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Diffusion in Fe-Mg olivine at elevated temperatures Misener, Donald James 1970

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DIFFUSION IN FE-MG OLIVINE AT ELEVATED TEMPERATURES by DONALD JAMES MISENER B.A. Sc., U n i v e r s i t y of Toronto, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of GEOPHYSICS We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g ree tha 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 r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Qi^pLuJ^ The U n i v e r s i t y o f B r i t i s h Co lumb ia Vancouve r 8, Canada Date TABLE OF CONTENTS Page • LIST OF FIGURES i ABSTRACT 1. RESEARCH OBJECTIVES 2. THEORETICAL FRAMEWORK . 2. I n f l u e n c e of G r a i n boundaries aid D i s l o c a t i o n s 9. Creep Rates 11. EXISTING EXPERIMENTAL DATA 14. APPARATUS 16. General D e s c r i p t i o n ' 16. Pressure C a l i b r a t i o n 17. 4 Temperature C a l i b r a t i o n 19. EXPERIMENTAL RESULTS 20. Specimens 21. Experimental Procedure 22. Experimental R e s u l t s 23. THEORETICAL RESULTS 24. CONCLUSIONS 28. APPENDICIES 33. REFERENCES 38. LIST OF FIGURES F i s h e r ' s Model f o r G r a i n Boundary D i f f u s i o n A c t i v a t i o n E n thalpy o f D i f f u s i o n p l o t t e d a g a i n s t A c t i v a t i o n Enthalpy of Creep Assembled Pressure I n t e n s i f i e r and Furnace Schematic Layout o f Temperature C o n t r o l l i n g and Recording Apparatus Pressure C a l i b r a t i o n Curve Experimental arrangement f o r Temperature C a l i b r a t i o n Temperature C a l i b r a t i o n Curve Schematic o f Specimen Holder Creep Rate of Dunite v s . S t r e s s ABSTRACT In the pro c e s s o f deformation o f a s o l i d a t low s t r a i n r a t e s , the r a t e o f atomic m i g r a t i o n i n the c r y s t a l l a t t i c e i s a c r i t i c a l f a c t o r . Experiments designed to measure the d i f f u s i o n c o e f f i c i e n t o f Fe i n o l i v i n e were u n s u c c e s s f u l . I t i s t h e o r e t i -c a l l y shown t h a t a t temperatures g r e a t e r than o n e - h a l f o f the m e l t i n g temperature the d i f f u s i o n c o e f f i c i e n t o f Mg i n o l i v i n e s i l i c a t e i s an e x p o n e n t i a l f u n c t i o n o f temperature and p r e s s u r e . D T p = 1.27 x IO - 4 exp(-38,000/RT) exp(-P 3.0/RT) The r e s u l t s a l s o i n d i c a t e t h a t a t T > .5T m and a t n o n h y d r o s t a t i c s t r e s s e s <500 bars the e x p e r i m e n t a l l y determined steady s t a t e creep o f the upper mantle rocks d u n i t e and p e r i d o t i t e may be governed by movement o f d i s l o c a t i o n s . I t i s shown t h a t the r a t e l i m i t i n g parameter i n the cre e p - e q u a t i o n i s pro b a b l y the i o n i c d i f f u s i o n c o e f f i c i e n t i n the o l i v i n e s i l i c a t e s t r u c t u r e . 2. RESEARCH OBJECTIVES The primary o b j e c t i v e of the p r e s e n t r e s e a r c h was to determine e x p e r i m e n t a l l y the c o e f f i c i e n t f o r chemical d i f f u s i o n o f Mg and Fe i n o l i v i n e and i t s temperature and p r e s s u r e depen-dence. Such data might then be a p p l i e d t o determine the d i f f u s i o n , mechanism and t o a r e - i n t e r p r e t a t i o n o f p r e s e n t l y known creep r a t e s i n o l i v i n e . A secondary o b j e c t i v e was to apply e x i s t i n g t heory to r e c e n t experimental r e s u l t s by other authors to d e t e r -mine i f t h e o r i e s developed f o r monotomic and d i a t o m i c systems c o u l d be extended to the more complicated s i l i c a t e s t r u c t u r e s . Due t o the l e n g t h o f time r e q u i r e d f o r each d i f f u s i o n experiment (100 to 150 hours) and the problems encountered w i t h o b t a i n i n g good samples (see Experimental R e s u l t s ) , the experimen-t a l scope of the work was reduced to the problem o f determing the d i f f u s i o n c o e f f i c i e n t o f Fe i n t o o l i v i n e and i t s dependence on temperature. The t h e o r e t i c a l work met with more success and i t was concluded t h a t e x i s t i n g t h e o r i e s developed f o r "simple" m a t e r i a l s c o u l d be a p p l i e d t o d a t a o b t a i n e d f o r o l i v i n e c r y s t a l s . THEORETICAL FRAMEWORK I o n i c d i f f u s i o n i n c r y s t a l s i s made p o s s i b l e by the presence of l a t t i c e i m p e r f e c t i o n s . The abundance o f these s t r u c -t u r a l d e f e c t s can be d e s c r i b e d by a Boltzman d i s t r i b u t i o n and hence the d e n s i t y o f d e f e c t s should be, and o r d i n a r i l y i s , s e n s i t i v e to temperature. D i f f u s i o n occurs when an i o n or vacancy t r a v e r s e s a f r e e energy b a r r i e r i n the c r y s t a l . The parameter o f i n t e r e s t i s the a c t i v a t i o n f r e e energy A G ' , r e q u i r e d to move the i o n or vacancy from a "ground s t a t e " l a t t i c e s i t e to an " e x c i t e d s t a t e " between two l a t t i c e s i t e s . I t w i l l be shown t h a t the temperature depen-dence o f the d i f f u s i o n c o e f f i c i e n t may be r e p r e s e n t e d by A H 1 , the e n t halpy o f a c t i v a t i o n and the p r e s s u r e dependence by A V' , the volume of a c t i v a t i o n . The necessary t h e o r e t i c a l development of the equations used i n l a t e r c a l c u l a t i o n s w i l l be g i v e n under headings of A G ' , A H ' , A v ' ' . A G ' ' The thermodynamic d e s c r i p t i o n o f the motion of a d e f e c t between l a t t i c e s i t e s was f i r s t g i v e n by Wert and Zener (1949). They showed t h a t the r a t e a t which a d e f e c t t r a v e r s e s a f r e e -energy b a r r i e r i s : 1 f - AG' V , N - = v exp | — | . (i) where: v = vibration frequency (lattice) in direction of defect motion AG' = free energy necessary to move from minimum energy site to the top of the barrier From Z e n e r 1 s (1952) f o r m u l a t i o n f o r the motion o f a l a t t i c e d e f e c t through a c u b i c c r y s t a l , the d i f f u s i o n c o e f f i c i e n t • becomes: D = \ \ r i <A V 2 <2> where: • i = index of the p o s s i b l e jump the defect cou. make r ^ = rate at which the i t h jump i s made A'X = co-ordinate change (s p a c i a l co-ordinates) In an orthorhombic m i n e r a l such as o l i v i n e , the d i f f u -s i o n c o e f f i c i e n t would become a second rank t e n s o r [Dij] due t o the d i f f e r e n c e i n the energy b a r r i e r s i n the t h r e e c r y s t a l -l o g r a p h i c d i r e c t i o n s . . F o l l o w i n g Zener (1952), we assume t h a t the most probable jump dominates the d i f f u s i o n p r o c e s s . A l l the r then become. 2 equal and the summation over (A X_^) reduces to a g e o m e t r i c a l f a c t o r , times the square o f the l a t t i c e parameter. For d i f f u s i o n o f a., s o l e atom (2) becomes: 9 - 1 D = f a T (3) where: a = L a t t i c e parameter • f = geometric f a c t o r (usually between 1 and 3) Combining e q 1 s (1) and (3) and assuming t h a t v a l u e s f o r f, a, and v are known a t the r e l e v a n t temperature and p r e s s u r e , a c a l c u l a t i o n of A Q ' may be made. A G ' = -RT l n f D \ (4) t f a 2 v J AH' The temperature dependence o f the d i f f u s i o n c o e f f i -c i e n t has been observed e x p e r i m e n t a l l y to be o f the form: D = Do exp |- AH/RTJ (5) where: D = d i f f u s i o n c o e f f i c i e n t at absolute temp. T R = gas constant AH' = enthalpy of a c t i v i t i o n By d i f f e r e n t i a t i n g eqn (5) p a r t i a l l y with r e s p e c t to 1/T a t con-s t a n t P, we o b t a i n : A H' = -R 3 f ^ - 1 3 j (6) (1/T) p Thus by measuring D as a f u n c t i o n o f T, AH' may be c a l c u l a t e d . U sing the e x p r e s s i o n f o r A G' and b a s i c thermodynamics Sears (1950), we o b t a i n the f o l l o w i n g t h e o r e t i c a l v a l u e A H = 3 (A G'/T) , ( 7 ) 3 (1/T) P S u b s t i t u t i n g i n eqn.(7) RT l n 3 (1/T) f _ 3 (InD) + 3 (In f a ^ D)-| L 3 (1/T) „ 3 (1/T) J AH' = 9 (- I D/fa2 D) (8) P „ ,  _,_ • 3 (In f a 2 D)'  p 3 (1/T) - p Wert and Zenner (1950) s t a t e t h a t the l a t t i c e parameter,a, and the frequency of v i b r a t i o n , v , are much l e s s s e n s i t i v e to changes i n temperature and p r e s s u r e than i s D. Thus A H' may be approx-imated by: r 3 (In D) A H = " R r 1 (9) s ( 1 / T ) P A V The p r e s s u r e dependence of the d i f f u s i o n c o e f f i c i e n t has been observed e x p e r i m e n t a l l y . t o be of the form: D = Do expj-P A V ' / R T J (10) where: . D = diffusion coefficient at temp. T and pressure P. R = gas constant A V 1 = activation value for diffusion By d i f f e r e n t i a t i n g eqn. (10) p a r t i a l l y w i t h r e s p e c t t o P a t c o n s t a n t T we o b t a i n : A V* = -RT r 9 1 n D" 3 P T Thus by measuring D as a f u n c t i o n o f P a t c o n s t a n t T, A V may be c a l c u l a t e d . U s i n g the thermodynamic r e l a t i o n f o r AG", Sears (1950) L S p J t (11) S u b s t i t u t i n g i n eqn. .(.11) A V '= -RT f 5 < l n D> _ 3 In (fa^v) "| I f the second term on the r i g h t - h a n d s i d e i s assumed to be of second o r der importance, (Rice and N a c h t r i e b , 1963), then we o b t a i n : v a t i o n volume was proposed by Keyes (1963) u s i n g the i d e a o f ac-t i v a t e d processes and an independent c a l c u l a t i o n of AG 1. Keyes adopts a s t r a i n - e n e r g y model i n which a torque L, i s a p p l i e d t o the two ends o f an i s o t r o p i c s o l i d and the s o l i d i s maintained a t a constant e x t e r n a l temperature and p r e s s u r e . The work done i s then assumed to be used i n c r e a t i n g v a c a n c i e s i n the c r y s t a l . Keyes (1963) concludes t h a t a measurement of the p r e s s u r e d e r i v a -t i v e o f the e l a s t i c shear modulus and an independent c a l c u l a t i o n o f AG 1 may be r e l a t e d to the a c t i v a t i o n volume: A V = (12) Another o f the methods used f o r c a l c u l a t i n g the a c t i -A V = AG » 3(ln u ) 3 (P) T " X (13) where: u = elastic shear modules X = Isothermal compressibility from b a s i c thermodynamics: fl(ln y) $(ln V) T (14) where: Y' th = thermal and thus: A V* = 2 AG'( Y t h - j ) X (15) 8. Keyes (1963) has a p p l i e d the formula to simple c r y s t a l s , such as-NaCl and KC1 and the a c t i v a t i o n volume turns out to be the same order o f magnitude as the atomic volume, although 2 to 5 times s m a l l e r . Employing a hard-shere model f o r the c r y s t a l ( p e r f e c t c r y s t a l w i t h atomic r a d i i equal t o the n e a r e s t neighbour d i s t a n c e ) , Lazarus and N a c h t r i e b (1963) conclude t h a t the a c t i v a t i o n volume f o r atomic d i f f u s i o n i n a c u b i c ' c r y s t a l should vary from .5 to 1.0 atomic volumes, f o r ot h e r c r y s t a l symetries they expect s m a l l e r a c t i v a t i o n volumes (. <. 5 atomic volume) due to the more "open" s t r u c t u r e s i n v o l v e d . Thus Keyes 1 theory seems to l i e w i t h i n the hard-sphere model range, although the a c t u a l mechanism o f d i f f u -s i o n ( i . e . i n t e r s t i t i a l vacancy, or i o n i c ) i s i m p o s s i b l e t o d e t e r -mine from t h i s approach. There have been o t h e r attempts to o b t a i n a v a l u e f o r A V through the use of t h e o r i e s r e l a t i n g d i f f u s i o n and m e l t i n g . R i c e and N a c h t r i e b (1959) w i t h t h e i r theory o f c o r r e s p o n d i n g s t a t e s have d e r i v e d the t h e o r e t i c a l r e l a t i o n s h i p : A V* = A V M AH' (16) AH m where A V M and A H M r e p r e s e n t the volume change and enthalpy of m e l t i n g . I t must be remembered t h a t t h i s A V r e f e r s t o the ac-t i v a t i o n volume f o r s e l f - d i f f u s i o n a t the m e l t i n g temperature and would be o f l i m i t e d v a l u e when d i s c u s s i n g o t h e r d i f f u s i o n mechan-isms a t lower temperatures. INFLUENCE OF GRAIN BOUNDARIES AND DISLOCATIONS -The t h e o r i e s r e f e r r e d to above have d e a l t w i t h the s i n g l e c r y s t a l assumption; however, the e f f e c t of g r a i n boundaries and d i s l o c a t i o n " b a r r i e r s " must not be n e g l e c t e d . The process o f g r a i n boundary d i f f u s i o n can be s t u d i e d t h e o r e t i c a l l y i f use i s made of work by F i s h e r (1951) on the r e l a t i o n between g r a i n boun-dary and bulk d i f f u s i o n . The c r y s t a l l i n e model i s shown i n f i g u r e 1. Assuming t h a t d i r e c t d i f f u s i o n from A to B i s o f secondary importance, F i s h e r d e r i v e s the. m o d i f i e d d i f f u s i o n e q u a t i o n : 3c' = D ' S 2 C ' + 2D 3_c q , where 3 t 3 y 2 oi 3 x x c = grain boundary concentration of solute c = grain bulk concentration of solute D ' = grain boundary diffusion coefficient D = grain bulk diffusion coefficient co = grain boundary thickness For most p e r i o d s o f time: 3 t 3 t max. a l l o w i n g f 3 C • n ' -i — 0 or c —•> max. value 3 t t = 0° t = 0 0 In t h i s case the approximate s o l u t i o n o f equ a t i o n (17) becomes c(x y t) = exp {— *— \ erfc | "I (18) 1 1 L 0) (H Dt) I (D'/D)I L 2 Dt J 2 2 90 - r y s f ^ 1 ft Sc/ur-e. Canc&n'trcLf/'on \\ /"V^ /- F'sher's . Made! -for.. G>rarn Boundary Di-jiu^io/i 10. The easiest way to obtain data for t h i s equation would be to measure the amount of solute diffused into a t h i n slab "dy" p a r a l l e l to the free surface. Thus S, the amount of solute i n : this slab becomes: r y+dy f t 0 0 • S = c(x i y it)dxdy (19) J y The value of D(x,y,t) may be d i r e c t l y obtained from measurement of C(x,y,t) where xy (or near the middle of the grain where boun-dary e f f e c t s are n e g l i g i b l e ) . Further work by Whipple (1954) has shown that for mean-i n g f u l r e s u l t s appreciable solute concentrations must, be obtained. In his solution of the d i f f e r e n t i a l equation (17) Whipple (1954) made use of a dimensionless parameter 3 where: 3 = is! _ a . (20) D 2 Dt Assuming d i f f e r e n t values for 3 / Whipple plotted solute concen-trations i n grain B versus 3 . For perceptible grain boundary d i f f u s i o n e f f ects (0 <85°, see figure 1) he found that 3 > 1.0 (21) For o l i v i n e , at a temperature of approximately 10 00°c, using the data of Jander and Stamm (1932) ' D * 7 x 10 - 1 1 cm2/sec t = 3.6 *x 105 sec to = 10"^ cm •r 3 = 1.0 10-5 l7 x 10-H J 2(7 x 10-H x 3.6 x .105)* D - 7 x 10 _ o (22) 11. Thus D' would have to be 1000 times l a r g e r than D to. s a t i s f y Whipple's c r i t e r i o n . In many.cases (Adda and P h i l i b e r t , 1966) >jj i s 10^ or 10 4; however, a v a l u e of 10 -^ f o r w i s q u i t e l a r g e (Adda and P h i l i b e r t , p.718) and would have to be assumed the maximum.allowable g r a i n boundary•thickness. Thus both D and D' may be c a l c u l a t e d , u s i n g eqn's (18) and (19) b e a r i n g i n mind i f the above l i m i t a t i o n s . S i nce D' is"much l a r g e r than D, g r a i n boundary phenomena w i l l dominate the s h o r t time, low temperature r e g i o n . The h i g h temperature, long time r e g i o n w i l l be dominated by bulk d i f f u s i o n , p r o c e s s e s , w i t h the boundaries a c t i n g as c o n s t a n t composition sources and s i n k s f o r the much slower bulk d i f f u s i o n p r o c e s s e s . CREEP RATES D i f f u s i o n c o e f f i c i e n t s once obtained may be a p p l i e d t o c a l c u l a t i o n s o f creep r a t e s and v i s c o s i t y o f m a t e r i a l s . .A g e n e r a l equation f o r creep r a t e i s (Weertman, 1970) e = f ( a ) D T >p (23) where f ( o) i s a f u n c t i o n o f the d i f f e r e n t i a l s t r e s s a and D^ p i s the p e r t i n e n t d i f f u s i o n c o e f f i c i e n t . The two creep mechanisms o f most i n t e r e s t i n geophysics ; are d i f f u s i o n o r Herring-Nabarro creep (low s t r e s s and High tem-perature) and creep r e s u l t i n g from the "climb" o r u n - p i n n i n g o f d i s l o c a t i o n s (higher s t r e s s and h i g h temperature). 12. I n H e r r i n g - N a b a r r o c r e e p ( N a b a r r o , 1948; H e r r i n g 1950), t h e g r a i n b o u n d a r i e s o r d i s l o c a t i o n s a r e : f i x e d and a c t o n l y a s s o u r c e s o r s i n k s f o r t h e d i f f u s i n g i o n s and v a c a n c i e s . The r e -s u l t i n g e q u a t i o n i s : ' • c D A v ' „ - . . „ . ; £ = — 0' (24) k T a where: £ '.= s t r a i n rate D = d i f f u s i o n c o e f f i c i e n t f o r vacancies T A V = • a c t i v a t i o n volume f or creep a =' mean grain radius O = d i f f e r e n t i a l shearing stress C = numerical constant I n d i s l o c a t i o n o r Weertman c r e e p (Weertman, 1957) , t h e d i s l o c a t i o n s h a v e moved a s f a r as p o s s i b l e and have become f i x e d i n p o s i t i o n . C r e e p o c c u r s when t h e d i s l o c a t i o n s " c l i m b " , o u t o f t h e i r p i n n e d p o s i t i o n s . T h i s movement i s a c c o m p l i s h e d b y t h e e x c h a n g e o f t h e d i s l o c a t i o n w i t h a v a c a n c y ; i t i s t h e v a c a n c y d i f f u s i o n w h i c h p e r m i t s t h e c r e e p . The r e s u l t i n g t h e o r e t i c a l e q u a t i o n d e r i v e d by Weertman (1957) i s : 4.5 Mi constant * — ^ 2 5 ^ where; M = density of Frank - Reed sources U =• shear modules D = d i f f u s i o n c o e f f i c i e n t f o r s e l f - d i f f u s i o n a = d i f f e r e n t i a l shearing s t r e s s Care must be taken when a p p l y i n g d i f f u s i o n data to the c a l c u l a t i o n o f creep r a t e s . In c a l c u l a t i o n s o f d i f f u s i o n c r e e p , the vacancy d i f f u s i o n c o e f f i c i e n t i s needed and f o r d i s l o c a t e creep the c o e f f i c i e n t o f s e l f d i f f u s i o n i s used. In m i n e r a l s such as o l i v i n e the r a t e l i m i t i n g d i f f u s i o n i s probably t h a t o f the c a t i o n s . One might expect the.O^ - anions t o be the r a t e l i m i t i n g ones due to t h e i r l a r g e r s i z e and thus s m a l l e r d i f f u s i o n c o e f f i - . c i e n t . I t has been observed however, Passmore e t a l , 1966) t h a t the d i f f u s i o n i s a g r a i n boundary phenomenum and would have l i t t l e e f f e c t over l o n g p e r i o d s o f time. (see eqn's (18). Experimental work by Sherby (1962), a t h i g h temperatures (T >.5Tm) has r e s u l t e d i n an e m p i r i c a l e quation o f the f o l l o w i n g form: ; - • £ = constant . D g 2 ^ _ j, (26) where: g = grain size ' D = diffusion coefficient for self-diffusion U = modules of rigi d i t y cr = differential shearing stress The "constant" and the power law (5.0) were determined e m p i r i c a l l y The use o f c a t i o n d i f f u s i o n d a t a i n the aforementioned v a c a n c y - m i g r a t i o n creep equations can l e a d to e r r o r s up t o a p p r o x i mately 20%, the reason b e i n g t h a t the vacancy p o p u l a t i o n i s a func tion. o f temperature w h i l e the base composition d i f f e r e n c e s which p r o v i d e the d i f f u s i o n are independent o f temperature. 14. INFLUENCE OF GRAIN BOUNDARIES ON CREEP In a p o l y c r y s t a l l i n e aggregate, the g r a i n boundary creep i s p r o b a b l y the dominant process (REE, Ree and E y r i n g , 1 960 ) . G r a i n boundary creep i s caused'by the p i l e up of d i s l o c a t i o n s on the boundaries and subsequent s l i d i n g o f the g r a i n s along the boundaries. Ree and E y r i n g (1960) developed the f o l l o w i n g theo-r e t i c a l e q uation f o r the creep of aggregates o f A l and Al-Mg a l l o y c r y s t a l s : £ = s i n h | - j - exp |-A.H'/RTJ> (27) °o The model was based on a mechanism o f t h e r m a l l y - a c t i v a t e d s e l f d i f f u s i o n and the a c t i v a t i o n energy f o r creep i s assumed t o be the 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 . EXISTING EXPERIMENTAL DATA R h e o l o g i c a l processes i n the mantle of the e a r t h seem most l i k e l y to be c o n t r o l l e d by creep i n the c r y s t a l l i n e s t a t e . Nabarro ( 1948) . S p e c i f i c c a l c u l a t i o n s o f the r h e o l o g i c a l p roper-t i e s r e q u i r e e x p erimental data and these are o u t l i n e d below. Sherby and Burke (1967) have presented an e x c e l l e n t r e -view o f the experimental d a t a on metals and a l l o y s . The e q u i v a -l e n c e o f the en t h a l p y o f a c t i v a t i o n f o r creep and t h a t o f d i f f u -s i o n was observed and i s i l l u s t r a t e d i n F i g u r e 2. A r e c e n t book by A s k i l l (1970) , extends the data o f Sherby and Burke (1967). and a l s o p r e s e n t s data on some o x i d e s . 14a. A hr CKccil./mol6) s z/f - J>ff as/ on 15. D i f f u s i o n c o e f f i c i e n t s and e n t h a l p i e s of a c t i v a t i o n are presented and where p o s s i b l e the experimental c o n d i t i o n s of the measurements. Since the advent of the n u c l e a r r e a c t o r and the genera-t i o n of r a d i o a c t i v e " t r a c e r s " , much work has been done on the r h e o l o g i c a l p r o p e r t i e s o f ceramic m a t e r i a l s used i n r e a c t o r con-s t r u c t i o n . MacKenzie (1968) has p u b l i s h e d a summation o f t h i s data and used the r e s u l t s to i n d i c a t e t h a t the r e g i o n o f the upper mantle pr o b a b l y e x h i b i t s the same creep pr o c e s s e s as evidenced by BeO, MgO, and A l 2 0 3 . D i f f u s i o n data on g e o l o g i c a l m a t e r i a l s i s scanty r e l a -t i v e to those on metals and a l l o y s . At p r e s e n t , data are a v a i l -a b l e f o r the d i f f u s i o n of v a r i o u s c a t i o n s (K,Na, and L i ) anto SiC>2 c r y s t a l s a t v a r i o u s temperatures and a l o n g the d i f f e r e n t c r y s t a l - ' l o g r a p h i c a x i s . F y f e and Verhoogen (1958) have p u b l i s h e d a s h o r t l i s t of d i f f u s i o n c o e f f i c i e n t s o f i n t e r e s t i n p e t r o l o g y . They d i s c u s s the v a r i o u s f a c t o r s (temperature, p r e s s u r e and d i f f e r e n -t i a l s t r e s s ) i n f l u e n c i n g the r a t e of d i f f u s i o n . Of s p e c i a l i n - -t e r e s t i n t h i s data i s the e a r l y work of Jander and Stamm (1932) . on the d i f f u s i o n o f Mg, N i , and Ge i n t o s i l i c a t e s , Mg2SiO^, .N^SiO^. and germanates, Mg2Ge04-The use o f the e l e c t r o n micro-probe i n measuring r e l a -t i v e abundances o f atomic s p e c i e s has become an i n v a l u a b l e t o o l i n d i f f u s i o n experiments. . 16. Varshneya and Cooper (1969) have used the e l e c t r o n micro probe i n t h e i r a n a l y s i s o f d i f f u s i o n i n s y n t h e t i c t e c t i t e s . They made a simple one-dimensional d i f f u s i o n couple and measured con-c e n t r a t i o n s of Fe along p r o f i l e s a t temperatures up to 1490°C. In the f i e l d o f M e t a l l u r g y , the e l e c t r o n micro-probe i s used e x t e n s i v e l y f o r d i f f u s i o n s t u d i e s and an e x c e l l e n t book has been p u b l i s h e d which o u t l i n e s some of these experiments (McKinley, H e i n r i c h , and W i t t r y , 1966). Recent measurements r e p o r t e d by Eaton (1968) and M i s r a and M u r r e l (1965) on the h i g h temperature creep of d u n i t e , appear to be the o n l y data a v a i l a b l e f o r comparison w i t h o l i v i n e d i f f u -s i o n d a t a . Eaton observed t h a t up to temperatures o f 950°C the s t r a i n r a t e versus temperature f o l l o w e d a m o d i f i e d Ree-Eyring equation and thus he was a b l e t o c a l c u l a t e an a c t i v a t i o n enthalpy f o r c reep. M i s r a and M u r r e l (1965) have made h i g h temperature creep measurements on Norwegian p e r i d o t i t e up to temperatures o f 750°C.' They observed a l i n e a r dependence between the l o g . o f the creep r a t e and the temperature. T h i s dependence was i n t e r p r e t e d as, an a c t i v a t i o n enthalpy f o r creep. APPARATUS: . GENERAL DESCRIPTION The b a s i c e x p e r i m e n t al apparatus c o n s i s t e d o f an ex-t e r n a l l y heated, p i s t o n - c y l i n d e r d e v i c e , which was p l a c e d i n a two-post p r e s s frame. A one-dimensional d i f f u s i o n c ouple was i n t r o d u c e d between the opposed p i s t o n s and h e l d a t c o n s t a n t tem-p e r a t u r e and p r e s s u r e f o r a s p e c i f i e d l e n g t h p f time. Upon 17. • completion of each experiment, the samples were c u t and p o l i s h e d ' and examined on the e l e c t r o n micro-probe. F i g u r e 3 i l l u s t r a t e s a c r o s s - s e c t i o n of the pressure, system as i t was assembled d u r i n g the experiments.. A d e t a i l e d d e s c r i p t i o n of the system may be found i n Appendix A. The sample, was c o n t a i n e d i n a s m a l l g r a p h i t e cap-and-sleeve which had the same t o l e r a n c e as the p i s t o n - t o - c y l i n d e r w a l l . The p r e s s u r e system was a c c u r a t e t o - 200 pounds over the range o f o p e r a t i o n . Heat was s u p p l i e d by a Nichrome r e s i s t a n c e furnace and the temperature was measured with standard sheathed chromel-alumel thermocouples. The temperature was c o n t r o l l e d by a con-t i n u o u s l y p r o p o r t i o n i n g b r i d g e w i t h one arm o f p l a t i n u m r e s i s t a n c e i n the f u r n a c e . The temperature was known to an a c c u r a c y of -5°C . up to 10 00°C. F i g u r e 4 i l l u s t r a t e s the temperature c o n t r o l l i n g and r e c o r d i n g apparatus as i t was d u r i n g the experiments. A more d e t a i l e d schematic of the temperature c o n t r o l l e r may be found i n Appendix B. CALIBRATION , PRESSURE CALIBRATION • The p r e s s u r e system was c a l i b r a t e d u s i n g a Baldwin Model 120 s t r a i n i n d i c a t o r and .a Bourdon s t r a i n gauge b r i d g e . Both i n -struments were s u p p l i e d by the M i n e r a l E n g i n e e r i n g Department, U.B.C - The gauge had r e c e n t l y been c a l i b r a t e d by the M i n e r a l E n g i n e e r i n g Department to i n d i c a t e 12.5 pounds a p p l i e d l o a d e q u i -v a l e n t ' t o 1.0 m i c r o - s t r a i n u n i t s on the b r i d g e . The b r i d g e d i a l 17b. F i g . 3 : C a p t i o n Item . . D e s c r i p t i o n 1. Press Frame (.8% hot r o l l e d s t e e l ) 2. Ram support r i n g 3. Enerpac RLC-100, 100 ton ram 4. I n t e n s i f i e r Ring (310 S t a i n l e s s S t e e l ) 5. I n t e n s i f i e r Ring (Inconnel X-750) 6. I n s u l a t i n g Rings (Asbestos) 7. Bomb C y l i n d e r (Rene R-41 S t e e l ) 8. P i s t o n s (Rene R-41) 9. Sample (See F i g . 9) 10. Temperature C o n t r o l l e r element (14.5 platinum) 11. Temperature Recording Thermocouple (Chrome1-Alumel) 12. Power l e a d s to h e a t i n g element 13. Ceramic Furnace Core 14. Furnace ( F i n e b r i c k and Asbestos Board) 15. Furnace Support 17c. •Is.O volt HOT- jt»->c riart CC-A ) 1 \—e>-~ ° ControlIer Po i~e s> ho rn £ ten ColcJ Jync+iea.CC-Ficj QrrOiOtj e.rr)Cjor~ of : I emperarure. Control jet OncJ : Temper afore* Recorder OjyifrafyS. 1 8 . • . was e a s i l y r e a d a b l e ' t o -.5 m i c r o - s t r a i n u n i t s . For the c a l i b r a t i o n a s m a l l c y l i n d e r o f Rene R-41 s t e e l o f the osame dimensions as a sample, was used i n the apparatus. The c a l i b r a t i o n curve up to 3 0,000 pounds i s shown i n F i g u r e 5; the gauge r e a d i n g r e f e r s to the o i l p r e s s u r e gauge . . -on the ram and the lo a d was o b t a i n e d from the Bordon meter. The b e s t f i t . l i n e y i e l d s the e q u a t i o n : A± = 540. + 19.24G . ( 2g) where: Aj_ = a p p l i e d l o a d i n pounds G = gauge r e a d i n g i n p . s . i . The zero p o i n t e r r o r i s due t o the compressing of the • i n s u l a t i n g r i n g s (see F i g u r e 3) b e f o r e the p i s t o n s take up the a p p l i e d l o a d . T h i s f a c t o r was r e a l i z e d and no runs were made be-low a gauge s e t t i n g o f 200 p . s . i . The e r r o r i n r e a d i n g the ram gauge was +_ 10 p . s . i . which would r e s u l t i n an e r r o r of 200 pounds on the a p p l i e d l o a d . Noting t h i s e r r o r , i t was found t h a t the experimental c a l i b r a t i o n agrees w i t h the curve s u p p l i e d by , Enerpac: A1 = 19. 625G (29) where the con s t a n t i s the s u r f a c e area o f the ram i n square i n c h e s . The i n s u l a t i n g r i n g s and the d i s t a n c e o f the ram from 4 the furnace maintained the ram a t a l l times below 60°C. Thus the p r e s s u r e was assumed a c c u r a t e to - 200 pounds over the range of o p e r a t i o n . . ' {p-'S. i) ~ too 18a. i / A-' Cob bfAtion 10 4* Pre. • 19. TEMPERATURE CALIBRATION Aschematic r e p r e s e n t a t i o n of the c a l i b r a t i o n o f the furnace temperature i s shown i n F i g u r e 6. Before the furnace was c a l i b r a t e d , the thermocouple to be used i n the p r e s s u r e v e s s e l was c a l i b r a t e d a g a i n s t a working standard thermocouple s u p p l i e d by Dr. H.J. Greenwood of the Geology Department, U.B.C. The W.S. . thermocouple had i t s e l f been c a l i b r a t e d by Dr. T.M. Gordon of the Geology Department, U.B.C. a g a i n s t U.S. N a t i o n a l Bureau of Standard! m e l t i n g p o i n t s and the c o r r e c t i o n s t o the W.S. were known. The • pr e s s u r e v e s s e l thermocouple was then p l a c e d i n a h i g h temperature furnace along with the W.S. thermocouple and both temperatures were recorded when the furnace reached e q u i l i b r i u m . Thus the p r e -ssure v e s s e l thermocouple was c a l i b r a t e d to N.B.S. m e l t i n g p o i n t s . The c a l i b r a t i o n o f the experimental furnace then p r o -ceeded by n o t i n g temperatures on the p r e s s u r e v e s s e l thermocouple, and the W.S. thermocouple at v a r i o u s c o n t r o l l e r s e t t i n g s . The f i n a l c o r r e c t i o n s to be a p p l i e d t o temperature r e a d -i n g s are summarized.in the f o l l o w i n g f ormula: T = T - A T (30) s m w ' where T^ = Sample temperature (W.S. thermocouple) T m = Pressure Vessel Temperature (Pres. Ves. Thermocouple) AT = -T + T s m 19a. Tern p era-rare Cal'braj'i'on • Schematic T r I Switch cxate r.i I'C re c t e P V Pressure V e s s e l T.Werrfto. Furnace. r i g '6.' fzvperimental- arrangement- for- 7cmp. Cal/'hraf/on 20. A l l v a l u e s of T s and T m were c o r r e c t e d to.N.B.S. melt-, i n g p o i n t s before they were used i n the c a l i b r a t i o n . E r r o r s i n these c o r r e c t i o n s were a second o r d e r e f f e c t and were neglected.'. An e r r o r o f - . 5°C was allowed f o r i n r e c o r d i n g the v a l u e s o f T s and T m . The thermal g r a d i e n t s w i t h i n the sample c a v i t y were not measured. The p o i n t a t which the c a l i b r a t e d temperatures were rec o r d e d i n the sample c a v i t y d i d not v a r y by more than .1 inches w i t h r e s p e c t to the l o c a t i o n of the d i f f u s i o n i n t e r f a c e d u r i n g the experimental runs. A s u b j e c t i v e estimate o f the tern- • p e r a t u r e e r r o r r e s u l t i n g from e r r o r s i n p o s i t i o n i n g of the sam-p l e s was -3°C. The f i n a l accuracy o f the temperature c a l i b r a t i o n was s e t a t -5°C. The curve o f T m versus AT i s p l o t t e d i n F i g u r e s 7 and was used d u r i n g the experimental runs to c a l c u l a t e the sample temperature. EXPERIMENTAL RESULTS • , . INTRODUCTION Three samples o f o l i v i n e were prepared arid "run" a t v a r i o u s temperatures t o t e s t the f e a s i b i l i t y o f the experimental arrangement and t o c o r r e l a t e the r e s u l t s w i t h t h e o r e t i c a l deve- -lopments. A l l experiments were of 150 hours d u r a t i o n and the same procedure was f o l l o w e d i n each experiment. Upon completion of the experiment, each sample was removed from the apparatus, im-bedded i n a b a k e l i t e h o l d e r and prepared f o r examination on the e l e c t r o n micro-probe. 20a. 2ir A T C X ) 2d-m— Mr /o Recorded Bom6 Temp. C°C) Fig 7 Temp. Di f-fenence: Bomb Temp. v s . SornpJ.a- Te.mf 9ao 21. SPECIMENS . The samples used were n a t u r a l o l i v i n e c r y s t a l s , taken from v o l c a n i c bombs or flows. The bulk composition was d e t e r -mined u s i n g the x-ray d i f f r a c t o m e t e r and the i n i t i a l c o n c e n t r a -t i o n , of Fe was determined by making random scans over a c r y s t a l . • w i t h the e l e c t r o n micro-probe and t a k i n g an average v a l u e . Micro probe scans were a l s o made along the g r a i n boundaries of some o f the c r y s t a l s and i t was noted t h a t the Fe c o n c e n t r a t i o n d i d not vary by more than 5% along the edges, r e l a t i v e t o the i n t e r i o r of the g r a i n s . I t was decided to d i s c a r d the i n i t i a l 10 microns i n the f i n a l d i f f u s i o n p r o f i l e i n o r d e r t o o b t a i n a more a c c u r a t e f i t of the experimental d a t a . A small c h i p of each sample was powdered and to each was added a " s p i k e " o f KBr standard. X-ray d i f f r a c t o m e t e r o s c i l -l a t i o n s were made over the d-j_3Q o l i v i n e peak and the 27.03°, 29 peak f o r KBr. Using the formula d e r i v e d e m p i r i c a l l y be Medaris and F i s h e r (1969): . . X= 15.8113 (3.0358-d 1 3 Q)* -7.2250 (31) where: X = fraction of forsterite i n the olivine The bulk composition o f each sample was c a l c u l a t e d (see Appendix i Each sample was then p o l i s h e d i n t o a c y l i n d r i c a l shape .2 i n c h e s i n h e i g h t and .3 inches i n diameter. The c o n t a c t s u r -face was then r e - p o l i s h e d i n o r d e r to minimize the c o n t a c t r e s i s -tance. The o t h e r h a l f o f the d i f f u s i o n couple ( a mixture of 22. e n s t a t i t e of non-constant Fe c o n c e n t r a t i o n ) was prepared i n the same way. The d i f f u s i o n . c o u p l e was then mounted i n the h o l d e r as shown i n F i g u r e 8. The Fe c o n c e n t r a t i o n on the g r a i n boundaries i n the -e n s t a t i t e was observed on the e l e c t r o n micro-probe to be a f a c t o r of from 1.5 to 2.0 g r e a t e r than i n the o l i v i n e c r y s t a l s and thus there was e s s e n t i a l l y an i n f i n i t e amount of Fe a v a i l a b l e f o r d i f f u s i o n . The s o l u t i o n s f o r the d i f f u s i o n e q u a t i o n i n a s e m i - i n f i n i t e s l a b (see Crank 1956, or' J o s t 1952) c o u l d then be a p p l i e d t o . t h e experimental c o n c e n t r a t i o n p r o f i l e s . EXPERIMENTAL PROCEDURE Each specimen was prepared as i n d i c a t e d i n F i g u r e 8. • A l l the p i e c e s o f the p r e s s u r e i n t e n s i f i e r were cleaned b e f o r e use and the p i s t o n s and c y l i n d e r w a l l were l i g h t l y rubbed with emery paper to remove any dust p a r t i c l e s on the s u r f a c e s . The p r e s s u r e system was then assembled and the furnace was p l a c e d around i t . The p r e s s u r e was then a p p l i e d and i n a l l cases 600, p . s . i . was used i n o r d e r to o b t a i n good s u r f a c e c o n t a c t a t a f a i r l y low p r e s s u r e . A t t h i s l o a d p r e s s u r e the sample p r e s s u r e (over 0.07 sg. i n . ) would be approximately 10 k i l o b a r s . The furnace c o n t r o l l e r was then s e t and the furnace turned on and l e f t to come to e q u i l i b r i u m ; i n a l l cases t h i s e q u i l i b r i u m was reached w i t h i n three hours. The p r e s s u r e was then r e - a d j u s t e d to 600 p . s . i . and the t i m i n g o f the experiment began. 2 2 a . 23. Three"runs" were made: one at 625°C, one a t 750°C and one a t 1000 °C. The temperature and p r e s s u r e were recorded a t r e g u l a r f o u r hour i n t e r v a l s and no subsequent adjustment was found to be necessary i n the temperature; however, the p r e s s u r e needed s l i g h t "boosts" as i t would s l o w l y decrease to 595 or 590 p . s . i . over the space of 50 hours. Thus the temperature and p r e s s u r e were assumed c o n s t a n t t o -5% over the d u r a t i o n of the experiments. A t the end of each "run" the p r e s s u r e was r e l i e v e d and then the temperature was reduced q u i c k l y ; the sample and h o l d e r were removed, vacuum s e a l e d i n Epoxy and mounted i n a B a k e l i t e c y l i n -der. T h i s c y l i n d e r was then c u t perpe'ndicular to the d i f f u s i o n s u r f a c e and p o l i s h e d f o r probe examination. A f t e r each run,, the p r e s s u r e apparatus was thoroughly cl e a n e d and the p r e s s u r e v e s s e l thermocouple was checked a g a i n s t the Working Standard. EXPERIMENTAL RESULTS E l e c t r o n micro-probe a n a l y s i s of the samples produced n e g a t i v e r e s u l t s . At the low temperatures (625°C, 750°C) the samples underwent' minor amounts of f r a c t u r i n g which d i s t u r b e d the c o u p l i n g s u r f a c e . An even more s e r i o u s problem was the f a c t t h a t the g r a i n s became separated by as much as 5 to 10 microns and thus s u r f a c e and not g r a i n boundary d i f f u s i o n became the dominant mechanism. The process o f s u r f a c e d i f f u s i o n i s much more complex and the t h e o r i e s presented f o r g r a i n bulk and g r a i n boundary d i f f u s i o n do not apply. 24. ••: At the h i g h e r temperature (1000°C) i t was observed t h a t the f r a c t u r i n g was s t i l l p r e s e n t . In a d d i t i o n , there was some d e t e r i o r a t i o n - o f the g r a p h i t e s l e e v e . The g r a p h i t e caps (see F i g u r e 8) had f r a c t u r e d and the s l e e v e was cracked the l e n g t h o f the sample. The sample was t e s t e d on the e l e c t r o n micro-probe but no d e t e c t a b l e d i f f u s i o n o f Fe had occured. THEORETICAL RESULTS ' • E x t r a p o l a t i o n o f e x i s t i n g d i f f u s i o n theory and data to estimate the a c t i v a t i o n parameters of d i f f u s i o n and the creep r a t e i n o l i v i n e i s presented below. The e a r l y work o f Jander and Stamm (1932) p r o v i d e s v a l u e s f o r the d i f f u s i o n o f Mg i n t o Mg 2Si04 a t two temperatures: D1070°C = 7 - 1 4 x 1 0 - 1 1 cm 2/sec -10 9 D12 00°C ~ 1*2. 2.6 x 10 cm z/sec S u b s t i t u t i n g these v a l u e s i n t o e quation (§) p r o v i d e s an estimate of AH'. The two v a l u e s o f D]_2oo0C r e s u-'- t f r o m a d i f f e r e n c e i n . the t r a n s f e r e n c e number. The v a l u e o f the t r a n s f e r e n c e number- . r e f e r s to the p r o p o r t i o n o f the flow c o n t r i b u t e d by the i o n i n q u e s t i o n : a t r a n s f e r e n c e number of 1.0 would imply t h a t the t o t a l amount of d i f f u s i n g m a t e r i a l would be Mg. I f the t r a n s -f e r e n c e number i s 1.0, the hi g h e r v a l u e of Di2 00°c l s a P P H c a ^ l e and i f the t r a n s f e r e n c e number i s 0.5 the lower v a l u e a p p l i e s . . The f o l l o w i n g v a l u e s f o r AH' were c a l c u l a t e d : t ='1.0 AH' = 38.0 Kcal/mole t = 0.5 AH' = 18.0 Kcal/mole . 2 5 . v From Anderson (1968) , the necessary d a t a on p o l y -c r y s t a l l i n e f o r s t e r i t e was o b t a i n e d . t o a l l o w a c a l c u l a t i o n of AG',. The Equation used was: A GT = -RT In {— i (4) •fwas,chosen to be 1.5; however, Changing t h i s c o n s t a n t by a f a c t o r o f 2 e i t h e r way w i l l o n l y change the c a l c u l a t i o n o f G.' by 5% (Keyes, 1963, p.74). The v a l u e of v was c a l c u l a t e d u s i n g the formula f o r the Debye frequency ( K i t t e l , 1966, p.175) 4 nv V 3 c±3 J where; - N = Avogadro's number V = Molar volume o f f o r s t e r i t e , c-j-,C]_ = Transverse and L o n g i t u d i n a l Wave V e l o c i t i e s i n the C r y s t a l The wave v e l o c i t i e s were assumed to be approximately equal and the v e l o c i t y a l o n g the " c " . c r y s t a l l o g r a p h i c a x i s i n f o r s t e r i t e was used. AG' was c a l c u l a t e d a t a temperature o f 13 43°C and the v a l u e s o f V and were c a l c u l a t e d a t t h a t temperature using; the formulae: • ' • 3 C T °t (1343°K) C273°K + ..„, A T ( 3 3 ) : P : . U1343 , . y 2 7 3 + (3 u/3 T)p' A T (34) P l 3 4 3 ° K " v s 2 " v + ( 3 v / 3 T ) A T 2 o Q e 1343 273 2 6 . • where: atomic wt v. = P 1 3 4 3 ' , The c a l c u l a t e d v a l u e of. A G 1 i s : A G ' - 5 4 . 0 Kcal/mole T h i s value of A G * may now be used i n e q u a t i o n ( 4 ) to c a l c u l a t e the a c t i v a t i o n volume of d i f f u s i o n . U s i n g .the thermal Gruneisen's constant (assumed independent of temperature, Anderson, 1 9 6 8 , p . 5 0 2 ) and a v a l u e o f v Q c a l c u l a t e d a t 1 3 4 3 ° C ; A V'becomes A V' ' • 2 A G ' ( y -h X - 5 4 ^ 8 5 ~ ' 3 3 ) * ^ * 4 ' 2 * 1 0 ?  T H . 8 . 1 x 1 0 5 x 1 0 6 - 3 cm3/mole . ( 3 5 ) The a c t u a l atomic volume to be used f o r comparison w i t h eqn. ( 3 5 ) must be c a l c u l a t e d from the i o n i c r a d i i o f Mg, as no va l u e s f o r A V have been e x p e r i m e n t a l l y determined. Since the Mg i s i n 6 - f o l d i o n i c c o - o r d i n a t i o n i n the o l i v i n e s t r u c t u r e , the i o n i c r a d i u s i s 0 . 6 6 A, which y i e l d s an I ' atomic volume of 0 . 7cm-Vmole • I f the Mg i s assumed to be i n an u n e x i t e d ground s t a t e d u r i n g the d i f f u s i o n jump (valence o f 0 . 0 ) . then the atomic volume becomes 1 3 . 9 cm^/mole. Since the c a l c u -l a t e d v a l u e of Av' i s 3.Ocm^/mole, we would expect an atomic volume of g r e a t e r than 6 . 0 (see page 8 ) . An e x p l a n a t i o n of the apparent d i s c r e p a n c y between the c a l c u l a t e d A V and a c t u a l atomic volumes may be t h a t the c a l c u l a t e d a c t i v a t i o n volume i s r e a l l y an average v a l u e c a l c u l a t e d over the t o t a l d i f f u s i o n jump d i s t a n c e . The Mg atom would have the i o n i c a c t i v a t i o n volume hear the' ground state, or low energy s i t e s and d u r i n g the time the Mg atom was near the. top of the Free-energy b a r r i e r i t would have the l a r g e r atomic a c t i v a t i o n volume. A simple average of the two valu e s of a c t i v a t i o n volume y i e l d s a v a l u e of 7.6 cm 3/mol which when compared wi t h the value of 3.0 c a l c u l a t e d from the combined t h e o r i e s of Keyes (1963) and Zenner (1952) i s w i t h i n the approximate l i m i t s s e t out on page 8 . From these c a l c u l a t i o n s i t appears t h a t t h i s s t r a i n - e n e r g y model may be a p p l i e d to the more complex d i f f u s i o n systems, with the same t h e o r e t i c a l l i m i t s . The most important t e s t o f these c a l c u l a t i o n s o f AH' and A V i s t o determine i f they i n t u r n are equal to the v a l u e s determined f o r high-temperature creep. Eaton (1968) determined a AH' f o r creep of d u n i t e of 35.1 Kcal/mole u s i n g the .Ree-Eyring equation. The samples were d u n i t e c o n s i s t i n g o f approximately 85% o l i v i n e . The o l i v i n e was quoted as b e i n g 93% f o r s t e r i t e . Not o n l y d i d the data f i t the assumption o f Ree-Eyring d i f f u s i o n c r e e p , as s t a t e d by Eaton, but a l s o the c a l c u l a t i o n s o f AH' based on d i f f u s i o n data presented here (38.0 Kcal/mole). M i s r a and M u r r e l l (1965) c a l c u l a t e d a AH' o f a p p r o x i -mately 31.0 Kcal/mole. The samples were p e r i d o t i t e c o n s i s t i n g ' of 80% o l i v i n e , 15% pyroxene and accesory i r o n o x i d e s . U n f o r t u -n a t e l y no c a l c u l a t i o n o f the composition o f the o l i v i n e was made.. M i s r a and M u r r e l l (1965) s t a t e t h a t t h e i r c a l c u l a t i o n o f AH 1 was based on o n l y one measurement a t 7 00°C and may have a l a r g e e r r o r C o n s i d e r i n g t h i s f a c t and the e r r o r o f 20% we have allowed i n our 2 3 . estimate o f AH' for. d i f f u s i o n , ' the a c t i v a t i o n enthalpy f o r p e r i -d o t i t e creep f i t s . t h e v a l u e we have o b t a i n e d f o r Mg d i f f u s i o n i n o l i v i n e . U n f o r t u n a t e l y , no estimates o f Av^ have been made ex-p e r i m e n t a l l y . Both Eaton (1968) and M i s r a and M u r r e l l (1965) performed t h e i r creep experiments under the c o n d i t i o n s o f no con-, f i n i n g h y d r o s t a t i c p r e s s u r e and thus o n l y the temperature and s t r e s s dependence of the creep r a t e c o u l d be c a l c u l a t e d . CONCLUSIONS EXPERIMENTAL From the three experiments on o l i v i n e samples, no p e r - . c e p t i b l e g r a i n bulk or g r a i n boundary d i f f u s i o n was observed. T h i s was due to the i n f e r i o r q u a l i t y of the specimens and t h e i r subsequent breakdown as o u t l i n e d i n the s e c t i o n on e x p e r i m e n t a l . r e s u l t s . A f t e r temperature and p r e s s u r e c a l i b r a t i o n , the equ i p -ment designed f o r the experiment was found to be a c c u r a t e t o +5°C up to temperatures of 1000°C and acc u r a t e to +200 pounds up to a pressume o f 30,000 pounds. The method o f u s i n g the e l e c t r o n m i c r o -probe to "scan" f o r v a r i o u s atomic c o n c e n t r a t i o n was found t o be a useable technique f o r d i f f u s i o n d i s t a n c e s g r e a t e r than 10 microns. THEORETICAL I t has been shown t h a t the enthalpy o f a c t i v a t i o n f o r d i f f u s i o n o f Mg i n t o Mg 2Si0^ was 38.0 Kcal/mole, d e r i v e d from.the 2 9 . data of Jander and Stamm (1932). The a c t i v a t i o n volume of d i f -f u s i o n f o r Mg i n t o Mg2SiO^ was c a l c u l a t e d u s i n g eqn' s (4), and (15) as being approximately 3.0 cm^ mole. A g e n e r a l d i f f u s i o n . . equation f o r Mg chemical d i f f u s i o n i n Mg2Si0^ may be. w r i t t e n : •' D = 1.27 x IO - 4 exp(-38,000/RT) exp(-P 3.0/RT) (36) where: R = gas constant (cal/g.mole deg.) o r (bar cm^/mole deg.) T = temperature (°K) • P = p r e s s u r e (bars) D = d i f f u s i o n , c o e f f i c i e n t (cm^/sec) The p r e - e x p o n e n t i a l constant was c a l c u l a t e d from the data o f Jander and Stamm (1932). Weertman (1970) i n h i s c a l c u l a t i o n s of the creep s t r e n g t h of the e a r t h ' s mantle uses a d i f f u s i o n c o e f f i c i e n t based on Shew-man's " r u l e of thumb" (Shewman, 1963). T h i s r u l e was e m p i r i c a l l y d e r i v e d and s t a t e s t h a t f o r most m a t e r i a l s a T m (melting tempera-— 8 t u r e ) , D i s approximately 10 and a t T m/2.0, D i s approximately 10~^-^. F o l l o w i n g Weertman's r e a s o n i n g , assuming the lowest p o i n t on the Mg0-Si02 e u t e c t i c r e p r e s e n t s the m e l t i n g p o i n t o f d u n i t e , r a t h e r than the m e l t i n g p o i n t o f pure f o r s t e r i t e , the m e l t i n g p o i n t becomes 1550°C. A c a l c u l a t i o n o f D i n e q u a t i o n (36) u s i n g T - 1048°K g i v e s a r e s u l t of •'' D1048°K = ! - 4 X 1 0 " 1 2 ' P=l atm. •, . 30. Weertmas s t a t e s t h a t he t r i e d to circumvent h i s lack-of knowledge of AH 1 and AV' by u s i n g the.aforementioned " r u l e o f thumb". I t appears from our estimate of AH' and AV' that. ' — 16 Weertman's v a l u e f o r d I O 4 8 ° K ° ^ ^ S A P P r o x ; ' - m a t e l Y f o u r o r d e r s of magnitude too s m a l l . CREEP RATES I f the steady s t a t e creep o f mantle m a t e r i a l s such as d u n i t e i s a d i f f u s i o n dominated p r o c e s s , the e n t h a l p i e s ' o f a c t i - • v a t i o n and d i f f u s i o n should be approximately e q u a l . Assuming the chemical d i f f u s i o n of Mg i n M ^ S i O ^ AH 1 was c a l c u l a t e d to be 38.0 Kcal/mole. W i t h i n the 20% estimated e r r o r , t h i s v a l u e agrees w i t h Eaton's (1968) AH^ of 35.1 f o r the steady s t a t e creep of d u n i t e and with M i s r a and M u r r e l l ' s (1965) of 31.0 Kcal/mole f o r the steady s t a t e creep of p e r i d o t i t e . The author knows o f no c a l c u l a t i o n , e i t h e r e x p e r i m e n t a l or t h e o r e t i c a l , f o r the a c t i v a t i o n volume o f creep. The a c t i v a -3 t i o n volume of d i f f u s i o n o f 3.0 cm /mole c a l c u l a t e d i n t h i s paper from the t h e o r i e s of Zener (1952) and Keyes (1963) agrees w i t h the hard-sphere c r y s t a l model (see page 26) and assuming t h a t the p r e s s u r e dependence of the creep r a t e occurs i n the d i f f u s i o n c o -e f f i c i e n t (see page l l ) the a c t i v a t i o n volume f o r creep w i l l be assumed equal t o 3.0 cm 3/mole. Weertman (1970) has c a l c u l a t e d t h e o r e t i c a l creep r a t e s based on a process of d i s l o c a t i o n movement (see F i g u r e g) and ' 31. . has compared h i s r e s u l t s w i t h those o b t a i n e d by Eaton (1968). I f .a r e c a l c u l a t i o n o f Weertman's eq u a t i o n . (1970, p.151, eqn.9) i s made wit h the d e t e r m i n a t i o n o f the d i f f u s i o n c o e f f i c i e n t i n -eqn (36) the creep r a t e s are seen to i n c r e a s e .and•within the e r r o r of 20% f o r AH' agree w i t h the v a l u e s determined by Eaton (1968) (see F i g . g ) . T h u s . i t appears t h a t the c a t i o n i c d i f f u s i o n i n o l i - : v i n e i s the dominant process i n de t e r m i n i n g the temperature and pr e s s u r e dependence of the high-temperature, low s t r e e creep o f d u n i t e - p e r i d o t i t e r o c k s . Gordon (1965) has made an estimate o f the v i s c o s i t y of the mantle based on a Herring-Nabarro creep model. He assumes an a c t i v a t i o n enthalpy of approximately 70 Kcal/mole and an a c t i -v a t i o n volume o f approximately 10 cirr/mole from the d i f f u s i o n o f 0 io n s i n a p e r i c l a s e s t r u c t u r e . . Below depths of 400 km. the p e r i c l a s e s t r u c t u r e may be the dominant one (Ringwood, 1966), however, Gordon's (1965) c a l c u l a t i o n s of the creep r a t e above t h i s depth seem dubious. I f an o l i v i n e s t r u c t u r e i s assumed f o r the upper mantle then Zharkov's (1960) estimate of oxygen i o n d i f f u -s i o n i n o l i v i n e should be used. Zharkov's estimate f o r anion d i f f u s i o n i n o l v i n e was 93.0 Kcal/mole and from the da t a presented i n t h i s paper t h i s v a l u e appears to be approximately 2.5 times too l a r g e . McKenzie (1968) has s t a t e d t h a t the anion d i f f u s i o n i s p r o b a b l y a g r a i n boundary e f f e c t and thus would not dominate a t temperatures above .5 T™. 31a. ] Creep f^ais (sec)'' . . . ' 1 i " ' : • :• ' • I I (7? 70) 32. I t has been shown t h a t the creep r a t e of upper mantle rocks • (above 400 km.) i s dependent on temperature and p r e s s u r e through the c a t i o n i c d i f f u s i o n i n the o l i v i n e s i l i c a t e s . The c a l c u l a - .. t i o n of creep r a t e s a t g r e a t e r depths must not be made u s i n g an e x t r a p o 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 p resented here. In ' the r e g i o n of the mantle below 400 km.'•, c a l c u l a t i o n s of c a t i o n i c d i f f u s i o n i n m i n e r a l s having the s p i n e l or r u t i l e s t r u c t u r e must be used. . . . APPENDICIES APPENDIX A. ENGINEERING SPECIFICATIONS OF APPARATUS APPENDIX B CIRCUIT DIAGRAM OF TEMPERATURE CONTROLLER APPENDIX C OLIVINE COMPOSITION CALCULATIONS. 34. APPENDIX A ENGINEERING SPECIFICATIONS OF APPARATUS r The frame was f a b r i c a t e d from 0.8% carbon, h o t - r o l l e d s t e e l . From standard engineering, c a l c u l a t i o n s , the e l o n g a t i o n o f the t e n s i l e members would be .002 i n . a t 100 tons a p p l i e d l o a d and the d e f l e c t i o n of the cross-members would be .003 i n . a t 100 tons. The y i e l d p o i n t o f the t e n s i l e members i f 70,000 p . s . i . and f o r the d e f l e c t i o n members i s 42,000 p . s . i . The maximum o p e r a t i n g values, were 8,00 0 and 6,0 00 p . s . i . r e s p e c t i v e l y . Thus, a s a f e t y f a c t o r o f g r e a t e r than f i v e was o b t a i n e d , even when o p e r a t i n g a t ambient furnace temperatures o f 1000°C. Threaded nuts were used a t the f o u r j o i n t s ; 4.5 i n . nuts had a t e n s i l e ; s t r e n g t h a f a c t o r o f n i n e over the a p p l i e d l o a d and thus assembly ease was gained a t no l o s s of p r e s s u r e s t a b i l i t y . The c e n t r a l p r e s s u r e v e s s e l and p i s t o n s were made from Rene-R-41 s t e e l ( C o b a l t - n i c k e l a l l o y ) w i t h a c l e a r a n c e of .001 i n . between the p i s t o n s and c y l i n d e r w a l l s . The p i s t o n s were backed on c i r c l e s of Inconnel X-750 ( n i c k e l - s t e e l a l l o y ) and these i n t u r n were backed on c i r c l e s of 310 s t a i n l e s s - S t e e l . The v a r i o u s a l l o y s were s e l e c t e d f o r s t r e n g t h as w e l l as r e s i s t a n c e to creep at the e l e v a t e d temperatures. 3JT. Boundary -of Printed C'rcoit Gerard. I20V , ~p/ 1 i _ ^nrrsijirfo SC. J t f IP,' To Oven . 169 1°. BY IZ3 , INHOOZ A A A -JOOJh ^isv I 2 V 2 9 7 9 .O0Hr7 SOW 3oV. '. j IK. SOW.C 2 3 8 9 . PL*. A / V V 82-TL / O O i l . • / K . sw. V70-f L-33K.SL. A A A / — | .66 ynn) IN HOO& 8 8 1> TP. ftppendiy. B- C i r c u i t Diagram •for ' 0 1/&r? Temp era fare* icjram Jor uv&n tempi APPENDIX C C a l c u l a t i o n of bulk composition of o l i v i n e c r y s t a l s : Samples I and I I From d i f f r a c t o m e t e r o s c e l l a t i o n s : 29 (degrees) d s p a c i n g (Angstroms) 3 2 . 2 8 5 . 2 . 7 7 3 3 2 . 2 8 0 • 2 . 7 7 3 3 2 . 3 0 5 2 . 7 7 1 3 2 . 2 8 0 2 . 7 7 3 d = 2 . 7 7 2 5 d S 2 . 7 7 3 From F i s h e r and M e d a r i s . ( 2 9 ) : X F o r s t e r i t e = 15 . 8 1 1 3 " . { 3 . 0 3 5 8 - 2 . 7 7 2 5 } - 7 . 2 2 5 ( x F o r . = . 8 8 7 8 Samples I and II are approximately 8 9% For, Sample I I I 20 (degrees) d(130) s p a c i n g (Angstroms) 32.220 2.778 32.230 2.777 32.230 2.777 32.250 2.77 6 d =.; 2..7770' x F o r . = 15.8113 ^3.0358 - 2.7770 j- - 7.2250 x F o r . = .82 88 Sample I I I i s approximately 83% F o r . 38. REFERENCES 1) Adda, Y, and P h i l i b e r t , J(1966). La. D i f f u s i o n dans l e s . S o l i d s , V o l . 2. Presses U n i v e r s i t a i r e s de France. 2) Anderson, O.L. e t . a l . (1968). 'Some E l a s t i c Constant Data of M i n e r a l s Relevant to Geophysics, Rev. Geophysics, V o l . 6 #4, p. 4.91. 3) A s k i l l , J (1970).' T r a c e r D i f f u s i o n Data f o r M e t a l s , A l l o y s and Simple Oxides, Plenum P r e s s . 4) . Crank, J (1956) Mathematics o f D i f f u s i o n , Oxford P r e s s . 5) Eaton, J.F.. (1968) The High Temperature Creep o f Dunite, Ph.D. T h e s i s , P r i n c e t o n U n i v e r s i t y , P r i n c e t o n , New J e r s e y . 6) F i s h e r , J.C. (1951) J o u r n a l o f A p p l i e d P h y s i c s , V o l . 22. 7) F y f e , W.S., Turner, F . J . and Verhoogen, J . (1958) Meta-morphic Reactions and Metamorphic F a c i s G.S.A. Memoir 73. 8) Gordon, R.B. (1965) D i f f u s i o n Creep i n the Ea r t h ' s Mantle J.G.R., V o l . 68, p.4967. 9) H e r r i n g , C. (1950). D i f f u s i o n a l V i s c o s i t y o f P o l y c r y s t a l l i S o l i d s , J . A p p l i e d Phys., V o l . 21, p.437. 10) . Jander, W. and Stamm, W. (1932) 2 e i t . Anorg. A l l e g . Chem. V o l . 207, #2. 11) J o s t , W. (1952) D i f f u s i o n , Academic Press 12) Keyes, R.W. (1963) Continium Models of the E f f e c t o f Pr e s s u r e on A c t i v a t e d Processes, i n S o l i d s Under P r e s s u r e , e d i t e d by W. Paul and P. Warshauer, McGraw-Hill Pub. Co.Ltd. 13) K i t t e l , C. (1966) I n t r o d u c t i o n t o S o l i d S tate P h y s i c s , 3rd Ed., John Wiley and Sons. 3 9 . 14) Lazarus, 0. and N a c h t r i e b , N.H. (1963) E f f e c t of High P r e s s u r e on D i f f u s i o n , i n S o l i d s Under P r e s s u r e , McGraw-Hill Pub. Co.Ltd. 15) McKenzie, D.P. (1968) Ge o p h y s i c a l Importance o f High Temperature Creep, i n H i s t o r y o f the Earth's" C r u s t , ed. by P.H. Phimery, P r i n c e t o n Univ. P r e s s . 16) McKinley, H e i n r i c h and Witty (1966) The E l e c t r o n M i c r o -Probe, John Wiley and Sons. 17) Medaris, L.G. and F i s h e r , G.W. (1969) C e l l Dimensions and D e t e r m i n a t i v e Curve f o r S y n t h e t i c Mg-Fe O l i v i n e , Amer. Min. V o l . 57, p.771 . 18) M i s r a , A.K. and M u r r e l l , S.A.F. (1965) E f f e c t of Tempera- ' t u r e and S t r e s s on Creep of Rocks, Geoph. Jour, of Royal A s t r . Soc. V o l . 9. 19) Nabarro, F.R.W. (1948) Deformation o f C r y s t a l s by Motion of S i n g l e Ions, Rept. Conf. S t r e n g t h o f S o l i d s , B r i s t o l , P. 75. 20) Passmore, E.M. e t . a l . (1966) Creep of Dense P o l y s r y s t a l l i n e MgO, Jour. American. Cer. Soc. Vol.46, p.133. 21) Ree, E.H., Ree J . and E y r i n g H. (1960) R e l a x a t i o n Theory of Creep i n ' M e t a l s , J . Eng. Mech. Div.; Proc. : Amer. Soc. C i v i l Eng. p.41. 22) . R i c e , S.A. and M a c h t r i e b , N.H. (1959) Dynamical Theory o f D i f f u s i o n i n C r y s t a l s , J . Chem. Phys. V o l 31, p. 139. 23) Ringwood, A.E. (19 66) Mineralogy of the Mantle i n Advances i n E a r t h S c i e n c e , e d i t e d by P.M.Hurley, p.357, M.I.T. P r e s s , Cambridge, Mass. 24) . Sherby, O.D. (1962) F a c t o r s A f f e c t i n g the High Temperature Strength o f S o l i d s , A c t a . Met. Vol.10, p.135.,' 25) Sherby, O.D. and Burke, P.M. (1967) Mechanical Behaviour of C r y s t a l l i n e S o l i d s a t E l e v a t e d Temperatures Prog, i n Mat. S c i . p. 325. 40. 2 6 ) Varsheneya and Cooper ( 1 9 6 9 ) T e c t i t e s , Inhomogeneities and Ion D i f f u s i o n , J.G.R. V o l . 7 4 , # 2 7 . 2 7 ) Wert, C.A., and Z e n e r / C . ( 1 9 4 9 ) I n t e r s t i t i a l Atomic D i f f u s i o n C o e f f i c i e n t s , Phys.' Rev. V o l . 7 6 , p. 1 1 6 9 . 2 8 ) Weertman,- J . ( 1 9 5 7 ) Steady S t a t e Creep Through D i s l o c a t i o n Climb, J . App. Phys. V o l 2 8 , - # 3 . 2 9 ) Weertman, J . ( 1 9 7 0 ) Creep S t r e n g t h o f the E a r t h ' s Mantle, Rev. Geoph. and Space S c i . , V o l . 8 , # 1 , p . 1 4 5 . . 3 0 ) Whipple,. R.I. ( 1 9 5 4 ) P h i l . Mag. V o l . - 4 5 , p. 1 2 2 5 . 3 1 ) Zener, C. ( 1 9 5 2 ) Theory of D Q f o r Atomic D i f f u s i o n i n Me t a l s , i n Imperfections i n Nearly P e r f e c t C r y s t a l s John Wiley and Sons, Inc., New York. 3 2 ) Zharkov, V.N. ( 1 9 6 0 ) Proc. of Geoph. I n s t . im.O. Yu. Schmidt. Academy o f Sc i e n c e s o f U.S.S.R., Moscow. 

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