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A note on the application of homogeneous nucleation theory to bubble nucleation in uranium metal
JOURNAL
OF NUCLEAR
31 (1969) 927-329.
MATERIALS
0 NORTH-HOLLAND
PUBLISHING
CO., AMSTERDAM
A NOTE ON THE APPLICATION OF HOMOGENEOUS NUCLEATION THEORY TO BUBBLE NUCLEATION IN URANIUM METAL J. W. Metallurgy
Division,
UKAEA,
Received
Using
relevant
values
of
observed
gas
HARRISON
25 November
irradiated
and
observed
gas
bubble
densities
Didcot,
Berke.,
1968
des bulles de gaz dans l’uranium
uranium. diffusionskoeffizienten
En utilisant
1.
les valeurs observes des coefficients de
gazeuse
en
uranium,
on
trouve
un
bon
2.
nucleation
stimmung
in metallic
zwischen
Gasblasenmengen a
Introduction
The purpose of this paper is to discuss recent measurements of gas bubble densities in thin foils of uranium irradiated to low burn-up and relate these measurements to selected experimental determination of rare gas diffusion coefficients using a simple theory of homoBecause of the internal geneous nucleation. consistency of the experimental data and theory it is suggested that the use of homogeneous nucleation theory is an adequate description of bubble
irradie.
in Mit den betreffenden
diffusion
UK
accord entre les concentrations theoriques et observes
diffusion
coefficients in uranium agreement is obtained between calculated
Harwell,
Werten
der beobachteten
im Uran find& berechneten
in bestrahltem
new nucleus, Greenwood
Gas-
man Uberein-
und
beobachteten
Uran.
derived the relation
Q= {3,!?/(SnWro2D,)}*, for the density e of gas bubbles, where j3 is the gas generation rate in gas atoms per second per atomic site, and a is the lattice parameter. Thus this simple theory of homogeneous nucleation predicts a simple relation between the density e of gas bubble nuclei and the gas diffusion coefficient D,. Since Hudson 2) has recently published measurements of gas bubble density made in low burn-up uranium at temperatures from 420 to 715 “C, it is possible to calculate D, from Greenwood’s relation and
uranium.
The observed bubble densities in low burn-up a and p uranium and the gas diffusion coeffcient
compare
the
resultant
values
of
D,
with
experimental measurements. When choosing experimental values of D, valid for comparison some care is needed. One of the earliest reviews of inert gas release from uranium metal 3, illustrated the variation of about 3 to 4 orders of magnitude in experimentally determined
Greenwood et al. 1) in discussing swelling mechanisms in uranium derived an expression for the bubble density to be expected from the homogeneous nucleation of rare gas atoms produced in fission to form gas bubbles. Under the assumption that the nucleation process ceases when gas bubbles reach a radius ro, at which gas atoms being generated in the lattice and diffusing at a rate determined by a diffusion coefficient D,, have a higher probability of encountering an existing bubble than of forming
value of D, at a given temperature. Savage a), reporting his own measurement of D, from uranium containing trace amounts of gas, pointed out that measurements previous to his fell into one or other of two classes characterised by the burn-up of the uranium on which experiments had been performed, and concluded 327
328
J.
W.
HARRISON
--I
Dg
cm*/sec
PERRAlLU)N IO-”
t
et
al
SAVAGE
using
HUDSON peak of
IO-‘*
bubble
using
HUDSON
observed radii at distributions for
r. -
258
IO_‘3
L
IO-‘0
IO -15
I0.9
I.1
I.0
1.2
I .3
x
Fig.
1.
Selected experimental
that high burn-up material gave incorrect values of D, since much of the-gas would be locked in bubbles
and not free to diffuse. His
own experiments were conducted on material irradiated from 0.0000650/o to 0.00025 at O,(, burn-up and even then it was concluded that gas-trapping was interfering with pure diffusional release in the highest burn-up specimens. Nagasaki and Kawasaki 5), following Savage, interpreted the large variations in value of D, derived from gas release measurements in a similar manner. More recently Perraillon et al. a), in studying rare gas diffusion in uranium, were careful in restricting their measurements to uranium irradiated to about 0.00006 at o/o burn-up and verified by thin-film transmission electron microscopy that no bubbles were present with diameters greater than 20 A, about
1.5
I .4
to; TK 0
values of D,.
the limit of resolution
of their technique.
It
seems then that only measurements made on very low burn-up uranium are of validity as far as the determination of, D, is concerned, and the relevant values of D, chosen for the present paper have been those reported by Savage at 500, 600 and 800 “C at a burn-up of 0.0000650/o burn-up and Perraillon et al. at temperatures of 700, 720, 750 and 810 “C. Fig. 1 contains the selected experimental values of D,. 3.
Calculation of D, from bubble density measurements
In calculating D, from observed bubble densities, the value of ra to be used is that at just when which Q reaches a maximum nucleation has ceased. Hudson 2) has measured bubble size distributions at four temperatures
BUBBLE
NUCLEATION
IN
URANIUM
between 420 and 715 “C and shows a widening
temperatures,
in the distributions
solid points
temperatures
with dose at the higher
of 620 and 715 “C. To calculate
329
METAL
then the vertical lines limited by are obtained.
Much better
agreement
is evident
D,, the values of e have been taken from the
the two sets of Dg values.
narrow low dose distributions (specimen numbers
not
4C, 4D, 6E, 6F, 7C and 7D) and values of ro from observed bubble radii at the distribution
higher than 420 “C nucleation
peaks,
high-temperature
viz. rc= 25, 50 and 60 A at 420, 620
and 715 “C respectively.
(The results at 500 “C
were ignored since no specimen burn-up rating was quoted and so @ could not be computed). The resultant values of D, obtained from these data are plotted as vertical barred lines in fig. 1, the bars indicating error limits due to Hudson’s quoted error limits on bubble counts. Comparison with the selected experimental values of D, shows a progressive disagreement as the temperature increases (the dashed line being a rough gide to the variation in the experimentally measured DB). However, if one assumes that the true value of rc should be M 25 A (the 420 “C value) that is that nucleation had in fact stopped at this bubble radius, and D, is recalculated at the higher
unreasonable
much
more
existing
since
rapidly
The assumption
at the
and
radii
between
will take place
Hudson’s
could
well after nucleation
is
temperatures
well
observed be
those
had ceased.
It
seems, then, that a simple theory of homogeneous nucleation is adequate in describing bubble formation in uranium. References
1)
G. W.
Greenwood,
Rimmer,
2) 3)
J. Nucl.
B. Hudson,
A. J. E. Foreman Mat.
J. Nucl.
Mat.
D. L. Gray, HW-62639
4) J. W.
Savage,
1 (1959)
22 (1967)
USAEC
NAA-SR-6761
and D. E.
305 121
Report
(1960)
USAEC
Report
Nihon
Genshi-
(1963)
5)
R.
Nagasaki
and
S.
Kawasaki,
ryoku Gakkai Shi 6 ( 1964) 646 ; available in English translation
9
as AERE-Trans
B. Perraillon, Rev.
V. Levy
Mkt. 63 (1966)
1034 (1965)
and Y.
227
Adda,
MBm. Sci.
OF NUCLEAR
31 (1969) 927-329.
MATERIALS
0 NORTH-HOLLAND
PUBLISHING
CO., AMSTERDAM
A NOTE ON THE APPLICATION OF HOMOGENEOUS NUCLEATION THEORY TO BUBBLE NUCLEATION IN URANIUM METAL J. W. Metallurgy
Division,
UKAEA,
Received
Using
relevant
values
of
observed
gas
HARRISON
25 November
irradiated
and
observed
gas
bubble
densities
Didcot,
Berke.,
1968
des bulles de gaz dans l’uranium
uranium. diffusionskoeffizienten
En utilisant
1.
les valeurs observes des coefficients de
gazeuse
en
uranium,
on
trouve
un
bon
2.
nucleation
stimmung
in metallic
zwischen
Gasblasenmengen a
Introduction
The purpose of this paper is to discuss recent measurements of gas bubble densities in thin foils of uranium irradiated to low burn-up and relate these measurements to selected experimental determination of rare gas diffusion coefficients using a simple theory of homoBecause of the internal geneous nucleation. consistency of the experimental data and theory it is suggested that the use of homogeneous nucleation theory is an adequate description of bubble
irradie.
in Mit den betreffenden
diffusion
UK
accord entre les concentrations theoriques et observes
diffusion
coefficients in uranium agreement is obtained between calculated
Harwell,
Werten
der beobachteten
im Uran find& berechneten
in bestrahltem
new nucleus, Greenwood
Gas-
man Uberein-
und
beobachteten
Uran.
derived the relation
Q= {3,!?/(SnWro2D,)}*, for the density e of gas bubbles, where j3 is the gas generation rate in gas atoms per second per atomic site, and a is the lattice parameter. Thus this simple theory of homogeneous nucleation predicts a simple relation between the density e of gas bubble nuclei and the gas diffusion coefficient D,. Since Hudson 2) has recently published measurements of gas bubble density made in low burn-up uranium at temperatures from 420 to 715 “C, it is possible to calculate D, from Greenwood’s relation and
uranium.
The observed bubble densities in low burn-up a and p uranium and the gas diffusion coeffcient
compare
the
resultant
values
of
D,
with
experimental measurements. When choosing experimental values of D, valid for comparison some care is needed. One of the earliest reviews of inert gas release from uranium metal 3, illustrated the variation of about 3 to 4 orders of magnitude in experimentally determined
Greenwood et al. 1) in discussing swelling mechanisms in uranium derived an expression for the bubble density to be expected from the homogeneous nucleation of rare gas atoms produced in fission to form gas bubbles. Under the assumption that the nucleation process ceases when gas bubbles reach a radius ro, at which gas atoms being generated in the lattice and diffusing at a rate determined by a diffusion coefficient D,, have a higher probability of encountering an existing bubble than of forming
value of D, at a given temperature. Savage a), reporting his own measurement of D, from uranium containing trace amounts of gas, pointed out that measurements previous to his fell into one or other of two classes characterised by the burn-up of the uranium on which experiments had been performed, and concluded 327
328
J.
W.
HARRISON
--I
Dg
cm*/sec
PERRAlLU)N IO-”
t
et
al
SAVAGE
using
HUDSON peak of
IO-‘*
bubble
using
HUDSON
observed radii at distributions for
r. -
258
IO_‘3
L
IO-‘0
IO -15
I0.9
I.1
I.0
1.2
I .3
x
Fig.
1.
Selected experimental
that high burn-up material gave incorrect values of D, since much of the-gas would be locked in bubbles
and not free to diffuse. His
own experiments were conducted on material irradiated from 0.0000650/o to 0.00025 at O,(, burn-up and even then it was concluded that gas-trapping was interfering with pure diffusional release in the highest burn-up specimens. Nagasaki and Kawasaki 5), following Savage, interpreted the large variations in value of D, derived from gas release measurements in a similar manner. More recently Perraillon et al. a), in studying rare gas diffusion in uranium, were careful in restricting their measurements to uranium irradiated to about 0.00006 at o/o burn-up and verified by thin-film transmission electron microscopy that no bubbles were present with diameters greater than 20 A, about
1.5
I .4
to; TK 0
values of D,.
the limit of resolution
of their technique.
It
seems then that only measurements made on very low burn-up uranium are of validity as far as the determination of, D, is concerned, and the relevant values of D, chosen for the present paper have been those reported by Savage at 500, 600 and 800 “C at a burn-up of 0.0000650/o burn-up and Perraillon et al. at temperatures of 700, 720, 750 and 810 “C. Fig. 1 contains the selected experimental values of D,. 3.
Calculation of D, from bubble density measurements
In calculating D, from observed bubble densities, the value of ra to be used is that at just when which Q reaches a maximum nucleation has ceased. Hudson 2) has measured bubble size distributions at four temperatures
BUBBLE
NUCLEATION
IN
URANIUM
between 420 and 715 “C and shows a widening
temperatures,
in the distributions
solid points
temperatures
with dose at the higher
of 620 and 715 “C. To calculate
329
METAL
then the vertical lines limited by are obtained.
Much better
agreement
is evident
D,, the values of e have been taken from the
the two sets of Dg values.
narrow low dose distributions (specimen numbers
not
4C, 4D, 6E, 6F, 7C and 7D) and values of ro from observed bubble radii at the distribution
higher than 420 “C nucleation
peaks,
high-temperature
viz. rc= 25, 50 and 60 A at 420, 620
and 715 “C respectively.
(The results at 500 “C
were ignored since no specimen burn-up rating was quoted and so @ could not be computed). The resultant values of D, obtained from these data are plotted as vertical barred lines in fig. 1, the bars indicating error limits due to Hudson’s quoted error limits on bubble counts. Comparison with the selected experimental values of D, shows a progressive disagreement as the temperature increases (the dashed line being a rough gide to the variation in the experimentally measured DB). However, if one assumes that the true value of rc should be M 25 A (the 420 “C value) that is that nucleation had in fact stopped at this bubble radius, and D, is recalculated at the higher
unreasonable
much
more
existing
since
rapidly
The assumption
at the
and
radii
between
will take place
Hudson’s
could
well after nucleation
is
temperatures
well
observed be
those
had ceased.
It
seems, then, that a simple theory of homogeneous nucleation is adequate in describing bubble formation in uranium. References
1)
G. W.
Greenwood,
Rimmer,
2) 3)
J. Nucl.
B. Hudson,
A. J. E. Foreman Mat.
J. Nucl.
Mat.
D. L. Gray, HW-62639
4) J. W.
Savage,
1 (1959)
22 (1967)
USAEC
NAA-SR-6761
and D. E.
305 121
Report
(1960)
USAEC
Report
Nihon
Genshi-
(1963)
5)
R.
Nagasaki
and
S.
Kawasaki,
ryoku Gakkai Shi 6 ( 1964) 646 ; available in English translation
9
as AERE-Trans
B. Perraillon, Rev.
V. Levy
Mkt. 63 (1966)
1034 (1965)
and Y.
227
Adda,
MBm. Sci.