Vacancies in binary alloys

ACTA

70

WZTALLURGICA,

VOL.

shaded

14, 1966

area represents

the antiphase

dashed line represents the boundary area on the slip plane.

area, and the

of the antiphase

The small arrows shown at

certain segments of the dislocation

lines (Figs. 2(b, e)

and 3(b, g)) indicate

the sign of that portion.

ments

sign will be attracted

with opposite

other and will annihilate,

Seg-

to each

allowing the dislocations

to

change their pinning sites. This interchange of pinning sit*es will occur twice for a given cycle in which a pair of dislocations of forming

is generated.

Fig. 2(a through curvature

The steps in the process

the loops are schematically

illustrated

g) and Fig. 3(a through

h).

in The

in some of the steps has been exaggerated

(for example

Fig. 3b) to bring out the fact that. dis-

location segments are of the opposite sign, although in the operation

of such soumes,

the inner dislocation

would not have to curve very much before t,he outer dislocation

will swing around and cause the reaction

shown in Fig. 3(c) to occur. This mechanism pinning

Frank-Read tures.

which will permit

sites shows

of

of pairs of

sources is possible in some ordered strw-

Whether two Frank-Read

produce

the change

that the operation

pairs of dislocations

sources cooperate to

or operate individually

will depend on their relative positions

and antiphase

energy. The author wishes to express his appreciation

of the

many helpful discussions with Drs. A. S. Argon, M. B. Bever,

J. F. Breedis,

Robinson.

J. S. LI. Leach

Support by the National

Space Adlninistration

and

under Contract

NsG-496

the Center for Space Research, Massachusetts of Technology,

P. M.

Aeronaut,&

and with

Institute

is acknowledged. P.

CHAUDIIARI

Department of Metallurgy Massachusetts Institute of Technology FIG. 3. Schematic, re~r~~en~tion of a Frzxnk-Read sxzrce generating superdisloc&tior~s.

It is therefore Frank--Read

of interest

to consider

sources can cooperate

so that

bounded

by the pair of dislocations

The dislocations

created

normally

area which

in the following

is not

is minimized.

The mechanism

shows how this problem

can

The two possible modes of generation of superdislocations by Frank-Read sources considered correspond to two extreme cases of pinning that may occur and are shown in Figs. 2 and 3. The Burgers vector, which has the same magnitude and direction for both is indicated

in binary

alloys*

of passing

be avoided.

the dislocations,

Vacancies

by one of it pair of such have the problem

the pinning sites of the other source. proposed

* Received June 8, 1965.

to give superdis-

locations

sources would

the antiphase

if a pair of

~a~br~~ge, Massac~~etts

by the large arrow.

The

&timat8es have been made of the ~oncent,ratiol~ of vacancies

in extremely

dilute

solutions.(l)

A more

detailed analysis will be presented here which provides an approximation that is no longer limited to extremely dilute solutions. In a regular binary substitutional

solid

solution

alloy of A and B atoms, let ni be the number of vacancies coordinated with ,i B atoms and (Z - ,i) A atoms such t,hat 0 < i < Z. Thus the to&l number

LETTERS

TO

THE

il

EDITOR

ways.

Having done so the remaining A and B atoms

for regular solutions

can be arranged

at’ random

on

the remaining lattice sites in 0

FREE

TYPE ENERGY

%a 0

ways.

A ATOMS B

ATOMS

?zR

-

m\7

1’

ioni(z- i))! (nh- jI, ‘ni iJI (‘)

Thus the total number of ways of obtaining the

specified statistical

state is given by w = 2~‘~. w2 . wg.

When there are no vacancies present UIreduces to the for random usual expression JVR = (nh + n,)!/n,!n,,!

VACANCIES

;

Flu.

(nA -

9%

g2b

{nL4+

w, =

c

0

mixing of atoms.

1. Arrangements of two A atoms and six B atoms about vacancies in the b.c.c. lattice.

Therefore the free energy above that for a regular solution without vacancies is

of vacancies

is nv = i$

ni and the total number

of

lattice sites is ns = n, + nB + n, where nA and nB are the number of A and B atoms respectively in the alloy. The free energy of formation

of a vacancy

gi will

7

G = The equilibrium established

5 nigi i=o

number

the surrounding

regardless of their arrangement sites.

On this basis there exists z + 1

values of gi. When i or z -

i are either zero or unity,

gi is the true free energy of formation But for other values of i or z -

of a vacancy.

i, gi represents some

average value for the various distinguishable ments.

on

arrange-

of such vacancies

aG- 0

(5)

consequently ni -= ns

z!(n,

-

three types of arrangements (a, b, c) of A and B atoms about a vacancy shown in Fig. 1 are possible. As each arrangement

formation

of a vacancy.

has a unique energy of

Although

tional effects are readily introduced the clarity, simplicity

such configurainto the analysis,

and utility of the analysis would

suffer by the presence of an extremely

is t’hen

---

j.

ni(z -

i))zpi(n,,

i)!i!

nA + n,

-

For example, in the b.c.c. lattice for i = 2 the

indicated,

K

by

be assumed to depend only on the atoms coordinated with the vacancy

kT In $

(2 -

-

;go

n&ii ecgJkT

$ ni

(6)

2

i _ 0

Since the number of vacancies is negligible relative to the number of A and B atoms, equation accurately

(6) is quite

given by

large number

of free energies, gij, wherej identifies the configuration. Therefore, priately

the values

weighted

of gi are taken

to be appro-

mean values for i or z -

i other

where

than zero or unity. The number of’ ways the different types of vacancies can be distributed at random on the lattice sites is given by

and where N, and N, are the mole fractions of A and B atoms and Bi the ith term of the binomial expansion of equation (8). Two terms contribute

n,!

w, =

z

(ns The number of vacancies

n,)!

IP ni! i-0

gi = hi -

n, will be assumed to be so

few relative to ns that the number of divacancies negligibly small. A and B atoms will be placed random about the vacancies

w, =

1;

;i

(z -

is at

i)!i!

ni

1

Tsti

(9)

where hi is the enthalpy and sti is the thermal entropy for the formation of a vacancy of the ith kind. Although approximate been

in

A

to gi, namely

suggested,‘2-s)

methods for calculating hi have all

are

known

unreliable. Consequently, at present hi are best determined experimentally. hand the factors contributing

to

be

rather

the values of On the other

to sti might be estimated

ACTA

Frc;. 2. Configurational

METALLURGICA,

entropy term for vacancies b.c.c. binary alloys.

in

in terms of the change in vibrational entropy as suggested by the Einstein theory of specific heats. When an A a,tom is coordinated with a vacancy one degree of its vibrational frequency changes from vA to v*’ where vA’ < Ye. ‘Jherefore the change in the thermal entropy for each A atom coordinated with a vacancy is til+ln~]-kjl+ln~i=kln~

(10)

where h is Planck’s constant and k(l + In kT/hv) is the Einstein thermal entropy(Q) per degree of freedom. A similar expression applies for each B atom coordinated with a vacancy. Consequently for each vacancy of the &h kind

VOL.

14, 1966

FIG. 3. Configurational entropy

dilute solutions where 0 < N, to write

-EO =: e--.hJkTest, %

(12)

for the number of vacancies in a pure metal. For very

<
in

it is possible

where the sum can be ended at that ith term where the i + 1 term is negligibly small relative to the ith term. Terminating the series at d = 1 gives a result which closely agrees with that suggested by Lamer for extremely dilute solutions.(i) In general equation (7) can be written as % _ =

&lR7

(14)

%

where R is the gas constant and 0, is the total free energy of formation of a mole of vacancies of the ith kind, which is given by Gi’ = Hi -

In general sti is not expected to vary much as i changes and it will always be positive. When N, = 0, equation (7) reduces to the well known expression

term for vacancies

f.c.c. binary alloys.

TX,, -

TS,,

(15)

where Sti is the thermal and S,$ the configurational entropy per mole of vacancies and

s.

-2

R

Z! = ln (z -_____ __

i)!,,

+

(z - 4 ln NA + i ln NFS(16)

The vaIues of #JR appropriate for binary alloys of b.c.c. and f.c.c. metals are given in Figs. 2 and 3. Since Xci is always negative Cl is always greater than Hi - TX,, by RT times the ordinate of Figs. 2 or 3.

LETTERS

Consequently -#JR vacancy f.c.c.

only

those

configurations

is small contribute concentration.

alloys,

relatively

where

Therefore

N,

of types

for

in concentrated there

will be

i = 0 or i :

with 12A or with 12B atoms.

the vacancies

which

to the total

= Ns = l/2.

few vacancies

coordinated

effectively

TO

12.

Most of

will be of the types i = 6, i = 5 or 7,

i = 4 or 8 etc., an issue of significance

in chemical

diffusion and related phenomena. The concept

of a binding

atom and a vacancy where the vacancy

energy between a solute

is useful for very dilute solutions is coordinated

exclusively

atoms or at most with only one B atom. Figs. 2 and 3 this concept fractions

with A

As shown in

is no longer useful at mole

of solute atoms greater than about 0.01.

The determination could

be

go and gl, vacancies concentrations

established

by

g, and g,, could then be ascertained vacancy

concent(rations

solid solutions.

in the two

In general, however,

Lattice parameter and structure of silvercopper alloys rapidly quenched from liquid state* Rapid They

found

terminal

to obtain all 13

quenching

limits and produces

dissolve each other completely.(l)

et a1.(1.2).

extends

the

non-equilibrium

However,

detailed

data seem to have not yet been published.

In order to study the structure of rapidly quenched alloy and precipitation from the non-equilibrium phase, the present

authors

have

Some

Specimen

carried

microscopy

various compositions below.

by determining

out by Pol Duwez

phases or amorphous allays.(3) For silver-copper alloys they produced a new phase in which silver and copper

state.

Similarly,

of metals and alloys from liquid

that the rapid

solid solubility

determining

dilute

quenching

state was first carried

For example

in pure A and B.

73

EDITOR

and electron

of the 13 values of gi in f.c.c.

metals demands detailed investigations.

THE

out X-ray

diffraction

of silver-copper

alloys

of

rapidly quenched from the liquid

of the results alloys

obtained

were prepared

are described

by melting

silver

(99.999% pure) and copper (99.99% pure) in vacua or in nitrogen using graphite crucibles and analysed chemically.

The rotating

cylinder

method

was em-

values of gi. the effect of temperature on the vacancy concentration in 13 different compositions may be

ployed to quench the alloys from liquid state.c3) Thin

required.

powder

Peak heights in appropriate

damping

capa-

strips were obtained patterns

city experiments might also help to resolve the details. This research was conducted as part of the activities

radiation.

of the Inorganic

thinning.t4)

Lawrence California.

Materials

Radiation

Research

Laboratory

Division

of the

of the University

of

Berkeley, and was done under the auspices

of the U.S. Atomic

Energy Commission. J. E. DORN J. B. MITCHELL

Inorgunic

Materials Research Division

Lawrence Radiation

Laboratory

specimens.

X-ray

were taken with Ni-filtered

as quenched

Cu-Ku

A 50 kV electron

microscope

was used to

observe the quenched structure without any specimen Figure 1 reproduces

an X-ray diffraction

pattern of

a Ag-53.6 at. y0 Cu alloy rapidly quenched from 1480°C. The pattern

shows the existence

phase with f.c.c. structure.

of three kinds of

Each phase is designated

as GC’,8’ and y’. Lattice parameters of these phases are plotted in Fig. 2 against composition for the specimens quenched

from

temperature

value for y’ decreases

of about

continuously

1500°C.

The

with increasing

University of California

copper content, while the values for CI’and B’ are con-

Berkeley, California

stant.

1. W.

2. 3. 4. 5. 6. 7.

31. LONER,

References “Point Defects and Diffusion in Metals

and Alloys”, Vacancies and Other Point Defects in Metals and Alloys, Institute of Metals Monograph and Report Series No. 23, p. 85 (1958). H. B. HUNTINGTON and F. SEITZ, Phys. Rev. 61, 315 (1942). H. I?. HUNTIXGTON and F. SEITZ, Phys. Rev. 76, 1728 (1949). H. B. H~JSTINGTON, Phys. Rev. 91, 1092 (1953). F. G. FUMI. Phil. Maa. 46. 1007 (1955). L. TEIVORD&. Phws. Rkv. 61. 109 i1958\. K. H. E:EN~E&NN und L. T&OR& 2. Naturf. l&b,

772 (1960\. 8. A. &EC+& and E. MANN,

J. Phys.

Chem. Solids 12, 326

Il96n). \-9. A. J. DEKKER, Solid State Physics, p. 65 (1959). 10. P. G. AHEWMON, Diffusion in Solids. D. 56. McGraw-Hill “” (1963). -I-

* Received June 1, 1965.

It is evident that cc’-phase is the non-equilibrium solid solution of silver and copper formed by the rapid quenching

described

by Pol Duwez et al.

parameters of $-phase rich ranges coincide quenched

with the values

for specimens

from the solid state by Ageev and Sachst5)

and by Ageev et uZ.c6),respectively. eter curve

The lattice

in the silver rich and the copper

for y’

deviates

Fig. 3 the deviation the corresponding

from

A = (Us, deviation

The lattice paramVegard’s law. In

a)/a is compared

of Au-Cu

with

alloys, where

a;,, is the lattice parameter of y’ and a is the one calculated according to the Vegard’s law. Both curves resemble each other quite well. The lattice parameter for cc’ and /l’ are a, = 4.025 & 0.001

3.668 f 0.001 A, which correspond and Cu-9

at.%

Ag alloys,

A and up, =

to Ag-16

respectively.

at.%

Cu

Since the