The mechanism of repeated precipitation on dislocations

THE

MECHANISM

OF REPEATED

PRECIPITATION E.

ON DISLOCATIOSS*

NESt

The mechanism of repeated precipitation on a climbing edge dislocation {or growth of planar precipitate colonies) has been analysed in terms of baiancing the rate at which vacancies are being absorbed The possibility of having the particles by the growing particles to the climb rate of the dislocation. dragged by the climbing dislocation have been considered and so has also the effect of varying the jog spacing and the core diffusion rate on the repeated climb process. Based on the suggested model various specific aspects are treated including (1) the size and distribution of colony particles, (2) the dislocation unpinning process, (3) the kinetics of colony growth, and (4) the dendritic dipole growth effect. JIECASISSIE

DE

LA PRECIPITATIOS

REPETEE

SUR

LES

DISLOC.~TIOSS

Le mecanisme de la precipitation rep&e sup une dislocation coin en tours de montee (ou de la croiasance de colonies planes de precipite) a et6 interprets en supposant que la vitesse it laquelle les lacunes sont absorbees par les particules qui grossissent et la vitesse de montee de la dislocation sont Pgales. L’auteur a Ctudie la possibilite d’entralnement des particules par la montee de la dislocation, ainsi que l’influence de la variation de l’espacement des crans et de la vitesse de diffusion dans le eoeur SUP le processus de montee repCtCe. Diff&ents aspects specifiques sont examines a partir du modele sugger-6, en particulier, (I} la taille et la repartition des colonies, (2) le processus de d6dpinglage des dislocations, (3) la cinetique de croisssnce des colonies et (4) l’effet de croissance dendritique de dipoles. DER

MECHASISMUS

\YIEDERHOLTER

AUSSCHEIDUSG

AX- VERSETZUSGES

Der Xechanismus wiederholter husscheidung an einer klettenden Stufenversetzung (oder das der Absorption van Wachstum ebener .~us~heid~gskolonien) wurde analysiert . Die Geschwindigkeit Leerstallen durch die wachsenden Teilchen wurde mit der ~lette~esch~vindigkeit in Beziehung geaetzt . AuOerdem wurden die Xultiglichkeit eines Xitziehens der Teilchen durch die kletternde Versetzung und der Bin&l3 einer veranderten Hohe des Versetzungssprungs und einer variablen Diffusionsgeschwindigkeit im Versetzungskern beim wiederholten Kletterprozel3 betrachtet. Auf der Grundlage des vorgeschlagenen Modells werden einige spezifische Gesichtspunkte diskutiert, u.a. (1) Gr6l3e und Verteilung der Teilchen in der Kolonie, (2) das Losreigen der Versetzungen, (3) die Kinetik des Boloniewachstums, und (4) das dendritische Dipolwaehstum.

INTRODUCTION

which this type of precipitation has been reported: (a) The decomposition products involved will cause the The appearance of precipitate colonies on stacking generation of a severe matrix strain unless vacancies faults in niobium containing austenitic stainless steels are being supplied in order to reduce the pa~iclel was first reported by Van Aswegen and Honeymatrix mismatch, (b) as a consequence the nucleation cornbe, and an interpretation of these configurations of particles occurs heterogeneously on dislocations, as being due to repeated precipitation of NbC on & and (c)the subsequent growth causes the dislocation to climbing a/3( 11l)-partial dislocation was first given climb in a direction consistent with feeding vacancies by Silcock and Tunstall.@) Recent work report on the to the particles. Repeated nucleation and growth on formation of planar precipitate colonies by this the same dislocation results in t,he formation of a repeated precipitrttion mechanism,+ in a. variety of planar colony which is partly or completely surrounded systems, i.e. different steeW7’ super alloys,(‘) copper alloy~(~*~)and semiconducting materials.(s-13) by an edge dislocation. All these features may be While the generation of precipitate colonies in accounted for by the model proposed by Silcock and Tunstall. However, other features are more difficult stainless steels and super alloys most frequently to explain in terms of this model. For instance, the involve the climb of Frank partials, this precipitation process is, in other systems (copperc4*5)andsilicon(11*13’, distribution of particles into widly spaced rows as observed in silicon,(g*ll’ or the tendency for copper also associated with the climb of a/2(110) and a (100) precipitate colonies in silicon to form dendrit’ic dipole Burgers vector dislocations. branches.(ili In a recent paper by Xes and Wasburn(ls~ The following aspects of the repeated precipitation phenomenon seem to apply to any of the systems in t~heseeffects were accounted for by assuming that the growing particles were dragged by the climbing dislocat.ion. This particle dragging effect was introdu* Received Yav 1, 1973; revised June 21, 1973. ced on the basis of the work by Nes and Solberg,04,15) t Central 1nst”itute for Industrial Research. Blindern. osio, sorway. shokng copper-silicide particles being dragged by z The description “repeated precipitation on dislocation” climbing dislocations in silicon. for the present precipitation phenomenon has been suggested by Xcholson. w The process has also been called discontinuous The Silcock-Tunstall theory may serve as a useful precipitation on dislocations (*.a’ in analogy with the grain quahtative model describing fundamental aspects as boundary precipitation reaction. ACTd 6

JIETALLURGICA,

VOL.

22, JASUARY

1974

81

the planar particle configuration and the presence of a However, a more surrounding edge dislocat,ion. refined theory is needed in order to account for such features as the size and distribution of particles within a colony and how these aspects are related to the particular precipitation system. The objective of this paper is to contribute to such an end. The present model will take account of the particie dragging effect and special emphasis will be focused on the dislocation climb process between consecutive nucleations. Systems having different activation energies both for the dislocation core diffusion and jog nucleation will be considered. In a final paragraph experimental observations of repeated preoipitation processes in different alloy systems will be discussed in terms of t,he present model. ANALYSIS OF THE DRIVING FORCE FOR THE DISLOCATION CLIMB PARTICLE DRAGGING PROCESS

The sketches in Fig. l(a-d) outline t,he principles of the Silcock-Tunstall model.(s) Precipitates nucleate on a Frank partial (Fig. la). Movement of jogs (J) provides vacancies to enable the particles to grow (b) with the consequent climb of the dislocation (c). Eventually the dislocation pinches off around the particles (d), and is available for fresh nucleation of precipitates. In this model the particles are considered fixed in the matrix and consequently the distance between rows of particles, d, should be about half the distance between particles within a row, L (Fig. 1). within copper precipitate colonies in However, silicon d frequent,ly is much larger than A.(‘@ With reference to Fig. 1 this effect can be accounted for if the particles are dragged along before unpinning of t,he dislocation. The movement of macroscopic defects as particles or voids have been observed in a variety of systems. Ashby and Centamore(16) report on the dragging of

a

b

C

d

FIG. 1. Suggested stages in the growth of a SbC precipitate colony from a Frank partial (after Silcock and Tunstall’z’).

particles pulled by moving grain boundaries during recrystallization in copper. Recently it was demonstrated by Yes and Solberg(15’ that solid inclusions can be pulled also by climbing dislocations. -1 theoretical analysis of the diffusional mobility of particles in solids has been presented by Niohoh;,‘rr’ and in analogy with atomic diffusion the diffusional velocity of a particle, vp, can be written 2’P = X,F, (11 where iv,, the particle mobility is dependent upon the rate controlling mechanisms by which the atoms move and are independent of the particular driving force imposed, F,. We are considering spherical particles only and in this case ZIf, has the form (3 where r is the particle radius, the diffusivity and concentration of interfacial vacancies are D, and cj respectively, 6 is the thickness of the particle-matrix interface and V, is the volume of a matrix atom. -1 general analytical expression for the force acting within the periods of repeatecl on the particles nucleations is difficult to derive. The driving force for the colony growth process is, as in any precipitation process, due t,o the supersaturated solid solution of the precipitating atoms. In this particular precipitation process, however, the growth kinetics is closely associated with the rate at which vacancies can be supplied to the growing particles. Thus in the present model the particle gro~~h~disIoeation climb sequence between consecut,ire nucleations will be analyzed in terms of balancing the supply rate of the vacancies to the climb rate of the dislocation. This is equivalent to saying that the driving force for the dislocation climb process is the osmotic force imposed on the dislocation generated by due to the local vacancy subsaturation the growing particles. This climb force will be of the form(l*) kTb c

per unit length of the dislocation, where c and c, are the local and equilibrium vacancy concentration respectively. Inorder toestablishtheforcesactingon the particles, let us consider the sketch in Fig. 2. According to the Silcock-Tunstall model the osmotic force causes the dislocation to born out between the particles, and as a consequence this is giving rise to elastic forces, F,, on the part,icles. In addition to these efastic line tension forces, the particles will also experience the osmotic force on the dislocation segments, a-b (Fig. 2) close to

SES:

REPE_ATED

PRECIPITATIOS

OS

DISLOCATIOSS

83

Edge

I

dislocation

T

Climb

I

\

direction

FIG. 3. . String of particles along an edge dislocation, c, IS the vacancy concentration in the particle interface and c,, is the equilibrium vacancy concentration in the core. \

I

FIG.

‘2. Alternative

dislocation-particle

the particles. Thus, according particle speed will be L’P = X,,

configurations.

to equation

ET6 c Fe + 2r * v In a CO

(1) the

(3)

where r is the particle radius. However, the contour of the colony growth front may not remain as illustrated in Fig. 2 throughout the time period of a precipitation sequence. As the particle mobility is a climb speed a function of r--4 and the dislocation function of Le2 (see the analysis below) then, if r < L, up may be larger than the average climb speed of the connecting dislocation segments. Thus in the initial stage of each precipitation sequence, the particles are in general expected to move ahead of the dislocation as illustrated by the broken line in Fig. 2 and it follows that in this case the line t’ension forces will This aspect will also be disoppose colony growth. cussed in more detail in the following section. ANALYSIS

w-ill appear as a straight line with the particles attached in a row as shown in Fig. 3. An expression for the climb speed may be obtained if we assume that local vacancy equilibrium is being attained at midpositions between the particles as indicated in Fig. 3. This assumption corresponds to treating the edge dislocation as a perfect source of vacancies. the rational of this will be discussed later in this paragraph. In can be this situation the dislocation climb speed. c’~: calculated by considering the linear diffusion of core vacancies towards the particles. The result is

4bQs vd =

(L -

242

(l -

z )

(4)

where D,, is the core self diffusivity, c, is the vacancy concentration in the particle matrix interface ancl cc0is the equilibrium vacancy concentration in the core. At this stage in the colony growth process no line tension forces will contribute to the particle motion and it follows from equation (3) that the particle speed is

OF THE DISLOCATIONPARTICLE MOTION

In the following, no attempts will be made to present a general theory describing the combined motion of the dislocation-particle configuration within the period of consecutive nucleations as the line tension forces will obscure any anal_tical keatment. However, in dealing with some aspects t,his difficulty may be overcome as exemplified by the simple considerations presented below. It is assumed that immediately prior to the unpinning both the particles and the dislocation will migrate at the same speed, i.e. the colony growth front

where D,, = csoVaDs is the interfacial self diffusivity and c,, is the equilibrium vacancy concentration in the particle interface. Our model is based on rd = P, and it is realized that this situation cannot prevail as a steady state condition. However, based on the assumption that the dislocation acts as a perfect source of vacancies, the unpinning of the particles will not occur before v, 1 up, i.e. before the particle radius r exceeds a critical value r* which can be obtained by equating the above relations for the particle and dislocation speed.

ACT-1

d-l

XETALLCRGICA,

Ke will non- consider the rational of treating the climbing edge dislocation as a perfect vacancy source. This assumption requires the following two conditions to be fulfilled. (1) The mesn free diffusion path for a core vacancy before evaporation into the lattice, -,’ must be larger than half the particle spacing, i.e. f 2 (L - 2r)/?. (2) Theequilibrium distance between thermal jog3 /.e a = n exp F,/kT, where a is an atomic distance along the dislocation line and Fj is the jog activation energ>-, must be less than the particle spacing. i.e. 2, 2 L - 2r. &cording to Hirth and Lothe;

where AK, is defined as the difference between the activation energies for self diffusion in bulk and in the core. While typical values of AR’, 21 3 (activation energy for bulk self diffusion) N O.i-1.2 eV and as (L - 2r)/:! is of the order a few 1008, ? is in general expected to be much larger than (L - 2)/2. However, the second condition, A, 2 L - 2r, is not, expected to be fulfilled in general as typical values of Fj is of the order l-3 e\‘. -4s a consequence the nucleation of jogs has to be considered in the climb process and the climb speed as presented in equation (A), will be reduced by a factor (L - 2r)l,.(18) The particles however, being propelled by the climbing segments close to them (marked a-b in Fig. 2) will st,ill move at a rate given by equation (5). This is because the dislocation segments due to elastic interactions with the particles mill contain a high jog density as a geometric necessity. Consequently the dislocation unpinning n-ill start n-hen r exceeds r*, where 47r _ r*3 = S(L _

3

In% zr*)j_,

e8

-5

gc,c”,

(1 _c;, \

*

@)

cc,/

The vacancy subsaburation c,]c,, will vary with the diffusivity of the precipitation species, i.e. if the activation energy for thermal diffusion of the precipitating atoms is larger than that of the core self diffusion, then the bulk diffusivity of these atoms will be rate controlling and c,/c,, will be close to unity. In that case n-e have ln c,/c,, N 1 - cJcc,,, and if we also assume D 8sz D,,, equation (6) reduces to 5 r*3 = d(L - 2r*) 1,. 3

(7)

If the activation energy for core diffusion is larger than that of atomic bulk diffusion for the precipitating atoms, a larger build up of the vacancy subsaturation

VOL.

22,

1954

b

(1

C

d

e

FIG, 4. Stages in the repeated precipitation process, (cl) and (e) illustrate alternative unpinning processes.

may be expected. However, as t’he particle speed (equation 5) is a function of c,/c$, In c,/c,, this will have a maximum for c,/c,, = l/e, and the right side of equation (7) needs to be multiplied by a factor l/(e - 1). The dislocation climb/particle gronth sequence between consecutive nucleations may now be described as schematically illustrated in Fig. 4. Particles nucleate on an edge dislocation (a) and after reaching a critical size, r, the effect of continued grotih will be a local vacancy subsaturation and consequently a chemical force causing particle migration. The value of r, depends both on the particle-matrix lattice mismatch and on the difference in chemical composition. Bs we in this initial stage assume r,, < r* the growing particles mill move ahead of the connecting dislocation segments (Fig. 4b), and the particles will continue to move increasingly further ahead unless a balance between line tension and osmotic forces is established. That is, the particles and the dislocation segments in contact with them are eschanging racancies at a rate which is higher than n-hat can be supplied from the connecting dislocat.ion se-men& i.e. 2’p’T>-

L 2

where T is the average time for a diffusing core atom to cover the distance L/B, i.e. 7 = Lz(SD,,)-1. However, having vD - T < L/2 the climbing dislocation will eventually catch up with the particles (Fig. AC). Alternative unpinning processes are illustrated in Figs. 4(d) and (e). The details of the unpinning w-ill be left for discussion in the following section where colony grow+h in various alloy systems will be

SES:

considered!

REPEATED

so n-ill also the physical

PRECIPITATIOS

consequences

of

having rgr > L/Z. TO

SPECIFIC

.iLLOY

SYSTEMS

above

precipitation

moclel on

can

be

climbing

applied

to

dislocations

growth) in any alloy system.

repeated

(or

colony

In the following, honerer?

the discussion n-ill be restricted to colony growth in austenitic steel and silicon makives as detailed experimental

results

in the literature. the

systems

detailed

of this

and Tunstall.“)

experimental

It was argued

on

partial n-as tist

this

t.he most

precipitation

partial

be estimated

in y-iron

can

considerations.

This

energy for jog nucleation

separation

has been

at which

i., N 300 x.

L is reported the particle

be estimated particle

distance,

d,

colony

growth

studied(2*lg’)

to equation

*I, which

diameter the

activation

eV which at in

gives

to be 100 ..I or less,

according

which

the

on line

and

size, r*, at the unpinning

4OA for L N 100

reported

based

0.4-0.5

most extensively

consequently gives 2r * -

of i, for a Frank

gives

about

iOO”C (the temperature

can

as a

jogs, il,, is less than the particle spacing, L, The value

steels

on

can only be treated

along the dislocation. tension

et ~2.“~)

the edge dislocation

nucleate

perfect source of vacancies if the equilibrium of thermal

in

as being

by Van Aswegen

above that

which the particles

colonies

process Howerer,

data

has been reported

are available

of SbC

with the climb of a Frank

given by Silcock system

on these

On the growth

interpretation

steels. associated

of

NbC

(7).

This

is close to the

35-50.k(1s)

particles

The

have

been

dragged before being left behind can also be estimated by

solving

equation

Lbd _” k * 4x/3 r*3

(7) (k

is

and a

by

observing

numerical

that

constant).

This gives for L > b; d = tki.,. The

constant,

cc (,lOO) While

hPPLICATION

The

OS

(9)

k, is for NbC in y-iron

about

unity,

giving

DISLOC_iTIOSS

dislocations the

colonies

dislocations

present

a characteristic in silicon

phology

distance

between

the

is

freshly

effect may be reflected appears to be constant configuration, varies.

while

independent nucleated

of

the

particles,

L.

in Fig. 15 in Ref.(ls), throughout both

L

and

spacing This

x-here d

the planar particle the

particle

size

In silicon, growth of planar precipitate

colonies has

elements

occurs

on b = n/2 (110)

dislocations.@Jl)

In the case of copper in silicon, colonies due to climbing

aspect’ of the decomposition

material.

direction.

of colonies

On the macroscopic

where

It is suggested

due to an anisotropy i.e. a higher

the

different

in

mor-

branches

(110) direction

that this eliptical

effect in the particle

particle

mobility

shape is mobility,

in the densely

will result in an eliptical

packed

configuration

as (110) type colonies contain only one such direction. The dislocation unpinning The

repeated

nucleation

silicon causes an inter-row reported

effect

activation

in silicon

may proceed case

of particles

as illustrated

of silicon,

to the

energy for jog nucleation

l(d)

where

sidered completely

or 4(d).

In

activation

is in the range 2.5-3

eV a dis-

must, be con-

unrealistic.

growth

front

shown in Fig. 4(b). continuously

unpinning,

the

bow out ahead of the particles

the colony

high

which according

in steels this process

in Figs.

however,

in

This strong part.icle

is attributed

(6) will suppress the dislocation

For the unpinning

location

as observed

much larger than

energy for jog nucleation,

to equation

the

process spacing

in other alloy systems,

dragging

Thusit is suggested that in silicon will always be as

In this state,

supplied

geometric

due to the elastic

jogs are

interaction

between the growing particles and the dislocation. Unpinning occurs first when new particles which are nucleated

in between the “old” one are taking oFer the

dragging-climb micrograph Another observed

process as illustrated

presented

consequence

in silicon is reflected

dendritic

in Fig. 4 (e). The

inFig. 5clearly

shows this effect.

of the high actiration dipole growth

in more details

energy for

in the frequently effect.

in a separate

this will be

section

below.

The kinetics of colony growth It is suggested for the repeated

that t.he rate controlling precipitation

process

self diffusion,

one has the highest actiration growth processes as observed

parameters

will be either,

(a) the atomic diffusion of the precipitating (b) the core (interface)

been reported to be associated with t,he precipitation of both oxygen and copper. The precipitation of these

from

prior to the pre-

are planar eliptically shaped agglomerates of coppersilicide particles.(llJO) The habit plane is most frequently {llO}with theelipticallongasisin the ‘110)

treated

dragging

reported.

generated

is the formation

configurations

effect

(9) is that. the

been

were

of these colonies they have the form of star

shaped

jog nucleation

from equation

also

in the matrix

cipitation,

d - 150-k at iOO”C. This theoretical row spacing fits t,he experimental value. An interesting to be deduced

hare

in steels

of copper

dislocation-free

s5

atoms,

depending

energy. in steels

or

rhich

The colony and silicon

illustrate these two different cases. For the precipitation of XbC in y-iron the diffusivity of niobium is close to that of self diffusion(21*Z) and

ACT.1

St?

METALLURGICA,

consequently much slower than the core self diffusion. Thus in this case the Sb diffusivity is expected to be rate controlling. An estimate for the colony gro!vth speed, I’, may be obtained by equating the rate at which vacancies are generated per unit length from the climbing dis!ocation (i.e. c/h’) to the flus of diffusing Sb atoms towards the growing particles. This flus is proportional to rrDsb(c - c’) n-here Dsl, is the diffusivity of niobium in y-iron and G and c’ are the actual and equilibrium niobium concentrations respectively. As DSb at iOO”C is about 2 . lo-l6 cm2/ se@‘) this gives a colony growth rate in an alloy containing about 1% Nb of about 1O-s-1O-1o cmjsec. This is in good agreement with experimental values

VOL.

22,

1974

6

1,: v = Colony

growth

rate

6 6 ;

j

x:0

Xrd

tzo

I.1

X)

FIG. 6. Sucleation (t = 0, x = 0) and unpinning (t = T, and r = d) of copper silicide precipitates in silicon.

copper colonies in silicon at 700°C, and the particle diameter at the unpinning can be estimated to about 10-j cm/set and MO-300-A respectivel-. giving an interface diffusivity of about 1O-s-1O-s cm’/sec. This may be compared to the interface diffusivity as est,imated in the usual way;(23) D surface

=

Doexp-&

where Do and &, relates to self diffusion values.“‘) giving lows cm2/sec. Thus the interface diffusivitv as calculated from the above model for the colony growth process appears to be of the right order of magnitude. Dipole growth

FIG. 5. TEM micrograph showing particles (arrows) moving ahead of the connecting dislocation segments (from the author’s laboratory).

which are reported to be in the range 10-8-10-s cm/sec.(lg) As the diffusion of copper atoms in silicon is very rapid, it is expected that t.he particle mobility or the interface self diffusivity will be the rate controlling parameter for the growth of copper colonies in silicon. Based on the sketch in Fig. 6 and with reference to equation (5), the particle speed can be witten dx _ 3D,J” 2’P =~13 dt

1 b__6 e (

The repeated precipitation process in silicon is also associated with a dendritic dipole growth effect.(11v13) This startling effect is schematically illustrated in Fig. 5. Particles nucleate on a dislocation (a) and, instead of following the growth pattern illustrated in Fig. 4, the particles and the dislocation segments in contact with them start to move ahead at a speed much faster than the connecting dislocation segments resulting in the formation of long dendritic dipole branches (Fig. ‘ib). It is suggested that this effect arises if the condition given in relation (8) above is established during the initial phase of the growth sequence. That is equiralent to saying that the supply rate of core vacancies

and !%,s,L C so

%o

e1

as 4~13 9 = kL bx, where k = 4nY3/Ldb, this equation can be integrated within the period of repeated nucleations and solved lvith respect to D,,, giving 4mP3 D,, = 9 * S” lvhere v is the colony growth rate. From a recent work by Ses and TVashburn(13) the =,gowth rate of

FIG.

i.

Schematic representation of the formation dendritic dipoles as observed in silicon.

of

SES:

from the dislocation for the initial

REPE_%TED

segments

particle

quence the particles

exchange located

segment given

rate,

and as a conse-

vacancies

only with the

in the interface.

particles grow their migration condition

is too low to account

growth

dislocation

PRECIPITdTIOS

in relation

As the

rate slows down and the (8) will not

be fulfilled

and it follows that the dipole configuration longer remain stable.

However,

the dipoles mai- be several attached,

continues

still after the condition valid.

growth

This

tendency

for the dendrites

lographic

directions,

colonies

and

It is suggested graphically

in

for

is at,tributed

{lOO}-type

straight

causing

apparently

to grow in certain

that jog nucleation

oriented

more difficult

effect

process

given in relation

i.e. in (112) directions

(110)

that

pm long, and wit,h large

particles

is no longer

this

would no

as it is observed

crystal-

for {llO}-

colonies.(11*13)

on these crystallo-

dislocation

the dipole

(8)

to the

segments

configurations

is left

behind to remain stable. Whilst only.*

this

effect

has

a similar mechanism

configurations in steel.

reported

These

precipitation

in silicon

may account for the dipole by

dipoles,

reaction,

been recognized Silcock

apparently

and

Denhamfe)

a result

also had particles

attached

of a to

one end.

* Sote added in proof. TEJI micrographs recently published by I-. K. HEIIXIXES and T. J. H_ZKARMXES (Phil. Nag. 28, 235, 1973) show that dendritic dipole growth may occur also in a low carbon iron-vanadium alloy.

OS

9;

DISLOC_1TIOSS ACKSOWLEDGEJIENTS

The author

is deeply grateful

and .J. D. Embury

to Professors

for many stimulating

J. Lothe

discussions.

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