r d e r t o d e t e r m i n e t h e v a l u e o f t h e exponent n, t h e w e i g h t l o s s d a t a g i v e n i n Appendix I , T a b l e s 1 and 2 were p l o t t e d as l o g ( A W) W  versus l o g t . 1  T h i s p l o t i s shown i n F i g u r e 7 f o r T i O i . 9 2  f o r TiOx.gg as e q u i l i b r i u m p r o d u c t s . v a l u e o f n = 0.5^ ± 0.1 f o r T i 0 i .  a  n  (  i  i'  n  F i g u r e 8.  The s l o p e o f t h e l i n e s produced a  and n = 0.48 ± 0.01 i n t h e case o f  9 2  TiOi.gs. 24  •A l i t e r a t u r e s u r v e y  r e v e a l e d t h a t such a r e d u c t i o n p r o c e s s  u s u a l l y f o l l o w s e i t h e r a l i n e a r o r a p a r a b o l i c l a w . T h e r e f o r e t h e average v a l u e o f n = O.52 * 0.01 s u g g e s t s a p a r a b o l i c r e l a t i o n .  E q u a t i o n (7)  can t h e n be w r i t t e n as AW  =  (A t ) ° '  5  (8)  W  The v a l u e s o f t h e f u n c t i o n A ° '  5  a t d i f f e r e n t temperatures are then the  s l o p e s o f t h e p l o t s o f t h e f r a c t i o n a l w e i g h t l o s s under i s o t h e r m a l c o n d i t i o n s v e r s u s t ° ' as shown i n F i g u r e s 9 5  a n £  l  10.  These f i g u r e s show t h a t t h e  weight l o s s reached t h e e q u i l i b r i u m v a l u e a f t e r a p e r i o d o f h e a t i n g o f about  Per Cent Weight Loss (-JJ- X  100)  2.0  F i g u r e 9.  P e r Cent Weight L o s s V e r s u s Square Root o f Time f o r Ti0 . z 1  g  - 27  F i g u r e 10.  P e r Cent Weight L o s s . V e r s u s Square Root o f Time f o r T i 0 i . g 8  -  -28  100  t o 300 minutes.  . F o r t h i s r e a s o n , o n l y - t h e d a t a b e f o r e 150 minutes o f  r e d u c t i o n were t a k e n i n t o account l e a s t squares 1 and  -  method.  f o r t h e c a l c u l a t i o n o f t h e s l o p e s by t h e  The r e s u l t s a r e t a b u l a t e d i n A p p e n d i x - I ,  Tables  2.  I n o r d e r t o determine  t h e oxygen p a r t i a l p r e s s u r e dependence  of t h e f u n c t i o n A, i t was assumed t h a t t h e r e l a t i o n s h i p took t h e form  A.=  or  A ' 0  K' 5  where K  P O  (9)  .  X 2  (10)  = K " .F02 ' 0  5  2  i n c l u d e s t h e s p e c i f i c r e a c t i o n r a t e c o n s t a n t and any  o t h e r v a r i a b l e s a f f e c t i n g t h e r a t e . - I n - l o g a r i t h m i c form, t h e r e l a t i o n s h i p w i l l be  ' logA°' ,= l  log K  5  +  x  2  ' l o g P0  .  2  (10)  (11)  2  o•5 The e x p e r i m e n t a l v a l u e s o f l o g A plotted against l o g P 0 shown i n F i g u r e 11.  2  (Appendix  (Appendix  I , Tables 1 and 2)  have been  I I I , Table 3) f o r each temperature  A l t h o u g h t h e s e curves a r e determined  and a r e  by two p o i n t s o n l y ,  t h e f o u r p l o t s can be c o n s i d e r e d as p a r a l l e l w i t h i n e x p e r i m e n t a l e r r o r .  The  v a l u e s o f x o b t a i n e d f r o m t h e s l o p e s o f t h e s e p l o t s a r e t a b u l a t e d i n Appendix I ( T a b l e k)  and g i v e a mean v a l u e o f - 1. 3  On t h i s b a s i s , K^/ can be d e r i v e d from the e x p e r i m e n t a l v a l u e o f A  o •5  1/  u s i n g e q u a t i o n (11).}  Tables laand .-  '  The c a l c u l a t e d values o f K  /  a r e shown i n Appendix I ,  2. •>  P r o v i d e d t h e o n l y temperature  /  dependent term i n K  i s the rate  c o n s t a n t , t h e s l o p e o f an A r r h e n i u s p l o t w i l l c o r r e s p o n d t o the a c t i v a t i o n energy f o r the r a t e d e t e r m i n i n g s t e p . both s e r i e s of experiments.  These a r e shown i n F i g u r e 12 f o r  -18  ^17  !  ^16  ~~ Log  F i g u r e 11.  ^ll" P0  ^13  2  / L Log-Log P l o t f o r A° as a F u n c t i o n of Oxygen P a r t i a l P r e s s u r e s a t D i f f e r e n t Temperatures 5  :  ^12  30  The c o r r e s p o n d i n g value' o f t h e a c t i v a t i o n energy  calculated  from t h e s l o p e i s 82 * 2 k c a l / m o l e f o r b o t h o f these e q u i l i b r i u m compositions.  G r a i n Growth Study The v a r i a t i o n - o f t h e g r a i n s i z e as -a f u n c t i o n o f the h e a t i n g time was  s t u d i e d under c o n t r o l l e d H2./H2O atmosphere over t h e temperature  o f 1000°C  t o 1300°C.  The photomicrographs  range  as shown i n F i g u r e 13a, b, c, d,  r e v e a l t h e d i f f e r e n t stages o f g r a i n growth d u r i n g t h e s i n t e r i n g p r o c e s s . These a r e v e r y s i m i l a r t o those shown i n F i g u r e 3 ( a f t e r C o b l e ) . v  The  exaggerated g r a i n growth was expected t o o c c u r a t o r above t h e Sauerwald temperature T  g  which f o r T i 0  2  (m.p. 1900°C) l i e s between 1250 and l425°C.  Exaggerated g r a i n growth was a c t u a l l y observed a t 1250°C a f t e r k-000 minutes o f s i n t e r i n g as shown i n F i g u r e 13d.  and a t 1300°C a f t e r 1200 minutes  T h e o r e t i c a l j u s t i f i c a t i o n f o r r e s u l t s obtained i n isothermal g r a i n 25 26 growth experiments was made i n i t i a l l y b y Beck e t a l . and by T u r n b u l l The average g r a i n diameter c o r r e l a t e d w i t h time a c c o r d i n g t o T u r n b u l l i s g i v e n as D  2  - D  2 Q  (12)  = K,£.V- t  where D i s t h e average g r a i n diameter a t time t , D  0  i s t h e average  o r i g i n a l diameter a t t = 0 , K, i s a r a t e c o n s t a n t and V i s t h e atomic volume. 27 Burke  ., however, deduced  t h e f o l l o w i n g e x p r e s s i o n f o r g r a i n growth on t h e  assumption t h a t t h e m o t i v a t i n g f o r c e f o r grain-boundary m i g r a t i o n d u r i n g g r a i n growth i s t h e s u r f a c e t e n s i o n o f t h e boundary, c u r v a t u r e o f t h e boundary D  - D  0  and t h a t t h e r a d i u s . o f  i s p r o p o r t i o n a l t o t h e g r a i n ..diameter = K  Q  t  exp  (-1$) -  (13)  - 31 -  (c)  x  (d)  1500  Figure 13.  x  600  Typical Micro-structures of Sintered T i 0 i .  9 2  .  (a) I n i t i a l Stage - f i r e d at 1150°C f o r 190 minutes, (b) Intermediate Stage Showing and Spherical Pores - f i r e d at H50°C f o r 280 minutes, (c) F i n a l Stage - f i r e d at 1200°C f o r 2800 minutes, (d) Beginning of Discontinuous Grain Growth - f i r e d at 1300°C f o r 1200 minutes.  Channel  -.52  where K n Q R T  Q  = = = = =  -  a rate constant t h e time exponent w i t h a t h e o r e t i c a l v a l u e o f u n i t y t h e a c t i v a t i o n energy f o r g r a i n growth gas c o n s t a n t a b s o l u t e temperature.  The g r a i n s i z e d a t a w h i c h a r e t a b u l a t e d i n A p p e n d i x - I I were examined b y p l o t t i n g t h e l o g o f t h e d i a m e t e r o f t h e g r a i n s v e r s u s l o g time i n F i g u r e lk'.  From t h e measurements o f t h e s l o p e s o f these l i n e s , i t i s  apparent t h a t t h e s l o p e s do n o t e q u a l . t h e t h e o r e t i c a l v a l u e o f l / 2 . 1  f o r e , a p l o t o f l o g (D  - D  Q  There-  ) v e r s u s l o g t was done i n F i g u r e 15, t h e s l o p e s  o f w h i c h i n d i c a t e an average v a l u e o f 0.6 * 0.1 f o r n. - I n o r d e r t o d e r i v e t h e v a l u e o f t h e a c t i v a t i o n energy f o r g r a i n growth, a f i n a l p l o t o f D v e r s u s t ° ' f o r each temperature 6  was made, ( F i g u r e 16),  - D  2 Q  the slope o f which  was t h e V a l u e o f t h e r a t e c o n s t a n t , K , w h i c h v a r i e s w i t h t h e a b s o l u t e 1  'temperature  (T) a s , K = K  Q  exp (- J^J-) . - The a c t i v a t i o n energy Q was t h e n  c a l c u l a t e d from t h e s l o p e o f t h e A r r h e n i u s p l o t ( F i g u r e 17), where l o g K was p l o t t e d a g a i n s t  - The d e r i v e d v a l u e f o r t h e a c t i v a t i o n energy was T'  78 k c a l / m o l e . Sintering  Study E x p e r i m e n t s were c a r r i e d out t o s t u d y t h e d e n s i f i c a t i o n o f r u t i l e  powder compacts i n . t h e temperature r e d u c i n g atmosphere. to  range o f 1000.to 1300°C i n a i r and i n a  The r e d u c i n g atmosphere; /was used i n t h e s i n t e r i n g ' s t u d y  o b t a i n t h e e q u i l i b r i u m compositions f o r n o n - s t o i c h i o m e t r i c r u t i l e o f  T i O i . 9 2 and'-TiOi-.ge-  The same atmosphere had a l s o been used p r e v i o u s l y f o r  weight l o s s and g r a i n growth measurements. The d e n s i f i c a t i o n d a t a a r e g i v e n i n Appendix I I I , Table 1, f o r the f i n a l composition of TiOi.92-  A d e t a i l e d study of the m i c r o s t r u c t u r e  -.35.-  i  i  I I  1  1  1  1—I—i—i  ii|—  '  :  r  Time(minutes) F i g u r e 15.  G r a i n Growth i n T i O ! . Temperature  9 2  Compacts w i t h  -35  -  - 36 -  CO >-  10^ T F i g u r e 17.  (°K)  L o g a r i t h m of G r a i n Growth Rate Versus R e c i p r o c a l Temperature  -•37  -  as a l r e a d y shown i n F i g u r e 13a-d,. r e v e a l e d t h a t t h e p o r e s were i n t h e form of c h a n n e l s , w h i c h p a r t i a l l y f i l l e d up i n t h e l a s t s t a g e .  The pores.became  a l m o s t s p h e r i c a l i n shape and were p r e s e n t i n t h r e e - o r f o u r - g r a i n - c o r n e r s , as was t h e case w i t h - t h e model proposed by C o b l e ^ .  The i s o t h e r m a l d e n s i f i c a t i o n ( o r s h r i n k a g e ) d a t a were c o n v e r t e d t o t h e r e l a t i v e d e n s i t y o f t h e compact w h i c h .-yjas; p l o t t e d a g a i n s t l o g . t i m e . 5 The f o l l o w i n g e q u a t i o n d e r i v e d by C o b l e dP dt  ND  =  V  o'.a  -j_3  £  3 Q  ,  .....(6)  T  y i e l d s on i n t e g r a t i o n t  •0  -P  ND ^ a V  P where P  Q  Q  .(lk)  In t  .A;E T  0  f  3c  i s t h e p o r o s i t y a t time t  J  t  Q  and z e r o p o r o s i t y a t time tf  3  and assuming 1  = A t , where 1 i s t h e g r a i n d i a m e t e r a t t i m e t . .The r e s u l t s  a r e shown i n F i g u r e l 8 a , b, c, d, e, f o r t h e f i n a l c o m p o s i t i o n o f T i O ; L . g . 2  S i m i l a r i n v e s t i g a t i o n on t h e d e n s i f i c a t i o n o f T i Q i . g s and T i 0 o u t , b u t no d e t a i l e d s t u d y was made on t h e i r g r a i n growth.  2  was  carried  - The r e s u l t s a r e  Shown i n F i g u r e A . I I I - l a , b, c, d, e, f o r T i O i i g e . a n d i n F i g u r e A-.III-2 f o r Ti0 . 2  The d a t a a r e t a b u l a t e d i n Appendix-'III, T a b l e 2.  The. v a r i a t i o n o f  t h e c o m p o s i t i o n o f n o n - s t o i c h i o m e t r i c f u t i l e as c a l c u l a t e d from t h e weight l o s s measurements was a l s o i n c l u d e d i n F i g u r e 18 and F i g u r e A - . I I I - l f o r the. f i n a l compositions o f T i G i - . g  2  and T i O i . g s r e s p e c t i v e l y .  From a l l t h e s e f i g u r e s , i t . i s q u i t e apparent t h a t t h e d e n s i f i c a t i o n b e h a v i o u r was c o m p l e t e l y changed a f t e r a c e r t a i n time o f h e a t i n g . . The d e n s i f i c a t i o n was stopped i n most c a s e s , r e a c h i n g an e n d - p o i n t d e n s i t y ,  .100  10 Time . Figure l 8 a .  1000  (minutes)  D e n s i f i c a t i o n o f Compacts f o r the F i n a l  Composition  of T i O i . g g a t T = 1000°C.  Time Figure l 8 b .  Densification  (minutes) o f Compacts f o r the F i n a l  o f T i O i . g s a t T =1050°C.  Composition  -•39 80 ..96  70 -p •H CQ  a  CD Q  t>  •H -P  -P-.92  60  cd H 0) «  50  1  I I I I I I 10  L  J  I 1 II III  Time (minutes) F i g u r e ,l8c.  100c,  I  I  I  I  I  I  -P-.90  II I  1000  D e n s i f i c a t i o n o f Compacts f o r t h e F i n a l C o m p o s i t i o n o f TiOx .. a t T = 1100°C. 92  100.  Time (minutes) F i g u r e l8d. D e n s i f i c a t i o n : o f Compacts f o r t h e F i n a l C o m p o s i t i o n of T i O i . g s a t T = 1150°C.  -.40 -  100  Time Figure  l8e.  (minutes)  D e n s i f i c a t i o n o f Compacts f o r the F i n a l Composition •of T i O i . 9 2 a t T = lB0'p°C  - kl -  a p p r o x i m a t e l y a t the same time for  ./as:.- t h e e q u i l i b r i u m c o m p o s i t i o n was  non-stoichiometric r u t i l e .  with.the  reached  The end-point d e n s i t y m o n o t o n i c a l l y i n c r e a s e d  i n c r e a s e of. temperature  of sintering.  The break i n the i s o t h e r m a l  d e n s i f i c a t i o n curves o c c u r r e d a f t e r a h e a t i n g time o f 100 t o 200.minutes f o r i non-stoichiometric r u t i l e  and between 10 t o 15 minutes f o r T I 0 . 2  In.order t o c a l c u l a t e the d i f f u s i o n c o e f f i c i e n t s D it  v  i n equation.(lk)  i s o n l y n e c e s s a r y t o know t h e v a l u e o f A e x p e r i m e n t a l l y as t h e v a l u e o f  the o t h e r c o n s t a n t s a r e known.  While  i n v e s t i g a t i n g the isothermal g r a i n  growth  ; ,of •: .an:-...: e q u i l i b r i u m c o m p o s i t i o n o f T i 0 i . g , i t was found, t h a t the f o l l o w i n g 2  e x p r e s s i o n s a t i s f i e d the e x p e r i m e n t a l  D  - D  2  T h i s e q u a t i o n i s almost  l neglecting D , 3  0  3  2 Q  =  K t ' 0  data.  (15)  6  equivalent t o  = D  3  = A t  .....(16)  as t h e i n i t i a l g r a i n s i z e D  Q  was v e r y s m a l l , and A i n  3/2 e q u a t i o n (16) i s e q u a l ' t o K • T h e r e f o r e , the p l o t o f D as shown i n F i g u r e v a l u e s o f A.  The  v e r s u s t i m e was made f o r a l l temperatures,  3  :  A.III-3 ( I n Appendix I I I ) , t h e s l o p e s o f which g i v e t h e  The r e s u l t s a r e r e c o r d e d i n Appendix I I I ,  Table,3.  s l o p e s , o f t h e p l o t o f t h e r e l a t i v e d e n s i t y versus l o g time a t  any. temperature  T aire e q u a l t o  . .,2.3.NDy % ; a 3  . ;  Q  . A K T U s i n g N = 10, #  = 1000 e r g s / c m , a 2  3 Q  = 1.57-  ergs/deg. and t h e v a l u e o f A (from F i g u r e  X 10" cm , % = I . 3 8 X 1 0 " 23  3  A.III-3) a t d i f f e r e n t  1 6  temperatures,  -k2  t h e c o r r e s p o n d i n g d i f f u s i o n c o e f f i c i e n t s (D ) were c a l c u l a t e d .  • The  r e s u l t s a r e g i v e n i n Table k, Appendix I I I .  (Figure  of l o g D . v e r s u s (^) produced y  diffusion.  An A r r h e n i u s p l o t  an a c t i v a t i o n energy o f 118  kcal/mole f o r  -  19)  - 43- -  -5  o -7  Ox \  Boundary D i f f u s i o n \  x  o  Model  \ \  \ \ \  o  \  \  S i n t e r i n g Neck Growth -P  a  <D •H O •H CH =H  -11  0)  O O  a  o  .-H CQ  <tH <H •H  Q Bulk D i f f u s i o n Model  -13  -15  6.6  J  I  6.8  l_  I  7.0  L  10  7-2  J  !  7-*  I  I  7.6  i  '  7-8  (°K)  T. F i g u r e 19.  D i f f u s i o n " C o e f f i c i e n t V e r s u s the R e c i p r o c a l o f the A b s o l u t e Temperature. D i r e c t l y measured oxygen d i f f u s i o n c o e f f i c i e n t s i n s i n g l e c r y s t a l s a r e compared w i t h v a l u e s c a l c u l a t e d from s i n t e r i n g e x p e r i m e n t s and models.  8.0  - kk DISCUSSION Defect S t r u c t u r e s o f R u t i l e The  reduction equation  i n v o l v i n g t h e c r e a t i o n o f t h e oxygen i o n  vacancy and i t s two t r a p p e d e l e c t r o n s may he d e r i v e d by u s i n g a model s i m i l a r t o t h a t a l r e a d y advanced by Greener,.Whitmore and-.Fi'heo Nb 0 2  11 5  and Whitmore and Kawai Of 2  ^  This equation i s  2  0  1  + V  2 ( g )  Q 2  -  +2  0  .  ..:..(17)  -  where 0-^ VQ2-  '  f o r Ti0 -.  i s an oxygen i o n i n a n o r m a l l a t t i c e p o s i t i o n and  i s an oxygen i o n vacancy i n w h i c h t h e two e l e c t r o n s were  i n t h e form, o f two T i The  for  + + +  ;ions..  l a w o f mass a c t i o n was a p p l i e d t o e q u a t i o n  result'  l/2 K  =  trapped  P  0  (17) w i t h t h e  2  [v 2-]-to]  .....(18)  Q  2  as t h e c o n c e n t r a t i o n o f t r a p p e d e l e c t r o n s i s t w i c e t h e c o n c e n t r a t i o n o f oxygen  vacancies '  The  equation  [0]  =  2  [V 2-] Q  (l8) becomes i n terms o f t h e oxygen vacancy K  :  [ V "  ]  1/3  = (I)  concentration  -1/6 p  o  .....(19)  2  T h i s e q u a t i o n shows t h a t under e q u i l i b r i u m c o n d i t i o n s t h e c o n c e n t r a t i o n o f vacancies • Equation  i s p r o p o r t i o n a l t o t h e ^ l / 6 power o f t h e oxygen p a r t i a l  pressure.  (19 ) can be r e w r i t t e n as [V 2-].= Q  0.63  -1/6 P  0  2  AHff exp-  [  - 3 %  +  AS°f . " R T  where A H ° f and A S°f a r e t h e e n t h a l p y and e n t r o p y  ]  (20)  of formation f o r  r e a c t i o n (17). A f t e r removing t h e temperature independent t e r m f r o m t h e  - ^  -  e x p e r i m e n t a l f u n c t i o n , e q u a t i o n (20) becomes  [V -]  =  o 2  C  "  P  V  (AZ1)  6  .....(21)  ^S°f exp—pg—). N  where C stands f o r (0.63  Weight Loss Study Buessem and B u t l e r  derived a s i m i l a r expression t o interpret  the t o t a l , weight l o s s of r u t i l e on e q u i l i b r a t i o n w i t h v a r i o u s oxygen .atmospheres, T h e i r equation,was. 2 Ti  4 +  ,+  0 ~  V 2-  ^  2  + 1 0  Q  2Ti  + 2  (22)  3 +  of which the e q u a t i o n c o n s t a n t i s H K  (T)  °  =  where H^., '  of T i  3+  '.  6  X  + 2H  A V  T i  a  +  +  1 H ° 0  2  P  k~ T  %ij_3+  and of oxygen gas  a  n  (  i  HQ  are the e n t h a l p y o f t h e a n i o n  i n the standard  vacancy  state.  From the e x p e r i m e n t a l d a t a , Buessem and B u t l e r o b t a i n e d an v a l u e o f 83• * 10 k c a l / m o l e . - B y . u s i n g a v a l u e of -53-75 kcal/mole  enthalpy  ( f o r the  change o f s t a n d a r d s t a t e ) t h e y o b t a i n e d f o r H  +  A Y  2H  T i  3+  .136.65 * 10 k c a l / m o l e . 3°  I n t e r p r e t i n g Cronemeyer s d a t a on c o n d u c t i v i t y o f r u t i l e  in  t h i s manner r e v e a l e d t h a t H . AY  +  2TI  3 +  •=  I27.O5  kcal/mole.  • A comparison of e q u a t i o n (21) ART or  ART  3 by e q u a t i o n  (21).  =  127.05 (or 136.65  = 4 2 . 3 5 kcal/mole  and t h e s e d a t a shows t h a t kcal/mole)  f o r the d e f e c t r e a c t i on r e p r e s e n t e d  -\e I n t h e p r e s e n t c a s e , no attempt was made t o determine t h e d e t a i l e d mechanism o f r e d u c t i o n o f T i 0  2  w i t h i n t h e scope o f t h i s i n v e s t i g a t i o n .  since t h i s d i d not f a l l However, t h e a c t i v a t i o n energy  o f 82 ± 2 k c a l / m o l e o b t a i n e d i s i d e n t i c a l t o t h e e n t h a l p y o f r e a c t i o n (22) 29  o b t a i n e d by Buessen and B u t l e r  . T h i s w o u l d suggest t h a t t h e r a t e -  d e t e r m i n i n g s t e p may be t h e f o r m a t i o n o f an oxygen-ion  vacancy.  G r a i n Growth The p r e d i c t e d i n c r e a s e o f g r a i n s i z e i n a n o r m a l g r a i n s i z e d i s t r i b u t i o n w i t h no i n h i b i t i n g second phase has been observed t o f o l l o w t h e square r o o t o f t i m e .  However, i m p u r i t i e s i n t h e l a t t i c e o r n o n u n i f o r m i t y i n  t h e g r a i n s i z e d i s t r i b u t i o n and shape i n t h e o r i g i n a l m a t e r i a l can cause t h e g r a i n growth t o be a f u n c t i o n o f a s m a l l e r power o f t i m e . be e x p r e s s e d as  This can  n  D  2  - D  2 Q  =  K t  0  w i t h n = 1 i n t h e t h e o r e t i c a l case.  < n 41  .....(15)  Another e x p l a n a t i o n f o r lower 27  t h a n t h e o r e t i c a l v a l u e s f o r n has been advanced b y Burke  . I n many g r a i n  growth i n v e s t i g a t i o n s , t h e o r i g i n a l m a t e r i a l c o n t a i n s a d i s p r o p o r t i o n a t e l y l a r g e number o f f i n e g r a i n s .  T h e r e f o r e t h e average r a d i u s o f c u r v a t u r e o f  t h e g r a i n boundary w i l l i n c r e a s e more r a p i d l y t h a n p r e d i c t e d b y t h e t h e o r e t i c a l e q u a t i o n where n has t h e v a l u e o f 1. I n t h i s s t u d y most o f t h e measurements p l o t t e d i n F i g u r e s 7  a  n  d 8  r e l a t e t o continuous c y l i n d r i c a l pores i n t e r c o n n e c t e d a t t h e g r a i n boundaries. The  second s t a g e o f pxure s h r i n k a g e , i . e . t h e i s o l a t e d pore phase, was observed  o n l y a t t h e h i g h e r temperatures  and a f t e r l o n g p e r i o d s o f h e a t i n g .  This i s  i l l u s t r a t e d b y F i g u r e 13d, o f a specimen heated a t 1300°C f o r 1200 m i n u t e s .  - >7  But this.-change i n p o r e shape does not i n f l u e n c e t h e r e l a t i o n :  i s . t h e same as l o n g as.exaggerated  g r a i n growth does.not  (15) which  occur.  Therefore  p a r t i c u l a r a t t e n t i o n was g i v e n a t t h e higher-temperatures, used, 1250 and 1300 C t o a v o i d t h e measurements, i n f l u e n c e d by exaggerated  g r a i n growth.  P  The  r e s u l t s o f the present i n v e s t i g a t i o n f o r TiOi.92 ±  can  Q  -.represented by t h e f o l l o w i n g e x p r e s s i o n  ,o-6  „  K  D  Q  t  . 78,000x  .exp (-  fo  )  The time exponent n h a s t h e v a l u e o f 0.6 i n s t e a d o f t h e t h e o r e t i c a l v a l u e n = 1.  F o r t h e n o n - s t o i c h i o m e t r i c c o m p o s i t i o n o f T i O i . g a ± o»oi>  while f o r T i 0  2  n  =  0.6  n = 1,/ ( A p p e n d i x ' I I I , • T a b l e 5) .k  The v a r i a t i o n o f t h e value, o f t h e exponent n w i t h t h e change o f s t o i c h i o m e t r y o f the compound has a l s o been .evaluated for. U0 31>32 2  value, o f n changed from n = 1.2 t o n = 0.8 f o r U0 compares, t h e v a l u e o f n f o r d i f f e r e n t ceramic  2  The  #  t o U0 + . . Table I I 2  x  oxides as determined  from  the g r a i n growth d a t a . Table I I . •  V a l u e s o f Exponent n f o r D i f f e r e n t Ceramic  :  Oxide  U0  .n  Reference  O.8/O.9  31  1.2  32  CaO  .1.0  33  MgO  .1.0  •3.3  U0  2 + x  2  A1 0 2  C  A  .0.66 (O.62-0.74)  3  O - 1 5  % -0'B5 O1.85 r  P r e s e n t Work T i 0  2  0.8 0.60  Oxides  12 34 -  - 48 -  Temperature Dependence o f G r a i n Growth B e f o r e d i s c u s s i n g the s i g n i f i c a n c e o f the p r e s e n t v a l u e o f the a c t i v a t i o n energy f o r the g r a i n growth o f T i 0 2  i t i s informative to  t X  c o n s i d e r the a c t i v a t i o n e n e r g i e s f o r the g r a i n growth p r o c e s s i n o t h e r ceramic  oxides.  For the purpose o f comparison,  the e n e r g i e s f o r the  s e l f d i f f u s i o n o f the components in- such systems are a l s o i n c l u d e d i n Table I I I . - Table I I I . A c t i v a t i o n Energy-  Data f o r D i f f e r e n t Ceramic  Activation - A c t i v a t i o n Energy f o r g r a i n growth Energy f o r s e l f d i f f u s i o n of the c a t i o n (Ref.) (Ref.) .114 . (35) 153 (12)  Oxides  :  . A1 0 2  3  MgO C a » 15 Zr 0 85 Q  U0  2  . Ti0  As corresponds of the two  2  Oxides  Activation Energy f o r s e l f d i f f u s i o n of the a n i o n (Ref.) 152 (36)  60'  •(53)  79  (37)  62.4  (38)  80 87  '(3^) (32)  109 f o r Ca Zr 88  (39) (41)  29.8 60  (40) (41)  78  -  74  (43)  Ci-85  2 +  4 +  -  i s e v i d e n t , the a c t i v a t i o n energy f o r g r a i n growth o f oxides t o the energy r e q u i r e d f o r the d i f f u s i o n a l p r o c e s s of e i t h e r  components.  On t h i s b a s i s , i t i s always c o n s i d e r e d t h a t the  one grain  growth o f o x i d e s i s c o n t r o l l e d by a d i f f u s i o n mechanism.  In t h e p r e s e n t i n v e s t i g a t i o n , the a c t i v a t i o n energy o b t a i n e d f o r the g r a i n growth study, i s 78 kcal/mole behaviour  for T i 0 i .  9 2  ±  o-oi°  The  temperature  o f the oxygen s e l f - d i f f u s i o n c o e f f i c i e n t has been determined  by  42 Haul and.Just technique.  u s i n g Linde r u t i l e  s i n g l e c r y s t a l s and 0  1 8  isotopic  T h e i r d i f f u s i o n d a t a can be r e p r e s e n t e d as f o l l o w s  exchange  -  D  N 2  0  -  1.6 exp  =  RT  v  ;  4  9  -  £H?  sec  between t h e temperature range 85O t o 1300°C  An e x p r e s s i o n f o r - t h e i n t r i n s i c  oxygen s e l f - d i f f u s i o n  coefficient  f o r n o n - s t o i c h i o m e t r i c . r u t i l e , - D Q " has been d e r i v e d i n Appendix IV, t h e 2  f i n a l form, o f which i s P 2-  =  0  where D  Q  D  Q  exp  AH°f U - (5-Rf~. W ) +  '  i s t h e u s u a l f r e q u e n c y f a c t o r , 4H°f i s t h e energy t o  form an oxygen vacancy, and U i s t h e a c t i v a t i o n energy f o r oxygen vacancy ;  migration., By comparing t h e e x p e r i m e n t a l v a l u e w i t h t h e thermodynamic calculation  A ° ;H  f  + U ;  =  74- kcal/mole  3  and by t a k i n g  AH°f = I27.O5 k c a l / m o l e from Cronemeyer's  d a t a , t h e energy  f o r vacancy m i g r a t i o n U i s about  74 -  127  =  31.6 kcal/mole  3 S i m i l a r c a l c u l a t i o n s w i t h the p r e s e n t g r a i n growth d a t a produced a v a l u e o f 35 k c a l / m o l e f o r t h e a c t i v a t i o n energy, f o r m i g r a t i o n o f t h e oxygen vacancy.  These r e s u l t s a r e i n good agreement  and i t appears r e a s o n a b l e  enough t o suggest t h a t oxygen i o n d i f f u s i o n i s t h e l i k e l y  rate-controlling  step i n t h e process- q f g r a i n growth o f n o n - s t o i c h i o m e t r i c  titania.  -  5Q-  Sintering The m i c r o s t r u c t u r e s as shown i n F i g u r e 13a-d, r e v e a l t h e d i f f e r e n t stages, o f d e n s i f i c a t i o n .  They a r e a l m o s t s i m i l a r t o those d e p i c t e d by Coble  ( F i g u r e 3 ) . -The i n t e r m e d i a t e stage i s shown i n F i g u r e s 13a and b, w h i c h a r e p h o t o m i c r o g r a p h s o f specimens, h e a t e d a t 1150°C f o r 190 minutes and 280 minutes respectively.  I n F i g u r e 13b, t h e p o r e s a r e a t t h e j u n c t i o n o f t h r e e — g r a i n  c o r n e r s , i n d i c a t i n g t h e presence o f c o n t i n u o u s p o r e s .  The f i n a l stage b e g i n s  when t h e pore phase i s e v e n t u a l l y p i n c h e d o f f . .The p r e s e n c e o f some p o r e s a t t h e f o u r - g r a i n c o r n e r s i n F i g u r e 13c, i n d i c a t e s t h a t t h e specimen had reached t h e f i n a l s t a g e o f s i n t e r i n g .  The m i c r o s t r u c t u r e s o f specimens  sintered  a t h i g h e r t e m p e r a t u r e s r e v e a l e d b o t h t r a p p e d pores i n t h e g r a i n s and p o r e s a t the  g r a i n boundary c o r n e r s as s^own i n F i g u r e 13d.  E x i s t e n c e of trapped pores  i n t h e g r a i n s i n a l l specimens, w h i c h were f i r e d a t o r above 1250°C suggests t h a t some degree o f d i s c o n t i n u o u s g r a i n growth o c c u r r e d i n a l l c a s e s .  This  i s a l s o e v i d e n t from.the d e n s i t y - t i m e c u r v e s ( F i g u r e l 8 a - e ) , where i n e v e r y case an e n d - p o i n t d e n s i t y was observed.  E x a g g e r a t e d g r a i n growth was c a r e f u l l y a v o i d e d i n t h i s  investigation  by c a r r y i n g out a l l e x p e r i m e n t s below 1300°C ( s i S a u e r w a l d t e m p e r a t u r e case o f T 1 0 ) . 2  in.the  A comparison o f F i g u r e s 13b and 13d shows/that t h e u n i f o r m i t y  i n g r a i n and pore s i z e w h i c h was p r e s e n t a t t h e b e g i n n i n g o f t h e s i n t e r i n g process l a t e r disappeared.  A t t h e l a t e r s t a g e s e v e r a l g r a i n s grew l a r g e r  at t h e expense o f s m a l l e r ones.  On f u r t h e r h e a t i n g , s h r i n k a g e o f t h e pores  i n t h e g r a i n b o u n d a r i e s was o b s e r v e d , w h i c h r e s u l t e d i n some i n c r e a s e i n r e l a t i v e d e n s i t y o f t h e compacts.  A n n e a l i n g t w i n s were observed i n t h e specimens a f t e r 2000 minutes of s i n t e r i n g a t o r above 1250°C; no s t u d y was. made t o t e s t t h e e f f e c t o f t h i s on t h e g r a i n growth o r s i n t e r i n g .  - 51 D e n s i t y - T i m e Curve I s o t h e r m a l d e n s i t y - t i m e curves  i n F i g u r e l8a-e, p r e s e n t t h e e f f e c t  o f t e m p e r a t u r e only,, as t h e e f f e c t s o f atmospheres were n o t d e t e r m i n e d , a l t h o u g h t h e oxygen p a r t i a l p r e s s u r e temperatures.  i n t h e system was v a r i e d a t d i f f e r e n t  The compacting p r e s s u r e was found t o have a s i g n i f i c a n t  e f f e c t on t h e i n i t i a l r e l a t i v e d e n s i t y and on t h e d e n s i f i c a t i o n r a t e and so was.kept c o n s t a n t .  The i n i t i a l r e l a t i v e d e n s i t y o f a l l specimens was  I.96 gm/em , p a r t i c u l a r l y f o r t h e n o n - s t o i c h i o m e t r i c e q u i l i b r i u m c o m p o s i t i o n 3  o f TiO-i..g . 2  The use o f compacts: o f such low h u l k d e n s i t y e l i m i n a t e d t o some  e x t e n t t h e p r e s e n c e o f t h e d e n s i t y g r a d i e n t produced by d i e w a l l f r i c t i o n , a l t h o u g h t h i s c o u l d n o t be c o m p l e t e l y a v o i d e d . . Large s h r i n k a g e  rate  a n i s o t r o p y was n o t o b s e r v e d .  The  i n i t i a l p a r t o f d e n s i t y - t i m e curves n e v e r extended t o t h e  f i n a l stage o f s i n t e r i n g a c c o r d i n g t o F i g u r e 15, cannot be a t t r i b u t e d t o t h e change o f pore shape.  so t h a t t h e change o f s l o p e In a d d i t i o n , according  12 to Coble  , t h e change o f pore phase c o n t i n u i t y s h o u l d occur a t about 95$  o f t h e t h e o r e t i c a l d e n s i t y o f t h e compact, w h i c h , i n t h i s i n v e s t i g a t i o n , was n e v e r a c h i e v e d  i n any specimen.  when t h e c o m p o s i t i o n  The change o f s l o p e i n a l l cases  occurred  o f t h e n o n - s t o i c h i o m e t r i c r u t i l e under t h e r e d u c i n g  atmosphere r e a c h e d an e q u i l i b r i u m v a l u e .  Thus, i t appears t h a t t h e change  o f r a t e i s n o t a f u n c t i o n o f change o f pore shape b u t i s r e l a t e d t o t h e f o r m a t i o n o f oxygen v a c a n c i e s i n t h e system d u r i n g t h e p r e - e q u i l i b r i u m s t a g e . , 10 0 B r y a n and P a r r a v a n o measured t h e r a t e o f neck growth between r u t i l e spheres i n an atmosphere o f H /H 0 (= 10) 2  2  i n t h e i r i s o t h e r m a l , neck growth r a t e c u r v e s .  and observed s i m i l a r b r e a k s  They e x p l a i n e d t h i s  observation  on t h e b a s i s o f p o l y g o n i z a t i o n o f m o n o c r y s t a l l i n e r u t i l e spheres and assumed t h a t t h e specimens r e a c h e d e q u i l i b r i u m a l m o s t i m m e d i a t e l y  during heating.  -.52  The  significance  c u r v e s i s n o t known.  -  o f t h e s l o p e s o f t h e d e n s i t y v e r s u s l o g time  The s l o p e may be a f f e c t e d  by t h e i n i t i a l p a r t i c l e  s i z e , p a r t i c l e shape and a l s o b y i n h o m o g e n e i t i e s i n i n i t i a l b u l k d e n s i t y , pore s i z e d i s t r i b u t i o n , and p a r t i c l e a l i g n m e n t .  The problem,then, i s t o  d e t e r m i n e whether t h e d i f f u s i o n model i s . supported, by j t h e - l i h e a r i t y c i o f .the curves.  Diffusion  Coefficient The  l i n e a r r e g i o n o f t h e d e n s i t y - t i m e curve i n t h e  stage o f d e n s i f i c a t i o n was u s e d . t o c a l c u l a t e The  pre-equilibrium  the d i f f u s i o n c o e f f i c i e n t .  observed i n c r e a s e i n d e n s i f i c a t i o n r a t e w i t h i n c r e a s i n g t t e m p e r a t u r e i s  t y p i c a l of thermally-activated  p r o c e s s e s i n c e r a m i c s . - Each curve has  r o u g h l y t h e same c h a r a c t e r . - The slopes, v a r y , b u t cannot be  interpreted  q u a n t i t a t i v e l y , and t h e r e f o r e t h e change i n s l o p e from one l i n e t o a n o t h e r i s not understood.  The  apparent d i f f u s i o n c o e f f i c i e n t s c a l c u l a t e d , from t h e r e s u l t s  are shown i n F i g u r e . 1 9 where t h e y were p l o t t e d as l o g D a g a i n s t l / T . d a t a p l o t t e d i n t h i s f i g u r e a r e H a u l and J u s t ' s  Other  d i r e c t l y measured d i f f u s i o n  c o e f f i c i e n t s f o r oxygen ( d e t e r m i n e d b y i s o t o p i c exchange t e c h n i q u e ) and Whitmore's  1 1  apparent d i f f u s i o n c o e f f i c i e n t s c a l c u l a t e d  f o r measurements  o f neck growth between two s p h e r e s .  The  diffusion coefficients . calculated  from ±i^£rm©'dd/a4e%ta£e'mea~sure-  .'..meats: 'are. lOwer than;_tnOse from neck - growth/'.:- measurements by an o r d e r o f magnitude b u t agrees w i t h t h e d i r e c t l y measured d i f f u s i o n c o e f f i c i e n t s . O n l y o r d e r - o f - m a g n i t u d e r e l i a b i l i t y m a y b e a t t a c h e d : t o t h e i n d i v i d u a l models o r r e s u l t s , and t h e r e f o r e t h e d i s c r e p a n c y does n o t d i s p r o v e t h e model in. t h i s case.  However, t h e c a l c u l a t e d  applied  a c t i v a t i o n energy o f 118 k c a l / m o l e i n  the p r e s e n t case i s n o t i n agreement w i t h o t h e r r e s u l t s .  This i s r e f l e c t e d  -53 i n t h e d i f f e r e n c e o f s l o p e s o f t h e l i n e s as shown i n F i g u r e 19.  This  d i s c r e p a n c y may be a t t r i b u t e d t o some o f the f a c t o r s , i n v o l v e d i n t h e c a l c u l a t i o n o f the d i f f u s i o n c o e f f i c i e n t s .  The e r r o r i n t h e measurement  o f g r a i n s i z e would a f f e c t t h e f a c t o r A ( e q u a t i o n (16)).  S i m i l a r l y , the  e r r o r i n . t h e measurement o f t h e s h r i n k a g e v a l u e s . w o u l d a f f e c t the. s l o p e of the density-time curve.  The c u m u l a t i v e e f f e c t o f t h e s e two on t h e  c a l c u l a t e d d i f f u s i o n c o e f f i c i e n t s may be c o n s i d e r a b l e . t r u e f o r t h e specimens s i n t e r e d below 1100°C. s i z e and i n t h e dimensions  This i s p a r t i c u l a r l y  The changQfin t h e g r a i n  o f t h e specimens w i t h time below 1100°C were  v e r y s m a l l . -Any e r r o r i n t h e measurement o f t h e s e two parameters would have a l a r g e e f f e c t on t h e v a l u e o f d i f f u s i o n c o e f f i c i e n t s a t l o w e r temperatures.  The e f f e c t o f t h e i n i t i a l shape o f t h e g r a i n on t h e r a t e i s n o t known. . C o o l e r s spherical grains.  shrinkage  o r i g i n a l model was based c o m p l e t e l y on.the  I n the present case, t h e i n i t i a l g r a i n s are l i k e  flakes.  . These a r e e x p e c t e d t o change i n t o a minimum s u r f a c e a r e a c o n f i g u r a t i o n i n t h e e a r l i e s t stage o f s i n t e r i n g , as t h e s e f l a k e s have l a r g e s u r f a c e energy a s s o c i a t e d w i t h them. . The s t e p s i n v o l v e d i n t h i s change o f c o n f i g u r a t i o n a r e n o t known a n d ^ t h e r e f o r e , i t s e f f e c t on t h e o v e r a l l s h r i n k a g e r a t e cannot be e v a l u a t e d .  • The e f f e c t o f oxygen p a r t i a l p r e s s u r e s on t h e d i f f u s i o n was  evaluated by the f o l l o w i n g procedure.  The i s o t h e r m a l g r a i n growth r a t e s  o f t h e e q u i l i b r i u m compositions'of TiOx.ge and T i 0 The d a t a were p l o t t e d as D The  coefficients  2  a t 1200°C were  determined.  (where D = average g r a i n d i a m e t e r ) v e r s u s t i m e .  s l o p e o f t h e l i n e s produced t h e v a l u e s o f A . f o r t h e s e two c o m p o s i t i o n s .  -  ^ -  U s i n g t h e s h r i n k a g e d a t a . a t 1200°C-the d i f f u s i o n c o e f f i c i e n t s f o r t h e equilibrium  compositions o f T i O i . g  and T i 0  8  2  were c a l c u l a t e d .  a r e t a b u l a t e d i n A p p e n d i x I I I i n T a b l e s . 5 and 6 . table  The r e s u l t s  I n the following  t h e d i f f u s i o n c o e f f i c i e n t s and t h e i r r e s p e c t i v e oxygen p a r t i a l  p r e s s u r e s a r e compared.  I t can be seen t h a t t h e d i f f u s i o n  were n o t s i g n i f i c a n t l y a f f e c t e d  coefficients  b y t h e change o f oxygen p a r t i a l p r e s s u r e  i n t h e system. . Table I V Calculated Diffusion Coefficients  at Different  Oxygen P a r t i a l  Pressures  Oxygen Diffusion Coefficient P a r t i a l Pressure ( D ) a t 1200°C i n Atmosphere cm /sec  Composition  v  2  4.32 X I O  .TiOi.es .  Ti0  10"  X  9M  - 1 1  1.2.£1 X, I O  2  1.41 X I O  10"  IX  1 1  - 1 5  1 3  0.21  - 1 1  I n v i e w o f t h e d i s a g r e e m e n t between t h e v a l u e o f t h e a c t i v a t i o n energy o b t a i n e d i n t h i s i n v e s t i g a t i o n studies,  attempts have been made t o a p p l y o t h e r a v a i l a b l e  densification. addition  and t h a t found i n o t h e r  Two o t h e r models a r e c u r r e n t l y  available.  sintering  models f o r t.t-. Coble i n 5  t o h i s b u l k d i f f u s i o n model, p r o p o s e d a g r a i n boundary d i f f u s i o n  model on.the u n d e r s t a n d i n g t h a t d e n s i f i c a t i o n might p r o c e e d by t h e m i g r a t i o n o f t h e g r a i n boundary.  Coble d e r i v e d t h e f o l l o w i n g  e q u a t i o n f o r the g r a i n  boundary d i f f u s i o n P =  where  P I>b W .;  I2  R  \ °a  D K W ^  ^ K 4  3  2  t ]  /  5  (23)  = volume f r a c t i o n pores a t t i m e t = d i f f u s i o n c o e f f i c i e n t o f atoms i n t h e g r a i n boundary = g r a i n boundary w i d t h , a n d o t h e r f a c t o r s s i m i l a r t o -i .:r t • e q u a t i o n ((.6). [  - 55 The values, o f boundary d i f f u s i o n c o e f f i c i e n t  were c a l c u l a t e d  u s i n g e q u a t i o n (23)and t h e p o r o s i t y d a t a . o f t h e e q u i l i b r i u m c o m p o s i t i o n of TiOi-92.  The r e s u l t s and c a l c u l a t i o n s a r e g i v e n i n Appendix V.  are p l o t t e d i n F i g u r e 19- t o compare w i t h t h e o t h e r r e s u l t s .  These  According  t o t h i s f i g u r e , t h e b u l k d i f f u s i o n model g i v e s a b e t t e r f i t f o r t h e present  result.  13 Johnson and C u t l e r  v e r y r e c e n t l y d e r i v e d an e q u a t i o n t o i n t e r p r e t  t h e i r l i n e a r s h r i n k a g e d a t a o f a l u m i n a compacts. f o l l o w i n g form  The e q u a t i o n has  the  1  AL L  °..  =  . K  (  K T  a  3  j)m  gp )  t  m  .(24)  where &L ' =-•• f r a c t i o n a l s h r i n k a g e Lo K = numerical constant yi< = s u r f a c e energy 'D = s e l f d i f f u s i o n c o e f f i c i e n t a = v a c a n c y volume r = p a r t i c l e radius t = time .m>p • = c o n s t a n t s 3  Q  In t h i s t : q q u a t i o n , t h e v a l u e o f t h e time exponent m v a r i e s between O.25  t o O.5O  a c c o r d i n g t o t h e geometry o f t h e c o n t a c t p o i n t s , such as  s p h e r i c a l , p a r a b o l o i d o r 160°  cone on a p l a n e e t c .  The f r a c t i o n a l l i n e a r  s h r i n k a g e data, i n the p r e s e n t i n v e s t i g a t i o n were a l s o p l o t t e d a g a i n s t t i m e i n a log-log scale.  The v a l u e o f m i s found t o be l e s s t h a n 0.1  in.this  c a s e . • The s i g n i f i c a n c e o f such a low v a l u e i s . not knowrj.and as a consequence no attempt was made t o c a l c u l a t e the d i f f u s i o n c o e f f i c i e n t u s i n g t h i s generalized equation.  -56  -  CONCLUSIONS  .• . • The r e d u c t i o n - of r u t i l e was c a r r i e d out t o two f i n a l o f T i O i ..92 and T i 0 i . g  u s i n g d i f f e r e n t H /H 0 atmospheres.  8  2  The f r a c t i o n a l  2  weight l o s s w i t h t i m e was found t o f o l l o w a . p a r a b o l i c A r r h e n i u s p l o t u s i n g r a t e s w h i c h were c o r r e c t e d  relationship.  An  f o r oxygen p a r t i a l p r e s s u r e  dependence produced an a c t i v a t i o n energy o f 82 ± 2 k c a l / m o l e . l o s s measurements u n d e r e q u i l i b r i u m c o n d i t i o n s  compositions  The weight  c a r r i e d out b y o t h e r ,  i n v e s t i g a t o r s produced an e n t h a l p y o f 83 ± 10 k c a l / m o l e . . t h a t t h e r a t e d e t e r m i n i n g s t e p i s t h e oxygen i o n vacancy  This.may suggest formation.  Grain growth data f o r the e q u i l i b r i u m composition o f T i 0 i . g 2 ± can be e x p l a i n e d  D  The  by t h e f o l l o w i n g  2  - D  2 Q  =  K  o  o-oi  expression  t°-°  ex  P  (  :,-2§^0)  v a l u e o f time exponent n was found, t o be d i f f e r e n t from t h e t h e o r e t i c a l  v a l u e o f u n i t y f o r ' T i O i - . 9 2 and T i 0 i - . g . 8  I t was e q u a l t o one f o r t h e g r a i n  growth o f s t o i c h i o m e t r i c r u t i l e . • The a c t i v a t i o n energy f o r oxygen i o n diffusion i nTi0  2  d e t e r m i n e d by oxygen I s o t o p i c exchange t e c h n i q u e was  found; t o be.74 k c a l / m o l e .  On t h i s b a s i s , i t i s j u s t i f i a b l e  t o suggest t h a t  oxygen i o n d i f f u s i o n i s t h e l i k e l y r a t e - c o n t r o l l i n g s t e p f o r t h e g r a i n growth of the T i 0 i . g  2  non-stoichiometric  composition.  D e n s i f i c a t i o n o f the f i n a l e q u i l i b r i u m composition of T i 0 i . g explained  u s i n g a s i n t e r i n g model proposed, by C o b l e .  was  2  The v a l u e s o f t h e  d i f f u s i o n c o e f f i c i e n t c a l c u l a t e d i n t h i s i n v e s t i g a t i o n were o f t h e r i g h t o r d e r o f magnitude b u t t h e a c t i v a t i o n energy f o r t h e d i f f u s i o n p r o c e s s c u l a t e d from, t h e s e d a t a d i d n o t agree w i t h t h a t o f o t h e r w o r k e r s . discrepancy i s explained  cal-  This  on t h e b a s i s o f e r r o r s i n t h e measurements, i n -  h o m o g e n e l t i e s i n t h e compact and o t h e r p o s s i b l e f a c t o r s w h i c h a f f e c t t h e d e n s i f i c a t i o n process.  -.57  -  RECOMMENDATIONS FOR FUTURE INVESTIGATIONS T h i s s t u d y on g r a i n growth and s i n t e r i n g o f r u t i l e compacts under r e d u c i n g atmospheres was t h e f i r s t attempt t o a p p l y a model o f d e n s i f i c a t i o n . o n r u t i l e powder compacts.  The e f f e c t o f i n i t i a l compacting p r e s s u r e s o n . g r a i n growth and d e n s i f i c a t i o n s h o u l d be c a r r i e d out u s i n g . r u t i l e powders o f known and s i m p l e geometry.  A l s o keeping.the  p a r t i c l e : s h a p e as s i m p l e as p o s s i b l e ,  i n i t i a l g r a i n s i z e e f f e c t on t h e d e n s i f i c a t i o n r a t e s h o u l d be i n v e s t i g a t e d . T h i s d a t a b y c o r r e l a t i n g w i t h t h e g r a i n growth-measurements. s h o u l d . h e l p t o i d e n t i f y t h e f a c t o r s w h i c h c o n t r o l t h e v a l u e o f t h e time exponent i n t h e g e n e r a l i z e d e x p r e s s i o n o f t h e g r a i n growth.  With t h e a v a i l a b i l i t y o f  a plasma, t h e powdered p a r t i c l e should-be. s p h e r o i d i z e d . b e f o r e o t h e r e x p e r i m e n t s are undertaken.  This should-eliminate the heterogeneity i n contact  angles  and i n number o f c o n t a c t s p e r p a r t i c l e . • An o b s e r v a t i o n o f t h e number o f c o n t a c t s p e r p a r t i c l e d u r i n g d e n s i f i c a t i o n , as w e l l as t h e measurement o f s u r f a c e a r e a o f t h e compacts would perhaps e x p l a i n t h e drop i n r a t e o f d e n s i f i c a t i o n once t h e e q u i l i b r i u m n o n - s t o i c h i o m e t r i c c o m p o s i t i o n has been reached  i nrutile.  The i n f l u e n c e o f g r a i n growth i n h i b i t i n g elements s h o u l d  be. i n v e s t i g a t e d i n o r d e r t o r e a c h t h e t h e o r e t i c a l d e n s i t y - i n t h e compacts.  To i s o l a t e t h e e f f e c t o f r e d u c t i o n on t h e d e n s i f i c a t i o n r a t e , . t h e powdered p a r t i c l e s should.be.reduced, t o a n o n - s t o i c h i o m e t r i c e q u i l i b r i u m c o m p o s i t i o n b e f o r e t h e s i n t e r i n g s t u d y i s c a r r i e d out.  - 8 5  BIBLIOGRAPHY G. C .. K u c z y n s k i , "Powder M e t a l l u r g y " , p . 11, Ed., W.. L e s z y n s k i , I n t e r s c i e n c e P u b l i s h e r s I n c . , New York and London, (I96I) W. J . Moore, " P h y s i c a l C h e m i s t r y " , p . 504, P r e n t i c e - H a l l I n c . ,  N.J., (1955). • W. D.,Kingery a n d M. Berg, J . A p p l . Phys.. 26, 1205 G.C.  (1955)-  K u c z y n s k i , "Powder M e t a l l u r g y " , p . 1-16, The I r o n and S t e e l I n s t i t u t e and the I n s t i t u t e o f M e t a l s , London, (1963).-  R...L.-Coble, J . Appl.-Phys. 3_2, 787 (1961). W. D. Jones "Fundamental P r i n c i p l e s o f Powder M e t a l l u r g y " , Edward A r n o l d P r e s s , London ( i 9 6 0 ) . C. Zener, see C - S. Smith,  Trans.-..AiirM.%%?blT$}(19^8).  G. C. K u c z y n s k i , " K i n e t i c s o f High Temperature'Processes", Ipl; 37^ E d . , W. D. K i n g e r y , Technology P r e s s a n d John W i l e y and Sons, (1958). . L. F. N o r r i s and G. • Parravano, J . Am.-Ceram... Soc., 46, 449 (1963). H. M. O'Bryan, J r . , and G. Parravano, "Powder M e t a l l u r g y " , p. 191, Ed., W. L e s z y n s k i ) ^ I n t e r s c i e n c e P u b l i s h e r s Inc.n.New York and London, (I96I). D. H. Whitmore and T o s h i h i k o Kawai, J , Am. Coram I. Soc., 4jj, 375 (1963) . R.. L. C o b l e , J . - A p p l . Phys.. 3 2 , 793 (1961). D. . L. Johnson and I . B.. C u t l e r , - J . Am.-Ceram. Soc. 46, 545 (1963). P. W. C l a r k , J . H. Cannon and J.-White, T r a n s . B r i t . Ceram.- Soc. 52, 1  (1953). A. H.-Webster and N.F.- H.- B r i g h t , - T h e E f f e c t s o f Furnace Atmospheres on t h e S i n t e r i n g Behaviour o f Uranium Oxide, Mines Branch Research Report R2, Department o f Mines and T e c h n i c a l Surveys, Ottawa, February 5, I 9 5 8 . • R. W.. G.- Wyckoff, C r y s t a l S t r u c t u r e Handbook, I n t e r s c i e n c e P u b l i s h e r s ,  New. York (1948).  ,  - M. E . Straumanis, T. E j i m a and W. J . James,-Acta. E. - A.. Gulbransen a n d K.-F.- Andrew, J . M e t a l s  7^1  C r y s t . 14, 493  • T r a n s . A.I.M.E.,  (I96I). 185,  (19^9).  W i l l y K i n n a and W i l l y Knorr, Z.-Metallkunde, 4j_, 594 ( I 9 5 6 ) . •M.H. Davies and C. E . B i r c h e n a l l , • J . Metals.3;; T r a n s . A.I.M.E., 191  877 (1951).  Bibliography Continued.  21.  -59  -  . K a r l H a u f f e , R e a c t i o n e n i n u n d an F e s t e h S t o f f e n , - I I , p..l35> - S p r i n g e r - V e r l a g , B e r l i n (1955).  22. a) J . S t r i n g e r , A c t a Met. 8, 758 (i960).. b j P. K o f s t a d , K. H a u f f e and H. K j o l l e r s d a l , A c t a . Chem.. Scand. 12,.239  (1958). j 23.  ,  . T e n t a t i v e Method f o r D e t e r m i n i n g t h e Average G r a i n S i z e o f M e t a l s , A.S.T-.M.. D e s i g n a t i o n E 112-55 T, A.S..T.M. S t a n d a r d s , P a r t 1,  (1955) P. 143324. 25.  W. H. McKewan,- T r a n s . A.I.M.E., 224,. 2 (1962). P.-A. Beck, J . C. Kremer, L. J . Demer, and M. L. H o l z w o r t h , - M e t a l s T e c h n o l . 14, (1947); Tech. Pub .Wo. 2280; • T r a n s . A. I-.M-.E. 17j?  372  26.  (1948).  D a v i d T u r n b u l l , J . M e t a l s \y T r a n s . A.I.M-.E., 191, 66l (1951).  27.  -J.-E. B u r k e , T r a n s . A.I.M.E. 180, 73 (1949).  28.  - E. H. G r e e n e r , D. H. tyhitmore and M. E. F i n e , J . Chem. Phys.. 34, 1017 (1961).  29.  - W. R. Buessem a n d S. - R. B u t l e r , " K i n e t i c s , o f HigheiTemperature P r o c e s s e s " , Ed., W. D.. Kingery,- Technology P r e s s and John W i l e y and Sons, p. 13  New: York, (1959).  30. 31. 32.  D. C. Cronemeyer, Phys. Rev.'87_, 876 (1952). - J . • R. MacEwan and V. B. Lawson, J . Am. Ceram.- Soc., 4jj, 42 ( I 9 6 2 ) . I . Amato, R.- L. Colombo and A. M. P r o t t i , J . Am. Ceram. S o c . 46, 407  (1963). 33 •  • A. V. - D a n i e l s , J r . , R. C .. L o w r i e , J r . , RnoL. Gibby and I v a n B C u t l e r , J.-Am. Ceram. Soc. 4^, 282 ( I 9 6 2 ) .  34.  T. Y. T i e n and E.. C. Subbarao, J . Am. Ceram. S o c , 46 , 489 (1963)..  35-  A. E. Paladin© a n d W.- D.. K i n g e r y , J... Chem. Phys. 32.,.. 957 (1962).  36.  Y. O i s h i and W. D. K i n g e r y , J . Chem..Phys. 33, 480(1960).  37-  R o l a n d - L i n d e r and G. D. P a r f i t t , J . Chem. Phys. 26, 182 (1957).  38.  Y. O i s h i a n d W. D . . K i n g e r y , J . Chem. Phys. 33, 905 (i960).  39-  W. H. Rhodes and R.-E... C a r t e r , " I o n i c S e l f - D i f f u s i o n i n C a l c i a S t a b i l i z e d Z i r c o n i a " presented a t t h e S i x t y - F o r t h Annual Meeting, The American Ceramic S o c i e t y , New Y o r k , A p r i l .30, 1962. - Symposium on K i n e t i c s o f Ceramic R e a c t i o n s No -1-25-63, Am. Ceram. S o c , B u l l . :  ;  41, 283 (1961).  Bibliography! Continued.  hO. . W.- D. K i n g e r y , J . Pappis,' M. E. Doty and D.. C . H i l l , J.- Am. Ceram.  Soc,  1+2, 393 (1959).  hi.  J . B e l l e / A.. B. A u s k e r n , W. A.. Bostrom and F. S. Susko, " D i f f u s i o n K i n e t i c s , o f Uranium D i o x i d e " , U.S. Atomic Energy Comm. R e p o r t No. WAPD-T-1155, - (I960).  h2.  R. Haul,'D. J u s t and G.- Dumbgen, " R e a c t i v i t y o f S o l i d s " , p. 6 5 , Ed., J . H. DeBoer, E l s e v i e r P u b l i s h i n g Company, Amsterdam, ( I 9 6 I )  - 61 APPENDIX I . - Table  1.  Weight Loss Data f o r ' T i O i . 9 2 ± Temperature A_W of Furnace W °C. % x  1000 (7-94)  1050' (7-67)  1100 (7-35)  1150 (7.12)  1200 (6.8$)  1 0  0 Time t ^ m  t  O-5  0.153 0.260 0.461 O.500 0.648 1.211 I.250  15 20 50 65 110 210 430  0.173O.270 0.430 0i646 O.740 1:170 1.970  10 15 35 75 100 24o' 1065  • AO-5 l o g A ° 0  (mtrr?  n  o-oi  6.116 3.87 4.52 X I O " 7.04.5 8.06 .10.53 14.51 20.50  5  5  log  P 0 - ° 5 S f 3 log K ° '  -1.214 -17.55  2  p  2  -2.92  -4.134  2  3.165 7.^677- ll 3.87 x i o ~ 2 5.90: 8.66 10.00 15.50 32.55  -1.1275 -16.765 -2.794  -3.9215  0.480 . 25 50 0.775 I.170 ,100 1.465 150 300 •1J730 1.745 1055  4 . 9 8 ' 12.641 7.04 X I O 10.00 12.22 14.50 32.45  -O.898  -16,16  -2.693  -3.591  0.678 O.871 0.990 ,1.450 1.750 1-775 •1,900 1.900  15 25 •35 75 l4o • 300 .470 .  15.536 X IO  -O.78I  -15.412 -2.568  -3.349  IO65  3.87 4.98 5.9O 8.66 11.82 17.30 .31.05 32.55  O.613 O.890 1.693 .1.740 1.750 1.800 1.972  10 25 55 70 130 250 780  31.65 4.98 7.40 9.23 11.42 15.85 28; 00  - 2  - 2  •  22.327  X IO"  2  -O.65I  -14.85  -2.475  0.126  5  - 62 Table 2. Weight Loss :Data. f o r TiGfe.gs.±' .QI ;  Temperature of Furnace  ioo  T i m e t  .  t  Q.5  .0.132 (7.94)  . 1050 • (7-67)  14^3  4.47 7.28 10.62 15.28 21.76 38.00  0.510 O.510 O.510  15 27.5 55 87 125 215 . 810  3.87 5.155 5.23 X 10 " 7.42 9.33 11.18 14.65 28.45  0.186 0.225 0.340 O.510 O.545 0-573 0.573  10 15 .35 55 105 490 1075 j  •3.165 5.667 3.87 X I O " 5.90 7.42 10.26 22.18 32.80  O.310 0.487 0.555 O.590  3.87 6.72 10.26 16.58 50.65 66.70  7.3358 X IO"  0.645 0.645  15 45 : 105 275 2555 ' 4440  0.140 0.255 O.380 0.534 0.534 O.598  24 30 38 6o 125 245  4.13 5.47 6.16 7.75 11.18 15.65  5.170 X 10  0.276 0.376 0.540 0.542 ! 0.542 0.203 0.265 0.415  0.486  1100 (7.35)  1150 (7-12)  1200 (6.86)  log-A°-5  • (mxrr?  ' °c. •  1000  .AO-5  20 53 113 233 473 !  ;  0  log P0  2  5  3.7428 2 xio~  -u.5 log 'log K ° ' P02- / 1  5  6  -1.4265  -16.376  -2.729  -4.2194  -1.288  -15.60  -2.60  -3.888  -1.2474  -14.236  -2.373  -3.6204  2  2  -1.135  -13.59  -2.265  -3.400  -0.287  -13too  -2.170  -2.457  2  _ 1  >  T a b l e 3. Oxygen P a r t i a l P r e s s u r e s Temperature Temperature p H 0. o f Furnace of t h e mm o f Mg °C. B u b b l e r °C.. 2  A.  For-.. TiOx  1000 '• 1050 . 1100 1150 1200 B.  '  9  2  *  P  H 0 2  1  O  S  p  o . 2  :  O-Ol  23 23 25 25 25  21.068 21.068 23-756 23.756 23.756  35-2 35-2 31.0 31.0 31.0  -17.53 -16.765 -16.16 -15.412 -14.85  71.88c 71.88 107.20 107.20 107.20  9.60 9.60 6.06 6.06 6.06  -16.376 -15.6  F o r T i O i •98. ± - . p i 0  1000 1050 1100 1150 1200  45 45 53 53 53  -14.236  -13.59 -13.00  - 64 Table  k.  D e t e r m i n a t i o n o f t h e Power F a c t o r o f ,P  (  Temperl o g A°-5 ature vTiOi.gs TiOi.92 °C |  log P - TiOi.gs  - Slopes  0 2  TiOi-,92  X 1 1  1000  -1.4265  -1.2140  -16.376  -17.53 -0.2125 =.o.i84 - 1. 1.159  1050  -1.2880  -1.1275  -15.600  -I6.765  1100  -1.2474  -O.898  -14.236  -16.160 -O.35 =_o.l82 1.93  1150  -1.135  -O.78I  -13.590  -15.412  1200  -O.287  -O.65I  -13.000  -14.850  1  I.I65  -  1  3  ± O.65  - 1 2.75  -0.354 =-0.1825 _ 1 1.82 2.7-5  1  •1  Average s l o p e i s assumed t o be  - 1 3.65  -0.16 =-0.137  1  - 65 -  APPENDIX I I . •Table 1, G r a i n Growth Study  Temperature . °C. 1100 (7.35)  1150 (7..12)  1200 (6.78)  1250 (6.50)  1300 (6.35)  Grain Size D  Time t min  . D2  90 380 1055 2555 5315  0.5329  D  2  -  v  t  0 - 6  > *  • '2 l o g K (cm/min " :  0  ^ —  0.73 0.93 1A0 1.66 2.08 1,365 1.50 1.91 2.50 2.96 3.34  120 290 705. ' 1980 3305 • 66O5  14.83  1.9600 2.7550 4,3260  0 0.3320 1.4271 2.2221 3.7031  35-4 65.3 110.8 171.5  •1.8630 .2.2500 3.6481 6.2500 8.7616 ll.1806  1.3301 1.7171 3.1152 5.7171 8.2287 10.5471  -17.7 30.2 51.3 95.1 .127.2 197.0  0.8649  (2.15; * 10" ) 10  -9.667  (5.8 X 1 0 ° ) _1  -9.237  1.90 2.57 3.52 5.50 5.93  110 280 : 1265 2820 4183  3.6100 6.6049 12.3904 30.2500 35.3000  3.0771 16.75 6.0720 29.5 73.0 11.8575 29^7171 117.3 34.9771 • 149.3  (23.2 X 10" )  2.92 3.4l 5.05 7.02 8.60  100 240 500 1270 2620  8.5329 11.6280 25.5329 49.2800 73.9600  8.0000 11.0951 25.0000 48.7471 73.4271  15.85 26.8 41.7 73-3 112.2  (60 X I O  7.30 7.80 12.23 8.98 9.80 10.30  • 90  53.2900  52.7571 60.3071  14.83  (250 X I O " )  149.0400  24.80  -7.600  -I5O  210 . 250 415 625  60.8400  149.5300 80.5329 96.6400 106.0900  80.0000 95.5071 105.5570  10  -8.645  - 1 0  -8.222  1 0  20.15  :  27.10 37.15 47.60  )  6  APPENDIX I I I . .Table 1. Temp. Time ° ' mm.  1000  0 15 20 4 50 65 .110 5  w gm  0.6911 O.6905 0.6893  430  0.6879 O.6876 0.6866 0.68275 0.6770  0 10 15 35 75 100 240 1065  0.7500 0.7487 0.7480 0.7468 0.7451 0.7445 0.7367 0.7360  0 10 .- . 10 30 50 80 100 150 i 300 1055  0.6909  240  1050  1100  0.6876 0.6856 0.68399 0.6828 0.6808 0.6790 0.67885  AW  w 100  0  v  x  ••*b,.  0:153 0.26 0.461 O.50  2  ~  0:2771 O.27615 0.2753 0.2739 O.276O  1.211 .2.05 0 O.173 O.27 O.43  0.3000 O.2987 O.2980 O.2968  0.74 1.77 1.97  0.29445 0.2867 0.2880  0.646  . 0 •0 0  0.48  0.775 1.00 1.17 1.465 1.73 1.745  x  gm  O.2726 0.26875 0.2630  0.648  TiO  O.295I  O..2769 O.2769 O.2769 0.2736  O.27I65  0.2699 0.2688 0.2668 O.2656 0.26485  Shrinkage Vol. $  Vol. at Time t cm 3  2 1.994 1.986 1L986 1.972 1.97 1.964 1.94 1.905  0 6.5 9 12 . 16.6 9 12 17.2 16,8  4.26 3.98  • 3.88  3-75 3-55  • 3.88  3.75 3-53 3.54  Weight a t Green Bulk Time t Density gm  8.3483 8.34 8.32 8.32 8.29 8.27 8.24 - 8.18  Density of Non-stoich. Ti0 Theo. 2  4.25  . P Po  2.222 2.34 2.34  2.14  4.238 4.238 4.227 4.225 4.22 4.20 4.174  0.462 • O.495 0.506 0.524 O.530 O.540 O.530 0.557 0.559  1.963 2.10 2,15 2.22 •2,34  4,244  0 2 21 1.99 ,: I.986 " '27.5 I.98 27 .8 I.968 28.6 1.964 "• 29.3 1.92 29.6 1.918 3°  4.26 3-36 3.08 3.07 3.02 3.005 2.97 2.955  8.3483 8.33 8.315 8.31 8.28 8.280 8.225 8.19  1.97  4.25  0.464  2.70 2.70 2.75 2.758 2.752.78 •  4.238 4.232 4.223 4.22 4.185 4.183  O.650  2 2' 2 1.972 1.96 1-95 1.94 1.926 1.918 1.916  4.28 3.09  8.3824 8.3824. 8.3824 8.34 8.32 8.305 . 8.28 8.27 8.26 8.255 ,  1.96 2.71 2.76 2.90 3.00 3.03 3,085 3:105 3-12 3.125  ;  0 27.8 28.9 32.8 35.3 35-6 37.4 • 37.8 38.3 38.5  3.04  2.88 2.77 2.76 2.69 2.67 2.65 2.695  2.48  continued,  4.24  4.25 4.25 4.25 4.226 4.217 4.209 4,201 4.19 ' 4.183 4.182  O.587 O.636 0.632 0.6§4 O.656-'  0.664  0.462 O.638  0.648  O.682 0.710 O.722 0.734 0.740 0.745 0.745  Table 1.Continued.  Temp ..Time °C. min.  1150  0 5 10 15 25 35 775 l4o 300 470  IO65 1200  0 5 10 20 25 .30 55 70 85 120 250 780  W gm  0.6663 0.6663 0.6663  O.66I78 0.66048 0.6597 0.65664 0.65465 0.65445 0.65365 0.6565  Aw w  v X  100  .0 0 0 0.678 0.871 0.99 1.45 1-75 1.775 1.90 1.90 •  0  2  .TiO  x  gm  0.2664 0.2664" 0.2664 " 0.26188 0.26058 O.2598 0.25674 0.25475 0.25455 0.25375 0.25375  0.8967 0.8967 0.8912  0 0  o.613"  0.3587 0.3587 0.3532 •  0.88875  0.89  0.35073  Shrinkage Vol.. ?  Vol. at .Time t cm  Weight a t Time t gm  3  2 2, 2; ... 1.965 •1.956 1.944 1.928 1.912 I.9IO 1.905 1.905 2 2 1.968 1.968 1.952  0.8815 0.8811  1.693 1.74  0.3435 0.3431  •1-913  0.8810 0,8806 0.8790  1.75 1.80 1.972  O.343O 0.3426 0.3410  1.913 1.910 1.910  1.916  0 . .. 35-2 36.5 36.5 38.3 38.3 4o.6 42.5 43.7 44.0 44.3  37-6 38.6 41.8 43.0 44.2 45.8 46.0 47.4 48.2 48.6  .  '4.28 2.78 2.725 2.73 2.65 2.65 2-55 2.46. •2.415 2.40 2,385 . 4.26 2.66 2.61 ' 2.47 2.43 2.375 2,30 2..30 2.24 2.21 2.185  '  •  Green Density of p Bulk N o n - s t o i c h i o . :"vp— D e n s i t y T i 0 Theo. 0  2  8.3824 8.3824 8.3'824 8.33 8.31 8.285 8.27 8.23 8.22. 8.22 8.22  -1.943-02 3-08 3.06 3.14 3.13 3.25 3.345 3.405 3.425 3.445  : 4.25 4.25 . " 4,25 4.221 4.214 4.204 4.192  8.3483 8.3483 8.28 8.27  1.965 3.145 3.18 3.345  4.25 4.25 4.224 4,217 4.211 4.211 4.186 4.18 4.18 4.18 . 4.178 4.178  8.255 8.255 8.21 8.20 8.20 8.20 8.18 8.18  3.4o 3.46 3.565 .3-57 3.665 3.71 3-75  4.179 4.176 4.174 4.174  0.71 0.725 0.725 0.745 O.745 0.775 0.80 " 0.816 0.820 0.825  0.74 0.75 0.795 0.81 0.825 0.845 0.852 0.875 O.89 . 0.90  I ON  -^1 1  APPENDIX I I I . i  Table.2. Temp . Time °C . m i n  1000  0 20 53 113233 1433  1050  1100  0 15 27.5 55 87 125 215 810 0 10 15 35 55 105 490 1075 2336 3650 5121 9910  w gm  AW  ..  Y  x  100  0  2  x  S h r i n k a ^ie V o l . a t Vol. $ Time t cm 3  0.8394 0.8292 0.8284 0.8270 0.8270 0.8270  0 0.132 0.276 0.376 0.542 0.542 0.542  0.3335 0.3324 0.3312 0.3304  0.8444 0.84269 0.84215 0.8409 0.8403 0.8401 0.8401 0.8401 . 0.8624 0.8608 0.8604 O.8595 O.8580 0.8577 0.8566 0.8566 0.8566 O.8566 0.8566 0.8566  O.83I5  TiO  gm  Weight a t Time t gm  Green D e n s i t y o f P Bulk Non-stoichio D e n s i t y T i 0 Theo. • ' po 2  0.3290 0.3290  2 1.994 1.992 1.984 1.974 1.974 1,97^  0 11]. 17 22.5 26 24.5 26  4.06 3.61 3.365 3.155 2.995 3.055 2.995  8.2556 8,2556 8.245 8,24 8.22 8.21 8.22  2.04 2.29 2.452 2.61 2.745 2.61 2.745  4.250 4.250 4.244 .4.243 4.236 4.228-  0.480 0.538 0.578 0.616 0.647 0.637 0.647  0 0.203 0.265 0.415 0.486 0.510 O.510 0.510  0.3379 0.33619 0.33565 0.3344 0.3338 0.3336 0,3336 0,3336  2 1.992 1.986 1.984 I.98 1.978 1.978 1.978  0 22 27 34 37-5 38.5 39 41  5.44 4.24 3.98 3.60 3,36 3-333' 3.32 .3.21  10.6531 10.62 10.62 10.62 10.61 10.60 10.60 10.60  1.96 2.50 2.67 2.945 •3.160 3.185 .3.19 3.19  4.250 4.24o 4.238 4.236 4.234 4.23I 4.231 4.231  0.461 0.588 0.630 0.695 0.745 0.752 0.753 0.743  0 0.186 0.225 0.340 0.510 0.545  0.3456 0,3438 0.3436 0.3427 0.3410 0.3407 0.3396 0.3396 0.3396 0.3396 0.3396 0.3396  2 1.99 1.985 1.98 .1.976 1.972 1.970 1.970 1.970 1.970 1.970 1.970  0 33 32 36 4o 44 • 45 45.5 45.5 46 46 46  5.15 3.80 •3.58 3.30 3.09 2.88 2.82 2.80 2.80 2.78 2.78 2.78  9.7650 9.7645 9.7645 9.7645 9.726 9-7 9-7 9-7 9-7 9-7 9,7 9-7  I.89 2.57 2.74 2.96 3.15 3.37 3.44 3.46 3.46 3.49 3.49 3.49  4.250 4,248 4.248 4.248 4.230 4.227 4.225 4.225 ,4.225 4.225 4.225 4.225  O.445 0.606 0.646 O.696 0.743 0,795 0.815 0.818 0.818 0.827 0.827 0.827  • P.573 0-573 0.573 0.573 0.573 0.573  O.329O  continued.  T a b l e 2. C o n t i n u e d .  Temp .. Time °C. min.  0  - W gm  w  100  2  TiO  x  gm  •Shrinkag ;e V o l . a t Time t V o l . fo cm 3  Weight a t Green D e n s i t y o f Time t Bulk Non-stoich, gm D e n s i t y T i 0 Theo.  P p  o  2  1150  0 15 45 105 275 2555 444o  0.8832 0.8802 0.8789 0.8783 0.8780 0.8775 0.8775  0 0.310 0.487 0.555 0.590 0.645 0.645  0,3532 0.3502 0.3489 0.3483 0.3480 0.3475 0..3475  2 1.986 1.978 1.974 1.972 1.97 1.97  0 38.2 40.0 41.8 45.0 50 50  5.4l 3.34 3-245 3.14 2.98 2.705 2.705  IO.5365 10.51 10.48 10.48 10.48 IO.47 10.47  1.95 3.14 3,23 3.345 3.52 3.87 3.87  4.25 4.248 4,226 4.225 4.228 4.225 4.225  0.458 0.74 0.764 0.792 0.833' 0.915 0.915  1200  0 22 30 38 60 125 245 755 1075 2035 3^15  1.7234 1.7209 1.7190 1.7168 1.7142 1.711*2 1.7142 1.7142 1.7142 1.7142 1.7142  0  0.6874 0.6849 0.6830 0.6808 0.6782 0.6782 0.6782 0.6782 0.6782 0.6782 0.6782  2 1.99 1.988  0 39 41 43 47 49.5 . 49.5 49.5 49.6 49.6 49.6  4.80 2.99 2.84 2.73 2.53 2.43 2.43 2.43 2.42 2.42 2.42  9-5779 9-5779 9.56 95'.'3~ 95-2 95-2 95.2 95-2 95.2 95-2 95-2  1.995 3.21 3.37 3.50 3.85 3.92 3.92 3.92 3.94 3.94 3.94  4.25 4.24 4.24 4.235 4.23 4.23 4.23 4.23 4.23 4.23 4,23  0.469 0.755 0.793 0.825 0.87 0.925 0.925 0.225 0.93 0.93 0.93  o.i4o 0.255 0,380 0.534 0.534 0.534 0.534 0.534 0.534 0.534  I.98O  1.976 1.976 1.976 1.976 1.976 1.976 1.976  I  OA  - 70 -  Table 3.  . Data.for D i f f u s i o n C o e f f i c i e n t C a l c u l a t i o n f o r the F i n a l Composition of-Ti.Ox-.g2  Temperature °C  1100  1150  1200 (6.78)  1250 (6.56)  1300 (6.35)  (7.67)  D3  F  3  l o g A(T)  ,t min.  . cm ' ( )sec 3  A  T  90 380 1055 2555 " "5315  3.13^X 10"•17  -16.505  2.5432 3-375 6.968 15.625 25.93^ 37-260  120 290 705 1980 3305  1.27 X. 10"16  -15.8955  1.90 2.57 3.52 5.50 5-93  6,859 16.9745 43.613 16.640 208.530  110 280 1265 2820 4183  8.25 X 10""16  -15.085  2.92 3.41 5.05 7.02 8.60  25.020 39.6514 128.500 345.945 636.056  100 240 500 1270 2620  4.25 X. 10""15  -14.37  7.30 7.80 12.23 8.92 9-80 10.30  389.017 474.552  90 150 210 250 415 625  .3.67 X 10""14  1.365 I.50 1.91 2.50 2.96 3.3^  (7.12)  1050  F 0.73 0.93 1.40 1.66 2.08  (7-35)  1000 : (7.9*0  D  O.389 0.8044 2.744 4,5733 8.998  708.000 941.190 1092.727  66O5  -13.4355  8.12 X 10"•19  -18.09  4,36 X 10" 18  -17.36  :  - 71 •Table 4. D i f f u s i o n C o e f f i c i e n t C a l c u l a t i o n s f o r the F i n a l Composition of T i O i , 9 2  Temperature °C.  A(T)  cm3  sec  • S l o p e s P y D = S l o p e : "A T  2,62° X 10"  (  fo  3  log D  log t  1000 1050 1100 1150 1200  (7-94) (7.67) (7.35) (7.12) (6,78)  8.12 X 10" 19 5.6 X 10~ 4.36 X- 10" 18 6 x io" 3.13 X 10" 17 6 x 10"  2  2.21 X 1 0 "  14  1.325 X 1 0 "  2  2  13  -13.656 -12.878  9.77 X.10"  13  -12.011  12  -11.3755  1.27 X 10" 16 6.1 X 10"  2  4.22 X 1 0 "  8.25 X io" 16 9.3 x 10"  2  4.32 X I O  - 1 1  -IO..366  - 72 -  Table 5 . Data f o r D i f f u s i o n C o e f f i c i e n t C a l c u l a t i o n s f o r T i O i . g  and T i 0  8  a t 1200°C  Final Composition TiOi.  Ti0  9 8  2  Time min.  P-  290  2.16  14.743  2625  4.50  91.125  5625  5.26  145.528  354  1.92  7.0779  489  2.26  11.5432  4.20  74.01  .5.10  133.20  •D  i4i4 • 2790 10640  vK  D3  .12.5  G  4.53 X 1 0  7.94 X  •1953.125  Table 6 . • Diffusion Coefficient Calculations a t 1200°C  Final Domposition  K  cm3 s  e  c  b  1 . Slope K f l ° P s (amO D -"2.623 X I O " e  cm Se"c~ 2  TiOx. Ti0  2  9 8  4.53 X I O  - 1 6  7.94 X 10"  1 6  0.23 0.27'  m  5.86 X I O 12.$1 X I O  - 1 1  -  1  1  3  sec  3  -  io"  1  6  1 6  2  - 73 -  100  1000 Time (minutes)  Figure A y I I I - l a .  D e n s i f i c a t i o n o f Compacts f o r t h e F i n a l C o m p o s i t i o n o f - T i 0 i , a t T = 1000°C. g a  10000  - 74 -  .100  1000  10000  Time (minutes) Figure  A.Ill-lb.  D e n s i f i c a t i o n o f Compacts f o r t h e F i n a l Composition o f T i 0 a t T = 1050°C. l i 9 8  100C-  1000 Time (minutes)  Figure A . I I I - l c .  D e n s i f i c a t i o n o f Compacts f o r t h e F i n a l Composition of T i 0 i > a t T = 1100°C. 9 8  .10000  - 75 100  70 I  I  I  I  I  100  1  Time  Figure A . I l l - I d .  !  I  I  I I II  l  1  I  I  I  I I II  .10000  (minutes)  D e n s i f i c a t i o n o f Compacts f o r the F i n a l Composition o f T i 0 i . a t T = 1150°C. 9  100  8  100Q( Time  Figure A . I l l - l e .  I  .1000  (minutes)  D e n s i f i c a t i o n o f Compacts f o r the F i n a l C o m p o s i t i o n o f T i O i . g e at.' T = 1200°C .  10000  - 76 -  Figure A . I l l - 2 .  D e n s i f i c a t i o n o f Compacts, o f S t o i c h i o m e t r i c R u t i l e T i 0 a t D i f f e r e n t Temperatures. 2  - 77  Figure A.III-3  -  V a r i a t i o n o f G r a i n S i z e w i t h Time and Temperature. T h i s i s . t o determine A o f e q u a t i o n (16).  - 78  -  APPENDIX- IV. D e f e c t E q u i l i b r i a and  Oxygen Ion D i f f u s i o n f o r  I f the d i f f u s i o n r a t e c o n s t a n t s measured i n d e p e n d e n t l y formation can be  Non-Stoichiometric-Rutile  f o r both 0 ~ 2  and V Q -  A  R  E  2  as f u n c t i o n s of temperature, the heat o f vacancy  and t h e . c o n c e n t r a t i o n  of vacancies  at d i f f e r e n t  temperatures  determined by the r e l a t i o n s h i p  V-  •  °"  [  2  ]  =  [ V  0 " 2  where D i n d i c a t e s the d i f f u s i o n c o n s t a n t s s i m p l i f y i n g and n o t i n g t h a t  D  02"  *  D  =  V  Q  i)  2  a  [0  2 _  ]  -  [ V  0 "]  2  ]  o f the r e s p e c t i v e s p e c i e s .  On  i n the l a t t i c e i s almost u n i t y ,  2  [V 2-]  ^f)  exp  Q  l  where  i s the v i b r a t i o n a l frequency  between a d j a c e n t  oxygen i o n s ,  oxygen vacancy through:the  of oxygen i o n and a, i s the  distance  U the a c t i v a t i o n energy f o r the motion o f  an  lattice.  :. S u b s t i t u t i n g t h e v a c a n c y c o n c e n t r a t i o n [ V Q - ] . °f e q u a t i o n -  2  (24)'-  the d i f f u s i o n c o e f f i c i e n t D Q ~ i s : 2  D  0 " 2  = P0 C  2  -1/6 *  4Hf u " 3RT - RT]  o a  where C  e x  /  This. e q u a t i o n ' c a n D  P  [  \  , •= C p  0 2  -1/6 exp- [- (AHf + ^ )  J  2  = C V a .  be f u r t h e r s i m p l i f i e d by u s i n g the u s u a l frequency 0 2  _=D  0  exp  -  factor  D  Q  +  I t i s s i g n i f i c a n t t o note t h a t the f r e q u e n c y  f a c t o r i n t h i s equation  depends  -1/6 on P 0  2  and. t h i s e x p r e s s i o n  f o r the oxygen i o n d i f f u s i o n c o e f f i c i e n t  o n l y at h i g h temperatures, so t h a t v i r t u a l l y a l l the. v a c a n c i e s ionized.  are  is valid  completely  APPENDIX. Vv Boundary D i f f u s i o n Model ( C o b i e ^ )  2/3  21^ Wffa 3 0  I;  k" T  ]_4. =  D  4  dP  3/2  2D  =  dt  B  = • -A(T), t  4  a  W y  Q  3  . ,  i  '  t  A k" T  3/2  2D  W  B  y  a  3  Int  Q  .A k T ;  A plot o f D  v e r s u s time produced  t h e v a l u e s o f A shown i n t h e  \3/2  Appendix V  T a b l e 1,  and a p l o t o f  v e r s u s l o g t produced t h e  slopes S a t d i f f e r e n t temperatures.  Using the following.known value of  t h e c o n s t a n t s t h e v a l u e s o f D^ o f d i f f e r e n t temperature shown i n Appendix V, T a b l e  1.  boundary d i f f u s i o n c o e f f i c i e n t 10  3  ergs/cm - 0 3  1.57  a r e c a l c u l a t e d and  X 10"  3  •  3  cm  k  1 , 3 8 X. 1 0 " e r g s / d e g  •W  25 A  1 6  cm / s e c  - 80 -  APPENDIX V. 1.  Table  Temperature  °C.  Time min.  D  D  2  ,4  x  /  M  A(T) 4 ifc  Slope S  \  n  = (S) AT 1.76 cm  90 (7<35) 380 • 1055 2555 5315  1150  120  :  0.5329 0.283 5.7 x 10 0,8649 0.748 1.96 3.85 2.755 7-55 4.326 19.7 1.863  3.47  (7.12) 290 705 1980 3305 6605  2.25 3.65 6.25 8,76 11.18  5.1 13.3 39 77 125  1200  110 (6.78) 280 1265 2820 • 4i83  3.61 6.6 12.4 30.25 35.3  13 43.5 154 915 1245  1250  240 (6.56)1270 2620  11.628 135 49.3 2420 74.0 5480  1300  90 150 210  415 625 1000  53.3 60.8 150.0  1050  2  ,  1 0  D  2  b  0.101  1  1.391  x 10"  8  •• -  -7.856  2  0.082  6 . 7 7 X 10~  B  -7.1695  5.0 x 1 0 "  1 9  0.132  1.71  6  -5.767  3.1  1 8  -  - 2 3  0.045  5.8 x i o "  1 1  -10.237  - 2 2  0.073  7.74  2820 3Y2L)  X 10"  x 10"  —  96.0 9200  106.0  11200  x io  :  4.56 x i o (7.67)  _  3.3 X 1 0 " °  5.75 (7-94)  10"  SRC  '  uoo  X  x 10"  1 0  -9.112  T H E UNIVERSITY O F BRITISH VANCOUVER  8,  COLUMBIA  CANADA  DEPARTMENT OF METALLURGY  Comments on T h e s i s and O r a l E x a m i n a t i o n Of Jacques P i e r r e J e a n T h i r i a r "SINTERING AND GRAIN GROWTH OF NON-STOICHIOMETRIC RUTILE" T h i s t h e s i s was s u b j e c t e d t o c r i t i c i s m due t o t h e t r e a t m e n t o f a c t i v a t i o n e n e r g i e s and t h e i r c a l c u l a t i o n s . The anomaly i s i n . t h e use o f a r a t e c o n s t a n t f o r A r r h e n i u s p l o t s t h a t i n v o l v e s t i m e t o a power o t h e r t h a n . u n i t y . i  M  On page 29, F i g u r e 12 t h e p l o t s a r e f o r w e i g h t l o s s i n terms o f l o g ( j g j ^ - ) v e r s u s l / T and t h e s l o p e s c o r r e s p o n d t o a p p r o x i m a t e l y 4-2 k i l o c a l o r i e s on t h e s e p l o t s . The quoted a c t i v a t i o n e n e r g y a t t h e t o p o f page 30 i s 82 * 2 k i l o c a l o r i e s p e r mole w h i c h i n d i c a t e s t h a t t h e a u t h o r has d o u b l e d t h e v a l u e i n r e c o g n i t i o n o f t h e e f f e c t o f u s i n g a l / 2 power i n t h e time u n i t o f h i s r a t e f u n c t i o n . T h i s t r e a t m e n t i s j u s t i f i a b l e s i n c e a t r u e r a t e law" based on f u n d a m e n t a l mechanisms has n o t y e t been o b t a i n e d a t t h i s p o i n t . On page 36, Figure.17 t h e p l o t i s f o r g r a i n growth and uses a r a t e f u n c t i o n of d i m e n s i o n ( ™ ^ ° ' - The s l o p e i s e s t i m a t e d a t 80 k i l o c a l o r i e s f r o m t h i s s l o p e . The a c t i v a t i o n energy quoted by t h e a u t h o r f o r t h i s i s 78 k i l o c a l o r i e s p e r mole (page 32 l i n e 13 o f t h e f i r s t p a r a g r a p h ) w h i c h i n d i c a t e s t h a t no c o r r e c t i o n f o r t h e time f u n c t i o n has been made i n t h i s c a s e . - T h i s i s n o t j u s t i f i a b l e s i n c e t h e A r r h e n i u s l a w i s based upon a p l o t o f a f u n d a m e n t a l r a t e c o n s t a n t and t h e t i m e exponent i s always -1 r e g a r d l e s s o f r e a c t i o n o r d e r o r c o m p l e x i t y o f t h e r a t e l a w . Time exponents o f o t h e r v a l u e may be used when t h e r a t e l a w i s n o t r e a l l y known, but t h e a c t i v a t i o n energy must be c o r r e c t e d by d i v i d i n g i t b y t h e t i m e exponent. In t h i s case t h e a c t i v a t i o n energy i s 78/0.s 1 3 0 . k i l o c a l o r i e s p e r mole. e  o r  A r r h e n i u s p l o t s a r e a g a i n shown on page 4-3 F i g u r e 19 and use a u n i t time d i m e n s i o n i n t h e r a t e c o n s t a n t . T h i s s h o u l d y i e l d v a l i d a c t i v a t i o n energy values. :  The a u t h o r has used c e r t a i n p r o c e d u r e s a p p a r e n t l y a c c e p t e d i n t h e c e r a m i c s l i t e r a t u r e i n h i s t r e a t m e n t o f F i g u r e 17 and s h o u l d t h e r e f o r e n o t be condemned, b u t the p r o c e d u r e i s r e c o g n i z a b l y wrong t o t h o s e f a m i l i a r w i t h t h e fundamentals o f k i n e t i c s p r o c e s s e s . I t i s p r o b a b l e t h a t many e r r o n e o u s c o n c l u s i o n s r e g a r d i n g mechanisms i n s o l i d s t a t e k i n e t i c t p r o c e s s e s . h a v e beemdrawnc f r o m .the vuse o f t h i s p r o c e d u r e , and i t i s recommended t h a t a t t e m p t s be made t o c o r r e c t them i n f u t u r e s t u d i e s where t h i s e r r o n e o u s p r o c e d u r e may n o r m a l l y be a p p l i e d .  E. P e t e r s , Associate Professor. EP/jmk March 3, 1964  

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