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Sintering and grain growth of nonstoichiometric rutile. 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 accept t h i s t h e s i s as conforming. to the standard r e q u i r e d from candidates f o r the degree of MASTER OF APPLIED 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 t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department o r by h i s representa t ives . It 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 f o r f i n a n c i a l ga in s h a l l not be allowed without my w r i t t e n permission. Department of M e t a l l u r g y , The U n i v e r s i t y of 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 pressed and heated at d i f f e r e n t temperatures (100G°C to 1300°C) under reducing ( H 2 / H 20) atmospheres to study the rate of weight l o s s , the g r a i n growth and the d e n s i f i c a t i o n . The weight l o s s measurements f o r reduct ion of r u t i l e to two non- s t o i c h i o m e t r i c compositions of 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 of 82 ± 2 k c a l / m o l e . No attempt was made to i d e n t i f y the ra te -determining s tep . Previous weight loss.measurements c a r r i e d out i n e q u i l i b r i u m condi t ions produced an enthalpy of 83 * 10 kcal /mole f o r the formation of an oxygen i o n vacancy. This could suggest that the r a t e - determining step might be the formation of an oxygen ion vacancy. The g r a i n growth study revealed that the n o n - s t o i c h i o m e t r i c composition of T i 0 i . g 2 d i d not obey the t h e o r e t i c a l r e l a t i o n of Burke. The r e s u l t s can be expressed by the f o l l o w i n g D 2 - D 0 2 = K t ° - 6 exp (- 2%^) This a c t i v a t i o n energy f o r g r a i n growth i s equal t o the a c t i v a t i o n energy f o r oxygen i o n d i f f u s i o n i n T i 0 2 . This suggests that the oxygen i o n d i f f u s i o n may be the r a t e - c o n t r o l l i n g step f o r g r a i n growth. The d e n s i f i c a t i o n on s i n t e r i n g was evaluated from l i n e a r shrinkage measurements of the compacts during reduct ion to T i 0 i . g 2 . A few. models were t r i e d , to f i n d the best f i t f o r the present data . While the photomicro- graphs suggest the Coble model f o r bulk d i f f u s i o n , and the values f o r the d i f f u s i o n c o e f f i c i e n t s are of the r i g h t order of magnitude, the a c t i v a t i o n energy f o r the rate determining step i s about 118 k c a l / m o l e , which i s not i n agreement with the previous s i n t e r i n g study on r u t i l e . i i i . • From g r a i n growth data f o r those compacts reduced to T l O i . g s at 1200°C and those s i n t e r e d i n open a i r , i t was seen that 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 discrepancy i n the a c t i v a t i o n energy value may be explained by a . p o s s i b l e e r r o r i n measurement and other unknown v a r i a b l e s which may c o n t r o l the d e n s i f i c a t i o n process . 1V-. • ACKNOWLEDGEMENT The author wishes t o g r a t e f u l l y acknowledge,the ass is tance given by members of the Department of M e t a l l u r g y . He i s e s p e c i a l l y g r a t e f u l t o ' D r . A . C D. Chaklader f o r h i s a d v i c e , guidance and ass is tance and to Mrs . A . M.-Armstrong 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 prepara t ion of the t h e s i s . The work was f inanced by a grant provided by the Defence Research .Board of Canada, D . R . B . 7501-02. V. • TABLE.OF CONTENTS Page INTRODUCTION 1 E a r l i e r Theories of S i n t e r i n g ( S p h e r i c a l Models) . . . . . . . . . 2 D e n s i f i c a t i o n of Powder Compacts . . . . . . . . . . 5 The End-Point Density 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 S i n t e r i n g of Oxides . . . . . . . . . . . . . . . . 9 a) Neck Growth Experiments . . . . . . .... . . . 9 b) D e n s i f i c a t i o n of Powder Compacts . . . . . . . . . . . . . . 11 C r y s t a l S t r u c t u r e of T i t a n i a . . . . . . . . . . . . 11 Defect Reaction i n R u t i l e . . . . . . . . . . . . . . . . 12 Aim of the Present Work 13 EXPERIMENTAL lk Titania-Powder 1̂ Compacting of the Powder . . . . . ... ... . . 16, The Furnace 16 C o n t r o l of the Furnace Atmosphere 18 D e s c r i p t i o n of a Run w i t h the Furnace 19 Measurements f o r Weight Loss and Dimensional Change . . . . . . . . 19 • Measurements f o r Grain Growth . . . . . . . ....... . . ... . . . .20 a) P o l i s h i n g and Etching. 20 b) Grain S i z e Measurements 21 RESULTS .22 X-Ray I n v e s t i g a t i o n . . . . .... . .22 Weight Loss Study .22 G r a i n Growth Study 30 S i n t e r i n g Study . . . . . . . . . 32 v i . Table of Contents Continued... Page DISCUSSION . kk Defect S t r u c t u r e of R u t i l e . . . . . . . . . . . . . . . . . . . . . kk Weight Loss -kf> G r a i n Growth " k6 Temperature Dependence of Grain Growth . k& S i n t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Density-Time Curve . . . . . . . . 51 D i f f u s i o n C o e f f i c i e n t s . . . . . . . . . . . . . . . . . . . . . . 52 CONCLUSIONS 56 RECOMMENDATIONS FOR FUTURE INVESTIGATION . . . . . . . . . . . . . . . . 57 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 58 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Appendix I . '/.Weight Loss' Study . . . . . . . . . . . . . . . . . 6 l Appendix I I . Grain Growth Study . . . . . . . . . . . . . . . . 65 Appendix I I I . S i n t e r i n g Study. Bulk d i f f u s i o n Model (Coble) . . . 66 Appendix IV. Defect 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 Non-stoichiometric R u t i l e . . . 78 Appendix V. Boundary D i f f u s i o n Model (Coble) . 79 V l l . LIST OF FIGURES Figure Page 1. Schematic Representation of the Contact Area Between Two P a r t i a l l y S i n t e r e d Spheres, (a). Center-to-center d i s t a n c e constant, (b) Center-to-center d i s t a n c e s h r i n k s ( a f t e r Kuczynski-'-) . . • 2 2. (a). I n i t i a l Stage of S i n t e r i n g , (b).Near End of I n i t i a l Stage. Spheres have bjegun t o coalesce, (c). I n t e r - mediate Stage. Dark g r a i n s have adopted shape of tetrakaidecahedron, e n c l o s i n g white pore.channels at g r a i n edges. ( d ) , F i n a l Stage. Pores are t e t r a h e d r a l i n c l u s i o n s a t corners where fo u r tetrakaidecahedra meet, ( a f t e r Coble^) . 10 3« (a) T y p i c a l Intermediate 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 at T h e o r e t i c a l D e n sity, (d) F i n a l Stage A f t e r Discontinuous Gr a i n Growth, ( a f t e r Coble^) 10 k. E l e c t r o n Micrograph of T i 0 2 Powders Showing the S i z e and Shape of the P a r t i c l e s , X 3000 15 5 . E l e c t r o n Micrograph of T i 0 2 Powders Showing the S i z e and Shape of the P a r t i c l e s , X U000 15 6. Schematic Diagram of the Furnace . 17 7. Per Cent Weight Loss as a Function of Time f o r T i 0 1 . g 2 . . . 8. Per Cent Weight Loss as a Function of Time f o r TiOx.gs . . . 25 9. Per Cent Weight Loss Versus Square Root of Time f o r T i 0 i . g 2 . 26 10. Per Cent Weight Loss Versus Square Root of Time f o r T i 0 ! . g 8 . 27 11. A Log-Log P l o t f o r A ° ' 5 as a F u n c t i o n of Oxygen P a r t i a l Pressures at D i f f e r e n t Temperatures . . . . 29 12. Temperature Dependence of Weight Loss 29 13- T y p i c a l M i c r o s t r u c t u r e s of S i n t e r e d T i 0 ! . g 2 . (a) I n i t i a l Stage. F i r e d at 1150°C f o r 190 minutes, X 1500. (b) .Intermediate Stage. Showing Channel; and S p h e r i c a l Bpres. F i r e d a t 1150°C f o r 280, minutes, X I5OO. (c) F i n a l S t a g e . . F i r e d a t 1200°C f o r 2800 minutes, X 1500. (d) Beginning of Discontinuous G r a i n Growth. F i r e d at 1300°C f o r 1200 minutes, X 600 31 lk. Log-Log P l o t f o r Average Gr a i n Diameter Versus Time 33 15. G r a i n Growth i n TIOX.QZ Compacts w i t h Temperature J>h L i s t o f Figures Continued. v i i i . • F i gure Page 16. Isothermal Grain Growth \ 35 17. Logarithm of G r a i n Growth Rate Versus R e c i p r o c a l Tempera- tu r e 36 18. D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition of.,. TiOx.92 (a) at T = 1000°C 38 (b) at'T = 1050°C 38 (c) at : T = 1100°C 39 (d) at T = ;1150°C I . . . 39 (e) a t T = 1200°C *f0 .19. D i f f u s i o n C o e f f i c i e n t Versus the : R e c i p r o c a l of the Absolute 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 values c a l c u l a t e d from s i n t e r i n g experiments and models k^> A - . I I I - l . D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition of TiOx.gs (a) at T ="1000°C 73 (b) at T = 1050°C 74 (c) at T = 1100°C 7k (d) at T = 1150°C 75 (e) at T = 1200°C 75 A . I I I - 2 . D e n s i f i c a t i o n of T i 0 2 at D i f f e r e n t Temperatures . . . . . . . 76 A . I I I - 3 . V a r i a t i o n of G r a i n S i z e w i t h Time and Temperature. This i s t o determine A of equation (6) 77 i x . . LIST OF TABLES Page Table I . Analyses of the Two R u t i l e Powder Samples lk Table I I . Values of 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 Energy Data f o r D i f f e r e n t Ceramic Oxides . . . . k& Table 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 . P a r t i a l Pressure ^>k i INTRODUCTION Because of the very h i g h m e l t i n g temperature of most oxide ceramics, d e n s i f i c a t i o n by s i n t e r i n g of powder compacts i s a common f a b r i c a t i o n t e c h - nique. Although t h i s technology may be as o l d as the a r t s of ceramics, an understanding of. the mechanisms i n v o l v e d during the s i n t e r i n g processes i s only a recent development. The term " s i n t e r i n g " as used by the 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 by which a mass of compacted powder i s transformed i n t o a more dense product by the a p p l i c a t i o n of heat alone. Observation o f the increased cohesion between the p a r t i c l e s suggests t h a t the process may be d i v i d e d i n t o stages. In the f i r s t stage, the growth of bridges between adjacent p a r t i c l e s occurs, but w i t h very l i t t l e d e n s i f i c a t i o n . I n the subsequent stages, the i n t e r p a r t i c l e necks grow bigger producing a n o t i c e a b l e shrinkage. F u r t h e r neck growth would r e s u l t i n the formation of i s o l a t e d .pores. The most s u c c e s s f u l determinations of the k i n e t i c s of s i n t e r i n g have been made u t i l i z i n g systems of s i m p l i f i e d geometry, where the study of the neck growth i s q u i t e f e a s i b l e by simple experimental techniques. . For t h i s reason, experiments i n t h i s f i e l d have been mostly confined t o the measurement o f the rate o f neck growth between a sphere and a plane, between two spheres, between a wire and a plane or between two wires as a f u n c t i o n of time and temperature. Such systems have the 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, but they have l i m i t e d a p p l i c a b i l i t y t o the d e n s i f i c a v t i o n of powder compacts. - 2 - E a r l i e r Theories of S i n t e r i n g ( S p h e r i c a l Models) 1 Kuczynski was the f i r s t t o attempt t o derive the r a t e expressions f o r the growth of the neck at the p o i n t of contact between two s p h e r i c a l p a r t i c l e s . • According t o him, the 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 the formation of 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 Representation of the Contact Area Between Two P a r t i a l l y S i n t e r e d Spheres. (a) Center-to-center distance constant, (b) Center-to-center distance s h r i n k s , ( a f t e r Kuszynski" 1") . T h i s can be brought about by one or more of the f o l l o w i n g processes: the viscous or p l a s t i c f l o w ; evaporation and condensation; and volume or surface d i f f u s i o n . The well-known r e l a t i o n s h i p between the r a d i i x of the neck and a, of the spheres, time t ; and temperature T can be described by one general equation n x_ .= F (T)t .....(1) m a where F (T) i s a f u n c t i o n of temperature only. -.5 - The d i f f e r e n t mechanisms i n v o l v e d i n the s i n t e r i n g process are r e l a t e d by the f o l l o w i n g numerical values: n = 2 m = 1 f o r v i s cous or p l a s t i c flow n- = •3 . m = 1 f o r evaporation.and condensation n = 5 m = 2 f o r volume d i f f u s i o n n = 7 • m = 3 f o r surface d i f f u s i o n . Although i t has been recognized t h a t the decrease of t o t a l surface energy of the 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 , t h i s excess surface energy i n a c t u a l compacts has never been very great. For i n s t a n c e , the net decrease i n f r e e energy occuring 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 corresponds t o an energy decrease of about 1 cal/gm. However, i n the neck area there e x i s t K s t r e s s e s due t o the curvature which may produce mass f l o w or a lowering of the vapour pressure i n . the neck r e g i o n . The vapour pressure over a f l a t surface p Q and ^ p , the lowering of the vapour pressure due t o the r a d i u s of curvature p^ are r e l a t e d , by 2 the 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. = - ^ Vo .(2) P 0 RTp where 2f i s the surface energy and V© the molar volume of the s o l i d . The evaporation-condensation mechanism f o r m a t e r i a l t r a n s p o r t during s i n t e r i n g can be e a s i l y understood from t h i s r e l a t i o n s h i p . I f the vapour pressure over the f l a t surface (or over the convex surface i n the s p h e r i c a l model, Figure 1) i s higher than t h a t of the 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 a t h i g h temperatures, evaporation from the convex r e g i o n and sub- sequent condensation i n the c a v i t y of the neck can be expected t o occur, r e s u l t i n g i n neck growth. - k - Another consequence of the e x i s t e n c e of the s t r e s s e s i n the neck can.be determined by c o n s i d e r i n g the vacuum surrounding the neck as a f l u i d composed of vacancies which evaporate i n t o the s o l i d . For these vacancies, the c a v i t y of the neck w i l l be a convexity and consequently the pressure of vacancies i n the neck area, w i l l be g r e a t e r than under the surfaces of the other p a r t s . Assuming t h a t the vacancy pressure 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 the s o l i d , equation (2) may be r e w r i t t e n as A c = V p , ( 3 ) Cb RTP . where C Q i s the e q u i l i b r i u m vacancy con c e n t r a t i o n under'a f l a t s u r f a c e . Due t o the excess c o n c e n t r a t i o n of vacancies A C i n the neck area, there e x i s t s a gradient of vacancies between t h i s area and the i n t e r i o r of the system. . This w i l l r e s u l t i n the vacancy m i g r a t i o n along t h e i r gradient accompanied by the volume or surface d i f f u s i o n of atoms i n the opposite d i r e c t i o n . The excess of vacancies have t o be removed from the system by d e p o s i t i n g them at the p o s s i b l e s i n k s , which may be the nearby surface or the g r a i n boundaries. The vacancies d e p o s i t e d - i n the g r a i n boundaries 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 the s o l i d vapour i n t e r f a c e , because the r a t e of d i f f u s i o n i s much l a r g e r i n the g r a i n boundary than i n the bu l k of t h e bodyj or i f the f l u i d i t y ' of the m a t e r i a l i n the g r a i n boundary i s h i g h (as i n the case of the v i s c o u s flow) the vacancies w i l l c o l l a p s e , as any v o i d w i l l . i n a l i q u i d of low v i s c o s i t y . In any case, the e l i m i n a t i o n of vacancies v i a g r a i n boundaries w i l l . r e s u l t i n the shrinkage accompanied by c e n t e r - t o - c e n t e r approach of the p a r t i c l e s . On the other hand, i f the e l i m i n a t i o n of vacancies, occurs at the f r e e surface,, the c e n t e r - t o - c e n t e r •distance of the spheres w i l l not change as would.be the case i n d e n s i f i c a t i o n -.5 - by the evaporation and condensation mechanism. i . From the d e r i v e d expression f o r AC, equation (3) and by u s i n g the f i r s t F i c k ' s d i f f u s i o n equation, the equation f o r volume d i f f u s i o n mechanism can.be obtained. I t s f i n a l form i s . .xf. = K A vQ r v t (k) a 2 RT • where D y i s the 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 numerical constant and has a value of about 1 0 0 . D e n s i f i c a t i o n of Powder Compacts The increase i n d e n s i t y of a powder compact d u r i n g s i n t e r i n g i s of the grea t e s t p r a c t i c a l importance,, but due t o the number of unknown v a r i a b l e s or p o o r l y d e f i n e d parameters i n v o l v e d i n the s i n t e r i n g process, i t i s d i f f i c u l t t o evaluate the b a s i c mechanisms of 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 of powder compacts c o n s i s t e d mostly of determining t h e i r d e n s i t y as a f u n c t i o n of temperature, although the 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 of view of the k i n e t i c s of the process. Kingery and Berg-^ t r i e d t o apply the volume d i f f u s i o n equation f o r neck growth between two p a r t i c l e s t o powder compacts of oxides. • They assumed t h a t the volume of the neck at any time i s equal t o the volume of the pore space removed.from the system, and obtained the f o l l o w i n g r e l a t i o n - s h i p by u s i n g equation (k) AV = - 3n V Q 8 kO g V n Py i V 5 V 5 a 3 k T. ••(5) where AV and V Q are the change of 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 the number of p o i n t s of contact 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 the experimental observations they have found t h a t t h i s equation has.very, l i m i t e d a p p l i c a b i l i t y . Although i t can be a p p l i e d f o r a volume shrinkage of up t o 2$>, i t f a i l e d completely k f o r a shrinkage g r e a t e r than Qfo. Kuczynski approached the problems of d e n s i f i c a t i o n of compacts from the p o i n t of view of pore shrinkage and considered 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 the'pores: v i s c o u s or p l a s t i c flow and volume d i f f u s i o n . The End-Point D e n s i t y 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 of powdered metals and oxides i n d i c a t e d t h a t i t was not p o s s i b l e t o reach the 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 alone and t h i s l e d the 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 there 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 the end-point d e n s i t y i s the r e s u l t of g r a i n growth, which takes place simultaneously 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 equation t o support t h i s hypothesis that as l o n g as the g r a i n growth i s continuous and the pores are connected by the g r a i n boundaries, complete e l i m i n a t i o n of pores i n the boundaries can be achieved, and t h i s w i l l r e s u l t i n complete d e n s i f i c a t i o n . Only when d i s - continuous g r a i n growth, took p l a c e i n a system would the pores be trapped i n s i d e the g r a i n s . These trapped pores would.not s h r i n k any f u r t h e r . In the l a t t e r case, the powder compact would reach an end-point d e n s i t y which would be below the 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 . Coble, u s i n g doped alumina ( t o c o n t r o l g r a i n growth) showed exp 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 achieve t h e o r e t i c a l d e n s i t y only by s i n t e r i n g . . On the other 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 reached an end-point d e n s i t y a f t e r a c e r t a i n p e r i o d of s i n t e r i n g . This phenomenon of exaggerated g r a i n growth and appearance of trapped pores a f t e r a few hours of s i n t e r i n g occurs only 6 at or above a c e r t a i n temperature c a l l e d the Sauerwald temperature , Tg. This i s g e n e r a l l y 2/3 t o 3 A 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 •Structure The change i n shape and s i z e of the pores i s very d i f f i c u l t t o study i n r e a l compacts as the pores are of d i f f e r e n t s i z e s and shapes. In the f i r s t stage, the necks grow between adjacent p a r t i c l e s and the g r a i n boundary does not move, because any displacement towards the center of the p a r t i c l e would mean an increase i n boundary area and thus, an increase i n g r a i n boundary energy. Afterwards, as s i n t e r i n g proceeds and as the vacancy 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 pore r a d i u s , the l a r g e r pores may increase i n s i z e as a r e s u l t of the condensation of vacancies o r i g i n a t i n g from the smaller ones. Thus, the pores may be c l a s s i f i e d as s m a l l ones, which s h r i n k , and l a r g e ones, which increase i n s i z e , d u r i n g s i n t e r i n g . The smaller pores are much more numerous and may i n c l u d e the b u l k of the p o r o s i t y i n the compact. When they disappear g r a d u a l l y the o v e r a l l d e n s i t y of the compact i n c r e a s e s . On the other hand, the pores of the second group may c o n t r o l the g r a i n growth by anchoring.the g r a i n boundaries. The c r i t i c a l diameter ('J ) of the pores of ra d i u s r , which 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 the volume f r a c t i o n p o r o s i t y f 7 i n the m a t e r i a l are r e l a t e d , according t o Zener by the 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 the l a r g e r pores- reach the c r i t i c a l diameter, exaggerated g r a i n growth w i l l occur and the trapped 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 formulate b u l k d i f f u s i o n models f o r the t o t a l course of shrinkage i n powder compacts, leading, t o t h e o r e t i c a l l y dense products. He assumed t h a t there are three stages of d e n s i f i c a t i o n . I n 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 t o 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 the neck growth stage, u s u a l l y i s accompanied by an increase i n r e l a t i v e d e n s i t y of powder compacts from 0.5 t o 0 .6 . This i s shown i n Figures 2a and 2b. During.the i n i t i a l ' stage of s i n t e r i n g , g r a i n growth can not occur, as i t would.require m i g r a t i o n of the g r a i n boundary from the minimum area p o s i t i o n which 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 area .and energy. In the second or intermediate stage, g r a i n growth begins and pore shape changes t o produce a matrix of pores and g r a i n boundary. The e q u i l i b r i u m angles formed between thgm are d i c t a t e d by surface t e n s i o n such t h a t the.three i n t e r a c t i n g surfaces form a s p a t i a l f o r c e balance. . This stage can be represented by Figure 2c . The pore phase i s very s i m i l a r t o a con- • t i n o u s channel 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 stage begins when the pore becomes discontinuous and the channels are r e p l a c e d by the g r a i n boundaries. The pores only occupy the , f o u r g r a i n corners and are n e a r l y s p h e r i c a l i n shape as shown i n Figure 2d. These pores at the four g r a i n corners w i l l g r a d u a l l y s h r i n k t o zero s i z e and the s i n t e r i n g w i l l proceed t o t h e o r e t i c a l d e n s i t y of the compact. An - 9 - a l t e r n a t i v e f i n a l stage would be when discontinuous g r a i n growth occurs before a l l the p o r o s i t y i s removed. I n t h i s case complete e l i m i n a t i o n of pores would be impossible. Coble supported h i s arguments by comparing h i s h y p o t h e t i c a l model w i t h the m i c r e s t r u c t u r e s of the specimens at d i f f e r e n t stages of s i n t e r i n g . These stages are shown i n F i g u r e . 3 . . The f i n a l equation r e l a t i n g the r a t e . o f pore shrinkage w i t h other parameters has the f o l l o w i n g form 3 dp = '. 'my{&n .(6) dt r k T where N = numerical constant: 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 = surface energy aQ = vacancy; volume 1 = average g r a i n diameter k =; Boltzmann's constaux T; = absolute temperature S i n t e r i n g of Oxides a) Neck Growth Experiments 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 of A l 2 0 3 , T i 0 2 and ZnO have been used by s e v e r a l workers. K i c z y n s k i ' p a r t i c u l a r l y used spheres of sapphire t o t e s t h i s volume d i f f u s i o n model of neck growth. . Parravano and N o r r i s ^ used spheres of ZnO t o study the r a t e of neck growth as a f u n c t i o n of temperature. T h e i r r e s u l t supported the model of evaporation and condensation 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 andParravano"^ s t u d i e d the s i n t e r i n g of s i n g l e c r y s t a l s of r u t i l e i n a i r and i n reducing atmosphere i n the temperature range of 900-1350°C, u s i n g a sphere-to-sphere model. T h e i r work i n d i c a t e d t h a t the predominant mechanism of 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 they (c) - 10 - Figure 2. (a) I n i t i a l Stage of S i n t e r i n g , (b) Wear End of the I n i t i a l Stage; spheres have begun t o coalesce, (c) Intermediate Stage; dark g r a i n s have adopted shape of tetrakaidecahedron, e n c l o s i n g white pore channels at g r a i n edges, (d) F i n a l Stage; pores are t e t r a h e d r a l i n c l u s i o n s at corners where fo u r tetrakaidecahedra meet, ( a f t e r C o b l e 5 ) . (b) Figure 3• ( a) T y p i c a l Intermediate 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 Stage at T h e o r e t i c a l Density (d) F i n a l Stage A f t e r Discontinuous G r a i n Growth ( a f t e r C o b l e 5 ) . - 11 - obtained an a c t i v a t i o n energy of 70 * h kcal/mole f o r the process. Kawai and Whitmore''""'" a l s o independently stu d i e d the sphere-to-plate bonding of vacuum-reduced monocry s t a l l i n e r u t i l e over the temperature range of 1200-1275°C. The r a t e law governing the 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 the volume d i f f u s i o n was the predominant mechanism of m a t e r i a l t r a n s p o r t i n the s i n t e r i n g process. v b), Dens i f i cat i o n of Powder Compacts The d e n s i f i c a t i o n of powder compacts of oxide was s t u d i e d by 12 Coble as dis c u s s e d p r e v i o u s l y . ;He used alumina as the standard m a t e r i a l 13 t o . t e s t h i s hypothesis. Very r e c e n t l y , Johnson and C u t l e r a l s o c a r r i e d out i n v e s t i g a t i o n s on the l i n e a r shrinkage r a t e of alumina powder compacts. Both of these 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 bulk d i f f u s i o n , and not the g r a i n boundary d i f f u s i o n was the basic- mechanism, of d e n s i f i c a t i o n . C l a r k lh and White used magnesia powder compacts t o study the r a t e of d e n s i f i c a t i o n and explained t h e i r r e s u l t s on.the b a s i s of a p l a s t i c f l o w model. The e f f e c t of non-stoichiometry on the r a t e of 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 workers w i t h U 0 2 + x . No r e s u l t s have been reported on the d e n s i f i c a t i o n of T i 0 2 powder compacts, although s i n t e r e d T i 0 2 d i s c s are b e i n g w i d e l y used i n the micro- wave guide tubes a t the, present time. 16 C r y s t a l S t r u c t u r e of T i t a n i a Titanium d i o x i d e can c r y s t a l l i z e i n three forms. M i n e r a l o g i c a l l y they are known as r u t i l e , anatase and b r o o k i t e . R u t i l e i s the s t a b l e form .1 above 820°C but metastable at room temperature and e x i s t s i n a l l commercial t i t a n i a 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 of p u r e l y i o n i c s t r u c t u r e , the rad i u s r a t i o of 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 c o o r d i n a t i o n of t i t a n i u m . w i t h , oxygen. The r u t i l e s t r u c t u r e may be described as b u i l t up from d i s t o r t e d T i 0 6 . octahedra > the octahedra forming chains i n the c - d i r e c t i o n and each octahedron sharing an edge w i t h the adjacent members of the chains. The c r y s t a l d e n s i t y ls-k.26 gm/cm3, as determined from the X-ray measurements. In the 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 are present i n t h e i r highest valence s t a t e +k and - 2 . Titanium i s a t r a n s i t i o n metal ©f the 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 the argon core. Defect Rea c t i o n i n R u t i l e 17 Straumanis et a l . r e c e n t l y measured the d e n s i t y and l a t t i c e parameter of r u t i l e powders f o r the oxygen d e f i c i e n c i e s from 0.5 t o 0.8 atomic per cent. .Their r e s u l t s i n d i c a t e t h a t the s i z e and shape of the u n i t c e l l i n t h i s range d© not change a p p r e c i a b l y so t h a t the change i n d e n s i t y can be a t t r i b u t e d t o oxygen vacancies alone. K i n e t i c s t u d i e s of the o x i d a t i o n of 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 the 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 covering e i t h e r the predominant p o i n t defect or the slower moving i o n i c species i n r u t i l e . The i n v e s t i g a t i o n s , of Gulbransen 18 19 and- Andrew- , and Kinna and Knorr support the case f o r i n t e r s t i t i a l c a t i o n 20 21 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 , w h i l e those of B i r c h e n a l l , 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 the o x i d a t i o n of t i t a n i u m . -.13 - From the semi-conducting behaviour of 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 metal excess n-type semi-conductor upon r e d u c t i o n . The predominant p o i n t d e f e c t s are oxygen i o n vacancies which are capable of t r a p p i n g e l e c t r o n s and thereby a c t i n g as a donor center. Aim.of the Present I n v e s t i g a t i o n This present i n v e s t i g a t i o n has been mainly concerned w i t h s i n t e r i n g and. g r a i n growth of 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 the experiments 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 pressure over a temperature range of .1000 t o 125G°C. -This was t o maintain a constant vacancy c o n c e n t r a t i o n of oxygen i n the system, i . e . a constant r a t i o of t i t a n i u m t o oxygen i n the 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 , the r a t e o f weight l o s s was a l s o determined i n the specimens used f o r the i n v e s t i g a t i o n of s i n t e r i n g a n d : g r a i n growth. - 14 - - EXPERIMENTAL .Titania Powder A l l experiments were performed u s i n g . r u t i l e powders supplied by the J . J . Baker Chemical Company, Phi1 1ipsburg, N.-J. • Two one-pound :samples were used, the chemical composition of which i s . given i n the following t a b l e . Table I. Analyses of the Two R u t i l e Powder Samples Lot 21303 Lot 28363 Water Soluble Salts 0.05 $ 0.02 % Arsenic 0 .0001 0.00005 Iron 0.002 0.002 Lead 0 .008 0.004 . Zinc 0.004 0.005 To determine the grain s i z e and shape of the powders, three methods were t r i e d : f i r s t , , the standard T y l e r sieves; second, a sedimentation technique using Andreasen's pipette; and, t h i r d s ; measurement5 of tl^.i.^rain.^sSgjegrdiBi'ived from p i c t u r e s taken by the ele c t r o n microscope. Figures 4 and 5 represent the pictures taken by the el e c t r o n microscope with the magnification of 3000 and 4000. These show that the grains were f l a k e d , having.two large dimensions but l i t t l e thickness. The p a r t i c l e s were a l l i n sub-sieve range. • According.to the measurements c a r r i e d out by the Andreasen's pipette technique,•over 80 $ of the p a r t i c l e s are less.than 4 microns. The p a r t i c l e s i z e was also determined by taking the average of the p a r t i c l e dimensions i n Figures 4 and 5 a n < i i s 1.5 J 1 * The l a r g e s t and smallest dimensions are 4 .25 M ^ d 0.25 p. r e s p e c t i v e l y . - 15 - Figure k. E l e c t r o n Micrograph of T i 0 2 Powders Showing the Size and Shape of the P a r t i c l e s . X 3000 Figure 5 . E l e c t r o n Micrograph of T i 0 2 Powders Showing the Size and Shape of the P a r t i c l e s . X 4000 - 1 6 - However, the average grain s i z e of smaller p a r t i c l e s , which are the l a r g e r fraction,have, an average diameter of about 0,8 u. Compaction of the Powder One c y l i n d r i c a l die having an ~ i n t e r n a l diameter of 0.5 inches and another one of rectangular shape, having the i n t e r n a l c r o s s - s e c t i o n a l dimensions of 0.5 X k inches were used f o r compacting the powder. The specimen thickness was kept approximately O.25 inches because t h i c k e r specimens tended to break down d u r i n g . f i r i n g . 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 of b r i t t l e ' m a t e r i a l s , such as oxides, used compacting pressures ranging from ^,000 t o 40,000 p s i . In the present i n v e s t i g a t i o n i t was found that an increase of pressure from 766O t o 40,000 p s i increased the i n i t i a l green density of the compacts from 1^3 to 2.30 gm/cm3. However, as the compacting pressure was increased, laminar cracks appeared on the specimens. These caused the specimens'to break i n t o pieces on s i n t e r i n g . As the specimens were a l s o used f o r weight l o s s studies, no l u b r i c a n t of any kind could be used. .Therefore, a compacting pressure of 12,000 p s i was adopted f o r a l l the weight l o s s , grain.growth and s i n t e r i n g measurements. The Furnace The furnace used f o r a l l experiments was e s s e n t i a l l y a h o r i z o n t a l type tube furnace heated by four globar;, heating elements. As shown i n Figure 6, the main tube consisted of a long Zircotube of 1 l / 8 inches i n diameter passing through the heating chamber, which was b u i l t with i n s u l a t i n g b r i c k s . Both ends of the tube were cooled by c i r c u l a t i n g water through copper jackets. Thus the specimen, while s t i l l i n s i d e the furnace was cooled qu i c k l y a f t e r i t was removed from the heating zone. A second Zircotube  - 18 - of 0.9 inches i n t e r n a l diameter was f i t t e d i n s i d e the main one t o a v o i d any thermal shock on the l a t t e r when the hot boat was p u l l e d i n t o the c o o l zone. The i n l e t o f the tube was connected t o a heated copper tube which was j o i n e d t o a bubbler and which c a r r i e d reducing atmosphere. The o u t l e t was d i r e c t l y connected t o a lon g g l a s s tube bent a t r i g h t angles, the end of which was dipped i n t o a water bath. By t h i s arrangement the furnace was e s s e n t i a l l y a c l o s e d system. With the help of a s t r i n g system as shown i n Figure 6 i t was p o s s i b l e t o move the boat from the c o o l zone t o the hot zone and v i c e - v e r s a . The temperature was measured by the useof a P t - P t 10$ Rh thermo- couple 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 pla c e d j u s t above the main Z i r c o t u b e . T h i s thermocouple was connected t o a Wheelco temperature r e g u l a t o r which c o n t r o l l e d the temperature w i t h i n * 10°C. There was a temperature g r a d i e n t along the l e n g t h of the main Zircotube and a l s o a temperature d i f f e r e n c e between the thermocouple and the specimen. For t h i s reason the p o s i t i o n of the boat was maintained i n the three inches of hot zone where the temperature was r e l a t i v e l y constant 1 ; and almost equal t o the recorded temperature i n the Wheelco r e g u l a t o r . The temperature of the specimens was o c c a s i o n a l l y measured w i t h a separate standardized P t - P t 10$. Rh thermocouple. C o n t r o l o f the Furnace Atmosphere The r a t i o of the pressures of H 2 and H 2 0 i n the i n l e t gas of the furnace was determined by p a s s i n g hydrogen s l o w l y i n t o two bubble r s , submerged i n a bath of water which was maintained at a f i x e d temperature. The flow of 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 ' - 1 9 - atmosphere i n s i d e , the butjblers. . From the 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 flows •. of l e s s than 7Q bubbles a minute gave r e p r o d u c i b l e • • r e s u l t s . The bubblers were connected w i t h the furnace by a copper pipe which was heated by a r e s i s t a n c e element a t a temperature of 70°C, s l i g h t l y h i g h e r than the bath, temperature t o a v o i d any condensation. D e s c r i p t i o n of a .Rim i n the Furnace The alumina specimen holder was loaded w i t h f i v e t o f i f t e e n specimens, and was put i n s i d e the. tube i n the c o o l zone. . The furnace atmosphere was g r a d u a l l y changed t o the 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 flow, of" argon f o r 10 minutes and subsequently by a flow, of predetermined H 2/H 2 0 mixture f o r 15 minutes. Afterwards the specimens holder was pushed i n t o the hotest zone of the tube. The recorded time f o r a l l runs began when the boats were f i r s t i n the hot zone. A f t e r a . d e f i n i t e time of f i r i n g . t h e reverse procedure was f o l l o w e d . The specimen holder was p u l l e d back i n t o the c o o l i n g zone but the same flow-was maintained f o r s e v e r a l minutes t o avoid any change i n the st o i c h i o m e t r y of r u t i l e . When the specimen temperature was s u f f i c i e n t l y low the tube was f l a s h e d w i t h argon f o r 10 minutes and f i n a l l y the specimens were taken out f o r measurements. - A f t e r the appr o p r i a t e measurements were taken the specimens were re p l a c e d i n the boat and the process was repeated. Measurements f o r Weight Loss and Dimensional Change The weight l o s s and shrinkage measurements.were c a r r i e d out on the same specimens, as both of these measurements were needed t o c a l c u l a t e the r e l a t i v e d e n s i t y . - 20 - From the weight: r a t i o of titanium to t i t a n i a which i s ^7-9/79S9J the weight of titanium i n the specimen before reduction was deduced. The weight of the oxygen was determined by subtracting the weight of the titanium from the weight of the i n i t i a l r u t i l e having a stoichiometric composition of T i 0 2 . The l o s s . o f oxygen a f t e r reduction was obtained by subtracting the i n i t i a l f i x e d weight of titanium from the weight of the specimen. These measurements were taken on a C h r i s t i a n Beckers manual balance. The shrinkage was calculated by measuring the length and width of the specimen with a micrometer having an accuracy of * 0.001 inches. The average of these two l i n e a r shrinkages was used to deduce the volume shrinkage from the f o l l o w i n g . r e l a t i o n .X 100 From the weight measurements and from the dimensions of the specimens a f t e r d i f f e r e n t periods of heat treatment, the volume and the bulk d e n s i t i e s were calculated and recorded. Measurements f o r Grain Growth a) P o l i s h i n g and Etching For the study of grain growth, the specimens were polished f i r s t using sandpaper of various sizes and then using wheels having suspensions of f i n e alumina powder. The sandpapers used were of the types 1, 0,; 00 and 000; the alumina powder had an average g r a i n size of 0.05 u. When the surfaces of the polished specimens were free from any v i s i b l e scratches, they were etched by immersing them i n a bath of concentrated b o i l i n g H 2 S 0 4 f o r 2 to - .21 - 3 minutes. Some specimens were subjected t o a. thermal etch f o r t h r e e y minutes a t 6^>0°C before a c i d e tching. .This i s t o make the g r a i n boundaries more apparent. In t h i s case however, the time f o r the H 2 S 0 4 immersion was reduced t o 30 seconds. b) Grain S i z e Measurements Se v e r a l p i c t u r e s of known m a g n i f i c a t i o n of these specimens were taken w i t h a Re i c h e r t metallographic microscope u s i n g r e f l e c t e d l i g h t . A m a g n i f i c a t i o n of I5OO was used f o r the 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 of 600 was used f o r those having l a r g e r g r a i n s . The average g r a i n s i z e measurements were made u s i n g 23 the i n t e r c e p t (or Heyn) procedure . - 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 the s t a b i l i t y range of r u t i l e on r e d u c t i o n the 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 out. Compacts of r u t i l e were reduced, t o a f i x e d . r a t i o of t i t a n i u m t o oxygen by heating i n the furnace a l r e a d y d e s c r i b e d and by u s i n g d i f f e r e n t r a t i o s of H 2/H 2 0 p a r t i a l pressures over the temperature range of 1 0 0 0 .to 1300°C. . The specimens were scanned i n a Norelco X-ray d i f f r a c t o m e t e r f o r the Bragg angles of 10° t o 80°. The "d" values were c a l c u l a t e d from.the corresponding d i f f r a c t o m e t r i c peaks and. compared with the standard 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 oxides present. I t was e s t a b l i s h e d t h a t the 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 composition of T±01.Q2. The compound Ti3p 5 was detected only i n reduced r u t i l e having a composition of T i O i . g i . In the same i n v e s t i g a t i o n , the r e l a t i o n s h i p between the bubbling r a t e of H 2 i n the water bath and the corresponding e q u i l i b r i u m r a t i o of t i t a n i u m t o oxygen obtained a t the d i f f e r e n t temperatures was determined. These r a t i o s were subsequently used during s i n t e r i n g , weight l o s s and- g r a i n growth measurements. Weight Loss Study The rates of r e d u c t i o n of T i 0 2 t o two w e l l d efined n o n - s t o i c h i o m e t r i c compositions having the r u t i l e s t r u c t u r e ( T i 0 i.g 2 ± o-oi a n ( i T i ^ . g a , * o-oi) were determined.by weight l o s s measurements. These experiments were c a r r i e d out at 1000, 1050, 1100,. II50 and 1200°C under the appropriate H 2/H 2 0 - 23 - atmospheres t o y i e l d these compositions at e q u i l i b r i u m . The f r a c t i o n a l weight l o s s ^r^- of the compacts can be r e l a t e d w t o the time of r e d u c t i o n by the f o l l o w i n g type of g e n e r a l i z e d equation. AW = ' , ( A t ) .....(7) w Z\W = weight l o s s at time t W = o r i g i n a l weight A = a f u n c t i o n of temperature, pressure and geometry of the sample. In order t o determine the value of the exponent n, the weight l o s s data given i n Appendix I , Tables1 and 2 were p l o t t e d as l o g ( A W) W versus l o g t 1. This p l o t i s shown i n Figure 7 f o r TiOi.92 a n ( i i ' n Figure 8. f o r TiOx.gg as e q u i l i b r i u m products. The slope of the l i n e s produced a value of n = 0.5^ ± 0.1 f o r T i 0 i . 9 2 and n = 0.48 ± 0.01 i n the case of Ti O i . g s . 24 •A l i t e r a t u r e survey revealed t h a t such a r e d u c t i o n process 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 or a p a r a b o l i c law. Therefore the average value of n = O.52 * 0.01 suggests a p a r a b o l i c r e l a t i o n . Equation (7) can then be w r i t t e n as AW = (A t)°' 5 (8) W The values of the f u n c t i o n A ° ' 5 at d i f f e r e n t temperatures are then the slopes of the p l o t s of the f r a c t i o n a l weight l o s s under isothermal c o n d i t i o n s versus t°' 5 as shown i n Figures 9 a n £ l 10. These f i g u r e s show th a t the weight l o s s reached the e q u i l i b r i u m value a f t e r a p e r i o d of heating of about  Per Cent Weight Loss (-JJ- X 100) 2 . 0 Figure 9. Per Cent Weight Loss Versus Square Root of Time f o r Ti01.gz - 27 - Figure 10. Per Cent Weight Loss.Versus Square Root of Time f o r T i 0 i . g 8 - 2 8 - 100 t o 300 minutes. .For t h i s reason, only-the data before 150 minutes of re d u c t i o n were taken i n t o account f o r the c a l c u l a t i o n of the slopes by the l e a s t squares method. The r e s u l t s are t a b u l a t e d i n Appendix-I, Tables 1 and 2. I n order t o determine the oxygen p a r t i a l pressure dependence of the f u n c t i o n A, i t was assumed t h a t the r e l a t i o n s h i p took the form A . = K ' P O 2 X . ( 9 ) or A 0' 5 = K 0" 5.F02 2' (10) where K inc l u d e s the s p e c i f i c r e a c t i o n r a t e constant and any other v a r i a b l e s a f f e c t i n g the r a t e . - I n - l o g a r i t h m i c form, the r e l a t i o n s h i p (10) w i l l be ' ' logA°' 5,= l l o g K + x l o g P0 2 . (11) 2 2 o • 5 The experimental values of l o g A (Appendix I , Tables 1 and 2) have been p l o t t e d a gainst l o g P 0 2 (Appendix I I I , Table 3) f o r each temperature and are shown i n Figure 11. Although these curves are determined by two p o i n t s only, the f o u r p l o t s can be considered as p a r a l l e l w i t h i n experimental e r r o r . The values of x obtained from the slopes of these p l o t s are t a b u l a t e d i n Appendix I (Table k) and give a mean value o f - 1. 3 On t h i s b a s i s , K̂/ can be d e r i v e d from the experimental value of 1 / o • 5 ' / A u s i n g equation (11).} The c a l c u l a t e d values of K are shown i n Appendix I , Tables laand 2. .- •> / Provided the only temperature dependent term i n K i s the r a t e constant, the slope o f an Arrhenius p l o t w i l l correspond t o the a c t i v a t i o n energy f o r the rat e determining step. These are shown i n Figure 12 f o r both s e r i e s of experiments. -18 ^17 ! ^16 ~~ ^ l l " ^13 : ^12 Log P0 2 Figure 11. / L Log-Log P l o t f o r A° 5 as a Function of Oxygen P a r t i a l Pressures at D i f f e r e n t Temperatures 30 The corresponding value' of the a c t i v a t i o n energy c a l c u l a t e d from the slope i s 82 * 2 kcal/mole f o r both of these equilibrium compositions. Grain Growth Study The variation- of the grain s i z e as -a function of the heating time was studied under c o n t r o l l e d H2./H2O atmosphere over the temperature range of 1000°C to 1300°C. The photomicrographs as shown i n Figure 13a, b, c, d, reveal the d i f f e r e n t stages of grain growth during the s i n t e r i n g process. These are very s i m i l a r t o those shown i n Figure 3 ( a f t e r C o b l e v ) . The exaggerated grain growth was expected to occur at or above the 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 grain growth was a c t u a l l y observed at 1250°C a f t e r k-000 minutes and at 1300°C a f t e r 1200 minutes of s i n t e r i n g as shown i n F igure 13d. 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 grain 25 26 growth experiments was made i n i t i a l l y by Beck et a l . and by Turnbull The average grain diameter c o r r e l a t e d with time according t o Turnbull i s given as D 2 - D Q 2 = K,£.V- t (12) where D i s the average grain diameter at time t , D0 i s the average o r i g i n a l diameter at t = 0, K, i s a rate constant and V i s the atomic volume. 27 Burke ., however, deduced the following expression f o r grain growth on the assumption that the motivating force f o r grain-boundary migration during grain growth i s the surface tension of the boundary, and that the radius.of curvature of the boundary i s proportional to the grain ..diameter D - D 0 = K Q t exp (-1$) - (13) - 31 - (c) (d) x 1500 x 600 Figure 13. Typical Micro-structures of Sintered T i 0 i . 9 2 . (a) I n i t i a l Stage - fired at 1150°C for 190 minutes, (b) Intermediate Stage Showing Channel and Spherical Pores - fired at H50°C for 280 minutes, (c) Final Stage - fired at 1200°C for 2800 minutes, (d) Beginning of Discontinuous Grain Growth - fired at 1300°C for 1200 minutes. - .52 - where K Q = a rate constant n = the time exponent w i t h a t h e o r e t i c a l value of u n i t y Q = the a c t i v a t i o n energy f o r g r a i n growth R = gas constant T = absolute temperature. The g r a i n s i z e data which are ta b u l a t e d i n Appendix-II were examined by p l o t t i n g the l o g of the diameter of the g r a i n s versus l o g time i n Figure lk'. From the measurements of the slopes o f these l i n e s , i t i s apparent 1 t h a t the slopes do not equal.the t h e o r e t i c a l value of l / 2 . There- f o r e , a p l o t of l o g (D - D Q ) versus l o g t was done i n Figure 15, the slopes of which i n d i c a t e an average value of 0.6 * 0.1 f o r n. - In order t o derive the value of the 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 of D - D Q 2 versus t°' 6 f o r each temperature was made, (Figure 16), the slope of which was the Value of 1 the r a t e constant, K , which v a r i e s w i t h the absolute 'temperature (T) as, K = K Q exp (- J^J-) . - The a c t i v a t i o n energy Q was then c a l c u l a t e d from the slope of the Arrhenius p l o t (Figure 17), where l o g K was p l o t t e d against - The derived value f o r the a c t i v a t i o n energy was T' 78 kcal/mole. S i n t e r i n g Study Experiments were c a r r i e d out t o study the d e n s i f i c a t i o n of r u t i l e powder compacts i n . t h e temperature range of 1000.to 1300°C i n a i r and i n a reducing atmosphere. The reducing atmosphere; /was used i n the s i n t e r i n g ' s t u d y t o o btain the 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 of TiOi.92 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 data are given 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 - . 3 5 . - i i I I 1 1 1 1—I—i—i i i | — : ' r Time(minutes) Figure 15. Grain Growth i n T i O ! . 9 2 Compacts with Temperature - 3 5 - - 3 6 - CO >- 10^ ( ° K ) T Figure 17. Logarithm of Grain 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 Figure 13a-d,. revealed t h a t the pores were i n the form of channels, which p a r t i a l l y f i l l e d up i n the l a s t stage. The pores.became almost s p h e r i c a l i n shape and were present 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 the case with-the model proposed by Coble^. The i s o t h e r m a l d e n s i f i c a t i o n (or shrinkage) data were converted t o the r e l a t i v e d e n s i t y of the compact which .-yjas; p l o t t e d against log. time. 5 The f o l l o w i n g equation d e r i v e d by Coble dP = ND V o'.a Q 3 , .....(6) dt y i e l d s on i n t e g r a t i o n -j_3 £ T - P •0 3 ND V^ a Q c I n t P 0 .A;E T J t Q t f .(lk) where P Q i s the p o r o s i t y at time t and zero p o r o s i t y a t time tf 3 and assuming 1 = A t , where 1 i s the g r a i n diameter a t time t . .The r e s u l t s are shown i n Figure l 8 a , b, c, d, e, f o r the f i n a l composition of Ti 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 the d e n s i f i c a t i o n of TiQi.gs and T i 0 2 was c a r r i e d out, but no d e t a i l e d study was made on t h e i r g r a i n growth. - The r e s u l t s are Shown i n Figure A . I I I - l a , b, c, d, e, f o r TiOiige.and i n Figure A-.III -2 f o r T i 0 2 . The data are t a b u l a t e d i n Appendix-'III, Table 2. The. v a r i a t i o n of the composition of 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 the weight l o s s measurements was a l s o i n c l u d e d i n Figure 18 and Figure A - . I I I - l f o r the. f i n a l compositions of TiGi-.g 2 and TiOi.gs r e s p e c t i v e l y . From a l l these f i g u r e s , i t . i s q u i t e apparent t h a t the d e n s i f i c a t i o n behaviour was completely changed a f t e r a c e r t a i n time of heating. . The d e n s i f i c a t i o n was stopped i n most cases, reaching an end-point d e n s i t y , 10 .100 1000 Time (minutes) . Figure l 8 a . D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition of TiOi.gg at T = 1000°C. Time (minutes) Figure l 8 b . D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition of TiOi.gs at T =1050°C. 80 70 -p • H CQ a CD Q t> • H - P cd H 0) « 60 -•39 ..96 -P-.92 50 I I I I I I 1 L J I 1 I I I I I I I I I I I I I I -P-.90 0 100c, 1000 Time (minutes) Figure ,l8c. D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition of TiOx ..92 at T = 1100°C. 100. Time (minutes) Figure l8d. 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 TiOi.gs at T = 1150°C. -.40 - 100 Time (minutes) Figure l 8 e . D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition •of T i O i . 9 2 at T = lB0'p°C - kl - approximately at the same time ./as:.- the equ i l i b r i u m composition was reached fo r non-stoichiometric r u t i l e . The end-point density monotonically increased with.the increase of. temperature of s i n t e r i n g . The break i n the isothermal d e n s i f i c a t i o n curves occurred a f t e r a heating time of 100 to 200.minutes f o r i non-stoichiometric r u t i l e and between 10 to 15 minutes f o r TI0 2 . In.order to calculate the d i f f u s i o n c o e f f i c i e n t s D v i n equation.(lk) i t i s only necessary to know the value of A experimentally as the value of the other constants are known. While i n v e s t i g a t i n g the isothermal grain growth ; ,of •: .an:-...: equilibrium composition of T i 0 i . g 2 , i t was found, that the following expression s a t i s f i e d the experimental data. D 2 - D Q 2 = K t 0 ' 6 (15) This equation i s almost equivalent to l 3 = D 3 = A t .....(16) neglecting D 0 3, as the i n i t i a l g r a in size D Q was very small, and A i n 3/2 equation (16) i s equal'to K •Therefore, the p l o t of D 3 versus :time was made f o r a l l temperatures, as shown i n Figure A.III-3 (In Appendix I I I ) , the slopes of which give the values of A. The r e s u l t s are recorded i n Appendix I I I , Table,3. The slopes, of the p l o t of the r e l a t i v e density versus l og time at any. temperature T aire equal to . .,2.3.NDy % ;a Q 3 . ; . A K T Using N = 10, # = 1000 ergs/cm 2, a Q 3 = 1.57- X 10" 2 3cm 3, % = I . 3 8 X 10" 1 6 ergs/deg. and the value of A (from Figure A.III-3) at d i f f e r e n t temperatures, -k2 - the corresponding 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 are given i n Table k, Appendix I I I . An Arrhenius p l o t (Figure 19) of l o g Dy. versus (^) produced an a c t i v a t i o n energy of 118 kcal/mole f o r d i f f u s i o n . -5 - 43- - -7 -P a <D •H O •H CH =H 0) O O a o .-H CQ <tH <H •H Q -11 -13 o O x \ Boundary D i f f u s i o n Model \ x o \ \ \ \ \ o \ \ S i n t e r i n g Neck Growth Bulk D i f f u s i o n Model -15 J I l_ I L J ! I I i ' 6.6 6.8 7.0 7-2 7-* 7.6 7-8 8.0 10 ( ° K ) T . Figure 19. D i f f u s i o n " C o e f f i c i e n t Versus the R e c i p r o c a l of the Absolute 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 are compared w i t h values c a l c u l a t e d from s i n t e r i n g experiments and models. - kk - DISCUSSION Defect S t r u c t u r e s of R u t i l e The r e d u c t i o n equation i n v o l v i n g the c r e a t i o n of the oxygen i o n vacancy and i t s two trapped e l e c t r o n s may he d e r i v e d by usin g a model s i m i l a r t o that a l r e a d y advanced by Greener,.Whitmore and-.Fi'heo f o r 11 Nb 2 0 5 and Whitmore and Kawai f o r T i 0 2 - . This equation i s O f ^ ' 1 0 2 ( g ) + V Q 2 - +2 0 . ..:..(17) 2 - where 0-̂  i s an oxygen i o n i n a normal l a t t i c e p o s i t i o n and VQ 2 - i s an oxygen i o n vacancy i n which the two e l e c t r o n s were trapped i n the form, of two T i + + + ; i o n s . . The law of mass a c t i o n was a p p l i e d t o equation (17) w i t h the resu l t ' l/2 2 K = P 0 2 [vQ2-]-to] .....(18) as the co n c e n t r a t i o n of trapped e l e c t r o n s i s twice the conc e n t r a t i o n of oxygen vacancies ' [0] = 2 [ V Q 2 - ] The equation (l8) becomes i n terms of the oxygen vacancy con c e n t r a t i o n K 1/3 -1/6 : [ V " ] = (I) p o 2 .....(19) This equation 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 the conc e n t r a t i o n of vacancies i s p r o p o r t i o n a l t o the ^l/6 power of the oxygen p a r t i a l pressure. • Equation (19 ) can be r e w r i t t e n as -1/6 A H f f + AS°f . [ V Q 2 - ] . = 0.63 P 0 2 exp- [ - 3 % " R T ] (20) where AH°f and A S°f are the enthalpy and entropy of formation f o r r e a c t i o n (17). A f t e r removing the temperature independent term from the - ^ - experimental function, equation (20) becomes [ V o 2 - ] = C P " V 6 (AZ1) .....(21) ^ S ° f N where C stands f o r (0.63 exp—pg—). Weight Loss Study Buessem and B u t l e r derived a s i m i l a r expression to i n t e r p r e t the t o t a l , weight loss of r u t i l e on e q u i l i b r a t i o n with various oxygen .atmospheres, Their equation,was. 2 T i 4 + , + 0 2~ ^ VQ2- + 1 0 2 + 2 T i 3 + (22) of which the equation constant i s K ( T ) = ° '. 6 X P H A V + 2 H T i a + + 1 H 0° 2 k~ T where H^., % i j _ 3 + a n ( i HQ are the enthalpy of the anion vacancy ' 3 + of T i and of oxygen gas i n the standard state. From the experimental data, Buessem and Butler obtained an enthalpy value of 83• * 10 kcal/mole. - By.using a value of -53-75 kcal/mole (for the change of standard state) they obtained f o r H A Y + 2H T i 3 + .136.65 * 10 kcal/mole. 3 ° Interpreting Cronemeyer s data on conductivity of r u t i l e i n t h i s manner revealed that H A Y. + 2 T I 3 + •= I27.O5 kcal/mole. • A comparison of equation (21) and these data shows that ART = 127.05 (or 136.65 kcal/mole) or ART = 42.35 kcal/mole f o r the defect r e a c t i on represented 3 by equation ( 2 1 ) . -\e - In the present case, no attempt was made t o determine the d e t a i l e d mechanism o f r e d u c t i o n of T i 0 2 since t h i s d i d not f a l l w i t h i n the scope of t h i s i n v e s t i g a t i o n . However, the a c t i v a t i o n energy of 82 ± 2 kcal/mole obtained i s i d e n t i c a l t o the enthalpy of r e a c t i o n (22) 29 obtained by Buessen and B u t l e r . This would suggest t h a t the r a t e - determining step may be the formation of an oxygen-ion vacancy. Grain Growth The p r e d i c t e d increase of g r a i n s i z e i n a normal 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 the square root of time. However, i m p u r i t i e s i n the l a t t i c e or nonuniformity i n the 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 the o r i g i n a l m a t e r i a l can cause the g r a i n growth t o be a f u n c t i o n of a sm a l l e r power of time. This can be expressed as n D 2 - D Q 2 = K t 0 < n 4 1 . . . . . ( 1 5 ) w i t h n = 1 i n the t h e o r e t i c a l case. Another e x p l a n a t i o n f o r lower 27 than t h e o r e t i c a l values f o r n has been advanced by Burke . I n many g r a i n growth i n v e s t i g a t i o n s , the o r i g i n a l m a t e r i a l contains 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 of f i n e g r a i n s . Therefore the average r a d i u s of curvature of the g r a i n boundary w i l l i ncrease more r a p i d l y than p r e d i c t e d by the t h e o r e t i c a l equation where n has the value of 1. I n t h i s study most of the measurements p l o t t e d i n Figures 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 interconnected at the g r a i n boundaries. The second stage of pxure shrinkage, i . e . the i s o l a t e d pore phase, was observed only a t the higher temperatures and a f t e r long periods of heating. This i s i l l u s t r a t e d by Figure 13d, of a specimen heated at 1300°C f o r 1200 minutes. - >7 But this.-change i n : p o r e shape does not influence the r e l a t i o n (15) which is.the same as long as.exaggerated grain growth does.not occur. Therefore p a r t i c u l a r a t t e n t i o n was given at the higher-temperatures, used, 1250 and 1300PC to avoid the measurements, influenced by exaggerated grain growth. The r e s u l t s of the present i n v e s t i g a t i o n f o r TiOi.92 ± Q can -.represented by the following expression D „ , o - 6 . 78,000x K Q t .exp (- fo ) The time exponent n h a s the value of 0.6 instead of the t h e o r e t i c a l value n = 1. For the non-stoichiometric composition of TiOi.ga ± o»oi> n = 0.6 while f o r T i 0 2 n = 1,/ (Appendix'III,•Table 5) . k The v a r i a t i o n of the value, of the exponent n with the change of stoichiometry of the compound has al s o been .evaluated for. U0231>32# The value, of n changed from n = 1.2 t o n = 0.8 f o r U0 2 to U02+x. . Table II compares, the value of n f o r d i f f e r e n t ceramic oxides as determined from the grain growth data. •: Values Table I I . of Exponent n f o r D i f f e r e n t Ceramic Oxides Oxide . n Reference U0 2 + x O.8/O.9 31 U02 1.2 32 CaO .1.0 33 MgO .1.0 •3.3 A1 20 3 .0.66 (O.62-0.74) 12 C A O - 1 5 %r-0'B5 O1.85 0.8 34 Present Work T i 0 2 0.60 - - 48 - Temperature Dependence of Grain Growth Before discussing the s i g n i f i c a n c e of the present value of the a c t i v a t i o n energy f o r the grain growth of T i 0 2 - t X i t i s informative to consider the a c t i v a t i o n energies f o r the grain growth process i n other ceramic oxides. For the purpose of comparison, the energies f o r the s e l f d i f f u s i o n of the components in- such systems are also included 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 Oxides Oxides - A c t i v a t i o n Energy f o r grain growth A c t i v a t i o n Energy f o r s e l f d i f f u s i o n of the cation A c t i v a t i o n Energy f o r s e l f d i f f u s i o n of the anion . A 1 2 0 3 153 (Ref.) (12) (Ref.) : .114 . (35) (Ref.) 152 (36) MgO 60' •(53) 79 (37) 62.4 (38) Ca Q» 15 Zr 0 U0 2 85 C i - 8 5 80 87 '(3^) (32) 109 f o r C a 2 + Z r 4 + (39) 88 (41) 29.8 (40) 60 (41) . T i 0 2 78 - - 74 (43) As i s evident, the a c t i v a t i o n energy f o r grain growth of oxides corresponds to the energy required f o r the d i f f u s i o n a l process of e i t h e r one of the two components. On t h i s basis, i t i s always considered that the grain growth of oxides 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 the present 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 obtained f o r the grain growth study, i s 78 kcal/mole f o r T i 0 i . 9 2 ± o-oi° The temperature behaviour of 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 using Linde r u t i l e single c r y s t a l s and 0 1 8 i s o t o p i c exchange technique. Their d i f f u s i o n data can be represented as follows - 4 9 - D N 2 - = 1.6 exp £H? 0 v RT ; sec between the temperature range 85O to 1300°C An expression for-the i n t r i n s i c 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 f o r non-stoichiometric. r u t i l e , - D Q 2 " has been derived i n Appendix IV, the f i n a l form, of which i s AH°f U P 02- = D Q exp - (5-Rf~.+ W ) ' where D Q i s the usual frequency f a c t o r , 4H°f i s the energy to form an oxygen vacancy, and ;U i s the a c t i v a t i o n energy f o r oxygen vacancy migration., By comparing the experimental value with the thermodynamic c a l c u l a t i o n A;H°f + ; U = 74- kcal/mole 3 and by taking AH°f = I27.O5 kcal/mole from Cronemeyer's data, the energy f o r vacancy migration U i s about 74 - 127 = 31.6 kcal/mole 3 Sim i l a r c a l c u l a t i o n s with the present grain growth data produced a value of 35 kcal/mole f or the a c t i v a t i o n energy, f o r migration of the oxygen vacancy. These r e s u l t s are i n good agreement and i t appears reasonable enough t o suggest that oxygen ion d i f f u s i o n i s the l i k e l y r a t e - c o n t r o l l i n g step i n the process- qf grain growth of non-stoichiometric t i t a n i a . - 5Q- 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 as shown i n Figure 13a-d, r e v e a l the d i f f e r e n t stages, of d e n s i f i c a t i o n . They are almost s i m i l a r t o those d e p i c t e d by Coble (Figure 3). -The intermediate stage i s shown i n Figures 13a and b, which are photomicrographs of specimens, heated at 1150°C f o r 190 minutes and 280 minutes r e s p e c t i v e l y . In Figure 13b, the pores are at the j u n c t i o n of three — g r a i n corners, i n d i c a t i n g the presence of continuous pores. The f i n a l stage begins when the pore phase i s e v e n t u a l l y pinched o f f . .The presence of some pores at the f o u r - g r a i n corners i n Figure 13c, i n d i c a t e s t h a t the specimen had reached the f i n a l stage of 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 of specimens s i n t e r e d a t higher temperatures revealed both trapped pores i n the g r a i n s and pores a t the g r a i n boundary corners as s^own i n Figure 13d. E x i s t e n c e of trapped pores i n the g r a i n s i n a l l specimens, which were f i r e d at or above 1250°C suggests t h a t some degree of discontinuous g r a i n growth occurred i n a l l cases. This i s a l s o evident from.the density-time curves (Figure l8a-e), where i n every case an end-point d e n s i t y was observed. Exaggerated g r a i n growth was c a r e f u l l y avoided i n t h i s i n v e s t i g a t i o n by c a r r y i n g out a l l experiments below 1300°C (siSauerwaldtemperature i n . t h e case of T 1 0 2 ) . A comparison of Figures 13b and 13d shows/that the u n i f o r m i t y i n g r a i n and pore s i z e which was present a t the beginning of the s i n t e r i n g process l a t e r disappeared. At the l a t e r stage s e v e r a l grains grew l a r g e r at the expense of smaller ones. On f u r t h e r h e a t i n g , shrinkage of the pores i n the g r a i n boundaries was observed, which r e s u l t e d i n some increase i n r e l a t i v e d e n s i t y of the compacts. Annealing twins were observed i n the specimens a f t e r 2000 minutes of s i n t e r i n g at or above 1250°C; no study was. made t o t e s t the e f f e c t of t h i s on the g r a i n growth or s i n t e r i n g . - 51 - Density-Time Curve Isothermal density-time curves i n Figure l8a-e, present the e f f e c t of temperature only,, as the e f f e c t s of atmospheres were not determined, although the oxygen p a r t i a l pressure i n the system was v a r i e d a t d i f f e r e n t temperatures. The compacting pressure was found t o have a s i g n i f i c a n t e f f e c t on the i n i t i a l r e l a t i v e d e n s i t y and on the d e n s i f i c a t i o n r a t e and so was.kept constant. 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/em3, p a r t i c u l a r l y f o r the 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 composition of TiO-i..g 2. The use of compacts: of such low hulk d e n s i t y e l i m i n a t e d t o some extent the presence of the d e n s i t y gradient produced by d i e w a l l f r i c t i o n , although t h i s could not be completely avoided. . Large shrinkage r a t e a n i s o t r o p y was not observed. The i n i t i a l p a r t of density-time curves never extended t o the f i n a l stage of s i n t e r i n g according t o Figure 15, so th a t the change of slope cannot be a t t r i b u t e d t o the change of pore shape. In a d d i t i o n , according 12 t o Coble , the change of pore phase c o n t i n u i t y should occur a t about 95$ of the t h e o r e t i c a l d e n s i t y of the compact, which, i n t h i s i n v e s t i g a t i o n , was never achieved i n any specimen. The change o f slope i n a l l cases occurred when the composition of the n o n - s t o i c h i o m e t r i c r u t i l e under the reducing atmosphere reached an e q u i l i b r i u m value. Thus, i t appears t h a t the change of r a t e i s not a f u n c t i o n of change of pore shape but i s r e l a t e d t o the formation of oxygen vacancies i n the system during the p r e - e q u i l i b r i u m stage. , 10 0 Bryan and Parravano measured the r a t e of neck growth between r u t i l e spheres i n an atmosphere of H 2/H 20 (= 10) and observed s i m i l a r breaks i n t h e i r isothermal, neck growth r a t e curves. They explained t h i s observation on the b a s i s of p o l y g o n i z a t i o n of mon o c r y s t a l l i n e r u t i l e spheres and assumed tha t the specimens reached e q u i l i b r i u m almost immediately during h e a t i n g . - .52 - The s i g n i f i c a n c e of the slopes of the d e n s i t y versus l o g time curves i s not known. The slope may be a f f e c t e d by the 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 by inhomogeneities 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 alignment. The problem,then, i s t o determine whether the 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. D i f f u s i o n C o e f f i c i e n t The l i n e a r r e g i o n of the density-time curve i n the p r e - e q u i l i b r i u m stage of d e n s i f i c a t i o n was used.to 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 . The observed increase i n d e n s i f i c a t i o n r a t e w i t h increasingttemperature i s t y p i c a l of t h e r m a l l y - a c t i v a t e d processes i n ceramics. - Each curve has roughly the same charact e r . - The slopes, vary, but cannot be i n t e r p r e t e d q u a n t i t a t i v e l y , and th e r e f o r e the change i n slope from one l i n e t o another 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 the r e s u l t s are shown i n Figure . 1 9 where they were p l o t t e d as l o g D against l / T . Other data p l o t t e d i n t h i s f i g u r e are Haul and J u s t ' s 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 (determined by i s o t o p i c exchange technique) 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 of neck growth between two spheres. The 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 ±i^£rm©'dd/a4e%ta£e'mea~sure- .'..meats: 'are. lOwer than;_tnOse from neck - growth/'.:- measurements by an order of magnitude but agrees w i t h the 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 . Only order-of-magnitude r e l i a b i l i t y maybe attached: t o the i n d i v i d u a l models or r e s u l t s , and t h e r e f o r e the discrepancy does not disprove the model a p p l i e d i n . t h i s case. However, the c a l c u l a t e d a c t i v a t i o n energy of 118 kcal/mole i n the present case i s not i n agreement w i t h other r e s u l t s . This i s r e f l e c t e d -53 - i n the d i f f e r e n c e of slopes of the l i n e s as shown i n Figure 19. This discrepancy may be a t t r i b u t e d t o some of the f a c t o r s , i n v o l v e d i n the c a l c u l a t i o n of 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 the measurement of g r a i n s i z e would a f f e c t the f a c t o r A (equation (16)). S i m i l a r l y , the e r r o r i n . t h e measurement of the shrinkage values.would a f f e c t the. slope of the density-time curve. The cumulative e f f e c t of these two on the 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 consi d e r a b l e . This i s p a r t i c u l a r l y t r u e f o r the specimens s i n t e r e d below 1100°C. The changQfin the g r a i n s i z e and i n the dimensions of the specimens w i t h time below 1100°C were ve r y s m a l l . -Any e r r o r i n the measurement of these two parameters would have a l a r g e e f f e c t on the value of d i f f u s i o n c o e f f i c i e n t s at lower temperatures. The e f f e c t of the i n i t i a l shape of the g r a i n on the shrinkage r a t e i s not known. .Coolers o r i g i n a l model was based completely on.the s p h e r i c a l g r a i n s . I n the present case, the i n i t i a l g r a i n s are l i k e f l a k e s . . These are expected t o change i n t o a minimum surface area c o n f i g u r a t i o n i n the e a r l i e s t stage of s i n t e r i n g , as these f l a k e s have l a r g e surface energy a s s o c i a t e d w i t h them. . The steps i n v o l v e d i n t h i s change of c o n f i g u r a t i o n are not known and^therefore, i t s e f f e c t on the o v e r a l l shrinkage r a t e cannot be evaluated. • The e f f e c t of oxygen p a r t i a l pressures on the d i f f u s i o n c o e f f i c i e n t s was evaluated by the f o l l o w i n g procedure. The isothermal g r a i n growth r a t e s of the e q u i l i b r i u m compositions'of TiOx.ge and T i 0 2 at 1200°C were determined. The data were p l o t t e d as D (where D = average g r a i n diameter) versus time. The slope of the l i n e s produced the values of A. f o r these two compositions. - ^ - Using the shrinkage data.at 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 the e q u i l i b r i u m compositions of T i O i . g 8 and T i 0 2 were c a l c u l a t e d . The r e s u l t s are t a b u l a t e d i n Appendix I I I i n Tables . 5 and 6. In the f o l l o w i n g t a b l e the 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 pressures are compared. I t can be seen t h a t the d i f f u s i o n c o e f f i c i e n t s were not s i g n i f i c a n t l y a f f e c t e d by the change of oxygen p a r t i a l pressure i n the system. . Table 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 at D i f f e r e n t Oxygen P a r t i a l Pressures Composition D i f f u s i o n C o e f f i c i e n t (D v) at 1200°C cm 2/sec Oxygen P a r t i a l Pressure i n Atmosphere 4.32 X I O - 1 1 1.41 X I O - 1 5 .TiOi.es 9M X 1 0 " 1 1 I X 1 0 " 1 3 . T i 0 2 1.2.£1 X, I O - 1 1 0.21 I n view of the disagreement between the value of the a c t i v a t i o n energy obtained i n t h i s i n v e s t i g a t i o n and th a t found i n other s i n t e r i n g s t u d i e s , attempts have been made t o apply other a v a i l a b l e models f o r t.t-. d e n s i f i c a t i o n . Two other models are c u r r e n t l y a v a i l a b l e . C o b l e 5 i n a d d i t i o n t o h i s bulk d i f f u s i o n model, proposed a g r a i n boundary d i f f u s i o n model on.the understanding t h a t d e n s i f i c a t i o n might proceed by the mig r a t i o n of the g r a i n boundary. Coble d e r i v e d the f o l l o w i n g equation f o r the g r a i n boundary d i f f u s i o n R D K W ^ a 3 2 / 5 P = I 2 ^ 4 K\ ° t ] (23) where P = volume f r a c t i o n pores a t time t I>b = d i f f u s i o n c o e f f i c i e n t of atoms i n the g r a i n boundary W = g r a i n boundary width,and other f a c t o r s s i m i l a r t o .; -i [.:r t • equation (( .6). - 55 - The values, of 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 using equation (23)and the p o r o s i t y data.of the e q u i l i b r i u m composition of T i O i - 9 2 . The r e s u l t s and c a l c u l a t i o n s are given i n Appendix V. These are p l o t t e d i n Figure 19- t o compare w i t h the other r e s u l t s . According t o t h i s f i g u r e , the bulk d i f f u s i o n model gives a b e t t e r f i t f o r the present r e s u l t . 13 Johnson and C u t l e r very r e c e n t l y d erived an equation t o i n t e r p r e t t h e i r l i n e a r shrinkage data of alumina compacts. The equation has the f o l l o w i n g form 1 . K a 3 j)m m AL = ( K T gp ) t .(24) L ° . . where &L ' =-•• f r a c t i o n a l shrinkage Lo K = numerical constant yi< = surface 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 Q 3 = vacancy volume r = p a r t i c l e r a d i u s t = time .m>p • = constants In thist:qquation, the value of the time exponent m v a r i e s between O.25 t o O.5O according to the geometry of the contact 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 or 160° cone on a plane e t c . The f r a c t i o n a l l i n e a r shrinkage data, i n the present 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 against time i n a l o g - l o g s c a l e . The value of m i s found t o be l e s s than 0.1 i n . t h i s case. • The s i g n i f i c a n c e of such a low value 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 g e n e r a l i z e d equation. CONCLUSIONS - 5 6 - .• . • 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 compositions of T i O i ..92 and T i 0 i . g 8 u s i n g d i f f e r e n t H 2/H 20 atmospheres. The f r a c t i o n a l weight l o s s w i t h time was found t o f o l l o w a . p a r a b o l i c r e l a t i o n s h i p . An Arrhenius p l o t u s i n g r a t e s which were c o r r e c t e d f o r oxygen p a r t i a l pressure dependence produced an a c t i v a t i o n energy of 82 ± 2 kcal/mole. The weight l o s s measurements under e q u i l i b r i u m c o n d i t i o n s c a r r i e d out by o t h e r , i n v e s t i g a t o r s produced an enthalpy of 83 ± 10 kcal/mole. This.may suggest .that the r a t e determining step i s the oxygen i o n vacancy formation. Grain growth data f o r the e q u i l i b r i u m composition of T i 0 i . g 2 ± o-oi can be explained by the f o l l o w i n g expression D 2 - D Q 2 = K o t°-° e x P (:,-2 § ^ 0 ) The value of time exponent n was found, t o be d i f f e r e n t from the t h e o r e t i c a l value of u n i t y f o r 'TiOi - . 9 2 and Ti0i-.g 8. I t was equal t o one f o r the g r a i n growth of 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 d i f f u s i o n i n T i 0 2 determined by oxygen I s o t o p i c exchange technique was found; t o be.74 kcal/mole. 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 the l i k e l y r a t e - c o n t r o l l i n g step f o r the g r a i n growth of the T i 0 i . g 2 n o n - s t o i c h i o m e t r i c composition. D e n s i f i c a t i o n of the f i n a l e q u i l i b r i u m composition of T i 0 i . g 2 was explained u s i n g a s i n t e r i n g model proposed, by Coble. The values of the 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 of the r i g h t order of magnitude but the a c t i v a t i o n energy f o r the d i f f u s i o n process c a l - c u l a t e d from, these data d i d not agree w i t h t h a t of other workers. This discrepancy i s expla i n e d on the b a s i s of e r r o r s i n the measurements, i n - homogenelties i n the compact and other p o s s i b l e f a c t o r s which a f f e c t the d e n s i f i c a t i o n process. - .57 - RECOMMENDATIONS FOR FUTURE INVESTIGATIONS This study on g r a i n growth and s i n t e r i n g of r u t i l e compacts under reducing atmospheres was the f i r s t attempt t o apply a model of 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 of i n i t i a l compacting pressures on.grain growth and d e n s i f i c a t i o n should be c a r r i e d out u s i n g . r u t i l e powders of known and simple geometry. A l s o keeping.the p a r t i c l e : s h a p e as simple 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 the d e n s i f i c a t i o n r a t e should be i n v e s t i g a t e d . This data by c o r r e l a t i n g w i t h the g r a i n growth-measurements. should.help t o i d e n t i f y the f a c t o r s which c o n t r o l the value of the time exponent i n the g e n e r a l i z e d expression of the g r a i n growth. With the a v a i l a b i l i t y of a plasma, the powdered p a r t i c l e should-be. spheroidized. before other experiments are undertaken. This should-eliminate the heterogeneity i n contact angles and i n number of contacts per p a r t i c l e . • An observation of the number of contacts per p a r t i c l e during d e n s i f i c a t i o n , as w e l l as the measurement of surface area of the compacts would perhaps e x p l a i n the drop i n r a t e of d e n s i f i c a t i o n once the 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 composition has been reached i n r u t i l e . The i n f l u e n c e of g r a i n growth i n h i b i t i n g elements should be. i n v e s t i g a t e d i n order t o reach the t h e o r e t i c a l d e n s i t y - i n the compacts. To i s o l a t e the e f f e c t of red u c t i o n on the d e n s i f i c a t i o n r a t e , . the powdered p a r t i c l e s should.be.reduced, t o a non-stoichiometric e q u i l i b r i u m composition before the s i n t e r i n g study i s c a r r i e d out. - 5 8 - BIBLIOGRAPHY G. C .. Kuczynski, "Powder Metallurgy", p. 11, Ed., W.. Leszynski, Interscience Publishers Inc., New York and London, (I96I) W. J . Moore, "Physical Chemistry", p. 504, P r e n t i c e - H a l l Inc., N.J., (1955). • W. D.,Kingery and M. Berg, J . Appl. Phys.. 26, 1205 (1955)- G.C. Kuczynski, "Powder Metallurgy", p. 1-16, The Iron and St e e l I n s t i t u t e and the I n s t i t u t e of Metals, 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 of Powder Metallurgy", Edward Arnold Press, London ( i960) . C. Zener, see C- S. Smith, Trans.-..AiirM.%%?blT$}(19^8). G. C. Kuczynski, "Kinetics of High Temperature'Processes", Ipl; 37^ Ed., W. D. Kingery, Technology Press and John Wiley and Sons, (1958). . L. F. Norris and G. • Parravano, J . Am.-Ceram... Soc., 46, 449 (1963). H. M. O'Bryan, J r . , and G. Parravano, "Powder Metallurgy", p. 191, Ed., W. Leszynski)^ Interscience Publishers Inc.n.New York and London, (I96I). D. H. Whitmore and Toshihiko Kawai, J , Am. Coram I. Soc., 4jj, 375 (1963) . R.. L. Coble, J.-Appl. Phys.. 32, 793 (1961). D. . L. Johnson and I. B.. C u t l e r , - J . Am.-Ceram. Soc. 46, 545 (1963). P. W. Clark, J . H. Cannon and J.-White, Trans. B r i t . Ceram.- Soc. 52, 1 (1953). A. H.-Webster and N.F.- H.- Bright,-The E f f e c t s of Furnace Atmospheres on the S i n t e r i n g Behaviour of Uranium Oxide, Mines Branch Research Report R2, Department of Mines and Technical Surveys, Ottawa, February 5, I 9 5 8 . • R. W.. G.- Wyckoff, C r y s t a l Structure Handbook, Interscience Publishers, New. York (1948). , - M. E. Straumanis, T. Ejima and W. J . James,-Acta. Cryst. 14, 493 (I96I). E. - A.. Gulbransen and K.-F.- Andrew, J . Metals • Trans. A.I.M.E., 185, 7 1̂ (19^9). W i l l y Kinna and W i l l y Knorr, Z.-Metallkunde, 4j_, 594 (I956). •M.H. Davies and C. E. Birchenall,• J . Metals .3;; Trans. A.I.M.E., 191 877 (1951). B i b l i o g r a p h y Continued. - 5 9 - 21. . K a r l Hauffe, Reactionen i n und an Festeh Stoffen,- 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 cta Met. 8, 758 (i960).. bj P. Ko f s t a d , K. Hauffe and H. K j o l l e r s d a l , Acta. Chem.. Scand. 12,.239 (1958). , j 23. . Tentative Method f o r Determining the Average Grain S i z e of Metals, A.S.T-.M.. Designation E 112-55 T, A.S..T.M. Standards, Part 1, (1955) P. 1433- 24. W. H. McKewan,- Trans. A.I.M.E., 224,. 2 (1962). 25. P.-A. Beck, J . C. Kremer, L. J . Demer, and M. L. Holzworth,-Metals Technol. 14, (1947); Tech. Pub .Wo. 2280; • Trans. A. I-.M-.E. 17j? 372 (1948). 26. David T u r n b u l l , J . Metals \y Trans. A.I.M-.E., 191, 66l (1951). 27. -J.-E. Burke, Trans. A.I.M.E. 180, 73 (1949). 28. - E. H. Greener, D. H. tyhitmore and M. E. Fi n e , J . Chem. Phys.. 34, 1017 (1961). 29. - W. R. Buessem and S. - R. B u t l e r , " K i n e t i c s , of HigheiTemperature Processes", Ed., W. D.. Kingery,- Technology Press and John W i l e y and Sons, p. 13 New: York, (1959). 30. D. C. Cronemeyer, Phys. Rev.'87_, 876 (1952). 31. - J . • R. MacEwan and V. B. Lawson, J . Am. Ceram.- Soc., 4jj, 42 (I962). 32. I . Amato, R.- L. Colombo and A. M. P r o t t i , J . Am. Ceram. Soc. 46, 407 (1963). 33 • • A. V. - D a n i e l s , J r . , R. C .. Lowrie, J r . , RnoL. Gibby and Ivan B C u t l e r , J.-Am. Ceram. Soc. 4^, 282 (I962). 34. T. Y. Tien and E.. C. Subbarao, J . Am. Ceram. S o c , 46 , 489 (1963).. 35- A. E. Paladin© and W.- D.. Kingery, J... Chem. Phys. 32.,.. 957 (1962). 36. Y. O i s h i and W. D. Kingery, J . Chem..Phys. 33, 480(1960). 37- Roland-Linder and G. D. P a r f i t t , J . Chem. Phys. 26, 182 (1957). 38. Y. O i s h i and W. D..Kingery, 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 at the S i x t y - F o r t h Annual Meeting, The American Ceramic Society, : New York, A p r i l .30, 1962. - Symposium on ; K i n e t i c s of Ceramic Reactions No -1-25-63, Am. Ceram. Soc, B u l l . 41, 283 (1961). Bibliography! Continued. hO. . W.- D. Kingery, J . Pappis,' M. E. Doty and D.. C . H i l l , J.- Am. Ceram. S o c , 1+2, 393 (1959). hi. J . B e l l e / A.. B. Auskern, 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. Report 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 of 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 ± o-oi Temperature A_W x 1 0 0 Time t of Furnace W m ^ n °C. % t O-5 • AO-5 (mtrr? l o g A 0 ° 5 5 l o g P 0 2 - ° p 5 2 S f 3 l o g K ° ' 5 1000 0.153 15 3.87 6.116 -1.214 -17.55 -2.92 -4.134 (7-94) 0.260 20 4.52 X I O " 2 0.461 50 7.04.5 O.500 65 8.06 0.648 110 .10.53 1.211 210 14.51 I .250 430 20.50 1050' 0.173- 10 3.165 7.^677- ll -1.1275 -16.765 -2.794 -3.9215 (7-67) O.270 15 3.87 x io~ 2 0.430 35 5.90: 0i646 75 8.66 O.740 100 10.00 1:170 24o' 15.50 1.970 1065 32.55 1100 0.480 . 25 4 . 9 8 ' 12.641 -O.898 -16,16 -2.693 -3.591 (7-35) 0.775 50 7.04 X I O - 2 I.170 ,100 10.00 1.465 150 12.22 •1J730 300 14.50 1.745 1055 32.45 1150 0.678 15 3.87 15.536 -O.78I -15.412 -2.568 -3.349 (7.12) O.871 25 4.98 X I O - 2 0.990 •35 5.9O ,1.450 75 8.66 1.750 l4o • 11.82 1-775 300 17.30 • •1,900 .470 . .31.05 1.900 IO65 32.55 1200 O.613 10 31.65 22.327 -O.65I -14.85 -2.475 0 .126 (6.8$) O.890 25 4.98 X I O " 2 1.693 55 7.40 .1.740 70 9.23 1.750 130 11.42 1.800 250 15.85 1.972 780 28; 00 - 62 - Table 2. Weight Loss :Data. f o r TiGfe.gs.±' ; 0 .QI Temperature of Furnace ' °c. • i o o T i m e t. t Q . 5 .AO-5 • (mxrr? 5 l o g - A ° - 5 log P0 2 -u.5 log P02- 1/ 6 'log K ° ' 5 1000 .0.132 20 4.47 3.7428 -1.4265 -16.376 -2.729 -4.2194 (7.94) 0.276 53 ! 7.28 x i o ~ 2 0.376 113 10.62 0.540 233 15.28 0.542 473 - 21.76 ! 0.542 14^3 38.00 . 1050 • 0.203 15 3.87 5.155 -1.288 -15.60 -2.60 -3.888 (7-67) 0.265 27.5 5.23 X 10 " 2 0.415 55 7.42 0.486 87 9.33 0.510 125 11.18 O.510 215 14.65 O.510 . 810 28.45 1100 0.186 10 •3.165 5.667 -1.2474 -14.236 -2.373 -3.6204 (7.35) 0.225 15 3.87 X IO" 2 0.340 .35 5.90 O.510 55 7.42 O.545 105 10.26 0-573 490 22.18 0.573 1075 j 32.80 1150 O.310 15 3.87 7.3358 -1.135 -13.59 -2.265 -3.400 (7-12) 0.487 45 : 6.72 X IO" 2 0.555 105 10.26 O.590 275 16.58 0.645 2555 ' 50.65 0.645 4440 66.70 1200 0.140 24 4.13 5.170 -0.287 -13too -2.170 -2.457 (6.86) 0.255 30 5.47 X 1 0 _ 1 O.380 38 6.16 0.534 6o; 7.75 > 0.534 125 11.18 O.598 245 15.65 Table 3. Oxygen P a r t i a l Pressures Temperature of Furnace °C. Temperature of the Bubbler °C.. p H 20. mm of Mg P H 2 0 1 O S  p o 2 . : A. For-.. TiOx ' 9 2 * O - O l 1000 '• 23 21.068 35-2 -17.53 1050 . 23 21.068 35-2 -16.765 1100 25 23-756 31.0 -16.16 1150 25 23.756 31.0 -15.412 1200 25 23.756 31.0 -14.85 B. For T i O i •98. ± 0 - . p i 1000 45 71.88c 9.60 -16.376 1050 45 71.88 9.60 -15.6 1100 53 107.20 6.06 -14.236 1150 53 107.20 6.06 -13.59 1200 53 107.20 6.06 -13.00 - 64 - Table k. Determination of the Power Facto r of ,P( Temper- ature °C | lo g A°-5 vTiOi.gs TiOi.92 l o g P 0 - TiOi.gs 2 TiOi-,92 - Slopes 1 X 1000 -1.4265 -1.2140 -16.376 -17.53 1 -0.2125 =.o.i84 1.159 - 1. 1050 -1.2880 -1.1275 -15.600 -I6.765 -0.16 =-0.137 I.I65 - 1 3.65 1100 -1.2474 -O.898 -14.236 -16.160 -O.35 =_o.l82 1.93 - 1 2.75 1150 -1.135 -O.78I -13.590 -15.412 -0.354 =-0.1825 1.82 _ 1 2.7-5 1200 1 -O.287 -O.65I -13.000 •1 -14.850 1 1 Average slope i s assumed t o be - 1 3 ± O.65 APPENDIX I I . •Table 1, Grain Growth Study - 65 - Tempera- t u r e . °C. Grain S i z e D Time t min . D 2 > * D 2 - v t 0 - 6 •: '2 l o g K (cm/min 0" 6 1100 ^ — 0.73 90 0.5329 0 14.83 (2.15; * 10" 1 0) (7.35) 0.93 380 0.8649 0.3320 35-4 1A0 1055 1.9600 1.4271 65.3 -9.667 1.66 2555 2.7550 2.2221 110.8 2.08 5315 4,3260 3.7031 171.5 1150 1,365 120 •1.8630 1.3301 -17.7 (5.8 X 10 _ 1°) (7..12) 1.50 290 .2.2500 1.7171 30.2 1.91 705. 3.6481 3.1152 51.3 -9.237 2.50 ' 1980 6.2500 5.7171 95.1 2.96 3305 8.7616 8.2287 .127.2 3.34 • 66O5 ll.1806 10.5471 197.0 1200 1.90 110 3.6100 3.0771 16.75 (23.2 X 10" 1 0) (6.78) 2.57 280 : 6.6049 6.0720 29.5 3.52 1265 12.3904 11.8575 73.0 -8.645 5.50 2820 30.2500 29^7171 117.3 5.93 4183 35.3000 34.9771 • 149.3 1250 2.92 100 8.5329 8.0000 15.85 (60 X I O - 1 0 ) (6.50) 3.4l 240 11.6280 11.0951 26.8 5.05 500 25.5329 25.0000 41.7 -8.222 7.02 1270 49.2800 48.7471 73-3 8.60 2620 73.9600 73.4271 112.2 1300 7.30 • 90 53.2900 52.7571 14.83 (250 X I O " 1 0 ) (6.35) 7.80 -I5O 60.8400 60.3071 20.15 12.23 210 149.5300 149.0400 24.80 -7.600 8.98 . 250 80.5329 80.0000 27.10 9.80 415 96.6400 95.5071 : 37.15 10.30 625 106.0900 105.5570 47.60 APPENDIX I I I . .Table 1. Temp. Time ° ' mm. w gm AW v w x 100 0 2 gm T i O x Shrinkage V o l . $ V o l . a t Time t cm 3 Weight a t Time t gm Green Bulk D e n s i t y D e n s i t y of Non-stoich. T i 0 2 Theo. . P Po 1000 0 0.6911 ••*b,. ~ 0:2771 2 0 4.26 8.3483 1.963 4.25 0.462 • 15 O.6905 0:153 O.27615 1.994 6.5 3.98 8.34 2.10 4,244 O.495 20 0.6893 0.26 0.2753 1.986 9 • 3.88 8.32 2,15 4.238 0.506 4 5 0.6879 0.461 1L986 12 . 3-75 8.32 2.22 4.238 0.524 50 0.2739 1.972 16.6 3-55 8.29 •2,34 4.227 O.530 65 O.6876 O.50 O.276O 1.97 9 • 3.88 2.14 4.225 O.540 .110 0.6866 0.648 O.2726 1.964 12 3.75 8.27 2.222 4.22 O.530 240 0.68275 1.211 0.26875 1.94 17.2 3-53 8.24 2.34 4.20 0.557 430 0.6770 .2.05 0.2630 1.905 16,8 3.54 - 8.18 2.34 4.174 0.559 1050 0 0.7500 0 0.3000 2 0 4.26 8.3483 1.97 4.25 0.464 10 0.7487 O.173 O.2987 1.99 ,: 21 3-36 8.33 2.48 4.24 O.587 15 0.7480 O.27 O.2980 I.986 " '27.5 3.08 8.315 2.70 4.238 O.636 35 0.7468 O.43 O.2968 I.98 27 .;8 3.07 8.31 2.70 4.232 0.632 75 0.7451 0.646 O.295I I.968 28.6 3.02 8.28 2.75 4.223 O.650 100 0.7445 0.74 0.29445 1.964 "• 29.3 3.005 8.280 2.758 4.22 0.6§4 240 0.7367 1.77 0.2867 1.92 29.6 2.97 8.225 2.75- 4.185 O.656-' 1065 0.7360 1.97 0.2880 1.918 3° 2.955 8.19 2.78 • 4.183 0.664 1100 0 0.6909 . 0 O..2769 2 0 4.28 8.3824 1.96 4.25 0.462 10 • 0 O.2769 2 ' 27.8 3.09 8.3824. 2.71 4.25 O.638 .- . 10 0 O.2769 2 28.9 3.04 8.3824 2.76 4.25 0.648 30 0.6876 0.48 0.2736 1.972 32.8 2.88 8.34 2.90 4.226 O.682 50 0.6856 0.775 O.27I65 1.96 35.3 2.77 8.32 3.00 4.217 0.710 80 0.68399 1.00 0.2699 1-95 35-6 2.76 8.305 . 3.03 4.209 O.722 100 0.6828 1.17 0.2688 1.94 37.4 2.69 8.28 3,085 4,201 0.734 150 0.6808 1.465 0.2668 1.926 • 37.8 2.67 8.27 3:105 4.19 0.740 i 300 0.6790 1.73 O.2656 1.918 38.3 2.65 8.26 3-12 ' 4.183 0.745 1055 0.67885 1.745 0.26485 1.916 38.5 2.695 8.255 , 3.125 4.182 0.745 continued, Table 1 .Continued. Temp °C. ..Time min. W gm Aw v w X 100 0 2 gm .TiO x Shrinkage Vol.. ? V o l . at .Time t cm 3 Weight a t Time t gm Green Density of p Bulk N o n - s t o i c h i o . :"vp— Density T i 0 2 Theo. 0 1150 0 0.6663 .0 0.2664 2 0 . .. ' 4 . 2 8 8.3824 -1.94- : 4.25 5 0.6663 0 0.2664" 2 , 35-2 2.78 8.3824 3-02 4.25 . 0.71 10 0.6663 0 0.2664 " 2; ... 36.5 2.725 8.3'824 3-08 " 4,25 0.725 15 O.66I78 0.678 0.26188 1.965 36.5 2.73 8.33 3 .06 4.221 0.725 25 0.66048 0.871 0.26058 •1.956 38 .3 2.65 8 .31 3.14 4.214 0.745 35 0.6597 0.99 O.2598 1.944 38 .3 2.65 ' 8.285 3.13 4.204 O.745 775 0.65664 1.45 0.25674 1.928 4o.6 2-55 8.27 3.25 4.192 0.775 l 4 o 0.65465 1-75 0.25475 1.912 42.5 2.46. 8.23 3.345 4.179 0.80 " 300 0.65445 1.775 0.25455 I . 9 I O 43.7 . •2.415 • 8 . 2 2 . 3.405 4.176 0.816 470 0.65365 1.90 0.25375 1.905 44.0 2.40 8.22 3.425 4.174 0.820 IO65 0.6565 1.90 0.25375 1.905 44 .3 2,385 . 8.22 3.445 4.174 0.825 1200 0 0.8967 0 • 0.3587 2 4.26 8.3483 1.965 4.25 5 0.8967 0 0.3587 2 37-6 2.66 8.3483 3.145 4.25 0.74 10 0.8912 o.613" 0.3532 • 1.968 38.6 2 .61 8.28 3.18 4.224 0.75 20 1.968 41.8 ' 2.47 8.27 3.345 4,217 0.795 25 0.88875 0.89 0.35073 1.952 8.255 4.211 .30 43.0 2.43 8.255 3 .4o 4.211 0.81 55 0.8815 1.693 0.3435 1.916 44.2 2.375 8.21 3.46 4.186 0.825 70 0.8811 1.74 0.3431 •1-913 45.8 2,30 8.20 - 3.565 4.18 0.845 85 46.0 2..30 8.20 .3-57 4.18 0.852 120 0.8810 1.75 O.343O 1.913 47.4 2.24 8.20 3.665 4.18 . 0.875 250 0,8806 1.80 0.3426 1.910 48.2 2.21 8.18 3.71 4.178 O.89 . 780 0.8790 1.972 0.3410 1.910 48.6 2.185 8.18 3-75 4.178 0.90 I ON -̂ 1 1 APPENDIX III. i Table.2 . Temp °C . Time . min w gm AW .. Y x 100 0 2 gm T i O x Shrinka^ V o l . $ ie V o l . at Time t cm 3 Weight a t Time t gm Green Bulk Density Density of N o n - s t o i c h i o T i 0 2 Theo. P '• p o 1000 0 O.83I5 0 0.3335 2 0 4.06 8.2556 2.04 4.250 0.480 20 0.8394 0.132 0.3324 1.994 11]. 3.61 8,2556 2.29 4.250 0.538 53 0.8292 0.276 0.3312 1.992 17 3.365 8.245 2.452 4.244 0.578 113- 0.8284 0.376 0.3304 1.984 22.5 3.155 8,24 2.61 .4.243 0.616 233 0.8270 0.542 O.329O 1.974 26 2.995 8.22 2.745 4.236 0.647 0.8270 0.542 0.3290 1.974 24.5 3.055 8.21 2.61 4.228- 0.637 1433 0.8270 0.542 0.3290 1,97^ 26 2.995 8.22 2.745 0.647 1050 0 0.8444 0 0.3379 2 0 5.44 10.6531 1.96 4.250 0.461 15 0.84269 0.203 0.33619 1.992 22 4.24 10.62 2.50 4.24o 0.588 27.5 0.84215 0.265 0.33565 1.986 27 3.98 10.62 2.67 4.238 0.630 55 0.8409 0.415 0.3344 1.984 34 3.60 10.62 2.945 4.236 0.695 87 0.8403 0.486 0.3338 I . 9 8 37-5 3,36 10.61 •3.160 4.234 0.745 125 0.8401 0.510 0.3336 1.978 38.5 3-333' 10.60 3.185 4.23I 0.752 215 0.8401 O.510 0,3336 1.978 39 3.32 10.60 .3.19 4.231 0.753 810 0.8401 . 0.510 0,3336 1.978 41 .3.21 10.60 3.19 4.231 0.743 1100 0 0.8624 0 0.3456 2 0 5.15 9.7650 I . 8 9 4.250 O.445 10 0.8608 0.186 0,3438 1.99 33 3.80 9.7645 2.57 4,248 0.606 15 0.8604 0.225 0.3436 1.985 32 •3.58 9.7645 2.74 4.248 0.646 35 O.8595 0.340 0.3427 1.98 36 3.30 9.7645 2.96 4.248 O.696 55 O.8580 0.510 0.3410 .1.976 4o 3.09 9.726 3.15 4.230 0.743 105 0.8577 0.545 0.3407 1.972 44 • 2.88 9-7 3.37 4.227 0,795 490 0.8566 • P.573 0.3396 1.970 45 2.82 9-7 3.44 4.225 0.815 1075 0.8566 0-573 0.3396 1.970 45.5 2.80 9-7 3.46 4.225 0.818 2336 0.8566 0.573 0.3396 1.970 45.5 2.80 9-7 3.46 ,4.225 0.818 3650 O.8566 0.573 0.3396 1.970 46 2.78 9-7 3.49 4.225 0.827 5121 0.8566 0.573 0.3396 1.970 46 2.78 9,7 3.49 4.225 0.827 9910 0.8566 0.573 0.3396 1.970 46 2.78 9-7 3.49 4.225 0.827 continued. Table 2. Continued. Temp .. Time - W 0 2 T i O x •Shrinkag ;e V o l . at Weight a t Green Density of P °C. min. gm w gm V o l . fo Time t Time t Bulk Non-stoich, p o 100 cm 3 gm Den s i t y Ti0 2 Theo. 1150 0 0.8832 0 0,3532 2 0 5 . 4 l IO.5365 1.95 4.25 0.458 15 0.8802 0.310 0.3502 1.986 38.2 3.34 10.51 3.14 4.248 0.74 45 0.8789 0.487 0.3489 1.978 40.0 3-245 10.48 3,23 4,226 0.764 105 0.8783 0.555 0.3483 1.974 41.8 3.14 10.48 3.345 4.225 0.792 275 0.8780 0.590 0.3480 1.972 45.0 2.98 10.48 3.52 4.228 0.833- 2555 0.8775 0.645 0.3475 1.97 50 2.705 IO .47 3.87 4.225 ' 0.915 444o 0.8775 0.645 0..3475 1.97 50 2.705 10.47 3.87 4.225 0.915 1200 0 1.7234 0 0.6874 2 0 4.80 9-5779 1.995 4.25 0.469 22 1.7209 o . i4o 0.6849 1.99 39 2.99 9-5779 3.21 4.24 0.755 30 1.7190 0.255 0.6830 1.988 41 2.84 9.56 3.37 4.24 0.793 38 1.7168 0,380 0.6808 I . 9 8 O 43 2.73 95'.'3~ 3.50 4.235 0.825 60 1.7142 0.534 0.6782 1.976 47 2.53 95-2 3.85 4.23 0.87 125 1.711*2 0.534 0.6782 1.976 49.5 . 2.43 95-2 3.92 4.23 0.925 245 1.7142 0.534 0.6782 1.976 49.5 2.43 95.2 3.92 4.23 0.925 755 1.7142 0.534 0.6782 1.976 49.5 2.43 95-2 3.92 4.23 0.225 1075 1.7142 0.534 0.6782 1.976 49.6 2.42 95.2 3.94 4.23 0.93 2035 1.7142 0.534 0.6782 1.976 49.6 2.42 95-2 3.94 4.23 0.93 3^15 1.7142 0.534 0.6782 1.976 49.6 2.42 95-2 3.94 4,23 0.93 I OA Table 3. - 70 - . 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 D F D 3 F 3 , t min. . cm3' A ( T ) s e c l o g A(T) 1100 0.73 O.389 90 3.13^X 10" •17 -16.505 (7-35) 0.93 0.8044 380 1 .40 2.744 1055 1.66 4,5733 2555 2.08 8.998 " "5315 1150 1.365 2.5432 120 1.27 X. 10" 16 -15.8955 (7.12) I.50 3-375 290 1.91 6.968 705 2.50 15.625 1980 2.96 25.93^ 3305 3.3^ 37-260 66O5 1200 1.90 6,859 110 8.25 X 10" "16 -15.085 (6.78) 2.57 16.9745 280 3.52 43.613 1265 5.50 16 .640 2820 5-93 208.530 4183 1250 2.92 25.020 100 4.25 X. 10" "15 -14.37 (6.56) 3.41 39.6514 240 5.05 128.500 500 7.02 345.945 1270 8.60 636.056 2620 1300 7.30 389.017 90 .3.67 X 10" "14 -13.4355 (6.35) 7.80 474.552 150 12.23 210 8.92 708.000 250 9-80 941.190 415 10.30 1092.727 625 1000 8.12 X 10" •19 -18.09 : (7.9*0 : 1050 4,36 X 10" 18 -17.36 (7.67) - 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 fo l o g t D = Slope "A T 2,62°:X 10" 3 l o g D 1000 8.12 X 10" 19 5.6 X 10~2 2.21 X 10" 1 4 -13.656 (7-94) 1050 4.36 X- 10" 18 6 x io"2 1.325 X 10" 1 3 -12.878 (7.67) 1100 3.13 X 10" 17 6 x 10" 2 9.77 X.10" 1 3 -12.011 (7.35) 1150 1.27 X 10" 16 6.1 X 10" 2 4.22 X 10" 1 2 -11.3755 (7.12) 1200 8.25 X io" 16 9.3 x 10" 2 4.32 X I O - 1 1 -IO..366 (6,78) - 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 8 and T i 0 2 at 1200°C F i n a l Composition Time min. •D P- D3 v  G m 3  K sec T i O i . 9 8 2 9 0 2 . 1 6 1 4 . 7 4 3 4 . 5 3 X 1 0 - 1 6 2 6 2 5 4 . 5 0 9 1 . 1 2 5 5625 5 . 2 6 1 4 5 . 5 2 8 T i 0 2 354 1 . 9 2 7 . 0 7 7 9 7 . 9 4 X i o " 1 6 4 8 9 2 . 2 6 1 1 . 5 4 3 2 i 4 i 4 • 4 . 2 0 7 4 . 0 1 2 7 9 0 . 5 . 1 0 1 3 3 . 2 0 1 0 6 4 0 . 1 2 . 5 • 1 9 5 3 . 1 2 5 Table 6 . • 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 at 1200°C F i n a l K cm3 1 . Slope K f Domposition s e c bl°P es (amO D -"2.623 X IO" 3 cm 2 Se"c~ T i O x . 9 8 4 . 5 3 X I O - 1 6 0 . 2 3 5 . 8 6 X I O - 1 1 T i 0 2 7 . 9 4 X 1 0 " 1 6 0 . 2 7 ' 12.$1 X I O - 1 1 - 73 - 100 1000 10000 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 the F i n a l Composition o f - T i 0 i , g a at T = 1000°C. - 74 - .100 1000 10000 Time (minutes) Figure A . I l l - l b . D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition of T i 0 l i 9 8 a t T = 1050°C. 100C- 1000 .10000 Time (minutes) Figure A . I I I - l c . D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition of T i 0 i > 9 8 at T = 1100°C. - 75 - 100 70 I I I I I 1 ! I I I I II I l 1 I I I I I I I 100 . 1 0 0 0 . 1 0 0 0 0 Time (minutes) Figure A . I l l - I d . D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition of T i 0 i . 9 8 at T = 1150°C. 100 100Q( 1 0 0 0 0 Time (minutes) Figure A . I l l - l e . D e n s i f i c a t i o n of Compacts f o r the F i n a l Composition of TiOi.ge at.' T = 1200°C . - 76 - Figure A . I l l - 2 . D e n s i f i c a t i o n of Compacts, of S t o i c h i o m e t r i c R u t i l e T i 0 2 at D i f f e r e n t Temperatures. - 77 - Figure A . I I I - 3 V a r i a t i o n of Grain S i z e w i t h Time and Temperature. This i s . t o determine A of equation (16). - 78 - APPENDIX- IV. Defect 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 Non-Stoichiometric-Rutile I f the d i f f u s i o n rate constants f o r both 0 2~ and V Q 2 - A R E measured independently as functions of temperature, the heat of vacancy formation and the.concentration of vacancies at d i f f e r e n t temperatures can be determined by the r e l a t i o n s h i p V- • [°2"] = [ V 0 2 " ] where D indicates the d i f f u s i o n constants of the respective species. On s i m p l i f y i n g and noting that [ 0 2 _ ] i n the l a t t i c e i s almost unity, D02" * D V Q 2 - [ V 0 2 " ] = i) a 2 [ V Q 2 - ] exp ^ f ) l where i s the v i b r a t i o n a l frequency of oxygen ion and a, i s the distance between adjacent oxygen ions, U the a c t i v a t i o n energy f o r the motion of an oxygen vacancy through:the l a t t i c e . :. S u b s t i t u t i n g the vacancy co n c e n t r a t i o n - [ V Q 2 - ] . °f equation (24)'- the d i f f u s i o n c o e f f i c i e n t DQ 2~ i s : -1/6 o 4 H f u , -1/6 D 0 2 " = C P 0 2 * a e x P [ " 3RT - RT] •= C p 0 2 exp- [- (AHf + ^ ) J / \ 2 where C = C V a . This. equation'can be f u r t h e r s i m p l i f i e d by using the usual frequency f a c t o r D Q D 0 2 _ = D 0 exp - + I t i s s i g n i f i c a n t to note that the frequency f a c t o r i n t h i s equation depends -1/6 on P0 2 and. t h i s expression f o r the oxygen ion d i f f u s i o n c o e f f i c i e n t i s v a l i d only at high temperatures, so that v i r t u a l l y a l l the. vacancies are completely ionized. APPENDIX. Vv Boundary D i f f u s i o n Model (Cobie^) 2 1 ^ W f f a 0 3 I 4; k" T 2/3 ]_4. = D 4 = • -A(T), t 3 / 2 dP = 2 D B W y a Q 3 . , i dt A k" T ' t 3 / 2 2 D B W y a Q 3 . A ; k T I n t A p l o t o f D versus time produced the values of A shown i n the \3/2 Appendix V Table 1 , and a p l o t of versus l o g t produced the slopes S at d i f f e r e n t temperatures. Using the following.known value of the constants the values of D^ of d i f f e r e n t temperature are c a l c u l a t e d and shown i n Appendix V, Table 1 . k •W boundary d i f f u s i o n c o e f f i c i e n t cm /sec 1 0 3 ergs/cm 3 - 0 3 • 3 1 . 5 7 X 1 0 " cm 1 , 3 8 X. 1 0" 1 6ergs/deg 25 A APPENDIX V. - 80 - Table 1. Temperature Time D 2 D A(T) 4 Slope n_ , °C. min. x , 4 i f c S \ = (S) AT 1.76 X 1 0 " 1 0 / M cm2 D b ' S R C uoo : 90 0.5329 0.283 5.7 x 10 2 1 0.101 1.391 x 1 0 " 8 •• - -7.856 (7<35) 380 0,8649 0.748 • 1055 1.96 3.85 2555 2.755 7-55 5315 4.326 19.7 1150 120 1.863 3.47 3.3 X 1 0 " 2 ° 0.082 6 .77 X 10~ B -7.1695 (7.12) 290 2.25 5.1 705 3.65 13.3 1980 6.25 39 3305 8,76 77 6605 11.18 125 1200 110 3.61 13 5.0 x 1 0 " 1 9 0.132 1.71 x 1 0 " 6 -5.767 (6.78) 280 6.6 43.5 1265 12.4 154 2820 30.25 915 • 4i83 35.3 1245 1250 240 11.628 135 3.1 X 1 0 " 1 8 - (6.56)1270 49.3 2420 2620 74.0 5480 1300 90 53.3 2820 — 150 60.8 3Y2L) 210 150.0 415 96.0 9200 625 106.0 11200 1000 5.75 x i o - 2 3 0.045 5.8 x i o " 1 1 -10.237 (7-94) : 1050 4.56 x i o - 2 2 0.073 7.74 x 1 0 " 1 0 -9.112 (7.67) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A V A N C O U V E R 8, C A N A D A DEPARTMENT OF METALLURGY Comments on Thesis and O r a l Examination Of Jacques P i e r r e Jean T h i r i a r "SINTERING AND GRAIN GROWTH OF NON-STOICHIOMETRIC RUTILE" This t h e s i s was subjected t o c r i t i c i s m due t o the treatment of a c t i v a t i o n energies and t h e i r c a l c u l a t i o n s . The anomaly i s i n . the use of a r a t e constant f o r Arrhenius p l o t s t h a t i n v o l v e s time t o a power other th a n . u n i t y . i M On page 29, Figure 12 the p l o t s are f o r weight l o s s i n terms of l o g ( jg j^ - ) versus l / T and the slopes correspond t o approximately 4-2 k i l o c a l o r i e s on these p l o t s . The quoted a c t i v a t i o n energy at the top of page 30 i s 82 * 2 k i l o c a l o r i e s per mole which i n d i c a t e s t h a t the author has doubled the value i n r e c o g n i t i o n of the e f f e c t of u s i n g a l / 2 power i n the time u n i t of h i s r a t e f u n c t i o n . This treatment 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 fundamental mechanisms has not yet been obtained at t h i s p o i n t . On page 36, Figure.17 the 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 dimension ( ™ ^ ° ' e - The slope i s estimated at 80 k i l o c a l o r i e s from t h i s s l o p e . The a c t i v a t i o n energy quoted by the author f o r t h i s i s 78 k i l o c a l o r i e s per mole (page 32 l i n e 13 of the f i r s t paragraph) which 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 the time f u n c t i o n has been made i n t h i s case. -This i s not j u s t i f i a b l e since the Arrhenius law i s based upon a p l o t of a fundamental r a t e constant and the time exponent i s always -1 r e g a r d l e s s of r e a c t i o n order or complexity of the r a t e law. Time exponents of other value may be used when the r a t e law i s not r e a l l y known, but the 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 by the time exponent. In t h i s case the a c t i v a t i o n energy i s 78/0.s o r 1 3 0 . k i l o c a l o r i e s per mole. Arrhenius p l o t s : are again shown on page 4-3 Figure 19 and use a u n i t time dimension i n the r a t e constant. This should y i e l d v a l i d a c t i v a t i o n energy v a l u e s . The author has used c e r t a i n procedures apparently accepted i n the ceramics l i t e r a t u r e i n h i s treatment of Figure 17 and should t h e r e f o r e not be condemned, but the procedure i s r e c o g n i z a b l y wrong t o those f a m i l i a r w i t h the fundamentals of k i n e t i c s processes. I t i s probable t h a t many erroneous conclusions regarding 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 from .the vuse of t h i s procedure, and i t i s recommended t h a t attempts 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 erroneous procedure may normally be a p p l i e d . E. P e t e r s , A s s o c i a t e P r o f e s s o r . EP/jmk March 3, 1964

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