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Comments on the determination of vacancy parameters in calorimetric studies
Scripta
METALLURGICA
V o l . 7, pp. 4 2 7 - 4 3 0 , 1973 P r i n t e d in the U n i t e d S t a t e s
COMMENTS ON THE DETERMINATION
* **
and
Press,
Inc,
OF VACANCY PARAMETERS
IN C A L O R I M E T R I C
G. M O Y A *
Pergamon
STUDIES
A. COUJOU * *
Laboratoire de Mbtallurgie - Universit~ de Provence, Centre de Saint J@rome - 13013 Marseille ,ffFrance) Laboratoire de Physique Electronique, Etude du m~tal - Universit~ Paul Sabatier- 31004 Toulouse (France)
(Received
February
8,
1973)
To date, few studies on quenched-in vacancies in metals are based upon calorimetric technique. However, among the various properties used to determine the vacancy parameters, stored energy measurements may be used to evaluate accurately the vacancy parameters and the concentrations of vacancies in metals. In the calorimetric method, the experimental procedure consists of rapidly cooling samples and then transferring them into a calorimeter for isothermal annealing. Thus the ener. gy released bE t during annihilation of the vacancies can be determined from the temperature rise /I/ (adiabatic calorimeter) or from the thermal flux measurement /2/ (conduction calorimeter). Such energy can be described experimentally by an Arrhenius plot in specimens quenched from various temperatures Tq : (I)
bEt = A exp(- B/Tq)
If the only quenched-in defects are monovacancies - in
concentration
Co
-
then the
energy stored bE e may be written in the form :
(2~)
~Ee
=
Co
E~v
where Co = exp(Si~k) exp(- E/~kTq). Horeover, if the energy released AEt, which is the quantity experimentally measured, represents exactly the energy stored, then : (2b)
AE e = bE t
From equations (2a) and (2b) : (3)
AE t = Co E fIv
427
C-OMMENTS
428
ON
VACANCY
Using identification monovaeancy
formation
these conditions, calculated
PARAMETERS
of equations
can be determined
the concentration
IN
CALORIMETRIC
(I) and (3)
'
STUDIES
of quenched-in
7, N o . 4
the energy E lfv and the entropy S fI v of
from the Arrhenius plot of in AE t versus vacancies
I/Tq. Under
is given by eq (3), or may be
f
from Eflv and Sty .
For gold, a detailed discussion
has recently be made /3/ of both the assumptions
herent in eqs (2a) and (2b), and this is to be published elsewhere. llke to emphasize that, in calorimetric tion (2 @.
Vol.
studies,
This may affect the determination
AE e # AEt, which is in contradiction
of the parameters
lysis, and in order to evaluate the error reported here an estimation
in-
In this note, we should to equa-
as derived from the above ana-
for these calorimetric
data, we shall give
of AE t.
I~termination of AEt
The quantities tric technique.
of heat involved during the annealing does limit in fact the calorime-
Since the bulk samples should be thin enough to give a sufficient
rate, it follows that a heavier comparatively
quenching
sample can not be used to increase the heat released.
high quench temperatures
Thus
are required to obtain a rather high concentration
of
vacancies. For the range of quench temperatures
used in the calorimetric
methods,
tion of vacancies
does not take place on fixed sinks such as dislocations,
etc : the theory
indicates that the vacancies
can generate their own sinks growing during
the annealing and leading to stable secondary defects ~hich can be observed by electron microscopy
(tetrahedra,
when annealing
dislocation
It is clear that the energy released by annihilation
is in agrement with the contribution
is L (tetrahedron
energy E~ff . The difference
of the vacancy to the energy of
is known, and
edge length or loop diameter
example), the total energy evolved on annealing can be described by the relation (4)
...)
of a vacancy in a cluster is not
the cluster under consideration. that If one asst~nes/the exact nature of secondary defects after annealing that the mean size of these aggregates
loops,
is completed.
to be taken as its own energy of formation but as an effective between these energies
the annihila-
grain boundaries,
for
:
AEt = C O E'iv - C T E(L)
where E(L) is the contribution
of one vacancy to the energy associated with aggregates
of
size L, CT
the total concentration
Further,
: C o -- C T
(5)
stored in these aggregates.
if it is assumed that essentially
led to these secondary defects investigated)
of vacancies
all of the quenched-in
(which is a reasonable
and then eq (4) may be expressed
A%
= co {E~- E(L)}
vacancies
have annea-
assumption under the atandard conditions in the form :
Vol.
7, No.
4
Therefore
COMMENTS ON VACANCY
eq(5)
instead of eq(3)
PARAMETERS
IN CALORIMETRIC
should be applied to calculate
STUDIES
the parameters
429
of for-
mation.
Suac~_~n~faulttetrahedraasseconda~defects In quenched metals, We shall consider tetraedra
several types o f secondary defects are experimentally
only metals
(SFT) are generally
of low stacking fault energy y. In these metals,
of SFT measured
E(L) = (ET)L n (ET)L represents b the nearest
with
It must be emphasized
and recent theorical locations,
n vacancies,
that E(L) is the mean contribution
isotropic,
infinite medium.
for E T in SFT /4/ depends on three terms
energy between dislocations,
In the above equation,
the contribution
(ET)L m a y be calculated
O is the shear modulus,
parameter which can be determine&
approximately
of a vacan-
from the elastic
The most complete : the energy of dis-
and the stacking fault energy.
Gb3 (ET)L = 12~(1--~) bL I 21n ~aL + ( ~2 + 21n 2 + 2 / ~ ( ~ 2
(6)
and
of a vacancy to the energy
- and does not represent
assuming a continuous,
expression
the interaction
:
n = L2/2b 2
into an SFT of edge length L. The value
theory of dislocations
its follows
atom spacing.
(ET)L of SFT - ie the energy per vacancy cy vanishing
after annealing,
the energy of such SFT containing
neighbour
- ~gl2
,,~)}1 +
~ is Poisson's
/~
~(
)2
ratio and a is a core
b y means of atomic calculations
( a ~ 0,5).
Applicationofabovea~lysistog~d For gold, computations
have been made for E T as a function
Taking M
0.3-
of L using eq (6).
:
G = 2,78.10 -3 N.m -2
¢ = 4
b = 2,88.10 -10 m
a = 0,42
Y = 45 .10-3 J.m -2 /5/
0,2-
the results
/l~/
,
are shown on FIGURE I.
0.1 FIGURE
I :VARIATION OF THE ENERGY PER VACANCY
AS A F U N C T I O N O F E D G E L E N G T H F O R S F T .
0 o
5Oo
fault
formed from v a c a n c y clusters.
If L Js the edge length
where
observed.
stacking
lOOO
430
COMMENTS
ON
VACANCY
PARAMETERS
IN C A L O R I M E T R I C
STUDIES
Vol.
7, No.
4
For the quenching temperature range used 1750 - 900 °Cl and for an annealing temperature of about 25 °C, the final edge lengths of SFT are practically constant /6/, /3/, for the specimen purity usually available (Johnson Matthey spectrographically pure gold). Thus if we identify eq. (I) with eq.( 5 ) instead of eq.(3), the determination of ~v from the slope of the experimental straight line in AE t =
f(I/Tq) is not changed. On the other hand, because
of tetrahedra, the entropy and the vacancy concentration determinations should be recalculated. For example the exact concentration C~ can be written as : CA = Co/(I - E(L)/E~) Thus since E(L) < E~v , C~ will be greater than Co. In our experiments, the SFT edge length observed after calorimeter annealing was L = 200 ~. According to the above analysis, our data suggest that the entropy ~
will be
enlarged by 50 % and the vacancy concentrations by about 20 % with regard to the values given by the classical calorimetric method. Under these conditions, the error is larger than the mean accuracy of the calorimetric measurements /3/. Therefore it should be emphasized, in summary, that an accurate determination of vacancy parameters cannot be obtained without a coordination of calorimetric and electron microscopy techniques.
REFERENCES
/1/
W. DE SORBO, Phys. Rev. 11.__77, 444 (1960)
/2/
G. MOYA, L. LAGARDE, Phys. Stat. Sol. 4 2 , 8 3 5 (1970)
/3/
G. MOYA, Th~se Marseille (1973)
/4/
T. JOSSANG, J.P. HIRTH, Phil. Mag. 1.~3 , 657 (1966)
/5/
P.J.C. GALLAGHER~'Symposium of measurement of stacking fault energy" in Metal. Trans. , 1 , 2429 (1970)
/6/
R.W. SIEGEL, Phil. Mag. 13 , 337 (1966)
METALLURGICA
V o l . 7, pp. 4 2 7 - 4 3 0 , 1973 P r i n t e d in the U n i t e d S t a t e s
COMMENTS ON THE DETERMINATION
* **
and
Press,
Inc,
OF VACANCY PARAMETERS
IN C A L O R I M E T R I C
G. M O Y A *
Pergamon
STUDIES
A. COUJOU * *
Laboratoire de Mbtallurgie - Universit~ de Provence, Centre de Saint J@rome - 13013 Marseille ,ffFrance) Laboratoire de Physique Electronique, Etude du m~tal - Universit~ Paul Sabatier- 31004 Toulouse (France)
(Received
February
8,
1973)
To date, few studies on quenched-in vacancies in metals are based upon calorimetric technique. However, among the various properties used to determine the vacancy parameters, stored energy measurements may be used to evaluate accurately the vacancy parameters and the concentrations of vacancies in metals. In the calorimetric method, the experimental procedure consists of rapidly cooling samples and then transferring them into a calorimeter for isothermal annealing. Thus the ener. gy released bE t during annihilation of the vacancies can be determined from the temperature rise /I/ (adiabatic calorimeter) or from the thermal flux measurement /2/ (conduction calorimeter). Such energy can be described experimentally by an Arrhenius plot in specimens quenched from various temperatures Tq : (I)
bEt = A exp(- B/Tq)
If the only quenched-in defects are monovacancies - in
concentration
Co
-
then the
energy stored bE e may be written in the form :
(2~)
~Ee
=
Co
E~v
where Co = exp(Si~k) exp(- E/~kTq). Horeover, if the energy released AEt, which is the quantity experimentally measured, represents exactly the energy stored, then : (2b)
AE e = bE t
From equations (2a) and (2b) : (3)
AE t = Co E fIv
427
C-OMMENTS
428
ON
VACANCY
Using identification monovaeancy
formation
these conditions, calculated
PARAMETERS
of equations
can be determined
the concentration
IN
CALORIMETRIC
(I) and (3)
'
STUDIES
of quenched-in
7, N o . 4
the energy E lfv and the entropy S fI v of
from the Arrhenius plot of in AE t versus vacancies
I/Tq. Under
is given by eq (3), or may be
f
from Eflv and Sty .
For gold, a detailed discussion
has recently be made /3/ of both the assumptions
herent in eqs (2a) and (2b), and this is to be published elsewhere. llke to emphasize that, in calorimetric tion (2 @.
Vol.
studies,
This may affect the determination
AE e # AEt, which is in contradiction
of the parameters
lysis, and in order to evaluate the error reported here an estimation
in-
In this note, we should to equa-
as derived from the above ana-
for these calorimetric
data, we shall give
of AE t.
I~termination of AEt
The quantities tric technique.
of heat involved during the annealing does limit in fact the calorime-
Since the bulk samples should be thin enough to give a sufficient
rate, it follows that a heavier comparatively
quenching
sample can not be used to increase the heat released.
high quench temperatures
Thus
are required to obtain a rather high concentration
of
vacancies. For the range of quench temperatures
used in the calorimetric
methods,
tion of vacancies
does not take place on fixed sinks such as dislocations,
etc : the theory
indicates that the vacancies
can generate their own sinks growing during
the annealing and leading to stable secondary defects ~hich can be observed by electron microscopy
(tetrahedra,
when annealing
dislocation
It is clear that the energy released by annihilation
is in agrement with the contribution
is L (tetrahedron
energy E~ff . The difference
of the vacancy to the energy of
is known, and
edge length or loop diameter
example), the total energy evolved on annealing can be described by the relation (4)
...)
of a vacancy in a cluster is not
the cluster under consideration. that If one asst~nes/the exact nature of secondary defects after annealing that the mean size of these aggregates
loops,
is completed.
to be taken as its own energy of formation but as an effective between these energies
the annihila-
grain boundaries,
for
:
AEt = C O E'iv - C T E(L)
where E(L) is the contribution
of one vacancy to the energy associated with aggregates
of
size L, CT
the total concentration
Further,
: C o -- C T
(5)
stored in these aggregates.
if it is assumed that essentially
led to these secondary defects investigated)
of vacancies
all of the quenched-in
(which is a reasonable
and then eq (4) may be expressed
A%
= co {E~- E(L)}
vacancies
have annea-
assumption under the atandard conditions in the form :
Vol.
7, No.
4
Therefore
COMMENTS ON VACANCY
eq(5)
instead of eq(3)
PARAMETERS
IN CALORIMETRIC
should be applied to calculate
STUDIES
the parameters
429
of for-
mation.
Suac~_~n~faulttetrahedraasseconda~defects In quenched metals, We shall consider tetraedra
several types o f secondary defects are experimentally
only metals
(SFT) are generally
of low stacking fault energy y. In these metals,
of SFT measured
E(L) = (ET)L n (ET)L represents b the nearest
with
It must be emphasized
and recent theorical locations,
n vacancies,
that E(L) is the mean contribution
isotropic,
infinite medium.
for E T in SFT /4/ depends on three terms
energy between dislocations,
In the above equation,
the contribution
(ET)L m a y be calculated
O is the shear modulus,
parameter which can be determine&
approximately
of a vacan-
from the elastic
The most complete : the energy of dis-
and the stacking fault energy.
Gb3 (ET)L = 12~(1--~) bL I 21n ~aL + ( ~2 + 21n 2 + 2 / ~ ( ~ 2
(6)
and
of a vacancy to the energy
- and does not represent
assuming a continuous,
expression
the interaction
:
n = L2/2b 2
into an SFT of edge length L. The value
theory of dislocations
its follows
atom spacing.
(ET)L of SFT - ie the energy per vacancy cy vanishing
after annealing,
the energy of such SFT containing
neighbour
- ~gl2
,,~)}1 +
~ is Poisson's
/~
~(
)2
ratio and a is a core
b y means of atomic calculations
( a ~ 0,5).
Applicationofabovea~lysistog~d For gold, computations
have been made for E T as a function
Taking M
0.3-
of L using eq (6).
:
G = 2,78.10 -3 N.m -2
¢ = 4
b = 2,88.10 -10 m
a = 0,42
Y = 45 .10-3 J.m -2 /5/
0,2-
the results
/l~/
,
are shown on FIGURE I.
0.1 FIGURE
I :VARIATION OF THE ENERGY PER VACANCY
AS A F U N C T I O N O F E D G E L E N G T H F O R S F T .
0 o
5Oo
fault
formed from v a c a n c y clusters.
If L Js the edge length
where
observed.
stacking
lOOO
430
COMMENTS
ON
VACANCY
PARAMETERS
IN C A L O R I M E T R I C
STUDIES
Vol.
7, No.
4
For the quenching temperature range used 1750 - 900 °Cl and for an annealing temperature of about 25 °C, the final edge lengths of SFT are practically constant /6/, /3/, for the specimen purity usually available (Johnson Matthey spectrographically pure gold). Thus if we identify eq. (I) with eq.( 5 ) instead of eq.(3), the determination of ~v from the slope of the experimental straight line in AE t =
f(I/Tq) is not changed. On the other hand, because
of tetrahedra, the entropy and the vacancy concentration determinations should be recalculated. For example the exact concentration C~ can be written as : CA = Co/(I - E(L)/E~) Thus since E(L) < E~v , C~ will be greater than Co. In our experiments, the SFT edge length observed after calorimeter annealing was L = 200 ~. According to the above analysis, our data suggest that the entropy ~
will be
enlarged by 50 % and the vacancy concentrations by about 20 % with regard to the values given by the classical calorimetric method. Under these conditions, the error is larger than the mean accuracy of the calorimetric measurements /3/. Therefore it should be emphasized, in summary, that an accurate determination of vacancy parameters cannot be obtained without a coordination of calorimetric and electron microscopy techniques.
REFERENCES
/1/
W. DE SORBO, Phys. Rev. 11.__77, 444 (1960)
/2/
G. MOYA, L. LAGARDE, Phys. Stat. Sol. 4 2 , 8 3 5 (1970)
/3/
G. MOYA, Th~se Marseille (1973)
/4/
T. JOSSANG, J.P. HIRTH, Phil. Mag. 1.~3 , 657 (1966)
/5/
P.J.C. GALLAGHER~'Symposium of measurement of stacking fault energy" in Metal. Trans. , 1 , 2429 (1970)
/6/
R.W. SIEGEL, Phil. Mag. 13 , 337 (1966)