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Plasticity of [Beta]'AuZn single crystals Schulson, Erland Maxwell 1967

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The University 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  -  ERLAND MAXWELL SCHULSON B.A.Sc. (Hons.), The University of B r i t i s h  Columbia  THURSDAY, JANUARY 4, 1968 AT 3:30 P.M. IN ROOM 201, METALLURGY BUILDING  COMMITTEE IN CHARGE Chairman: C A . Brockley L.C. Brown E. Teghtsoonian  B.N.Moyls D.L. Williams N.R. Risebrough D. Tromans  External Examiner: Dr. R.E. Smallman Department of Physical Metallurgy and Science of Materials The University of Birmingham Birmingham, England  Research Supervisor:  Dr. E. Teghtsoonian  THE  PLASTICITY OF Q> 'AuZn SINGLE CRYSTALS  ABSTRACT  Single c r y s t a l s of the CsCl type i n t e r m e t a l l i c compound 'AuZn were prepared and tested i n tension over a wide range of temperatures, s t r a i n rates and orientations for three compositions, Au-rich (51.0 at .% Au), stoichiometric and Zn-rich (51.0 a t . % Zn). ;  S l i p surfaces are generally non-crystallographic planes in the zone of the s l i p d i r e c t i o n [001], and are temperature, s t r a i n rate and orientation s e n s i t i v e . A model based on thermally activated s e s s i l e - g l i s s i l e transformations of screw dislocations has been proposed to explain non-crystallographic s l i p . Multi-stage work-hardening i s observed over the temperature range 0.2*^ T/T ^ 0 . 3 5 . In stage I the work-hardening rate i s low \^/*. /1000 to *A./5000) but r i s e s sharply during stage II ( Q - Q ^ _/A/500) . Stage III i s characterized by a r a p i d l y decreasing hardening rate coincident with the onset of profuse large-scale c r o s s - s l i p . Surface s l i p line studies revealed that the end of easy glide i s coincident with the onset of localized s l i p on non-crystallographic planes i n the [lOd] zone. /  Thin f o i l electron microscopy was c a r r i e d out on c r i t i c a l l y chosen crystallographic sections from annealed and deformed c r y s t a l s . At the beginning of stage I clusters of edge d i s l o c a t i o n dipoles were revealed, forming walls perpendicular to the glide plane. The d i s l o c a t i o n density of the walls increases during easy g l i d e . During testing at intermediate temperatures ('v.3 to .4 T ) serrated y i e l d i n g was detected i n non-stoichiometric c r y s t a l s and was attributed to  dislocation-solute  atom interactions.  Under special testing conditions (77°K or near<001> orientations) s l i p occurs i n <111> d i r e c t i o n s . The associated work-hardening rates are very high and d u c t i l i t y i s low. Thermal a c t i v a t i o n studies were made to determine the d i s l o c a t i o n mechanism responsible for the temperature s e n s i t i v i t y of y i e l d i n stoichiometric crystals below <"v 220 K. Activation volume measurements are consistent with both the Peierls-Nabarro and c r o s s - s l i p mechanisms below ^ 1 5 0 K.  GRADUATE STUDIES  F i e l d of Study:  Physical Metallurgy  M e t a l l u r g i c a l Thermodynamics  C. S. Samis  Structure of Metals  E. Teghtsoonian  Advanced Physical Metallurgy  E. Teghtsoonian L.C. Brown  Diffusion  L.C. Brown  Electron Microscopy  D. Tromans  Related Studies: Quantum Theory of Solids Theory of P l a s t i c i t y  R. Haering H. Ramsey  THE PLASTICITY OF ^ ' AuZn ;  SINGLE CRYSTALS  by  . ERLAND .-MAXWELL'. SCHULSON B.A.Sc, ..University of British Columbia, 1964  A 'THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE.REQUIREMENTS' FOR'THE DEGREE OF DOCTOR OF PHILOSOPHY in- the Department of METALLURGY  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA ' December, I967  In p r e s e n t i n g  for  thesis  an a d v a n c e d d e g r e e  that  the  Study.  thesis  Library  for  agree  scholarly  or  publication  without  shall  I further  Department  or  this  at  of  make i t  that  freely  for  for  permission.  Metallurgy  1968  Columbia  It  is  financial  of  British  avai1ab1e  permission  thesis  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a A p r i l 1,  of  p u r p o s e s may be g r a n t e d  this  my w r i t t e n  fulfilment  the U n i v e r s i t y  by hils r e p r e s e n t a t i v e s .  Department of  Date  in p a r t i a l  for  the  Columbia,  I  reference  and  extensive  by  the  requirements  copying  this  Head o f my  understood  gain  of  agree  shall  that  not  be  copying  allowed  Supervisor:  i  Dr. E. Teghtsoonian ABSTRACT  Single crystals of the CsCl type i n t e r m e t a l l i c compound  Q*AuZn  were prepared and tested i n tension over a wide range of temperatures, s t r a i n rates and orientations f o r three compositions,,Au-rich  ( 5 1 . 0 at.$> Au),  stoichiometric and Zn-rich ( 5 1 > 0 at.$ Zn).  S l i p surfaces.are generally non-crystallographic planes i n the zone of the s l i p d i r e c t i o n orientation s e n s i t i v e .  [ 0 0 1 ] ,  .and are temperature, s t r a i n rate and  A model based on thermally activated s e s s i l e - g l i s s i l e  transformations of screw dislocations has been proposed to explain noncrystallographic  slip.  Multi-stage work-hardening i s observed over the temperature range 0 . 2  ^T/T  / f / 5 0 0 0 )  'Z.  O..35.  In stage I the work-hardening rate i s low'(  but r i s e s sharply during stage TI (On  characterized by a r a p i d l y decreasing hardening onset of profuse large-scale c r o s s - s l i p .  ~  /'/5OO).  A/lOOO to  Stage III i s  rate coincident with the  Surface s l i p l i n e studies revealed  that the end of easy glide i s coincident with the onset of l o c a l i z e d on non-crystallographic planes i n the [ 1 0 0 ]  Thin f o i l electron microscopy was  slip  zone.  carried out on c r i t i c a l l y chosen  crystallographic sections from annealed and deformed c r y s t a l s .  At the  beginning of stage I clusters of edge d i s l o c a t i o n dipoles were revealed, forming walls perpendicular to the glide plane.  The d i s l o c a t i o n density  of the walls increases during easy g l i d e .  During t e s t i n g at intermediate temperatures ( ~ . 3 to .h T ) m serrated y i e l d i n g was  detected i n non-stoichiometric crystals and  attributed to d i s l o c a t i o n - s o l u t e atom i n t e r a c t i o n s .  was  i i  •Under s p e c i a l testing conditions s l i p occurs in <111>  directions.  very high and d u c t i l i t y i s  The  (77°K or near K. OOl"/* orientations)  associated  work-hardening rates  are  low.  Thermal a c t i v a t i o n studies were made to determine the d i s l o c a t i o n mechanism responsible f o r the temperature s e n s i t i v i t y of y i e l d i n stoichiometric  c r y s t a l s below <^220°K.  Activation volume measurements are  consistent with both the Peierls-Nabarro and ">-150 K. o  cross-slip.mechanisms below  ACKNOWLEDGEMENT The author g r a t e f u l l y acknowledges the advice and encouragement given by his d i r e c t o r , Dr. E. Teghtsoonian.  Helpful  discussions with various members of the f a c u l t y and fellow graduate students are also acknowledged.  F i n a n c i a l assistance was received  from the International Nickel Company of Canada Limited i n the form of a post-graduate fellowship.  iv  TABLE OF CONTENTS Page THE PLASTICITY OF  @' AuZn SINGLE CRYSTALS  1.  INTRODUCTION AND OBJECTIVES  2.  DEFORMATION CHARACTERISTICS-OF (3»' AuZn SINGLE CRYSTALS...  7  2.1.  EXPERIMENTAL PROCEDURE  7  2.2.  2.3.  2.k.  1  .  ;  2.1.1.  A l l o y Preparation.and C r y s t a l Growth  2.1.2.  Tensile Specimen Preparation  2..1..3.  Testing Procedure  .7 •  8 9  GENERAL DESCRIPTION OF THE SHEAR STRESS-SHEAR STRAIN CURVES  11  2.2.. 1.  Experiments .....  .11  2.2.2.  D e f i n i t i o n of Work-Hardening Parameters  11  2.2.3.  Temperature and S t r a i n Rate Dependence  .12  2.2.4.  E f f e c t of Deviations from Stoichiometry  .18  2.2.5.  Orientation Dependence  21  2k  SERRATED FLOW  2k  2.3.1.  Occurrence  2.3.2.  Origin  2.3.3-  Dislocation-Solute Atom Interactions  29  2.3.4.  Segregating Species  32  . .  .28  .36  DEFORMATION MODES 2.4.1.  Introduction  2.4.2.  Procedure  2.4.3.  Definitions  43  2.4.4.  S l i p Direction  .44  2.4.5.  Primary-Slip Plane 2.4.5.1.  .36 .  Temperature Dependence  ......  43  47 .47  TABLE OF CONTENTS  (continued) Page  2.4.6..  2.5.  2.4.5.2.  S t r a i n Rate Dependence  56  2.4.5-3•  Orientation Dependence  57  2.4.5.4.  Composition Dependence  64 64  Discussion 2.4.6.1.  < 0 0 1 > Zonal S l i p  65  2.4.6.2.  { hkl] < 1 1 1 > S l i p  72 78  WORK-HARDENING'BEHAVIOUR 2.5.1.  2.5.2.  .Flow Parameters  .78  2.5.1.1.  Y i e l d Stress  .78  2.5.1.2.  The Work-Hardening Rate in-Stage I , Q  2.5.1.3.  ..The End of Stage I  86 88  2.5.1.4.  The Work-Hardening Rate i n Stage I I , On  92  2.5.1.5.  Stage I I I  .97  2.5.1.6.  Maximum Shear Stress and D u c t i l i t y  ..  S l i p Line Variation During Deformation  ..98 .100  2.5.2.1.  Procedure  101  2.5.2.2.  Observations  .101  2.5.2.3.  Deformation Bands  .110  2 . 5 . 2 . 3 . 1 . Characteristics  110  2 . 5 . 2 . 3 . 2 . Crystallographic Nature...  .111  2 . 5 . 2 . 3 . 3 . Mechanism of Formation....  112  Microcracks  113  2.5.2.4. 2.5.3.  ±  Transmission Electron Microscopy of.Thin Films.  115  2.5..3.1.  Introduction  115  2.5.3.2.  Procedure  116  2.5.3.3.  Observations  117  vi  TABLE OF CONTENTS  (continued) .' Page 2 . 5 . 3 . 3 . 1 .  2 . 5 . 3 . 3 . 2 .  As-Grown Structure  .  Variation i n Dislocation Structure During Deformation, at 2 9 3 ° K .  . 2 . 5 . 3 . . 3 . 2 . I .  2 . 5 . . 3 - 3 . 2 . 2 .  2 . 5 . 3 . 3 . 2 . 3 .  1 1 9  (hko) Section  1 1 9  Section  1 2 4  ( 1 1 0 )  Perpendicular (hko) 1 2 6  Section 2 . 5 . 3 . 4 .  2 . 5 . 4 .  2.6.  1 1 7  -Discussion  Discussion  1 2 6  . ....  . .  1 3 1  2 . 5 . 4 . 1 .  Y i e l d Stress V a r i a t i o n with Orientation  1 3 1  2 . 5 . 4 . 2 .  Work-Hardening  1 3 3  l  THERMALLY .ACTIVATED YIELD 2 . 6 . 1 .  Introduction  2 . 6 . 2 .  A c t i v a t i o n Volume  2 . 6 . 3 .  A c t i v a t i o n Energy  2 . 6 . 4 .  Discussion  4  0 140 l  4  4  I5O 1 5 3  2 . 6 . 4 . 1 .  Impurity Obstacles  1 5 3  2 . 6 . 4 . 2 .  Peierls-Nabarro  1 5 3  2 . 6 . 4 . 3 .  Cross-Slip  3 .  SUMMARY AND CONCLUSIONS  4.  SUGGESTIONS FOR FUTURE WORK  5.  APPENDICES  (PN) Mechanism  . 1 5 8  1 6 2  . .  1.  C r y s t a l Homogeneity  2.  Evaluation of Machining Damage  3.  Equations f o r Resolved Shear Stress and Resolved Shear-Strain  I 6 5  1 6 6 . 1 7 0  1 7 3  vii  TABLE OF CONTENTS  (continued) Page  6.  k.  Taylor Rotation. Axes  175  5.  Shear Modulus as a Function of S l i p System  .178  BIBLIOGRAPHY  l8h  viii • LIST OF FIGURES Page 1.  Common types of s u p e r l a t t i c e s i n which c r y s t a l structure does not change upon the formation of long-range order  2.  E q u i l i b r i u m diagram f o r the Au-Zn system.  3.  Diagram of a t e n s i l e specimen  h. 5.  Showing specimen g r i p p i n g arrangement Schematic resolved shear stress-shear s t r a i n curve  6.  .1 6 10 10 .12  Resolved shear stress-shear s t r a i n curves of Au-rich (3* • AuZn c r y s t a l s as a f u n c t i o n of temperature between 77°K and 488°K ( f t = 2 . 5 x 1 0 " / s e c ) . 3  7.  Resolved shear stress-shear s t r a i n curves o f ' s t o i c h i o m e t r i c AuZn c r y s t a l s as a f u n c t i o n of temperature between 77°K and 443°K ( Tf = 2 . 5 x 10 / s e c )  15  Resolved shear stress-shear s t r a i n curves of Z n - r i c h AuZn c r y s t a l s as a f u n c t i o n of temperature between 7 7 ° and 473°K ( ft = 2 . 5 x 1 0 / s e c )  16  3  8.  K  _3  9.  .Resolved shear stress-shear s t r a i n curves of Au-rich AuZn c r y s t a l s as a function of s t r a i n rate from 2.5 x 10" /sec t o 2 . 5 x 10" /sec (T = 293°K)....  17  Resolved shear stress-shear s t r a i n curves of c r y s t a l s as a f u n c t i o n composition. 10.1. a t 293°K and'373°K. 10.2. a t 77°K and ~220°K  19 20  4  10.  11.  .14  2  ^J' AuZn  Resolved shear stress-shear s t r a i n curves of Au-rich (5,' AuZn c r y s t a l s as a f u n c t i o n of o r i e n t a t i o n . (T - 295°K; = 2 . 5 x 10" /sec)...-  22  12.  Schematic representation of a serrated flow curve  25  13.  Photograph of segments of a serrated load-elongation curve during stage I and stage I I deformation  27  14.  Showing the v a r i a t i o n i n c r i t i c a l s t r a i n with s t r a i n rate  15.  Showing the e f f e c t of temperature on c r i t i c a l s t r a i n  .34  16.  A (001) stereographic p r o j e c t i o n showing the parametersc h a r a c t e r i z i n g the specimen o r i e n t a t i o n and the s l i p plane r e l a t i v e t o (110)  .45  "Stereographic representation of specimen a x i s r e o r i e n t a t i o n during p l a s t i c deformation as a function of temperature.....  .46  3  17.  critical  ix  LIST OF FIGURES  (continued) Page  18.  C r y s t a l orientations used i n s l i p plane-temperature study....  .48  19.  Photomicrographs of t y p i c a l s l i p traces on.orthogonal faces A and B where A /^(2"01) -and B ~ (010)  .49  20.  21.  22.  -Replicas of surface s l i p traces on orthogonal faces A and B  48  -A sketch of s l i p trace development on orthogonal surfaces •when one surface i s p a r a l l e l to the Burgers vector of the mobile d i s l o c a t i o n s  . 51  •Back-reflection Laue X-ray pattern from surface shown i n Figure I9.A.I ..  53  23.  Showing the v a r i a t i o n i n the s l i p plane parameter with temperature  55  24.  Showing specimen orientations used i n s l i p plane analysis....  .57  25.  Showing the v a r i a t i o n of s l i p plane parameter with orientation  59  26. 27.  Showing duplex s l i p i n c r y s t a l oriented - along 101 boundary  -  111  60  -A (001) stereographic projection showing the most highly stressed system of the form {lio} C 0 0 l ) . as a function of orientation  28.  Multiple s l i p observed near [ 0 0 l ]  29.  -Schematic i l l u s t r a t i o n of continual c r o s s - s l i p on . orthogonal ^110}  62 63  orientations  68  planes  30.  Schematic representation, of d i s s o c i a t i o n reactions  31.  -A sketch of the s e s s i l e to g l i s s i l e transformation  32.  'Schematic i l l u s t r a t i o n of the continual c r o s s - s l i p cycle defining the s l i p plane parameter. Showing the v a r i a t i o n of y i e l d stress with temperature f o r  33-  68 sequence..  .70  70  ^>AuZn single c r y s t a l s . . .  83  34.  Showing the resolved y i e l d stress dependence on orientation..  85  35-  "Showing e f f e c t of specimen geometry on i n h i b i t i n g (hko) slip Showing the v a r i a t i o n i n stage I hardening rate with temperature  Au-rich,stoichiometric and Zn-rich  36.  [001] 86 87  X  LIST OF FIGURES  (continued) Page  37• 38. 39-  kO. kl. k2. 43.  Showing the v a r i a t i o n i n stage T work-hardening with orientation  rate 88  Showing the e f f e c t of temperature and composition.on the extent of easy glide  89  Showing the effects of temperature and composition on the stress at the end of easy glide  .90  Showing the v a r i a t i o n on the extent of easy glide with orientation  .92  Showing the v a r i a t i o n of stage. II work-hardening with temperature  .93  rate  Showing the orientation dependence of stage II workhardening rate .. Showing the e f f e c t of temperature  .95  on the stress at the .98  end of stage II kk.  Showing the v a r i a t i o n of maximum shear stress with temperature  45. Showing the v a r i a t i o n of t o t a l d u c t i l i t y with temperature 46, 47, V a r i a t i o n i n s l i p l i n e structure with s t r a i n at 77°K,  48, 49, l40°K, 293°K, 398°K and 473°K 50. 51. 52.  Stereographic projection of deformation band poles versus c r y s t a l orientation and test temperature  99 99 .102  ... 106 I l l  Schematic representation of dislocations i n deformation bands 1 1 2 Electron micrographs of d i s l o c a t i o n structure:  53 • 54.  i n as -grown crystals . at the beginning of stage I, (hko) section  1 1 8 1 2 0 ,  55-  at the end of stage I, (hko) section  1 2 2 ,  1 2 1  1 2 3  56. 5758.  at the beginning at the beginning . a t the beginning perpendicular  of of of to  stage I, ( 1 1 0 ) section..... .. 1 2 5 stage I, section perpendicular t o (hko).. 1 2 7 stage I, section approximately (110) . 128  59-  Showing the experimental compared with the predicted values of c r i t i c a l resolved y i e l d stress r a t i o versus s l i p plane parameter 1 3 2  60.  I l l u s t r a t i n g the athermal and thermal components of the y i e l d stress  1 4 3  xi  LIST OF FIGURES  (continued) Page  61.  62.  Schematic r e p r e s e n t a t i o n o f changes i n the flow curve accompanying s t r a i n r a t e change t e s t s A c t i v a t i o n volume a g a i n s t shear s t r a i n a t temperatures  lk6  between 77°K and 213°K 63.  IU5  -Showing the v a r i a t i o n i n e f f e c t i v e s t r e s s w i t h temperature..  6k.  Showing the v a r i a t i o n o f a c t i v a t i o n volume w i t h e f f e c t i v e  65.  s t r e s s f o r ^ AuZn and bcc metals Showing the v a r i a t i o n , o f a c t i v a t i o n e n t h a l p y w i t h e f f e c t i v e stress. .. 1  lk&  iky I52  66.  Schematic i l l u s t r a t i o n o f the P e i e r l s - N a b a r r o mechanism  154  67.  Schematic i l l u s t r a t i o n o f the s i n u s o i d a l Peierls " h i l l " profiles  156  and  quasi-parabolic  68.  Showing the v a r i a t i o n i n a c t i v a t i o n e n t h a l p y w i t h temperature  157  69.  Showing the v a r i a t i o n i n a c t i v a t i o n volume w i t h temperature..  .159  70.  Showing the f u n c t i o n a l effective stress  Al.  dependence o f a c t i v a t i o n volume on l6l  Showing c o m p o s i t i o n g r a d i e n t s i n as-grown single crystals  ^>'AuZn  A2.1, Showing the v a r i a t i o n i n degree o f a s t e r i s m w i t h r e d u c t i o n A2.2, o f diameter o f machined t e n s i l e specimens .-A2.-3. A2.4. Showing t e n s i l e s t r e n g t h o f machined  168  171  c r y s t a l v e r s u s amount  removed from the specimen diameter  172  A3.1.  .  174  Ak.l.  S k e t c h o f a s t e r i s m from F i g u r e 22  175  A4 .2.' S t e r e o g r a p h i c - p r o j e c t i o n o f a l l £L10) and (21lj p o l e s w i t h r e s p e c t t o the indexed d i f f r a c t i o n from F i g u r e 22  177  A5.I.-Showing the x^ r e f e r e n c e frame r e l a t i v e t o the x^ frame  179  A5.2.  182  xii  LIST OF TABLES No.  Page  1.  Common Super-lattices  3  2.  Showing the v a r i a t i o n i n c r i t i c a l s t r a i n with temperature and s t r a i n rate for non-stoichiometric  3. k. 5. 6. 7. 8.  (3' AuZn c r y s t a l s . . .  S l i p systems i n Cs'Cl type compounds Comparison of l i n e energies and mobilities of low energy dislocations i n CuZn, NiAl, CsBr and AuZn -Results of s l i p l i n e analysis of temperature e f f e c t on s l i p plane parameter  kO 52 .56  Results of s l i p plane analyses of orientation.effect on s l i p plane parameter  58  Correlation of s l i p d i r e c t i o n with heats of formation and electronegativity differences i n CsCl type compounds........  75  Ratio of the stress at the end of easy glide to the y i e l d stress.as a function of temperature, composition and orientation  lh.  V a r i a t i o n i n stage I I hardening rate with s t r a i n rate  15.  Comments on s l i p l i n e v a r i a t i o n during deformation  16.  37  Results of s l i p trace analyses of s t r a i n rate effect on s l i p plane parameter  9,.10, Work-hardening parameters as a function of temperature, 11,12. composition and orientation. 13.  26  -Activation parameters / i H and v * at temperatures between-77°K and 175°K for stoichiometric crystals...  79 -82  91 .9^ 107,  108, 109  0  • A l . l . Chemical analysis of as-grown crystals A2..1. -Effect of annealing temperature and time on the strength of two t e n s i l e specimens r e l a t i v e to the unannealed condition A5.I.  Non-zero suffixes  A5.2.  Shear moduli! for-various s l i p systems i n cubic structures  .151 167  170 l8l 183  1  1.  •INTRODUCTION AND OBJECTIVES Intermetallic compounds can be defined as intermediate phases  in binary or higher order metal - metal systems.  Such compounds may possess  long range order at a l l temperatures.in the s o l i d state or undergo an orderdisorder transformation at a c r i t i c a l temperature above which the compound adopts a nearly random structure.  In the l a t t e r case, short range order  p e r s i s t s i n a decreasing degree up t o the melting point.  When highly ordered,  i n t e r m e t a l l i c compounds have lower symmetry than the corresponding disordered a l l o y , leading t o extra r e f l e c t i o n s , termed superlattice r e f l e c t i o n s , i n t h e i r d i f f r a c t i o n patterns.  The four most common superlattices i n which the c r y s t a l structure does not change upon the formation of long range order are shown in Figure 1 and t h e i r common names, corresponding disordered structure and examples of each are given i n Table 1.  <  Figure 1.  *  >  The most frequently occurring super-  I  Common types of superlattices i n which c r y s t a l structure does not change upon the formation of long-range order: ( a ) B 2 , fb) L l fc).D0i9 (d),D0 . ( a f t e r S t o l o f f and D a v i e s ! ) 1  2  3  3  l a t t i c e type i s the B2 or CsCl type structure which takes the form of two interpenetrating simple cubic l a t t i c e s .  The symmetry of the B2 structure i s  2  lowered from a bcc i n which a l l sites are equivalent to a simple cubic i n which the ( 0 , 0, 0) and ( 1 / 2 , 1 / 2 , . 1 / 2 ) sites are d i f f e r e n t .  The consequence  of reduced symmetry i s that two types of regionsrnay occur within a c r y s t a l : where the A atoms and B atoms are i n t h e i r respective <=<. ( 0 , 0 , 0 ) and (1/2,  l / 2 , 1 / 2 ) sites and where the A and B atoms interchange sites such  that A occupy  sites and B occupy cX. s i t e s .  The boundary, between two  such  regions w i l l contain bonds between l i k e atoms and w i l l be a surface of higher energy.  The regions, are known as antiphase domains (APD) and the  boundary as an antiphase boundary (APB).  Because a stable domain structure . 114  *  requires at least four sublattices, Bragg  suggested that APB  s i n the  B 2 structure(having two sublattices) which form during an ordering process should disappear as a result of domain growth. -Recent transmission electron microscopy  studies on the B 2 compound NiAl  5  (ordered up to i t s melting  point) led to the conclusion that' A P D ' S do not exist i n annealed or deformed >  *  samples, substantiating Bragg s suggestion.  In s i m i l a r studies APB  s have  been detected i n the B 2 compound Q-brass quenched from above the c r i t i c a l •116*117 ordering temperature. However, experiments were not carried out to *  determine whether or not the APB  s would disappear on prolonged annealing.  The second most common superlattice type i s the L l structure, 2  related to fee i n much the same way as the B 2 superlattice i s to bcc.  The  face centered s i t e s are occupied by A atoms and the corner s i t e s by B atoms. Symmetry i s again lowered to simple cubic and antiphase domains can exist since of the four i n i t i a l l y equivalent sites ( 0 , 0 , 0 ) , ( 0 , 1 / 2 , l / 2 ) , (l/2,  0, l / 2 ) and ( l / 2 , l / 2 , 0 ) , any one may be occupied by a B atom and the  other three by A atoms. APB^s have been detected i n - L l compounds 118* u s * 1 2 0 i i Cu Au and Ni Mn . Less commonly observed superlattices are the 2  2  3  DO3 and D O 1 9 types.  3  The most complex i s the D0 .type which i s b u i l t up of 3  eight bcc unit c e l l s and may be considered as being composed of four i n t e r -  3 TABLE 1 Common Superlattices  Structure Type  L2  0  Ll  or B2  C ommon • Name  Examples  CsCl  bcc  CuZn AgZn AuZn AuCd AgCd NiTi  AgMg NiAl CuAl FeAl FeCo  Cu Au  fee  Cu Au Au Cu Ni Mn Ir Cr  Ni Fe Ni Al •Pt Fe  Fe Al Fe Si  Fe Be Cu Al  MgaCd Cd Mg  Ti Al Ni Sn  3  2  Disordered Structure  3  3 3  3  3  3  3  D0  •Fe Al  bcc  3  3  3  3  3  D0  . Mg3Cd  19  hep  3  3  3  penetrating  fee l a t t i c e s .  For the D O 1 9 structure, the ordered unit c e l l may  be compared to four unit c e l l s of the disordered  In some instances structure.  3  hep structure.  ordering also effects a change i n c r y s t a l  The most common superlattice of t h i s type i s t y p i f i e d by the  CuAu structure i n which alternate layers of Cu and Au atoms form on ( 0 0 1 ) planes of the fee disordered The  solution, d i s t o r t i n g i t into a f c t structure.  fourfold axis of symmetry i s normal t o the alternating planes of Au and  Cu atoms and the a x i a l r a t i o i s usually between 0.9 and 1.0. of the f c t ordered structure ( c l a s s i f i e d as L l  0  Other examples  type superlattice) are  CoPt, FePt and FePd.  Deformation studies on intermetallics have centered almost wholly on the behaviour of p o l y c r y s t a l l i n e material under various  conditions  k  of temperature, s t r a i n rate, grain size, defect structure created by departure from stoichiometry and degree of long range order.  In a comprehensive  122  treatment of the subject p r i o r to  1 9 5 9 *  Westbrook  reviewed a large number  of papers devoted to f a b r i c a t i o n , testing procedures and the properties of very high melting point compounds as well as a few concerned with surface s l i p l i n e structure and antiphase boundary observations. in associated d i s l o c a t i o n structure and behaviour was time.  General interest  not apparent at that  In recent years with the improvement of f a b r i c a t i o n and t e s t i n g  techniques and the advent of sophisticated transmission electron microscopy contrast theory, much progress has been made from both the experimental and t h e o r e t i c a l approaches i n understanding the p l a s t i c behaviour and fracture of i n t e r m e t a l l i c s . Also many papers have appeared i n the l i t e r a t u r e r e l a t i n g strength and degree of long-range order. 113  In t h e i r recently published review, 'Stoloff and Davies point out that the mechanical behaviour of a l l o y s that form superlattices at a c r i t i c a l temperature below the melting point can be understood mainly in terms of changes i n d i s l o c a t i o n configuration with degree of order.  The  ordered materials deform by the movement at r e l a t i v e l y low stresses of superlattice dislocations which consist generally of c l o s e l y spaced pairs of unit d i s l o c a t i o n s .  Since the dislocations are constrained to move as a  group to preserve the ordered arrangement of the l a t t i c e ,  cross s l i p i s 123*124  hindered  thereby leading to high work-hardening rates (Cu Au 3  and b r i t t l e fracture (FeCo  ).  1  FeCo  2  5  )  Decreasing the degree of long range  order brings about an increase i n the separation of superlattice p a r t i a l s which can explain the peak i n y i e l d stress manifested near the c r i t i c a l 7  ordering temperature by many superlattices (Cu Au, 3  6  _  n  • 125*127  ieuo,  75*128  CuZn  127*129*13 0  and F e A l 3  ).  Variations i n long range order, however, cannot  explain s i m i l a r intermediate-temperatures strengthening  effects i n a l l o y s  5 1  0  ordered up t o the melting point (AgMg,  131>132>>133  1  Ni Al  134  and NiAl  3  ).  The resolution of t h i s phenomenon necessitates detailed single c r y s t a l studies.  In CsCl type superlattices (including i n t e r m e t a l l i c compounds as well as ionic materials such as C s l and CsBr) a d i s t i n c t i o n may be made between compounds on the basis of s l i p d i r e c t i o n . When bonding i s of ionic character, e l e c t r o s t a t i c forces prevent l i k e ions from becoming nearest neighbors and K001^slip  occurs, but when bonding i s of metallic character  and the two kinds of "ions" that alternate along close packed rows are p r a c t i c a l l y i n d i f f e r e n t to t h e i r neighbors, <111> s l i p occurs.  The poly-  c r y s t a l l i n e mechanical consequence of s l i p d i r e c t i o n i s r e f l e c t e d i n the extreme b r i t t l e n e s s of the " i o n i c " compounds compared to the r e l a t i v e d u c t i l i t y of the "metallic" compounds.  Further discussion of s l i p modes i n  CsCl type superlattices with direct reference to the l i t e r a t u r e w i l l appear i n the appropriate sections of the t h e s i s .  Perhaps the major obstacle i n the path of more widespread i n d u s t r i a l use of intermetallies i s t h e i r extreme b r i t t l e n e s s at low tern13 R  peratures.  Origin of b r i t t l e n e s s i n AgMg  13P5  ,,NiAl  136  ' and NiGa  has been  attributed t o the segregation of i n t e r s t i t i a l impurities to grain boundaries. Grain boundary contamination, ness.  however, cannot be the only reason f o r b r i t t l e -  In NiAl f o r instance, single c r y s t a l studies  occurs on £L10} <"00lj> systems.  i110]  l  <"001>modes  29  show that s l i p  Since there are only three independent  , general p o l y c r y s t a l l i n e d u c t i l i t y i s not possible, and  hence, even i n the absence of grain boundary contamination, b r i t t l e at low temperatures.  NiAl would be  This example i l l u s t r a t e s the value of single  c r y s t a l studies i n uncovering the nature of the deformation behaviour of intermetallics. The object of the work presented  i n the thesis was to study the  p l a s t i c deformation of an intermetallic compound i n single c r y s t a l form. The B2 compound (l)  'AuZn was chosen f o r three reasons: The moderate and congruent melting temperature of 725°C  would ease the preparation of homogeneous single c r y s t a l s ; . (2)  The s o l u b i l i t y range from  ..5 to 5I.O at. % Au, Figure 2 ,  would permit an evaluation of the effects of deviations from stoichiometry on the p l a s t i c behaviour; (3)  The c r y s t a l structure i s r e l a t i v e l y simple and remains 1OT)13T  highly ordered to the melting point,  thereby simplifying subsequent  analysis of the r e s u l t s . Because the work represents one of the f i r s t deformation studies of i n t e r metallic c r y s t a l s , the thesis is' not limited to the investigation of a single phenomenon.  Instead the effects were noted of a wide range of  variables that include temperature, s t r a i n rate, orientation and deviations from stoichiometry.  J u s t i f i c a t i o n f o r the project stems from an academic .  interest i n the fundamental deformation behaviour of ordered a l l o y s .  Figure 2 .  Equilibrium diagram f o r the Au-Zn system.  7 2.  DEFORMATION CHARACTERISTICS OF  @' AuZn'SINGLE CRYSTALS  2.1  EXPERIMENTAL PROCEDURE  2.1.1  A l l o y Preparation and C r y s t a l Growth The gold and zinc used i n t h i s investigation was of 99-999$  p u r i t y and was supplied by.the Cominco Ltd., T r a i l , B.C. i n the form of gold splatter and one-half inch diameter zinc rod. Alloys weighing approximately 75 grams  were prepared by encapsulating gold and zinc of  accurately weighed amounts.under reduced pressure i n 11 mm.  diameter  c a r e f u l l y cleaned fH S0 -Cr 03 hot solution) quartz tubes then fusing at 2  800°C.  4  2  The melts were repeatedly agitated to ensure thorough mixing  then quenched i n cold water to minimize segregation on s o l i d i f i c a t i o n .  To  remove the de-zinced surface zone, castings were electrochemically polished i n 5$ KCN solution (12 v o l t s , approximately -l^ amp/cm ,.40°C). 2  The piped  end was cropped o f f each casting. The b i l l e t s were reduced to 0.108 inch wire f i r s t by hot swaging at 300°C to 0.16J inch rod then cold drawing to the f i n a l size. The wire was straightened while held i n the die by heating with a propane torch a f t e r the f i n a l draw.  Segments to be grown into single crystals  approximately 16 inches i n length, were thoroughly cleaned (degreased, abraded, and electrochemically polished i n 5$ KCN solution), charged into pre-cleaned, close f i t t i n g 0.113 inch quartz q u i l l s reduced pressure.  then sealed under  Single crystals were grown i n the standard Bridgman  manner by superheating the melt 50°C to 775°0 then lowering the charge at 7cm/hr. through a temperature gradient of 25°C/cm. subsequently removed i n HF.  The quartz sheaths were  C r y s t a l orientations were determined at three  points along t h e i r lengths from back-refleetion-Laue X-ray d i f f r a c t i o n  8 patterns.  I t was  found t h a t t h i s t e c h n i q u e y i e l d e d c r y s t a l s b e a r i n g random  a x i a l orientations,  and t h a t a common o r i e n t a t i o n c o u l d be p r e s e r v e d  adopting a standard seeding  by  technique.  D u r i n g p r e l i m i n a r y . i n v e s t i g a t i o n s of the c r y s t a l growth conditions,  i t was  found t h a t c r y s t a l s h a v i n g smooth s u r f a c e s would be  o b t a i n e d o n l y i f the w i r e were c l o s e l y charged  i n t o the s i l i c a  quills.  Loose f i t t i n g , charges y i e l d e d c r y s t a l s w i t h s e v e r e l y c a v i t a t e d s u r f a c e s .  • C r y s t a l s were a n a l y z e d f o r composition v a r i a t i o n s r e s u l t i n g solidification-,^.:.for i n t e r s t i t i a l The  a n a l y t i c a l procedure  purposes  and  accepted.  q u i t e c l o s e t o the composition i n the t h e s i s w i l l  impurities.  r e s u l t s are r e p o r t e d i n Appendix 1.  of o b t a i n i n g f a i r l y u n i f o r m  of as-grown c r y s t a l s was  2.1.2  c o n t e n t , .and f o r t r a c e element  compositions,  from  For the  o n l y the f i r s t t w o - t h i r d s  Because the a c t u a l composition  was  i n t e n d e d on a l l o y i n g , a l l compositions s t a t e d  r e f e r t o the i n i t i a l a l l o y  composition.  T e n s i l e Specimen P r e p a r a t i o n  "Dumb-bell" shaped t e n s i l e by c a r e f u l l y machining diameter was  reduced  specimens, F i g u r e 3., were prepared  single crystals in a jewellers lathe.  from  0.113  l e s s than 0.0005 i n c h deep.  The  gauge  i n c h t o 0.090 i n c h through a s e r i e s of cute  Specimens were subsequently hand-polished  in  the l a t h e w i t h w e l l - l u b r i c a t e d 0 and .3/0 emery papers.  Machining i n Appendix 2.  I t was  i n c h from the abraded  damage was  e v a l u a t e d and the r e s u l t s are r e p o r t e d  found t h a t damage was surface.  minimized  upon removing 0.005  N o n - e l l i p i t i c a l c r o s s - s e c t i o n s and  f r e e gauge s e c t i o n s were e f f e c t e d by r a p i d l y r o t a t i n g and the specimen end f o r end d u r i n g p o l i s h i n g .  taper-  repeatedly turning  Gauge diameters were a c c u r a t e  9 +  to -0.0005 inch.  Machining damage was completely eliminated by subsequently  annealing specimens i n evacuated pyrex capsules at 300°C f o r one hour. 2.1.3  Testing Procedure Experiments were performed by straining specimens i n a Floor  Model.Instron t e n s i l e machine at s t r a i n rates varying from 2.5 to  2.5  x l O ~ / s e c and temperatures ranging from 77°K to 488°K. 2  x 10~ /sec 4  Load-  elongation curves were autographically recorded during straining. t e s t i n g environments accurate to ^2°  Liquid  included nitrogen (77°K), oxygen  (90°K), petroleum ether cooled with nitrogen (133 to 293°K) and heated s i l i c o n e o i l (293  t o 488°K).  Specimen dimensions were c a r e f u l l y measured using a Gaertner t r a v e l l i n g microscope with, a 10X eyepiece.  Diameters, accurate to -O.OOO5  inch were averaged from s i x reading along the gauge section and across two perpendicular diameters.  Specimens were successfully gripped i n a s e l f - a l i g n i n g p i n chuck and threaded c o l l e t system., Figure k..  11  2.2  GENERAL DESCRIPTION OF THE SHEAR STRESS-SHEAR STRAIN CURVES  2.2.1  Experiments Specimens of three compositions (Au-rich = 5 1 . 0 at.$> Au,  Stoichiometric = 5O.O at.$ Au and Zn-rich = 4 9 . O a t A u ) oriented near the centre of the standard stereographic triangle were prepared and tested i n tension at temperatures from  77°K  to 488 K. C  Cross-head speed was con-  stant at O.O5 inch per minute corresponding to a strain-rate of  2 . 5 x 1 0  3  /sec.  Experiments were also performed to determine the effects of strain rate and orientation on the room temperature deformation behaviour of 5I.O at.$> Au•crystals.  Excluding the s t r a i n rate study, a l l . t e s t s were done i n duplicate  and found to be reproducible i n most cases to within five percent. An IBM computer was programmed to calculate resolved shearstresses and shear-strains from equations presented i n Appendix 3 . For these calculations i t was necessary to know the s l i p system operative under the experimental conditions investigated. tion modes i s presented i n section 2 . 4 that f o r specimens oriented within the  A thorough analysis of deforma-  of the thesis where i t i s shown [ 0 0 l ] - [ l 0 l ]  - [ i l l ] stereographic  t r i a n g l e , s l i p occurs on planes belonging to the [ 0 0 1 ] orientations very near the [ 0 0 1 ] operative.  zone, except f o r  corner where [ i l l ] zonal planes become  The results of the s l i p mode study have been incorporated i n the  computations of resolved shear stress and shear s t r a i n . 2.2.2  Definition, of Work Hardening  Parameters  A schematic resolved shear-stress shear-strain curve t y p i c a l of crystals oriented near the centre of the primary stereographic triangle and deformed at room temperature i s shown i n Figure 5 -  The curve w i l l be divided  1 2  according to a scheme given by M i t c h e l l et a l .  Stage 0, a region of  decreasing hardening rate i s followed by two stages of l i n e a r hardening, I and I I , then by a second region, of decreasing hardening rate, stage I I I .  stage I I I  03 W <U U -P CO  u CO  Shear S t r a i n Figure 5-  *X  0  Schematic resolved shear stress-shear s t r a i n curve,  i s defined as the y i e l d stress at the f i r s t detectable departure from  l i n e a r i t y ; " C i i s the i n i t i a l flow stress obtained by extrapolating stage I to zero s t r a i n . V), , TV  and  Stresses T„ , '"fc^/ and V),|  m  f  ^"u - "Ci  Total  The work-hardening rate i n stage I i s defined  and during Stage II as 0  n  =  TTii  2.2.3  to shear strains  j f . i s defined as the.maximum flow stress.  d u c t i l i t y - i s given as Y . as Ox =  correspond  ^.n Ti., -  r„'  Temperature and S t r a i n Rate Dependence The temperature dependence of the shear stress-shear s t r a i n  1 3  curves for the Au-rich, stoichiometric and Zn-rich crystals i s i l l u s t r a t e d in Figures 6 , , 7<  a  n  &  8 respectively.  tures from approximately  to  2 0 0  In the range of intermediate tempera-  350°K  i t can be seen that the flow curves are  very similar to those c l a s s i c a l l y observed for face-centered cubic metals ' '' 2  and those more recently reported for body-centered cubic m e t a l s . A low work-hardening,  1 , 5 , 6 , 7 , 8 , 9 , 1 <  easy glide stage I i s followed by a higher work-  hardening stage II region. strain).  3  Total d u c t i l i t y i s very large  f-300$>  shear  As the temperature i s either increased or decreased, the length  of stage I i s decreased u n t i l a parabolic type of flow curve i s obtained; .ductility i s also reduced i n both cases. been observed i n Nb,  1  Ta  9  and Fe 1  Similar temperature effects have  single c r y s t a l s .  1  The extent of stage II  and I I I decrease and increase respectively with increasing temperature above approximately  2°3°K.  At temperatures below approximately  150°K,  rapid  hardening i a terminated i n b r i t t l e fracture, while at temperatures greater than approximately  <375°K,  i n i t i a l hardening i s followed by general work-  ^softening u n t i l chisel-edge type d u c t i l e fracture occurs.  Over the tempera-  ture range i n which stage III flow occurs, the maximum flow stress decreases with increasing temperature. In the range of intermediate temperatures stage 0 tends to decrease with increasing temperature. the Au-rich c r y s t a l s , Figure 6.  This i s p a r t i c u l a r l y noticeable f o r  It i s also apparent that the slow t r a n s i t i o n  zone between stages I and T I tends to decrease with increasing temperature.  The e f f e c t of s t r a i n rate on the room temperature p l a s t i c i t y of Au-rich c r y s t a l s . i s shown i n Figure 9 . s t r a i n rates investigated 2 . 5  x 10  4  It i s apparent that over the range of  to 2 . . 5 x 1 0 ~ / s e c , .the general flow  behaviour i s rather s t r a i n rate i n s e n s i t i v e .  2  There i s , however, an effect  on stage II hardening which w i l l be discussed accordingly.in section  2 . 5 . 1 .  -f=-  Figure 6  Resolved shear stress-shear strain curves of' Au-rich $'AuZn crystals as a function of temperature between 7 7 ° K and 488°K. (Y = x 10" /sec) 2 . 5  3  Figure 7.  Resolved shear stress-shear s t r a i n curves of stoichiometric AuZn c r y s t a l s as a function of temperature between 77°K and khyYL. ( If = 2.5 x 10" /sec) 3  5 0  1 0 0  1 5 0  2 0 0  Resolved Shear S t r a i n T Figure 8 .  2 5 O  3 0 0  3 5 0  (#)  .Resolved shear stress-shear s t r a i n curves of Zn-rich (2>'.AuZn crystals as a function of temperature between 7 7 ° K and kj3°K. ("fl* = x 10" /sec) 2 . 5  3  4 0 0  25  50  100  150 Resolved  F i g u r e 9-  200 Shear S t r a i n V  250  JOO  350  ($>)  R e s o l v e d s h e a r " s t r e s s - s h e a r s t r a i n curves o f A u - r i c h C^'AuZn c r y s t a l s as a f u n c t i o n o f s t r a i n r a t e from 2.5 x 10" /sec t o 2.5 x 10" /sec. (T = 293°K) 4  2  400  1 8  2.2.4  E f f e c t of Deviations from'Stoichiometry  To i l l u s t r a t e the effect of deviations from stoichiometry on the shear stress-shear s t r a i n curves. at 77°K, 223°K, 293°K and 373°K,curves from Figures 6 , . 7 and 8 are superimposed on Figure 10,where the shortened notation' +Au,  St. and +Zn refer to the Au-rich, stoichiometric and  compositions respectively.  zinc-rich  I t can be seen that at a l l temperatures, .yield ~  and flow stresses are higher for the non-stoichiometric material with approximately equal hardening on both sides of stoichiometry, similar to the behaviour of p o l y c r y s t a l l i n e AuZn,  12  AgMg,  13  NiAl ', and C u A u . 14  8  15  3  appears that the difference i n flow stress between stoichiometric and stoichiometric material i s more pronounced during easy glide. intermediate temperature  It non-  In the  range deviations from stoichiometry increased the  length of .stage I, with the most pronounced lengthening occurring f o r the Au-rich c r y s t a l s .  Similar stage I lengthening has been observed with  is increasing Zn content i n «X-brass-and-in Ni-Co a l l o y s . too, that stage I I I begins, at lower temperatures  1 7  It-is.-observed,  for the stoichiometric crys-  tals. . It i s also evident during straining at intermediate temperatures that flow i n Zn-rich crystals occurs i n a wavy manner p r i o r to the high hardening rate, stage II region (for.instance, at 293°K and 348°K, Figure 8).Wave "frequency" and "height" were observed to increase and decrease respectively with increasing temperature ~-3k8°K.  then disappear just above  By c a r e f u l l y observing the specimen during straining at room  temperature,  i t could be seen.that each "wave" was  one dimensional thinning within the gauge section. was uniformly thinned, stage II hardening began. crystals deformed uniformly throughout,.the  coincident with l o c a l i z e d Once the gauge section Stoichiometric and Au-rich  gauge at a l l temperatures.  25  Figure -1.0.1. Resolved shear stress-shear s t r a i n curves, of composition at 293°K and 373°K.  ^'AuZn crystals as a function of  Figure 10.2.  Resolved shear stress-shear s t r a i n curves of composition at 77°K and — 220°K.  AuZn crystals as a function of  21 It appears that work-hardening rates during stage I and II are not greatly affected by deviations from stoichiometry, but at 77°K the i n i t i a l hardening  rate appears to be higher for the non-stoichiometric  crystals. Deviations from stoichiometry seem to have l i t t l e effect on t o t a l d u c t i l i t y , except at 77°K where Zn-rich material i s considerably less • 12  d u c t i l e , similar to p o l y c r y s t a l l i n e AuZn behaviour.  It i s possible that  Zn-rich material undergoes a low temperature phase transformation similar 18  to that reported i n p o l y c r y s t a l l i n e AuZn relative brittleness.  which could account for the  Because the transformation temperature decreases  with increasing Au, stoichiometric and Au-rich crystals may parent structure at 77°K.  Upon metallographic examination  c r y s t a l s , no unusual markings were observed Zn-rich c r y s t a l s .  retain the of deformed  on the surface of the b r i t t l e  The p o s s i b i l i t y of transformation induced b r i t t l e n e s s  was not further investigated during the course of t h i s work. .2.2.5  Orientation Dependence The orientation dependence of the s t r e s s - s t r a i n curves i s shown  i n Figure 11.  Within the stereographic triangle i t i s apparent that as the  p o s i t i o n of the specimen axis i s varied from the [001]  - [ i l l ] boundary  towards the centre, the flow curve changes from two stages of l i n e a r hardening to three—stage hardening at the expense of both stages I and II. .The work hardening rate  appears unchanged but On  With increasing distance from the [101]  decreases  slightly.  - [ i l l ] boundary, both stages I and  II and the t r a n s i t i o n region are shortened and t o t a l d u c t i l i t y i s decreased. This behaviour i s somewhat analogous to the orientation dependence of easy 3  glide i n fee single crystals  1  and bcc niobium  where the extent of easy  S3  2 3  glide decreases as the orientation approaches the symmetry boundaries [001][111] and [001]-[101] respectively. Whereas specimens oriented well within the stereographic triangle give r i s e to multi-stage work-hardening  curves,.those oriented  along the [ 1 0 1 ] - [ i l l ] boundary near the •[101] corner display parabolic type hardening, and near [ i l l ] , semi two-stage hardening.  Curves from these  orientations are very much l i k e the most commonly observed flow curves f o r 19 24 _  bcc single crystals  and that observed for the bcc ordered a l l o y NiAl  deformed i n compression along the [101] d i r e c t i o n .  2 5  An exception to the general multi-stage behaviour f o r orientations within the triangle has been observed very near the [001] corner (orientation 10 Figure 11) .  P l a s t i c flow.is. characterized by an extremely  high hardening rate and fracture occurs a f t e r very l i t t l e deformation, somewhat s i m i l a r to the low temperature behaviour of AuZn single c r y s t a l s .  2k  2 ..3  • SERRATED FLOW  2.3.1  Occurrence An interesting feature disclosed by t e n s i l e t e s t s , not shown  in Figures 6, 7> 8 , 9 and 11  i s the serrated nature of the p l a s t i c flow  in non-stoichiometric crystals at intermediate temperatures. .The same 12  phenomenon has been reported by Causey t i o n of p o l y c r y s t a l l i n e  (jJ'AuZn.  during room temperature deforma-  Au-rich crystals displayed serrated  load-elongation curves from 260°K to 403°K and Zn-rich crystals from 293°K to 408°K.  (These temperatures must be taken as approximate  limits; i.e.  Au-rich c r y s t a l s , f o r instance, when tested at 260°K display serrated flow curves whereas at 217°K show smooth curves; at high temperature  serrations  are observed at ir03°K but not at f33°K.) i  A t y p i c a l l y serrated resolved shear-stress shear-strain curve i s shown schematically in-Figure 12.  Serrations begin a f t e r a few percent  pre-strain i n the stage 0 region and become f u l l y developed at the onset of stage I.  The c r i t i c a l strain, V* u  jerk was  i n Figure 12, necessary f o r t h e - f i r s t c  observed to decrease with increasing temperature and decreasing  s t r a i n rate, Table 2 .  During stage I, the amplitude and frequency of the  load-drops fluctuate s l i g h t l y , but on the average remain approximately constant. • It was also observed that both the frequency and amplitude increase s l i g h t l y with increasing temperature.  Perhaps the most s t r i k i n g  observation i s that the amplitude i s decreased abruptly during the t r a n s i t i o n region.  During stage II the amplitude i s greatly reduced and the  frequency tends to zero near the onset of stage I I I . .No serrations are observed during stage I I I deformation. Well-defined serrated y i e l d i n g was observed during stage I deformation f o r A u - r i c h crystals at a l l orientations within the stereo1  Tc  .Figure 1 2 .  Shear S t r a i n  Schematic representation of a serrated flowcurve t y p i c a l f o r non-stoichiometric c r y s t a l s tested around 300°K. i s the c r i t i c a l s t r a i n before f i r s t jerk?  2 6  TABLE 2 Showing the Variation i n C r i t i c a l S t r a i n with Temperature and S t r a i n Rate f o r Nonstoichimetric ft AuZn Crystals 1  Composition 5 1 . 0  •Test Wo.  at.i Au  84  it  Temp. '. °K 2 6 0  • S t r a i n Rate  If/sec 2 . 5  x  1 0 "  0 . 1 7  -  0 . 2 3  0 . 2 0  -  0 . 2 6  0 . 0 7  -  0 . 1 3  0 . 0 7  -  0 . 1 3  "  0 . 0 2  -  0 . 0 6  "  0 . 0 2  -  0 . 0 5  "  0 . 0 1 8  "  0 . 0 1 6  0 . 0 9  -  0 . 1 1  0 . 0 7  -  0 . 1 1  0 . 0 4  -  0 . 0 6  0 . 0 4  -  0 . 0 6  0 . 0 3  -  0 . 0 4  0 . 0 3  -  0 . 0 4  3  11 8 5  11  6 0  it  6 1  2 9 3  11  11 7 5  11  3 7 3  11 7 6  ti  8 2  11  4 0 3  11 8 3  it-9.0  at.i  Au  11  8 9  9 0  11 1 0 3  11  2 9 3  2 . 5  x  1 0 "  3  11  348  "  1 0 4  11 9 1  11  3 7 3  ti 9 2  11  5 1 . 0  IT  C r i t i c a l Strain  a t . i Au.  1 0 5  4 0 8  1 1 2  2 9 3  1 1 0  11  0 . 0 1 6  2 . 5  x  1 0 ~  4  0 . 0 5  -  0 . 0 7  2 . 5  x  1 0 "  2  0 . 2 0  -  0 . 2 8  27 \  graphic triangle except near the [OOl] apex where fracture occurred a f t e r very l i t t l e deformation (section 2 . 2 ) .  For orientations along the [ l O l ] -  [ l l l ] boundary, however, minor serrations were observed a f t e r very l i t t l e prestrain (2% shear strain) but were not detected past the high hardening region, disappearing at shear strains greater than approximately ^>Of>.  This  behaviour i s i d e n t i c a l with the stage II damping out and stage III absence of serrated flow reported above. A s t r a i n rate cycling experiment was performed at room temperature on an Au-rich ( 5 1 - 0 of the Instron between  a  t $ Au) c r y s t a l by cycling the cross-head speed  0 . 0 0 2  to  0 . 2 0  inch per minute, corresponding to a  s t r a i n rate v a r i a t i o n from 1 x 1 0 / s e c to 1 x 1 0 ~ / s e c . 4  strain-rate-change flow curve was photographed  2  Part of the  and i s shown i n Figure 1 3 .  It can be seen that both the amplitude and frequency of the serrations decrease  Figure 13.  with increasing s t r a i n rate.  Photograph of segments of a serrated load-elongation curve during stage I and stage II deformation.  28  Metallographic examination of an Au-rich specimen strained 1  w e l l into stage I revealed a very even d i s t r i b u t i o n of s t r a i n markings throughout the gauge which coincided with the operative s l i p plane.  Since  the s t r a i n markings would not reappear when electro-chemically polished i n 5$ KCN and etched i n the same solution ( f o r one minute at 1.5 V p o t e n t i a l ) , they were assumed to be s l i p traces.  From these observations i t appears that three conditions must be met f o r the occurrence  of well-defined serrated flow i n ,@*AuZn single  crystals: (1)  .The composition must deviate from stoichiometry;  (2)  The resolved shear stress-shear s t r a i n curve must  display a well-developed (3)  .The test temperature must l i e within a c r i t i c a l range  centered approximately T  m  easy glide region;  about 325°K which i s equivalent to 0.33 T  m  where  i s the absolute melting point.  2.3.2  Origin  Serrated flow has been observed during p l a s t i c  deformation  of many i n t e r m e t a l l i c compounds of various superlattice types. compression t e s t i n g from. 77°K t o 350°K'Fe Be 3  During  single crystals (DO3 super-  l a t t i c e ) undergo continual mechanical, twinning, giving r i s e to serrated 72  flow curves.  Load drops during t e n s i l e experiments have been detected  in the B2 compounds N i T i  6  and Cu-saturated  (60 at $ Cu)  and were attributed to strain-induced transformations. deformation  (^-brass c r y s t a l s  -During t e n s i l e  at intermediate temperatures, serrated y i e l d i n g observed i n 13  non-stoichiometric AgMg (3-brass.  (B2) and Cu Au 3  (B2) and N i A l  1 4  (B2) and i n crystals of  ( L l ) was attributed to dislocation-solute 2  7  2 9  77  atom interactions, commonly termed the Portevin-LeChetelier e f f e c t . It i s believed that the present phenomenon can also be explained in terms of solute was  interactions with moving dislocations.  Since serrated  yielding  r e s t r i c t e d to non-stoichiometric compositions, i t appears that  the  12  solutes  interacting are either excess Au or Zn atoms.  that a s u b s t i t u t i o n a l type defect structure  Causey  has shown  exists on both sides  of  stoichiometry suggesting that the segregating species diffuses through a vacancy-type mechanism.  The  interaction mechanism w i l l be discussed i n  the next section. 2.3.3  .Dislocation-Solute  Atom Interaction  It i s generally believed the density  that p l a s t i c deformation may  increase  of mobile dislocations ^ according to the empirical r e l a t i o n -  ship:  ^ = KiY* where Y  (1)  i s the p l a s t i c shear s t r a i n and K  x  and m are constants.  During  deformation at a constant s t r a i n rate 0 , the product of the average d i s l o c a t i o n velocity.v  and mobile d i s l o c a t i o n density remains constant,  according to: - f ^ b v  ( 2 )  where b i s the Burgers vector of the mobile dislocations. i s a decreasing function of "8*.  ..Therefore v  Under suitable conditions of temperature  and s t r a i n rate, the d i s l o c a t i o n v e l o c i t y approaches a c r i t i c a l value v (at a c r i t i c a l s t r a i n ), s u f f i c i e n t l y low to enable solute atoms to c c segregate towards the mobile dislocations; the s t r a i n fields, of the segregating species and the d i s l o c a t i o n interact, atmospheres are formed and the d i s l o c a t i o n i s either slowed down or pinned.  The 79  conditions for pinning were f i r s t expressed by C o t t r e l l  critical in the  relationship  3 0  where D i s the s e l f - d i f f u s i o n c o e f f i c i e n t of the segregating species and I is the atmosphere radius.  In order to maintain the applied s t r a i n rate,  the v e l o c i t y of the unrestricted dislocations must increase, necessitating a l o c a l i z e d increase i n stress. -At a s u f f i c i e n t l y high stress l e v e l , the pinned dislocations either break-away from t h e i r atmospheres and/or fresh dislocations are generated.  At t h i s point, the f i r s t load drop occurs.  •The process of solute locking followed by d i s l o c a t i o n break-away and/or generation i s repeated during subsequent deformation, giving rise to the observed serrated flow curve.  From t h i s description of serrated y i e l d i n g , the experimentally observed effects, of temperature,  s t r a i n rate and strain can be explained.  Because of the marked effect of temperature on d i f f u s i o n c o e f f i c i e n t s , v >  4_D at low temperatures I  for a l l strains beyond y i e l d .  Because of the  low D values,atmospheres are not formed around mobile dislocations, motion i s not impeded and consequently serrated y i e l d i n g does not occur. At high temperatures  4p_ >• v I  at a l l strains and atmospheres are formed,  but because of the larger D values,move along with dislocations and do not 'impede their motion. In the intermediate temperature range where serrated flow occurs,  v^ increases proportional to the increase i n D with tem-  perature necessitating decreasing values of prestrain before f i r s t To understand  the effect of s t r a i n rate on the  jerk.  critical  s t r a i n , equation ( 3 ) i s rewritten by incorporating equation ( 2 ) and 82  equating I to 4b, to give:  y Because vacancy-type  c  = D  (4)  3  defects are created during i n i t i a l straining the 80  value of D i s believed to increase according to: -Q/kT D = a 0 Z Ce a  (5)  3 1  where a i s the l a t t i c e parameter, O the Debye frequency, Z the  coordination  number, C the concentration of vacancy-type defects created during ing, Q the a c t i v a t i o n energy f o r motion of the segregating Boltzmann constant, and T the absolute temperature. given by the empirical r e l a t i o n s h i p :  strain-  species, k the  The parameter C i s  83  n (6)  C = Kt 2  where K  2  By substituting ( 6 )  and n are constants.  into ( 5 )  a n  d  Cl)  into  7 8  ( 4 ) , i t i s seen that at constant temperature: (n + m) Yc where K  3  i s a constant.  =K  3  ^  C  ^  () 7  I t i s immediately apparent from (7) that the  c r i t i c a l s t r a i n increases with s t r a i n rate, i n l i n e with the experimental observations. In discussing s t r a i n rate effects on amplitude and frequency of serrations, one must be c a r e f u l to d i s t i n g u i s h between true s t r a i n rate effects and apparent s t r a i n rate e f f e c t s .  Apparent effects are noted  d i r e c t l y from the autographically recorded  load-elongation  are recorded  i n the t h e s i s .  curves,  and  It i s quite possible that at high s t r a i n  rates, because of poor pen response on the t e s t i n g machine, a stress drop which i s coincident with a burst of dislocations w i l l not be  detected.  Consequently, an apparent large decrease i n amplitude with s t r a i n rate does not necessarily mean that the number of dislocations released or generated per burst has been greatly reduced.  Because of the uncertainty i n r e l a t i n g  •the apparent with the. true e f f e c t , s t r a i n rate dependence on serrated flow w i l l not be discussed further. During stage I deformation, the serration amplitude remains r e l a t i v e l y constant,  suggesting that an equilibrium exists between the  mobile d i s l o c a t i o n density and the vacancy-type defect  concentration.  32  During stage I I , however, because of increased d i s l o c a t i o n - d i s l o c a t i o n interactions  (section 2 . 5 . 2 ) , the defect production  upsetting the equilibrium.  rate i s increased  D i f f u s i o n rates may then be large enough to  allow atmospheres to move with dislocations which would account f o r the gradual decay of serrated y i e l d i n g .  Some basis f o r t h i s b e l i e f i s derived  from the following rough c a l c u l a t i o n s .  Assuming that the mobile d i s l o c a t i o n  9 /  2  density at the end of stage I i s ~ 1 0 /cm , the average d i s l o c a t i o n v e l o c i t y v' necessary to maintain the applied s t r a i n rate of ~ 1 0 / s e c was calculated 3  from equation (2) to be v ~ 1 0  cm/sec where b was taken as 3 x 10  cm.  Furthermore, assuming that the defect concentration C as a function of -4  s t r a i n may be given roughly as C ~-10  .  <y  I l l  , then the d i f f u s i o n c o e f f i c i e n t  D* at the end of easy glide calculated from equation (5) i s given as D ' ~ 10 cm /sec, where a was taken as 3 x 10 cm, as 1, Z as 8, 0 as I 0 / s e c and Q as "~0.25 ev. (see next section). On substituting D ' 13  82  into equation (3) and l e t t i n g I = 4b  _  3  , i t i s seen that 4 0 ^ 1 0  cm/sec  .1  and therefore somewhat greater than v , consistent with the suggestion that atmospheres d i f f u s e along with moving dislocations at the end of stage I. I t must be emphasized that these calculations are only crude approximations 9 /  since they are based on the questionable  assumption that ^ mobile  2 m  ~4 .  and that C ~ 1 0  j at large s t r a i n s .  Since specimens oriented along the  [ l 0 l ] - [ l l l ] boundary deform by multiple s l i p (section 2 . 4 . 5 . 3 ) the explanat i o n based on an increased vacancy production  rate can probably account f o r  the greatly reduced amplitude of serrations f o r these orientations. 2.3.4  Segregating Species The problem remains to i d e n t i f y the segregating  species. • An  analysis w i l l be carried out to determine the a c t i v a t i o n energy of motion, Q, with the view that knowledge of such a parameter i s valuable regard.  i n this  Although successful i n estimating Q, the analysis does not  permit an unambiguous i d e n t i f i c a t i o n , of the species.  Since the v a r i a t i o n  33 T with "o was not determined f o r Zn-rich c r y s t a l s , the energy Q " c be evaluated f o r Au-rich c r y s t a l s only. in  By equating D i n equations t i o n (1)  (6)  and  for  s t r a i n rate:  (4).and (5)  will  and substituting equa-  and G respectively, i t can be shown that at constant m  +  Q/kT  n  (8)  = K .e 4  c where K  4  i s constant.  against log  Q  1 0  C  The sum  (equation 7)  fm + n) may be obtained by p l o t t i n g and reciprocating the slope.  d i f f i c u l t i e s i n obtaining accurate values of curves, i t was  l o g  1  0  c  Because of  from load-elongation  decided to p l o t the range of s t r a i n , Table 2 ,  i n which ~Y  f e l l , . F i g u r e ih, where the bracketed points locate the c r i t i c a l s t r a i n range.  At strains less than those depicted by the lower brackets, serra-  tions were not detected whereas at strains greater than those specified by the upper brackets, serrated flow had d e f i n i t e l y begun.  To determine the  value (m + n) from the points i n Figure Ik, two extreme l i n e s were drawn and the corresponding  slopes measured then reciprocated to give an average  value of (m + n) with a deviation term. able -With  the values 2 . 2  -0.1  and 1 . 9  It was  found that (m+n)=3.6 - 0.9,compar-  - 0 . 2 determined f o r Cu-Sn  71  and  84  Cu-Zn  a l l o y s respectively. It i s now  possible t o determine Q. from t h e slope o f the graph  In "ft against the r e c i p r o c a l o f t h e absolute temperature, Figure c  15.  Again, t h e c r i t i c a l s t r a i n range i s plotted and the bracketed points have the same s i g n i f i c a n c e as described above. points i n Figure 1 5 ,  To determine t h e slope from the  t h e two-extreme l i n e technique was  again used,then  to obtain the maximum and minimum, and hence most probable  activation  energy, the highest and lowest slopes were m u l t i p l i e d by the highest and lowest  (m+  n) values.  The most probable a c t i v a t i o n energy Q obtained i n  Figure iK.  -0.5  Showing the v a r i a t i o n . i n C r i t i c a l with C r i t i c a l Strain Rate (T = 293°K; 5 1 . 0 at 4 Au) °  Strain  Y c  3^  -  o H  - 1 . 0  IS  -p  CO  03  o  •H  •p  -1.5  •H  in  O  -3.5  • -3.0  -2T5  -2.0  -r.5  C r i t i c a l Strain Rate ( l o g m ^ )  Figure 1 5 .  Showing the effect of temperature on c r i t i c a l s t r a i n ( ft = 2 . 5 x 1 0 " / s e c , 5 1 . 0 at.#Au) 3  t h i s way was found to be: Q = 0 . 2 6 - 0 . 0 8 ev. The a c t i v a t i o n energy f o r motion of vacancies i n material of the same 85  composition  was given by Mukherjee et a l  ,  as 0 . ^ 7 - 0 . 0 5 ev., .determined  from isothermal annealing studies on quenched-in defects i n J mm. single c r y s t a l s .  diameter  I t i s apparent that the energy for motion of vacancies  does not f a l l within the range determined f o r the present  activation  energy. Due to the paucity of data i n the l i t e r a t u r e l i s t i n g the a c t i v a t i o n energies f o r the motion of defects i n intermetallic compounds in general, and AuZn i n p a r t i c u l a r , i t i s impossible  at t h i s juncture  to associate Q = 0 . 2 6 - 0 . 0 8 ev. with the motion of any defect species. The i d e n t i t y of the segregating species, therefore, remains unknown.  3 6  2.4  DEFORMATION MODES  2.4.1  Introduction The operative s l i p systems i n CsCl type' superlattices :are  strongly dependent on the degree of ionic bonding between the component atoms. to  In considering the question how large must the ordering energy be  change the s l i p d i r e c t i o n from the usual bcc "metallic" < 1 1 1 > type to  the " i o n i c "  <001>-type,,  Rachinger and C o t t r e l l  (RC) begin with Nabarro's  2 6  27  .postulation that <11I> s l i p should not occur unless the t o t a l a < 1 1 1 ^ dislocation-  dissociates into two superlattice p a r t i a l dislocations according  to: a < 1 1 1 7 - = a  < 1 1 1 >  + a<lll>  •2  ( 9 )  2  Nabarro s postulation i s based.on the fact that t o t a l a < 1 1 1 > dislocations are able to dissociate into three perfect a  type dislocations with  < 0 0 1 >  no change i n e l a s t i c energy, and consequently, an applied stress, that acts strongly.on one of the three a independent of the others. occurs,  <001y  < 0 0 1 / >  components w i l l move t h i s component  RC point out that even i f dissociation ( 9 )  s l i p 'may' s t i l l be favourable i f the stacking fault energy  l i n k i n g the two superlattice p a r t i a l s i s high enough so that t h e i r e q u i l i brium, spacing i s only ~ a , one l a t t i c e spacing.  The c r i t i c a l factor, then,  in determining whether or not < 1 1 1 > s l i p occurs i s the c r i t i c a l stacking fault energy CO  Q  above which, < 0 0 1 ^ i s the favoured mode and below which  the equilibrium separation of the p a r t i a l s i s greater than a so that ^ 1 1 1 > is favoured.  As a quantitative c r i t e r i o n , RC assume that 0)  exerts a  force on the dislocation equal to the t h e o r e t i c a l shear strength of the l a t t i c e , and hence may be expressed as: •v uJ  c  where  =°<.f/\>  i s the t h e o r e t i c a l shear strength (  Burgers vector.  ( 1 0 ) ~ l / 3 0 )  CX  Giving t y p i c a l values of ' 3 x 1 0  u  and b i s the  dynes/cm  and 2.5 x 1 0  cm  37  to /f and b respectively, uJ  was calculated as ~ 2 5 O ergs/cm .  Then  2  considering the atomic density on  ( 1 1 0 )  planes f27 2 N  a  a  n  d  t  h  a  t  e  a  c  n  a  t  o  m  forms bonds with two nearest neighbours i n p a r a l l e l layers, RC calculate that on the nearest neighbour bond approximation: V  AB " i ( AA V  +  V  BB)  S ! _ ^ - - 0 6 ev.  =  2  c  2 Y  _  2 ~  which i s the c r i t i c a l ordering energy per atomic,bond, charge of only - 0 . 1  electron on each"ion'.'  ( 1 1 )  A  equivalent to a  They conclude that since ionic  character as small as t h i s i s quite possible, even i n highly "metallic" alloys,  <  1  0  0  >  should be the common mode of s l i p i n CsCl type structures.  In an attempt to evaluate t h e i r hypothesis, RC studied s l i p modes i n several CsCl compounds and compared the s l i p d i r e c t i o n with that expected on the basis of bond type.  In compounds undergoing an order-disorder  reaction, k T was taken as a measure of the ordering energy per bond, c  where T  c  i s the ordering temperature i n degrees Kelvin.  given i n Table 3 .  Their results are  For the i o n i c compounds noted, < C l 0 0 ^ i s the s l i p For CuZn, <"111> s l i p was observed, again i n agree-  d i r e c t i o n , as predicted.  -TABLE 3 S l i p Systems i n CsCl Type Compounds (Rachinger and C o t t r e l l ) 2 6  Compound  Plane  Direction  Bond Character T (°K) Ordering Energy (ev.) = kT /4 c  c  CuZn AgMg TIBr, T i l , T1C1 • TIBr, L i T l , MgTl AuZn AuCd  1 1 0  1 1 1  7 3 8  3 2 1  1 1 1  1 0 9 3  0 . 0 1 5  110  0 0 1  1 1 0  0 0 1  1 1 0  0 0 1  ionic compounds 9 9 8 (Tm.p.) 9 0 0 (Tm.p.)  (Tm.p.)  0 . 0 2 3  0 . 0 2 2 0 . 0 2 0  5 8  raent with t h e i r hypothesis, since the low ordering energy of 0 . 0 1 5 less than the c r i t i c a l energy  0  compounds.AuZn. and: AuCd  appear to invalidate the hypothesis since the  .  0  6  ev.  ordering energies are less than 0 . 0 6  But, s l i p directions  ev. i s  ev.  <"001/  >  i n the  However, as RC point out, the  fact that AuCd and AuZn are ordered up to t h e i r melting points suggests that the use of:T p as T m  i n computing the ordering energy probably results  c  in a figure less than the true value. 29  •Ball and Smallman  (BS) have recently reported s l i p systems  in the B2 compound NiAl to be of the form {llo}<001>.  Since NiAl, l i k e AuCd,  AgMg and AuZn, i s ordered up to i t s melting point, BS calculated that the ordering energy per atomic bond^from the relationship kTmpjis 0.0k ev, somewhat less than the c r i t i c a l 0.06 ev. for <100? s l i p according to 'RC. •.BS conclude that the RC c r i t e r i o n f a i l s again to s a t i s f a c t o r i l y predict s l i p modes i n CsCl type compounds.  The problem of s l i p systems i n CsCl  compounds i s then reconsidered i n terms of the d i s l o c a t i o n e l a s t i c energy E 30  which i s given by the relationship.derived E = Kbf_ In R kir r o  by Foreman: ( 1 2 )  where R and r are the outer and inner cut-off r a d i i respectively, b the Q  Burgers•vector and K the energy factor which i s a function of the e l a s t i c constants c  1 ; L  ,c  1 2  ,c  4 4  and the d i r e c t i o n cosines  location l i n e with respect t o the cube axis. dislocations l y i n g along ^ l l ^ and f o r screws along and ^ H O ,  < " 1 0 0 >  by Foreman.  30  ,^ ,  of the d i s -  Energy factors for screw  directions have been calculated by Head  and (HOy  as well as edges l y i n g along ^lOO^  Since the r e l a t i v e mobility-S  ( i . e . the r a t i o of  the stress required to move a d i s l o c a t i o n to that necessary to make the atomic planes move r i g i d l y over one another) of the lower energy dislocations in possible glide planes w i l l determine t h e i r rate of m u l t i p l i c a t i o n and  39  hence the predominant s l i p system, the S values of the lower energy 33  dislocations must also be compared. determine S:  Eshelby  has proposed an equation to  -21T? S. =  k-rr ±_e  (13)  b  where  i s the width of the d i s l o c a t i o n and can be calculated from the  expression:  33  J?j=  b  IK  d  2/5,  b  (Ik)  where K i s the energy factor, ^  the shear modulus i n the s l i p d i r e c t i o n  on the glide plane and d the spacing between the glide planes. To compare the experimentally observed s l i p modes i n some C s C l compounds with those predicted by the e l a s t i c energy-dislocation mobility c r i t e r i o n , BS calculated E and S f o r screw and edge dislocations l y i n g along <• 1117,  <1017 and <^100>directions  i n CuZn, N i A l and CsBr.  For  comparison the author c a l c u l a t e d the corresponding terms f o r s i m i l a r d i s l o c a t i o n arrangements i n the i s o - s t r u c t u r a l compound AuZn. energy c o n f i g u r a t i o n s , the r e l a t i v e m o b i l i t i e s and the  The lowest  experimentally  .determined s l i p modes are given i n Table k. Although a ^100>>appears t o be the most favourable t o t a l d i s location i n CuZn, BS show that a<lll>' dissociates into two a O - l l > 2  superlattice p a r t i a l s having t o t a l energy lower than that f o r the aO-00> dislocation.  The predicted system [ l i o )  with experimental r e s u l t s .  (lllXs  therefore consistent  In CsBr, energetically both  ^ l i o ] ( 001>and  [lOO) ('OOl^are possible systems, but because of the favourable mobility term, [ l i o ] ('OOl^is predicted, again i n agreement with experiment. (001>  Clearly,  i s the favourable s l i p d i r e c t i o n i n NiAl and the energetics suggest  that s l i p should occur on {lOOJ planes.  However, because the mobility term  4o  TABLE.k Comparison of Line Energies E and M o b i l i t i e s 'S of Low.EnergyDislocations i n CuZn, NiAl, CsBr (after B a l l and'Smallman ) and-AuZn 29  Slip System 110  110  110  110  Ref. 111  001  001  001  26  29  26  70  C ompound  Plane 010 110 110 110  100 111 110 100  e s s e  3.24  100 110 110 100  100 100 1.00 100  e e s s  7-69 8.839.29 9.29  • 71 .513  100 110 110 110  100 100 110 100  e e s s  2.3  .431 .276 .426 .482  110 100  100 100  s s  CuZn  NiAl  AuZn  CsBr  E - l i n e energy where 1 % e = edge d i s l o c a t i o n s = screw d i s l o c a t i o n  In R r  Dislocation Character  E*x 10 " 4 (ergs/cm)  Burgers Vector  k.k k.k  4.86  2.64 2.78 3-0 .825 .825  S .723 • 7 .086 .626  .46  .67  .46 •67  i s taken as unity. Q  i s more favourable on {ll6} planes, the predicted system i s {lio} ^ 0 0 1 ^ consistent with the experimental observations. c l e a r l y the preferred s l i p d i r e c t i o n .  In• AuZn  as well, < 0 0 1 ^ i s  Since mobility appears to dominate  in low energy configurations, {lip} <001> i s the predicted s l i p system, i n 2 6  agreement with the results, of'Rachinger and C o t t r e l l  and consistent with  the observations to be presented i n sections 2 . 4 . 3 and 2 . 4 . 4 of the thesis. The l a t e s t discussion of s l i p systems i n CsCl type compounds suggests that the atom size r a t i o R^/Rg may be an important factor governing  kl  the choice of s l i p system.  Lautenschlager et a l  grouped B2 compounds  into three classes: .A, mainly i o n i c with some degree of covalent bonding; • B, m e t a l l i c a l l y bonded and reinforced by a strong covalent component; C, wholly metallic with a s l i g h t covalent tendency. •In class 'A, .ionic c h a r a c t e r i s t i c s are believed to dominate the s l i p d i r e c t i o n and  < 0 0 1 / '  i s the preferred mode, while i n class C, an orientation  c r i t e r i o n dominates and on the basis of Schmid factor calculations f o r the  {lio}  systems [lOO}  ^ 0 0 1 } ,  <112>,  $ 2 1 1 }  < 1 1 1 > ,  { ' 3 2 l }  < 1 1 1 > ,  (lio) £LLO>, {lio}  the most favoured s l i p d i r e c t i o n i s along  <001?and  (It must be  < 1 1 1 ? .  pointed out that the orientation c r i t e r i o n necessarily assumes that the c r i t i c a l resolved shear stress i s the same on a l l possible s l i p systems.) In class B, though, Lautenschlager et a l state that 'R^/^B S of s l i p mode.  o v e r n  s the choice  The c r i t e r i o n f o r c l a s s i f y i n g a p a r t i c u l a r compound was not  stated. Hard sphere CsCl models were c o n s t r u c t e d atom size r a t i o from  1 . 0 0 0  to  0 . 7 3 2  34  and the effect of  ( i . e . range i n which CsCl structure i s  s t a b l e ) on the choice of s l i p system was evaluated i n the following manner. 3 5  Choosing either  < 1 0 0 > ,  < 1 1 0 >  or  < 1 1 1 >  as the s l i p d i r e c t i o n , possible s l i p  planes were either accepted or rejected i f the maximum displacement to the shear plane, d  x  the l a t t i c e parameter. (lOO} <  0 0 1 ) ,  (HO) (001},  were less than or greater than 0 . 3 a ,  normal  where a i s  The only systems f u l f i l l i n g t h i s c r i t e r i o n were { 1 1 0 }  £L10>, (lio)  < 1 1 1 > ,  { 2 1 l }  ^lll>and  £2l]  < 1 1 1 > .  For a s p e c i f i c R^/Rg r a t i o , the most favourable system from these s i x p o s s i b i l i t i e s i s associated with the smallest  value.  On this basis i t  was shown that < 1 1 1 > i s the favourable s l i p direction as R^ tends t o 1 . 0 0 0 R~B while  < 0 0 1 >  i s preferred when the r a t i o tends to  0 . 7 3 2 .  For values  k2  between 1.000 and 0.732, however, the d in .its choice f o r s l i p d i r e c t i o n .  x  c r i t e r i o n i s less discriminative  Also, i t appears that the d j _  criterion  i s not able to distinguish unambiguously between the possible s l i p planes in the <11L>  or ^OOl? zone. •A second important deformation mode that i s often encountered  i n bcc metals and a l l o y s i s that produced by mechanical twinning. ordering, however, Laves  41  Upon  pointed out that a c r y s t a l may lose i t s twinning 42  ability.  Marcinkowski  and Fisher  considered the number of nearest  neighbor A-B bonds broken and A-A and B-B bonds formed during mechanical twinning i n a CsCl type compound, assuming the twin mechanism to be that invariably observed i n bcc l a t t i c e s , namely £211} ^ U l X  Unlike the case  for s l i p , a suitable p a i r of dislocations that might create a twin without leading to any subsequent disorder does not e x i s t , because i f the .a < 1 1 1 > 6 twin d i s l o c a t i o n dissociates or combines with another, the necessary atom movements f o r twin formation would not be provided.  Consequently the stress  necessary to move the twinning d i s l o c a t i o n , and thereby disorder the i s believed to be very high.  lattice,  Marcinkowski and Fisher conclude, therefore  that < 1 1 1 > s l i p i s generally preferable to twinning i n the B2 l a t t i c e . alloys that exhibit < 0 0 1 > s l i p , , twinning on the £211~J ^ l l l ^  In  system i s  probably even less l i k e l y than i n systems undergoing <111> s l i p . • As noted, considerable information i s available on the deformation modes i n the various CsCl type intermetallic compounds. no detailed account has appeared  However,  i n the l i t e r a t u r e concerning the effects  of temperature, s t r a i n rate and c r y s t a l orientation (on deformation modes). •Such a study has been carried out during the course of the present i n v e s t i gation with the view that such information would aid in understanding the general deformation c h a r a c t e r i s t i c s of  ^'AuZn and possibly elucidate the  underlying d i s l o c a t i o n mechanisms responsible f o r the,observed behaviour.  43  2.4.2  Procedure For  the purposes of trace analysis, highly polished f l a t  surfaces were prepared by spark-machining square cross-sections on the J>mm. diameter crystals over a 2 cm. gauge length then electrochemically removing approximately 0.004 inch from the eroded surface i n fresh yjo KCN solution. A l l specimens were subsequently encapsulated under reduced pressure and annealed one hour at 300°C t o remove any s l i g h t residual strains that may have been incurred from the spark-machining operation. A l l specimens were examined under the o p t i c a l microscope p r i o r to straining.  In the  experiments designed-to study temperature effects on deformation modes, the same p a i r of crystallographic faces were exposed on a l l specimens. This procedure i s , of course, not possible when studying orientation e f f e c t s .  In determining the orientation dependence of the s l i p plane, specimens were strained i n tension u n t i l well-defined s t r a i n markings could be seen under the o p t i c a l microscope. percent were found to be adequate.  Shear strains from 5 "to 10  For metallographic examination, speci-  mens were supported i n . a s p e c i a l l y designed " j i g " then examined under greenf i l t e r e d oblique l i g h t i n g using a Reichert metallograph. on adjacent surfaces were c a r e f u l l y paired, noted and then  Strain markings photographed.  The orientation of the surfaces examined was determined from back-reflection Laue X-ray patterns taken a f t e r straining.  To minimize experimental errors  extreme care was taken to a l i g n the c r y s t a l face p a r a l l e l t o the X-ray f i l m plane. -The operative s l i p plane was determined by a two-surface trace 43  analysis according to a procedure.given by Barrett  . -Reproducibility was  found to be within three degrees. 2.4.3  Definitions In studying s l i p systems i t was useful to characterize the  kk  orientation of the specimens by the angles p  s  44  proposed by Taylor, studies on T a ,  Nb  4 8  45  and % • i n the manner f i r s t  shown i n Figure 16,.and recently employed  i n similar  and Fe-Si a l l o y s .  i s the angle  4 6  '  Accordingly,  4 7  ^  between the s l i p d i r e c t i o n b (shown l a t e r to be [ 0 0 1 ] ) and the t e n s i l e axis Cfand }C i s the angle between a reference plane i n the [ 0 0 1 ] zone, taken as ( 1 1 0 ) ,  and the maximum resolved shear stress plane M i n the zone,which  i s normal to the plane containing b and C~„  In Figure 1 6 , '\f/'is  defined as  the angle between the macroscopic s l i p plane and the reference plane  (110)  and w i l l subsequently be termed the s l i p plane parameter. 2.k.k  .Slip Direction  The Burgers vector of the mobile dislocations was determined from changes i n c r y s t a l orientation caused by p l a s t i c deformation.since . i t can be shown  49  that the d i r e c t i o n i n the glide plane to which the l o n g i -  t u d i n a l axis moves i s the s l i p d i r e c t i o n .  The changes i n orientation f o r a  s l i g h t l y A u - r i c h c r y s t a l (~50.3 at # A u ) during deformation at 77°K, 293°K, 398°K and if73°K are shown i n Figure 17.  lkO°K,  .The corresponding load-  elongation curves are also shown so that a rough idea may be gained of the amount of s t r a i n induced p r i o r to each re-orientation.  I t i s evident that  the s p e c i f i c d i r e c t i o n of specimen axis re-orientation i s temperature dependent.  A t 77°K, two experiments were performed from which  d i f f e r e n t r e s u l t s were o b t a i n e d .  I t i s observed  slightly  t h a t t h e specimen a x i s i n  one case r o t a t e d towards [100] t h r o u g h o u t d e f o r m a t i o n ( F i g u r e 17.1.a) b u t i n t h e second case r o t a t e d toward [100]  i n i t i a l l y t h e n towards [001]  the l a t e r s t a g e s o f d e f o r m a t i o n ( F i g u r e 1 7 . I . b ) . a x i s f o l l o w e d a great c i r c u l a r r o u t e to e i t h e r  during  I n b o t h cases t h e specimen  [100] or [00l],  During  s t r a i n i n g at temperatures lU0°K, 298°K and 398°K (Figure 17.2.a; Figure 1 7 . 3 . a ; .Figure 17.4.a) the specimen axis rotated d i r e c t l y towards  [001]  0 0 1  Figure 1 6 .  A ( 0 0 1 ) stereographic projection showing the parameters characterizing the specimen orientation and the s l i p plane r e l a t i v e to  ( 1 1 0 ) .  7? = angle between s l i p d i r e c t i o n [ 0 0 1 ] t e n s i l e axis 0"~.  and  angle between the ( 1 1 0 ) reference plane and most highly stressed plane (of pole M) in the [ 0 0 1 ] zone. = s l i p plane parameter; angle between the reference plane and the observed s l i p plane of pole S. ' ( 1 1 0 ) ' "  17.5a Extension (inch) .Figure 17.  Stereographic representation of specimen axis reorientation during plastic deformation as a function of temperature. (0 denotes i n i t i a l orientation)  47  throughout deformation,  while at 473°K (Figure 17.5.a) towards  [OOl]  i n i t i a l l y then towards [lOO].  From these r e s u l t s , i t could be inferred that the re-orientation along two d i r e c t i o n s during straining i s indicative of a change i n the operative s l i p system.  S l i p systems have been studied during  and the r e s u l t s are reported i n section 2 . 5 . 2  where i t i s shown that the  most prominent system at 473°K changes during deformation. i t may be concluded  deformation  At t h i s juncture,  that the most prominent s l i p d i r e c t i o n i n  (^'AuZn i s  the < 0 0 1 > type, i n agreement with the observations of Rachinger and  Cottrell  2  and the predictions based on e l a s t i c i t y theory given i n section 2 . 4 . 1 .  .2.4.5  Primary S l i p  2.4.5.1  Temperature Dependence  Planes  The primary s l i p plane dependence on temperature from 77°K to 473°K was  investigated f o r two orientations within the stereographic  t r i a n g l e , Figure 1 8 ,  using slightly. Au-rich ( ~ 5 ° - 5  at $ Au)  crystals.  Typical s l i p traces from orientation 1 observed at low magnification are shown in-Figure 1 9 ,  and at high magnification, Figure 2 0 .  The high magnifi-  cation structures were obtained from transmission electron microscopy studies of replicated surfaces using chromium-shadowed carbon r e p l i c a s taken from cellulose-acetate impressions  of the c r y s t a l surfaces.  It i s seen that s l i p traces on faces A are generally short and wavy while those on faces B are long and r e l a t i v e l y straight. i s p a r t i c u l a r l y evident i n Figure 2 0 . above 77°K i s [ 0 0 1 ] ,  i t was  This differenc  Since the operative s l i p d i r e c t i o n  possible to calculate that the i n c l i n a t i o n of  the s l i p vector to faces A and B(which were oriented from back-reflection photographs) i s 28° and 1 0 °  respectively. .The short wavy traces,.therefore,  h8  001  20.B  1  Figure  20.  micron  Replicas of surface f a c e s A and B.  s l i p t r a c e s on  orthogonal  50  were created by edge dislocations t r a c i n g the paths of screws and the long straight traces were formed by screw dislocations t r a c i n g the motion of edges.  S l i p trace development i s sketched i n Figure 21..  These 1;  observations 45  are very s i m i l a r to s l i p traces observed i n Wb single crystals  and  appear analogous to those of slip-bands in Fe-3.2$> S i crystals °revealed 5  by etching.  The presence of the short, wavy traces suggests that the screw  dislocations t r a v e l over r e l a t i v e l y short distances before they either c r o s s - s l i p onto other planes or stop, while the long, straight traces  imply  that the edges t r a v e l over quite long distances. . At 77°K i t i s apparent that three systems are operative, although system 1,dominates; :.Figiire ;19;1, while at higher temperatures, only one system operates.  Schematic p a i r i n g i s i l l u s t r a t e d under each photo-  micrograph in Figure 19. i t was  Because of the profuse waviness of face A traces  decided to characterize the respective markings by a narrow wedge  rather than a l i n e ; .in t h i s way  they could be r e a l i s t i c a l l y paired with  the straight B traces and then analyzed. will'be discussed i n section 2.5.2  Markings l a b e l l e d 2 i n Figure  under the t i t l e of deformation bands.  At temperatures above 77°K> i t was faces  found that the s l i p sur-  are non-crystallographic planes ( i . e . high index) i n the [001]  and that the macroscopic s l i p plane varied with temperature. of these.analyses,  19.2  zone  The results  including the range, are given i n Table 5 under  column Ajy . At 77°K (Figure 19-1)  the dominant s l i p plane was. found to be  a non-crystallographic plane i n the [TOO] zone l y i n g within 3 or k degrees of (Oil) and therefore could not be expressed in terms of AJT" as i t i s presently defined; trace - 2 ,  i s a non-crystallographic plane in the  [001]  zone which can be expressed i n terms of the s l i p plane parameter \|/" and i s given i n Table 5-  Since the s l i p d i r e c t i o n must be p a r a l l e l to the zone  51  5 2  axis, .the Burgers vectors of dislocations giving r i s e t o primary deformation are p a r a l l e l to [ 1 0 0 ]  at 7 7 ° K a n d p a r a l l e l . t o [ 0 0 1 ]  at higher temperatures.  These results are i n agreement with the primary s l i p d i r e c t i o n determined from a x i a l rotations during p l a s t i c deformation, section  2.k.k.  TABLE 5  Results of S l i p Trace Analyses of Temperature E f f e c t on S l i p Plane ParameterAf/"  Orientation (Figure 1 8 ) NO. 1  2  Test No.  y.  ?• •'  2 0  2k  6  k2  1 5 3  ( -j* = 2 . 5  x lQ-3/sec.)  (deg.)  Test Temp. °K  7 7  5  -  6  -  1 7  1 5 5  iko  1 3  146  2 9 3  2k - 29  1 3 5  2 9 3  2 2  -  2 7  1 5 7  3 9 8  2 9  -  3 7  1 5 6  q-73  ko -  5 0  1 3 0  7 7  1  2  1 3 2  2 1 0  3  -  5  1 2 7  . 4 5 3  1 1  -  1 5  1 3 3  4 7 5  1 3  -  1 7  P r i o r to discussing the v a r i a t i o n i n ~y with temperature, one further feature of the  77°K  deformation traces must be noted.  The pole of  t r a c e - 3 i n Figure 1 9 . 1  l i e s within two degrees of ( O i l ) , the most highly  stressed plane of the { l i o } ^ l l l ^ s y s t e m f o r the given orientation.  An  examination, of the back-reflection 'Laue photograph obtained from the surface i n Figure I9.A.I revealed two d i s t i n c t branches of asterism, -Figure 2 2 ,  53  Figure 22.  Back-reflection Laue X-ray pattern from surface shown i n Figure I9.A.I. Note two branches of asterism.  5  i n d i c a t i n g that deformation on two systems has occurred.  The  4  Taylor  r o t a t i o n a l axis f o r each branch i s determined in Appendix k where i t i s shown that the axes are consistent with s l i p on the two systems, ( 0 1 1 ) [ l 0 0 ] which i s very close to the primary s y s t e m ! and probably defines system 3.  (Oil) [ i l l ] , which  I t appears, therefore, that at 77°K, s l i p  wanders s l i g h t l y i n the [ i l l ] d i r e c t i o n from s t r i c t  [100]  slip.  An  examination of Figure 19.A.I reveals that the straight trace-3 l i n e s are in fact branches of the wavy trace -1  lines.  The v a r i a t i o n i n the macroscopic s l i p plane with temperature can now be considered  further.  It i s observed that i f the s l i p plane  parameter ~y i s plotted against the absolute temperature T, a f a i r l y good straight l i n e connects the points and goes through the o r i g i n , Figure •It can be seen, too, that i f an error of - 3 degrees was  23.  incurred in  indexing the s l i p planes, suitable adjustment would give an even better straight l i n e .  In any case several features are worth noting:  (1) plane in the [ 0 0 1 ] (2) in the [ 0 0 1 ]  The s l i p surface i s generally a non-crystallographic zone; The s l i p surface i s not the most highly stressed plane  zone, except at approximately 2 2 0 ° K , where "VJ/ = X  f°  r  both  orientations 1 and 2 . C l e a r l y , more information must be gained before i t i s known with any certainty whether or not the temperature at which "ty" ="X i s unique and independent of orientation; (3)  The temperature s e n s i t i v i t y of the s l i p plane parameter  i s 'an increasing function of In section  2 . 4 . 5 - 3  .  i t w i l l be shown that the s l i p plane i s not a function  of <j° , thereby v a l i d a t i n g comparison  (3).  To the author's knowledge, the above results represent  the  55  0  100  200  300  4oo  Temperature T°K  Figure 2 3 .  Showing the v a r i a t i o n of s l i p plane parameter with temperature T. ( T = 2 . 5 x 10" /sec) 3  500  56  f i r s t documented study of temperature effects i n a material exhibiting non-crystallographic s l i p . on planes i n the (111"}  Although i t i s known that bcc metals deform  zone and tend to s l i p on {llOJ planes at low  51  the "\y(T) curves have never been reported.  temperatures,  A possible  reason f o r t h i s oversight may be that because of the wavy nature of noncrystallographic traces, authors believed that s l i p traces close to £licty £32l}  and £ 2 1 l ] were a c t u a l l y these planes, since the angle between the  planes i s quite small ( 1 9 ° and 1 1 ° respectively). 2A.5„2  S t r a i n Rate Dependence Specimens from orientation 1 were prepared from the same as-  grown c r y s t a l used i n the temperature  study and strained a few percent at  room temperature at two a d d i t i o n a l s t r a i n rates, 0  g = 1 x 10  /sec and  = 2.5 x 1 0 / s e c , corresponding to cross-head speeds of 0 . 0 0 0 2 and 5 - 0 x  inch per minute respectively.  The s l i p traces were analyzed and the  r e s u l t s are l i s t e d in.Table 6 which includes f o r comparison sake the room temperature r e s u l t s from the previous tests where f  = 2 . 5 x 10 /sec.  •It i s r e a d i l y apparent that the s l i p plane parameter i s s t r a i n rate sensitive.  As  increases,  decreases, s i m i l a r to the e f f e c t of decreasing  temperature. TABLE 6 Results of Slip.Trace Analyses of S t r a i n Rate E f f e c t on S l i p Plane Parameter "u/ (T = 293°K) Orientation (Figure 18) No. X f 1  20  24  Test No. 180 146 155 143  "^(sec  X  )  1 x 10"5 2 . 5 x 10 3 2.5 x 10"3 2.5 x 10"1  53 24 22 20  - 36 - 29 - 27 - 23  57  S l i p plane dependence on s t r a i n rate has been observed i n ,  52  Fe- -3% S i single crystals bending.  tested at room temperature under three-point  At s t r a i n rates above 10/sec. only  {llO) s l i p was  observed  whereas at lower s t r a i n rates, s l i p occurred on the most highly stressed plane i n the <111>  zone.  In-AuZn i t i s apparent that the most highly  stressed plane i n the <001> zone i s the s l i p plane only.at intermediate  -i , s t r a i n rates of approximately 2.5 x 10 2.4.5.3  /sec.  Orientation Dependence The s l i p plane dependence on orientation has been studied at -3  room temperature and at a s t r a i n rate of 2.5 x 10  ,  /sec for the specimen  orientations shown in'Figure 24. 00.1  101  .Figure 24.'  Showing specimen orientations used in slip plane analysis.  For a l l orientations except No. -1, the primary slip surface was a non-crystallographic  plane in the*[00l] zone.  found to be  The results of the  two-  surface analyses are given i n Table 7. On comparing the s l i p plane parameters f o r two orientations of constant ~)C but d i f f e r i n g $  values (for instance Wo. 6 and 9> Figure 24) i t  appears that ~\jf i s independent of ^  , implying that the stress normal to the  macroscopic s l i p plane has n e g l i g i b l e effect on the s l i p surface. i n Table 7 were therefore plotted as ~\y versus 'X- , Figure 25.  The results  The dashed l i n e  represents an i d e a l case of non-crystallographic s l i p where the macroscopic s l i p surface i s the plane of highest resolved shear stress i n the [001]  zone.  It'appears.that for orientations near both the [ 0 0 l ] - [ l l l ] and [OOl]-[lOl] boundaries, i d e a l behaviour i s approached whereas for other orientations (approximate l i m i t s :  5 ° ^ ^ < 4 0 ° ) macroscopic s l i p occurs on less highly stressed  planes. S l i p plane dependence on orientation i n both tension and compression has been studied in-Fe-3$ and Fe-6.5$ S i single crystals'at room temperature, ' 46  53  i n Fe-3$ S i at 77°K  v  and i n Nb single crystals  With reference to s l i p planes i n the ^111> reference  {llOJ  at 295°K.  zone inclined at an angle T^'to a  plane, the Fe-Si alloys displayed ljf ( ]C ) curves similar to  those observed i n AuZn but Wb showed preference for either (21lj  or {lio}  slip.  TABLE 7 Results of S l i p Plane Analyses of Orientation 'Effect on S l i p Plane Parameter (T=293°K; - ^ 2 . 5 x 10" /sec.) 3  Test  Orientation (Ref. Fig..24)  147  2 53 •4 4 8 6 9 7 10  -  144 135 146 152 123 138 148 150  X-Extrapolated from Figure 23.  Y  2 6 . 15 20 20 28 32 32 38 4i  (cleg.)  |  •18 42 23 24 24 34 44 22 39 19  (deg.)  Y  (deg  2 - <k 7* 22 - 28 24 - 29 22 - 27 30 - 36 3^ - 39 32 - 37 38 - 40 4l  Figure 2^.  Showing the v a r i a t i o n of s l i p plane parameter ~\y with orientation "X. (T.= 293°K; Y .2.5 x 10" /sec) s  3  Two s l i p systems, shown i n Figure 2 6 , were operative i n o r i e n t a t i o n 6 which i s close t o the [ l O l ] - [ l l l ] boundary.  On a n a l y s i s i t  was found.that trace - 2 belonged to the [ 0 0 1 ] zone and consequently the corresponding s l i p plane parameter, "U/ = J>k to 3 9 degrees, was l i s t e d i n Table 7 ; trace - 1 belonged to the [ 1 0 0 ] zone, f a l l i n g ^ 3 9 (Oil) and  6  degrees from ( 0 0 1 ) .  Figure 2 6 .  degrees from  The reason both [ 0 0 1 ] and [ 1 0 0 ] zonal s i  Showing duplex s l i p i n c r y s t a l oriented along 1 0 1 - 1 1 1 boundary, ( o r i e n t a t i o n 6, Figure 2k)  61  occurs along the boundary'[101]-[ill] and single [001] s l i p occurs along [ 0 0 l ] - [ l l l ] becomes apparent when one considers the v a r i a t i o n i n Schmid factor on systems of the form £L10} ^00l]>.  Although t h i s system i s a  special case of non-crystallographic <001> s l i p , i t can be considered to i l l u s t r a t e the point at hand i f i t i s assumed that the fundamental planes on which dislocations move are of t h i s form. Unlike the case of {ill} O l O ^ s l i p i n fee metals a s p e c i f i c system of the form  ( l i o ) <001> i s not limited to orientations within one  of the 2k primary stereographic t r i a n g l e s ; .instead, each system operates within two adjacent triangles equivalent to a stereographic quadrangle, Figure 2 7 .  I t can be seen that along the [ 0 0 l ] - [ l l l ] t i e - l i n e ,  (110).[001]  i s the most highly stressed system giving r i s e to the observed traces of that form. • Along. [ 1 0 1 ] - [ i l l ] both (110)[001] and (011)[100] are equally stressed which i s consistent with the observation that both [00l] and [100] zonal s l i p i s active i n these orientations. The remaining primary s l i p systems to be reported are those active i n the near'[001]  orientations.  Multiple s l i p shown i n Figure 2 8  has been observed -in orientation l a n d i s of a completely d i f f e r e n t nature from that observed i n other orientations. not  Strain markings are coarse and  as evenly spread throughout the gauge section as were those i n other  orientations.  The markings shown i n Figure 2 8 are t y p i c a l of a l o c a l i z e d  area near the middle of the gauge.  From two-surface analysis, i t was found  that the most prominent set, trace -1, was within 2 degrees of (112). Because of uncertainties i n p a i r i n g , the extra traces were indexed from single surface analyses; trace -2 was found 'to be consistent with either {21l}  or [ l i o } and trace - 3 , with {321} .  To determine the s l i p d i r e c t i o n  associated with the most prominent markings, a c y l i n d r i c a l specimen was  62  100  110/  DI /  /  ^  101(E)  1  E2 A3  B3  / E2  J  Oil  010  /301^1  C2  B3 _  ii:  \/  / B 5  \ ^  A3  A3  \  1011  \ w  )  C2  Dl  i i o \ (B)  j  \sd  101  \  010  [2]  111 no  \  I  (F)  no  \  \  /  A3 Nv  E 2  \  /  \ . Ill  * \  \ .  Fl  Dl.  /  E2  I  N^llO  Fl  /  (CO  \  D1  Fl  /  Fl /  /  \  110  (A)  •100 [1]  .Plane s  Directions  A  •1 2  B  C D E F Figure  27.  (110)  (iio)  (101) (Oil) (101) (Oil)  3  [100] [010] [001]  A (001) stereographic p r o j e c t i o n showing the most h i g h l y stressed system of the form {llO} <001> as a function of o r i e n t a t i o n .  Figure 2 8 .  Multiple s l i p observed i n near [ 0 0 1 ] orientations. (No. 1, Figure 2k)  64 prepared and strained u n t i l ( 1 1 2 ) traces were e a s i l y detected.  The position  round the specimen axis at which the s l i p traces disappeared was noted then indexed from a back-reflection Laue X-ray pattern since i n t h i s position the Burgers vector l y i n g i n ( 1 1 2 )  i s p a r a l l e l to the specimen  The cross product of the slip-plane normal  surface.  [ 1 1 2 ] and the specimen-surface  normal [ 1 1 0 ] gave the s l i p d i r e c t i o n , i . e . [112]  x [llO] .= [ I l l ]  It i s therefore concluded that the most prominent tion 1 i s ( 1 1 2 ) [ i l l ] .  s l i p system i n orienta-  I t should be noted that these traces are rather  wavy suggesting that- ( l l 2 ) may not be an elementary plane on which d i s locations move but a macroscopic s l i p plane made up of composite s l i p on other, more fundamental, , [ l l l ] zonal planes.  2.4.5.4  Composition Dependence  Although a complete study of composition effects on the s l i p plane parameter "Vj/ was not undertaken, a few experiments were performed on Au-rich ( 5 1 . 0 at $ Au) and Zn-rich ( 4 9 . 0 at $> Au) c r y s t a l s .  Essentially  no composition dependence was detected, and therefore, further experiments were abandoned.  Composition e f f e c t s have been reported in'Fe-Si a l l o y s  where increased S i content from 5 . 0 to 6.5 wt. for  £llo) O-H^  crystallographic s l i p .  5 3  .increased the tendency  . However, because of an ordering ,  reaction reported i n Fe-Si at approximately 5-5% S i , :  5 4  the nature of the  composition e f f e c t i s not known. 2.4.6.  Discussion  It i s now possible to compare the primary deformation modes in  ^ A u Z n with those operative i n systems of d i f f e r e n t and similar  structures.  .The non-crystallographic nature of the <001^  zonal s l i p i s  65 d i s t i n c t l y • d i f f e r e n t from the octahedral {ill} <110> s l i p systems, observed i n fee metals and ordered alloys Cu Au  and N i A l ,  3  3  d i f f e r e n t from cube  (l00)^110^ s l i p observed i n A l a n d N i A l , and different from the generally 5 7  5 6  3  observed { 3 2 l } < l l l > , ' ' 2 6  3 8  3 9  {21l)<lll>  s l i p i n the bcc ordered a l l o y AgMg.  4 0  and ( l i o ) <lll} ° crystallographic 4  The observed non-crystallographic s l i p  closely resembles the <lll"> zonal s l i p i n bcc metals CuZn,  Ta ' , 9  Cr  4 8  2 3  59  W,  5  Fe,  Fe-Si a l l o y s , ' p a r t i a l l y ordered FeCo  36  and ordered Fe Al., and i s 3  very much the same as the {210} , {310) and {100} d i s l o c a t i o n traces observed 29  in t h i n films of NiAl during transmission electron microscopy studies. The systems {21l} < l l l > and {lid] <111> observed i n AuZn under the special testing conditions mentioned are t y p i c a l of the more commonly reported s l i p modes i n bcc metals and ordered a l l o y s .  Both <^001>and <111> zonal s l i p  w i l l now be considered i n more d e t a i l . .2.4.6.1  <001> Zonal S l i p . The immediate task a r i s i n g from the observation of non-  crystallographic s l i p . i s t o distinguish between the macroscopic s l i p planes, which are the planes observed under the o p t i c a l microscope, and the fundamental s l i p planes, which should be the planes on which .individual locations move.  In bcc metals three views are prevalent; f i r s t ,  dis-  elementary  60  slip.planes are generally non-crystallographic; {llO}  second, they are only  planes and d i f f e r e n t macroscopic s l i p planes are the result of  composite s l i p on \110J planes;  . and t h i r d , the elementary planes are  [lid} as well as {21l} planes and d i f f e r e n t macroscopic s l i p planes are the 62*51  result of composite s l i p on both types of crystallographic planes. 51  Adopting the t h i r d approach Kroupa and Vitek  have quite successfully  calculated the~Vj/Q3 curve f o r an F e - 3 $ . S i a l l o y based on a thermally activated c r o s s - s l i p model of dissociated screw dislocations.  Considering  66 screw d i s l o c a t i o n p a r t i a l s l y i n g i n sessile configurations on \llO~] and {21l] planes, they show that the thermal energy necessary t o transform the s e s s i l e configurations into g l i s s i l e ones, bowed out onto {211} and {llOJ glide planes decreases as the force on the p a r t i a l s , due to an applied shear stress, increases. In t h i s way, as the position of the t e n s i l e axis varies, so does the force on individual p a r t i a l s and as a r e s u l t , some recombinations are more l i k e l y than others.  Consequently, the macroscopic  s l i p plane, which .is believed t o be governed by the motion of recombined screw d i s l o c a t i o n s , i s orientation sensitive. 29  B a l l and SmalLman  have suggested that the \210}, {310} and  {IOC } s l i p traces-observed i n NiAl are the result of continual c r o s s - s l i p 1  of screw dislocations on orthogonal ( l l O ) planes, shown schematically i n Figure 29.  Combining t h i s general description of non-crystallographic <Q01^  s l i p with the dissociated screw d i s l o c a t i o n concept of Kroupa and Vitek, an attempt w i l l be made to obtain a more detailed account of ^001^ zonal s l i p in  ^J'AuZn.  That -Q-IO) planes are believed t o be the elementary -  slip  planes was deduced from the observations that\J/ degenerates to zero at very low temperatures  (and•presumably high s t r a i n rates) and small "X values;  .that thermally activated c r o s s - s l i p of screw dislocations governs the macroscopic s l i p plane i s concluded from the pronounced influence of temperature and deformation rate on the s l i p plane parameter The p o s s i b i l i t y of s p l i t t i n g of <.001> dislocations i n bcc metals  63 has r e c e n t l y been discussed by. V i t e k .  Using i s o t o p i c e l a s t i c i t y theory,  he found that d i s s o c i a t i o n s of the d i s l o c a t i o n <001>according  t o the  reaction: a <001> =a/ <011> + a / <017> 8  8  (15)  are stable f o r an a r b i t r a r y o r i e n t a t i o n of the d i s l o c a t i o n l i n e i n a l l bcc metals s t u d i e d , ranging f r o m ' L i and  (3 brass w i t h anisotropy r a t i o s Adhere  67  as the r a t i o o f t h e shear modulus on t h e { 1 0 0 ) < 0 0 1 > system  A i s defined to  t h a t i n {llO~)<ilO> and i s g i v e n as 2 C 4 / ( C i - C i 2 ) ) o f 9.1+ and 8.8 r e s 4  p e c t i v e l y t o Nb w i t h a r a t i o o f 0 . 5 .  1  Furthermore, V i t e k found  that  d i s s o c i a t i o n s of the type: a <001> = a / <110> + a / ^ l l 2 > + a / ^ 1 1 2 ? are s t a b l e o n l y f o r an edge d i s l o c a t i o n ;  an e x t r e m e l y  anisotropy factor f o r  (16)  f o r an a r b i t r a r y o r i e n t a t i o n o f t h e  d i s l o c a t i o n l i n e they are s t a b l e only i n L i , metals which p o s s e s s  + a/Q<110>  ^ - b r a s s , Na and K; i . e . , those  high anisotropy factor.  Since the  ^'AuZn i s 3 - 3 ( c a l c u l a t e d from e l a s t i c  stiffness  64 constants  reaction(16)for  t h e case  a n d ( l 6 ) a r e sketched  o f screw d i s l o c a t i o n s i s d o u b t f u l . - R e a c t i o n s ( 1 5 )  i n F i g u r e J>0.  Considering reaction(15)in extended d i s l o c a t i o n d e f i n e s a f a u l t s l i p plane  ), r e a c t i o n ( 1 5 ) a p p l i e s b u t  g i v e n by Schwartz and Muldawer  more d e t a i l i t i s seen t h a t t h e  on a ( l O O ) p l a n e .  i s b e l i e v e d t o be a £ 1 1 0 } p l a n e ,  S i n c e the fundamental  dissociation(15)renders  dis-  l o c a t i o n s s e s s i l e w i t h r e s p e c t t o ( l l O ) motion. ' In order t o f i n d t h e average motion o f screw d i s l o c a t i o n s which governs t h e s l i p then n e c e s s a r y  t o study t h e t r a n s f o r m a t i o n o f t h e s e s s i l e d i s l o c a t i o n i n t o a  dislocation glissile stress f i e l d .  geometry i t i s  on [llO]  The important  p l a n e s as a t h e r m a l l y a c t i v a t e d event  in a  parameters t h a t must be determined a r e t h e  a c t i v a t i o n energy f o r t h e s e s s i l e - g l i s s i l e t r a n s f o r m a t i o n and t h e d i s t a n c e t r a v e l l e d by t h e d i s l o c a t i o n d u r i n g an a c t i v a t i o n  event.  • I f the s e s s i l e - g l i s s i l e t r a n s f o r m a t i o n i s considered as the  reverse of reaction(15)plus the bowing out of the recombined length on a (llOJ  g l i d e p l a n e , then the a c t i v a t i o n energy i s the sum of the c o n s t r i c -  t i o n energy U , t h e r e c o m b i n a t i o n c  energy U  energy due t o bowing o f t h e d i s l o c a t i o n  r  and t h e i n c r e a s e , i n t h e l i n e  A U L ^ i n u s t h e work done b y t h e  6 8  Figure 2 9 .  Schematic representation of continual cross-slip on orthogonal (llO"} planes. (After B a l l and Smallman ) 29  a £ 0 0 l } .  a [OOI3  I  >  a<n; k<lld .a  a  < 1 1 0 >  8  < 1 1 0 >  A  \  < 0 1 7 >  a  < 0 1 1 >  >  8  {ii°} { 1 0 0 }  Reaction  a < 0 0 1 >  =  a 8  •Figure 3 0 .  < 0 1 1 > +  a 8  (15]  < 0 1 7 >  Reaction a  < 0 0 1 >  (16)  =a<110>+a<112>+a<112> 8  k  +  k  Schematic representation of dissociation reactions (P5)and  a  <  1  1  8  (16)  0  >  69  l o c a l stress "C=  (X.  a  -T^ ) acting on the { l i o } plane in the d i r e c t i o n of  the Burgers vector, where t"  a  i s the resolved  back stress acting on the glide plane.  shear stress and 7^  Entropy assistance  The transformation sequence i s i l l u s t r a t e d in Figure 3 1 . g l i s s i l e d i s l o c a t i o n i s believed  i s the  i s neglected. Because the  to be a <0(XL> unstable d i s l o c a t i o n , i t i s  assumed that the distance of glide on the {_110} plane w i l l be equal to the bowing-out distance.  Consequently i t i s assumed that the transformations  of the s e s s i l e into the g l i s s i l e configuration  and vice versa are  continually  repeated for the occurrence of p l a s t i c flow.  In the  [ 0 0 1 ] zone, the most probable elementary glide plane,  either ( 1 1 0 ) or ( 1 1 0 ) , w i l l be determined by the activation energy necessary for cross s l i p from the  ( 1 0 0 ) and  ( 0 1 0 ) planes onto the  ( 1 1 0 ) and  planes which i n turn depends on the e f f e c t i v e stress acting on the planes.  The p r o b a b i l i t y p for a c t i v a t i o n i s given by:  (110) glide  65  p = e where AG  (1 ) 7  i s the Gibbs free energy of a c t i v a t i o n and k and T have t h e i r  usual meaning.  AG  i s that energy which must be supplied by a thermal  f l u c t u a t i o n of the d i s l o c a t i o n before the transformation i s completed. average number of a c t i v a t i o n events N on the  The  ( 1 1 0 ) plane before' an event  occurs on the ( 1 1 0 ) plane i s then given by the p r o b a b i l i t y r a t i o : .  where subscripts .The  _ N = Piio Pilo 1 1 0 and  -[AG  1 1 0  - A Gi Io 3 A m  •= e  . (18)  K i  110 refer to the two planes of interest.  average motion of the screw dislocations can then be  considered to be composed of N units of s l i p on the by one unit of s l i p on the  ( 1 1 0 ) plane.  ( 1 1 0 ) plane followed  The unit of s l i p on each plane i s  70  a[00l] (100)  *a[017T ,8  .[oil]  Extended Figure ^>1.  Constricted  Recombined  Bowed-out  A sketch of the s e s s i l e to g l i s s i l e transformation sequence.  [001]  '1  J  .Figure J>2.  Schematic i l l u s t r a t i o n of the continual c r o s s - s l i p cycle defining the s l i p plane parameter \J7 . The c i r c l e d regions represent the transformation sites ( i . e . sessile to g l i s s i l e and vice versa).  71  given by the bowing-out distance d, which i s related to the l o c a l stress on the two planes.  This "cycle" i s shown diagramatically  in Figure 3 2 , where  i t i s apparent that the s l i p plane p a r a m e t e r ^ can be given by  the  relationship: tan V  (19)  = dxlo N.d  1 1 0  In order to account for the dependence of  on temperature, s t r a i n rate  and orientation, the parameters N and d must be expressed in.terms of T, "o and}£.  (Note: X  specifies the orientation dependence since £  was  observed to have n e g l i g i b l e effects.) Because the effects of temperature and s t r a i n rate are believed to be s i m i l a r , "\|/ need be expressed only as a function of temperature and  orientation.  It can be shown that the free energy of activation for the s e s s i l e - g l i s s i l e transformation c r i t i c a l for the nucleation of cross i s given by the expression: AG  slip  65  1_ ^»b  = U**  2/(fg-  r f rV  (20)  g  s  /  f  where b i s the Burgers vector erf the screw d i s l o c a t i o n [001 ] , l i n e tension of the perfect d i s l o c a t i o n and  i s the  B  o  f" i s the l i n e tension of the s  extended d i s l o c a t i o n . •It should be noted that since an entropy term does not appear i n equation ( 2 0 ) ,  AG  i s r e a l l y an activation enthalpy.  corresponding value of d i s given by the expression:  a = r On substituting ( 1 8 ) ,  (20)  g  -  and  re  (21)  tanV :;T*,o(X) e  T  into (19)  L^iio  (21)  i t i s found that:  T?io J  (22)  TTio( %) where' A i s a constant equal to ( 2 [ r U  i s the constriction energy  g  -  T] s  3  The  65  1  [  g  )  2  and  72 M  T110OO  where  =T  ''-'  f 23)  1 1 D  M  F  and  Cos ^ . - T~ y  a  T-iio(X) =?;  S i n  /* "  £  x£o  (24)  M  i n which (J  a  i s defined as the resolved shear stress acting on the most  highly stressed plane on the [001] zone. and  x£  0  Since the back stresses i„  1 1 0  are not known, i t i s d i f f i c u l t to test the v a l i d i t y . o f equation  (22) quantitatively.  Q u a l i t a t i v e l y , however, i t i s i n agreement with the  experimental observations. -As temperature  decreases, tan "^f and hence  tends to zero f o r a constant value of^jL; the exact  (^") behaviour  Y  cannot  be stated since the v a r i a t i o n of the l o c a l stresses with temperature i s not known.  As "X increases and approaches- 45° the applied stresses and hence  the l o c a l e f f e c t i v e stresses *^"no  and T ^ i l o become equal so that tan "\|^  approaches unity and ~\y approaches 45°.  Since'Tj" i£o tends to zero as  r~  approaches zero, equation (22) becomes undefined i n t h i s l i m i t and hence cannot be compared with the observation that ~\J/ tends to zero. The e s s e n t i a l feature i n t h i s description i s the assumption of dissociated screw dislocations.  Since stacking f a u l t energies i n bcc metals  are considered generally to be higher than those i n fee metals, i t i s l i k e l y that f a u l t energies i n ordered bcc alloys w i l l be even higher.  Consequently  the extent of p a r t i a l dislocation separation may be very small, i n which case an anisotropic d i s l o c a t i o n core would develop.instead of a true faulted plane.  Whether or not an anisotropic core offers s i g n i f i c a n t resistance to  d i s l o c a t i o n motion i s not known; .the above description assumes i t does.  -The  a t t r a c t i v e feature, however, i s that the description appears to account f o r the observations. 2.4.6.2  -Chkl)  <lll>Slip  Using a method given by Groves and K e l l y number of independent  66  for determining the  s l i p systems i n crystals, B a l l and Smallman  29  have  75  recently shown that the continual c r o s s - s l i p process on orthogonal  {llO}  planes of the <001> zone does not increase the number of independent s l i p systems i n N i A l , leaving only three independent ^ l l O ^ O O l ^ systems to operate.  .The same result applies to-AuZn. -Whereas the t o t a l d u c t i l i t y to  fracture i n p o l y c r y s t a l l i n e NiAl decreases abruptly to l e s s than three percent at temperatures below 0.4-5' Tm, AuZn remains i n excess of ten percent  total ductility in polycrystalline at temperatures as low as 77°K  12  (0.077-Tm).  On single surface analyses  of s l i p traces i n AuZn grains,  general "thko} traces were evident within the grains, and traces were observed near the boundaries.  £21]}  and  {52l}  Since general p l a s t i c i t y of a  p o l y c r y s t a l l i n e aggregate necessitates the operation of at least f i v e Q Q  independent deformation modes,  the d u c t i l i t y of p o l y c r y s t a l l i n e AuZn was .  attributed to s l i p on the extra { 2 1 l } The  and { 5 2 l )  single c r y s t a l observations  operation of { 2 1 l } < 1 1 1 > and {llO) <111>  planes.  are d i r e c t proof f o r the  s l i p systems i n AuZn; the  trace i s also believed to be associated with <C 1U>  slip.  {52l}  Examination of  NiAl single c r y s t a l s compressed along < 0 0 l " ^ axis revealed no extra s l i p 25  modes; instead, evidence, of kinking was iso  - s t r u c t u r a l compounds C s l  6 9  observed  and C s B r . ' 6 9  7 0  s i m i l a r to that i n the It appears, therefore, that  for c r y s t a l orientations i n which the Schmid factors on the ^ l l o J ^ O O l ^ systems are near zero ( i . e . .the near <001> orientations) CsCl type compounds either s l i p on <111> the corresponding  zonal planes- or undergo kinking.  I f <111>  p o l y c r y s t a l l i n e aggregates are d u c t i l e , but i f kinking  occurs, the aggregates are  brittle.  It remains to be shown whether or not <111^ explained  s l i p occurs  s l i p can be  i n terms of the three models reviewed e a r l i e r , v i z . those based  on ordering energy per atomic bond, d i s l o c a t i o n l i n e energy and atom size  74 25  ratio.  As pointed out by Rachinger and C o t t r e l l  (RC) the fact that  <Clll>  s l i p occurs i n a CsCl type superlattice implies that a <111^ superlattice 2 p a r t i a l s are present connected by a ribbon of stacking fault wider than about one l a t t i c e spacing.  Obviously, AuZn i s a near borderline case  between "metallic" and " i o n i c " bonding nature since the c r y s t a l s l i p s i n both the "metallic" <111^  and " i o n i c " (001>  conditions (77°K; near <001>orientations).  directions under special testing AuZn, then, i s an almost i d e a l  compound i n which to evaluate the RC c r i t e r i o n which states that compounds with ordering energies greater or less than 0.06 ev. per atomic bond s l i p along <001> or <'111> respectively and conversely, those with energies of ^0.06 (not  ev. s l i p i n both directions.  However since accurate ordering energies  estimates using kT p) for-AuZn are not known at present, the true m  ~4~~ evaluation of the RC c r i t e r i o n remains to be performed. impossible to state whether.or not <111>  Therefore, i t i s  s l i p i n the AuZn superlattice can  be explained i n terms of the c r i t i c a l ordering energy c r i t e r i o n . It i s interesting to compare heats of formation, e l e c t r o negativity differences and s l i p directions for several CsCl type compounds, Table §.  I f i t i s assumed that the heats of formation and the e l e c t r o -  negativity difference between component species i s an indication of the strength of the A-B bond, then i t can be seen that bond strength has a direct e f f e c t on the s l i p d i r e c t i o n .  As a very rough estimate, i t appears  that compounds having heats of formation greater than "-6000 cal/mole and electronegativity differences (on the A-R  scale) i n excess o f ~ 0 . 2 4 ,  slip  along <0017 l a t t i c e directions whereas those with lower heats of formation and associated electronegativity differences s l i p along <111>. t i o n range i n which both <001>and <111>  The t r a n s i -  s l i p may occur appears to l i e  somewhere between formation energies, of 4500 to 6000 cal/mole and e l e c t r o negativity differences of 0.21 to 0.24.  Anomalous i n t h i s respect i s AuCd  75 TABLE 8 Correlation of S l i p Direction with Heats of Formation and Electronegativity . Differences i n CsCl Type Compounds  Compound  Electronegativity Difference A X  Heat of Formation (cal/mole) ref.  •Slip Direction uvw ref.  14000  105  001  29  M  6400  106  001  26  AuZn  ..24  6150  107  001 , 111 26, present work  MgTl  .21  106  001  AuCd  .Ok  4660  106  AgMg  .19  4380  108  111  26, .39, ko  CuZn  .09  2900  106  111  26  NiAl  .28  LiTl  5000 to 65OO  26 26  001 and possibly 111  ~Allred-Rochow E l e c t r o n e g a t i v i t i e s f o r component species used to determine A x .  with  A X ^'0.04.  Since the Allred-Rochow electronegativity scale i s only  one of many (used here because i t gives the best  A X -slip direction  c o r r e l a t i o n ) , . i t was found that on other scales ( P a u l i n g , A X f o r AuCd f e l l close t o AuZn.  1 0 9  Mullikeri  110  )  Qualitative though i t i s , the above  comparison does lend some support to the basic idea of Rachinger and C o t t r e l l that bond strengths play a very important role i n determining s l i p directions in CsCl type compounds. 29  I f the l i n e energy model of B a l l and Smallman  i s to account  for both <001 > and <111/' s l i p d i r e c t i o n s , then i t must be shown that the t o t a l energy of the superdislocation  ( i . e . self-energies of both a X l l l >  2 p a r t i a l s plus the interaction energy between the p a r t i a l s plus the energy of the stacking fault l i n k i n g the p a r t i a l s ) as well as mobility are very  76 near t h e v a l u e s f o r t h e ( O O l ) d i s l o c a t i o n s ( T a b l e 4).  A c c u r a t e computa-  t i o n s n e c e s s a r i l y r e q u i r e a knowledge o f s t a c k i n g f a u l t energy and w i d t h , w h i c h a r e not known f o r AuZn. .Again, q u a n t i t a t i v e assessments between t h e o r y and experiment cannot be performed.  As noted e a r l i e r , t h e r e l a t i v e atom s i z e model o f 34  :  Lautenschlager et a l  .  .  p r e d i c t s O O l / s l i p d i r e c t i o n s when R^ .HI  0.732, but (111/ when the r a t i o approaches u n i t y .  approaches  C l e a r l y t h e r e i s an  i n t e r m e d i a t e r a t i o where t h e model p r e d i c t s b o t h <00l) and ( i l l ) s l i p . • L a u t e n s c h l a g e r e t a l c a l c u l a t e d R^ f o r a s e r i e s o f C s C l t y p e compounds, i n c l u d i n g AuZn, i n t h e f o l l o w i n g manner.  G i v e n the a t o m i c and i o n i c  radii  of each c o m p o n e n t , l a t t i c e parameters f o r the o r d e r e d bcc s t r u c t u r e s may  be  c a l c u l a t e d assuming b o t h s p e c i e s , i n one c a s e , are p r e s e n t as i o n s and i n a second c a s e , as atoms.  The measured l a t t i c e parameter i s then compared  w i t h t h e c a l c u l a t e d ' " i o n i c " and "atomic" parameters. r e l a t i o n s h i p between R^ and l a t t i c e determined s i n c e  R^ Rg  f a s h i o n .Rzn/p " " /  A  s  .Assuming a l i n e a r  parameter, the e f f e c t i v e r a t i o i s  and R^ are known. C a l c u l a t e d i n t h i s ionic Rg atomic From t h e i r measured v a r i a t i o n o f d (defined i n  s e c t i o n 2.4.1) a g a i n s t R^y^  A  , i t appears t h a t b o t h ( i l l ) and (001)  d i r e c t i o n s a r e f a v o u r a b l e when t h e r a t i o i s i n the v i c i n i t y  of 0.9.  slip The  o c c u r r e n c e o f b o t h ( O O l ) a n d ( i l l ) s l i p i n AuZn i s t h e r e f o r e c o n s i s t e n t w i t h the  p r e d i c t i o n s based on the atom s i z e One  noted.  f u r t h e r c r y s t a l l o g r a p h i c f e a t u r e o f ( i l l ) s l i p must be  A t b o t h 77°K and i n near ( O O l ) o r i e n t a t i o n s a t room temperatures  ( c o n d i t i o n s under w h i c h <111> was  ratio.  s l i p was d e t e c t e d ) t h e o p e r a t i v e s l i p p l a n e  c l o s e t o the maximum r e s o l v e d shear s t r e s s p l a n e i n the ( i l l ) zone.  The  Schmid f a c t o r was~0.49 f o r b o t h t h e ( O i l ) [ i l l ] system o p e r a t i v e a t 77°K and  77 the (112)  [ i l l ] system operative at room temperature.  Combined with the  fact that the (112) traces were somewhat wavy (Figure 28)  these observa-  tions t e n t a t i v e l y suggest that s l i p occurs on non-crystallographic planes in the <111>  zone.  78 2.5  WORK-HARDENING BEHAVIOUR  2.5.1  Flow Parameters The work-hardening parameters as defined i n section 2.2.2 are  compiled i n Tables 9 > -10 and .11 f o r Au-rich, stoichiometric and Zn-rich crystals respectively (Figures 6 , 7 * and 8 ) and Table 12 for Au-rich crystals tested at room temperature i n various orientations (Figure 11). The following discussions centre on.the v a r i a t i o n of the hardening parameters with temperature, composition and orientation. 2.5.1.1 Y i e l d Stress The temperature dependence of*f  0  i s i l l u s t r a t e d i n Figure 3 3 •  Points denoted as crosses were obtained from multiple tests on one c r y s t a l by subtracting the t o t a l work-hardening due to straining at p r i o r temperatures from the y i e l d stress.at the test temperature.  There i s good agreement  between y i e l d stresses determined from the work-hardening curves and those determined from multiple tests on one specimen. T "  0  for stoichiometric:AuZn  decreases s l i g h t l y above 220°K but approximately t r i p l e s between 220°K and  77°K.  For the non-stoichiometric c r y s t a l s , the general shape of theT*0-T  curve i s d i f f e r e n t . above approximately  Two regions of weak temperature-sensitivity exist,  300°K  and below about 140°K.  • F r o m . 3 0 0 ° K t o lkO°K  f  0  increases about three times f o r the Zn-rich crystals but less than two times for  the Au-rich a l l o y .  I t i s also seen that y i e l d stress for Zn-rich crystals  displays a second region of temperature s e n s i t i v i t y above approximately k00°K. Similar temperature dependence of y i e l d -stress has been found i n polycryst a l l i n e AuZn.  12  .If the data shown i n Figure 3 5 are used to compare the  TABLE 9 Work-Hardening Parameters f o r 5 1 . 0 a t . $Au$'AuZn Single ( T = 2.5 x 1 0 " / s e c )  Crystals  3  Tercp  to  T11  T i  ^111"  Test  °K  71 72  77  6700 6400  -  79 80  lkl  6000 6500  -  87 88  •181  5650 5l4o  10150 10150  77 86  217  3340 5080  8800 9800  12800 14200  13900 15300  84  260  4150 4020  8500 8500  11900 11900  60 61  293  4220 3720  7600 7300  10500 10300  75 76  373  3230 34io  7100 7100  8300 8300  8600 8500  82 83  403  3430 3800  6300 6600  7350 7700  7550 7800  81  433  3130  .-  -  73 74  488  3100 3400  •-  -  85  Ti  psi  TTii  iTin fo  ^~  Tm  -  -  I78OO 21700  -  -  -  -  -  -I63OO 17700  44 50  106 116  -  _  _  20700 23200  56 44  160 152  202 190  -  -  -  13000 13000  -  22800 21700  46 46  148 148  198 198  -  11600 11700  -  18000 17500  42 40  152 158  211 232  -  11300 11100  29 30  116 126  146 150  9300 8900  :_9550 9250  14 20  43 45  -  7808  8000  -  -  -  5700 5900  -  -  -  -  11100 11000  Q  -  !  •21100 18600  —  14100 14300  T n  i  QiT psi  50 42  - .. 2 0 0 0 0 17500  4 72  -  152 178  3750 3550  _  270 280  2500 2930  10000 8800  -  308 292  2300 2300  8730 914 0  _  302 312  1900 1900  7000 7180  194 198  209 209  220 228  1000 1000  5160 5260  52 49  76  64  85 83  144 97  2370 2500  7200 7000  -  -  -  -  47  -  -  -  -  -  -  60 50  -  -  —  -  5  -10800 9300  TABLE 1 0 Work-Hardening Parameters f o r Stoichiometric ( IT = 2 . 5  Test  7  Temp. °K  7 7  psi  r  Ti  -  5 9 0 0  8  7 7  5 8 0 0  9  1 5 3  2 8 8 0  5 0 0 0  9 0 0 0  1 5 3 .  2 5 3 0  i+ooo  8 0 0 0  1 8 3  1 8 3  2 3 0 0  1 + 8 0 0  7 1 5 0  3  2 2 3  i 9 6 0  3 6 0 0  1+1+00  5 3 0 0 5 2 0 0  1 0  in  3  m  r  1  1 6 0 0 0  _  _  r  87OO  (5 ' AuZn'Single Crystals  x 10" /sec)  _  1 9 7 0 0  -  -II+5OO  r  11  . 11  -  -  Y  m  in -  -  56OO  -  1 9 5  2 5 0 0  3 1 6  2 0 0 0  7 7 0 0  2 6 0  -  .31+0  2 2 0 0  8 5 0 0  -  2 9 2  2 9 0 0  5 1 7 0  3 5 2  3 0 0 0  5 1 2 0  1 2 7 0 0  3 0  8 6  1 8 7 0 0  2 1 1 + 0 0  1 6  6 0  9 0  2 5 8  1 7 0 0 0  2 0 9 0 0  2 0  9 0  1 2 0  ll+7  -  5  2 9 3  1 8 8 0  3 3 0 0  1 + 5 0 0  5 1 0 0  9 1 + 0 0  1 3 1 0 0  11+  5 0  6 5  11+1+  6  2 9 3  1 5 4 0  2 1 + 0 0  1 + 5 0 0  5 0 0 0  8 7 0 0  1 2 5 0 0  1 0  7 0  8 7  1 5 8  2 0  3 7 3  1 8 9 0  5 8 0 0  7 5 0 0  1 2  5 7 0 0  8 0 0 0  1 0  1 7 8 0  1 8  4 4 3  1 8 3 0  1 9  1+1+3  1 8 2 0  -  -  1 + 6 0 0  -  -  91+  2 5 0  2 9 0  8 8  2 2 0  21+3  8 0  1 2 2  8 0  1 6 0  -  -  1 + 9 0 0  -  -  -  -  1 + 7 0 0  2 5 0 0  3 7 3  2 2 1 + 0 0 - 2 3 2 0 0  1 1 0  6 0  1 1 + 1 0  2 1  -  -151+  7 2  2 0  2 2 3  -  1+6  -  3 0  1+  -  -  7 0  1 + 6 0 0  -  psi  _  1 2 0 0 0  -  On  61  -  2 3 0 0  -  2 1 + 0 0  -  Co  o  TABLE 11 Work-Hardening Parameters f o r 5 1 - 0 a t Z n @ ' AuZn Single C r y s t a l s ( T = 2 „ 5 x 10" /sec) 3  Test  Temp. t o °K -  95 96  77 77  8200 8850  -  101 102  i4o  140  8510 7670  -  99 100  192' 192  5190 4130  10600  97 98  243 2^3  4420  89 90  293 293  3220 4000  78OO  103 104  348 348  2780 3040  6600  91 92  373 373  2510 3320  5200  105 106  408 408  2840 3250  93 9k  473 473  2510 2240  4370  ^111  fix psi  m  -  -  -  _  _  _  -  -  -  14700 14300  _  _  10400  -  10000 12000  10600 12500  8200  65OO 65OO -  -  24500 24600  ^m  -  -  -  -  -  18 18 80 50  _  28300 23800  _  -  -  -  -  -  52 52  110 98  _  _  _  -  I97OO I65OO  -  -  10800 -  16200 17800  17700 19000  40  94 -  _  50  _  14300 15100  26 20  70 70  _  -  13100 14700  7600 7400  7600 7400  9450 9450  10200 985O  28 42  5900 7100  6050 7400  8200 9700  9200 10000  _  _  _  -  -  -  _  _  _  -  -  85OO  85OO  e  i p s i.On  -  -  -  I67OO I65OO _  -  190 3700 140 3890  180 l4o  247 200  256 2000 222 -  7300 10000  -  218 224  260 240  274 245  290 420  3870 4050  79 78  80 80  104 106  138 122  156 1200 132 1100  7780 7700  14 12  40 38  52 55  64  90 90  150 1620 146 1640  17000 13600  6200 5700  _  _  _  -  -  -  175 152  _  -  22 20  -  -  3450 3500  _  _  _  _  _  -  -  -  225 218  _  -  12 12  -  72 _  -  -  TABLE 1 2 Work-Hardening Parameters as a Function of Orientation f o r ^> AuZn Single Crystals (T = 2 9 3 ° K ; ~fT= x 10" /sec; at.#Au) 3  2 . 5  Test 1 1 5 1 1 6  Orientation (Fig.ii) "X l  6  1 2 1 2  1 3  1 2 2  • 1 1 9  3  3 0  To k6  k9  k6  4  3 6  kk  1 1 8  6 0 5  1 4  1 3 7  6 9  8 3 0 0  1 1 7 0 0  i 4 o o o  4120  85OO  1 2 2 0 0  i 4 o o o  -  4 i 6 o 3 9 1 0  -  -  3 6 5 0  _  -  4 0 0 0 3 7 4 0  -  -  -  4 2 2 0  7 6 0 0  1 0 5 0 0  1 1 6 0 0  3 7 2 0  7 3 0 0  1 0 3 0 0  1 1 7 0 0  1 1 2 0 0  3 9 0 0  7 8 O O  1 0 6 0 0  7  3 2  2 3  3 0 0 0  7 4 0 0  9 4 0 0  7 5 0 0  1 0 1 0 0  1 0 2 0 0  1 2  12  3 9 8 0  8  3 9 8 0  7 4 0 0  1 0 0 0 0  1 0 1 0 0  3 7 0 0  6 4 0 0  8 0 0 0  1 0  1 6  10  3 3 5 0 0 1 5  —  5 3 6 5 0 0  -  -  9 8 0 0  8 3 0 0  -  m  5 6  1 2 0  2 1 0  5 6  1 4 0  2 1 0  20400 2 1 4 0 0  1 2 0 0 0  1 7 1 0 0  -  -  1 2 8 0 0  1 9 4 0 0  -  II7OO  1 6 2 0 0  -  1 1 8 0 0  -  —  7 9 2 0  -  3 0 0  -  2 5 0  -  3 5  -  1 2 4  -  -  2 4  -  1 5 2  2 4  -  1 6 0  3 5  -  1 8 5  -  -  -  3 0 2  1 9 0 0  7 0 0 0  -  - 3 1 2  1 9 0 0  7 1 8 0  -  2 8 0  1 8 0 0  5 8 0 0  2 6 0  2 8 5  1 8 5 0  5 0 0 0  -  1 5 8  3 9 0 0  5 8 5 0  -  1 5 8  3 9 0 0  5 8 5 0  1 6 4  3 6 0 0  5 6 0 0  8 6  -  2 1 1  1 5 8  2 3 2  1 7 0 0 0  5 4  1 4 8  1 8 0  1 4 4 0 0  3 4  1 1 0  1 3 2  1 5 3 0 0  2 1 4  3 0  6 2  6 6  1 5 2 0 0  3 0  6 2  6 6  1 2 5 0 0  1 4  5 0  6 0  -  7 9 2 0  2 7 4 0  -  1 5 2  -  2 7 2 0  1 2 6  40  4 9 5 0 0  Q11  2 2 0  4 2  4 5 0 0 0  3 4 0  psi  1 0 0  1 7 5 0 0  • 1 8 0 0 0  -  0i  -I5O  8 0  -  7f>  Xn  - 340  -  1 9 8 0 0  1 3 6 0 0  "fill T f i n £  2 4 0 0 0  1 6 4 0 0  -II3OO  Y11  Yi  2 3 8 0 0  1 8 5 0 0  -  3 1  9  T  -  1 8  6 8  7 0  ^111  6  6 7  1 3 9  T11  2 6  6 1  2 5  psi  3 7 0 0  3 7 0 0  1 2 0  • 1 1 7  T11  fx  f  , 5 1 . 0  -  -  1 5 4  1 2 6  -  -  1 0  -  1 8  3 8 5 0 0  CO  ro  4-9.0 50.0 51.0  •  o A  I  0  I  50  I  100  I  150  1  200.  1  250  1  300  at. % Au at. io Au at. i Au  1  350  J  400  i  450  Temperature T°K Figure 3 3 .  Showing the v a r i a t i o n of y i e l d stress with temperature f o r Au-rich,Stoichiometric and Zn-rich @ AuZn single c r y s t a l s . ( t = 2 . 5 x lO^/sec; ? ~ 26°, X ~~ 2 0 ° ) x  1  500  8k  strengthening e f f e c t of the excess Au and Zn, i t i s seen that at temperatures above approximately whereas below For  example, at  both species impart similar hardening  200°K  200°K  Zn appears to be the more potent strengthening agent. .At"/at.  150°K,  deviation i s  crystals and jkOO p s i f o r Au-rich while at only 5 0 0 p s i f o r Au-rich. hardening above with  A0~  A  ~  9  0  0  133°K  0  psi/at.  o f ~ 1 7 0 0  ^  5  1  0  is  77°K,  p s i f o r Zn-rich  0  ^  2  7  0  for Zn-rich and  0  In p o l y c r y s t a l l i n e material, Causey  12  found that  was approximately equal on both sides of stoichiometry  psi/at  for'Zn-rich alloys and  /V.75OO  p s i / a t . i f o r Au-rich  C  a l l o y s ; at  77°K  a pronounced minimum exists at  ^ 5 0 - 5  at. $ Au so that the  hardening per at. "jo deviation from stoichiometry i s considerably less f o r Au-rich than Zn-rich a l l o y s .  It appears, therefore, that the effects of  deviations from stoichiometry on y i e l d stress are the same i n single and p o l y c r y s t a l l i n e AuZn. Since the absolute melting point of AuZn i s  9 9 8 ° K  temperature i s d i r e c t l y proportional to temperature i n degrees K. same range of homologous temperatures 0 . 0 8 • I O O * 1 0 1  stoichiometric AgMg  to 0 . 5 T  m  the homologous Over the  single crystals of  2 5 * 3 7  and NiAl  were observed to display y i e l d stress  variations s i m i l a r to AuZn increasing approximately three times i n the temperature i n t e r v a l . 2 5 T and NiTi  6 7  to . 0 8 T . P o l y c r y s t a l l i n e AgMg ' , N i A l 13  ffl  38  exhibited comparable y i e l d stress temperature dependence.  2 B  '  In  94  general the behaviour i s similar to bcc metals (for review see Conrad although the dependence i s not quite as strong.  6 7  )  The temperature dependence  of y i e l d w i l l be further considered in.the l i g h t of thermally activated deformation mechanisms,  Section 2 . 6  of the thesis.  The orientation dependence s l i p plane i n the [ 0 0 1 ]  o  f  T  0  resolved on the macroscopic  d i r e c t i o n i s shown i n Figure  . Although some  scatter i n the results i s evident, i t appears that'CTn i s approximately  8  5  001  3 -65 3 -67  Figure  5 4 .  Showing the  resolved  ( 5 1 . 0  Au;  at.  %  T  independent of o r i e n t a t i o n and o f t h e s e o b s e r v a t i o n s and parameter"V*/ w i t h  operative  may  be  g i v e n as  value o f f a  nine times h i g h e r than T o  f  o  {hko}  r  i n o r i e n t a t i o n 1 0 because o f the  do not  allow  I t can be  stress c a l c u l a t i o n s that system, namely system.  (OOl)  ( o k l ) [ l 0 0 ]  5  0  i n section  slip.  psi.  The  significance  [OOl]  1 1 ) was  ~ 3 5 0 0 0  resolved  p s i which i s about  It i s believed  that  s l i p for orientations  seen, t o o ,  was  not  as  ^  In  f a v o u r a b l e as  slip  r a t i o of 0 . 1  is 5 ° , [ 0 0 1 ]  ( 0 1 1 ) [ 1 0 0 ]  i s O.O5  w h i l e the  is slip  resolved  is  shear-  stressed- { h k o j ^ O O l ) on the  g i v e n o r i e n t a t i o n 1 0 the Schmid f a c t o r on a  system taken as  Figure  i n which *f  from'Schmid f a c t o r and  second most h i g h l y  the  suppressed  t e s t specimens.  I n o r i e n t a t i o n 1 0 where  plane  2 . 5 . 4 . 1 .  expected t o operate was  geometry o f the  s l i p on the  ( o k l ) [ l 0 0 ] ,  For the  8  f o r the g i v e n specimen diameter/gauge l e n g t h  l e s s than a p p r o x i m a t e l y 6 ° . thus r e s t r i c t e d .  3  i n o r i e n t a t i o n 1 0 (Figure  (hko)[OOl] d e f o r m a t i o n mode t h a t might be  grip constraints  ^  (112)-[ill] system and found t o be  3 5 ' i t i s shown t h a t  orientation.  t h e i r r e l a t i o n t o the v a r i a t i o n i n s l i p  o r i e n t a t i o n is. discussed  The on the  s t r e s s dependence on  yield 293°K)  =  observed  representative  f a c t o r on  ( 1 1 2 ) - [ i l l ]  86  Figure 35 -  Showing effect of specimen geometry on i n h i b i t i n g (hko)[00l] s l i p .  i s 0 . 4 8 , a difference of 9-5 times.  This i s greater than the difference  of about.9 times i n c r i t i c a l resolved shear stress for s l i p on the two systems.  Although (112)-[lll] i s the most prominent system i t was observed  in Figure 28 that some -TllOj s l i p may have occurred as w e l l . .The occurrence of t h i s extra mode can most l i k e l y be accounted f o r by the fact that stress on the (011)[l00] system probably becomes c r i t i c a l i n the i n i t i a l hardening stages past y i e l d . 2.5.1.2  The Work-Hardening Rate i n Stage I, Q The work-hardening  t  rates during stage I deformation 0  are  1}  shown i n Figure 36 f o r the three compositions tested where the results are plotted as Q±/M , where  i s the shear modulus characteristic of the operat modulii have been calculated as a function  of s l i p system i n Appendix 5-  -It i s apparent that Q  x  i s a minimum at i n t e r -  mediate temperatures of fv210°K, 295°K and 370°K f o r the stoichiometric, •Zn-rich and Au-rich crystals respectively.  The v a r i a t i o n of Q with temx  perature appears to be l i n e a r for the non-stoichiometric c r y s t a l s .  In fee  metals, stage T hardening rates have been observed to either increase 16*83  slightly,  remain constant  13'F> j 13 B  or decrease s l i g h t l y  140  in a monotonic  87  150  200  250  300  350  400  450  Temperature T°K Figure 36.  Showing the v a r i a t i o n i n stage I work-hardening with temperature.  rate Qj  fashion with increasing temperature, while i n bcc crystals Nb  and Ta , Q  1  was  found to increase with increasing temperature, from near zero at ^ . 0 8 T . The slope of the Ox>T curves i n AuZn crystals i s similar to Nb and Ta and considerably stronger than fee metals.  The occurrence of a minimum hardening  rate at intermediate temperature appears to be unique.  The very low values of 0  X  about room temperature  f o r Zn-rich  crystals i s believed responsible for the wavy nature of the corresponding flow curves (Figure 8 ) .  What probably happens i s that early i n the t e s t ,  some part of the gauge section deforms at a s l i g h t l y faster rate than the rest.  The cross-sectional area i n this region becomes s l i g h t l y smaller than  in the rest of the specimen giving rise to a corresponding increase i n the shear stress.  Since the rate at which s l i p planes harden with increasing  s t r a i n i s low, then continued s l i p i n t h i s reduced section w i l l occur more e a s i l y than i n the remainder of the specimen.  In t h i s manner l o c a l i z e d  88  thinning occurs along the gauge- analogous to a necking mechanism.  Once the  primary s l i p planes harden to such an extent that further s l i p requires an applied t e n s i l e stress greater than i s necessary to promote new s l i p packets, deformation i n the l o c a l i z e d area- ceases.  A new  "neck, i s formed and the  procedure repeats i t s e l f u n t i l the gauge section i s uniformly thinned. The dependence of 0  X  Figure 37-  on i n i t i a l orientation i s shown i n  While i n s u f f i c i e n t experiments were performed to permit unambiguous  comments on orientation e f f e c t s , i t i s noted that the hardening rate near the  001  Figure 57-  [001]  Showing the v a r i a t i o n i n stage I work-hardening with orientation. (51.0 at. i Au; T = 293°K)  rate  corner i s approximately two times greater than that near the middle of  the stereographic t r i a n g l e . 2.5.1.3  The End of Stage I The s t r a i n at the end of easy glide ~$  38.  1 X  i s shown i n Figure  The extent of stage I i s considerably greater f o r non-stoichiometric  compositions by the approximate  ratios of  stoichiometric crystals respectively.  :1 f o r Au-rich, Zn-rich and  Similar effects have been observed on  8  9  2001-  •  49.0  O  5 0 . 0  A  5 1 . 0  a t . % Au a t . $ Au a t . i Au  1 5 0  %  1 0 0  •H  H  0/-  M  W CH O  5 0  -P  c <u -p X  _L 1 5 0  2 0 0  2 5 0  3 0 0  400  3 5 0  450  Temperature T°K  Figure 3 8 .  Showing the e f f e c t o f temperature e x t e n t o f easy g l i d e .  a d d i n g s o l u t e elements the  t o fee m e t a l s .  1 6  '  1 7  '  1 4 1  and composition on the  An e x p l a n a t i o n  1 4 1  i s based  on  assumption t h a t easy g l i d e ends when the s t r e s s c o n c e n t r a t i o n around  c l u s t e r s o f d i s l o c a t i o n s on the p r i m a r y s l i p plane are s u f f i c i e n t l y l a r g e t o initiate yield  slip  on c r y s t a l l o g r a p h i c a l l y s i m i l a r secondary systems.  S i n c e the  s t r e s s i s r a i s e d on b o t h s i d e s o f s t o i c h i o m e t r y , the c l u s t e r s need  g r e a t e r s t r e s s f i e l d s t o move secondary d i s l o c a t i o n s and s i n c e the work h a r d e n i n g r a t e s a r e about the same, t h e n easy g l i d e must be more e x t e n s i v e to  e f f e c t the l a r g e r c l u s t e r s .  studies  (section  2 . 5 - 3 )  T h i n f o i l t r a n s m i s s i o n e l e c t r o n microscopy  v e r i f y the assumption t h a t d i s l o c a t i o n c l u s t e r s are  p r e s e n t on p r i m a r y s l i p p l a n e s and suggest t h a t c l u s t e r s g i v e r i s e t o the m a c r o s c o p i c a l l y d e t e c t e d d e f o r m a t i o n bands ( s e c t i o n of which l o c a l i z e d  secondary s l i p  2 . 5 . 2 )  i n the  vicinity  i s d e t e c t e d near the end o f stage I.  9 0  Except at temperatures below ~ 2 0 0 ° K , the extent of easy glide increases with decreasing temperature, which i s similar to the effect of deviations from stoichiometry and can be explained i n a similar manner. Since the y i e l d stress increases with decreasing temperature (Figure  33)  larger clusters and hence more extensive easy glide i s necessary to i n i t i a t e secondary s l i p .  The effects of temperature and composition on the stress at the end of easy glide  are shown i n Figure 3 9 -  For the  stoichiometric  •  Temperature T°K Figure 3 9 -  Showing the e f f e c t s of temperature and composition on the stress at the end of easy g l i d e .  crystals i t i s seen that lTi\ increases  ~  2  .  but remains constant at temperatures above temperature v a r i a t i o n .  5  times between 220°K,  220°K  and  150°K  similar to the y i e l d stress  Non-stoichiometric a l l o y s display a monotonic v a r i a t i o n  91  in T i i with temperature, decreasing by ~2 times over the range l80°K t o 400°K, . An interesting result i s obtained when one calculates the stress r a t i o s T " n / ^ f o r the three compositions at various temperatures and orientations, Table 13.  It i s found that the average r a t i o i s 2.4 - 0 . 2 ,  2.9 - 0.2 and 2.7 - 0.4 f o r the Zn-rich, stoichiometric and Au-rich a l l o y s respectively: within the l i m i t s statedT"-^/  appears to be independent of  CO  temperature,composition and orientation • suggesting that the work hardening mechanism i s unchanged over t h i s range of temperature, composition and orientation. TABLE 13 '^11/7 - as--a Function of Temperature, ^° Composition and Orientation s  Table  15.1 T°K Zn-Rich  192 2.8 3-5  243 2.4  293 2.4 2.1  348 2.7 2.4  155 5-1 3-1  183 5.1  -  223 2.2 5.2  293 2.4 2.9  181 2.5 2.8  217 3.8 2.8  260 2.9 2.0  293 2.5 2.8  373 2.6 2.4  403 2.2 2.0  1 3-1  •5  6 2.7  7 3-1  8 2.5 2.5  9 2.2  -  375 2.5 2.1  Avg  2.4 ? .2  Table 15.2 T°K Stoichiometric  .tii/r  Avg,  2.9 t .2  0  Table 13.3 T°K Au-Rich  Avg  2.7 + .4  Table 13-4 Orientation (Figure 11)  Wtb  Au-Rich  •3.0  2.5 2.8  -  -  -  Avg  •W r  2.7 t .4  The effect of orientation on the extent of easy glide i s shown i n Figure 40.  I t i s apparent that "jTn increases as ^  increases,  0  92 c o n s i s t e n t w i t h the observations t h a t 0 •TV-  while C  X 1  X  decreases w i t h  increasing  remains a p p r o x i m a t e l y unchanged.  001  101 Figure 4 0 .  2.5.1.4  Showing t h e v a r i a t i o n i n t h e e x t e n t o f easy g l i d e w i t h o r i e n t a t i o n . (51.0 a t . •% A u ; T = 293°K)  The Work-Hardening Rate i n Stage I I ,  0  X 1  - The temperature and c o m p o s i t i o n dependence o f stage I I  work-  h a r d e n i n g r a t e i s shown i n F i g u r e kl where the r e s u l t s are g i v e n i n terms of Q . X1  Sample f l o w curves from which 0  a l s o shown.  n  v a l u e s were r e l i a b l y t a k e n are  S i n c e p a r a - l i n e a r t y p e h a r d e n i n g was observed a t  temperatures  below ~150°K i t was q u e s t i o n a b l e whether o r not the s l o p e 0^ o f the segment of the f l o w curve c o u l d be r e l a t e d t o stage I I  hardening.  This  d i f f i c u l t y was p a r t i a l l y r e s o l v e d by s t u d y i n g the v a r i a t i o n i n s l i p structure during straining.  linear  line  The r e s u l t s o f t h i s s t u d y a r e r e p o r t e d i n a  l a t e r s e c t i o n o f t h e t h e s i s where i t w i l l be shown t h a t m u l t i p l e s l i p c o n t i n u a l l y a t 77°K, but o n l y i n minor amounts d u r i n g stage I I temperatures.  at  occurs  higher  P a r a - l i n e a r h a r d e n i n g was t h e n b e l i e v e d t o be r e l a t e d , i n p a r t  a t l e a s t , t o stage I I  hardening rates  since both types of deformation are  m a n i f e s t a t i o n s of m u l t i p l e s l i p processes.  The c o r r e s p o n d i n g l i n e a r work-  9 3  I 1 0 0  I 1 5 0  _J_ 2 0 0 .  ;  1 2 5 0  J 3 0 0  1  1  35O  1 + 0 0  Temperature T°K Figure 41.1.  Showing the v a r i a t i o n of stage II work-hardening Q with temperature.  X1  rate  .Strain Figure 4l.2.  Schematic flow curves. .Type A analyzed f o r l i n e a r hardening rates and Types C and D f o r stage II hardening rates; • Type B not analyzed".  9k  hardening rates at 77°K are plotted, therefore, i n Figure kl where they are connected with the Oix values, by a dashed l i n e . higher work-hardening  An explanation of the  rates during para-linear flow i s given i n section 2 . 5 . 4 . range - -225°K to  Over the temperature  350°K i n which 0 i X  could be r e l i a b l y evaluated, i t i s apparent that hardening rates during stage II decrease by  2.5  times from  M  to  /••  with increasing  temperature.  1200  500  This behaviour i s d i f f e r e n t from the very minor decrease with temperature in fee metals and a l l o y s , but resembles the quite pronounced decrease above 142*143 0.27 T i n Cd. I t i s also•interesting to note that 0 is strain m n  rate sensitive, decreasing with decreasing s t r a i n rate.  The flow curves i n  Figure 9 were analyzed f o r t h i s feature and the results are given i n Table TABLE 14 V a r i a t i o n i n Stage II Hardening Rate with S t r a i n Rate (T = 295°K, g.= 26°, X= 20°)  Yfsec." )  ©ii (psi)  112  2.5 x 10"  4  5900  1.36  60, 61  2.5 x 10"  3  7100  1.63  110  2.5 x 10"  2  9100  2.09  Test  1  Oil//'  x<10  3  It appears, therefore, that thermally activated recovery processes can take place concurrent with p l a s t i c deformation. Below  350°K, the e f f e c t of composition on S  lx  i s very small,  16*17*141 similar to fee systems.  '  At higher temperatures, however, non-  stoichiometric crystals exhibit an unusual sudden r i s e i n hardening rate, which i s somewhat similar to behaviour of bcc metals These results suggest that•the work-hardening  1*9  .1 T . • m mechanism i s probably the same 1  near  Ik,  95  for a l l compositions below 550°K, but d i f f e r s for stoichiometric and nonstoichiometric . crystals above t h i s temperature. The dependence of 0 Figure k-2.  I t can be seen that O n  approaches the [001]-[101] boundary  n  on c r y s t a l orientation i s shown i n decreases as the specimen orientation ( i . e . as "X-increases) which i s s i m i l a r  to the e f f e c t of increasing temperature and decreasing s t r a i n rate.  001  Figure k-2.  Showing the orientation dependence of stage II workhardening rate (51.0 at. $ Au;.T = 293°K)  •An intimate relationship between temperature, s t r a i n rate and orientation was noted e a r l i e r i n the discussion on deformation modes. • It was shown that the s l i p plane parameter "\|/. increases with increasing temperature, increasing"X and decreasing s t r a i n rate which was  interpreted  in terms of an increasing tendency f o r c r o s s - s l i p of screw dislocations. It i s once more apparent that these variables are related through t h e i r combined effect on stage II hardening rates. increasing ")£ and decreasing s t r a i n rate, 0 the observed v a r i a t i o n i n Q  1±  n  With increasing temperature, decreases.  i s simply a manifestation  It i s not known i f of the changing s l i p  plane and hence a property of the macroscopic s l i p plane per se, or i f i t  96 suggests that c r o s s - s l i p plays a dual role i n the deformation of AuZn, active as a dynamic recovery mechanism as well as governing the choice of s l i p plane. Work-hardening rates i n specimens oriented near the [OOl] corner (Figure 1 1 , orientation . 1 0 ) must be discussed separately. As already noted, i n these orientation {hko}  < 0 0 L >  systems no longer serve as the  primary s l i p modes; instead s l i p occurs on a systems, probably { 3 2 l }  { 2 1 l } -  (ill)system.  Secondary  < 1 1 1 > and (llO) (OOl), operate as well (Figure 2 8 )  but observations at fracture show that t h e i r contribution to the t o t a l d u c t i l i t y i s negligible with respect to the { 2 1 l } < l l l ) primary s l i p .  It i s  to be concluded therefore that the very high work-hardening rate and limited d u c t i l i t y are d i r e c t results of < 1 1 1 > s l i p .  The l i n e a r hardening slopesOg  wereanalyzed giving values of  In terms of shear modulus,  ^  3  8  ,  5  0  0  psi.  t h i s i s equal to Si w h e r e w a s calculated from the relationship (ill)  [ 2 1 1 }  ,-  6 0  ..  = 1(c -ci n  and c  4 4  • +  2  c 4 ) , Appendix 54  The e l a s t i c constants c  were taken from the data of Muldawer and Schwartze  .  c 2 X  1 1 ;  It is  immediately apparent that hardening .rates resulting from ( 1 1 1 ) s l i p are over 1 0 times greater than those associated with general £hko} ( 0 0 1 ) modes.  slip  The reason f o r t h i s unusually high value must i n some way be asso-  ciated with antiphase boundaries created by the motion of l a ( 1 1 1 ) type  2 superlattice p a r t i a l d i s l o c a t i o n s .  Possible superdislocation hardening  mechanisms are discussed i n section 2..5..U. It i s s i g n i f i c a n t that the l i n e a r hardening rate  0 ^ / l 5  12  observed i n p o l y c r y s t a l l i n e AuZn  i s considerably nearer the value //  in single crystals undergoing O-ll "> s l i p than the values  to /<Y  .^ 5 0 0  6 0  1 2 0 0  i n crystals deforming along ^ 0 0 1 > d i r e c t i o n . (It should be noted that quoted i n reference 1 2 was obtained using the shear modulus f o r { 2 1 1 } s l i p rather than f o r (hko} (.001)1)  While s l i p on the [hko)  <T001)  (111>  systems i s  97 believed to account for most of the p l a s t i c i t y observed i n p o l y c r y s t a l l i n e material, { 2 1 1 }  and { 3 2 l ]  traces were observed i n the v i c i n i t y of grain  boundaries, suggesting that < 1 1 1 > s l i p does occur.under rather high stress conditions.  In view of the very high hardening rate associated with < 1 1 1 >  s l i p in single crystals i t appears as. though.the rapid hardening i n polyc r y s t a l l i n e material below {hkl| < 1 1 1 > systems.  <v-300°K  i s controlled by d i s l o c a t i o n motion on  Hence, grain boundary hardening, rather than hardening  within the grain i s believed to control the flow stress.  This i s consistent  ' 12 with the Hall-Petch behaviour of AuZn  , where the Petch slope k^.. relating  the flow stress to the r e c i p r o c a l of the square root of the grain size i s observed to increase with increasing s t r a i n . . 2 . 5 . 1 . 5  Stage I I I Stage III i s characterized by rapidly decreasing  work-hardening rate with increasing s t r a i n . the stress T i n  a  In Figure  values of  i t i s shown that  "t the end of stage "II decreases with increasing temperature,  suggesting that a thermally activated recovery mechanism i s responsible f o r the breakdown of l i n e a r hardening.  Since only a few orientations investigated  gave.rise to three-stage work-hardening curves, i t i s d i f f i c u l t to comment with certainty on the quantitative e f f e c t s of specimen orientation o n T j i i i There i s some evidence to suggest  thatT"m  decreases as  X  increases,  but  u n t i l further experiments can be carried out, t h i s must remain as a tentative observation  only.  98 20  • o  r  A  49.0 at. $ Au 50.0 at. <?o Au 51.0 at. io Au  15  10  •H [fl  CO I  H H  200  250  3OO Temperature  Figure 1+3.  350  1+00  1+50  T°K  Showing the, effect of temperature on the stress at the end of stage I I .  2.5-1.6 .Maximum Shear Stress and D u c t i l i t y The maximum shear-stress (~ and t o t a l shear-strain to fracture m 2(j are shown i n Figures 1+1+ and 1+5 as functions of temperature and composition. tends to decrease with increasing temperature except over the intermediate temperature range 200°K to 300°K where a peak i s detected.  • 12  Polycrystalline  AuZn "' shows exactly the same v a r i a t i o n i n ultimate t e n s i l e strength versus temperature.  In both p o l y c r y s t a l l i n e and single c r y s t a l material, the  maximum flow stress decreases, by about 5 times between 77°K and 500°K. Although?"^ increases with deviations from stoichiometry i n both materials, the effect i s not nearly as pronounced as i t i s on the y i e l d stress. The shear s t r a i n to fracture V V  may be taken as a measure of  99  30 CO  5 0 . 0  P.  o  5I.O  25  I  CO CO  at. i Au at. # Au at. i Au  4 9 . 0  •H  20  cu  u  -P CO  u  15  o3  JB CO  10  X  I 1 1 0 0  200  300  400  500  Temperature"T°K Figure 44,  Showing the v a r i a t i o n of maximum shea'r stress with temperature.  at. % Au 50.0 at. % Au 0 at. <f> Au 49.0  200  300  500  Temperature T°K Figure 4 5 .  Showing the v a r i a t i o n of t o t a l with temperature.  ductility  100  d u c t i l i t y i n single c r y s t a l s .  From Figure 45 i t can be seen that the  temperature dependence of d u c t i l i t y may be divided into three d i s t i n c t regions: (A) from 77°K to 250°K, i n which Y  increases by about 6 to 10  times depending on composition; (B) from 250°K to 400°K where Y  decreases  b y ^ 2 times f o r Zn-rich crystals and about 5 times for stoichiometric and Au-rich a l l o y s ; (C) above 400°K where d u c t i l i t y appears to increase i n both Zn-rich and Au-rich c r y s t a l s .  Over the temperature ranges A and B, s t o i -  chiometric crystals are most ductile and Zn-rich crystals are the least ductile,.but i n region C, Zn-rich alloys display greatest d u c t i l i t y .  The  temperature range of maximum d u c t i l i t y . i s coincident with the peak flow stress. • 12 Polycrystalline d u c t i l i t y  exhibits i d e n t i c a l behaviour over the same  temperature range suggesting that similar mechanisms may be responsible for fracture.  U n t i l some crystallographic aspects of single c r y s t a l fracture  are presented i n section 2.5-2, further'discussion i s terminated. 2.5-2  •  S l i p Line Variation'During Deformation In addition to the direct evaluation of work-hardening rates,  supplementary studies are often performed to arrive at possible hardening models.  S l i p . l i n e studies on the surfaces of deformed c r y s t a l s , transmission  electron microscopy i n t h i n films, X-ray d i f f r a c t i o n , e l e c t r i c a l r e s i s t i v i t y and magnetic properties of. ferromagnetic materials are useful experiments from t h i s viewpoint. led to the development  Indeed detailed studies of the f i r s t two types have of two very prominent work-hardening theories of  stage II deformation, the long range hardening theory of Seeger  144>145  and  146  the pile-up theory of Hirsch.  Consequently information from both surface  s l i p l i n e v a r i a t i o n during deformation and the corresponding dislocation structure i n thin films i s believed to be of utmost importance i n under-  stand ing "the gsnersil dsfopina-'tion behaviour of metal c r y s t a l s .  Both, surface  101 s l i p l i n e studies and transmission electron microscopy experiments were carried out during the present investigations and the results are reported i n t h e i r respective sections 2.5-2  and 2.5-3•  While these studies were  not intended to give a detailed account of the work-hardening mechanisms, they were designed to add information which would a i d i n understanding  the  p l a s t i c behaviour of ^ AuZn single crystals. 2.5-2.1  Procedure Specimen preparation f o r s l i p l i n e analyses has been given i n  section 2.4.  Crystals of orientation 1 i n Figure 18 previously used to  determine primary s l i p traces versus temperature have been used to study •the v a r i a t i o n of s l i p l i n e s during t e n s i l e deformation at temperatures between 77°K and 473°K. oblique f i l t e r e d 2.5.2.2  Specimen surfaces were examined o p t i c a l l y under  l i g h t i n g with the Reichert  metallograph.  Observations In presenting photomicrographs showing the development of s l i p  l i n e s , i t was decided f o r the purposes of comparison to include the series already presented i n section 2.4  since they are characteristic of the s l i p  l i n e appearance during i n i t i a l flow. B ( i d e n t i f i e d i n section 2.4)  S l i p . t r a c e s on orthogonal faces A and  are shown as a function of strain at 77°K,  l40°K, 293°K, ,398°K and'473°K i n Figures 46 to 50 respectively. Schematic flow curves noting the strains at which observations were made are also shown.  Because of the non-crystallographic nature of s l i p traces, systems  are indexed with respect to approximate -{hkoji planes i n the (001) zone. For easy comparison, comments on the observations are summarized i n Table The results are discussed i n the context of a general discussion on workhardening,  section 2 . 5 - 4 . 2 .  15.  1 0 2  46.A.l  46.A.2  46.A.3  Figure 4 6 .  V a r i a t i o n i n s l i p l i n e s t r u c t u r e w i t h s t r a i n a t 77°K.  X  1 0 0  20  F i g u r e 4 7•  Variation in slip  120  line  structure with strain  a t l40°K. X 100  Figure  48.  Variation i n slip line structure with X 100 ( e x c e p t w h e r e n o t e d ) .  strain  at  293°K;  1 0 5  h  r  Figure  4 9 .  Ico  ^5  a)  Variation i n s l i p l i n e structure with s t r a i n at X 1 0 0 (except where noted)  398°K.  1 0 6  5 0 .  A.l  50.B.l  5 0 . A . 2  Figure  5 0 .  Variation i n s l i p l i n e structure with s t r a i n at  473°K.  X  1 0 0  TABLE 15 Comments on S l i p Line Variation During Deformation  Temp. °K  77  ll+O  293  Ref. Fig.  1*6  47  1+8  System (approximate with respect to plane) (Oil)I100J  (210)1001J  (310)L001J  598  1*9  f510)I001J  it-73  50  (100)[001]  Primary S l i p System Description Initial face A-coarse, slightly wavy; face B-fine, straight on both A and B, banded s l i p lines wavy on A and straight on B; relatively fine  increased waviness of face A traces; straight traces on face B and finely spaced increased waviness of face A traces; face B traces remain straight and finely spaced  See note, p.XC9  Secondary Slip System Description  Variation with Strain coarsen on both faces A and B  Yes No Yes  development of fine traces between "bands"  Yes  face A traces coarsen,become wavy on a largescale with wave traces lying close to (100); traces B adopt slight wavy appearance both traces A and B coarsen while traces A become profusely wavy  Yes  tt  First Detected at yield  during last few percent strain at end of stage I  No  -  Yes  at ~50jt  strain (i.e. at peak flow stress)  Systems (Oil)I111J  (110)[001]  {321} and  {211) (021)U00j  -  Initial minor amount  11  TI  short, straight detected near shadowed areas p a r a l l e l to microcracks short, curved appearance, occurring in localized regions i n v i c i n i t y of deformation hands -  (010)[100] long,  slightly wavy, coarse  Contribution to Total Strain -)f Variation with Strain coarsen slightly coarsen and increase i n number remain fine, straight, widely spaced coarsen, new traces develop around smalle r deformation hands  -  remain app rox imat ely constant i n number, but coarsen  £ f i n a l (degree) Measured Calculated  (from  [100])  51  (from  Amount  ~xaf,  [100])  lh  15  11  10  -  -  -  -  Negligible  Negligible  -  TABLE 15  (Continued) .Fracture Appearance  Other Crystallographic. Features Temp. °K  77  iko  Type  F i r s t Detected  microcracks  at  microcracks  during l a s t fewpercent s t r a i n  deformation bands  293  398  s l i p line clusters v i s i b l e on face B deformation bands s l i p line clusters v i s i b l e on face B deformation bands  kO&  •Plane  (001) within 5 degrees  yield . onset of stage I I I  Variation with S t r a i n apparent increase i n length  (001) within .5 degrees Section  -  (100) within 5 degrees  coarsen  2.5.2.3 hQPJo  (near maximum flow stress)  (100) within 5 degrees Section  yield  fracture surface p a r a l l e l to microcracks  2.5.2.3. -LJ-U..  Section yield  fracture surface p a r a l l e l to microcracks  2.5.2.3  coarsen; at fracture gauge section "saturated" with clusters  fracture surface p a r a l l e l to f i s s u r e s , which appear at begininj of stage III i n v i c i n i t y of s l i p clusters fracture plane p a r a l l e l to cluster  Note re Table 15: Contribution of secondary systems to t o t a l s t r a i n was estimated by calculating specimen reorientations assuming various amounts of secondary s l i p , then comparing the calculated with the measured values. .Reorientation calculations were performed using the relationship: 49  Sin  jrt .= lo 1.  Sin <£  0  1  where the suffixes o and i refer to the o r i g i n a l and instantaneous values of gauge length 1 and angle £ between the t e n s i l e axis and the s l i p direction.  110  • 2.5.2..3  Deformation Bands  2 . 5 . 2 . 3 . 1 Characteristics A f a i r l y common feature of the surface appearance of the deformed c r y s t a l s i s the occurrence of a series of markings traversing the  primary s l i p traces, termed deformation bands.  The bands can be des-  cribed as s t r i a t i o n s which can be seen under very low magnification; at high magnification, they seem to disappear.  Deformation bands were  detected i n a l l orientations at room temperature, except those near the [001] corner where ( 1 1 2 ) [ i l l ] s l i p predominates, and at a l l temperatures between l40°K and 398°K f o r a constant orientation near the middle of the stereographic t r i a n g l e .  Bands form ir. the very early stages of deformation.  Their  v a r i a t i o n i n appearance during straining i s best i l l u s t r a t e d i n the room temperature observations,-Figure 48. -Although the distance between any two i s not quite uniform, the average spacing i s approximately 0.2 to 0 . 3 mm. and the average width about 0 . 0 3 to 0.04 mm.  As s t r a i n increases, the bands  increase i n i n t e n s i t y and become wavy (Figure 48.B.2),while the average separation appears unchanged.  I t can be seen that the primary s l i p traces  crossing deformation bands change t h e i r d i r e c t i o n , s l i g h t l y at f i r s t (Figure 48.B.l),but quite markedly at higher strains (Figure 48;B.':2). The appearance of the s l i p l i n e s suggests a change of elevation at the band s i t e and the change i n s l i p l i n e d i r e c t i o n indicates that the material within the band lags behind the matrix during l a t t i c e reorientations. These markings closely resemble similar structures observed i n aluminum single c r y s t a l s .  1 4 7  Ill  2 . 5 . 2 . 3 . 2 Crystallographic Nature During the early stages of deformation, the boundaries of the bands may be defined as plane surfaces.  Two surface trace analyses were  subsequently performed and the results are shown stereographically i n Figure 51101  Figure 5 1 .  Stereographic projection of deformation band poles versus c r y s t a l orientation and test temperature.  It i s apparent that the poles cluster round the [OOl] s l i p d i r e c t i o n to within, an average o f ~ 8 ° , suggesting that the s l i p d i r e c t i o n plays an important role i n deformation band formation. orientation or temperature were detected.  No systematic effects of  112 2 . 5 . 2 . 3 . 3 Mechanism of Formation Deformation bands are a f a i r l y common feature of the deformati of cubic c r y s t a l s .  Two explanations have been advanced to account f o r 147*148*149  t h e i r formation.  From a macroscopic aspect  i t i s believed that  bands are caused by bending which occurs as a result of constraints at the specimen loading grips, or by inhomogeneous l a t t i c e rotations where one section of the c r y s t a l s l i p s more than i t s neighbours.  The band planes  produced move i n the d i r e c t i o n of s l i p u n t i l two of opposite sign meet and become stuck.  .A planar obstacle i s consequently formed against which l a t e r  dislocations p i l e up and create the deformation band.  This explanation  accounts f o r the observed scale, but does not explain the fundamental dislocation processes by which bend planes are formed. 150  . Mott band formation.  has proposed a more detailed theory to account f o r  His idea of the d i s l o c a t i o n arrangement within a band i s  shown i n Figure 52. -Walls of positive edge dislocations are pushed by the applied shear stress and arrive from the right while walls of edge d i s l o c a -  Figure 52.  Schematic representation of dislocations i n deformation bands, (after M o t t ) 1 5 0  115  tions of opposite sign and roughly equal strength a r r i v e from the l e f t , thereby forming a deformation band.  The positive w a l l causes the  lattice  to t i l t downward while the negative w a l l effects.an upward curvature.  The  mutual a t t r a c t i o n of p o s i t i v e and negative dislocations makes these con151  figurations f a i r l y stable. •Mott's edge d i s l o c a t i o n wall-model necessarily implies that the deformation bands are perpendicular to the s l i p d i r e c t i o n .  In  •I4yy148  deformation bands l i e along (110)  aluminum  the s l i p d i r e c t i o n i n support of Mott's theory.  planes perpendicular to In-AuZn i t was  although the bend planes are not quite perpendicular to the [ 0 0 1 ]  shewn that slip  d i r e c t i o n , they tend to adopt orientations centered about t h i s pole.  The  view i s taken therefore, .that a mechanism similar to that proposed by Mott can account f o r the band formation i n .2.5-2.4  j^'AuZn.  Microcracks  The nucleation and propagation of microcracks along  {lOO}  planes i s a commonly observed fracture mode i n bcc metals deformed at low 152*153*154  temperatures.  In considering the i n i t i a t i o n  structures, C o t t r e l l with  1 5 5  of a crack i n bcc  has suggested that the formation of dislocations  a[OOl] Burgers vectors .is an important step.  According to the  reaction: ,la[lll]-+ la[111] = a[00l] 2  (25)  2  an edge d i s l o c a t i o n of Burgers vector a [ 0 0 l ] can form from a combination of two dislocations with Burgers vectors l a [ i l l ] and l a [ i l l ] g l i d i n g i n the 2  2  (101)  and (101)  [010]  and i s a pure edge with glide plane ( 1 0 0 ) .  planes respectively.  The product d i s l o c a t i o n l i e s along  are normally not mobile i n bcc l a t t i c e s ,  Since  a [ 0 0 l ] dislocations  Cottrell'suggests that they act as  b a r r i e r s against which other dislocations p i l e up and eventually nucleate a  Ilk  crack.  Cottrell's mechanism is not likely to be responsible for crack  nucleation in  AuZn since slip trace analyses show that [OOl] dislocations  are highly mobile. The crystallographic similarity between the high temperature deformation bands and the low temperature microcracks (i.e. both are approximately coincident with (001) planes) suggests that similar edge dislocation wall mechanisms may be responsible for their formation. Whereas stress concentrations resulting from wall formation may be relieved through the operation of secondary {hko} (00]) slip systems at 293°K (Figure k&.B.2-), they appear to be relieved through crack formation at low temperatures, presumably because the increased c r i t i c a l shear stress for {hko] (001) slip renders secondary slip unfavourable.  •  The increasing susceptibility  for crack formation at low temperatures, therefore, probably accounts for the decreasing ductility, region A in Figure k^>.  115  2.5-3  Transmission Electron•Microscopy of Thin Films  2 ..5.3 .1  Introduction Considerable dispute has arisen i n the l i t e r a t u r e concerning  the degree to which dislocation arrangements observed in.thin films are 144*172*173  representative of configurations i n the bulk material.  144  Seeger  has c r i t i c i z e d the use of t h i n f i l m microscopy observations to support work-hardening theories, on the grounds that long-range stress f i e l d s present i n the cold-wbrked  state and extending over distances large with  respect to the usual f o i l thickness are to a large extent relaxed during the process of preparing a f o i l from the bulk, since the surface of the f o i l s must be stress-free.  The relaxation i n stresses must then effect a  change i n the d i s l o c a t i o n arrangement.  The degree of rearrangement i s  believed to depend•on the stacking f a u l t energy of a material being greater 173  .the higher the energy.  Hirsch,  on the other hand, takes the view that  since d i s l o c a t i o n distributions obtained from etch-pit studies are i n s u f f i c i e n t l y good agreement with the results of thin f i l m microscopy, the l a t t e r may. be regarded as being reasonably representative of the bulk, at least as regards the o v e r a l l d i s l o c a t i o n d i s t r i b u t i o n . In recent years more elaborate specimen preparation techniques have been established i n attempts to preserve the d i s l o c a t i o n arrangements that are t r u l y c h a r a c t e r i s t i c of crystals undergoing p l a s t i c deformation. 174  Dislocation arrays have been pinned by both p r e c i p i t a t i o n techniques 175*176  and neutron i r r a d i a t i o n exposures.  In the case of Cu crystals  i r r a d i a t i o n pinning did not s i g n i f i c a n t l y affect the d i s l o c a t i o n arrangement since similar structures were observed i n f o i l s prepared i n the more 164  conventional manner.  U n t i l more experiments  of this nature are carried  116  out, i t must "be assumed that the unpinned structure i s f a i r l y characteristic of the pinned bulk arrays.  What must also be evaluated i s the effect of  i r r a d i a t i o n and subsequent point defect creation on d i s l o c a t i o n structure.  The purpose of the present study i s to examine the d i s l o c a t i o n d i s t r i b u t i o n i n annealed and deformed' AuZn single crystals tested at room temperature.  I r r a d i a t i o n experiments were not carried out and i t i s  therefore not possible to estimate the degree to which the f o i l structure i s c h a r a c t e r i s t i c of the bulk.  Recently, martensitic transformation  products have been observed near the edges of Zn-saturated (53.9 at. $ Zn) AuZn f o i l s  177  s i m i l a r to the twin-like marking observed by Causey  1  2  . In  the present study, however, s i m i l a r markings were not detected f o r any f o i l orientation.  Any rearrangements that occurred during thinning, therefore,  could not be attributed to stress f i e l d s of a transformation product. 2.5-3-2  Procedure Because of d i f f i c u l t i e s  from.3 mm. diameter c r y s t a l s , large 5  experienced i n preparing thin f o i l s m m  - diameter Au-rich (51-0 at.$> Au)  crystals oriented near the middle of the stereographic t r i a n g l e were employed to study d i s l o c a t i o n d i s t r i b u t i o n .  The room temperature work-  hardening behaviour of the larger crystals agreed well with the small c r y s t a l behaviour.  Specimens were mounted .in epoxy resin and strained i n  the usual manner. To obtain information about the three-dimensional nature of dislocation distributions i n deformed c r y s t a l s , i t i s necessary to examine f o i l s of several orientations.  Specimens of three orientations were  obtained from deformed crystals by spark machining discs /"ulmm. i n thickness: (1) p a r a l l e l to the (hko) glide plane and p a r a l l e l to the [001]  117 •Burgers vector, termed (hko)  section;  (2) p a r a l l e l to the (110) plane and p a r a l l e l to the Burgers vector, called (110) and  section;  (3) perpendicular to the glide plane.and p a r a l l e l to the Burgers vector, termed perpendicular  (hko).section.  Discs were cut using surface s l i p traces as a guide; the orientation of sections (2) and  (3) was  checked with the back-reflection Laue X-ray  technique. Thinning was  achieved by repeatedly jet-machining  electro-chemically p o l i s h i n g  1 7 8  the discs in a mixture of \yjo hydrochloric  acid, 50fo ethyl alcohol and 5$ glycerine at -20°C and 12 volts.. e l e c t r o l y t e was  contained  alcohol and s o l i d C0 . 2  and  The  i n a pyrex beaker cooled in a bath of methyl  -To avoid s t r a i n i n g the f o i l during thinning the  usual technique of lacquering the outer rim of the specimen was abandoned. After thinning, specimens were thoroughly washed i n d i s t i l l e d water then rinsed i n ethyl alcohol'and kept i n a dessicator u n t i l examined. Observations were made on a Hitachi Hu 11A electron microscope operated at 100 KV.  Contrast was  through~10° during examination. was  obtained  varied by t i l t i n g the specimens  A selected area d i f f r a c t i o n pattern  from each area photographed to permit subsequent analyses of  d i s l o c a t i o n arrangements r e l a t i v e to prominent c r y s t a l directions.  2.5.3.3  Observations  2.5-3.3.1- As-Grown Structure Dislocation structures observed i n f o i l s prepared from asgrown c r y s t a l s are shown i n Figure 53-  Dislocation density i n as-grown  1 1 8  3&000X  ZS'ooox 5 3 . 1  Figure 5 5 •  5 3 - 2  E l e c t r o n micrographs of d i s l o c a t i o n structure i n as-grown crystals.  7  /  3  c r y s t a l s i s estimated as ~10 cm/cm  since on the whole only a few d i s l o c a -  tions were v i s i b l e i n any photomicrograph corresponding t o a f o i l area of -7  *»10  P  cm . Figure 5 5 - 1 represents the most commonly observed as-grown  structure, showing rather long, s t r a i g h t d i s l o c a t i o n s , while Figure  55-2  represents a less commonly observed zig-zag d i s l o c a t i o n structure s i m i l a r n  to e q u i l i b r i u m configurations found i n  p -brass.  179*180  The unusually  high density seen i n Figure 55.2 i s not t o be associated w i t h zig-zag d i s l o c a t i o n s since s i m i l a r shapes were observed i n other low density areas. In an e l a s t i c a l l y i s o t r o p i c c r y s t a l w i t h no applied s t r e s s , the concept of d i s l o c a t i o n l i n e tension implies that the e q u i l i b r i u m p o s i t i o n of a d i s l o c a t i o n running between pinning points ( f o r instance, f o i l surfaces) i s a s t r a i g h t l i n e , s i m i l a r t o the structures i n Figure 5 3 . 1 . On the other hand, zig-zagged d i s l o c a t i o n s are considered t o be a d i r e c t 179  r e s u l t of c r y s t a l anisotropy.  A s t r a i g h t d i s l o c a t i o n which i s i n a  high energy d i r e c t i o n may be unstable w i t h i t s t o t a l energy decreasing i f  1 1 9  '  i t changes to a zig-zag shape. A  =  31  Calculated from Zener s  2 C and using that data of Schwartz and Muldawer, (Cli-C )  relationship 64  4 4  the degree of  1 2  e l a s t i c anisotropy A i n (5AuZn i s 3 - 3 >  which i s probably high enough to  render certain dislocations unstable, and hence account f o r their zig-zag shape. 2 . 5 . 3 - 3 . 2  V a r i a t i o n i n Dislocation Structure During Deformation at  2 . 5 . 3 . 3 . 2 . 1  295°K  fhko) Section The effect of room temperature deformation on dislocation  structure i n f o i l s cut approximately p a r a l l e l to the glide plane and containing the [ 0 0 1 ] Burgers vector i s shown i n Figures 5 ^ d 5 5 a n  The  structures shown are t y p i c a l of those observed i n several f o i l s prepared from crystals strained 3 5 $ "to the beginning, of easy g l i d e , Figure ^k, and 1 3 0 $ to the end of easy glide, Figure 5 5 -  The most outstanding characteris-  t i c of the d i s l o c a t i o n structure i s the long, generally straight arrays of d i s l o c a t i o n bundles.  With increasing deformation, the dislocation content  of the bundles increases while the average distance between any two to decrease s l i g h t l y  from~2  to  1 . 5  microns.  appears  At the end of stage I, several  areas along neighbouring bundles appear bridged by dislocations p a r a l l e l to those within the bundle, giving rise to a rectangular shaped c e l l structure. •Structural developments were not followed into stage II because of the increasing d i f f i c u l t y  experienced i n preparing s a t i s f a c t o r y f o i l s from the  more heavily work-hardened c r y s t a l s .  Crystallographic reference directions are shown on most electron micrographs.  I t i s readily apparent that the d i r e c t i o n of the  bundles i s perpendicular to the operative s l i p d i r e c t i o n [ 0 0 1 ]  suggesting  that the dislocations comprising the arrays possess predominantly edge  120  Figure  Figure 5 4 .  54.1  E l e c t r o n micrographs of d i s l o c a t i o n s t r u c t u r e at the b e g i n n i n g o f easy g l i d e ; (hko) s e c t i o n s p a r a l l e l t o m a c r o s c o p i c s l i p p l a n e . ( Y = 35$)  121  Figure 5*1.2  122  Figure 5 5 - 1 Figure 5 5 .  Electron micrographs of d i s l o c a t i o n structure at the end of easy g l i d e ; (hko) section p a r a l l e l to macroscopic s l i p plane. ( V = 1 3 0 $ )  1 2 3  124  character. graphs.  The absence of screw dislocations i s noticeable i n a l l micro-  Analysis of dark f i e l d patterns employing the g_'b = 0 i n v i s i b i l i t y 181  criterion  ,where g i s the d i f f r a c t i o n vector responsible for the d i s -  location contrast and b i s the Burgers vector, showed that the Burgers vector of the dislocations i n the bundles was  consistent with [OOl].  However, detailed Burgers vector•analysis to determine possible non-[00l] dislocations i n the clusters was  not carried out since high resolution dark  f i e l d photographs could not be obtained with the existing f a c i l i t i e s . 2.5.3.3.2.2  (110) Section Because of the non-crystallographic nature of the macroscopic  s l i p plane, sections p a r a l l e l to the fundamental (110) glide plane containing the s l i p d i r e c t i o n [001] were also examined. seen a f t e r 35$ deformation  are shown i n Figure 5 6 .  Typical structures I t can be seen that no  major differences e x i s t between the d i s l o c a t i o n structures on the macroscopic and fundamental glide planes, although the length of the clusters (bundles) on the (110) plane appears s l i g h t l y greater than on the glide plane suggesting that the edge array may  (hko)  l i e along [ l l O ] d i r e c t i o n s .  The observation that the bundles penetrate the f o i l should not be taken as evidence against t h i s suggestion, since even i n f i n i t e l y  long bundles would  pass through the f o i l which .deviated s l i g h t l y h>k°) from (110).  Because  of this deviation the length of the bundle in.the parent c r y s t a l i s uncertain.  During examination i n the microscope, f o i l s t i l t e d to bring many  (110) r e c i p r o c a l l a t t i c e  spots into r e f l e c t i o n showed considerably longer  bundles than are shown i n Figure 5 6 , often strung over distances of suggesting that the bundles are at least t h i s long.  ,  Figure  %.  E l e c t r o n metallograph of d i s l o c a t i o n s t r u c t u r e at the b e g i n n i n g o f s t a g e I ; (11.0) s e c t i o n c o n t a i n i n g t h e  Burge.rs v e c t o r  [00l].  1 2 6  2 . 5 . 3 . 3 . 2 . 3  Perpendicular (hko) Section Sections perpendicular to the macroscopic s l i p planes and  containing the [OOl] s l i p d i r e c t i o n were examined to determine the extent to which these two-dimensional bundles formed walls perpendicular to the glide plane.  T y p i c a l structures observed a f t e r 5 5 $ s t r a i n are shown i n  Figure 5 7 - In comparison with the bundled arrays on the glide planes, contrast e f f e c t s from the perpendicular sections are due to d i s l o c a t i o n s passing almost v e r t i c a l l y through the f o i l s .  The piercing character of the  dislocations was v e r i f i e d by t i l t i n g the specimen stage and observing the decreasing projected length of d i s l o c a t i o n onto the f i l m plane, Figure 5 8 , as the f o i l approached a r e f l e c t i n g position e f f e c t i v e l y perpendicular to the ( 1 1 0 )  glide plane.  From these sections d i s l o c a t i o n density at the /  9  start of easy glide i s estimated as ~ 1 0  3  cm/cm .  In Figure 5 7 i t can be seen that the [OOl] s l i p d i r e c t i o n i s perpendicular to the dislocations, again suggesting that edge d i s l o c a t i o n s are present i n the f o i l .  predominantly  In Figure 5 7 - 1 dislocations are  observed to be along rows approximately.parallel to both [ 2 1 0 ] •The same trend i s evident i n Figure 5 7 . 2 ,  and  [OOl].  although the rows appear shorter.  These observations suggest that the two-dimensional bundle arrays are i n fact low walls ( ^ 2 ^ high) of edge dislocations.  Since the projected d i s -  location length decreased near the ( 1 1 0 )  r e f l e c t i o n s i t i s believed that  the walls are perpendicular to the ( 1 1 0 )  fundamental glide plane.  The  [OOl] rows suggest that some of the bundles may be narrow carpets of dislocations l y i n g i n the glide plane.  2 . 5 . 3 . 4  Discussion Three points must be explained: ( 1 ) the nature of the bundled  1 2 7  5 7 - 2  Figure 5 7 .  E l e c t r o n micrographs of d i s l o c a t i o n structure at the beginning of stage I ; s e c t i o n p e r p e n d i c u l a r t o the m a c r o s c o p i c s l i p p l a n e and c o n t a i n s t h e [ 0 0 1 ] Burgers vector.  57.1  57-2 Figure 5 7 .  E l e c t r o n micrographs of d i s l o c a t i o n structure at the b e g i n n i n g o f s t a g e Ij s e c t i o n p e r p e n d i c u l a r t o t h e m a c r o s c o p i c s l i p p l a n e and c o n t a i n s t h e [OOl] Burgers vector.  128  129 array, (2) the mechanism of the wall formation, and (3) the absence of screw dislocations.  These points w i l l be considered  in.order.  Edge d i s l o c a t i o n bundles ("clusters", "strands" or "braids") are a common feature of the stage I d i s l o c a t i o n structure on the primary 144;  164;165)182  s l i p plane i n many metal c r y s t a l s of fee hep  1 6 7  '  1 8 3  structures.  dislocations.  7,8>160  , bcc  and  Characteristic,.too, i s the absence of screw  The edge clusters are usually arrays of d i s l o c a t i o n dipoles,  i . e . close p a i r s of dislocations of opposite sign lying along roughly paral l e l planes that may be frequently linked t o form narrow closed loops, of dislocation line.  The bundles observed i n AuZn are also believed to be  comprised almost e n t i r e l y of d i s l o c a t i o n dipoles which often l i n k to form closed loops as i s seen p a r t i c u l a r l y well i n Figure ^>k.  The bundles are  comprised of approximately equal numbers of edge dislocations of opposite sign, as no net contrast changes are observed' across the arrays, even at the higher stresses near the end of stage I. A model f o r dipole formation  i n Mg c r y s t a l s has been given by  167  Hirsch and L a l l y .  Assuming that approaching edge dislocations of oppo-  s i t e signs on p a r a l l e l s l i p planes trap one another, bands of dipoles form i f the distance between the s l i p plane i s less than a certain c r i t i c a l distance.  This model i s believed to account for dipole formation  i n AuZn  as w e l l . -Assuming that s l i p occurs i n part through the operation of Frank-Read sources, then at higher stresses when more sources operate, the distance between active s l i p planes decreases.  Once the applied stress  reaches a c r i t i c a l value, thought to be near the stress at the onset of stage I, s u f f i c i e n t s l i p planes are active and edge trapping may begin, forming the f i r s t dipoles.  With increasing deformation, clusters of  dipoles. are expected to develop i n the v i c i n i t y of the o r i g i n a l s , giving  130 rise  to an increased density of dislocations within the c l u s t e r s , and  subsequent wall growth.  The similarity- between the crystallographic nature of the previously reported macroscopic deformation bands and the microscopically observed edge d i s l o c a t i o n walls suggests that the mechanism of formation both structures may  be s i m i l a r .  of  I f edge d i s l o c a t i o n trapping can account  for both structures,,as suggested, the difference i n scale between the bands and clusters may  be explained  i f i t i s assumed that p e r i o d i c a l l y extra-  heavily populated clusters form which can give r i s e to an o v e r a l l l a t t i c e t i l t , large enough to be detected  optically.  Whether or not screw d i s l o c a t i o n a n n i h i l a t i o n by c r o s s - s l i p 1 6 7  as suggested by-Hirsch and L a l l y  can account for the apparent absence  of screws i n AuZn i s not known since sections inclined to the Burgers vector were not studied.  Because of the ease with which c r o s s - s l i p occurs  in AuZn and since the f o i l s examined were a l l p a r a l l e l to the Burgers vector of the primary d i s l o c a t i o n s , i t i s highly probable that screw d i s locations escape by c r o s s - s l i p during the thinning process as suggested by 144  Seeger. The foregoing discussion presupposes that the dipoles are formed i n the bulk of the c r y s t a l during deformation.  Evidence f o r t h i s  17=;  has been observed i n Cu crystals- where i t was dipoles. are present  shown  that d i s l o c a t i o n  i n f o i l s prepared from specimens subjected to neutron  i r r a d i a t i o n pinning while s t i l l under stress.  Clearly, similar irradiation  under load of AuZn c r y s t a l s i s necessary before i t can be ascertained whether or not the dipole structure i s t r u l y representative of the bulk arrangements, or whether i t forms, i n larger part, due to a relaxation of stresses within the f o i l s during thinning.  131 2.5-4  Discussion  2.5.4.1  Y i e l d Stress Variation with Orientation It was shown that the y i e l d stress resolved on the macroscopic  s l i p system {hko} (001) i s apparently independent of orientation (section 2.5-1-1); while the s l i p plane i s orientation sensitive (section 2.4.5-3)However, the s l i p plane i s not always the most highly stressed plane i n the (001) zone suggesting that the y i e l d stress on non-crystallographic planes should be a continuous function of t h e i r position i n the zone. To resolve t h i s apparent paradox, an attempt w i l l be made to predict the r e l a t i v e v a r i a t i o n of y i e l d stress with orientation and then to note to what extent the apparent y i e l d stress "invariance" with orientation i s consistent with the predictions.  This treatment i s based on the Taylor  44  analysis 4R  used recently to analyze non-crystallographic  , rp 4 8  ana i a  s l i p i n Fe-3%  single c r y s t a l s .  The  resolved shear stress  necessary f o r the onset of  p l a s t i c flow on the potential macroscopic s l i p plane i s given by the expression (Appendix .3) :  T ("y0 where  0~  Y  = <r s i n £ cos  i s the t e n s i l e stress at y i e l d  section 2.4.3).  cos ("X-V)  0  Assuming that  tj"("l/)*  ("l|/",  "y.  and ^  (26)  are defined ( i n  i s independent of % , as suggested  by Figure 34, then on d i f f e r e n t i a t i n g (26) with respect t o * y and rearranging, i t i s found that:  .d  T  = mn  tan CX-T|/)  Y  which gives the v a r i a t i o n of the shear resistance with the angle Integrating  (27)  (27) between the l i m i t s 0 and  gives the espression:  .  132  ( y  tan  In  - y  (28)  )<ay  no* i s the resistance to shear on the ( 1 1 0 )  where  (V)*  From the  (p£ )j relationship, Figure 2 5 , equation ( 2 8 ) was  experimentally determined solved f o r T  plane.  hy employing- Simpson' method f o r graphical integration. s  HO  curve i s shown i n Figure 59 and i s compared with the  The r e s u l t i n g  experimentally determined values of y i e l d stress versus orientation from Figure 3 4 . for  (110)  The experimental values are given r e l a t i v e to the y i e l d stress s l i p , taken as  3 8 5 O  psi.  O  1 10  o n V ( X ) data  O  1 05  0  0  iio  1 00  95  Experimental Data  O  -  0  °  0  —  O  8  0  90  85 0  80  1  5  1  10  1 •15  1  20  1  25  I 30  I  I 35  ! 40  45  Figure 5 9 • Showing the experimental compared with.the predicted values of c r i t i c a l resolved y i e l d stress r a t i o versus ' W .  Since the predicted orientation dependence of the r e l a t i v e y i e l d stress varies by less than 5 $ over the orientation range 0 4 "Vi/^.45 0 i t i s d i f f i c u l t to ascertain whether or not the somewhat scattered experi-  .133 mental results display a similar trend.  The apparent "invariance" i n y i e l d  stress with orientation i s therefore not i n disagreement with the predictions. C l e a r l y , before d e f i n i t e statements can be made concerning the y i e l d stress dependence on orientation, experiments giving highly reproducible results must be performed. .If i t i s assumed that the fundamental glide system i s (110)[001] as suggested e a r l i e r i n the discussion of non-crystallographic s l i p  (section  2 . 4 . 6 . 1 ) , then the present results imply that as X increases the c r i t i c a l For constant X and  resolved shear stress f o r (110)[001] s l i p decreases.  varying ^ , the c r i t i c a l resolved shear stress on (110)[001] i s constant. This may be explained by considering again the concept of dissociated screw dislocations contained on the (100) planes (section 2 . 4 . 6 . 1 ) .  As "X-  increases, so does the r a t i o of the shear stress resolved on (100)[001] to that on (110)[001] implying an increasing tendency f o r d i s l o c a t i o n  recombina-  t i o n r e l a t i v e to that f o r bowing-out i n the s e s s i l e to g l i s s i l e transformat i o n process (Figure 31)-  Consequently, i f recombination i s the mechanism  controlling the y i e l d stress, then the c r i t i c a l resolved shear stress f o r (110)[001] s l i p should decrease with increasing X >  a  s  observed.  Since  t ( 100)[001]/T( 110)[001] i s independent of £ , then the tendency f o r recombination r e l a t i v e to the bowing-out process .is unchanged and the y i e l d stress resolved on (110)[001] should be constant, again i n agreement with experiment. 2.5.4.2.  Work Hardening It was shown that the form of the work hardening curves of  ^J'AuZn single crystals changes markedly over the temperature range 77°K to 475°K from para-linear at low temperatures (below ~-.15 T ) to multi-  134 stage at intermediate temperatures  ( ^ . 2 0 to .35 T ) m  followed by work-softening at higher temperatures  a n (  ^ then to parabolic  (above'*-.4 T ) .  In this  respect AuZn i s e n t i r e l y d i f f e r e n t from fee and hep crystals which are known to display multi-stage work-hardening  curves at temperatures as low l  as 4°K. Ta  Recently deformation studies on crystals of bcc metals Nb have shown that the form of the work-hardening  and  curves i n these  materials i s also highly temperature sensitive, varying i n much the same 9  fashion as AuZn. corresponds to T  The temperature range of three-stage hardening i n Ta m  ^ 0.10 to 0 . 1 8 while i n Nb  1  t o ^ 0 . 1 0 to 0 . 2 5 , only s l i g h t l y  lower than the multi-stage hardening range i n AuZn.  The apparent likeness  in the deformation behaviour of the ordered bcc compound and ordinary bcc metals suggests that similar hardening mechanisms may be controlling the flow stress i n both instances. The occurrence of c r o s s - s l i p immediately upon the onset of p l a s t i c flow and the formation of d i s l o c a t i o n dipole clusters on the primary glide planes during easy glide are features common to both AuZn and bcc metals, s i g n i f y i n g a fundamental s i m i l a r i t y i n dislocation behaviour during deformation.  Since dipole clusters have been observed i n fee and  hep crystals as w e l l then i t i s believed that the fundamental AuZn-bcc metal relationship i s linked mainly through the common continual c r o s s - s l i p of screw d i s l o c a t i o n s . thermally activated.  The motion of screw dislocations i s known to be Consequently, the exact path along which screws  move w i l l depend on temperature and the extent to which temperature i n fluences t h i s motion could very well be the c r i t i c a l factor i n subsequent work-hardening  behaviour.  Since observations of three-stage hardening i n bcc metals are r e l a t i v e l y new, authors are concerned primarily with noting the rather  135 s p e c i f i c conditions of temperature, strain rate, orientation and impurity l e v e l under which multiple stage flow occurs.  E f f o r t i s also being  directed at studying the general work-hardening  characteristics and attempts  are being made to correlate d i s l o c a t i o n structure observed i n t h i n - f o i l s with stage I and stage II hardening rates  J  >  B  :  i  l  e  c  "  l  e  However, detailed  l  models to account f o r stage I and stage T I hardening such as those presented for the c l a s s i c a l l y studied fee and hep metals have not been proposed. Based predominantly on electron metallography observations i n thin f o i l s , the general concensus of opinion seems to be that work-hardening  i n bcc  metals can be explained i n a similar way to fee metals since analogous d i s l o c a t i o n clusters and subsequent c e l l formation are common structural features of the work-hardened state.  However because 0 i,  U  )  f o r instance,  2*7*8  i s approximately h a l f that f o r fee metals (Nb ~- /»/600  9 ,  1  , .Ta ~/V/600  162  /* /k-00 to /f /900  Fe  compared with 0  1 X  ~ft /200 t o / * /300)  and because  i t i s quite strongly temperature dependent, .it i s probable that some important differences exist i n hardening mechanisms f o r the two structures. Certainly, any model f o r stage T I hardening i n bcc metals must d i f f e r from fee models by including temperature sensitive parameters. Flow curves from crystals of the highly ordered bcc compounds 4  AgMg  25*37  0  and NiAl  show e s s e n t i a l l y para-linear type hardening, although 37  crystals of NiAl  compressed at 25°C i n < l l l > a n d <112>  to deform i n a two-stage manner.  orientations tend  Work-hardening was not discussed. Some  discussion, however, has arisen on the work-hardening mechanism i n poly38  c r y s t a l l i n e bcc ordered AgMg  1  and FeCo  2  5  which undergo (111) s l i p .  Based on the creation of anti-phase boundaries resulting from the motion of superlattice p a r t i a l dislocations 1 a <111^ these mechanisms predict  2 an unusually high hardening rate, since as well, as interacting with each other, superlattice dislocations, on intersecting, disorder the l a t t i c e  0  136 125*127  which alone i s enough to increase the flow stress substantially. This mechanism probably accounts f o r the exceptionally high work-hardening rate observed in.AuZn c r y s t a l s oriented near the [OOl] corner which deform on the { 2 1 l )  <111>  system ( 0  "~ /f/60 compared with On  tations near the middle of the t r i a n g l e . )  It may  ^-/f /5OO i n orien-  also account for the  higher l i n e a r hardening rate i n specimens tested at 77°K (0^ which are believed to undergo a minor amount of <111)  ^/f/200)  slip.  The disordering  mechanism, however, i s not applicable to AuZn tested under conditions giving r i s e to a ^ O O l ^ s l i p  since t h i s vector does not disorder the l a t t i c e .  The mechanisms of work-hardening i n ordered bcc crystals undergoing < 0 0 1 )  s l i p . a r e i n a l l l i k e l i h o o d , as complex, i f not more so  because of the two atom types, as i n bcc metals and close-packed structures. To avoid speculation, a model w i l l not be proposed. shown where the present hardening theories.  observations  It w i l l simply  be  f i t the framework of e x i s t i n g  As shown stage I deformation i s characterized by  hardening rates i n the order of ///5OOO to fl /lOOO which are not greatly d i f f e r e n t from those i n easy glide of fee c r y s t a l s  6  ( ^  10  . ) .  A  Slip  occurs on a single non-crystallographic system giving r i s e to 'dislocation dipoles which form walls perpendicular to the primary (110)  glide plane.  With increasing deformation the d i s l o c a t i o n density within the walls increases and the average w a l l spacing tends to decrease s l i g h t l y .  These  144*164*165  observations  are very similar to those reported i n fee metals  7*8*160  and bcc Nb  suggesting  that s i m i l a r mechanisms may  be  responsible  f o r the observed hardening. Two p r i n c i p a l theories have been proposed to explain work1.66  hardening during easy glide i n close packed crystals,that of Seeger et a l 167  and of Hirsch and'Lally.  The e s s e n t i a l difference between the theories  137 i s that of d i s l o c a t i o n d i s t r i b u t i o n .  Based on s l i p . l i n e studies, Seeger  et a l consider that hardening arises c h i e f l y from the long range i n t e r actions of the stress f i e l d s of i n d i v i d u a l dislocations randomly arranged in the l a t t i c e , while Hirsch and -Lally r e l y on electron metallography observations and take the opposite view that stress f i e l d s associated with d i s l o c a t i o n clusters present the dominant b a r r i e r to d i s l o c a t i o n motion. I f the clustered arrays observed i n  (jj'AuZn can be taken as t r u l y represen-  t a t i v e of the work-hardened structure, then i t would appear that the l a t t e r description i s consistent with the observations. .The disputed  nature of stage TI hardening i s perhaps one of the most  topics i n deformation theory today.  Several mechanisms have  been proposed i n which the flow stress i s controlled either by long range 1441168  stresses from dislocations p i l e d up at insurmountable obstacles, interactions with forest dislocations, dislocations,  170  1 6H  by  by s e s s i l e jogs on g l i d i n g  by bowing-out of d i s l o c a t i o n loops  171  .  , .  .,  , .  or by dislocations  piling-up at long continuous barriers whose effectiveness varies with 172  distance  from them.  • The view i s taken here that any of these models  could probably account for the flow stress i n AuZn, since e s s e n t i a l to them a l l are e l a s t i c interactions between primary and secondary d i s l o c a tions which are manifested in.AuZn through s l i p on secondary systems detected at the end of stage I.  .The detailed differences between d i s l o c a -  t i o n interactions probably accounts for the quite low values of Q AuZn compared with fee metals'( /i /5OO to  /f /300).  n  to // /l200 compared with /y  in /200  E s s e n t i a l to a deeper understanding of hardening mechanisms  in. AuZn i s a knowledge of the Burgers vectors  of dislocations comprising  the c l u s t e r s . "Since unequivocal Burgers vector analyses were not performed during these investigations possible d i s l o c a t i o n interactions w i l l not speculated upon.  be  138  Stage III hardening also remains to be explained. decrease i n work-hardening  The  rate at the onset of stage III i n fee metals i s  explained by the occurrence of thermally.activated c r o s s - s l i p of screw dislocations.  Although thermal a c t i v a t i o n does play a role as evidenced  by T ^ i i decreasing with increasing temperature,  i t i s not known whether  the c r o s s - s l i p explanation applies to AuZn i n which c r o s s - s l i p occurs continually throughout deformation.  I t i s possible, as suggested by  1  M i t c h e l l et a l  f o r Nb, that either large-scale c r o s s - s l i p or the breakdown  of d i s l o c a t i o n b a r r i e r s causes the onset of stage I I I . S l i p l i n e observations did, i n f a c t , show that large-scale c r o s s - s l i p occurs quite early .in the deformation of AuZn crystals ( i . e . near the end of stage I ) , and becomes profuse during stage I I I , suggesting that t h i s may be the dynamic recovery mechanism. It was  shown that specimens oriented along the  [lOl]-[lll]  boundary of the stereographic t r i a n g l e work harden i n a parabolic or semi-twostage manner, s i m i l a r to the stage II-stage I I I region of flow exhibited by c r y s t a l s oriented w e l l within the t r i a n g l e .  Subsequent metallographic  examination, of deformed [ l O l ] - [ l l l ] crystals revealed that duplex  slip  occurs throughout deformation, thereby accounting f o r the absence of easy glide.  Since the two systems operating are of the same general form as  the primary and secondary stage T I systems, v i z . ( h k o ) [ 0 0 l ] and  (okl)[lOO],  s i m i l a r d i s l o c a t i o n interaction mechanisms are probably responsible f o r the high hardening rates i n both the early stages of parabolic flow and stage II deformation. of  Whether the hardening results from long-range interactions  a [ 0 0 l ] and a[lOO] dislocations or from obstacles created by reactions  of the type a [ 0 0 l ] + a[lOO] i s not known.  a [ l O l ] against which dislocations p i l e up,  I t might be expected, though, that since the Burgers  139 vectors of the mobile dislocations are mutually perpendicular, the longrange interaction may be weak and hence not an e f f e c t i v e hardening mechani  140  2.6  THERMALLY ACTIVATED YIELD  2.6.1  Introduction P l a s t i c deformation i s now generally recognized as a dynamic,  i . e . time dependent, process that may be thermally activated.  Since  p l a s t i c flow occurs through the movement of dislocations, i t i s believed that during the course of passage through the l a t t i c e , dislocations p e r i o d i c a l l y contact obstacles that, unless.overcome, cause the d i s l o c a t i o n to stop.  Obstacles are e s s e n t i a l l y of two types, short range, extending  over distances i n the order of ten atomic diameters and long range that possess stress f i e l d s of the order ten atomic diameters or greater. Thermal energy i s able to a s s i s t the applied stress i n pushing the d i s l o c a t i o n past short range obstacles but, because of t h e i r extent, cannot a i d i n getting dislocations past the stronger long range obstacles.  Hence the  names thermal and athermal.barriers to flow. -At s u f f i c i e n t l y high temperatures, a l l thermal barriers.become transparent and dislocations pass through the l a t t i c e unimpeded once the applied stress has overcome the athermal b a r r i e r s .  At lower temperatures, however, thermal b a r r i e r s are  present and must be overcome by the assistance of stress and thermal energy i f an applied s t r a i n rate i s to be maintained.  Usually several thermal,  b a r r i e r s are encountered, the strongest of which determines the rate at which dislocations can move under the given conditions of temperature, stress and s t r a i n rate.  Assuming that thermally activated processes occur sequentially, i . e . one a f t e r another, and assuming that the same event i s rate controlling throughout the l a t t i c e , then i t i s generally accepted that the macroscopic shear s t r a i n rate  t  may be expressed by the r e l a t i o n s h i p :  87  lkl AG  = Vo  f where 0  e  (2 )  k T  9  i s a parameter which depends on the number and arrangement of  the dislocations and t h e i r v i b r a t i o n a l frequency,  A G i s the change in  Gibbs free energy of the system during an activated event and k and have t h e i r usual significance. energy"  AG  therefore  T  The process having the highest "activation  controls the s t r a i n rate.  I f , on the other hand,  thermally activated processes occur independently and the rate c o n t r o l l i n g step i s not the same at a l l points i n the l a t t i c e then the shear s t r a i n .  .  .  rate i s given as:  65  AG•  f  ^  f  .  ^  Y  .  e  (30)  k T  01  : th where i refers to the i  — kind of mechanism.  In t h i s case the  application  of a c t i v a t i o n theory f o r the purposes of i d e n t i f y i n g the rate c o n t r o l l i n g mechanism i s not possible  since one  can no longer extrapolate macroscopic  thermodynamic measurements to a single activated event occurring s p e c i f i c region of the Schoeck  lattice. has  recently obtained an expression for  has been modified s l i g h t l y by.Risebrough AG  =  AH  in a  + T  1 - T  ^/i d T i  and  stands presently  AG  which  as;  T*v*  (31)  yv  where^-j i s the shear modulus on the s l i p plane in the s l i p d i r e c t i o n , AH  i s the a c t i v a t i o n enthalpy  AH = - k T ^ j l n t / f  Uxr"  and  , given by:  0  \  /  ) P* T  \ -  )  , and  f  (32)  /To  v * i s the a c t i v a t i o n volume defined as: v*  =  _ / ^ AG  I  --w/dm  t/1To\  (53)  lk2  and  £"ls the  s t r e s s component e f f e c t i v e  i n a s s i s t i n g the thermal energy i n 88  p u s h i n g the d i s l o c a t i o n past the  l a r g e s t t h e r m a l o b s t a c l e and  as the d i f f e r e n c e between the a p p l i e d s t r e s s t " T ^ a c t i n g i n the v i c i n i t y of the b a r r i e r . volume a r i s e s when one  considers  a c t i v a t i o n event, by the length 1  a distance  The  and  a  the l o n g range s t r e s s  concept o f a c t i v a t i o n  the work W done on the system d u r i n g  stress  p u s h i n g a d i s l o c a t i o n segment of  C  b" 1  d  (jlj.)  where b i s . the Burgers v e c t o r o f the d i s l o c a t i o n . the term  A c t i v a t i o n volume,  b i d .  •To account f o r the temperature dependence o f y i e l d p o s t u l a t e d t h a t the a p p l i e d s t r e s s C components "£"'*and  a  -  c o n s i s t s of two ( p r e v i o u s l y  a  T*.  (  T  +  .For an i d e a l system i n which the  A  3  change below a c r i t i c a l temperature T  s c h e m a t i c a l l y , F i g u r e 60.  Since  temperature, d e c r e a s i n g .Below T , c  through the  however, the d e c r e a s i n g  c  (above which, m a  y  be  v a r i e s through changes i n the  / c>ln  '  t / f'o  c9 tr *  \  IT  L  shear  an  shear modulus.  the p a r t i a l  a t a constant  on  and hence a r i s e i n y i e l d s t r e s s ,  the a c t i v a t i o n parameters  i s necessary t o evaluate  illustrated  amount o f thermal energy a v a i l a b l e  a s s i s t i n g the overcoming of s h o r t range b a r r i e r s n e c e s s i t a t e s  .To e v a l u a t e  )  s m a l l temperature changes i n the  i n c r e a s e i n the e f f e c t i v e s t r e s s term  and  5  y i e l d i s o n l y s l i g h t l y dependent  c  it  defined)  the r a t e c o n t r o l l i n g mechanism i s no  l o n g e r t h e r m a l l y a c t i v a t e d above T ,  for  8 8  rate c o n t r o l l i n g thermally activated  t h e r m a l o b s t a c l e s are t r a n s p a r e n t ) , .the components o f T/  modulus.  Seeger  such t h a t :  r  mechanism does not  an  d : W =  then, i s s i m p l y  i s defined  AG,  AH  and  v  differentials  s t r u c t u r e , • i ,e. d e n s i t y , arrangement  and  143  i  ff  CO CO  0)  u -p  CO  cu  •H  Temperature Figure' 60.  I l l u s t r a t i n g the athermal and thermal components of the y i e l d stress ' f . (after S e e g e r ) 88  a  the number of mobile dislocations remaining e s s e n t i a l l y constant.  Assuming  the y i e l d stress to be governed by the same thermally activated mechanism over the range of temperatures of interest, then one can consider that the structure i s r e l a t i v e l y constant at y i e l d . from C slope  (Figure 60) a plot of  C  8 6  Hence,on subtracting  TT/y  against temperature i s realized and the  /A't"* | may be taken as / j ' t " * ] . / T 1 c)T /T  Likewise plots of ^  versus  \A T 1  0 at constant temperature can.be adjusted (by subtracting C/f which i s not s t r a i n rate sensitive, assuming a constant mechanism) to give against  o  and hence allow  / e)ln \  / " j ^ n |_ to be evaluated.  dxr*  X  Instead of  /T  studying s t r a i n rate effects on y i e l d stress i t i s common practice to perform d i f f e r e n t i a l s t r a i n rate change tests on a single specimen during flow at a fixed temperature then to extrapolate, to zero s t r a i n to evaluate the second p a r t i a l .  lkk .The object of the experiments reported here was to measure the activation parameters i n AuZn single crystals then to suggest a possible rate c o n t r o l l i n g mechanism responsible f o r the temperature dependence of y i e l d (section 2.5.T.I).  .The discussion w i l l be limited to  stoichiometric crystals oriented near the middle of the stereographic triangle. 2.6.2  Activation Volume and E f f e c t i v e Stress Flow stress differences accompanying instantaneous changes  of s t r a i n rate may be taken as variations i n effective stress A t * w i t h i f i t is- assumed that the structure remains constant during the instant of change. TO  4  to 10~  D i f f e r e n t i a l tests corresponding to strain rate changes from 2  per sec were carried out by varying the cross-head speed on  the Instron from 0.002 to 0.20 inch per.minute, and vice versa,.using a push-button speed selector.  Experiments were performed at temperatures  ranging from 77°K to 213°K, i . e . over the temperature i n t e r v a l corresponding to temperature sensitive y i e l d (Figure -33)-  ..Typical flow curve variations  accompanying a s t r a i n rate change are shown schematically in Figure 6 l . Tn p r i n c i p l e , both stress increments and decrements may be taken as  A C  However, because of less uncertainty i n measurement, stress increments were taken i n t h i s work, by subtracting the flow stress s t r a i n rate from the flow stress  at the higher rate.  at the lower 2  was taken at  the f i r s t deviation from l i n e a r i t y , following Basinski and C h r i s t i a n .  90  145 A l l T.  1  AT Low Strain  High Strain  #.  i d  T4 145°K< T< 215°K  T A  Figure 6 l .  i  T  Schematic representation of changes i n the flow curve accompanying s t r a i n rate change t e s t s . From the relationship presented i n 2.6.1, activation volumes  were calculated from: y *  =kT/m {  V* /i\  (36)  a  A C  and are plotted i n Figure 62 (in units of b ) as a function of shear 3  s t r a i n ; b i s the Burgers vector of the mobile [001] dislocations and was ,  -8  taken as the l a t t i c e parameter:3.14 x 10 volume at y i e l d , v deformation.  i s ~-30b  3  12  At 77°K the activation  cm.  and i s constant within 5b  As the temperature increases,  v  n  3  throughout  also increases and at the  same time, becomes somewhat more s t r a i n (stress) sensitive, decreasing with increasing s t r a i n .  At 215 K f o r instance,  v  D  decreases from 525b  at  Figure 62.  Activation volume against shear s t r a i n at temperatures between 77°K and 213°K.  lA7  y i e l d to 130b  3  at fracture.  In Figure 62  i t can be seen that activation volumes determined  just past y i e l d at a l l temperatures are s l i g h t l y lower than those obtained by extrapolating l a t e r values to zero s t r a i n .  In a l l l i k e l i h o o d , the  i n i t i a l l y lower volumes are associated with uncertainties i n determining A T * since the departure from l i n e a r i t y with a strain rate increase just past y i e l d i s not nearly as marked as i t i s at higher strains (Figure 6 l ).  For the purposes.of comparison and for calculating a c t i v a t i o n energies  i n section 2.6.3  a c t i v a t i o n volumes  v  D  extrapolated to zero s t r a i n w i l l be  employed. A c t i v a t i o n volumes are commonly plotted against the e f f e c t i v e To separate (  from the applied stress, the  athermally  attributed l i n e a r portion of the y i e l d stress-temperature curve f o r stoichiometric crystals  (Figure 33 ) was  extrapolated to 0°K and at each  temperature of interest below the c r i t i c a l temperature T  c  '—220°K the  corresponding, athermal stress component'was subtracted from the yield. stress.  The subsequent  --T results are plotted i n Figure 63  and extra-7-*  polated to 0 K f o r future reference. •• Activation volumes, then, versus <• are shown i n Figure 6k .  Also shown are the variations f o r p o l y c r y s t a l l i n e  AuZn and the bcc metals (V, ,Nb, . Ta, Cr,Mo;W,, Fe) over the same range of 91  taken from.the compiled data of Conrad and Hayes.  It i s apparent that  single and p o l y c r y s t a l l i n e AuZn display exactly the same v behaviour that f a l l s within.the suggesting  D  -  C  range for the bcc t r a n s i t i o n metals,  that s i m i l a r mechanisms may be rate c o n t r o l l i n g .  t  148  Figure  6$.  Showing the v a r i a t i o n i n e f f e c t i v e with temperature. v  stress  1^9  Figure 6k .  Showing the v a r i a t i o n of a c t i v a t i o n volume with e f f e c t i v e stress f o r Q AuZn and bcc metals. 1  150  2.6.3  A c t i v a t i o n Energy The expression given for the a c t i v a t i o n free energy  AG  in equation(31)when rewritten to include the terms for A H and v , gives the r e l a t i o n s h i p : A G = -kT  2  I"/ a t ' L . .  (^inT/To) \  <s>r»  H I  a T  -4*  /»  /V^o  1 - T  (57)  T * l -I  ^  It i s assumed that the temperature v a r i a t i o n of the e l a s t i c s t i f f n e s s constants i n AuZn i s small implying that changes i n shear modulus with temperature are also small. that  This assumption i s j u s t i f i e d on the basis  7^, , which i s related to the shear modulus v a r i a t i o n with temperature, 10 percent from 300°K to 0°K.  increases only approximately  term d/r T**in equation(37)will to the larger term temperature.  A G may  Hence, the  be small and may be neglected with respect  giving the effective stress v a r i a t i o n with then be approximated as  A G ~ -  V*T  / ar* \ ^ , . I aT / / o  A H ; i.e.:  =  (  A H  5 8  )  y  .As a measure of a c t i v a t i o n energy, therefore, a c t i v a t i o n enthalpies were calculated rather than a c t i v a t i o n free energies and are reported i n Table. 16 The term / <5£** \ -fr/^A at d i f f e r e n t temperatures was slope of the  T" - T curve, Figure  evaluated from the  63.  .To determine the height of the rate c o n t r o l l i n g thermal 91-94  barrier,  A H ^ i t i s necessary to measure  Figure 65 shows AH  0  ~ 0.9' ev.  A H versus  A H when  C  i s zero.  C and, when extrapolated to zero stress, gives  For bcc metals,  91-94  A H  0  i s approximately  3  0.1^ b .  On  t h i s basis,,the c o n t r o l l i n g energy b a r r i e r i n AuZn crystals i s •^-0.15 ^b /  (where b =  a <00T> = 3.14  x 10  8  cm:and f, = C  4 4  = 3.0 x 1 0  1 1  dynes/cm ) 2  3  151  and, therefore, of the same order of magnitude as the pure bcc metals, again suggesting that similar mechanisms may be rate c o n t r o l l i n g . TABLE 16 Activation Parameters AH and v at Temperatures between 77°K and 175°K f o r Stoichiometric Crystals  T (°K)  vo/b (from -Fig.69)  r*(psi) (from Fig.65)  G  A H (ev.) (from •Equation(38)  77  30  4100  • 23  90  33  3200  .29  100  50  2600  -35  125  76  1600  .46  132  90  1300  • 49  143  125  1000  .66  150  137  800  • 73  172  170  400  .72  175  200  350  • 79  In p o l y c r y s t a l l i n e AuZn, i t was found about h a l f of the single c r y s t a l value.  that  A H ' ' 0 . 4 3 ev., o  v  Since a c t i v a t i o n volumes for both  systems are i n good agreement, .part of the discrepancy may l i e i n the certainty of measuring  / A'C'* \ .  1 AT  In both instances the p a r t i a l was  /T  measured from the slope of the corresponding y i e l d stress-temperature curve.  In polycrystals only f i v e temperatures were used to establish the  shape of the y i e l d stress curve over the range of interest below f 250°K, w  whereas i n single crystals y i e l d stresses were measured more frequently giving a t o t a l of 13 points over the same temperature range, i n t e r j e c t i n g a greater degree of certainty i n the measured  / AT*  \ terms.  152  .Figure 65.  Showing the v a r i a t i o n of activation enthalpy with e f f e c t i v e stress.  153  2.6.4  Discussion Since the present a c t i v a t i o n parameters are i n accord with the  results for bcc metals suggesting that similar mechanisms may be responsible for y i e l d of AuZn, the commonly discussed bcc rate c o n t r o l l i n g mechanisms w i l l be reviewed.  2.6.4.1.  Impurity  Obstacles  In view of the rather high i n t e r s t i t i a l content (Oxygen plus Nitrogen ^-.300 ppm Appendix 1),  i t i s possible that d i s l o c a t i o n - i n t e r s t i t i a l  . interactions could be c o n t r o l l i n g the y i e l d stress.  Assuming that the  d i s l o c a t i o n i s bent to make nearest neighbour contacts with the impurity atom, then the distance 1  between obstacles i s related to the impurity ST  concentration C through the Burgers vector content of 300 ppm  1*  For an impurity  the shear stress necessary to push the d i s l o c a t i o n  between the obstacles, given by V equivalent to <~-90,000 p s i . (Figure 63) i t was  2 , b ~ 'C .  =/* b, i s approximately/*/50, which i s  I**  On extrapolating the tT - T curve to 0°K  found that C.  0  ^ 1 5 , 0 0 0 p s i which i s considerably less  than the stress necessary to push dislocation through impurity obstacles spaced 50bin.theslip plane.  Assuming that the d i s l o c a t i o n makes three1 3  98  dimensional nearest-neighbour  contacts, then  b ""C  and the  corresponding  1* stress necessary to push the dislocations through the obstacle f i e l d i s /*/l5  and i s greater than the t h e o r e t i c a l shear strength of the l a t t i c e .  In the l i g h t of an impurity model, i t would also be d i f f i c u l t to explain the increase i n a c t i v a t i o n volume with increasing temperature. 2.6.4.2 • Peierls Nabarro (PN) Mechanism It has been the conclusion of a great many workers that the rate c o n t r o l l i n g mechanism i n bcc metals  91*94;95*96*99  and ordered a l l o y  154 AgMg  lOO;101  i s the overcoming of the Peierls-Nabarro barrier to flow, i . e .  the inherent l a t t i c e energy b a r r i e r that a d i s l o c a t i o n , l y i n g along close packed rows of atoms, overcomes as i t moves from one equilibrium to the next.  "valley"  In surmounting the Peierls " h i l l s " the configuration of  atoms at the d i s l o c a t i o n core i s altered which therefore increases the d i s l o c a t i o n energy.  Hence the energy of a straight d i s l o c a t i o n i s a function  of i t s displacement from the bottom of a "valley" and has the p e r i o d i c i t y of the spacing between the p a r a l l e l rows of atoms.  The c r i t i c a l step i n the PN mechanism i s the thermally a c t i vated nucleation of a p a i r of kinks that w i l l subsequently move apart bringing  the d i s l o c a t i o n into i t s next equilibrium  Figure 66.  position, Figure 66.  Schematic i l l u s t r a t i o n of the PN mechanism.  155  The  d i s l o c a t i o n l i n e i n i t i a l l y l i e s along A BoC 0  in the Peierls "valley".  0  i n an equilibrium  Under the action of an e f f e c t i v e stress  the d i s l o c a t i o n moves from i t s equilibrium position to a new ABC  part way  up the " h i l l " .  supply  i n pushing a d i s l o c a t i o n segment AB C over the  ' h i l l " thereby creating a double kink, AB* brings the d i s l o c a t i o n into a new activated  (.  position  Thermal fluctuations occasionally  enough energy to a s s i s t C  position  and B'C, which on moving apart,  position A B C  ready for the next  event. On the basis of a l i n e energy model of a dislocation, Dorn  and Rajnak  103  have recently derived an expression for the energy U  nucleate a kink.  Assuming that the Peierls " h i l l " i s sinusoidal: U  = 2f  a  n  rr where  p  to  I  - l 2  \  (\ a b  \  r  /  r  0  (59)  2  i s the d i s l o c a t i o n l i n e energy per unit length i n the  low  energy position A BoCo, a i s the spacing between p a r a l l e l rows of closely 0  spaced atoms i n the s l i p plane, T"  the Peierls stress (the stress necessary,  i n the absence of thermal energy, to push the d i s l o c a t i o n over the  barrier) 102.  and b the Burgers vector.  For a quasi-parabolic  " h i l l " , Guyot and Dorn  have derived the following expression for U : 1  3/  U The  k  = TT  sinusoidal and quasi-parabolic  2  To a / 2 Tp a b \ 8 1 ^ j  2  p r o f i l e s are shown i n Figure 67 .  (kO )  156  Figure 67-  Schematic i l l u s t r a t i o n of the sinusoidal and quasi-parabolic Peierls " h i l l " p r o f i l e s .  To evaluate the kink energy i n AuZn, values for :  \~ and  T  Q  P are necessary.  Assuming that  ["~- — 5 . 0 0  x 10  system) and that 7j" at 0°K may be approximated  ergs/cm (Table 4 ;  (llOJCOOl)  as 15,000 p s i (obtained by  1? L - T to 0°K) the kink energy i s then given as ~- 0.24  extrapolating  ev.  When C = 0 thermal fluctuations must supply enough energy to nucleate a double kink, i.e.- .2  ~~ 0.48 ev.  I f i t i s assumed that the PN mechanism  i s rate c o n t r o l l i n g over the whole range of temperature y i e l d , i . e . below 220°K, then to ~ 0 . 4 8  s e n s i t i v i t y of  ( i . e . at 220°K) should be equivalent  AHo  C l e a r l y the value 0.9 ev. obtained e a r l i e r i s considerably  ev.  larger than the energy estimated to nucleate a double kink.  I t appears  therefore, that i f the PN mechanism acts as a rate controlling process, i t operates over;a lower temperature  range.  It i s assumed that the PN mechanism i s rate controlling below a c r i t i c a l temperature temperature  Tp C  where T p may be estimate'd by noting the C  at which the experimentally determined activation enthalpy i s  approximately 0.48 ev.  T, Figure 68.  From the data .in Table 16,  AH  i s plotted against  Assuming that a straight l i n e through the o r i g i n joins the  points then T  nTS  —' 125°K.  The p l a u s i b i l i t y of a -PN mechanism controlling y i e l d at tempera-  157  Figure 68.  Showing the v a r i a t i o n i n a c t i v a t i o n enthalpy with temperature.  158 tures below '^"125°K becomes apparent when one examines the activation 102  volumes.  Guyot and Dorn  have, given an expression f o r the width w of  the c r i t i c a l sized loop forming the embryonic kinks: I_ w • = jrf  2a F  \  0  S U T  (la )  2  )  P  where a l l parameters have the same meaning as defined e a r l i e r .  The a c t i v a (42)  t i o n volume f o r the PNv process = w a b i s then expressed as:  * which upon substitution f o r [<•, T . a and b gives to the a c t i v a t i o n volume i n •AgMg  100  '  101  v  below 250°K.  0  , 3 ~ 40b , similar On comparing the  calculated with the experimentally determined values plotted against temperature  i n Figure 69, i t i s apparent that below ^ 120°K, v* ~  30-l+0b  3  and i s not strongly temperature  sensitive, agreeing well with the dictates •  of a PN process.  0  Above 120 K, v  increases l i n e a r l y with temperature  suggesting that a d i f f e r e n t mechanism becomes rate c o n t r o l l i n g . noted that the c r i t i c a l temperature  It i s  for a PN process estimated from the  a c t i v a t i o n volume measurements i s i n good agreement with that deduced from the a c t i v a t i o n enthalpy calculations.  That PN may be rate controlling  below ' -'125°K i s further evidenced by the fact that activation volume N  remains approximately constant during deformation (Figure 62).  Analysis of  12  p o l y c r y s t a l l i n e data  was consistent with the dictates of a PN mechanism  controlling y i e l d below a c r i t i c a l temperature a c r i t i c a l temperature /  of approximately 150°K,  s l i g h t l y higher than the present temperature  of  ^125°K but i n quite good agreement i n view of the uncertainties i n  determining T p. 2.6.4.3 Cross S l i p C  In view of the extensive cross s l i p occurring continually during the deformation of AuZn, the p o s s i b i l i t y that i t may control the  159  !  0  .  I  50  I  100  L_  150  I  200  Temperature T°K  Figure 69.  Shewing the v a r i a t i o n i n activation volume with temperature.  I  250  160  deformation rate w i l l be considered.  The analysis w i l l be a qualitative  one, based on the experimentally determined activation volumes. activation volume as the product of the activated length 1,  Considering  the activated  distance d * and the Burgers vector b and employing expressions given by 65  Dorn  *  *  f o r the parameters d- and 1-  for the cross s l i p process,, one  arrives at an expression r e l a t i n g activation volume to e f f e c t i v e stress: v* = -• A .(43) (r*) 2  where A i s a constant incorporating the l i n e tension of the bowing-out d i s l o c a t i o n and i t s Burgers vector.  T<\1  Since  > then, q u a l i t a t i v e l y ,  (r*) a c t i v a t i o n volume should increase with decreasing temperature), as observed.  On p l o t t i n g In v  Q  ^  2  ( i . e . increasing  against In  C  i n Figure 70,  two regions of behaviour are apparent, from 77°K to ~ 150°K i n which v  0  V l  and from ^150°K to 213°K over which v *°<-l 0  . It  appears, therefore, that below 150°K the activation volume v a r i a t i o n follows quite closely the dictates of the cross s l i p mechanism suggesting •that cross s l i p may be an alternative to the PN mechanism controlling y i e l d at low  temperatures.  161  Figure 7 0 .  Showing the functional dependence of a c t i v a t i o n volume on e f f e c t i v e stress.  162  3. 1.  SUMMARY AND CONCLUSIONS The primary s l i p surface i s a non-crystallographic plane (hko) i n the zone of the s l i p d i r e c t i o n [OOl].  With increasing temperature,  decreasing s t r a i n rate and increasing distance of the t e n s i l e axis from the [ 0 0 l ] - [ l l l ] boundary, the s l i p plane varies from (110) This phenomenon was  interpreted in terms of continual c r o s s - s l i p of  screw d i s l o c a t i o n s on orthogonal planes (110) 2.  to (100).  S l i p l i n e observations  and  (110).  indicate that edge dislocations can s l i p over  large distances on {lio} planes while screw dislocations s l i p over much smaller distances.  These observations are consistent with the  continual c r o s s - s l i p mechanism proposed to explain non-crystallographic slip. 3.  •Multi-stage work-hardening and associated large d u c t i l i t y as high as 300$ shear s t r a i n has been observed for Au-rich (51-0 stoichiometric and Zn-rich (51.0  at.$ Au),  at.fo Zn) c r y s t a l s i n the approximate  temperature range 0.2 2 T / T 2 0.35 and for a l l orientations within m the standard stereographic t r i a n g l e except near the [001] corner.  k.  Following a short t r a n s i t i o n region (stage 0 ) ,  stage I easy glide  begins, characterized by r e l a t i v e l y low hardening rates (/</l000 to Sf/'j000)  .  Transmission  electron microscopy studies revealed walls of edge  dislocations perpendicular to the primary glide plane which are believed to be the main cause of stage T hardening. 5.  The p r i n c i p a l effects of deviations from stoichiometry are to increase the flow stress and lengthen the extent of stage I.  6.  The end of easy glide i s coincident with s l i p on secondary systems of  163  the form (okl)[lOO] i n l o c a l i z e d regions of the c r y s t a l near deformat i o n bands.  The contribution of the secondary system to the o v e r a l l  deformation i s n e g l i g i b l e . 7.  Stage II deformation commences a f t e r a f a i r l y slow t r a n s i t i o n and the development of the secondary system probably accounts for the high workhardening rate.  0  X1  i s nearly independent of composition, but i s some-  what sensitive to the other experimental variables,increasing  with  decreasing temperatureIncreasing s t r a i n rate and decreasing distance of the specimen axis from the [ 0 0 l ] - [ l l l ] boundary.  ©11A/500,  about half that f o r fee metals. 8.  The onset of stage III i s coincident with s l i p l i n e cluster formation a l l along the gauge section which i s believed to result from largescale c r o s s - s l i p of screw dislocations.  9.  At temperatures below ^-150°K, para-linear hardening i s observed. The onset of minor amounts of ( O i l ) [ i l l ] s l i p at 77°K (as detected from Taylor analysis of asterism i n a back-reflection Laue photograph) .is believed to be responsible, for the high l i n e a r hardening rate.  10.  .Increasing s u s c e p t i b i l i t y . t o crack formation along (001)  planes i s  believed to be responsible f o r decreasing ductility.below — 250°K. 11.  Above ~350°K, parabolic type hardening occurs followed by work softening and t o t a l s t r a i n to fracture decreases.  This behaviour was  explained i n terms of increasing propensity f o r large-scale c r o s s - s l i p . 12.  D u c t i l i t y increases with temperature above ^ 450°K, coincident with the operation, to about the same extent, of two s l i p systems of the general forms (hko)[00l] and  (okl)[l00].  164 In orientations near the [001] ( i l 2 ) [ l l l ] s l i p occurs. d u c t i l i t y i s low  corner of the stereographic  Hardening rates are very high ( ///60)  (^20$  shear s t r a i n ) , presumably due to the  ing of the l a t t i c e r e s u l t i n g from the motion of partials a [ i l l ] .  2  triangle, and  disorder-  superdislocation  The y i e l d stress i s about ten times greater than  f o r (hko)[00l] s l i p . Serrated  flow was  detected at temperatures between  for non-stoichiometric  293°K and 400°K  crystals and i s attributed to a d i s l o c a t i o n -  solute atom interaction. Deformation bands roughly p a r a l l e l to (001) a l l orientations at temperatures above the bands are a manifestation  planes were detected i n  140°K.  I t i s suggested that  of the formation.of edge dislocation  walls of opposite sign. It was  found that the y i e l d stress i s approximately orientation  insensitive and was  shown to be consistent with predictions based on  the observed v a r i a t i o n of s l i p plane with orientation.  These results  were interpreted i n terms of decreasing c r i t i c a l resolved shear stress for glide on (110)[001] with increasing distance  from the  [00l]-[lll]  boundary, a r i s i n g from the increasing ease of recombination of dissociated screw dislocations l y i n g on (100)  planes.  The temperature s e n s i t i v i t y of y i e l d i n stoichiometric crystals below *-220°K was  subjected to thermal a c t i v a t i o n analysis.  either the Peierls-Nabarro  The dictates of  mechanism or c r o s s - s l i p of screw d i s l o c a -  tions are consistent with the experimentally determined activation volumes and a c t i v a t i o n enthalpies below  150°K.  165 •k.  -SUGGESTIONS FOR FUTURE WORK The following questions arise from t h i s work and can be  considered as the most natural extensions of i t : 1.  -What are the effects, of a wider range of temperature and orientation on non-crystallographic s l i p and hence on the shape of the A// (~X.) •and "\J/(T)  curves?  At constant strain rate,, i s the temperature  at which the s l i p surface coincides, with the most highly stressed plane i n the <001^ zone independent of ~)C ? 2.  Assuming fundamental (110)[OOl] s l i p , how w i l l the c r i t i c a l resolved shear stress-orientation relationship T* ("ty/) vary with temperature and s t r a i n rate?  3.  What are the d i s l o c a t i o n reactions leading to stage II work-hardening as revealed by detailed Burgers vector analysis of d i s l o c a t i o n clusters using high resolution dark f i e l d techniques. Also, does ( i l l ) s l i p contribute to general deformation f o r orientations other than those near (OOl) and at temperatures above 77°K?  k.  For near (001). orientations, i s the s l i p plane i n the zone of the s l i p d i r e c t i o n ( l l l ^ temperature sensitive?  How does the c r i t i c a l resolved  shear stress for ( i l l ) s l i p vary with temperature and strain rate? 5.  Is serrated y i e l d i n g a phenomenon inherent in.the deformation of nonstoichiometric  (^'AuZn at intermediate temperatures and strain rates  or i s i t a manifestation of dislocation-impurity atom interactions?  166 APPENDIX I C r y s t a l Homogeneity A.1.1  Chemical Analysis and Composition  Gradients  Specimens of approximately 300 mg. were accurately weighed then dissolved i n .15 remove N0  2  cc of warmed aqua regia.  The solution was boiled to  then diluted to 200 cc with d i s t i l l e d water and warmed again.  Hydrazine sulphate was  c a r e f u l l y added to precipitate gold.  The solution  was l e f t over night then subsequently -filtered and the residue ashed at 600°C a f t e r drying at 150°C. .The gold was c a r e f u l l y weighed and the zinc was determined by difference. Crystals of three different nominal compositions, 4-9.0, 50.0 and 51.0  at $ Au were grown and analyzed chemically f o r A u at five points  along t h e i r length. in Figure A l . l . ~0.04  at $.  The results are given i n Table A1..1 and are plotted  ..All analyses were i n duplicate and are accurate to within  To a f i r s t approximation, the average composition over the  f i r s t two-thirds of the c r y s t a l to s o l i d i f y deviates only s l i g h t l y from the nominal  composition. In a l l crystals a region of high zinc content i s noted over  approximately the l a s t t h i r d to s o l i d i f y . difference between gold and zinc ( ^ Au  =  In view of the large density  ^9«3 g/oc; ^.£n  the high vapour pressure of zinc at melt temperatures  =  T-l g/cc) and  (775°C), i t i s not  surprising that the last end to freeze should be zinc r i c h .  Attempts to  reduce the composition gradients by homogenizing the as-grown crystals i n reduced pressure at a temperature  s u f f i c i e n t l y high for rapid d i f f u s i v i t y  of zinc vapour met with no success.  TABLE A l . l  Chemical Analysis of As-Grown Crystals  Nominal Comp. (at. % Au)  51.0 '  Distance along the c r y s t a l • (inch)  0-5 ,3.6 . 9-8 • ik. 2  18.0 50.0  0.8 2.4. 7.0 11.6 14.2  49.. 0  0.3 3-5 9-8 14.4 17.5  Composition (Vft.i Au)  75.90 75.89 75-88 75.85 75.85 75.84 75-73 75.70 75.38 75.45 75-00 75.15 75-16 75-12 75.13 74.99 75-06 74.97 74.42 74.33 74.63 74.61 74.45 74.62 74.38 74.32 74.32 74.29 73.90 73-88  I  5  I  1  10  15  — — '  20  Distance From F i r s t End t o ' S o l i d i f y (inch)  Figure A l .  Showing composition  gradients i n as-grown  l^'AuZn single c r y s t a l s .  _ o>  |  1  CD  169  A.1.2  Interstitials  and Trace Elements  Results are given in-Table A.1.2 element content i n a t y p i c a l  f o r i n t e r s t i t i a l and trace  i^'AuZn t e n s i l e specimen, analyzed by Ledoux  Analysts, Teaneck, New Jersey-..  .TABLE  A.1.2  Impurity Analysis i n Typical ft'AuZn Tensile Specimen  Interstitial Element' Carbon Oxygen Nitrogen  Content (ppm)  •  8 -•  195  . 127  Hydrogen  . Trace Elements  l  Not Detected  170 APPENDIX 2 Evaluation of Machining Damage Evidence that the cold-worked surface layers of the t e n s i l e specimens could be completely removed by polishing and annealing was obtained from back-reflection Laue d i f f r a c t i o n patterns and room temperature t e n s i l e tests on stoichiometric specimens. fresh 5$ KCN solution. the surface,  Ad,  strength versus  The surface layers were removed i n  The v a r i a t i o n i n asterism with amount polished from  i s shown i n Figures A2.1,. A2.2 and A2.5.  Ad  f o r two series of experiments i s shown  Tensile i n Figure A2.4  and the e f f e c t of annealing time and temperature on the y i e l d strength of two d i f f e r e n t crystals i s reported i n Table A..2.1. -For these l a t t e r t e s t s , each specimen was strained just past y i e l d then re-annealed and tested again.  It i s apparent that removing a surface layer 0.005 inch and  annealing at 300°C f o r one hour renders the t e n s i l e specimens free from machining damage.  TABLE A.2.1 E f f e c t of annealing temperature and time on the strength of two t e n s i l e specimens r e l a t i v e to the unannealed condition  Specimen E-l-7  D-1-5  Ad  (inch) 0.0100 11 11  0.0094 it  11 11  Annealing Temp.(°C)  -  Annealing Time (hr.)  -  '300 •300  1 h. 3 h.  300 300  1 h. 3 1/2 1  -  425  -  .Tensile Load at Yield (lb.)  129 103  104 42  35 35 34  Ad -  0  A2.1 Figures A2.1, A2.2, A2.5.  Ad =0.0033 inch A2.2  A d - O.OO98 inch  A2.3  Showing the v a r i a t i o n i n degree of asterism with reduction of diameter of machined t e n s i l e specimens.  H -^1  172  Figure  k2.k.  Showing t e n s i l e strength of machined c r y s t a l versus amount removed from the specimen diameter.  173  APPENDIX 3 Resolved Shear Stress T~ ( V )  Equations:  and Resolved Shear S t r a i n  The equations. used i n computing resolved shear stress (^(Tj/ ) and resolved shear s t r a i n parameter ~\lf (section 2.k)  Yflj/') on  the operative s l i p plane defined by the  were obtained  i n the following manner.  From  49  Schmid and Boas  , the resolved shear stress and resolved shear s t r a i n acting  on the operative s l i p system are given as: 1  f  Sin (90  = P_ Ao  -  - Sin  [(1)  0) O  t  2  £  and s.2  ^  2  - Sin  -i  )= c-  1  Cos  (2)  Po  Sin(9O-0 ! o  where P i s the instantaneous t e n s i l e load on the crystal., A cross s e c t i o n a l area of the specimen, 1 gauge lengths respectively, ^ and the t e n s i l e axis and and the t e n s i l e axis. X  aid  of Figure A3.1  AN<  ^ F  (  sec  the  initial  and 1 the i n i t i a l and instantaneous  the i n i t i a l angle between the s l i p d i r e c t i o n  (9O-0o) "the i n i t i a l angle between the s l i p plane  The angle 0 may  " t i o n 2.k)  Y>  0  G  0  be written i n terms of the parameters  characterizing s l i p modes i n  (^AuZn.  With the  and applying the cosine law from spherical trigonometry,  i t i s found that: Cos  0  O  Therefore, from (1)  (y  = Cos  - Y ) Sinf o  (3)  i n terms of the known parameters ^ , ~)C and l]/, ( Q  (2)  and  -C- (l|r ) = P A  may  be rewritten i n the form:  Sin £ o Cos 0  and If  {-%  - \f)  l  ,2 c  - Sin"  1  and -I  1-  2  CO  174  001  100 Figure A ^ . l .  175 APPENDIX k Taylor Rotation Axes '(•Re: Asterism at 77°K, Figure  22)  The Taylor rotation axis l i e s i n the s l i p plane and i s perpendicular to the s l i p d i r e c t i o n .  I t i s that axis about which the  rotates during s l i p on a given system.  lattice  The i d e n t i f i c a t i o n of rotation axes  from asterism on back-reflection X-ray photographs of deformed crystals i s a 156*157*158  commonly used technique shown i n Figure 22 was  f o r identifying s l i p systems. assumed to arise from.the operation of two  systems operating simultaneously.  159  from deformation bands (for review, see Hirsch) out since bands' were not detected at 77°K. i s i l l u s t r a t e d schematically i n Figure X  Ak.l.  slip  I t i s known that asterism. may also arise '  Figure  Asterism  , but t h i s o r i g i n was ruled  A t y p i c a l branch from Figure 22  A4.1. Y  Sketch of asterism from Figure 22. V i s the v e r t i c a l axis in the f i l m plane; X-X and Y-Y are associated with branches 1 and 2 respectively and are the projections of the rotation axes Ri and R onto the f i l m plane. 2  The primary s l i p system at 77°K was crystallographic plane i n the [100]  zone l y i n g ^ 6 °  found to be a nonfrom ( O i l ) ; a secondary  176  trace was coincident with the most highly stressed plane of the -£L10} ( i l l ) system, namely ( O i l ) .  Representing the primary system as (011)[l00] and  assuming that the secondary system i s ( O i l ) [ i l l ] , the Taylor rotation axes are then given as [Oil] -and [211]respectively. all  {llO} and {211}  To check the assumption  poles are plotted stereographically i n Figure Ak.2  r e l a t i v e to the indexed d i f f r a c t i o n spots from Figure 22.  Also plotted are  X-X and Y-Y, the f i l m plane projections of the rotation axes R respectively, along which the poles of the axes must l i e .  x  and R  2  I t i s seen that  [ O l l ] l i e s on the Y-Y projection implying that s l i p on the (011)[l00] system gave r i s e to asterism branch 2.  The pole [211]  i s seen to l i e along X-X,  and thereby consistent with the assumption that one of the branches of asterism i n Figure 22 ( i ..e. -branch 1) i s the result of (Oil) [ i l l ] s l i p .  177  Figure AU.2.  Stereographic projection of a l l { l i o ) and {21l} poles plotted with respect to indexed d i f f r a c t i o n spots from Figure 22. The base c i r c l e represents the film plane. Rotation axes "R± a n d R l i e on the great c i r c l e s X-X and Y-Y respectively. 2  0  178  APPENDIX 5 Shear Modulus as a Function of S l i p System The work-hardening parameters < , 0 and 0 0  expressed  i n terms of the u n i t l e s s quantities  X  1  X  0j__ and Q  1X  the shear modulus c h a r a c t e r i s t i c of a given s l i p system.  are often where /i i s  Expressed  in this  way, the quantities can be r e a d i l y compared with similar terms characterizing work-hardening i n other systems.  During the course of the present  investi-  gations, s i m i l a r quantities were obtained only a f t e r a general method had •been established f o r evaluating ^  {uvwj.  as a function of s l i p system ^hklj  The method employed w i l l be outlined i n i t s general and hence most useful form using standard second order tensor notation and the repeated  suffix  convention. The components of the stress tensor T handed set of orthogonal axes x i , x , x 2  ^  =  £ mnpq  c  n  n  mn  3  relative to a rightmn i s given by the expression:  ; ^ , n = 1, pq  2,  .3)  (i)  where C represent the e l a s t i c s t i f f n e s s constants of the material and mnpq €pq i s the corresponding s t r a i n tensor.  Equation (1) defines  the stress state r e l a t i v e to the X ^ ( i = l , 2 , 3 )  reference frame.  completely In c a l -  culating shear modulii for a s l i p system defined by the unit vectors n and (given i n the s l i p plane and  frame) where n i s p a r a l l e l to the normal of the {hkl} i s p a r a l l e l to the s l i p direction <uvw> , i t i s necessary  to change the reference frame fromx^ to a new set of orthogonal axes Sc^The new frame i s oriented r e l a t i v e to the o l d frame i n such a manner that x  x  i s p a r a l l e l t o Q_ and x i s p a r a l l e l t o n; .x is then given by the vector 2  _g x n to form a right-handed  3  system, Figure A5.1« Relative to the new  system of axes the stress tensor 'Ci ^ i s given by the second order tensor  179  Figure A ^ . l .  Showing the X  reference frame r e l a t i v e to the X frame. i  i transformation law: r.  1.  im  a. T jn ran  ( 2)  where a. . i s the transformation matrix which relates the new frame to the old, In order to expand equation (2) into a term consisting of e l a s t i c constants and s t r a i n , i t i s necessary to express the s t r a i n tensor £  i n terms of a PO.  s t r a i n tensor i n the new frame  €. ^  by employing the reverse transformation  method f o r second order tensors f pq v  where a  = a, a. p kp l q * k l  (3)  denotes the transformation matrix r e l a t i n g the old frame to the new.  Combining equations  (1),  (2) and ( 3 ) , the stress tensor i n the new system i s  given by the expression:  F. ,  <  = a. a a a C (L , Ik) J 1m j kp l q mnpq C k i \' The component of which i s of special interest i n determiningyc- (n, (3 ) ij _ _ ~ corresponds to the stress acting on face x i n d i r e c t i o n x ; i . e . f~ which n  X  n  2  x  21  i s given by: ?** = a a a a C £. 21 2m m kp l q mnpq k l  /  (5]  180 Equation (5) w i l l be considered again. I f i t i s assumed that l a t t i c e rotations within a body can be neglected as i n f i n i t e s i m a l , .then the stress tensor i s symmetric regardless = T~ and nm  of the reference frame, i ,.e. f .mn  . =?"..• ij J  Symmetry then  1  effects a reduction i n the number:of e l a s t i c constants c h a r a c t e r i s t i c of a general material from 8 l to J>6.  The stress state can then be written i n  terms of a shortened 'notation: T Vm The r e l a t i o n s h i p between  Tmn - Tzi  K  € m  n  n  mn mn  ':  '  (n=l,2,...6)  ^23  ^12  7-  32  l  33  (6)  n i s seen by comparing the tensor *7" • mn  r  12  ^22  L  C  and  m with the vector i n six-space  T*• i i T  =  r f  '-1  i2  ^22  '23  23  33  6  5  -  %  r r 4  r  m 3  Equation (5) w i l l now be evaluated f o r the special case of Use w i l l be made of the fact that a l l but 12 of the J6  cubic symmetry.  are zero, and of the 12 non-zero terms, only 3 are  e l a s t i c constants C mn  independent, namely c , c x l  given as: Cii C mn  C  1 2  0  0 0  0  c  3 2  3 3  0  0  0 0  C  1 2  0  0  0  Cn  C  1 2  0  0  0  i2  C  X 1  0  0  0  0  '0  4 4  0  C  C12  .00 0  0  C12  0  0  0  0  0  0  0  0  0  0  0  0  0  4 4  .o  0. c  0  1 2  / Cn  0  c  0  0 \  The e l a s t i c constant matrix i s then  0 .' 0  !  c  4 4  CO. 0  1 3  C21  C31  and c .  1 2  0  5 5  c  6 6  G  C  4 4  C  0  C  \  4 4  From the greatly reduced number of non-zero e l a s t i c constants i t can be s£en that C  mnpq  i s non-zero only f o r the suffixes given i n Table  A5.1.  181  TABLE A5.I Non-Zero Suffixes  C  C  Cn  O1111  1  1  1  1  c  0n22  1  1  2  2  1133  1  1  3  3  1 2  C  Suffixes  c  2 1  C2211  2  2  1  1  c  2 2  c  2  2  2  2  2 2 2 2  023  C  33  2  2  3  3  C31  C3311  y  3  1  1  032  C3322  3  3  2  2  033  C3333  3  3  3  3  C 3 3  2  2  3  3  3  =03232 =02332 =03223  C  c  4 4  2 2  2  2  2  3  C 1313  5 5  ^66  1  C3131  3  =Cl331 =03113  1  =  3  3  3 3  2  2  3  1  3  2  3  2 2  3  3 3  1  1  3  1  1 1  0 l l =0 121  1  2  1  2  2  1  2  1  =01221 =02112  1  2  2  1  2  1  1  2  2  2  2  By substituting values of C and the appropriate suffixes from Table A5.I mnpq into equation (5) an expression f o r ^ 2 1 i obtained i n terms of the s  independent e l a s t i c constants c between the X found that  2  1  i  and x  i x  ,c  1 2  and c  4 4  and the d i r e c t i o n cosines  reference frames. Following t h i s procedure i t was i i s given by the relationship:  ~~  Z*  '21  =Cn  P  -  + 2C  L_  12  2  (ana i)  + (a a )  2  £ a  C  1 1  a  1 2  a  2 1  a  2 2  2 2  1 2  •+ a  1 1  2  1 3  2 2  2  1 2  1 2  + a  2 3  ;  2 2  a  1 3  a  1 2  .+ ( a a )  2 3  2 1  a  2 3  :+ 2 a a a a i x ^ ] ^ 2 1 2 1  a  ~1  J  -J  a  1 3  a  a  2 2  2 3  "J  2  + (a a )  2  2 2  1 3  2  2 3  2  + (a a ) + 2a 1 1  1 3  a ia  2  (a a )  + (a a )  +  1 2  + (a a )  1 2  2a  2 1  2 1 3  a  1 3  a  2 3  a  2 1  2  + (a a ) l x  2 3  1 1  other strain-terms  +  2 2  (7)  k/-2,.l^l) v  kl  The shear m o d u l u s ( n , , | _ ) =  T.gi  c) 6 cients of £  2 1  computed d i r e c t l y from the coeffi-  i s  2 1  i n equation (7) Consider the shear modulus f o r the ^ . l o ) (001/*system.  transformation matrix i s given as (Figure _  x  The  A5:2)  3  Figure A5.2.  (1,  0  0  0  1 \T2 -1  1  0 1  n f  1 V2  (It should be noted that the components of the f i r s t row of a^j, namely a i j are the components of a unit vector along x  1  r e l a t i v e to the x  frame.  k  Likewise a . and a . are the components of unit vectors along x and x J J respectively. In a similar fashion the components of the f i r s t , second and 2  3  2  3  183  t h i r d columns of a.jj, namely vectors along •x , x 1  2  and x  3  a  a n c i  2  ^ i3' a  a  r  ^  e  e  c o m  P  o n e n  "  t s  or  " unit  respectively, r e l a t i v e to the x^ frame.) On  substituting values of a. . from the above matrix into equation (7) i t i s seen that:  \llo) <001> = C  yy  (8)  4 4  For the sake of completion, shear modulii were calculated on other possible s l i p systems i n cubic structures and are given i n Table A 5 . 2 . TABLE A3.2 . Shear Modulii f o r Various S l i p Systems i n Cubic Structures  Cubic S l i p System  A  (n,  <3> )  [hko] <001> .  C44  [001]  <110>  C44  (110]  <110>  1/2  £.10}  <111>  1/3 ( C n - C  1 2  ^ l l l ] <Xio>  1/3 ( C n - C  1 2  •+  (211) <±1X>  1/3 ( C n - C  1 2  '•+ C 4 4 )  (Cn-'ds) + C44) C44)  Since the transformation matrix i s symmetrical with respect t o n and the modulus characteristic, of plane n and d i r e c t i o n J}_ i s the same as that c h a r a c t e r i s t i c of plane _@ i n d i r e c t i o n n.  Note added i n proof: Equation (k) could be obtained d i r e c t l y from equation (1) by noting that C i s a fourth order tensor and subjecting i t t o the transmnpq 0  formation law f o r fourth order tensors, .i.e. '"mn _ ,.mnpq ^ pq =  C  ijkl T k l  C . ., , = a. a , a, a, C ljkl 1m j 'kp l q mnpq n  184  BIBLIOGRAPHY 1  T.E.Mitchell, .R.A.Foxall and P.B.Hirsch, P h i l . Mag. 8, I895, (1963).  2  A.Seeger, "Dislocations and Mechanical Properties of Crystals, (Wiley, New York) p.243, (1957).  3  L.M.Clarebrough and M.E .Ha rgr eaves, • Prog. Met. 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