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Growth and deformation of copper whiskers Shetty, Mangalore Nagappa 1964

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GROWTH AND DEFORMATION OF COPPER WHISKERS by MANGALORE NAGAPPA SHETTY A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILMENT OF FOR THE DEGREE OF PHILOSOPHY in the Department of METALLURGY We accept this thesis as conforming to the standard required from candidates for the degree of DOCTOR OF PHILOSOPHY Members of the Department of Metallurgy THE UNIVERSITY OF BRITISH COLUMBIA June, 196A In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Bri t i sh Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that per-mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that, copying or publi-cation of this thesis for financial gain shall not be allowed without my written permission*. Department of Metallurgy The University of Brit ish Columbia, Vancouver 8, Canada D a-te August 11 . 1964 The University of British. Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of MANGALORE NAGAPPA SHETTY B . S C o , Madras University, 1956 D.I,I.Sc., Indian I n s t i t u t e of Science, 1958, Bangalore M.Sc, Uni v e r s i t y of Utah, 1960 FRIDAY, AUGUST 7, 1964 at 2;00 P.M. IN ROOM 201, MINING BUILDING COMMITTEE IN CHARGE Chairman; I„ McT« Cowan W,M. Armstrong J.A.H.. Lund R. Ba r r i e E. Teghtsoonian L.C. Brown R„M. Thompson External Examiner; R.L, F l e i s c h e r Research S c i e n t i s t General E l e c t r i c Research Laboratory Schenectady, N,Y. GROWTH AND DEFORMATION OF COPPER WHISKERS ABSTRACT Copper whiskers were grown by the hydrogen reduction of the halide vapours. The substrate and the whiskers were examined for growth morphologies. Whiskers of (100), (110) and (111) orientations i n the diameter range 30-400jLX. were tested i n tension* Differences i n the work hardening c h a r a c t e r i s t i c s of (111) and (100) whiskers were discussed i n terms of the d i f f e r e n t kinds of jogs formed i n the two orien t a t i o n s . Analysis of the diameter dependence of the y i e l d stress resulted i n a 1/d r e l a t i o n based on the assumption of surface nucleation of d i s l o c a t i o n s . Temperature and s t r a i n rate change experiments were made on (1.10) whiskers. A c t i v a t i o n distance and a c t i v a t i o n energies were used to determine a rate con-t r o l l i n g mechanism. At low temperatures, cross s l i p and i n t e r s e c t i o n processes were ind i s t i n g u i s h a b l e , while at higher temperatures, cross s l i p i s rate con-t r o l l i n g . From the calculated a c t i v a t i o n distance and for a given rate c o n t r o l l i n g d i s l o c a t i o n mechanism, stacking f a u l t energies were estimated for copper and other FCC metals. A twinning model was proposed based on the idea of f a i l u r e by recombination of a Lomer-Cottrell b a r r i e r . GRADUATE STUDIES F i e l d of Study: Metallurgy M e t a l l u r g i c a l Thermodynamics C. S„ Samis Structure of Metals I I I E. Teg'htsoonian Properties of Ceramic Materials . _ „ „ , , , A,C.D„ Chaklader Topics i n Physical Metallurgy J.A„H„ Lund Transformation and D i f f u s i o n W.M. Armstrong Related Studies: Theory and Applications of D i f f e r e n t i a l Equations C,A, Swanson S t a t i s t i c a l Mechanics R,F. Snider Physics of the S o l i d State R. ,Barrie ABSTRACT Goppor whiskers were grown hy the hydrogen reduction of the halide vapours* The substrate and the whiskers were examined for growth morphol-ogies; Whiskers of [lOO] , [lio] and [ill] orientations in the diameter range 30 - ^ OOLlwere tested in tension^ Differences in the work hardening characteristics of [ill] and [lOO] whiskers were discussed in terms of the different kinds of jogs formed in the two orientations. Analysis of the diameter dependence of the yield stress resulted in a 1 relation based d on the assumption of surface nucleation of dislocations. Temperature and strain rate change experiments were made on [lio} whiskers. Activation distance and activation energies were used to deter-mine a rate controlling mechanism. At low temperatures, cross slip and intersection processes were indistinguishable, while at higher tempera-tures, cross slip is rate controlling. From the calculated activation distance and for a given rate controlling dislocation mechanism, stacking fault energies were estimated for copper and other FCC metals. A twinning model was proposed based on the idea of failure by recombination of a Lomer-Gottrell barrier. ACKNOWLEDGMENTS My grateful acknowledgments are to Dr. E. Teghtsoonlan for his continued help, despite his busy and multifarious pre-occupations. I also wish to thank him for having suggested the problem on the temperature and strain rate dependence of whiskers, and his critical evaluation of the manuscript. Grateful acknowledgments are also due to Prof.F,A.Forward for having provided me with an opportunity to be in the University of British Columbia, and the stay was a very pleasant one indeed. I wish to recall some of the very useful discussions which I held with several of the staff members in the department, espec-ial l y with Dr. E.Peters on the strain rate equation. Mrs. Armstrong brought to my notice the review paper on surface energies, which was of great assistance in the quantitative discussion on size dependence. I sincerely thank Mr. R.Butters and R. Richter for their technical help. Generous and timely cooperation of my fellow graduate students during the course of this project is gratefully appreciated. I remain indebted to Mr. K.R. Mally for his help and encouragement without which my visit to Canada would not have been possible. Financial assistance was provided by the Defence Research Board and the National Research Council of Canada. TABLE OF CONTENTS Page 1. GROWTH 1 1* 1 Introduction • . • • • • 1 1. 2 Experimental Procedure • 3 1. 3 Results? U (1) Examination of the Substrate o e « o « . « o « e e o e A (2) Examination of Whisker Tips and Surfaces „ « « » e » » U 1. U Discussion • • » » . < > • • • • • . » . • • • • • • • • • • • 16 2. STRESS - STRAIN CURVES 25 2. 1 Introduction, • • 25 (1) Characteristics of Stress - Strain Curves 26 (2) Size Effects 27 (3) Deformation Characteristics in Whiskers and Bulk Single Crystals. • • • • • • • • • • • • • • • • • • • 28 2. 2 Experimental Procedure • • • • • • • • • • • • • • • 30 (1) Testing method . • • • • • • • • 30 (2) Load Cycling 33 (3) Testing . 33 {U) Orientation Determination 35 2. 3 Results • • • • • • • • • • • • • • 35 (1) Stress - Strain Curves • • • « • • • • • • • • • • • • 36 (2) For Whiskers of [lOO] and [ l l l l Orientation 44. TABLE OF CONTENTS CONTINUED Page (3) For Whiskers of [lio] Orientation* 47 (4) Metallography and Fine Structure • • • • • • • • • 47 (5) Diameter Dependence of Yield Stress. 50 2.4- Discussions 56 (1) Hardening in [ i l l ] and [lOol whiskers . . . . . « 56 (2) Diameter Dependence, o s o e o e e o s e i o ^ a o s 62 3. TEMPERATURE AND STRAIN RATE DEPENDENCE 68 3. 1 Introduction 68 3. 2 Experimental Procedure. . . . . . . . 70 (1) Temperature Change Tests » • • • • 71 (2) Strain Rate Change Tests . . . . . . . . . . . . . 72 3. 3 Results .72 (1) Temperature Change Tests 72 (2) Strain Rate Dependence of the Flow Stress 74 (3) Activation Energy, Activation Volume, Frequency Factors, etc. 79 (4) Activation Energy. 82 (5) Activation Volume • 83 (6) In ^ o 83 C (7) Activation Distance • • • • • 84 TABLE OF CONTENTS CONTINUED Page 3 . 4 Discussion. - Mechanisms • 90 (1) Formation of Jogs • • • 90 (a) Forest Intersection Mechanism. 90 (b) Non Conservative Motion of Jogs in Screw Dislocations • o o s s e . o o . o e . e . o . 91 (c) Conse rva t ive M o t i o n o f J o g s e a e « « s « o o < i 92 (2) Cross - Slip of Screw Dislocations 92 (3) Impurity Dislocation Interaction • • • • • • • • • 94. (4) Peierls Mechanism » « . . • • . < > . <>•.••••. 94 3 . 5 . Discussion of Experimental Results • • • • • < » • . . . . . 98 (1) Hardening at 295°K and 273°K 98 (2) Hardening at 200 K , . . . . . . . . . . 9 . . . . X00 (3) Hardening at 77°K 0 . . . . . o 100 (4) Hardening Characteristics of [106] and [ill] Whiskers 102 (5) The Quantity In J o 102 € 3.6 Twinning . . . . .103 3.7 Dependence of ^ on V ; 109 4- SUMMARY AND CONCLUSIONS 112 5 SUGGESTIONS FOR FUTURE WORK 118 TABLE OF CONTENTS CONTINUED Page 6 APPENDICES I Stress - Strain Curves. • • • • • • • • • • • • • 119 II On Jogs » 137 III L — C Reactions O o o e < j o o c » « . s « o o « « « > o 14.2 IV Image Force o e o o c c o o o e o © o o o » o o » 14-5 V Load Measurement. • • . . . . . . . . . . . . . . IAS VI Activation Energy Calculations, 150 VII Double Slip Conditions and Schmid Factor 154 VIII Estimation of Errors. .<,.....-.<>•••• 158 IX Forces Between Dislocations • • . . . » . • • • . 161 1. Introduction. • 161 2. Theory and Application 161 7 BIBLIOGRAPHY 174-FIGURES No, Page 1. Etch Pits on Copper Reduced from GuCl^.Etched Briefly in Conc.HN03 3000X. . . / 5 2a to Spiral Etch Pits on Copper Reduced from Cu01o. Etched 2f Briefly in ConcHNO.^. 3000X i 6=8 3. Tips of as Grown Copper. Whiskers, Sketch shows the Orientation of the Facets, 800X » . 0 , • • • « • • 10 4. Growth Stops on as Grown Whisker Tips e Sketch Shows Orientation of the Facets e 800X 0 , e 9 © o » , < > o 11 5. Surface of a Whisker Exposed to Air . 1600X . . . . . . 12 6. Tip of a Whisker Exposed to A i r . 1600X. 12 7. Growth Steps on the (001) Surface of a [llO] Whisker. Original Magnification 110X . . . . . . . . . . . . 13 8. Growth Steps on the (110) Surface of the Whisker in F i g . 7. 110X. . 14 9. Growth Steps on the ; (110) Surface of the Same Whisker in F ig , 7 . 600X. . . 15 10. A possible Growth Spiral on a (110) Plane 16 11. Growth Spiral With a Twinned Segment, , , , , 17 12. Form of the Spiral in Fig.10 After a Complete Sequence of Twinning , , , , 17 13. Twinning within a Grain 18 14. A Dislocation Node in a [ l l l l Whisker, [ m ] is parallel to the Axis, and \11& are Normal to the Facets at the T i p . . . , 19 15. Illustrating Horizontal Glide Traces Left on the (001) Plane or the (110) Plane of the Whisker in F ig . 7 due to the Egress of % <011> Dislocations on (111) Plane. 20 16. Sequence Showing the Form of the Steps Left by Gliding Screw Dislocations. One of the Glide Planes is Parallel to the Axis. • • « • « • , , , • • • • , • • 22 FIGURES - Cont'd. No. Page 17. Sequence Showing the Form of the Steps Left on (110) Plane, Due to Gliding Screw Dislocations on Intersecting Planes. . . . . . . . . . . . . . . . . 23 18. Yielding of a Fine Whisker (after Brenner) . . . . . . 26 19. Repeated Yielding of a Fine Whisker (after Brenner). . 27 20. Schematic Shear Stress - Strain Curve for Single 21. Suspension Used for Whisker Testing. 0 a e « « « » * « 31 22. Mounting of a Whisker. • • • . . » • • • • • . . . . . 32 23. Mounted Whisker Ready for Test 3k 24. Stereogram Showing the Whisker Orientations. • • • • • 36 25. Load Elongation Curve (a) [ l i o l Whisker and (b) [lOO] Whisker. Cross-head Speed 0.002n/min 37 25. Load Elongation Curve for [ill] Whisker- Cross-head cont'd. Speed 0.002n/min. 38 26. Tensile Stress - Strain Curves. (Use Left Scale for No. 16) . . . . . . . . . o e . . . . . © . . . « . 39 27a to Comparison of Stress - Strain Curves for Coarse 27c Whiskers, Fine Whiskers and Single Crystals. . . . 4 0 - 4 2 28. Diameter Dependence of Easy Extension. • • • • • • • • 45 29. Temperature Dependence of Easy Extension for Coarse Whiskers In a Narrow Range of Diameters (60 - 80yU) 46 30. A [ i l l ] Whisker Deformed at Liquid Nitrogen Tempera-ture Showing Deformation Twins. Sketch Shows Twin Trace. 110X 48 31. Load Elongation Curve at the Onset of Twinning for a [ill] Whisker Deformed at Liquid Nitrogen Temperature 4-9 32. A [lOO] Whisker Deformed at Liquid Nitrogen Temperature Showing (a) Double Slip and (D) "Cellular» Slip-Line Markings. 210X • • • • • 51 FIGURES - Cont'd. No. Page 32. The [lOO] Whisker Deformed at Liquid Nitrogen Temp-cont'd. erature Showing Non-octahedral Slip. 110X. • • . • • 52 33. A [loo] Whisker Deformed at Liquid Nitrogen Tempera-ture Showing Failure by Necking. 110X • • • • • • • 53 34. (HO) Plane of a [lio] Whisker Deformed at Q°C Showing Beth Primary and Secondary Slip o o e o . . . . . . . 54 35. A Fractured [lio] Specimen Deformed at Liquid Nitrogen Temperature. 110Xe e o a o o o O i > o o o o o o o s > » 55 36. Diameter Dependence of Yield Stress for Copper Whiskers 57 37. Oriented Unit Tetrahedron to Show the Slip Systems in [ill] and [lOO] Whiskers. • • < , . « . . . . • • . • • 58 38. Stereographic Projection for Cubic Crystal in Standard Orientation, (Notation after Rosi and Mathewson). . • 59' 39. Gliding of a Dislocation PQ Across a Whisker Leaving a Surface Step of Area 6* » • • • « < > • > « • • • • • . 63 40. Comparison of Yield Stress Data for Iron Whiskers • • • 67 41. Temperature Dependence of the Flow Stress in Copper (after S e e g e r ) . . . . . . . . . . . . . . . . . . . . 68 42. Stereogram Showing the Specimen Orientations. . • • 70 43. Temperature Dependence of the Flow Stress Ratio for Coarse Whiskers and Bulk Single Crystals of Copper. • 73 44» Temperature Dependence of the Flow Stress Ratio for Copper Single and Poly Crystals. The High Tempera-ture region is From Poly Crystals . • • • 75 45. Strain Rate Dependence of the Flow Stress of Copper Single Crystals, (from Ref. 34) 76 46a. A f v s V (Shear)Curves for [lio] Whiskers 77 46b. A f v s T (Shear) Curves for [lioj Whiskers 78 47. Form of A f v s t Curve for [ i l l ] and [lOO] Whiskers . . 79 FIGURES - Cont'd. No. Page 48. A rVs t Curves for [ill] Whiskers 80 49. ATvs f Curves for [loo] Whiskers. 81 50a. Activation Distance d as a Function of Strain. . . . . 85 50b. Activation Distance d as a Function of Strain 86 51. Activation Distance as a Function of Strain for Single Crystal Copper at 295°K (from Ref.34)«> • » . « » . . 89 52. Intersection of Two Extended Dislocations« . . . . . . 91 53. Sequence Showing Cross-Slip 93 54-. Activation Energy for Cross-Slip (From Ref.16) . . . . 95 55. Kink Formation in a Dislocation Due to Thermal Fluctuations • • • • • • • • • • • . . . 96 56. Dislocation Intersection at Low Temperatures . . . . • 101 57. Failure of an L-C Barrier (a) Recombination, (b) Dissociation (after Strolr ^ ) . • » . . . • • • • 104. 58. An L-C Barrier . . . . . . . . . . . . . . . . . . . . 105 59. Sequence Showing Failure by Recombination of an L-C Barrier and Subsequent Slip on (001) Plane (Schematic) 660a Formation of a Twin Source Du  to the Operatio of Partial Dislocations AP and BQ in Fig.59. The Twinning Dislocation Will Climb Down with the Help of the Screw Segments AD and BG 107 60b. Dislocation Dipoles Along the Axis of a Whisker. . . . 115 60c. Probable Dislocation Distribution in Heavily Deformed Whisker Compared to Bulk Single Crystal. . . . .. • 116 61. Unit Tetrahedron1 AB, BC, etc. are BVs and c<,fc. • are Slip Planes 137 62. Illustrating Vacancy Formation by Moving Jogs. . • • • 138 63. Formation of Interstitials by Moving Jogs . . » . • • 139 FIGURES - Cont'd. No. Page 64.. Sequence Showing the Formation of Dislocation Dipoles and Loops. 140 65. Unit Tetrahedron Showing Different BVs 142 66. Image Dislocation at a Curved Surface. . . . . . . . . 14-6 67- Image Force Exerted on a Dislocation 1 by a whisker Surface. . o s a o « e . s o « o e o o . « o e 0 0 147 68. Load Measurement Using Fine Balance Control© . » o . » 14-9 69. Force Distance Curve • 151 70. Stereogram Showing Choice of Glide Systems (a) and Resultant Lattice Rotations (b) (from Ref. 61) . . • 154 71 . Stereogram Showing Lattice Rotation for a [lio]Whisker 155 72. Coordinates for Resolved Shear Stress Calculations • • 156 73. Coordinate Systems! x i is for Dislocation I and x£ is for Dislocation I I . . . . . . . . . . . . . . . . . 164 74.. Unit Tetrahedron Showing BVs and Dislocation Directions 169 75 . Two Edge Dislocations on Parallel Slip Planes. „ . . . 172 76. Variation of Force Due to Interaction Along an Edge Dislocation Line. Interaction Results Due to the Presence of a Second Non Parallel Edge Dislocation Lying on a Parallel Slip Plane at a Fixed Distance x 9 n. 173 TABLES No. Page I Conditions of Growing Different Kinds of Whiskers 3 II Comparison of Yield and Maximum Stresses for Whiskers and Bulk Single Crystals. . . . . . . . 43 III Number and Type of Jogs. . . . . . . . . . . . . . 61 IV Comparison of Yield Stress Data. . . . . . . . o e 66 V Temperature Baths . . . s e e . « . . . . . . . < > 71 VI Activation Volume, Activation Energy and Frequency Factors . . . . . . . . . . . . . . . . . . . . . S3 VII Stacking Fault Energies for Copper . . . . . . . . 88 VIII Dislocation Mechanisms and Activation Energies • • 97 IX Stacking Fault Energies for Au, Ni,Al, Ag and Cu . 99 X ^^YioXd «<> « e • # a o e e e • • • • • e o « • * • X20 XI Schmid Factor as a Function of Strain. 157 XII Force Components for Edge-Edge Interaction . e • • 170 XIII Force Components for Screw-Screw Interaction . . . 171 1. GROWTH 1 1. 1 Introductions The growth of crystals has been the subject of intensiye research during the last decade or two. The basic formalism forthe growth of perfect crystals was laid by Willard Gibbs* as early as 1878,and i t was developed extensively by Volmer, Sranski, Becker and Doring during the past thirty 2 years, Frenkel introduced the idea of stepped surfaces with a high concen-tration of kinks for the growth of perfect crystal layers. Since real crystals grew at supersaturations far below that required for the growth of 3 perfect crystals, i t was concluded by Frank that real crystals are not per-fect. In his classic paper he proposed that real crystals grew by screw dislocations which produce perpetual growth steps. Furthermore, he pointed out that impurities are mainly responsible for initiating these dislocations. In such eases growth can persist either by one dislocation or by two dislocations operating in cooperation. In the f irst case the growth steps wi l l be spirals and in the latter case they wi l l be closed loops. These can be either circular or polygonal in shape depending on the anis-tropy of the surface. Polygonal shaped steps have their segments parallel to close packed directions. Experimental verification was immediately followed by using modern optical techniques, an excellent account of which is given by Verma^ and an extensive discussion of the subject is contained in a recent symposium 5 report "Growth and Perfection of Crystals", and in a review article by 6 Hirth and Pound. 2 The basic premises of the theory have been applied successfully only to a very few cases of whisker growth. Since whiskers grow under a diversity of conditions i t is not possible to postulate a unified theory which is universally applicable. However, some of the salient features of the pertinent theories wi l l be recalled, (1) Eshelby's theory speculates on the presence of a Frank-Read source in the substrate in a plane normal to its Burgers Vector (BV), The result i s , the dislocation loops so generated wi l l effectively extrude a whisker from the substrate, (2) Hirsch and Amelinckx theory explains the growth and kink-ing of whiskers by invoking the idea of climb of a dislocation node from the substrate, (3) Some unusual whisker forms were explained by Webb who speculated on the possibility of dislocation climb near the whisker t i p . This principle was extended much more satisfactorily by Amelinckx to ex-plain kinking in FCC lattices. In this a growth dislocation of BV 'b• is expected to be in neutral equilibrium according to b ^ b ^ • b^, and i f b^ is parallel to one of the growth faces unlike b2> growth wi l l take place in a direction parallel to b 2 resulting in a kinked whisker, 10 Formation of such whiskers with twinned orientation was explained by a similar mechanism along with the repeated formation of a stacking fault. In the present work an examination of the substrate as well as whisker surfaces wi l l be presented. 1, 2 Experimental Procedure; 5 Table I is a summary of the captions used for growing different kinds of whiskers by hydrogen reduction of the respective halide. In the present work copper whiskers were grown both from dehydrated c u C l 2 and Cul by reduction with hydrogen at about 550*C. With dehydrated CuGl2, even though there was profuse whisker growth the whiskers were very thin compared to those obtained from Cul, The substrate was much more coherent in the case of CuGl 0, CONDITIONS OF TABLE I GROWING DIFFERENT KINDS OF WHISKERS Metal Halide Temperature Range *G Copper ( CuCl ( CuBr ( Cul 430 - 650 - 850 Silver v ( AgCl . ( Agi • 700 - 800 - 900 Iron ( FeCl 2 ( FeBr2 730 - 760 Nickel NiBr 2 740 Cobalt CoBr2 650 - 730 - 735 Manganese £ j}) MnCl2 940 Gold AuCl 550 Platinum PtCl 800 Palladium PdCl 2 960 1. 3 Results s (1) Examination of the Substrate: The reduction product consists of a thin copper deposit containing whiskers. The coherent deposit was peeled from the porcelain boat used for reduction from regions where dense whisker growth had taken place. These pieces were then mounted in plasticine, just by pressing a ball of plasticine against the whiskers, thus exposing the bare face free of whiskers for examination. This face was etched very briefly in cone. HNO^ , washed and dried in air and examined in an optical microscope. Radically different mierostructures were seen than what one would expect from melt grown specimens. Apart from the usual etch pits, large grains with spiral steps were observed (Figs.l and 2 ). An attempt was made to examine the copper deposit by transmission electron microscopy. But i t was not successful as i t was extremely difficult to prepare thin enough specimens by electropolish, because the deposit, while seemingly coherent was very cellular. ( 2 ) Examination of Whisker Tips and Surfaces: Unetched whiskers were also similarly examined. Whiskers with very well developed polygonal cross-sections did not usually show growth steps or hil l s on the bounding faces* Nevertheless, i t was not difficult to pick out whiskers of very odd growth surfaces* However, since i t was difficult to come to any reasonable conclusion from such whiskers, attention was focussed on whisker tips* Numerous whisker tips were carefully surveyed under an optical micro-scope. These whiskers were previously mounted on plasticine on a glass 5 F i g . l Etch Pits on Copper Reduced From CuCl 2 . Etched Briefly in Con. HNO_. 3000X (a) (b) Fig.2 Spiral Etch Pits on Copper Reduced From CuCl ? . Etched Briefly in Conc.HNOo. 3000X 9 slide. The tips were very well developed pyramidal surfaces which were free of any surface markings as seen in Fig, 3, However, these were a few whiskers which in fact did show growth steps as shown in Fig,4-» A schematic sketch of each whisker is also included. Finding those whiskers was merely fortuitous and chances were about 1 in 20, Whiskers exposed to air for a period of one week usually showed some surface markings, as shown in Fig, 5«> Fig ,6 is the tip of one such whisker of hexagonal cross-section, A very rare whisker produced from the reduction of Cul, is shown in Figs ,7 to 9, The axis of the whisker was [ll6\ and "the bounding surfaces were (001) and (llO), A series of photographs showing the growth steps on one of the (001) surfaces is shown in Fig. 7. Figs. 8 and 9 show two selected areas from the (llO) surface. Fig.3 Tips of as Grown Copper Whiskers. Sketch Shows the Orientation of the Facets. 800X Fig .4 (c) Growth Steps on as Grown Whisker Tips. Orientation of the Facets. 800Z Sketch Shows 12 F i g . 6 T i p o f a Whisker Exposed to A i r . 1600X Fig.8 Growth Steps on tho (110) Surface of the Whisker in Fig.7. 1101 Fig.9 Growth Step* on the (110) Surface o f the Same W h i i k e r in F ig . 7. 600X 16 1. U Discussion; Hoping that the observed etch steps in Fig. 2 are parallel to the original growth steps, the form of these etch spirals can be explained as follows. For this we consider the growth of a (110) plane by a screw dis-location of £[ID] BV. The growth spirals will then assume the anisotropy of the surface, and in this case we can expect them to be rectangular. Such a surface is shown in FigoID and the angular relationships are as denoted. The plane of the paper is (110). The two (111) planes shown in Fig. 10 are below the plane of the paper, and their traces on the (110) plane are [ l l 2 ] and [T12] • The experimentally observed microstructure can be [ l l 2 ] 72^  [lib] [112J (111) [001] (111) Fig.10 A Possible Growth Spiral on a (110) Plane. Fig.12 Form of the Spiral in Fig.10 After a Complete Sequence of Twinning. 18 obtained from the above, by repeated twinning say on ( i l l ) plane in a direction such as OA. For example, consider only one twin along OA. This produces the structure shown in Fig.11. Continued twinning along segment OD will result in Fig.12. The growth steps will now be parallel to the sides of the pentagon. Such sequence of twinning can give rise to a whole series of microstructures. If twinning took place on planes not passing through 0, we get microstructures of the type shown in Fig.13. Fig. 13 Twinning Within A Grain. The growth steps observed on the faoets of some whisker tips must be due to growth dislocations ending on the surface. In the case of the [ill] whisker in Fig. 4. we need to explain the formation of the (100) and (110) facets at the ti p . In order to explain this we consider a dislocation node as shown in Fig.lA. All the dislocations are marked such that the sum of the BVs at the node is zero. Of the three £ ^HO^ dislocations, \ [lOl] and £[oii] could give rise to the growth of (101) 19 and (Oil) facets. The dislocation £[lld] can be expressed in neutral equilibrium as, £[110] £ [100] • £[010] . Energy<X£ £ £ Thus the growth of {lOO} facets can be expected. Then a question arises why the whisker should grow without any branches and have just one axis. £ [lio] ^ [ l i i ] Fig. 14 t A dislocation node in a [ill] whisker, [ i l l ] is parallel to the axis, and <CllO>> are normal to the facets at the tip. In order to explain this we consider the dislocation line energy above and below the node. Above, i t is proportional to 3b and below i t is proportional to 2 b (where b « £[llo] ). This considerable reduction 3 in the line energy will force the node to climb as the whisker grows so that OA, OB, etc. remain constant in length. This will result in a net growth parallel to the [il l ] axis. It now remains to explain the surface markings shown by the whisker in Figs. 7 to 9. These could be due to the gliding of screw dislocations which are not parallel to the axis of the whisker. In this case, movement would be from the tip towards the base. 20 An analysis of these surface steps showed that in most bases, they are not the common octahedral planes• The horizontal traces on both the (001) and the (HO) surfaces are probably due to the gliding of screw dislocations with BV £ OlQ^ on one of the {ill} planes shown in Fig. 15. (110) growth plane (111) glide plane Trace of (111) glide plane (001) growth plane [llO] whisker axis Fig.15 Illustrating Horizontal Glide Traces Left on the (001) or the (110) Plane of the WhiskerJLn Fig. 7 dualto the Egress of ^^Olpi Dislocations on (111) Plane. All the other intersecting steps are attributable to the gliding of screw dislocations on non octahedral planes. On the (001) growth plane the observed stops are found to be parallel to [210] direction. Similarly on the (llO) growth plane, the steps are parkllel to a [22l] direction. 21 The steps as such seem to be normal to the respective growth planes• This suggests that the BVs of the associated screw dislocations must be [001]and ^ - [ l i o ] . Accordingly the respective glide planes are (l20) and ( i l l ) . The form of the steps may be easily explained as follows. The gliding of a series of dislocations on the ( i l l ) plane will result in a structure of the type shown in Fig016a.Subsequent gliding on planes of (120) or (114.) type will give rise to Fig.16b. This configuration will produce the block-like structure seen in Fig, 7. As may be seen from these pictures, complete growth has taken place where the dislocation f i r s t intersected the surface. The selected area from the (110) surface shown in Fig. 9 is probably due to the gliding of screw dislocations on the two intersecting (114.) and (114-) type planes only, leaving traces parallel to [221 ] and f_22l] respectively. The form of the steps may be obtained as follows. Gliding on one of the planes will give rise to Fig.17a. Further opera-tion of dislocations on intersecting planes will result in Fig. 17b. If now the dislocation B glides out of the crystal along the glide plane of A, the step OP will vanish, because A and B are of opposite sense. The final structure is shown in Fig.l7c. By similar operations the observed microstructure can be derived. Alternately, the structure in Fig. 9 can be derived without the introduction of dislocation A provided the dislocation B after having reached the point 0, changed its glide plane and moved in the direction OP*. Such a change of glide plane is possible because the BV -^[llO] is common to both (114-) and (114.) planes. The latter operation seems to be much more likely in t 1 2 2 < < whisker axis (a) glide direction glide < direction Fig. 16 Sequence Showing the Form of the Steps Left by Gliding Screw Dislocations. One of the Glide Planes is Parallel to the Axis. 2 3 Fig. 17 Sequence Showing the Form of the Steps Left on (110) Plane, due to Gliding Screw Dislocations on Intersect-ing Planes. obtaining the observed microstructure 24 The origin of such dislocations is unknown. As previously mentioned they might be either due to the thermally activated motion of screw dislocations not parallel to the axis or due to those formed by the breakdown of a Frank network previously present in the whisker. 2. STRESS - STRAIN CURVES 25 2. 1 Introduction; Dislocation theory has exerted a profound impact on the study of the plasticity of bulk single crystals. The best calculations show that a perfect crystal should possess a strength of about 3 percent of its shear modulus. Although bulk single crystals do not possess this unusually high strength, the historic experiment of Gait and Herring^" showed that filamentary crystals or "whiskers" do possess strengths of this order of magnitude. In spite of t$Ls extreme close-ness in value of the shear strength, the detailed nature of whiskers is s t i l l unknown. 12,13 H Experiments by Brenner, Gorsuch and others tended to con-firm the idea that whiskers are essentially perfect crystals. From these experiments, one can conclude that the only major deviation from perfection l ies in the possible existence of axial screw dislocations. 15 Price investigated the deformation of zinc platelets in an electron microscope, and observed that dislocations are generated in otherwise 5 5 perfect regions. Amelinckx by using decoration methods, and Webb by using x-rays, have conclusively shown that some whiskers do contain axial screw dislocations. However, since these are not expected to contribute much to the i n i t i a l nucleation of s l ip, the manifestation of high yield strength by whiskers is not very well understood. Whether whiskers are strong due to the complete absence of dislocation sources, or due to Frank-Read sources of extremely short length, is s t i l l un-certain. 26 At present there exists considerable data on the yield strengths 12 13 16 IV of different metal whiskers ' ' ' up to about 3 0 ^ in diameter. On 18 the other hand Suzuki investigated melt grown bulk single crystals from about 1500jU to about 60/^ in diameter. The present work is solely devoted to vapour reduced copper whiskers of about 30 - 400jJ- in diamet-er. In this i t is hoped that the results will bridge the gap between whisker behaviour and bulk single crystals. In what follows whiskers <^ 30JJ. in diameter are called 'fine whiskers ' and those y 30^X 'coarse whiskers'• (1) Characteristics of Stress - Strain Curves % Fine whiskers exhibit a remarkable yield drop followed by a region in which the flow stress remains almost constant as shown in leld stress s __J flow stress 7J~. Fig. 18. The ratio yi x may be as high as 80 »1. f l Fig.18 Yielding of a Fine Whisker (after Brenner). 27 Brenner however, points out that this yield drop should not be identif-ied with the yield point phenomena shown by Fe-C alloys. For example, as he indicates the whisker behaviour might be due to the spontaneous generation of dislocations followed by a region in which the strain rate is very high. That is why some whiskers might f a i l immediately after tha yield drop. In the region where the flow stress is constant whiskers deform by Iideraband propagation. This is characterized by the nucleation of slip in one system at some part of the crystal. This region then traverses across the specimen until the whole specimen is f i l l e d with slip lines. This is then followed by a rapid hardening region. If Nucleation took place In more than one point in the speci-men then the stress - strain curve would be as shown in Fig. 19. Fig.19 Repeated Yielding of a Fine Whisker (after Brenner). (2) Size Effects; Whiskers show a prominent size dependence in their mechanical 12,13,16 properties. According to the existing data a variation of yield stress proportional to 1 Is observed for fine whiskers. The 18 i ' only data by Suzuki on size dependence measurements for bulk single 28 crystals of copper do not however show a striking diameter dependence of the yield stress* A deformed whisker when retested excluding the deformed region, has a yield stress which is not only increased con-19 siderably, but also shows in turn a size dependence. No such work has been done on bulk singlecrystals. (3 ) Deformation Characteristics in Whiskers and Bulk  Single Crystalss It is Interesting to distinguish the deformation modes in whiskers and single crystals. As previously mentioned, fine whiskers are characterized by a conspicuous yield drop followed by a constant stress region in which the whisker deforms by Luders band propagation. Bulk single crystals of copper have stress - strain curves characterized by three different work hardening regions, as shown in F ig . 20. The form of the curve is extremely sensitive to orientation. Fig.20 Schematic Shear Stress Crystal Copper. - Strain Curve for Single The region of easy glide (Stage I) is one in which the work hardening rate is very low. In Stage II the work hardening rate is high but linear followed by parabolic hardening in Stage III, which is also ex-hibited by fine whiskers to some extent. The deformation characteristics of single crystals wil l not be discussed in detail here. The existing theories are rather controv-ersial in nature. However i t wi l l be pointed out that according to one of the theories the region of easy extension (Stage I) is marked by sl ip starting at longest available Frank-Read sources and progress-ing unti l a l l such sources are exhausted. In Stage II rapid hardening sets in due to increased interaction of dislocations from the primary and secondary systems which resultsin the formation of dislocation 20 35 barriers, (see for example, Friedel and Soeger ) . Stage III is marked by the break down of these dislocation barriers and resulting 21,35 in the pronounced cross-slip of screw dislocations • Some of the work hardening characteristics in coarsewhiskers .\ wil l be discussed in the light of the current theories and using the modern concepts of dislocations, in this and in the next section. It is worth mentioning that whiskers tested by Brenner and several other investigators were pulled in a soft beam tensile testing machine. The tests are extremely discontinuous in nature and do not provide a continuous variation of strain. In the present work, the testing method permitted not only a wide load range but also a continu-30 ous variation of strain. In addition, temperature change tests could be conducted very easily, using a standard fixture mounted below the cross-head. The procedure is identical with that used for bulk single crystals, except for the slight differences in the i n i t i a l handling precautions. 2. 2 Experimental Procedure; (1) Testing Method? Copper whiskers of suitable dimensions were successfully tested in a floor model Inst ron testing machine using low strain rates and very sensitive load cells, A suspension schematically shown in Fig, 21 was used for mounting whiskers. It consists of a mild steel rod approximately £" in diameter, and is as short as possible so as to minimize lateral deflection and also to bo consistent with the conven-ience of test procedure. At the bottom end i t is split to accommodate the mounting rod, which is 2" long, and is threaded at its upper end. In the earlier room temperature tests the mounting agent used was diphenyl carbazide. However, for liquid N2 tests this was found to be unsuitable because i t not only contracted considerably, but also detached from the mounting stage. Solder was found to be very suitable for mounting and a l l the whiskers for these tensile tests were mounted in solder. The mounting technique is very simple. A carefully selected whisker is cut to a reasonable length and placed horizontally on a thoroughly clean glass plate. Cleanliness is essential for i f the glass plate is not clean enough, the whisker will stick to i t and f a i l -V ure becomes inevitable. The final arrangement for mounting is shown in Fig. 22. 31 to the load cell mounting agent whisker Fig.21 Suspension Used for Whisker Testing. 32 glass plate Fig.22 Mounting of a Whisker. It is very convenient to hold the mounting rod firmly against the table by means of plasticine* As the solder melts, surface tension draws the whisker horizontally into the molten solder* Over-heating the whisker was easily prevented by not heating the mounting stage after the solder had melted and by gently blowing cool air* The system almost instantly attains the ambient temperature* The whisker can be further examined at very high magnifications directly to detect any possible surface damage, and the whisker discarded i f necessary* It can also be directly used in the orientation determination of the whiskers * The suspension ready for test is threaded on to the universal fixture from the load c e l l * The bottom grip simply consists of a rigid block with a small pool of molten solder melted in situ. This is gradually raised until a reasonable length of the whisker is Inside the pool* The whole operation is done with a copious flow of cold hydrogen onto the whisker. The system is then allowed to cool to room temperature* A mounted whisker is shown in Fig, 23 . (2) Load Cycling? During mounting and the introduction of any temperature baths, the whisker experiences a load due to thermal expansion effects. By careful adjustment of the load cycling doVice in the Instron i t is possible to maintain the load constant within + 0,2% f u l l scale. (3) Testing? All whiskers tested for size dependence measurements were 34 Fig.23 Mounted Whisker Ready for Test. 35 strained at a basic cross-head speed of 0.002n/min , the lowest available in the Instron. Some of the Ini t ia l tests were done at a rate of 0.01"/ min. Yield stress values and stress - strain curves were also extracted from strain rate change experiments deformed alternately at G.002n/min and 0.02w/min. The gauge length of the whisker could not be determined before the experiment. Since the total elongation is always known from the load elongation curves, the original length is deduced from accurate measurements of the final length. Since the i n i t i a l length of the speci-men cannot be fixed very accurately, the strain rate (£) is not identical from one test to another (4.) Orientiatlon DetanBJhations Usually orientation determination could be made by examining the cross-section under a microscope, for the axes are already known for familiar cross-sections. Nevertheless X-ray orientation determina-tions were made for a l l whiskers used in strain rate and temperature change tests. 2. 3 Results 8 Whiskers tested in this work belonged to three prominent orientations corresponding to the corners of the stenographic triangle in F i g . 24. Whiskers of D-1Q] orientation were thin blades. (1) Stress - Strain Curvest Some typical load - elongation curves are shown in Fig. 25. These curves are serrated for most whiskers. The serrations tend to smooth out with whiskers of very large diameter. With whiskers around 50JJ- (• 20JJ-) in diameter the stress - strain curves were very similar to those of fine whiskers, (of course excluding the i n i t i a l yield). A region of Luders band propagation followed by Stage III hardening could be seen (Fig. 26 ). However i t could not be decided whether the absenoe of the conspicuous yield drop in these whiskers was due solely to the characteristic of the whisker used or due to mounting. For larger whiskers the form of the stress strain curve is shown in Fig. 27 for a l l the three prominent orientations. For the sake of comparison in Fig.27 curves are given for whiskers and bulk single crystals for the three orientations. It may be noted that in Fig.27c a divergence is shown in the stress - strain curves. The divergence in the single crystals is also such that the coarse 10-* 2 2 Cross-sectional arear 35x10 y* 0.002 0.004. 0.006 Elongation (in) (a) 0.008 2 2 Cross-sectional aroa= 35x10 JJ-0.002 0.004 0.006 Elongation (in) (b) 0.008 Fig . 25 Load-elongation Curve for (a) [lio] Whisker and (b) [100] Whisker. Cross-head Speed 0.002"/min. 38 Fig.25 Cont'd. Load-elongation Curve for [ill] Whisker. Cross-head Speed 0.002n/min. 100 —i 1 r 0<l 0-Z 0-3 Fig. 26 Tensile Stress - Strain Curves. (Use left scale for No.16). 40 Fig. 27 Comparison of Stress - Strain Curves for Coarse Whiskers, Fine Whiskers and Single Crystals. 41 Fig. 27 Cont'd. Comparison of Stress - Strain Curves for Coarse Whiskers, Fine Whiskers and Single Crystals. 4 2 Fig. 27 Cont'd. Comparison of Stress - Strain Curves for Coarse Whiskers, Fine Whiskers and Single Crystals. 43 whiskers and the single crystals merge at the limits. However, this is found to be unimportant in other orientations. (Tensile stress - strain curves for single crystals are obtained from the shear stress - strain curves of Ref.18 for selected orientations close to the corners of the stereographic triangle). From these curves i t is evident that there is a wide range of behaviour from fine whiskers to bulk single crystals*. The yield stress is extremely large for fine whiskers. Subsequent to yielding the max-imum stress is larger for coarse whiskers, as may be seen from Fig. 27. It is also clear that these quantities show a strong dependence on orientation. Some typical values for yield and maximum stresses are given in Table II for the three orientations considered. TABLE II COMPARISON OF YIELD, AND MAXIMUM STRESSES FOR WHISKERS AND BULK SINGLE CRYSTALS. Tensile 2Stress kg/mm Orientation Fine Whisker Coarse Whisker Single Crystal 70 1 - 2 o.i ^yield [ioo] 50 1 0.1 _liio] 50 1 0.1 "[ i l l ] 15 100 10 - 20 T L max. [lOQJ 10 20 10 - 20 _ [llO] (thin 10 - 20 10 - 20 blades) From a large number of stress - strain curves (see App.I and Fig• 26) we may summarize the results as followjs: (2) For Whiskers of [ i l l ] and [lOO] Orientations (a) With increase in diameter the region of Liiders band propa-gation decreases rapidly and results in a small region of easy glide, which may not be present in some whiskers. For [100] whiskers, this "easy extension" (Liiders Band or easy glide) is shown in Fig. 28. (b) For a given orientation the region of easy extension sojamed to increase with decrease in temperature, (see S16, S38 for [i l l ] and S23' and S35 for [lOO] ). However, the complete absence of easy glide at low temperatures was not uncommon. Due to lack of specimens at fixed diameter, a narrow range of diameters is chosen for comparison in Fig.28 for [lOO] whiskers. (c) The hardening rates are a l l similar at a l l temperatures and diameters. The effect of temperature may be masked by the varia-tion of impurity content for whiskers grown from different batches of iodide. (d) Ductility does not change appreciably with diameter and temperature. (e) The hardening rate in [il l ] whiskers is about five times larger than in [lOO] whiskers at a l l temperatures. 0 0 8 _ 0-06 Easy Extension 0-04-. [100] • • • • • 1 1 1 , : , , 50 [OO 150 ZOO 2.50 300 Temperature °K Fig . 29 Temperature Dependence of Easy Extension for Coarse Whiskers in a Narrow Range of Diameters (60 - 80^-0. (3) For Whiskers of fllOl Orientations 47 (a) These show a l l the three stages of work hardening* (b) Stage III hardening rates are similar to those for O-OOl orientation* (o) The easy glide region increases with decrease i n temperature. (d) Ductility i s considerably higher compared to both [ l l l l and [lOO] orientations. (e) Deformation i s mostly on one system. The divergence i n the curves of similar orientation at a given temperature i s partly due to the effect of diameter and as previously mentioned partly due to the minor differences i n composi-tion that might prevail from one batch of whiskers to another. Composition of a typical batch of whiskers i s given i n the next section. (4) Metallography and Fine Structures (a) Whiskers of [ l l l l orientation deformed on two s l i p systems. At l i q u i d nitrogen temperature these whiskers deformed also by twinning. A l l the six faces of a twinned whisker are shown i n F i g . 30. A schem-atic sketch i s also included. Twinning was confirmed by taking a back reflection Laue picture, using a s l i t collimator, which showed distinct s p l i t t i n g of reflections. A load elongation curve at the onset of twinning i s reproduced i n Fi g . 31. (See next section for a mechanism of twinning). Micrograph i n F i g . 30 also shows deformation on non octahedral planes. An analysis of the s l i p traces on the two adjacent non octahedral s l ip I F ig . 30 twin trace A {ill] Whisker Deformed at Liquid Nitrogen Temperature Showing Deformation Twins. Sketch Shows a Twin Trace. 110X 49 200 WW***9 150-Load (gms) 100 2 2 Cross-sectional Area=18.5xl0 y-Cross-head Speeds0.002"/min 50 -0.0594 0.06M 0.0634- 0.0654 0.0674-Elongation(in) 0.0694-Fig. 31 Load Elongation Curve at the Onset of Twinning for a [l l l j Whisker Deformed at Liquid Nitrogen Temperature surfaces shoved that this plane is of the (110) type. (b) Whiskers of [100] orientation show different kinds of slip line markings. The predominant type corresponds to "double slip" in which two slip planes symmetrically oriented with respect to 21 the tension axis operate alternately. Less common is one in which the slip lines are fragmented resulting in the formation of a "oellular" structure. This terminology will be further used in discussion. Occasionally a third type is observed due to non octahedral slip. Fig.32-is from the same whisker shewing a l l the different kinds of slip markings. Fig.33 is from a whisker where failure is imminent. mostly on one slip plane. Secondary slip systems seem to have come into operation after some deformation. However, in this case second-ary slip takes place earlier than predicted by the Von Gbler and Sacis condition (see Appendix VII), Fig.34- shows both primary and secondary sl i p . Fig. 35 Is from a fractured specimen tested at liquid nitrogen temperature. (5) Diameter Dependence of the Yield Stress? The dependence of yield stress on the diameter of fine whisk-12 13 16 ers has been obtained by various workers ' ' in the form (c) Whiskers of [lid] orientation (thin blades) deformed d(/0 (b) 210X Fig. 32 A £LOCu Whisker Deformed at Liquid Nitrogen Temp-erature Showing (a) Double Slip and (b) "cellular" Slip Line Markings. 52 Fig.32 Cont'd. The [lOO] Whisker Deformed at Liquid Nitrogen Temperature Shoving Non Octahedral Slip. 110X Fig. 33 A [lOcH Whisker Deformed at Liquid Nitrogen Temperature Showing Failure By Necking. 110X (b) 210X Fig. 34 (HO) Plane of a [lio] Whisker Deformed at 0°C Showing Both Primary and Secondary Slip. 55 Fig.35 A Fractured Q-lo] Specimen Deformed at Liquid Nitrogen Temperature. 110X 56 B (^20) are constants. Fig, 36is a plot of ^ v s 1 from different J d sources. The above straight line is also shown. Yield stress values from the present work are as indicated. They f a l l along the same straight line and show a similar scatter. 2, U Discussion; (1) Hardening in [ill] and [lOo] Whiskers The work hardening characteristics of bulk single crystals of copper of orientations close to [ill] and [lOO] has been discussed 21 in some detail in a review article by Clareborough and Hargreavos . 23 Recently Saimoto related small differences in the work hardening characteristics of [lllj and [loo] bulk single crystals of copper to the different kinds of jogs formed in the two orientations. In the present discussion different sources of work hardening in the two orientations will be considered in some detail using the properties of jogs developed in Appendix II. In order to facilitate discussion we consider, the unit tetrahedron ABGD shown in Fig.37a. In Fig.3'tothe tetrahedron is re-oriented with respect to the two orientations. In principle, for exact [ill] orientation, three slip planes JB , */*, and 8 will be i n operation, resulting in 6 slip systems} and for exact [lOO] orientation a total of eight systems will be in operation. In such cases, the rate of work hardening for [lOO] would be slightly larger. This is due to the fact that when a l l the slip planes are in operation (as in [lOO] ) four different Lomer - Cottrell (L= C) barriers can be formed, Fig. 36 Diameter Dependence of Yield Stress for Copper Whiskers. Fig . 37 Oriented Unit Tetrahedron to Show the Slip Systems in Lm] and [lOO] Whiskers. whereas with only six systems ((jLlll) a maximum of three L - C barriers will be produced, (see Appendix III), 59 In practice, only two slip planes operate for both orienta-tions - sayjB and 6 for [ill) and ~( and 8 for [lOO] • Therefore, since equal numbers of L - C barriers are formed in the two orientations, the difference in work hardening characteristics cannot be attributed to the number of L - C barriers alone. It should also be noticed that at the start of the deformation, the Schmid factor is zero on the Conjugate Plane [ill] [no] Cross-Plane[ill] [111] Critical Plane [i l l ] Primary Plane Fig,38 Stereographlc Projection for Cubic Crystal in Standard Orientation;(notation after Rosi and Mathewson), {100} <110> system, for any combination of the primary and secondary system in either of.the orientations. Hence, the L - C barriers i f at a l l formed are stable. ZL Using Rosi and Mathewson's notation for the slip planes (FIg .38), we now consider the different dislocation interactions and the types of Jogs formed in gliding screw dislocations. For [ill] orienta-tion, the active slip systems will lead to dislocation intersections of the primary cross type, J^ eg. (Ill) [lOl]/(lll) • In each of 60 the three cases a 60° jog will be formed. Intersections of the type (111) [101] / ( l i l ) [llO] and (111) [oil] /(111) [lOll win give rise to equal numbers of 60° interstitial and vacancy jogs. (Ill) Q)ll] / (l i l ) [lio] will produce 60° interstitial jogs only. In the [lOO] orientation the intersection is of the primary conjugate type? ie., (Ill) [lOl]/(111) [lOl] and (111) [lio]/(lll) [lio] . o In both cases a 90 interstitial jog is formed which is sessile especially at low temperatures and will give rise to the formation of dipoles (see Appendix II). Intersections of the type (111) [llo ] / ( l l l ) [lOl] and (111) [lOlj /(111) [lio] will give rise to the formation of . o 60 interstitial type jogs. For the two active slip planes considered (cf* j3 for [ i l l ] and for [lOO] ), the number and the type of defect formed is listed in Table III. This table is meaningful only at the beginning of de-formation and the numbers will change with the density and distribu-o tion of dislocations. However i t should be noticed that 90 inter-s t i t i a l jogs are formed much more readily in [lOO] than in [ i l l ] orientation. TABLE III NUMBER AND TYPE OF JOGS Orientation System Intersection No. of Jogs. 60° 0 90 AB(6 ) G - G 1 G - F 2 1 [ill] AC(J3,C5) G G - G and/or - F 6 2 G - G 1 -G - F 2 1 BD(f) G - G 1 1 G - F 3 ADK) G - G 1 1 [100] G - F 3 -BC.(<5") G - G 1 1 G - F 3 -AC(<f) G - G 1 1 G - F 3 -[jJotes (i) G - G refers to intersection between two gliding screw dislocations, where only interstitial type Jogs are formed. G - F refers to intersection between a glide and a forest dislocation where equal number of vacancy and interstitial type jogs are produced. For notation see Fig. (ii) Intersections resulting in kink formation are ex-cluded here. 62 (i i i ) A dislocation of a given BV, eg. AD may be both an active glide dislocation in and a non active forest dislocation in S The properties of 60 "and 90 jogs are discussed at some length in Appendix II. o The predominance of 90 jogs is at least in part responsible for a lower work hardening rate in [lOO] orientation. 90° jogs can also give rise to the formation of "dislocation clusters". In the present context, "dislocation clusters" refer to agglomerates of dislocations lying parallel to <(ll2)> directions, resulting from screw dislocations acquiring sessile jogs. Dislocation clusters have in fact been observed by Fourie 25 et a l . in deformed bulk single crystals of copper. Formation of such clusters is probably responsible for the cellular structure of slip lines observed in Fig. 32b. (2) Diameter Dependences If the equilibrium density of dislocations in these whiskers 6 8 , 2 is less than 10 - 10 /cm then for whiskers less than about 10JJ-in diameter, the following inverse diameter dependence may be derived. Consider a whisker of circular cross - section as in Fig. 39. Slip has taken place by the motion of a dislocation PQ, currently of length '1* which was nucleated at some point S on the surface. Its motion produces a surface step of area 6 (PBQSP), and a slipped area A (PSQP). Let ^ be the stress acting on the dislocation which moves i t from the point S. Then we can write, ^ e f f *A - -f£ + Ejjl 63 ID Fig. 39 Gliding of a Dislocation PQ Across a Whisker Leaving a Surface Step of Area <§ . where the term on the left hand side is the work done, ' 1 ' is the BV, "C is the surface energy, E is the dislocation line energy/unit s D length. The effective stress may be written as, eff " a int. (2) where ^ a is the applied stress, and ^" i n t i s t o t h o inagQ force acting on the dislocation (see Appendix IV for estimation). From eiqn (1) we have, r 8 = £ £ . + _ V _ + r ( 3 ) bA bA i n t The largest contribution in the right hand side of eqn (3) comes from the dislocation line energy term, which reaches a maximum when 1 a 2R. Approximating^ = 2Rb, we get -T2Rb E_2R r = -^-z * -2—2- + C i n t a h 7TR b JTfi 2 2 — ^ - J — int (4) R 7TbR For copper "f • 2200 ergs/cm2at 300 PK 2 6' 6 5 and in 2 _g ED 3 — l n £ > using a value of r » r x 10 cms (r —b) we get, 4 Ti x ° ° o ^(kg/mm2) = 17 • 423__ + ? d (/^) d(yw) int - /SO + 1 1 kg/mm2 _(5) The above analysis is applicable to larger whiskers provided that surface nucleation of dislocations is assumed to prevail. The presence of internal defects such as impurities, dislocations etc., will give rise to another internal stress "C^. The effective stress can then be written as T i f f = " ^ i n t " ^ i ^ e cin (5) will now become, 'a(kg/mm ) = 480 + (11 kg/mm + T*) _(6) T" may be expected to be slightly lower than the yield stress of the ordinary single crystals. The above analysis seems to be in good agreement with the experimentally determined values, (see Fig, 3 6 ) , If the material is very surface active, the theoretically de-rived constants may not coincide with the experimentally determined ones • 12 16 Measurements also exist for Fe, Co and Ni whiskers up to ^15JUL in diameter. Using a value of 3500 ergs/cm at room 26 8 , 2 temperature andjU^>7,5 x 10 gm/cm for these metals, the above analysis gives, 2 2 (kg/mm ) s 8Q0 + 18 (kg/mm ) a -d(jLt) The experimental and the theoretical expressions are listed in Table IV. In Table IT the expression for was obtained by taking the arithmetic mean of the yields stress values at different diam-eters and plotting as a function of diameter and measurements were made on whiskers <^ 20jLb . Accordingly observe the limited applic-ability of the ^" a v expressions for Fe and Go, because at large diameters the average stress ( f&v,) values become negative which is contrary to expectation. In order to compare the experimental and the calculated ex-pressions, Brenner's values for Fe are replotted in Fig, Ifi. Both the straight lines are as shown. The agreement is good within the observed scatter. TABLE IV COMPARISON OF YIELD STRESS DATA 66b Metal Applied Stress (T" )kg/mm2 Source Experimental Calculated Cu (a) = 308 + 16.8 Bokshtein (b) V = £L<L + 36 Brenner av d Co Fe r av 1630 - 50 Brenner d(jLL) « 1200 - 1£ T - 800 -f 18 Bokshtein d ( j U ) d l j l ) av Ni r = 620_ + 23 3. TEMPERATURE AND STRAIN RATE DEPENDENCE 3.1 Introduction? 27 It was f i r s t pointed out by Seeger that the flow stress of a metal could be split into two componentss V3 which is strongly temperature dependent and the component Tg which Is independent of temperature. The variation is as shown in Fig.41. He attributed this behaviour to different rate controlling mechanisms. For example, in metals with low stacking fault energies, jogs may be created in edge dislocations more easily than point defects at jogs in screw dislocations. Hence the f i r s t process will predominate at extremely low temperatures. The strong temperature dependence of the flow stress up to a critical temperature T c is due to the increasing ease with which point defects are oreated at jogs in screw dislocations. Due to the increased mobility of Fig.41 Temperature Dependence of the Flow Stress in Copper, (after Seeger)• 69 the point defects, above the temperature T this is no more a source of . . . . . . c hardening and the temperature dependence is the same as the shear modulus. An experimental investigation of the temperature dependence of 28 the flow stress by Cottrell and Stokes using aluminum, and by Cottrell 29 and Adams using copper, resulted in what is known as the Cottrell -Stokes law. They observed that i f the same specimen is deformed altern-ately at two different temperatures, then the fractional change In flow stress accompanying a change in temperature reaches a constant value after some amount of in i t i a l deformation. This constancy of the frac-tional flow stress change which is now known as the Cottrell - Stokes law, demands that a proportionality be maintained between the long range and the short range stresses opposing dislocation motion. Identifying these stresses with those due to glide and forest dislocations respect-ively, Cottrell and Stokes suggested that their observations were con-sistent with the idea that during deformation, the dislocation pattern remained unchanged, the only change being in scale. Anticipating a significant difference in pattern for whiskers compared with bulk cryst-als, the present Cottrell - Stokes experiments were undertaken. 3 0 Later Basinski showed that temperature and strain rate change experiments are equivalent, and strain rate change tests have an added advantage that the yield point effects inherent in the temp-erature change tests are avoided. The hitherto existing experimental results have been a l l on melt grown single or poly crystals. In the present work results of similar experiments on copper whiskers produced as described earlier, will be discussed 70 3.2 Experimental Procedure? Single crystal copper whiskers of [lOO], [lio] and [ i l l ] orientations were used (Fig. 42). A spectrographs analysis of a typical batch of whiskers showed the following composition: Spectrographs Analysis of Copper Whiskers Al • - 0.009% Mn « - 0.008$ Au B - 0.001$ Ni -- 0.0004$ Nb Co • - 0.004$ Si • - 0.04$ Ag Fe -- 0.03$ Ti • - 0.001$ Others Mg • - 0.0003$ V • - 0.001$ Fig. 42 Stereogram Showing the Specimen Orientations. (1) Temperature Change Tests 8 71 Due to the extensive deformation (^50%) shown by whiskers of [lio] orientation only these whiskers were used for temperature cycling. The change in the flow stress as a result of deformation at two differ-ent temperatures was measured for several specimens at the following series of temperatures. TABLE V TEMPERATURE BATHS 0 Temp K Bath 77 Liquid Nitrogen 136 Controlled level of Liquid N 2 200 + 2 Solid Co2 " Acetone 273 ± 2 Water - ice 373 + 1 Boiling water 423 + 1 Silicone o i l The temperature was always changed from higher to lower temp-29 erature to avoid work softening effects . A copper jacket was used to protect the whisker from the liquid baths. During any change of bath and until the desired test temperature was reached, the specimen was always maintained at zero load by using the load cycling device. At o o 136 K and 200 K tests were conducted by lowering the specimen temper-o o ature to Iquid nitrogen. For tests at 273 K and 373 K the temperature o o was respectively lowered to 200 K and 273 K. At higher temperatures, o temperature cycling became more and more cumbersome. Hence at 423 K a direct temperature change was not made to a lower temperature. o Instead, the same bath was cooled to about 373 K by blowing cold air. 72 In both cases the temperature of the bath was accurately controlled by using precision Microset thermo-reigulators previously set such that the specimen always reached the desired test temperatures. Accordingly the baths had to be maintained at temperatures slightly higher than the test temperatures. The specimen temperature was always checked by a thermocouple during the course of the experiment. In a l l cases the temperatures could be accurately controlled within 1 to 2 ° . A basic cross-head speed of 0.002"/min was used for a l l temperature change tests. (2) Strain Rate Change Testss Strain rate change tests were made at temperatures 77°K, 200°K, o o 273 K and 295 K. The increase in the flow stress was measured by chang-ing the cross-head speed from 0.002n/min. to 0.02n/min. In order not to lose the sensitivity in the Instron for small load increment measurements, a change of scale to higher scales was avoided. Instead, the fine balance was used to control the pen swing (for details see Appendix V)• 3.3 Results: (a) Temperature Change Tests; r The constant flow stress ratio 3L is plotted as a func-r ? 7 7~ tion of temperature in F ig . 43. Wherever necessary, the ratio 3L in the curve is obtained by using the transformation, T \ T C77 \ T i ; (Note that 77°K is used as the reference temperature). ^77 .(1) Fig.43 Temperature Dependence of the Flow Stress Ratdo for Coarse Whiskers and Bulk Single Crystals of Copper. 74 31 Shear modulus correction was applied using Overton and Gafney's meas-urements, which were extrapolated for higher temperatures. The curve consists of a temperature independent region followed by a region in which the flow stress ratio is strongly temperature depond-ent up to a cr i t ica l temperature T ( ^ 300 K) . For temperatures greater than T c , the ratio is again independent of temperature. Temperature dependence sets in again at s t i l l higher temperatures. 29 Superimposed in F ig . 43 are the results of Cottrell and 32 Makin for bulk single crystals of copper. Tho results in the present work are found to be similar to those obtained for bulk single crystals. T The form of the jT curve, differs from that obtained by Cottrel and ^77 by Makin in that the temperature dependent region is considerably inr creased, and the temperature independent region is contracted. However the form of the curve is in f u l l agreement with that obtained by these workers. 33 Hirsch made measurements on the variation of the flow stress ratio T up to about 1273 K using poly crystalline copper. Using -273 equation (1) and the data of Cottrell and Adams, his ratio of J— are T 273 transformed to T and plotted in conjunction with Cottrell 29 77 and Adams data for single crystals in Fig.44.. The high temperature transition occurs around 500°K compared to about 370°K (Fig. 43) for copper whiskers. (2) Strain rate Dependence of the Flow Stress? 34 Hirsch, et a l discuss in detail the strain rate dependence 0.2 4 200 S o 600 800 1000 1200 1400 Temperature K _^  Fig. 44 Temperature Dependence of the Flow Stress Ratio for Copper Single and Poly Crystals. The High Temperature Region is from Poly Crystals. 76 of copper single crystals of various orientations and of polycrystalline copper specimens. For single crystals Hirsch observed the dependence shown in F ig . 45. A V r Fig.45 Strain Rate Dependence of the Flow Stress of Copper Single Crystals (from Ref.34-). In the present work for the [HQ] orientation the scatter is such that regions a and b are obscured. However' region c is distinctly present. Typical,, curves are presented in F ig . 4.6. Even though second-ary systems operate in most of these specimens towards the end of de-formation, no change in slope in the Afvs "V curves is observed. For the [ i l l ] and fiooj orientation, the possible form of the curve is shown in F ig . 47. Regions 'a' and »d 1 are not distinctly shown by most whiskers. A few selected curves are presented in Figs. 48 and49. For a l l orientations the linear region f c 1 could be expressed in the form A t : A + B f , The intercept A is negative in some cases • —1 1 ; r 1 1 T 1 2 A 6 8 10 12 H TZ (shear) (kg/mnr) F ig . 4.6a A T V S V(shear) Curves for [lio] Whiskers. 0.15 A S6A - 200 K © S79 - 77°K 0.10 H 2 (kg/mm ) 0.05-1 6 8 (shear) (kg/mm ) 10 Fig . 46b Afvs T (shear) Curves for [lio] Whiskers 79 AT r Fig. 47 Form of A T V s T Curve for [ill] and [lOO] whiskers. (3) Activation energies, Activation volume, Frequency factors.etc. The strain rate is given by o k T where, €. q = NAb~? , with .(2) N = no. of activation sites per unit volume A s the area swept by a dislocation segment in one activated jump. b m BV of the dislocation ^ = vibration frequency of the dislocation. AQ is the thermal component of the activation energy expressed as, Q a H - v .(3) where H is the total activation energy and v is the activation volume, 7Ta is the applied shear stress. 2.Ch 20 40 60 80 100 120 ljo r (kg/mm ) F ig . 48 ^TvsT Curves for [ill) Whiskers. 82 Results of the temperature change experiments on [lio] whiskers were used to calculate the activation energy, activation o o o o volume and the frequency factors at 77 K, 200 K, 273 K and 295 K. Details of the calculations are given In Appendix VI , Previous meas-urements show that Cottrell - Stokes ratio has no striking orientation dependence. Hence, although no temperature change tests were made on whiskers of [lOO] and [ill] orientations, values from [lio] whiskers were used to determine activation energies etc,, for these orientations, (A) Activation Energy? Whiskers of [lio] orientation usually deformed up to about 50 - 60J6, and a continuous variation in activation energy is observed o o for these whiskers. For specimens tested at 295 K and 273 K, the total activation energy H  A AQ + V^(SEE ;Appendix VI for description) in -creased gradually from about 0,5 ev and reached an almost constant value of 5.5 ev (+ 0,3ev) towards the third stage of the stress - strain o curve. At 200 K these values are 0,5 ev and 3,8 ev \± 0,2), Whiskers of [ i l l ] and [loo] orientation always deformed less than 25$, Although some specimens showed an i n i t i a l scatter in the total activation energy, for the most part they deformed with an almost 0 0 constant j; activation energy. For specimens tested at 295 K and 273 K o o the latter value is about 5.2 ev(± 0,3 ev). At 200 K and 77 K this is about 3,8 ev(+ 0,2 ev) and 1,8 ev(+ 0,2 ev) respectively. (5) Activation Volumes 83 For a l l specimens the activation volume decreased slowly -20 3 r n with increasing stress. This value is ~^10 cm for [2101 whiskers, -20 -21 3 r i -21 -22 3 r ,n 10 -10 cm for [_10Ql whiskers and 10 -10 cm for [1111 whiskers* It also showed a small temperature dependence, decreasing slightly with decreasing temperature, (6) The quantity In ^  : A striking feature of the quantity In %P IS its strong temperature dependence. The average value for specimens of a l l orient-ations is — 35 at 273°K, ^20 at 200°K and 8 at 77°K. A summary -of the results is presented in Table VI, TABLE VI ACTIVATION VOLUME, ACTIVATION ENERGY AND FREQUENCY FACTORS Orient-ation Total Deformation Activatio^ Volume,cm v - bdl Temp. °K Total Activation Energy ev H B A Q + vT In Co £ [110] 50% -20 -20 10 -0.5x10 295 273 0.5 to 5.5(±0.3) 35 [lOO] -20 -21 10 - 10 200 77 295 273 0.5 to 3.8(+0.2) 0.5 to 2.5(±0.1) 5.2(f 3) 20 8 35 [HI] 25% -21 -22 10 - 10 200 77 3.8(±0.2) 2.2(+0.2) 20 8 (7) Activation Distance? 84 For the purpose of discussion, the quantity 'd' which Is the activation distance expressed in , v = b d 1 (4) wi l l be considered, where v is the activation volume, b is Burgers vector and 1 is the activation length, 'd' could be calculated for any value of the flow stress with a knowledge of the activation volume. For this consider the equation T - oLjUb^S (5) where the flow stress "V is assumed to arise due to dislocations distributed in a random pattern, with density N per unit AREA.JJ- and b have the usual meaning, oC is a constant depending on the character of the dislocation considered ( <X - 1 for edge, and 277" ( l - V ) ° ( s 1 for screw dislocations). If the mean spacing between 2 IT dislocations is 1, then N > _1_, and from (5) l 2 T s CXUb (6) 1 Identifying this '1' with the activation length, we can write d = v = vT _ (7) b 1 CXLLb2 •d' values for some [llOJ specimens are plotted in F ig , 50 with oLm 1 , For specimens deformed at 0°C and room temperature, 27T •d' values tend to level 'cfaf in,the third stage. Suppose the rate controlling process is the cross-slip of Fig . 50a Activation Distance d as a Function of Strain. 87 screw dislocations (see F i g . 53 i n Discussion), then this constant value of 'd' permits us to accurately estimate the stacking fault energy. The average value of 'd' at these two temperatures i s 15 A°(+ 1A°). In the equilibrium equation, ~f « Ma2 (8) 24 JT d where " f i s the stacking fault energy, 'a 'is the l a t t i c e parameter and *d1 i s now taken to be the equilibrium separation of the two partials. A small stress dependence of 'd' exists, which has to be ignored however for lack of precise calculations. The value of stacking fault energy 2 2 i s then found to be 48 ergs/cm with p - s 4.4 x 10" dynes/cm , -8 _8 o a s 3.54 x 10 cms, and d « 15 x 10 cms for copper at 0 C and room temperature. Current values for copper are given i n Table. VII. 36 However, i f Hirschs • mechanism of constriction and jog formation i s assumed to be the rate controlling process (see F i g . 52) then the activation distance i s equal to 2d, where as before, d i s the equilibrium separation of the pa r t i a l s . Accordingly the stacking fault energy w i l l be twice the value obtained earlier assuming Seeger's 35 mechanism. As before the shear stress i s given by equation (6)• Stacking fault energies at lower temperatures are also calculated based on a precise dislocation mechanism determined from the observed activation energy. From equation (7) i t may be noted that the d vs £ curves 88 i n the present work are equivalent to the v T v s £ curves obtained by Hirsclit »d' values for a specimen close to the [lib] orientation aro extracted from Hirsch's data i n F i g . 51. At higher temperatures they tend to reach a constant value towards the third stage of the stress -strain curve. However at lower temperatures !d' is not essentially a constant i n this region. TABLE VII STACKING FAULT ENERGIES FOR COPPER o Temp. K Stacking Fault 2 energy (ergs/cm ) Comments Source 295 40 Taken twice the twin boundary energy 37 Fullman 295 4D Using anisotropic e l - Shoeck as t i c i t y for interaction between partial disloc- and ations Seeger 295 40 assumed 35 Hirsch 80 295 4-5 <57 <80 70 ± 10 From dislocation nodes II 39 "et.al 295 163 - Berner^ 295 67 + 17 From stacking fault probabilities Vassa-mallet & ^ Massalski 295 70 From l i n e shift meas-urements during pre-ferred orientation Small- /2 man,et.al" 295 - 273 200 77 48 + 3 62 + 3 8^+3 See text Present work 90 3. 4 Discussion - Mechanisms? In order to establish the rate controlling mechanism a knowledge of the total activation energy H a A Q + vT"is essential. This Is usually done by comparing the experimentally determined H values to the theoretically calculated energies for specific disloca-tion mechanisms. For a single rate controlling process the activation energy wil l be a constant,, Accordingly, a spectrum of activation energies wi l l be observed for several rate controlling processes. For such systems a precise determination of the exact rate controlling mechanisms becomes extremely di f f icult . The following specific dislocation mechanisms have been suggesteds (1) Formation of Jogs? This mechanism may have several variations (and has wide applicability in FCC lattices). (a) Forest intersection mechanisms In this a glide and a forest dislocation acquire jogs by mutual intersection. The activation energy required is twice the energy of Jog formation. Taking the energy of a single jog as about 3 42 0.2yib , this gives about 0.9ev, with an activation energy of about l.Sev for the intersection process. However, i f the dislocations are dissociated,twice the constriction energy has to be added (see F ig . 52). The constriction energy for copper is about 0.7 to 0.8ev for screw dislocations. This results in an activation energy of about 3.5ev for the intersection of dissociated screw dislocations in copper. Fig . 52 Intersection of Two Extended Dislocations. For edge dislocations constriction energy is very high. For copper this is about 4ev giving a total activation energy of — 9.5ev. (b) Non Conservative motion of .logs in screw dislocations; Screw dislocations with jogs are greatly inhibited from conservative motion. However, they may move either by producing inter-st it ials or vacancies. The shear stress necessary to move the jogged dislocation is given by, ^" s b , where '1 1 is the average J 1 jog spacing, and jB is 0.2 for vacany jogs and equal to one for inter-s t i t i a l jogs. The activation energy for producing interstitials is 3 about Ub ( ^ 4»5ev for Cu). For non conservative motion the re-92 quisite activation energy must be provided. In copper vacancies are relatively easily produced requiring an activation energy of 1 ev. At low temperatures this mechanism might be rate controlling. At higher temperatures the vacancy thus formed wi l l diffuse away from the parent dislocation. Thus the rate controlling process at higher temperatures wi l l require a total activation energy equal to that of self diffusion (2.3ev for copper). This wi l l result 33 43 in a temperature dependence of the flow stress.* (c) Conservative motion of .loess A variant of the above mechanism is the conservative motion of the jog along the screw dislocation. This requires long range move-ment of the jogs along the dislocation l ine . The activation energy 45 required is of the order of several tenths of ev. (2) Cross ° Slip of Screw Dislocations; This mechanism wi l l be discussed with special reference to FCC structures. In FCC lattices with sufficiently low stacking fault energies a dislocation is dissociated producing a stacking fault ribbon of width ' d ' . During deformation, the two partial dislocations bound-ing the stacking fault ribbon wi l l recombine over a length '11 of the fault producing the original dislocation. The total dislocation so produced wi l l cross-slip from the primary slip' plane, and redissociate producing another set of partial dislocations bounding a ribbon of stacking fault which wi l l now spread in the cross - s l ip system. The sequence of events is shown in F ig . 53 .In this the activation dis-93 Fig . 53 Sequence Shoving Cross - S l ip . 94-tanc© wil l bo equal to the width of the stacking fault ribbon, and the activation energy is then E o E - E (1) , where E is the cons ° cons contribution from the two constrictions and E Q(1) is the energy decrease when a total screw dislocation of length '1* dissociates into two par-t ia l s . Seeger gives the curve reproduced in F ig . 54 in which the total activation energy for cross s l ip appears as a function of stress. 7- 2 X -3 Using a typical value of C 3 » 200 kg/cm J s 0,5 x 10 for whiskers JJ-of [lio] orientation, we find from F ig . 5 4- that the total activation energy for cross s l ip is about 5.2ev. The curve in F i g . 54. was obtained / 2 by Seeger, using a stacking fault energy of 40 ergs/cm for copper. (3) Impurity - dislocation interaction'; This mechanism is of considerable importance in the temper-4-7 ature dependence of the flow stress of BCG metals. In FCC metals however, this has not yet been found to be the deciding factor. For an Impurity atom directly below an edge dislocation the interaction 3 energy is U ^^jj-b £ . When the size of tho impurity atom is about 1.05 times that of the parent lattice, £ = 0.05 and U » 0.9ev for 4S copper. Screw dislocations also interact with impurity atoms. In any case the experimentally observed activation energies in the pres-ent work are higher and this mechanism wi l l not be considered any further. (4-) Peierls Mechanisms In this, dislocations are believed to be parallel to close packed directions and assume low energy configurations in their po-tential valleys. Thermal fluctuations wi l l give rise to the formation of kinks which depending on their direction of motion wi l l either anni-hilate or give rise to a small displacement from the equilibrium 49 position as shown in F ig . 55. The activation energy for the process is the energy required for the formation of two kinks P and Q in a 50 dislocation such as AB. This mechanism has also been used in explain-ing the behaviour of BGG metals. In FCC metals this has been used in 51 explaining internal friction measurements from which the energy of kink formation is determined to be about 0.04ev in copper. As before, i t is evident that this cannot be a rate controlling mechanism in the temperature dependence of the flow stress of copper. F ig . 55 Kink Formation in a Dislocation Due to Thermal Fluctuations. 97 Table VIII is a summary of the different rate controlling mechanisms. Activation distances are given wherever possible, based 2 on a stacking fault energy of 50 ergs/cm . TABLE VIII DISLOCATION MECHANISMS AND ACTIVATION ENERGIES Mechanism Activation Enerev • §v Activation Dist-ance d A° 1 (a) Jog formation by dissoc-iated screw-screw inter section 3,5 30 Jog formation by dissoc-iated edge-edge inter-section 9.5 30 (b) Non conservative motion of jogs in screws 1,0 for vacancies 4.5 for interstit-ials 5.0 for single defects (c) Conservative motion of jogs in screws 0.5 to 1.5 -2 Cross-slip of screw dis-locations 5.2 15 3 Impurity - dislocation interaction 1.0 - 2.0 2.5 - 5.0 U Peierls Mechanism <0.1 -3. 5 Discussion of Experimental Resultss Work hardening characteristics in stage I, II and III may only be discussed for whiskers of [liol orientation. Activation length ' 1 ' and the activation distance 'd' wil l be considered for the purpose of discussion. (1) Hardening at 295 and 273°K:8 The activation energies observed at these two temperatures are very similar. However due to the continuous variation in the total activation energy in stage I and II a. precise dislocation mech-anism cannot be decided. Haitd/ening results from the gradual decrease of the activation length ' 1 ' of the dislocation line and an almost linear increase in the activation distance *d'. In tho Hostage the activation distance and the total activation energy reach an almost constant value. This is no doubt suggestive of a single dislocation mechanism as the rate controlling process. Of the mechanisms mentioned in Table VIII intersection of the glide and the forest dislocations (both extended) and the cross-s l ip of extended screw dislocations are the two mechanisms worth considering. We observe that the experimental activation energy and activation distance are not in agreement with the intersection meek an ism. For the cross - sl ip process i f we assume that the total activation energy is high enough as envisaged by Seeger (Fig. 54) then i t is not necessary to demand that an extensive pile up should be present for this mechanism to be operative. The theoretical 99 calculation in F ig . 54. is based on a stacking fault energy of \ : v 2 4.0 ergs/cm for Cu. The experimentally measured stacking fault energy is of this order of magnitude i f cross - slip is assumed to be the controlling mechanism. Cross - s l ip mechanisms without the presence of extensive pile ups have been recently suggested by Hirsch • In a similar manner stacking fault energies may be calculated for any metal with a knowledge of the activation distance and a spec-i f i c dislocation mechanism as the rate controlling process. Using the 7- 36 average v v- values given by Hirsch the stacking fault energies are determined for Au, Ni, Al and Ag as described earlier (Table IX). 42 The most recent estimates by Smallmann,et al are also indicated. TABLE IX STACKING FAULT ENERGIES FOR Au, Ni, A l , Ag and Cu. Metal T 8 ? v T d(A°) (ergs/cm^) 42 Small-K ev Intersection Cross-slip mann Au 300 3.85 17.3 78.8 34.4. IS Ni 5.6 11.6 211.0 110.5 225.0 Al n 0.61 2.76 420.0 213.0 150.0 Ag n 3.8 15.8 81.0 40.5 25 Cu 300 (pres-ent work) 5.5 15.0 96.0 48.0 70 0 100 (2) Hardening at 200 K.s The observed activation energy is about 3.8ev. Variation in *1? and 'd* is similar to that at room temperature. In most cases there is a tendency for the activation distance to reach a constant o o value of approximately 12A , compared to that of 15A at room temper-ature. This suggests that the stacking fault energy has a temper-ature dependence in this range. Observe that even i f the activation energy is quite close to that required for an intersection process, the activation distance is very small. On the contrary i f the observed temperature dependence of the stacking fault energy is accepted, a proportionately lower activation energy is required for cross-slip. This temperature dependence of the stacking fault energy is independent of the rate controlling mechanism. o (3) Hardening at 77 K8 o Different specimens tested at 77 K showed activation ener-gies varying from 2 to 2.5ev towards the third stage of the stress-strain curves. As before no specific mechanism can be attributed to the early stages. There is no striking difference in the ductility or the form of the stress strain curves, compared to those obtained at higher temperatures. The activation distance reached is low, having values <\8.5 A°. For the intersection process we see that agreement with the observed activation energy and activation distance can be obtained, i f intersection is assumed to take place between partially constricted dislocations. The different contributions are about 1.8ev from the two jogs and about 0.8ev from the two half constricted dislocations (see Fig,56), Hence the separation of the two partials is taken as 8.5A°. F i g , 56 Dislocation Intersection at Low Temperatures, In considering the cross-slip process, i t is noteworthy that the observed activation distance and activation energy are about half of that observed at room temperature. Cross-slip is possible, because now the stacking fault energy is almost twice that at room temperature. As before 8,5A° is taken as the equilibrium separation of the partials. Note that for aluminum with a stacking fault energy 2 of ^ 200 ergs/cm , activation energy for cross-slip is only lev. Hence at liquid nitrogen temperature intersection and cross-slip pro-cesses are indistinguishable. 2 This leads to a stacking fault energy of about 84. ergs/cm , 102 and a temperature coefficient of the stacking fault energy of 0.16(+ 0.1) ergs/cm /°K. This is about three times larger than the value given by 36 2 Hirsch (0.054 ergs/cm /°K) calculated from the entropy considerations of the split dislocations. (4) Hardening Characteristics of LlOQl and [ l l l l Whiskers s The activation volumes and activation energies are summar-ized in Table VI. Most of these whiskers showed only the third stage of the work hardening curve. From the activation energies we can con-clude that the rate controlling mechanisms are the same as in [lid] whiskers. The rapid work hardening characteristics of these whiskers is discussed in the previous section. (5) The quantity In s The frequency factors In decreased with decrease in temperature. Values of 35> 28 and 8 are obtained (see Table VI) o o respectively at room temperature, 200 "K and 77 ~K. This temperature dependence comes from the strong temperature dependence of the activa-tion energy. The higher value is closer to the theoretical estimate of 32 by Seeger^- derived from a random forest model. These values, o except the one at 77 K, are higher than the previously reported value of 11 by Basinslc? for bulk single crystals of eoppor. The temper-ature dependence of the activation energy and the frequency factors 52,53 has been discussed in detail by Conrad and wi l l not be considered any further. 3 . 6 Twinning; Since the stress levels reached at least in the case of [ i l l ] whiskers are extremely high i t is interesting to examine the strength of L-C barriers under these conditions. An L-C barrier can f a i l either by recombination or by dissociation under high stress concentrations . F ig , 57 provides the necessary information. Assuming that there are no pile ups present, F ig , 57 shows that the stress necessary to break such a 7 2 7 barrier is about 1,8 x 10 gm/cm by recombination and 4 x 10 gms/ cm2 by dissociation at 77°K. The tensile stress reached by a whisker just before twinning is 1,34. x 10 gms/cm at 77°k ( ^  7.0 6 2 7 x 10 gms/cm - resolved shear stress) which drops to 1.2 x 10 2 6 2 6 2 gms/cm (6 x 10 gms/cm - resolved). This value of 7 x 10 gms/cm is comparable to the stress necessary to break an L-C barrier by recombination.(Stroh himself points out that his calculations should not be taken too l i t era l ly ) . This twinning stress is about 5 to 10 times larger than the value for bulk single crystals of copper 6 2 55 ( ^ —> 1.0 to 1.5 x 10 gms/cm ) # Twinning models based on Cottrell - Bilby's pole mechanism 56 are discussed in detail by Venables • In the present work a mechan-ism based on the above idea of failure of a Lomer-Cottrell barrier by recombination wi l l be proposed. Since the external stress is high enough, i t is not necessary to assume the presence of pile ups against the barrier. 104-Vig. 39. Stress at which a Lomer-CoitreU sessile di.nl by recombination, plotted as a function of temporal ton values of the dislocation width {; (a) copper ifa l/a •* 0-3, (cj aluminium {/a = 0-2, (d) aluminium ( tU> number of dislocations piled up under a resolved i G is the rigidity modulus (a) Fig. 40. Strew at which a Lomel^Cottrell sessile dielocM. by dissociation, plotted as a function of temperature, for value* of the dislocation width C; (<•) copper £/o •> 0' - 0-S, (c) aluminium £/« — 0 2, W aluminium {/o •» number ot dislocations piled up under a resolved ah ear O ia the rigidity modulus (b) F i g . 57 Failure of an L - C Barrier (a) by Recombination and (b) by Dissociation (after StroJr*). 105 An L-C barrier is shown in F ig . 58. It consists of a wedge shaped stacking fault bounded by three partial dislocations. The respective BVs are as shown. Failure by recombination wi l l result when the partial dislocations recombine to form tho original total dislocation ^ ^0-ig] J which wil l then glide in the (001)plane. We consider a segment of the L-C dislocation which has re-combined (AB in F i g . 59) over a length *1' to form the total disloca-tion which then glides in the (001) plane. Fig.59b shows the result-ing dislocation configuration. Now any of the uncombined partial dislocations can act as twin sources. Because the screw dislocations AD and BC with BV -|-[I1Q] on (001) can now be expected to be sessile and have a component displacement equal to the spacing between the ( i l l ) Fig . 58 An L-C Barrier 106 107 ( i l l ) planes. It is not necessary that extensive slip take place in the (001) plane. A macroscopic twin can however he formed i f DC glides away sufficiently from AB, and partial dislocations AP and BQ rotate around the poles A and B. After one complete rotation, the (111) plane wi l l contain a stacking fault and a segment AB of the partial dislocation wi l l have climbed down by ^-JllQland operate on the next parallel (111) plane as shown in Fig.60a.Similarly, operation of the partials AS and BT wi l l result In twinning In the (111) plane. Twinning wi l l then take place on two conjugate planes. 57 This model Is similar to a model proposed by Suzuki , based on the propagation of a double - fault plane that results from the interaction between a dislocation pile up and an L-C barrier. This F ig . 60a Formation of a Twin Source Due to the Operation of Partial Dislocations AP and BQ in F ig . 59, The Twinning Dislocation Will Climb Down With the Help of the Screw Segments AD and BC. 108 interaction produces a pair of twinning dislocations which then results in the build up of a twin due to their spiralling around a perfect dis-location which has a BV component perpendicular to the twin plane. Suzuki 's model gives the shear stress necessary for twin propagation as T - ~f + J^Sl _ (9) 2bx where ~f is the stacking fault energy, jl is the shear modulus, b^ is the BV of the partial dislocation and '1' is its length. We wi l l apply equation (9) to the present model by identify-ing '1* with AB in F ig . 59. The length of AB can be obtained from the 6 , 2 maximum shear stress reached just before twinning - 7.00 x 10 gms/cm in the present case. This stress is that necessary to cause failure of the L-C barrier as shown in F ig . 59b. Using the ideal Frank-Read expression, *U a .^b , where '1 1 is now the BV of the total disloca-tion AB(or DC), we find that 1 = AB a 160*31. With ~f * 80 ergs/cm2 at o 77 k and b 1 s b , w e f i n d that, ^ 6 6 , , 2 C : 2.8 x 10 + 4-.0 x 10 (gms/cm ) T 6 / 2 s 6.8 x 10 gms/cm Thus the model gives twinning stress very close to the experimentally 6 2. observed value of 6.00 x 10 gms/cm * This mechanism is consistent with the idea that a- lowered work hardening rate in the third stage is due to the break down of L-C barriers. In bulk single crystals however, a stress concentration factor of about 5 is required, which necessit-ates the presence of a pile up of strength 5. This wi l l proportionately 109 reduce the observed twinning stress, as is the case with bulk single crystals. 3.7 Dependence of A ^ o n ^ ; Significance of B in A V = A + B V, From the strain rate equation, H -vT ^ - ^ oe V kT J (10) we have, * ^ l n jL ' - J L + 3L-L — __(H) e o k T k T and . v . kT / °2sS. d r k T / ^ ^ ) _(12) T Eliminating v from (10) and (12) and rearranging the terms, we obtain, .(13) In € t H k T a At any temperature we observe that H ^ ln £ k T 7 / 2: ^ o Hence we can write, * AT - f M i n i s . \ <£• ) j u ) H no which reduces to, A ^  = 2.303 kT T (15) H for JL 2 - = 10, then B = 2.303 kT . - £ i H At room temperature, with H > 5ev, B = 0.008 This value of B compares very well with the actual slope of AT~vs and i t further justifies that the approximations involved in obtaining H are reason-able. A l l the whiskers tested in this work show typical parabolic hardening in the third stage. No effort has been made to determine the exact work hardening exponent. 4 A multi-staged ATvs TZ curve was interpreted by Hirsch as due to a change in the dislocation geomentry with deformation. Notice that specimen S44 which has deformed considerably more than the rest of the [lOO] specimens shows stage 'd' deep in the third stage of the work hardening curve. The final scatter of points is due to impending fai lure. In the light of the present discussion this multi-staged i t TS T curve may also be explained as due to a stress dependent activation energy. In equation (15) we observe that when A'T'is a constant independent of stress, H is linearly dependent on stress. When A C~is linearly dependent on stress, and the Cottrell-Stokes law I l l is strictly obeyed, H is a constant as i t is for a single rate con-trolling process. Hence, at any given temperature i t should be possible to predict the rate controlling mechanism from the Cottrell-Stokes ratio, or vice versa. A SUMMARY AND CONCLUSIONS 112 I 1. Twinned etch spirals are observed in halide reduced copper sub-strate. Frequent occurrence of these, in regions where dense whisker growth had taken place, may in fact be responsible for nucleating whiskers. 2. Under unusual circumstances screw dislocations not parallel to the axis of a whisker may escape out of the whisker leaving glide traces. The block-like structure which is the result of egress of such dislocations is then eliminated by the addition of material at the ledges in the block structure, leading ultimately to smooth sur-faces • 3. The presence of growth steps on facets at the tip of a whisker is due to the presence of dislocations with a screw component normal to a facet. II 1. The difference in the work hardening characteristics of [ill] and [100] whiskers is attributed to the difference in the character-o istics of the jogs formed in the two orientations. Predominantly 90 interstitial jogs are formed in [100] orientation and 60° interstitial jogs in [ill] orientation. 2. If surface nucleation is assumed the 1 diameter dependence of d the yield stress of whiskers arises from three prominent contributions, (a) The largest contribution is from the strain energy of the slip dislocation which reaches a maximum when the dislocation passes through the centre of cross-section of the whisker. 113 (b) The sl ip step gives rise to a surface energy term, and (c) an image force term when the dislocation is very close to the surface. I l l 1. . Activation energies along with activation distances have been used to determine a rate controlling mechanism for the deformation of copper whiskers from the temperature dependence of the flow stress. 2. Measurements show that in copper at liquid nitrogen temperature, cross-slip and intersection processes are indistinguishable. At higher temperatures cross-slip in fact is the rate controlling process. 3. Knowing the mechanism and the corresponding activation distance, stacking fault energies have been estimated. A temperature coefficient of 0.16 ergs/cm /°K is deduced for copper from such stacking fault energy measurements. Ut, A twinning mechanism for whiskers of [ i l l ] orientation is pro-posed based on the idea of failure by recombination of the partial dislocations of a Lomer-Cottrell barrier. 17 An upper estimate of the dislocation density in whiskers tested in the present work can be made from the i n i t i a l yield stress. 8 2 This is about 10 /cm a value which is comparable or slightly larger than the density in bulk single crystals. Dislocation densities of about 6 2 1A 10 /cm have been reported for iron whiskers. In fine whiskers surface nucleation of slip is much more likely than internal nucleation. Since impurities lower surface energies, they have a greater tendency to segregate on the whisker surface giving rise 114. to regions of small stress concentrations, which might act as sites for dislocation nucleation. In large whiskers as well as hulk single crystals internal as well as surface nucleation is possible. In a whisker which deforms by single slip i t is interesting to determine the mean distance travelled by a glide dislocation. This may be easily determined from the amount of easy extension for the whisker and knowing that there is no substantial change in the dislocation density in this region. Strain is given by £ = bAN — 1 where A Is the area swept by a dislocation of BV 'b' and N is the number 3/2 of activation sites per unit volume (^ ~, P ' where P is the dislocation density). Considering a dislocation ring of raditts R> equation (1) may be written as 2 £. = bTT R N — - 2 from which B - f H . - 3 Vb7T N with € —0 . 3 and f = 10 8 /cm 2 , R-20yUand with » 1 0 6 / cm2, R^oOO^U. This means that for whiskers in this diameter range, most of the dislocations escape out of tho crystal in the early stages of the deformation. Observe that R is a very sensitive function of the dislocation density. In the early stages of deformation "oriented dislocations" can be Glide dis locations F ig , 60b Dislocation Dipoles Along the Axis of a whisker. expected in whiskers compared to the random networks in hulk single cryst-als. In the present context, the term "oriented dislocations" specifies edge dislocation dipoles uniformly distributed along the axis of a whisker due to the presence of axial screw dislocations. This is schematically shown in F ig . 60b. Formation of such dislocation dipoles wi l l lead to a radically different dislocation structure towards the end of deformation. A uniform distribution of dislocations along the axis of the whisker wi l l result compared to a quasi uniform distribution observed in single cryst-66 als (see for example Basinski ) . This is schematically shown in Fig.oOc. i 116 The cross-hatched region corresponds to a region of high dis-location density separated by almost dislocation free regions. This might be the reason why the fracture strength reached by whiskers is much great-er than in bulk single crystals. Large whiskers exhibit serrated load elongation curves. A thorough investigation of this phenomenon is beyond the scope of this thesis. It may be said that If they are not due to impurities, then they might be due to Frank - Read sources which are discontinuously distributed, i e . , sources which require stresses, T0+ A T ^ » + ^2 ' T Q + A f • ° t c . for operation compared to T Q + / dZ" \ de f o r bulk single crystals, where T is the yield stress and / d T | is the rate ° U € of work hardening. 118 5. SUGGESTIONS FOR FUTURE WORK Electron microscopic work, especially on thin blade-like whiskers before and after plastic deformation should resolve most of the existing Speculation, Dislocation structures mentioned in the previous section may be expected. Even though the effect of hydrogen on copper whiskers is unknown, i t would be interesting to test copper whiskers which have been maintained in an atmosphere of hydrogen for long times. Deformation followed by a vaccum anneal must also be made. Determination of activation distances may be extended to liquid helium temperatures. At this temperature the activation distance must be ^ b . This wi l l also justify the use of a particular flow stress equation. APPENDICES APPENDIX I STRESS STRAIN CURVES TABLE X 120 YIELD Specimen No. Orientation Dia(jJ-) Test Temper-ature °K eYield(kg/mm ) 8 [110] 50 295 0.5 10 [100] 35 n 1.07 H [loo] 68 n 0.6 16 [nil 35 tt 21 [100] 61 n 1.3 XI [100] 59 77 4.1 X4- [no] 57 n 3.2 X6 [100] 60 it 1.9 X7 [Hi 28 n 12.0 X9 [100] 70 n 5.1 X10 [H| 43 n 13.75 X13 [no] 79 n 7.2 XM [nil 39 n 24..0 X23 [100] 92 n 7.0 X24 [no] 81 n 2.26 S7 [no] 170 295 1.6 S8 CLOO] 153 n 0.11 S16 [nl 101 77 0.3 S18 [IH] 92 n 1.47 S19 [100] 90 it 0.3 TABLE X - Cont'd.. 121 Specimen No. Orientation Dia(j^) Test Tgmper-attire K ^Yield(kg/mm2; S22 [110] 172 77 1.25 S23 [100] 291 II 0.15 S24. [111] 97 n 0.3 S25 [no] 134 n 6.0 S26 [100] 173 ti 5.1 S28 liooD 226 273 1.03 S31 [no] 290 1.00 S33 [100] 82 N 0.04. S34- [no] 201 n 0.27 S35 [100] 137 n 0.08 S36 [m] 83 it 0.19 S37 [no] 316 n 0.8 S38 , [m] 72 n 0.56 S39 [111] 101 200 4..00 SA1 [nil 107 ti 11.00 SA2 &00O 268 a o . H S43 [in] . 95 it 5.5 s u [loo] 197 n 0.39 S45 86 ii 0.38 SA6 [ni] 105 n 0.3 S48 [100] 179 n O . H S49 [no] 195 n 0.20 S51 [no] 202 295 0.35 122 TABLE X - Cont'd.. Specimen No. Orientation Dia(^) Teat Tgmper-atxrre K rYield(kg/mm2) S52 [110] 210 295 0.49 S55 [llO] 188 273 2.6 S56 [110] 197 273 0.42 S62 [no] 180 200 0.80 S63 [no] 348 n 0.94 S64 [110] 400 n 0.90 S76 [110] 171 n 1.84 S57 [no] 216 77 0.65 S59 [110] 187 n 2.1 S79 [no] 280 n 0.8 S80 [no] 242 n 1.7 Note; (1) Specimens from 8 to 21 and XI to X24 are deformed at a fixed cross-head speed of 0.01"/min. and 0.002"/min. respectively. Specimens from 37 to S80 are deformed alternately at 0.002"/kLn and 0.02"/min. The curves presented are at the lower strain rate. (2) yield values are reduced to room temperature wherever necessary as described in section three. O'l e O'Z 0-3 128 APPENDIX II 137 ON JOGS We consider only the jogs formed in screw dislocations. As may be seen from the unit tetrahedron in F ig . 61 three different kinds of jogs can result by mutual intersection of screw dislocations. (a) BC (c<) / AD (JB) - a 90° jog is formed. For conservative motion of the jog, the slip plan is (100) whi,ch is not a common slip plane, o (b) BC (ex.) / AC (JE>) - a 60 jog is formed. For conservative motion the slip plane is 6. (c) BC (cx) / CD (p) - a 60° kink, which wi l l vanish as the dislocation glides in its s l ip plane. The number of point defects (interstitials or vacancies) produced by the non conservative motion of a jog is given by the 58 scalar triple product N « h1<C1 x b 2) t where V is the atomic Fig,62 Illustrating Vacancy Formation by Moving Jogs. 3 volume ( V s b ) and b2 are the BV and the direction of the Jog respectively, and is the distance moved by the jog in the direc-tion of motion of the parent dislocation (see F ig . 62 ) , It is also possible to determine the type^of defect for a given orienta-tion of the dislocations as may be easily seen from Fig . 62. In F i g . 62 is shown a dislocation with a jog. If the jog moved along with the dislocation l ine, the material above wi l l be 139 interstit ial F ig . 63 Formation of Interstitials by Moving Jogs. displaced with respect to the one below in the direction of the BV of the dislocation b^. We observe that depending on the direction of (or C^), either vacancies (Fig.62) or interstitials (Fig.63) wi l l be produced. A 90 jog wil l emit a vacancy or an interst it ial for every I I C . | , where C, • a <112>JGJ = b ('a • is the lattice para-V f - 1 1 6 1 W meter and b • a (llti} )• In the absence of formation of such defects, 2 edge dislocation dipoles wi l l be formed in a direction parallel to (Fig.6*Va). These dipoles wi l l act as obstacles for dislocations in the primary system and wi l l not be effective obstacles for dis-locations in the secondary system. For example consider the inter-section of another screw dislocation with such a dipole (Fig. 64b). The two jogs A and B in the screw bg are equal and opposite and can annihilate and the dislocation is free to move. Also, since 140 Fig . 64 Sequence Showing the Formation of Dislocation Dipoles and Loops. there is no short range interaction between edge and screw disloca-tions, this intersection wi l l not be a major source of hardening. For the 60° jog excluding the possibility of conservative motion on a different s l ip plane, we find that for non conservative motion a vacancy or an interst i t ial wi l l be emitted for every 2 \ C 1 34 It has been suggested that interstit ial type jogs can move conservatively along the dislocation line by stress aided aotiva-o tion. Of the two kinds of jogs formed we see that a 60 jog can in fact move conservatively along the dislocation line because the plane of conservative motion contains the dislocation line also, o For the 90 jog the plane of conservative motion is not a common sl ip plane. 59 Gottrell has shown that only interst i t ial jogs are prod-uced by the intersection of screw dislocations moving under equal resolved shear stresses. However, when one of the BVs is common to two sl ip planes, both vacancy and interst it ial type jogs are formed. The latter is very similar to the Intersection of a gliding screw dislocation intersecting a stationary forest, where equal number of vacancy and interst it ial jogs are produced. APPENDIX III L - C REACTIONS Fig.65 Unit Tetrahedron Showing Different BVs. AD = £ [lio] AB - £ [Oil] AC s [lOl) DB BC DC [lOl] [no] i[oii] i . £[101] AC (111) £[oii] CD (111) i[no] AD (001) 2. i[ioi] AC (111) iliio] BC (111) £[oif) AB (100) 3. i[ioi] AC (111) i[oii] BA (111) i[iio] CB (001) 5. 6. 7. 8. £ [oil] CD (Til) £ Loii] CD (111) £ [oil] cp (in) £ [no] BC (111) £ [iio] BG (111) £ [101] BD (HI) £LiTo] CB (111) £ [101] DB (111) £ [iio] DA (111) £ [lOl] BD (111) £ [oil] AB (111) £[oii] AB (111) £[ioi] BD (010) £ [101] BE (010) £ [ioi] CA (010) £ [oil] CD (100) £ [ioi] GA (010) £ [no] AD (001) 143 10. i [101] + i [no] —* i [on] BD DA BA i (111) (111) (100) 11. i[oii] + *[iio] — iLloil AD AD DB (111) (111) (010) 12. •£- [lio] + £ [lOl] — £ [pll] AD CA CD (111) (111) (100) APPENDIX IV IMAGE FORGE H5 Consider a dislocation close to the surface in a circular cross-section (Fig, 66), When r^ ^> b, where b is the BV of the dis-location, the image force exerted by the surface is large. The image dislocation is not straight, and accordingly the expression for image force exerted by planar surfaces cannot be used. In this case the image dislocation could be taken approximately as a circular loop with BV same as the glide dislocation,, Notice that the strain energy contribution to the yield stress (see p»63) when the dis-location is at position such as A, close to the surface, is very small. No precise calculations exist for image forces exerted by non planar surfaces. However in the above problem, If the image dislocation is treated as a circular loop encircling the glide dis-location, then the interaction energy for such a system may be ob-60 tained from Kroupa's calculations to be, E r . jU.b2r (1) where *r' refers to 2 ^ in F ig . 66. The attractive force per unit length on the dislocation line segment of length '1' is given by r s U^ET\ ~ W ; (2) T h S n r i n t * !E S (3) From Fig . 66 we have, 14-6 Image dislocation Fig* 66 Image Dislocation at a Curved Surface. 8 r l r 2 r^(d » r t) where 'd' is the diameter of the whisker. When r ^ ^ b eqn.(4) reduces te 1 « 2 \liA .,, (5) from eqn.CJ) "<T. int J i L . 2y/pT .(6) Notice that this alone gives rise to a 1 dependence. This depend-ence is aegleeted in p. 64 compared to the large 1 dependence from d the strain energy term. For d ^  10 JJ- , 11 kg/mm for copper. This is the maximum value of f° r * n e chosen diameter. When the dislocation is at position A in Fig. 66 the attractive force by the surface S., is neglected. ^^n+, decreases monotoni-cally and reaches zero when the dislocation passes through the centre 0, and is negative when i t approaches S 2. An approximate Fig. 67 Image Force Exerted on a Dislocation 1 by a Whisker Surface. plot is shown in Fig.67. APPENDIX V LOAD MEASUREMENT When the load scale In the Instron is at position 1, the chart is calibrated to 100 gms f u l l scale. With deformation i t becomes necessary that the load scale be inoreased to 2, 5> 10, etc., so that now the total pen swing corresponds to 200 gms, 500 gms, 1000 gms, respectively. For direct load measurements this procedure does not introduce considerable error. However, i t correspondingly decreases the sensitivity in load increment measurements which is very small for whiskers, and in the i n i t i a l stages of deformation i t is within the sensitivity limit (loads up to 0.2 gms could be measured when the load scale is at 1)• In order to keep the load scale at position 1, the fine bal-ance control is used. Whenever the pen is about to reach the top of the load scale, i t is lowered to some convenient position at the bottom of the chart by merely turning the fine balance control knob. The load drop can always be measured by extrapolation as shown in Fig.68. Between changes of strain rate i t is also necessary that the load is either reduced to zero whenever possible or lowered by about 50 - 100 gms. This is done to minimize irreversible effects. Fig . 68 Load Measurement Using Fine Balance Control APPENDIX VI ACTIVATION. ENERGY 150 The strain rate is expressed by the usual Arrehnius equation, I = £ - AQ (1) e . o® kT where, 1 = NAb V , with o N - Number of activation sites per unit volume, A » is the area swept by a dislocation segment in one activated jump, b s BV of the dislocation, "V a vibration frequency of the dislocation, AQ is the thermal component of the activation energy expressed as, Q s H - v r a _ _ _ ( 2 ) where H is the total activation energy, and v is the activation volume, T' is the applied shear stress. These quantities are schem-atically represented in Fig,69. The required quantities are AQ, H and v. From equation (1) we have, In € = In C - Ag = In C " S + • T. 0 kT ° kf a kT Differentiating this at constant temperature, we get, 5 r a / T M l a t J T kT k T ( ^ J T 151 Force Distance Fig,69 Force - distance Curve If 4<9,«H, we have, / dH \ , f A r ~ ) » - _„(5) ' a /rp ^ ^ T Assuming that € c is independent of stress, from (A) and (5) we get 'dln€ d r a / T T T or, v s kT ^ d l n € \ (6) Equation (6) wheti-written* in terms of experimental quantities becomes, v = kT In £, - ln£2 . (7) a l - a2 Thus from strain rate change experiments activation volume may be determined. 152 Differentiating equation (3) at constant strain rate and as before assuming that € 0 Is independent of temperature, we get kT I dra I l fdx \ + v » ^& fdv \ k T \draJ. k T kT ^ r A V ^a/^T \ S o k T ^ r J i .(8) With the earlier assumptions s t i l l holding, equation (8) reduces to, H kT V ^ a / d r \ kT _y_ kT k T' .(9) Transforming as before to experimental quantities, we write, H - v r - Tv / £ 2 a , a i 1 a 2 2 a. .(10) H may then be determined from a temperature change test, 153 Since v appears as a function of stress, and the stress values for any two whiskers do not in general coincide at a given temperature equation (10) is modified as follows in order to suit the experimental results. For this we write, r a T 2 . T 1 TT 2 H " ^ a1 (11) v r a n T - T T At any temperature T^ (X^ > T 2 ) the average value of a 2 is used T. a l from a Cottre -Stokes type of experiment. Since Cottrell-Stokes law is obeyed, ~ -~ appears as a constant independent of stress. Then from v r a equation (2), A Q 8 H ~ 1 (12) v Z~ v 7Z a l v a1 Using —S_ from equation (11) AQ can be determined. Hence both H v V c a l and AQ can be obtained as a function of stress. Knowing AQ, In ^ may be directly obtained from the strain X rate equation (1), For whiskers of [lio] orientation, shear stress values are used with Schmid factor correction given in Appendix V I I , For [ill] and [lOO] whiskers tensile stress values are used. APPENDIX VII DOUBLE SLIP CONDITIONS 154 The stereograms in Fig* 70 show the possible choice of glide systems and the resultant lattice rotations* For exact [lio] orientation, there are in principle two potential slip planes with four different slip directions, two on each of the slip planes* The Schmid factor for this system is 0.408. However, for whiskers of [llOj orientation, there seems to be only one active slip system. Let us assume that the system is (111) with [lOl) as the slip direction. The specimen axis will rotate along the great circle passing through the poles [lOll and [Oil] (see Fig. 71 ) until i t reaches the [ooi] - [llll symmetry plane, 22 when double slip sets in • The normal strain 1 £ 1 is related to the Fig. 70 Stereogram Showing Choice of Glide Systems (a) and Resultant Lattice Rotations (b)(from Ref.61). 155 100 F i g . 71 Stereogram Showing Lattice Rotation for a [lio] Whisker. 61 1 5 6 axial rotation as follows % 1 + € s sin Tip (1) sin A where, A - 0 and X are the angles between the specimen axis and the slip direction respectively before and after a strain € . For |llQ) orientation, the required rotation for double slip to start is 30°. With A 0 - 60° and X a 30° 1 + £ - 10732. Hence double slip will set in for a l l whiskers of [lid] orientation at about 73% strain. Experimentally we observe that double slip has occurred in specimens which have failed before this strain is reached. Resolved Shear Stress Calculations s axis slip direction Fig. 72 Coordinates for Resolved Shear Stress Calculations. The cr i t ica l resolved shear stress is given hy, 5 T s ^cos9 cos X ^ cr o o where 0O andXQ are as shown in Fig»72 and refer to the in i t ia l values. T is the tensile stress. During deformation both 9 and A. change according as eqn (1). The true resolved shear stress is then T a = r / l - s i n i o T " / 1 - s in 2 ^ _ ^ X / ( 1 - ) 2 / T J T - T 2 The Schmid factor (cC) is given as a function of strain in Table for the fllO] orientation. TABLE XI SCHMID FACTOR AS A FUNCTION OF STRAIN Strain € cL 0 0.408 0.1 0.47 0.2 0.51 0-3 0.636 0.4 0.684 0.5 0.744 0.6 0.758 0.7 0.788 0.8 0.813 0.9 0.833 1.0 0.85 1.1 0.855 1.2 0.877 1.3 0.894 APPEND n VIII ESTIMATION OF ERRORS 158 The main source of error is in A d measurements. High sensitivity in load measurements is however achieved by methods described in Appendix V. Further, when the f a l l load scale in the Instron is calibrated to 100 gms. pen deflections are also re-corded for 0.1 gm, 0.2 gm. etc., and used as a scale for accurate load measurements« This way loads can be easily measured with an accuracy of 0.2 gms or at best 0.1 gm. The former value will be used in estimation of errors. At the start of tho deformation, the Afvalues for [lio] and [loo] specimens are comparable to the above limit. For [ i l l ] specimens the A ^ a l n e s are usually high. In what follows, errors will be estimated in activation volume (v), activation distance (d) and the total activation energy (H) using a typical value of aT~ S 5 gms. for a [lio] specimen. (1) Activation volumes Activation volume is given by, v * (i) AT* where oi. is a constant, whose magnitude is unimportant. Taking logarithms and differentiating eqn (1), we get, <fv a . S ( A T ) _J2) Using the above mentioned values for A C~ and 8 AT~, we find that, 159 — S •:.. °- 2..-. (3) v 5 " W / which gives an error of IS in v. ( 2 ) Activation distance. Activation distance is expressed as d - / v t (4) where ft is a constant« Taking logarithms and differentiating eqn (4.) we get, Sd ,. s jTy. + (5) d v ^ Since T >^> cT<?*, we find after using eqn ( 3 ) , that the error in d is also U$o (3) Total activation energy (H)8 From Appendix VI we can express H as, H a / v t (6) wherep is a constant. Following the same procedure as for 'd 1 we find that the error in H is I&. We observe that the error in a l l these quantities decreases with deformation, because ATincreases with stress. Actually in most {ll6] specimens the error in v, d and H is smaller than the above calculated value, because A T i s usually about 7 gms or larger in the 3rd stage. The error in [loo] is comparable to that in Clip] spec-imens or smaller. The error in [ill) specimens is much smaller than either of these two orientations and unimportant for the purpose of discussion. APPENDIX IX 161 FORCES BETWEEN TWO DISLOCATIONS OF ARBITRARY ORIENTATION AND BV 1. Introduction; Tho nature of interaction between two dislocations is of consid-erable importance in the theory of work hardening of metals. Interaction be-tween parallel dislocations with parallel BVs has been established from fi r s t 62 63 principles early in the theory of dislocations, Saada has calculated the stability of a glide dislocation as i t approaches a forest dislocation and 60 concluded that the change in form is very small. Later, Kroupa had investigated in detail the interaction between a glide dislocation and a dislocation loop. In this a similar analysis will be given to calculate the interaction between two straight dislocations under the assumption that they are rigid and infinite, 2, Theory The force F on a dislocation due to an arbitrary stress field 64-is given by, F s k x (<Hb), 1 where k is a unit vector tangent to the dislocation line,CT is the stress dyadic and b is the BV of the dislocation. Choosing dislocation I parallel to the x^ axis in the x.^ system, and Its BV in the x^ ~ xj plane making angle cx' with the x^  axis (Fig.73), i f the stress tensor is referred to the same set of axes, then in the dyadic notation, 162 CT • i i 0 ^ + IJ (7J2 -r ik c r " * ki c r 3 1 + kj C7 2^ + kk C7J3 , i , j , k, are the unit vectors parallel to the x-^ , Xgj x^  axes respectively. The BV b is given by b • ibsincX - kbcos o< and 0™.b * bsino^iCr^ + j CJ^ + k (7J1) + Then, bcoscx(i (T£ + j + k (jy^ F 3 kx bsincx(i CT^ + J + k a^) * bsin (i c r 1 3 + 3 ^ + k a^) s bsina(j (T1:L - i 0^) + bcosoc(J CT^ - i C£ 3 ) a - i (bsinc* CJ"31 + bcoscx (7^) 4 j (bsincx OZ^ •+ bcoscx (T^ ) The components are, F^ m -(bsincx CT"^  + bcoscx 0~^y) F 2 s (bsino< C7^1 * bcoscx cr^y) Referring dislocation II to the x' system, with its BV in the x^  - xj plane making an angleyB with the xj axis, the components of Its stress f ield are, 163 2' 0~ (3: .2 12 l l l l cr, 22 v %i «2  X 2 ( X 1 • 2 , <*1 33 mm^L-mm ( x « 2 + x ' 2 ) 1 2 y E ,2 .2 c r - D 2 x i K -*2> 12 - - — — - t 2 — ^ r 2 (x + x r 1 2 cr_ 13 x . + ) c r s D, 23 *1 7 T X l + T2-where D 2 a J^ao sini? D 2 8 7^ b2 c o s 2TT (1^) s -164-In Fig.73is shown the relative disposition of the two dislocations, r , 9,0 are the spherical polar coordinates. The necessary transformations are, x! s ^ i j C x j - x 4 J and, 0 . i 2 i j ck ex cr mi nj mn Fig . 73 Coordinate Systems: x. is for Dislocation I and i l is for Dislocation II. The superscript refers to the second dislocation, and °^±^ is the direc-tion cosine between the x'^ and x^  axes. From the cartesian force components the radial and other angular components could at once be ob-tained by using the transformation, F :9 F, J cos9 sin0 sin9 sin(2C cosJZf sin© sin0 cos© sinJ2f 0 cos© cos0 sin© cos^ -sin^f 165 F, with tho following definitions? where E is the interaction energy* r dQ ' F-, » 1 ( ) What is of most interest is the radial and the angular components in the direction of motion of the second dislocation,, This is calculated as follows: Equation 2 gives the force per unit length of dislocation I at points (x-^ , X£> x^ ) for a given position of the second dislocation, after the appropriate transformations given in equation U» Along dislocation I, s X2 a 0, and by proper chare of r , 9 and 0f, the direction of motion of dislocation II can always be made to pass through the origin of disloca-tion I„ This can be easily done by expressing (x 1 r t, x„_ , x__) in the * • v..... 10 <fJ JV 1 transformed equation 2 in terms of r , © and 0 such that ' r ' is always parallel to the direction of motion of dislocation II• When r is parallel to x , we have, x^  s - r , Xg s 0, x^  s 0. Then the transformations for the stress tensor in A reduce to, 0~ -12 <*12 ° k + ^22 ° l l ) ( T ? , 12 0~. 11 11 21 12 166 2 • r r - „ ( C ( O/ l O ( CX \rr^ 23 " 2 2 3 3 3 2 2 3 23 2 2 i 0".. s + <**i ° W 13 2 1 3 3 3 1 2 3 23 6 cent >d The force components are, E 2 F i = " b i ^ a i E 2 P 2 * b l ^11 for pure edge-edge, and F l = - b ?*-23 v W3 2 2 1 °13 8 for pure screw - screw interaction,, There is no interaction between sin edge and a screw dislocation (because of the assumption that the medium is isotropic). From the above expressions the necessary radial and angular components are then obtained using equation 5. Two examples are presented below for interactions in FCC metals. (1) Edge - Edge interactions Dislocation I: BV £ [ l l o ] Dislocation*Us BV \ [loll x 2 / / [no] x^ / / Liol] x 2 / / [ n i l x2' / / [ill] x 3 / / [ll2] x^  / / frill] The following table of direction cosines is obtained; 167 x i X 2 X 3 1 x l I 2 1 ~ 2~/31 t x 2 1 3 v T 3 i 1 - 2751 \l2 3 6 Then, and. cos© sin0 s sin© sin0 " cos 0 * ^12 0 ~ 11 1 I I ' 2*V3 , c r 2.1 fa o~ 2 V3 12 12 E ? 5 L 2 ^ 12 2 E 2r The cartesian forqe components are: E E F | s -P D2 2r (2) Screw - Screw interaction: Dislocation I , BV £ [lio] xi / / [ll2] x 2 / / [ i l l ] x, / / [no] F, E E 2' b, D 2 using 5 . Dislocation II BV £ [pit] x i / / [ 2 1 l ] X2 / / [ill] x3' / / [Oil] cos9 3±n0 1 , sin© sin^ > 2J2? , cos0 6° 3 273 Then, 23 23 6r cr. 13 3 2 2\/3JU 23 2 r-^  £ 3 ^ CT-23 3 — i and, S S b l D2 6r F 2 • V f bfp| from which, F_ S S b l P 2 . 12r S S F 9 s " b l D2 , F 3\/2r S S b , D2 Thus from the cartesian force components, the radial and the angular force components are determined for any orientations of the two dislocations. Several different orientations are possible. 169 Referring to Thompson's tetrahedron in Fig.74., the edges correspond to BVs for an FCC crystal. Pure edge dislocations l i e parallel to <^ 112> with <^ 110J> as their direction of motion. Pure screw dislocations l i e parallel to <^110£> , with <^ 112^ > for their direction of motion. Every ( i l l ) plane has three orientations of each kind giving rise to twelve different orientations. Keeping one of the dislocations fixed, we can examine the F ig . 1L, Unit Tetrahedron Showing BVs and Dislocation Directions• interaction due to eleven other orientations, the twelfth one being parallel to the f irst one. TablesXLTandXIIE arise as a result of this calculation. Since we know that parallel dislocations of the same kind with parallel BVs repel from one another, the nature of the force compon-ents for other orientations could be established. However, there are certain other limitations. These expressions do not hold after the dis-locations have intersected or associated to give rise to some other dislocation. The following case is of some special interest. Two dislocations (edge - edge) on parallel sl ip planes, BVs not parallel (Fig.75)* TABLE XII 170 FORCE COMPONENTS FOR EDGE - EDGE INTERACTION Dislocation Dislocation I II F l F2 F r F„ [I10][llj[ll2] [oiijfiiilfcii] 2 0 c* A 0 u [io:D[in][i2l] _<X 2 0 o< 0 [ l l 0 ] [ l l 2 [ l l 2 ] 3 0 0 21/2" cx 3/3' 0 [ioi][ni[i2il 2 -/?« . J i cX 12 - i f -jpiij [211] c* 5 °< 12 cX 2 ^ 12 [iio]|in][ii2] _ 2 c * 3 0 0 - 2 (Joe 3l/3 . 0 [ l 0 j [ l l l ] [ l 2 l ] cX - VL ex 4 3 cX 2 ^ CX 2v^ [on][iifl[2iil cX 2 12 - i f - 12\/IT [no][ii3[ii2] 3 0 <X 3 3 0 [01^[ui|[21l] cX 0* 0 _ °< 12 v T CX 12/J1 2*wir [lof)[lTi[l2ll ~ 6 0 _ °< 12 x/21 cv CX 12^T [110] [11J [112] ex 0 0 0 °< - b ED r E 2 , E D 2 E 3 ^ b 2 2TT(i-V) 171 TABLE XIII FORCE COMPONENTS FOR SCREW » SCREW INTERACTION Dislocation I Dislocation II F l F2 F r F8 [ll2][lll][ll0] [2l3[lll| fell] J& 2 0 I 0 u [121] [iii] [101] 2 0 Z 0 u [112] [ill] [110] -2 5 3 ^ 2 £ 0 f > [121] [ill] [101] P " 2 0 J l A 12 7 3J2 [211I [nil [011] P 6 3J 12 ~ 2s/2" [112] [ill] [lio] -\> - 1 > • 3 J [12O [111] [101] 0 - f > - 12 3JTJ 12&?J [211] [111] [01) 0 0 0 0 0 [112] [ifi] [110] 3 ~tP * =» ^ 0 0 [211] [111} [on] 6 0 [121] [111] [101] 6 & £ 3 9 12J3 [112] [111] [110] fi> 0 0 0 P « b i s D2 S r 2 2 TT Fig.75 Two Edge Dislocations on Parallel Slip Planes x 3 / / [ l l 2 ] . Dislocation I, x-> Dislocation II, x^ //[lOHj , x^ //jlU] , x^ // [ l 2 l ] . 1 2., "2 0 2 0 1 2 From equation 4> 12 and 11= 2 E D„ ( 4 ,2x2 Xo ) X • (X 2 - X 3 0) 173 As before we have x ( •» 0 m x 2 . Choosing the o r i g i n of the second dis-location along x 2 , we have x^0 s 0 » X ^ Q * NOW, for a given x 2 Q we can calculate the force per unit length at any point along the dislocation I . From equation 9> „ 2 .2 2 3 • ( * 3 + * ^ ) 2 and F 2 An approximate plot of F( , along the d i s l o c a t i o n line is shown in Fig.76. Fig.76 Variation of Force Due to Interaction Along an Edge Dislocation Line. Interaction Results Due to the Presence of a Second Non Parallel Edge Dislocation Lying on a Parallel Slip Plane at a Fixed Distance x 2 0 . 7. BIBLIOGRAPHY 174 1. Gibbs, J.W., "On the Equilibrium of Heterogeneous Substances", Collected Works, Longmans, Green and Co.,New York (1928). 2. Frenkel, J., J.Phy., USSR, % 392, (1945). 3. Frank, F.C. et al, Phil. Trans. 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Read, W„T<,, "Dislocations in Crystals",(1953) Swalin, R.Ao, "Thermodynamics of Solids", John Wiley, p.192, (1962) Basinski, Z.S., Phil9Mage, % 51, (196/V) 

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