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Recovery in cadmium 1970

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RECOVERY IN CADMIUM by EDMOND CHARLES HAMRE B.A.Sc., University 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 this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1970 In presenting th i s thes is in pa r t i a l fu l f i lment o f the requirements fo r an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee l y ava i l ab le for reference and study. I fur ther agree tha permission for extensive copying o f th i s thes is for scho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Department of Metallurgy The Un ivers i ty o f B r i t i s h Columbia Vancouver 8, Canada Date January 12, 1970 ABSTRACT The recovery of mechanical properties following deformation of single c r y s t a l cadmium has been studied. Such recovery has been observed above 0.26 T (-120°C). M Crystals covering a range of orientation were deformed i n .tension at -196°C and recovered at elevated temperatures. Transmission electron microscopy to relate tensile and recovery behaviour to dislo 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 portion of the easy glide region i s completely recoverable. At higher strains i n easy g l i d e , a portion of the work hardening was not recoverable. I t i s believed that i n this l a t t e r section, dislocations are generated on the second order pyramidal system {1122} <1123>. These dislocations w i l l combine with basal dislocations to form stable obstacles i n the l a t t i c e which w i l l be. responsible for the non-recoverable work hardening. The end of easy glide was found to occur at x = 20°, independent of recovery or i n i t i a l orientation. 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 this point, resulting i n a higher work hardening rate. Recovery i n stage I I was observed to increase the amount of s t r a i n attainable. I t was also observed that while recovery up to intermediate i i strains i n stage I I affected only basal dislocations, both basal and pyramidal dislocations appear to be recovered at high strains . Pyramidal dislocations may recover by the processes observed by Price. The rate controlling mechanism for y i e l d and flow of cadmium single crystals i s thought to be one of the non-conservative motion of jogs. An attempt was made to calculate an activation energy for the recovery process, but the data did not y i e l d any meaningful numbers. This may be a result of the d e f i n i t i o n of recovery adopted for this work. ACKNOWLEDGEMENT The author wishes to thank his research director, Dr. E. Teghtsoonian, for his h e l p f u l advice and encouragement. He also wishes to thank fellow graduate students for many stimulating discussions. Financial assistance i n the form of a National Research Council Studentship, National Research Council operating grant A-2452 and a. hard-working wife i s gratefully acknowledged. , i v TABLE OF CONTENTS PAGE 1. INTRODUCTION 1 2. EXPERIMENTAL PROCEDURE 4 2.1 Sample preparation 4 2.2 Tensile testing 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 plots 8 3.2 Recovery results 12 3.2.1 Method (1) 14 3.2.2 Method (2) 20 3.2.3 Method (3) 22 3.3 Strain rate change tests 28 3.3.1 Activation 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 glide 44 3.4.1.4 Recovery effects 49 3.4.2 Stage I I 49 3.4.2.1 Orientation effects 49 3.4.2.2 Recovery effects 49 3.4.3 Stage I I I 50 PAGE 4. DISCUSSION 54 4.1 Recovery results 54 4.1.1 Activation energy 54 4.1.2 Comparison of recovery methods (1), (2), (3) 59 4.2 Work hardening parameters 66 4.2.1 Easy glide 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 dislocations 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 Strain rate change tests 78 4.3.1 Stage I 80 4.3.1.1 Activation volume behaviour 80 4.3.1.2 Cottrell-Stokes 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 Electron microscopy techniques 95 6.1.1 Cutting a thin section 95 6.1.2 Thinning 96 6.1.3 Examination of thinned specimens 97 REFERENCES 98 v i LIST OF FIGURES PAGE Fig 1. Orientation range of single crystals 6 Fig 2. Typical resolved shear stress - shear s t r a i n curve at -196°C 9 Fig 3. Typical resolved shear stress - shear s t r a i n curve at 20°C 10 Fig 4. Typical tensile kinks formed at 20°C 11 Fig 5. De f i n i t i o n of recovery 13 Fig 6. The var i a t i o n of method (1) recovery at 30°C with s t r a i n 15 Fig 7. The variation of method (1) recovery at -20°C with s t r a i n 16 Fig 8. The var i a t i o n of method (1) recovery at -70°C with s t r a i n 17 Fig 9. The var i a t i o n of method (1) recovery at -100°C with s t r a i n 18 Fig 10. The var i a t i o n of method (1) recovery with time at y ~ 0.25 19 Fig 11. The va r i a t i o n of saturation recovery with s t r a i n 21 Fig 12. The va r i a t i o n of method (2) recovery at 20°C with s t r a i n 23 Fig 13. The var i a t i o n of method (2) recovery at -30°C with s t r a i n 24 Fig 14. The variation of method (2) recovery at -70°C with s t r a i n 25 Fig 15. The var i a t i o n of method (2) recovery at -90°C and -110°C with s t r a i n 26 Fig 16. The effect of temperature on easy glide deformation 27 Fig 17. The determination of load change with s t r a i n rate change 29 Fig 18. Comparison of activation volume data obtained from s t r a i n rate up-change and down-change 30 Fig 19. The s t r a i n dependence of activation volume 31 Fig 20. The effect of saturation recovery on activation volume i n easy glide 33 Fig 21. The effect of saturation recovery on activation volume i n stage I I 34 Fig 22. The va r i a t i o n of A T with T 35 Fig 23. The var i a t i o n of A T - T with saturation recovery i n easy glide 36 v i i PAGE F i g 24. The v a r i a t i o n of A T - T with saturation recovery i n stage I I 37 F i g 25. Or i e n t a t i o n dependence of the c r i t i c a l resolved shear stress 40 Fig 26. Temperature dependence of the c r i t i c a l resolved shear stress 41 Fig 27. Orientation dependence of stage I work hardening rate 42 Fig 28. Orientation dependence of the length of easy g l i d e (schematic). 43 F i g 29 Temperature dependence of stage I work hardening rate , 45 Fig 30. The e f f e c t of o r i e n t a t i o n ( s p e c i f i c example) on the length of stage I 47 F i g 31. Twins formed i n the stage I - stage I I t r a n s i t i o n (x 200) 48 Fig 32. The e f f e c t of saturation recovery on the length of stage I I 51 Fig 33. Relative shapes of c r y s t a l s deformed at 20°C and -196°C 52 Fig 34. The v a r i a t i o n of log (1~R) with time, 56 F i g 35. The v a r i a t i o n of rog (slopi e) with r e c i p r o c a l absolute temp. 58 F i g 36. Comparison of saturation recovery with deformation at -196°C and 20°C 60 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 63 Fig 38. T y p i c a l method (2) recovery at -30°C and -70°C 65 F i g 39. Comparison of present work to that of Bocek and Kaska regarding the temperature dependence of the c r i t i c a l resolved shear stress 67 Fig 40. Comparison of shear stress with that on basal plane on second order pyramidal plane 74 Fig 41. The v a r i a t i o n of saturation recovery with o r i e n t a t i o n 76 Fig 42. The v a r i a t i o n of flow stress following saturation recovery i n stage I I 90 v i i i LIST OF TABLES PAGE Table I. The effect of i n i t i a l orientation and recovery on the length of stage I 46 1 1. INTRODUCTION Recovery may be defined generally as the reversion 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 this study, this d e f i n i t i o n may be stated more s p e c i f i c a l l y . In this case, recovery occurs when the flow stress of a cadmium single c r y s t a l decreases towards the i n i t i a l y i e l d stress when thermal energy i s added to the system. This decrease i n flow stress-is. probably caused by a decrease i n dislocation density as a result of egress of dislocations from the system. When recovery occurs concurrently with 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 for at least part of the temperature s e n s i t i v i t y of some work hardening parameters. This recovery takes place at intermediate temperatures - high enough so that dislocations may rearrange themselves, but not so high as to promote r e c r y s t a l l i z a t i o n . Dynamic recovery probably accounts for most of the difference i n the work hardening rate 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 thoria dispersion i s thought to be responsible for the high strength at high temperatures i n T.D. n i c k e l . Most studies of recovery per se have not dealt with the change of mechanical properties, but rather have been concerned with 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 either 43 by deformation or radiation. This type of recovery usually takes place at temperatures below that where the mechanical properties are s i g n i f i c a n t l y affected, and the p r i n c i p a l method for measuring such recovery i s the 44 change i n e l e c t r i c a l r e s i s t i v i t y . -, 2 One study which looked at the change i n flow stress 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 single vacancy migration. T„ = .25 i s the same temperature at which dynamic recovery becomes apparent. These results are comparable 2 to those of a similar study by P e i f f e r and Stevenson. i Electron microscope studies on evaporated p l a t e l e t s of zinc 3 by Kroupa and Price associated dynamic recovery with conservative,; climb of prismatic dislocation loops. In this mechanism, the area inside 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 thin f o i l s of Mg was due to cross-slip and subsequent annihilation of; screw dislocations. Risebrough^ thinks that this mechanism would ,not be operative i n Cd due to the lack of a suitable cross-slip 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 and by Liicke et a l ^ . . The former was concerned with the thermal activation characteristics of recovery i n aluminum single c r y s t a l s , and the l a t t e r looked at the effect of recovery on various work hardening parameters i n zinc. Neither of these studies concluded anything about the dislocation arrangements or the effect of recovery on dislocations. :; Other studies which were concerned with the activation parameters 8 9 related to load decay after s t r a i n (Oelschlagel on zinc; Lukac or\ cadmium; Feltham"^ on Mg) did not draw any conclusions with respect to dislocation motion. I t was the aim of this study to investigate the recovery of mechanical properties i n single c r y s t a l cadmium with s p e c i f i c reference to the dislocation 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 single crystals 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 single crystals has been studied so that the effects of recovery may be better understood. ; Cadmium was chosen as the material to be used i n this work primarily because of i t s c r y s t a l structure. In the past, most dislocation theories have been concerned with the simplest c r y s t a l structure, namely face-centered cubic, and as a result r e l a t i v e l y l i t t l e i s known of the dislocation mechanisms i n hexagonal close-packed metals. A second reason for choosing cadmium i s the ease with which single crystals may be produced i n quantity. Other hexagonal metals such as zirconium and titanium ,do not possess t h i s quality. F i n a l l y , cadmium was chosen because of the d u c t i l i t y i t exhibits at temperatures below the recovery range. Zinc, which i s similar to cadmium i n most other respects has a tendency to cleave at low temperatures. - , . Other characteristic 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 factors. i Single crystals were used instead of p o l y c r y s t a l l i n e material to exclude the complicating grain boundary constraints. With single crystals i t i s possible to calculate the shear stress on any p a r t i c u l a r plane.at any time. 4 2. EXPERIMENTAL 2.1 Sample preparation The material used for this study was 99.999% Cd as supplied by Cominco Ltd., T r a i l , B.C. This material was received i n the form of one-half inch bars, and was extruded to 0.2 inch rods for subsequent growth into single c r y s t a l s . Single crystals were grown using a modified Bridgman i n evacuated 5 mm. inside diameter pyrex tubes which had previously been coated on the inside 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. If such wetting did take place, the surface of the resultant single c r y s t a l was marked by many craters not unlike bubbles. The tubes were lowered at a rate of 6 cm/hr. through a furnace with a thermal gradient of 25°C/cm. Randomly oriented crystals were produced by this method, Wjhile crystals oriented for long easy g l i d e , which were required for most recovery tests, were produced with a standard seeding technique once a suitable , orientation had been obtained. The pyrex tubes were removed from the c r y s t a l by di s s o l u t i o n , i n concentrated HF, following which the crystals were etched i n concentrated HC1. This etch revealed any grain boundaries which may have been present, and also removed any Aquadag which may have adhered to the specimen surface. Crystals were chemically polished i n a fresh solution of: ,; 320 gm. Cr0 3 ;| , 40 gm. Na2S0^ 1000 ml. H20 . • to remove approximately 0.002 inches from the surface. The orientation 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 orientation range of crystals used i n this study i s shown i n Fig. 1. The majority of samples had an orientation of 40° < x < 46° where y i s the i n i t i a l angle between the basal (0001) plane and the te n s i l e axis. ,• 2.2 Tensile testing > Tensile deformation of the crystals was carried out on a fl 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 tests being done at 1.3 x 10 sec Crystals 10 cm. long were soldered into aluminum grips for testing Since there was no reduced gauge section on the specimens, the length between the grips constituted the gauge length. This length was 6 to 8 cm., and with a diamter of 0.5 cm., the length to diameter rat i o n always f exceeded 10 to 1. Test temperatures were maintained by immersing the specimen into an appropriate l i q u i d held at the required temperature. The baths used and their 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 Electron microscopy An attempt to correlate mechanical properties and recovery behaviour to dislocation 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) Fig 1. Orientation range of single cry 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 tests were carried out on cadmium single crystals i n three different ways: 1) Crystal 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 raised to allow recovery for a fixed time. The temperature was lowered to -196°C, and deformation resumed for an arbitrary s t r a i n increment. Referred to i n the following text as either method (1) or type (1) recovery. 2) Crystal 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 for some fixed time, then deformation was resumed for a suitable s t r a i n increment, a l l at a ; constant temperature. Referred to i n the following text as method (2) or type (2) recovery. ,( 3) Crystal was deformed to a predetermined s t r a i n , the Instron crosshead was stopped, and the load allowed to decay during recovery for a fixed time, then deformation was resumed for an arbitrary s t r a i n , increment, a l l at a constant temperature. Referred to i n the following, text as method (3) recovery. ;, 8 3. RESULTS 3.1 Resolved shear stress - shear s t r a i n plots Load-elongation data were transformed to resolved shear stress- resolved shear s t r a i n plots using the following relationships"'""'": p l l 2 i - A° T = J sinxo [( ) ~ s i n^0 1 2 Yl £ 1 Y = I i n ^ { [ ( £ i ) 2 - s i n X 0 1^-cosXo > (2) * Calculation and p l o t t i n g of the results 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 for a c r y s t a l deformed at -196°C i s shown i n Fig 2. The c r i t i c a l resolved shear stress has been determined by extrapolating the line a r easy glide back to zero s t r a i n . For comparison between various tests, the length of easy glide (jj) w a s defined as the s t r a i n at the intersection of the extrapolated line a r stage I and •, stage I I sections. In general, fracture 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 ess 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 realized that equations (1) and (2) are v a l i d only for s l i p on a single s l i p system. While this i s probably not true for cadmium following easy gli d e , the calculations have been extended to f a i l u r e to allow- comparison of present results to other work.  5 0 C H 400-J 0"T 1 1 1 1 1 1 1 1 r- 0 -5 10 1-5 2 0 2-5 3-0 3-5 4 0 4 5 Fig 3. Typical resolved shear stress - shear s t r a i n curve at 20°C. than i n the f i r s t as shown i n Fig. 3. These work hardening rates are substantially lower than those obtained at -196°C. Fig. 3 also shows that the work hardening curve i s not as linea r as i t i s at -196°C. During the i n i t i a l deformation at 20°C, the crystals developed many te 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 Fig. 4. At -196°C the deformation was more homogeneous, with fewer less sharply defined kinks formed. Twinning was also a feature at 20°C. In this case, the twinning was not operative u n t i l very high strains (^400% shear s t r a i n ) . Fig 4. Typical tensile 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 for 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. Thus recovery must begin i n the range , -140° < T < -110°C. Accordingly, most attention has been focussed on results 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) involve 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 this study, recovery has been defined as that ; fraction of the work hardening which i s removed by a given anneal. In terms of flow stress parameters,this i s : T . , - T . 1-1 1 r . , R = = softening due to anneal .-, . work hardening T . , - crss ° and i s described graphically i n Fig. 5. This d e f i n i t i o n allows for a range of recovery from zero for no change i n flow stress to 100%, for a recovery which results i n a flow stress equal to the i n i t i a l c r i t i c a l resolved shear stress. Recovery, as defined, i s independent of s t r a i n or absolute flow stress values. This d e f i n i t i o n i s also consistent with that used by other workers^'^. . 13 SHEAR STRAIN y Fig 5. Definition of recovery. A second method of measuring recovery which may be applicable, but which has not been used i n this study would be to compare the flow stress under test conditions to that at -196°C. In this way both s t a t i c and dynamic recovery would be measured and accounted for i n tests i n which both occurred. I t would be d i f f i c u l t to interpret such data since there are variables 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, this approach has not been applied to- the data. j i 3.2.1 Method (1) The results obtained for method (1) recovery are shown i n Figs. 6 through 9, which are plots of recovery (R) vs. s t r a i n (y)• Comparing these p l o t s , i t i s readily seen that recovery increases with increasing temperature and time. R, however, also varies with s t r a i n , and this must be taken into account when comparing recovery under various conditions. With a l l deformation at -196°C, the resolved shear stress - shear s t r a i n curves were si m i l a r . I f recovery i s a si m i l a r function of s t r a i n at a l l times and temperatures investigated, the recovery values may be compared at any arbitrary 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 structure and so provide a constant activation entropy. Such a comparison at an ;J arbitrary s t r a i n of y = 0.25 i s shown i n Fig. 10. Some of the points on this plot have been determined by extrapolation of the i n d i v i d u a l graphs (Figs. 6-9), but this 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 overall effect of time and temperature was the same, (i. e . the slope of the plot of recovery vs. time was the same at most temperatures). At -100°C, the values were obtained by maintaining the, difference between the two curyes which i s present at y^ (y = 1.8), -and extrapolating the 40 min. curve to y = 0.25. This operation i n effect neglects the i n i t i a l value at 90 min., but i t was f e l t that the difference at i s a more r e l i a b l e value than the single 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 this ,case, i t was found that recovery was essentially independent of time at the end of stage I as shown i n Fig. 6. A possible reason for this anomaly . w i l l Fig 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 Fig 9- The va r i a t i o n of method (1) recovery at -100°C with s t r a i n . Fig 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 saturation recovery. Fig. 11 shows the results of type (1) recovery at various strains i n the high temperature range 50°C to 100°C. These points are from a variety of experimental annealing conditions, yet form no systematic variation with either time or temperature. Although Fig. 11 shows only a few different annealing conditions, within each l i s t e d condition . there may have been a va r i a t i o n of as much as ± 10°C i n temperature and/or +30 . , ... . . . . _q mm. i n time and there was s t i l l no systematic va r i a t i o n i n recovery. I t i s believed that each in d i v i d u a l point represents the maximum recovery attainable for i t s par 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 (Fig. 6), i t i s f e l t that the conditions employed were more than adequate to achieve the maximum recovery attainable. 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 scatter i n the data, this scatter does occur i n a reasonably narrow band. The curve i n Fig. 11 \ represents the upper l i m i t of recovery, and because i t does depict -( the maximum recovery attained, i t i s termed the "saturation recovery". This curve shows that recovery i s essentially complete at low s t r a i n s , but at strains i n excess of 75%, i t decreases markedly. The o v e r a l l shape of this curve i s very si m i l a r to those of Fig. 6, and may also show a minimum at Yj- 3.2.2 Method (2) :, Results of method (2) recovery are shown i n Figs. 12 to 15. As was the case for method (1), the amount of recovery i s a function of Fig 11. The v a r i a t i o n of saturation recovery with s t r a i n . N3 annealing time, temperature and s t r a i n . The general shape of the curves i s also si m i l a r . The primary differences are i n the magnitude of recovery and i n the regularity of data. The regularity of data with this method as compared to method (1) may be due to the fact that method (2) tests were much easier to perform experimentally than were those for 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 results of this itest 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 Fig. 16. I t can be seen that the flow curve of the f i r s t c r y s t a l i s i d e n t i c a l to that of the second following the recovery anneals. Thus i t i s concluded that at least 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 results are very si 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 effect of the applied load on recovery was negligible. Whenever any deviation did occur between the two methods i t was such that the recovery was enhanced by the applied load, but always i n negligible amounts.  Fig 13. The v a r i a t i o n of method (2) recovery at -30°C with s t r a i n . 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 . - 9 0 ° C 15 min. - 110 Q C 15 min. 0 -5 1 0 1-5 2-0 2-5 3 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 effect of temperature on easy glide deformation. 28 3.3 Strain rate change tests Strain rate change tests were performed on cadmium single crystals at -196°C. These crystals were s i m i l a r to those used for recovery experiments i n that they were oriented to show a long easy glide. The 12 results of such experiments should show the v a r i a t i o n i n activation volume with s t r a i n , and also whether or not this material obeys the C o t t r e l l - 13 Stokes law, since both of these parameters are calculated from s t r a i n rate change data. In addition to s t r a i n rate changes, recovery anneals wer,e performed intermittently during some tests to determine the effects of recovery on both activation volume and Cottrell-Stokes behaviour. The t y p i c a l shape of the load-elongation plot during a s t r a i n rate change cycle i s shown i n Fig. 17. Also shown i n this plot i s the method employed to measure the change i n flow stress 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 rate (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 lag associated with a s t r a i n rate decrease on the Instron tensi l e machine. An example of the difference between these two measurements i s shown i n Fig. 18. This i s a plot of activation volume (which is-propor- t i o n a l to AP as a function of s t r a i n . As a result of the scatter i n AP , only AP values have been used for the following r e s u l t s . 29 LOAD ELONGATION Fig 17. The determination of load change with s t r a i n rate change. 3.3.1 Activation 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 activation volume. This i s defined as: V = kT In  E l / e 2 Activation volume determination should give some idea as to the rate controlling processes of deformation. The variation i n activation volume with s t r a i n i s shown i n Fig. 19. I t i s seen that v i s a steadily decreasing function of s t r a i n throughout the entire range. •, The effect of saturation recovery (60 min. at 75°C) on activation Fig 18. Comparison of activation volume data obtained from s t r a i n rate up-change and down-change. u> o 5 C H r Fig 19. The s t r a i n dependence of activation volume. (The relevant portion • ... of: :the-stress-strain-'curve i s " showni for comparison.) u> volume i n easy glide i s shown i n Fig. 20. I t i s seen that at low s t r a i n , recovery increases the value of activation volume back to the value that was found at y i e l d . At higher strains (>70%) i t was found that recovery s t i l l increased the value of activation 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 activation volume as shown i n Fig. 21. In this range, the change i n v with recovery i s essentially constant, and not a function 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 plot A T , rather than v which i s proportional to A T \ versus T rather than s t r a i n . { I f ;such a plot i s l i n e a r , and passes through the o r i g i n then the Cottrell-Stokes 5 16 law i s considered to be obeyed. Risebrough and Davis found that the Cottrell-Stokes law was obeyed i n cadmium single crystals at -196°C during stage I I deformation. The obeyance or non-obeyance of this law i s not clea r l y understood with respect to hexagonal metals; however the plot does provide much useful information. The plot of A T vs. T for single 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 this plot at the end of linea r easy glide, but the Cottrell-Stokes law i s not s t r i c t l y . obeyed i n either region. When saturation recovery anneals are performed i n stage I, i t i s seen i n Fig. 23 that the f i r s t anneal has no effect on the A T - T relationship, but after 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 diff e r e n t effect than i t did i n stage I as shown i n Fig. 24. After 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-  Fig 22." The v a r i a t i o n of A T with T . Fig 23. The v a r i a t i o n of Ax - x with saturation recovery i n easy glide. ON IOOH 9CH 8 0 - A T gm/mm 704 6 0 - O — Crigiicl .'clues £ — After I Annecl • — After 2 Annccts V - After 3 Annecls © — A'lcr 4 Anne ills 50- 6 0 0 1000 1400 T gm/mm' 1800 2200 F i R 24. The va 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 following this 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 since although A T did. decrease at this point, i t did not decrease as much as i t had increased from the previous recovery. Again, the slope of the plot 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 stress on the system reached , 2 2000 gm/mm . .. 3.4 Work hardening parameters To determine the effects of c r y s t a l orientation on various work hardening parameters, a study was carried 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 effects which are caused by the changing orientation of a c r y s t a l during a tens i l e test and those which are ch a r a c t e r i s t i c of recovery. Crystals with i n i t i a l orientation X q ranging from 25° to 48° were investigated. , In a l l cases X q was very nearly equal to XQ- 3.4.1 Stage I . Stage I has been defined as the f i r s t linear stage of hardening during the deformation of a cadmium single 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 stress or c r i t i c a l resolved shear stress, the work hardening rate and the amount of linear s t r a i n . These factors, and the effects of orientation and recovery on them, are discussed below. 39 3.4.1.1 C r i t i c a l resolved shear stress 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 essent i a l l y independent of i n i t i a l orien- tation as shown i n Fig. 25. This i s i n agreement with Schmid's shear stress law"'"''" which states that there should be no orientation dependence of this stress. The va r i a t i o n i n c r i t i c a l resolved shear stress with temperature i s shown i n Fig. 26. The stress values have been corrected for the. temperature dependence of the shear modulus by dividing each by the, shear modulus at i t s par t i c u l a r temperature. Values of shear modulus (C^) 14 were obtained from Gerland and Silverman . I t i s seen that at lowi; 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 scatter 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 this behaviour. 3.4.1.2 Work hardening rate The work hardening rate i n easy glide (9j) was found to increase about 50% with X Q decreasing from 45° to 25° as shown i n Fig. 27. The effect of recovery on this parameter was to increase 0 to a value comparable to that for a v i r g i n c r y s t a l with i n i t i a l orientation equal to the orientation 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 Fig. 28 where sample S39A was deformed f romT̂ ah" iniTial"".^orientation of 45° to 35°. Sample M4A had an i n i t i a l r . orientation of 35°,and i t i s seen that following recovery of sample>S39A, the work hardening rate of the two crystals i s the same. , Fig. 29 shows the v a r i a t i o n of 6̂. with temperature at a n ' i n i t i a l -3 -1 str a i n rate of 1.3 x 10 sec . Shown for comparison are the results of 30H O 25H O O o 2 C H c r s s gm/mm 15H o o 9 ) o o o o o o ° 9 2 0 ! 24 — i — 28 32 36 degrees — i — 4 0 — i — 4 4 ! 48 Fig 25. Orientation dependence of the c r i t i c a l resolved shear stress. 12 150- I4CH gm/rnm2 130- \ 120- o \ oo o no- 100- 18 O O o V •s. o' 8 o —I— 42 i 46 — i — 5 0 — i — 22 2 6 — i — 30 V „  3 4 A o degrees — i — 38 Fig 27. Orientation dependence of stage I work hardening rate. 2000- 1600- V deqrees -:Fig. 28. Orientation, dependence of the length, of easy glide (schematic). 44 Risebrough^. I t i s seen that the two sets of data are comparable with the exception of the present point at -110°C. This point i s high and may result from the fact that i t was obtained very early i n the course of this work at which time production, handling and testing 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 lat 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 glide The data i n Table I, which i s shown schematically i n Fig. 30, shows that the end of easy glide occurs when x = 20.2 ± 1.2°, irrespective of i n i t i a l orientation or recovery. This figure i s a st r e s s - s t r a i n plot on which s t r a i n has been converted to orientation by the r e l a t i o n : s l n X - o s i n x 1 T The data i n Table I and from Fig. 27 show that recovery has an effect such that a cr y s t a l deformed to an orientation x i n stage I and completely recovered has the work hardening rate and new stage I length i equal to a v i r g i n c r y s t a l of orientation x- I t was also noted that the end of easy glide at -196°C was marked by twinning i n the sample. These twins are shown i n Fig. 31.; The stress at which twinning occurred was s l i g h t l y higher than the stress at 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 stress was 2 300 to 400 gm/mm , and independent of i n i t i a l orientation. At 20° C,, twinning 2 occurred at an equivalent stress of 310 - 380 gm/mm . This stress 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. Specimen S39B S39D S39C S41B S37C S37B S36D S39A M9B M12C L3C S37A M15B M11C L3B M8A Ml 2 A M11B S29B M4A M2C S42A U1C M6B L2A The effect of i n i t i a l orientation and recovery on the length of stage I. 48 48 48 47 46 46 46 45 45 45 43 42 42 42 42 41 40 40 36 35 34 29 27 26 25 20.3 20.5 21.5 20.3 19.2 20.5 20.9 20.7 21.0 22.5 23.0 19.3 19.6 19.6 20.4 18.5 19.0 20.5 19.7 19.8 20.1 18.7 22.0 18.4 19.0 Recovery i n stage I 20 min. -20°C ' 90 min. -100°C 40 min. -100°C 30 min. +50° C 20 min. -70°C 70 min. -70°C 10 min. +30°G 10 min. -20°C none saturation saturation - 20 min. + 30°C none none saturation none \ saturation none saturation none saturation none none none none 5 0 0 i 48 44 40 36 32 28 24 20 16 X degrees Fig 30. The effect of orientation on the length of stage I (specific example). 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 effects of recovery on the various stage I work hardening parameters, i t has been found that recovery has no effect on the length of stage I; i t increases the work hardening rate. Recovery has no effect on the s t r a i n at which twinning occurs. 3.4.2 Stage II , Stage I I i s the second l i n e a r hardening portion 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 discussion, 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 essenti a l l y the same i n a l l specimens. 3.4.2.1 Orientation effects •; A l l crystals tested i n this study had an i n i t i a l orientation X Q greater than 25°. Consequently, a l l showed some easy glide before stage I I began at an angle of approximately 20°. Therefore there was no var i a t i o n i n orientation with which to compare stage I I work hardening parameters. There was no effect of i n i t i a l orientation i n stage I on stage II 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 effects , While there was no noticeable effect on the length of stage I, i t was found that recovery had a considerable effect on the length of stage I I . 50 Fig. 32 i s a-plot of the resolved shear stress - shear s t r a i n data from two comparable crystals. Crystal M4A was deformed continuously at -196°C. 2 Crystal M15B was deformed to a stress of 2000 gm/mm at -196°C and then recovered at 75°C for 30 min. These conditions are i n excess of the minimum requirements for saturation recovery. Following this recovery, 2 the specimen was again strained at -196°C to a stress 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 glide i n specimen M4A was 80%. Crystal M15B exhibited 180% shear s t r a i n from the end of easy glide. The t o t a l s t r a i n achieved i n this 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. Fig. 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. I t i s seen that the two crystals are s i g - n i f i c a n t l y ' d i f f e r e n t . Crystal 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 crystals M4A and M15B; up to the f i r s t anneal as shown i n Fig. 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 exhibited only two stages of hardening, with f a i l u r e occurring while 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 crystals with i n i t i a l orientation less than 30°, a th i r d stage with a   substantially lower non-linear work hardening rate was often observed. Sample M15B was annealed extensively i n stage I I . After the f i r s t few anneals, the work hardening rate increased, but at higher strains i t decreased and deviated from l i n e a r i t y as shown i n Fig. 32. This may be due to a t h i r d stage of hardening. 54 4. DISCUSSION 4.1 Recovery results 4.1.1 Activation energy As a f i r s t step i n the evaluation of the recovery data, an attempt was made to calculate an activation energy for tests i n which a l l deformation was performed at -196°C, with 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 dislocation configuration should be similar preceding the various anneals, which would not be the case for 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 orientation results.. Comparing recovery results as a function of time (Fig. 9) i t was seen that with the exception of+30°C, the difference i n recovery with time i s . not p a r t i c u l a r l y s t r a i n sensitive. That i s , the shape of the recovery-strain curves was consistent at a given temperature, so that a change i n recovery time changes the recovery by a constant amount, irrespective of s t r a i n . With respect to the results at 30°C, F i g . 10 showed that work hardening i n crystals 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 possible 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 saturation may explain the s i m i l a r i t y i n recovery values for various times at 30°C close to Yj- Since saturation recovery may be s i g n i f i c a n t at Yj at 30°C, the comparison of recovery has been limited to s t r a i n less than 50%. The value of s t r a i n chosen a r b i t r a r i l y for this comparison was 25%, and i n some cases this required the back extrapolation of recovery curves. A t r i a l and error technique of curve f i t t i n g was employed to try to establish a recovery-time relationship. By this method, i t was found that recovery i s not a power function of time. i. e . R ^ A t K Rath et a l ^ found that for aluminum single c r y s t a l s , recovery was proportional 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 relationship holds. However, i f i t i s assumed that recovery i s proportional to log time for cadmium single c r y s t a l s , the calculation of a meaningful activation energy was found to be impossible from this data. During the ca l c u l a t i o n , the derivation of the recovery rate constant caused the temperature dependence of recovery to be e f f e c t i v e l y cancelled so that the resultant activation energy was zero. Following the analysis of Rath et a l ^ did not produce any meaningful values for activation energy from present r e s u l t s . For values of recovery from R = 0.1 to R = 0.5, activation energy values ranged from almost zero to 20,000 cal/g. atom, and the Arrhenius plots from which these values were calculated were not l i n e a r . This method of analysis i s somewhat dubious because i t does not use a true rate constant which i s independent of time or recovery for calculation of activation energy. Also, this r e l a t i o n predicts that recovery w i l l exceed.unity which i s impossible by d e f i n i t i o n . Consequently, the relationship R a log t i s rejected for the present work. TIME min. F i g . 34. The v a r i a t i o n of log (1-R) with time. 57 The relationship between recovery and time which has been adopted for this work i s : log (1-R) = At This 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 plot of the present data i s shown i n Fig. 34,. I t i s seen that the plot at each temperature may be made to pass through a common point on the time axis at log (1-R) = 0. The fact that this time i s negative i s not a serious drawback to this analysis since analogies to physical 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 nuclei decreases with time would exhibit a similar curve when the growth i s extrapolated back to zero. Since the rate law used i n this analysis i s f i r s t order, the slope of the plots at each temperature represents the rate constant, and as such may be used to calculate the activation energy. This i s shown i n Fig. 35. The activation energy found from this plot i s 2500 cal/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 controlling process i s f i r s t order, and i n the s o l i d state this would probably be di f f u s i o n . Since the value i s so small i t i s not possible to determine whether this would be bulk d i f f u s i o n or pipe d i f f u s i o n . The activation energy for bulk d i f f u s i o n i n single c r y s t a l cadmium i s about 18,500 cal/b. atom, and that for pipe d i f f u s i o n i s about half of t h i s . -2- -2- log (slope) -3>t F i § - 3 J - T h e v a r i a t i o n of log (slope) with reciprocal 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 for 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 addition to the.measured s t a t i c recovery. In order to compare the results 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 following calculation shows that they are equivalent only at low st r a i n s . Fig. 11 defined a value of saturation recovery which i s dependent only on s t r a i n . This curve i n Fig. 11 i s now used to determine the minimum attainable flow stress of a c r y s t a l deformed at -196°C. This flow stress 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 after each i n f i n i t e s i m a l increment of s t r a i n . The results of this calculation are shown i n Fig. 36. In Fig. 36, the s o l i d l i n e labelled "Saturation Recovery" shows the minimum flow stress as calculated 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 Fig. 11. This curve represents that portion of  the flow stress which i s not recoverable. The dashed l i n e i n Fig. 36, labelled "+20°C corrected", i s the s t r e s s - s t r a i n curve for a c r y s t a l deformed at 20°C, and as such represents high temperature deformation (20°C = .5T^). In this case dynamic recovery should proceed at such a rate that the c r y s t a l i s essentially recovered at a l l times. For comparison between this and the previously calculated curve, the flow stress values i n this case have been corrected to their equivalent at -196°C by a. r a t i o of the shear moduli at -196°C and 20°C. It i s evident from comparison between this 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 saturation recovery starts to decrease fronu 100% (Fig. 11). This observation i s also v e r i f i e d by the results of a single test at 70% shear s t r a i n as shown i n Fig. 16. • Thus, i t i s concluded that i n i t i a l easy glide deformation at low temperature i s i d e n t i c a l to deformation at high temperature with the addition of work hardening which i s completely recoverable by the conditions employed. At strains i n excess of 70%, up to the end of easy glide^ the two curves i n Fig. 36 deviate s l i g h t l y , and i t i s thought that t h i s ; i deviation 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 orientation results. , The deviation between the two curves i s small, 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 this 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 unlikely that a .second system would operate. The only possible explanation for the discrepancy 62 as shown i n Fig. 36 other than s t r u c t u r a l differences between the two crystals would be that the flow stress at 20°C was not completely recovered at high strains . This was not v e r i f i e d experimentally. This discrepancy should not, however, influence the conclusion that i n i t i a l easy glide at low temperature i s equivalent to that at high temperature with the addition of completely recoverable work hardening. I t should be possible to compare the results 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 results of method (2) i n which the crystals were deformed and then recovered with the load removed at the same temperature. As was seen with respect to Figs. 6-9 the primary difference between this type of test and that i n which deformation was at low temperature was i n the magnitude of recovery. One reason for this 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 this d e f i n i t i o n , recovery i s the fr a c t i o n of the work hardening recovered, and since at temperatures 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 Fig. 37 shows the same flow stress 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 explain the results at high recovery temperatures, where saturation i s approached, but at lower temperatures (-100° to 0°C) this i s not the case. In the lower range of test temperatures (0°C) where saturation recovery, either by s t a t i c or dynamic means, i s not a factor, the above argument i s not v a l i d . In this range, the recovered flow stress at ,pnet Fig 37. Schematic diagram of recovery to a similar stress l e v e l with deformation — --- at different temperatures. temperature i s always s i g n i f i c a n t l y different from the flow stress at another temperature. This i s i l l u s t r a t e d i n Fig. 38, at test temperatures of -70°C and -30°C. The difference i n flow stress between these two curves i s due primarily to the difference 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) with deformation at -196°C, the comparison should be made of flow stress . following recovery at each value of s t r a i n , as suggested with respect to the d e f i n i t i o n of recovery at the beginning of this thesis. However, , this 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 . This has not been determined. Consequently no direct comparison of the results of method (2) with those of method (1) could be drawn. Comparison of the results of method (3), i n which the load was not removed from the sample, to either method (1) or method (2) was not made because of the negligible differences between this method and method (2). Since recovery rates were found to increase with 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 acting i n conjunction with the available thermal energy. The effects observed, however, were ne g l i g i b l e . In conclusion, the results of the three types of recovery test have indicated that recovery i s thermally activated, and i s the res u l t of two or more processes which change with temperature. I t remains for other types of test to show what these processes might be. Saturation recovery has shown that deformation at low temperature i s related d i r e c t l y to, deformation at high temperature i n the i n i t i a l easy glide region, but that this i s not the case at strains 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 saturation recovery, i s a function of s t r a i n . Since s t r a i n i s d i r e c t l y related to orientation, i t was thought that perhaps the above observation could be explained on the basis of c r y s t a l orientation. I t was with this i n mind that the effect of orientation upon various work hardening parameters, both by themselves and i n conjunction with recovery, was studied. , , 4.2.1 Easy glide 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 stress. Comparison of this factor between this work and others should give an idea of the comparability of the quality of the crystals 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 exhibited a c r i t i c a l resolved shear stress of 25 gm/mm . Gibbons found c r i t i c a l stresses of 9.8 and 17.1, 2 15 gm/mm for 99.9994% Cd at 20°C. Bocek et a l tested the temperature dependence of various work hardening parameters i n 99.99% Cd. Their , results for crss vs. temperature along with those of the present work are shown i n Fig. 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 1 6 Davis for 99.9% and 99.9999% Cd show very l i t t l e difference of crss 2 for such a large difference i n pu r i t y , with the crss being about 20-30 ,gm/mm . <, A more probable explanation for the discrepancy would be the presence of more substructure i n Bocek's cryst a l s . In comparison to the present, work,  4 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 found that easy glide i n Mg single crystals was reduced from 250% to 50% for crystals with substructure. The presence of substructure would also explain 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 crystals used i n this work must have been of high purity 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 orientation of either work hardening rate or length of easy glide. Liicke et a l ^ observed that zinc single crystals at 20°C exhibited 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 va r i a t i o n of the work hardening rate with orientation i n hexagonal metals. >; The results 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 orientation of the length of easy g l i d e , since easy glide terminates at a s p e c i f i c orientation (see Fig. 30; Table I ) . To a lesser degree, the work hardening rate also depends on orientation (see Fig. 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 i n zinc i n the temperature range 50° to 100°C. Bocek, Hdtzsch and Simmin also quote the results of Wolr on Cd at 20°K and 90°K which show si m i l a r trends for the work hardening rate. Liicke et a l ^ found that recovery has a d i s t i n c t effect on the length of stage I. Their results showed that a zinc single 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 after a recovery of 24 hrs. at 25°C. 69 Thus, the t o t a l length of easy glide increased by 70%. For cadmium crystals deformed at -196°C and recovered this was not the case. As shown i n Fig. 28 and Table I , recovery had no effect at a l l on the length of easy glide. A possible reason for this discrepancy would be that the operation of second order pyramidal s l i p i s not related to orientation 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 this study i s the recovery of mechanical properties of cadmium, i t i s necessary to know something of the dislocation arrangements present during deformation to understand what might happen during recovery. As a consequence of t h i s , the following section i s concerned with the hardening characteristics of single c r y s t a l s , and the possible dis 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 single 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 line a r stage. Various theories have been proposed to explain this t r a n s i t i o n i n hexagonal metals. The p r i n c i p a l alternatives are: 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 basal 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 alternative i s thought to be most important to this 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 ascribe the t r a n s i t i o n from easy glide 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 into immobile dislo c a t i o n rings. Present experimental results are inconsistent with this theory. F i r s t l y , i t i s d i f f i c u l t to imagine vacancies having adequate mobility at -196°C to form s e s s i l e loops. If such loops did manage to form, then saturation 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 dislocation loops formed by non basal glide i n Cd annealed out by volume di f f u s i o n at temperatures above -40°C. No effect 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. j 4.2.2.2 Non-active basal dislocations ., - 24 Kratochvil and Koutnick , from shape change measurements: during the deformation of cadmium single c r y s t a l s , concluded that secondary dislocations i n the basal plane must operate. By comparison with other 4 work, p r i n c i p a l l y that of Hirsch and L a l l y on magnesium, they conclude that the interaction of primary and secondary basal dislocations 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 . Hirsch 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 for Mg and that hardening mechanisms i n other - hexagonal close-packed metals may be di f f e r e n t . I t i s possible that secondary basal dislocations operate, but i t i s doubtful that they are ; responsible for 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 electron 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 sub- boundaries and twinning. They observed that at the onset of stage I I , the stress on the prismatic plane was close to the experimentally determined value for prism s l i p . However i n comparison of thei r work on Mg with face- centered cubic materials, they think that twins act as barriers to s l i p 25 lines i n Mg, whereas secondary dislocations provide the barriers i n : f . c . c . It i s stress 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 state that the above argument may not be applicable,to hexagonal metals other than magnesium and the results of 26 5 B e l l and Cahn , Risebrough , and the present work agree with t h i s . ; B e l l and Cahn noted the presence of {1122} <1123> s l i p traces i n zinc crystals p r i o r to twinning, and associated the intersection of dislocations on this system with basal dislocations to produce stress 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 stress concentrations which i n turn caused secondary (prism) s l i p . In cadmium, the present work and tfyat of Risebrough have noted that twinning i s always present i n stage II.= However, Risebrough has explained that since a considerable portion 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 stress associated with the end of easy glide i s r e l a t i v e l y , temperature independent, that twinning must be an "after the fact " consider- ation. On this basis, i t must be some dislocation configuration which i s responsible for the end of stage I. 4.2.2.4 Second order pyramidal s l i p ^ 27 Bocek, Svabova and Hotzsch , i n a study of single 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 result of flow on the second order pyramidal system {1122} <1123>. Their results show, however, that at this point there i s a considerable v a r i a t i o n i n the shear stress on this sytem with respect to orientation. This i s i n disagreement with Schmid's law that the c r i t i c a l shear stress be independent of orientation. To compensate for this discrepancy, Bocek et a l have . calculated an in t e r n a l stress which may be present on the secondary system as a result of pile-ups of dislocations on the basal system. The results of this calculation show this i n t e r n a l stress to be of the order of (the flow stress on the basal system. When this stress i s added to the applied stress on the secondary system, the result i s that the t o t a l stress i s , now essentially orientation independent. Also, this t o t a l stress i s 28 comparable to the stress found by Stoloff and Gensamer for the appearance of second order pyramidal s l i p traces on cadmium single crystals oriented to suppress basal s l i p . Thus Bocek et a l conclude that the s t a r t of the linear stage I I i s associated with the onset of flow on the second order pyramidal system. Interaction of these dislocations with basal dislocations w i l l produce sessiles which contribute to work hardening, and so explain 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 arise when i t i s compared to the results of the present study. T h e j f i r s t problem i s with respect to the v a r i a t i o n of the stress at the beginning of stage I I with orientation. 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 stress at this point was also essen 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 dislocations are contained i n the supposed pile-ups, and this number 73 should s i g n i f i c a n t l y affect the magnitude of the i n t e r n a l stress on the secondary system. The major problem arises 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 orientation study, and from s t r a i n rate change tests have shown that i t i s not the onset of stage I I but rather the s t a r t of deviation 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 material i s extensively twinned at the s t a r t of stage I I , and Bocek has not considered this point. 4.2.3 Present model The basis for 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 for 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 this sytem i n the l a t t e r part of stage I and that this i s responsible for the non-recoverable work hardening as shown?-by the drop i n saturation recovery with s t r a i n i n Fig. 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 this secondary system and this i s then responsible for 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 results from the development of a dynamic equilibrium between basal and pyramidal dislocations. This w i l l be discussed i n following sections. At this point, 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 factor as discussed by Bocek et a l , . The results of the present work show that i t i s possible;.: that second order pyramidal s l i p could be responsible for various work hardening and recovery phenomena. Fig. 40 shows the shear stress on the second order pyramidal system compared to the flow stress on the basal plane for a c r y s t a l  75 with i n i t i a l orientation XQ = 45°. Fig. 41 i s a replot of the data on Fig. 11, with s t r a i n transformed to orientation. I t i s seen from this plot that non- recoverable work hardening starts i n the range 30° < x < 35°. This range corresponds to .5<y<.7 on Fig. 40, and i n this range, the stress on the 2 second order pyramidal system i s 75 to 100 gm/mm . I t i s probably only coincidental that this i s the range i n which the stress on the secondary system becomes greater than the stress on the basal system. At the end of easy g l i d e , at which x ~ 23° (see Fig. 28), the stress on the pyramidal 2 system i s ̂  260 gm/mm . ,. . ' 28 Stoloff 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 crystals not oriented for basal s l i p . Their material, however, had a c r i t i c a l resolved shear 2 stress for basal s l i p of 85 gm/mm , while the material used i n this work 2 had a comparable stress of 20 gm/mm . Consequently the c r i t i c a l stress 2 for flow on the pyramidal system i s probably less than 500 gm/mm i n this material, but since the effect of impurities and substructure on .pyramidal glide i s not known, i t i s impossible to assign a defi n i t e value. However, 2 this value of 500 gm/mm was given for the appearance of s l i p traces which would imply massive s l i p on this sytem, but to end easy glide r e l a t i v e l y few dislocations 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 stress. :, ,, Assuming that second order pyramidal dislocations are active at orientations < 30°, i t i s possible to explain non-recoverable work ^ w ..29 hardening as shown i n Fig. 41. Bocek, Lukac and Svabova have shown that energetically favourable reactions between different second order pyramidal systems and between second order pyramidal and basal dislocations can occur. The resultant dislocations 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 dislocations i s probably the most important. Once these sessiles form, they would probably not be amenable to annealing, and so cause non-recoverable work hardening. Substantiating evidence for this was found i n tests which 39 combined recovery with 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 stress i n zinc at 20°C i s a c r i t i c a l function of the pyramidal dislocation density. In zinc i t appears that; second order pyramidal dislocations may also cause non-recoverable work hardening. 5 , I t was also seen i n Fig. 36 that there i s a small deviation 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 stress measurement, and i n the re p r o d u c i b i l i t y between two different specimens are too large to permit the calculation of the work> hardening produced by pyramidal dislocations and their interaction with. basal dislocations from x = 30° to x ~ 23° i f this secondary system operates at -196°C and not at 20°C. When x reaches 23°, the difference i n Schmid factors on the two systems i s such that there would be substantially. more a c t i v i t y on the second system. At this point the stress on thejpyramidal 2 plane i s 260 gm/mm which may be very near the macroscopic c r i t i c a l stress. The present results show that this stress, at the end of easy glide,., i s independent of i n i t i a l orientation. I f this i s the y i e l d stress for second order pyramidal, then the massive a c t i v i t y on this sytem would explain , the termination of line a r easy glide. 78 Thus i t may be concluded, that while a c r y s t a l with x o = 45° deforms entirely i n basal glide 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 dislocation 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 that macroscopic s t r a i n i s achieved on this system, but merely that:disloc- ations are generated which w i l l subsequently interact with basal dislocations to cause poor recoverable work hardening. When x reaches 23° macroscopic flow may occur on the pyramidal system thus terminating linear easy, glide. Since the end of the easy glide 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 this angle as a normalizing factor for a l l „ tensile results. However, since x varies quite slowly with s t r a i n at low x values, stress curves have been plotted versus s t r a i n , with the intercept of extrapolated easy glide and stage I I slopes taken at x = 20.2° , (Y = 1.8 for X Q = 45°). 4.3 Strain rate change tests ^ Activation volume measurements, which are derived from s t r a i n rate change tests, may indicate 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 interpretation of activation volume data i s that the density of mobile dislocations remains constant during a s t r a i n rate change. I f this i s the case, then i t ; : i s assumed that the change i n flow stress resulting from a change i n s t r a i n rate i s due only to the deformation rate controlling process or processes. I t i s known that the deformation made during easy glide i n cadmium single crystals i s s l i p on the basal system. Consequently, i t i s assumed that the flow stress i n this region i s determined only by the 79 density of basal dislocations and by the way i n which they move through the l a t t i c e . In stage I I deformation, i t i s assumed that the flow stress i s s t i l l controlled only by the density of basal dislocations which are assumed to be the mobile dislocations. For face-centered cubic materials, 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, Fig. 33 shows that while . 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 . (Fig. 33 compares a c r y s t a l deformed we l l into stage I I at -196°C to a c r y s t a l deformed at 20°C where essenti a l l y a l l deformation was on the basal plane). Since the width did 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 wel l as for face-centered cubic materials, 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 dislocation density i n stage I I as i t i s i n stage I. From the results of the present work on orientation, and from the results of Risebrough^, i t i s expected that any secondary a c t i v i t y which takes place w i l l be on the second order pyramidal system. Consequently i t i s assumed i n the following discussion that the dislocation forest i s composed ent i r e l y of second order pyramidal dislocations. I t may be that other types are present as .grown-in dislocations, but these should not change during deformation. I t has also been assumed that the contribution to s t r a i n of the pyramidal dislocations i s negl i g i b l e . . 80 4.3.1 Stage I 4.3.1.1 Activation volume behaviour Fig. 19 shows that activation volume during easy glide i s a steadily decreasing function of s t r a i n . At y i e l d , the activation volume -20 3 3 i s 40 x 10 cm , which i s equivalent to 15,000 b . This value drops to 3 about 5500 b at the end of easy glide. Activation volume values i n this range are ind i c a t i v e of ..a rate controlling mechanism of either forest intersection by the basal dislocations or the non-conservative motion of jogs i n the basal dislocations. I t i s not possible on the basis of rate parameter measurements alone to distinguish between these two mechanisms. Risebrough^ has claimed that the mechanism con t r o l l i n g y i e l d and flow i n cadmium single crystals i s one of forest intersection. ;He found no major inconsistencies i n this argument whereas his experimental data did not agree with the limitations on the jog mechanism. He believed that the jog mechanism was unacceptable primarily because the flow stress 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 activated. Mott , has stated that this vacancy nucleation i s completely athermal, and;, , secondly that single vacancies w i l l not migrate at appreciable rates below T„ = 0.5. . n " The present results find inconsistencies i n the forest intersection mechanism. In the i n i t i a l stages of s t r a i n (up to 70%) , i t was found that the activation volume decreased s i g n i f i c a n t l y . I f the forest intersection mechanism were rate c o n t r o l l i n g , this decrease would imply an increase i n the forest density. However, when the crystals were recovered i n this-range, 81 i t was found that the activation volume recovered back to the value found at y i e l d (see Fig. 20). Since i t i s very unlikely that both primary and forest dislocations would recover i n exactly the same manner, as the forest intersection mechanism would require, i t has been assumed that the forest dislocations do not recover at a l l . I t has been shown that second order pyramidal dislocations w i l l combine with basal dislocations to form energetically stable 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 basal d i s - locations. This i s made more apparent when compared to the behaviour of pyramidal dislocations at higher strains as discussed i n a l a t e r section. Thus, i t appears that the forest intersection mechanism i s not applicable to the present system. The other possible rate co 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 this i s the rate controlling mechanism, while Risebrough"' has rejected this 33 mechanism on the basis of the arguments of Mott , but some of these arguments are questionable as to their v a l i d i t y . F i r s t , Mott states that single 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 T̂ . Secondly, Mott does not adequately explain, why the process of vacancy nucleation at a jog should be completely athermal. If such nucleation were thermally activated, then the combination of this plus vacancy migration could explain the temperature dependence of the flow stress i n cadmium. This would then remove any objection to the assumption that the non-conservative motion of jogs i s the rate controlling mechanism. Thus, since doubt does exist with respect to the Mott theory, and since experimental results preclude the acceptance of the forest intersection mechanism, i t w i l l be assumed that the non-conservative motion of jogs controls flow i n easy glide i n cadmium. Having made the above assumption, i t i s now possible to explain the behaviour of activation volume with s t r a i n i n easy glide. During the i n i t i a l stages of s t r a i n , there remains a constant density of forest dislocations. Activation volume decreases i n this range due to the decreased inter-jog spacing along the mobile basal dislocations. On recovery, the jogs i n these basal dislocations anneal out, probably, by diff u s i o n along the dislocation l i n e . When a jog of one sign meets one of the opposite sign, the two annihilate and so decrease the jog density. Under conditions of saturation recovery, some equilibrium concentration of jogs i s attained which would be the same concentration as existed at y i e l d . Therefore, the activation volume following recovery w i l l be the same as i t was at y i e l d . I f this equilibrium concentration of jogs i n basal dislocations i s such that the inter-jog spacing along the dislocations i s larger, than the spacing between forest dislocations, then the activation volume measured under such conditions should be indicative of the forest density. This should be true both at y i e l d and following recovery. Howeveronce the basal dislocation has moved through the f i r s t set of forest dislocations that i t encounters, the inter-jog spacing w i l l be less than the forest spacing, and so i t w i l l be the jog mechanism which i s measured. I f f i t i s assumed that the inter-jog spacing i s larger than the average forest spacing, then on the basis of the previous argument i t i s possible to , calculate the forest density from activation volume data following recovery. Activation volume i s defined as-: v = b£d ,) j where b = Burger's vector , £ = activated length of dislocation ; d = distance over which dislocation i s moved. 83 If i t i s assumed that the activation distance i s equal to the Burger's vector, then If forest intersection i s the rate co n t r o l l i n g mechanism as has been assumed at y i e l d and following recovery, then £ i s the average spacing between the forest dislocations. The forest density i s then: P = 1 = •a 2 b it On the basis of the above assumption, the forest density'at ; 6 2 y i e l d has been calculated as 4.7 x 10 lines/cm . Close to the end of easy g l i d e , the pyramidal density has increased by about one order of 7 -2 magnitude to 4 x 10 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 density 9 -2 i s about 2 x 10 cm . • ( These above values are consistent with the results from ,the orientation analysis. Activation volume data show that the forest does not change up to about 70% s t r a i n . From this point to the end of easy glid e , i t i s found that the forest density increases from 4 x 10 to 7 -2 4 x 10 cm and i t i s this 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 glide to the middle of the t r a n s i t i o n region an increment from the end of easy glide to the middle of the t r a n s i t i o n region an increase i n pyramidal dislocations 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 glide. Thus, i t i s concluded from an unrelated type of test that the conclusions regarding work hardening parameters are v a l i d i n that there i s microactivity on a secondary system 84 i n the l a t t e r part of easy glide followed by macroactivity i n the tr a n s i t i o n region. 4.3.1.2 Cottrell-Stokes behaviour The Cottrell-Stokes behaviour of a metal i s the v a r i a t i o n i n flow stress resulting 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 v a r i a t i o n i s l i n e a r , passes through the o r i g i n , then the Cottrell-Stokes law i s considered to be obeyed. The interpretation of the results and the significance of obeyance or non-obeyance i s not cle a r l y understood, and i s usually only applied to face-centred cubic materials. Many tests, however, have been performed on many different materials to test this 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 single crystals 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 results show that nowhere during the deformation of single c r y s t a l cadmium at -196°C i s the Cottrell-Stokes law obeyed. !, , The p l o t t i n g of A T vs. T as i n a Cottrell-Stokes test gives useful information besides checking the obeyance of the Cottrell-Stokes law. Since activation volume i s representative of the rate controlling mechanism, and i s determined d i r e c t l y from the measurement of A T , then A T i t s e l f must also be indicat i v e of the rate controlling process. In the present case, A T at y i e l d i s - a direct measure of the number of forest intersections taking place at the time of the s t r a i n rate change. Ap strains past y i e l d , A T i s a measure of the density of jogs present i n j 85 basal dislocations. As mentioned previously, i t i s believed that the flow stress T i s dependent on the density of basal dislocations. Therefore, a plot of A T vs. T i s , i n stage I , e s s e n t i a l l y a plot of the density of jogs vs. the density of basal dislocations. Fig. 23 shows that the plot of A T vs. T i s l i n e a r i n the . i n i t i a l portion of stage I. Using the above argument, this s i g n i f i e s that the basal jog density increases proportionately to the density of basal .dis- locations. In this 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 s the same as i t was during s t r a i n p rior to recovery. Thus, the increase i n the density of jogs with respect to the increase i n basal density i s unchanged. Following the second recovery, which i s the point at which secondary , a c t i v i t y 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 plot. This i s explained i f i t i s assumed that the pyramidal dislocation density i n this range i s increasing s l i g h t l y . With a higher pyramidal density, there w i l l now be a larger number of jogs being formed for a given increase i n the basal dislocation density (T ) ., At the end of linear easy gli d e , Fig. 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 plot. I t i s expected that there i s a macroactivity on the pyramidal system at this point • causing a substantially higher forest density. (As mentioned previously, the density of forest dislocations was found to increase by two orders of magnitude from the end of easy glide 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 result i n a substantially higher number of jogs formed, and soythe slope of the A T - T plot 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 controlling mechanism changes i n any way. However, for the purpose of s i m p l i f i c a t i o n of the following discussion, i t w i l l be assumed that this mechanism i s now one of forest intersection 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 simi l a r i n their behaviour. Thus i n the following discussion, A T i s assumed to be representative of forest density rather than jog density. During stage I I deformation, where the forest density i s large, probably of the same order as the basal density, the basal dislocations w i l l ;;become highly jogged very quickly. Consequently these basal dislocations w i l l probably not move appreciable distances. New dislocations w i l l be generated and become jogged i n i t i a l l y at a rate determined by the forest density., Therefore, the measured values of Ax and activation volume would not be s i g n i f i c a n t l y different i f forest intersection were the rate co n t r o l l i n g mechanism. The plot of A T V S . T for deformation i n stage I I (Fig. 22) i s ; essentially l i n e a r . This indicates that the factors leading to both ; T and A T increase proportionately. j, Comparing cadmium to face-centered cubic metals, i t i s seen that the 36 above conclusion i s consistent with the results of Steeds and Basinski 37 and Basinski . These workers found that the density of secondary d i s - locations i n copper single crystals i s comparable to the density of primary dislocations throughout stage I I , i n spite of the fact 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 further 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 proportional increase i n both pyramidal (forest) and basal (primary) dislocations rather than- an, increase i n basal jogs with no s i g n i f i c a n t increase i n forest density. 87 The behaviour of A T vs. T with respect to recovery (Fig. 2 4 ) i s s i g n i f i c a n t l y different i n stage I I than i t was i n stage I. While i n stage I, recovery brought about a substantial 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 essentially no change i n A T . Strain following this recovery showed that the slope of A T vs. T i s s t i l l the same as i t was for s t r a i n preceding recovery. l( That T changes markedly indicates that the basal disl 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 , the ;flow stress i s dependent primarily on basal density. Because A T does not change during this anneal, the density of pyramidal dislocations must remain constant during recovery. The fact that the slope of A T vs. T i s the same following recovery as i t was preceding recovery indicates , that the densities of pyramidal and basal dislocations are s t i l l increasing i n the same r e l a t i v e proportion. However, since the basal density decreased while the pyramidal density remained constant during recovery, there must be a net increase of pyramidal dislocations to the system whenr. i t i s compared at the same stress l e v e l . There are more pyramidal dis r locations at point B i n Fig. 2 4 than at point A. , The second recovery anneal gives a further s h i f t to the A T - T plot as shown i n Fig. 2 4 . In this case, however, A T as w e l l as T does decrease s i g n i f i c a n t l y . On the basis of the above argument, this indicates that some pyramidal dislocations must be l o s t at this time. Again, s t r a i n following recovery shows that the increase of both basal and pyramidal dislocations remains i n the same proportion as that p r i o r to any recovery since the slope of the plot i s unchanged. In this s t r a i n increment^between recovery anneals, there i s a further increase to the system of pyramidal dislocations, as shown by comparison of si m i l a r stress levels A,B, and C i n Fig. 2 4 . f. Following further recoveries, the A T - T plots are coincident with that plot obtained following the second recovery. Thus, a condition must have been reached i n which both pyramidal and basal dislocations are recovered. The proportion of each type of dislocation recovered i s the same as that i n which i t i s generated during s t r a i n , therefore j establishing some sort of equilibrium i n the structure. This equilibrium condition prescribes the density of basal and pyramidal dislocations at any stress l e v e l . 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 this equilibrium condition. I f experimental conditions were different ( 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 equilibrium might change, although i t i s thought that the end result would be the same. The recovery of pyramidal dislocations as found i n the above 23 results may occur by the mechanism as observed by Price . He found that second order pyramidal dislocations 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 into 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 result of the findings of this section on Cottrell-Stokes behaviour, i t i s concluded that for a hexagonal close-packed metal such as cadmium, the Cottrell-Stokes law has no r e a l significance. I f i t i s found to be obeyed i n any hexagonal system, i t i s probably only coincidental. 4.4 Flow stress following recovery ;. The flow stress measured following recovery i s probably determined primarily by basal dislocation density at this point. In stage I I , the basal density present following saturation recovery appears to be a function of the pyramidal dislocation density which i s also present. Fig. 42 i s a plot of the flow stress achieved after saturation recovery as a function of s t r a i n . In this p l o t , stage I I l i n e a r i t y begins i n a l l tests at a s t r a i n of about 1.75. I t i s seen that the basal s, density rises to a maxiumum value at approximately 60% s t r a i n following the onset of stage I I l i n e a r i t y , and then remains constant. This i s very si m i l a r to the behaviour deduced for pyramidal dislocations i n the previous section. Fig. 42 shows that there i s an increase i n the basal density 2 2 (flow stress) by a factor of 3 from 250 gm/mm to 800 gm/mm . Fig. 21 showed that the pyramidal density ( A T ) i n this same stress range also increases by a factor of 3. Thus, i t appears that there i s a direct relationship between the density of basal dislocations and the density • of pyramidal dislocations present following recovery. This relationship i s probably caused by the way i n which ;the basal dislocations are trapped by the pyramidal dislocations, or perhaps by a stress f i e l d associated with the pyramidal dislocations. That the pyramidal density i s the governing factor i n this relationship i s more apt to be the case than the converse, i n which the basal density prescribes the pyramidal density, since i t has been shown i n the previous section that the pyramidal density i s insensitive to recovery. 4.5 Work hardening i n stage I I These activation volume and A T vs. T data also give some insight into the work hardening behaviour of cadmium single crystals i n stage I I .  The data f i t w e l l with the arguments of Hirsch and M i t c h e l l based on face-centered cubic materials. The p r i n c i p a l points i n their theory, together with comparison to the present work are as follows: 1) At the end of stage I , there are long continuous obstacles which form, barriers to newly formed s l i p l i n e s . Dislocations are stopped by e l a s t i c i nteraction with both the primary and secondary dislocations i n the obstacles. In cadmium, i t i s proposed that these barriers are the s e s s i l e dislocations formed by the interaction of basal and second order pyramidal dislocations.' 2) Hardening' i s due to the hardening of potential primary sources by the increased density of 'primary and secondary dislocations. 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 following 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) In 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 dislocations to new basal dislocations generated increases rapidly, u n t i l the regions within which secondary sources operate extend throughout the c r y s t a l . At this point, the proportion of secondary d i s - locations depends on the i n t e r n a l stress pattern. For a given arrangement of primary dislocations* the density of secondary dislocations will ; ;,be the maximum possible compatible with the in t e r n a l stress, leading to the maximum possible 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 similitude develops i n which the r a t i o of secondary to primary dislocations remains constant throughout stage I I , and only the scale of.the. structure decreases as the stress increases. The present, results are compatible with this theory up to the point of recovery,, i n , stage I I . At this 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 did. On further 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 possible to maintain this condition, and consequently give a higher work hardening rate. This was observed experimentally (see Fig. 32). This, too, i s consistent with Hirsch and Mitchell's theory that the work hardening rate i s dependent, among other factors, on the density of secondary dislocations i n the region of the pile-ups. ,, 4) The flow stress i s determined partly by forest density, l i n e tension, and long range stress i n the softest region between pile-ups. The r e l a t i v e contributions have yet to be determined. In the present work, i t appears that the flow stress i s dependent primarily on the basal density, which would be equivalent to Hirsch and Mitchell's long range stress. Thus the present results 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 this theory could be extended to include hexagonal close-packed metals as w e l l as face- centered 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 relationship does not change i n this range, dynamic recovery should affect both basal and forest dislocations. 93 5. SUMMARY AND CONCLUSIONS 1) Deformation i n easy glide at -196°C i s by basal glide only when X > 35°. In this range, the work hardening i s completely recoverable by annealing at 75°C for 30 min. 2} Deformation i n easy glide at -196°C from x - 35° to the end,of easy glide i s primarily by basal glide with some a c t i v i t y on the second order pyramidal s l i p system. Work hardening i n this region i s not completely recoverable. This i s probably due to the formation of stable obstacles by the interaction of basal and pyramidal dislocations. • 3) The end of easy glide occurs at x = 20.1° ± 1.2°. This i s probably a result 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 primarily on the basal systemjalthough the densities of basal and pyramidal dislocations are probably about the same. Recovery early i n stage I I causes a decrease i n basal density with no corresponding decrease i n pyramidal density. As a resul t of t h i s , the work hardening rate following recovery i s higher than that preceding recovery. At higher strains i n stage I I , both basal and pyramidal ( dislocations are recovered. . -; 5) The s t r a i n attainable i n stage I I at -196°C i s increased, by recovery. 6) Deformation i s probably controlled i n stage I by the non- conservative motion of jogs. The data from stage I I may also be interpreted i n terms of this mechanism. 7) The Cottrell-Stokes law i s not s t r i c t l y obeyed through any portion of the deformation of cadmium single 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 this law i n hexagonal metal would be coincidental. 8) I t was found to be impossible to calculate a meaningful , activation energy. 95 6. APPENDIX 6.1 Electron microscopy techniques Ah attempt was made to thin sections of cadmium single crystals for observation i n the electron microscope. In this way, i t was hoped to observe the dislocation structure of the specimens, and to determine the effects of deformation and recovery upon this structure. Unfortunately, a l l attempts at thinning were unsuccessful, and no structures were observed. The techniques employed i n this study, and the problems . encountered, are l i s t e d below. i; , 6.1.1 Cutting a thin section . The f i r s t problem i n this 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 specified orientation from a bulk single c r y s t a l . I t was also, necessary to ensure that no extraneous dislocations were introduced into the s l i c e during the cutting operation. There was no d i f f i c u l t y inovlved i n determining the desired orientation 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 cutting a thin s l i c e from the 5 mm. diameter bulk single crystal.; I t was decided that cutting by any mechanical means such as a jeweller's saw. would be.unacceptable due to the amount of deformation introduced to the sample during the cutting operation. Spark erosion was the f i r s t method t r i e d to obtain the desired s l i c e . 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 negligible deformation introduced into the sample. This method worked wel l for the f i r s t , cut, but when a second cut was attempted to produce a thin s l i c e , problems arose. During this second cut, the s l i c e which was 1-2 mm. thick was bent by the forces developed during the .cutting 96 procedure up towards the sparking t o o l . When this occurred, sparking took place between the side of the tool 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 into a wedge. Experiments i n shielding the side of the tool to prevent this from happening were unsuccessful. Even i f i t could be assumed that the deformation invoied i n the bending was i n s i g n i f i c a n t , i t was impossible to thin the wedge-shaped s l i c e s for subsequent observation i n the electron microscope. I As an alternative to spark erosion-, acid cutting was t r i e d to produce the desired s l i c e s . In this method, a counter-balanced specimen was held l i g h t l y against a reciprocating polyester thread. The thread • was kept saturated with d i l u t e n i t r i c acid by a drip feed arrangement so that a l l cutting was produced by the disso l u t i o n of the cadmium by the n i t r i c acid. The vel 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 . One cut took approximately 8 hours. . • , The s l i c e s produced by this method varied i n thickness throughout the section by as much as 1 mm. since the thread could not be prevented from moving s l i g h t l y out of i t s intended plane during the cut. 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 ne g l i g i b l e deformation and were the best available. , 6.1.2 Thinning The s l i c e s produced by acid cutting were thinned by standard electro- polishing techniques. I t was found that chromic-acetic electro-polishing solution used at the plateau voltage gave excellent polishing r e s u l t s , producing a clean, shiny surface. The major problem involved during thinning was to produce a regular, 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 did not have perfectly smooth surfaces. To try to overcome this problem, a j e t polishing technique was employed, i n which a thin stream of polishing solution i s directed towards the specimen surface during electro-polishing. This produced the desired concavity but i t did 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 perforation, the included angle of the specimen, leading up to the perforation was quite high. Chemical polishing in,- d i l u t e n i t r i c acid p r i o r to electro-polishing 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 electron microscope. I t was found that areas adjacent to perforations were always too thick for electron transmission. Consequently, i t was impossible to determine any dislo c a t i o n structures i n these samples. Since this work was attempted, two transmission electron 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 for the present material. 98 REFERENCES 1 Sharp J.V., M i t c h e l l A., and Christian J.W. Acta Met. 13, 965 (1965). 2 P e i f f e r H.R. and Stevenson F.R. Phys. Stat. Soc. _4, 411 (1964). 3 Kroupa F. and Price P.B. P h i l . 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Recovery and Rec r y s t a l l i z a t i o n of Metals p . l Interscience, New York (1963). 45 Wolf, P. Diplomarbeit, Freiberg (1965).

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