Geometry and kinetics of the polygonization of sodium chloride

GEOMETRY

AND

KINETICS

OF THE

SODIUM

POLYGONIZATION

OF

CHLORIDE*

S. AMELINCKXt

and

R.

STRUMANE:

The geometry and kinetics of the domain growth during polygonization of sodium chloride is studied. It is found that the breaking up of ending walls is the dominating process in the initial stages, whilst the Y-shaped junctions dominate the growth process in the 6nal stages. The interaction between growth in subboundaries and thenewlyformed polygon walls is also considered. In an appendix the interaction of isolated edge dislocations with a wall of edge dislocations is discussed in some detail. The kinetics of the polygonizstion process was studied by means of isothermal annealing curves. Two different activation energies are found: one for the low temperature range (49 kcal/mol) and one for temperatures above 375°C (30 kcal/ mol). The difference between these two activation energies is interpreted as the activation energy for jog formation; its value is 0.6-0.8 eV. GEOMETRIE

ET

CINETIQUE

DE

LA

POLYGONISATION

DU

CHLORURE

DE

SODIUM

Les auteurs ont etudie la geometric et la cinetique de croissance des domaines de polygonisation du chlorure de sodium. 11s ont trouve que le processus essentiel dans les stades initiaux fait apparaitre des parois semi-infinies. Par contre, des jonctions en forme d’Y caracterisent les stades finaux. 11s ont aussi consider6 l’interaction entre les dous-joints existants et les parois polygonales nouvellement formees. En appendice, ils discutent en detail l’interaction de dislocations-coin isolees avec une paroi de telles dislocations. La cinetiqe de pyolgonisation a et& etudiee a l’aide de courbes isothermes de restauration. Deux energies #activation ont et& trou Bes: l’une Q basse temperature (49 kcal/mole) et l’autre pour des temperatures superieures a 375°C (30 kcal/mole). La difference entre ces deux energies d’activation est interpret&e comme l’energie d’activation necessaire a la formation de crans; sa valeur est de 0.6 B 0.8 eV. GEOMETRIE

UND

KINETIK

DER

POLYGONISATION

VON

NATRIUMCHLORID

Die Geometrie und die Kinetik des Wachstums von Subkornern wahrend der Polygonisation von Natriumchlorid wird untersucht. Wie sich zeigt, l&en sich in den ersten Stadien vorzugsweise die Enden von Versetzungswiinden auf, wahrend sich in den letzten Stadien des Wachstumsvorgangs iiberwiegend Y-formige Verzweigungen bilden. Die Wechselwirkung zwischen urspriinglichen Kleinwinkelkorngrenzen und neugebildeten Versetzungswiinden wird ebenfalls betrachtet. In einem Anhang wird die Wechselwirkung einer einzelnen Stufenversetzung mit einer Stufenversetzungswand ausftihrlich diskutiert. Die Kinetik des Polygonisationsvorgangs wurde durch isotherme Erholungskurven untersucht. Es ergaben sich zwei Aktivierungsenergien: Eine ftir den Bereich tiefer Temperaturen (49 kcal/Mol) und eine ftir Temperaturen oberhalb 375°C (30 kcal/Mol). Die Differenz dieser beiden Aktivierungsenergien wird als Aktivierungsenergie fur die Bildung von Versetzungsspriingen gedeutet; sie betragt 0,6-0,s eV.

INTRODUCTION

Polygonization of crystals.

The majority

bardu2)

the kinetics

aspects.

and by means of etchpits.(@

No systematic study of the geometry of polygonization

Observations

are due to Dunn

of NaCl has been observed previously

by means of X-raysc5)

of these studies are mainly

concerned with the geometrical concerning

Polygonization

has been observed in a large number

has so far been made by the use of decoration

and Hib-

although

for silicon iron and to Gilmanc3) for zinc.

occasionally

been observed.(7)

polygonized

The purpose

methods,

structures

have

of this paper

is to

The methods used in these studies were mainly etching and X-ray diffraction. A method suitable for

present the results of such observations in NaCl, KC1 and KBr as well as the results of a study of the

the study of crystals presenting a pronounced

kinetics of polygonization

plane

cleavage

in NaCl.

has been

described for NaCl by one of the present authorst4) and has been used successfully by GEOMETRICAL

Gilmanc3) for his study of zinc. The same method is used here for the observations concerning the kinetics of the phenomenon.

(a) Methods of observation The methods of decoration have been described previously.(‘s*) They essentially consist in doping the crystal during growth from the melt with 0.75 wt.% of AgCl or AgBr according to the type of alkali halide.

*

Received March 15, 1959; revised July 15, 1959. t Laboratorium voor Kristalkunde, Rozier 6, Gent, Belgium. Now at S.C.K., Mol-Donk, Belgium. $ S.C.K., Mol-Donk, Belgium. ACTA

METALLURGICA,

VOL.

5, MAY

1960

ASPECTS

312

AMELIKCKX

The decoration

STRUMANE:

AND

of the dislocations

is then achieved

POLYGONIZATION

by

an anneal in hydrogen. The method is well suited for the study of annealing

is larger than the anneal required treatment. particles

in air precedes

The which

decoration

for decoration,

the decoration

consists

can be observed

a

heat

of small

silver

in ultramicroscopic

illumination. For the study of the first stages of polygonization use has

been

recently.(s)

made

of

The crystal

cent of AgNO, decoration

a method

is now doped

by weight addition

is achieved

described

at 40 kV, 20 mA, the specimen This

to the melt, and

leaves

as could

dislocation

3-4

tube operated

being placed

the dislocation

be proved

on the

containing

pattern

by showing

half loops are maintained

In this case the decoration NO,-

during

followed by an anneal in air of 1 hr at 500°C.

method

turbed

more

with 0.4 per

by X-irradiation

hr by means of a copper target X-ray window,

(iioo).U sually

(a/2)[110],

undis-

that

small

in the crystal.

is due to small cavities

the gaseous decomposition

products

of the

group.oO)

time. very

The outer regions often

The specimens large crystal Czochralski

of specimens are cleaved

of KCl,

KBr

parallelopipeds or NaCl,

The specimens

are then annealed

short annealing

periods

for various the annealing

grown

from a by the

this then gives rise, after annealing, structure,

indented

case the crystal

somewhere

dimensions

of the photograph.

The

are of the type

bent,

but

corner.

Focussing

the crystal tells us that the dislocations

The

deeper into are inclined

at an angle of about 45” with respect to the observation plane.

They

glide planes.

are further

deformation.

found

to be still in their

We assume thereforethat

were in the observed

configuration

The photograph

the dislocations

immediately

shows

that

after

the dis-

locations have a definite tendency to adopt positions “one on top of the other” which is their equilibrium The first stage in polygonization takes place already

at room

in KC1

temperature;

by means of etching.

We will now illustrate successive stages in the formation of domain walls during increasing annealing The annealing temperature is always 700°C k the annealing periods were 2 hr, 4 hr, 8 hr and

12 hr. Typical

represent

(i) Specimens deformed on a single glide system.

was not

in the left bottom

in their

traces of the glide planes are parallel to the largest

regions of these crystals are represented

photograph

glide systems in the NaCl structure

Figs. 3-6;

is perpendicular dislocation

observation.

to the bending axis, dots

lines intersecting

After annealing

structure of N&l (doped with bent about an axis normal to the plane of the photograph. x 150

in

the plane of the the plane of

at higher temperatures

the dislocations are very nearly perpendicular to the plane of observation; they have nearly pure edge

FIG. 1. Polygonized AgCI),

of

this we refer to Fig. 2. In

To illustrate

the series of photographs,

(c) Results

consists

glide plane.

this particular

of 30 mm, and

same time used for decoration.

to the “chevron”

of the dislocations

10°C;

time is at the

It very

as shown e.g. in Fig. 1.

The very first stage of polygonization

periods.

For the

are however

only.

course in a rearrangement

are bent under times.

of the crystal

on one system

this was also confirmed

a water film to a radius of curvature they

deformed

position.

method.

two systems withglide planes

often happens that the regions on both sides of the neutral plane are deformed on a different glide system ;

therefore (b) Preparation

313

parallel to the axis of bending are active at the same

structures as it is already inevitably accompanied by an annealing treatment. If the total annealing time heat treatment

OF NaCl

FIG. 2. Crystal of KC1 (doped with AgNO,) deformed by indentation in the left bottom corner. x 400

ACTA

314

METALLURGICA,

VOL.

8,

1960

Fig. 4.

Fig. 3.

Fig. 5.

Fig. 6.

Fros. 3-6. Crystals of KC1 (doped with AgCl), bent to a radius of 3 cm and annealed for respectively 2, 4, 8 and 12 hr. x 600

character. The photographs represent at the same time the typical configurations which illustrate the

deformation, differs, one gets a situation as shown in Fig. 5. The first effect of the annealing will now be a

way in which domains

rearrangement

coarsen.

These characteristic features are those already described earlier for silicon iron”*‘) and zinc(3): the Y-junctions

and the semi-infinite

walls.

We find that

in the glide planes,

and in the given

situation this will give rise to a configuration in Fig. results.

8(b);

it is clear

that

as shown

a semi-infinite

wall

in the polygonization of alkali halides the initial stage is characterized by the frequent occurrence of semiinfinite walls (see Fig. 7a,b,c); in the latter stages the Y-junctions become more important.

The behaviour of annealing treatment Due to the mutual wall the dislocations

It is easy to visualize how the semi-infinite walls are generated in the crystal. If the number of

top dislocation is subject to a stress field due to all other dislocations, it will climb the greatest distance;

dislocations

the following

in parallel glide planes, immediately

after

the semi-infinite walls under an has been considered by FriedeP). repulsion of dislocations in the will be induced to climb. As the

dislocation

will climb less as it is subject

AMELINCKX

AND

STRUMANE:

POLPCONIZATION

OF

315

NaCl

(b)

(4 FIG. 7. Semi-infinite disIocation

walls.

Note the regnIar variation

in spacing.

i: 600

= = 4, e FIG. 8, The generation

51;==== --0

of semi-infinite walls (a) immediately

d(arWraryunits)

after bending (b) after rearrangement.

d(arbitrary units)

(b) (a) FIG. 9, relationship between 26, and d, (ti, = distance between dislocations with index p and p + 1; d, = distance of dislocations with index p from leading dislocation).

to the back stress of the first and to the stress field of all following dislocations. The resulting rearrangement will produce a configuration where distances between dislocations gradually increase towards the end of the wall. The distances between suocessive dislocations u, have been measured on enlarged photographs. Friedel’s

interpretation leads to a linear relationship between l/u, (u,-distance between dislocations with indexpand p + 1) and d, (distance of dislocation with index 23 from the leading dislocation). This is approximately verified by our observation, as shown by Fig. 9(a,b).

316

ACTA

METALLURGICA,

VOL.

8,

1960

(b)

(a) FIG. 10. (a) Intersecting

will

now

consider

characteristic

the

dislocation walls. Note the depletion in the region adjacent to the wall with large orientation difference. (b) Intersecting high angle walls. x 600

geometrical

for specimens

systems with mutually perpendicular Mutually perpendicular form.

dislocation

tilt boundary

Burgers vectors. walls then usually

intersecting

tilt

misorientation smallest

(7);

boundaries

tilt boundaries Examples

one example different have

The boundary

tilt angle then usually

from the intersection

shown

up into

slightly

the

wall

asymmetrical.

schematically

in Fig.

XY,

is also

boundary region

then

12.

The width

is

of the

of dislocations

upon the angle of misorient&ion

XY.

of the

The larger this angle the smaller the

of influence:

behaviour

which

This mechanism

between

this explains

the difference

in

the two ends of a finite wall as

visible e.g. in Fig. 10(a).

is somewhat

angles.

be taken

becomes

will depend

visible in Fig. 10(b). The behaviour

will

region which is in this way depleted

this is usually

angles of misorientation.

of this are shown in Ref.

repelled by all the others of that wall, this dislocation

points a small section of

the case when the two intersecting have comparable

on two glide

is formed;

At some intersection

asymmetrical

configurations

deformed

if the two

very

different

which has the

ends some distance

point (Fig. lOa).

In a few cases

(d) Interaction Whereas

of walls with pre-existing

certain

pre-existing

without

formed

walls, this is not always

in Figs.

subboundary

interaction

14. both

and

are newly

result as visible

As a consequence the

by

the case and very

shaped subboundaries

13 and

decoration”(i2)

visible

subboundaries

crossed

often zig-zag

any

subboundaries

original

of “double

position

the final position

of

the

can easily

be

recognized. The straight lines correspond to the original position of the subboundary, the zig-zag lines indicate FIG. 11. The interaction of a single edge dislocation with a wall of edge dislocations. x 600

this may be due to a lack of decoration;

the boundary

acting

as a sink of impurities

and vacancies

drain

its

prevent

neighbourhood

and

the final position.

The interaction

as in Figs. 15 and 16. We now discuss these two

may be pictured

cases separately.

may

decoration.

In the cases shown (Fig. lOa), this is however not the case. To explain this behaviour, we consider the interaction of a single edge dislocation with a wall of edge dislocations of the kind shown in Fig. 11. This calculation is given in the appendix. From this calculation it is clear that the dislocation A will be induced to climb towards the wall XY, and as moreover the end dislocation of the finite wall U V is

I

I

I

L

i I L

L I I (4

FIG. 12. The formation

(b) of a slightly asymmetrical in a wall.

part

In

AMELINCKX

STRUMANE:

AND

POLPGONXZATION

OF

Fig. 13. FIG. 13 and 14:

1

T T r-t-: TT +IT

317

NaCl

Fig. 14. Interaction

of polygonization

wall with preexisting

subboundary.

x 400

;

t’ 7”

T

7

T

T ‘-:,

T

iaf

fW

Fro. 15. Fo~n%tion of zig-zeg shaped boundary as a consequence of the interaction of PolYgoni~tion walls and a preexisting boundary. l?he original boundary contains a number of dislocations of the same sign with respect. t,o dislocations in the wall. T

T

:*

T T

1 L

‘A;

-‘l” T.l. T*. T T T 5 T T 7 T T T T

T T T T T T -II r LT +l Tl

“I

T’L T 1 T -i T T H

T T T 7,-t-447 7 T T T T 7” T 7 T T

T T T -if-l T T T T T T T

T 1

T

I

T T T T T T T

T

+ * -if

tb)

Fro. 16, Formation of zig-zag shaped boundary as a consequence of the interaction of polygonization walls and % preexisting boundary. The original boundary contains a number of dislocations of opposite sign with respect to the dislocations in the wall.

ACTA

318

the

first example,

boundary opposite

shown

in Fig.

METALLURGICA,

14, the

8,

1960

original

contains a number of dislocations to the sign of the dislocations

VOL.

of a sign

forming

the

polygon wall. Anneal now results in a mutual annihilation of dislocations of opposite sign. The resulting

situation

is pictured

the original configuration

in Fig.

16(b),

The second possibility

is pictured

the original subboundary

in Fig. 15;

contains dislocations

same sign as those of the intersecting

walls.

sections

and

of subboundary

now

result

figuration

of 15(b) is formed.

An observed

is shown

in Fig.

decoration

13.

while

was as shown in Fig. 16(a).

Double

here

0 P

of the Mixed

the

con-

example has again

revealed the original position of the subboundary

2

I

(the

460

IO'

d

t(h)

straight lines) and the final shape (zig-zag lines).

FIG. 17. Isothermal domain growth curves for N&I. KINETICS

(a) Experimental

methods

To derive the activation

energies of the process we

measured the growth rate of the polygon different temperatures, of specimens. following

The specimens

way :

domains

at

for the same radius of curvature

Small

were prepared

cleavage

in the

parallelopipeds

of

25 x 10 x 4 mm were all bent to the same radius of curvature

(30 mm)

cylindrical

metal piece of this radius.

by

pressing

were wet during the deformation

them

against

a

The specimens

The log of the growth rate dO/dt versus the reciprocal of temperature is plotted in Fig. 18. It is clear that the curve

consists

of two

break corresponding

rectilinear

to a temperature

corresponding

activation

of

are respectively

the

curve

temperatures Gilmanc3)

with

similarly

49 kcal/mol

at low

at high temperatures.

two

different

energies at high and low temperatures

activation

for zinc.

and the metal piece

was covered with wet filter paper as well, in order to minimize

damage to the crystal.

that

specimens

all

were

It should be noted

cleaved

crystal, grown in this laboratory method.

from

the

in air for various

by the Czochralski

temperatures.

time

intervals

and

plane has then polygonal can be observed

character

in reflected

light.

and the domains The method

of

cleavage plane is thus seen as a set of parallel strips of colour.

Photographs

of the domains

batch of specimens. The isothermal growth Fig.

17.

Vertically

are taken is deduced

and

IO_'5 _

lo-”

0 t ,

lo-'9

e -

10-2’

in

the mean orientation

<

\

10-23

0 D lo-25

E E I

I

lO-27

difference between walls which is deduced from the mean width of domains. Horizontally we plot the logarithm of time. It is readily visible that the

IO-=

curves level off for long annealing

10-3'

times,

the more

rapidly the higher the temperature. The slope of the initial rectilinear part is used to derive the growth rate.

/'v2=49Kcol/mol

\

the

for every

rate curves are plotted

is plotted

IO-

the cleavage

is the Franpon method of phase contrast This method “translates” in polarized light.(i3) orientation differences into colour differences. The

width

lcr”

at different

observation

mean

10-g

After cooling they are cleaved following

a plane parallel to the axis of bending;

different

10-7

same

Batches of ten specimens were then annealed

1.0

II 1.5

I/T

a

The

energies for the two parts

and 30 kcal/mol

found

parts

of 375°C.

(lO-3 OK-' )

FIG. 18. Plot of the log of growth rate versus the inverse of temperature.

AMELINCKX

of the kinetics

(b) Discussion According

STRUMANE:

AND

to Mott(i4) and Friedelul)

(i) the difference in activation and below

quantity

Ufi;

the

we can use the of

jog

energy at temperatures

break

gives

Utj = 19 kcal/mol

(ii) the quantity

directly

the

= 0.8 eV.

Ufj is given by the formula Ufj = kT, In (Z/b)

where

T,

is the temperature

at the break

curve and 1 is the size of the domains perature.

Substituting

in the

at that tem-

for T, 647°K and 1 = 2 - 1O-3

cm, gives llfj = 13.4 kcal/mol

two values is not too unsatisfactory. the case for zinc, using Oilman’s according

with the activation

activation

which is to be compared

energy deduced from conductivity

energy

comparison

depends

of the specimens;

and 47.200 cal/mol, temperature

is difficult

on

the

for low

and high

range are given in the literature.

Measure-

respectively,

specimens,

stances, indicated

prepared

of the ionic conductivity in the same

kcal/mol

circum-

that almost for the entire range of

temperatures impurity

as

impurity

values of 20.600 cal/mol

ments made in this laboratory of NaCl

of 2

energy in the high temperature

measurements. It is clear that a direct contents

data, and Friedelul)

to the method used to derive it.

For the activation

this

of the

This was not

that the jog energy differs by a factor

range we find 30 kcal/mol

(Institute

used in the polygonization The conductivity prevails.

experiments, value of 30

found here from the polygonization

data is

pour

Scientifique

Dr. J. Goens,

the jog

as a small

segment

energy can be estimated the Burgers

vector

of dislocation

line, its

as 0.1 ,u b12b,,(11) where b, is

and b, the “length”

of the jog.

One can now consider two kinds of jogs;

neutral ones

with

b2 = a/2 or

b, = ad?

or charged

ones

with

a/22/2.

The value computed on the assumption of a neutral jog and taking p = 1500 kg/mm2 fits best the observed

value.

The measurements

that the majority

319

therefore suggest

of jogs is probably

neutral.

This

may be due to the fact that dissociating a neutral jog into two oppositely charged jogs requires an extra amount of energy, due to coulomb interaction. ACKNOWLEDGMENTS We wish to thank Professor W. Dekeyser for his continuous interest and useful discussions. This work

Director

de

la

Recherche

et 1’Agriculture;

Comite

and by S.C.K. (StudieWe are also indebted to

of S.C.K.,

for permission

to

publish these results. 1. 2 3: 4. 5. ;:

8. 1:: 11. 12. 13. ::: 16. 17.

REFERENCES G. G. DUNN and W. R. HIBBARD, Acta Met. 3,409 (1955). W. R. HIBBARDand G. G. DUNN, Acta Met. 4,306 (1956). J. J. GILMAN, Acta Met. 3, 277 (1955). S. AMELINCKX,Nature, Lond. 173, 993 (1954). S. KONOBEJEVSEYand I. MIRER, 2. Kristallogr. 81, 69 (1932). S. AMELINCKX,Acta Met. 2, 848 (1954). S. AMELINCKX,in Dislocations and Mechanical Properties of Crystals (Eds. J. C. FISHER, W. G. JOHNSTON, R. THOMSON and T. VREELAND, Jr.). Wiley, New York (1957). W. VAN DERVORSTand W. DEKEYSER,Phil. Mag. 1, 882 (1956). S. AMELINCKX,Phil. Mag. 3, 653 (1958). S. AMELINCKX, W. MAENROUT-VAN DER VORST and W. DEKEYSER, Acta Met. 7, 8 (1959). J. FRIEDEL, Lee dislocations. Gauthiers-Villars, Paris (1956). S. AMELINCKX,Acta Met. 6, 34 (1958). M. FRAN~ON,Rev. Opt. (thQor. instrum.) 31, 65, 170 (1952). N. F. MOTT, Proc. Roy. Sot. A 220, 1 (1954). J. M. BURGERS, PTOC. Akad. Sci. Amst. 42, 293 (1939). A. H. COTTRELL,Dislocations and Plastic Flow in Crystals p. 98. Clarendon Press, Oxford (1953). F. R. N. NABARRO,Adv. Phys. 1, 271 (1952).

APPENDIX In this appendix we describe, in some more detail than has been done previously, the interaction of isolated dislocations with a wall of edge dislocations. The stress component rszv”due to an infinite wall of edge dislocations has been calculated by Burgens( the other components can be found in the same way by summation over all dislocations in the wall .‘i6) The results are given by the equations: o,,

(&b

b

-

z

D&z cash 2rrq cos 2~ p - 1 2h2 (sinh2qr + sin2 pn)-

UVUb =

(1)

~

TD qv sinhZqn + sin2pr + sinh2 qn --~ sin 2n p (sinh2q= + sin2prr)’ 2h

(2)

-

-

TO sin2prr + sinh2 qsr - qrr sinh2 qrr __ sin 25rp (sinh2qrr + sin2~sr)~ 2h

(3)

therefore not unreasonable. Taking into account line energy only and considering

l’encouragement

dans 1’Industrie

d’etude de 1’Etat Solide) centrum voor Kernenergie).

= 0.6 eV.

It is clear that in our case the correspondence

showed

OF NeCl

ispart of aresearch programmesupportedbyI.R.S.1.A.

polygonization data to obtain the energy formation lJfj by the use of two procedures: above

POLYGONIZATION

-

where D = bp/27~(1 - Y) h = distance between dislocations in the wall P = Ylh q = z/h. The signs of these stresses in the neighbonrhood of the wall are represented in Figs. 19 and 20, which shows also the lines along which the stresses vanish. From these diagrams it is easy to deduce the sign of the force to which an isolated edge dislocation is subject. (a) Edge dislocations of the same Burgers vector as those of the wall (i) Same sign. From Fig. 19(a) it is clear that dislocations will tend to glide towards the wall in the cross hatched region, whilst they will be repelled in the other regions. This was already discussed in a quantitative way by Nabarro’i”. The component of force in the glide plane is given by Fe = b a,b.‘4’ If climb is an allowed procedure one has however also to take into account that the dislocation is subject to a force perpendicular to the glide plane given by F, = -b u,,*‘~’ and which induces climb. From Fig. 20 it is clear that the sign of this force is such

320

ACTA

METALLURGICA,

VOL.

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1960

FIG. 20. The stress crab due to the wall of edge dislocations.

/

y7-i

IA--___~________

\

1

_J I, 1 -----------__ +J!______ f yykh

y’

( 2k+l)

h/2

(b) FIG. 19. (a) The stresses 0~2 due to the boundary. (b) The stresses oyybdue to the boundary. that it tends to bring the edge dislocation into the planes ?/ = (Sk + 1) h/2 in which they can glide off towards the wall. Consequently, all edge dislocations in the region adjacent to the boundary will finally be taken up into the boundary. The paths the dislocations will follow during anneal are schematized in Fig. 21. (ii) Opposite sign. The dislocations will now be repelled in the cross hatched regions of Fig. 19(a) and attracted in the others. From Fig. 19(b) it is clear, on the other hand, that they will be induced to climb towards the planes y = MC in which they are attracted towards the wall. They will finally be annihilated by a dislocation of the wall. (b) Edge dislocations having a Burgers vector perpendicular to the one of the dislocations in the wall (i) Positive dislocations (Fig. 19(a), dislocation A). The

FIG. 21. Paths followed by isolated edges, during their migration towards the wall. equilibrium positions with respect to glide are given by the intersection points of the curve represented by a solid line and the glide plane; the magnitude of the force causing glide is TV = b omb. In these positions the dislocations are however subdect to a force in the x-direction Fz = ba,” and which induces them to climb towards the wall Fig. 19(b). Both force components vanish, near to the dislocation in the wall, on a line forming an angle of 45” with the glide plane, which is such that regions of compression and dilatation overlap. The result is again that dislocations in the neighbourhood of the wall will be pulled into the wall, which now however acquires asymmetrical character. (ii) Negative dislocations (Fig. 19(a), dislocation B). The same considerations as above apply apart from a change in the sign of the forces; the dotted lines now represent equilibrium positions for glide in Fig. 19(a).