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Solid solution strengthening of magnesium Akhtar, Ainul 1968

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SOLID SOLUTION STRENGTHENING OF MAGNESIUM by AINUL AKHTAR B.Sc- (Hons), Utkal University, India, 1962 B„E., Indian I n s t i t u t e of Science, India, 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 July , 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h.i;s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Metallurgy  T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , C a n a d a D a t e September 18, 1968 ABSTRACT -1-Solid solution strengthening i n magnesium polycrystals containing Zn, A l , Cd, In and Pb as solute has been investigated over the temperatures between 78° and 513°K with p a r t i c u l a r emphasis on the d i l u t e a l l o y s . The va r i a t i o n of y i e l d stress with concentration occurs i n either two or three stages. In stage I , the y i e l d stress increases rapidly and l i n e a r l y with concentration; i n stage I I , the rate of increase of y i e l d stress i s very much less than i n stage I; i n stage I I I , the y i e l d stress decreases with solute additions. The solution hardening rates and the t r a n s i t i o n concentrations from stage I to stage I I (C T) depend on the size-difference between Mg and the solutes. The results are discussed i n terms of the v a r i a t i o n with concentration of the CRSS for both basal and prismatic s l i p . I t i s proposed that at concentrations less than C^, the increase i n CRSS for prismatic s l i p i s the dominant factor; beyond C^, y i e l d i s governed by a balance between basal hardening and prismatic softening. The effect of solute on the d u c t i l i t y of magnesium at elevated temperatures i s discussed i n terms of a stress induced polygonization process and the d u c t i l i t y maxima observed i n the Mg-Al alloys are explained. Single crystals of Mg-Zn alloys oriented for basal s l i p have been deformed i n tension over the temperature range from 78°K to 423°K. The v a r i a t i o n of the basal d i s l o c a t i o n density caused by the addition of solute has been studied using transmission electron microscopy of thin f o i l s . The increase i n disloc a t i o n density which was found to be proportional to the square root of the solute concentration, cannot account for the observed increase i n the athermal stress. A dislo c a t i o n etch p i t technique has been developed and used to measure the va r i a t i o n i n the forest d i s l o c a t i o n density with solute concentration. The forest density increases l i n e a r l y and rapidly up to a certain minimum solute concentration, beyond which i t remains almost constant. The results are i n good agreement with the observed thermally activated flow stress for low solute concentrations. The observed v a r i a t i o n i n the athermal component of CRSS has been discussed i n the l i g h t of an increased f r i c t i o n stress a r i s i n g due to a random d i s t r i b u t i o n of solute. Using rate theory, i t has been shown that the forest i n t e r -section remains the rate c o n t r o l l i n g mechanism up to a certain low concentration of solute beyond which the single solute atom pinning of dislocations becomes the rate determining process. The solute dependence of the work hardening parameters are also reported and examined i n the l i g h t of the existing theories of work hardening. Single crystals of Mg - Zn and Mg- A l alloys have also been deformed so as to suppress basal s l i p and {1012} twinning and to induce prismatic s l i p . The results have been explained i n terms of an increasing athermal stress and a decreasing P e i e r l s stress with the addition of solute. P e i e r l s stress has been shown to be the rate controlling mechanism below room temperature. The observed v a r i a t i o n of CRSS for prism s l i p with solute concentration accounts adequately for the concentration dependence of y i e l d stress i n the p o l y c r y s t a l l i n e aggregate. The results also suggest that the decrease i n P e i e r l s stress with solute addition i s not necessarily associated with a decreasing c/a r a t i o and the monovalent nature of the solute. i i i ACKNOWLEDGEMENT The author g r a t e f u l l y acknowledges the advice and encouragement given by his research director, Professor E. Teghtsoonian. Thanks are also extended to other members of the faculty and the graduate students for helpful discussions. Financial assistance i n the form.of Lead-Zinc research fellowship and the National Research Council studentship i s g r a t e f u l l y acknowledged. i v TABLE OF CONTENTS Page. 1. Solution Strengthening i n Polycrystals 1 1.1. Introduction and Objectives 1 1.2. Experimental Procedure 4 1.2.1. Materials and allo y preparation 4 1.2.2. Preparation of poiyerystalline specimens 4 1.2.3. The growth of single crystals 5 1.2.4. Standard sections used 7 1.2.5. Spark erosion damage 8 1.2.6. Preparation of single c r y s t a l specimens for tensil e tests 9 1.2.7. Testing procedure 10 1.3. Deformation characteristics of poiyerystalline aggregates 12 1.3.1. Nature of the st r e s s - s t r a i n curves 12 1.3.2. The y i e l d stress 1.3.2.1. The temperature dependence of y i e l d 18 1.3.2.2. The concentration dependence of y i e l d 21 1.3.2.3. Solution strengthening i n stage I 24 1.3.2.4. Strengthening above Cj 28 1.3.2.5. The temperature dependence of the solution strengthening parameters 28 1.3.3. The c r i t i c a l t r a n s i t i o n concentration 31 1.3.4. The maximum stress v a r i a t i o n with temperature and alloying 35 1.3.5. Flow stress i n r e l a t i o n to temperature and alloying 38 1.3.6. D u c t i l i t y 44 1.3.6.1. Alloying effect 44 1.3.6.2. The temperature dependence of d u c t i l i t y 44 1.4. Discussions 49 1.4.1. Deformation modes i n Magnesium 49 1.4=1.1. Crystallographic s l i p 50 Table of Contents (Cont) v -Page 1.4.1,2= P l a s t i c deformation by twinning 54 1.4.1.3. Grain boundary deformation 57 1.4.1.4. C e l l formation 58 1.4.2. Fracture of Magnesium polycrystals 59 1.4.3. Solution hardening 60 1.4.3.1. The var i a t i o n of O y p and d u c t i l i t y with solute concentration , 64 1.4.3.2. The solution hardening rate (|£.) 64 1.4.3.3; Hardening beyond.Cj 66 1.4.3.4. The t r a n s i t i o n concentration C T 67 1.4.3.5. The flow stress 68 1.4.4. The effect of temperature on d u c t i l i t y 69 1.4.4.1. The d u c t i l i t y maxima 70 .1.4.5. Strengthening effects i n multicomponent s o l i d solutions 72 2. Solution hardening i n allo y single crystals 75 2.1. Introduction and Objectives 75 2.2. Stress-Strain Relationships i n Basal S l i p 76 2.2.1. The st r e s s - s t r a i n curves 76 2.2.2. The effect of substructure on work hardening 78 2.2.3. The st r e s s - s t r a i n curves of the Mg-Zn alloys 80 2.2.4. Yi e l d points . 84 2.2.5. The c r i t i c a l resolved shear stress 88 2.2.6. The rate of solution hardening 91 2.2.6.1, The temperature dependence of Sj 91 2.2.6.2. The v a r i a t i o n of STT with temperature 93 2.2.7. Work hardening 93 2.2.7.1. The work hardening rate i n Stage A, 0 A 93 2.2.7.2. The extent of easy glide 96 2.2.7.3. Stress at the onset of Stage B, rx B 100 2.2.7.4. The work hardening rate i n stage B, 9 B 100 2.2.7.5. Deformation i n stage C 103 2.3. The effect of solute on the dis l o c a t i o n densities 103 2.3.1. The basal dislocation density 105 2.3.1.1. F o i l preparation 106 2.3.1.2. Observations 107 Table of Contents (Cont) v i Page 2.3.1.3. The technique of disloc a t i o n d i s -location density measurement and i t s l i m i t a t i o n s 108 2.3.1.4. The effect of solute on the basal dislocation density 112 2.3.2. The forest d i s l o c a t i o n density 114 2.3.2.1. Crystal orientation and the p i t characteristics 114 2.3.2.2. The study of spark erosion damage i n Mg using the disloc a t i o n etch p i t method 115 2.3.2.3. Variation of the.etch p i t density with alloying 118 2.4. Discussions . 119 2.4.1. The c r i t i c a l resolved shear stress 119 2.4.2. Solution strengthening i n the athermal region 121 2.4.2.1. The basal dislocation density 122 2.4.2.2. E l a s t i c interaction 125 2.4.2.3. Chemical interaction 126 2.4.2.4. Short range order 128 2.4.2.5. Strengthening due to a random d i s t r i b u t i o n of the solute 130 2.4.2.6. Strengthening at low solute concentrations. 133 2.4.2.7. Valency effect 135 2.4.2.8. Other hardening mechanisms 135 2.4.3. Solution strengthening at, low temperatures 136 2.4.3.1. Thermally activated deformation 140 2.4.3.2. The effect of solute on the apparent activation parameters 143 2.4.3.3. The activation volume at y i e l d 145 2.4.3.4. Solution effect on the apparent activation energy 148 2.4.3.5. Zinc alloying and the thermally activated flow 149 2.4.3.5.1. Cross s l i p 151 2.4.3.5.2 P e i e r l s , Pseudo Pe i e r l s and Recombination mechanisms 152 2.4.3.5.3. The intersection model 153 a) intersection model, at low solute concentration 155 2.4.3.5.4. Dislocation pinning by the solute atoms 156 v i i Table of Contents (Cont) Page 2.4.3.6. Strengthening mechanisms i n d i l u t e alloys 159 2.4.3.6.1, Deviations from square array 163 2.4.4. Work hardening of Mg s o l i d solutions 164 2.4.4.1.1. The easy glide i n Mg 165 2.4.4.1.2. Easy glide i n a l l o y c y r s t a l 167 2=4.4.2. The temperature dependence of 9 A i n all o y crystals 168 2.4.4.3. The extent of easy glide 169 2.4.4.4. Work hardening in.stage B 171 2.4.4.5. The stage 'C' of deformation 172 2.4.5. The Effect of solute on the ease of prismatic s l i p 173 2.4.5.1. Introduction and objectives 173 2.4.5.2. Experimental results 174 2.4.5.2.1. The st r e s s - s t r a i n curves 174 2.4.5.2.2. D u c t i l i t y 181 2.4.5.2.3. The CRSS for prismatic s l i p 181 2.4.5.2.4. The CRSS of Mg-Zn alloys 183 2.4.5.2.5. The effect of solute on the flow stress 187 2.4.5.3. Discussions 189 2.4.5.3.1. The disloca t i o n mechanism for prismatic s l i p 189 2.4.5.3.2, The effect of solute 190 2.4 s5.3.3. The results of the presgnt work 192 2.4.5.3.4. The Origin of T g and T 195 2.4.5.3.4.1. The athermal stress 195 2.4.5.3.4.2. The v a r i a t i o n of T* with alloying 196 3. Summary of Conclusions 198 4. Suggestions for Future Work 201 5. ' Appendices 202 A. Determination of solute concentration i n the alloys 202 B. The Preferential nature of the solute-dislocation i n t e r -action 206 C. The determination of Ar from s t r a i n rate change tests 209 Bibliography 211 v i i i LIST OF FIGURES No. Page 1. Standard sections for obtaining t e n s i l e specimens from the as-grown c r y s t a l . . . . . . . . . . . . . . . . . . 6 2. Gripping arrangement for basal s l i p specimens . . . » . 11 3. Stress-strain curves for Mg-Zn. alloys tested at 295°K . 13 4. Stress-strain curves for Mg-Al alloys tested at 295°K . 14 5. Stress-strain curves for 65y grain size Mg . . . . . . 15 6. Stress-strain curves for Mg + 0.055 at.% A l a l l o y . . . 16 7. Stress-strain curves for Mg + 0.53 at.% A l a l l o y . . . . 17 8. Yi e l d stress-temperature relationships for Mg-Al alloys 19 9. Yi e l d stress-temperature relationships for Mg-Cd alloys 20 10. The composition dependence, of Tt i n Mg-^ -Al and Mg-Cd alloys 22 11. Y i e l d stress-concentration plot for Mg^Zn alloys . . . . 23 12. Y i e l d stress-concentration plot for Mg-Cd alloys . . . . 23 13. Stage I solution hardening at 295°K for various alloys . 26 14. Stage I slope vs size m i s f i t parameter 27 15. Stage I I slope vs size .misfit parameter . . . 27 16. The effect of temperature on the solution hardening curve for Mg-Al alloys . . . . . . . . . . . 29 17. The effect of temperature on the solution hardening curve for Mg-Cd alloys . . . . . . . 30 18. C r i t i c a l . t r a n s i t i o n concentration, C^, from stage I to stage I I vs temperature for Cd solute . . . . . . . . . 32 19. Cx vs temperature for A l solute. . . . . . . . . . . . . 32 20. a) Stage I slope vs temperature for Mg-Cd alloys . . . . . 33 b) Stage I I slope vs temperature.for Mg-Cd alloys . . . . . 33 21. Stage I slope vs temperature.for A l solute . . . . . . . 34 i x L i s t of Figures (Cont) No, Page 22. C r i t i c a l t r a n s i t i o n concentration, C T from stage I to stage I I vs size m i s f i t parameter . . . . . . . . . . . 34 23. Maximum stress vs. temperature for Mg-Al alloys . . . . 36 24. Fracture stress vs. composition for Mg-Cd alloys tested at 295°K . . . . . . . . . . . . . . . . . 37 25. Flow stress vs. temperature for 65y Mg . . . . . . . . 39 26. Flow stress vs. temperature for Mg + 0.055 at.% AI al l o y 40 27. Flow stress vs. temperature for Mg + 0.53 at.% AI al l o y 41 28. Transition temperature from stage I to stage I I i n the flow stress-temperature curves of Mg + 0.055 at.% AI all o y vs. s t r a i n . . . . . . . . . . . . . . 42 29. Flow stress vs. composition for Mg-Zn al l o y . . . . . . 43 30. D u c t i l i t y vs. concentration for various solutes . . . . 45 31. True s t r a i n to fracture vs. temperature for Mg-Al alloys 46 32. True strain,to maximum stress vs. temperature for Mg-Al alloys . . . . . . . . . . 48 33. True s t r a i n associated with negative work hardening vs. temperature for Mg-^Al alloys . . . . . . . . . . . . . 48 34. CRSS for basal s l i p vs. concentration (After Levine et al(74)) 63 35. Revised curves drawn for data from ref. 74 63 36. Solution hardening curve for Mg-Zn-In, ternary alloys tested at 295°K with Zn concentration fixed at 0.004 at.%. . . . 74 37. Solution hardening curve for Mg-Zn-In-ternary alloys tested at 295°K with Zn concentration fixed at 0.007 at.%. 74 38. Resolved shear stress-shear s t r a i n curves for Mg single crystals oriented for basal s l i p . . . . . . . 77 39. Schematic resolved shear stress-shear s t r a i n curve . . 79 X L i s t of Figures (Cont) No. Page 4 0 . Sub-boundary.running p a r a l l e l to the tensile-axis i n a Mg t 0 . 0 5 4 at.%.Zn.alloy c r y s t a l deformed i n easy glide at room temperature . . . . . . . . . . . . . . . 7 9 4 1 i Resolved.shear stress-shear s t r a i n curves for Mg single crystals deformed by various workers at room,temperature 8 1 4 2 . Resolved shear stress vs. shear s t r a i n curves for Mg + 0 . 0 1 9 at;% Zn allo y single crystals . . . . . . . 8 2 4 3 . Resolved shear stress vs. shear s t r a i n curves for Mg + 0 . 0 1 9 at.% Zn.alloy single crystals . . . . . . . 8 3 4 4 . Resolved shear stress vs. shear s t r a i n curves for Mg-Zn single crystals deformed at 3 7 3 ° K . . . . . 8 5 4 5 . Resolved shear stress vs. shear s t r a i n curves for Mg-Zn alloy single crystals deformed at 2 9 5 ° K . . . . . . . . 8 6 4 6 . Resolved shear stress vs. shear s t r a i n curves for Mg-Zn alloy single crystals deformed at 1 9 5 ° K . . . . . . . 8 7 4 7 . CRSS for basal s l i p vs. temperature for Mg-Zn.single crystals 8 9 4 8 . Solution strengthening i n basal s l i p vs. concentration for Mg-Zn alloys 9 0 4 9 . Solution strengthening i n basal s l i p vs. square root.of the solute concentration . . . . . 9 0 5 0 . The v a r i a t i o n of the solution strengthening parameter S^ with temperature . . . . . . . . . . . . . 9 2 5 1 . The v a r i a t i o n of the solution strengthening parameter SJJ with temperature . . . . . . . . . . . . . . . . . 9 2 5 2 . The work hardening rate i n stage A vs. temperature for Mg-Zn single crystals . . . . 9 4 5 3 . The temperature dependence of the work hardening rate i n the easy glide of Magnesium as reported by various workers . . . . . . . . . . . . . . . 9 5 5 4 . The increase i n work hardening rate i n easy glide vs. Zn concentration . 97 x i L i s t of Figures (Cont) No. Page 55. The Increase i n work hardening rate i n easy glide vs. square root of the Zn concentration . . . . . . . . . . 98 56. The extent of easy glide as a function of temperature for Mg-Zn single crystals . . . . . . . . . . . . . . . 99 57. The decrease i n the extent of easy glide as a function of the solute concentration . . . . . . . . . . . . . . 99 58. The effect of temperature and solute concentration on the stress at the onset of stage B . . . . . . . . . . 101 59. The effect of temperature on the work hardening rate of Mg-Zn crystals i n stage B . . . . . . . . . . . . . . . 102 60. The increase i n the work hardening rate i n stage B vs. Zn concentration . . . . . . . . . . . . . . . . . 102 61. The stress at the onset of stage C vs. Zn concentration 104 62. Typical disl o c a t i o n structure i n as.grown Mg c r y s t a l . . 107 63. Electron micrograph showing dislocations surrounding a p a r t i 64. Typical disl o c a t i o n structure i n an undeformed Mg + 0.18 ate % AI a l l o y single c r y s t a l . . . . . . . . . . . . . 110 65. Typical disl o c a t i o n structure i n an undeformed Mg + 0.38 at. % AI al l o y single c r y s t a l I l l 66. Basal-dislocation density vs. square root of solute concentration for Mg-Al alloys . . . . . 113 67. a) Hexagonal shape of the etch p i t s . . . . . . . . . . . 116 b) D i s t r i b u t i o n of etch p i t s i n a Mg + 0.019 at.%. Zn all o y c r y s t a l . . . . . . . . . . . . . . . . . . . . . 116 68. Dislocation etch p i t s on the {0001} plane of Mg single C l T y S _ , _ i l • o o o a e « o « * e e o e « O B O O O o e > e o H 7 69. Etch p i t density vs. Zn concentration . . . , . , , . . 117 70. The temperature dependence of y i e l d i n terms of the stress components 120 71. Comparison of the observed solution strenthening with that expected from the increase i n the basal d i s l o c a t i o n L i s t of Figures (Cont) x i i No. Page 1/2 72„ The increase-in.CRSS of Mg vs. ( L i concentration) i n Mg-Li a l l o y single crystals . . . . . . 124 73. r vs. concentration for Ag s o l i d solutions . . . . . 134 74. The effect of solute on the T - T curve (schematic) . . 136 ft 75. T vs. temperature for Mg-Zn single crystals . . . . . 138 * 76. (T^- ) vs. solute concentration for Mg-Zn single dT . , . Y / Y 0 . crystals . . . . . . . . . . . . . . . . . . . . . . . 138 77. Activation volume vs. shear s t r a i n for Mg-Zn.single crystals . . . . . . . . . . . . . . . 144 78. Activation volume at y i e l d vs. Zn concentration . . . 146 79. Activation volume at y i e l d vs. temperature for Mg and Mg + 0,045 at.% Zn a l l o y single crystals . 146 * 80. Activation volume at y i e l d vs. x for Mg and Mg + 0.45 at.% Zn all o y single crystals , . . . . . 147 81. Apparent activation energy vs. Zn concentration . . . 150 82. Dislocation pinning by solute atoms giving r i s e to low temperature f r i c t i o n stress (schematic) . . . . . . . 156 83. Comparison.of the experimental results with the composition dependence of activation volume as predicted by Friedel's model 158 84. Concentration.dependence of CRSS extrapolated to 0°K for Ag-In and Ag-Al single. crystals 158 85. Resolved shear stress vs. shear strain,curves for Mg single crystals oriented for prismatic s l i p . . . . . 175 86. Resolved shear stress vs. shear s t r a i n for Mg + 0.019 at.% Zn a l l o y single crystals oriented for prismatic 87. Resolved shear stress vs. shear s t r a i n curves for Mg + 0.258 at.% Zn all o y single crystals oriented for prismatic s l i p 177 x i i i L i s t of Figures (Cont) No. Page 88. Resolved:shear stress vs. shear s t r a i n curves for Mg + 0.45 at,% Zn all o y single,crystals oriented for prismatic s l i p . . . . . . . . . . . . . . . 178 89. Resolved shear stress vs. shear s t r a i n curves for Mg-Zn-single crystals deformed at 423°K i n prism s l i p orientation . . . . . . . . . . . . . . . . . . . . . 179 90. Resolved shear stress vs. shear s t r a i n curves for Mg-Zn single crystals deformed at 78°K i n prism s l i p orientation . . . . . . . . . . . . . . . . . . . . . 180 91. Shear-strain to-fraeture vs. temperature for Mg-Zn single.crystals oriented for prismatic s l i p . . . . . 182 92. CRSS for prismatic s l i p vs. temperature for Mg-Zn single crystals . . . . . . . . . . . . . . . . . . . 184 93. CRSS for prismatic s l i p vs. Zn concentration . . . . . 185 94. CRSS for prismatic s l i p vs. temperature for Mg-^Al single crystals . . . . . . . . . . . . . 186 95„ Flow stress for prismatic s l i p vs. concentration for Mg-Zn single crystals tested at 78°K . . . . . . . . . 188 96. Flow stress vs. concentration for Mg-Zn single crystals tested at 423°K . . . . . . . . . . . . . 188 97. Activation volume at y i e l d vs. temperature for Mg-Zn single crystals deformed i n prism s l i p orientation . . 191 98. CRSS for prismatic s l i p vs. temperature for Mg-Li al l o y single crystals . . . . . . . . . . . . . 193 99. Schematic representation of the effect of solute on the various components of flow stress i n prismatic s l i p . 194 100. Absorbance vs. Zn concentration 205 101. Solution hardening curves for Mg-Zn and Mg-In binary systems 207 102. The nature of the flow stress observed during strain.rate change tests i n Mg-Zn single c r y s t a l s . (Oriented for basal s l i p ) . . . . . . . . . . . . . . . 210 X I V LIST OF TABLES Page I Solution hardening parameters for polycrystals tested 3.t 293 K o o o Q O o e > o ' o o o » o o o o o o o o o o 25 II S l i p systems i n hexagonal metals o » o o o o o e o o o 54 III Comparison of solution hardening rates „ » « « , » „ « 65 IV Strengthening due to increased forest density = , 0 0 0 162 PART I 1. Solution Strengthening i n Polycrystals L L INTRODUCTION & OBJECTIVES : The systematic study of s o l i d solution strengthening mechanisms has lagged considerably behind i t s applications. This i s especially the case for materials with hep structure. Solid solutions with fee structure have been considered i n some d e t a i l during the past (1 2) decade ' . However, the limited understanding of the deformation characteristics of hep pure metals has inhib i t e d investigations of solution strengthening i n this class of material. Of a l l hexagonal metals, titanium and magnesium have been studied i n the greatest d e t a i l because of their favourable strength to weight r a t i o . The pure metal magnesium i s not used i n many technological applications, mainly due to i t s embrittlement during cold working. Therefore,,the problem of determining the deformation mechanisms which (3-13) control flow i n magnesium has been a subject of considerable interest As i n the engineering of other metals alloying of magnesium has been used to obtain strength, d u c t i l i t y , workability, corrosion resistance, low density and c a s t a b i l i t y . In r e l a t i o n to deformation behaviour much of the s o l i d solution work i n magnesium and i n hexagonal structures i n general has been confined to the Mg-Li a l l o y s ' ^ . Interest i n this system of alloys was aroused mainly because these alloys showed an unusual lowering of the c r i t i c a l resolved shear stress for prismatic s l i p with increasing solute addition. This anomalous s o l i d solution effect was attributed to the decrease of c/a r a t i o i n these a l l o y s , compared to that of pure magnesium. I t was concluded that only those 2 alloying elements which decrease the c/a r a t i o of magnesium can lower. the CRSS for prismatic s l i p . One of the objectives of the present investigation was to evaluate the significance of the c/a r a t i o i n the ease of prismatic s l i p and also to evaluate the effect of r e l a t i v e valence of the solute on the properties examined. For this purpose the solutes chosen were Zn, Cd, AI, In and Pb. Of these Zn and Cd belong to the same valency group as Mg and have the same c r y s t a l structure as w e l l , but d i f f e r i n the atomic si z e . Zn has l i t t l e effect on the c/a r a t i o , whereas Cd increases i t ^ ^ . AI, In and Pb belong to higher valency groups, and a l l three increase the c/a r a t i o . These elements cover a wide range of size differences with respect to magnesium. A second objective of the work presented i n th i s thesis was to examine the effects of alloying at low concentrations. Work by ( 1 8 ) Hardie and Parkins using hardness measurements had indicated the p o s s i b i l i t y of a very high strengthening effect at low concentrations i n Mg base alloys containing a wide variety of solutes. Many of the previous s o l i d solution strengthening studies have been carried out on polycrystals. I t i s possible to derive quantitative information concerning the dislocation-solute interaction i n face centred cubic a l l o y s . However, i n the case of hexagonal s o l i d solutions such studies may at best be considered to provide a qu a l i t a t i v e understanding of the subject. The difference i n the two groups of close packed structures arises mainly due to the different ways i n which their p o l y c r y s t a l l i n e aggregate undergoes p l a s t i c deformation. 3 A necessary condition for a polycrystal to deform p l a s t i c a l l y and yet retain continuity i s that the number of independent deformation systems be f i v e ^ ^ 22,60)^ ^ t^ e face centred cubic structures the primary single s l i p mode, i s capable of giving r i s e to f i v e independent s l i p systems. Since the s l i p systems are crystallographically equivalent, they are affected equally by the addition of solute. In contrast, no single s l i p mode i s capable of giving r i s e to f i v e independent s l i p systems i n hexagonal metals. (The second order {1122} <1123> pyramidal s l i p i s capable of giving r i s e to f i v e independent modes, but i t operates as a secondary system at high temperatures i n the case of Cd, Zn and T i . This system, however, does not operate i n magnesium). The p l a s t i c deformation of p o l y c r y s t a l l i n e magnesium involves ( 1 4 ) more than two sets of crystallographically nonequivalent s l i p systems The solute dislocation i n t e r a c t i o n , therefore, on the two sets of planes cannot be expected to be i d e n t i c a l . In view of this l i m i t a t i o n of the polycrystal study, the investigation has been extended to include the effect of solute on single crystals as w e l l . The f i r s t part of t h i s thesis i s concerned with the effect of solute on the deformation behaviour of the p o l y c r y s t a l l i n e aggregates. The second part deals with studies on single crystals i n making quantitative estimates of the dislocation solute interaction for various s l i p systems. 4 1.2. EXPERIMENTAL PROCEDURE: 1.2.1. Materials and Alloy Preparation: High purity Mg (99.995), obtained from the United Minerals and Chemicals Corporation, N.Y. , i n the vapour deposited form, and alloying elements having an impurity l e v e l below 0.005% were used i n the present investigation. Melting was carried out at 680®C i n a c y l i n d r i c a l graphite mould, housed inside a v e r t i c a l resistance heated furnace. Alloying additions were made i n the form of. pre^-weighed pure metals for compositions greater than 0.2 at.%. However, master alloys were used for lower.concentrations. After s t i r r i n g , the melt was s o l i d i f i e d i n the melting pot by sharp unidi r e c t i o n a l cooling i n order to avoid pipe formation. For alloys containing high vapour pressure solutes (at the melting temperature) l i k e Zn and Cd, the graphite pot along with a tapered graphite plug was enclosed i n a stainless s t e e l bomb. This procedure minimized the loss of the more v o l a t i l e component from the melt, The compositions of the alloys were checked and found to be close to the nominal values. Details of the chemical analyses are described i n appendix (A). 1.2.2. Preparation of Poiy e r y s t a l l i n e Specimens: 1 " The castings approximately 1 /Q i n diameter and 5" long were homogenized and subsequently hot r o l l e d at 430°C to 0.032" thickness. Reductions of 10 - 15% per pass were made i n the beginning to break down the cast structure, followed by 2 0 - 25% reduction per pass, which controlled the f i n a l grain size of the material. The material was given intermittent annealing for 5 minutes at 430°C before every pass. The r o l l e d sheet was sheared into 3" long by 0.75" wide 5 s t r i p s , which i n turn were punched into te n s i l e test specimens having a reduced gauge length of 0.8" with a width of 0.2". The ten s i l e axis of the specimen was kept p a r a l l e l to the r o l l i n g d i r e c t i o n . Edges were removed by 3/0 emergy paper and the specimens were given 20 mins. anneal at 430°C. The l a s t annealing operation removed a l l residual stresses due to punching, and gave a consistently uniform grainsize of 60 - 65p. The test pieces were pickled i n 20% HNO^  which removed the oxide layer.from the surface and also gave s l i g h t grain boundary grooving, which helped examining the microstructure subsequently. 1.2.3. The Growth of Single Crystals: Single crystals were grown from the melt using a modified Bridgeman technique. Alloys were made following the procedure described 4 it " e a r l i e r . The pol y c r y s t a l l i n e rod approximately \ /% i n diameter, arid 5" long was cleaned and kept i n a graphite mould, which was covered with a tight f i t t i n g tapered graphite plug, as shown i n f i g . (1). The mould was then lowered i n a v e r t i c a l tube resistance furnace at the rate of 0.4"/hr. This procedure gave consistently good quality single. c r y s t a l s . A shallow temperature gradient was used i n order to minimize the solute segregation i n the al l o y c r y s t a l s . For a constant degree of supercooling, and temperature gradient, the apex angle of the nucleation t i p was found to influence the orientation of the c r y s t a l with respect to the direc t i o n of growth. The orientation of the crystals grown i n th i s manner could be controlled without seeding. A l l crystals used i n the present work were grown such that the basal plane remained inclined at nearly 45° to the growth di r e c t i o n . This procedure was 6 l'3 0-12" M—0-5"—H Specimen dimensions PRISM s l i p specimen Basal s l i p specimen from section I I from section I Fig. 1. Standard sections for obtaining t e n s i l e specimens f the as grown c r y s t a l . 7 found to be most convenient i n obtaining large sections having the desired orientation. Subsequent chemical analyses on samples taken from the two ends of the r e l a t i v e l y short length of the as grown cr y s t a l did not show a detectable v a r i a t i o n i n composition. For each solute concentration investigated, only one c r y s t a l was grown from which a l l the ten s i l e test specimens as well as samples for dislocation density evaluation could be obtained. 1.2.4. Standard Sections Used: The orientation of the as grown c r y s t a l was determined using X-ray back r e f l e c t i o n Laue technique. The c r y s t a l was mounted on a "V" block which i n turn was clamped on to the base plate of a spark erosion machine. Using a goniometer, the c r y s t a l was rotated with respect to i t s growth axis and the spark erosion tool - a f l a t copper strip~was adjusted such that the desired section could be obtained on subsequent spark erosion. Two sets of s l i c e s having a thickness of 0.1" each were used i n the present investigation. Section I: These s l i c e s were cut p a r a l l e l to the d i r e c t i o n of growth, keeping the {0001} plane perpendicular to the f l a t face. Tensile test specimens oriented for basal s l i p were obtained from t h i s section. Section I I : The surface of this series of s l i c e s was kept p a r a l l e l to the basal {0001} plane within 2 degrees. Samples for dislocation density evaluations as well as t e n s i l e test pieces to induce prismatic s l i p were obtained from this section. 1.2,5. Spark Erosion Damage: In recent years spark erosion has become a more att r a c t i v e alternative to the acid machining methods, i n the shaping of suitable test specimens from larger single c r y s t a l s . The main reasons for the preference of spark erosion over the acid machining methods are the r e l a t i v e l y faster cutting rate and the f l a t t e r surfaces obtainable by the former technique. However, the uncertainty i n the depth of the p l a s t i c a l l y deformed layer, due to spark erosion, has been a drawback. Lately interest has been shown i n determining the depth and d i s t r i b u t i o n of spark damage. Much of this work has been based on the (23) revealing of dislocations by suitable etching techniques. SamiMels has shown that the depth of damage i n p o l y c r y s t a l l i n e 70-30 brass i s v ' "(24) limited to 45y below the surface. The work of Sestak and Libovicky (25) (26^ on Fe-4%Si a l l o y , of Steeds on copper and of Platnik on the (111) plane of Sb and B i and {0001} plane of Zn have demonstrated that with a spark energy of the order of 1C? ergs the damage i s limited to (27) a depth of 150 y from the surface. Hazzledine working with Cu-Al alloys concluded that the damage was 300 y deep for si m i l a r spark energy. (28) More recently Turner et a l have evaluated the effect of p l a s t i c anisotropy on spark damage i n Zn single c r y s t a l s . 3 Spark energies of the order of 10 ergs have been used i n the present work. Using similar spark energies and transmission electron (29) microscopy for revealing dislocations,, Hirsch and L a l l y have found that the basal dislo c a t i o n density remains unaffected i f s l i c e s thicker than 1 mm are cut. However,an accurate check on the depth of damage could not be made by these authors because of the n o n - a v a i l a b i l i t y of a 9 suitable etchant to reveal dislocations i n magnesium. In the course of the present investigation, an etchant has been developed to reveal the non basal dislocations i n Mg and i t s d i l u t e a l l o y s . The d e t a i l s of the method w i l l be described i n a l a t e r chapter. Using this method of etching i t was found that the depth of spark damage was limited to a surface layer 300M thick. I t i s important to note that the present etch p i t t i n g technique reveals the effect of spark erosion on the nonbasal edge dislocation density alone. The effect of spark erosion at and below a depth of 300y on the basal dislocations was checked as follows. Two t e n s i l e test specimens were taken from the same s l i c e . One was chemically polished to a depth of 300jj and the other to 500y . The i n i t i a l flow stresses and the i n i t i a l work hardening rates of the two specimens were found to be equal within experimental error, indicating that the spark damage on the basal planes was i n s i g n i f i c a n t below a depth of 300p . 1.2.6. Preparation of Single Crystal Specimens for Tensile Tests; A copper tool was used to spark erode test pieces whose f i n a l shape and dimensions after chemical polishing are shown i n fig=(1). The material used for obtaining the test pieces was i n the form of s l i c e s cut from the as grown c r y s t a l . After shaping the specimens were chemically polished i n 10-20% HNO^  to remove the damaged layer. The orientation of each specimen was checked by the back r e f l e c t i o n Laue method. No asterism was observed, indicating that the crystals were s t r a i n free. The orientation of the s l i p plane with respect to the t e n s i l e axis i s shown i n f i g . (1) for basal and prismatic s l i p respectively. 10 In the case of specimens oriented for basal s l i p the basal plane made an angle of 45 i 8° with the tensile axis whereas those oriented for prismatic s l i p had the basal plane p a r a l l e l to the tensi l e axis within 2 degrees. In the l a t t e r case one of the 1st.order prism planes of the form {1010} was kept at an angle close to 45° to the tensi l e axis, so as to have the maximum shear stress acting on th i s plane. The alignment of the basal plane with the wider face of the test piece f a c i l i t a t e d the i d e n t i f i c a t i o n of the prism s l i p markings on the narrow face. A l l specimens were given a 2 hr. anneal at 450°C i n order to remove any straining caused during handling. The specimens were chemically polished again i n order to remove the surface layer for easier examination of the deformation markings. 1.2.7. Testing Procedure: A l l specimens were deformed on a Floor Model Instron. The poiyerystalline specimens were tested at a s t r a i n rate of 2 x 10 sec ; -4 -1 whereas the single crystals were deformed at 1.66 x 10 sec except when otherwise stated. Testing media included Liquid Nitrogen 78°K Petroleum ether + Dry ice 195°K Petroleum ether cooled with l i q u i d nitrogen 133°K to 295°K A i r 295°K Silicone O i l 295°K to 513°K In each case the temperature was controlled to + 2°. For poiyer y s t a l l i n e specimens wedge grips were used which provided a r i g i d testing apparatus. Single crystals oriented for prismatic s l i p were much stronger than the poiyerystalline specimens. Hence the s p l i t jaw grips were used i n this case with increased precaution i n the alignment 1 1 Fig. 2. Gripping arrangement for basal s l i p specimens. 12 procedure. For sing l e . c r y s t a l specimens oriented for basal s l i p a separate set of grips w a s used as shown i n f i g . (2). In a l l cases cycle control was used i n order to prevent overloading of specimens during gripping. The data on mechanical properties of the polycrystals are expressed i n FPS units, while those for single crystals are given i n CGS units. This i s done i n order to f a c i l i t a t e comparison with previous work. 1.3. DEFORMATION CHARACTERISTICS OF POLYCRYSTALLINE AGGREGATES: 1.3.1. Nature of the Stress Strain Curves: The true stress vs true s t r a i n curves for pure magnesium and representative sets of Mg-Zn and Mg-Al alloys deformed at room temperature are shown i n f i g s . (3) and (A). I t i s readily observed that at low solute concentrations the st r e s s - s t r a i n curves for the alloys l i e above that for pure magnesium. At higher concentrations of solute, however, a knee appears and the stress s t r a i n curves, f a l l below those of the alloys containing lower amounts of solute. Similar results were obtained for Mg-Cd, Mg-Pb and Mg^In a l l o y s . These results are consistent with the observations of Yoshinaga and H o r i u c h i ^ ^ and of Hauser et a l ^ " ^ on Mg-Li al l o y s . The lowest concentration of lithium studied by Yoshinaga was 0.4 at%. The effect of temperature on the st r e s s - s t r a i n curves of magnesium i s shown i n f i g . (5), and for two of the Mg-Al alloys i n f i g s . (6) and (7). I t i s observed that i n the early stages of straining the work hardening curve i s parabolic at high effe c t i v e temperatures, however, the rate of i n i t i a l work hardening increases more rapidly with s t r a i n at lower temperatures. Reference to f i g s . (5), (6) and (7) indicates a large amount of s t r a i n beyond the point of maximum stress 40 Fig. 4/ Stress-strain curves for Mg-Al alloys tested at 295°K. 0 I _ i 1 1 1 I I I L I 0 4 8 12 16 20 24 28 32 36 40 True s t r a i n % Fig. 6. Stress-strain curves for Mg + 0.055 at. % AI a l l o y . 0 8 16 24 32 40 48 True s t r a i n % Fig. 7. Stress-strain curves.for Mg + 0.53 at. % A l a l l o y . 18 i n specimens tested above 373°K. The Mg-Cd alloys also exhibited a similar behaviour. This behaviour was observed not to be associated with necking. 1,3.2, The Y i e l d Stress: The d e f i n i t i o n of a y i e l d stress i n p o l y c r y s t a l l i n e magnesium i s d i f f i c u l t because of the gradual nature of the y i e l d process. Therefore,in the present work the stress at 0.2% offset s t r a i n w i l l be called the y i e l d stress Oyp' .1.3.2.1. The Temperature Dependence of Yield? The temperature dependence of the y i e l d stress for various alloys i s shown i n f i g s . (8) and ( 9 ) for both aluminum and cadmium solutes. The curves may be divided conveniently into three stages. At low temperatures there exists a region of moderate and linear decrease i n with increasing temperature (stage I ) , followed by stage I I , i n which the y i e l d stress decreases more rapidly with temperature. Stage I I ends at approximately 400°K [ f i g . (8)] and a thi r d l i n e a r stage follows where the rate of decrease of avp with temperature i s similar to that i n stage I. Y i e l d stress-temperature relationships have been obtained by Hauser, Landon and Dorn^*"^ , for p o l y c r y s t a l l i n e pure Mg of varying grain sizes; however, thei r study was not s u f f i c i e n t l y extensive as to establish the various stages shown i n f i g . (8). Temperature i n *K Fig. 8. Yield stress-temperature relationships for Mg-Al a l l o y s . 100 200 300 400 Temperature i n "K Fig. 9. Y i e l d stress-temperature relationships for Mg-Cd all o y s . 21 I t i s also interesting to note that the temperature at which t r a n s i t i o n from stage I to stage I I occurs ( T j i s a sensitive function of the solute concentration i n the low a l l o y s . However, beyond a certain minimum concentration of solute, which we w i l l designate C , the t r a n s i t i o n temperature increases only s l i g h t l y with c increasing solute content. This i s shown i n f i g . (10), where T i s plotted against the solute concentration for both aluminum and cadmium solutes. The numerical values of C are 0.17 and 0.054 at % for c Mg-Cd and Mg-Al alloys respectively. 1,3.2.2. The Concentration Dependence of Y i e l d : The y i e l d stress at room temperature i s plotted against solute concentration for both Zn and Cd solutes as shown i n f i g s . (11) and (12), The curves may be divided into three stages. Stage I at the lowest solute concentrations i s a region of rapid and li n e a r increase i n y i e l d stress with increasing solute. At a concentration C T the stage I ends and the stage I I strengthening begins, which i s also approximately linear but has a slope approximately an order of magnitude lower than that i n stage I. Stage I I ends with a gradual decrease i n q^ p with further increase i n solute concentration during stage I I I . This type of multistage solution strengthening has also been observed with AI, Pb, Cd and In solutes, although, the stage I I I region has not been observed i n a l l cases. ( 1 8 ) E a r l i e r work by Hardie and Parkins using hardness measurements had indicated the p o s s i b i l i t y of a high strengthening effect at low solute concentrations followed by a less rapid increase i n hardness at higher concentrations, i n alloys of Magnesium containing a variety of (14) solutes. Work by Hauser, Landon and Dorn , had indicated a three 22 23 Fig. 12. Y i e l d stress-concentration plot for Mg-Cd al l o y s . 24 stage strengthening effect i n Mg-Li a l l o y s , similar to that observed i n the present work. However, neither study was s u f f i c i e n t l y complete to provide the necessary information. 1.3.2.3. Solution Strengthening i n Stage I: The stage I regions for a l l the f i v e systems examined at room temperature are shown i n f i g . (13), and c l e a r l y reveal the r e l a t i v e strengthening effect of the various solutes i n the order-Zn, A I , Pb, da ' Cd, In. The solution strengthening rate (•—) i n stage I for various solutes are l i s t e d i n Table I along with other relevant data. The stage I slope has been plotted as a function of the size difference, Ar between magnesium and the various solutes as shown i n f i g . (14). I t i s seen that the strengthening effect of the solute increases as the difference i n size between the solvent and the solute becomes larger. In attempts to correlate the hardening effect to the size di f f e r e n c e ^ i t i s important to note that the concept of an atomic size i n an a l l o y i s problematic and hence care must be exercised while making quantitative interpretations involving the use of such a parameter. There are two methods i n use for the evaluation of the size m i s f i t parameter. The one involving the use of the change i n l a t t i c e parameter due to alloying i s the more commonly used method, which relates the strengthening effect of the solute as follows: dc ' K C a dc ; where a i s the l a t t i c e parameter, da/(j c, i t s v a r i a t i o n with solute con-centration, K and n are adjustable parameters. The changes i n the l a t t i c e parameters of magnesium due to alloying have been tabulated by B u s k ^ ^ , TABLE I Solution..har.deninR„-paramafeeES-for- polyerystals..tested at 295°K SOLUTE SOLUBILITY CRITICAL STAGE I SLOPE STAGE II SLOPE YIELD STRESS ELEMENT at % CONCENTRATION d(T 1 d(T " AT C T Crj, dc dc (J"T Max. at 25° C at % psi/at % solute psi/at % solute in psi Zn 4.0 0.5 - 0.01 345 x 103 9 x 103 11,950 Al 11.5 1.6 0.067 59.5 x l 0 3 0.9 x l 0 3 12,500 Pb 8.0 1.5 0.068 42.6 x l 0 3 0.6 x l 0 3 11,400 Cd Complete 12 0.16 22.7 x 103 0.254 x 103 12,360 In 19.0 13 0.25 11.0 x 103 0.65 x 103 11,250 F i g . 14. Stage I slope vs size m i s f i t parameter. Fig. 15. Stage I I slope vs size m i s f i t parameter. 28 (31) (32) (33) Hardie and Parkins , von Batchelder and Rauchle and King However, no single combination of n and K could give a satisfactory f i t for the present results. A l t e r n a t i v e l y ? t h e difference i n size between the pure metals can be used as a size m i s f i t parameter. The la s t mentioned approach has been taken i n the present work. Atomic (34) r a d i i tabulated by Van Vlack have been used here. I t i s important to note that the change i n the shear modulus of the material due to alloyi n g i s also a contributing factor i n solution strengthening. In the absence of experimental data r e l a t i n g to the v a r i a t i o n of the shear modulus with solute content, however, th i s factor has not been considered here. 1.3.2.4. Strengthening Above C-: The solution hardening rates i n stage I I , at room temperature are l i s t e d i n table I for a l l solutes examined. Fig. (15) shows the dc " dependence of (~J^) on the size m i s f i t parameter, which follows the same general trend as O^ -) against Ar. Although stage I I extends over a much wider range of compositions the hardening rate i s an order of magnitude lower compared to that i n stage I. Stage I I merged with stage I I I with a gradual decrease i n 0 Y p . Stage I I I was observed only i n Mg-Al and Mg-Zn al l o y s . 1.3.2.5. The Temperature Dependence of the Solution Strengthening  Parameters: The solution hardening curves for Mg-Al and Mg-Cd alloys deformed over a range of temperatures are shown i n figS.(16) and (17) respectively. The multistage nature of strengthening prevails at a l l temperatures. The extent of stage I increases with decreasing temperature 29 0 . 1 0 . 2 0 . 3 A t o m i c % A l 0 . 4 0 . 5 F i g . 1 6 . T h e e f f e c t o f t e m p e r a t u r e o n t h e s o l u t i o n h a r d e n i n g c u r v e f o r M g - A l a l l o y s . 30 r 1 2 3 4 5 A t o m i c % C d F i g . 1 7 . T h e e f f e c t o f t e m p e r a t u r e o n t h e s o l u t i o n h a r d e n i n g c u r v e f o r M g - C d a l l o y s . 31 below 250°K. Above this temperature, however, C_ remains independent of temperature. The temperature dependence of C_ for the Mg-Cd and Mg-Al alloys i s shown i n f i g s . (18) and (19) respectively. The v a r i a t i o n of the solution hardening rate with temperature i s shown i n f i g . (20) for the Mg-Cd a l l o y s . I t i s observed that between dc 78°K and 140°K (-j-^ ) ' i s affected very s l i g h t l y "by temperature, whereas a pronounced increase i s apparent between 140 and 260°K. attains a maximum i n the v i c i n i t y of room temperature and starts to decrease do " thereafter with increasing temperature. (^ jjr) » o n t n e o t n e r hand, decreases rapidly from 78°K, reaches a minimum around room temperature and increases thereafter with increasing temperature. Fig. (21) shows the va r i a t i o n of 0^ -) with temperature i n the Mg-Al alloys which i s do " similar to that i n the Mg-Cd al l o y s . However, (^ -) was found to be independent of temperature i n the Mg-Al a l l o y s . 1.3.3. The C r i t i c a l Transition Concentration: The temperature v a r i a t i o n of C^ , has already been described e a r l i e r . The numerical values of C_ for the f i v e a l l o y systems examined at room temperature are l i s t e d i n table I . The t r a n s i t i o n concentration could be related to the size m i s f i t as can be seen from the Ar-C^, plot i n f i g . (22). For a large value of Ar, C_ i s small. As the.size m i s f i t parameter Ar i s decreased, C_ increases, indicating a very large value as Ar approaches zero. This i s as expected, for the addition of a solute with Ar = 0 i . e . magnesium i t s e l f . From the C T - Ar relationship i n f i g . (22), the t r a n s i t i o n concentration for lithium as solute turns out to be s0.2 at.%. A 32 0 . 9 -0 . 7 " 0 . 5 0 . 3 -0 . 1 1 0 0 2 0 0 3 0 0 T e m p e r a t u r e ° K 4 0 0 F i g . 1 8 . C r i t i c a l t r a n s i t i o n c o n c e n t r a t i o n , C T f r o m s t a g e I t o s t a g e I I v s . t e m p e r a t u r e f o r C d s o l u t e . 0 . 1 3 0 . 1 1 -0 . 0 9 A 0 . 0 7 0 . 0 5 1 0 0 2 0 0 T e m p e r a t u r e . ° K 3 0 0 4 0 0 F i g . 1 9 . C^, v s t e m p e r a t u r e f o r A l s o l u t e . 33 1 0 0 2 0 0 3 0 0 4 0 0 T e m p e r a t u r e ° K F i g . 2 0 ( b ) . S t a g e I I s l o p e v s t e m p e r a t u r e f o r M g - C d a l l o y s . 34 Fig. 22. C r i t i c a l t r a n s i t i o n concentration, C^  from stage I to stage I I vs size m i s f i t parameter. 35 d i r e c t e x p e r i m e n t a l v e r i f i c a t i o n o f t h i s e s t i m a t e i s n o t p o s s i b l e f r o m t h e p r e s e n t r e s u l t s , s i n c e l i t h i u m h a s n o t b e e n i n c l u d e d i n t h e p r e s e n t s t u d y . E a r l i e r e x p e r i m e n t s o n t h e M g - L i a l l o y p o l y c r y s t a l s h a v e n e v e r b e e n c a r r i e d o u t a t s u f f i c i e n t l y l o w c o n c e n t r a t i o n s o f l i t h i u m a s t o v e r i f y t h e a b o v e c o n c l u s i o n e i t h e r . I n d i r e c t e v i d e n c e i n s u p p o r t o f t h e c o r r e c t n e s s o f t h e a b o v e e s t i m a t e , h o w e v e r , c o m e s f r o m t h e s i n g l e c r y s t a l d a t a o n M g - L i a l l o y s . T h i s w i l l b e d i s c u s s e d i n a l a t e r c h a p t e r . 1 . 3 . 4 . T h e M a x i m u m S t r e s s V a r i a t i o n w i t h T e m p e r a t u r e a n d A l l o y i n g : T h e t e m p e r a t u r e d e p e n d e n c e o f t h e f r a c t u r e mode i n p o l y c r y s t a l l i n e m a g n e s i u m i s o u t l i n e d i n c h a p t e r ( 1 . 4 . 2 . ) . I n t h e " b r i t t l e f r a c t u r e " r e g i o n t h e max imum s t r e s s i s e q u a l t o t h e f r a c t u r e s t r e s s , h o w e v e r , a t h i g h e r t e m p e r a t u r e s Mg d o e s n o t u n d e r g o f r a c t u r e a t t h e max imum s t r e s s . T h e r e f o r e , a t h i g h e r t e m p e r a t u r e s t h e t e r m m a x i m u m s t r e s s r e f e r s t o t h e s t r e s s a t t h e o n s e t o f n e g a t i v e w o r k h a r d e n i n g . T h e t e m p e r a t u r e d e p e n d e n c e o f t h e m a x i m u m s t r e s s f o r Mg a s w e l l a s a s e r i e s o f M g - A l a l l o y s i s s h o w n i n f i g . ( 2 3 ) . T h e t r a n s i t i o n t e m p e r a t u r e f r o m b r i t t l e t o d u c t i l e t r a n s i t i o n i n p u r e Mg h a v i n g a n a v e r a g e g r a i n s i z e o f 65 u i s 2 2 5 ° K i n t h e p r e s e n t w o r k . U n l i k e t h e t r a n s i t i o n i n o^p - t e m p e r a t u r e r e l a t i o n s h i p t h e t r a n s i t i o n t e m p e r a t u r e i n t h e max imum s t r e s s t e m p e r a t u r e p l o t was f o u n d t o b e u n a f f e c t e d b y s o l u t e . T h e b r i t t l e f r a c t u r e s t r e s s o f a l l o y s c o n t a i n i n g s o l u t e l e s s t h a n C j was f o u n d t o b e l o w e r . t h a n t h a t o f m a g n e s i u m . B e y o n d C^,, t h e b r i t t l e f r a c t u r e s t r e s s i n c r e a s e d w i t h i n c r e a s i n g s o l u t e c o n c e n t r a t i o n . T h e v a r i a t i o n o f max imum s t r e s s w i t h s o l u t e c o n c e n t r a t i o n i s s h o w n i n f i g . ( 2 4 ) f o r t h e M g - C d a l l o y s d e f o r m e d a t r o o m t e m p e r a t u r e . 37 Fig. 24. Fracture stress vs. composition for Mg-Cd alloys tested at 295°K. 38 T h e m i n i m u m i n f i g . ( 2 4 ) w h i c h i s a t a c o n c e n t r a t i o n o f s o l u t e s l i g h t l y l e s s t h a n C T d i s a p p e a r s a t h i g h e r t e s t i n g t e m p e r a t u r e s ( n o t s h o w n h e r e ) . 1 . 3 . 5 . F l o w S t r e s s i n R e l a t i o n t o T e m p e r a t u r e a n d A l l o y i n g : F l o w s t r e s s t e m p e r a t u r e r e l a t i o n s h i p s f o r p u r e Mg a n d a r e p r e s e n t a t i v e s e t o f M g - A l a l l o y s a r e s h o w n i n f i g s . ( 2 5 , 26 a n d 2 7 ) . I t i s o b s e r v e d t h a t t h e f l o w s t r e s s - t e m p e r a t u r e c u r v e s f o r a n y s p e c i f i e d s t r a i n a r e o f t h e same s h a p e a s t h e y i e l d s t r e s s - t e m p e r a t u r e c u r v e s , t h e r e f o r e , t h e t e r m i n o l o g y u s e d i n d e f i n i n g t h e t h r e e s t a g e s i n t h e ° Y P ~ ^ c u r v e s w i l l a l s o b e a p p l i e d t o t h e f l o w s t r e s s c u r v e s . A n o t i c e a b l e f e a t u r e o f f i g s . ( 2 5 , 26 a n d 2 7 ) i s t h e e f f e c t o f i n c r e a s i n g s t r a i n o n t h e t r a n s i t i o n t e m p e r a t u r e f r o m s t a g e I t o s t a g e I I . I n t h e c a s e o f p u r e m a g n e s i u m a n d a l l o y s c o n t a i n i n g s o l u t e s i n l a r g e e x c e s s o f C ^ , t h e t r a n s i t i o n t e m p e r a t u r e T was o b s e r v e d t o b e a f f e c t e d o n l y s l i g h t l y w i t h i n c r e a s i n g s t r a i n , h o w e v e r , a n a l l o y c o n t a i n i n g s o l u t e c l o s e t o C T ( f i g . ( 2 6 ) ) e x h i b i t e d a p r o n o u n c e d d e c r e a s e i n T w i t h i n c r e a s i n g s t r a i n a s s h o w n i n f i g . ( 2 8 ) . T h e c o n c e n t r a t i o n d e p e n d e n c e o f t h e f l o w s t r e s s i n M g - Z n a l l o y s t e s t e d a t r o o m t e m p e r a t u r e i s s h o w n i n f i g . ( 2 9 ) . A t a n y f i x e d s t r a i n t h e f l o w s t r e s s i n c r e a s e s r a p i d l y w i t h a l l o y i n g u p t o a c o n c e n t r a t i o n o f s o l u t e e q u a l t o C^,. B e y o n d C T a c o n t i n u o u s d e c r e a s e i s o b s e r v e d w i t h i n c r e a s i n g s o l u t e c o n t e n t . T h e d e c r e a s e i n f l o w s t r e s s a t 0 . 2 % s t r a i n w a s o b s e r v e d t o c o m m e n c e a t 0 . 3 a t % Z n , h o w e v e r , a t h i g h e r s t r a i n s a l l a l l o y s c o n t a i n i n g s o l u t e i n e x c e s s o f h a d f l o w s t r e s s e s l o w e r t h a n t h a t a t C^,. S i m i l a r r e s u l t s h a v e b e e n o b t a i n e d f o r t h e M g - A l a l l o y s t e s t e d b e t w e e n 7 8 ° K a n d 4 2 3 ° K . Fig. 25.. Flow stress vs.- temperature for 65y Mg. 28 100 200 300 400 Temperature °K Fi g . 27. Flow stress vs. temperature for Mg + 0.055% AI a l l o y . 500 Fig. 28. Transition temperature from stage I to stage. I I i n the flow stress-temperature curves of Mg + 0.055 at.% A l a l l o y vs. s t r a i n . Fig* 29. Flow stress vs. composition for Mg-Zn al l o y . 44 1 . 3 . 6 . D u c t i l i t y : . P e r c e n t . e l o n g a t i o n o r t r u e s t r a i n b a s e d o n t h e u n i f o r m e x t e n s i o n o f t h e g a u g e - l e n g t h was u s e d a s a m e a s u r e o f d u c t i l i t y , a s i t was n o t p r a c t i c a l t o o b t a i n r e d u c t i o n i n a r e a v a l u e s b e c a u s e o f t h e s p e c i m e n d i m e n s i o n s . 1 . 3 . 6 . 1 . A l l o y i n g E f f e c t : A p l o t o f d u a t i l i t y a g a i n s t c o n c e n t r a t i o n f o r e a c h s o l u t e e x a m i n e d a t r o o m t e m p e r a t u r e i s s h o w n i n f i g . ( 3 0 ) . I t i s r e a d i l y o b s e r v e d t h a t t h e s t r a i n t o f r a c t u r e d e c r e a s e s u n t i l a c o n c e n t r a t i o n c l o s e t o i s r e a c h e d , a n d t h e n i n c r e a s e s w i t h f u r t h e r i n c r e a s e i n t h e a l l o y c o n t e n t . I t i s i n t e r e s t i n g t o n o t e t h a t t h e s o l u t e s w e r e e f f e c t i v e i n i n c r e a s i n g t h e d u c t i l i t y i n t h e o r d e r Z n , A l , C d a n d I n p a r a l l e l i n g t h e i r r e l a t i v e s t r e n g t h e n i n g e f f e c t s . 1 . 3 . 6 . 2 . T h e T e m p e r a t u r e D e p e n d e n c e o f D u c t i l i t y : B e c a u s e o f t h e l a r g e a m o u n t o f s t r a i n a s s o c i a t e d w i t h t h e n e g a t i v e w o r k h a r d e n i n g a t t e m p e r a t u r e s a b o v e 3 7 3 ° K , i t w a s n e c e s s a r y t o i n d i c a t e t h e s t r a i n t o max imum s t r e s s , i n a d d i t i o n t o t h e t r u e s t r a i n t o f r a c t u r e . T h e e f f e c t o f t e m p e r a t u r e o n t h e s t r a i n t o f r a c t u r e f o r m a g n e s i u m a n d a s e r i e s o f b i n a r y M g - A l a l l o y s i s s h o w n i n f i g . ( 3 1 ) . U p t o a p p r o x i m a t e l y 3 0 0 ° K , t h e d u c t i l i t y o f m a g n e s i u m i s a f f e c t e d v e r y s l i g h t l y w i t h c h a n g e s i n t e m p e r a t u r e a n d a l l o y i n g . A l s o i r r e s p e c t i v e o f t h e t e s t i n g t e m p e r a t u r e , a l l o y i n g u p t o t h e t r a n s i t i o n c o n c e n t r a t i o n h a s l i t t l e e f f e c t o n t h e d u c t i l i t y o f m a g n e s i u m , a l t h o u g h ' ° Y J P l n c ^ e a s e s d r a m a t i c a l l y i n t h e s ame r a n g e o f c o n c e n t r a t i o n s . B e t w e e n 3 0 0 ° K a n d 4 0 0 ° K , c < 7 70 100 200 300 400 500 Temperature °K Fi g . 32. True s t r a i n to fracture vs. temperature for Mg-Al alloys. 47 however, alloying additions i n excess of C T have a profound effect i n increasing the d u c t i l i t y . Magnesium and alloys containing solute less than C T show a progressive increase i n d u c t i l i t y with temperature i n th i s i n t e r v a l . The higher alloys show maxima and minima i n . d u c t i l i t y i n the temperature i n t e r v a l between 400° and 500°K. The shape and position of the maxima show a systematic v a r i a t i o n with solute concentration. In constant s t r a i n rate test s , the d u c t i l i t y peak becomes broader and s h i f t s to lower temperatures with increasing aluminum addition. The temperature dependence of the true s t r a i n to maximum stress and that associated with the negative work hardening are shown i n f i g s . (32) and (33) respectively. The s t r a i n associated with the negative work-hardening i s very small below room temperature. Above room temperature, however, the s t r a i n to maximum stress and that associated with the negative work hardening follow a trend similar to the temperature va r i a t i o n of the s t r a i n to fracture, with the exception that the peaks do not broaden. The maxima i n the case of s t r a i n to maximum stress were observed at about 4503K, whereas those i n the case of the s t r a i n associated with the negative work hardening are i n the v i c i n i t y of 420°K. D u c t i l i t y maxima similar to those i n the present (35) work have been observed e a r l i e r by Greenwood et a l i n Mg-0,2% Pb al l o y and i n Mg-0.8% A l a l l o y by S'tacey^ 3 6\ True s t r a i n to maximum stress p-O Q CO ho < H i-l • C CD rt n> CO a rt • d i-t n> 03 ri H» fu 3 rt C rt i-t O m 3 Hi fa o i-t H-3 S C O Q 3 > CD rt i-i &1 CO 1—1 CO M CO O •<! CO ho o o o ho o LO O e % •P» o C n o I Co rt C i-l ft) CO o o -p-o o Ul o o 0 • o • •0 o O o O s • • • • Ul hO o U) . cn cn > > > > O Q CO CO 2 S H O Q O i-( I M C > 7? CD M EJ" CO » (B rt H 1 M i-t H U B O (D H -^ 3 3 H -3 (U O Q CO CO < CO 03 o o r t r t n> m 3 a. 1 3 n> i-< PJ rt 3* C * 3 CD n> 0 Q Hi 0) O r t < CD (e f.-e t) % Co O o -p-o o H CD 3 1 3 CD H CO c n CD C n O o o CJN O O O -n 8* 49 1.4. DISCUSSIONS In order to understand the mechanism of solution strengthening i n poiyerystalline aggregates, i t i s essential to have a knowledge of the crystallographic deformation modes available, and their effect on the macroscopic.flow and fracture characteristics of the material. Keeping this i n mind i t appears l o g i c a l to give a brief review of the e a r l i e r work on the deformation and fracture of magnesium followed by a discussion of the effects of solute on the various flow parameters. 1.4.1. Deformation Modes i n Magnesium: (19) In 1928 von Mises f i r s t pointed out that a necessary condition for a polycrystal to be duc t i l e when i t deforms by crystallographic s l i p i n i t s grains i s that the number of independent s l i p systems available be f i v e . The t o t a l number of s l i p systems i s given by the cr y s t a l structure. Usually there i s one s l i p mode and a number of s l i p systems for this mode as determined by the point group symmetry. Groves and K e l l y a n d Kocks^"^ have examined the number of independent s l i p systems i n various c r y s t a l structures, von Mises' finding has been used i n the studies of face centred cubic metals i n finding the number of (37) s l i p systems found to operate near the grain boundary and also i n theoretical attempts to deduce the poiye r y s t a l l i n e stress s t r a i n curve (22) from that of single c r y s t a l . In many systems, however, other mechanisms such as twinning, grain boundary shear and c e l l structure formation may occur thereby reducing the number of independent s l i p systems required. I t i s helpful to remember that the von Mises c r i t e r i o n i s taken to be a necessary condition for any polycrystal d u c t i l i t y whatsoever, not as one that determined different degrees of d u c t i l i t y . 50 1.4.LI. Crystallographic S l i p : Magnesium undergoes s l i p predominantly on the basal system {0001} <1120 > . The shortest s l i p vector which preserves the ABAB basal or close packed plane stacking i s of magnitude a and dir e c t i o n < 1120> . The t o t a l s t r a i n energy i s lowered by the dissociation i n the basal plane of the disloc a t i o n into two p a r t i a l s having the Burgers vector -f- <1010> A stacking f a u l t exists i n a ribbon between the two p a r t i a l s . Although no direct way of measuring the fa u l t energy i s available for magnesium, 2 (39) Seeger estimates i t to be i n the range 200 - 400 ergs/cm . This leads to a predicted separation of the p a r t i a l s at room temperature of the order of an interatomic distance. S l i p vectors of the form other than <1120> have never been observed i n magnesium. Non-basal s l i p vectors of the form <1123> have been observed i n addition to the. — (40 41) basal <1120> vector i n other hexagonal close packed metals l i k e Cd ' (42) (43) Zn and Be , The temperature dependence of the CRSS for basal s l i p has been studied by Sheely and Nash^"^ , Conrad and Robertson^^ and (3) by Basinski Small amounts of non-basal s l i p are seen after room temperature (14) deformation of pure Mg . I t has been a popular b e l i e f that i f and when non-basal s l i p does occur i t w i l l do so much more readily i n systems which have c/a r a t i o equal to or less than.the ideal value of 1.633. This (44) argument follows from r e l a t i v e close packing considerations. Seeger has suggested that i n addition to the c/a r a t i o , the stacking f a u l t energy w i l l have a strong influence on the deformation modes in.as much (45) as i t controls the cross s l i p process. Stoloff and Davies using hexagonal close packed Zn-Ag alloys of s l i g h t l y varying a x i a l r a t i o s have 51 shown that the e/a r a t i o i s not the only c r i t e r i o n for non-basal s l i p . Zirconium and beryllium serve as good examples to show that the occurrence of non basal s l i p i s not determined e n t i r e l y by c/a r a t i o and the stacking f a u l t energy of the material. Although both these metals have approximately the same c/a r a t i o (less than ideal) and have high (47) stacking fa u l t energies, whereas basal s l i p predominates i n beryllium Zr deforms mainly by prismatic {1010}<1120> s l i p ^ \ The e a r l i e s t observations of non-basal s l i p i n magnesium were made by Schmid^^ who concluded that the {101l}<1120> pyramidal system became operative at 225°C and above. Later work on the compression of (49) thin single c r y s t a l wafers by Bakarian and Mathewson confirmed the significance of this temperature and contributed the f i r s t extensive evidence on the waviness of elevated temperature s l i p l ines on this system. Burke and Hibbard^"^ observed only basal s l i p i n h i g h purity single crystals of magnesium at room temperature except i n the region close to the grip where {1011}<1120> system was operative due to the grip constraints. Thus, up u n t i l 1955 there was an acceptance of 225GC as the temperature below which no s l i p system except {0001}<1120> would operate i n Mg without the imposition of unusual stress systems. In 1955, Dorn et a l ^ " ^ began to examine rather c r i t i c a l l y the deformation i n tension of poiyerystalline pure magnesium. Small amounts of non-basal s l i p were seen at room temperature and the temperature dependence of non-basal s l i p was found to be.not as simple (12) — — as thought before. Hauser et a l observed prism {1010}<1120> basal {0001} <1120> cross s l i p between 78°K and 298°K at t r i p l e points which are susceptible s i t e s for stress concentration. 52 Reed-H'Ul and Robertson^"^ studied the te n s i l e deformation of magnesium single crystals with the stress axis close enough to the basal plane so that basal s l i p and mechanical twinning on {1012> were suppressed. This orientation i s of considerable p r a c t i c a l significance i n the tensi l e deformation of po l y c r y s t a l l i n e sheet and extruded metal because the texture of these materials i s such that the basal plane tends to a l i g n i t s e l f with the dir e c t i o n of r o l l i n g or extrusion. Reed-Hill found that {loTo}<1120> s l i p occurred at 93°K and 298°K. At low temperatures the s l i p lines had a minimum spacing of less than 5 x 10 ^  cm. Fine prism s l i p l ines were seen to be cross slipped by basal s l i p at both these temperatures. The va r i a t i o n of the c r i t i c a l resolved shear stress for prismatic s l i p over the temperatures ranging between 78°K and 800?K (9) has been investigated by Ward-Flynn, Mote and Dorn and by Horiuchi (52) and Yoshinaga . These results are supplementary to the data obtained by Reed-Hill and are i n excellent agreement with those of Hauser et a l . F i r s t order pyramidal s l i p {1011}<1120> has been observed i n (51) magnesium single crystals deformed at elevated temperatures. Reed-Hill found that with the geometry of his crystals the s l i p l i n e s were so irregular and diffuse that direct trace analysis of thei r system was not possible. However, a careful study of the asterism i n Laue X-ray photograms made from the deformed crystals showed them to be interpretable as the result of predominantly {1011}<1120> pyramidal s l i p . Large discrete {1011} pyramidal s l i p bands were also observed on specimens strained i n tension at 298°K with the te n s i l e axis i n the basal plane and approximately 17° from a {1010} di r e c t i o n . However,the high loads to fracture, the limited d u c t i l i t y at 298°K and the apparent r a r i t y of 53 {1011} s l i p bands a l l point to the fact that pyramidal s l i p i s not an re (48) (53) important mode of p l a s t i c deformation at room temperature . There appears to be no c o n f l i c t between these results and those of Schmid and B a k a r i a n ^ ^ except that the significance of the 225°G i s l o s t . Chaudhri et a l ^ ^ ' " ' " ^ have investigated the high temperature deformation of coarse grained po l y c r y s t a l l i n e magnesium i n t e n s i l e creep at 533°K. From a careful trace analysis coupled with the waviness of the bands of s l i p l ines they concluded that although <1120> dir e c t i o n was invariant, s l i p occurred with microscopic alternation on prism and pyramidal planes. Second order pyramidal {1122} s l i p bands have also, been (53) observed by Reed-Hill on single c r y s t a l specimens strained i n tension at 83°K with the t e n s i l e axis i n the basal plane and approximately 2° from a <1010> direct i o n . The asterism found i n a Laue back r e f l e c t i o n photograph and microscopic observation of s l i p l i n e s indicated that the s l i p direction i s probably <1010> . Although Cd, Zn and Be show {1122} <1123> pyramidal s l i p bands, non basal s l i p vectors have never, been observed i n magnesium. In hexagonal metals the operation of the second order {1122} <1123 >pyramidal s l i p i s a necessary as well as s u f f i c i e n t condition for the a v a i l a b i l i t y of f i v e independent s l i p systems. As reviewed (56^ by Dorn and M i t c h e l l this system i s the only one which can promote extensive deformation p a r a l l e l to the c axis. Table I I shows the number of independent systems for each of the prominent s l i p systems. 54 Table I I Sl i p Systems i n Hexagonal Metals (After D o r n ( 5 6 ) ) No. Sli p Systems Burgers Vector Number of Ir»depanden Systems 1 {0001} <1120> a 2 2 UOlo} <1120> a 2 3 {lOll} <1120> a 4 4 {1122} <1123> c + a 5 5 1 + 2 + 3 a 4 No combination of {0001} <1120>, {1010}<1120> and {101l}<1120> can provide the f i v e independent systems required. Since the second order pyramidal s l i p of the type {1122}<1123> i s absent i n magnesium, p l a s t i c deformation of the poiyerystalline aggregate i s not possible, i n p r i n c i p l e , from s l i p alone. However, considerable p l a s t i c i t y i s observed. This i s due to c e l l formation, twinning, compression banding and grain boundary deformation, which supplement s l i p i n order to s a t i s f y von Mises c r i t e r i o n . 1,4.1.2. P l a s t i c Deformation by Twinning Twinning systems are often not included as possible deformation modes for the purpose of sa t i s f y i n g compatibility conditions, because of the limited amount of deformation achieved even when the entire volume has been twinned. Also twinning can.accommodate either c-extension or c- contraction but not both. However, the profusion of two or more twinning systems, some leading to an extension and the others leading 55 to a contraction i n the c d i r e c t i o n , can assume the role of independent deformation modes. For instance R e ^ ^ , T i ^ ^ and Z r ^ ^ ' ~ ^ exhibit essentially unlimited d u c t i l i t y although (c + a) s l i p has never been i d e n t i f i e d i n any of them. Kocks and Westlake^*^ have attributed the large d u c t i l i t y i n these materials to the profusion of {112l}<ll26>and {1122} <L123> twinning. The former leads to an extension and the l a t t e r to a contraction i n the c-direction. The twinning system most common to magnesium i s the {1012} <1011:' type. {1012} twinning results i n an extension i n the c-direction and i s favoured by compression p a r a l l e l to the basal plane and tension perpendicular to i t . The complete formation of a twin under conditions where the parent c r y s t a l has no res t r a i n t to the resulting shear i s a simple case, not usually met i n practice. In general accommodation to the shear.is accomplished by a phenomenon known as accommodation kinking. The plane of accommodation i s usually called a bend or a kink plane. Accommodation may be expected on both {1010} and <1120> planes. Non-crystallographic (11) (12) boundary formation has also been observed i n magnesium. Dorn ' found that i n many cases they crossed grain boundaries. He postulated that these boundaries formed because of the bending of the l a t t i c e associated with the non-homogenous deformation of the underlying grains. The temperature dependence of {1012} twinning has never been studied thoroughly under conditions where the variables of s t r a i n rate and orientation were fixed . Twinning i n p o l y c r y s t a l l i n e magnesium becomes less important at elevated temperature, but whether this i s i n t r i n s i c to the twinning mechanism or not cannot be said, since the c r i t i c a l stress 56 for nonbasal s l i p and grain boundary deformation decrease rapidly with increasing temperature. From the studies of deformed poiyerystalline specimens Roberts reported the existence of a twinned structure apparently on the habit of {3034} . Couling and Roberts i d e n t i f i e d the habit as [3034}. This was confirmed on magnesium single crystals by Reed-Hill The twins are uniquely narrow and they exist i n bands forming interesting arrays at low s t r a i n l e v e l s . These twins are formed when tension i s applied perpendicular to the c axis. The r o l l i n g texture of magnesium e f f e c t i v e l y places the grains i n an orientation unfavourable to both basal s l i p and {1012} twinning. However {3034} twins are formed when tension i s applied i n the d i r e c t i o n of r o l l i n g . Also, since they are responsible for b r i t t l e fracture of Mg they have been studied i n somewhat greater d e t a i l than the {1012} twins. The {3034} lamellae are formed as a re s u l t of a primary twinning on the {1011} plane followed by a second order {1012} twin formed i n the primary twin. Using electron microscope replicas Hartt ( 64^ and Reed-Hill have shown that the i r r a t i o n a l habit of these twins i s probably the direct result of a need to accommodate the second order twinning shear. A model has been presented which explains the accommodation resulting i n both an external shear i n the matrix and an in t e r n a l shear i n the lamella. Similar twins have been observed by Wonsiewicz and (66) Backofen^"^ i n the complex straining of magnesium single crystals at higher temperatures and i n some alloys of Mg by Hosford, J r . et a l at room temperature. The detailed understanding of these twins explains the compression banding and the r o l l i n g texture i n magnesiun/^^. However, the established twin systems i n magnesium cannot assume the role of an 57 independent deformation mode mainly because they are formed at places of special stress concentrations and are incapable of accommodating both expansion and contraction i n the c-direction. Twinning on .{1013},{1014}. , { 1124} , {10l5 } ( 6 3 ) as well ("62} as on {1121 } have been reported. However, thei r existence i s not well established. 1.4.1,3, Grain Boundary Deformation: The operation of grain boundary shearing under tension at room temperature was proved i n the p o l y c r y s t a l l i n e aggregate by Hauser et ( 1 1 ) a l . The importance of the process i n the creep of p o l y c r y s t a l l i n e magnesium at elevated temperatures has been demonstrated by Couling and Roberts , They found that the high temperature grain boundary deformation i n p o l y c r y s t a l l i n e magnesium i s a two stage process involving alternate boundary shearing and migration. Increasing test temperatures and decreasing s t r a i n rates favour larger contributions to the over a l l s t r a i n from boundary deformation, In the l i m i t i n g case i t appears that a l l the deformation would be localized at the boundaries. A mechanism which explains the observations i s the alternation of anelastic boundary shears with the capture of these shears when the boundaries migrate to new positions, The number of cycles necessary to produce the measured shear has been calculated and found to agree q u a l i t a t i v e l y with metallographic observations. 58 1.4.1.4. C e l l Formation: C e l l formation results from a stress-accelerated polygonizatipn (54 55) (54) process caused by non»homogenous d i s t o r t i o n ' . Chaudhri et a l have noticed i n the case of Mg (especially coarse grained aggregates) that the breakdown of the metal into c e l l s i s marked i n the grain boundary region. Polygonization can be so pronounced i n the grain boundaries that i t actually leads to r e c r y s t a l l i z a t i o n . Due to the lack of a v a i l a b i l i t y of f i v e independent modes, c e l l formation plays an important role i n the deformation of magnesium. For example, the t r a n s i t i o n i n fracture stress and d u c t i l i t y of magnesium i n the v i c i n i t y of room temperature i s thought to be associated with the onset of c e l l formation (69) (64) and r e c r y s t a l l i z a t i o n . Recently, however, Reed-Hill has associated this t r a n s i t i o n with the operation of {1011}- {1012} double twinning. Suiter and Wood^^ have suggested a mechanism for the formation of c e l l structures. This depends on the r e l a t i v e movements of the grain boundaries causing rumpling and twisting of the structure. Dislocation movements reli e v e the s t r a i n associated with the deformation and lead to the formation of sub-boundaries which develop into the observed c e l l structures. (35) C e l l formation has also been observed i n Mg-rPb alloys and (36) Mg-Al alloys d e t a i l s of which w i l l be discussed l a t e r . 59 1.4.2, Fracture of Magnesium Polyerystals: Studies on the flow and fracture stress characteristics of high purity magnesium carried out by Hauser et a l ^ * ^ and by Toaz and R i p l i n g ^ ^ have revealed that fine grained Mg fractures at low but rather constant stress below about 250°K. Above this range, the fracture stress decreases rapidly with increasing temperature and the d u c t i l i t y increases markedly. The temperature at which this change occurs i s regarded by Hauser et a l as the t r a n s i t i o n point from b r i t t l e to d u c t i l e fracture, A true t r a n s i t i o n temperature refers to the situation where a change from b r i t t l e to du c t i l e behaviour occurs with no thermal i n s t a b i l i t y of the microstructure and where negl i g i b l e deformation occurs prior to fracture i n the b r i t t l e range. The use of the term b r i t t l e fracture range i s not s t r i c t l y j u s t i f i e d i n the case of magnesium, since a f a i r amount of p l a s t i c deformation precedes fracture. However, the characteristics of the low temperature fracture of magnesium are i n many ways similar to those of b r i t t l e fracture. For example, the fracture stress increases l i n e a r l y with the reciprocal of the square root of grain diameter i n accordance with the Hall-Petch r e l a t i o n ^ " ^ , the t r a n s i t i o n temperature increases with increasing grain size and the low temperature fracture stress as well as fracture s t r a i n are found to be r e l a t i v e l y independent of the s t r a i n rate. On the basis of some metallographic studies Toaz and Ripling conclude that the discontinuity ( i n the fracture stress-temperature relationship) i n the case of pure magnesium results from the entrance of r e c r y s t a l l i z a t i o n during def o r m a t i o n ^ ^ . I t has been observed by (72) Risebrough that the negative work hardening i n the stress s t r a i n curves 60 of poiyerystalline Zn and Cd i s associated with r e c r y s t a l l i z a t i o n . Metallographic examination of the test samples before and after the onset of the negative work hardening i n the present work suggests r e c r y s t a l l i z a t i o n to be responsible for the work-softening of magnesium. The s t r a i n associated with the negative work hardening, which was taken as the s t r a i n to fracture less the l i m i t i n g s t r a i n to maximum stress, was.found to decrease with decreasing temperature, becoming vanishingly low at about 250°K. This observation tends to support the conclusion of Toaz and Ripling on the nature of the t r a n s i t i o n temperature. The fracture of Mg and i t s alloys near and below room temperature results primarily from the j o i n i n g of the intragranular cracks by a moderate amount of intergranular cracking. The twinned structures which have been i d e n t i f i e d as t 3034 } twins and the higher order planes such as {1014 },{1015} and {1124} are the crack nucleation s i t e s i n intragranular fracture. At elevated temperatures, however, the grain boundaries play an important role i n the fracture of magnesium and i t s a l l o y s . This w i l l be considered at a l a t e r stage. 1.4.3. Solution Hardening: E a r l i e r assessments of solution strengthening i n magnesium alloys have considered the possible effects of various solute species on such quantities as the cohesive strength, surface tension and flow (14) stress . However, such considerations are rarely h e l p f u l i n defining the detailed processes involved. Accordingly, i t i s more profi t a b l e to attempt a direct assessment of the deformation modes and the way i n which solute additions affect them. 61 As outlined i n the la s t section the deformation of p o l y c r y s t a l l i n e Mg occurs predominantly by s l i p on the basal {0001} <1120> system with additional s l i p on the prismatic {1010}<1120> system accompanied by a double twinning i n the sequence {1011} - { 1012} and some grain boundary s l i d i n g . Prism s l i p occurs predominantly at grain corners which are si t e s of high stress concentration, and thus acts as a stress r e l i e f mechanism contributing to increased d u c t i l i t y . Furthermore 5 the y i e l d strength of po l y c r y s t a l l i n e magnesium w i l l be governed by the CRSS for both basal and prismatic s l i p . The most comprehensive data available are based on studies on the Mg-Li system by Dorn and his (1^ 15 73) colleagues ' ' . These experiments have shown that addition of lith i u m to magnesium causes an increase i n the CRSS for s l i p on the basal system, but a decrease i n the CRSS for s l i p on the prism system. They have also shown that, for polyerystals, the y i e l d stress of these a l l o y s , varies with concentration of lithium roughly i n a manner similar to that found i n the present work, and that a similar v a r i a t i o n of d u c t i l i t y with con-centration was also found. Yoshinaga and H o r i u c h i ^ ^ have also investigated the deformation behaviour of Mg-Li polyerystals as a function of lithium concentration. Although both groups of workers found similar v a r i a t i o n i n y i e l d stress with solute concentration, the d u c t i l i t y minimum was observed at 0.4 at.% L i by Horiuchi ;and Yoshinaga while Dorn et a l found the minimum at 4 at. % L i . The interpretation offered by Dorn et a l i s that the y i e l d stress of the po l y c r y s t a l l i n e aggregate i s the combined res u l t of the CRSS for both basal and prismatic s l i p . The i n i t i a l rapid increase was attributed to the dominant solution hardening for basal s l i p , while the 62 subsequent reduced rate of hardening and increased d u c t i l i t y was attributed to the increasing importance of prismatic s l i p as an available deformation mode. Twinning and boundary shear were considered to be of minor importance. P a r a l l e l i n g these conclusions i s the observation that increasing lithium content results i n a reduction of the c/a r a t i o , and hence the decrease of CRSS for prismatic s l i p i s the expected result of the decreasing (73) Peierls stress. The work of Ahmadieh and Dorn has shown Peie r l s stress to be the rate co n t r o l l i n g mechanism for prismatic s l i p at low temperatures. These ideas cannot account adequately for the present r e s u l t s . Consider, f i r s t , the dependence of CRSS for basal s l i p on concentration. (74) Levine, Sheely and Nash have measured the CRSS for basal s l i p i n single crystals of d i l u t e alloys of magnesium with Zn, A l , Tl, Cd and In and their results are reproduced i n f i g . (34). These authors conclude that the solute strengthening effect i s linear with concentration for Zn, A l and T l and that for Cd and In there i s no hardening at low concentrations followed by a linear hardening at higher concentrations. A closer examination of these data indicates that i f l i n e s of t r u l y best f i t are drawn through the points, then i n no case do they extrapolate back to the CRSS for pure magnesium. Such li n e s are drawn i n f i g . (35), and show that for a l l solute species, the CRSS for basal s l i p i s unaffected by solute addition up to a concentration, approximately the same as C T i n the present work. This conclusion has been confirmed i n Mg-Zn a l l o y s , d e t a i l s of which are discussed i n a l a t e r chapter. Clearly, then the present observation of the very rapid increase i n a v with solute addition from pure Mg to C T cannot be Fig. 35. Revised curves drawn for data from ref. 74. 64 attributed to a change i n CRSS for basal s l i p . I t would appear l o g i c a l therefore, to conclude that this increase i n must be due to an increase i n the CRSS for prismatic s l i p . So f a r , there are no reports i n the l i t e r a t u r e on the : v a r i a t i o n of CRSS for prismatic s l i p at very low solute concentrations. Accordingly, such measurements have been carried out i n the present work and the results for Mg-Zn alloys are shown i n f i g . (93). These confirm the conclusion that the CRSS for prismatic s l i p does, i n f a c t , increase with increasing solute concentration up to C^. Details of this work are discussed i n a la t e r chapter. 1.4.3.1. The Variation of and D u c t i l i t y with Solute Concentration: i The v a r i a t i o n of both and d u c t i l i t y with solute concentration are now more readily understandable. The y i e l d stress i n p o l y c r y s t a l l i n e material i s determined by the stress necessary to activate both basal and prismatic s l i p . The i n i t i a l high rate of solution hardening i s due to the rapid increase i n the CRSS for prismatic slip,-while the CRSS for basal s l i p remains unchanged upto C_ (at and above room temperature). Due to s l i p on the basal system, stress concentrations w i l l arise which may be relieved either by crack formation or by p l a s t i c flow on the prismatic s l i p system. As the CRSS for prismatic s l i p increases from pure magnesium to C-, less p l a s t i c deformation i s possible prior to -fracture and the d u c t i l i t y i s reduced. da ' 1.4.3.2. The Solution Hardening Rate 0^ -) : The c r i t i c a l resolved shear stress for basal s l i p i n magnesium i s approximately two orders of magnitude lower than that for 65 prismatic s l i p . For example at room temperature the numerical values 2 2 for the two systems are 51 gm/mm and 4600 gm/mm respectively. Thus i t may seem rather unlikely that prismatic s l i p should operate at the y i e l d stress of the p o l y c r y s t a l l i n e aggregate. However, i n view of the strong r o l l i n g and r e e r y s t a l l i z a t i o n texture of magnesium and i t s alloys the operation of prismatic s l i p i s not at a l l surprising. These materials have a texture such that the basal plane remains inclined to the r o l l i n g or extrusion d i r e c t i o n within 10°. The Schmid factor for basal s l i p i n this orientation i s about 4-5 times lower than that for prism s l i p plane. Thus the deformation behaviour of the p o l y c r y s t a l l i n e aggregate should show a closer s i m i l a r i t y to that of single crystals oriented for Prismatic s l i p rather than for basal s l i p . Q u a l i t a t i v e l y , the.stress-s t r a i n curves of the polyerystals do show a closer resemblance to the st r e s s - s t r a i n curves of the single crystals oriented for prismatic s l i p . I t would be interesting at t h i s juncture to compare the observed i n i t i a l solution hardening rate of the polyerystals with that of single crystals i n prism s l i p orientation. Single crystals and polyerystals of both Mg-Al and Mg-Zn systems have been deformed at room temperature i n the present work. This i s shown i n Table I I I . For converting the shear stress to t e n s i l e stress the following relationship a = . 2 r has been used. Table I I I Comparison of Solution Hardening Rates i n Single Crystals With Polyerystals System .da/ . For polyerystals at 295°K (dc"} l n P S 1 For prismatic s l i p at 295°K Mg-Zn 3.45 x 10 5 4.74 x 10 5 Mg-Al 5.95 x 10 4 6.24 x 10 4 66 I t i s apparent from the table that the solution strengthening rate of the polyc r y s t a l l i n e aggregatesis close to.that of single crystals oriented for prism s l i p . No attempt w i l l be made here to compare the effect of temperature da ' on (-j^ -) i n the polyerystals with that i n single c r y s t a l s . The effect of grain boundary s l i d i n g and c e l l formation i n po l y c r y s t a l l i n e material above room temperature (0.31 T ) makes such a comparison d i f f i c u l t . Below room temperature on the other hand i t i s not the prismatic s l i p system alone which i s affected by the addition of solute up to C_. This amount of solute i s effective also i n increasing the CRSS for basal s l i p below room.temperature. These considerations do not permit a quantitative comparison at temperatures below room temperature. 1.4.3.3. Hardening Beyond C^ ,: For a l l the allo y systems considered here the c/a r a t i o increases with increasing solute concentration; that i s , the effect i s opposite to that for Mg-Li system. Yet reference to f i g . (93) shows that for increasing concentrations beyond C^ ,, the CRSS for prismatic s l i p decreases. This decrease cannot be accounted f o r , then, i n terms of a.decreasing Pe i e r l s stress resulting from a reduction i n the a x i a l r a t i o , as i n the Mg-Li case. The decreasing CRSS for prismatic s l i p can account for both the decreased rate of solution hardening and the increasing d u c t i l i t y beyond C^. The reduced rate of hardening and increased d u c t i l i t y i n stage I I can, be attributed to the increasing importance of prismatic s l i p as an operative deformation mode. The operation of prismatic s l i p i n the polyerystals i s made s l i g h t l y easier due-to the increase i n CRSS for basal s l i p i n the alloys containing concentrations of solute 67 i n excess of C^ . Thus y i e l d i n these alloys w i l l be achieved by a balance between basal and prismatic s l i p . The i n i t i a l work hardening rate w i l l become lower with increasing solute concentration (beyong C^), because of the increasing ease of stress r e l i e f by prismatic s l i p . This l i n e of argument can be extended further to explain . the observed lowering of i n stage I I I . The y i e l d stress was evaluated at 0.2% p l a s t i c s t r a i n . An allo y containing solute concentration close to w i l l have a high i n i t i a l work hardening rate due to the d i f f i c u l t operation of prismatic s l i p . This w i l l lead to a large flow stress at 0.2% s t r a i n . On the other.hand an allo y containing solute i n large excess of w i l l have a lower i n i t i a l work hardening rate and as a r e s u l t , the flow stress at 0.2% s t r a i n w i l l become smaller than at a lower solute concentration. 1.4.3.4. The Transition Concentration C^l Above 300°K the CRSS for basal s l i p remains unchanged i n stage I of solution strengthening, whereas that for prismatic s l i p increases. Hence i n polycrystals the existence of a c r i t i c a l t r a n s i t i o n concentration i s a manifestation of a t r a n s i t i o n i n the CRSS - concentration relationship for prismatic s l i p . This conclusion i s supported by the equality of the tra n s i t i o n concentration observed i n poiyerystalline aggregate at room temperature (0.01 at.% Zn) to that obtained i n crystals oriented for prismatic s l i p (0.006 a t . % ) . I t should be noted that at a l l temperatures the t r a n s i t i o n concentration i n crystals oriented for prismatic s l i p remains constant. The s i t u a t i o n i s , however, rather complicated at low temperatures, as discussed e a r l i e r . Therefore,the c r i t i c a l t r a n s i t i o n 68 concentration, as obtained from the a of the poiyerystalline aggregates tested below room temperature, i s rather a complex quantity which cannot be discussed at the present. 1.4.3.5. The Flow Stress: Toaz and R i p l i n g ^ ^ have found that the t r a n s i t i o n i n the maximum stress-temperature curve results from the entrance of 5 r e c r y s t a l l i z a t i o n . Staceys work on magnesium and i t s alloys containing small amounts of A l has indicated that c e l l formation precedes r e c r y s t a l l i z a t i o n Metallographic studies on pure Mg and Mg-Al alloys deformed between 200 and 300°K up to a s t r a i n of 2%, i n the present work, have f a i l e d to reveal r e c r y s t a l l i z e d grains, although the t r a n s i t i o n i n the flow stress from stage I to stage I I occurs i n the same temperature i n t e r v a l . It i s proposed, therefore, that the onset of c e l l formation (sub grains) i s responsible for the t r a n s i t i o n i n the flow s t r e s s " temperature curves. This can be better understood as follows. When deformation proceeds at a fixed s t r a i n rate there i s stress concentration at the grain corners. This stress concentration w i l l not be relieved by s l i p i n the adjacent grains, due to von Mises' c r i t e r i o n not being s a t i s f i e d . The operation of prismatic s l i p s a t the grain corners, w i l l only partly relieve the stresses because basal and prismatic s l i p combined cannot provide f i v e independent modes of deformation. At lower temperatures this stress concentration i s f i n a l l y relieved by crack nucleation and subsequent fracture, whereas at higher temperatures stress accelerated polygonization w i l l r elieve the stress concentration and deformation w i l l proceed at a lower flow stress than at lower temperatures. 69 This explanation involving polygonization and c e l l structure formation i s consistent with the observed s t r a i n s e n s i t i v i t y of the t r a n s i t i o n temperature i n the flow stress-temperature curves for 0.055% A l all o y ( f i g . (28)). The corresponding t r a n s i t i o n temperature i n pure Mg and Mg - 0.53% A l alloy are observed to be less sensitive to the s t r a i n at which the flow stress i s measured. (0„055 at % A l i s close to C T and therefore the CRSS for prismatic s l i p i n t h i s a l l o y w i l l , be higher than i n pure Mg or the higher a l l o y s ) . Theref ore^, at a fixed s t r a i n rate the extent to which stress r e l i e f w i l l be achieved by the operation of prismatic s l i p w i l l be small, thus making c e l l formation an important mechanism. Since c e l l formation i s a stress accelerated polygonization process, i t w i l l operate at a lower temperature, the higher the applied s t r a i n , as i s the case with 0.05 at, % A l a l l o y . The s t r a i n s e n s i t i v i t y w i l l be lowered with the increasing ease of prismatic s l i p , because more stress r e l i e f w i l l be achieved through the operation of prismatic s l i p ; thereby making c e l l formation a less important process than i n the 0.05 at. % A l a l l o y . 1.4.4. The Effect of Temperature on D u c t i l i t y : The effect of solute on d u c t i l i t y at constant temperature has been rationalized i n terms of the r e l a t i v e ease of prismatic s l i p . However, there are two observations which are s t i l l unexplained. They are 1) The increase i n d u c t i l i t y of Mg as well as i t s alloys.with temperature^ above room temperature. 2) The observed maxima and minima i n the d u c t i l i t y - temperature curves of the Mg-Al al l o y s . 70 The d u c t i l i t y t r a n s i t i o n i n mangesium has been attributed to the onset of r e c r y s t a l l i z a t i o n ^ ^ . As the temperature increases the CRSS for prismatic s l i p decreases and also the ease of polygonization i s enhanced. Both these processes lead to an increased stress r e l i e f and make longer time available for r e c r y s t a l l i z a t i o n to proceed. Therefore^ the d u c t i l i t y of magnesium increases continuously with temperature beyond 295°K. The addition of solute up to C T increases the flow stress for prismatic s l i p and therefore pthe d u c t i l i t y of these alloys i s observed to be s l i g h t l y lower than that of magnesium. At higher solute concentrationsj however, the effectivensss of the stress r e l i e f increases due to the lower flow stress for prismatic s l i p and higher flow stress for basal s l i p leading to an increased d u c t i l i t y i n these a l l o y s . I t i s important to note that up to 420°K pyramidal s l i p {1011} <1120> does not operate . (51) i n magnesium 1.4.4.1. The D u c t i l i t y Maxima: D u c t i l i t y maxima have been observed i n Mg-alloys containing 0.2% (35) Pb and 0.8% A l . Green Wood et a l have found extensive grain boundary cavitation i n the Mg~0.2 % Pb all o y i n the temperature i n t e r v a l i n which the loss i n d u c t i l i t y i s encountered. I t was suggested that grain boundary cavitation i s responsible for the early fracture. A s h i f t i n the d u c t i l i t y maximum to lower.temperatures was also observed with decreased s t r a i n rate, which could be accounted for i n terms of the longer time available for vacancies to d i f f u s e , coalesce and form intergranular voids at a slower s t r a i n rate. 71 Stacey reported the occurrence of d u c t i l i t y peaks i n Mg-0.8% A l alloy and explained his results i n terms of a combined effect of cavitation and the formation of c e l l structure adjacent to the grain boundaries. Since the grain boundary movements increase with increasing temperature, the s t r a i n to be accommodated by l a t t i c e bending should accordingly decrease. Thus the formation of c e l l structure, which according to Stacey i s at least i n part responsible for the d u c t i l i t y peak, may be expected to be well developed at slower s t r a i n rates and at an intermediate temperature range, which i n turn w i l l be determined by the s t r a i n rate. The present results on the high temperature d u c t i l i t y of Mg-Al alloys can now be better understood. The effect of allo y i n g up to C_ i s to increase the CRSS for prismatic s l i p . Therefore^in these alloys the attainment of stress r e l i e f through prismatic s l i p i s even less than i n pure magnesium. The s t r a i n rate used i n the present experiments (75%/hr) i s rather high so that there i s neither much time available for r e c r y s t a l l i z a t i o n to proceed, nor can the vacancies coalesce to form intergranular voids. This explains the lower d u c t i l i t y and the absence of maxima i n these al l o y s . Increasing the alloying content i n excess of C^, i s i n effect equivalent to reducing the s t r a i n rate. This equivalence can be understood as follows. The flow stress for prismatic s l i p decreases with increasing solute,addition beyond C_. Since prismatic s l i p acts as one of the mechanisms causing stress r e l i e f at the grain corners, the effective-length of time available for the r e c r y s t a l l i z a t i o n to proceed, therefore, increases at higher solute concentrations. As a 72 r e s u l t , the fine grained structure adjacent to the boundaries i s better developed when the solute concentration i s high. This explains the increased d u c t i l i t y with solute concentration. However, the coalescence of vacancies to form intergranular voids, being a time dependent phenomenon, i s also enhanced with increasing solute content thereby leading to a s h i f t i n the d u c t i l i t y peak to lower temperatures. 1.4.5. Strengthening Effects i n Multicomponent Solid Solutions: It has been frequently noted that the strength of a multicomponent s o l i d solution can be obtained by adding up the strengthening effects of the corresponding binary s o l i d solutions. The above conclusion has been empirically established for face centred cubic a l l o y s . However, even i n the absence of comprehensive data on binary and ternary systems such behaviour should not be expected i n p o l y c r y s t a l l i n e s o l i d solutions having hep structure. Unlike the p o l y c r y s t a l l i n e aggregates of fee structure, which deform by the operation of one s l i p mode (having more than f i v e s l i p systems and hence s a t i s f y i n g von Mises c r i t e r i o n ) , the hep materials undergo p l a s t i c flow by the simultaneous operation of two or more deformation modes. Since the various deformation modes are affected d i f f e r e n t l y by the addition of solute, the y i e l d stress of the p o l y c r y s t a l l i n e aggregate represents a balance between these deformation modes and the additive nature of solutes i s unlikely to be observed i n hep materials. In the present work t h i s conclusion was v e r i f i e d on a Mg-In-Zn allo y tested at room temperature. The strength calculated from the Mg-In and Mg-Zn binaries was found to be much higher than the observed strength of this a l l o y . The Mg-In-Zn ternary was chosen because i t involved the component responsible for the maximum strengthening effect (Zn) and also the one giving the minimum strengthening effect (In), of a l l the solutes examined. The: ternary a l l o y examined had concentration of each solute i n excess of the tr a n s i t i o n concentration for the corresponding binary system. A c r i t i c a l examination of the observed strength of this a l l o y suggested that the stage I strengthening i s achieved by one component only and the solute i n excess of C^ , for th i s component as well as the t o t a l amount of solute of the other component, have additive effect i n stage I I . The calculated value using Mg-In and Mg-Zn binaries was found to agree closely with the experimental value of a^p of the ternary, i f Zn i s assumed to be the preferred solute responsible for stage I strengthening. Details of t h i s are given i n appendix (B) . In order to gain some insight, the investigation of the ternary systems was carried further. Two series of Mg-Zn-In alloys were prepared containing 0.004 at.% and 0.007 at»% Zn respectively (both less than C^ , for Zn) and In additions up to 1.1 at.% were made. The results are shown i n f i g s . (36,37). Since Zn concentration i s lower than C , In should contribute partly to the stage I strengthening. Experimentally then, a three stage solution strengthening curve i s obtained, as expected. The results show that the stage I slope (lb i n ternary) i s reduced, the stage I I slope i s increased and the t r a n s i t i o n concentration i s reduced from the corresponding values for the Mg-In binary. The implication may be that solution strengthening results from a type of solute atom disloca t i o n interaction with preference for prism plane dislocations and large size difference solutes to be involved. 74 0 0.2 0.4 0.6 0.8 1.0 Atomic % In 75 PART I I 2. Solution Hardening In Alloy Single Crystals 2.1. INTRODUCTION AND OBJECTIVES: The study of single crystals has the advantage of avoiding the d i f f i c u l t problem of dealing with the constraining conditions imposed by the grain boundaries during the deformation of the p o l y c r y s t a l l i n e aggregates. This s i m p l i c i t y , which enables one to obtain a better knowledge of shear stress conditions on part i c u l a r s l i p planes, has disadvantages due to the poor re p r o d u c i b i l i t y of the flow stress values of different single c r y s t a l s , because of the effects of s l i g h t changes i n such factors as substructure, impurity concentration, orientation and i n i t i a l d islocation density. Additional complications arise i n allo y single crystals grown from the melt, due to the solute segregation. Single crystals of d i l u t e s o l i d solutions of magnesium containing Zn i n amounts up to 0.45 at % have been studied i n the present investigation. " The study of the d i l u t e alloys was undertaken because of three main reasons. The solution hardening rate i n the polyerystals i n Mg d i l u t e solutions i s extremely high. The f i r s t l o g i c a l step i n pursuit of an explanation of the hardening i n polyerystals i s to study the solution hardening i n single crystals of low al l o y s . Another objective of the d i l u t e solution study was to determine the indirect strengthening effect of solute a r i s i n g due to a change i n the dislocation density. Strengthening i s possible i n close packed structures through an increase i n the disloc a t i o n density, caused by the presence of the solute. This i n d i r e c t effect of the solute i s believed to be s i g n i f i c a n t at low concentrations of solute 7 6 ( 7 5 ) L a s t l y , much of the current theory of solution strengthening assumes either an i d e a l or a. regular solution model. Since these models are better descriptions at low solute concentrations, they can be tested more exactly i n d i l u t e solutions only. The choice of Zn as solute was made because of i t s strong hardening effect as observed i n the Mg-Zn polycrystals. This strong hardening effect was hoped to make the study of d i l u t e solutions easier. Zn i s a hexagonal close packed metal which constitutes a s u b s t i t u t i o n a l solute i n magnesium at the Mg r i c h end of the phase diagram. Additional reasons for the choice of Zn are outlined i n the section on prismatic s l i p . The experimental procedure for specimen preparation i s given e a r l i e r . 2.2. STRESS-STRAIN RELATIONSHIPS IN BASAL SLIP: 2.2.1. The Stress-Strain Curves: It i s now w e l l recognized that the s t r e s s - s t r a i n curves of a variety of c r y s t a l structures show certain s i m i l a r i t i e s . Thus the three stages i n the basic work hardening curve are common to the face centred cubic metals and a l l o y s ^ ^ , 7 7 ) ^ t^ e &Y\aa.li. h a l i d e s ^ ^ , some intermetallics , body centred N i o b i u m ^ ^ and Germanium^"^ . However, the work hardening curve of hexagonal metals consists of two or three (3,29,82,83) stages ' ' ' . The shear stress-vs-shear s t r a i n curves of pure Mg crystals deformed at temperatures between 78°K and 423°K at a constant s t r a i n rate are shown i n f i g . (38). I t i s observed that the stress increases i n a parabolic manner at low s t r a i n s ; a minimum rate of work hardening i s 1400 01 I I I I I I L 0 1 2 3 4 5 6 7 Shear Strain Fig. 38. Resolved shear stress-shear s t r a i n curves for Mg single crystals oriented for Basal s l i p . 78 attained at intermediate s t r a i n s , followed by a region of rapid hardening. A t h i r d stage with a decreasing work hardening rate i s also observed i n crystals deformed at and above room temperature. Following the common tendency to break s t r e s s - s t r a i n curves into sections of similar work hardening rates, the three stages of deformation w i l l be defined as shown schematically i n f i g . (39). I t i s interesting to note that whereas the low i n i t i a l -4 -5 work hardening rate ( 10 -10 G) and an extensive range of easy glide are common to many hexagonal metal c r y s t a l s , the "Sigmoidal Shape" i s a characteristic feature of magnesium only. The i n i t i a l stage of work hardening becomes sub-divided into two linear regions i n Z n ^ ^ - and C d ^ ^ crystals deformed at room-temperature, however, th i s was not observed i n Mg crystals i n the present work. These differences i n the work hardening behaviour place Mg i n a category quite different from most of the other common hexagonal metals. 2.2.2. The Effect of Substructure on Work Hardening: In order to make a quantitative comparison of the work hardening parameters i n d i l u t e alloys possible, i t i s important to keep the effect of other variables on these quantities to a minimum. One such variable i s a low angle boundary running p a r a l l e l to the t e n s i l e axis of the c r y s t a l . Ordinary etchants and X-ray back r e f l e c t i o n technique w i l l not detect a single sub-boundary i n a large c r y s t a l . However, o p t i c a l micrographs of specimens, exhibiting a high rate of work hardening extended only small amounts, show boundaries of twinning and enhanced deformation not observed i n the other specimens. An example of such a boundary i n Mg-0.054 at.% Zn a l l o y deformed at room temperature i s shown i n f i g . (40). 79 Fig. 40. Sub-boundary running p a r a l l e l to the tensile-axis i n a Mg + 0.054 at.% Zn a l l o y c r y s t a l deformed i n easy glide at room temperature. 140X 80 (29) It has been suggested e a r l i e r by Hirsch and L a l l y that this high rate of work hardening i s associated with the said sub-boundaries. These investigators have observed that the presence of a single sub-boundary of the order of a degree leads to a reduction i n the extent of easy glide by a factor of f i v e , an increase i n the i n i t i a l work hardening rate by an order of magnitude and a loss i n the sigmoidal shape of the stre s s - s t r a i n curve. However, the work hardening rate i n stage B remains unaffected by the presence of sub-boundaries. The stress«=strain curves of Mg crystals deformed by various workers at room temperature are summarized i n f i g . (41). The inconsistency apparent i n f i g . (41) i s not surprising i n view of the dramatic effect of sub-boundaries on work hardening. In most of the e a r l i e r investigations the growth rate has been > l " / h r . In the present work, however, i t was observed that l"/hr i s about the l i m i t i n g growth rate beyond which the crystals contain many sub-boundaries, as observed by the s p l i t t i n g of the back r e f l e c t i o n spots. Therefore, i n order to obtain good crystals a growth rate of 0.4"/hr was used. A l l t e n s i l e test specimens for one composition were taken from one single c r y s t a l and tests were conducted i n duplicate. Repeat tests were considered necessary when the extent of easy glide i n the f i r s t two crystals differed by more than 50%. 2.2.3. The Stress-Strain Curves of the Mg-Zn All o y s : Stress-strain curves of a representative set of Mg-Zn alloys are shown as a function of the test temperature i n f i g s . (42,43). In order to f a c i l i t a t e comparison of the work hardening behaviour as a function of solute, at a constant temperature, the st r e s s - s t r a i n curves Shear s t r a i n Fig. 41. Resolved shear stress-shear s t r a i n curves for Mg single crystals deformed by various workers at room temperature. 1 4 0 0 Shear s t r a i n F i g . 43. Resolved shear stress vs. shear s t r a i n curves for Mg + 0.258 at.% Zn alloy single c r y s t a l s . 84 of alloys containing different amounts of solute are grouped together as shown i n f i g s . (44,45, and 46). It i s observed that the "S" shape of the curves i s retained up to a concentration of solute equal to 0.258 at» % Zn. The sigmoidal shape i s most pronounced i n an intermediate temperature range between 143°K and 295°K. Both above and below t h i s temperature i n t e r v a l the region of easy glide tends to become l i n e a r . The allo y containing 0.45 at.% Zn i s an exception i n that y i e l d points were observed i n this a l l o y at and below room temperature, leading to the absence of the sigmoidal shape of i t s s t r e s s - s t r a i n curves. 2.2.4. Yi e l d Points: Y i e l d points have been observed i n Mg-Zn alloys i n the present work at low temperatures. When tested at 78°K, only a s l i g h t y i e l d point was observed i n 0.054 at* % Zn a l l o y , however, a well defined y i e l d drop became apparent on increasing the solute concentration to 0.45 at.%. The y i e l d points continued to appear at increasingly higher temperatures, as the solute content was increased. For example, the appearance of y i e l d points was limited to 78°K only i n the case of 0.054 at. Zn a l l o y , to temperatures below 200°K i n the case of 0.258 at.% Zn and i n the case of Mg-0.45 at.% Zn al l o y y i e l d points could be seen i n specimens tested at room temperature. In addition to the i n i t i a l y i e l d points, multiple y i e l d points have been observed i n alloys containing 0.019 at*% Zn when tested at 423°K. The serrated yielding starts immediately after the material undergoes p l a s t i c flow and continues up to a certain s t r a i n , which increases with increasing solute addition. Beyond th i s s t r a i n the st r e s s - s t r a i n J I '. L 0 1 2 3 4 5 Shear Strain Fig. 45. Resolved shear stress vs. shear s t r a i n curves for Mg-Zn al l o y single crystals deformed at 295QK. 0 1 2 3 4 5 Shear Strain F i g . 46„ Resolved shear stress vs . shear s t r a i n curves for Mg-Zn a l l o y single c r y s t a l s deformed at 195SK. 88 curve remains smooth t i l l fracture occurs. 2.2.5. The C r i t i c a l Resolved Shear Stress: The CRSS of Mg crystals are shown as functions of composition and testing temperature i n f i g . (47). Gross p l a s t i c flow was considered to begin at the f i r s t departure from l i n e a r i t y . Sometimes CRSS i s obtained by the extrapolation of stage I to zero s t r a i n , however, such a procedure was not p r a c t i c a l because of the "S" shape of the str e s s - s t r a i n curves. It i s clear from f i g . (47) that the CRSS of Mg decreases l i n e a r l y from 74 gm/mm2 at 78°K to 41 gm/mm2 at 330°K. Beyond 330°K the CRSS remains essent i a l l y independent of temperature, varying s l i g h t l y due to a change i n shear modulus with temperature. The present trend of the temperature (6 13 89) dependence of CRSS i n Mg i s i n agreement with the e a r l i e r findings ' ' The strengthening effect of Zn on Mg i s seen to be strongly dependent on temperature. For example, the difference i n CRSS between pure Mg and 0.258 at. % Zn c r y s t a l tested at 78°K i s approximately 4-5 times higher than that at room temperature and above. I t i s interesting to note that beyond 330°K, the alloys containing Zn i n amounts up to 0.02 at, % have the same CRSS as that of Mg. However, the same alloys when tested at 78°K show a substantial increase i n CRSS over that of Mg. The strengthening effect of Zn as a function of the solute concentration i s shown i n f i g . (48) for various testing temperatures. The data at 0°K have been obtained by the extrapolation of the curves i n f i g . (47) to absolute zero. I t i s apparent from f i g . (48) that the strengthening effect of the solute on Mg i s not a simple linear function of the solute 5 0 0 400-CM | 300j-co CO o 200 100-Mg-Zn Single Crystols BASAL SLIP 0 100 Mg o — Mg •'0-006 At .%Zn A — Mg+0 019 • Mg + 0 054 9 Mg + 0 15 o • Mg + 0 258 o— Mg + 0-45 " 2 0 0 3 0 0 T E M P E R A T U R E 9 K F i g . 47. CRSS for Basal s l i p vs. temperature for Mg-Zn single cr y s t a l s . 4 0 0 Fig. 48c. Solution .strengthening i n basal s l i p Fig. 49. Solution strengthening i n basal vs. concentration .for. Mg-Zn all o y s . s l i p vs. square root of the solute, concentration. 91 (74) concentration as reported e a r l i e r by Levine, Sheely and Nash Fig. (49) shows a plot of the increase i n CRSS of the al l o y over that of magnesium against C , where C i s the concentration-of the solute i n at.%. I t i s clear that the solution hardening occurs i n two linea r stages, I and I I . Stage I continues up to a concentration of solute equal to 0.025 at.% Zn, followed by stage I I which has a slope approximately 3 times higher than i n stage I. The tr a n s i t i o n concentration from stage I to stage I I i s observed to be independent of temperature. The occurrence of a two stage h solution hardening behaviour i n the CRSS-C plot s i m i l a r to that i n the present work has been recently reported i n the d i l u t e Ag-base (face centred (85} cubic) s o l i d solutions containing In solute 2.2.6. The Rate of Solution Strengthening:. The non-linearity of solution hardening with respect to zinc con-centrations makes i t d i f f i c u l t to define a single parameter representing the strengthening rate over, a range of compositions. However, solution hardening rate can be conveniently described by the slopes S^ and S^ of the two linear parts of the curves i n f i g . (49). I t i s important to note that the solution hardening rate at any solute concentration i s related to the solute con-h centration and the corresponding slope of the CRSS-C curve as follows: , 9 T _ \ = 1 8 T • = S 3C c loFz ' 3C^ 2C^ (1) 2.2.6.1, The Temperature Dependence of S-j-: The temperature dependence of S^.is shown i n f i g . (50). I t i s apparent that Sj decreases l i n e a r l y from 316 gm/mm / ( a t . % ) 2 at 0°K to zero at.330°K. In alloys containing solute up to 0.025 at.% Zn ( i . e . stage I) no solution hardening i s observed above 330°K. The c r i t i c a l resolved shear stresses, of Mg and the alloys containing solute i n amounts up to 0.025 at.% Zn become athermal at 330°K. 92 Q O — I - — o 0 100 200 300 400 Temperature.°K Fig . 51. The va r i a t i o n of the solution strengthening parameter S T with temperature. 93 2.2.6.2. The Variation of.S With Temperature: The effect of temperature on i s shown i n f i g . (51). The 2 1/2 solution hardening parameter decreases from 960 gm/mm (at.%) at 2 1/2 0°K to 225 gm/mm /(at.%Zn) at 250°K. Beyond 250°K, S decreases very s l i g h t l y with increasing temperature. 2,2.7. Work Hardening: The work hardening parameters have been defined i n an e a r l i e r section. The effect of solute on these quantities w i l l be considered here. Because of the inconsistency i n the work hardening parameters of Mg reported e a r l i e r i n the l i t e r a t u r e , i t was considered necessary to investigate the s t r a i n hardening behaviour of Mg over a range of temperatures , 2.2.7.1. The Work Hardening Rate i n Stage A, QA : The work hardening rate during stage A deformation, 9/^ , i s shown i n f i g . (52) as a function of composition and temperature. I t i s conventional to express the work hardening rate as 9/G where G i s the shear modulus; however, i n the absence of data r e l a t i n g to the concentration dependence of G, 9 i s reported here without taking G into account. In order to make comparison with other metals possible, 9/G values at a few selected temperatures and compositions are indicated i n f i g . (52). The values of shear modulus for this purpose have been taken from the work of Slutsky and Garland and Koster The temperature dependence of 9 ^ for Mg i s shown i n f i g . (53). The values of 9 ^ reported in.the l i t e r a t u r e e a r l i e r are also included for comparison. The present results indicate that 9 ^ decreases rapidly 500 400 -S 0 0 (3 •H < CD 300 200 -100 -O 0.45 at.%. Zn ^ 0.258 at.% Zn Q 0.054 at.% Zn • 0,019 at.% Zn (9.5xl0~ 5) O -O (0.78X19""5) o -D 100 200 300 400 Pig. 52, Temperature °K The work hardening rate i n stage A vs. temperature for Mg-Zn single crystals, 1 2 0 0 Temperature Fig» 53. The temperature dependence of the work hardening rate i n the easy glide of Magnesium as reported by various workers. 96 with increasing temperature from a value of 6 /G equal to 1.38 x 10~^ at 78°K to 1.07 x 10~ 5 at about 250°K. Beyond 250°K, however, 6 /G A remains independent of temperature. The large scatter i n d a t a ^ ^ and (QQ\ the high values of 0^ reported e a r l i e r may have been a result of the sub-boundaries, which are l i k e l y to be present i n these crystals (because of the fast rate of growth used by these workers). The existence of a temperature insensitive 9^ below 200°K as reported by Schmid^ 8 8\ Bocek^ 8 9^ and C o n r a d ^ i s , however, surprising. The concentration dependence of 6^ for the Mg-Zn alloys i s shown i n f i g . (54) for a representative set of temperatures. These curves have been constructed by taking isothermal sections from f i g . (52). It i s clear that the work hardening rate increases i n a parabolic 1/2 fashion with increasing Zn content. The linear dependence of 8^ on C i s apparent from f i g . (55). 2.2.7,2. The Extent of Easy Glide: The s t r a i n at the end of easy g l i d e , Y^.is shown i n f i g . (56) as a function of temperature for a representative set of Mg-Zn al l o y s . In pure Mg, y increases from approximately 230% shear s t r a i n at 78°K to 470% at room temperature. Beyond room temperature y g decreases with increasing temperature. Although the effect of Zn i s to decrease Y B ) the trend i n the temperature dependence of y D remains the same i n the a alloys as i n Mg. A plot of t h e [ ( t B ) M g _ a l l o y ] a g a i n S t s o l u t e concentration for different temperatures ( f i g . 57) indicates that the decrease i n the extent of easy glide caused by alloying i s independent of temperature between 195° - 373°K. 0 0.1 0.2 0.3 0,4 Atomic % Zn Fig. 54. The increase i n work hardening rate i n easy glide vs. Zn concentration. Fig.55. The increase i n work hardening rate i n easy glide vs. square root of the Zn concentration. 9 9 6.0 100 200 300 Temperature °K 400 Fig. 56. The extent of easy glide as a function of temperature for Mg-Zn single c r y s t a l s . J. 0.1 0.2 0.3 Atomic % Zn 0.4 0.5 Fi g . 57, The decrease i n the extent of easy glide as a function of the solute concentration.' 100 2.2.7.3. Stress at the Onset of Stage B, T : Stage B was considered to begin at a stress determined by the intersection of the extrapolated linear stages A and B. However^since stage B was not c l e a r l y defined below room temperature, the values of i below room temperature are not r e l i a b l e and hence w i l l not be reported B here. The variation of x_ with Zn concentration i s shown i n f i g . (58). B I t i s apparent that x„ increases with increasing solute concentration. n Also i t i s interesting to note that [(x„) - (T„),. ] remains ° B a l l o y B Mg independent of temperature above 295°K for any fixed solute concentration. 2.2.7.4. The Work Hardening Rate i n Stage B, 0 p: For most of the crystals tested below room temperature, the work hardening rate i n stage B was found to increase with s t r a i n t i l l fracture occurred. Under such circumstances 9 cannot be evaluated accurately. o Therefore, the results of the tests conducted above room.temperature w i l l be presented here. The temperature dependence of 9 for Mg and four Mg-Zn alloys i s shown i n f i g . (59). The work hardening rate 9^  decreases with increasing temperature. The effect of alloying on 9 i s shown i n f i g . (60). It i s D apparent that the work hardening rate increases i n a near parabolic manner with increasing Zn content and that the effect i s more pronounced the lower the temperature. F i g . 58. The effect -of temperature and solute concentration on the stress at the onset of Stage B. 102 Atomic % Zn Fig. 60. The increase i n the work hardening rate i n Stage B vs. Zn concentration. 103 2.2.7.5. Deformation i n Stage "C": Stage C immediately follows the stage B, and i s characterised by a decreasing work hardening rate with increasing s t r a i n . Stage C was c l e a r l y observed above room temperature only. The stress corresponding to the f i r s t deviation from l i n e a r i t y at the end of stage B w i l l be termed T^,. The v a r i a t i o n of with Zn concentration i s shown i n f i g . (61) for 373°K and 423°K. It i s observed that increases with alloying i n a near parabolic manner. 2.3. THE EFFECT OF SOLUTE ON THE DISLOCATION DENSITIES: It has been pointed out by Seeger^"^ that solution hardening i s possible i n close packed structures, as a result of a change i n the d i s l o c a t i o n density through the presence of a solute. This indirect hardening effect i s thought to be most pronounced at low solute concentrations, especially (90 91) when the difference i n size between the solvent and the solute i s large ' Bearing i n mind that the difference i n size between Mg and Zn i s large and that the alloys investigated i n this work contain low concentrations of the solute, an examination of the dislocation densities i n these alloys deserves attention. \ There are two ways possible through which strengthening may be achieved due to an increase i n dislocation density. The glide dislocation density may increase leading to an increase i n the long range stress f i e l d i . e . the athermal component being affected. Alternatively^or simultaneously, the forest dislocation density may vary thereby changing the short range dislocation-dislocation interaction leading to a change i n the thermally activated component of the flow stress. Fig. 61 The stress at the onset of Stage C vs. Zn concentration. 105 In the face centred cubic structures a knowledge of the dislocation density on the glide plane provides information on the forest dislocation spacing as w e l l , since the glide dislocation and the forest dislocations l i e on crystallographically similar planes. However, differences are to be expected i n hexagonal metals, because the two sets of dislocations are not contained on crystallographically similar planes. Thus i t has been decided to examine the two sets of dislocations independently of one another. 2.3.1. The Basal Dislocation Density: Transmission electron microscopy of thin f o i l s has been used i n the present investigation to measure the basal dislocation density. There exists considerable controversy i n the l i t e r a t u r e concerning the degree to which dislocation arrangements observed i n t h i n f o i l s (92-94) are representative of the bulk material . I t i s believed that the degree of rearrangement due to stress relaxation increases^the higher (95) the stacking f a u l t energy. In fact, experiments on Al-Ag s o l i d solutions have revealed that more than 60% of the dislocations are l o s t during the process of thinning. In the present study the easy glide plane {0001} was kept p a r a l l e l to the f o i l surface, thus lowering che probability of losing dislocations through c r o s s - s l i p . This f o i l orientation has been used e a r l i e r by Hirsch (29) and L a l l y i n examining the v a r i a t i o n of d i s l o c a t i o n density as a function of deformation i n magnesium. Through the examination of a c a r e f u l l y chosen set of nonbasal sections, these authors have come to the conclusion that the arrangement and densities of dislocations i n the basal sections are t y p i c a l of the bulk material. 106 2.3.1.1. F o i l Preparation: Considerable d i f f i c u l t y was encountered i n obtaining f o i l s , free from oxide f i l m , of the Mg-Zn alloys. Therefore,it was decided to study Mg-Al crystals instead, at a comparable concentration l e v e l . The choice of A l as solute was mainly due to the large size difference between Mg and A l atoms, which i s comparable to that between Mg and Zn, The d i s t r i b u t i o n coefficients of the two a l l o y systems at low concentration levels are also comparable. The f o i l s were prepared by the chemical polishing of 0.1" c r y s t a l sections p a r a l l e l to the {0001} plane. The polishing solution consisted of 10% n i t r i c acid i n water. Shallow cavities were created on each face of the s l i c e by impinging a j e t of acid-water mixture. The f i n a l thinning was done by immersing the specimen i n the centre of a beaker containing the thinning solution. Perforations were seen f i r s t at the c a v i t i e s . The narrow region between two perforations was found most suitable for the electron microscopy work. 2.3.1.2. Observations: A representative portion of the dislocation structure i n Mg i s shown i n f i g . (62). The main features are the long.dislocation dipoles at A, B and C. The presence of unidentified dark p a r t i c l e s i n the electron (29) micrographs of Mg have been reported e a r l i e r . In the present work such p a r t i c l e s have been observed i n varying amounts i n a l l specimens ( f i g . (62)) and are often surrounded by dislocation loops. On t i l t i n g the specimen the dislocations go out of contrast but the dark spots remain i n contrast suggesting them to be p a r t i c l e s . However, th e i r s ize was too small for i d e n t i f i c a t i o n using selected area d i f f r a c t i o n . A F i g . 62, Typical d i s l o c a t i o n structure i n as grown Mg crystals- F o i l p a r a l l e l to {0001} plane. Note p a r t i c l e s at D, E, and F, and the high density of dipoles. 108 higher magnification micrograph of such a p a r t i c l e and the sourrounding dislocations i s shown i n f i g . (63). Sometimes these p a r t i c l e s have been observed to emit dislocation s p i r a l s under the heat of the electron beam. Typical dislocation structures i n alloys containing 0.18 at. % and 0.36 at.% A l are shown i n f i g s . (64)and(65) respectively. I t i s observed that the closely spaced dipoles become fewer, and the dislocations become more evenly distributed as the alloying content i s increased. 2.3.1.3. The Technique of Dislocation Density Measurement and i t s Limitations: (97 98) A modification of the random l i n e intersection method ' of Ham was used i n the present work. The random li n e s consisted of s i x c i r c l e s with random centres drawn on a transparent template. The observed length of the dislocation l i n e becomes the actual length (not a projection) i f i t i s assumed that the dislocations run very nearly p a r a l l e l to the basal plane. Under these conditions the dislocation density p_ w i l l be given by P G = M/(2Lt) (2) where N i s the number of intersections of the random c i r c l e s with the dislocation l i n e s . L i s the t o t a l length of the circumference of the c i r c l e s . t i s the thickness of the f o i l i n the region under examination. Much d i f f i c u l t y was encountered i n determining the f o i l thickness. Because s l i p traces cannot be obtained i n basal sections, the f o i l thickness had to be estimated by counting the number of the extinction 1 0 9 Fig. 63. Electron micrograph showing dislocations surrounding a p a r t i c l e . F i g . 64. T y p i c a l d i s l o c a t i o n structure i n an undeformed o Mg + 0.18 at.% A l a l l o y single c r y s t a l . F o i l surface p a r a l l e l to {0001} plane. I l l F i g . 65. Typical disl o c a t i o n structure i n an undeformed Mg + 0.38 at. % A l a l l o y single c r y s t a l . F o i l p a r a l l e l to {0001} plane. 112 fringes from the f o i l edge. In order to obtain an average dislocation density, counts were made on at least three different areas on each specimen. The t o t a l random l i n e length used on each specimen was at least 200 y. Burgers vectors of kind other than <1120> have never been observed i n magnesium, neither have dislocations sharply inclined to (29) the basal plane been seen . Thus with the f o i l orientation chosen i n the present work, a l l the dislocations were i n contrast. This eliminated the need for correcting the measured density for dislocations out of contrast. The number of short lengths of dislocations meeting the surface at closely separated points (indicating that they are i n c l i n e d to the basal plane) was much smaller than the long segments l y i n g i n the basal plane, hence the measured density may be taken as the basal dislo c a t i o n density. 2.3.1.4. The Effect of Solute on the Basal Dislocation Density: 8 2 In pure magnesium a density of 2.4 x 10 lines/cm was observed, 8 (29) which i s close to 2.2 x 10 reported e a r l i e r by Hirsch and L a l l y (96) 3 6 2 Tsui , however, has obtained a density of the order of 10 -10 /cm i n Mg single c r y s t a l s . k I t i s seen from a plot of p against C i n f i g . (66) that the d i s l o c a t i o n density increases i n a parabolic manner with A l concentrations, Numerically p increases by a factor of three due to an addition of solute equal to 0.38 at.%. 1 1 3 8 Mg-Al Single Crystols Dislocation Density in Sections Parallel to (OOOI) Plane P c * [ 2 - 4 * 8 3 5 \fc ] XIO 8 lines/cm 2 0-2 0-3 0-4 05 06 JC (At .%) 1 7 2 Fig. 66. Basal disl o c a t i o n density vs. square root of solute concentration for Mg-Al a l l o y s . 1 1 4 h From the straight l i n e plot of C against i n f i g . (66) the following relationship i s obtained. (p ) = [2.4 + 8.35^ ] x 10 8 lines /cm2 (3) G C Assuming t h i s relationship holds good up to the l i m i t of s o l i d s o l u b i l i t y (1.6 a t . % ) , the dislocation density i n the highest a l l o y w i l l be only 5 times higher than i n Mg. This i s i n agreement with the e a r l i e r findings i n fee s o l i d solutions that the disloc a t i o n density i n the alloys i s less than an order of magnitude higher than i n the solvent (2) pure metal 2.3.2. The Forest Dislocation Density: The forest dislocations were revealed by an etch p i t t i n g technique, which was developed i n the course of the present investigation. 2.3.2.1. Crystal Orientation and the P i t Characteristics: The {0001} plane of magnesium i s the most suitable surface for etching, since i t w i l l not reveal the basal dislocations, whereas the nonbasal dislocations of edge character, which constitute the forest dislocations can be p r e f e r e n t i a l l y revealed. A l l screw d i s -locations w i l l be p a r a l l e l to the {0001} plane and hence w i l l not be revealed by etching. Crystals of Mg and Mg-Zn alloys were spark eroded to obtain 0.1" thick s l i c e s p a r a l l e l to the {0001} plane within.2 degrees. The spark damaged layer was removed by chemical polishing i n 10% HN0~ and the specimen was washed i n ethanol and dried. 115 The etchant consisted of a 20% solution of HNO^  i n water and the etching was carried out by dipping the polished specimen for about 10 sees, i n warm n i t r i c acid solution kept at 30-35°C. The etch p i t s obtained i n this manner were c i r c u l a r i n shape. Hexagon shaped etch p i t s were, however, produced by etching i n a 30% HNO^  solution maintained at room temperature. Such geometric shapes are characteristic of the disl o c a t i o n etch p i t s . Some of the hexagon shaped p i t s are shown i n f i g . (67). For counting the number of d i s -locations , however, i t was found convenient to use the c i r c u l a r p i t s . The etched surfaces of a representative set of Mg-Zn alloys are shown i n f i g s . (67(a),(b) and (c)). 2.3.2.2. The Study of Spark Erosion Damage i n Mg Using the Dislocation  Etch P i t Method: In order to examine the damaging effect of spark erosion a sample was etched without removing the entire damaged layer. A micrograph of such a specimen i s shown i n f i g . (68). I t i s apparent that small twins are formed at the edge of the specimen and the nonbasal edge dislocations can be seen at the twin boundaries. On further chemical polishing and subsequent etching the twins disappear, suggesting that they are confined to a narrow surface layer. The etch p i t density decreased t i l l a surface layer approximately 150u i n thickness was removed from the surface. Below th i s depth the density remained constant. 116 480 x 480 X Fig. 67(a). Hexagonal shape of the etch p i t s , Fig. 67(b). D i s t r i b u t i o n of etch p i t s i n a Mg + 0.019 at.% Zn c r y s t a l . 84X. 117 Fig. 68. Dislocation etch p i t s on the {0001} plane of Mg single c r y s t a l . Note non-basal edge dislocations at the twin boundaries. 75X Fig. 69. Etch p i t density vs. Zn concentration. 118 2.3.2.3. Variation of the Etch Pit. Density with A l l o y i n g : The concentration dependence of the number, of etch p i t s per unit area i s shown i n f i g . (69). In order to avoid l o c a l differences, counts were made at a magnification such that at least 800 p i t s were i n the f i e l d of view at one time. Three different regions were examined on each specimen. The var i a t i o n from specimen to specimen i s shown i n f i g . (69) by the scatter bars. I t i s apparent that the etch p i t density increases rapidly with alloying up to approximately 0.05 at.% Zn, followed by a very s l i g h t increase up to 0.45 at, % Zn. It i s interesting to note that the etch p i t density i n 4 2 magnesium i s 3.5 x 10 /cm . From activation volume measurements on the (4) other hand Conrad has arrived at a forest density of the order of 7 8 2 10 - 10 /cm . Transmission electron microscope studies on non-basal sections (basal plane almost p a r a l l e l to f o i l normal) made by Hirsch (29) and L a l l y , however, indicate that the forest d i s l o c a t i o n density i s 8 2 unlikely to be as high as 10 /cm . 119 2.4. DISCUSSIONS: Much progress has been made i n the understanding of the s o l i d solution strengthening mechanisms i n metals after the development of the dislocation theory. Most of the important mechanisms of interaction were proposed within a few years after the study of the behaviour of dislocations i n s o l i d solutions had been undertaken. In the past decade, however, the major problem has been the one, of deciding which mechanisms are important i n a given experimental situ a t i o n and unravelling the contributions of the various mechanisms. The results of the present study on Mg-Zn alloys are examined i n the l i g h t of the various strengthening mechanisms, followed by a discussion of the effect of solute on the work hardening parameters of magnesium. 2.4,1. The C r i t i c a l Resolved Shear Stress: The CRSS of single crystals i s perhaps the only satisfactory parameter which can be used for a quantitative comparison with the theoretical predictions without any adjustable parameters. In the case of p o l y c r y s t a l l i n e aggregates such adjustable parameters become important, because of the constraining conditions imposed by the grain boundaries. The experimental results can be best viewed against the background of the theory of temperature dependence of the flow stress (38) (99^ as developed by Seeger and Friedel . According to this theory the applied stress can be considered as the sum of two components such that T(T) = TG + T* (4) 120 T * i s referred to as the thermal component- of the applied stress and i s associated with short range obstacles which can be overcome with the aid of thermal energy. T i s the athermal component that arises due to the long range e l a s t i c interactions such as those between p a r a l l e l glide d i s -locations at distances large compared with b, the Burgers vector. Such obstacles cannot be overcome with the aid of thermal energy. The consequence of Seeger's postulate i s schematically shown i n f i g . (70), for the sit u a t i o n where the mechanism of y i e l d does not change with temperature. rn varies with temperature only through a change i n the shear modulus G such that TG/Q becomes independent of temperature. Seeger has modified his d e f i n i t i o n of l a t e r to include (92) a short range e l a s t i c interaction term The effect of solute on the two components of the CRSS w i l l now be considered separately. Temperature Fi g . 70. The temperature dependence of y i e l d i n terms of the stress components (after Seeger'^ 5'). 121 2.4.2. Solution Strengthening i n the Athermal Region: The c r i t i c a l resolved shear stresses of magnesium and i t s alloys containing Zn i n amounts up to 0.45 at.% Zn become athermal above 330°K ( f i g . 47). I t i s apparent from fig.(49) that the addition of Zn i n amounts up to 0.025 at.% has no effect on the CRSS of Mg above 330°K ( i . e . S T - 0), however, x increases l i n e a r l y with the square root of Zn concentration above 0.025 at.%. Another ch a r a c t e r i s i t c feature of the solution hardening i n the athermal region i s the constancy of the strengthening parameter S^j.. As mentioned e a r l i e r x i s the stress required to overcome the long range obstalces i n the c r y s t a l . The long range stress f i e l d can be affected by one or more of the following interactions involving the solute and the dislocations: 1) A change i n the basal dislocation density. 2) E l a s t i c interaction. 3) Chemical interaction. 4) Short range order. 5) Long range order. 6) F r i c t i o n stress due to random d i s t r i b u t i o n of solute. Of these,the change i n dislocation density i s an ind i r e c t effect of solute leading to strengthening. The remaining f i v e arise due to a direct interaction between the solute and the di s l o c a t i o n . 1 22 2.4.2.1. The Basal Dislocation Density; Seeger^^^ has shown that the long, range stress f i e l d x^ i s related to the glide dislocation density p as follows: T Q = a b G / p (5) where a i s a constant, b i s the Burgers vector, and G i s the shear modulus i n the s l i p d i r e c t i o n . The dislocation density of magnesium has been evaluated as (29) a function of shear stress by Hirsch and L a l l y . These authors came to the conclusion that the flow stress of Mg at room temperature (the thermally activated component of the flow stress i s very small at room temperature and hence the CRSS may be taken to be equal to T ^ ) can be related to the disloc a t i o n density by the following expression T .= x 0 + 2.7 x 10~ 3/p~ (6) where x i s the flow stress 2 and X q i s a constant equal to 40 gm/mm . It has been shown e a r l i e r i n th i s thesis that the basal d i s -location density can be expressed as a function of A l concentration by the r e l a t i o n p_ - [2.4 + 8.35 /C] x 10 8 cm"2 (3) where C i s the concentration of solute. The subscript G has been dropped for convenience. 123 Combining equations (6) and (3) A T = 40.5 [ 1 - V 1+3.48 C% ] (7) where A T i s the strengthening at a concentration C. The values of A T obtained using equation (7) are plotted i n f i g . (71) along with the experimental results of Sheely, Levine (74) and Nash on the CRSS of Mg-Al alloys tested at room temperature. It i s seen that equation (7) gives a constantly increasing value of x with solute concentration. However, the present results have established that x remains unaffected i n Mg-Zn alloys upto a certain minimum concentration of solute. Also e a r l i e r i n this thesis i t has been shown that this concentration independence of x i s common to a host of Mg-solid solutions including the Mg-Al al l o y s . Thus at low solute concentrations equation (7) i s a poor description of the effect of solute, and i t i s apparent that at higher solute concentrations the predicted values of x„ are far smaller than those observed experimentally. I t should be noted here that p i n equation (6) comprises the grown i n dislocations plus those generated due to work hardening, whereas equation (3) describes only the grown i n disloc a t i o n density v a r i a t i o n due to solution effect. Thus^whereas i n the former case dislocation-dislocation interaction i s a contributing factor, i t i s absent i n the l a t t e r case. In view of this fact the combination of equations (6) and (3) to y i e l d equation (7) i s not f u l l y j u s t i f i e d . However, equation (7) w i l l be considered as a close approximation for the sake of comparison with the experimental r e s u l t s . 124 .0.1 0.2 At. % A l 0.3 0.4 Fig. 71. 400 Comparison of the observed solution strengthening with that expected from the increase i n the basal d i s l o c a t i o n density. 300 ~ 200 -100 2 ,1/2 ,. ,1/2 i n (at. % L i ) ' Fig. 72. The increase i n CRSS of Mg v s ( L i concentration)in Mg-Li al l o y single crystals (Data from Horiuchi and Yoshinaga 125 Therefore,it i s reasonable to conclude that the change i n the basal dislocation density i s not an important factor i n solution strengthening of magnesium. Similar conclusion has been arrived at by Adams, Vreeland and Wood i n the case of Zn base s o l i d solutions ^ . , A,(118) containing A l 2.4.2.2. E l a s t i c Interaction: The mechanism of e l a s t i c interaction was o r i g i n a l l y proposed by C o t t r e l l ^ ^ ^ to explain the yielding of Fe-C al l o y s . Strengthening arises from the e l a s t i c interaction between the stress f i e l d of a dislocation and the stress f i e l d due to the l a t t i c e d i s t o r t i o n r e s u l t i n g from the m i s f i t of the solute atom. The problem has been treated by the methods of c l a s s i c a l e l a s t i c i t y by several authors 104) ^ a s been successfully applied to many cases of i n t e r s t i t i a l solutes. An analysis for the substitutional solutions has been given by Suzuki using the concept of. continuously varying composition and l a t t i c e parameter i n the neighbourhood of the di s l o c a t i o n . As pointed out by Suzuki, the solution hardening i n substitu t i o n a l alloys due to t h i s mechanism decreases markedly with temperature by thermal activation and becomes p r a c t i c a l l y negligible at higher temperatures. The present results on Mg-Zn alloys at higher temperatures, however, do not conform to the requirements of this mechanism. The CRSS of Mg-Zn alloys i s almost independent of temperature above about 330°K, yet solution strengthening effect i s quite pronounced above 330°K ( f i g . 47). Therefore,the e l a s t i c interaction theory.cannot be applied to the case of solution hardening i n Mg-Zn alloys tested above room temperature. 126 2.4.2.3. Chemical Interaction: It i s now well established that the stacking f a u l t energy can be a function of composition for alloys i n which the dislocations are extended. Evidence for this has been obtained from X-ray measurements of stacking fault occurrences i n deformed a l l o y s f r o m direct electron microscopic / 1 / \ observations of extended dislocations and from the temperature and s t r a i n (141 142) rate s e n s i t i v i t y of tJ_J_J_ ' i n t n e stress s t r a i n curves of single crystals having fee structure. As suggested by S u z u k i } this v a r i a t i o n i n stacking f a u l t energy with composition should result i n a difference i n composition between the faulted area of extended dislocations and the bulk material. A surface thermodynamic.treatment of the problem has also been given by Guard and F i n e ^ ^ ^ . Experimentally, Spreadborough^"'"^"^ has shown that a difference i n the chemical r e a c t i v i t y between the faulted area and the bulk material exists which i s evidence that segregation does occur. This segregation has a pinning effect on the dislocations. Suzuki's theory has l a t e r been extended to regular solutions and i n i t s f i n a l form stands as follows: R T bt 9m* RT A where T i s the increase i n the c r i t i c a l shear stress of the a l l o y over that of the pure metal, n^r 1 1^ a r e. the mole fractions of the solute and the solvent respectively, V i s the molar volume of the a l l o y , b i s the Burgers vector, t i s the thickness of the faulted region, Y i s the stacking f a u l t energy, 1 2 7 f m i s the mole fraction of the solute i n the faulted region, AH i s the integral heat of mixing, R i s the gas constant, and T the absolute temperature at which equilibrium i s attained. I t i s seen from equation (8) that a knowledge of the v a r i a t i o n of stacking f a u l t energy with solute concentration (in the stacking fault) i s essential i n determining the a p p l i c a b i l i t y of the equation to a particular s i t u a t i o n . In the case of pure magnesium the stacking f a u l t energy has been estimated, from theoretical considerations, to be equal 2(39) to 2 0 0 ergs/cm . Quantitatively, however, this value of Y has never been v e r i f i e d using electron microscopy. Because of the rather long extinction distance i n magnesium and i t s low shear modulus, i t i s d i f f i c u l t to observe the alternate extension and contraction of the nodes i n the hexagonal network formed by the interaction- of two sets of (29) screws with different basal Burgers vectors . Stacking f a u l t energy can also be estimated by having a knowledge of the stress required to induce cross s l i p . The method has been applied successfully to the case of crystals with face centred cubic structure.. However, this method has never been applied to .the hep materials mainly because the stress to induce cross s l i p i s not apparent from the stress s t r a i n curves of t h i s . group of cry s t a l s . Bearing i n mind that (4^-f) for Mg-Zn. alloys i s not d mA at hand and that the thermodynamic functions of the system are also unknown, an attempt w i l l be made to calculate the extent of Suzuki hardening under certain assumptions. F i r s t , we w i l l assume an i d e a l mixing and 3 ^ AH 3 therefore -—•—• - 0. Secondly, (^~f\ w i l l be calculated using the . 3mA2 V 3m,/ A A 128 stacking f a u l t energies of pure Zn and pure magnesium. Also assuming the stacking fa u l t to be two atomic layers thick equation (8) reduces to the following: T = — — ™ S m Q2 ( 9 )  2 3 RT V where c/a i s the a x i a l r a t i o of the c r y s t a l , and Q = V < T*» - Y ^ (10) 2c the other symbols having their usual meaning. 2 a 2 Substituting Y, • = 200 ergs/cm Me y' = 70 ergs/cm Zn 3 and V = 14 cm /mole T = 458°K i . e . 0.5 T m. This value i s reasonable i n that the d i f f u s i o n of solute atoms to the stacking f a u l t s w i l l be negligible below 0.5 T . x for 0.45 at.% Zn allo y comes out to be-2 21 gm/mm . On the other hand the experimental value of ( O Q ^^rC ^ M g = 2 105 gm/mm . Thus i t i s seen that the chemical hardening mechanism cannot adequately account for the observed strengthening. Additional discrepancy i s encountered i n the composition dependence of solution strengthening. At low solute concentrations (upto 0.45 at.% Zn) equation (9) predicts an almost linear relationship between x and C whereas the experimental h data follow a C trend. 2.4.2.4. Short Range Order: The energy of a s o l i d solution varies l i n e a r l y with the degree of l o c a l order while the entropy varies roughly logarithmically; therefore, unless the interaction between l i k e and unlike neighbours i s i d e n t i c a l , the equilibrium state w i l l be one with a non zero.local o r d e r , 129 As pointed out by Fisher^09) this order need not be very great to make a si g n i f i c a n t contribution to the strengthening. According to F l i n n ^ ^ ^ the short range order interaction would lead to a temperature independent f r i c t i o n stress i . e . x should increase with alloying i f SRO interaction i s present. Applying the theoretical result of Fl i n n to the case of basal s l i p i n hep metals, 2 J L ( mA mB) . v 2 (11) a 3kT where v i s the interaction energy for l i k e bonds. When the value of v/kT i s small, v can be calculated using the following approximate formula a kT v = ( 1 2 ) Here i s the l o c a l order parameter which can be determined by X-ray or neutron d i f f r a c t i o n experiments. However, such experiments have not been carried out with Mg-Zn all o y s . In the absence of experimentally determined values of the short range order parameter equation (11) cannot be used to calculate the strengthening due to short range order i n the 2 Mg-Zn system. Nevertheless^the fact that equation (11) predicts a C dependence of the f r i c t i o n stress speaks against SRO hardening. In the h present work the observed hardening follows a C r e l a t i o n . Lately i t has been shown by Koppenal^"''^ that neutron i r r a d i a t i o n does not influence the strength of Cu-14% A l crystals upto a dose where radiation induced short range ordering i s already complete. This also i s against the idea that SRO can strongly contribute to the long range stress f i e l d . 130 2,4.2.5. Strengthening.Due to a Random Distr i b u t i o n of the Solute: The e a r l i e s t theory of solution strengthening based on a random (102 112) d i s t r i b u t i o n of the solute was propounded by Mott and Nabarro ' This theory computes the average magnitude of the stress i n the matrix of a d i l u t e s o l i d - s o l u t i o n due to replacing atoms of the matrix by ones that are larger or smaller, ( i . e . size effect) thus straining the l a t t i c e . Although considering sign, the average stress on a d i s l o c a t i o n i s zero, r e l a t i v e l y short lengths experience stresses of the order of the average stress magnitude and hardening r e s u l t s . The theory cannot adequately account for the athermal solution strengthening i n general due to two reasons. As pointed out by F l e i s c h e r ^ ^ " ^ the theory of Mott and Nabarro applies only to dislocations whose stress f i e l d s have d i l a t i o n associated with them, and hence pure screw dislocations are able to move freely. However, the main objection to the theory was the experimental f a i l u r e ^ Q f the parameter 6 (defined as 1 db 6 = " , where b i s the l a t t i c e parameter) to be a proper index of the r e l a t i v e hardening when different solute atoms are added to a metal. I t has been observed that the valency of the added element i s a strong factor. Fleischer has extended the analysis further to include two interactions responsible for solution s t r e n g t h e n i n g > H 6 ) ^  Q n g ^ g caused by the hydrostatic component of the stress f i e l d of a d i s l o c a t i o n interacting with the volume change produced, by inserting a foreign atom , (size or 6 interaction) and the other occurs because the inserted atom represents a small region of different e l a s t i c properties from the matrix, so that a dislocation interacts because i t must do more (or less) work 131 than usual i n e l a s t i c a l l y shearing the impurity atom (modulus or n interaction). This second interaction i s f i r s t order for either edge or screw dislocations since both have shear strains. The size interaction with a screw di s l o c a t i o n , however, i s zero to a f i r s t order, since a screw dislocation has no hydrostatic stress according to linear e l a s t i c i t y . On the other hand there i s a second order e l a s t i c interaction caused by an expansion, which i n essence arises from the atomic roughness of the planes i n a c r y s t a l . These planes are forced apart as they are sheared across one another; thus there exists a hydrostatic component of stress around a screw dislocation. Taking the 6 and the n interactions into account, the dependence of the room temperature plateau stress on concentration of solute comes out as follows T„ - Z.G.C"2 . c J / " (13) & 3/2 -= A . I * . I . C G where Z i s a constant calculated by Fleischer to be equal to 1/760 for screws, n - a6 (14) <5 i s already defined. ? where n = (15) x] = . , G being the shear modulus, G dC The value of a l i e s between 3 and 16 for interaction with screw dislocations whereas i t i s greater than or equal to 16 for interaction with edge dislocations. For p o l y c r y s t a l l i n e Cu-solid solutions a i s estimated to be about three implying a screw dislo c a t i o n solute atom i n t e r a c t i o n ^ " ^ . Recently S a x l ^ " ^ has extended Fleischer's calculation of the d i s l o c a t i o n solute interaction starting from f i r s t p r i n c i p l e s and gets rather similar r e s u l t s . 132 The present results conform to the predicted square-root C dependence of the f r i c t i o n stress ( f i g . 49). However,in order to make a quantitative v e r i f i c a t i o n possible, a knowledge of the concentration dependence of the l a t t i c e parameters and the shear modulus i s necessary. The concentration dependence of the l a t t i c e parameters i n the case of (31 119) Mg-Zn alloys has been determined by a number of workers ' . The (33) •linear size factor of 0.2 as computed by King w i l l be used i n the present calculations. The parameter n has never been evaluated for Mg-Zn a l l o y s . However, e l a s t i c constants of some of the Mg-base alloys have been evaluated as a function of solute concentration i n the case of Ag, In and Sn solutes(1^0)^ p r o m these results i t appears that those elements belonging to group I of the periodic table tend to increase the shear modulus of magnesium whereas those belonging to group I I I and IV decrease i t . Therefore, since Zn belongs to the same group i n the periodic table as magnesium, i t would be l o g i c a l to conclude that n w i l l be small i n the case of Mg-Zn al l o y s . Now i f we neglect n i n comparison with dx 6 and substitute the values of — r , G and e after d i f f e r e n t i a t i n g dC*! ' screw \, 1 equation (13) with respect to C 2 , Z comes out to be ^-jj > whereas i f Eedge ^ S u s e c* then Z becomes ^ \QQ • ^ t ^ s apparent that edge dis l o c a t i o n solute atom interaction does not give a satisfactory f i t . The value of Z can be improved i f we assume a linear dependence of T on e c . By 3/2 1 substituting £ g instead of E g - the best f i t i s obtained with Z = " J Q Q -This value of Z would become closer to (as estimated by Fleischer) i f 1 i s p o s i t i v e ; however, this speculation cannot be j u s t i f i e d u n t i l n i s evaluated experimentally. The linear dependence of on E g instead of 3/2 (121) e has,however,been observed i n the case of Ag base s o l i d solutions 133 The only other hexagonal al l o y system in.which the room temperature plateau has been measured i s the Mg-Li system. Yoshinaga' and H o r i u c h i ^ ^ have attributed the hardening to the Suzuki chemical strengthening me chani sin. although the experimental data conform to the theoretical prediction only upto 6 at.% L i . Above this concentration of L i the observed hardening i s much smaller than i s required by the Suzuki hardening mechanism. Yoshinaga and HoriU.chi's data has been plotted by h the present author against C as shown i n f i g . (72). I t i s apparent that C dependence holds good i n the case of Mg-Li alloys too. Thus i t i s seen that the f r i c t i o n stress mechanism does explain adequately the observed va r i a t i o n of the room,temperature plateau— stress with solute concentration i n the Mg-Zn and Mg-Li s o l i d solutions. 2.4.2.6. Strengthening at Low Solute Concentrations: An interesting feature of f i g s . (49) and (72) i s that the straight l i n e through the experimental points i n the against C plot does not pass through the o r i g i n . The intersection on the C'2 axis corresponds to a composition equal to 0.025 at.% i n the Mg-Zn system and 0.29 at.% i n the case of Mg-Li all o y s . A c r i t i c a l examination of Rogausch's^ 2^ data reveals that t h i s feature i s common to the Ag-base s o l i d solutions too. Rogauschs data are reproduced i n f i g (73). This effect has,however,not been explained e a r l i e r . The concentration independence of x^ w i l l be better understood i n the l i g h t of a knowledge of the low temperature rate c o n t r o l l i n g mechanisms i n the Mg-Zn all o y s . This w i l l be discussed i n a l a t e r chapter. 134 2 3 c inAt-% Fig. 73. T vs concentration for Ag s o l i d solutions (af Rogausch (2)). 135 2.4.2.7. Valency Effect: I t has been observed by several investigators that the r e l a t i v e valency of the solvent and the solute i s an important factor contributing i fc -u • (114,115,122) „ • -v. to solution strengthening . However, since the present results are best explained i n terms of the size and modulus interactions, the valency effect i s of no consequence. Valency effect i s i n fact incorporated i n the modulus interaction term. 2.4.2.8. Other Hardening Mechanisms: The only other -hardening mechanism not covered so far i s the long range order hardening. This mechanism i s of importance when dealing with the deformation of the super l a t t i c e s only and i s of no consequence i n the present study, which deals with d i l u t e solutions. 1 3 6 2.4.3. Solution Strengthening at Low Temperature: Before entering into the detailed discussions of the low temperature strengthening mechanismsit i s profitable to examine the various possible ways i n which the r-T curve can be altered by the addition of solute. P r i n c i p a l l y this can happen i n three ways as shown schematically i n f i g . (74). CD CD u 0 ) . - r l >< Temperature (a) Effect of solute on x only . Temperature (b) Effect of solution on x * only • Temperature (c) Increasing obstacle strength .• Fig. 74. The effect of solute on the x-T curve (Schematic) a) By increasing i n which uniformly elevates the x-T curve • b) By increasing the density of the short range obstacles. If this i s done strengthening w i l l be limited to temperatures below . c) By introducing stronger obstacles. This w i l l s h i f t -T to higher temperatures, 137 Experimentally the alloys containing 0.006 and 0.019 at.% Zn conform to the situ a t i o n i n (b). At higher Zn concentrations both x and x increase, however, the x-T curve i s not uniformly elevated. * This i s apparent from f i g . (75), where only x i s plotted against temperature for various solute concentrations. Here x has been obtained by subtracting x from the measured CRSS (after applying the shear modulus correction to x ). I t i s observed that the temperature s e n s i t i v i t y of x increases with increasing solute content. This again suggests that the short range order i s not an important hardening mechanism i n these alloys. Short range hardening i s inherently athermal, since the energy for nucleating forward s l i p never reaches a maximum value. As a segment of dislocation bows out the energy continuously increases due p r i n c i p a l l y to the disordering that i s induced across the s l i p plane. For t h i s mechanism, therefore, deformation must be induced exclusively mechanically by the application of a s u f f i c i e n t l y high stress to cause disordering. Therefore, i n order that short range ordering i s the 3x * p r i n c i p a l strengthening mechanism, (y^-) at low temperatures should be unaffected by the amount of solute present, which i s different from what i s observed experimentally. I t i s recognized that some mechanisms are thermally aetivatable under some conditions and athermal under others. For example, . weak C o t t r e l l atmosphere locking of dislocations i s thermally aetivatable, but more concentrated and stronger Cottrell-atmosphere locking can be athermal. Suzuki locking -constitutes a second example: weak Suzuki locking of p a r t i a l s having a narrow stacking f a u l t ribbon i s thermally aetivatable, but under otherwise similar conditions, i n alloys that have 138 0 100 200 300 400 Temperature °K * Fig. 75, vs. temperature for Mg-Zn single c r y s t a l s . 139 low stacking f a u l t energies, the unlocking mechanism i s athermal. The t r a n s i t i o n of C o t t r e l l locking from a thermally activatable to an athermal mechanism i s p r i n c i p a l l y due to the height of the activation energy ba r r i e r , whereas this t r a n s i t i o n for Suzuki looking arises from the fact that high thermal fluctuations i n energy can only occur over small volumes of the c r y s t a l . The fact that the width of the stacking fau l t i n magnesium is.rather narrow (due to high stacking f a u l t energy) makes Suzuki locking a p o s s i b i l i t y and secondly the low solute concentrations involved may be taken to include C o t t r e l l locking as a possible athermal mechanism i n the Mg-Zn al l o y s . However, with the help of arguments, similar to those described in.connection with the v a r i a t i o n of T _ , i t can be shown that neither i n trend nor quantitatively does Suzuki hardening explain the observed low temperature strengthening. Moreover,no single mechanism can explain the observed break i n the x-C plot at 0.025 at.% Zn. Therefore, at this juncture the i d e n t i f i c a t i o n of the short range obstacles i n Mg and the Mg-Zn alloys may be helpful i n determining the mechanism of solution strengthening at low temperatures. The j u s t i f i c a t i o n for such a procedure comes from the following analysis. I t i s now widely accepted that the obstacles giving r i s e to the short range stress T * i n close packed metals are the forest d i s -locations. Therefore, the low temperature strengthening i n Mg can arise from an indirect effect of solute, namely the change i n the.forest d i s -location density. From the present etch p i t counts i t i s apparent that the.forest dislocation density increases with increasing amounts of solute. However, experimentally there i s l i t t l e change i n the forest 140 * spacing beyond 0.04 at.% Zn, although the observed (g^ -) ( f i g . 76) continues to increase rapidly even at concentrations beyond 0.04 at.% Zn. This implies that the change i n the forest spacing may be the important factor i n solution strengthening at low solute concentrations, but at higher solute concentrations the short range obstacles must be different from the forest dislocations. The rate theory w i l l be used i n an effo r t to id e n t i f y the short range obs-tacles which determine the thermal component of the flow stress at higher solute concentrations. The possible role of the change i n the forest dislocation spacing as a solution strengthening mechanism can also be ascertained by comparing the observed activation length with the measured etch p i t spacing. 2.4.3.1. Thermally Activated Deformation: It has been pointed out e a r l i e r that obstacles are ess e n t i a l l y of two types, short range obstacles are effec t i v e over distances less than ten atomic diameters and long range obstacles possess stress f i e l d s of the order of ten atomic diameters or greater. Thermal energy i s able to assist the applied stress i n pushing the dislocation past the short range obstacles, but cannot aid i n getting dislocations past the stronger long range obstacles. Above T c a l l thermal barriers become transparent and dislocations move through the l a t t i c e unimpeded orice the applied stress has overcome the athermal bar r i e r s . Below T c, however, thermal barriers must be overcome by the assistance of stress and thermal energy i f an applied strain-rate i s to be maintained. Of the several thermal barriers encounted, the strongest determines the rate at which dislocations can move under given conditions of stress, s t r a i n rate and temperature. 141 Provided the thermally activated processes occur i n sequence and that the same event i s rate controlling throughout the l a t t i c e , the macroscopic s t r a i n rate y may be expressed as - AF/kT Y = V 6 (16) where Y q depends on the number and arrangement of the dislocations and their v i b r a t i o n a l frequency. AF i s the change i n Gibbs' free energy of the system i n surmounting the strongest obstacle. On the other hand, i f the thermally activated processes occur independently and the rate controlling step i s not the same throughout the l a t t i c e , then the s t r a i n rate w i l l follow the relationship^24^» Z - AF/kT Y o i e" (17) where the subscript "i" refers to the i * " * 1 kind of mechanism. Under such circumstances the application of the rate theory for the i d e n t i f i c a t i o n of the rate controlling mechanism i s not possible, since the extrapolation of the macroscopic thermodynamic measurements to a single activated event occurring i n a s p e c i f i c region of the l a t t i c e i s no longer possible. The change i n free energy for the thermally activated event can be expressed as j AH + I |S . I i - ( „ ) AF = 2i £ 1 - T . 3G G 3T where G i s the shear modulus and AH i s the activation enthalpy, given by AH . . (19) ^ Ho 142 V* i s i d e n t i f i e d as the activation volume and i s equal to v* The concept of activation volume arises when one considers the work W done on the system during an.activation event by the effective stress T * pushing a dislocation segment of length 1 a distance d (the activation distance) where b i s the Burgers vector of the dislocations. The term (b'dl) i s called the activation volume. The evaluation of the activation parameter AF, AH and V* involves g T* din y/y an evaluation of the p a r t i a l d i f f e r e n t i a l s ( . ) and I 12. at y/y T constant density, arrangement and number of mobile dislocations. For different alloys these f a c t o r s , w i l l be d i f f e r e n t , however, here the constancy refers to the situ a t i o n when one al l o y i s examined over a range of temperatures. Assuming the y i e l d stress to be governed by the same thermally activated mechanism over the range of temperatures of in t e r e s t , one can consider that the structure i s r e l a t i v e l y constant at y i e l d ^ ^"^ . Hence, on subtracting T from x ( i . e . CRSS) a plot of x* against Ax temperature i s obtained ( f i g . (75)) and the slope Og^ r) may be taken as ^ In *^  / ° ( y j — ) . The second p a r t i a l — ^ Q i s usually evaluated from d i f f e r e n t i a l s t r a i n rate change tests on a single specimen during flow at a fixed temperature. The flow stress difference accompanying instantaneous * changes of s t r a i n rate may be taken as va r i a t i o n i n effec t i v e stress Ax with y i f i t i s assumed that the structure remains constant during the instant of change. D i f f e r e n t i a l tests corresponding to a s t r a i n rate change by 2 a factor of 10 were carried out by varying the cross head speed on the instron from 0.002 to 0.2 ipm and vice versa using a push button speed selector. Experiments were performed at temperatures ranging from 143 78CJK to 295°K covering the entire range of temperature senitive y i e l d i n g . The details of the s t r a i n rate change tests are described i n appendix (C). 2.4.3.2. The Effect of Solute on the Apparent Activation Parameters: activation volumes were calculated for a series of Mg-Zn al l o y s . The effect of the applied stress and the solute concentration on the activation volume i s shown.in f i g . (77). These tests were, conducted at 78°K and the 3 activation volumes are expressed i n terms of b where b i s the Burgers vector of the glide dislocations. The measured activation volumes have been plotted against the resolved shear s t r a i n as shown i n f i g . (77) from which the activation volume at y i e l d can be obtained by an extrapolation to zero s t r a i n . I t i s observed that i n the case of magnesium the activation volume decreases rapidly at low s t r a i n s , but at higher strains the rate of decrease i s much smaller. The allo y containing .0.019 at.% Zn exhibits e s s e n t i a l l y similar behaviour as Mg except that the magnitude of V* i s smaller than that of the pure metal at low s t r a i n l e v e l s . The difference i n V* between magnesium and the low a l l o y , however, decreases rapidly so that at strains beyond 1.5 the two have the same activation volume at a constant s t r a i n l e v e l . The higher alloys containing from 0*15 to 0.45 at.% Zn, on the other hand, are associated with activation volumes which are less sensitive to both,stress and s t r a i n . This i s apparent from the fact that whereas i n Mg the act i v a t i o n volume 3 3 decreases from 4500 b at y i e l d to 400 b at fracture, the corresponding 3 change i n the case of an allo y containing 0.45 at.% Zn i s from 320 b to 230 b 3. From the relationship V* = kT (20) V=4500 b M g - Z n Single Crystals BASAL SLIP 7 8 ° K ^ o - o ^ ^ . r — ^ u _ OOI9Zn - o - ^ - o ^ ^ ^ ^ ^ o ^ ^ ^ ^ 0 15 Zn V ^ 0-45 Zn ^ 0 258 Zn SHEAR STRAIN F i g . 77. Activation volume vs. shear s t r a i n for Mg-Zn single crystals, 145 2.4.3.3. The Activation Volume at Y i e l d : A direct measurement of the activation volume at y i e l d i s rather d i f f i c u l t because of the uncertainty i n measuring Ax at low strains . In order to minimize error, the extrapolated value of V to zero s t r a i n has been used. The concentration dependence of the * activation volume at 78 K i s shown i n f i g . (78). I t i s clear that V 3 decreases dramatically from 4500 b i n pure magnesium to less than 1500 3 b by the addition of as l i t t l e as 0.05 at.% Zn. Between 0.05 and 0.45 at.% Zn, on the contrary, the activation volume decreases very s l i g h t l y . From f i g s . (77) and (78) i t appears that the alloys containing solute i n amounts less than 0.05 at.% belong to a category different from the one which the higher alloys belong to. The temperature dependence of the activation volume at y i e l d i s shown i n f i g . (79). Two compositions were chosen for t h i s purpose, Mg being the representative of the group.containing upto 0.05 at.% solute while 0.45 at.% Zn all o y belonging to the l a t t e r category. The activation volume at y i e l d was once again obtained by the method of extrapolation to zero s t r a i n . The activation volume i s observed to increase rapidly with temperature i n the case of magnesium, whereas the temperature s e n s i t i v i t y of V i n the case of 0.45 at,% Zn a l l o y i s considerably lower; especially at temperatures below 200°K. Profitable information may be obtained i f V i s plotted against x . This has been done.for Mg and 0.45 at.% Zn all o y as shown i n f i g . (80). The activation volume decreases d r a s t i c a l l y with increasing values of x i n the case of magnesium whereas there i s only a moderate decrease i n the case of 0.45 at.% Zn a l l o y . The values F i g . 79. Activation volume at y i e l d vs. temperature for Mg and Mg + 0.45 at. % Zn a l l o y single c r y s t a l s . 147 cn > e o > a o • U > O < 18,000 16,000 14,000 -12,000 10,000 8,000 6,000 4,000 2,000 Eig. 80, 100 200 300 * . / 2 i T i n gm/mm * N Activation volume at y i e l d vs. T for Mg and Mg + 0.45 at. % Zn al l o y single c r y s t a l s . 148 * * 3 of V at T - 0 for Mg comes out to be approximately 16,000 b and that 3 ( 5) for 0.45 % Zn a l l o y i s 3,300 b . Conrad, using an intersection model for Mg single c r y s t a l s , has shown that the nature of the force-distance curve may vary s l i g h t l y with temperature ( i . e . with x ) due to the influence of stress on the amount the g l i d i n g dislocatiois bow out on the s l i p plane thereby changing the effective forest spacing "1". This decrease i n activation volume with s t r a i n has been interpreted by (3) Basinski as a r i s i n g due to an increase i n the forest dislocation density Therefore, i f the activation volume i s to be used i n determining the forest spacing at y i e l d , then i t i s essential that the value at x = 0 be chosen. In this connection i t i s worth noting that the temperature v a r i a t i o n of the activation volume can be caused by the temperature dependence of the stacking f a u l t energy, which i n turn determines the activation distance "d". However, i n materials having high stacking f a u l t energy (e.g. Mg) the distance between the p a r t i a l s i s unlikely to be altered much through changes i n temperature and hence the activation distance "d" i s not l i k e l y to be the factor responsible for the temperature dependence of V . 2.4.3.4. Solution Effect on the Apparent Activation Energy: * Substituting for AH and V i n equation (18) the activation Gibbs free energy becomes AF = -kT 2 | - L I V 3 T / - ^ • -7T- I (21) 3x* The term ( -r-^ ; . j i s small and hence w i l l be neglected compared to \ 9 T • Gj (3X*.\ ~~vf J> AF can then be approximated to AH i.e. AF : AH : - V* T (|V)Y /y • (22) 1 4 9 I t i s important to note that when AH i s substituted for AF, i t i s AS /fc assumed that the entropy term e i s incorporated i n the pre-exponential term y . This approximation i s v a l i d only when the entropy change does not have s i g n i f i c a n t contribution to the over a l l free energy change. The term AH calculated using equation ( 1 9 ) does not include the work done by the effective stress during thermal activation. Correcting for this work done the t o t a l activation energy (enthalpy) AH 0 i s then expressed as AH0 AH + V * T * ( 2 3 ) where V T i s the work done during thermal activation. The variation of AHo with Zn content has been calculated using equations ( 1 9 , 2 0 ) and ( 2 3 ) and i s shown i n f i g . ( 8 1 ) . The term AHo should be labelled "apparent" activation enthalpy, since the effective stress before the activation event i s unknown. AH -o * would be the true activation enthalpy only when T = 0 . However, this has.not been done i n the present work. 2 . 4 . 3 . 5 . Zinc Alloying and the Thermally Activated Flow: In s o l i d solutions the mechanism of thermally activated de-formation can be any one of the following: 1 ) Intersection of forest dislocations, 2 ) Dislocation pinning by single solute atoms, 3 ) Tetragonal s t r a i n centres, 4) P e i e r l s , Pseudo P e i e r l s , and Recombination mechanism, 5 ) Cross s l i p , 6) Cottrell-Lomer Dissociation, 151 7) Fisher locking, and 8) Suzuki locking. Some of these mechanisms have been discussed e a r l i e r . For example Fisher and Suzuki locking mechanisms do not apply to the present case as shown i n the e a r l i e r part of this section. The mechanism of Cottrell-Lomer dissociation does not apply to the case of hexagonal structures. Also the case of Tetragonal s t r a i n centres i s inapplicable since i t applies to i n t e r s t i t i a l impurities only, while Zn i s a substitutional solute i n Mg. The remaining four w i l l be discussed i n the following pages. 2.4.3.5.1. Cross S l i p : (73) The activation volumes associated with the cross s l i p process are of the same order as those for intersection, both.being greater than 3 * 100 b . The experimentally determined V i n the Mg-Zn alloys are also 3 greater than 100 b . However, activation volume alone cannot be taken to decide between the cross s l i p and the intersection mechanisms. Additional information may be obtained from the following considerations. • In the range of temperature investigated, the only established cross s l i p planes i n magnesium are the basal {0001} and the prism {1010} planes. The CRSS for prismatic s l i p i n each of these alloys i s about two orders of magnitude higher than that for basal s l i p . Therefore, i t i s unlikely that cross s l i p would be possible at the onset of p l a s t i c flow. This conclusion i s supported by the absence of cross s l i p markings on the surface of the deformed c r y s t a l s , although the s l i p l i n e observations should not be taken as conclusive when o p t i c a l microscopy i s used for their 152 examination. Cross s l i p does play a s i g n i f i c a n t role i n the work hardening (29) of magnesium i n the stage A of deformation , however, th i s i s of l i t t l e consequence when CRSS i s the parameter under consideration. Thus i t may be concluded that cross s l i p cannot be the rate c o n t r o l l i n g mechanism i n basal s l i p . 2.4.3.5.2. P e i e r l s , Pseudo-Peierls and Recombination Mechanisms: Peier l s mechanism concerns the thermal activation of disloc a t i o n motion past linear obstacles. A rather complete review of this mechanism has been given recently by Guyot and D o r n ^ 2 ^ . The pseudo-Peierls mechanism was o r i g i n a l l y proposed by Dorn to explain the thermally activated flow of Mg-Li a l l o y s . I t i s a s l i g h t modification of the Peie r l s mechanism for the case where the two p a r t i a l s are very close together (0.5 to 2b). The stacking fa u l t energy plays a s i g n i f i c a n t role i n the Pseudo Peie r l s mechanism whereas the Peierls-Nabarro theory does not take t h i s into consideration^"*" 2^ . These mechanisms w i l l be discussed i n greater d e t a i l i n connection with prismatic s l i p . Recombination i s thought to be a successful mechanism i n explaining e f f e c t i v e l y the orientation dependence and the i n i t i a l rates of s t r a i n hardening i n bcc metals ^ 2 ^ . Without going into any further d e t a i l s of the above three mechanisms, i t should be pointed out that the activation volume associated 3 with each of these mechanisms is less than 80 b , whereas the * 3 experimentally determined values of V are greater than 300 b . Therefore, these mechanisms are not applicable to the present case. 153 2.4.3.5.3. The Intersection Model: * I t i s now well recognized that the obstacles determining x i n the basal glide of magnesium are the forest dislocations. In view of the fact that the grown i n dislocation density does change i n many instances of close packed structures by the addition of solute, there i s reason to believe that the indirect solution strengthening due to such a change can be of importance i f the strengthening due to direct interaction i s r e l a t i v e l y small. In the past attempt has been made by Hendrickson and Fine to explain the observed strength of the Ag-Al alloys i n terms of increased grown i n dislocation densities. However, 10 -2 such an explanation requires the presence of dislocations upto 10 cm i n the as grown c r y s t a l , which i s too high to be met i n practice. The case of Mg-Zn w i l l now be examined i n the l i g h t of the grown i n forest dislocation density (the increase i n basal dislocation density can contribute to r only and therefore w i l l not be considered here since x* i s being dealt with at the present). The activation volume at 78°K for each of the Mg-Zn alloys 3 tested i s greater than 100 b ( f i g . 78) thus forest intersection can be the rate controlling mechanism i n these al l o y s . ft ft From f i g . (80) the values of V at x = 0 can be used to calculate the forest dislocation density at y i e l d i n Mg and 0.45 at.% Zn alloy using the following expression. V* = b.d.l. (24) If a simple assumption i s now made that the activation distance "d" can be approximated by the Burgers vector "b" then * 9 V = l> .1 (25) 154 Again, assuming a square array, the forest dislocation density p= —_ (26) 1 6 —2 Such calculations i n the two cases y i e l d p = 3.8 x 10 cm 7 -2 and p„ ,,- „ = 9 x 1 0 cm . A similar calculation i n Mg by Conrad 0.45 Zn —A (68) -4 yielded 1 = 1.5 x 10 cm i n one set of experiments, ' and 1 = 3.0 x 10 cm J o o i n another ^ \ Conrad has attributed this inconsistency i n his results to the difference i n growth conditions, impurity content and the effect of recovery treatment on 1 q . The calculated p w =3.8 x 10^ i s two orders of magnitude Mg higher than the etch p i t density. This discrepancy has been discussed already. Here we w i l l compare the r a t i o p anQy/ P ^ g with the corresponding etch p i t figure. From the activation volume measurements P Jo =24 whereas the corresponding r a t i o of etch p i t s = 4. 0.45 Zn M M g This implies that either the assumption that "d" remains constant i s not true or the basic assumption that the rate c o n t r o l l i n g mechanism i n Mg 0.45 at<>% Zn a l l o y i s one of intersection i s wrong. Now l e t us consider the v a r i a t i o n of "d" with solute concentration. If i t i s the width of the stacking fa u l t that i s determining "d'^then "d" cannot decrease to a value less than 0.5 - 2 b, which i s the distance between.the p a r t i a l s i n Mg. Therefore, the only other p o s s i b i l i t y i s that "d" increases. I f i t does then the r a t i o P a l l o v -p ' becomes even,larger than 24, making the discrepancy worse. Therefore, the f i r s t assumption that the rate c o n t r o l l i n g mechanism i n 0.45 at.% Zn a l l o y i s one of intersection of forest trees must be wrong. / 155 a) Intersection Model at Low Solute Concentrations: The etch p i t density increases rather rapidly by the addition 1/2 of Zn upto 0.04 at.% and the break i n the C - x plot i s at 0.025 at.% Zn. Therefore, i t i s l i k e l y that the addition of solute.in such small quantitites w i l l not a l t e r the rate controlling mechanism of magnesium and hence strengthening could be due to increased forest density. In order to put P0 025 Z this suggestion to test l e t us compare '• derived from Mg activation volume measurements with the r a t i o of etch p i t s at a comparable * * situa t i o n . In the absence of the value of V for 0.025 Zn al l o y at T = 0, the activation volumes at 78°K w i l l be used. This w i l l not be too * erroneous because the temperature effect on V w i l l be approximately of the same order i n the two cases. .0»025 Zn evaluated i n t h i s manner i s P Mg equal to 3.2, the corresponding etch p i t r a t i o being 2.5. In view of the experimental error involved i n the measurement of the activation volume th i s may be regarded as a good agreement. Hence the thermally activated mechanism i n Mg and the alloys containing upto 0.025 at.% Zn i s one of intersection. What remains to be seen i s which of the rate controlling mechanisms can account for the flow i n the higher a l l o y s . The only mechanism not discussed so far i s the single solute atom dislocation pinning mechanism. This w i l l be applied to both low and high Zn alloys i n order to see i f i t can provide an alternative to the intersection model and to test i t s a p p l i c a b i l i t y at higher concentrations of solute. 156 2.4.3.5.4. Dislocation Pinning by the Solute Atoms: A model for the low temperature short range solute dislocation interaction has been formulated b y ' F r i e d e l ^ ^ ^ . According to this model the dislocation moves i n a zigzagging fashion from one position of maximum core interaction energy to the next. The si t u a t i o n i s schematically re-presented i n f i g . (82). Fig. 82. Dislocation pinning by solute atoms giving r i s e to low temperature f r i c t i o n stress (Schematic). The stress x° to do this at T = 0 i s determined by the interaction energy and the concentration C of the solute according to the equation: For T= 0: x° bLx = U (27) ' o m In the case of hexagonal close packed metals, 2 b 3 V = bLx = Activation Volume = — . — , and (28) 0 /3 " C 3 i n fee metals bLx = b /Q (note C i s atom f r a c t i o n , not at.%) T ° U o = =H . (29) o /3 Um " kTlh-(? /Y) for T>0 x (T.) •= 4 • c — , ° (30) 157 At f i n i t e temperatures the f r i c t i o n stress i s decreased by thermal activation expressed i n i t s simplest form i n equation (30). In order to make a comparison between the theory and the experiment we w i l l f i r s t calculate the activation volume as a function of the. solute concentration according to equation (28). Since the measured activation volume varies with temperature i t i s essential to extrapolate these values to 0 ° K and then compare with the V q obtained from equation (28). This has been done i n f i g . (83). The s o l i d l i n e represents the calculated value whereas the dashed l i n e follows the extrapolated values of the activation volume to 0 ° K . I t i s apparent that the theroetical prediction agrees with the experimental results only at concentrations of solute greater than 0.3 at.%. There i s marked deviation at low concentrations specially at concentrations below 0.05 at.% Zn, implying that the pinning of dislocation by single solute.atoms i s an unlikely mechanism at low concentrations, although at higher concentrations such a mechanism i s possible. Additional support for the dislocation pinning by single solute atoms comes from the fact that the higher temperature flow stress i s adequately explained by the f r i c t i o n stress mechanism i n these a l l o y s , which again involves the concept of single solute atoms interacting with d i s -location l i n e s . Using x° for 0.45 at.% Zn all o y and the activation volume at 0 ° K the value of U i s found to be 0.48 e.V., which i s much too low m compared to that obtained by the s t r a i n r a t i o change tests at 78 ° K . (H Q = 1.237 e.V.). A similar discrepancy was encountered by Rogausch^ "''2'''^  i n the case of Ag-In a l l o y s . ) 7000 Comparison of Friedel's Model with the Experimental Results. — Extrapolated to 0°K — Calculated using eqn. 0 2 0-3 ATOMIC% Zn 0-4 0 5 F i g . 83. Comparison cf the. experimental-results.with, the composition.dependence of activation volume as predicted by-. Friedel^s model. 2000 1600 «M 6» 1200 .c 800 400 1 V K / <-> * Silver-Al o Silver -Ir uminium dium 0 2 4 6 c in At.-% Fig. 84. Concentration dependence of CRSS extrapolated to 0°K for Ag-rln and Ag-Al single crystals (after Haasen^ 2'). oo 159 F i n a l l y a remark may be made on the predicted l i n e a r i t y of T° on C as represented by equation (30). The present values of T° do not.show a l i n e a r relationship with C. E a r l i e r experiments of Rogausch on Ag-In and those of Hendrickson and Fine on Ag-Al alloys reveal that T° i s a linear function of the solute concentration. The two sets o of data plotted together are reproduced i n f i g . (84). A closer examination of these data shows that the straight l i n e when extrapolated to zero con-o 2 centration of solute gives T q for Ag = 380 gm/mm whereas the actual o / 2 T from Rogauschs data comes out to be less than 150 gm/mm . What this o implies i s that x° i s not a linear function of C at low solute concentrations. The present experiments were performed on d i l u t e Mg-Zn alloys and this i s why T° i s non-linear with respect to C. The reason for such a non l i n e a r i t y being that i n this concentration range there exists a t r a n s i t i o n from the intersection mechanism to the solute atom dislo c a t i o n pinning mechanism. (85) Recent experiments of Hendrickson on Ag s o l i d solutions suggest that the disloc a t i o n density i s not affected much by solute addition. However, t n e s c a t t e r i n the dislocation density measurements was too high i n Hendricksons experiments to distinguish the change of density by a factor of two or three. 2.4.3.6. Strengthening Mechanism i n Dilute Alloys: The solution strengthening parameter Sj when plotted against temperature, shows a linear-decrease upto 330°K, above this temperature, however, no solution strengthening is,observed. The most obvious conclusion would be that the addition of solute upto 0.025 at.% Zn does not contribute to the long range stress f i e l d . 160 Moreover, since solution strengthening i s observed at low temperatures, i t must be due to some sort of a short range interaction process. I t was shown i n a previous section that the rate c o n t r o l l i n g mechanism i n Mg and the low alloys under consideration i s one of intersection of forest dislocations. Also, since the experimental results have shown that the forest dislocation density does increase rapidly with alloying i n the low a l l o y s , i t i s most l i k e l y that solution strengthening i s caused by the presence of a larger number of forest dislocations. The constancy of T £ i n the CRSS-temperature plot further substantiates the above conclusion. In the presence of available data on the extent of increase i n the forest density with solute concentration, i t i s possible to make a quantitative comparison between the observed strength of the allo y with that predicted by the higher forest density. Assuming the forest dislocations to form a regular "square" array,Seeger^^ 8- has developed an equation which relates the stress required for the intersection process to the forest spacing 1 as follows: 2 , r ar-d kT - pb v " i . . . . T = TG I bT~d" ~ bT~d l n I L ~ M ( 3 1 ) L s s f y J where p i s the density of the glide dislocations j f i s the fr a c t i o n of the s l i p plane over which the maximum average amplitude i s TQ v i s the Debye frequency , T i s the average l i n e energy , Y i s the s t r a i n rate d i s .the effective diameter of the forest dislocations , a i s a strength factor such that, aT i s the strength of the obstacle, the rest of the symbols have their usual meanings• 161 Moreover, x = T g at T c < T (32) therefore , ard = kT In p b (33) fy where i s the c r i t i c a l temperature above which thermal fluctuations i n energy greater than ard occur as frequently as i s needed to maintain * the imposed s t r a i n rate even when r = 0. To this approximation the mechanical behaviour for cutting a.simple type of localized obstacle, the stress to induce flow at the absolute zero i s given by To = V + bl o s On the basis of this model the behaviour.of the d i l u t e Mg-Zn alloys w i l l now be examined. I t i s clear from the CRSS-temperature relationships i n f i g . (47) that the flow stress of magnesium becomes athermal at 330°K. Also this value of T i s unaffected by solute i n c J the 0.006 and 0.019 at.% Zn a l l o y s . This constancy of T £ implies that the strength of the obstacles remains unchanged by the addition of solute i. e . T g i s independent of zinc concentration. Under such circumstances the thermally activated component of flow stress can be affected only through a change i n 1 (bearing i n mind here that T i s constant). In S ( j other words a change i n the forest dislocation density i s responsible for solution strengthening at low concentrations. Now equation (34) can be rewritten as follows: T Q = A + f - (35) 2 where A = T^ 0= 40 gm/mm , independent of Zn concentration and ar 0 B = — — . 162 B w i l l also be taken as independent of Zn concentration since the var i a t i o n i n b and the shear modulus with concentration are much smaller than that i n 1 ( T i s a function of the shear modulus). s o Now expressing T q i n terms of forest dislocation density To = A + B / i 7 (36) or, T 0 = A + B /p^ T" (37) where p i s the forest dislocation density, r However, at low solute concentrations p^ = 3.4 x 10 4 [ 1 + 5.35 x 104C] cm 2 (3a) r Substituting i n equation (37) the value of p F 4 1 / 2 x 0 = A + 184.4 x B (1 + 5.35 x 10 C) (38) Using the data for pure magnesium, the constants A and B can be determined. A and B come out to be 2 A = 48.8 gm/mm (taking G into account) B = 0.196 gm/mm Therefore, T q * 48.8 + 36,2 (1 + 5.35 x 1 0 4 C ) 1 / 2 gm/mm2 (39) The CRSS of 0.006 and 0.019 at.% Zn al l o y calculated using equation (39) are shown i n the table below. TABLE IV Strengthening due to increased forest density . At. % Zn T° Extrapolated T° Calculated i n gm/mrn^  i n gm/mm^  0.006 106 122 0.019 130 168 163 I t i s seen from the Table IV that the calculated values come out to be higher than the CRSS extrapolated to 0°K. However, the extrapolation i s done over too large a temperature range to be considered accurate. A quantitative comparison w i l l be meaningful only when experimental data are available close to absolute zero. Nevertheless, . qu a l i t a t i v e l y the C 2 dependence of the CRSS i s i n agreement with the dictates of the forest intersection mechanism. Further evidence i n support of t h i s conclusion comes from the activation volume measurements. This has been discussed e a r l i e r . I t i s worth while examining at this juncture the assumptions involved i n the idealized prototype model of Seeger which has been used for calculating x . Deviations from the predicted value can occur due to the following reasons. 1) the force displacement diagram can d i f f e r from the simple case assumed i n calculating equation (35). However, i n the case of magnesium th i s approximation i s j u s t i f i e d , because i t i s a case of intersection of undissociated basal glide dislocations with unreactive forest dislocations. 2.4.3.6.1. Deviations from Square Array: Obstacles never form a regular array as assumed i n formulating equation (35). Enough progress has been made on the s t a t i s t i c a l treatment of the problem to suggest that randomly dispersed obstacles give much lower stresses over the lower temperature range than i s obtained * «.!, (129,130) _ from the square arrays • Over the higher temperature range, A however, the stress x for the random d i s t r i b u t i o n of obstacles l i e s above that predicted for a square array model and decreases very slowly with increasing temperature. 164 Whereas i t was assumed that a single cutting would permit 2 the dislocations to move only over the average area I i n the square array, i t has been demonstrated by K o c k s ^ 3 ^ and by the computerized (129) experiments of Foreman and Makin , that a larger area i s swept out. This arises because once a cutting has been achieved there i s a certain probability the dislocation can unzip past the next neighbour etc. In view of these limitations inherent i n the theoretical formulation of equation (35) the observed values of x° may be considered to be i n good agreement with the theoretical predictions. 2.4.4. Work Hardening of Mg Solid Solutions: In the past much of the work i n solution strengthening has been concerned with the understanding of the fundamental strengthening mechanisms governing the onset of p l a s t i c flow. However, attempts to correlate the stress s t r a i n curves of single crystals to those.of the corresponding poly c r y s t a l l i n e aggregates have revealed that relationships (131-135) do exist between the two sets of work hardening parameters Keeping i n mind that the ultimate objective of the single c r y s t a l study i s to lead to an understanding of the macroscopic flow characteristics of the polyerystals, a study of solution strengthening would be incomplete without a discussion of the work hardening behaviour i n allo y single c r y s t a l s . The work hardening characteristics of the Mg-Zn al l o y single, c r y s t a l oriented for basal s l i p are discussed below. 165 2.4 .4 .1c The Easy Glide i n Magnesium: I t was f i r s t pointed out by Mott that the deformation mechanisms i n face centred cubic crystals i n stage I and i n Hexagonal close packed crystals i n stage A are si m i l a r . The work hardening rate -4 -5 i n easy glide i s t y p i c a l l y 10 - 10 G and i n both cases i t appears that the crystals deform almost.entirely on one s l i p plane. Two p r i n c i p a l theories have been proposed to explain the (137) work hardening i n easy g l i d e : that of Seeger et a l and of Hirsch and (29) (138) L a l l y . Hirsch's theory has been extended further by Hazzledine Essentially both the theories assume that Frank-Read sources are present i n the c r y s t a l and that they emit dislocations under applied stress. In Seeger's model i t i s assumed that a certain number of sources remain active throughout stage I and emit dislocations gradually during deformation. Seeger's model i s a close approximation to the ease of Cu, Zn, Ni-Co alloys (29) etc. On the other hand, s l i p l i n e studies i n Mg indicate that only a f r a c t i o n of the sources are active i n any small s t r a i n i n t e r v a l , the density of active sources i s independent of s t r a i n and dislocations are (1 38^ emitted i n bursts from the sources . Hirsch's theory was o r i g i n a l l y , designed for the case of Mg and hence w i l l be discussed i n b r i e f . In Hirsch's model the dislocations from sources operating simultaneously trap one another and form dipole bands for edges and screws; the screws annihilate by cross s l i p leaving edges and an excess of screws of one sign. The flow stress i s controlled by the i n t e r n a l stress from the excess edge.and screw dislocations and from those with non primary basal Burgers vectors. Since the dislocations are paired off or annihilate, t h e i r contribution to work hardening i s expected to be low. 166 In this model the dislocation arrangement i s s t a b i l i z e d by a f r i c t i o n a l force of some kind and the prism segments or the jogs are believed to constitute the Frank-Read sources. The increase i n the hardening rate i n Mg with decreasing temperature could be explained as follows. Since the jogs are more d i f f i c u l t to move at lower temperatures, the effective source length w i l l become smaller as the temperature decreases. The number of potential sources i s also l i k e l y to increase, because the minimum height of a jog which can serve as anchoring point for a source would be smaller (the stable jog height being temperature dependent). Moreover, since the CRSS for prismatic s l i p increases rapidly with decreasing temperature, the screws are. less l i k e l y to annihilate and w i l l probably form dipole bands. The increase i n f r i c t i o n stress due to the presence of jogs, which are d i f f i c u l t to move, w i l l lead to a s i t u a t i o n where the dipole bands can support a larger number of excess dislocations of one sign. (3) Basinski has observed that Cottrell-Stokes' law i s obeyed i n magnesium below 60°K. However, at higher temperatures deviations occur; suggesting the presence of two short-range processes. This observation has been rationalized i n the l i g h t of Hirsch's theory as follows. The density of the prism segments produced by cross s l i p increases with increasing s t r a i n . Intersection of these segments leads to the formation of kinks, which i s a thermally activated process and, therefore, contributes to the decreasing activation volume. The o r i g i n of the observed Cottrell—Stokes law can be attributed to the proportionality between the prism segments produced by c r o s s . s l i p , and the density of the screw dislocations (before a n n i h i l a t i o n ) , when annihilation occurs. 167 The present results are i n f a i r l y good agreement with the above theory. 2.4.4.1.2. Easy Glide i n Alloy Crystals: It i s now well established that i n the case of s o l i d solutions having fee structure, the extent of easy glide increases with solute addition, the work hardening rate i n stage I remains unaffected i n systems showing limited s o l i d s o l u b i l i t y , and decreases where the (2 139) s o l u b i l i t y i s extensive ' . In body centred cubic metals the effects are ir r e g u l a r , possibly because of uncontrolled i n t e r s t i t i a l impurities ^  . The work hardening behaviour of hexagonal a l l o y single c r y s t a l s , however, has not been investigated thoroughly. The early work of Schmid and co-workers had indicated that the work hardening rate i n Mg increases with solute concentration i n the case of Zn and ( 88) A l solutes . On the other hand Zn exhibits a lowering i n 0^ with + ^(88,147) increasing impurity content An attempt w i l l be made here to explain q u a l i t a t i v e l y the observed dependence of O^on Zn concentration in the l i g h t of Hirschs theory of work hardening i n Mg^ 2^\ The dislocation sources i n Mg are the prism segments or jogs. The effect of solute upto 0.025 at.% Zn i s to increase the forest dislocation density. This leads to an increase i n the number of Frank-Read sources resulting i n a larger number of dipole bands being formed, which i n turn increase the long range stress f i e l d and a more rapid hardening r e s u l t s . Moreover, since the CRSS for prismatic s l i p 1 6 8 becomes higher, the screws are less l i k e l y to annihilate and w i l l probably form dipole bands, although the contribution from the l a t t e r (29) source would be small as pointed out by Hirsch An increase i n the solute content above 0.025 at.% Zn results i n a rather rapidly increasing f r i c t i o n stress arising from the size and modulus interactions. In the presence of a higher f r i c t i o n stress the dipole clusters become d i f f i c u l t to move. In f a c t , the magnitude of the f r i c t i o n stress determines the way i n which the excess dislocations d i s t r i b u t e themselves. Thus the presence of solute w i l l lead to a higher work hardening rate through a change i n the f r i c t i o n stress. 2,4.4.2. The Temperature Dependence of 9^ i n Alloy Crystals: The.low temperature work hardening i n the easy glide of al l o y crystals containing upto 0.025 at.% Zn, can be explained i n much the same manner as i n pure. Mg. As shown e a r l i e r , the rate c o n t r o l l i n g mechanism at,the onset of flow i n Mg and the low alloys i s one of intersection. The increase i n 9^ i n these a l l o y s , however, arises due to the following reason. The presence of a larger number of jogs i n these alloys increases the low temperature f r i c t i o n stress, enabling the dipole bands to support a larger number of excess dislocations of one sign. The increased s t a b i l i t y of the dipole bands, leads to a more rapidly increasing flow stress with s t r a i n . When a dislocation moves at a low temperature on the basal plane of an a l l o y c r y s t a l , containing solute i n excess of 0.025 at.% Zn^ the long range f r i c t i o n stress i s overcome (as at higher temperatures) and also the forest dislocations are intersected. In addition a short range obstacle must also be surmounted. The f r i c t i o n stress a r i s i n g due to this short 169 range solute dislocation interaction increases with decreasing temperature. Therefore, the work hardening rate 0 A should increase with decreasing temperature as the solute content i s increased. Q u a l i t a t i v e l y this i s i n agreement with experiment as can be seen from f i g . (54). The parabolic nature of the concentration dependence of 9^ arises probably due to a C relationship between the increase i n forest density with solute content at low concentration and that between the f r i c t i o n stress as the Zn content i s increased. 2.4.4.3. The Extent of Easy.Glide: The face centred cubic metals Cu, Ni and A l show a continuous decrease i n Y J J with increasing t e m p e r a t u r e . On the other hand the hexagonal metals Cd^ 2^ and Z n ^ 4 ^ exhibit an increase i n the extent of easy glide with increasing temperature. The present results on magnesium indicate an increase i n Y B with temperature upto 295°K. Above.this temperature, however, easy glide i s followed by a parabolic hardening, which masks stage B. Hence the decrease i n Y-g beyond 295°K as shown i n f i g . (56) should be accepted with reservation. The reason for the termination of easy glide and the onset of the rapid hardening i s not the same i n a l l materials. For example s l i p on the secondary systems i s known to be responsible for the onset of stage I I i n the fee metals and a l l o y s . In hep Zn and Cd, the condensation of vacancies into se s s i l e loops i s considered responsible for the rapid u A • d37) hardening 1 7 0 No well established theory, however, exists for the case of (29) magnesium. I t has been suggested by Hirsch that the density of dipole bands becomes so large at the end of stage A that they trap the new dislocations formed. The stress concentration,therefore,increases leading to twins being nucleated, which subsequently act as barriers to the new s l i p l i n e s . The pairing of the newly formed dislocations i s largely hindered because of the twins. These dislocations p i l e up, leading to a stress concentration. The stress concentration i s relieved by s l i p on the basal plane with non-primary Burgers vectors, prismatic s l i p and further twinning. The observed decrease i n Y-g with alloying can be rationalized i n terms of an increase i n 9 w i t h alloying. Since the work hardening rate increases with a l l o y i n g , the stress to nucleate twins i s reached e a r l i e r i n the al l o y than i n Mg, provided the shear stress for twin formation remains unaffected by solute. However, even i f the stress . required for twinning increases with solute content, Yg can s t i l l decrease with alloying provided the increase i n 9-^ i s s u f f i c i e n t l y large so as to a t t a i n the stress l e v e l necessary to nucleate twins at a s t r a i n less than Yg i n Mg. The increase i n Yg with temperature i n these alloys can perhaps be rationalized i n a similar manner considering the high temperature dependence of 9. and a low temperature s e n s i t i v i t y of the twinning stress. 171 2.4.4.4. Work Hardening i n Stage B: The work hardening rate i n stage I I of fee metals i s -3 t y p i c a l l y 9.^ = 3 x 10 G at room temperature. The work hardening rate 9 for Mg at room temperature i s -4 3.7 x 10 G, which i s approximately an order of magnitude lower than i n fee metals. The low work hardening rate i s thought to be due to the high stress necessary to nucleate twins, which can subsequently act as (29) barriers to the formation of s l i p l ines The var i a t i o n of 9 „ with temperature has never been explained a i n the past. However, on the basis of a twin model, such a decrease i s to be expected. Stress r e l i e f at the dipole bands i s attained not only through twinning, but also through slip on the basal plane with non-primary Burgers vectors and by prismatic s l i p . Between room temperature and 423°K the CRSS for basal s l i p remains unchanged^therefore^excess stress r e l i e f at higher temperature i s not expected to be due to s l i p on basal plane with non-primary Burgers vectors. However, i n th i s temperature range the CRSS for prismatic s l i p decreases rather rapidly. Thus the extent of stress concentration becomes smaller due to the^operation of prism s l i p . This leads to a slower process of twin nucleation and hence the work hardening rate decreases. The Effect of Solute on 9 C : The CRSS for basal s l i p increases with solute content i n alloys containing solute i n excess of 0.025 at.% Zn, whereas the operation of prismatic s l i p becomes more d i f f i c u l t i n the lower a l l o y s . Both these processes would lead to a higher stress concentration at the dipole bands 172 than i n Mg. The stress concentration w i l l result i n a larger number of twins being nucleated and thus the work hardening rate w i l l be higher. 2.4.4.5. The Stage 'C' of Deformation: The stage I I I of work hardening i n face centred cubic s o l i d T I I I C a n (141-142) solutions i s associated with cross s l i p and therefore T ^ j _ c a n be.used to calculate the stacking f a u l t energy of the material The o r i g i n of stage C i n magnesium, however, i s not well understood. Crystals deformed into stage C do not show cross s l i p markings when examined under o p t i c a l microscope. The s l i p l i n e s appear i n coarse clusters. The exsistence of fine cross s l i p markings, however, cannot be ruled out. Replica studies of the s l i p l ines may resolve t h i s problem. A discussion of the effect of solute on x^ , . i n Mg w i l l be premature, u n t i l the reason for the dynamic recovery i s established. 173 2.4,5. The Effect of Solute on the Ease of Prismatic S l i p : 2.4.5.1. Introduction and Objectives: The role of prismatic s l i p i n the deformation of the po l y c r y s t a l l i n e aggregates of Mg and i t s alloys has been pointed out e a r l i e r i n this thesis. However, the study of single crystals oriented for prismatic s l i p , which should constitute the ground work for the understanding of the macroscopic deformation characteristics of Mg polyerystals has not attracted as much attention as i t deserved. A rather comprehensive l i t e r a t u r e i s available on the prismatic s l i p of Mg-Li a l l o y s , which exhibit the so called " s o l i d solution softening e f f e c t " ^ 4 16,73)^ ^ has been the conclusion of Dorn and his coworkers that only those elements which decrease the c/a r a t i o of magnesium would be effective i n decreasing the CRSS for prismatic s l i p when dissolved i n magnesium. Incidentally,the elements which decrease the a x i a l r a t i o of Mg belong to group I of the periodic table. In contrast to this conclusion the present results on the deformation of the po l y c r y s t a l l i n e s o l i d solutions of Mg containing solutes which belong to group I I , I I I and IV of the periodic table (and hence increase the c/a r a t i o of Mg), suggest that irrespective of the type of solute added the CRSS for prismatic s l i p decreases with increasing amounts of solute beyond a c r i t i c a l concentration C^ ,. Thus one of the objectives of the investigation of prismatic s l i p i n single crystals was to evaluate the significance of c/a r a t i o (and hence valency) on the ease of prismatic s l i p . The second.objective was to examine the effects of alloying at very low concentrations, where a rapid rate of solution hardening i s observed i n the p o l y c r y s t a l l i n e a l l o y s . 174 For this purpose the solute chosen, was Zn. The choice of Zn was mainly due to the reason that the poiyerystalline alloys of the Mg-Zn system exhibit a l l the three stages of solution hardening d i s t i n c t l y . Also the c/a r a t i o of Mg does not decrease by the addition of Zn. Some alloys containing A l as solute were also investigated. Aluminum belongs to a higher valency group (III) and increases the c/a r a t i o of Mg. 2.4.5.2. Experimental Results: 2.4.5.2.1. The Stress-Strain Curves: The resolved shear stress vs shear s t r a i n curves of a representative set of Mg-Zn alloys are shown as a function of. temperature for various compositions i n f i g s . (85-88). I t i s apparent that the d u c t i l i t y i s considerably lower i n prismatic s l i p than i n basal s l i p . For example pure magnesium undergoes fracture at a shear s t r a i n of 580% i n basal s l i p whereas f a i l u r e occurs at a s t r a i n less than 2% i n prism s l i p . In prismatic s l i p , the shear stress increases i n a parabolic manner with s t r a i n at low s t r a i n levels followed by a linea r hardening at higher strains. In some cases a well defined t h i r d stage with a decreasing work hardening rate followed the linear hardening. In order to make the effect of the solute apparent, the stress-s t r a i n -curves of the alloys deformed at fixed temperatures are placed together and are shown i n f i g s . (89-90). I t i s observed that at lower temperatures (78-200°K) the st r e s s - s t r a i n curves of a l l the al l o y single crystals f a l l below that of magnesium. But at temperatures above.200°K small additions of solute (up to 0.019 at.% Zn) l i f t the curves above that of pure magnesium. On increasing the alloying content beyond 0.02 at.%, however, the stress - s t r a i n curves f a l l below those for the lower a l l o y s . 175 0 0.04 0.08 0.12 Shear s t r a i n Fig. 85. Resolved shear stress vs. shear s t r a i n curves for Mg single crystals oriented for prismatic s l i p . 176 0 0.04 0.08 0.12 Shear s t r a i n Fig. 86. Resolved shear stress vs. shear s t r a i n for Mg + 0.019 at. % Zn all o y single crystals oriented for prismatic s l i p . F i g . 87. Resolved shear stress vs. shear s t r a i n curves for Mg +0.258 at. % Zn a l l o y single c r y s t a l s oriented for prismatic s l i p . F i g . 88. Resolved shear stress vs. shear s t r a i n curves for Mg + 0.45 at. % Zn all o y single crystals oriented for prismatic s l i p . 179 2 I • I I I l _ L 0 0.2 0.3 0.6 Shear s t r a i n . F ig. 89. Resolved shear stress vs. shear s t r a i n curves for Mg-Zn single crystals deformed at 423 °K i n prism s l i p orientation. I I I L_ I 0 0.04 0,08 0.12 0,16 Shear s t r a i n Fig. 90. Resolved,shear stress, vs. shear s t r a i n curves for Mg-Zn single crystals deformed at 78°K i n prism s l i p orientation. 181 An interesting feature of the crystals tested at 423°K i s the large amount of s t r a i n associated with work-softening. Similar work softening has been observed above room temperature i n the compression of single crystals of Mg oriented such as to suppress basal s l i p ^ " ^ . Backofen et a l have attributed the softening to the onset of {1011} {1012} double twinning. However, i n the present tens i l e tests at 423°K the crystals showed necking, therefore, i t cannot be concluded whether or not twinning i s responsible for the loss i n work hardening rate. 2.4.5.2.2. D u c t i l i t y : The shear s t r a i n to fracture i s plotted against the testing temperature i n f i g . (91) for a l l the Mg-Zn alloys tested i n orientations favourable to prismatic s l i p . I t i s observed that d u c t i l i t y decreases with increasing temperature i n the range 78°-250°K and increases again at higher temperatures. A similar trend has been observed by Reed-Hill and Robertson^ i n pure Mg. In spite of the l^rge scatter i n data i t can be seen that d u c t i l i t y decreases by small additions of solute at temperatures above 250°K, followed by a rapid increase at higher solute concentrations. 2.4.5.2.3. The CRSS for Prismatic S l i p : The f i r s t deviation from l i n e a r i t y was once again used as the y i e l d c r i t e r i o n . The temperature and al l o y dependence of the CRSS for prismatic s l i p i n Mg-Zn alloys i s shown i n f i g . (92). The temperature dependence of the CRSS may be divided roughly into three regions. A r e l a t i v e l y thermal insensitive region exists at temperatures between 78°K and 170°K, followed by a strongly temperature dependent stage. The temperature dependent stage merges gradually into a F i g , 91. Shear s t r a i n to fracture vs. temperature for Mg-Zn single crystals oriented for prismatic s l i p . 183 region of lesser temperature s e n s i t i v i t y at temperatures i n the v i c i n i t y of 400<iK. The numerical values of CRSS for Mg obtained i n the present (143) work are i n agreement with the values reported i n the l i t e r a t u r e 2.4.5.2.4. The CRSS of Mg-Zn Alloys: The composition dependence of CRSS for various temperatures i s shown i n f i g . (93). From f i g s . (92 and 93) i t i s clear that at low temperatures the CRSS of magnesium remains unaffected by small additions of solute followed by a rapid decrease at higher concentrations. For example at 78°K the CRSS of Mg as well as an allo y containing 0.019 at.% 2 2 Zn a l l o y i s 9.95 kg/mm , whereas that for 0.45 at.% Zn al l o y i s 6.7 kg/mm . At higher temperatures, on the other hand, the CRSS increases by small additions of Zn, followed by a decrease at higher concentrations. Above approximately 350,K the CRSS of a l l the alloys are higher than that of Mg. The results of the Mg-Al alloys are shown i n f i g . (94). Higher alloys could not be investigated because of the d i f f i c u l t y encountered i n growing single crystals containing more than 0.2% A l . The trend i n the composition 1 dependence of CRSS nevertheless i s the same i n both Mg-Al and Mg-Zn al l o y s . I t i s important to note at this juncture that the i n i t i a l increase i n CRSS for prismatic s l i p followed by a decrease at higher concentrations accounts adequately for the multistage solution hardening observed i n the po l y c r y s t a l l i n e material. Solid solution softening has previously been observed i n Mg-Li (73) alloys . However, the study of single crystals oriented for prismatic s l i p was confined to concentrations.in excess of 6 at.% L i . The existence 2 0 0 300 T E M P E R A T U R E ° K 4 0 0 Fig. 92, CRSS for prismatic s l i p vs. temperature for Mg-Zn single crystals, ATOMIC % Zn F i g . 93. CRSS for prismatic s l i p vs. Zn concentration. Fig. 94. CRSS for prismatic s l i p vs. temperature for Mg-Al single cry s t a l s . 187 of an i n i t i a l rapid solution strengthening rate i n the polycrystals of this system, however, suggests an increasing value of CRSS for prismatic s l i p i n these alloys at low L i concentrations at room temperature. 2.4.5.2.5. The Effect of Solute on the Flow Stress: The process of yielding i n single crystals of Mg oriented for prismatic s l i p i s rather gradual. Therefore, the interpretations based on the measurements of CRSS, which i s taken as the f i r s t deviation from l i n e a r i t y can be erroneous, especially when the differences i n the values of CRSS being examined are small (as i s the case with the d i l u t e a l l o y s ) . In order to avoid this d i f f i c u l t y the flow stress values at various s t r a i n levels have been plotted against solute concentrations for a representative set of temperatures as shown i n f i g s . (95 and 96). Fig. (95) has been constructed for crystals deformed at 78°K. It i s observed that the CRSS at low solute concentrations i s much the same as that of Mg, but the flow stresses at higher strains decrease rapidly with Zn concentration. At the higher concentrations of solute on the other hand, the flow stress decreases less rapidly than does the CRSS. The flow stress concentration relationship at 423°K ( f i g . 96) shows that the shape of the flow curves remains similar to that i n the CRSS-concentration plot. The CRSS increases with solute content i n the low alloys followed by a decrease at higher concentrations. These results tend to confirm the v a l i d i t y of the interpretations based on the measurements of CRSS. Mg-Zn Single Crystals PRISMATIC SLIP 6 L 1 1 — _ 1 1 L_ 0 01 0 2 0 3 04 0 5 ATOMIC % Zn Ftg. 95. Flow stress for prismatic s l i p vs. concentration for Mg-Zn single crystals tested at 78°K. Fig. 96. Flow stress vs. concentration for Mg-Zn single crystals tested at 423°K, 189 2.4,5.3. Discussions: An understanding of the dislocation mechanisms for s l i p i n the pure metal i s often helpful i n analyzing the nature of the solute-dislocation interaction. With this view i n mind the dislo c a t i o n mechanisms responsible for prismatic s l i p i n magnesium w i l l be discussed followed by a discussion of the effect of solute. 2.4.5.3 .1 . The Dislocation Mechanism for Prismatic S l i p : Attempts have been made e a r l i e r to explain the dislo c a t i o n mechanisms for prismatic s l i p i n pure m a g n e s i u m ^ 2 4 . I t has been the conclusion of Ward-Flynn, Mote and Dorn^ 4"^ that the rate controlling mechanism i n prismatic s l i p above 450°K i s the thermally activated cross slipping of screw dislocations extended i n the basal plane as proposed by F r i e d e l ( 1 0 4 ) . The low temperature rate controlling mechanism has never been systematically investigated i n the pure metal. However, from the trend (73) of the T - T curve^Dorn suggests that Peierls mechanism must be controlling below room temperature. In the present work a limited number of s t r a i n rate change tests were conducted to test Dorns prediction of the mechanism of low temperature prismatic s l i p . The activation volumes were measured by changing the s t r a i n rate by a factor of ten. However, since Mg shows rather limited d u c t i l i t y at low temperatures i t was d i f f i c u l t to conduct d i f f e r e n t i a l s t r a i n rate tests. This d i f f i c u l t y was overcome by deforming crystals of i d e n t i c a l orientations to fracture using two d i f f e r e n t . s t r a i n rates. * The difference i n y i e l d stress was then used for calculating V . 190 (73 Such a procedure has been used i n the past on Mg-Li all o y single crystals The results of such tests on Mg, as we l l as a series of Mg-Zn alloys are shown i n f i g . (97). It i s seen that i n Mg as well as i t s alloys the 3 activation volumes, remain below 80 b at temperatures up to 295°K. This i s consistent with the idea of Pei e r l s mechanism as the rate controlling step. I t i s important to point out at this juncture that the activation 3 volume associated with the cross.slip process i s > 100 b . In the a l l o y 3 containing 0.006 at.% Zn, the activation volume was found to be 205 b at room temperature which i s higher than that associated with P e i e r l s mechanism. No explanation, however, could be afforded for this exception. A quantitative v e r i f i c a t i o n of the Pei e r l s mechanism cannot be made here since magnesium does not show a d i s t i n c t t r a n s i t i o n temperature from the the :rmally activated to an athermal mechanism. I t i s interesting to note that i n the case of magnesium an athermal region does not exist i . e . The Friedel cross s l i p mechanism, which i s a thermally activated process, takes over the low temperature Pei e r l s mechanism, which again i s thermally activated. 2.4.5.3.2. The Effect of Solute: The effect of solute on the strength of c r y s t a l l i n e solids i n general i s best understood i f the contribution of the solute to the athermal and the thermally activated components of the flow stress are considered separately. The experimental results on s o l i d solutions available,to date indicate that the athermal flow stress never decreases with the addition * of solute, although T increases or decreases depending on the rate F i g . 97. A c t i v a t i o n volume at y i e l d vs. temperature for Mg-Zn single crystals deformed i n prism s l i p orientation. 192 controlling mechanism operative i n the al l o y . From the results of (73) the experiments by Ahmadieh and Dorn on Mg-Li.alloys i t i s seen that at s u f f i c i e n t l y high concentrations of lithium the athermal stress l e v e l becomes apparent and that i t increases with increasing solute content, as would be expected of the athermal stress i n r e l a t i o n to solute. The appearance; of the athermal stress l e v e l deserves a c r i t i c a l examination. For the time being the experimental observation that the Peie r l s stress decreases with increasing L i content w i l l be taken for granted. Since * the athermal stress increases and the T term decreases with increasing lithium addition, T (from Pe i e r l s to athermal) i s expected to decrease c also with lithium concentration as i s observed ( f i g . 98). The consequence of such a decrease i n T i s that the thermal.energy available at T , i s c c not high enough for cross s l i p to occur, therefore, the dislocations have to move under the long range stress f i e l d (athermal). J . 4 . 5 . 3 . 3 . The Results of the Present Work: The results of the Mg-Zn alloys can be explained i n terms of * an increasing xn and a decreasing x . Note that i n the case of Mg-Zh * alloys the increase i n x_ and the decrease i n x are not s u f f i c i e n t to make the athermal stress l e v e l appear i n the x -T curve. At lower * temperatures, where the x term i s dominant over x„, the increase i n x i s of l i t t l e importance and therefore the CRSS decreases continuously with Zn concentration. At higher temperatures, however, a peak i n the CRSS-concentration curve appears. This can be explained better with the help of a sketch., Fig. (99) i l l u s t r a t e s the si t u a t i o n schematically. We w i l l assume for the sake of argument that x i n the alloys arises due to a f r i c t i o n stress 193 0 100 200 300 400 500 600 700 - T, °K Fig. 98. CRSS for prismatic s l i p vs. temperature for Mg-Li a l l o y single crystals (after Ahmadieh and Dorn<7 3>). 194 Fig. 99. Schematic representation of the effect of solute on the various components of flow stress i n prismatic s l i p . 195 mechanism involving size and modulus effect. This would lead to a C dependence of x . This i s shown,as curve (1). Since x changes.only s l i g h t l y (through a change i n shear modulus) with temperature curve (1) should be v a l i d to a f i r s t approximation at a l l temperatures.. On the * contrary the v a r i a t i o n of x with concentration i s a strongly temperature dependent function. Curves(2) and (3) represent the effect of solute on x at a low temperature (T<< Tc) and at a high; temperature (T £S Tc) respectively. The combination of (1) and ( 2 ) and (1) and (3) are shown as curves (4) and (5). Curves (4) and (5) are q u a l i t a t i v e l y i n good agreement with the observed concentration dependence of CRSS at low and high temperatures respectively. A quantitative v a r i f i c a t i o n of such a proposition, however, i s not possible at the present, because x cannot be separated from x^ i n the absence of a well defined T c. The anomalous CRSS-concentration relationships i n bcc Ta-base . (144) alloys have also been explained by M i t c h e l l and Raffo i n a si m i l a r manner. 2.4.5.3.4. The Origin of x^ and x : In the above explanation there has been no mention of the * o r i g i n of T„ and x and their v a r i a t i o n with solute concentration. This w i l l be done.in the following pages. 2.4.5.3.4.1. The Athermal Stress: The observed athermal stress .in the Mg-Li alloys.was attributed (73) by Dorn to the short range ordering in.these alloys . However, a determination of the short range order parameter by Averbach and Herbstein^"^"^ i n these alloys indicates that short range ordering cannot account for 196 the observed athermal stress- A f r i c t i o n stress mechanism similar to the one i n basal s l i p of Mg-Zn alloys could be c o n t r o l l i n g , however, such a guess would, be premature i n the absence of s u f f i c i e n t experimental, data on prismatic s l i p . * 2.4.5.3.4.2. The Variation of x With Allo y i n g: From the results of the p o l y c r y s t a l l i n e Mg-Li alloys Hauser, (14) Landon and Dorn. proposed that the CRSS for prismatic s l i p i n these alloys decreases with increasing solute and that this decrease i s a result of the decreasing a x i a l r a t i o (c/a) with a l l o y i n g . Subsequently the rate controlling mechanism was i d e n t i f i e d i n single crystals oriented for (73) prismatic s l i p by Ahmadieh, M i t c h e l l and Dorn to be one of P e i e r l s stress. However, no explanation was given as to why the P e i e r l s stress decreases with lithium a l l o y i n g . Recently Dorn,has modified his explanation further,to explain the observed lowering ,of C R S S . This i s the so called "Pseudo-Peierls mechanism", which i s an intermediate sit u a t i o n between Peie r l s stress and the F r i e d e l cross s l i p mechanisms. 7 Dorn argues that Friedels derivations,do not hold good for cases where the stacking f a u l t energy i s so high that the distance between the p a r t i a l s i s 0.5 - 2b as i s the case.with Mg. In such situations the stacking f a u l t energy must be taken into account. These considerations led Dorn to the conclusion that the stacking f a u l t energy of magnesium increases with the addition of solute I t i s not very clear from his theory as to whether i t i s meaningful to talk about a higher stacking f a u l t energy when the distance between the p a r t i a l s i s already of the order of 0.5 - 2b, especially when the present knowledge of the interatomic forces i s so l i m i t e d . Even i f the 197 theory,is taken for granted, certain other inconsistencies are encountered. The increase i n the stacking f a u l t energy with lithium alloying i s considered by Dorn to be a result of the decreasing electron to atom r a t i o (e/a). Therefore, i n the case of Zn solute where e/a remains constant and A l where e/a increases, there should not be lowering of CRSS with solute. The present results do not conform to these conclusions. Also i f we take the c/a to be the c r i t e r i o n as was (14) proposed e a r l i e r by Hauser and Dorn. then also the results of Mg-Zn alloys i n which the c/a r a t i o remains constant and the Mg-Al alloys where c/a increases cannot be accounted for. Lately, however, i t has been suggested by M i t c h e l l and (144) Raffo that the nucleation of a pair of kinks becomes easier i n the presence.of small amounts of (up to 4% Re i n Ta) solute thus.lowering the P e i e r l s stress. The effect of solute can a l t e r n a t i v e l y be v i s u a l i z e d as. being perturbation of the p e r i o d i c i t y of the P e i e r l s h i l l , thus lowering the effective value of the height of the b a r r i e r . The acceptance or rejection of this idea w i l l probably depend on whether a l l solutes * decrease T at a l l concentrations of solute or not. S u f f i c i e n t experimental data are not available at the present to take such a decision. I t should be borne i n mind here that even i f the CRSS data are available, there must be a method of obtaining T from the experimentally measured CRSS. From the results on Mg-solid solutions oriented for prismatic s l i p i t appears that the decrease i n P e i e r l s stress with increasing solute concentration i s not necessarily associated with a decreasing c/a r a t i o or with the valency of the solute. The results of the p o i y e r y s t a l l i n e s o l i d solutions i n the present work i n fact suggest this to be a general solution effect i n Mg. 198 3» Summary and Conclusions: 1) In binary a l l o y polyerystals of Mg containing Zn,. A l , Pb, Cd or In as solute, solution hardening occurs i n a two or three stage fashion. In stage I at the lowest solute concentrations, the hardening i s very high and,linear. Beyond a t r a n s i t i o n concentration, the hardening is.very much.reduced. 2) A t h i r d stage characterized by a "solution softening" effect was observed i n the case of Zn and A l solutes. 3) The strengthening observed i s i n the order l i s t e d , and this i s also the order of size mismatch with Zn being the greatest. The d u c t i l i t y decreases for solute additions upto C^ and then increases. 4) The tr a n s i t i o n concentration i s a function of the size difference between.the solvent and the solute^being smaller for larger Ar. 5) The predominant factor during stage.I hardening i s the increase i n the.CRSS for prismatic s l i p . During stage I I , the CRSS for prismatic s l i p decreases with increasing solute additions. 6) The disc o n t i n u i t i e s observed i n the d u c t i l i t y temperature plot of the Mg-Al alloys arise due to a combined effect of r e c r y s t a l l i z a t i o n and grain boundary c a v i t a t i o n , the former being related to the CRSS for prismatic s l i p i n the alloy.. 7) The experiments on the ternary Mg-In-Zn alloys suggest that stage I hardening results from a.type of solute atom di s l o c a t i o n int e r a c t i o n , with preference for prism plane dislocations and large size difference solutes to be involved. 199 8) In single crystals of magnesium-Zn alloys oriented for basal s l i p , the athermal component of the CRSS i s independent of Zn concentration upto h 0.025 at.% beyond which a C dependence i s obeyed. 9) Transmission electron microscopy of thin f o i l s of Mg-Al crystals shows that the grown i n basal dislocation density increases i n a parabolic manner with increasing solute content. This increase, however, cannot account for the observed variation of T with solute concentration. 10) The athermal component of the CRSS inMg s o l i d solutions arises due to an increase i n the f r i c t i o n stress with increasing solute concentration. 11) An etch p i t technique for revealing the forest dislocations inMg has been developed and used to evaluate the increase i n the forest density with solute concentration. The results show that the increase i s high and linear upto 0.04 at.% Zn, beyond which the change i s rather small. 12) The low temperature solution hardening occurs i n two stages, h the two stages being lin e a r functions of C having different slopes, with the t r a n s i t i o n at 0.025 at.% Zn. 13) Stage I hardening arises due to an increase i n the forest dislo c a t i o n density with solute concentration whereas i n stage I I a change i n the thermally activated mechanism occurs from intersection to the single solute atom dislocation pinning mechanism. 14) The effects of Zn on the work hardening parameters of Mg single crystals i n basal s l i p are to increase the stage A and stage B slopes and to decrease the extent of easy glide. 200 15) Tensile tests conducted on oriented single crystals to suppress basal s l i p and {1012} twinning and to induce prismatic s l i p show that for Zn and A l solutes at low temperatures the CRSS for prismatic s l i p decreases with increasing solute addition. At high temperatures , however,the CRSS f i r s t increases with solute concentration followed by a decrease. 16) The low temperature rate controlling mechanism i n prismatic s l i p i s the overcoming of the Peie r l s stress b a r r i e r . 17) The observed concentration dependence of the CRSS i n prismatic s l i p can be accounted for i n terms of a decreasing P e i e r l s stress with increasing solute concentration. 18) The decrease i n the Peie r l s stress of magnesium with increasing solute addition i s not necessarily associated with either a reduction of the a x i a l r a t i o or the r e l a t i v e valency of the solute. 201 4. SUGGESTIONS FOR FUTURE WORK: The present study can lead to several l i n e s of investigation. The ones that deserve immediate attention include: 1. An extension of the study of the effect of solute on the ease of prismatic s l i p to include other solute elements and testing temperatures down to 4.2°K, i n order to determine the nature of the athermal and the thermally activated components of the flow stress. 2. A thorough study of the d i l u t e ternary al l o y single crystals i n order to gather more information on the pre f e r e n t i a l nature of solute-dislocation interaction on the prism planes and to determine the significance of CT. 3. An investigation of the dependence of the forest dislocation. Spacing on pre-strain, using the etch p i t t i n g technique developed i n the present work can lead to a better understanding of the work hardening behaviour,of magnesium. 4. A systematic study of the y i e l d points and the serrated yielding phenomenon i n the Mg-Zn al l o y s . 5. An investigation of solution strengthening in.basal s l i p at higher solute concentrations and lower.temperatures (up to 4.2°K) i n order to supplement the present data and to determine the nature of the solute dislocation interaction. 6. An experimental study of the effect of a variety of solutes on the work hardening parameters of Mg coupled with a thorough s l i p l i n e study and transmission electron microscopy of thin f o i l s to determine the microstructural aspects.of work-hardening i n the all o y s . 202 APPENDIX - A DETERMINATION OF SOLUTE CONCENTRATION IN THE ALLOYS: Much of the present work has been carried out on materials containing small amounts of alloying elements. Atomic absorption spectrophotometry has been used to determine the concentration of solute. The attributes which made atomic absorption a t t r a c t i v e are the r e l a t i v e l y few a n a l y t i c a l interferences and the high precision of the technique. PRINCIPLE OF THE METHOD: In p r i n c i p l e the sample i s dissociated from i t s chemical bonds (in solution) and placed into an unexcited, unionized ground state. It i s then capable of absorbing radiation at discrete frequencies of narrow band width. Dissociation i s achieved by burning the sample i n a flame; An effective burner for atomic absorption i s designed with a "premix" system, i n which f u e l , oxidant and the sample are combined i n a mixing chamber before being burned. A combination of a i r and acetylene was used for Cd, Zn and Pb. In the case of A l solute, however, a special n i t r o u s -oxide acetylene flame was used, since the air-acetylene mixture does not ( 1 4 9 ) give a flame hot enough to dissociate aluminum compounds The radiation to be absorbed by the sample i s provided by a hollow cathode lamp, which emits only the spectrum of the desired element, together with that of the f i l l e r gas. In order to screen out the undesired emission, the radiation i s passed through a f i l t e r or monochromator which i s tuned to pass the l i n e of interest but screens out others. 2 0 3 INTERFERENCE: For a given design of the burner, o p t i c a l system and the / fuel-oxidant combination, the two most serious errors i n making a quantitative estimate of elements by atomic absorption are brought about by the presence of another i n t e r f e r i n g element i n solution and due to a difference i n the v i s c o s i t y of the solutions under examination. The v i s c o s i t y effect otherwise known as the Bulk or Matrix effect arises due to the v a r i a t i o n i n the amount of dissolved material i n solution. In order to minimize the error due to v i s c o s i t y e f f e c t , the _ | | concentration of the anion (NO^ i n the present case) and the Mg ion were kept constant i n a l l the solutions of each set of unknowns and the corresponding standards. Since the alloys under examination contained only small amounts of alloying elements, the change i n the v i s c o s i t y a r i s i n g due to the difference i n the amount of the solute element present i n the samples was considered n e g l i g i b l e . Another type of interference known as the chemical interference arises when the presence of a second ion i n solution either depresses or enhances absorption due to the ion under examination. E a r l i e r work by B e l l ^ " ' ^ has indicated-that the presence of magnesium does not interfere i n the analysis of Zn and Cd. In the presence of aluminum, however, the analysis of Mg i s d i f f i c u l t . Although i n the present case A l »as analysed i n Mg matrix, , ++ i t was considered safe to keep the Mg concentration i n the sample very nearly constant. Such matching techniques have been used by e a r l i e r workers i n order to overcome chemical i n t e r f e r e n c e . 204 PROCEDURE: Preweighed samples were dissolved in.10% HNO^ . The _ [ j concentration of NO^  ion and Mg were adjusted to a constant value and absorption tests were carried out at the following w a v e l e n g t h s . Metal Wavelength A 0 A l 3093 Cd 2288 Pb 2170 Zn 2138 A t y p i c a l concentration absorbance plot for Mg-Zn alloys i s shown i n f i g . (100). I t i s seen that absorption i s a linea r function of the Zn concentration up to approximately 10 ppm. The linear part of the curve was used for determining the composition of the unknown. RESULTS AND LIMITATIONS: Atomic absorption was found to be most s e n s i t i v e i n the case of Zn and Cd where 1 ppm of the element i n solution could be detected eas i l y . In the case of aluminum, however, the s e n s i t i v i t y was about an order of magnitude lower, which necessitated upgrading of the solutions. The compositions determined using atomic absorption were found to be lower by up to 20% of the nominal composition i n the case of Zn and Cd solutes. This i s to be expected since the vapour.pressure of these solutes i s rather high at the melting temperature of the a l l o y . In the case of Pb and A l solutes, the determined compositions were within - 10% of the nominal compositions . 1.0 0 2 4 6 8 l a 12 14 Zn Concentration i n ppm., F i g . 100, -Absorbance vs. Zn concentration. 206 APPENDIX - B THE PREFERENTIAL NATURE OF THE SOLUTE-DISLOCATION INTERACTION: A ternary Mg a l l o y containing 0.07 at.% Zn and 1 at.% In was tested i n tension at room temperature. The y i e l d stress-composition curves for the binary Mg-Zn and Mg-In alloys are shown together i n f i g . (101). I t i s apparent that the concentration of each solute i n the ternary i s higher than C T for the corresponding binary system. The y i e l d stress of pure Mg i s 8,500 p s i . From the figure cfyp for Mg-0.07 at.% Zn a l l o y i s 12,580 p s i and that for Mg-1 at. % In i s equal to 11,430 p s i . Assuming that the strength of a multi-component s o l i d solution can.be obtained by adding up the strengthening effects of the corresponding binary solutions, the strength of the Mg-Zn-In binary a l l o y examined should equal 15,510 p s i . Experimentally, however o^p for the ternary a l l o y i s observed to be.12,800 - 200 p s i , suggesting that the strengthening effect of the various solute:species present simultaneously i n Mg i s non-additive. An alternative p o s s i b i l i t y i s the pre f e r e n t i a l interaction of one solute species with dislocations giving r i s e to hardening which i s equivalent to that i n stage I i n the corresponding binary system. The excess of the preferred solute above C^, ( i n the binary) as well as the t o t a l amount of each of the other solute species present i n the ternary w i l l have additive strengthening contributions equivalent to those i n stage I I of thei r corresponding binary systems. In the case of Mg-Zn-In ternary al l o y whether In or Zn i s the preferred species can be decided as f o l l o w s . Assuming that In atoms are p a r t i c i p a t i n g in.the stage I interactions 207 Fi g . 101. Solution hardening curves for Mg-Zn and Mg-In binary systems. 208 Strength due to In present i n an amount equal to C, (0.25 at.%) = 11,250 p s i (A) 0.75 at„% In w i l l participate i n stage I I interaction giving r i s e to,another 187 psi (B) 0,07 at.% Zn w i l l also contribute an equivalent of stage I I of Mg-Zn binary which i s 630 p s i (C) The strength of the ternary a l l o y with In as the preferred species would then be = (A) + (B) + (C) = 11,250 + 187 + 630 p s i =12,067 p s i A similar calculation taking Zn.to be the preferred species yields 12,750 ps i to be the strength.of the ternary a l l o y . Keeping in.mind that the re p r o d u c i b i l i t y of the present results are + within - 250 p s i , a comparison of the calculated strength values with the experimental result suggests Zn to be the preferred species i n stage I interaction. This conclusion i s further substantiated by the experiments on a series of ternary a l l o y s . of solute at concentrations above. C^ . i s not s t r i c t l y v a l i d , because at s u f f i c i e n t l y high Zn concentrations solution softening i s to be expected rather than,strengthening, whether or not - the l i m i t i n g con-centration above.which softening occurs would be affected by the presence, of In cannot be decided u n t i l further experiments are performed. It should, however, be noted here that the additive effect 209 APPENDIX - C TIE DETERMINATION OF A T FROM STRAIN RATE CHANGE TESTS: The most convenient method of determing the v a r i a t i o n of A T with s t r a i n rate i s through instantaneous changes i n strain rate during the tensile test. I t i s , however, important to remember that the flow stress difference accompanying a change i n s t r a i n rate can be equated to A T only i f i t i s assumed that the dislocation structure remains constant during the instant of change. In the present work the s t r a i n rate was changed by a factor 2 of 10 (from a cross head.speed of 0,002 to 0=2 ipm) during the tensil e / ' test using a push button speed selector. In pr i n c i p l e both stress increments and decrements may be taken as A T •„ However, a considerable delay was observed to be associated with the change from high to low st r a i n rate. For this reason, i t was decided that the error a r i s i n g from y i e l d point phenomenon (in the alloys) would be less than that resulting from the time delay during the decrease i n s t r a i n rate. Therefore, a l l A T values were obtained during an increase i n s t r a i n rate. Much of the s t r a i n rate change tests were conducted at 78°K. Fig. (102) i l l u s t r a t e s schematically the flow curves during s t r a i n rate change tests. : Because of the abrupt nature of y i e l d after a change i n st r a i n rate, i t was r e l a t i v e l y easy to obtain A T i-n t n e case of Mg. 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