\f ,

 

ANNEALING or DEFORMATION Twms m zmc
SINGLE CRYSTALS

Dissertation for the Degree of Ph. D.‘
MICHIGAN STATE UNIVERSITY
WEI HSIUNG KAO
1975 .

 

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3 1293 00

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This is to certify that the

thesis entitled

Annealing of Deformation Twins in
Zinc Single Crystals
presented by

Wei Hsiung Kao

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Metallurgy

 

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62/11th 1t ’44-’“9‘ [1.4+

Major professor

 

Date fizz-3021 /’?,, & 7r

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I.
I)“ ABSTRACT

ANNEALING OF DEFORMATION TWINS IN ZINC SINGLE CRYSTALS
BY

Wei Hsiung Kao

The thermal stability of the {lOlZ} <10ll> type deformation twins
in hexagonal close-packed zinc single crystals is investigated. A dis-
location model is proposed to explain the receding of the twin tip upon
annealing. The interaction between the twinning dislocations at the
twin tip is analyzed. The results show that the coherent twin boundary
energy Vt must lie between 1.4 1:0.4 and 2.8 i 0.8 ergs/cmz. The
receding process is described as a polygonization process assisted by

the tension of the twin interfaces.

The dislocations in the trace left behind by a receding twin
observed by other research workers as well as the author are found to
polygonize after prolonged annealing and form a wall parallel to the
basal plane of the matrix. Attempts were made to identify the disloca-
tions in the trace. The results seem to suggest that their Burgers
vector is '3 +IZ. Each of them would combine with a (43) dislocation

O .., O
to become one with Burgers vector c and then polygonize.

The present experimental results also show that small twins are
thermally unstable. Twins of sizes smaller than 1.4 microns would

disappear completely with no surface tilt remaining. Those twins with

Wei Hsiung Kao

thickness lying between 2 to 35 microns would disappear but the surface
tilt would partially remain. The magnitude of the change in the sur-
face tilt is found to be approximately 45 to 50 minutes of arc. The
fact that smaller twins less than 1.4 micron thickness can be annealed
out completely offers an explanation to the outstanding problem that no

recrystallization twins have ever been observed in zinc.

ANNEALING OF DEFORMATION TWINS

IN ZINC SINGLE CRYSTALS

by

Wei Hsiung Kao

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics and Materials Science

1975

To

My Parents

ii

ACKNOWLEDGMENTS

The author wishes to express his gratitude to Professor C. T. Wei,
who suggested the problem and gave valuable guidance throughout the
project. He sincerely thanks him also for his encouragement during the
course of this study. Re is indebted to the other members of his

guidance committee, Dr. K. Subramanian, Dr. H. Womochel, and Dr. J. Bass.

Special thanks are due to my wife, Marina, for her understanding,
help, and encouragement throughout the course of this study. Without

her help this work would not have been finished.

iii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . .

LIST OF FIGURES . . . . . . . . . . . . . . .

I.

II.

III.

INTRODUCTION . .

l.
1.
l.
l.
l.

l
2
3
4
5

Receding of twins during annealing
Accommodation region . . . .

Emissary dislocations . . . . . . . . . . .
Energy of the twin boundary . . . . .

Temperature coefficient of Yt .

EXPERIMENTAL PROCEDURE .

NNNNNN
C‘U‘l-l-‘UDNH

Preparation of zinc single crystals .
Introduction of deformation twins .
Heat treatment of the specimens .
Polishing and etching .
Interferometry . . . . . . . . . .

Scanning electron microscopy

RESULTS . . . . . . . . . .

wwwwwuw
\IO‘UlJ-‘UDNH

Receding of the twin

Dislocations left behind by receded twins .
Polygonization of dislocations in the traces
Characteristics of dislocations in the trace
Scanning electron microscopy

Interferogram . . . . . . . . . . . .

Annealing of small twins . . . . . . . . .

iv

Page

vi

. vii

\ONO‘U'IN

ll

11
16
16
l9
l9
19

21

21
27
27
32
32
32
43

TABLE OF CONTENTS (Continued)
Page

IV. THEORY AND DISCUSSION . . . . . . . . . . . . . . . . . . . 46

4.1 The receding of the twin tip and the twin boundary
energy . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.1 A simplified two-dislocation model . . . . . . . 47
4.1.2 Dislocation model of a many-layer twin . . . . . 55
4.1.3 Change in shape of the twin during annealing . . 72

4.2 Characteristics of dislocations left by the receded
twin . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3 Surface tilt left by the receded twins . . . . . . . . 79
4.4 Polygonization of the dislocations in the trace . . . . 83

V. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . 87
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
APPENDIX B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

APPENDIX C O O O O O C O O O O O O O O O O O O O O O O O O O O O O 96

LIST OF TABLES
Page

1. Equilibrium positions of dislocations in the single-slip-
band mOdel O O O O O O O O I O O O O O O O O O I O O O O O O O 97

vi

10.

11.

12.

LIST OF FIGURES

Zinc specimen sizes and orientations . . . . . . . . . . . . .
Graphite crucible for crystal growing . . . . . . . . . . .

Furnace set-up (schematic) . . . . . . . . . . . . . . . . . .
A (1102)[1IOI] twin in two specimens of different orientations
Furnace set-up for heat treatment . . . . . . . . . . . . . .

A (loiz)[loii] twin was annealed at 200°C for 1 hour.
Slightly polished. 100x . . . . . . . . . . . . . . . . . . .

(1i02)[1ioi] twins on the (loi0) plane of the specimen. 145x
(a) Freshly introduced twin, not polished . . . . . . . . . .
(b) 30 minutes at 200°C, polished . . . . . . . . . . . . .'.
(c) 90 minutes at 200°C, polished . . . . . . . . . . . . . .
(d) 2 1/2 hours at 200°C, polished . . . . . . . . . . . . . .
(e) 24 hours at 200°C, polished . . . . . . . . . . . . . . .
(f) 48 hours at 200°C, polished . . . . . . . . . . . . . . .

The shape of a receding twin tip (schematic) . . . . . . . . .

(1102)[1iol] twins on the (1010) plane. 145x . . . . . . . .
(a) Fresh twins, not golished . . . . . . . . . . . . . . . .
(b) 90 minutes at 200 C, polished . . . . . . . . . . . . . .
(c) 2 1/2 hours at 200°C, polished . . . . . . . . . . . . . .
(d) 24 hours at 200°C, polished . . . . . . . . . . . . . . .
(e) 48 hours at 200°C, polished . . . . . . . . . . . . . . .

A trace of etch pips left by a receded twin; specimen was
polished and etched after the annealing. 145x . . . . . . .

A polygonized trace of dislocations. The specimen with a
(10I2)[10II] twin was annealed at 2000 C for 24 hours and
was then etched. 140X . . . . . . . . . . . . . . . . . . .

Polygonized traces left by the receded twins. 24 hrs. at
20000, etChedo 145x 0 O O O O I O O O O O O O O O O O O 0

vii

Page
12
13
15
17

18

22

23
23
23
23
24
24
24

25
26
26
26
26

26
26

28

29

30

LIST OF FIGURES (Continued) Page

13. Pictures showing the sequence of the polygonization process.
150x . . . . . . . . . . . . . . . . . . . 31
(a) 2 hours at 200°C, polished and etched . . . . . . . . . . 31
(b) same twin as (a), the twin was polished off first, and
then the specimen was annealed for another 12 hours,
polished and etched. . . . . . . . . . . . . . . . . . . . 31

14. A (lI02)[lIOI] twin after annealing at 200°C for 12 hours,
then polished and etched. 150x . . . . . . . . . . . . . . . 33
(a) twin trace on (1010) . . . . . . . . . . . . . . . . . . . 33
(b) same trace on (1010) . . . . . . . . . . . . . . . . . . . 33

15. Twins on (lIOO) prism plane. Specimen was annealed at 200°C
for 2 hours, stress was then applied and specimen was etched.
130x . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
(a) (01I2)[01II] twins . . . . . . . . . . . . . . . . . . . . 34
(b) (Oll2)[0IlI] twin . . . . . . . . . . . . . . . . . . . . 34

16. Scanning electron micrographs of the etch pips pattern of a
receded twin . . . . . . . . . . . . . . . . . . . . . . . . 35
(a) 2,200X . . . . . . . . . . . . . . . . . . . . . . . . . . 35
(b) 5,000X . . . . . . . . . . . . . . . . . . . . . . . . . . 35

17. Scanning electron micrographs of the etch pips pattern of a
'receded twin. 1,800X . . . . . . . . . . . . . . . . . . . . 36

18. Scanning electron micrographs of two receded twin tips and the
traces left behind. 5,000X . . . . . . . . . . . . . . . . . 37

19. Scanning electron micrograph of the etch pips pattern of a
polygonized trace which was left behind by a receded twin.
2’200x . O C O C O O O O O O O C O O O O O O O O O O O C O O 38

20. A (lI02)[lIOI] twin with a polygonized trace. 200°C, 4 hours
and etched . . . . . . . . . . . . . . . . . . . . . . . . 39
(a) Optical microscopy, 150x . . . . . . . . . . . . . . . . . 39
(b) Scanning electron micrograph, 2,000X . . . . . . . . . . . 39

21. Scanning electron micrographs of the same twin as in Fig. 20.
2,000X . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
(a) Twin tip . . . . . . . . . . . . . . . . . . . . . . . . . 40
(b) Bent trace . . . . . . . . . . . . . . . . . . . . . . . . 40

22. Scanning electron micrOgraph_ of the etch pips pattern of a
freshly introduced (1I02)[1IOI] twin on the (lOIO) plane.
1, 800x O O O C O O O O I O O O O C O O I O O I O O O O O O O 41

23. (a) A freshly introduced twin on the (0001) surface. 300x . . 42
(b) Interference pattern of the same twin. 300x . . . . . . . 42

(c) Interference pattern of the same twin after it was
annealed at 200°C for 24 hours . . . . . . . . . . . . . . 42

viii

LIST OF FIGURES (Continued) Page
24. Same twin as in Fig. 23, but viewed from the (lOIO). 300x . . 44
(a) Fresh twin, corresponding to Fig. 23(a) . . . . . . . . . 44
(b) After annealing, corresponding to Fig. 23(c), etched . . . 44
25. Small twins on (0001) plane. 350x . . . . . . . . . . . . . . 45
(a) Before annealing . . . . . . . . . . . . . . . . . . . . . 45
(b) After annealing at 200°C for 10 hours . . . . . . . . . . 45

26. Dislocation model of a two-layer twin . . . . . . . . . . . . 48

27. Interaction force between two parallel dislocations in the
edge orientation. X-component . . . . . . . . . . . . . . . 52

28. Interaction energy between two parallel dislocations in the
edge orientation . . . . . . . . . . . . . . . . . . . . . . 54

29. The polygonization process of a two-layer twin . . . . . . . . 56
30. Dislocation model of a many-layer twin . . . . . . . . . . . . 57

31. Interaction forces between the (m+l) Eh dislocation and the
[99-(m+l)] dislocations in the boundary. X-component . . . . 60

32. Interaction forces between the (m+l) Eh dislocation and the
j £11- diSlocatIOn. X-Component o o o o o o o o o o 0 ~ 0 o o o 64

33. Interaction forces between the (m+l)£h_dislocation and the
m-dislocation wall. X-component . . . . . . . . . . . . . . 65

34. Burgers vectors in hexagonal close-packed lattice . . . . . . 76
35. Slip systems and twin system in zinc crystal . . . . . . . . . 77
36. Reference twin systems . . . . . . . . . . . . . . . . . . . . 78

37. The polygonization process of the dislocations in the trace
left by a receded twin (schematic) . . . . . . . . . . . . . 80

38. Dislocation reactions along the boundary of a (1I02)[1IOI]
twin and the matrix crystal . . . . . . . . . . . . . . . . . 81

39. Polygonization process of the dislocations_in the trace
(schematic). Plane of the drawing is (1120) . . . . . . . . 84

40. A twin on (1120) plane (schematic) . . . . . . . . . . . . . . 95

ix

I. INTRODUCTION

Twinning is one of the two modes by which a crystalline solid can
be plastically deformed. The other mode, slip, has been studied exten-
sively. On the other hand, due to the complexity of the twinning pro-
cess, many of its associated phenomena remain to be explored. Twinning
is important since experimental evidence seems to link it with fracture

in certain cases.

As defined by Friedell, a twin is a "polycrystalline edifice built
up of two or more homogeneous portions of the same crystal species in
juxtaposition and oriented with respect to each other according to well
defined laws." Except for a few rare casesz, the relationship between
two parts of a twinned crystal must have a symmetry so that one part can
be brought into coincidence with the other part, or the operation of
either (1) reflection in a lattice plane of low indices, or (2) rotation
through 60°, 90°, 120° or 180° about a lattice row of low indices. The
above description of twinning can be applied to all forms of twins,
namely, (1) growth twins, which form during solidification at the solid-
liquid interface, (2) annealing twins, which form during recrystalliza-
tion of a cold-worked material, and (3) deformation twins, which form by
a shear process in response to an applied stress. In general, growth
and annealing twins have only a secondary effect on the deformation
behavior of solids, whereas deformation twinning is an important

mechanism of plastic deformation in many crystalline solids.

2

The problems associated with deformation twinning are usually cate-
gorized as:
1. Criteria of the active twinning system in a particular type of
crystal.
2. Crystallography of the twinning system.
3. Nucleation mechanism.

~ 4. Growth mechanism.

There is yet another aspect of deformation twinning, the thermal
stability of the twins, that is also important for understanding the
twinning phenomenon. So far, most of the research work with respect to
the twinning has been concentrated on the four categories listed above.
The problems associated with annealing of deformation twins remain mostly
unsolved. It is, therefore, the purpose of the present work to investi-
gate the behavior of deformation twins in zinc single crystals during

annealing.

Zinc crystals are known to twin readily when they are oriented
unfavorably for the basal slip system to operate under particular
loading conditions. The active twin system in zinc. the {10I2}<10II>

system, has a twin shear of v = 0.139 determined by its crystallography.

l.l Receding_of twins during annealing

 

In their study of zinc crystals, Mathewson3’4 and Phillips4 found
that twins contracted upon annealing. The same phenomenon was also
observed in the annealing of twinned iron crystalss’6’7. Gindin and
Startsev8 annealed twinned bismuth crystals and found that wedge-shaped
twins of son in thickness were not annealed out after many hours at

250°C, but twins of Zn or less were removed completely. Furthermore,

wedge-shaped twins started to vanish at the thin end. Garber9 studied

twinned sodium nitrate crystals and concluded that annealing at 100°C
caused the twins to shorten, and they could be completely lost within 15
minutes at 306°C. The thinner the twins, the sooner they were removed

during the annealing.

The receding of the twins happens in almost all the hexagonal close-
packed (hcp) crystals and also in some of the body-centered cubic (bcc)
crystals. Deformation twins in orthohombic a-uranium were investigated
by Cahn10 and Calais et alll. They found that the twins were all less
than 10 microns in size. The lengthwise absorption of the twins by the
matrix began when the crystals were annealed at 630-6500C for 16 hours.
The twins eventually became parallel-sided with blunt tips. Klassen-
Neklyudova12 pointed out that the lens-shaped regions in.o-uranium were
twins of the second kind with irrational K1 planes. These twin inter-
faces straightened out upon annealing in an orientation of lower free
energy. It was suggested by Cahn13 that the contraction of the twin tip
was determined by the relative states of strain of the twin and the
matrix. If the matrix has less strain energy than the twin has, the

twin will contract. The swelling of the twin can occur only when the

matrix is further strained by slip after the twin is formed.

Using X-ray diffraction method, Lavrent'yev et a114 studied
annealing of the twins by observing the twin traces on (111) plane of
some antimony crystals. Their results showed that annealing of the
crystals at 600°C caused the disappearance of those twin layers of 5 to
20M in size, and restored the monocrystallinity of the specimen. The
time needed was less than 5 hours. Churchman15 pointed out that
annealing of twinned titanium single crystals at 800°C would cause the

twin lamellae to disappear internally, but the surface step originally

generated by the twin lamellae would remain "unaltered." This scienti-
fically puzzling problem was described by Cahn13 as "The mystery of the
smile without the Cheshire cat." Not all sizes of the disappeared twins
would leave behind a surface tilt. Lavrent'yev and Startsev16 found
that upon annealing of zinc crystals at 310°C for 3 hours, most twins
disappeared internally and left behind a surface tilt. For those twins
whose accommodation regions were smaller than 15H: both the twin and its

accommodation zone would disappear with no surface tilt left behind.

Associated with the surface tilt there is another puzzling pheno-
menon. Rosebaum17 found that after the annealing and etching of a zinc
single crystal, a partially contracted twin on (0001) cleavage surface
might leave behind one or two rows of dislocations in the space pre-
viously occupied by the twin lamella. The dislocations can be revealed
as etch pits by using a proper etching technique. However, the shear
strain produced by the twin was not removed much, if at all. He sug-
tested that a twin which did not go through the entire crystal would
cause a rotation about its composition plane. The amount of rotation
would be proportional to the maximum thickness of the twin. Upon
annealing, the rotation might remain in the crystal and, if it does, it
must be accommodated by a twist boundary of screw dislocations. However,
there has been no experimental evidence to substantiate this theory.
The characteristics of these dislocations, as well as the mechanism
through which they are left by the receding twin, remain to be deter-

mined.

Rosenbaum also investigated the possibility of the dislocations at
the tip of the twin in zinc being the result of an elastic relaxation.

Observations were made of twins on the stage of a microscope both before

and after the bending of the crystal. No evidence of any relaxation was
found. Kosevitch and Bashmakov18 also failed to observe any elastic
relaxation of twins in either annealed zinc, antimony, or bismuth
crystals. Price19 used the transmission electron microscopy for
studying zinc whiskers and platelets. He noticed that when the beam
current was turned on, the twin tip sometimes receded. No dislocations

were reported to have been left behind by the receded twin.

1.2 Accommodation region

 

In zinc crystals, the twin shear causes a change of slope of 3058’
of the basal plane, thus making the twin visible in an optical micro-
scope. This twin shear is often accompanied by an accommodation effect
which appears on the (0001) surface in the form of a change in the shape
of the surface relief around the twin. Since the (0001) slip plane is
bent in the accommodation zone during twinning, dislocations of opposite
signs form in this plane. Judging from the interference pattern,
Lavrent'yev et al16 have established the fact that the boundary between
the accommodation zone and the matrix is convex. It consisted of a wall
of the positive parts of dislocation loops. The negative parts of the
loops are connected partially with the twin and partially with the slip

plane of the accommodation region.

By using interferometry, Moore20 examined the traces of twins on
the cleavage surface of zinc crystals. The traces and the accompanying
accommodation kinks were compared with the metallographic sections cut
normal to the twin traces. Based on his experimental results, Moore
pointed out that the boundary between the accommodation region and the
matrix crystal consisted of a number of edge dislocations of the same

sign. Lavrent'yev et a116 made observations of twins in zinc after

successive periods of annealing, X-ray diffraction patterns and inter-
ferograms showed that polygonization took place in those twin accommoda-
tion regions which were larger than 100 microns. A scheme to account for
the observed accommodation pattern was deduced from the results. Direct
observation of the polygonization inside the accommodation region of

zinc has not been reported.

1.3 Emissary dislocations

Ahead of the advancing tip of a twin there is, in general, a region
of stress concentration. In many cases, slip was observed in front of

21’22. Hull23 studied single crystals of 3.25%

the advancing twin tip
silicon iron deformed at 20°K in tension along the '<OlL> axis. He
indicated that a twin-matrix boundary of a/6[lll] twinning disloca-
tions can be present only when the dislocations are widely spaced, i.e.,
the twin-matrix boundary is at a very small angle to the twin composi-
tion plane. When the twinning dislocation is more closely spaced, the
boundary is modified by the occurrence of slip around the end of the
twin due to local stress concentration. Sleeswyk24 proposed the so-
called "emissary dislocation” model based on his work with o-iron. He
suggested that for a growing twin, where twinning dislocations are piled
up behind the advancing tip, every third twinning dislocation -a/6[lll]
will dissociate into an a/3[lll] complementary dislocation and an

- a/2[lll] emissary dislocation. The emissary dislocation can glide
only on the {112} twin composition plane ahead of the twin and pro-
duces a homogeneous shear equivalent to the twinning shear without pro-
ducing a twinned region. The residual incoherent boundary will then

consist of - a/6[1ll] and a/3[lll] dislocations and have low energy

with no long-range stress. This accounts for the obtuse twin tip angle

in.o-iron. No direct experimental evidence to substantiate Sleeswyk's

theory has been reported so far.

Cahn13 theorized that the phenomena associated with receding twins
can be explained in terms of Sleeswyk's emissary dislocation model. He
indicated that the contraction of a twin lamella becomes possible only
when the blunted end becomes an incoherent boundary. This boundary
will behave like an ordinary grain boundary, and be able to migrate
without concomitant shear. In the case of zinc, Cahn suggested that
the twin lamella would emit slip dislocations and then contract. The
emitted dislocations would carry the macroscopic shear and could be
locked in position by vacancies or tangle with the dislocations in the
accommodation region. The thin twins in zinc, which disappear without
leaving their shear behind, could be considered to have been removed by

the reverse shear due to internal stress.

1.4 Energy of the twin boundary

 

Twin boundary energy Yt has been determined for some fcc and bcc
metals. The boundary energy can be obtained experimentally through the

measurement of the ratio of either yt/Yb or Yt/Ys’ where and

Yb
ys represent the grain boundary energy and surface tension, respec-

tively. vs is usually measured by the zero-creep method using either

thin foils38 or wires of small diameter25’26’27. Pullman28 obtained

Vt/Vb in copper by measuring the dihedral angles formed at the inter-
sections of twin boundaries and grain boundaries. The mean value of

yt/Yb was found to be 0.035 : 0.006. Taking Yb as 600 ergs/cmz,

Yt of copper is then approximately 21 ergs/sz. In a similar attempt29,

y of the incoherent twin boundary energy in copper was found to be

t
480 ergs/cmz.

31’30 measured the grooves

Using interference microscopy, Mykura
formed at the intersections of annealing twin boundaries and the sur-
faces of some nickel sheets. The orientation dependence of Vt was
then determined by applying Herring's equation32. Kurdman and Cadek33
applied the same method to sixteen different fcc crystals and alloys for
determining the coherent twin boundary energies. The ratios were found

to lie between 0.015 and 0.022 with Vt estimated to lie between 26.8

and 37.6 ergs/cmz.

Bhandari et al34 used thermal grooving and interferometric tech-
niques for measuring the interfacial energy of coherent and incoherent
deformation twin boundaries in relation to several crystallographic
surfaces of some rhombohedral-hexagonal a-aluminum oxide crystals.
yt/YS for the coherent boundary was found to be 0.31 with no signifi-
cant variation from one sample to another. For the incoherent twin
boundary, Yt/YS'E 0.58 for the (lOIO) surface plane. By taking
905 ergs/cm2 as the average value of vs of alumina at 185000, Vt
was then found to be 282 ergs/cm2 for the coherent twin boundary energy.

All the values of Vt mentioned above were obtained by indirect
methods. The mean values of Ys and Yb were used for computing Vt
at high temperatures. The room temperature Yt was then obtained by
extrapolation from the high temperature values. In such a method, the
grain boundary torque in a twin boundary-grain boundary intersection
and the variations in the surface tension with crystallographic orienta-
tion were assumed either constant or negligible. The errors introduced

by these assumptions may be insignificant when Vt is large, as in

alumina, but they may become appreciable in metals with small Vt.

Thus the values of Vt so obtained may not always be reliable for some

metals.

Nilles and Olson35 have obtained a relative energy value of 0.081
.i 0.019 for yt/Ys for the deformation twins in bcc iron crystals.
The corresponding coherent twin boundary energy should then be
160 ergs/cmz. No direct measurements of deformation twin energies are
known for hcp metals. Y00 and Wei36 considered the growth mechanism of
twins in zinc single crystals and estimated that Yt should be 1-4.i

0.4 ergs/cmz.

1.5 Temperature coefficient of y;

 

Murr37 has shown, from Gibbs fundamental equation, that the tem-
perature coefficient of twin boundary energy of pure metals should be
negative. By comparing the values of stacking fault energy st
measured indirectly using the zero-creep method with the values measured
directly from the extended stacking-fault nodes in an electron micro-
sc0pe of many fcc crystals, it appears that dvt/dT decreases with yt.
For instance, in silver, Vt = 11 ergs/cm2 and dvt/dT =
-0.003 ergs-cm-Z-deg-l. In the case of platinum, Vt = 161 ergs/cm2 and
dvt/dT = -0.040 ergs-cm-Z-deg-l. McLean et al38 measured dyt/dT of
cobalt and concluded that its value is small and negative. Kurdman and
Cadek33 also pointed out that the temperature dependence of Yt was

small and hardly measureable for some fcc crystals. Thus, in the case

of zinc, the temperature dependence of Yt may also be negligible.

In the present work an attempt is made to investigate the various

phenomena that take place during the annealing of the (10I2)[10II]

10

deformation twins in zinc crystals, both theoretically and experimen-

tally; namely,

1. The receding of the twins.

2. The energy of the twin interface.

3. Characteristics of the dislocations left behind by a receded
twin.

4. The surface tilt associated with a twin before and after

annealing.

II. EXPERIMENTAL PROCEDURE

2.1 Preparation of zinc single crystals

Zinc crystals were grown from the melt in a soft mold by using a
modified Bridgman technique. The specimen blanks were machined from
commercial zinc ingots of 99.99+% purity. The shape and size of the
specimen blank is shown in Fig. l. A single crystal seed with known
orientation was then welded to the lower end of the specimen blank with
an acetylene torch using ammonium chloride as a flux. Both the specimen
blank and the seed were cleaned in diluted hydrochloric acid before and
after the welding. Next, the orientation of the seed with respect to
the blank was checked by using the back-reflection Laue method. The
portion of the blank above the welding joint could be bent when neces-
sary in order to obtain the desired orientation after the crystal-
growing Operation. The seed end of the specimen was polished for a few
minutes in a polishing solution suggested by Vreeland et al39 to remove
the sharp corners at the welding joint and any surface imperfections

introduced during handling.

The specimen blank was then placed in a graphite crucible with the
seed end of the blank placed in the slot at the top face of the lower
thermal block T1 as shown in Fig. 2. A mixture of fine alumina and
graphite powder was then packed around the specimen blank. The slot of

the upper thermal block T was placed over the top of the specimen

2

11

12

Specimen blank

Ii“:—

 

Ifi-IchviJ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5
VP
fl
K /
“f
Seed—9 ‘1; b. Final specimen
._.._L .1_L_

 

 

.H.

a. Specimen blank and seed

Fig. l. Zinc specimen sizes and orientations.

II

...-......"-

I
V-
4

 

“"08
a.”
3m

....-.".'”l”1

 

GRAPHITE CRUCIBLE

13

 

QT
2cm'
.__l_

 

 

-II 'v-OJ c m
2,1,; holes
1 for thermocouples

 

 

 

 

UPPER THERMAL BLOCK 72

/;I k\ 2?! em

-| |-0.1cm

u I;
V °,

 

town THERMAL stocx T1

Fig. 2. Graphite crucible for crystal growing.

14

blank. Two thermocouples were placed inside the crucible, passing
through the holes in T2. One thermocouple was used for monitoring the
temperature of the welding junction, the other was positioned 1.5 cm

below the junction. This arrangement made it possible to control the

position of the solid-liquid interface during the crystal growth.

The crucible was then hung in a furnace, as shown in Fig. 3.
After soaking for three hours, and the lowest position of the solid-
liquid interface reaching about 1.5 cm below the welding joint, the
crucible was slowly lowered through the furnace at the rate of 2.5 cm
per hour for growing the crystal in the desired orientation. During
the growing process, dry nitrogen or hydrogen flowed through the tube

to provide a non-oxidizing atmosphere.

The crystals were grown with either the <1010> or the <1120>
direction parallel to the longitudinal axis of the specimen.4 They were
then cleaned in concentrated hydrochloric acid and visually examined.

An off-set method40 was used to eliminate the sub-boundaries, if
necessary. The crystal was then checked with the back-reflection Laue
method to insure the right orientation. The orientation can also be
checked by cleaving the crystal along the basal plane in liquid nitrogen
and then examining the orientation of the twin traces on the cleaved
plane. The uncertainty in the orientation of the crystals was within
:20. Etch pip study showed that the dislocation density in the crystals

was approximately 105 cm-z.

The crystals were then cut with a wire saw and cleaved along the

basal plane in liquid nitrogen into small pieces of 2 cm x 0.3 cm x 0.2 cm

15

 

@‘
W

stoblizin wire
weight

 

 

wire suspension rod

 

a t ermocoup es

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A
\
. T
.'_. .:.'N .
:3} 5 ~77: olumuno powder
:13? I — specimen
0 ' o
graphite §gn_:};: 1 : O
N-J, ,1. ‘
crucible g .1- . I.’\ g -—-——- heating element
0 ' 0
seed
T1

 

 

 

 

furnace tube

 

 

I inert gas inlet

Fig. 3. Furnace set-up (schematic).'

16

in size as shown in Fig. 1. Each piece was chemically polished to
remove any surface damage and deformation caused by cutting and

cleaving.

2.2 Introduction of deformation twins

 

Deformation twins were introduced into the zinc crystals either by
bending or by point loading applied on the (0001) surface. A twin

viewed from the (1120) and (lOIO) surface respectively is shown in Fig. 4.

In the case of simple bending, the specimen was simply supported on
a bending fixture by two knife edges in an Instron testing machine. The
load was applied through another knife edge fixed on the crosshead of
the testing machine. The knife edges were engaged on the (0001) plane
in such a way that either the 1<10I0> or the <112Q> direction was

parallel to the bending axis.

In the case of point loading, the load was applied on the (0001)
surface in the [OOOI] direction by using a pin indenter that was fixed

on the crosshead of the Instron machine.

2.3 Heat treatment of the specimens

All the heat treatments were carried out at 200°C. During the
heat treatment the specimens were placed in a horizontal tube furnace
as shown in Fig. 5. The tube was evacuated first and then a stream of
high purity hydrogen was allowed to flow continuously through the tube.
The hydrogen.was purified by passing through copper turning at 650°C
and anhydrous CaSOA. The temperature of the specimen was monitored by

a thermocouple in contact with it. The heat-treated specimens were

furnace cooled to minimize the thermal stress.

17

(0001)

 

 

a:

 

 

 

k—_-

 

I X (1130)

(Geo)

 

 

 

 

--—---‘-- _

 

 

sol—o )

 

(line)

Fig. 4. A (1I02)[1IOI] twin in two specimens
of different orientations.

 

///////

‘ 399“???“ ”E

i ////////;//a my
. \ V////7////z///A/

Flmmeter Tl I
Sanple holder

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

19
2.4 Polishingrand etching

For polishing and etching undecorated high purity zinc single
crystals and revealing the dislocation sites, solutions developed by
Vreeland et a139 (Appendix A) were found to be suitable. The polishing
solution could be used successfully on all planes. However, the etching
solution could only be used effectively on the (lOIO) prism plane. Thus

all the etch pip studies were made on the prism plane of the specimen.

The formation of the etch pips is due to the rapid diffusion of
mercury to the dislocation sites over the polished surface, and the
retardation of the rate of polishing by the mercury. During the course
of this study, it was found that the etch pips were very difficult to
remove from the surface once they were formed. It seems that the P-l

solution can remove them more effectively than P-2 solution (Appendix A).

2.5 Interferometry

 

To measure the flatness of the surface, a Leitz Double Beam.Mirau
Interferometer was used. This system has the advantage of making the
measurements without the need of contacting the specimen surface with
the reference mirror in the interferometer. To insure accuracy, the
measurements were carried out with a monochromatic light beam emitted

from a sodium vapor lamp.

2.6 Scanninggelectron microscopy

The topological features of the crystal surface developed by
etching were observed with an Advanced Metals Research Scanning Electron
Microscope operated at 17 RV. No coating of the crystal surface was

used. The specimens were examined in the emissive mode with a tilting

20

angle of 41° of the crystal surface with respect to the incident

electron beam.

III. RESULTS

3.1 Receding of the twin

 

In order to observe how the twins might have receded, the speci-
mens were polished slightly after annealing. Both the original position
of the twin tip and its position after annealing can be seen in Fig. 6.
Further polishing will remove the residual surface tilt left behind by

the receded twin tip and only the blunted twin tip will show.

Fig. 7 shows the trace of the twin on the (lOIO) surface of the
specimen after annealing at 200°C for various lengths of time. The
specimen was polished heavily after each annealing to remove the resi-
dual surface tilt, and then photographed. Fig. 7(f) represents the
final shape of the twin tip after 24 hours annealing at 200°C. The
final height of the twin tip was found to be within 35 x 10.4 cm. Fur-
ther annealing did not change the size or shape of the twin. Fig. 8 is

a schematic drawing of a receding twin tip.

Fig. 9 shows the change of the twins in a specimen upon annealing
at 200°C for various periods of time. It shows that the receding rate
of the twin is different from one twin to another and appears to be

irregular.

21

22

 

Fig. 6. A (10I2)[10II] twin was annealed at 200°C
for 1 hour. Slightly polished. lOOX

23

(a)

 

(b)

(e)

 

 

Fig. 7. (1I02)[1IOI] twins on the (lOIO) plane of the specimen. 145x
(a) Freshly introduced twin, not polished.
(b) 30 minutes at 200°C, polished.
(c) 90 minutes at 200°C, polished.

24

 

Fig. 7.

 

(Continued) 145x

(d) 2 1/2 hours at 200°C, polished.
(e) 24 hours at 200°C, polished.
(f) 48 hours at 200°C, polished.

25

 

(a)

(b)

(C)

Fig. 8. The shape of a receding twin tip (schematic).

Fig. 9. (1I02)[1IOI] twins on the

    

(1010) plane. 145x
(a) Fresh twins, not polished.
(b) 90 minutes at 200°C,

polished.

(c) 2% hours at 200°C, polished.
(d) 24 hours at 200°C, polished.
(e) 48 hours at 200°C, polished.

27

3.2 ‘Qislocations left behind by receded twins

For observing the dislocations left behind by the receded twins,
only those twins which extended over the entire thickness of the specimen
were selected. It could be assumed that all the twinning dislocations on

the twin boundaries in such specimens were of edge character.

Fig. 10 shows the twins after annealing and etching. The etch pips
represent the intersecting points of the dislocations with the prism
plane. It clearly indicates that a row of dislocations is left behind
by the receded twin. The density of the dislocations along the trace

was found to be non-uniform.

3.3 Polygonization of dislocations in the traces

In an effort to identify the nature of the dislocations in the
traces, a twin introduced by bending was annealed for a prolonged period
of time. Fig. 11 shows such a trace of dislocations after annealing at
200°C for 24 hours. The row of dislocations was observed to bend and
eventually polygonize. The segment of polygonized dislocation boundary
formed at the tip of the row of dislocations was parallel to the trace
of the basal plane. Numerous polygonized boundaries of basal disloca-
tions can be seen over the entire specimen. Fig. 12 shows that the
polygonization of the dislocations in the trace was stopped when the

trace met another boundary.

Fig. 13 shows two intermediate stages of polygonization of the dis-
locations in the trace. Fig. 13(a) shows the trace after 2 hours of
annealing at 200°C. The specimen was then polished to remove all the
etch pips and annealed for another 12 hours. The etch-pip pattern of

the same specimen after the second annealing is shown in Fig. 13(b).

28

/‘ /99
<9 9?
«g, /

 

(1010)

Fig. 10. A trace of etch pips left by a receded twin;
specimen was polished and etched after the
annealing. 145x.

29

 

(1010)

Fig. 11. A polygonized trace of dislocations. The specimen
with a (1012)[10111 twin was annealed at 200°C
for 24 hours and was then etched. 140X

30

210/

/1~

 

(1010)

Fig. 12. Polygonized traces left by the receded twins.
24 hrs. at 200°C, etched. 145x

31

.r. u flail-3.1 “1910..
1‘ 3

ins

 

[0001]

 

(b)

 

Pictures showing the sequence of the polygonization

Fig. 13.

150x
(a) 2 hours at 200°C, polished and etched.

Process 0

(b) same twin as (a), the twin was polished off

first, and then the specimen was annealed for

another 12 hours, polished and etched.

32

3.4 Characteristics of dislocations in the trace

Fig. 14 shows the trace of a (lI02)[1IOI] twin on both the front and
the back surfaces of the specimen after annealing and etching. These

figures appear to be similar.

A specimen with a (01I2)[01II] twin was annealed at 200°C for 2
hours and pulled in the [01I0] direction parallel to its basal plane.
Fig. 15 shows the etch-pip pattern on the (1I00) plane of the specimen.
The applied stress introduced many second-order pyramidal dislocations
in front of the receded twin tip. However, the row of dislocations left
behind by the receded twin tip were not removed by the applied stress
and remained to be a distinctive group. This is an indication that they

are not second-order pyramidal dislocations.

3.5 Scanning electron microscopy

 

Figs. 16 through 19 show the scanning electron micrographs of some
receded twin tips after etching. Fig. 20(a) shows an optical micrograph
of a twin with a bent trace of etch pips. Fig. 21 is the scanning elec-
tron micrographs of the etch pips of the same specimen as in Fig. 20(a)
at high magnifications. Fig. 22 is an etch-pip pattern of a freshly
formed twin. These micrographs show the detail of the etch pips in the

traces and in the twin boundaries.

3.6 Interferogram

 

The surface tilt of a twin and its change after annealing were
examined with the interferometer as described previously. Fig. 23(a)

shows a twin and its accommodation zone on the (0001) surface before the

 

Fig. 14.

33

A (1I02)[1IOI] twin after annealing at 200°C for
12 hours, then polished and etched. 150x

(3) twin trace on (1010).

(b) same trace on (1010).

34

[0001]

[1210]

 

Fig. 15. Twins on (1I00) prism plane. Specimen was annealed
at 200°C for 2 hours, stress was then applied and
specimen_was etched. 130X
(a) (0112)[0111] twins.

(b) (0112)[0111] twin.

35

[1210]

 

[0001]

 

Fig. 16.

Scanning electron micrographs of the etch pips
pattern of a receded twin.

(a) 2,2oox.

(b) 5,000x.

36

 

Fig. 17. Scanning electron micrographs of the etch pips
pattern of a receded twin. 1,800X

37

 

 

 

 

 

Fig. 18. Scanning electron micrographs of two receded twin
tips and the traces left behind. 5,000X

38

0
’z

./

 

(1010)

Fig. 19. Scanning electron micrograph of the etch pips
pattern of a polygonized trace which was left
behind by a receded twin. 2,200X

39

 

Fig. 20.

A (1I02)[1IOI] twin with a polygonized trace.

200 0°C 4 hours and etched.
(a) Optical microscopy, 150x
(b) Scanning electron micrograph, 2,000X

 

Fig. 21.

40

(b)

Scanning electron micrographs of the same twin as
in Fig. 20. 2,000X

(3) Twin tip.
(b) Bent trace.

41

 

[0001]

[1210]

Fig. 22. Scanning electron micrograph of the etch _pips
pattern of a freshly introduced (1I02)[1IOI]
twin on the (10I0) plane. 1,800X

42

 

(b) (C)

Fig. 23. (a) A freshly introduced twin on the (0001) surface.
300x
(b) Interference pattern of the same twin. 300x
(c) Interference pattern of the same twin after it
was annealed at 200°C for 24 hours.

43

heat treatment. Fig. 23(b) shows the interference pattern of the same
twin. After annealing at 200°C for 24 hours, another interference pat-
tern was taken as shown in Fig. 23(c). In Fig. 23(b) the angle a
between the surface tilt of the twin and the basal plane was calculated
from the fringe pattern and found to range from 3045' to 3050' along the
twin (Appendix B). These values are within the experimental error of the
theoretical value of 3058'. The angle In changed to approximately 3°

after annealing as measured from the fringe pattern shown in Fig. 23(c).

The trace of the same twin on the (10I0) surface was also observed.
Fig. 24(a) shows the twin before annealing. After annealing at 200°C for
24 hours, the twin disappeared completely. The specimen was then etched.
A trace of dislocations was found to have been left behind by the twin,

as shown in Fig. 24(b).

3.7 Annealing of small twins

Twins ranging from 1 to 5 micron in thickness were generated by
scratching the (0001) surface of the specimen with a razor blade. Fig.
25(a) shows many of such freshly introduced small twins. Fig. 25(b) was
taken without polishing the specimen after it was annealed at 200°C for
10 hours. It shows that one of the twins along with its surface tilt has
disappeared completely. The thickness of this twin was approximately

1.4 micron.

44

 

Fig. 24. Same twin as in Fig. 23, but viewed from the (lOlO).
300X
(a) Fresh twin, corresponding to Fig. 23(a).
(b) After annealing, corresponding to Fig. 9‘

iric), etched.

45

 

(b)

Fig. 25. Small twins on (0001) plane. 350x
(3) Before annealing.
(b) After annealing at 200°C for 10 hours.

IV. THEORY AND DISCUSSION

Deformation twin in zinc is a compound twin. In other words, the
indices of K1 and K2 planes, and H1 and n2 directions are all
rationa141. In principle, a compound twin can be formed either by
glide of dislocations on K1 with the rational direction “1 parallel
to the Burgers vector .Bt of the twinning dislocation (type I), or by
glide of dislocations on K2 with Burgers vector parallel to H2
(type II). These two twinning modes are alternative ways of forming
the same twin. Raisaz, Startsev et a143 and Bengus44 conducted a series
of studies on the etch pit pattern of twinned calcite. They observed
that there were pile-ups of dislocations at the twin interfaces and
concluded that the twins were of type I, as introduced by Frank45. On
the other hand, Bullogh46 suggested that in substances with diamond
cubic or hcp structures a twin interface is of the type II. Cahn13
also pointed out another drawback of the type II twin model, in that it
is inconsistent with the observation that in calcite the dislocation
distribution along a twin interface changes as the interface migrates.
The above objections raise some doubt in one's mind as to whether type
II twins actually exist in crystals. Thus, in the following discussion

of the twins in zinc, the type I model will be adopted.

46

47
4.1 The receding of the twin tip and the twin boundgry energy

Deformation twins in zinc are observed to shrink, even at room
temperature, and the rate of receding of the twin tip increases with
increasing temperature. It is further observed that the twins in zinc
recede lengthwise from the tip rather than transversely, as shown in
Fig. 6. Figs. 7 and 8 show the sequence of the receding process. It
can be seen clearly that, as the twin recedes, the tip becomes blunt at
first. The receding rate of the tip will then decrease with the
increase in height of the tip, and eventually become zero. The shape

of the twin tip will become parallel-sided after it stops receding.

The receding of the twin tip during annealing can be described as
a thermally activated polygonization process assisted by the twin inter-
face energy. It appears that the energy of the coherent twin interface
is the most important factor with regard to either the nucleation of
the deformation twins43 or the receding of the twin tips in zinc crys-
tals. Y00 and Wei36 estimated that the coherent twin boundary energy
of the (1012) [1011] type twins in zinc is of the order of
1.4 i.°-4 ergs/cmz, based on an idealized model of the twin with
pointed tip. Using such a low value, the ease of the nucleation of
deformation twins in zinc crystals was partially explained. An attempt
is made here to give a more rigorous treatment of the twin interface
energy, and to explain the thermodynamic instability of the deformation

twins in zinc crystals.

4.1.1 A simplified two-dislocation model

Much of the behavior of the tip of a twin can be understood with a

drastically simplified two-dislocation model. As shown in Fig. 26, let

48

 

 

 

 

Fig. 26. Dislocation model of a two-layer twin

49

A represent a hypothetical crystal lattice in which a two-layer twin B
is being formed by the successive production of two pairs of twinning
dislocations in the edge orientation lying in two neighboring {10I2}
atomic planes. When a shear stress T is applied, the dislocations of
the opposite sign will move in opposite directions resulting in the

expansion of the twinned region B.

When there is no obstacle in front of dislocation #1, its forward
movement is hindered mainly by the tension of the two interfaces
dragging behind it. Dislocation #2 is not under such restrictive
forces, therefore it will soon catch up with dislocation #1. A dynamic
equilibrium is then attained, and the two dislocations will move as a

group in the direction dictated by the applied shear stress.

Considering a case in which the moving dislocation #1 is halted by
an obstacle, let Ht be the Burgers vector of the twinning dislocation.

The forces acting on dislocation #1 per unit length are:

pbt + 2yt = Tbt + F2,1 (1)

where pbt is the Peierls-Nabarro force resisting the motion of the
dislocation in terms of an equivalent stress p; Yt is the coherent
twin interface energy per unit area and is assumed to be independent of

the twin thickness or the temperature; and F is the component of

2,1
the force per unit length acting on dislocation #1 by dislocation #2 in
the twinning direction. It is further assumed that the velocity of the
dislocation is low, so the energy dissipated by the moving dislocation

can be neglected, and the stress field of the moving dislocation and

its elastic strain energy remain the same as that of the dislocation at

50

rest. The forces acting on dislocation #2 are:

pbt + Yt + F -= Tbt + y

1,2 t . (2)

where F1 2 = F2 1 is the component of the force acting on #2 by #1.
3 O

The sum of (1) and (2)

2vt = th (T-p) (3)

gives the external forces acting on the two dislocations. The differ-

ence of (l) and (2) gives

t 2,1 (4)

provided that the maximum repulsive force (F2 1)max between the two
’ O

dislocations is numerically greater than yt. If, however, (F2,1)max.
is smaller than yt, the two dislocations will travel in a configura-
tion of minimal interaction energy, with dislocation #2 atop of disloca-

tion #1.

Let a set of perpendicular axes be chosen so that the X-axis is in
the [1011] direction, and Y-axis is perpendicular to the (1012) glide
plane. Let the position of dislocation #1 be at the origin, and dislo-
cation #2 be at (x,d), where d is the distance between neighboring
{10I2} atomic planes and is equal to 1.687 X in zinc. Let x2,1 be
the distance between dislocation #1 and dislocation #2 along the x-axis.

The X—component of the interaction force is:

 

G b2 x (x2 -y2)
r = t 2,1 2,1 (5)
x 2fl(1-v) 2

2 2
(“2,1 + y )

Substituting X = x2 1/d and y = d into (5), one obtains
’

51

2
_ G bt xgx2 - 1)
x 2nd(1-v) (X2 + 1)2

in which G = 3.72 x 1011 dyne/cm is the shear modulus of zinc. The

maximum repulsive force per unit length between dislocation #1 and dis-

location #2 is reached when x2 1/d = 2.41 and is equal to:
’

G b2

t
(F2,l) 0.250

max. 2vd(l-v) ° (5b)

Fig. 27 is a plot of F2,1 as a function of X when Yt < (F2,1)max.

The angle 0 of the tip of the twin is then a measure of Yt' Such a
configuration may or may not be thermodynamically stable, depending on
is compared with F

how V For instance, if the shaded area C

t 2,1’

in Fig. 2 is greater than the area D, the configuration of these two
dislocations is then metastable. Enough thermal activation may cause
dislocation #2 to overcome the maximum repulsive force between them and
ride above dislocation #1. The energy of the system is then reduced by
(C-D). This reduction in the interaction energy of the two disloca-
tions is the "driving force" needed for such a process, essentially a
polygonization process, to happen. The twin will then no longer have

a pointed tip, but a blunt one. On the other hand, if C<D, there
will be no driving force to push the two dislocations together, and
hence they will be kept apart by the repulsion between them. The twin

1

will have a pointed tip with an apex angle of e = cot- x2 1M.
3

The interaction energy between the two dislocations as a function

of X can be calculated by integrating eq. (5a),

52

.uausoqaounx .doaumudufiuo uwvo on”. a«
maowumuonHv 33¢qu 25 cassava ouuow noauueuouaH KN .me

 

 

 

 

Auuémv P I 1 n6.
1 a N6.
. "x ’1
F "x I’ll; H.°l
n_e._a_u___o_o an o a so « Any—ts.
_ _u—d _ _ a _ . 4 4 a _ a 4 a 4’4 (_M—1
lI I s lflrl "so
- I.J.|-~.- ~.s--
N 07.1 a a a 3.1 e.........0.....\
a. . . : ..... - «a
nus—0 Iii:
P 1 M6
Amman v I! quATNu .
. v0

 

)r‘.".:‘

P A'Ihtz
:99

was, (

53

X x

E = ‘ .Io F2,1 “-21% (5°)

and is shown in Fig. 28, curve A.

The polygonization process can be helped by pinning down disloca-
tion #2 and removing the applied shear stress. In such a case, dislo-
cation #1 will be pulled back toward dislocation #2 by a force per unit
length numerically equal to 2Yt instead of Yt' The interaction
energy between the two dislocations is then represented by curve B in
Fig. 28. The angle between such a pair of parallel-edge dislocations
has to be smaller than cot—1X1 for polygonization to take place.

This initial condition for polygonization is achieved either by an

applied stress, or by the presence of other dislocations, or both.

In Fig. 28, X1 is approximately equal to 3.2, corresponding to
9’? cot.1 3.2 = 17°. In other words, for a two-layer twin with a
pointed tip, the apex angle of the tip must be smaller than 17°.

Otherwise, the tip will become blunt when there is enough thermal acti-

vation.

When the applied shear stress T is gradually reduced, both dislo-
cations will gradually slow down, stop, and eventually move backward.
Let T1 be the applied stress at which the dislocations start to move

backward. One may analyze the situation as before, and obtain
Yr = F2,1 = “1*” be (Sd)

This means that unless these two dislocations polygonize, they will
remain at the same configuration regardless of whether they are moving
forward, or backward, or standing still, held by a stress 72 = Yt/bt

in an otherwise perfect crystal lattice.

54

,.cofiumuaowuo uwoo onu a“

 

msowumuoamwm Hmfiaeumm o3u coo3umn Swedes coauumuouaH .wN .wwm
AI ladle Ix
x
n N u o u- N- M- Q-
”— u - . . . . .
Fun” Fvnl
. ”-Ollt

P(,4-l)1:z
3‘19

 

JO(

3%

 

55

The Peierls stress p is of the order of 10.4 to 10.2 G; the

higher value is associated with covalent crystals, and the lower value,
close-packed meta1848. The Peierls force for the twinning dislocations

in zinc is therefore on the low side. If we take p’¥ 10.4 G, then

11 -4

pb 3.72 x 10 x 10 x 0.235 x 10'.8 erg/cm2

t

0.087 dyne/cm .

In the presence of enough thermal activation, the polygonization
process probably will start by forming a pair of kinks along the dislo-
cation line #1, as shown in Fig. 29. This pair of kinks will move
apart over the entire length of dislocation #1 and cause it to move
toward dislocation #2. The process will continue until these two
dislocations reach their equilibrium position, with one riding on top
of the other. By so doing, the twin recedes, with a decrease in the
interaction energy of the two dislocations. The shape of the twin tip
will change from pointed to blunt. The process described above is
essentially a polygonization process assisted by the tension of the
twin interfaces. It will happen only when the twinning dislocations
are purely or mainly of an edge character. Higher temperature reduces
the Peierls potential by blurring the lattice periodicity, increases
the amplitude of the vibrating dislocation line, increases the proba-

bility of forming the kinks, and hence increases the pologonization rate.

4.1.2 Dislocation model of a many-layer twin

Since an observable twin must have many layers, a more general
model is needed. A dislocation model representing the tip of a receded

many-layer twin is shown in Fig. 30. Assuming that there are m twinning

56

—-><10Tf>

 

 

I I
lg}::;?f’/:y ’i/'

 

I
_i’ 1 I’ ’ ’ ’ Dislo.#1

Peie r I 8 energy we I lay

Fig. 29. The polygonization process of a two-layer twin.

57

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uuo.ce.e_v_3h 2.9.0.30 .ALEX ll—
:25 c 2.2: fowl-log 1 «Ii

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

s._.
c E... A m“ a...
w. a
Wfiuho! ‘ e Ammoav

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:98: IY

. cl_l u.
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00° 5025— Bmzh .CULOp—OU

58

dislocations arranged in a vertical wall followed by (n - m) disloca-
tions forming a sort of "pile-up" on successive atomic planes, let a
set of rectangular Cartesian axes be chosen so that the X-axis is
parallel to the [lOII] direction, and Y-axis is perpendicular to the
twinning plane. Let d be the spacing of the (10I2) plane and x1,j
be the distance between the igh and jgh dislocations measured along the
X-axis. Let F.

1.1
experienced by the jEE dislocation in the stress field of the igh.dislo-

be the X-component of the interaction force

cation, and 6 be the angle between the tangent of the incoherent twin
interface at the tip and the coherent twin interface. Furthermore, the
twin is assumed to be of unit thickness measured normally to the plane
of the drawing, and the crystal is perfect except for the presence of
the twin. It is also assumed that the linear, isotropic, elasticity

theory is valid.

Because the angle of the twin tip is small and the spacing of the
glide plane is also small, the distance between the twinning disloca-
tions in the incoherent boundary measured along the X-axis can be
considered as being equal to the corresponding distance in a linear
dislocation pile-up as a first approximation. Since the distance
between neighboring dislocations in such a pile-up increases as the
dislocations are further and further away from the leading dislocation,
the incoherent interface at the tip will be curved as shown in Fig. 30.
In order to examine the interaction forces between the (m+l) £h_dislo-
cation and the [n - (m+1)] dislocations following it, certain assump-

tions will have to be made.

Eshelby, Frank and Nabarro49 investigated the equilibrium position

of n dislocations in a piled-up group under a constant applied stress.

59

Mitchell, Hecker and Smialek50 calculated the equilibrium position for
n = l to 99 by equating the first derivatives of the ugh Laguerre poly-
nomial to zero. The values of the equilibrium positions of the disloca-

tions in such a pile-up are listed in Appendix C.

The resolved shear stress acting on the twin plane in the twinning
direction when the twin grows was measured by Chyung and Wei47 to be
130 i 50 g/mmz. The low value of 80'g/mm2 is used for calculating the
equilibrium position of the dislocations in a linear pile-up. As
defined earlier, the position of dislocation #l is at the origin of the
rectangular Cartesian coordinate, dislocation #2 is at (x2;1, d), the
qgh dislocation is then at [xq’1, (q-l)d], etc. Thus the distance
between the qgh and the (q+l) Eh dislocation along the X-axis will be

(x - xq 1), and the distance along the Y-axis will be d.
9

q+l
Fig. 31 shows the total force exerted by the dislocations in the
pile-up on the (m+l) Eh dislocation as a function of m. It shows

that E
J=m+2

F. drops 86% when m increases from 1 to 10.
j,m:l

It is assumed that the system in Fig. 30 is on the verge of moving
forward under an applied shear stress T acting on the {10I2} twinning
plane in the <10II> direction. Let all the dislocations be straight,
parallel to each other, extend over the entire thickness of the crys-
tals, and be in the edge orientation. It is also assumed that the
energy of the twin interface is, in general, a function of the local
height of the twin, qd, or yt = Yt (qd). The X-components of the

forces acting on the qth dislocations are:

60

me

O?

.udmaOQEooux .mumvcson can s.“ 333.303.; 21.87am“
on”. use soguooonwv Mm. 3+5 ecu smegma moouow coHuooumuoH .3 .ma&

T...) 102.3022. 2: e. neo:ouo_u_1 055:). .o ceaEaeV 2

on on on

32°

("’Auxp)

333

66

zhhlusl
I-I-lll‘!i g

61

Q’1 ____ n
2’. F. + 2v (qd) + pb = Tb + 2)! (q-ld) + )3 F.
or
q-l n
2y (qd) - 2y (q-ld) = (T - p)b - E F. + E F (6)
t t t j=1 qu j=q+1 jsq

The sum of the X-components of the forces acting on the leading group of

m dislocations gives:

n m
2y (md) = m(T - p)b + E E F. . (7)
t t j=m+1 k=1 J’k
For the (mnl) Eh dislocation, one obtains:
m n
2yt(m+ld) - 2vt(md) = (T - p)bt - .§ Fj,m+1 + . Z ,Fj,m+1 (8)
j—l j=m+2

For the last dislocation in the system, the nrh, the sum of the forces

gives:
2yt(nd) - 2yt(E?1d) = (r - p)bt - r F. . (9)

There are altogether (n-m+l) such equations. The sum of all the
(n-m+l) equations gives again the total external forces acting on the

system:

2yt(nd) = n(T - p)bt . (10)

If n is sufficiently large so that yt(nd)’¥ ytcn), the true coherent

twin interface energy, one obtains:

Yt(°°) ’2 321(1 - p)bt . (11)

62

and

 

becomes very small.

When 7 is gradually reduced to r at which point the twin

1

starts to shrink due to the tension of the twin interfaces, one obtains:

2vt(nd) = n(T1 + p)bt . (13)

Solve for yt(nd) from eq. (10) and (13) assuming n is large:

yt(m) ’y £’(T + T1)bt . (14)

It may appear that yt(m) can be evaluated from this equation by

applying stresses T and T and observing the movement of the twin

1
interface. However, one may expect unsurmountable difficulties in such

an experiment. One may try to obtain some limiting values of Vt based

on the following analysis. The difference of eq. (7) and (8) gives:

2vt(md) - zlvto‘fid) - vt(md)]

 

n m m n
= (m-1)(T-p)b + 2 2 F + 2 F. - 2 F. (15)
t j=m+1 k=1 j,k j=1 j,m+l j=m+2 j,m+1
Since
2 2 -——-72 2
Chi: fjjm+1 (x ,m+l ' “n+1” d )
F = - ' ______ . (16)
j,m+l 2n(1 v) (x? m+1 + m+1_J.2dz)2
3
Let
th
D = """" s (17)

2n(l-v)

63
x = ._1_l-1. (18)

and substituting into eq. (16) one obtains

 

2 -————-2
j,m+l d _ 2 ——-2 2 '
j=l j—l (xm’m1 + m+l-j )

According to NabarroSI, the force between a dislocation and an infinite

wall when the dislocation glides out of the wall is:

 

Dbt fi2x2 2 fix
Fx = -'f;- (l - d2 / sinh I? ) .

In the present case, when m'e m, the force between this wall and the

(m+l) Eh. dislocation can be written as:

rain Db

-_1_ =____t__22,2
:1 Fj,m+l 2 Fx 2dX (1 n X / s1nh. nX) . (19a)
F1g. 32 shows Fm+l,j as a function 0: xm+l,j for j = m,m-l,m-2,m-3
and m-4 respectively. Fig. 33 shows 2 F as a function of
j=1 m+1,j
Xm+l,j’ for m = 1,2,3,10,50,500,1000 and as m«~ w.

When m is much smaller than n, one can neglect (m-l)(T-p)bt in

eq. (15) and write:

1 m
Sm = - z Fj,m+1+6 (20)
.181 .
where
Sm = yt(md) +'n (21)

T1 = vt(md) - Yt(;I;—1-d) (22)

64
0.3

0.2
0.1
M 234567391011121314
-o.1
-02

- 0.3

 

 

 

 

 

 

 

 

- 0.1

Xhm-o-l
d

Fig. 32. Interaction forces between the (m+l) _t_h_ dislocation
and the j _t_11 dislocation. X-component.

)dyn%m

61:2
23(14):!

PI

0.4
0.3
0.2
0.1
0.0

- 0.2
‘ 0.3
- 0.4
0.3
0.2
0.1

1

Fig. 33.

3 4

65

 

fi——

56 7 091011121314

 

34
34

I8)

7

lb)

7

(CI

X/d-e

 

8 9 10 11 12 13 14
I‘b
x/d

8 9 1011121314

X/d ->

Interaction forces between the (m+l)£h dislocation

and the m -dislocation wall.

X-component.

 

 

66

 

 

 

l

 

‘ m=io
3 4 5 6 7 8 9 1011121314
)( -I-
/d
(d)
L
_ m=so

     
 
 

I
A

I I
I’ 1’ y”
II I 1 1

I
67891011121314

I
l
I A
l
i

1

is)

Fig. 33. (Continued)

)dyne/cm

652
znil-i’ld

 

FC

0.5
0.4
0.3
0.2
0.1
0.0

$ I I
. C3 C’
u: 10 he

l
55
r)

I I
.0 9
05 (I

0.3
0.2
0.1
0.0

- 0.1

- 02
- 0.3
- 0.4
- as
- o.s

 

1

234

Fig. 33.

67

7891011121314

If)

(9)

(Continued)

m=1oo

 

 

X/d_,

m =soo

x/d->

13

68

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nun..

 

 

III: P‘fl-t)”:
AIMP( 3‘19 ):1

8":—
A g
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2.3 u E 3
«d

69

5 = l ‘2‘ g F - '3 F (23)
2 j==m+1 k=1 j'k j=m+2 33m”
One may also obtain from eqs. (15), (21), (22):
m
SIn = jEI FJ’.m+1 - e (24)
where
1 n m
e = E j=§+2 [FJ’mH 1:1 FJ’k] (25)
and
Fi,j = Fj,i
Also, eqs. (20) through (24) give
1 m ,
6 + e = 5"? Fj,m+1 . (26)
J—l
When a growing twin has a pointed tip or m=1, equations (20)
through (26) then become:
8 = '1 F + 6 (208)
1 2 1,2
51 = Yt(d) +11 (21a)
T1 = vt(d) - vt(2d) (22a)
6 = l' g F - g F (23a)
2 j=2 j,1 j=3 3,2
S = F - c (24a)

70

1 n
e = 2- g [Fj,2 - Fj,l] (25a)
j-3
5+ = _1.F (26a)
6 2 1,2 ‘
n n
Since 2 F >’ 2 F and e >’O, one obtains
.=2 j,l .=3 j,2
J J
o < a < lF (27)
2 1,2
0 < < 1F (28)
e 2 1,2 °
From eqs. (24a) and (21a) one obtains
lF + 5 = s = F - e
2 1,2 1 1,2 ’
which leads to
1I: <lF +5=2 (d)- (2d)-F - <F (29)
2 1,2 2 1,2 Vt Vt ‘ 1,2 6 1,2 °
If the angle 8 at the twin tip can be measured experimentally, F1 2
3

can then be read out from Fig. 27. S1 = 2yt(d) - yt(2d) is definitive

within a factor of two.

It may be assumed that yt(m) <1yt(2d) < yt(d), meaning that the
coherent twin interface becomes more and more stable as the number of

layers in the twin increases. It follows that
= - >
n vt(d) vt(2d) 0
and

vt(°°) < vt(d) = 81-11 = Flz-e-n

9

01‘

71

Yt(m) ‘< F1,2 ' e ' fl °

From eq. (29)

S1 = 2Yt(d) - vt(2d) = vt(d) +-fi
so that

vt(w) <1 vt(d) <1 F1,2 - e j n (30)
and

yt(m) <: yt(d) >’ %-F1’2 + a - n . (31)

Neglecting (T - p)bt, equation (8) becomes

n
_ _ .l - v
vt(d) - vt(2d) - fl 2 (F1,2 .: Fj 2) (32)
J-3
n
and therefore n must be smaller than l/2 [(F1,2)max - .é Fj,2]
n 3—3
. _ V . .
If n is greater than 1/2 [(Fl,2)max j:3 Fj,2] , equ1libr1um

cannot be established for m = l. The interface at the tip of the twin
will pull the second dislocation forward until it rides above the
leading dislocation. The case becomes one of m = 2. Since most of the
deformation twins in zinc crystals are observed to have pointed tips,
one can then set a limit of the values of the coherent twin boundary
energy to lie between 1/2 F and F based on equations (30) and

1,2 1,2
(31), i.e.,

1.4 i 0.4 ergs/cm2 5 Yt g 2.8 _-_i_- 0.8 ergs/cm2

for the (ll02)‘<llOi> twin system in zinc crystals.

72

4.1.3 Change in shape of the twin during annealing

When the twin begins to recede during annealing, the twin inter-
face energy is the main driving force. The dislocation at the tip of
the twin is pulled backward by the two interfaces with a force numeri-
cally equal to 2yt. It is a polygonization process assisted by the
twin interfaces, in addition to the interaction of the twinning disloca-
tion at the tip, and, as a result, there is a net decrease in the
energy of the system. _The backward pull of 2yt is essentially con-
stant, but the resistance of the lattice to the backward movement of
the polygonized wall is proportional to m. It will stop when 2yt'y

mpbt. Using Yt = 1.4rv 2.8 ergs/cmz, one obtains m = 30r~ 60.

When m increases, the interaction between the (m+l) Eh. and the
group of m dislocations in the wall becomes important. As shbwn in
Figs. 32 and 33, the interaction is a short range attraction and a long
range repulsion when m is small. When m increases, the repulsion
between the (m+l) Eh dislocation and the dislocation wall will gradually
decrease and finally change into an attractive force. In the meantime,
the range of the attractive interaction and the magnitude of the attrac-

tive force will both increase.

When the angle of the twin tip lies between 4° and 13°, as normally
observed in zinc crystals, Fig. 33 indicates that, as the number of dis-
locations in the wall reaches 500 or more, the (m+l) £h_dislocation
will experience an attractive force due to the stress field of the dis-
location wall at the tip. It will move forward and polygonize if this
attractive force is larger than the lattice resistance. Between
m = 30rv 60 and m = 500, even though the interaction between the m

dislocations in the polygonized wall and the (m+l) Eh dislocation is

73

repulsive at x/d = 13, polygonization can still proceed by thermal
activation if there is a net decrease in the total energy of the
system. As shown in Fig. 33(e), the shaded areas C and D are approxi-
mately equal. Polygonization is possible if the position of the

(m+l) Eh dislocation x/d g 13. In reality, the forward repulsion
acted on the (m+l) Eh dislocation by the subsequent dislocations in the
pile-up as shown in Fig. 31 will facilitate the polygonization process.
This process will continue and eventually the neighboring region of the
wall will be depleted of the twinning dislocations. The process will
stop when the distance between the mtg and the (m+l) £h_dislocation
becomes so large that the attractive force between them is insufficient
to overcome the lattice friction to the motion of the (m+l) Eh dislo-
cation. A metastable configuration is then reached, and the twin will

have a blunt and parallel-sided tip as shown in Fig. 7.

4.2 Characteristics of dislocations left byithe receded twin

As pointed out earlier in Chapter I, section 1.2, basal disloca-
tions must have been formed in the accommodation zone with their nega-
tive parts being adjacent to the twin boundary during the twinning. In
Fig. 11, a large number of polygonization walls consisting of basal
dislocations can be observed. Fig. 22 shows a scanning electron micro-
graph of a freshly twinned crystal. Many dislocations can be seen

near the tip region of the twin.

Fig. 14 shows the trace left by the receded twin on both the
front and back surfaces of the specimen. The similarity of these two
pictures indicates that the dislocations in the trace must extend over

the entire thickness of the specimen and lie on a set of parallel planes.

74

The dislocations in the trace cannot be basal dislocations since
they do not polygonize in the direction perpendicular to the basal
plane. Fig. 15 shows that under the specific loading conditions des-
cribed previously to produce a larger number of (1122) [2023] second
order pyramidal dislocations near the trace and the twin tip, the trace
can still be seen as a distinctive line among the second order pyramidal
dislocations, not moved by the applied stress. It is, therefore,
unlikely that the dislocations in the trace are second order pyramidal

dislocations.

The scanning electron micrographs shown in Figs. 16 through 19
indicate that the trace is really a continuation of dislocation in the

neighborhood of the twin boundary.

Figs. 16 and 18 show that the density of the dislocations in the
trace is different from that in the twin boundary. Since the shape of
the etch pips is related to the character of the dislocations, the dif-
ferent types of etch pips existing in the twin boundary would indicate
that there are more than one type of dislocations in the boundary. The

dislocations left behind by the receded twin are of one particular type.

Dislocations can only polygonize into a wall perpendicular to the
Burgers vector of the dislocations. Figs. 11 and 12 show that the tip
of the trace has polygonized after prolonged annealing. The polygonized
wall revealed by the etch pips on such a prism plane is in a direction
parallel to the trace of the basal plane. The same phenomenon can also
be seen in Figs. 12, 13 and 20. Fig. 13 also shows the sequence of the

polygonization process.

75

The above observations point out the fact that the Burgers vector
of the dislocations in the polygonized wall must be ‘3' These disloca-
tions are neither basal nor second order pyramidal dislocations. Since
the slip traces in the twin were observed to be parallel to the trace
of the basal plane in the twin, one can assume that the most probable
slip dislocations in the twin are basal dislocations. An a/3[2110] or
(-a3) basal dislocation in a (1102)[1101] twin can penetrate the twin
boundary and be incorporated into the matrix to become a a/3[1123] or
(c + a3) dislocation as described by Y00 and Weisz. Such a dislocation
is constrained to move on the (1100) plane in the matrix. It can not
move far away from the twin boundary because (1100) is not an active
slip plane. Since there are many basal dislocations near the twin
boundary, as described earlier, the annealing will assist the reaction
between the basal dislocations and the prismatic dislocations near the
twin boundary. Thus, the following reaction could have happened during
the annealing:

(3 + Z3) + (433) , 8 (37)

 

or

 

a/3[1123] + a/3 [1120] .7 a[0001] . (38)

As pointed out by Y00 and Wei52, the a/3[1123] dislocation would
lower its energy if it dissociamusinto two partial dislocations. Thus,

reaction (38) can be written as:
2 x a/e[iiz3] + a/3[ll20] ———————» a[0001]. (39)

.)
Furthermore, a[c] dislocation is also likely to dissociate into two

partial dislocations:

a[0001] -—————-—~+ a/z [0001] + a/2 [0001] . (40)

76

01

 

 

 

 

 

 

 

 

 

Fig. 34. Burgers vectors in hexagonal close-packed lattice.

77

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782
A>v

ANNE
A2.

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WNONQW ANNE £3

 

 

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fl xxfl. 5. issues
V“ F

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78

<;:—(|012)
4

 

 

 

 

 

 

 

Fig. 36.

[
[

(T1.tl):
(T4,t4)7
(T2.t4):
(T5.t5);
(T3.t3);

(736.1:6):

 

(1012)

(1102)

(0112)
(0112)
(1102)

(1012)

Reference twin systems.

[1011]
[1101]
[0111]

[01111

[1101]

[10111

79

The true representative reaction can be obtained by adding equations (39)

and (40):

 

2 x a/6[1123] + a/3[1120] a/2[OOOl] + a/2 [0001] (41)

or
2 x 1/2 (E? + Z) + (£3) —— 2 x ("8/2) (42)

The resulting dislocations would lie on the prism plane of the
matrix and have Burgers vector 'Z/2[0001]. This incorporation process
is shown in Fig. 38. Fig. 37 shows schematically the dislocations in
the trace before and after polygonization. By using the criterion of
the square of magnitude of the Burgers vector, there is a reduction in
energy from

2 x 1.0542 a2 + a2 '2 2 x 0.928 a2

E - E
_ 2 1 _ 1.72 - 3.22 _ _ .

 

thus equation (41) is energetically favorable.

The dislocations lying in the prismatic plane with Burgers vector
1/2 2 are known to be inactive in zinc crystals at room temperature.

53
The present results substantiate such observations in the literature .

4.3 Surface tilt left by the receded twins

In principle, deformation twinning is crystallographically rever-
sible. If no accompanying slip is involved, twinning itself will produce
a surface tilt angle a of 3058’ as it is fixed by the crystallography
of zine. When the twin recedes during annealing, and if it is completely
annealing out without accompanying slip process, the surface tilt should

disappear with the twin, as is the case of small twins less than some

80

(1010)

 

 

 

 

 

/_

_\
(1010)

 

 

 

 

 

 

 

(0001)

 

 

 

(1120)

 

 

 

 

 

 

 

 

 

 

 

 

 

the trace left by a receded twin (schematic).

(a) Before annealing.

The polygonization process of the dislocations in
(b) After annealing.

Fig. 37.

 

_-.__.-__M)

 

 

 

 

.L _ __ LT : Bgsal disiocation loop in the matrix,
-a and a.

Basal dislocation :3 in the twin.

“-1

Dislocation in the matrix along the
twin boundary as a result of disloca-
tion incorporation.

Fig. 38. Dislocation reactions along the boundary of a
(1102)[1101] twin and.the matrix crystal.

82

1.40 in thickness. Since such small twins are thermally unstable,
therefore recrystallization twins do not exist in cold worked and

annealed zinc crystals.

The angle a of the surface tilt associated with the deformation
twins, observed previously by other research workers and in the present
work, was not always exactly equal to the theoretical value of 3°58’.
Since slip in the matrix as well as in the twin usually takes place
during twinning, and since slip dislocations if arranged in some speci-
fic way such as those in the accommodation kink may give rise to a sur-
face tilt, the surface tilt angle a could be the result of the slip

dislocations.

As the twin recedes during annealing, a polygonized wall of
twinning dislocations is formed at the tip of the twin. Such a segment
of dislocation wall inside the crystal will have a strong stress field
around it. In bcc crystals the (112) twin plane is also a secondary
slip plane. The high stress field in the neighborhood of such a twin
tip may produce the so-called emissary dislocations, which may in turn
explain the surface tilt that remains after the twin is completely
annealed out. However, in zinc crystals, the [1102] twin plane is not
an active slip plane. Emission of dislocations in the twinning plane is
not likely to happen. The present work also shows that the dislocations
left behind by the receded twin are most likely those slip dislocations
resulting from a dislocation incorporation process. To account for the
surface tilt that remains after the twin is annealed out, there must be
some other dislocation mechanism or mechanisms involved. A possible

explanation is that the stress field of the polygonized twinning

83

dislocations wall may draw neighboring basal slip dislocations with 43
Burger vector to the twin interface to account for the bend that is

Opposite to the main accommodation kink.

The final tilt angle that remains after the twin is annealed out
would depend on the number of basal dislocations that are drawn to the
original twin interface and thus would not be constant. The change in
the surface tilt of some 45' to 50' as a result of the annealing could
be the portion carried away by the disappeared twin not accountable by
the slip dislocations drawn to the twin interface. During prolonged
annealing, some of the 4: basal dislocations might combine with the
I: 4: dislocations to produce the [Z dislocations which polygonize as
described previously. A positive proof of the above theory is yet to be

found.

4.4 Polygonization of the dislocations in the trace

 

Fig. 11 shows that the trace which was left behind by the receded
twin is bent at the tip after prolonged annealing, and the bent portion
of the trace is parallel to the trace of the basal plane. This indi-
cates that the dislocations in the trace have polygonized and formed a
boundary perpendicular to the [0001] direction. Since these disloca-
tions were the products of the dislocation incorporation process along
the twin boundary, they formed an array parallel to the receded twin
.54,55

boundary before polygonization. This array makes an angle of 46°59

with the basal plane of the matrix as shown in Fig. 39.

The array of dislocations is not necessarily in a low energy con-
figuration. Its energy can be lowered by a rearrangement of the dislo-

cations into a low angle boundary that has no long range stress. The

84

/["i’1o1]

9-3'5 0’

 

 

_LQLLQJJm

 

 

 

 

 

 

 

A—fi;
1 ' 2
I’ll
-_.§‘1

(1102) [1101‘] Twin

Fig. 39. Polygonization process of the dislocations
in the trace (§chematic). Plane of the
drawing is (1120).

85

forming of this low angle boundary may require dislocation climb and
glide, a polygonization process. Thermal activation is needed for the
dislocations in the array to polygonize. As shown in Fig. 39, the angle
between dislocation #l and #2 is 46059,, and hence there is a small
repulsion between them. The polygonization is possible only when dislo-
cation #1 can climb into the attractive stress field of dislocation #2.
Both dislocations then will move together to dislocation #3. This
process continues until the dislocations form a wall. The Peierls-
Nabarro force which hinders the movement of the wall will become larger
and larger as the height of wall increases, and eventually the movement
of the wall will stop. This process is eesentially similar to the poly-
gonization process described in Section 4.1, except for the absence of
the twin interfaces. Since the number of dislocations in the trace
should be related to the number of slip dislocations in the twin incor-
porated into the matrix through the twin boundary, the density of the
dislocations in the trace would not necessarily be uniform, as was

observed to be the case.

During the course of this research, it was found that the poly-
gonization phenomena associated with the dislocations in the trace were
related to the loading conditions under which the twins were introduced.
When the twins were introduced by the three-point-bending method,
prolonged annealing would cause the traces to polygonize readily. But
when the twins were introduced by point loading, in most cases the
traces did not polygonize. This difference probably lies in the number
of 4: dislocations generated in the matrix by the two loading condi-
tions. The three-point-bending method tended to bend the (0001) slip

planes of the crystal severely, and hence many basal dislocations with

86

Burgers vector 2' were generated. On the other hand, the deformation
was localized in the point loading method and few basal dislocations
were generated. Since ‘2 dislocations were essential in producing the
.2 dislocations in the trace, as described previously, it was reason-
able that polygonization of the dislocations in the trace was more
likely to occur when the twins were generated by the three-point-
bending method. The traces left by the twins which were generated by
the point loading method probably consisted of '2 +'Z dislocations
that have not combined with basal dislocations to form the '2 disloca-

tion required for the polygonization process.

At the present, one can only say the energy of the polygonized
boundary is lower than that of the dislocation array. The detailed
process is not exactly known. Further experimental work to clarify this

point is needed.

87

V. CONCLUSIONS

Based on a simplified dislocation model, the energy of the coherent
twin interface of the {1102} <1101> twins in zinc crystals has been

calculated to lie between 1.4 i 0.4 and 2.7 i 0.8 ergs/cmz.

The receding of the tip of such a deformation twin in zinc during
annealing has been treated as a polygonization process assisted by
the twin interfaces. The theory predicts that when the polygonized
wall of twinning dislocations reaches a certain height the process

should stop, and the tip of the twin should become parallel-sided.

Annealing of zinc single crystals containing deformation twins

"introduced by either the bending or indentation method at 200°C for

ten hours or more shows that those twins whose thickness was less

than 1.4 x 10“4 cm would disappear completely with no surface tilt

remaining. Those twins with thickness lying between 2 and 35 x 10.!+
cm,depending upon the apex angle of the twin, would disappear, but
the surface tilt would partially remain. The surface tilts before
and after annealing were measured by using an interferometric method
and the monochromatic light of the sodium of 589 millimicron in wave-

length, and were found to have changed by 45' to 50' due to the

annealing for those twins of 2 to 35 micron in thickness.

By using an etching method suggested by Vreeland et al to bring out
the dislocation etch pips, it was confirmed that rows of disloca-

tions were left behind by the receding twin tips after prolonged

88

annealing. The experimental results also showed that those rows
left behind by the twins introduced by bending, polygonized readily
to form a dislocation boundary parallel to the basal plane, but

those left behind by twins introduced by indentation did not.

Attempts were made to identify the character of the rows of disloca-
tion by observing the shape of the etch pips using a scanning elec-
tron microscope. The results seem to suggest that the Burgers vec-
tors of the dislocations in the row be : +-Z, each of which might
combine with a neighboring dislocation with a (-2) Burgers vector

r)
to become one with c, and then polygonize.

Since small twins are thermally unstable, annealing twins may not

nucleate in zinc crystals.

BIBLIOGRAPHY

10.

ll.

12.

13.

140

15.

16.

BIBLIOGRAPHY

Friedel, G., Lecons de Cristallographie, p. 421, Berger Levzault,
Paris, 1926.

 

Cahn, R. W., Adv. in Physics, 3, 396, 1954.
Mathewson, C. H., Trans. Amer. Inst. Min. Met. Eng., 16, 554, 1928.

Mathewson, C. H., and Phillips, A. 3., Trans. Amer. Inst. Min.
Metall. Engrs., 15, 143, 1927.

Krivobok, N. V., Amer. Inst. Met. Eng., Preprint No. 1557-E, 1926.

Garber, R. I., Girdin, I. A., Konstantinovskii, M. G., and Startsev,
V. 1., Doklady Akad. Nauk SSSr 74(2), 343, 1950.

Hausner, H. H., and Pinto, N. P., Trans. Amer. Soc. Metals, 43,
1152, 1951.

Gindin, I. A., and Startsev, V. I., Zh. EKSper. Teoret. Fiz., 29,
738, 1950. '

Garber, R. I., Zhur. EKsptl. i Teoret. Fiz., 17(1), 63, 1947B.
Cahn, R. W., Adv. in Physics, 3, 363, 1954.

Calais, D., Lacombe, P. and Simenel, N., J. Nucl. Materials, 1, 325,
1959.

Klassen-Neklyudova, M. V., Mechanical Twinning, Consultants Bureau,
New York, p. 72, 1964.

 

Cahn, R. W., Deformation Twinning, p. 8, A.I.M.E. Met. Soc. Conf.,
Gordon and Breach, New York, 1964.

 

Lavrent'yev, F. F., Soifer, L. M., and Startsev, V. 1., Kristallo-
grafiya, _3, 3, 472, 1960.

Churchman, A. T., J. Inst. Metals,_§3, 39, 1955.

Lavrent'yev, F. F., and Startsev, V. I., Fiz. Metal. Metalloved, 13,
3, 441, 1962.

89

l7.

l8.

19.

20.

22.

23.

24.

25.

26.

27.

28.

29.

33.

34.

35.

36.

37.

38.

39.

90

Rosenbaum, H. 3., Acta Met., 2, 742, 1961.

Kosevich, V. M., and Bashmakov, V. L., Kristailografica, 3, 749,
1959.

Price, P. 8., Phil. Mag., 2, 873, 1960.

Moore, A. J. W., Acta Met., 3, 163, 1955.

Hull, D., Acta Met., 2, 909, 1962.

McHargue, C. J., Trans. Met. Soc. AIME, 224, 334, 1962.

Hull, D., Deformation Twinning, p. 121, A.I.M.E. Met. Soc. Conf.,
Gordon and Breach, New York, 1964.

 

Sleeswyk, A. W., Acta Met., 29, 705, 1962.
Udin, H., Shaler, A. J., and Wulff, J., Metals Trans., p. 186, 1949.
Udin, H., Buttner, F. H., and Wulff, J., J. Metals, p. 1209, 1951.
Udin, H., Funk and Wulff, J., J. Metals, p. 1206, 1951.

Fullman, R. L., J. of Applied Physics, 22, 4, 448, 1951.

Fullman, R. L., J. of Applied Physics, 22, 4, 456, 1951.

Mykura, H., Acta Met., 2) 570, 1961.

Mykura, H., Acta Met., 9, 595, 1961.

Herring, C., The Physics of Powder Metallurgy, McGraw-Hill, p. 143,
1951.

 

Kurdman, J., and Cadek, J., Phil. Mag., 29, 413, 1969.

Bhandari, O. P., Bertolott, R. L., and Scott, W. D., Acta Met., 22,
1515, 1973.

Nilles, J. L., and Olson, D. L., J. of Applied Physics, 42, 531,
1970.

Yoo, M. H., and Wei, C. T., Phil. Mag., 22) 759, 1966a.
Murr, L. H., Scripta Met., Q, 203, 1972.
McLean, M.,and Mykura, H., Phil. Mag., 2Q, 1191, 1967.

Brandt, R. C., Adams, K. H., and Vreeland, T. Jr., J. Appl. Phys.,
23, 587, 1963.

40.

41.
42.

43.

44.
45.
46.
47.
48.

49.

50.

91

Chyung, C. K., On the Improvement of Lattice Perfection in High
Purity Zinc Crystals Grown from the Melt, M. S. Thesis, Michigan
State University, 1962.

 

Jillson, D. S., J. Metals, 188, 1009, 1950.
Rais, G. B., Kristallografiya, 23 325, 1958.

Bengus, V. Z., Komnik, S. N., and Startsev, V. 1., Doklady Akad.
Nauk., SSSR, 141, 607, 1961.

Bengus, V. Z., Soviet Physics Crystallography, §, 322, 1963.
Frank, F. C., Phil. Mag., 42) 809, 1951.

Bullough, R., Proc. Roy. Soc., 2242, 568, 1957.

Chyung, C. K., and Wei, C. T., Phil. Mag., 123 161, 1967.
Yoo, M. H., and Wei, C. T., Phil. Mag., 22, 759, 19663.

Eshelby, J. D., Frank, F. C., and Nabarro, F. R. N., Phil. Mag., 42,
351, 1951.

Mitchell, T. B., Hecker, S. 8., and Smialek, R. L., Phys. State.
801., 223 585, 1965.

Nabarro, F. R- N., Theory of Crystal Dislocations, p. 75, Oxford
Press, 1967.

 

Yoo, M. H., are Wei, C. T., Phil. Mag., 14, 573, 1966b.

p...

Gilman, J. j., Trans. Amer. inst. Min. (Meta

1.) Engrs., 206, 1326,
1956.

Taylor, A., and Leber, Sam., J. of Metals, Feb., 190, 1954.

Salkovitz, E. 1., Trans. A I.M.E., 191, 64, 1951.

APPENDICES

APPENDIX A

Polishing and Etching of Dislocations

Present experiment requires polishing and etching solutions which

are suitable for undecorated high purity zinc single crystals. It has

39

been found that the solutions developed by Vreeland et a1 were suit-

able for the purpose. The solutions are:
P—l 160 gr CrO3

20 gr NaZSO4

500 ml distilled water

P-2 Equal parts of: Methanol

BOA H202

conc. HNO3
P-3 160 gr CrO3

500 ml distilled water

E One part: 1 gr Hg(N03)2
1 m1 conc. HNO3
500 m1 H20
Two parts: distilled water

92

93

The polishing procedure:
1. Dip with mild agitation in solution P-l for 20—100 sec.
2. Dip occasionally in solution P-2 to accelerate the polishing
process.

3. Dip finally in solution P-2 to remove the CrO3 film.

The etching procedure:
1. Dip with mild agitation in solution E, 5-6 sec.
2. Dip with mild agitation in solution P-l, 5-6 sec.
3. Dip with mild agitation in solution P-3, 2-3 sec.
4. Rinse in running tap water.
5. Rinse in running distilled water.

6. Dry in an air blast.

Best results are obtained when the polishing procedure is imme-

diately followed by the etching procedure.

The formation of etch pips is due to the rapid diffusion of mercury
to dislocation sites over the polished surface, and the diffused mercury

retards the rate of polishing.

APPENDIX B
Double Beam (Mirau) Interferometer

To measure the surface tilt caused by the twinning process, the
Mirau Interferometer was used. This system, contrary to the Multiple
Beam Interferometer, is designed to provide for the measurement of
surface steps without making contact with the specimen. A light source
of low coherence such as a tungsten filament lamp can be used because
reference and measuring rays travel exactly the same distance. But to
insure.accuracy, measurement should be carried out in monochromatic
filter or a monochromatic light source. In the present work, a sodium

vapor lamp was used as light source.

When the interferometer is tilted against the specimen surface, a
fringe system of equidistance fringes is produced. An elevation or
depression in the specimen surface causes a displacement of the fringe
at this location. The size of this displacement is a direct measure

for the height or depth of the steps.

d = 1072 (1)
in which d = step height or depth (9):
K = relative fringe displacement expressed in parts of one
fringe to fringe distance,
1 = used wavelength (589np for sodium lamp).

94

95

To measure the surface tilt angle 6, as shown in the figure below,
eq. (1) can be modified as d = w/cot 9, in which w is the width of

the twin on (0001) surface. Therefore

0 = cot.1 w/d = cot.1 2w/K1. (2)

(”L

Fig. 40. A twin on (1120) plane (schematic).

APPENDIX C
Equilibrium positions of dislocations in the single-slip-band model

Eshelby et al49 analyzed the equilibrium positions of n disloca-
tions in a piled-up group under an applied stress. The equilibrium
positions X = X(i) (i = l,2,..., n) of the dislocations are determined

from the equations

 

X(O) = O
n 0
120 X(j) _ X(i) + T = O (J = l,2,...,n)
i=j
where D = ————£fll——' (i otro 1 e1 ti it theor )
2n(1 _ v) s p c as c y y
0:53-
2n (anisotropic elasticity theory)

By considering X(i) as the zeros of the polynomial
f(X) = (X - X(1)) (X - X(2))--.(X - X(n)) .

Eshelby et al49 found that f(X) is given by the first derivative of

the n-th Laguerre polynomial

21X
f(X) - L D

I

n
. . . . . . . 50

The equilibrium p031t10ns for n = 1 to 99 obtained by Mitchell et al

are listed in the following Table.

96

97

TABLE 1

Equilibrium positions of dislocations in the single-slip-band model.

Equilibrium positions of dislocations in pile-ups are obtained by multi-
plying by D/ZT.

 

 

 

n x1, x2, x3, x4, x5, x6, x7, x8 ,...., xn

1 2.000

2 ‘1.268, 4.732

3 0.935, 3.305, 7.759

4 0.743, 2.572, 5.731, 10.95

5 0.617, 2.113, 4.611, 8.399, 14.26

6 0.527, 1.796, 3.877, 6.919, 11.23, 17.65

7 0.451, 1.564, 3:352, 5.916, 9.421, 14.19, 21.09

8 0.409, 1.385, 2.956, 5.182, 8.162, 12.07, 17.25, 24.59

9 0.368, 1.243, 2.646, 4.617, 7.222, 10.57. 14.84, 20.38,...., 28.12
10 0.334, 1.128, 2.396, 4.167, 5.487, 9.428, 13.10, 17.70,...., 31.68
19 0.183, 0.616, 1.301, 2.240, 3.440, 4.911, 6.561, 8.706,...., 64.65
24 0.146, 0.493, 1.039, 1.786, 2.738, 3.899, 5.273, 6.868,...., 83.37
29 0.122, 0.410, 0.864, 1.485, 2.275, 3.236, 4.370, 5.682,...., 102.3
39 0.091, 0.307, 0.647, 1.112, 1.702, 2.417, 3.260, 4.231,...., 140.4
49 0.073, 0.246,.0.517, 0.888, 1.359, 1.930, 2.501, 3.374,..... 178.8
59 0.061, 0.205, 0.431, 0.740, 1.132, 1.607, 2.165, 2.807,. . 217.4
69 0.052, 0.175, 0.369, 0.634, 0.970, 1.376, 1.854, 2.403,...., 256.1
79 0.045, 0.153, 0.323, 0.555, 0.848, 1.204, 1.622, 2.101....., 295.0
89 0.040, 0.136, 0.287, 0.493, 0.754, 1.070, 1.441, 1.867,...., 334.0
99 0.036, 0.123, 0.258, 0.444, 0.678, 0.962, 1.296, 1.680,. .., 373.0

 

 

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