- Email: [email protected]
On the in-pile nucleation and growth of grain-boundary bubbles
320
of Nuclear Materials II8 (1983) 320-324 North-Holland Publishing Company
Journal
ON THE IN-PILE NUCLEATION
AND GROWTH
OF GRIN-BOUNDARY
BUBBLES
M.H. WOOD and K.L. KEAR Theoremal
Received
Physics Division,
14 February
AERE
1983; accepted
Hanveil,
Didcot,
11 April
Oxon.,
0x11
ORA,
UK
1983
The paper considers the nucleation and growth of gas bubbles on the grain boundaries of structural materials during irradiation and under an applied stress. A simple expression is derived for the gas-driven time-to-rupture that is inversely proportional to the product of the square of the stress and the square root of an effective diffusion coefficient for gas atoms on the grain boundaries. A comparison between calculated and experimentally measured bubble spacings and times-to-rupture demonstrates the overall validity of our model.
1. In~~uction
2. Gas bubble growth
In a recent paper Bullough, I-Tarries and Hayns [l] considered the effect of stress on the growth of gas bubbles in stainless steel during irradiation. They argued that the growth of gas bubbles on the grain boundaries orthogonal to an applied stress is driven by the transmutation gas produced during irradiation and that the residual interstitials form suitably orientated interstitial platelets that extend the grains by the “jacking mechanism” of Harris [2]. This gives rise to irradiation creep and, eventually, to the rupture of the material along the grain boundaries. Using the ideal gas law to simplify their analysis, they deduced a time-to-rupture inversely proportional to the square of the stress. In this paper we extend the work of Bullough et al. [l] to predict the density of bubbles nucleated on the grain boundaries. This results in a simple expression for the time-to-rupture. It shows the same dependence on stress as the previous model and includes an exponential dependence on irradiation temperature that originates from the bubble nucleation model. in section 2 of the paper we review the work of Bullough et al. (11 on the gas-driven growth of bubbles on the grain boundaries, and in section 3 we present our model for the nucleation of the bubbles, In section 4, we compare predictions from our model with experimentally observed bubble spacings and times-to-rupture. Finally, we draw together the conclusions of the paper in section
We consider a spherical bubble in a stress-free body lying within a grain. Assuming the ideal gas law, for simplicity, and equilibrium between the pressure of the gas contained in the bubble and the surface tension restraining force, the number of gas atoms. n,. in a bubble of radius r, is
J.
0022-3 115/83/0000-0000/$03.00
no = %rryri/3k
BT,
(1)
where y is the surface energy, k, is Boftzmann’s wnstant and T the absolute temperature. Then the critical hydrostatic tensile stress, CT,“,required for the bubble to grow in an unstable fashion is
where rc = (3)“‘ra
= (P~~k~T/S~~)“2
(3)
is the critical radius when
is a maximum.
Substituting “*
=p
for rC in eq. (2) gives
4(3)“*y 9r,
’
This is the so-called Hyam-Sumner [3] critical hydrostatic stress required to grow an equilibrium gas bubble of radius r,, and gas content n,. The corresponding
0 1983 North-Holland
M. H. Wood, K. L. Kear / Nucleation and growth of grain
expressions for a bubble on a grain boundary lying orthogonal to a uniaxial tensile stress, a,, are obtained [l] by replacing u,” by uC in eq. (5). This is a static result and shows the stress at which a gas bubble containing a fixed number of gas atoms will grow unstably. The equivalent result [l] for a stress, e,“, imposed during irradiation, when the gas content of the bubble increases in time, is that the bubble will grow unstably when its gas content, n(t), reaches no. For an area concentration, cb, of bubbles lying on the grain boundaries orthogonal to an applied tensile stress, it can easily be deduced from eq. (5) that the time-to-rupture, t,, is given by
ta=
[
128 ?rry3c, Kk,TK
1 1 2’
where K is the gas generation rate boundary, assuming no trapping structure within the grains, and equation can only be solved for centration, cb. is known. In the derive a simple model for the boundary bubbles.
per unit area of grain of gas at the microu is the stress. This ta if the bubble confollowing section we nucleation of grain-
-boundarybubbles
321
nuclei being the only sinks, k* is given by [6] k*=8(1
-r&‘s2)2{s2[(r&‘s2-
1)(3-ri/s*)
+4 ln(s/r,)]}-‘.
S=(7Tcb)-“2.
(9)
Now, the number of jumps, n, taken by a gas atom in the boundary to meet another (mobile) gas atom is given by: n =
(2za3c)~’
= Dk2/(2ra3K),
(10)
where a3 is the atomic volume, and z is the number of sites explored per jump. The corresponding average time, r, to meet another gas atom is given in terms of the gas-atom jump frequency, v = a2S/4D, by r = n/v = k*6/8raK.
(11)
the average time for a gas atom to diffuse existing bubble nucleus is given by Now,
rb = S/Dk*.
to an
(12)
Equating r and rt,, as given by eqs. (11) and (12) respectively, we obtain the approximate nucleation density from the solution of
3. Gas bubble nucleation
k* = (8raK/D)“‘,
We assume, as we have for bubble growth, that during the bubble nucleation phase there are sufficient vacancies available to ensure that the nucleation rate during irradiation is controlled by the generation rate and mobility of the gas atoms. Moreover, we assume that grain boundary bubbles nucleate homogeneously on those grain faces suitably orientated with respect to the applied stress. Our approach for the nucleation of grain-boundary bubbles in steels is similar to those followed previously to describe the nucleation of fission-gas bubbles within fuel grains [4,5]. We postulate that bubble nuclei of radius rb are produced until such time that a gas atom is more likely to be captured by an existing nucleus than to meet another gas atom and form a new nucleus. We employ homogeneous rate theory (61 to obtain the mean volume concentration, c, of gas atoms on a grain boundary. In steady state, when the production of new nuclei drops off, we find c = K/Dk=,
(8)
where
(7)
where D is the product of the diffusion coefficient for gas atoms on the grain boundary and the grain boundary width, 6, and k* is the total strength of the grain boundary sinks for migrating gas atoms. For bubble
with k* given by eq. (8). An approximate cb
(13)
result for cb is
-
Eq. (6) then yields for the time-to-rupture tR-
[g$(~)“*]$.
(15)
For the eq. (15) to hold, it is necessary that ta z+ z,,, where t, is the time to reach the end of the nucleation phase. For a stress of 100 MPa and the data of table 1, we estimate t,/f, - 106. Hence, the condition t, z+ t, is easily satisfied. Implicit in the use of eqs. (14) and (15) to calculate bubble spacings on the grain boundaries and times-to-rupture is the assumption that, after time t “, nucleation virtually ceases and gas atoms go only into existing bubbles. Although this assumption is not strictly valid, the results presented in the following section indicate that it is a reasonable one for the present application. Finally, the presence of any second-phase precipitates on the grain boundaries could markedly affect the gas atom mobility, as well as provide preferred sights for the nucleation of gas bubbles. We take account of
M. H. Wood, K. L. Kear / Nucleatron and growth of gram
322 Table I Parameters
used in calculations k, = 1.38x 1O-23 J/K r=4 u~=O.~XIO~~~~~ y = 2 J/m* K = 5 x IO” atoms/m* s D, = 2 x lo-l2 exp( - 2.78 x lo- ‘9/.k,r) E =1,87~1O-‘~J c,“= 2X lOI4 mm2
Boltzmann’s constant Number of sites explored per jump Atomic volume Surface energy [ 1l] Gas atom production rate per unit area of grain boundary Grain boundary self-diffusion coefficient [ 121 Trap binding energy Trap concentration per unit area of grain boundary [ 131
the effect of such precipitates by considering D, in eqs. (14) and (15), as an effective diffusion coefficient for gas atoms on the grain boundaries. This is defined by
171 D = D,/(
1 + D,kf~,/‘8),
(16)
where D, is the product of the grain-boundary self-diffusion coefficient and the grain-boundary width 6, k: is the strength of the precipitate traps for migrating gas atoms which we take to be given approximately by kf = mt with c, the concentration of precipitate traps per unit area of grain boundary, and T, is the average time a gas atom spends at a trap. This time may be written 7, = exp( -%/k,T)/v,
(17)
where Y is the gas atom jump frequency given above, and E, is the binding energy of a gas atom to a precipitate trap. The data in table 1 are used in all the calculations discussed in the next section: this data set includes a trap concentration, c,, of 2 x 10’4/m2 and a binding energy, E,, of - 1.2 eV.
Table 2 Comparison Experimental
-boundan; bubbles
of calculated
and experimental
grain boundary
bubble
4. Comparison with experiment We have used the simple model derived above for the concentration of grain boundary bubbles, eq. (14), and the time-to-rupture, eq. (15), to obtain results for stainless steel. In this section we compare the calculated results with those from a variety of experiments [&lo]. Harries et al. [8] measured the spacings of grainboundary bubbles for steels containing 2, 43 and 97 ppm boron after irradiation at 650°C in a primarily thermal neutron flux. Using the expression (14) we calculated a bubble spacing of - 0.6 pm for the steel with a low boron content, which may be compared with the experimental observation of 0.53 pm. The ratios of the bubble spacings in the higher boron content steels to that in the 2 ppm boron steel have been calculated, and are compared with the observations by Harries et al. [8] in table 2. The agreement is very good and would seem to confirm the validity of our gas-bubble nucleation model. The second set of comparisons between experiment and theory relate to a number of in-pile creep rupture experiments [9,10] performed on steel between 650 and 850°C in a mainly thermal neutron flux. In fig. 1, we show calculated and experimental results for time-to-
spacings Calculated
’
Sample no., n
Boron content,
C,
Bubble spacing,
(ppm)
(rm)
2 3
2 43 97
0.53 0.20 0.16
a Ref. [8]. b Calculated (S,/S,)
= (C,/C,)“.25.
1
m3/s
S,
Relative spacing, S, /S,
Relative spacing b, S, /S,
1
1 0.46 0.37
0.38 0.30
hi. H. Wood, K. L. Kear / Nucleation and growth of grain
I
!
from
ref
l
75O’C
from
ref.9
A
850%
from
ref
9
0
720%
from
ref
10
x 65O’C
-boundarybubbles
323
I
9 20125
INb
1 1970
steel
steel
.
I
I
I
10
102
IO3
Time
to
rupture
( hours
4
IO'
1
Fig. 1. Comparison of experimental and calculated time to rupture as a function of stress for steel.
rupture as a function of stress over the entire temperature range of the experiments. The calculated stress dependence is, from eq. (15) um2. The experiments indicate a slightly higher stress exponent over most of the stress range, that is at stresses ,< 150 MPa. The calculated u- 2 dependence arises directly from our use of the ideal gas law to simplify the analysis. Numerical calculations using Van der Waals gas law give an exponent of - 2.2. At stresses > 150 MPa the experiments hint at a very rapid drop-off in the time-to-rupture. This is probably due to a changeover in mechanism from gas-driven rupture, the model described above, to creep-driven rupture characterised by a high stress exponent [lo]. The calculated temperature dependence of the timeto-rupture originates from the nucleation model derived in section 3, and reproduces the dependence shown by the experiments with some success. Overall, the model gives times-to-rupture that agree with the experimental results within a factor of - 2.
5. Conclusions We have extended a previous [l] model of the gasdriven growth of grain boundary cavities to include a simple model to describe the gas-driven nucleation of such grain boundary bubbles. Whereas the previous model gave the stress but not the temperature dependence of the time-to-rupture, the new model gives both dependencies. Results from the model are found to be in good agreement with experimental measurements of the spacings of bubbles on the grain boundaries of irradiated steels containing different quantities of boron. We have also compared model predictions with measured timesto-rupture during in-pile creep rupture experiments. The ue2 stress dependence from the model seems to slightly underestimate the experimental dependence. However, this model result depends on the gas law used: for example, whereas u -’ is found using the ideal gas law, the Van der Waals gas law gives u-2,2. The measured
324
M. H. Wood, K. L. Kear / Nucleatron and growth of grain boundary bubbles
temperature dependence is also well reproduced model. The overall level of agreement between ment and theory is within a factor of - 2.
by the experi-
References
[II R. Bullough,
D.R. Harries and M.R. Hayns, J. Nucl. Mater. 88 (1980) 312. 121J.E. Harris, Proc. Conf. Vacancy ‘76, Bristol, Eds. R.E. Smallman and J.E. Harris, Metals Sot. (1976) 170. I31 E.D. Hyam and G. Sumner, Proc. Symp. on Radiation Damage in Solids, Venice, IAEA Vienna 1 (1962) 323. A.J.E. Foreman and D.E. Rimmer, J. (41 G.W. Greenwood, Nucl. Mater. 4 (1959) 305. 151A.D. Whapham, Phil. Mag. 23 (1971) 987.
[6] J.R. Matthews and M.H. Wood, J. Nucl. Mater. 91 (1980) 241. [7] A.C. Damask and G.J. Dienes, Phys. Rev. 120 (1960) 99. 181 D.R. Harries, A.C. Roberts, G.T. Rogers, J.D.H. Hughes and M.A.P. Dewey, Radiation Damage in Reactor Materials, IAEA-SM-120/G-5 Vienna II (1969) 357. [91 J.T. Venard and J.R. Weir, ASTM Special Technical Publication 380 (1965) 269. C. Wassilew, L. Schafer and K. Anderko, Proc. Reaktortagung, Hannover, Deutsches Atomforum E.V., ( 1978) 609. L.E. Murr, Interfacial Phenomena in Metals and Alloys (Addison-Wesley, London, 1975). A.F. Smith, CEGB Report No. RD/B/N2270 (1972). H. Trinkaus and H. Ullmaier, Phil. Mag. 39 (1979) 563.
of Nuclear Materials II8 (1983) 320-324 North-Holland Publishing Company
Journal
ON THE IN-PILE NUCLEATION
AND GROWTH
OF GRIN-BOUNDARY
BUBBLES
M.H. WOOD and K.L. KEAR Theoremal
Received
Physics Division,
14 February
AERE
1983; accepted
Hanveil,
Didcot,
11 April
Oxon.,
0x11
ORA,
UK
1983
The paper considers the nucleation and growth of gas bubbles on the grain boundaries of structural materials during irradiation and under an applied stress. A simple expression is derived for the gas-driven time-to-rupture that is inversely proportional to the product of the square of the stress and the square root of an effective diffusion coefficient for gas atoms on the grain boundaries. A comparison between calculated and experimentally measured bubble spacings and times-to-rupture demonstrates the overall validity of our model.
1. In~~uction
2. Gas bubble growth
In a recent paper Bullough, I-Tarries and Hayns [l] considered the effect of stress on the growth of gas bubbles in stainless steel during irradiation. They argued that the growth of gas bubbles on the grain boundaries orthogonal to an applied stress is driven by the transmutation gas produced during irradiation and that the residual interstitials form suitably orientated interstitial platelets that extend the grains by the “jacking mechanism” of Harris [2]. This gives rise to irradiation creep and, eventually, to the rupture of the material along the grain boundaries. Using the ideal gas law to simplify their analysis, they deduced a time-to-rupture inversely proportional to the square of the stress. In this paper we extend the work of Bullough et al. [l] to predict the density of bubbles nucleated on the grain boundaries. This results in a simple expression for the time-to-rupture. It shows the same dependence on stress as the previous model and includes an exponential dependence on irradiation temperature that originates from the bubble nucleation model. in section 2 of the paper we review the work of Bullough et al. (11 on the gas-driven growth of bubbles on the grain boundaries, and in section 3 we present our model for the nucleation of the bubbles, In section 4, we compare predictions from our model with experimentally observed bubble spacings and times-to-rupture. Finally, we draw together the conclusions of the paper in section
We consider a spherical bubble in a stress-free body lying within a grain. Assuming the ideal gas law, for simplicity, and equilibrium between the pressure of the gas contained in the bubble and the surface tension restraining force, the number of gas atoms. n,. in a bubble of radius r, is
J.
0022-3 115/83/0000-0000/$03.00
no = %rryri/3k
BT,
(1)
where y is the surface energy, k, is Boftzmann’s wnstant and T the absolute temperature. Then the critical hydrostatic tensile stress, CT,“,required for the bubble to grow in an unstable fashion is
where rc = (3)“‘ra
= (P~~k~T/S~~)“2
(3)
is the critical radius when
is a maximum.
Substituting “*
=p
for rC in eq. (2) gives
4(3)“*y 9r,
’
This is the so-called Hyam-Sumner [3] critical hydrostatic stress required to grow an equilibrium gas bubble of radius r,, and gas content n,. The corresponding
0 1983 North-Holland
M. H. Wood, K. L. Kear / Nucleation and growth of grain
expressions for a bubble on a grain boundary lying orthogonal to a uniaxial tensile stress, a,, are obtained [l] by replacing u,” by uC in eq. (5). This is a static result and shows the stress at which a gas bubble containing a fixed number of gas atoms will grow unstably. The equivalent result [l] for a stress, e,“, imposed during irradiation, when the gas content of the bubble increases in time, is that the bubble will grow unstably when its gas content, n(t), reaches no. For an area concentration, cb, of bubbles lying on the grain boundaries orthogonal to an applied tensile stress, it can easily be deduced from eq. (5) that the time-to-rupture, t,, is given by
ta=
[
128 ?rry3c, Kk,TK
1 1 2’
where K is the gas generation rate boundary, assuming no trapping structure within the grains, and equation can only be solved for centration, cb. is known. In the derive a simple model for the boundary bubbles.
per unit area of grain of gas at the microu is the stress. This ta if the bubble confollowing section we nucleation of grain-
-boundarybubbles
321
nuclei being the only sinks, k* is given by [6] k*=8(1
-r&‘s2)2{s2[(r&‘s2-
1)(3-ri/s*)
+4 ln(s/r,)]}-‘.
S=(7Tcb)-“2.
(9)
Now, the number of jumps, n, taken by a gas atom in the boundary to meet another (mobile) gas atom is given by: n =
(2za3c)~’
= Dk2/(2ra3K),
(10)
where a3 is the atomic volume, and z is the number of sites explored per jump. The corresponding average time, r, to meet another gas atom is given in terms of the gas-atom jump frequency, v = a2S/4D, by r = n/v = k*6/8raK.
(11)
the average time for a gas atom to diffuse existing bubble nucleus is given by Now,
rb = S/Dk*.
to an
(12)
Equating r and rt,, as given by eqs. (11) and (12) respectively, we obtain the approximate nucleation density from the solution of
3. Gas bubble nucleation
k* = (8raK/D)“‘,
We assume, as we have for bubble growth, that during the bubble nucleation phase there are sufficient vacancies available to ensure that the nucleation rate during irradiation is controlled by the generation rate and mobility of the gas atoms. Moreover, we assume that grain boundary bubbles nucleate homogeneously on those grain faces suitably orientated with respect to the applied stress. Our approach for the nucleation of grain-boundary bubbles in steels is similar to those followed previously to describe the nucleation of fission-gas bubbles within fuel grains [4,5]. We postulate that bubble nuclei of radius rb are produced until such time that a gas atom is more likely to be captured by an existing nucleus than to meet another gas atom and form a new nucleus. We employ homogeneous rate theory (61 to obtain the mean volume concentration, c, of gas atoms on a grain boundary. In steady state, when the production of new nuclei drops off, we find c = K/Dk=,
(8)
where
(7)
where D is the product of the diffusion coefficient for gas atoms on the grain boundary and the grain boundary width, 6, and k* is the total strength of the grain boundary sinks for migrating gas atoms. For bubble
with k* given by eq. (8). An approximate cb
(13)
result for cb is
-
Eq. (6) then yields for the time-to-rupture tR-
[g$(~)“*]$.
(15)
For the eq. (15) to hold, it is necessary that ta z+ z,,, where t, is the time to reach the end of the nucleation phase. For a stress of 100 MPa and the data of table 1, we estimate t,/f, - 106. Hence, the condition t, z+ t, is easily satisfied. Implicit in the use of eqs. (14) and (15) to calculate bubble spacings on the grain boundaries and times-to-rupture is the assumption that, after time t “, nucleation virtually ceases and gas atoms go only into existing bubbles. Although this assumption is not strictly valid, the results presented in the following section indicate that it is a reasonable one for the present application. Finally, the presence of any second-phase precipitates on the grain boundaries could markedly affect the gas atom mobility, as well as provide preferred sights for the nucleation of gas bubbles. We take account of
M. H. Wood, K. L. Kear / Nucleatron and growth of gram
322 Table I Parameters
used in calculations k, = 1.38x 1O-23 J/K r=4 u~=O.~XIO~~~~~ y = 2 J/m* K = 5 x IO” atoms/m* s D, = 2 x lo-l2 exp( - 2.78 x lo- ‘9/.k,r) E =1,87~1O-‘~J c,“= 2X lOI4 mm2
Boltzmann’s constant Number of sites explored per jump Atomic volume Surface energy [ 1l] Gas atom production rate per unit area of grain boundary Grain boundary self-diffusion coefficient [ 121 Trap binding energy Trap concentration per unit area of grain boundary [ 131
the effect of such precipitates by considering D, in eqs. (14) and (15), as an effective diffusion coefficient for gas atoms on the grain boundaries. This is defined by
171 D = D,/(
1 + D,kf~,/‘8),
(16)
where D, is the product of the grain-boundary self-diffusion coefficient and the grain-boundary width 6, k: is the strength of the precipitate traps for migrating gas atoms which we take to be given approximately by kf = mt with c, the concentration of precipitate traps per unit area of grain boundary, and T, is the average time a gas atom spends at a trap. This time may be written 7, = exp( -%/k,T)/v,
(17)
where Y is the gas atom jump frequency given above, and E, is the binding energy of a gas atom to a precipitate trap. The data in table 1 are used in all the calculations discussed in the next section: this data set includes a trap concentration, c,, of 2 x 10’4/m2 and a binding energy, E,, of - 1.2 eV.
Table 2 Comparison Experimental
-boundan; bubbles
of calculated
and experimental
grain boundary
bubble
4. Comparison with experiment We have used the simple model derived above for the concentration of grain boundary bubbles, eq. (14), and the time-to-rupture, eq. (15), to obtain results for stainless steel. In this section we compare the calculated results with those from a variety of experiments [&lo]. Harries et al. [8] measured the spacings of grainboundary bubbles for steels containing 2, 43 and 97 ppm boron after irradiation at 650°C in a primarily thermal neutron flux. Using the expression (14) we calculated a bubble spacing of - 0.6 pm for the steel with a low boron content, which may be compared with the experimental observation of 0.53 pm. The ratios of the bubble spacings in the higher boron content steels to that in the 2 ppm boron steel have been calculated, and are compared with the observations by Harries et al. [8] in table 2. The agreement is very good and would seem to confirm the validity of our gas-bubble nucleation model. The second set of comparisons between experiment and theory relate to a number of in-pile creep rupture experiments [9,10] performed on steel between 650 and 850°C in a mainly thermal neutron flux. In fig. 1, we show calculated and experimental results for time-to-
spacings Calculated
’
Sample no., n
Boron content,
C,
Bubble spacing,
(ppm)
(rm)
2 3
2 43 97
0.53 0.20 0.16
a Ref. [8]. b Calculated (S,/S,)
= (C,/C,)“.25.
1
m3/s
S,
Relative spacing, S, /S,
Relative spacing b, S, /S,
1
1 0.46 0.37
0.38 0.30
hi. H. Wood, K. L. Kear / Nucleation and growth of grain
I
!
from
ref
l
75O’C
from
ref.9
A
850%
from
ref
9
0
720%
from
ref
10
x 65O’C
-boundarybubbles
323
I
9 20125
INb
1 1970
steel
steel
.
I
I
I
10
102
IO3
Time
to
rupture
( hours
4
IO'
1
Fig. 1. Comparison of experimental and calculated time to rupture as a function of stress for steel.
rupture as a function of stress over the entire temperature range of the experiments. The calculated stress dependence is, from eq. (15) um2. The experiments indicate a slightly higher stress exponent over most of the stress range, that is at stresses ,< 150 MPa. The calculated u- 2 dependence arises directly from our use of the ideal gas law to simplify the analysis. Numerical calculations using Van der Waals gas law give an exponent of - 2.2. At stresses > 150 MPa the experiments hint at a very rapid drop-off in the time-to-rupture. This is probably due to a changeover in mechanism from gas-driven rupture, the model described above, to creep-driven rupture characterised by a high stress exponent [lo]. The calculated temperature dependence of the timeto-rupture originates from the nucleation model derived in section 3, and reproduces the dependence shown by the experiments with some success. Overall, the model gives times-to-rupture that agree with the experimental results within a factor of - 2.
5. Conclusions We have extended a previous [l] model of the gasdriven growth of grain boundary cavities to include a simple model to describe the gas-driven nucleation of such grain boundary bubbles. Whereas the previous model gave the stress but not the temperature dependence of the time-to-rupture, the new model gives both dependencies. Results from the model are found to be in good agreement with experimental measurements of the spacings of bubbles on the grain boundaries of irradiated steels containing different quantities of boron. We have also compared model predictions with measured timesto-rupture during in-pile creep rupture experiments. The ue2 stress dependence from the model seems to slightly underestimate the experimental dependence. However, this model result depends on the gas law used: for example, whereas u -’ is found using the ideal gas law, the Van der Waals gas law gives u-2,2. The measured
324
M. H. Wood, K. L. Kear / Nucleatron and growth of grain boundary bubbles
temperature dependence is also well reproduced model. The overall level of agreement between ment and theory is within a factor of - 2.
by the experi-
References
[II R. Bullough,
D.R. Harries and M.R. Hayns, J. Nucl. Mater. 88 (1980) 312. 121J.E. Harris, Proc. Conf. Vacancy ‘76, Bristol, Eds. R.E. Smallman and J.E. Harris, Metals Sot. (1976) 170. I31 E.D. Hyam and G. Sumner, Proc. Symp. on Radiation Damage in Solids, Venice, IAEA Vienna 1 (1962) 323. A.J.E. Foreman and D.E. Rimmer, J. (41 G.W. Greenwood, Nucl. Mater. 4 (1959) 305. 151A.D. Whapham, Phil. Mag. 23 (1971) 987.
[6] J.R. Matthews and M.H. Wood, J. Nucl. Mater. 91 (1980) 241. [7] A.C. Damask and G.J. Dienes, Phys. Rev. 120 (1960) 99. 181 D.R. Harries, A.C. Roberts, G.T. Rogers, J.D.H. Hughes and M.A.P. Dewey, Radiation Damage in Reactor Materials, IAEA-SM-120/G-5 Vienna II (1969) 357. [91 J.T. Venard and J.R. Weir, ASTM Special Technical Publication 380 (1965) 269. C. Wassilew, L. Schafer and K. Anderko, Proc. Reaktortagung, Hannover, Deutsches Atomforum E.V., ( 1978) 609. L.E. Murr, Interfacial Phenomena in Metals and Alloys (Addison-Wesley, London, 1975). A.F. Smith, CEGB Report No. RD/B/N2270 (1972). H. Trinkaus and H. Ullmaier, Phil. Mag. 39 (1979) 563.