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Recovery in cadmium Hamre, Edmond Charles 1970

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RECOVERY IN CADMIUM by EDMOND CHARLES HAMRE B.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1964  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of METALLURGY  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA January, 1970  In p r e s e n t i n g t h i s t h e s i s an advanced degree at the L i b r a r y I  for  freely  f u l f i l m e n t o f the of B r i t i s h  available  for  requirements f o r  Columbia, I agree  that  reference and study.  p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s  thesis  s c h o l a r l y purposes may be granted by the Head o f my Department o r  by h i s of  the U n i v e r s i t y  s h a l l make i t  f u r t h e r agree tha  in p a r t i a l  this  representatives.  It  thesis for financial  i s understood that copying o r p u b l i c a t i o n gain s h a l l  written permission.  Department o f  Metallurgy  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  J a n u a r y 12, 1970  not be allowed without my  ABSTRACT  The recovery of mechanical p r o p e r t i e s f o l l o w i n g deformation of s i n g l e c r y s t a l cadmium has been studied. Such recovery has been observed above 0.26 T  (-120°C). M  C r y s t a l s covering a range of o r i e n t a t i o n were deformed i n .tension at -196°C and recovered at elevated temperatures.  Transmission e l e c t r o n  microscopy to r e l a t e t e n s i l e and recovery behaviour to d i s l o c a t i o n structures was found to be impossible.  I t was observed that work hardening during the i n i t i a l p o r t i o n of the easy g l i d e region i s completely recoverable. At higher s t r a i n s i n easy g l i d e , a p o r t i o n of the work hardening was not recoverable.  It is  believed that i n t h i s l a t t e r s e c t i o n , d i s l o c a t i o n s are generated on the second order pyramidal system {1122} <1123>.  These d i s l o c a t i o n s w i l l combine  with b a s a l d i s l o c a t i o n s to form stable obstacles i n the l a t t i c e which w i l l be. responsible f o r the non-recoverable work hardening.  The end of easy g l i d e was found to occur at x = 20°, of recovery or i n i t i a l o r i e n t a t i o n .  independent  This phenomenon i s associated with  flow on the second order pyramidal system which w i l l produce a much higher density of obstacles at t h i s p o i n t , r e s u l t i n g i n a higher work hardening rate.  Recovery i n stage I I was observed to increase the amount of strain attainable.  I t was also observed that while recovery up to intermediate  ii  s t r a i n s i n stage I I affected only b a s a l d i s l o c a t i o n s , both b a s a l and pyramidal d i s l o c a t i o n s  appear to be recovered at high s t r a i n s .  Pyramidal d i s l o c a t i o n s may recover by the processes observed by P r i c e .  The rate c o n t r o l l i n g mechanism f o r y i e l d and flow of cadmium s i n g l e c r y s t a l s i s thought to be one of the non-conservative motion of jogs.  An attempt was made to c a l c u l a t e an a c t i v a t i o n energy f o r the recovery process, but the data d i d not y i e l d any meaningful numbers. This may be a r e s u l t of the d e f i n i t i o n of recovery adopted f o r t h i s work.  ACKNOWLEDGEMENT  The author wishes to thank h i s research d i r e c t o r , Dr. E. Teghtsoonian, f o r h i s h e l p f u l advice and encouragement.  He also  wishes to thank f e l l o w graduate students f o r many s t i m u l a t i n g discussions.  F i n a n c i a l assistance i n the form of a N a t i o n a l Research Council Studentship, National Research Council operating grant A-2452 and a. hard-working wife i s g r a t e f u l l y acknowledged.  ,  iv TABLE OF CONTENTS  PAGE  1. INTRODUCTION  1  2. EXPERIMENTAL PROCEDURE  4  2.1 Sample preparation  4  2.2 Tensile t e s t i n g  5  2.3 Electron microscopy  5  2.4 Recovery tests  7  3. RESULTS  8  3.1 Resolved shear stress - shear s t r a i n p l o t s 3.2 Recovery r e s u l t s  8 12  3.2.1 Method (1)  14  3.2.2 Method (2)  20  3.2.3 Method (3)  22  3.3 S t r a i n rate change tests  28  3.3.1 A c t i v a t i o n volume  29  3.3.2 A T versus T  32  3.4 Work hardening parameters  38  3.4.1 Stage I  38  3.4.1.1 C r i t i c a l resolved shear stress  39  3.4.1.2 Work hardening rate  39  3.4.1.3 Length of easy g l i d e  44  3.4.1.4 Recovery e f f e c t s  49  3.4.2 Stage I I 3.4.2.1 Orientation  49 effects  3.4.2.2 Recovery e f f e c t s 3.4.3 Stage I I I  49 49 50  PAGE 54  4. DISCUSSION 4.1 Recovery r e s u l t s  54  4.1.1 A c t i v a t i o n energy  54  4.1.2 Comparison of recovery methods (1), (2), (3)  59  4.2 Work hardening parameters  66  4.2.1 Easy g l i d e parameters  66  4.2.2 Theories of stage I I  69  4.2.2.1 Condensation of vacancies  70  4.2.2.2 Non-active basal d i s l o c a t i o n s  70  4.2.2.3 Twinning  70  4.2.2.4 Second order pyramidal s l i p  71  4.2.3 Present model  73  4.3 S t r a i n rate change tests  78  4.3.1 Stage I  80  4.3.1.1 A c t i v a t i o n volume behaviour  80  4.3.1.2 C o t t r e l l - S t o k e s behaviour  84  4.3.2 Stage I I  86  4.4 Flow stress following recovery  88  4.5 Work hardening i n stage I I  89  5. SUMMARY AND CONCLUSIONS  93  6. APPENDIX  95  6.1 E l e c t r o n microscopy techniques  95  6.1.1 Cutting a t h i n s e c t i o n  95  6.1.2 Thinning  96  6.1.3 Examination of thinned specimens  97  REFERENCES  98  vi LIST OF FIGURES PAGE F i g 1.  O r i e n t a t i o n range of s i n g l e c r y s t a l s  6  F i g 2.  T y p i c a l resolved shear s t r e s s - shear s t r a i n curve at -196°C  9  F i g 3.  T y p i c a l resolved shear s t r e s s - shear s t r a i n curve at 20°C  10  F i g 4.  T y p i c a l t e n s i l e kinks formed at 20°C  11  F i g 5.  D e f i n i t i o n of recovery  13  Fig 6.  The v a r i a t i o n of method (1) recovery at 30°C w i t h s t r a i n  15  F i g 7.  The v a r i a t i o n of method (1) recovery at -20°C with s t r a i n  16  Fig 8.  The v a r i a t i o n of method (1) recovery at -70°C w i t h s t r a i n  17  F i g 9.  The v a r i a t i o n of method (1) recovery at -100°C with s t r a i n  18  Fig 10. The v a r i a t i o n of method (1) recovery w i t h time at y ~ 0.25  19  Fig 11. The v a r i a t i o n of s a t u r a t i o n recovery with s t r a i n  21  F i g 12. The v a r i a t i o n of method (2) recovery at 20°C with s t r a i n  23  F i g 13. The v a r i a t i o n of method (2) recovery at -30°C w i t h s t r a i n  24  Fig 14. The v a r i a t i o n of method (2) recovery at -70°C with s t r a i n  25  Fig 15. The v a r i a t i o n of method (2) recovery at -90°C and -110°C with s t r a i n  26  F i g 16. The e f f e c t of temperature on easy g l i d e deformation  27  F i g 17. The determination of load change with s t r a i n r a t e change  29  F i g 18. Comparison of a c t i v a t i o n volume data obtained from s t r a i n rate up-change and down-change Fig 19. The s t r a i n dependence of a c t i v a t i o n volume F i g 20. The e f f e c t of s a t u r a t i o n recovery on a c t i v a t i o n volume i n easy g l i d e Fig 21. The e f f e c t of s a t u r a t i o n recovery on a c t i v a t i o n volume i n stage I I  30 31 33 34  F i g 22. The v a r i a t i o n of A T with T  35  F i g 23. The v a r i a t i o n of A T - T w i t h s a t u r a t i o n recovery i n easy g l i d e  36  vii  PAGE Fig  24.  The v a r i a t i o n of A T -  T with s a t u r a t i o n recovery i n stage I I  Fig  25.  O r i e n t a t i o n dependence of the c r i t i c a l  r e s o l v e d shear  stress  40  Fig  26.  Temperature dependence of the c r i t i c a l  r e s o l v e d shear  stress  41  Fig  27.  O r i e n t a t i o n dependence of s t a g e I work hardening  Fig  28.  O r i e n t a t i o n dependence of the l e n g t h of easy g l i d e  Fig  29  Temperature dependence of stage I work h a r d e n i n g  rate  Fig  30.  The e f f e c t of o r i e n t a t i o n on the l e n g t h of s t a g e ( s p e c i f i c example)  I  rate  42  (schematic). ,  47  31.  Twins formed i n the s t a g e I  Fig  32.  The  Fig  33.  R e l a t i v e shapes of c r y s t a l s deformed a t 20°C and  Fig  34.  The v a r i a t i o n of l o g  (1~R) w i t h  Fig  35.  The v a r i a t i o n of rog  (slopie) w i t h r e c i p r o c a l a b s o l u t e temp.  Fig  36.  Comparison of s a t u r a t i o n r e c o v e r y w i t h d e f o r m a t i o n -196°C and 20°C  37.  - stage I I t r a n s i t i o n  (x  48 I I  -196°C  time,  Schematic diagram of r e c o v e r y to a s i m i l a r s t r e s s w i t h d e f o r m a t i o n a t d i f f e r e n t temperatures  52  (2) r e c o v e r y a t -30°C and  58  at 60 level  T y p i c a l method  Fig  39.  Comparison of p r e s e n t work to t h a t of Bocek and Kaska r e g a r d i n g the temperature dependence of the c r i t i c a l r e s o l v e d shear s t r e s s Comparison of shear s t r e s s on second w i t h t h a t on b a s a l p l a n e  51  56  38.  40.  200)  e f f e c t of s a t u r a t i o n r e c o v e r y on the l e n g t h of s t a g e  Fig  Fig  43 45  Fig  Fig  37  63  -70°C  order pyramidal  65  67 plane 74  Fig  41.  The v a r i a t i o n of s a t u r a t i o n r e c o v e r y w i t h  orientation  Fig  42.  The v a r i a t i o n of flow s t r e s s f o l l o w i n g s a t u r a t i o n r e c o v e r y i n stage I I  76  90  viii LIST OF TABLES PAGE Table I .  The e f f e c t of i n i t i a l o r i e n t a t i o n and recovery on the length of stage I  46  1 1. INTRODUCTION  Recovery may be defined generally as the r e v e r s i o n towards an i n i t i a l state by a meta-stable structure when s u f f i c i e n t energy i s present. For the purposes of t h i s study, t h i s d e f i n i t i o n may be stated more specifically.  In t h i s case, recovery occurs when the flow s t r e s s of a  cadmium s i n g l e c r y s t a l decreases towards the i n i t i a l y i e l d s t r e s s when thermal energy i s added to the system.  This decrease i n flow s t r e s s - i s .  probably caused by a decrease i n d i s l o c a t i o n density as a r e s u l t of egress of d i s l o c a t i o n s from the system.  When recovery occurs concurrently w i t h  deformation, i t i s termed dynamic recovery, whereas i f i t takes place under s t a t i c conditions i t i s termed s t a t i c recovery. Dynamic recovery i s an important phenomenon i n that i n many instances i t i s responsible f o r at l e a s t part of the temperature of some work hardening parameters.  sensitivity  This recovery takes place at intermediate  temperatures - high enough so that d i s l o c a t i o n s may rearrange but not so high as to promote r e c r y s t a l l i z a t i o n .  themselves,  Dynamic recovery probably  accounts f o r most of the d i f f e r e n c e i n the work hardening r a t e of cadmium between l i q u i d nitrogen and room temperatures.  The suppression of both,  s t a t i c and dynamic recovery by the t h o r i a d i s p e r s i o n i s thought to be responsible f o r the high strength at high temperatures i n T.D. n i c k e l . Most studies of recovery per se have not d e a l t with the change of mechanical p r o p e r t i e s , but rather have been concerned w i t h the behaviour of point defects which have been introduced to the c r y s t a l l a t t i c e e i t h e r 43  by deformation or r a d i a t i o n .  This type of recovery u s u a l l y takes place  at temperatures below that where the mechanical p r o p e r t i e s are s i g n i f i c a n t l y a f f e c t e d , and the p r i n c i p a l method f o r measuring such recovery i s the change i n e l e c t r i c a l r e s i s t i v i t y .  44  -,  2 One study which looked at the change i n flow s t r e s s i n conjunction with e l e c t r i c a l measurements was that of Sharp, M i t c h e l l and Christian.''" They found an annealing peak i n Cd at T^ = .25 which they associated with s i n g l e vacancy migration.  T„ = .25 i s the same temperature  at which dynamic recovery becomes apparent.  These r e s u l t s are comparable 2  to those of a s i m i l a r study by P e i f f e r and Stevenson.  i  E l e c t r o n microscope studies on evaporated p l a t e l e t s of zinc 3 by Kroupa and P r i c e  associated dynamic recovery with conservative,; climb  of p r i s m a t i c d i s l o c a t i o n loops.  In t h i s mechanism, the area i n s i d e the  loop i s conserved, and the loop climbs by the transfer of vacancies around 4 i t by pipe d i f f u s i o n .  Hirsch and L a l l y  found that dynamic recovery i n  t h i n f o i l s of Mg was due to c r o s s - s l i p and subsequent a n n i h i l a t i o n of; screw d i s l o c a t i o n s .  Risebrough^  thinks that t h i s mechanism would ,not be  operative i n Cd due to the lack of a s u i t a b l e c r o s s - s l i p system.  :;  Studies which looked s p e c i f i c a l l y at the change i n mechanical 6 properties due to recovery were performed by Rath et a l al^..  and by Liicke et  The former was concerned with the thermal a c t i v a t i o n  characteristics  of recovery i n aluminum s i n g l e c r y s t a l s , and the l a t t e r looked at the e f f e c t of recovery on various work hardening parameters i n z i n c .  Neither  of these studies concluded anything about the d i s l o c a t i o n arrangements or the e f f e c t of recovery on d i s l o c a t i o n s .  :;  Other studies which were concerned with the a c t i v a t i o n parameters 8 9 related to load decay a f t e r s t r a i n (Oelschlagel on z i n c ; Lukac or\ cadmium; Feltham"^ on Mg) d i d not draw any conclusions w i t h respect to d i s l o c a t i o n motion. I t was the aim of t h i s study to i n v e s t i g a t e the recovery of mechanical properties i n s i n g l e c r y s t a l cadmium with s p e c i f i c reference to the d i s l o c a t i o n behaviour during recovery.  This work covers the range  3  of temperature from the f i r s t observance of recovery up to temperatures at which strained s i n g l e c r y s t a l s would r e c r y s t a l l i z e . In the course of t h i s work, the o v e r a l l work hardening behaviour of cadmium s i n g l e c r y s t a l s has been studied so that the e f f e c t s of recovery may be b e t t e r understood.  ;  Cadmium was chosen as the m a t e r i a l to be used i n t h i s work p r i m a r i l y because of i t s c r y s t a l s t r u c t u r e .  In the past, most d i s l o c a t i o n  theories have been concerned with the simplest c r y s t a l s t r u c t u r e , namely face-centered cubic, and as a r e s u l t r e l a t i v e l y l i t t l e i s known of the d i s l o c a t i o n mechanisms i n hexagonal close-packed metals.  A second reason  for choosing cadmium i s the ease with which s i n g l e c r y s t a l s may be produced i n quantity.  Other hexagonal metals such as zirconium and titanium ,do not  possess t h i s q u a l i t y .  F i n a l l y , cadmium was chosen because of the d u c t i l i t y  i t e x h i b i t s at temperatures below the recovery range.  Z i n c , which i s  s i m i l a r to cadmium i n most other respects has a tendency to cleave at low temperatures.  -  ,  .  Other c h a r a c t e r i s t i c properties of cadmium such as i t s strength, largest of a l l c/a r a t i o s , medium stacking f a u l t energy, and anisotropy were not s i g n i f i c a n t f a c t o r s .  i  Single c r y s t a l s were used instead of p o l y c r y s t a l l i n e m a t e r i a l to exclude the complicating g r a i n boundary c o n s t r a i n t s .  With s i n g l e c r y s t a l s  i t i s possible to c a l c u l a t e the shear s t r e s s on any p a r t i c u l a r plane.at any time.  4  2. EXPERIMENTAL  2.1 Sample preparation The m a t e r i a l used f o r t h i s study was 99.999% Cd as supplied by Cominco L t d . , T r a i l , B.C.  This m a t e r i a l was received i n the form of  one-half inch bars, and was extruded to 0.2 inch rods f o r subsequent growth i n t o s i n g l e c r y s t a l s . Single c r y s t a l s were grown using a modified Bridgman i n evacuated 5 mm. i n s i d e diameter pyrex tubes which had previously been coated on the i n s i d e surface with Aquadag.  The Aquadag, which i s a suspension of c o l l o i d a l  graphite i n water, prevented the p a r t i a l wetting of the pyrex by molten cadmium.  I f such wetting d i d take p l a c e , the surface of the r e s u l t a n t  s i n g l e c r y s t a l was marked by many craters not u n l i k e bubbles.  The tubes  were lowered at a r a t e of 6 cm/hr. through a furnace with a thermal gradient of 25°C/cm. Randomly oriented c r y s t a l s were produced by t h i s method, Wjhile c r y s t a l s oriented f o r long easy g l i d e , which were required f o r most recovery t e s t s , were produced with a standard seeding technique once a s u i t a b l e , o r i e n t a t i o n had been obtained. The pyrex tubes were removed from the c r y s t a l by d i s s o l u t i o n , i n concentrated HF, following which the c r y s t a l s were etched i n concentrated HC1.  This etch revealed any g r a i n boundaries which may have been present,  and also removed any Aquadag which may have adhered to the specimen surface. C r y s t a l s were chemically polished i n a fresh s o l u t i o n of: 320 gm. C r 0  3  ,  ;  ;|  ,  40 gm. Na S0^ 2  1000 ml. H 0 2  to remove approximately 0.002 inches from the surface.  . •  The o r i e n t a t i o n of each specimen was determined to an accuracy of ± 1° using the back r e f l e c t i o n Laue technique.  The o r i e n t a t i o n range  of c r y s t a l s used i n t h i s study i s shown i n F i g . 1. had an o r i e n t a t i o n of 40° < x  < 46° where y  The majority of samples  i s the i n i t i a l angle between  the basal (0001) plane and the t e n s i l e a x i s .  ,•  2.2 T e n s i l e t e s t i n g  >  Tensile deformation of the c r y s t a l s was c a r r i e d out on a f l o o r _2  model Instron (Model TTM) at i n i t i a l s t r a i n rates varying from 1.3 x 10 sec -5 -1 -3 -1 to 1.3 x 10 sec , with the majority of t e s t s being done at 1.3 x 10 sec C r y s t a l s 10 cm. long were soldered i n t o aluminum g r i p s for t e s t i n g Since there was no reduced gauge s e c t i o n on the specimens, the length between the g r i p s constituted the gauge length.  This length was 6 to 8 cm.,  and with a diamter of 0.5 cm., the length to diameter r a t i o n always f exceeded 10 to 1. Test temperatures were maintained by immersing the specimen i n t o an appropriate l i q u i d held at the required temperature.  The baths used  and t h e i r respective temperature ranges were: l i q u i d nitrogen  -196°C  petroleum and ether  -140 to -70°C  ••  ethanol  -70 to 0°C  ,  water  0 to 100°C  2.3 E l e c t r o n microscopy An attempt to c o r r e l a t e mechanical properties and recovery behaviour  to d i s l o c a t i o n d i s t r i b u t i o n was made by means of transmission  electron microscopy.  Unfortunately, experimental d i f f i c u l t i e s and the  opacity of cadmium to electrons rendered  this investigation f r u i t l e s s .  6  (0001)  (1010)  F i g 1. O r i e n t a t i o n range of s i n g l e c r y s t a l s .  7 An account of the techniques employed and the problems encountered i s given i n Appendix 1.  2.4 Recovery tests Recovery t e s t s were c a r r i e d out on cadmium s i n g l e c r y s t a l s i n three d i f f e r e n t ways: 1)  C r y s t a l was deformed at -196°C to a predetermined s t r a i n , the  load was removed from the specimen, and the temperature was r a i s e d to allow recovery f o r a f i x e d time.  The temperature was lowered to -196°C,  and deformation resumed f o r an a r b i t r a r y s t r a i n increment.  Referred to i n  the following text as e i t h e r method (1) or type (1) recovery. 2)  C r y s t a l was deformed to some predetermined s t r a i n , the load was  removed, and the c r y s t a l allowed to recover f o r some f i x e d time, then deformation was resumed f o r a s u i t a b l e s t r a i n increment, a l l at a constant temperature.  Referred to i n the f o l l o w i n g text as method (2) or  type (2) recovery. 3)  ;  ,  (  C r y s t a l was deformed to a predetermined s t r a i n , the Instron  crosshead was stopped, and the load allowed to decay during recovery f o r a f i x e d time, then deformation was resumed f o r an a r b i t r a r y s t r a i n increment, a l l at a constant temperature. text as method (3) recovery.  ,  Referred to i n the following, ,  ;  8 3. RESULTS  3.1 Resolved shear stress - shear s t r a i n p l o t s  Load-elongation data were transformed to resolved shear s t r e s s resolved shear s t r a i n p l o t s using the f o l l o w i n g relationships"'""'":  T  =  p J sinxo [(  Y  =  £1 Iin^ [ £i {  l  (  i- ° ^ 0 1 Yl A  l  2  )  )  2  ~  -  s  s i n  i  n  2  X  0  1^-cosXo >  (2) *  C a l c u l a t i o n and p l o t t i n g of the r e s u l t s was performed on a IBM 7044 and l a t e r on a IBM 360/67 computer. A t y p i c a l curve f o r a c r y s t a l deformed at -196°C i s shown i n F i g 2.  The c r i t i c a l resolved shear stress has been determined by  extrapolating the l i n e a r easy g l i d e back to zero s t r a i n . between various t e s t s , the length of easy g l i d e (jj)  w a s  For comparison defined as the  s t r a i n at the i n t e r s e c t i o n of the extrapolated l i n e a r stage I and •,, stage I I sections.  In general, f r a c t u r e occurred i n the stage I I region  of the curve at -196°C, and as a consequence a t h i r d stage to the ,work hardening curve with a lower work hardening rate was not observed.. Twinning generally occurred during the t r a n s i t i o n from stage I to stage I I , as evidenced by load drops on the load-elongation p l o t s , and by metallographic examination. At 20°C, the curve s t i l l showed e s s e n t i a l l y two stages of work hardening, with the work hardening rate higher i n the second stage * I t i s r e a l i z e d that equations (1) and (2) are v a l i d only f o r s l i p on a s i n g l e s l i p system. While t h i s i s probably not true f o r cadmium f o l l o w i n g easy g l i d e , the c a l c u l a t i o n s have been extended to f a i l u r e to allow- comparison of present r e s u l t s to other work.  50CH  400-J  0"T  1  1  1  1  1  1  1  1  0  -5  10  1-5  20  2-5  3-0  3-5  4 0  F i g 3.  T y p i c a l resolved shear stress - shear s t r a i n curve at 20°C.  r4  5  than i n the f i r s t as shown i n F i g . 3.  These work hardening rates are  s u b s t a n t i a l l y lower than those obtained at -196°C.  F i g . 3 also shows that  the work hardening curve i s not as l i n e a r as i t i s at -196°C.  During  the i n i t i a l deformation at 20°C, the c r y s t a l s developed many t e n s i l e kinks (10 to 20 i n an i n i t i a l 8 cm. length) as shown i n F i g . 4.  At -196°C  the deformation was more homogeneous, with fewer less sharply defined kinks formed.  Twinning was also a feature at 20°C.  twinning was not operative u n t i l very high s t r a i n s  In t h i s case, the (^400% shear s t r a i n ) .  F i g 4. T y p i c a l t e n s i l e kinks formed at 20°C.  12 3.2 Recovery  results  Risebrough"' has found that dynamic recovery i n cadmium does not take place at temperatures less than -120°C.  This has been confirmed  i n the present study, where i t was found that f o r an anneal of 60 min. at -140°C, no recovery was detectable, whereas 15 min. recovery at -110°C showed s i g n i f i c a n t recovery. -140° < T < -110°C.  Thus recovery must begin i n the range ,  Accordingly, most a t t e n t i o n has been focussed on  r e s u l t s from method (1), i n which a l l deformation i s at -196°C, and.recovery takes place i n an unloaded specimen at elevated temperatures.  Under  •  these circumstances, recovery takes place only during the anneal cycles and not during deformation.  Both methods (2) and (3) i n v o l v e recovery  during deformation. One problem involved with any study of recovery i s the d e f i n i t i o n of recovery i t s e l f .  For t h i s study, recovery has been defined as that  ;  f r a c t i o n of the work hardening which i s removed by a given anneal. In terms of flow stress parameters,this i s : T .  R  =  ,  1-1  -  T1 .  T . , - crss  =  . , due to anneal softening work hardening ° r  and i s described g r a p h i c a l l y i n F i g . 5.  .-, .  This d e f i n i t i o n allows f o r a  range of recovery from zero f o r no change i n flow stress to 100%, f o r a recovery which r e s u l t s i n a flow stress equal to the i n i t i a l resolved shear s t r e s s .  critical  Recovery, as defined, i s independent of s t r a i n or  absolute flow stress values. that used by other workers^'^.  This d e f i n i t i o n i s also consistent with .  13  SHEAR  STRAIN  y  Fig 5. D e f i n i t i o n of recovery.  A second method of measuring recovery which may be a p p l i c a b l e , but which has not been used i n t h i s study would be to compare the flow stress under test conditions to that at -196°C.  I n t h i s way both s t a t i c  and dynamic recovery would be measured and accounted f o r i n tests i n which both occurred. I t would be d i f f i c u l t to i n t e r p r e t such data since there are v a r i a b l e s involved such as stress and s t r a i n which are d i f f i c u l t to standardize.  Consequently, t h i s approach has not been applied to-  the data. j i  3.2.1 Method (1) The r e s u l t s obtained f o r method (1) recovery are shown i n Figs. 6 through 9, which are p l o t s of recovery (R) vs. s t r a i n (y)• Comparing these p l o t s , i t i s r e a d i l y seen that recovery increases with increasing temperature and time.  R, however, also v a r i e s with s t r a i n ,  and t h i s must be taken i n t o account when comparing recovery under various conditions. With a l l deformation at -196°C, the resolved shear s t r e s s shear s t r a i n curves were s i m i l a r .  I f recovery i s a s i m i l a r f u n c t i o n  of s t r a i n at a l l times and temperatures  i n v e s t i g a t e d , the recovery values  may be compared at any a r b i t r a r y value of s t r a i n .  The same value of  s t r a i n should provide comparison of recovery at a constant s t r u c t u r e and so provide a constant a c t i v a t i o n entropy. a r b i t r a r y s t r a i n of y = 0.25  Such a comparison at an  i s shown i n F i g . 10.  ;J  Some of the points on  t h i s p l o t have been determined by e x t r a p o l a t i o n of the i n d i v i d u a l graphs (Figs. 6-9), but t h i s should not introduce any s i g n i f i c a n t errors since comparison at the end of stage I showed that while the values of recovery were reduced, the o v e r a l l e f f e c t of time and temperature was the same, ( i . e . the slope of the p l o t of recovery vs. time was the same at most temperatures).  At -100°C, the values were obtained by maintaining t h e ,  difference between the two curyes which i s present at y^ extrapolating the 40 min. curve to y = 0.25.  (y = 1.8), -and  This operation i n e f f e c t  neglects the i n i t i a l value at 90 min., but i t was f e l t that the d i f f e r e n c e at  i s a more r e l i a b l e value than the s i n g l e point which was neglected.  A second case i n which the behaviour of recovery with time was not the j same at y = .25 and y = y  was with the+30°C set of data.  In t h i s ,case,  i t was found that recovery was e s s e n t i a l l y independent of time at the end of stage I as shown i n F i g . 6.  A p o s s i b l e reason f o r t h i s anomaly . w i l l  F i g 6. The v a r i a t i o n of method (1) recovery at 30°C with s t r a i n .  20  F i g 9-  The v a r i a t i o n of method (1) recovery at -100°C with s t r a i n .  F i g 10. The v a r i a t i o n of method (1) recovery with time at y =  0.25.  be discussed l a t e r with respect to s a t u r a t i o n recovery. F i g . 11 shows the r e s u l t s of type (1) recovery at various s t r a i n s i n the high temperature range 50°C to 100°C.  These points are from a  v a r i e t y of experimental annealing c o n d i t i o n s , yet form no systematic v a r i a t i o n with e i t h e r time or temperature.  Although F i g . 11 shows only  a few d i f f e r e n t annealing c o n d i t i o n s , w i t h i n each l i s t e d c o n d i t i o n . there may have been a v a r i a t i o n of as much as ± 10°C i n temperature +30 _q  . mm.  ,  ...  .  and/or  . . .  i n time and there was s t i l l no systematic v a r i a t i o n i n recovery.  I t i s believed that each i n d i v i d u a l point represents the maximum recovery a t t a i n a b l e f o r i t s p a r t i c u l a r c r y s t a l at the s t r a i n shown.  Individual  points were not, however, tested to determine whether or not maximum recovery had been reached, but since a l l recovery values are not  ,  s i g n i f i c a n t l y higher than those obtained i n 40 min. at 30°C ( F i g . 6 ) , i t i s f e l t that the conditions employed were more than adequate to achieve the maximum recovery a t t a i n a b l e .  I t was found that higher  temperatures  (> 100°C) i n i t i a t e d r e c r y s t a l l i z a t i o n . While i t i s apparent that there i s some s c a t t e r i n the data, t h i s s c a t t e r does occur i n a reasonably narrow band.  The curve i n F i g . 11 \  represents the upper l i m i t of recovery, and because i t does depict (  the maximum recovery a t t a i n e d , i t i s termed the " s a t u r a t i o n recovery". This curve shows that recovery i s e s s e n t i a l l y complete at low s t r a i n s , but at s t r a i n s i n excess of 75%, i t decreases markedly.  The o v e r a l l shape of  this curve i s very s i m i l a r to those of F i g . 6, and may a l s o show a minimum at Y j 3.2.2  Method (2)  ,  :  Results of method (2) recovery are shown i n F i g s . 12 to 15. As was the case f o r method (1), the amount of recovery i s a f u n c t i o n of  F i g 11. The v a r i a t i o n of s a t u r a t i o n recovery with s t r a i n .  N3  annealing time, temperature and s t r a i n . i s also s i m i l a r .  The general shape of the curves  The primary d i f f e r e n c e s are i n the magnitude of recovery  and i n the r e g u l a r i t y of data.  The r e g u l a r i t y of data with t h i s method  as compared to method (1) may be due to the f a c t that method (2) tests were much easier to perform experimentally than were those f o r method (1). In order to compare the dynamic recovery present i n method (2) tests with the s t a t i c recovery i n method (1), a c r y s t a l was deformed  .  i n i t i a l l y at -196°C, then allowed to completely recover (60 min. at 75°C; i.e.  method (1)) and f i n a l l y deformed at 20°C.  The r e s u l t s of t h i s i t e s t  are compared to a c r y s t a l deformed at 20°C, recovered under the same conditions and at the same s t r a i n ( i . e . method ( 2 ) ) , and then deformed at 20°C i n F i g . 16.  I t can be seen that the flow curve of the f i r s t  i s i d e n t i c a l to that of the second following  crystal  the recovery anneals. Thus  i t i s concluded that at l e a s t up to the s t r a i n at which recovery was performed that the s t a t i c recovery measured i n method (1) i s equivalent to s t a t i c and dynamic recovery measured i n method (2).  3.2.3 Method (3)  ~  In method (3), where the load was not removed during recovery, the r e s u l t s are very s i m i l a r to method (2), both i n magnitude of recovery and i n the general shape of the recovery vs. s t r a i n curves.  Direct,  comparison of the curves showed that i n most cases, the e f f e c t of the applied load on recovery was n e g l i g i b l e .  Whenever any d e v i a t i o n d i d occur  between the two methods i t was such that the recovery was enhanced by the applied load, but always i n n e g l i g i b l e amounts.  F i g 13. The v a r i a t i o n of method (2) recovery at -30°C w i t h s t r a i n .  F i g 14.  The v a r i a t i o n of method (2) recovery, at. -70°C with s t r a i n .  -90  - 110  0  -5  10  1-5  2-0  2-5  °C  Q  3  15  C  15  min.  min.  0  FiR 15. The v a r i a t i o n of method (2) recovery at -90°C and -110°C with s t r a i n .  250  2 0  0H  Fig_16. The e f f e c t of temperature on easy g l i d e deformation.  28 3.3 S t r a i n rate change t e s t s  S t r a i n rate change t e s t s were performed on cadmium s i n g l e c r y s t a l s at -196°C.  These c r y s t a l s were s i m i l a r to those used f o r recovery  experiments i n that they were oriented to show a long easy g l i d e .  The 12  r e s u l t s of such experiments should show the v a r i a t i o n i n a c t i v a t i o n volume with s t r a i n , and also whether or not t h i s m a t e r i a l obeys the C o t t r e l l 13 Stokes  law, since both of these parameters are c a l c u l a t e d from s t r a i n  rate change data. In a d d i t i o n to s t r a i n rate changes, recovery anneals wer,e performed i n t e r m i t t e n t l y during some t e s t s to determine the e f f e c t s of recovery on both a c t i v a t i o n volume and C o t t r e l l - S t o k e s behaviour. The t y p i c a l shape of the load-elongation p l o t during a s t r a i n rate change cycle i s shown i n F i g . 17.  Also shown i n t h i s p l o t i s the  method employed to measure the change i n flow s t r e s s accompanying a change i n s t r a i n rate.  I t was found experimentally that the values measured from  an increase i n s t r a i n r a t e (AP^) were much more reproducible than those associated with a decrease i n s t r a i n rate (AP^).  This was  probablyjdue  to the time l a g associated with a s t r a i n rate decrease on the Instron t e n s i l e machine.  An example of the d i f f e r e n c e between these two measurements  i s shown i n F i g . 18. t i o n a l to AP AP , only AP  This i s a p l o t of a c t i v a t i o n volume (which is-propor-  as a f u n c t i o n of s t r a i n .  As a r e s u l t of the s c a t t e r i n  values have been used f o r the f o l l o w i n g r e s u l t s .  29  LOAD  ELONGATION  F i g 17. The determination of load change with s t r a i n r a t e change.  3.3.1 A c t i v a t i o n volume The f i r s t method of presentation of the s t r a i n rate change data i s i n the form of a c t i v a t i o n volume. V  = kT In  E l  /e  This i s defined as:  2  A c t i v a t i o n volume determination should give some idea as to the rate c o n t r o l l i n g processes of deformation. The v a r i a t i o n i n a c t i v a t i o n volume with s t r a i n i s shown i n F i g . 19. I t i s seen that v i s a s t e a d i l y decreasing f u n c t i o n of s t r a i n throughout the e n t i r e range.  •,  The e f f e c t of s a t u r a t i o n recovery (60 min. at 75°C) on a c t i v a t i o n  F i g 18. Comparison of a c t i v a t i o n volume data obtained from s t r a i n rate up-change and down-change.  u> o  5CH  r  F i g 19. The s t r a i n dependence of a c t i v a t i o n volume. (The relevant p o r t i o n • ... of: :the-stress-strain-'curve i s " showni f o r comparison.)  u>  volume i n easy g l i d e i s shown i n F i g . 20. I t i s seen that at low s t r a i n , recovery increases the value of a c t i v a t i o n volume back to the value that was found at y i e l d .  At higher s t r a i n s (>70%) i t was found that recovery  s t i l l increased the value of a c t i v a t i o n volume considerably, but not to a value as high as that at y i e l d .  In stage I I , recovery s t i l l increased the  value of a c t i v a t i o n volume as shown i n F i g . 21. I n t h i s range, the change i n v with recovery i s e s s e n t i a l l y constant, and not a f u n c t i o n of s t r a i n . i  3.3.2  A T versus T Another method of presenting the same data i s to p l o t A T , rather  than v which i s p r o p o r t i o n a l to A T \ versus T rather than s t r a i n .  {  I f ;such  a p l o t i s l i n e a r , and passes through the o r i g i n then the C o t t r e l l - S t o k e s 5 law i s considered to be obeyed.  Risebrough  16 and Davis  found that the  C o t t r e l l - S t o k e s law was obeyed i n cadmium s i n g l e c r y s t a l s at -196°C during stage I I deformation.  The obeyance or non-obeyance of t h i s law  i s not c l e a r l y understood with respect to hexagonal metals; however the p l o t does provide much u s e f u l information. The p l o t of A T vs. T f o r s i n g l e c r y s t a l cadmium i s shown i n Fig.  22. I t i s seen that there i s a d i s t i n c t change i n t h i s p l o t at the  end of l i n e a r easy g l i d e , but the C o t t r e l l - S t o k e s law i s not s t r i c t l y  .  obeyed i n e i t h e r region. When s a t u r a t i o n recovery anneals are performed i n stage I , i t i s seen i n F i g . 23 that the f i r s t anneal has no e f f e c t on the A T - T r e l a t i o n s h i p , but a f t e r the second anneal there i s a s l i g h t increase i n slope. Saturation recovery i n stage I I has a much d i f f e r e n t e f f e c t than i t d i d i n stage I as shown i n F i g . 24. A f t e r the f i r s t anneal there i s a large s h i f t i n the curve to the l e f t .  This i s because recovery  reduces  50-  F i g 22." The v a r i a t i o n of AT w i t h T .  F i g 23. The v a r i a t i o n of Ax - x with saturation  recovery i n easy g l i d e .  ON  IOOH  9CH  Crigiicl  .'clues  O  —  £  — After  I  Annecl  •  — After  2  Annccts  V  -  After  3  Annecls  ©  — A'lcr  4  Anne ills  80AT gm/mm  704  60-  50-  1400  1000  600  T  F i R  1800  2200  gm/mm'  24. The v a r i a t i o n of A T - T with saturation recovery i n stage I I .  x s i g n i f i c a n t l y but leaves A T e s s e n t i a l l y unchanged. The slope of the curve f o l l o w i n g t h i s recovery i s the same as i t was p r i o r to recovery.  A second  anneal i n stage I I caused a further s h i f t to the l e f t s i n c e although A T did. decrease at t h i s p o i n t , i t did not decrease as much as i t had increased from the previous recovery.  Again, the slope of the p l o t was the same.  .Further recoveries show that both A T and T  decrease by the same amount  that they increased during s t r a i n from the previous recovery.  Note.,that  each recovery was performed when the s t r e s s on the system reached 2 2000 gm/mm .  3.4 Work hardening  , ..  parameters  To determine the e f f e c t s of c r y s t a l o r i e n t a t i o n on various work hardening parameters, a study was c a r r i e d out using a wide range of i n i t i a l orientations.  This study should also help to d i f f e r e n t i a t e between  e f f e c t s which are caused by the changing o r i e n t a t i o n of a c r y s t a l during a t e n s i l e test and those which are c h a r a c t e r i s t i c of recovery. with i n i t i a l o r i e n t a t i o n X In a l l cases X  q  q  Crystals  ranging from 25° to 48° were i n v e s t i g a t e d . ,  was very nearly equal to X Q  3.4.1 Stage I  .  Stage I has been defined as the f i r s t l i n e a r stage of hardening during the deformation of a cadmium s i n g l e c r y s t a l .  The primary work  hardening parameters to be considered are the i n i t i a l flow s t r e s s or c r i t i c a l resolved shear s t r e s s , the work hardening rate and the amount of l i n e a r s t r a i n .  These f a c t o r s , and the e f f e c t s of o r i e n t a t i o n and recovery  on them, are discussed below.  39  3.4.1.1 C r i t i c a l resolved shear s t r e s s The c r i t i c a l resolved shear stress at -196°C was found to be 2  2  19.4 gm/mm ± 4.5 gm/mm , and e s s e n t i a l l y independent of i n i t i a l o r i e n t a t i o n as shown i n F i g . 25.  This i s i n agreement with Schmid's shear  stress law"'"''" which states that there should be no o r i e n t a t i o n dependence of t h i s s t r e s s . The v a r i a t i o n i n c r i t i c a l resolved shear s t r e s s with temperature i s shown i n F i g . 26.  The stress values have been corrected f o r the.  temperature dependence of the shear modulus by d i v i d i n g each by the, shear modulus at i t s p a r t i c u l a r temperature.  Values of shear modulus 14  were obtained from Gerland and Silverman  (C^)  . I t i s seen that at low  i;  temperatures, there i s no change i n crss with temperature, while at higher temperatures there appears to be a s l i g h t decrease.  There i s so much  s c a t t e r i n the data that i t i s s t a t i s t i c a l l y d i f f i c u l t to make any d e f i n i t e statements about t h i s behaviour. 3.4.1.2 Work hardening rate The work hardening rate i n easy g l i d e (9j) was found to increase about 50% with X decreasing from 45° to 25° as shown i n F i g . 27. Q  The e f f e c t of recovery on t h i s parameter was to increase 0 to a value comparable to that f o r a v i r g i n c r y s t a l with i n i t i a l o r i e n t a t i o n equal to the o r i e n t a t i o n which had been achieved by s t r a i n to the point of recovery. This i s i l l u s t r a t e d i n F i g . 28 where sample S39A was f romT^ah" iniTial"".^orientation of 45° to 35°.  deformed  Sample M4A had an i n i t i a l  r .  o r i e n t a t i o n of 35°,and i t i s seen that following recovery of sample>S39A, the  work hardening rate of the two c r y s t a l s i s the same. Fig.  ,  29 shows the v a r i a t i o n of 6^. with temperature at a n ' i n i t i a l  -3 -1 s t r a i n rate of 1.3 x 10 sec . Shown f o r comparison are the r e s u l t s of  30H  O O  O  25H  o  o o  9)  2CH crss  o o  o  gm/mm  o o  15H  20  !  24  —i— 28  32  36  degrees  —i— 40  —i— 44  F i g 25. O r i e n t a t i o n dependence of the c r i t i c a l resolved shear s t r e s s .  o  ° 9  !  48  12  150\  I4CH  gm/rnm  2  130-  o \  oo o  120-  O  no-  O  o  V  8 •s.  o' o  100-  18  —i— 22  F i g 27.  26  —i— 30  V o A  „degrees 3  4  —i— 38  —I—  i  42  46  O r i e n t a t i o n dependence of stage I work hardening rate.  —i— 50  2000-  1600-  V  -:Fig. 28.  deqrees  Orientation, dependence of the length, of easy g l i d e  (schematic).  44 Risebrough^.  I t i s seen that the two sets of data are comparable w i t h  the exception of the present point at -110°C.  This point i s high and  may r e s u l t from the fact that i t was obtained very early i n the course of t h i s work at which time production, handling and t e s t i n g procedures had not been f u l l y developed.  Consequently,  the c r y s t a l may have contained  undiscovered substructure or other flaws which would have caused it,,to be discarded at a l a t e r date.  .•  The i n i t i a l work hardening rate at 20°C at a s t r a i n rate of j -5 -1 2 1.3 x 10 sec was found to be 7.9 gm/mm .  ; ,  3.4.1.3 Length of easy g l i d e The data i n Table I , which i s shown schematically i n F i g . 30, shows that the end of easy g l i d e occurs when x 20.2 ± 1.2°, i r r e s p e c t i v e =  of i n i t i a l o r i e n t a t i o n or recovery.  This f i g u r e i s a s t r e s s - s t r a i n p l o t  on which s t r a i n has been converted to o r i e n t a t i o n by the r e l a t i o n : s l n  X -  T  o sinx  1  The data i n Table I and from F i g . 27 show that recovery has an e f f e c t such that a c r y s t a l deformed to an o r i e n t a t i o n x i  n  stage I and  completely recovered has the work hardening r a t e and new stage I length i  equal to a v i r g i n c r y s t a l of o r i e n t a t i o n xI t was also noted that the end of easy g l i d e at -196°C was marked by twinning i n the sample.  These twins are shown i n F i g . 31. The ;  stress at which twinning occurred was s l i g h t l y higher than the s t r e s s a t which the s t r e s s - s t r a i n curve deviated from l i n e a r i t y . This s t r e s s was 2 300 to 400 gm/mm , and independent of i n i t i a l o r i e n t a t i o n . At 20° C,, twinning 2 occurred at an equivalent s t r e s s of 310 - 380 gm/mm . This s t r e s s was not reached u n t i l very high s t r a i n s , approximately 3.5 to 4.0.  .  ;  ;  o  Table I .  The e f f e c t of i n i t i a l o r i e n t a t i o n and recovery on the length of stage I .  Specimen  Recovery i n stage I  S39B  48  20.3  S39D  48  20.5  ' 90 min.  -100°C  S39C  48  21.5  40 min.  -100°C  S41B  47  20.3  30 min. +50° C  S37C  46  19.2  20 min. -70°C  S37B  46  20.5  70 min.  S36D  46  20.9  10 min. +30°G  S39A  45  20.7  10 min.  M9B  45  21.0  none  M12C  45  22.5  saturation  L3C  43  23.0  saturation  S37A  42  19.3  20 min. + 30°C  M15B  42  19.6  none  M11C  42  19.6  none  L3B  42  20.4  saturation  M8A  41  18.5  none  Ml 2 A  40  19.0  saturation  M11B  40  20.5  none  S29B  36  19.7  saturation  M4A  35  19.8  none  M2C  34  20.1  saturation  S42A  29  18.7  none  U1C  27  22.0  none  M6B  26  18.4  none  L2A  25  19.0  none  20 min. -20°C  -70°C  -20°C  -  \  500i  48  44  40  36  32  X  F i g 30.  28  24  20  degrees  The e f f e c t of o r i e n t a t i o n on the length of stage I ( s p e c i f i c example).  16  Fig 31.  Twins formed i n the stage I - stage I I t r a n s i t i o n (x 2 0 0 ) .  49 3.4.1.4 Recovery  effects  Summarizing the p r i n c i p a l e f f e c t s of recovery on the various stage I work hardening parameters, i t has been found that recovery has no e f f e c t on the length of stage I ; i t increases the work hardening rate. Recovery has no e f f e c t on the s t r a i n at which twinning occurs.  3.4.2 Stage I I  ,  Stage I I i s the second l i n e a r hardening p o r t i o n of the work hardening curve and i s characterized by a work hardening rate s i g n i f i c a n t l y higher than stage I . In the ensuing d i s c u s s i o n , the t r a n s i t i o n region from stage I to stage I I has been ignored since i t was e s s e n t i a l l y the same i n a l l specimens.  3.4.2.1 O r i e n t a t i o n e f f e c t s  • ;  A l l c r y s t a l s tested i n t h i s study had an i n i t i a l o r i e n t a t i o n X greater than 25°.  Q  Consequently, a l l showed some easy g l i d e before stage I I  began at an angle of approximately 20°.  Therefore there was no v a r i a t i o n  i n o r i e n t a t i o n with which to compare stage I I work hardening parameters. There was no e f f e c t of i n i t i a l o r i e n t a t i o n i n stage I on stage I I parameters. At -196°C, the stress l e v e l at the beginning of the t r a n s i t i o n 2 region from stage I to stage I I was found to be 150 to 250 gm/mm  for  a l l tests. 2 The work hardening rate i n stage I I was approximately 4250 gm/mm . 3.4.2.2 Recovery e f f e c t s  ,  While there was no n o t i c e a b l e e f f e c t on the length of stage I , i t was found that recovery had a considerable e f f e c t on the length of stage I I .  50 Fig.  32 i s a-plot of the resolved shear s t r e s s - shear s t r a i n data from  two comparable c r y s t a l s .  C r y s t a l M4A was deformed continuously at -196°C. 2  C r y s t a l M15B was deformed to a s t r e s s of 2000 gm/mm at -196°C and then recovered at 75°C f o r 30 min. These conditions are i n excess of the minimum requirements f o r s a t u r a t i o n recovery.  Following t h i s recovery, 2  the specimen was again s t r a i n e d at -196°C to a s t r e s s of 2000 gm/mm and then recovered.  This cycle was continued u n t i l the specimen broke.*  The t o t a l shear s t r a i n obtained from the end of easy g l i d e i n specimen M4A was 80%. C r y s t a l M15B e x h i b i t e d 180% shear s t r a i n from the end of easy g l i d e .  The t o t a l s t r a i n achieved i n t h i s c r y s t a l was  comparable to the s t r a i n achieved during deformation at 20°C.  I t i s not  suggested that deformation plus recovery i n stage I I i s equivalent to , deformation at 20°C.  F i g . 33 shows the f i n a l shape of c r y s t a l M15B,.and  a c r y s t a l deformed at 20°C. nificantly' different.  I t i s seen that the two c r y s t a l s are s i g -  C r y s t a l M15B i s considerably more rumpled than .  the other.  ,  ;  The work hardening rate i n stage I I at -196°C was found to be 2 about 4250 gm/mm . This i s the rate shown by c r y s t a l s M4A and M15B; up to the f i r s t anneal as shown i n F i g . 32. Following the f i r s t anneal, 2 2  the work hardening rate increased by 1250 gm/mm to 5500 gm/mm .  3.4.3 Stage I I I i  The majority of tests at -196°C e x h i b i t e d only two stages of hardening, w i t h f a i l u r e occurring w h i l e the specimens were s t i l l in,stage I I . 2 The stress at f a i l u r e i n these specimens was approximately 3500 gm/mm . The c r y s t a l s with i n i t i a l o r i e n t a t i o n less than 30°, a t h i r d stage w i t h a  s u b s t a n t i a l l y lower non-linear work hardening rate was often observed. Sample M15B  was annealed extensively i n stage I I . A f t e r the  f i r s t few anneals, the work hardening rate increased, but at higher s t r a i n s i t decreased and deviated from l i n e a r i t y as shown i n F i g . 32. may be due to a t h i r d stage of hardening.  This  54 4. DISCUSSION  4.1 Recovery r e s u l t s  4.1.1  A c t i v a t i o n energy As a f i r s t step i n the evaluation of the recovery data, an  attempt was made to c a l c u l a t e an a c t i v a t i o n energy f o r t e s t s i n which a l l deformation was performed at -196°C, w i t h recovery anneals at higher temperatures  (Figs. 5-8).  These data were used i n preference to those  i n which deformation was at elevated temperature because they should provide consistent boundary conditions for the anneals.  That i s , the  d i s l o c a t i o n c o n f i g u r a t i o n should be s i m i l a r preceding the various anneals, which would not be the case f o r high temperature  deformation  since i t has been shown that dynamic recovery takes place at temperatures greater than -120°C. The recovery data were normalized to a common Yj> the reasons for which w i l l be discussed with respect to the o r i e n t a t i o n r e s u l t s . . Comparing recovery r e s u l t s as a f u n c t i o n of time ( F i g . 9) i t was seen that with the exception of+30°C, the d i f f e r e n c e i n recovery w i t h time i s . not particularly strain sensitive.  That i s , the shape of the r e c o v e r y - s t r a i n  curves was consistent at a given temperature, so that a change i n recovery time changes the recovery by a constant amount, i r r e s p e c t i v e of s t r a i n . With respect to the r e s u l t s at 30°C, F i g . 10 showed that work hardening i n c r y s t a l s was completely recoverable up to about 50% shear s t r a i n . At higher s t r a i n s , some work hardening occurred which was not recoverable. Thus i t was not p o s s i b l e to achieve high recovery values close to Y J -  Since  i t was observed that the majority of recovery took place at short times (Fig.  5 ) , the approach of s a t u r a t i o n may e x p l a i n the s i m i l a r i t y i n recovery  values f o r various times at 30°C close to Y j Since s a t u r a t i o n recovery may be s i g n i f i c a n t at Y j at 30°C, the comparison of recovery has been l i m i t e d to s t r a i n l e s s than 50%. The value of s t r a i n chosen a r b i t r a r i l y f o r t h i s comparison was 25%, and i n some cases t h i s required the back e x t r a p o l a t i o n of recovery curves. A t r i a l and error technique of curve f i t t i n g was employed to t r y to e s t a b l i s h a recovery-time r e l a t i o n s h i p .  By t h i s method,  i t was found that recovery i s not a power f u n c t i o n of time. i.e.  R ^ At  K  Rath et a l ^ found that f o r aluminum s i n g l e c r y s t a l s , recovery was p r o p o r t i o n a l to the logarithm of time.  For the present  work, there i s i n s u f f i c i e n t data to prove whether or not such a r e l a t i o n s h i p holds.  However, i f i t i s assumed that recovery i s  proportional to log time f o r cadmium s i n g l e c r y s t a l s , the c a l c u l a t i o n of a meaningful a c t i v a t i o n energy was found to be impossible from t h i s data.  During the c a l c u l a t i o n , the d e r i v a t i o n of the recovery  rate constant caused the temperature dependence of recovery to be effectively  cancelled so that the r e s u l t a n t a c t i v a t i o n energy was  zero.  Following the a n a l y s i s of Rath et a l ^ d i d not produce any meaningful values f o r a c t i v a t i o n energy from present r e s u l t s .  For  values of recovery from R = 0.1 to R = 0.5, a c t i v a t i o n energy values ranged from almost zero to 20,000 c a l / g . atom, and the Arrhenius p l o t s from which these values were calculated were not l i n e a r .  This method  of a n a l y s i s i s somewhat dubious because i t does not use a true rate constant which i s independent of time or recovery for c a l c u l a t i o n of a c t i v a t i o n energy.  A l s o , t h i s r e l a t i o n p r e d i c t s that recovery w i l l  exceed.unity which i s impossible by d e f i n i t i o n .  Consequently,  r e l a t i o n s h i p R a log t i s r e j e c t e d f o r the present work.  the  TIME F i g . 34.  min.  The v a r i a t i o n of log (1-R) with time.  57 The r e l a t i o n s h i p between recovery and time which has been adopted f o r t h i s work i s : log This  (1-R) = At  rate law i s not defensible on the basis of present  data, however i t i s consistent i n that recovery may never exceed one, and i s a f i r s t order rate law. A p l o t of the present data i s shown i n F i g . seen that the p l o t at each  34,. I t i s  temperature may be made to pass through  a common point on the time axis at log (1-R) = 0.  The f a c t that t h i s  time i s negative i s not a serious drawback to t h i s analysis since analogies to p h y s i c a l and chemical systems may be drawn i n which such a phenomenon would be expected.  For instance, a nucleation and  growth system i n which the number of n u c l e i decreases w i t h time would e x h i b i t a s i m i l a r curve when the growth i s extrapolated back to  zero. Since the rate law used i n t h i s analysis i s f i r s t order,  the  slope of the p l o t s at each temperature represents the rate  constant, and as such may be used to c a l c u l a t e the a c t i v a t i o n energy. This i s shown i n F i g .  35. The a c t i v a t i o n energy found from t h i s  p l o t i s 2500 c a l / g . atom.  This value i s too small to be meaningful,  but i f log (1-R) = At does describe the rate of recovery, then the rate c o n t r o l l i n g process i s f i r s t order, and i n the s o l i d s t a t e t h i s would probably be d i f f u s i o n .  Since the value i s so small i t  i s not p o s s i b l e to determine whether t h i s would be bulk d i f f u s i o n or pipe d i f f u s i o n .  The a c t i v a t i o n energy f o r bulk d i f f u s i o n i n s i n g l e c r y s t a l  cadmium i s about 18,500 cal/b. atom, and that f o r pipe d i f f u s i o n i s about h a l f of t h i s .  -2-  -2log (slope)  -3>t  F i  §-  3  J-  T h e  v a r i a t i o n of l o g (slope) with r e c i p r o c a l absolute temperature.  59  4.1.2 Comparison of recovery methods ( 1 ) , ( 2 ) , (3) In method (1) recovery, a l l recovery was obtained under s t a t i c conditions but experimental conditions f o r types (2) and (3) were such that there was s i g n i f i c a n t dynamic recovery during deformation i n a d d i t i o n to the.measured s t a t i c recovery.  In order to compare the r e s u l t s of a l l  methods d i r e c t l y , i t must f i r s t be determined whether or not dynamic and s t a t i c recoveries are equivalent.  The f o l l o w i n g c a l c u l a t i o n shows that  they are equivalent only at low s t r a i n s . F i g . 11 defined a value of s a t u r a t i o n recovery which i s dependent only on s t r a i n .  This curve i n F i g . 11 i s now used to determine  the minimum a t t a i n a b l e flow s t r e s s of a c r y s t a l deformed at -196°C.  This  flow s t r e s s i s the value which would be obtained from a c r y s t a l which was f u l l y recovered a f t e r each i n f i n i t e s i m a l increment of s t r a i n .  The r e s u l t s  of t h i s c a l c u l a t i o n are shown i n F i g . 36. In F i g . 36, the s o l i d l i n e l a b e l l e d "Saturation Recovery" shows the minimum flow s t r e s s as c a l c u l a t e d from a t y p i c a l s t r e s s - s t r a i n curve at -196°C and from F i g . 11.  This curve represents that p o r t i o n of  the flow s t r e s s which i s not recoverable. The dashed l i n e i n F i g . 36, l a b e l l e d "+20°C corrected", i s the s t r e s s - s t r a i n curve f o r a c r y s t a l deformed at 20°C, and as such represents high temperature deformation (20°C = .5T^).  I n t h i s case dynamic recovery should proceed at such a rate  that the c r y s t a l i s e s s e n t i a l l y recovered at a l l times.  For comparison  between t h i s and the previously c a l c u l a t e d curve, the flow s t r e s s values i n t h i s case have been corrected to t h e i r equivalent at -196°C by a. r a t i o of the shear moduli at -196°C and 20°C. I t i s evident from comparison between t h i s and the previously calculated curve that they are equivalent up to ^ 50% - 70% shear s t r a i n . This i s the point at which s a t u r a t i o n recovery s t a r t s to decrease fronu 100% ( F i g . 11). This observation i s also v e r i f i e d by the r e s u l t s of a s i n g l e test at 70% shear s t r a i n as shown i n F i g . 16.  •  Thus, i t i s concluded that i n i t i a l easy g l i d e deformation at low temperature i s i d e n t i c a l to deformation at high temperature with the a d d i t i o n of work hardening which i s completely recoverable by the conditions employed. At s t r a i n s i n excess of 70%, up to the end of easy g l i d e ^ the two curves i n F i g . 36 deviate s l i g h t l y , and i t i s thought that t h i s  ;i  d e v i a t i o n may be due to a c t i v i t y on s l i p systems other than the basal system.  This w i l l be discussed l a t e r , with respect to the o r i e n t a t i o n  results.  , The d e v i a t i o n between the two curves i s s m a l l , but i t appears  to be the opposite of what would be expected i f a secondary s l i p system operated at low temperature.  I f t h i s were the case, the recovered curve  at -196°C should have a higher flow stress than the equivalent curve at +20°C since the stress at 20°C i s so low that i t i s u n l i k e l y that a .second system would operate.  The only p o s s i b l e explanation f o r the discrepancy  62 as shown i n F i g . 36 other than s t r u c t u r a l d i f f e r e n c e s between the two c r y s t a l s would be that the flow s t r e s s at 20°C was not completely recovered at high s t r a i n s .  This was not v e r i f i e d experimentally.  This discrepancy  should not, however, i n f l u e n c e the conclusion that i n i t i a l easy g l i d e at low temperature  i s equivalent to that at high temperature with the a d d i t i o n  of completely recoverable work hardening. I t should be p o s s i b l e to compare the r e s u l t s obtained from recovery methods (2) and (3), where deformation was at intermediate ito , high temperatures  to those of method (1) where deformation was at low  temperature. Figures 12-15 showed the r e s u l t s of method (2) i n which the c r y s t a l s were deformed and then recovered w i t h the load removed at the same temperature.  As was seen with respect to F i g s . 6-9  the primary  d i f f e r e n c e between t h i s type of t e s t and that i n which deformation was at low temperature was i n the magnitude of recovery.  One reason f o r t h i s  discrepancy l i e s i n the d e f i n i t i o n of recovery as given i n equation (3). In t h i s d e f i n i t i o n , recovery i s the f r a c t i o n of the work hardening and since at temperatures  recovered,  greater than -120°C recovery takes place during  deformation, there i s a lower t o t a l work hardening measured than would be at temperatures  less than -120°C.  The schematic diagram i n F i g . 37  shows the same flow s t r e s s i s achieved by a recovery of .5 at elevated temperature, while a recovery of .75 i s required at low temperature.;  This  w i l l e x p l a i n the r e s u l t s at high recovery temperatures, where s a t u r a t i o n i s approached, but at lower temperatures  (-100° to 0°C) t h i s i s not the case.  In the lower range of t e s t temperatures  (0°C) where s a t u r a t i o n  recovery, e i t h e r by s t a t i c or dynamic means, i s not a f a c t o r , the above argument i s not v a l i d .  In t h i s range, the recovered flow s t r e s s at ,pne  t  F i g 37.  Schematic diagram of recovery to a s i m i l a r stress l e v e l with deformation  —  at d i f f e r e n t  ---  temperatures.  temperature  i s always s i g n i f i c a n t l y d i f f e r e n t from the flow s t r e s s at  another temperature. of -70°C and -30°C.  This i s i l l u s t r a t e d i n F i g . 38, at test  temperatures  The d i f f e r e n c e i n flow s t r e s s between these two curves  i s due p r i m a r i l y to the d i f f e r e n c e i n the amount of dynamic recovery taking place i n each case.  Thus, to compare d i r e c t l y such curves along  with the measured recovery values to data obtained from method (1) w i t h deformation at -196°C, the comparison should be made of flow s t r e s s . f o l l o w i n g recovery at each value of s t r a i n , as suggested w i t h respect to the d e f i n i t i o n of recovery at the beginning of t h i s t h e s i s .  However, ,  t h i s would require a knowledge of the absolute amount of dynamic recovery which has taken place up to any given s t r a i n . Consequently  This has not been determined.  no d i r e c t comparison of the r e s u l t s of method (2) w i t h those  of method (1) could be drawn. Comparison of the r e s u l t s of method (3), i n which the load was not removed from the sample, to e i t h e r method (1) or method (2) was  not  made because of the n e g l i g i b l e d i f f e r e n c e s between t h i s method and method (2). Since recovery rates were found to increase w i t h  temperature,  i t i s reasonable to assume that the recovery processes are thermally activated.  Consequently,  i t was expected that there would be some increase  i n recovery due to the applied load a c t i n g i n conjunction w i t h the a v a i l a b l e thermal energy.  The e f f e c t s observed, however, were n e g l i g i b l e .  In conclusion, the r e s u l t s of the three  types of recovery test  have indicated that recovery i s thermally a c t i v a t e d , and i s the r e s u l t of two or more processes which change with temperature.  I t remains f o r other  types of test to show what these processes might be.  S a t u r a t i o n recovery  has shown that deformation at low temperature i s r e l a t e d d i r e c t l y to, deformation at high temperature i n the i n i t i a l easy g l i d e region, but that t h i s i s not the case at s t r a i n s i n excess of 70%.  66 4.2 Work hardening parameters  I t was observed that recovery, and i n p a r t i c u l a r s a t u r a t i o n recovery, i s a f u n c t i o n of s t r a i n . o r i e n t a t i o n , i t was  Since s t r a i n i s d i r e c t l y r e l a t e d to  thought that perhaps the above observation could be  explained on the basis of c r y s t a l o r i e n t a t i o n . I t was w i t h t h i s i n mind that the e f f e c t of o r i e n t a t i o n upon various work hardening parameters, both by themselves and i n conjunction with recovery, was studied.  4.2.1  , ,  Easy g l i d e parameters One of the most common parameters measured i n any s i n g l e , c r y s t a l  study i s the c r i t i c a l resolved shear s t r e s s .  Comparison of t h i s f a c t o r  between t h i s work and others should give an idea of the comparability of the q u a l i t y of the c r y s t a l s used.  This knowledge should then f a c i l i t a t e  the comparison of other parameters. Schmid and Boas''""'" i n some of the e a r l i e s t work on cadmium found that 99.996% Cd at room temperature e x h i b i t e d a c r i t i c a l resolved shear stress of 25 gm/mm . 2  Gibbons  gm/mm f o r 99.9994% Cd at 20°C.  found c r i t i c a l stresses of 9.8 and 17.1, 15 Bocek et a l tested the temperature  dependence of various work hardening parameters i n 99.99% Cd.  Their  ,  r e s u l t s f o r crss vs. temperature along with those of the present work are shown i n F i g . 39.  I t i s seen that there i s a large discrepancy between  the two sets of data which might be thought to be due to impurity content since the present work used 99.999% Cd.  However, the graphs presented by  16 Davis  f o r 99.9% and 99.9999% Cd show very l i t t l e d i f f e r e n c e of crss  2 for such a large d i f f e r e n c e i n p u r i t y , with the crss being about 20-30 ,gm/mm . <,  A more probable explanation f o r the discrepancy would be the presence of more substructure i n Bocek's c r y s t a l s .  In comparison to the present, work,  they showed s i g n i f i c a n t l y shorter easy g l i d e , and Hirsch and L a l l y  4  found  that easy g l i d e i n Mg s i n g l e c r y s t a l s was reduced from 250% to 50% f o r c r y s t a l s w i t h substructure.  The presence of substructure would also  e x p l a i n the higher work hardening rate found by Bocek.  Since the work  by Bocek i s one of the most comprehensive studies of cadmium i n the l i t e r a t u r e , the favourable comparison of the present work shows that the c r y s t a l s used i n t h i s work must have been of high p u r i t y and have contained r e l a t i v e l y l i t t l e substructure. 17 18 Early work of J i l l s o n and Deruyttere detected no systematic dependence on o r i e n t a t i o n of e i t h e r work hardening rate or length of easy g l i d e . Liicke et a l ^ observed that zinc s i n g l e c r y s t a l s at 20°C e x h i b i t e d 130% shear s t r a i n i n easy g l i d e , independent of the i n i t i a l 19 orientation.  Diehl  observed no v a r i a t i o n of the work hardening rate with  o r i e n t a t i o n i n hexagonal metals.  >;  The r e s u l t s of the present study are contradictory to those , of the above workers.  I t was found that there i s a d e f i n i t e dependence  on o r i e n t a t i o n of the length of easy g l i d e , since easy g l i d e terminates at a s p e c i f i c o r i e n t a t i o n (see F i g . 30; Table I ) .  To a l e s s e r degree,  the work hardening rate also depends on o r i e n t a t i o n (see F i g . 27). ,,These „ 20 observations are more i n accord with those of Bocek and Kaska  , who  noted an increasing work hardening rate and decreasing length of stage I with decreasing X  q  Hdtzsch and Simmin  i n zinc i n the temperature range 50° to 100°C. Bocek, also quote the r e s u l t s of Wolr  on Cd at 20°K and  90°K which show s i m i l a r trends f o r the work hardening rate. Liicke et a l ^ found that recovery has a d i s t i n c t e f f e c t on the length of stage I . Their r e s u l t s showed that a zinc s i n g l e c r y s t a l ,which had been deformed 70% at room temperature showed a s t r e s s - s t r a i n curves i d e n t i c a l to that of a v i r g i n c r y s t a l a f t e r a recovery of 24 h r s . at 25°C.  69 Thus, the t o t a l length of easy g l i d e increased by 70%.  For cadmium  c r y s t a l s deformed at -196°C and recovered t h i s was not the case.  As  shown i n F i g . 28 and Table I , recovery had no e f f e c t at a l l on the length of easy g l i d e .  A possible reason f o r t h i s discrepancy would be that the  operation of second order pyramidal s l i p i s not r e l a t e d to o r i e n t a t i o n i n the same way i n zinc at 20°C as i t i s i n cadmium at -196°C. i 4.2.2  Theories of stage I I While the topic of t h i s study i s the recovery of mechanical  properties of cadmium, i t i s necessary to know something of the d i s l o c a t i o n arrangements present during deformation to understand what might happen during recovery.  As a consequence of t h i s , the f o l l o w i n g s e c t i o n i s  concerned w i t h the hardening c h a r a c t e r i s t i c s of s i n g l e c r y s t a l s , and the possible d i s l o c a t i o n mechanisms which may occur during deformation.\ Perhaps one of the most c r i t i c a l points during the deformation of a s i n g l e c r y s t a l i s the t r a n s i t i o n from the f i r s t l i n e a r stage of hardening to the second l i n e a r stage.  Various theories have been proposed  to explain t h i s t r a n s i t i o n i n hexagonal metals. are:  The p r i n c i p a l a l t e r n a t i v e s R.  :  •  1) the formation of a c r i t i c a l density of obstacles as a [result of the condensation of vacancies  (:  2) the i n i t i a t i o n of a c t i v i t y of the two previously non-active Burger's vectors i n the b a s a l plane 3) the onset of twinning 4) the i n i t i a t i o n of a c t i v i t y on the second-order pyramidal system. The l a s t a l t e r n a t i v e i s thought to be most important to t h i s study; however a l l theories w i l l be b r i e f l y discussed.  70 4.2.2.1 Condensation of vacancies 22 Seeger and Trauble  a s c r i b e the t r a n s i t i o n from easy g l i d e to  stage I I to a c r i t i c a l density of obstacles. These obstacles have formed by the condensation of vacancies i n t o immobile d i s l o c a t i o n r i n g s . experimental r e s u l t s are i n c o n s i s t e n t with t h i s theory.  Present  Firstly,  i t i s d i f f i c u l t to imagine vacancies having adequate m o b i l i t y at -196°C to form s e s s i l e loops.  I f such loops did manage to form, then s a t u r a t i o n 23  recovery should be s u f f i c i e n t to allow them to anneal out.  Price  •• found  that s e s s i l e d i s l o c a t i o n loops formed by non b a s a l g l i d e i n Cd annealed out by volume d i f f u s i o n at temperatures above -40°C.  No e f f e c t of annealing  on the t r a n s i t i o n s t r a i n was observed i n the present work. 4.2.2.2 Non-active b a s a l d i s l o c a t i o n s  j ., -  24 K r a t o c h v i l and Koutnick  , from shape change measurements:  during the deformation of cadmium s i n g l e c r y s t a l s , concluded that secondary d i s l o c a t i o n s i n the basal plane must operate. 4 work, p r i n c i p a l l y that of Hirsch and L a l l y  By comparison with other  on magnesium, they conclude that  the i n t e r a c t i o n of primary and secondary b a s a l d i s l o c a t i o n s i s responsible for s t r a i n hardening and the t r a n s i t i o n to stage I I . I t appears that such comparison i s u n j u s t i f i a b l e .  H i r s c h and L a l l y emphasize that their-model  i s s p e c i f i c a l l y designed f o r Mg and that hardening mechanisms i n other hexagonal close-packed metals may be d i f f e r e n t .  I t i s p o s s i b l e that  secondary b a s a l d i s l o c a t i o n s operate, but i t i s doubtful that they a r e responsible f o r the t r a n s i t i o n to stage I I .  ;  ;  4.2.2.3 Twinning 4 Hirsch and L a l l y , i n a transmission e l e c t r o n microscope,,study of magnesium, found that stage I I i s accompanied by s l i p of the two previously  non-active Burger's vectors i n the basal plane, the formation of subboundaries and twinning.  They observed that at the onset of stage I I , the  stress on the p r i s m a t i c plane was close to the experimentally determined value f o r prism s l i p .  However i n comparison of t h e i r work on Mg w i t h face-  centered cubic m a t e r i a l s , they think that twins act as b a r r i e r s to s l i p 25 l i n e s i n Mg, whereas secondary d i s l o c a t i o n s provide the b a r r i e r s i n : f . c . c . I t i s s t r e s s concentrations a r i s i n g from the twins which may now produce prism s l i p .  , ; Hirsch and L a l l y s t a t e that the above argument may not be  a p p l i c a b l e , t o hexagonal metals other than magnesium and the r e s u l t s of 26 B e l l and Cahn  5 , Risebrough , and the present work agree with t h i s .  ;  Bell  and Cahn noted the presence of {1122} <1123> s l i p traces i n z i n c c r y s t a l s p r i o r to twinning, and associated the i n t e r s e c t i o n of d i s l o c a t i o n s on t h i s system with basal d i s l o c a t i o n s to produce s t r e s s concentrations which i n turn would cause twinning.  This i s opposite to magnesium i n which Hirsch  and L a l l y thought that twins produced s t r e s s concentrations which i n turn caused secondary (prism) s l i p . Risebrough have noted  In cadmium, the present work  and tfyat of  that twinning i s always present i n stage II.= However,  Risebrough has explained that since a considerable p o r t i o n of the c r y s t a l remains untwinned  i n the t r a n s i t i o n region from stage I to stage II,, and  that the s t r e s s associated with the end of easy g l i d e i s r e l a t i v e l y , temperature independent, that twinning must be an " a f t e r the f a c t " consideration.  On t h i s b a s i s , i t must be some d i s l o c a t i o n c o n f i g u r a t i o n which  i s responsible f o r the end of stage I . 4.2.2.4 Second order pyramidal  slip  ^ 27 Bocek, Svabova and Hotzsch , i n a study of s i n g l e c r y s t a l cadmium at 20°K, assume that the onset of l i n e a r i t y i n stage I I i s a r e s u l t  of flow on the second order pyramidal system {1122} <1123>.  Their r e s u l t s  show, however, that a t t h i s point there i s a considerable v a r i a t i o n i n the  shear s t r e s s on t h i s sytem w i t h respect to o r i e n t a t i o n .  This i s i n  disagreement with Schmid's law that the c r i t i c a l shear s t r e s s be independent of o r i e n t a t i o n . calculated  To compensate f o r t h i s discrepancy, Bocek et a l have .  an i n t e r n a l s t r e s s which may be present on the secondary system  as a r e s u l t of pile-ups of d i s l o c a t i o n s on the b a s a l system.  The r e s u l t s  of t h i s c a l c u l a t i o n show t h i s i n t e r n a l s t r e s s to be of the order of (the flow s t r e s s on the basal system.  When t h i s s t r e s s i s added to the applied  stress on the secondary system, the r e s u l t i s that the t o t a l s t r e s s i s , now e s s e n t i a l l y o r i e n t a t i o n independent.  Also, this t o t a l stress i s 28  comparable to the stress found by S t o l o f f and Gensamer  f o r the appearance  of second order pyramidal s l i p traces on cadmium s i n g l e c r y s t a l s oriented to  suppress basal s l i p .  Thus Bocek et a l conclude that the s t a r t of the  l i n e a r stage I I i s associated with the onset of flow on the second order pyramidal system.  I n t e r a c t i o n of these d i s l o c a t i o n s w i t h basal d i s l o c a t i o n s  w i l l produce s e s s i l e s which contribute to work hardening, and so e x p l a i n the  r e l a t i v e l y high work hardening rate i n stage I I .  , 27  A number of problems with respect to the work of Bocek et a l a r i s e when i t i s compared to the r e s u l t s of the present study.  Thejfirst  problem i s w i t h respect to the v a r i a t i o n of the s t r e s s at the beginning of stage I I with o r i e n t a t i o n . The present r e s u l t s , which encompass considerably more tests than appear to have been done by Bocek, show that the  angle at which stage I I begins i s constant at x  =  20.2° ± 1.2°(see Fig.28).  2 The flow s t r e s s at t h i s point was also e s s e n t i a l l y constant at 750 gm/mm . Another problem i s that as yet no experiments have been able to show how many d i s l o c a t i o n s are contained i n the supposed p i l e - u p s , and t h i s number  73 should s i g n i f i c a n t l y a f f e c t the magnitude of the i n t e r n a l s t r e s s on the secondary system.  The major problem a r i s e s from the s t r a i n at which Bocek  considers flow on the secondary system to be e f f e c t i v e . Results from the present o r i e n t a t i o n study, and from s t r a i n rate change t e s t s have shown that i t i s not the onset of stage I I but rather the s t a r t of d e v i a t i o n from stage I which i s the c r i t i c a l point i n the t r a n s i t i o n .  Also, the present  study as w e l l as that of Risebrough^ has shown that the m a t e r i a l i s extensively twinned at the s t a r t of stage I I , and Bocek has not  considered  t h i s point.  4.2.3  Present model The basis f o r the present model i s s i m i l a r to that of Bocek.in  that second order pyramidal s l i p i s thought to be responsible f o r the t r a n s i t i o n from stage I to stage I I . However, i t i s believed that there i s some a c t i v i t y on t h i s sytem i n the l a t t e r part of stage I and  that  t h i s i s responsible f o r the non-recoverable work hardening as shown?-by the drop i n s a t u r a t i o n recovery with s t r a i n i n F i g . 11.  The end of stage I  l i n e a r i t y i s believed to be caused by macroscopic flow on t h i s secondary system and t h i s i s then responsible f o r the increasing work hardening rate i n the t r a n s i t i o n region. The onset of stage I I l i n e a r i t y probably r e s u l t s from the development of a dynamic e q u i l i b r i u m between basal and pyramidal dislocations.  This w i l l be discussed i n f o l l o w i n g s e c t i o n s .  At t h i s p o i n t ,  i t i s s u f f i c i e n t to note that stage I I l i n e a r i t y i s not as s i g n i f i c a n t a 27 f a c t o r as discussed by Bocek et a l  ,  .  The r e s u l t s of the present work show that i t i s possible;.: that second order pyramidal s l i p could be responsible f o r various work hardening and recovery phenomena. F i g . 40 shows the shear s t r e s s on the second order pyramidal system compared to the flow stress on the basal plane f o r a c r y s t a l  75 with i n i t i a l o r i e n t a t i o n X  Q  =  45°.  F i g . 41 i s a replot of the data on F i g . 11,  with s t r a i n transformed to o r i e n t a t i o n .  I t i s seen from t h i s p l o t that non-  recoverable work hardening s t a r t s i n the range 30° < x < 35°. corresponds to .5<y<.7  on F i g .  This range  40, and i n t h i s range, the s t r e s s on the 2  second order pyramidal system i s 75 to 100 gm/mm . I t i s probably only c o i n c i d e n t a l that t h i s i s the range i n which the s t r e s s on the secondary system becomes greater than the s t r e s s on the basal system.  At the end  of easy g l i d e , at which x ~ 23° (see F i g . 28), the s t r e s s on the pyramidal 2 system i s ^ 260 gm/mm . ,. . ' 28 S t o l o f f and Gensamer found that the f i r s t appearance of 2 pyramidal s l i p traces occurred at 500 gm/mm i n Cd c r y s t a l s not oriented for basal s l i p . Their m a t e r i a l , however, had a c r i t i c a l resolved shear 2 stress f o r basal s l i p of 85 gm/mm , while the m a t e r i a l used i n t h i s work 2 had a comparable s t r e s s of 20 gm/mm . Consequently the c r i t i c a l s t r e s s 2 for flow on the pyramidal system i s probably less than 500 gm/mm i n t h i s m a t e r i a l , but since the e f f e c t of i m p u r i t i e s and substructure on .pyramidal g l i d e i s not known, i t i s impossible to assign a d e f i n i t e value. 2  However,  t h i s value of 500 gm/mm was given f o r the appearance of s l i p traces which would imply massive s l i p on t h i s sytem, but to end easy g l i d e r e l a t i v e l y few d i s l o c a t i o n s would be required, so i t would be reasonable to assume that these would be generated at a s i g n i f i c a n t l y lower s t r e s s .  , ,,  :  Assuming that second order pyramidal d i s l o c a t i o n s are a c t i v e at o r i e n t a t i o n s < 30°, i t i s p o s s i b l e to e x p l a i n non-recoverable work ^ ..29 w  hardening as shown i n F i g . 41.  Bocek, Lukac and Svabova  have shown  that e n e r g e t i c a l l y favourable reactions between d i f f e r e n t second order pyramidal systems and between second order pyramidal and basal d i s l o c a t i o n s can occur.  The r e s u l t a n t d i s l o c a t i o n s are s e s s i l e and w i l l produce  work hardening.  These reactions are:  77 1/3 [2113] + 1/3 [1123] •* 1/3 [3030]  (a)  1/3 [2113] + 1/3 [1123] -y 1/3 [1210]  (b)  1/3 [2113] + 1/3 [2110] •*• 1/3 [0003]  (c)  Reaction (c) between second order pyramidal and basal d i s l o c a t i o n s i s probably the most important. Once these s e s s i l e s form, they would probably not be amenable to annealing, and so cause non-recoverable work hardening.  Substantiating evidence f o r t h i s was found i n tests which 39  combined recovery w i t h s t r a i n rate changes.  Laurent'ev et a l  have found  that the c r i t i c a l resolved shear s t r e s s i n z i n c at 20°C i s a c r i t i c a l function of the pyramidal d i s l o c a t i o n density.  In zinc i t appears that;  second order pyramidal d i s l o c a t i o n s may also cause non-recoverable work hardening. I t was also seen i n F i g .  5  ,  36 that there i s a small d e v i a t i o n from  a room temperature s t r e s s - s t r a i n curve of a s t r e s s - s t r a i n curve completely recovered at i n f i n i t e s i m a l s t r a i n increments. The errors involved i n the  flow s t r e s s measurement, and i n the r e p r o d u c i b i l i t y between two  d i f f e r e n t specimens are too large to permit the c a l c u l a t i o n of the work> hardening produced by pyramidal d i s l o c a t i o n s and t h e i r i n t e r a c t i o n with. basal d i s l o c a t i o n s from x = 30° to x ~ 23° i f t h i s secondary system operates at -196°C and not at 20°C. When x reaches 23°, the d i f f e r e n c e i n Schmid factors on the two systems i s such that there would be s u b s t a n t i a l l y . more a c t i v i t y on the second system. 2  At t h i s point the s t r e s s on thejpyramidal  plane i s 260 gm/mm which may be very near the macroscopic c r i t i c a l s t r e s s . The present r e s u l t s show that t h i s s t r e s s , at the end of easy glide,., i s independent of i n i t i a l o r i e n t a t i o n .  I f t h i s i s the y i e l d s t r e s s f o r second  order pyramidal, then the massive a c t i v i t y on t h i s sytem would e x p l a i n , the  termination of l i n e a r easy g l i d e .  78 Thus i t may be concluded, that w h i l e a c r y s t a l with x  = o  45°  deforms e n t i r e l y i n basal g l i d e i n the i n i t i a l stages of deformation, when the s t r a i n i s such that x i s of the order of 30° to 35°, second order pyramidal d i s l o c a t i o n a c t i v i t y becomes s i g n i f i c a n t .  This i s not to say  t h a t macroscopic s t r a i n i s achieved on t h i s system, but merely t h a t : d i s l o c ations are generated which w i l l subsequently i n t e r a c t w i t h b a s a l d i s l o c a t i o n s to cause poor recoverable work hardening.  When x reaches 23° macroscopic  flow may occur on the pyramidal system thus terminating l i n e a r easy, g l i d e . Since the end of the easy g l i d e i s determined by the angle between the basal (or second order pyramidal) plane and the t e n s i l e , a x i s , i t i s reasonable to use t h i s angle as a normalizing f a c t o r f o r a l l „ tensile results.  However, since x v a r i e s quite slowly with s t r a i n at low x  values, stress curves have been p l o t t e d versus s t r a i n , with the i n t e r c e p t of extrapolated easy g l i d e and stage I I slopes taken at (Y = 1.8  for X  Q  x  =  20.2°  ,  = 45°).  4.3 S t r a i n rate change tests  ^  A c t i v a t i o n volume measurements, which are derived from s t r a i n rate change t e s t s , may i n d i c a t e the nature of the rate c o n t r o l l i n g mechanism during deformation.  The p r i n c i p a l assumption made regarding i n t e r p r e t a t i o n  of a c t i v a t i o n volume data i s that the density of mobile d i s l o c a t i o n s remains constant during a s t r a i n rate change.  I f t h i s i s the case, then i t i s ; :  assumed that the change i n flow s t r e s s r e s u l t i n g from a change i n s t r a i n rate i s due only to the deformation rate c o n t r o l l i n g process or processes. I t i s known that the deformation made during easy g l i d e i n cadmium s i n g l e c r y s t a l s i s s l i p on the b a s a l system.  Consequently, i t  i s assumed that the flow stress i n t h i s region i s determined only by the  79  density of b a s a l d i s l o c a t i o n s and by the way i n which they move through the lattice. In stage I I deformation, i t i s assumed that the flow s t r e s s i s s t i l l c o n t r o l l e d only by the density of b a s a l d i s l o c a t i o n s which are assumed to be the mobile d i s l o c a t i o n s .  For face-centered cubic m a t e r i a l s , 30  i t has been shown by X-ray d i f f r a c t i o n measurements of Ahlers and Haasen 31 32 and M i t c h e l l and Thornton and by shape change measurements of Kocks that most of the deformation i n stage I I takes place by s l i p on the primary system.  For the present system, F i g . 33 shows  that w h i l e .  deformation i s not homogeneous during stage I I , the width of the c r y s t a l does not decrease s i g n i f i c a n t l y .  ( F i g . 33 compares a c r y s t a l deformed  w e l l i n t o stage I I at -196°C to a c r y s t a l deformed at 20°C where e s s e n t i a l l y a l l deformation was on the b a s a l plane).  Since the width d i d not decrease  s i g n i f i c a n t l y , i t i s not unreasonable to assume that during stage I I i n cadmium, as w e l l as f o r face-centered cubic m a t e r i a l s , the majority ,of the deformation occurs on the primary system.  Therefore, since flow s t i l l  occurs on the basal system, i t i s assumed that the flow stress i s s t i l l a measure of the basal d i s l o c a t i o n density i n stage I I as i t i s i n stage I . From the r e s u l t s of the present work on o r i e n t a t i o n , and from the r e s u l t s of Risebrough^, i t i s expected that any secondary  activity  which takes place w i l l be on the second order pyramidal system.  Consequently  i t i s assumed i n the f o l l o w i n g d i s c u s s i o n that the d i s l o c a t i o n f o r e s t i s composed e n t i r e l y of second order pyramidal d i s l o c a t i o n s .  I t may be  that other types are present as .grown-in d i s l o c a t i o n s , but these should not change during deformation.  I t has also been assumed that the c o n t r i b u t i o n  to s t r a i n of the pyramidal d i s l o c a t i o n s i s n e g l i g i b l e .  .  80 4.3.1  Stage I  4.3.1.1 A c t i v a t i o n volume behaviour F i g . 19 shows that a c t i v a t i o n volume during easy g l i d e i s a s t e a d i l y decreasing f u n c t i o n of s t r a i n . i s 40 x 10  -20  At y i e l d , the a c t i v a t i o n volume  3 3 cm , which i s equivalent to 15,000 b .  This value drops to  3 about 5500 b  at the end of easy g l i d e .  A c t i v a t i o n volume values i n t h i s range are i n d i c a t i v e of ..a rate c o n t r o l l i n g mechanism of e i t h e r f o r e s t i n t e r s e c t i o n by the basal d i s l o c a t i o n s or the non-conservative motion of jogs i n the b a s a l d i s l o c a t i o n s . I t i s not p o s s i b l e on the basis of rate parameter measurements alone to d i s t i n g u i s h between these two mechanisms. Risebrough^ has claimed that the mechanism c o n t r o l l i n g y i e l d and flow i n cadmium s i n g l e c r y s t a l s i s one of f o r e s t i n t e r s e c t i o n . He ;  found no major i n c o n s i s t e n c i e s i n t h i s argument whereas h i s  experimental  data did not agree with the l i m i t a t i o n s on the jog mechanism.  He believed  that the jog mechanism was unacceptable p r i m a r i l y because the flow s t r e s s i n zinc and cadmium was  temperature dependent below T  = 0.25.  This  H phenomenon would require that the nucleation of a vacancy i n conjunction 33 with the non-conservative motion of jogs be thermally a c t i v a t e d . Mott , has stated that t h i s vacancy nucleation i s completely secondly below T„ = . n  athermal, and;,  that s i n g l e vacancies w i l l not migrate at appreciable  ,  rates  0.5. " The present r e s u l t s f i n d i n c o n s i s t e n c i e s i n the f o r e s t i n t e r s e c t i o n  mechanism.  In the i n i t i a l stages of s t r a i n (up to 70%) , i t was  the a c t i v a t i o n volume decreased s i g n i f i c a n t l y .  found that  I f the f o r e s t i n t e r s e c t i o n  mechanism were rate c o n t r o l l i n g , t h i s decrease would imply an increase i n the f o r e s t density.  However, when the c r y s t a l s were recovered i n this-range,  81 i t was found that the a c t i v a t i o n volume recovered back to the value found at y i e l d (see F i g . 20). Since i t i s very u n l i k e l y that both primary  and  f o r e s t d i s l o c a t i o n s would recover i n e x a c t l y the same manner, as the f o r e s t i n t e r s e c t i o n mechanism would r e q u i r e , i t has been assumed that the f o r e s t d i s l o c a t i o n s do not recover at a l l .  I t has been shown that  second order pyramidal d i s l o c a t i o n s w i l l combine with b a s a l d i s l o c a t i o n s to form e n e r g e t i c a l l y s t a b l e obstacles, so i t should be reasonable to assume that these w i l l not anneal out i n the same manner as mobile b a s a l d i s locations.  This i s made more apparent when compared to the behaviour of  pyramidal d i s l o c a t i o n s at higher s t r a i n s as discussed i n a l a t e r s e c t i o n . Thus, i t appears that the f o r e s t i n t e r s e c t i o n mechanism i s not a p p l i c a b l e to the present system. The other p o s s i b l e rate c o n t r o l l i n g mechanism i s the nonr conservative motion of jogs.  In the present system i t i s believed that t h i s  i s the rate c o n t r o l l i n g mechanism, while Risebrough"' has r e j e c t e d t h i s 33 mechanism on the basis of the arguments of Mott arguments are questionable as to t h e i r v a l i d i t y .  , but some of these F i r s t , Mott s t a t e s that  s i n g l e vacancy migration i s not appreciable below .5  i n Cu.  However,  Sharp, M i t c h e l l and Christian"'" have shown that i n Cd vacancy migration occurs at about .25 T^.  Secondly, Mott does not adequately explain, why  the process of vacancy n u c l e a t i o n at a jog should be completely athermal. If such n u c l e a t i o n were thermally a c t i v a t e d , then the combination of t h i s plus vacancy migration could e x p l a i n the temperature dependence of the flow stress i n cadmium.  This would then remove any o b j e c t i o n to the assumption  that the non-conservative motion of jogs i s the rate c o n t r o l l i n g mechanism. Thus, since doubt does e x i s t with respect to the Mott theory, and since experimental r e s u l t s preclude the acceptance of the f o r e s t i n t e r s e c t i o n mechanism, i t w i l l be assumed that the non-conservative motion of jogs  c o n t r o l s flow i n easy g l i d e i n cadmium. Having made the above assumption, i t i s now possible to explain the behaviour of a c t i v a t i o n volume with s t r a i n i n easy g l i d e .  During the  i n i t i a l stages of s t r a i n , there remains a constant density of f o r e s t dislocations.  A c t i v a t i o n volume decreases i n t h i s range due to the  decreased i n t e r - j o g spacing along the mobile basal d i s l o c a t i o n s .  On  recovery, the jogs i n these basal d i s l o c a t i o n s anneal out, probably, by d i f f u s i o n along the d i s l o c a t i o n l i n e .  When a j o g of one sign meets one  of the opposite s i g n , the two a n n i h i l a t e and so decrease the j o g density. Under conditions  of s a t u r a t i o n recovery, some e q u i l i b r i u m concentration  of jogs i s attained which would be the same concentration as e x i s t e d at yield.  Therefore, the a c t i v a t i o n volume f o l l o w i n g recovery w i l l be the  same as i t was at y i e l d . I f t h i s e q u i l i b r i u m concentration of jogs i n basal  dislocations  i s such that the i n t e r - j o g spacing along the d i s l o c a t i o n s i s larger, than the spacing between f o r e s t d i s l o c a t i o n s , then the a c t i v a t i o n volume measured under such conditions  should be i n d i c a t i v e of the f o r e s t  This should be true both at y i e l d and f o l l o w i n g recovery.  density.  Howeveronce  the basal d i s l o c a t i o n has moved through the f i r s t set of f o r e s t d i s l o c a t i o n s that i t encounters, the i n t e r - j o g spacing w i l l be l e s s than the f o r e s t spacing, and so i t w i l l be the j o g mechanism which i s measured.  If i t f  i s assumed that the i n t e r - j o g spacing i s l a r g e r than the average f o r e s t spacing, then on the basis of the previous argument i t i s p o s s i b l e to , c a l c u l a t e the f o r e s t density from a c t i v a t i o n volume data f o l l o w i n g recovery. A c t i v a t i o n volume i s defined as-: v = b£d where b = Burger's vector £ = activated length of d i s l o c a t i o n d = distance over which d i s l o c a t i o n i s moved.  ,) j , ;  83 I f i t i s assumed that the a c t i v a t i o n distance i s equal to the Burger's vector, then  I f f o r e s t i n t e r s e c t i o n i s the r a t e c o n t r o l l i n g mechanism as has been assumed at y i e l d and f o l l o w i n g recovery, then £ i s the average spacing between the f o r e s t d i s l o c a t i o n s . The f o r e s t density i s then: P  =  1  =  b it  •a 2  On the basis of the above assumption, the f o r e s t d e n s i t y ' a t 6 y i e l d has been c a l c u l a t e d  as 4.7 x 10  2 lines/cm .  ;  Close to the end of  easy g l i d e , the pyramidal density has increased by about one order of 7 magnitude to 4 x 10  -2 cm  .  Recovery i n the middle of the t r a n s i t i o n  region between stage I and stage I I shows that the pyramidal 9 i s about 2 x 10  density  -2 cm  .  •  (  These above values are consistent with the r e s u l t s from ,the orientation analysis.  A c t i v a t i o n volume data show that the f o r e s t does  not change up to about 70% s t r a i n .  From t h i s point to the end of easy  g l i d e , i t i s found that the f o r e s t density increases from 4 x 10 7 4 x 10  to  -2 cm  and i t i s t h i s increase which i s associated with microj  a c t i v i t y on the second order pyramidal system.  For a small s t r a i n increment  from the end of easy g l i d e to the middle of the t r a n s i t i o n region an increment from the end of easy g l i d e to the middle of the t r a n s i t i o n region an increase i n pyramidal d i s l o c a t i o n s of two orders of magnitude was found.  This i s consistent with the idea of macroactivity on the second  order pyramidal at the end of easy g l i d e .  Thus, i t i s concluded from an  unrelated type of t e s t that the conclusions regarding work hardening parameters are v a l i d i n that there i s m i c r o a c t i v i t y on a secondary system  84 i n the l a t t e r part of easy g l i d e followed by macroactivity i n the t r a n s i t i o n region.  4.3.1.2 C o t t r e l l - S t o k e s behaviour The C o t t r e l l - S t o k e s behaviour of a metal i s the v a r i a t i o n i n flow s t r e s s r e s u l t i n g from a change i n s t r a i n rate (or temperature) as a function of the t o t a l stress on the system.  I f this variation i s linear,  passes through the o r i g i n , then the C o t t r e l l - S t o k e s law i s considered to be obeyed.  The i n t e r p r e t a t i o n of the r e s u l t s and the s i g n i f i c a n c e of  obeyance or non-obeyance i s not c l e a r l y understood, and i s usually only applied to face-centred  cubic materials.  Many t e s t s , however, have been  performed on many d i f f e r e n t materials to t e s t t h i s law. Most materials which obey the law are face-centered  cubic, but obeyance has been reported 34  on some hexagonal metals.  Basinski  has found obeyance i n magnesium  s i n g l e c r y s t a l s below 47°K; Davis"^ and Risebrough^ found obeyance during V-35  stage I I deformation of cadmium at -196°C and Bocek and Lukac  claim  zinc i n stage I I at 20°C also obeys.  ,  The present r e s u l t s show that nowhere during the deformation of s i n g l e c r y s t a l cadmium at -196°C i s the C o t t r e l l - S t o k e s law obeyed. !, , The p l o t t i n g of A T vs. T as i n a C o t t r e l l - S t o k e s t e s t gives useful information besides checking the obeyance of the C o t t r e l l - S t o k e s law.  Since a c t i v a t i o n volume i s representative of the rate c o n t r o l l i n g  mechanism, and i s determined d i r e c t l y from the measurement of A T , then AT i t s e l f must also be i n d i c a t i v e of the rate c o n t r o l l i n g process.  In  the present case, A T at y i e l d i s - a d i r e c t measure of the number of f o r e s t i n t e r s e c t i o n s taking place at the time of the s t r a i n rate change.  Ap  s t r a i n s past y i e l d , A T i s a measure of the density of jogs present i n j  85 basal d i s l o c a t i o n s .  As mentioned p r e v i o u s l y , i t i s believed that the flow  stress T i s dependent on the density of basal d i s l o c a t i o n s .  Therefore,  a p l o t of A T vs. T i s , i n stage I , e s s e n t i a l l y a p l o t of the density of jogs vs. the density of basal d i s l o c a t i o n s . Fig.  23 shows that the p l o t of A T vs. T i s l i n e a r i n the . i n i t i a l  p o r t i o n of stage I.  Using the above argument, t h i s s i g n i f i e s that the  basal jog density increases proportionately to the density of b a s a l .dislocations.  In t h i s i n i t i a l s t r a i n region, the behaviour of A T vs. T i s  unaffected by recovery i n that r e l a t i o n of A T to T i t was  i s the same as  during s t r a i n p r i o r to recovery. Thus, the increase i n the density  of jogs with respect to the increase i n b a s a l density i s unchanged. Following the second recovery, which i s the point at which secondary activity  ,  i s expected to begin, i t i s seen that there i s a s l i g h t increase  i n the slope of the p l o t .  This i s explained i f i t i s assumed that the  pyramidal d i s l o c a t i o n density i n t h i s range i s i n c r e a s i n g s l i g h t l y .  With  a higher pyramidal density, there w i l l now be a l a r g e r number of jogs being formed f o r a given increase i n the basal d i s l o c a t i o n density (T) ., At the end of l i n e a r easy g l i d e , F i g . 22 shows that there i s a s i g n i f i c a n t increase i n the slope of the A T - T p l o t .  I t i s expected  that there i s a macroactivity on the pyramidal system at t h i s point • causing a s u b s t a n t i a l l y higher f o r e s t density. the density of f o r e s t d i s l o c a t i o n s was  (As mentioned p r e v i o u s l y ,  found to increase by two  orders  of magnitude from the end of easy g l i d e to the middle of the t r a n s i t i o n region.)  Under these circumstances,  a given increase i n basal density  w i l l r e s u l t i n a s u b s t a n t i a l l y higher number of jogs formed, and soythe slope of the A T - T p l o t w i l l increase.  i {  86 4.3.2 Stage I I In stage I I deformation,  there i s no evidence to suggest that the  rate c o n t r o l l i n g mechanism changes i n any way.  However, f o r the purpose  of s i m p l i f i c a t i o n of the f o l l o w i n g d i s c u s s i o n , i t w i l l be assumed that t h i s mechanism i s now one of f o r e s t i n t e r s e c t i o n rather than the non-conservative motion of jogs.  This assumption i s probably not true, but i t should  not introduce s i g n i f i c a n t errors since the two mechanisms are so s i m i l a r i n t h e i r behaviour.  Thus i n the f o l l o w i n g d i s c u s s i o n , A T i s assumed to  be representative of f o r e s t density rather than j o g density. During stage I I deformation, where the f o r e s t density i s l a r g e , probably of the same order as the basal density, the basal d i s l o c a t i o n s w i l l become ;;  highly jogged very q u i c k l y .  Consequently these b a s a l d i s l o c a t i o n s w i l l  probably not move appreciable distances.  New d i s l o c a t i o n s w i l l be generated  and become jogged i n i t i a l l y at a r a t e determined by the f o r e s t density., Therefore, the measured values of Ax and a c t i v a t i o n volume would not be s i g n i f i c a n t l y d i f f e r e n t i f f o r e s t i n t e r s e c t i o n were the rate c o n t r o l l i n g mechanism. The p l o t of A T V S . T f o r deformation essentially linear.  i n stage I I ( F i g . 22) i s ;  This i n d i c a t e s that the factors leading to both  T and A T increase proportionately. Comparing cadmium to face-centered  ;  j, cubic metals, i t i s seen that the 36  above conclusion i s consistent with the r e s u l t s of Steeds  and B a s i n s k i  37 and B a s i n s k i  . These workers found that the density of secondary d i s -  locations i n copper s i n g l e c r y s t a l s i s comparable to the density of primary d i s l o c a t i o n s throughout stage I I , i n s p i t e of the f a c t that p l a s t i c s t r a i n on the secondary system i s very small.  This i s f u r t h e r evidence for-, the  statement that the l i n e a r i t y of A T vs. T i s due to a p r o p o r t i o n a l increase i n both pyramidal  ( f o r e s t ) and basal (primary) d i s l o c a t i o n s rather than- an,  increase i n basal jogs with no s i g n i f i c a n t increase i n f o r e s t density.  87 The behaviour of A T vs. T with respect to recovery ( F i g . 2 4 ) i s s i g n i f i c a n t l y d i f f e r e n t i n stage I I than i t was i n stage I .  While i n  stage I , recovery brought about a s u b s t a n t i a l decrease i n both A T and T , the i n i t i a l recovery anneal i n stage I I caused a large drop i n T and e s s e n t i a l l y no change i n A T .  S t r a i n f o l l o w i n g t h i s recovery showed that  the slope of A T vs. T i s s t i l l the same as i t was f o r s t r a i n preceding recovery.  l(  That T changes markedly i n d i c a t e s that the b a s a l d i s l o c a t i o n density decreases s i g n i f i c a n t l y since as was explained e a r l i e r , t h e f l o w ;  stress i s dependent p r i m a r i l y on b a s a l density.  Because A T does not  change during t h i s anneal, the density of pyramidal d i s l o c a t i o n s must remain constant during recovery.  The f a c t that the slope of A T vs. T  i s the same following recovery as i t was preceding recovery i n d i c a t e s , that the d e n s i t i e s of pyramidal and b a s a l d i s l o c a t i o n s are s t i l l i n c r e a s i n g i n the same r e l a t i v e proportion. However, since the b a s a l density decreased while the pyramidal density remained constant during recovery, there must be a net increase of pyramidal d i s l o c a t i o n s to the system when . r  i t i s compared at the same stress l e v e l .  There are more pyramidal d i s r  locations at point B i n F i g . 2 4 than at point A.  ,  The second recovery anneal gives a further s h i f t to the A T - T p l o t as shown i n F i g . 2 4 . decrease s i g n i f i c a n t l y .  In t h i s case, however, A T as w e l l as T does  On the basis of the above argument, t h i s i n d i c a t e s  that some pyramidal d i s l o c a t i o n s must be l o s t at t h i s time.  Again, s t r a i n  f o l l o w i n g recovery shows that the increase of both b a s a l and pyramidal d i s l o c a t i o n s remains i n the same proportion as that p r i o r to any recovery since the slope of the p l o t i s unchanged.  In t h i s s t r a i n increment^between  recovery anneals, there i s a further increase to the system of pyramidal d i s l o c a t i o n s , as shown by comparison of s i m i l a r s t r e s s l e v e l s A,B, and C in Fig.  24.  f.  Following  further recoveries, the A T - T p l o t s are coincident with  p l o t obtained f o l l o w i n g the second recovery.  that  Thus, a condition must  have been reached i n which both pyramidal and basal d i s l o c a t i o n s are recovered.  The proportion of each type of d i s l o c a t i o n recovered i s the  same as that i n which i t i s generated during s t r a i n , therefore e s t a b l i s h i n g some s o r t of e q u i l i b r i u m i n the s t r u c t u r e . condition prescribes any stress l e v e l .  j  This e q u i l i b r i u m  the density of basal and pyramidal d i s l o c a t i o n s at  This condition was  found to be maintained i n the c r y s t a l  from the second anneal i n stage I I to f a i l u r e of the specimen. I t i s probably coincidence that i t took exactly two recovery anneals i n stage I I to reach t h i s e q u i l i b r i u m c o n d i t i o n .  I f experimental  conditions were d i f f e r e n t ( i . e . i f the stress at which recovery was  carried  out were d i f f e r e n t ) , then the number of anneals required to reach e q u i l i b r i u m might change, although i t i s thought that the end r e s u l t would be the same. The recovery of pyramidal d i s l o c a t i o n s as found i n the above 23 r e s u l t s may  occur by the mechanism as observed by P r i c e  .  He found that  second order pyramidal d i s l o c a t i o n s which had formed long loops by  the  cross-glide of screw segments f i r s t annealed by these loops s p l i t t i n g up i n t o rows of c i r c u l a r loops.  These c i r c u l a r loops then annealed out of  the sample by means of volume d i f f u s i o n .  i  As a r e s u l t of the findings of t h i s s e c t i o n on C o t t r e l l - S t o k e s behaviour, i t i s concluded that for a hexagonal close-packed metal such as cadmium, the C o t t r e l l - S t o k e s law has no r e a l s i g n i f i c a n c e .  If i t is  found to be obeyed i n any hexagonal system, i t i s probably only c o i n c i d e n t a l . 4.4 Flow stress following recovery  .  ;  The flow stress measured f o l l o w i n g recovery i s probably determined p r i m a r i l y by basal d i s l o c a t i o n density  at t h i s p o i n t .  In stage I I , the b a s a l density present f o l l o w i n g s a t u r a t i o n recovery appears to be a function of the pyramidal d i s l o c a t i o n density which i s also present.  F i g . 42 i s a p l o t of the flow s t r e s s achieved a f t e r s a t u r a t i o n  recovery as a f u n c t i o n of s t r a i n .  In t h i s p l o t , stage I I l i n e a r i t y begins  i n a l l t e s t s at a s t r a i n of about 1.75.  I t i s seen that the b a s a l, s  density r i s e s to a maxiumum value at approximately 60% s t r a i n f o l l o w i n g the onset of stage I I l i n e a r i t y , and then remains constant.  This i s very  s i m i l a r to the behaviour deduced f o r pyramidal d i s l o c a t i o n s i n the previous section. F i g . 42 shows that there i s an increase i n the b a s a l density 2  2  (flow s t r e s s ) by a f a c t o r of 3 from 250 gm/mm to 800 gm/mm .  F i g . 21  showed that the pyramidal density ( A T ) i n t h i s same stress range also increases by a f a c t o r of 3.  Thus, i t appears that there i s a d i r e c t  r e l a t i o n s h i p between the density of b a s a l d i s l o c a t i o n s and the density • of pyramidal d i s l o c a t i o n s present f o l l o w i n g recovery. This r e l a t i o n s h i p i s probably caused by the way i n which the ;  basal d i s l o c a t i o n s are trapped by the pyramidal d i s l o c a t i o n s , or perhaps by a s t r e s s f i e l d associated with the pyramidal d i s l o c a t i o n s .  That the  pyramidal density i s the governing f a c t o r i n t h i s r e l a t i o n s h i p i s more apt to be the case than the converse, i n which the b a s a l density p r e s c r i b e s the pyramidal d e n s i t y , since i t has been shown i n the previous s e c t i o n that the pyramidal density i s i n s e n s i t i v e to recovery. 4.5 Work hardening i n stage I I  These a c t i v a t i o n volume and A T vs. T data also give some i n s i g h t i n t o the work hardening behaviour of cadmium s i n g l e c r y s t a l s i n stage I I .  The data f i t w e l l with the arguments of H i r s c h and M i t c h e l l face-centered cubic m a t e r i a l s .  based on  The p r i n c i p a l points i n t h e i r theory,  together with comparison to the present work are as f o l l o w s : 1)  At the end of stage I , there are long continuous obstacles which  form, b a r r i e r s to newly formed s l i p l i n e s .  D i s l o c a t i o n s are stopped by  e l a s t i c i n t e r a c t i o n with both the primary and secondary d i s l o c a t i o n s i n the obstacles.  I n cadmium, i t i s proposed that these b a r r i e r s are the s e s s i l e  d i s l o c a t i o n s formed by the i n t e r a c t i o n of b a s a l and second order pyramidal dislocations.' 2)  Hardening' i s due to the hardening of p o t e n t i a l primary sources  by the increased density of 'primary and secondary d i s l o c a t i o n s .  Since i t  i s expected that there i s s i g n i f i c a n t secondary a c t i v i t y f o l l o w i n g l i n e a r easy g l i d e , there should be s u f f i c i e n t density to harden primary sources i n cadmium. 3)  ,• I n the t r a n s i t i o n region between stage I and stage I I , the r a t i o  of new secondary d i s l o c a t i o n s to new b a s a l d i s l o c a t i o n s generated increases r a p i d l y , u n t i l the regions w i t h i n which secondary sources operate extend throughout the c r y s t a l .  At t h i s p o i n t , the p r o p o r t i o n of secondary d i s -  locations depends on the i n t e r n a l s t r e s s pattern.  For a given arrangement  of primary d i s l o c a t i o n s * the density of secondary d i s l o c a t i o n s will ,be ;;  the maximum p o s s i b l e compatible with the i n t e r n a l s t r e s s , leading to the maximum p o s s i b l e hardening rate.  Thus, Hirsch and M i t c h e l l have assumed  that a s i t u a t i o n of s i m i l i t u d e develops i n which the r a t i o of secondary to primary d i s l o c a t i o n s remains constant throughout stage I I , and only the scale of.the. s t r u c t u r e decreases as the s t r e s s increases. The present, r e s u l t s are compatible w i t h t h i s theory up to the point of recovery,, i n , stage I I . At t h i s point i t was found that the secondary d i s l o c a t i o n  density did not decrease whereas the primary density d i d .  On f u r t h e r  deformation, the secondary density was higher than the previously maximum possible density.  Since recovery has permitted an increased secondary  density with respect to primary density, i t i s p o s s i b l e to maintain c o n d i t i o n , and consequently give a higher work hardening r a t e . observed experimentally  this  This was  (see F i g . 32). T h i s , too, i s consistent with  Hirsch and M i t c h e l l ' s theory that the work hardening rate i s dependent, among other f a c t o r s , on the density of secondary d i s l o c a t i o n s i n the region of the p i l e - u p s . 4)  ,,  The flow s t r e s s i s determined p a r t l y by f o r e s t density, l i n e  tension, and long range s t r e s s i n the s o f t e s t region between p i l e - u p s . The r e l a t i v e contributions have yet to be determined.  I n the present work,  i t appears that the flow s t r e s s i s dependent p r i m a r i l y on the b a s a l density, which would be equivalent to Hirsch and M i t c h e l l ' s long range stress.  Thus the present r e s u l t s are consistent with the theory of Hirsch and M i t c h e l l , and i t i s reasonable to assume that t h i s  theory  could be extended to include hexagonal close-packed metals as w e l l as facecentered  cubic. At very high s t r a i n s , i t was found that the work hardening rate  decreased.  This may be comparable to a stage I I I which i s often  associated with dynamic recovery.  Since the A T vs. T r e l a t i o n s h i p  does not change i n t h i s range, dynamic recovery should a f f e c t both b a s a l and f o r e s t d i s l o c a t i o n s .  93 5. SUMMARY AND CONCLUSIONS  1)  Deformation i n easy g l i d e at -196°C i s by b a s a l g l i d e only when  X > 35°. In t h i s range, the work hardening i s completely recoverable by annealing at 75°C f o r 30 min.  2}  Deformation i n easy g l i d e at -196°C from x - 35° to the end,of  easy g l i d e i s p r i m a r i l y by b a s a l g l i d e with some a c t i v i t y on the second order pyramidal s l i p system.  Work hardening i n t h i s region i s not  completely recoverable. This i s probably due to the formation of s t a b l e obstacles by the i n t e r a c t i o n of b a s a l and pyramidal d i s l o c a t i o n s .  3)  •  The end of easy g l i d e occurs at x = 20.1° ± 1.2°. This i s  probably a r e s u l t of exceeding the y i e l d stress on the second order; pyramidal system, thus forming a much higher density of obstacles. ,  4)  Deformation i n stage I I i s p r i m a r i l y on the b a s a l systemjalthough  the densities of b a s a l and pyramidal d i s l o c a t i o n s are probably about the same.  Recovery early i n stage I I causes a decrease i n b a s a l density with  no corresponding decrease i n pyramidal density. As a r e s u l t of t h i s , the work hardening rate following recovery i s higher than that preceding recovery.  At higher s t r a i n s i n stage I I , both b a s a l and pyramidal  d i s l o c a t i o n s are recovered.  5) recovery.  (  . -;  The s t r a i n a t t a i n a b l e i n stage I I at -196°C i s increased, by  6)  Deformation i s probably c o n t r o l l e d i n stage I by the non-  conservative motion of jogs.  The data from stage I I may also be  i n t e r p r e t e d i n terms of t h i s mechanism.  7)  The C o t t r e l l - S t o k e s law i s not s t r i c t l y obeyed through any  p o r t i o n of the deformation of cadmium s i n g l e c r y s t a l s .  On the basis  of the present r e s u l t s , i t appears that obeyance of t h i s law i n hexagonal metal would be c o i n c i d e n t a l .  8)  I t was found to be impossible to c a l c u l a t e a meaningful ,  a c t i v a t i o n energy.  95 6. APPENDIX  6.1 E l e c t r o n microscopy techniques Ah attempt was made to t h i n sections of cadmium s i n g l e c r y s t a l s for observation i n the e l e c t r o n microscope.  In t h i s way, i t was hoped  to observe the d i s l o c a t i o n s t r u c t u r e of the specimens, and to determine the  e f f e c t s of deformation and recovery upon t h i s s t r u c t u r e . Unfortunately,  a l l attempts at thinning were unsuccessful, and no structures were observed. The techniques employed i n t h i s study, and the problems . encountered, are l i s t e d below.  ,  i;  6.1.1 Cutting a t h i n s e c t i o n .  The f i r s t problem i n t h i s study was to obtain a t h i n , p a r a l l e l -  sided s l i c e of a s p e c i f i e d o r i e n t a t i o n from a bulk s i n g l e c r y s t a l .  I t was  also, necessary to ensure that no extraneous d i s l o c a t i o n s were introduced i n t o the s l i c e during the c u t t i n g operation.  There was no d i f f i c u l t y  inovlved i n determining the desired o r i e n t a t i o n since X-ray techniques are w e l l established and s u f f i c i e n t l y accurate. Problems were encountered i n c u t t i n g a t h i n s l i c e from the 5 mm.  diameter bulk s i n g l e crystal.;  I t was decided that c u t t i n g by any mechanical means such as a j e w e l l e r ' s saw. would be.unacceptable due to the amount of deformation introduced to the sample during the c u t t i n g operation. Spark erosion was the f i r s t method t r i e d to obtain the desired slice.  I t was hoped that by using a r e l a t i v e l y low energy spark that  there would be n e g l i g i b l e deformation introduced i n t o the sample.  This  method worked w e l l f o r the f i r s t , cut, but when a second cut was attempted to produce a t h i n s l i c e , problems arose. which was 1-2 mm.  During t h i s second cut, the s l i c e  t h i c k was bent by the forces developed during the .cutting  96 procedure up towards the sparking t o o l .  When t h i s occurred, sparking  took place between the side of the t o o l and the s l i c e , and the o r i g i n a l l y p a r a l l e l - s i d e d s l i c e was eroded i n t o a wedge.  Experiments i n s h i e l d i n g  the side of the t o o l to prevent t h i s from happening were unsuccessful. Even i f i t could be assumed that the deformation i n v o i e d i n the bending was i n s i g n i f i c a n t , i t was impossible to t h i n the wedge-shaped s l i c e s f o r subsequent observation i n the e l e c t r o n microscope.  I  As an a l t e r n a t i v e to spark erosion-, a c i d c u t t i n g was produce the desired s l i c e s .  In t h i s method, a counter-balanced  was held l i g h t l y against a r e c i p r o c a t i n g polyester thread.  t r i e d to specimen  The thread •  was kept saturated with d i l u t e n i t r i c a c i d by a d r i p feed arrangement so that a l l c u t t i n g was produced by the d i s s o l u t i o n of the cadmium by the n i t r i c acid.  The v e l o c i t y of the thread was kept below 20 ft/min to-  reduce the p o s s i b i l i t y of damage from f r i c t i o n . 8 hours.  One  cut took  approximately  .  •  ,  The s l i c e s produced by t h i s method v a r i e d i n thickness throughout the s e c t i o n by as much as 1 mm.  since the thread could not be  from moving s l i g h t l y out of i t s intended plane during the cut.  prevented These  s l i c e s were used i n subsequent thinning experiments even though they were not of uniform thickness since they had sustained n e g l i g i b l e  deformation  and were the best a v a i l a b l e .  ,  6.1.2  Thinning The s l i c e s produced by a c i d c u t t i n g were thinned by standard e l e c t r o -  p o l i s h i n g techniques.  I t was  found that chromic-acetic e l e c t r o - p o l i s h i n g  s o l u t i o n used at the plateau voltage gave e x c e l l e n t p o l i s h i n g r e s u l t s , producing a clean, shiny surface. was  The major problem involved during thinning  to produce a r e g u l a r , s l i g h t l y concave surface on each side of the .sample.  !  97 This was d i f f i c u l t since the o r i g i n a l s l i c e s d i d not have p e r f e c t l y smooth surfaces.  To t r y to overcome t h i s problem, a j e t p o l i s h i n g technique  was employed, i n which a t h i n stream of p o l i s h i n g s o l u t i o n i s d i r e c t e d towards the specimen surface during e l e c t r o - p o l i s h i n g .  This produced  the desired concavity but i t d i d not s a t i s f a c t o r i l y smooth out the i r r e g u l a r i t i e s i n the surface.  Consequently, when the specimen was sub-  sequently thinned to p e r f o r a t i o n , the included angle of the specimen, leading up to the p e r f o r a t i o n was quite high.  Chemical p o l i s h i n g in,- d i l u t e  n i t r i c acid p r i o r to e l e c t r o - p o l i s h i n g was also found to be unsuccessful in,removing surface i r r e g u l a r i t i e s .  =  6.1.3  ,  Examination of thinned specimens  The samples thinned as described above were examined i n the e l e c t r o n microscope.  I t was found that areas adjacent to p e r f o r a t i o n s were always  too thick f o r e l e c t r o n transmission.  Consequently, i t was impossible to  determine any d i s l o c a t i o n structures i n these samples. Since t h i s work was attempted, two transmission e l e c t r o n micrographs 40 41 of cadmium have appeared  '  .  However, due to the lack of c l a r i t y , and  obvious d i f f i c u l t i e s involved i n obtaining these photographs, the techniques used were not subsequently attempted f o r the present m a t e r i a l .  98  REFERENCES  1  Sharp J.V., M i t c h e l l A., and C h r i s t i a n J.W.  2  P e i f f e r H.R. and Stevenson F.R.  3  Kroupa F. and P r i c e P.B.  4  H i r s c h P.B. and L a l l y J.S.. P h i l . Mag. 12, 595 (1965).  5  Risebrough N.R.  6 (  i  .  .  PhD Thesis, U n i v e r s i t y of B r i t i s h Columbia .  .  Phys. Stat. Soc. _4, 411 (1964).  P h i l . Mag. 6, 243 (1960).  Rath B.B., Nakada Y., and Hu H. .  Acta Met. 13, 965 (1965).  (1965).  To be published.  7  Llicke K. , Masing G. , and Schroder K.  Z. M e t a l l k . 46, 792 (1955)  8  O e l s c h l a g e l D.  9  Lukac P.  10  Feltham P.  11  Schmid E. and Boas W.  12  Dorn J.E.  13  C o t t r e l l A.H. and Stokes R.J. Proc. Roy. 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