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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 PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department of METALLURGY  We accept this thesis as conforming to the 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 i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t available for reference and study.  freely  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 i s 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 B r i t i s h Columbia, Vancouver 8, Canada Da  -te  August 11 . 1964  The U n i v e r s i t y  of B r i t i s h . 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 U n i v e r s i t y , 1956 D.I,I.Sc., I n d i a n I n s t i t u t e of Science, 1958, Bangalore M . S c , U n i v e r s i t y o f Utah, 1960  FRIDAY, AUGUST 7, 1964 a t 2;00 P.M. IN ROOM 201, MINING BUILDING  COMMITTEE IN CHARGE Chairman; W,M. Armstrong R. B a r r i e L.C. Brown External  I„ McT« Cowan J.A.H.. Lund E. T e g h t s o o n i a n R„M. Thompson  Examiner; R.L, F l e i s c h e r Research S c i e n t i s t G e n e r a l E l e c t r i c Research L a b o r a t o r y Schenectady, N,Y.  GROWTH AND DEFORMATION OF COPPER WHISKERS ABSTRACT  Copper whiskers were grown by t h e hydrogen of  the h a l i d e vapours.  reduction  The s u b s t r a t e and t h e w h i s k e r s  were examined f o r growth m o r p h o l o g i e s . Whiskers the  of (100), (110) and (111) o r i e n t a t i o n s i n  diameter range 30-400jLX. were t e s t e d i n t e n s i o n *  D i f f e r e n c e s i n t h e work hardening c h a r a c t e r i s t i c s of (111) and (100) w h i s k e r s were d i s c u s s e d i n terms of t h e d i f f e r e n t k i n d s o f jogs formed  i n the two o r i e n t a t i o n s .  A n a l y s i s o f the diameter dependence of t h e y i e l d resulted  stress  i n a 1/d r e l a t i o n based on t h e assumption o f  surface nucleation of d i s l o c a t i o n s . Temperature  and s t r a i n r a t e change experiments were  made on (1.10) w h i s k e r s .  A c t i v a t i o n d i s t a n c e and  a c t i v a t i o n e n e r g i e s were used t o determine a r a t e cont r o l l i n g mechanism.  At low temperatures, c r o s s  slip  and i n t e r s e c t i o n p r o c e s s e s were i n d i s t i n g u i s h a b l e , w h i l e a t h i g h e r temperatures, c r o s s s l i p trolling. for  i s r a t e con-  From t h e c a l c u l a t e d a c t i v a t i o n d i s t a n c e and  a g i v e n r a t e c o n t r o l l i n g d i s l o c a t i o n mechanism,  stacking fault  e n e r g i e s were e s t i m a t e d f o r copper and  other FCC m e t a l s . A twinning model was proposed based on t h e i d e a o f f a i l u r e by r e c o m b i n a t i o n of a L o m e r - C o t t r e l l  barrier.  GRADUATE STUDIES  Field  of Study:  Metallurgy  M e t a l l u r g i c a l Thermodynamics S t r u c t u r e of M e t a l s  III  C. S„ E.  Teg'htsoonian  P r o p e r t i e s of Ceramic M a t e r i a l s . _ „  „  A,C.D„ Topics i n Physical Metallurgy Transformation Related  and D i f f u s i o n  Samis  ,, ,  Chaklader  J.A„H„ Lund W.M.  Armstrong  Studies:  Theory and A p p l i c a t i o n s of D i f f e r e n t i a l Equations S t a t i s t i c a l Mechanics P h y s i c s of the S o l i d S t a t e  C,A, R,F.  Swanson Snider  R. , B a r r i e  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 [ i l l ] orientations i n the diameter range 30 - ^OOLlwere tested i n tension^  Differences i n the work hardening  characteristics of [ i l l ] and [lOO] whiskers were discussed i n terms of the different kinds of jogs formed i n the two orientations. Analysis of the diameter dependence of the yield stress resulted i n 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 determine a rate controlling mechanism. At low temperatures, cross s l i p and intersection processes were indistinguishable, while at higher temperatures, cross s l i p i s 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 preoccupations.  I also wish to thank him for having suggested the  problem on the temperature and strain rate dependence of whiskers, and his c r i t i c a l evaluation of the manuscript. Grateful acknowledgments are also due to Prof.F,A.Forward for having provided me with an opportunity to be i n 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 i n the department, especi a l 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 i n 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 i s gratefully appreciated. I remain indebted to Mr. K.R. Mally for his help and  encouragement without which my v i s i t 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  Introduction  1. 2  Experimental Procedure  1. 3  Results?  1. U  • . •  •••  1  •  3 U  A  (1)  Examination of the Substrate  (2)  Examination of Whisker Tips and Surfaces „ « « » e » »  Discussion  o  e  «  o  «  .  «  o  «  e  e  o  e  • • » » . <>•••••.». • • • • • • • • • • •  2. 2. 1  1  STRESS - STRAIN  16  CURVES  25  Introduction,  ••  2. 3  25  (1)  Characteristics of Stress - Strain Curves  26  (2)  Size Effects  27  (3)  Deformation Characteristics i n Whiskers and Bulk Single Crystals. • • • • • • • • • • • • • • • • • • •  2. 2  U  Experimental Procedure (1)  Testing method . • • • •  (2)  Load Cycling  (3)  Testing  {U)  Orientation Determination  Results  28  • • • • • • • • • • • • • • •  30  ••••  30 33  .  • • • • • • • • • •  33 35  ••••  35  (1)  Stress - Strain Curves • • • « • • • • • • • • • • • •  36  (2)  For Whiskers of [lOO] and [ l l l l Orientation  44.  TABLE OF CONTENTS  CONTINUED Page  2.4-  (3) For Whiskers of [lio] Orientation*  47  (4)  Metallography and Fine Structure • • • • • • • • •  47  (5)  Diameter Dependence of Yield Stress.  50 56  Discussions (1)  Hardening i n [ i l l ] and [lOol whiskers  (2) Diameter Dependence,  3. TEMPERATURE AND 3. 1  Introduction  3. 2  Experimental Procedure.  3. 3  o  s  STRAIN  o  e  o  e  e  o  s  e  . . . . .«  56  i  62  o  ^  a  o  s  68  RATE DEPENDENCE  68 . . . . . . .  70  ••••  71  (1)  Temperature Change Tests »  (2)  Strain Rate Change Tests . . . . . . . . . . . . .  72  Results  .72  (1)  72  Temperature Change Tests  (2) Strain Rate Dependence of the Flow Stress  74  (3) Activation Energy, Activation Volume, Frequency Factors, etc.  79  (4)  82  Activation Energy.  (5) Activation Volume  •  83  (6) In ^ o (7)  83  C  Activation Distance  •  ••••  84  TABLE  OF CONTENTS  CONTINUED Page  3. 4  Discussion. - Mechanisms •  90 90  (1) Formation of Jogs • • • Forest Intersection Mechanism.  (b)  Non Conservative Motion of Jogs i n Screw Dislocations •  (c)  3.5  90  (a)  o  o  s  s  e  .  o  Conservative Motion of J o g s  o  e  .  a  e  o «  e «  . s  e «  . o  o  o  .  91  <  i  92  (2)  Cross - Slip of Screw Dislocations  (3)  Impurity Dislocation Interaction  (4)  Peierls Mechanism » « . . • • . < > . < > • . • • • • . 94  92 •• • • • • • • •  . Discussion of Experimental Results • • • • • < » • .  94.  . . . . 98  (1)  Hardening at 295°K and 273°K  98  (2)  Hardening at 200 K , . . . . . . . . . . . . . . X00  (3)  Hardening at 77°K  (4)  Hardening Characteristics of [106] and [ i l l ] Whiskers 102  (5)  The Quantity In J o  9  3.6 Twinning 3.7 Dependence  . . . . .o  0  102  €  of ^  on V  4-  SUMMARY  5  100  ; AND CONCLUSIONS  SUGGESTIONS FOR FUTURE WORK  . . . . .103 109 112 118  TABLE  OF CONTENTS  CONTINUED Page  6 I II III IV V VI  APPENDICES  Stress - Strain Curves. • • • • • • • • • • • • •  119  On Jogs  137  »  L — C Reactions Image Force  o  e  O  o  o  o  o  e  c  <  c  j  o  o  o  o  c  o  »  e  «  .  o  ©  s  Load Measurement. • • . . . . . . .  «  o  o  o  o  o  «  »  «  «  o  >  o  o  14.2  »  14-5  . . . . . . .  IAS  Activation Energy Calculations,  150  Double Slip Conditions and Schmid Factor  154  VIII  Estimation of Errors. . < , . . . . . - . < > • • • •  158  IX  Forces Between Dislocations • • . . . » . • • • .  161  VII  1.  Introduction. •  161  2.  Theory and Application  161  7  BIBLIOGRAPHY  174-  FIGURES No, 1.  Page Etch Pits on Copper Reduced from GuCl^.Etched Briefly i n Conc.HN0 3000X. . . /  5  3  2a to 2f  Spiral Etch Pits on Copper Reduced from Cu01 . Etched Briefly i n ConcHNO.^. 3000X i  6=8  3.  Tips of as Grown Copper. Whiskers, Sketch shows the Orientation of the Facets, 800X » . , • • • « • •  10  o  0  4.  Growth Stops on as Grown Whisker Tips Orientation of the Facets  e  800X  0  Sketch Shows  e  ,  e  9  ©  o  »  ,  <  >  o  5.  Surface of a Whisker Exposed to A i r .  6.  Tip of a Whisker Exposed to A i r .  7.  Growth Steps on the (001) Surface of a [llO] Whisker. Original Magnification 110X . . . . . . . . . . . . Growth Steps on the (110) Surface of the Whisker i n F i g . 7. 110X. .  8. 9.  1600X . . . . . .  1600X.  11 12 12 13 14  Growth Steps on the ; (110) Surface of the Same Whisker in Fig, 7 .  600X. . .  15  10.  A possible Growth Spiral on a (110) Plane  11.  Growth Spiral With a Twinned Segment, , ,  , ,  17  12.  Form of the Spiral i n Fig.10 After a Complete Sequence of Twinning , , , , Twinning within a Grain  17 18  A Dislocation Node i n 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  Illustrating Horizontal Glide Traces Left on the (001) Plane or the (110) Plane of the Whisker i n F i g . 7 due to the Egress of % <011> Dislocations on (111) Plane.  20  Sequence Showing the Form of the Steps Left by Gliding Screw Dislocations. One of the Glide Planes is Parallel to the Axis. • • « • « • , , , • • • • , • •  22  13. 14.  15.  16.  16  FIGURES  - Cont'd.  No. 17.  Page 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.  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 0.002 /min  37  0  a  e  «  «  «  »  *  31  «  n  25. Load Elongation Curve for [ i l l ] Whiskercont'd. Speed 0.002 /min.  Cross-head 38  n  26.  Tensile Stress - Strain Curves. (Use Left Scale for No. 16) . . . . . . . . . o e . . . . . © . . . « .  39  27a to 27c  Comparison of Stress - Strain Curves for Coarse Whiskers, Fine Whiskers and Single Crystals. . . .  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  A [ i l l ] Whisker Deformed at Liquid Nitrogen Temperature Showing Deformation Twins. Sketch Shows Twin Trace. 110X  48  Load Elongation Curve at the Onset of Twinning for a [ill] Whisker Deformed at Liquid Nitrogen Temperature  4-9  A [lOO] Whisker Deformed at Liquid Nitrogen Temperature Showing (a) Double Slip and ( ) "Cellular» S l i p Line Markings. 210X •••••  51  30.  31. 32.  40-42  D  FIGURES - Cont'd. No.  Page  32. The [lOO] Whisker Deformed at Liquid Nitrogen Tempcont'd. erature Showing Non-octahedral Slip. 110X. • • . • • 33. 34.  A [loo] Whisker Deformed at Liquid Nitrogen Temperature Showing Failure by Necking. 110X • • • • • • •  35.  53  (HO) Plane of a [lio] Whisker Deformed at Q°C Showing Beth Primary and Secondary Slip . . . . . . . o  o  e  52  54  o  A Fractured [lio] Specimen Deformed at Liquid Nitrogen Temperature. 110X  e  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  37.  Oriented Unit Tetrahedron to Show the Slip Systems i n  57  [ill] and [lOO] Whiskers. • • < , . « . . . . • • . • • Stereographic Projection for Cubic Crystal i n Standard Orientation, (Notation after Rosi and Mathewson). . • 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 i n Copper  38. 39.  (after S e e g e r ) . . . . . . . . . . . . . . . . . . . . 42.  Stereogram Showing the Specimen Orientations. . •  43.  Temperature Dependence of the Flow Stress Ratio for Coarse Whiskers and Bulk Single Crystals of Copper. • Temperature Dependence of the Flow Stress Ratio for Copper Single and Poly Crystals. The High Temperature region i s From Poly Crystals . • • •  44»  45.  •  58 59'  68 70 73 75  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 [ i l l ] 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 i n 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.  AnBarrier L-C Barrier . . . . . .Slip . .on . .(001) . . .Plane . . .(Schematic) .... 105 and Subsequent 106  59. 60a  Sequence Showing Failure byDue Recombination of an L-C Formation of a Twin Source to the Operation of Partial Dislocations AP and BQ i n 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 i n 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. 64..  Page 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 . •  154  71.  Stereogram Showing Lattice Rotation for a [lio]Whisker  155  72.  Coordinates for Resolved Shear Stress Calculations • •  156  73.  Coordinate Systems! x  Resultant Lattice Rotations (b)  i  (from Ref. 61).  i s 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 .  173  9n  TABLES  No. I II  III IV V VI  Page Conditions of Growing Different Kinds of Whiskers  3  Comparison of Yield and Maximum Stresses for Whiskers and Bulk Single Crystals. . . . . . . .  43  Number and Type of Jogs. . . . . . . . . . . . . .  61  Comparison of Yield Stress Data.  66  Temperature Baths  ..  .  s  e  e  . . . . . . . o e  . «  .  .  .  . . . . < >  71  Activation Volume, Activation Energy and Frequency . . . . . .  S3  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 « • * •  Factors . . . . . . . . VII  XI XII XIII  . . . . . . .  Schmid Factor as a Function of Strain.  X20 157  Force Components for Edge-Edge Interaction .  e  • •  Force Components for Screw-Screw Interaction . . .  170 171  1. 1. 1  GROWTH  1  Introductions The growth of crystals has been the subject of intensiye research  during the last decade or two.  The basic formalism f o r t h e 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 fect.  that real crystals are not per-  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 i n cooperation.  In the f i r s t case the growth  steps w i l l be spirals and i n the latter case they w i l l be closed loops. These can be either circular or polygonal i n shape depending on the anistropy 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 i s contained i n a recent symposium 5 report "Growth and Perfection of Crystals", and i n 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 w i l l be recalled, (1)  Eshelby's  theory speculates on the presence of a Frank-  Read source i n the substrate i n a plane normal to i t s Burgers Vector (BV), The result i s , the dislocation loops so generated w i 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 explain kinking i n FCC l a t t i c e s .  In this a growth dislocation of BV 'b•  i s expected to be i n neutral equilibrium according to b ^ b ^ • b^, and i f b^ i s parallel to one of the growth faces unlike b2> growth w i l l take place i n a direction parallel to b  2  resulting i n 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 f a u l t . In the present work an examination of the substrate as well as whisker surfaces w i l l be presented.  1, 2 Experimental Procedure; 5 Table  I  i s 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 u C l  and Cul  c  by reduction with hydrogen at about 550*C.  2  With dehydrated CuGl , even 2  though there was profuse whisker growth the whiskers were very thin compared to those obtained from Cul, The substrate was much more coherent i n the case of CuGl , 0  TABLE I CONDITIONS OF  GROWING DIFFERENT  KINDS OF WHISKERS  Metal  Halide  Temperature Range *G  Copper  ( CuCl ( CuBr ( Cul  430  -  650  -  850  700  -  800  -  900  Silver  v  Iron  ( AgCl . ( Agi • ( FeCl ( FeBr  730  2  -  2  Nickel  NiBr  2  Cobalt  CoBr  2  Manganese £ j})  MnCl  2  Gold  AuCl  550  Platinum  PtCl  800  Palladium  PdCl  2  760  740 650  - 730 940  960  -  735  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 i n plasticine, just by pressing a  b a l l of plasticine against the whiskers, thus exposing the bare face free of whiskers for examination. This face was etched very briefly i n cone. HNO^, washed and dried i n a i r and examined i n 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 d i f f i c u l t 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 h i l l s on the bounding faces*  Nevertheless, i t was not d i f f i c u l t to pick  out whiskers of very odd growth surfaces*  However, since i t was d i f f i c u l t  to come to any reasonable conclusion from such whiskers, attention was focussed on whisker tips* Numerous whisker tips were carefully surveyed under an optical microscope. These whiskers were previously mounted on plasticine on a glass  5  Fig.l  Etch Pits on Copper Reduced From C u C l . Briefly in Con. HNO_. 3000X 2  Etched  (a)  (b) Fig.2  Spiral Etch Pits on Copper Reduced From C u C l . Briefly i n Conc.HNOo. 3000X ?  Etched  9  slide.  The tips were very well developed pyramidal surfaces which were  free of any surface markings as seen i n Fig, 3, However, these were a few whiskers which i n fact did show growth steps as shown i n Fig,4-»  A schematic  sketch of each whisker i s also included. Finding those whiskers was merely fortuitous and chances were about 1 i n 20, Whiskers exposed to a i r for a period of one week usually showed some surface markings, as shown i n Fig, 5«>  Fig,6  i s the t i p of one such  whisker of hexagonal cross-section, A very rare whisker produced from the reduction of Cul, i s shown i n Figs ,7 to 9,  The axis of the whisker was [ll6\  were (001) and (llO),  and "the bounding surfaces  A series of photographs showing the growth steps  on one of the (001) surfaces i s shown i n Fig. 7. two selected areas from the (llO) surface.  Figs. 8 and 9  show  Fig.3  Tips of as Grown Copper Whiskers. Orientation of the Facets. 800X  Sketch Shows the  (c) Fig .4  Growth Steps on as Grown Whisker Tips. Sketch Shows Orientation of the Facets. 800Z  12  Fig.6  T i p o f a W h i s k e r Exposed t o A i r .  1600X  Fig.8  Growth Steps on tho (110) Surface of the Whisker in Fig.7. 1101  Fig.9  Growth Step* on t h e W h i i k e r i n F i g . 7.  (110) S u r f a c e o f t h e Same  600X  16 1.  U  Discussion; Hoping that the observed etch steps i n F i g . 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 dislocation of  £[ID] BV. The growth spirals w i l l then assume the anisotropy  of the surface, and i n this case we can expect them to be rectangular. Such a surface i s shown i n FigoID and the angular relationships are as denoted. The plane of the paper i s (110).  The two (111) planes shown i n F i g . 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  [ll2] (111)  72^  [lib] (111) [001] [112J  Fig.10  A Possible Growth Spiral on a (110) Plane.  Fig.12  Form of the Spiral i n Fig.10 After a Complete Sequence of Twinning.  18 obtained from the above, by repeated twinning say on ( i l l ) plane i n a direction such as OA.  For example, consider only one twin along OA.  This produces the structure shown i n Fig.11. Continued twinning along segment OD  w i l l result i n Fig.12. The growth steps w i l l 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 i n Fig.13.  F i g . 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 i n F i g . 4.  we need to explain the formation of the  (100) and (110) facets at the t i p .  In order to explain this we consider  a dislocation node as shown i n Fig.lA. A l l the dislocations are marked such that the sum of the BVs at the node i s 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 i n neutral equilibrium as, £[110] Energy<X£  £ [100]  •  £[010] .  £  £  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]  ^[lii]  F i g . 14 t  A dislocation node i n a [ i l l ] whisker, [ i l l ] i s parallel to the axis, and <CllO>> are normal to the facets at the t i p .  In order to explain this we consider the dislocation line energy above and below the node. proportional to 2 b  Above, i t i s proportional to 3b  and below i t i s  (where b « £[llo] ). This considerable reduction  3  i n the line energy w i l l force the node to climb as the whisker grows so that OA, OB, etc. remain constant i n length.  This w i l l result i n  a net growth parallel to the [ i l l ] axis. It now remains to explain the surface markings shown by the whisker i n Figs. 7 to 9.  These could be due to the gliding of  screw dislocations which are not parallel to the axis of the whisker.  20 In this case,  movement would be from the t i p towards the base.  An analysis of these surface steps showed that i n 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 £ O l Q ^  on one of the {ill} planes shown i n F i g . 15.  Trace of (111) glide plane (001) growth plane  (110) growth plane  [llO] whisker axis  (111) glide plane Fig.15  Illustrating Horizontal Glide Traces Left on the (001) or the (110) Plane of the WhiskerJLn F i g . 7 dua to the Egress of ^^Olpi Dislocations on (111) Plane. l  A l l 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 (l20) and  ^-[lio].  Accordingly the respective glide planes are  (ill) . The form of the steps may be easily explained as follows. The  gliding of a series of dislocations on the ( i l l ) plane w i l l result i n a structure of the type shown i n Fig 16a.Subsequent gliding on planes 0  of (120) or (114.) type w i l l give rise to Fig.16b.  This configuration  w i l l produce the block-like structure seen i n F i g , 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 i n F i g . 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 w i l l give rise to Fig.17a. Further operation of dislocations on intersecting planes w i l l result i n F i g . 17b. If now the dislocation B glides out of the crystal along the glide plane of A, the step OP w i l l vanish, because A and B are of opposite  sense.  The f i n a l structure i s shown i n Fig.l7c. By similar operations the observed microstructure can be derived.  Alternately, the structure i n  Fig. 9 can be derived without the introduction of dislocation A provided the dislocation B after having reached the point 0, changed i t s glide plane and moved i n the direction OP*. Such a change of glide plane i s possible because the BV -^[llO] i s common to both (114-) and (114.) planes. The latter operation seems to be much more l i k e l y i n  t 22  1  <  <  whisker axis  (a)  glide direction  glide < direction  F i g . 16  Sequence Showing the Form of the Steps Left by Gliding Screw Dislocations. One of the Glide Planes i s Parallel to the Axis.  23  F i g . 17  Sequence Showing the Form of the Steps Left on (110) Plane, due to Gliding Screw Dislocations on Intersecting Planes.  obtaining the observed microstructure  24  The origin of such dislocations i s 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 i n the whisker.  2.  2. 1  25  STRESS - STRAIN CURVES  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 i t s 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 i n value of the shear strength, the detailed nature of whiskers is s t i l l unknown. 12,13 Experiments by Brenner, Gorsuch  H 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 i e s i n the possible existence of axial screw dislocations. 15 Price investigated the deformation of zinc platelets i n an electron microscope, and observed that dislocations are generated i n 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 i p , the manifestation of high yield strength by whiskers i s 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 uncertain.  26 At present there exists considerable data on the yield strengths of different metal whiskers 18 the other hand Suzuki  12 13 16 IV ' ' ' up to about 3 0 ^ i n diameter. On  investigated melt grown bulk single crystals  from about 1500jU to about 60/^ i n diameter. The present work i s solely devoted to vapour reduced copper whiskers of about 30 - 400jJ- i n diameter.  In this i t i s hoped that the results w i l l bridge the gap between  whisker behaviour and bulk single crystals.  In what follows whiskers  <^30JJ. i n 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 i n which the flow stress remains almost constant as shown i n F i g . 18. The ratio  Fig.18  l e l d stress yield flow stress  s  __Jx 7J~. fl  may be as high as 80 »1.  Yielding of a Fine Whisker (after Brenner).  27 Brenner however, points out that this yield drop should not be identified with the yield point phenomena shown by Fe-C alloys. as he indicates  For example,  the whisker behaviour might be due to the spontaneous  generation of dislocations followed by a region i n which the strain rate i s very high. That i s why some whiskers might f a i l immediately after tha yield drop.  In the region where the flow stress i s constant  whiskers deform by Iideraband propagation. This i s characterized by the nucleation of slip i n one system at some part of the crystal. This region then traverses across the specimen until the whole specimen i s f i l l e d with s l i p lines. region.  This i s then followed by a rapid hardening  If Nucleation took place In more than one point i n the speci-  men then the stress - strain curve would be as shown i n Fig. 19.  Fig.19  (2)  Repeated Yielding of a Fine Whisker (after Brenner).  Size Effects;  Whiskers show a prominent size dependence i n their mechanical properties.  12,13,16 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 i s not only increased con19  siderably, but also shows i n turn a size dependence.  No such work has  been done on bulk singlecrystals. (3)  Deformation Characteristics i n Whiskers and Bulk Single Crystalss  It i s Interesting to distinguish the deformation modes i n whiskers and single crystals.  As previously mentioned, fine whiskers  are characterized by a conspicuous yield drop followed by a constant stress region i n 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 i n F i g . 20. The form of the curve i s extremely sensitive to orientation.  Fig.20  Schematic Shear Stress - Strain Curve for Single Crystal Copper.  The region of easy glide (Stage I) is one i n which the work hardening rate is very low.  In Stage II the work hardening rate is high but  linear followed by parabolic hardening i n Stage III, which i s also exhibited by fine whiskers to some extent. The deformation characteristics of single crystals w i l l not be discussed i n detail here. ersial i n nature.  The existing theories are rather controv-  However i t w i l l be pointed out that according to  one of the theories the region of easy extension (Stage I) i s marked by s l i p starting at longest available Frank-Read sources and progressing u n t i l a l l such sources are exhausted.  In Stage II rapid hardening  sets i n due to increased interaction of dislocations from the primary and secondary systems which resultsin the formation of dislocation 20  barriers, (see for example, Friedel  35  and Soeger  ).  Stage III i s  marked by the break down of these dislocation barriers and resulting 21,35 i n the pronounced cross-slip of screw dislocations • Some of the work hardening characteristics i n coarsewhiskers .\  w i l l be discussed i n the light of the current theories and using the modern concepts of dislocations, i n this and i n the next section. It i s worth mentioning that whiskers tested by Brenner and several other investigators were pulled i n a soft beam tensile testing machine.  The tests are extremely discontinuous i n 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 i s identical with that used for bulk single crystals, except for the slight differences i n 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 i n a floor model Inst ron testing machine using low strain rates and very sensitive load c e l l s ,  A suspension schematically shown i n  Fig, 21 was used for mounting whiskers.  It consists of a mild steel  rod approximately £" i n diameter, and i s as short as possible so as to minimize lateral deflection and also to bo consistent with the convenience of test procedure.  At the bottom end i t i s split to accommodate  the mounting rod, which i s 2" long, and i s threaded at i t s upper end. In the earlier room temperature tests the mounting agent used was diphenyl carbazide. However, for liquid N tests this was found to be 2  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 i n solder. The mounting technique i s very simple. A carefully selected whisker i s cut to a reasonable length and placed horizontally on a thoroughly clean glass plate.  Cleanliness i s essential for i f the  glass plate i s not clean enough, the whisker w i l l stick to i t and f a i l V ure becomes F i g . 22.  inevitable.  The f i n a l arrangement for mounting i s shown i n  31  to the load c e l l  mounting agent whisker  Fig.21  Suspension Used for Whisker Testing.  32  glass plate  Fig.22  Mounting of a Whisker.  It i s 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 a i r *  almost instantly attains the ambient temperature*  The system  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 i n the orientation determination of the whiskers * The suspension ready for test i s 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 i n s i t u .  This  i s gradually raised u n t i l a reasonable length of the whisker i s Inside the pool*  The whole operation i s done with a copious flow of cold  hydrogen onto the whisker. The system i s then allowed to cool to room temperature*  (2)  A mounted whisker i s shown i n Fig, 2 3 .  Load Cycling?  During mounting and the introduction of any temperature baths, the whisker experiences a load due to thermal expansion effects. careful adjustment of the load cycling doVice i n the Instron possible to maintain the load constant within (3)  +  By  i t is  0,2% f u l l scale.  Testing?  A l l whiskers tested for size dependence measurements were  34  Fig.23  Mounted Whisker Ready for Test.  35 strained at a basic cross-head speed of 0.002 /min , the lowest available n  i n the Instron. min.  Some of the I n i t i a l tests were done at a rate of 0.01"/  Yield stress values and stress - strain curves were also extracted  from strain rate change experiments deformed alternately at G.002 /min n  and 0.02 /min. w  The gauge length of the whisker could not be determined before the experiment.  Since the total elongation i s always known from the  load elongation curves, the original length is deduced from accurate measurements of the f i n a l length.  Since the i n i t i a l length of the speci-  men cannot be fixed very accurately, the strain rate (£) i s 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 i n strain rate and temperature change tests. 2. 3 Results 8 Whiskers tested i n this work belonged to three prominent orientations corresponding to the corners of the stenographic triangle i n F i g . 24.  Whiskers of D-1Q] orientation were thin blades.  (1)  Stress - Strain Curvest Some typical load - elongation curves are shown i n F i g . 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-) i n diameter the stress - strain curves were very similar to those of fine whiskers, (of course excluding the i n i t i a l y i e l d ) . 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 i n 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 i s shown i n F i g . 27  for a l l the three prominent orientations. For  the sake of comparison i n Fig.27 curves are given for whiskers and bulk single crystals for the three orientations.  It may be noted  that i n Fig.27c a divergence i s shown i n the stress - strain curves. The divergence i n the single crystals i s also such that the coarse  10-* 2 2 Cross-sectional  0.002  0.004. 0.006 Elongation (in) (a)  arear 35x10 y*  0.008  2 2 Cross-sectional aroa= 35x10 JJ-  0.002  0.004  0.006  0.008  Elongation (in) (b)  F i g . 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 0.002 /min. n  100  —i  0<l  F i g . 26  Tensile Stress - Strain Curves.  1  0-Z  r  0-3  (Use l e f t scale for No.16).  40  F i g . 27  Comparison of Stress - Strain Curves for Coarse Whiskers, Fine Whiskers and Single Crystals.  41  F i g . 27  Cont'd.  Comparison of Stress - Strain Curves for Coarse Whiskers, Fine Whiskers and Single Crystals.  42  F i g . 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 l i m i t s . found to be unimportant i n other orientations.  However, this i s  (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 i s evident that there i s a wide range of behaviour from fine whiskers to bulk single crystals*. The yield stress i s extremely large for fine whiskers.  Subsequent to yielding the max-  imum stress i s larger for coarse whiskers, as may be seen from Fig. 27.  It i s also clear that these quantities show a strong dependence on orientation. Some typical values for yield and maximum stresses are given i n Table II for the three orientations considered. TABLE  II  COMPARISON OF YIELD, AND MAXIMUM STRESSES FOR WHISKERS AND BULK SINGLE CRYSTALS. Tensile Stress kg/mm 2  Orientation  Fine Whisker 70  ^yield  T L  max.  Coarse Whisker 1-2  Single Crystal o.i  [ioo]  50  1  0.1  _liio]  50  1  0.1  "[ill]  15  100  10 - 20  [lOQJ  10  20  10 - 20  10 - 20  10 - 20  _ [llO] (thin 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 i n diameter the region of Liiders band propagation decreases rapidly and results i n a small region of easy glide, which may not be present i n some whiskers. For [100] whiskers, this "easy extension" (Liiders Band or easy glide) i s shown i n F i g . 28. (b) For a given orientation the region of easy extension sojamed to increase with decrease i n 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 i s chosen for comparison i n 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 variation 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 i n [ i l l ] whiskers i s about five times larger than i n [lOO] whiskers at a l l temperatures.  008 _  [100]  0-06  Easy Extension  •  0-04-.  •  • 1  50  1  [OO  1  150  ,  ZOO  :  ,  2.50  •  • ,  300  Temperature °K F i g . 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 I I I hardening rates are similar to those f o r O-OOl  orientation* (o)  The easy g l i d e region increases with decrease i n  temperature. (d)  D u c t i l i t y i s considerably higher compared to both  [llll  and [lOO] o r i e n t a t i o n s . (e)  Deformation i s mostly on one  system.  The divergence i n the curves of s i m i l a r o r i e n t a t i o n at a given temperature i s p a r t l y due to the e f f e c t of diameter and as previously mentioned  p a r t l y due to the minor differences i n composi-  t i o n that might p r e v a i l from one batch of whiskers to another. Composition of a t y p i c a l batch of whiskers i s given i n the next s e c t i o n .  (4)  Metallography and Fine Structures (a)  At All  Whiskers of [ l l l l  o r i e n t a t i o n deformed on two s l i p systems.  l i q u i d nitrogen temperature these whiskers deformed also by twinning. the s i x faces of a twinned whisker are shown i n F i g .  a t i c sketch i s a l s o included.  30.  A schem-  Twinning was confirmed by taking a  back r e f l e c t i o n Laue picture, using a s l i t collimator, which showed d i s t i n c t s p l i t t i n g of r e f l e c t i o n s .  A load elongation curve at the onset  of  twinning i s reproduced i n F i g .  31. (See next section f o r 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 i p I  twin trace F i g . 30  A {ill] Whisker Deformed at Liquid Nitrogen Temperature Showing Deformation Twins. Sketch Shows a Twin Trace. 110X  49  200  WW***  9  150Load (gms)  2  Cross-sectional Area=18.5xl0 yCross-head Speed 0.002"/min  100  s  50 -  0.0594  0.06M  0.0634-  0.0654  0.0674-  0.0694-  Elongation(in) F i g . 31  Load Elongation Curve at the Onset of Twinning for a [ l l l j Whisker Deformed at Liquid Nitrogen Temperature  2  surfaces shoved that this plane i s of the (110) type. (b) Whiskers of [100] orientation show different kinds of s l i p line markings.  The predominant type corresponds to "double  s l i p " i n which two s l i p planes symmetrically oriented with respect to 21 the tension axis operate alternately.  Less common i s one i n which the  s l i p lines are fragmented resulting i n the formation of a "oellular" structure. This terminology w i l l be further used i n discussion. Occasionally a third type i s observed due to non octahedral s l i p . Fig.32-is from the same whisker shewing a l l the different kinds of s l i p markings. (c)  Fig.33 i s from a whisker where failure i s imminent.  Whiskers of [lid] orientation (thin blades) deformed  mostly on one s l i p plane. Secondary s l i p systems seem to have come into operation after some deformation. However, i n this case secondary s l i p takes place earlier than predicted by the Von Gbler and Sacis condition (see Appendix VII), secondary s l i p .  Fig.34- shows both primary and  F i g . 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 whisk12 13 16 ers has been obtained by various workers ' ' i n the form  d(/0  (b) Fig. 32  210X  A £LOCu Whisker Deformed at Liquid Nitrogen Temperature 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)  F i g . 34  210X  (HO) Plane of a [lio] Whisker Deformed at 0°C Showing Both Primary and Secondary S l i p .  55  Fig.35  A Fractured Q-lo] Specimen Deformed at Liquid Nitrogen Temperature. 110X  56 B (^20) are constants. F i g , 36is a plot of ^ v s 1 from different d sources. The above straight line i s also shown. Yield stress values  J  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 i n [ i l l ] and [lOo] Whiskers  The work hardening characteristics of bulk single crystals of copper of orientations close to [ill] and [lOO] has been discussed 21 i n some detail i n a review article by Clareborough and Hargreavos . 23 Recently Saimoto  related small differences i n the work hardening  characteristics of [lllj and [loo] bulk single crystals of copper to the different kinds of jogs formed i n the two orientations.  In the  present discussion different sources of work hardening i n the two orientations w i l l be considered i n some detail using the properties of jogs developed i n Appendix I I . In order to f a c i l i t a t e discussion we consider, the unit tetrahedron ABGD shown i n Fig.37a. In Fig.3'tothe tetrahedron i s reoriented with respect to the two orientations.  In principle, for  exact [ill] orientation, three s l i p planes JB , */*, and 8 w i l l be i n operation, resulting i n 6 s l i p systems} and for exact [lOO] orientation a total of eight systems w i l l be i n operation. In such cases, the rate of work hardening for [lOO] would be slightly larger.  This i s due to  the fact that when a l l the s l i p planes are i n operation (as i n [lOO] ) four different Lomer - Cottrell (L= C) barriers can be formed,  F i g . 36  Diameter Dependence of Yield Stress for Copper Whiskers.  F i g . 37  Oriented Unit Tetrahedron to Show the Slip Systems i n L m ] and [lOO] Whiskers.  59 whereas with only six systems ((jLlll) a maximum of three  L-C  barriers w i l l be produced, (see Appendix I I I ) , In practice, only two s l i p planes operate for both orientations - sayjB and 6 for [ill) and ~( and 8 for [lOO] • Therefore, since equal numbers of L - C barriers are formed i n the two orientations, the difference i n 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 i s zero on the  [111] C r i t i c a l Plane  Conjugate Plane [ill]  [no]  [ i l l ] Primary Plane  Cross-Plane[ill] Fig,38  Stereographlc Projection for Cubic Crystal i n Standard Orientation;(notation after Rosi and Mathewson),  {100} <110>  system, for any combination of the primary and secondary  system i n 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 s l i p planes  (FIg.38), we now consider the different dislocation interactions and the types of Jogs formed i n gliding screw dislocations. For [ill] orientation, the active s l i p systems w i l l lead to dislocation intersections of the primary cross type, J^eg.  (Ill) [lOl]/(lll)  •  In each of  60 the three cases a 60° jog w i l l be formed.  Intersections of the type  (111) [101] / ( l i l ) [llO] and (111) [oil] /(111) [lOll w i n give rise to equal numbers of 60° i n t e r s t i t i a l and vacancy jogs.  (Ill) Q)ll] /  ( l i l ) [lio] w i l l produce 60° i n t e r s t i t i a l jogs only. In the [lOO] orientation the intersection i s of the primary conjugate type? i e . , (Ill) [lOl]/(111) [lOl] and (111) [ l i o ] / ( l l l ) [lio] . o In both cases a 90 i n t e r s t i t i a l jog i s formed which i s sessile especially at low temperatures and w i l l give rise to the formation of dipoles (see Appendix I I ) . [lOl]  Intersections of the type (111) [ l l o ] / ( l l l )  and (111) [lOlj /(111) [lio] w i l l give rise to the formation of  .o 60 i n t e r s t i t i a l 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 i s listed in Table III. This table i s meaningful only at the beginning of deformation and the numbers w i l l change with the density and distribuo 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 i n [lOO] than i n [ i l l ] orientation.  TABLE III NUMBER AND TYPE OF JOGS Orientation  System  Intersection  No. of Jogs. 60°  AB(6 )  [ill]  AC(J3,C5)  BD(f)  ADK) [100] BC.(<5")  AC(<f)  [jJotes  90  G - G  1  G - F  2  1  G - G and/or G - F  6  2  G - G  1  -  G - F  2  1  G - G  1  1  G - F  3  G - G  1  1  G - F  3  -  G - G  1  1  G - F  3  -  G - G  1  1  G - F  3  -  0  (i) G - G refers to intersection between two gliding screw  dislocations, where only i n t e r s t i t i a l type Jogs are formed. G - F refers to intersection between a glide and a forest dislocation where equal number of vacancy and i n t e r s t i t i a l type jogs are produced. For notation see F i g . (ii) cluded here.  Intersections resulting i n kink formation are ex-  62 (iii)  A dislocation of a given BV, eg. AD may be both an  active glide dislocation i n  and a non active forest dislocation i n S  The properties of 60 "and 90 jogs are discussed at some length i n Appendix II. o The predominance of 90 jogs i s at least i n part responsible for a lower work hardening rate i n [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 i n fact been observed by Fourie 25 et a l .  i n deformed bulk single crystals of copper. Formation of  such clusters i s probably responsible for the cellular structure of s l i p lines observed i n F i g . 32b. (2)  Diameter Dependences  If the equilibrium density of dislocations i n these whiskers 6 8,2 is less than 10 - 10 /cm then for whiskers less than about  10JJ-  i n diameter, the following inverse diameter dependence may be derived. Consider a whisker of circular cross - section as i n F i g . 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  63  which moves i t from the point S. Then we can write, ^ e f f *A  F i g . 39  - -f£  +  ID  Ejjl  Gliding of a Dislocation PQ Across a Whisker Leaving a Surface Step of Area <§ .  where the term on the l e f t hand side i s the work done, ' 1 ' i s the BV,  "C  s  i s the surface energy,  length.  The effective stress may be written as, eff "  where ^  E i s the dislocation line energy/unit D  a  a  (2)  int.  i s the applied stress, and ^ "  i s  t o  i n t  t h o  inagQ  force acting on the dislocation (see Appendix IV for estimation). From eiqn (1) we have,  r  8  = £ £ . bA  +  _ V _ bA  +  r  ( 3 ) i n t  The largest contribution i n 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  r  -T2Rb = -^-z h 7TR 2  a  *  D  3  — 4 Ti l  ^(kg/mm ) 2  n  £  x  2200 ergs/cm t 2  a  int  300 K ' P  > using a value of r »  r  o  = 17 • d (/^) -  /SO  int  (4)  7TbR  2 E  C  - J —  R •  +  b  — ^  For copper "f  E_2R -2—2JTfi 2  423__ + d(yw)  26  65  and i n  _g x 10 cms (r — b ) we get, °  °  ? int  + 1 1 kg/mm  _(5)  2  The above analysis i s applicable to larger whiskers provided that surface nucleation of dislocations i s assumed to prevail. The presence of internal defects such as impurities, dislocations etc., w i l l give rise to another internal stress "C^. The effective stress can then be written as  Tiff =  " ^int "^ i  ^  ec  i n (5)  w i l l 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 i n good  agreement with the experimentally determined values, (see Fig, 3 6 ) , If the material i s very surface active, the theoretically derived constants may not coincide with the experimentally determined ones • 12 16 Measurements also exist for Fe, Co and Ni whiskers up to ^15JUL i n 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, a  2 (kg/mm ) s 8Q0 + -d(jLt)  2 18 (kg/mm )  The experimental and the theoretical expressions are listed i n Table IV. In Table IT the expression for  was obtained by taking  the arithmetic mean of the yields stress values at different diameters and plotting as a function of diameter and measurements were made on whiskers <^20jLb . a b i l i t y of the ^ "  a v  Accordingly observe the limited applic-  expressions for Fe and Go, because at large  diameters the average stress ( f ,) values become negative which &v  i s contrary to expectation. In order to compare the experimental and the calculated expressions, Brenner's values for Fe are replotted i n F i g , Ifi.  Both  the straight lines are as shown. The agreement i s good within the observed scatter.  TABLE  66b  IV  COMPARISON OF YIELD Metal  Cu  (T" )kg/mm Calculated  Applied Stress Experimental  (a)  = 308  (b) V = £L<L av d  Fe  r av  Co  Ni  STRESS DATA  av r  «  +  2  16.8  Source  Bokshtein  36  Brenner  1630 d(jLL)  50  Brenner  1200  1£  +  -  d(jU)  = 620_  +  23  T  - 800 dljl)  -f 18  Bokshtein  3. 3.1  TEMPERATURE AND  STRAIN  RATE DEPENDENCE  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 V  3  which i s strongly  temperature dependent and the component Tg which Is independent of temperature. The variation  i s as shown i n Fig.41. He attributed  this behaviour to different rate controlling mechanisms. For example, i n metals with low stacking fault energies, jogs may be created i n edge dislocations more easily than point defects at jogs i n screw dislocations. Hence the f i r s t process w i l l predominate at extremely low temperatures. The strong temperature dependence of the flow stress up to a c r i t i c a l temperature T  c  i s due to the increasing ease with which point defects are  oreated at jogs i n screw dislocations.  Due to the increased mobility of  Fig.41 Temperature Dependence of the Flow Stress i n 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 Stokes law.  using copper, resulted i n what is known as the Cottrell They observed that i f the same specimen i s deformed altern-  ately at two different temperatures, then the fractional change In flow stress accompanying a change i n temperature reaches a constant value after some amount of i n i t i a l deformation.  This constancy of the frac-  tional flow stress change which i s 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 respectively, Cottrell and Stokes suggested that their observations were consistent with the idea that during deformation, the dislocation pattern remained unchanged, the only change being i n scale.  Anticipating a  significant difference i n pattern for whiskers compared with bulk crysta l s , the present Cottrell - Stokes experiments were undertaken. 30  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 i n 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,  70  w i l l be discussed 3.2  Experimental Procedure? Single crystal copper whiskers of [lOO], [lio] and [ i l l ]  orientations were used  (Fig. 4 2 ) . 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  Ni -- 0.0004$  Nb  Co •- 0.004$  Si •- 0.04$  Ag  Fe -- 0.03$  Ti  •- 0.001$  Mg •- 0.0003$  V  •- 0.001$  - 0.001$  Fig. 42  Others  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 i n the flow stress as a result of deformation at two d i f f e r ent temperatures was measured for several specimens at the following series of temperatures. TABLE  V  TEMPERATURE BATHS 0 Temp K 77  Bath Liquid Nitrogen  136  Controlled level of Liquid N  200 + 2  Solid Co  273 ± 2  Water - ice  373 + 1  Boiling water  423 + 1  Silicone o i l  2  2  " Acetone  The temperature was always changed from higher to lower temp29 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 u n t i l 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 tempero 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 a i r .  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 i n the flow stress was measured by changing the cross-head speed from 0.002 /min. to 0.02 /min. n  n  In order not to lose the sensitivity i n 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 i s plotted as a func?7 7~ tion of temperature i n F i g . 43. Wherever necessary, the ratio 3L ^77 i n the curve i s obtained by using the transformation, r  T C  77  T \ T  .(1)  \  i ;  (Note that 77°K is used as the reference temperature).  Fig.43  Temperature Dependence of the Flow Stress Ratdo for Coarse Whiskers and Bulk Single Crystals of Copper.  74  Shear modulus correction was applied using Overton and Gafney's  31  meas-  urements, which were extrapolated for higher temperatures. The curve consists of a temperature independent region followed by a region i n which the flow stress ratio is strongly temperature depondent up to a c r i t i c a l temperature T ( ^ 300 K ) .  For temperatures greater  than T , the ratio i s again independent of temperature. Temperature c  dependence sets i n again at s t i l l higher temperatures.  Superimposed i n F i g . 43 are the results of Cottrell 32 Makin  for bulk single crystals of copper.  29  and  Tho results i n 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 i n that the temperature dependent region is considerably inr creased, and the temperature independent region is contracted. However the form of the curve is i n f u l l agreement with that obtained by these workers. 33 Hirsch made measurements on the variation of the flow stress -273 ratio T(1) up to the about 1273 using poly copper. equation and data of K Cottrell and crystalline Adams, his ratio of Using J— are transformed to 29 and Adams  T  T  and plotted i n conjunction with Cottrell  273  77 data for single crystals i n 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 i n detail the strain rate dependence  0.2  4  200 Fig. 44  So  600  800 Temperature  1000 K  1200  1400 _^  Temperature Dependence of the Flow Stress Ratio for Copper Single and Poly Crystals. The High Temperature Region i s from Poly Crystals.  76 of copper single crystals of various orientations and of polycrystalline copper specimens.  For single crystals Hirsch observed the dependence  shown i n F i g . 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 i s such that regions a and b are obscured. However' region c is distinctly present.  Typical,, curves are presented i n F i g . 4.6. Even though second-  ary systems operate i n most of these specimens towards the end of deformation, no change i n slope i n the A f v s "V curves is observed. For the [ i l l ] and f i o o j orientation, the possible form of the curve i s shown i n F i g . 47. Regions 'a' and »d are not distinctly 1  shown by most whiskers.  A few selected curves are presented i n F i g s . 48 and49.  For a l l orientations the linear region c f  i n the form A t  :  A +B f ,  1  could be expressed  The intercept A is negative i n some cases •  —1  2  1  A  ;  r  6  1  8 TZ (shear) (kg/mnr)  1  10  F i g . 4.6a A T V S V(shear) Curves for [lio] Whiskers.  T  12  1  H  A S6A - 200 K © S79 - 77°K 0.15  0.10 H 2 (kg/mm ) 0.05-1  6  F i g . 46b  Afvs  8 (shear) (kg/mm )  10  T (shear) Curves for [lio] Whiskers  79  AT  r  F i g . 47  (3)  Form of  A T V s T Curve for [ill] and [lOO] whiskers.  Activation energies, Activation volume, Frequency factors.etc.  The strain rate i s given by o where,  €.  kT  .(2)  q  = NAb~? , with  N  = no. of activation sites per unit volume  A  s the area swept by a dislocation segment i n one activated jump.  b  m BV of the dislocation  ^  = vibration frequency of the dislocation.  AQ i s the thermal component of the activation energy expressed as, Q  a H -v  .(3)  where H i s the total activation energy and v i s the activation volume, 7T  a  i s the applied shear stress.  2.Ch  20  40  60  80  100  r (kg/mm ) F i g . 48  ^ T v s T Curves for [ill)  Whiskers.  120  ljo  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 V I , Previous measurements 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 i n activation energy i s observed o o for these whiskers. activation energy H  For specimens tested at 295 K and 273 K, the total A  AQ + V^(SEE ;Appendix VI for description) i n -  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 i n the  total activation energy, for the most part they deformed with an almost 0  constant j; activation energy.  0  For specimens tested at 295 K and 273 K o o  the latter value is about 5.2 e v ( ± 0,3 ev).  At 200 K and 77 K this  i s about 3,8 ev(+ 0,2 ev) and 1,8 ev(+ 0,2 ev) respectively.  (5)  83  Activation Volumes For a l l specimens the activation volume decreased slowly -20  n with increasing stress. This value i s ~^10 cm for [2101 whiskers, -20 -21 3 i -21 -22 3 , 10 -10 cm for [_10Ql whiskers and 10 -10 cm for [1111 r  whiskers*  3  r  r  n  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 i t s strong temperature dependence. The average value for specimens of a l l orientations i s —  35 at 273°K, ^20  at 200°K  -of the results i s presented i n Table  and 8 at 77°K. A summary  VI,  TABLE  VI  ACTIVATION VOLUME, ACTIVATION ENERGY AND FREQUENCY FACTORS Orientation  [110]  Total Deformation  50%  Activatio^ Volume,cm v - bdl -20 -20 10 -0.5x10  -20 10 -  [lOO]  -21 10  25%  [HI]  -21 10  - 10  -22  Temp. Total Activation °K Energy ev H BAQ + v T  In Co  £  295 273  0.5 to 5.5(±0.3)  35  200 77 295 273  0.5 to 3.8(+0.2) 0.5 to 2.5(±0.1)  20 8  5.2(f 3)  35  200  3.8(±0.2)  20  77  2.2(+0.2)  8  (7)  84  Activation Distance?  For the purpose of discussion, the quantity 'd' which Is the activation distance expressed i n , v  = b  d  1  (4)  w i l l be considered, where v i s the activation volume, b i s Burgers vector and 1 i s 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 i n a random pattern, with density N per unit AREA.JJ- and b have the usual meaning, oC i s a constant depending on the character of the dislocation considered ( <X °(  s  1  for screw dislocations).  1 for edge, and 277" ( l - V ) If the mean spacing between  2 IT  dislocations i s 1, then  N > _1_, l  and from (5)  2  T  s  CXUb  (6)  1 Identifying this  '1' with the activation length, we can write d  =  b  v 1  •d' values for some [llOJ  = CXLLb vT 2  _  (7)  specimens are plotted i n F i g , 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  F i g . 50a  Activation Distance d as a Function of Strain.  87 screw dislocations (see F i g . 53 i n Discussion), then t h i s constant value of 'd' permits us to accurately estimate the stacking f a u l t energy.  The average value of 'd' at these two temperatures i s  15 A°(+ 1A°).  In the equilibrium equation,  ~f  «  (8)  Ma  2  24 JT d where " f i s the stacking f a u l t energy, *d  1  'a 'is the l a t t i c e parameter and  i s now taken to be the equilibrium separation of the two p a r t i a l s .  A small stress dependence o f 'd' e x i s t s , which has to be ignored however f o r lack of precise c a l c u l a t i o n s . i s then found to be 48 ergs/cm  2  The value of stacking f a u l t  with p -  s  2  4.4 x 10" dynes/cm ,  -8 _8 o a s 3.54 x 10 cms, and d « 15 x 10 cms f o r copper at 0 C temperature.  energy  and room  Current values f o r copper are given i n Table. V I I . 36  However, i f Hirschs • mechanism of c o n s t r i c t i o n and jog formation i s assumed to be the r a t e c o n t r o l l i n g process (see F i g . 52) then the a c t i v a t i o n distance i s equal to 2d, where as before, d i s the equilibrium separation of the p a r t i a l s . Accordingly the stacking f a u l t energy w i l l be twice the value obtained e a r l i e r assuming Seeger's 35 mechanism. As before the shear stress i s given by equation (6)•  Stacking f a u l t energies at lower temperatures  are also  calculated based on a precise d i s l o c a t i o n mechanism determined  from  the observed a c t i v a t i o n 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 f o r a specimen close to the [ l i b ] orientation aro extracted from Hirsch's data i n F i g . 51.  At higher temperatures  they  tend t o reach a constant value towards the t h i r d stage of the stress s t r a i n curve.  However at lower temperatures  !  d ' is  not e s s e n t i a l l y a  constant i n t h i s region. TABLE STACKING o Temp. K  FAULT  VII  ENERGIES  FOR  COPPER Source  Stacking Fault energy (ergs/cm )  Comments  295  40  Taken twice the twin boundary energy  295  4D  Using anisotropic e l - Shoeck a s t i c i t y for interaction between p a r t i a l d i s l o c - and ations Seeger  295  40  80  2  70 ± 10  295  163  295  67 + 17  295  70  48 + 3 62 + 3 8^+3  37  35 Hirsch II  4-5 <57 <80  295  295 - 273 200 77  assumed  Fullman  From d i s l o c a t i o n nodes  -  39 "et.al Berner^  From stacking f a u l t probabilities  Vassamallet & ^ Massalski  From l i n e s h i f t measurements during preferred orientation  Small- /2 man,et.al"  See text  Present work  90 Discussion - Mechanisms? In order to establish the rate controlling mechanism a  3. 4  knowledge of the total activation energy H A Q + vT"is essential. a  This Is usually done by comparing the experimentally determined H values to the theoretically calculated energies for specific dislocation mechanisms.  For a single rate controlling process the activation  energy w i l l be a constant,,  Accordingly, a spectrum of activation  energies w i l l be observed for several rate controlling processes.  For  such systems a precise determination of the exact rate controlling mechanisms becomes extremely d i f f i c u l t . The following specific dislocation mechanisms have been suggesteds (1)  Formation of Jogs? This mechanism may have several variations (and has wide  applicability i n FCC l a t t i c e s ) . (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 42 3  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 i g . 52). The constriction energy for copper is about 0.7 to 0.8ev for screw dislocations.  This results i n an activation energy of about 3.5ev  for the intersection of dissociated screw dislocations i n copper.  F i g . 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 i n screw dislocations; Screw dislocations with jogs are greatly inhibited from  conservative motion. However, they may move either by producing inters t i t i a l s or vacancies. dislocation is given by,  The shear stress necessary to move the jogged ^" s  b J  ,  where '1 is the average 1  1  jog spacing, and jB is 0.2 for vacany jogs and equal to one for inters t i t i a l jogs. 3 about U b ( ^  The activation energy for producing i n t e r s t i t i a l s is 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. be rate controlling.  At low temperatures this mechanism might  At higher temperatures the vacancy thus formed  w i l l diffuse away from the parent dislocation.  Thus the rate controlling  process at higher temperatures w i l l require a total activation energy equal to that of self diffusion (2.3ev for copper).  This w i l l result  33 43  i n a temperature dependence of the flow stress.* (c)  Conservative motion of .loess A variant of the above mechanism i s the conservative motion  of the jog along the screw dislocation.  This requires long range move-  ment of the jogs along the dislocation l i n e .  The activation energy 45  required is of the order of several tenths of ev. (2)  Cross ° Slip of Screw Dislocations; This mechanism w i l l be discussed with special reference to  FCC structures.  In FCC lattices with sufficiently low stacking fault  energies a dislocation i s dissociated producing a stacking fault ribbon of width ' d ' . During deformation, the two partial dislocations bounding the stacking fault ribbon w i l l recombine over a length '1 of the 1  fault producing the original dislocation.  The total dislocation so  produced w i l l cross-slip from the primary slip' plane, and redissociate producing another set of partial dislocations bounding a ribbon of stacking fault which w i l l now spread i n the cross - s l i p system.  The  sequence of events i s shown i n F i g . 53.In this the activation dis-  93  F i g . 53  Sequence Shoving Cross - S l i p .  94tanc© w i l 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 ( 1 ) is the energy decrease Q  when a total screw dislocation of length '1* dissociates into two partials. Seeger  gives the curve reproduced i n F i g . 54 i n which the  total activation energy for cross s l i p appears as a function of stress. 72 X -3 Using a typical value of C 3 » 200 kg/cm J 0,5 x 10 for whiskers JJ-  s  of [lio] orientation, we find from F i g . 5 4- that the total activation energy for cross s l i p is about 5.2ev. The curve i n F i g . 54. was obtained / 2 by Seeger, using a stacking fault energy of 40 ergs/cm  for copper.  Impurity - dislocation interaction'; This mechanism is of considerable importance i n the temper4-7 ature dependence of the flow stress of BCG metals. In FCC metals (3)  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 i s U ^^jj-b £ . When the size of tho impurity atom is about 1.05 times that of the parent l a t t i c e , copper.  £ = 0.05 and U » 0.9ev for 4S  Screw dislocations also interact with impurity atoms.  In  any case the experimentally observed activation energies i n the present work are higher and this mechanism w i 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 i n their potential valleys. Thermal fluctuations w i l l give rise to the formation of kinks which depending on their direction of motion w i l l either annihilate or give rise to a small displacement from the equilibrium 49  position as shown i n F i g . 55. The activation energy for the process is the energy required for the formation of two kinks P and Q i n a  50 dislocation such as AB. This mechanism has also been used ing the behaviour of BGG metals.  i n explain-  In FCC metals this has been used i n  51 explaining internal f r i c t i o n measurements  from which the energy of  kink formation is determined to be about 0.04ev i n copper.  As before,  i t i s evident that this cannot be a rate controlling mechanism i n the temperature dependence of the flow stress of copper.  F i g . 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 DISLOCATION MECHANISMS AND Mechanism  VIII ACTIVATION ENERGIES Activation Enerev • §v  Activation Distance d A°  1 (a) Jog formation by dissociated screw-screw inter section  3,5  30  Jog formation by dissociated edge-edge intersection  9.5  30  (b) Non conservative motion of jogs i n screws (c) Conservative motion of jogs i n screws 2  Cross-slip of screw dislocations  3  Impurity - dislocation interaction  U  Peierls Mechanism  1,0 for vacancies 4.5 for i n t e r s t i t ials 0.5 to 1.5 5.2  5.0 for single defects  15  1.0 - 2.0  2.5 - 5.0  <0.1  -  3.  5 Discussion of Experimental Resultss Work hardening characteristics i n stage I, II and III may  only be discussed for whiskers of [liol orientation.  Activation  length ' 1 ' and the activation distance 'd' w i l 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 i n the  total activation energy i n stage I and II a. precise dislocation mechanism cannot be decided. Haitd/ening results from the gradual decrease of the activation length ' 1 ' of the dislocation line and an almost linear increase i n the activation distance *d'. In tho H o s t a g e 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 i n Table VIII intersection of the glide and the forest dislocations (both extended) and the crosss l i p of extended screw dislocations are the two mechanisms worth considering.  We observe that the experimental activation energy and  activation distance are not i n agreement with the intersection meek an ism. For the cross - s l i p process i f we assume that the total activation energy is high enough as envisaged by Seeger (Fig. 54) then i t i s not necessary to demand that an extensive pile up should be present for this mechanism to be operative.  The theoretical  99  calculation i n F i g . 54. \ 2  is based on a stacking fault energy of  :  v  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. of extensive pile ups  Cross - s l i p mechanisms without the presence 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 speci f i c dislocation mechanism as the rate controlling process. Using the 736 average v v- values given by Hirsch the stacking fault energies are determined for Au, N i , Al and Ag as described earlier (Table IX). 42 The most recent estimates by Smallmann,et al are also indicated. TABLE STACKING FAULT Metal  Au  T  8? K 300  Ni  vT  IX  ENERGIES FOR Au, N i , A l , Ag and Cu. d(A°)  ev  (ergs/cm^) Intersection  Cross-slip  42 Smallmann  3.85  17.3  78.8  34.4.  5.6  11.6  211.0  110.5  225.0  420.0  213.0  150.0  IS  Al  n  0.61  Ag  n  3.8  15.8  81.0  40.5  25  5.5  15.0  96.0  48.0  70  Cu 300 (present work)  2.76  100  0  (2)  Hardening at 200 K.s The observed activation energy is about 3.8ev.  i n *1? and 'd* i s similar to that at room temperature.  Variation 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 temperature. This suggests that the stacking fault energy has a temperature dependence i n 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. (3)  o 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 stressstrain curves.  As before no specific mechanism can be attributed to  the early stages.  There is no striking difference i n 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 processes 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 i n Table V I .  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 i n [lid] whiskers.  The rapid work hardening characteristics of these whiskers  is discussed i n the previous section. (5)  The quantity In  s  The frequency factors temperature.  In  decreased with decrease i n  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 activation energy.  The higher value is closer to the theoretical estimate  of 32 by Seeger^ derived from a random forest model. o -  These values,  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 i n detail by Conrad any further.  and w i l l not be considered  3.  6  Twinning; Since the stress levels reached at least i n 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 . the necessary information.  F i g , 57 provides  Assuming that there are no pile ups  present, F i g , 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/ cm by dissociation at 77°K. 2  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 gms/cm (6 x 10 gms/cm - resolved).  6 2 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 e r a l l y ) .  This twinning stress is about 5 to 10  times larger than the value for bulk single crystals of copper 6 5 ( ^—> 1.0 to 1.5 x 10 gms/cm ) 5  2  #  Twinning models based on Cottrell - Bilby's pole mechanism 56 are discussed i n detail by Venables •  In the present work a mechan-  ism based on the above idea of failure of a Lomer-Cottrell barrier by recombination w i 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  F a i l u r e of an L - C Barrier (a) by Recombination and (b) by D i s s o c i a t i o n (after S t r o J r * ) .  An L-C barrier is shown i n F i g . 58.  105 It consists of a wedge  shaped stacking fault bounded by three partial dislocations. respective BVs are as shown.  The  Failure by recombination w i l l result  when the partial dislocations recombine to form tho original total dislocation  ^^0-ig] J  which w i l l then glide i n the  (001)plane.  We consider a segment of the L-C dislocation which has recombined (AB i n F i g . 59) over a length *1' to form the total dislocation which then glides i n the (001) plane. ing dislocation configuration.  Fig.59b shows the result-  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  (ill)  F i g . 58  An L-C Barrier  106  107 ( i l l ) planes.  It is not necessary that extensive s l i p take place i n  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 w i l l contain a stacking fault and a segment AB of the partial dislocation w i l l have climbed down by ^-JllQland operate on the next parallel (111) plane as shown i n Fig.60a.Similarly, operation of the partials AS and BT w i l l result In twinning In the (111) plane. Twinning w i 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.  F i g . 60a  This  Formation of a Twin Source Due to the Operation of Partial Dislocations AP and BQ i n F i g . 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 i n the build up of a twin due to their spiralling around a perfect dislocation which has a BV component perpendicular to the twin plane. Suzuki 's model gives the shear stress necessary for twin propagation as  T  -  ~f 2b  + J^Sl  _  (9)  x  where ~f is the stacking fault energy, jl is the shear modulus, b^ is the BV of the partial dislocation and '1' is i t s length. We w i l l apply equation (9) to the present model by identifying '1* with AB i n F i g . 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 i n the present case.  This stress is that necessary to cause failure  of the L-C barrier as shown in F i g . 59b. Using the ideal Frank-Read expression, *U a .^b  ,  where '1 is now the BV of the total disloca1  tion AB(or DC), we find that 1 = AB a 160*31. With ~f * 80 ergs/cm o 77 k and b  1  s  b , ^ C T  w :  f that, 2.8 x 10 + 4-.0 x 10 e  i n d  6  6 s  6.8 x 10  6  2  at  , , 2 (gms/cm )  / 2 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 necessitates the presence of a pile up of strength 5. This w i l l proportionately  109 reduce the observed twinning stress, as is the case with bulk single crystals. 3.7  Dependence of A ^ o n ^ ; B  Significance of  i n A V = A + B V,  From the strain rate equation, H  we have,  -vT  - ^o V kT  ^  e  J  * l n  jL e  (10) ^  —  - J L + 3L-L  '  o  kT  and  kT  __(H)  .  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 kT a  At any temperature we observe that  H kT  Hence we can write, AT  ^ 7  ln  /  £ 2: ^o  * - f  M i n i s . H  \ <£• )  ju)  no which reduces to,  A^  for JL 2  -  £  = 10,  =  then  2.303 kT T H  (15)  B = 2.303 kT .  i  H  At room temperature, with H  > 5ev,  B = 0.008  compares very well with the actual slope of AT~vs  This value of B and i t further  justifies that the approximations involved in obtaining H are reasonable. 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 H i r s c h 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 i n the third stage of the work hardening curve.  The final scatter of points i s  due to impending f a i l u r e . 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  Ill i s s t r i c t l y obeyed, H is a constant as i t is for a single rate controlling 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  I  1.  SUMMARY  112  AND CONCLUSIONS  Twinned etch spirals are observed i n halide reduced copper sub-  strate. Frequent occurrence of these, i n regions where dense whisker growth had taken place, may i n 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 i s the result of egress of  such dislocations i s then eliminated by the addition of material at the ledges i n the block structure, leading ultimately to smooth surfaces • 3.  The presence of growth steps on facets at the t i p of a whisker  i s due to the presence of dislocations with a screw component normal to a facet. II  1.  The difference i n the work hardening characteristics of [ i l l ]  and [100] whiskers i s attributed to the difference i n the charactero i s t i c s of the jogs formed i n the two orientations.  Predominantly 90  i n t e r s t i t i a l jogs are formed i n [100] orientation and 60° i n t e r s t i t i a l jogs i n [ill] orientation. 2.  If surface nucleation i s assumed the 1 diameter dependence of d  the yield stress of whiskers arises from three prominent contributions, (a) The largest contribution i s from the strain energy of the s l i p dislocation which reaches a maximum when the dislocation passes through the centre of cross-section of the whisker.  113 (b)  The s l i p step gives rise to a surface energy term, and  (c)  an image force term when the dislocation i s very close to the surface.  Ill  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 i n copper at liquid nitrogen temperature,  cross-slip and intersection processes are indistinguishable. At higher temperatures cross-slip i n 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 i s deduced for copper from such stacking fault energy measurements. Ut,  A twinning mechanism for whiskers of [ i l l ] orientation i s 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 i n whiskers tested i n the present work can be made from the i n i t i a l yield stress.  8 2 This i s about 10 /cm  a value which i s comparable or slightly larger  than the density i n bulk single crystals.  Dislocation densities of about  6 2 1A 10 /cm have been reported for iron whiskers. In fine whiskers surface nucleation of s l i p is much more l i k e l y 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 s l i p 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 i s no substantial change i n 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 ' density).  where P is the dislocation  Considering a dislocation ring of raditts R> equation (1) may  be written as £.  2 = bTT R N  —-2  from which B  with € — 0 . 3 and f = 1 0 / c m 8  2  ,  -  .-3  f H Vb7T N  R-20yUand with  » 1 0 / cm , 6  2  R^oOO^U.  This means that for whiskers i n this diameter range, most of the dislocations escape out of tho crystal i n 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 i g , 60b  Dislocation Dipoles Along the Axis of a whisker.  expected i n whiskers compared to the random networks i n hulk single crystals.  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 i s schematically  shown i n F i g . 60b. Formation of such dislocation dipoles w i 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 w i l l result compared to a quasi uniform distribution observed i n single cryst66 als (see for example Basinski ) .  This is schematically shown i n Fig.oOc.  i  116  The cross-hatched region corresponds to a region of high dislocation density separated by almost dislocation free regions.  This might  be the reason why the fracture strength reached by whiskers is much greater than i n 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, T + A T ^  »  0  T  Q  + Af  • ° t c . for operation compared to  T  Q  + ^2  '  + / dZ" \ de  f  o r  bulk  single crystals, where T of work hardening.  °  i s the yield stress and  / d T | is the rate  U€  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 i n 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 i n 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. be ^ b .  At this temperature the activation distance must  This w i l l also justify the use of a particular flow stress equation.  APPENDICES  APPENDIX  I  STRESS STRAIN CURVES  TABLE  120  X  YIELD Specimen No.  Orientation  Dia(jJ-)  Test Temperature °K  e  Yield(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 Specimen No.  Orientation  Dia(j^)  X -  121  Cont'd..  Test Tgmperattire K  ^Yield(kg/mm ; 2  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  S33  [100]  82  N  0.04.  S34-  [no]  201  n  0.27  S35  [100]  137  n  0.08  S36  83  it  0.19  S37  [m] [no]  316  n  0.8  S38  , [m]  72  n  0.56  S39  [111]  101  200  4..00  SA1  [nil  107  ti  SA2  &00O  268  S43  [in]  su  [loo]  .  95 197 86  S45  1.00  11.00  a  o.H  it  5.5  n  0.39  ii  0.38  n  0.3  SA6  [ni]  105  S48  [100]  179  n  O.H  S49  [no]  195  n  0.20  S51  [no]  202  295  0.35  122 TABLE  Specimen No.  X -  Cont'd..  Teat Tgmperatxrre K  Yield(kg/mm )  Orientation  Dia(^)  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)  r  2  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 i n section three.  O'l  e  O'Z  0-3  128  APPENDIX  137  II  ON JOGS  We consider only the jogs formed i n screw dislocations. As may be seen from the unit tetrahedron i n F i g . 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 s l i p plane, o (b)  BC (ex.) / AC (JE>)  -  a 60 jog is formed.  For conservative  motion the s l i p plane is (c)  BC (cx) / CD (p)  -  6.  a 60° kink, which w i l l vanish as the dislocation glides i n its s l i p plane.  The number of point defects (interstitials or vacancies) produced by the non conservative motion of a jog i s given by the 58 scalar t r i p l e product N « h <C x b ) where V is the atomic 1  1  2  t  Fig,62 Illustrating Vacancy Formation by Moving Jogs.  3 volume ( V  s  b )  respectively, and  and b2 are the BV and the direction of the Jog is the distance moved by the jog i n the direc-  tion of motion of the parent dislocation (see F i g . 62  ),  It i s  also possible to determine the type^of defect for a given orientation of the dislocations as may be easily seen from F i g . 62. In F i g . 62 is shown a dislocation with a jog.  If the jog  moved along with the dislocation l i n e , the material above w i l l be  139  interstitial  F i g . 63  Formation of Interstitials by Moving Jogs.  displaced with respect to the one below i n 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)  w i l l be produced. A 90  jog w i l l emit a vacancy or an i n t e r s t i t i a l for every  I I C . | , where C, • a <112>JGJ = b ('a • i s the lattice paraV f - 1 6 W meter and b • a (llti} )• In the absence of formation of such defects, 2 1  1  edge dislocation dipoles w i l l be formed i n a direction parallel to (Fig.6*Va).  These dipoles w i l l act as obstacles for dislocations  i n the primary system and w i l l not be effective obstacles for dislocations i n 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 i n the screw bg are equal and opposite and can annihilate and the dislocation is free to move.  Also, since  140  F i g . 64 Sequence Showing the Formation of Dislocation Dipoles and Loops.  there is no short range interaction between edge and screw dislocations, this intersection w i l l not be a major source of hardening. For the 60° jog excluding the possibility of conservative motion on a different s l i p plane, we find that for non conservative motion a vacancy or an i n t e r s t i t i a l w i l l be emitted for every 2 \ C 1 34  It has been suggested that i n t e r s t i t i a l type jogs can move conservatively along the dislocation line by stress aided aotivao t i o n . Of the two kinds of jogs formed we see that a 60 jog can i n 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  s l i p plane. 59 Gottrell  has shown that only i n t e r s t i t i a l 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 s l i p planes, both vacancy and i n t e r s t i t i a l 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 i n t e r s t i t i a l jogs are produced.  APPENDIX  III  L-C  REACTIONS  Fig.65 Unit Tetrahedron Showing Different BVs. AD  =  £ [lio]  DB  [lOl]  AB  -  £ [Oil]  BC  [no]  [lOl)  DC  i[oii]  AC  i.  2.  3.  £[101]  s  £[oii]  i[no]  AC  CD  AD  (111)  (111)  (001)  iliio]  £[oif)  i[ioi] AC  BC  AB  (111)  (111)  (100)  i[oii]  i[iio]  AC  BA  CB  (111)  (111)  (001)  i[ioi]  £ [oil] CD  (Til) 5.  £ Loii] CD  £LiTo]  £[ioi]  CB  BD  (111)  (010)  £ [101]  £ [101]  DB  BE  (111)  (010)  £ [iio]  £ [ioi]  (111) 6.  £ [oil]  DA  CA  (in)  (111)  (010)  £ [no]  £ [lOl]  £ [oil]  cp  7.  8.  BC  BD  CD  (111)  (111)  (100)  £ [iio]  £ [oil]  £ [ioi]  BG  AB  GA  (111)  (111)  (010)  £ [101]  £[oii]  £ [no]  BD  (HI)  AB  AD  (111)  (001)  143  10.  i [101]  +  BD  12.  i[oii]  DA  BA  (111)  (100)  i  (111)  11.  i [no] —* i [on]  +  *[iio] —  iLloil  AD  AD  (111)  (111)  (010)  £ [lOl] —  £ [pll]  •£- [lio]  +  DB  AD  CA  CD  (111)  (111)  (100)  APPENDIX IV  H5  IMAGE FORGE Consider a dislocation close to the surface i n a circular cross-section (Fig, 66), When r^^> b,  where b i s the BV of the d i s -  location, the image force exerted by the surface is large.  The image  dislocation i s 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 i s at position such as A, close to the surface, i s very small. No precise calculations exist for image forces exerted by non planar surfaces.  However i n the above problem, If the image  dislocation i s treated as a circular loop encircling the glide dislocation, then the interaction energy for such a system may  be ob-  60  tained from Kroupa's  calculations to be, E  . jU.b r  (1)  2  r  where *r' refers to 2 ^ i n F i g . 66. The attractive force per unit length on the dislocation line segment of length '1' i s given by r  T  h  S  n  From F i g . 66  r  int  s  U^ T\  * !E  we have,  E  S  ~  W;  (2)  (3)  14-6  Image dislocation  Fig* 66  Image Dislocation at a Curved Surface.  8  r  l 2 r  r^(d » r ) t  where 'd' i s the diameter of the whisker. When r ^ ^ b reduces te  from eqn.CJ)  1  «  "<T. int  eqn.(4)  2 \liA  .,, (5)  JiL.  .(6)  2y/pT  Notice that this alone gives rise to a  1  dependence. This depend-  ence i s aegleeted i n p. 64 the strain energy term.  compared to the large 1 dependence from d For d ^ 10 JJ- ,  11 kg/mm for copper. This i s the maximum value of  f° * r  When the dislocation i s at position A i n Fig. 66 force by the surface S., i s neglected.  ^^ +,  n e  chosen diameter.  the attractive  decreases monotoni-  n  cally and reaches zero when the dislocation passes through the centre 0, and i s negative when i t approaches S . 2  An approximate  Fig. 67  Image Force Exerted on a Dislocation 1 by a Whisker Surface.  plot i s shown i n Fig.67.  APPENDIX V LOAD MEASUREMENT When the load scale In the Instron i s at position 1, the chart i s 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 i n load increment measurements which i s very small for whiskers, and i n the i n i t i a l stages of deformation i t i s within the sensitivity limit (loads up to 0.2 gms could be measured when the load scale i s at 1)• In order to keep the load scale at position 1, the fine balance control is used. Whenever the pen i s 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 i n Fig.68.  Between changes of strain rate  i t i s also necessary that the  load i s either reduced to zero whenever possible or lowered by about 50 - 100 gms. This i s done to minimize irreversible effects.  F i g . 68  Load Measurement Using Fine Balance Control  150  APPENDIX VI ACTIVATION. ENERGY  The strain rate i s expressed by the usual Arrehnius equation,  I  = £ - AQ . o® kT  e  where, 1  (1)  =  NAb V , with  N  -  Number of activation sites per unit volume,  A  »  i s the area swept by a dislocation segment in one activated jump,  b  s  BV of the dislocation,  o  "V a  vibration frequency of the dislocation,  AQ i s the thermal component of the activation energy expressed as, Q  s  H -vr  _  a  __(2)  where H i s the total activation energy, and v i s the activation volume, T'  i s the applied shear stress.  These quantities are schem-  atically represented i n Fig,69. The required quantities are AQ, H and v. From equation (1) we have, In € = In C  - Ag kT  0  = In C  °  "  S kf  + • T.a kT  Differentiating this at constant temperature, we get,  5  r  a /  M T  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 ~ )  ' a /rp  ^  »  -  _„(5)  ^T  Assuming that € i s independent of stress, from (A) and (5) we get c  'dln€ d  r  a/  or,  TT  T  v  s  kT  (6)  ^dln€\  Equation ( 6 ) wheti-written* i n terms of experimental quantities becomes, v  =  kT  In £, - ln£ a  l -  a  2  .  2  Thus from strain rate change experiments activation volume may be determined.  (7)  152 Differentiating equation (3) at constant strain rate and as before assuming that €  kT  I dr  I  a  k  Is independent of temperature, we get  0  l  fdx \ + v  T  \ aJ.  » ^& fdv \  kT  dr  kT ^ r A  V  ^a/^T \  S  kT^rJi  o  .(8) With the earlier assumptions s t i l l holding, equation (8) reduces to, H kT  ^a / d r \  V  kT  _y_ kT  .(9)  k T'  Transforming as before to experimental quantities, we write, H  -  v r  -  a  Tv / £ 2 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 i n general coincide at a given temperature equation (10) is modified as follows i n order to suit the experimental results.  For this we write, r T 2 1 TT " ^ a  H v  .  a  r a  1  T  n  T  2  (11)  - T T  At any temperature  T^ (X^ > T ) the average value of 2  a  T.  2  i s used  l Since Cottrell-Stokes law is a  from a Cottre -Stokes type of experiment. obeyed, ~ - ~ v  appears as a constant independent of stress.  equation (2),  A Q v Z~ a  Using — S _ v V  c a  Then from  a  r  H  8  l  vv  7Z a  ~ 1  (12)  1  from equation (11) AQ can be determined.  Hence both H  l  and A Q can be obtained as a function of stress. Knowing A Q ,  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 i n Appendix V I I , [lOO] whiskers  tensile stress values are used.  For [ill] and  APPENDIX VII  154  DOUBLE SLIP CONDITIONS  The stereograms i n Fig* 70  show the possible choice of glide  systems and the resultant lattice rotations* For exact [lio] orientation, there are i n principle two potential s l i p planes with four different s l i p directions, two on each of the s l i p planes*  The Schmid factor  for this system i s 0.408. However, for whiskers of [llOj  orientation,  there seems to be only one active s l i p system. Let us assume that the system i s (111) with [lOl) as the s l i p direction.  The specimen axis  w i l l rotate along the great circle passing through the poles [lOll and [Oil]  (see F i g . 71 ) until i t reaches the [ooi] - [llll 22  when double s l i p sets i n  Fig. 70  • The normal strain  1  symmetry plane,  £ i s related to the 1  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.  axial rotation as follows 1 + € where, A -  0  and  61 %  1 5 6  sin Tip sin A  s  (1)  X are the angles between the specimen axis and the  s l i p direction respectively before and after a strain € . For |llQ) orientation, the required rotation for double s l i p to start is 30°. With A  0  - 60°  and  X  a  30°  1 + £  -  1 732. Hence double s l i p 0  w i l l set i n for a l l whiskers of [ l i d ] orientation at about 73% strain. Experimentally we observe that double s l i p has occurred i n specimens which have failed before this strain i s reached.  Resolved Shear Stress Calculations s axis  slip direction  Fig. 72 Coordinates for Resolved Shear Stress Calculations.  The c r i t i c a l resolved shear stress is given hy, T s ^ cr where 0  andX  O  Q  5  ^cos9 cos X o o  are as shown i n Fig»72 and refer to the i n i t i a l values.  T i s 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 - sinioT" / (1- ) 2  The Schmid factor (cC)  /1  -  /  sin ^ TJT-T2 2  _  ^ X  is given as a function of strain i n 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 ESTIMATION  158  VIII OF  ERRORS  The main source of error i s i n A d  measurements. High  sensitivity i n load measurements i s however achieved by methods described i n Appendix V.  Further, when the f a l l load scale i n  the Instron i s calibrated to 100 corded for 0.1  load measurements« accuracy of 0.2  0.2  gm,  gms. pen deflections are also re-  gm. etc., and used as a scale for accurate  This way loads can be easily measured with an  gms or at best 0.1  gm.  The former value w i l l be  used i n estimation of errors. At the start of tho deformation, the Afvalues for [lio] and [loo] specimens are comparable to the above l i m i t . specimens the A ^ a l n e s are usually high.  For [ i l l ]  In what follows, errors  w i l l be estimated i n 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 i s given by, v  (i)  *  AT* where oi. is a constant, whose magnitude i s unimportant. logarithms and differentiating eqn (1),  <fv  a  .  S  Taking  we get,  _J2)  (AT)  Using the above mentioned values for A C ~ n d a  8 AT~, we find that,  159  —  •:..°- ..-.  S  (3)  2  v  5  "  W  /  which gives an error of IS i n v. (2)  Activation distance. Activation distance is expressed as d  -  where ft is a constant« eqn  (4.)  (4)  / v t Taking logarithms and differentiating  we get,  Sd ,. s d  jTy. + v  ^  (5)  Since T^>> cT<?*, we find after using eqn ( 3 ) , that the error i n  d  i s also U$o (3)  Total activation energy (H)8 From Appendix VI we can express H as, H  wherep is a constant.  a  (6)  / v t  Following the same procedure as for ' d we 1  find that the error i n H i s I&. We observe that the error i n a l l these quantities decreases with deformation, because ATincreases with stress. most {ll6]  specimens the error i n  Actually i n  v, d and H is smaller than the  above calculated value, because A T i s usually about 7 gms or larger i n the 3rd stage.  The error i n [loo] i s comparable to that i n Clip] specimens or smaller.  The error i n [ill)  specimens is much smaller  than either of these two orientations and unimportant for the purpose of discussion.  161  APPENDIX IX FORCES BETWEEN TWO DISLOCATIONS OF ARBITRARY ORIENTATION AND BV  1.  Introduction; Tho nature of interaction between two dislocations i s of consid-  erable importance i n the theory of work hardening of metals.  Interaction be-  tween parallel dislocations with parallel BVs has been established from f i r s t 62  principles early i n the theory of dislocations,  63  Saada  has calculated the  stability of a glide dislocation as i t approaches a forest dislocation and 60  concluded that the change i n form i s very small.  Later, Kroupa  had  investigated i n detail the interaction between a glide dislocation and a dislocation loop.  In this a similar analysis w i l l be given to calculate the  interaction between two straight dislocations under the assumption that they are rigid and i n f i n i t e , 2,  Theory The force F on a dislocation due to an arbitrary stress f i e l d 64-  i s given by, F  s k x (<Hb),  1  where k i s a unit vector tangent to the dislocation line,CT i s the stress dyadic and b i s the BV of the dislocation. Choosing dislocation I parallel to the x^ axis i n the x.^ system, and Its BV i n the x^ ~ xj plane making angle cx' with the x^ axis (Fig.73), i f the stress tensor i s referred to the same set of axes, then i n the dyadic notation,  162  CT  • ii 0 ^  ki c r  3 1  + IJ (7J  -r  2  ik c r "  *  + kj C7^ + kk C7J 2  3  ,  i , j , k, are the unit vectors parallel to the x-^, Xgj x^ axes respectively. The BV b i s given by b and  •  ibsincX -  kbcos o<  0 ™ . b * bsino^iCr^  + j CJ^ + k (7J ) 1  bcoscx(i (T£  +  + k (jy^  +j  Then, 3 kx bsincx(i CT^ + J  F  bsin  (i c r  1 3  s  b s i n a ( j (T  a  - i (bsinc* CJ"  1:L  - i 0^)  31  j (bsincx OZ^  +3 ^  + k a^)  *  + k a^) + bcosoc(J CT^ - i C £ )  + bcoscx (7^)  3  4  •+ 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 i t s BV i n the x^ - x j plane making an angleyB with the xj axis, the components of Its stress f i e l d are,  163 2' 0~  l l l l  12  %  cr,  .2  (3:  v  X  22  2  i «  ( X  2  • , 2  1  <*1  mm^L-mm  33  (x«  2  1  cr  12  -  2  2  2  E D  + x' )  ,2  .2  K -*2>  i  x  y  - — — - 2 — ^ r  (x  t  1  + x  2  2  r  x.  cr_  13 +  cr  s  D,  *1 7T  23 X  where  D  2  a  l  )  T2-  +  J^ao sini? 2  TT (1^)  D  s  2  8  7^ 2 b  -  c  o  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! i and,  2 0 . ij  F i g . 73  s  ^ijCxj - x J 4  ck ex cr mi nj mn  Coordinate Systems: x. is for Dislocation I and i l i s for Dislocation I I .  The superscript refers to the second dislocation, and °^±^ is the direction cosine between the x'^ and x^ axes.  From the cartesian force  components the radial and other angular components could at once be obtained by using the transformation,  165 F :  9  F,  cos9 sin0  sin9 sin(2C  cosJZf  sin© sin0  cos© sinJ2f  0  cos© cos0  sin© cos^ -sin^f  F,  J with tho following definitions?  r  dQ '  F-,  »  1 (  )  where E i s the interaction energy* What i s of most interest i s the radial and the angular components in the direction of motion of the second dislocation,,  This i s 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 i n 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 dislocation I„ This can  be easily done by expressing (x , x „ _ , x__) i n the 1rt  * •.....  10  v  1  <fJ  JV  transformed equation 2 i n terms of r , © and 0 such that ' r ' i s always parallel to the direction of motion of dislocation II• When r i s parallel to x , we have, x^ s - r , Xg 0, x^ s 0. Then the transformations for s  the stress tensor i n A reduce to, 0~-  12  0~. 11  <*12 ° k  11  +  21  ^22 ° l l ) ( T ?12 ,  12  166 2  • „  rr-  23  (C(  "  2  O/ 2  3  3  \rr^  CX  l O (  3  2  2  23  3  2 2  0".. 13  + <**i ° W  s 2  1  3  3  3  1  2  3  i 23  6 cent >d  The force components are, E F  P  2  i  = " i ^ai  2  *  b  E l  b  2 ^11  for pure edge-edge, and  F  l  =  - ?*-23  v  b  W 1 °13 3  2  2  8  for pure screw - screw interaction,, There i s 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 i n FCC metals. (1)  Edge - Edge interactions Dislocation I:  BV £ [ l l o ]  Dislocation*Us  BV \  [loll  x  2  / / [no]  x^ / / Liol]  x  2  / / [nil  x' / /  x  3  / / [ll2]  x^ / / frill]  The following table of direction cosines is obtained;  2  [ill]  167 i  x  1  x  X  3  1 ~ 2~/3  2  t 2  1  1 - 275  1  vT  3  i  Then,  2  I  l  x  X  3  \l2  cos© sin0  s  sin© sin0  "  cos 0  *  6  3  1  1 I  I  ' 2*V3  ,  and. cr  ^12  0~  12  E 2r  E ? 5 L  2.1 fa o ~ 2 V3 12  11  2 ^ 12 2  The cartesian forqe components are:  F  (2)  |  s  -P  E E 2 2r  F,  D  2'  E E b, D 2  using 5 .  Screw - Screw interaction: Dislocation I ,  BV £ [lio]  Dislocation II  BV £ [pit]  [21l]  xi  / /  [ll2]  xi  / /  x  / /  [ill]  X2  / / [ill]  / / [no]  x'  / / [Oil]  2  x,  3  cos9  3±n0  1 , 6°  sin© sin^ > 2J2? , 3  cos0  273  Then,  23  23  cr.  3 2  13  6r  2 3 ^ CT-23  2\/3J 23 U  r-^ £ 3 —i  and, b  S S l 2 6r D  F  •  2  Vf  bfp|  from which, F_  S S l 2. 12r  b  P  S S F  9  s  " l 2 3\/2r b  D  ,  F  b  S S , 2 D  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 i n Fig.74., for an FCC crystal.  Pure edge dislocations l i e parallel to <^112>  <^110J> as their direction of motion. to  the edges correspond to BVs with  Pure screw dislocations l i e parallel  <^110£> , with <^112^> for their direction of motion.  Every  (ill)  plane has three orientations of each kind giving rise to twelve different orientations.  Keeping one of the dislocations fixed, we can examine the  F i g . 1L, Unit Tetrahedron Showing BVs and Dislocation Directions• interaction due to eleven other orientations, the twelfth one being parallel to the f i r s t one. calculation.  TablesXLTandXIIE arise as a result of this  Since we know that parallel dislocations of the same kind  with parallel BVs repel from one another, the nature of the force components for other orientations could be established. certain other limitations.  However, there are  These expressions do not hold after the dis-  locations have intersected or associated to give rise to some other dislocation. The following case i s of some special interest.  Two dislocations  (edge - edge) on parallel s l i p planes, BVs not parallel (Fig.75)*  TABLE XII FORCE COMPONENTS FOR Dislocation  Dislocation  I  II  F  [I10][llj[ll2] [oiijfiiilfcii]  _<X 2  [ll0][ll2[ll2]  3  [ioi][ni[i2il  EDGE - EDGE INTERACTION  F  2  [io:D[in][i2l]  jpiij  l  170  F  2  0  c* A  0  0  o<  0  0  0  21/2" cx 3/3'  -/?«  2 c* 5  [211]  [iio]|in][ii2]  _ 2c* 3  0  [l0j[lll][l2l]  cX  - VL ex 43  [on][iifl[2iil  cX 2  [no][ii3[ii2]  -  0  ~6 ex  [110] [11J [112]  °<  0  E b D 2 r  cX 2^  0  - 2 (Joe 3l/3 .  ,  0 CX  2 ^  2v^ -if-  _ °< 12  v T CX 12/J  _ °< 12  x/2 cv  1  1  E 3  ^ 2 2TT(i-V) b  12\/IT 0  3  0  E D 2  12  cX  0  E  0  - i f -  <X 3  0*  u  °< 12  0  cX  [lof)[lTi[l2ll  . J i cX 12  12  3  [01^[ui|[21l]  F„  r  2*wir CX  12^T 0  TABLE  171  XIII  FORCE COMPONENTS FOR SCREW » SCREW Dislocation I  Dislocation II  F  [ll2][lll][ll0] [2l3[lll| fell]  l  F  F  2  2  F  r  0  J&  INTERACTION  8  0  I  [121] [iii] [101] 2 [112] [ill] [110] -23 ^5  0  [121]  0  Jl A 7  12  3J2  6  3  12  ~ 2s/2"  -\>  -1>  •  - f >  -  [ill]  [211I  [101] "  [nil  [011]  [112] [ill] [lio]  P 2  0  [211] [111] [011) 0 [112] [ifi] [110] 3 [211] [111} [on] 6 [121] [111] [101]  «  b  J  0 ~tP  fi>  s i D r  2  12 0 ^  & 0  0  3T J  J  0  0 0 £ 3 9  0 S 2  u  f >  3  *=»  6  [112] [111] [110] P  2£  P  [12O [111] [101]  0  Z  u  2 TT  J  12&?  J  0 0  12J3  0  Fig.75 Two Edge Dislocations on Parallel Slip Planes Dislocation  x  I, x->  Dislocation II,  x^ //[lOHj ,  x^ / / j l U ] ,  1 2.,  0  2  1 2  From equation 4> E D„  11= 2  12  ( 4  and  X  •  (X - X ) 2  3 0  //[ll2].  x^ // [ l 2 l ] .  0  "2  3  ,2x2 Xo )  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 , we have x^ s 0 » X ^ Q * 2  0  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 ( *3 +  .2 2 * ^ )  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 i n 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  174  7.  BIBLIOGRAPHY  1.  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