Inhomogeneous ordering due to excess vacancies: An order-strengthening mechanism

INHOMOGENEOUS AN

ORDERING

DUE

TO

ORDER-STRENGTHENING P.

EXCESS

VACANCIES:

MECHANISM*

S. RUDMANt

It is noted that quenched-in excess vacancies decay more rapidly in the vicinity of dislocations, and It is shown that in partially ordered this gives rise to a lower long range ordering rate near dislocations. systems, dislocations will be locked-m by an order gradient and that a yield point is to be expected. The magnitude of the effect is calculated for B-brass on the basis of quasi-chemical binding and a yield stress comparable with the observed is obtained. It is also shown that this model of inhomogeneous ordering yields a simple explanation of an anomalous resistivity effect observed in Cu,Au by Dugdale. ORDRE

NON

UN

HOMOGENE

MECANISME

DU

DE

A

DES

LACUNES

DURCISSEMENT

PAR

EN

EXCES:

ORDRE

11 est note que les laounes en exces apes trempe evoluent plus rapidement au voisinage des dislocations et ce phenomene donne lieu a une vitesse d’ordre a grande distance plus faible au voisinage des dislocations. L’auteur montre que dans les systemes partiellement ordonnes, les dislocations seront bloquees 11 calcule l’importance de I’effet par un gradient d’ordre et que l’on doit s’attendre a un “yield-point”. pour le laiton-@, sur la base d’une liaison quasi chimique et obtient une tension au “yield-point” com11 montre egalement que ce models d’ordre non homogene fournit parable a celle qui est observee. une explication simple d’un effet anormal de resistivite observe dans la Cu,Au par Dugdale. INHOMOGENE

ORDNUNG

EIN

INFOLGE

MECHANISMUS

DER

VON

UBERSCHUBLEERSTELLEN:

ORDNUNGSVERFESTIGUNG

Im UberschuB eingefrorene Leerstellen heilen in der Nachbarschaft von Versetzungen schneller aus, und daher ist die Geschwindigkeit der Fernordnung in der N&he von Versetzungen geringer. In teilweise geordneten Systemen werden Versetzungen durch einen Gradienten des Ordnungsgrades festgehalten, so da13 man eine Streckgrenze zu erwarten hat. Die GroBe des Effektes wird fur B-Messing auf der Grundlage einer quasichemischen Bindung berechnet, die erhaltene FlieDspannung ist vergleichbar mit Dies Model1 inhomogener Ordnung gibt such einc einfache Erklarung tines anomalen der beobachteten. Widerstandseffektes. der von Dugdale in Cu,Au beobachtct wurde.

INTRODUCTION

Several

forms

and

in systems that undergo a change in lattice symmetry

mechanisms

of formation

of

on ordering as in CuAuc2) and in CoPt(3).

inhomogeneous distributions of order in ordered alloys have been identified. We wish here to propose a new

also occurs, at least for temperatures

form and mechanism

first-order

of its physical

of formation

consequences.

and consider some

Let

us first

briefly

the critical temperature,

in all systems that undergo a

transformation

barrier that reduces

because

the nucleation

review the currently recognized forms of order inhomo-

evidence

geneity.

presented by Guinier and Griffoulc5).

If we consider crystal regions of the size of an atom and its immediate neighbors, then ordered alloys are

have shown that such partially

intrinsically

inhomogeneous.

environment

is called short-range

discussed short-range On a scale dislocation

This variation order.

Alloys

Fisher(l) has

order as a strengthening

dimensions comparable to interpartially long-range ordered distances,

to precipitation that undergo

formations

factor.

of

to be homogeneous,

growth.

than by the growth

type of order inhomogeneity

progressive

alloy will be commatrix.

This

2

METALLURGICA,

VOL.

occurs most prominently

10, MARCH

1962

ordered systems are systems.

second-order

ordering

trans-

etc.) have been assumed

on this larger scale, even in the

is apparently

perfection

there are contiguously

well described

by the

of order within nuclei(4) rather of nuclei. ordering

However,

whenever

regions there is the

possibility of anti-phase boundaries, that is, planes of finite thickness, of a lesser degree of order,(e) between

* The research reported in this document has been sponsored in part by the Wright Air Development Center of the Air Research and Development Command, United States Air Force through its European Office. Received August 23, 1961. t Department of Physics. Technion, Israel Institute of Technology, Haifa, Israel. Present address: Battelle Memorial Institute, Columbus, Ohio. ACTA

has been

Newkirk et uZ.(s)

partially ordered state. The nucleation rate is so high that contiguous nuclei exist from the start and the transformation

In this case an ordering

rate.(sp4) X-ray

hardening

(eg. CuZn, Fe,Al,

systems can be inhomogeneously ordered if the ordering reaction takes place by coarse nucleation and posed of ordered regions in a disordered

of a free-energy

for such ordered nuclei in Cu,Au

analogous

in local

It probably

not too far below

ordered regions whose lattice sites are related to each other by fractional unit cell vector translations. Cottrellc7) and Brownc6) have discussed strengthening mechanisms due to anti-phase boundary type inhomogeneity. 195

ACTA

196

METALLURGICA,

VOL.

10,

1962

ANT’ -pHA:._%‘uy --\..*

,i

+FR.+rFR-+rf FIG. 1. Idealized

MODEL FOR EXCESS INHOMOGENEOUS

dislocation

VACANCYORDERING

The presently proposed form of order inhomogeneity is also a non-equilibrium will occur

state of a partially

that

ments.

Let us consider an ordering alloy that has been

held at a temperature

after

certain

ordered

system

T,, obtaining

thermal

treat-

a degree of long-

range order S, (if TI > T,, S, = 0), and a vacancy concentration 12i. Now let us rapidly transfer to, and then hold the alloy at, another temperature the order tends to the equilibrium vacancy To

concentration

be specific,

equilibrium

has been obtained

value na.

at TI so that S, < S,, If the

atomic interchange

process in ordering is by a vacancy

mechanism,

Vineyard(*)

has

shown

that

vacancy

(1)

concentration.

value, na.

of interest here is that the vacancy vacancy

presumably

The point

lifetime is a func-

being short in the neighborhood

sinks and long when distant from sinks.

of The

vacancy concentration will thus relax from n, to n2 more rapidly near sinks. By equation (l), we have for

are found in pairs,(‘) single dislocations

sinks.

However,

After Blandin

where

given by the vacancy sinks are probably

tions,(g) and their tangled geometry

with a

sink distribution. climbing

disloca-

does not permit

a rigorous calculation of the vacancy concentration distribution. In addition, even were the vacancy dis(1) would present a formid-

problem. Accordingly. to obtain a feeling for the effect of this form of

order inhomogeneity, we shall employ a highly idealized model of a periodic distribution of dislocations in the slip plane and consider the one dimensional modulations representative

present

idealization,

the

and FriedeloO) we can approximate distribution

An = n, - n2 7 = average vacancy

Introducing integrating

this expression

as lsin n-x/RI

(2)

lifetime

into

we obtain a function

equation

(1) and

that can be approxi-

mated as S = X1 + AS jsin TX/RI

(3)

It can also be shown that

in the degree of order in the slip plane as of the three dimensional

modulations

f(t) = 0,

lim

t=o (4)

( t=oo For present purposes the explicit time dependence

of

AS is not important,

the essential part is the propor-

tionality

As a measure

of the degree of

we can take AS/S,,

and it is easily

AS cc An.

inhomogeneity, seen that

t=o AS/S,

= An/n,F(t),

That AS/S,

able integration semi-quantitative

the

n = n2 + An exp [-t/T]

recent studyol’

The order will thus be inhomogeneous

tribution known, equation

for

the inter-pair vacancy

sinks.

The vacancy

on

scheme of Fig. 1 will suffice.

the present example of TI > T,, that the ordering rate will be greater in regions removed from vacancy modulation

by

While all mobile dislocations

the slip plane can also occur that can serve as vacancy

AS oc Ant(t),

Immediatelyafter the transfer, T,-+ T,, the vacancy concentration will be ni, and eventually it will every-

tion of position,

boundaries.

in Fig. 1.

where X1 = degree of LRO at the dislocation.

dS/dt CCnf (Ti, S)

where relax to the equilibrium

anti-phase

the

ordering rate at Ti is given as

where n = instantaneous

are arranged in pairs connected

that

TI > T,,

and n, > n2, where ni = N exp [-Ef/kTi]. then

This model is illustrated

The dislocations

value S, and the

that

model.

in real crystals.

T,, where

to the equilibrium

let us assume

distribution

F(t) = 0, lim

may be appreciable

( t=*

(5)

is evidenced

by the

of ordering kinetics in PeaAl where it

was shown that cold work greatly retards the ordering rate because

of the associated

vacancy sink density.

increased

dislocation

Further, even without explicitly

calculating F(t) of equation (5), we can expect appreciable inhomogeneity if Anlnz > 1. We can write An/n2 = exp [-E,/k(l/T,

-

Taking: TI = 4OO”C, T, = 200°C. obtain An/n, = 103.

l/T,)]

-

1

E, = 1 eV,

It is interesting that only relatively 1954, account of the fact that excess

(6) we

recently, in quenched-in

vacancies can play a significant, and even dominant role in ordering kinetics has been noted.

RUDMAN:

INHOMOGENEOUS

ORDERING

DUE

TO

EXCESS

107

VACANCIES

THE FLOW STRESS CONTRIBUTION OF INHOMOGENEOUS ORDERING

Let

us consider

the slip-plane

point of view of quasi-chemical

energy

from

the

bonds for the modu-

lated order of Fig. 2, assuming the distribution of equation (3). After slip of a distance x, that is after m dislocation

pairs

have

glided

the

crystal

length,

2mb = x, the slip plane energy is then given ast6) E(x) = y (S2) + const

(7)

where E(x) = energy/unit area y = &N’z’w( + for anti-phase,

-

for

no

anti-phase) N’ = number of plane area

x’ = inter-sublattice,

DISTANCE FIG.

2.

The distribution of long range order (schematic)

We can thus expect the order modulations with time as presented shall be primarily

of such

schematically

interested

a distribution

other properties anomalous model,

to vary

in Fig. 2.

motion.

will manifest

We

However,

themselves

is readily

effect observed

explained

Let

us consider

from

by the present

dimensional sponding

an ordering alloy that after downwhose one-

to time t, in Fig. 2. Now suppose that the

T,, such that the equilibrium

to some temperature,

degree of order at T,, S,,

lies as illustrated in Fig. 2: S1 < S, < S1 + AS. Then referring, to Fig. 2, with isothermal holding at region

I will disorder the vacancy

and region

concentration

II will order. in region

I is

schematically

in Fig. 3.

sin n-x/R + (R - 2x) x cos TX/R] + const

= --rry( AS)2/2R[(1

-

2x/R) sin TX/R]

omax which from equation

(10)

(10) is

o max = y( AS)2/2R

(II)

It is to be noted from equation (10) that ~(2 = 0) = 0, occurs at x E R/6, thus, as noted by and that a,,,

REGION

II-

ORDERING

The

REGION

havior in CusAu for Tl = 37O”C, T, = 23O”C, T, = 250°C. Dugdale proposed a short range order model explanation of this effect. It will require further experimentation to decide between Dugdale’s and the

(9)

As a measure of the flow stresscontributionweoantake

point of interest is the initial rise in resistivity. Dugdale(13) has observed just such resistivity be-

present model.

x)/R

(8) yields

o = dE(x)/dx

orders. If we were to measure the resistivity as a function of time at T, we would expect to obtain as illustrated

of equation

and, after Fisher,(l) the flow stress is given as

greater than in region II, and thus we can expect region I to disorder more rapidly than region II

behavior

(8)

S = S1 + AS sin 71-l/R S’ = ~‘3~+ AS sin ~(1 -

analog is given, say, by the curve corre-

alloy in this state is up-quenched

T,,

where

E(x) = y(AS)2/2xR[2R

EFFECT DUE ORDERING

an order distribution

that r/R < 1, then

s0

for its

Tl to T,, and being held at T, for

some time, obtains

co-

atom bond

(P) = 1/R “SSW

Integration

quenching

However,

Assuming

plane,

in

existence. AN ORDERING KINETICS TO INHOMOGENEOUS

cross-slip

slip

number

by Dug-

and hence we can take it as evidence

sites/unit

21= (E, + %,)/2 - EB E AA, etc. = nearest neighbor A atom-A energy etc.,

and we shall first briefly consider an

ordering kinetics

dale(13) which

ordination

here in the consequences

ondislocation

such order modulations

sublattice

I-DISORDERING

TIME

Fro.

3. Resistivity behavior during ordering of an inhomogeneously ordered alloy (schematic).

ACTA

198

Bro~n,‘~)

this order strengthening

contribute

METALLURGICA,

mechanism

will not

to a yield point, but only to a flow stress.

To see if this mechanism

can contribute

significantly

to the flow stress let us numerically

estimate o,,,.

calculation

among

will differ somewhat

ordered structures, fi-brass structure,

the several

but to fix ideas let us consider the assuming

X’ = 1/1/2as, z’ = 2

(110) [ill]

slip, with

observed flow stress6 in p-brass is of the order of lo*very high, say 1012/cm2, the presently range order modulations

density is

considered

will not contribute

long

signifi-

cantly to the flow stress. YIELD POINT CONTRIBUTION INHOMOGENEOUS ORDERING

When

we take into

account

OF

then

anti-phase S,,

boundary

with S, > S,.

in material of degree of dislocation

in material

eliminates

an

of degree or order,

After Brownt6), the shear stress to

move the pair is then

AE,

= energy/unit

+ AE,) area to

phase boundary

(12) create

an anti-

in material

of degree

of order S, AE,

= energy/unit

area to eliminate

phase boundary of order S, and from equation

Substituting

in material

an antiof degree

and not adequately

apparently

mechanisms

several

operative,

order-

including

the

present proposal, and it does not seem to be worthwhile it should

be possible

contributions.

How-

to isolate

the present

order effect by experiments

on partially

There is a further mechanism modulations

can

interact

by which the order

with

dislocations.

In

general the lattice parameter is a function of the degree of order and thus inhomogeneous

ordering may give

rise to stress fields that can interact with the dislocation

stress fields,

mechanism.

giving

rise to a further

We shall not perform

locking

this calculation

here.

SUMMARY

When

ordering

occurs

elevated temperature, interchanges cies.

OF RESULTS

after

quenching

a significant excess

decays

from

an

part of the atomic

can be due to quenched-in

The vacancy

excess vacan-

inhomogeneously,

faster near vacancy sinks, slower distant from vacancy sinks.

In vacancy

rate is proportional

mechanism

ordering,

to the vacancy

the ordering

concentration,

and

so when ordering is carried out at a lower temperature

vacancy sinks).

will be inhomogeneous

= 2N’z’SL2v

(13a)

AE,

= -2N’z’Sp2v

(13b) (12) and em-

(being

sinks, and greater more distant

less near from such

It has been shown here that, assuming that the

vacancy

sinks are dislocations,

the dislocations

will

be locked in an order gradient and a yield point effect is possible. calculated observed

AE,

(13) in equation

the order

The magnitude

of the effect

parison

yield

experimental

to the experimentally

stress was obtained.

between

attempted

has been

for P-brass on the basis of quasi-chemical

binding and a value comparable

(7) we have

Equation

is complex are

following a quench from a more elevated temperature,

0 = l/2b(AEL where

strengthening

dis-

locking effect.

boundary

order, S,, and the following

There

action considered

that the mobile

Referring to Fig. 2, we see that for a small displacement of the dislocation pair, the leading dislocation creates in its wake an anti-phase

behavior

explained.

here, but only note that this stress field effect may be comparable with the quasi-chemical bonding inter-

locations are anti-phase boundary connectedpairs, we obtain a dislocation-order

temperature

ordered alloys.

The experimentally

log dyn/cm2, and thus unless the dislocation

and in FesAl by Cahn

et al.(lQ. In all of these alloys the strength-quenching

modulated

density of 101°/cm2),

10’ dyn/cm2.

1962

et aZ.04), in CuZu by Browno5)

ever,

a = 2.95 d, (AS)2 = 0.2

0 mitxg

10,

to speculate as to the respective

v = kT,, T, = 740”K,

,

R = 1O-5 cm (i.e. a dislocation we obtain:

This

VOL.

the theory

results

from

the

because no “clean”

other order-strengthening

Detailed

and presently literature

experiments

mechanisms

com-

available was

not

free from

are available.

ploying the constants for p-brass used previously, with SL2 - SF2 = 0.1, we obtain a E 10s dyn/cm2 and we

It has also been noted that an additional

are thus led to expect a strong yield point contribution from the presently considered long range order modulations.

dislocation stress field and a stress field in the lattice because of the dependence of the lattice parameter on the degree of long range order.

There are several studies

in the literature

of the

strength of ordering alloys that have undergone thermal treatments of the type required to produce the presently described effect: in CusAu by Kuczynski

order

gradient

interaction

may

exist

dislocationbetween

the

Evidence for the present model of order-inhomogeneity has been found in an anomalous resistivity effect observed by Dugdale in CusAu, which is simply accounted for by the present theory.

RUDMAN:

INHOMOGENEOUS

ORDERING

REFERENCES 1. J. C. FISHER, Acta Met. 2, 9 (1954). 2. G. BORELIUS, J. Inst. Met. 74, 17 (1947). 3. J. B. NEWKIRK, A. H. GEISLER, D. L. MARTIN and R. SMOLUCHO~SKI, Trans. Amer. Inst. Min. (Met&) Engrs

188,1249

(1950).

4. G. DIENES, Acta Met. 3, 549 (1955). 5. A. GIJINIERand R. GRIFFOUL,C.R. Acad. Sci. Paris, 224, 1168 (1947). 6. N. BROW, Phil. Mas. 4. 693 (1959). 7. A. H. COTTRELLin Re~atibn of &rope&es to Microstructure. American Society for Metals, Cleveland (1953). 8. G. VINEYARD, Phys. Rev. 102, 981 (1956).

DUE

TO

EXCESS

VACANCIES

199

9. J. BARDEEN and C. HERRING in Atom Movements. American Society for Metals, Cleveland (1951). 10. A. BLANDIN and J. FRIEDEL, Acta Met. 8, 384 (1960). 11. J. BRANSKY and P. S. RODMAN, USAF Sci. Note No. 5, Conk. No. AF 61(052)-122 (June 1961). Submitted to Acta

Met.

12. R. A. DU~DALE and A. GREEN, Phil. Mug. 45, 163 (1954). 13. R. A. DUGDALE, Phil. Mug. 1,537 (1956). 14. G. C. KUCZYNSPI, M. DOYAMA and M. E. FINE, J. AppZ. Phys. 27, 651 (1956). 15. N. BROW, Acta Met. 7,210 (1959). 16. A. LAWLEY, E. A. VIDOZ, and R. W. CAHN, Actn Met. 9, 287 (1961).