Analysis of dislocation loops in relation to slip, twinning and phase transformations

ANALYSIS

OF DISLOCATION LOOPS IN RELATION TO PHASE TRANSFORMATIONS*t K.

SADANANDAS

and

M.

SLIP,

TWINNING

AND

J. MARCINKOWSKI:

A detailed numerical analysis has been made of the nucleation of dislocation loops in ordered alloys as well as in f.c.c. metals. The results show that the nucleation of superlattice dislocations becomes increasingly more difficult at low temperatures, with the result that deformation occurs by the generation of antiphase boundaries (APB) at these low temperatures. This nucleation difficulty with respect to super-lattice dislocations is even more pronounced with a decrease in APB energy resulting from a change in either the degree of order or the composition. Application of the above analysis to f.c.c. metals shows that twinning can be enhanced by decreasing the stacking fault energy of these metals. It is further shown that the concepts developed in the present analysis can be used to determine the kinetics of nucleation associated with phase transformations. LES

BOUCLES

DE DISLOCATIONS, MACLAGE ET LES

LEUR RELATION TRANSFORMATIONS

AVEC LE GLISSEMENT, DE PHASES

LE

Une analyse numerique detaillee de la germination des boucles de dislocations a 6th effect&e dans les alliages ordonnes ainsi que dans les metaux c.f.c. Les resultats montrent que la germination des superdislocations devient de plus en plus difficile aux basses temperatures, si bien qu’a une temperature assez basse, la deformation resulte de la formation de parois d’antiphase. Cette difficult& de germination est m&me encore plus prononcee si l’energie des parois d’antiphases decroit par suite d’une variation du degre d’ordre ou de la composition. L’application de oette analyse aux metaux c.f.c. montre que le maclage peut augmenter si l’energie de faute d’empilement de ces metaux decroit. Les auteurs montrent en outre que les idles qu’ils ont present&es peuvent etre utilisees pour determiner les cinetiques de germina. tion associees aux transformations de phases. ANALYSE

VON VERSETZUNGSRINGEN ZWILLINGSBILDUNG UND

IM ZUSAMMENHANG PHASENUMWANDLUNGEN

MIT

GLEITUNG,

Sowohl an geordneten Legierungen als such an f.c.c. Metallen wurde eine ausfiihrliche numerische Analyse der Keimbildung von Versetzungsringen durchgefiihrt. Die Ergebnisse zeigen, da3 die Keimbildung von Uberstrukturversetzungen bei tieferen Temperaturen immer schwieriger wird; das hat zur Folge, daB die Verformung bei diesen tiefen Temperaturen iiber die Bildung von Antiphasengrenzen erfolgt. Diese Keimbildungsschwierigkeit. fur die Uberstrukturversetzungen ist bei abnehmender Energie der Antiphasengrenzen (d.h. bei Anderungen des Ordnungsgrades oder der Zusammensetzung) noch ausgepriigter. Die Anwendung der obigen Analyse auf f.c.c. Metalle zeigt, da8 man durch Verminderung der Stapelfehlerenergie dieser Metalle vermehrte Zwillingsbildung erreichen kann. AuBerdem wird gezeigt, da8 die in der vorliegenden Arbeit entwickelten Konzepte auf die Bestimmung der Keimbildungskinetik bei der Phasenumwandlung angewandt werden k&men.

INTRODUCTION

It is well known that many metallurgical phenomena such as slip, twinning or phase transformation involve both nucleation and propagation processes. And the kinetics of these processes depend sensitively on the relative magnitudes of the activation energies for nucleation. Many experimental observations also indicate that nucleation of slip, twinning or even of a second phase occurs preferentially at regions of internal stress concentration such as grain boundaries, inclusions, deformation bands, etc. In the following paragraphs, an analysis of homogeneous nucleation of dislocation loops in ordered alloys, as well as in f.c.c. metals and their alloys, is presented. The analysis is then extended to account for the yield stress-temperature curves associated with ordered alloys as well as to the nucleation of twinning in f.c.c. metals. While no detailed analysis of the kinetics of nucleation * Received September 6, 1973. 7 The present research effort was supported by the United Commission under contract No. AT-(40-1)3935. $ Engineering Materials Group, and Department of Mechanical Engineering University of Maryland, College Park, Maryland 20742, U.S.A. ACTA

METALLURGICA,

VOL.

22, JULY

1974

during phase transformations is made, it will be shown that concepts similar to those discussed with reference to the nucleation of slip and twinning can be extended even to phase transformations. Analysis of dislocation loops in ordered alloys It is instructive first to examine the nucleation of dislocation loops in ordered alloys. The purpose here is two-fold; most importantly, it will be shown that the simple concepts developed in the present analysis can adequately explain the yield stress-temperature curves of many ordered alloys. Secondly, the concepts developed here are shown to be quite general and can easily be extended to an understanding of the nucleation of twins in f.c.c. metals and their alloys. The variation of the yield stress with a decrease in temperature for many ordered alloy&4) can be represented schematically as shown in Fig. 1. While the yield stress is relatively independent of temperature in the temperature region of BB’ in Fig. 1, it increases sharply for temperatures less than B. Such sharp increases in yield stress were observed in Fe,Si(1-4) and Fe,A1(2*3)alloys. In addition, marked

863

ACTA

864

I

I

o! I2

METALLURGICA,

I

c

I

\

i

I

1

B’L______B,

FIG. 1. Schematic illustration showing the effect of APB energy on yield stress-temperature curves. In this figure y1 > yz.

increases in yield stresses begin to occur in polycrystalline FeCo alloys(Q tested at 77 K. With a further decrease of temperature below B, the yield stress of Fe,A1(2,3) and Fe3Si(1*6)varies less rapidly, as represented by AA’ in Fig. 1. The variation of yield stress with temperature in the region of AA’, however, is identical to the yield stress variation with temperature of the corresponding disordered alloy or flow stress variation with temperature of the ordered alloy in the Stage III work hardening region.(l) From these observations it can be concluded that the dislocations in the ordered alloys behave as uncoupled dislocations in the temperature range AA’, while they behave as superlattice dislocations in the temperature range BB’. The above conclusion is further supported by the fact that the stress level corresponding to A’ is equal to the stress necessary for the propagation of antiphase boundaries (APB’s) in the ordered alloy, which is given by 7 = y/b where y is the APB energy and b is the Burgers vector of the dislocation bounding the APB. Figure 1 also shows the effect of y on the yield-stress temperature curves. In addition to the decrease in the level of AA’ with decreasing y, the transition from AA’ to BB’ is shifted to higher temperatures. The decrease in stress level, A’, is to be expected due to the lower stress needed for the propagation of APB’s.(I-~) In the limit of y = 0, corresponding to a fully disordered alloy, the yield stress variation with temperature is represented by the AA’ curve alone for all temperature ranges in agreement with the previous discussion. While it is clear that the propagation of superlattice dislocations in ordered alloys is much easier than that of ordinary dislocations, since there is no net production of APB’s, the same cannot be said concerning the nucleation of superlattice dislocations. In the following sections the analysis of the nucleation of superlattice dislocations in B2 type

VOL.

22,

1954

alloys is carried out in much the same way as was done earlier.@) However, in the present analysis, emphasis is placed on an understanding of the effect of y on the nucleation processes. The importance of the present analysis need not be stressed further if it is recognized that the APB energy can be significantly affected either by changing the degree of order or by altering the composition. or by both (see Review by Marcinkowskic’)). For simplicity, the present analysis is restricted to B2 alloys. However, the conclusions obtained from the present analysis are quite general and can be applied to all types of ordered alloys. Formulation of the problem Figures 2(a and b) represent successive dislocation configurations in B2 alloys if the nucleation of each dislocation loop is independent of the other. On the other hand, Fig. 2(c) represents an alternate mode of generation involving a complete superlattice dislocation while Fig. 2(d) represents the dislocation configuration in a corresponding disordered alloy. The simplest case to consider is the nucleation of single loop configurations corresponding to A, B and D, where the letter designation corresponds to the respective loops in Figs. 2(a, b and d). The activation energy for the nucleation of loop D, for example, is given by the maximum in the total energy of the 10op(~)where the total energy is given by ET

=

E,,,

-

Er

(1)

where ERss is the self energy of the loop of radius R, which has been obtained by Kroner,@) and E, is the

FIG. 2. Dislocation loop configurations which may:be generated in ordered and disordered I32 alloys.

SADANANUA

DISLOCATION

~~ARCI~KO~SKI:

AND

The total

work done by the shear stress 7 given by E7 = nR%b. The variation

(2)

of the activation

stress for loop D is represented

energy

with

shear

by the dashed curve

in Fig. 3. In obtaining

this curve, the material

stants for the particular

FeCo alloy were employed,(s)

i.e. ,u = 7 x 10~1d~~es/cm2, b = (~/2){111} IO-* cm, w-hile the core parameter

LOOPS

con-

= 2.47 x

activation

superlattice

SLIP,

8BB

TN’INNING

energy for the nucleation

dislocations(E)

activation

energy-stress

dislocation

is represented

fully disordered

curve

for

the

The

superlattice

in Fig. 3 by A + B.

For a

A + B is equal to twice

alloy,

of

is then given by the sum

energies of the A and B loops.

of the activation

activation

the

energy of the single loop D, and is repre-

sented by the dashed line labeled 20 in Fig. 3. It is immediately

E = (1/2)u{lll)~

AND

apparent

in Fig.

3 t,hat for

stresses

[2e( 1 -

Y)]. Using curve D as a reference, it is a simple

greater than li: A + B is equal to A.

matter

to

this is that at these high stresses there is essentially no

determine

the

activation

curves for A and B loops.

energy-stress

In particular,

the total

GA

= E,,,

-

E, -t- E;,

(3)

and

respectively.

= E,,,

-

E, -

E,

(4)

The last term in the above expression

is

the energy due to the APB and is given by

point to note in Fig. 3 is that with a

of y, the energy

A comparison

of equations

APB energy contribution Hence the activation

that

is similar to the shear stress.

energy-stress

B loops ean be obtained

curves for A and

by a horizontal

translation

of the D curve to the left or to the right, respectively, by y/b stress units. The solid A curve in Fig. 3 represents the activation energy-stress

curve for a fully ordered FeCo alloy in

which the APB energy on (111) planes is taken as(S) 157 ergs/cm2. represents

curve labeled A, however,

The dash-dot

the activation

energy-stress

curve

for a

ordered FeCo alloy with y = 37.4 ergs/cm2.

%or clarity, the corresponding

of the dislocation rate

curves for B loops are

in Fig. 3.

A and

between

In order to determine loops that contribute

of a eryst,al, the following

the nature

to the strain

equation

may

be

considered.f2) exp (---E&CT)

(5)

(2) and (5) shows

difference

A + B increases and in the limit of y = 0, A is equal

c’: = NAb,v

E, = ryR2.

omitted

decrease

to half of A + B. E,”

partially

energy barrier for the inner loop B.

activation

An important

energies for A and B loops are given by

The reason for

+ NAbkBp

(6)

exp ( -EA+GT)

where N is the total number of dislocation

nucleation

sites per unit volume, A is the area swept out by each loop,

v is the Debye

constant,

K is Boltzmann’s

frequency,

T is the absolute

t,emperature.

The sub-

scripts A and A + B in the above equation A and A + B loops, respectively.

refer to

The contribution

to the total strain rate from A loops is negligible if the applied

stress is less than y/b. Let us suppose

there are a sufficient number of dislocation sites, N, to maintain

the imposed

crystal, if the activation and temperature

of A loops in the partially stress at the nucleation the activation

superlattice

ordered

and the available locations

alloy, the shear

(Fig.

3).

At this stress

energy for the nucleation

dislocation

tJo nucleate

is 1 eV

For the nucleation

sites has to be raised to at

least 885 x lOa dynes/cm2 level,

strain rate in a

energy for nucleation

of the test is T,.

t,hat

nucleation

(A + B)

of a

is nearly

1.5 eV

thermal energy may be insufficient

a sufficient

number

of superlattice

to maintain the imposed strain rate.

a crystal tested at temperature

dis-

Hence

T, could only deform

by the propagation

of A loops and the external stress has to be raised to the level of y/b. At a higher temperature

T,,

however,

the

number

of superlattice

dislocations nucleated could be sufficient to maintain the imposed strain rate. Since the propagat,ion of A

STRESS

t

108dyneo/cm2)-

FIG. 3. Vari&m of activation energy with shear stress for the different dislocation loops in Fig. 2. 5

loops may not be necessary to maintain strain rate at such high temperature, stress falls below the level of y/b. Assuming nucleation

that sites

the exist

same in

number

hhe fully

the imposed the applied of dislocation ordered

alloy

ACTA

866

(y = 157 ergs/cm2),

Fig.

3 shows

METALLURGICA,

that the internal

at’ress level has to be raised to at least 935 x lO* dynes/cm2,

existence

1974

of such transition

in yield

stress.

A few

alloys, such as Cu3Au,(1Q-1Q)do not show such transi-

The

tion, though the APB energy is relatively low and the reason for this is not clear. The present analysis also neglects the orientation dependence which is con-

the

nucleation

close to the activation of superlattice

22,

activation energy for the nucleation dislocations is 1.06 eV, which is very

for

corresponding of superlattice

VOL.

A

of

loops.

energy for A loops.

distocaOions nucleated

The number

at temperature

siderably

important

for

single

alloys that possess high y.(“)

crystalline

However,

ordered

most of the

7’, should be sufficient to maintain the imposed strain

B2 alloys that possess

rat,e. Hence the applied stress falls below y/b.

show a large ipcrease in yield stress with a decrease in

ever, with decrease in temperature, superlatt’ice

dislocations for

becomes

temperatures

How-

the nucleation increasingly

and

less

than

imposed

st,rain rate could only be maintained

the

by the

The present analysis can also be used to understand the variation

of yield

stress with degree

of order.

Figure 4(a) shows the effect of a decrease of X on the

pr(~pagation of A loops and the external stress has to

yield stress-t,emperai,ure

be raised to y/b.

8,

Because of the smaller difference in

seem to

temperature.‘JQ)

more

T,,

dificult

of

(111) slip directions

the

APB

energy

curves.

With a decrease in

decreases,(T)

which,

in turn,

the activation

energies of A and A + B loops in the

markedly

fully

alloy

4(a) also shows that the yield stress increases with a

ordered

as compared

ordered alloy, the transition at a lower t’emperature. of reducing

Indeed,

partially

in the yield stress occurs

y is t,o increase

spondingly

the

Hence, the important the energy

A and A + B loops

between

to

to increase

in Fig.

it was observed

difference

could

to t,he extent

y in Fe-Si

to lower temperatures may not occur

In such cases the BB’ curve in

even at absolute zero.

Fig. 1 extends all the way down to 0 K. of t,he transition

This absence

in yield stress could also be under-

stood by the effect

ofy on

to t’he presence

which acts as a frictional

the stress level R in Fig. 3.

In the above analysis,

the dislocations

type

and the yield

of the outer loop.

transition

This however, may not be realistic,

since the outer loop A may still be considerably

close

to the source so as to influence the nucleation

process

of the inner loop B.

difficult

However,

a system

techniques,

crossed

hence is unstable.

it is extremely

involving

both

loops by static

since t,he outer loop

the activation This difficulty

energy

A has

barrier

can be overcome

and by

assuming

are of the superlattice

creases to zero at large distances away from the source.

stress varies

in Ni,Si,(ll)

of the inner

of the position

Thus, for all ranges of

along BB’ in Fig. 1. This accounts such

in order

the nucleation

loop, B, is assumed to be independent

aIready

to rx3,the inner loop nucleates spontaneously

Such a

Fe,A1’21p27) and CuZn(28-30) alloys.

equilibrium

of this is that as y tends

in Fig. 4(b).

been observed in FeCo,(21) I?eCo-2V,(P2*23) Ni,Mn,(Q*+2Q)

increases.

temperature,

dis-

of yield stress wit811the degree of order has

in t,urn the energy at which A and A i_ B are equal,

the APB energy.

be

orde+Q)

T, the yield stress

ture T,) as shown schematically variation

t!o analyse

to minimize

may

stress on superlattice

At a test temperature

With an increase in y, the stress level E decreases, or The implication

This

of short, range

from Fig. 4(a) varies with X (or with quench tempera-

increases the transition

that such a transition

S at high temperatures.

attributed locations.

that decreasing

be shifted

decreasing

Figure

temperature,

temperature.(4) On the other hand, if y is very high, the transition temperature

temperature.

3 and corre-

the transition

alloys by varying composition

effect

affects the transition

with

that the internal

stress at the source de-

temperature

for the absence of

Ni,A1(12) and AgMg(lQ)

alloys which are believed to have high APB energies.c7) The transition,

however,

can be observed

ducing the degree of order or by changing position.

For example,

with temperature polycrystalline parison to compounds.

by re-

t,he com-

sharp increases in yield stress

were observed in non-stoichiometric

AuZn(14) and NiA1(15) alloys in comyield stresses of their stoichiometric These increases could be attributed to a

decrease of y in these alloys, which has an effect of increasing the transition temperature (Fig. 1). In many other ordered alloys, tests have not been carried out at s~~~~ientIy Iow temperatures

to disprove

the

FIG. 4. Schematic illustrations showing (a) effect of long range order parameter, S, on the yield stress-temperature curves, (b) variation of yield stress with S or with quench temperature, T,.

SiSDANANDA

AND

MARCINKOWSKI:

DISLOCATION

This, in fact, is a realistic assumption since the internal stress due to inclusions, etc. are Iocalized.@lJ2) Before examining the effects of internal stress on the nucleation process, further considerations are in order. Curve A, in Fig. 5, for example, shows the effect of shear stress on the critical radius of the A loop, where the critical radius is defined as t,he radius for which t,he total energy of the loop is a maximum.(6) Under constant stress conditions, the dislocation loop accelerates after its nucleation. On the other hand, the dislocat’ion loop can expand with zero acceleration if the internal stress decreases homogeneously along the curve A. The reason for this is that curve A represents the locus of points for which dE,/dR = 0. As R -+ co the stress approaches y/b as is to be expected. It is also possible to stabilize the outer loop, A, by a proper selection of an internal stress and an applied stress. Consider, for example, an internal stress that is constant for R less than x and zero for R great,er than x. Such a stepped function for the internal stress is represented schematically in Fig. 6. If the dislocation loop A has to be nucleated, it is obvious that x should be greater than the critical radius for the nucleation under the conditions of total shear stress (TV+ TV). As the dislocation loop expands following 6he nucleation, it finds a stable equilibrium at R = x only if the applied stress, fa, in Fig. 6, is less than the shear stress given by the A curve at R = z (Fig, 5). Using this stable equilibrium position as a reference state, the activation energy for the inner loop, B, can be determined. The calculations involve the determination of the position of the outer loop for a

LOOPS

l

i

AND

Y67

TWINXING

._I

DISTANCE R-

Fm. 6. Schematic ill~tr&tion showing the form of the internal st.ress function used in t.he present osIculations.

given position of the inner loop by minimizing the total energy of the entire system as represented by ET

=

Em -t E,,,

-I- E, -

rh(R2 + p2) i y?r(R2 -

p2).

(7)

where the first two terms on the right of the equal sign represent the self energies of the outer and inner loops respectively, while the last two t’erms are the lvork done by the tot,al shear stress (TV+ TV)and the work expended in creating the APB’s, respectively. The interaction energy, E,, has been given by Kriiner.@) After the nucleation of the inner loop, the entire superlattice dislocation could expand spontaneously in the presence of the applied stress. There are cases, however, where the superlattice djslocat~ion finds an additional energy barrier. Figure 7, for esample, shows the nature of this second energy barrier fos TV= 300 x lo8 dynes/cm2 and x: = 100 :< lo-* cm. The dashed curves in Fig. 7 are obtained using the radius of the inner loop, p, as an independent variable. 10*CI-.-~__.__ .._. 1 I -R ----

Fro. 5. Variation of c&ical loop radius with shear stress for the different loops shown in Fig. 2.

SLIP,

p

I

FIG. 7. Variation of the total energy of e, superlattice dislocation in FeCo with two different independent variables, R and p.

ACTA

868

METALLURGICA,

VOL.

22,

1974

When p = 0, the outer loop has a stable equilibrium With

as R=x.

an increasing

p, the equilibrium

position of the outer loop is altered.

The total energy

of the whole system, however, goes through a maximum at some critical radius of the inner loop. This maximum inner

corresponds

loop.

The

to the activation

system,

lattice dislocation,

now

and an ambiguity

to t*he selection of an independent calculations. could

In principle,

expand

along

7 describes

the lowest

two energy paths, variable,

dislocation

curve until a minimum For

goes through

corresponding variables.

along

the superlattice

stress to

of applied

stress where

represents

the

solid

curves

variable.

in Fig.

The curves are

defined from R = x, since it corresponds equilibrium calculations

activated.

involve

brium position

In the present

the determination

case, the

of the equili-

of the inner loop, for a given radius of

the outer loop, by minimizing system

superlattice

to the first

position for the outer loop when the inner

is being

given

by

dislocation

the total energy of the

equation becomes

(7).

The

unstable

entire

when the

presence

stability

of

superlattice

of an applied

dislocations

in the

stress can be understood

by

reference t’o the R, and R,, curves in Fig. 5. Under a homogeneous

stress, Rcl, corresponds

to the minimum

radius of the outer loop for which the inner loop is stable.

If the radius of the outer loop is less than Rcl,

the inner loop collapses. loop increases

further,

As the radius of the outer superlattice

dislocations

beyond

R, is defined

minimum

radius

which the superlattice

spontaneously.

con-

sisting of both outer and inner loops become unstable

in Fig. 5, it

of the

dislocation

The existence

outer

of the second

For the suppression

R, for the range of applied

defined and x should be greater than R,, for the range of applied stresses for which R, is not defined. been

chosen : x = 500 x lo-* cm

lo8 dynes/cma.

For this applied

parameters have and

7, = 50 x

stress, x is clearly

greater bhan the R, curve in Fig. 5 and hence no second energy barrier exists for superlattice tions.

The activation

energies

for both

maxima

exist

at R = R,.

disloca-

outer

and

inner loops are determined as a function of the internal stress and are represented in Fig. 9 for a fully ordered FeCo.

The activation

energy of the outer loop is the

same as that given by the A curve in Fig. 3. The curve denoted by B’ represents the activation energy for the inner loop in the presence of the outer loop.

energy

R, is

stresses for which

of the addition

stresses,

of

the second energy barrier, x should be greater than

stress is represented applied

energy

that given by the R, curve or R,, curve in Fig 5 for all the applied stresses selected.

when R is greater than R,. The energy of the superlattice dislocation in the presence of a homogeneous more clearly in Fig. 8. For small

loop

can expand

barrier in Fig. 7 is due to the fact that x is less than

For further analysis, the following

outer loop reaches some critical radius, R,. The

FIG. 8. Stability of superlattice dislocation with radius of the outer loop for various applied stresses.

7

The

dynes/cm2.

selected as an independent

whole

I

dis-

a second energy barrier which of the applied

I

1

the dashed

the change in the total energy when R is

100 x lo*

loop

in the

the energy of the

decreases

with an increase

represent

dislocation path

in the energy occurs at p = x.

7, = 75 x lOa dynes/ems,

disappears

energy

by the R and p coordinates.

For p as an independent

location

variable for further

of p and R as independent

to the selection superlattice

of a super-

arises with regard

the superlattice

energy surface determined Figure

energy of the

consists

of a positive

interaction

Because

energy term

the B’ curve is higher than B,

to the total energy,

Wit’h an increase in the applied stress R, approaches

where B represents the independent

RcI, and for the stresses greater than 75 x 10s dynes/ cm2 R, is not defined in Fig. 5. For such high stresses

inner loop.

However,

nucleation

the activation

of the

energy for the

superlattice dislocation, A + B’ is not significantly different from A + B. The effect of larger differences

the superlattice dislocation becomes unstable at the minimum radius R,,. The energy maximum in Fig. 8 should not be interpreted as the activation energy for

in energies between A and A + B’ in comparison to A and A + B, is to shift the transition temperature

the superlattice

to higher temperatures

the activation

dislocation

since it does not include

energy for the inner loop.

For the range

Fig. 3.

The above

as discussed with reference t’o

calculations

could

also be done

SADAKANDA

1 %i;,-

v

DISLOCATION

AND MARCTNKOWSKI:

,

900

STRESS

t 950

‘%:. 1000

,\ 1050

] II00

[x tO*dynes/cm2 1

FIG. 9. Variation of activation energy with shear stress for different dislocation loops in fully ordered F&o.

using any other internal stress function. The condition for the suppression of the second energy barrier in such cases is that the first stable equilibrium position for the outer loop should be greater than that given by the R, curve in Fig. 5. For high internal stresses represented in Fig. 9, the activation energy for the inner loop, B’, is found to be nearly independent of the exact nature of the internal stress fun&ion. Since the effect of the incorporation of an intera&ion energy in the nucleation process is then seen to be negligible, it will be sufficient for further analysis t’o consider the nucleation of the two loops as independent of one another.

Many f.c.c. metals deform by twinning at low temperatures and by slip at high temperatures.(33) Also, with a decrease in stacking fault energy, the alloys show an increased tendency to deform by twinning at low temperatures. Because of the close analogy between stacking faults and APB’s, nudeation of the Schockley partials can be treated in much the same way as the nucleation of partials with respect to superlattice dislocations. In particular, dislocation loop configurations similar to those shown in Fig. 2 can be imagined with the following alterations. The Burgers vector of the Schockley partials is taken as (a/6)(112) while y, here, represents the stacking fault energy. As in the ordered alloys, the stacking fault energy can be signi~cantly affected by changing the composition in f.c.c. alloys.(M) The effect of composition on twinning was first studied in the Ag-Au system.@5) It was found that lower stacking fault energy metals such af Ag, were much more prone to twinning than higher stacking fault energy metals such as Au. The energy for nucleation of dislocation loops in these metals is

LOOPS

AND

SLIP,

TWINSIXG

869

represented in Fig. 10. In obtaining this figure, the following pararnete~(36) have been used for Ag : ,U= 3.38 x loll dynes/cm 2, b ::= (a/6)(112) = 1.67 x 1O-8 cm and y = 17 ergs/cm%; for Au : p =: 3.1 x 10L1 dynes/cmz, b == (a/6)(112) = 1.67 x lo+ cm, and y I= 55 ergs/cm2. The curves in Fig. 10 are obtained by first determining the energy-stress curves for D loops in both metals. The corresponding energies for A and B loops can be obtained by simply translating D curves by y/b stress units. The modnlus of Ag is slightly greater than Au and this has the effect of shifting D curves of Ag to larger stresses. The reason for this shift is due to a larger contribution from the self energy to the t,otal energy in equation (I). For clarity, botSh D and B curves are not represented in Fig. 10. A more important point to note in Fig. 10, however, is the larger difference in the activation energies between A and A + B loops in Ag compared to that in Au. The reason for this larger difference is again due to the difference in st,acking fault energies of the two metals. The results in Fig. 10 are analogous to those of Fig. 3. With a decrease in temperat#ure,it becomes increasingly more difficult to nucleate A + B loops in comparison to A loops and this difficulty is even more pronounced in the low stacking fault energy metal, Ag. Hence if the imposed st,rain rate is to be met, then the metal has to deform by the expansion of A loops that propagate stacking faults. The nucleation of the A loop can be considered as the basic step for the nucleation of a twin, since for small thickness, a twin reduces to a stacking fault. The thickening of the twin can be treated as the nucIeatGon of simple (u/6)(1 12) loops that do not have any stacking faults and this could occur without any further activation in the presence of the high internal stresses.‘“i’) Hence, in analogy with the ordered alloys, the twinning stress should be clearly related to the st,ress necessary to propagate the stacking fault. Indeed, in many alloys

FIG. 10. Vtlrietion of activation energy with shear stress for different dislocation loops in f.c.c. metals.

870

ACTA

METALLURGICA,

it was observed that the twinning stress increases with an increase in stacking fault energy.(33) The observed(35) increase of twinning stress in Ag with the addition of Au could also be related to the increase in stacking fault energy of Ag. The increased twin formation at low temperatures in many metals and alloys has generally been attributed t,o the decrease ofstacking fault energy with a decrease in temperature. This hypothesis however can be ruled out since many metals and allays(4) show the twinning stress to be nearly independent of temperature. The inoreased tendency to form twins at low temperatures can easily be accounted for on the basis of Fig. 10. In particular, with a decrease in temperature, there is less thermal energy available and hence the nucleation of A -i_ B loops becomes increasingly more difficult in comparison to t,he nucleation of A loops. Kence the imposed deformation rate at these temperatures could only be maintained by the propagation of A loops which lead to twinning. Similarly, an increase in strain rat8e also favors the formation of twins. Smitl~,(3B)for example, observed twinning in Cu, Fe and Ctu-Zn at room temperatures at high strain rates, while these materials twin at)normal strain rates only at sub-zero temperatures. This may be attributed to the fact that additional activation is necessary for A + R loops while A loops can readily propagate after their nucleation, in order to maintain the high strain rates. Thus, the increase of strain rate has the effect of shifting the transition temperature to high temperatures. In analogy, t,he transition temperatures even in ordered alloys could also be increased by deforming the alloys at higher strain rates. Such experiments, however, have not been reported thus far. Another irnport,ant experimental observation in support of the present analysis is the effect of composition on the transition temperature from slip to twinning. For example, twinning first occurred at 280 K in Ag whereas it occurred below 100 K in Au.(aa) This is in accordance with the previous discussion that the transition occurs at lower temperatures for higher stacking fault energy metals. Similarly, the increase in transition temperature(QQ) with the addition of IT to Cb could also be attributed t.o the decrease of stacking fault energy. It is assumed here t,hat nucleation of twins in b.c.c. metals and their alloys can also be treated in much the same way as in f.c.c. metals and their alloys. The stability of a twin lamella has been analyzed earlier assuming that twins can be represented in much tbo same way as a Griffith crack.(**) In the high stress regions when the twin lamella contains only one

VOL.

22,

1074

partial dislocation the previous and the present calculations are identical. However at low stresses, the twin lamella contains more than one dislocation on parallel planes and the energy-stress curves represented earlier(4Q)are similar t,o R, curves in Figs. 5 and 8. They represent the stability of whole lamella ratlzer than the actual act,ivation energy for all the dislocation loops, comprising the t’win lamellae. In comparison to the previous analysis, the present analysis stresses the importance of the difference in activat,ion energies of A and A + B loops and of the kinetics of deformation that determines which of the two loops nucleated contribute to deformation. The present analysis also assumes that nucleation of twins occurs at regions of stress concentration. Indeed, many experimental observations show t,hat#twins are preferentially nucleated at stress concentration regions(41*42)such as grain boundaries, inclusions, deformation bands, etc.

If it is recognized that stacking faults in f.c.c. metals are simply ribbons of an h.c.p. phase, the nucleation of an h.c.p. phase in a f.c.c. metal can be treated in the same way as the nucleation of partial dislocations. Hence, the concepts that are used in the nucleat,ion of t8wins can be readily extended to allotropic phase transformations. The presence of innumerable stacking faults in the initial stages of t,he transformation of p (f.c.c.) La to M. (43) hexagonal La or Co (f.c.c.) to Co (h.c.p.)(44**5)gives added support to the concept that t,he kinetics of phase trailsformation could be related to the kinetics of dislocation nucleation and their propagation. Existence of partial dislocations bounded by stacking faults was also observed during stress induced y (f.c.c.) to E (1l.c.p.) transformations(46-Ps) in 18-8 stainless steel. In fact, it was observed that the nucleation of F phase in 18-8 stainless steel occurs preferentially at regions of st’ress conoentration(4g) such as grain boundaries or inclusions, etc. The free energy of activat,ion for all shear transformations can be represented in general by

A(W) = AG, + E, + E, - El + AC, (8) where AG is the volume free energy change associated with the phase t.ransformation, Es, E, are the sums of the self energies and interaction energies, respectively, of the dislocation loops comprising t’he interface. and AGs is the surface energy. For a crack or a twin, AG, is zero and AG, reduces to the surface energy and twin boundary energy, respectively. In eases involving volume free energy, the analysis can be done in much the same way as that for twins except that all of the

SADANANDA

~xn

partial dislocations associated energy already

comprising

with stacking

if AC,

DISLOC-iTION

MARCINKOWSKI:

the nucleus are to be

faults

is negative.

that

have

negative

Such calculations

have

been done for the /3 -+ ct transformation

La.‘431 The ET term in the above expression the work done by both internal stresses in the expansion prising t,he interface, volume

free energy

in

corresponds

to

as well as external

of dislocation

loops

com-

and hence it is similar to the term.

Thus it is important

realize that both external

to

as well as internal stresses

could alter the free energy of activation even in phase To the authors knowledge, this transformations. factor has not been emphasized in earlier theories of of this phase transformatiolls. t50) The significance term can be realized if it is recognized so-called

coherent

comprised originally

of

interfaces

interphase

described

dislocations

as virtual

nature of these dislocations each interphase complicat,ed

which

dislocatio~ls.(51)

of as were The

has to be determined

in order to understand

phase transformations.

that even the

could be thought

The situation

for

the kinetics of is even more

if the phase transforlnation

involves

long

range diffusion.

It is not clear at this stage how such

diffusion

the activation

affects

free energy

and tho

kinet,ics of nucleation. SUMMARY

A detailed alloys

The analysis

metals

loops in ordered

and their alloys is

was carried out in order to

for the effects of composition

of atomic relation

of dislocation

as well as in f.c.c.

presented. account

analysis

CONCLUSIONS

order on the yield

in alloys.

stress vs temperature

In particular,

with a decrease in temperature, lattice dislocations

it was shown that nucleation

becomes increasingly

and hence the deformation

and the degree

of super-

more difficult,

ordinary dislocat,ion loops.

occurs by the expansion of Similar concepts were next

exbended to the nucleation

of dislocation

loops in f.c.c.

metals and alloys, in order to account for the increased

similar to t,hose employed

for twinning,

the nucleation

could be used

of a second phase.

The comput,er time for this investigat~ion was made available through the facilities Science Center of the University

of the Computer of Maryland. The

present! research effort was supported States Atomic Energy Commission AT-(40-I)-3935

by the United

under contract No.

REFERENCES 1. G. E. LAKSO

1111 (1969).

and ZVf. J. IMARCINROWSKI, TMS-AIME245,

871

TWLKNING

KAYSER and M. J. MARCINKOWSKI,

6. M. J. MARCIKKOU’SKI and H. J. LEA&I?;, Phy. Sfntus Solidi 24, 149 (1967). 7. M. J. MARCI~KOWSBI, Treat. on &1&. s%i. Terhnol. ttt be published. der versetzungen. und 8. E. KRBNER, Kontinuuwwtheorie Eigenqxznnungen. Springer (1958). and D. E. CAMPBELL, Ordered 9. M. J. MARCINKOWSKI

Allogs-Structural Application and Physical Metallurgy, edited by B. H. KEAR, C. 1'.Srxs, N. S. STOLOFF und W. H. WESTBROOK, p. 331. Clait,or (1970). 10. R. DEWIT, Solid State Pky<9. 10, 249 (1960). 11. P. H. THORNTON and R. G. DAVIES, Met. Trans. 1, 549 119701. 12. &. E.‘Po~ov,

E. V. KOZLOV and I. V. TERESHKO, Phys. 26 (4), 129 (1968). 13. J. C. TERRY and R. E. SMALLMAN, Phil. Mug. 8, 1837 Met. Met&. (1963).

14. k. R.‘CAUSEY and E. TEGHTSOONIAN, Met. Trans. 1, 1177 (1970). 15. R. T. PASCOE and C. W. A. NEWEY, Metal Sci. J. 2, 138 16. 17. 18. 19. 20.

21. 22.

(1968). T. G. LANGDON and J. E. DORN, Phil. Mag. 17,999 D. P. POPE, Phil. Mug. 25, 917 (1972). D. P. POPE, PhiL Mag. 27, 541 (1973).

(1968).

J. B. MITCHELL, OSAMA ABO-EL-FOROW and J. E. DORN, ivet. Trans. 2, 3256 (1971). P. S. RUDMAN, Acta Met. 10, 253 (1962). M. J. MARCINKOWSKI and H. CHESSIN, Phil. Mug. 10, 837 (1964). N. S. STOLOFF and R. G. DAVIES, Acta Met. 12,473 (1964). P. MOIXE, J. P. EYMEBY and P. GROSRRAS, Phys. Statws

Solidi 46b. 177 f19711. 24. Bl. J. MARCINK~WSK~ and 1). S. MILLER, Phil. _&fag. 6, 871 (1961). 25. F. M. C. BESAG and R. E. SMALLMAN, Ordered AlloysStructwal Application and Physical MetaElurgy, edited by B. H. REAR, C. T. SIMS, N. S. STOLOFP and J. H. IVEST-

BROOK, p. 259. ClaitOr (1970). 26. F. M. C. BESAG and R. E. SMALLMAN, Acta Met. 18, 249

(1970). 27. A. LAWLEY, E. A. VIDOZ and R. W. CAHCI‘,Acta Met. Q, 287 (1961). 28. H. &EI&

and

(1956). 29. N. BROWN,

Mechanical

r\‘. BROWN,

Trans.

Properties

AIME

197,

of Intermetallic

1240 Com-

powds, edited by J. W. WESTBROOK, p. 177. Wiley (1960). 30. N, BROWX-, Phil. Msg. 4, 693 (1959). 31. N. I?. MOTT and F. R. N. NABARRO, E’roc. Roy. 800~.

Londma 52, 86 (1960). 32. J. D, ESHELLY, Proe. Roy. Sot. .London, 241, 381 (1957). 33. J. A. VENABLES, Deformation Twinning, odited by R. E.

34.

35. 36.

ACKNOWLEDGEMENTS

SLIP,

Phil. Mug. 20, 763 (1969). 3. H. I’. LEAMY, F. X. KAYSE~ and M. J. MARCINKOWSKI, Phil. Mug. 20, 769 (1969). 4. G. E. LAKSO and M. J. ~~ARCINKOW~~~I, Xet. Trans. to be published. MARCXNKOWSKI, 5. SHENG-TIFoNG,K.SADANANDA~~~X.J. Met. Trans. to be published.

tendency of these metals to deform by twinning at low t,empckratures. It was also shown that concepts, to understand

ASD

2. H. J. LEADIY, F. X.

23,

AND

LOOPS

37.

REED-HILL, J. P. HIRTH and H. C. ROGERS, Metallurgical Society Conference, Vol. 25, p. 77. Gordon & Broach (1964). P. 1%. TROR~TON, T. E. MITCHKLL and P. B. HIRSH, Phil. Mug. 7. 1349 (1962). H. &&KI ani C. s. BARRETT, Acta Met. 6, 156 (1958). J. P. HIRTH and J. LOTEE, Theory of Dislocations. MoGraw-Hill (1968). M. J. MARCINKOWSEI and K. 8. SREE HARSHA, J. a&.

Pkys. 39, 6063 (1968). 38. C. S. SMITH, TMS-AIME 212, 574 (1958). 39. D. 0. Hossox and C. J. MCHARGUE, J. Metals 15, 91 (1963). 40. SRENG-TI FONG, Iif.J. MAIKXNKOWSKI and Ii. SADANAKDA, Acta Met. 21, 799 (1973). 41. P. B. PRICE, Proc. Roy. Sot. London A280, 251 (1960). 42. D. HULL, Deformation Twinning, edited by R. E. REEDHILL, J. P. HIRTH and H. C. ROGERS, p. 121. Breach (1964).

Gordon &

872

ACTA

METALLURGICA,

43. M. J. MARCINKOWSKI and E. N. HOPKINS, TMS-AIME 242, 579 (1968). 44. T. R. ANMiTmRAnrAN and J. W. CHRISTIAN, Acta Cry&. 9, 479 (1956). 45. C. R. HOUSKA and B. L. AVERBACH, Acta Met. 9, 479 (1956). 46. H. M. OTTE, Acta Met. 5, 614 (1957). 47. B. CINA, Acta Met. 6, 748 (1958).

VOL.

22,

1974

48. J. A. VENABLES, Phil. Mug. 7, 35 (1962). 49. H. FUJITA and S. UEDA, Acta Met. 20, 759 (1972). 50. J. W. CHRISTIAE;, Theory of Transformations in Metals and Alloya. Pergamon Press, New York (1965). 51. M. J. MARCINKOWSKI, in Fundamental Aspects of Dislocation Theory, edited by J. A. SIMMONS, R. DEW~T and R. BCLLOUGH, Vol. 1, No. 513. NBS Special publication 317 (1970).