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The quantitative analysis of loop-growth and void-swelling in nickel
Journal of Nuclear Materials North-Holland, Amsterdam
79
120 (1984) 79-87
THE QUANTITATIVE
ANALYSIS
OF LOOP-GROWTH
AND VOID-SWELLING
IN NICKEL
B. COCHRANE a, S.B. FISHER b, KM. MILLER b and P.J. GOODHEW B ’ Dept. of Metallurgv and Materials Technology, Uniuersity of Surrey, Guildford GlJ2 ZXH, UK ’ Cenlral Electricity Generating Board, Berkeley Nuclear Laboratories, Berkelqv, Gloucestershire, GLl3 9PB, UK Received
17 January
1983; accepted
10 September
1983
Dislocation loop and void growth experiments at several temperatures in the HVEM have been used to investigate point defect behaviour in nickel. It has been found that the dislocation bias is about 5% for interstitials and that voids also have a
preference for interstitial defects. The value of the void bias is much less than the dislocation bias, being about 0.576, so that swelling still occurs.
The vacancy
migration
energy,
Ez, has been confirmed
to be 1.1 +O.l eV.
1. Introduction
of the function
In the absence of an applied stress, irradiation-induced void growth can only occur in the presence of point-defect sinks which are biased and which therefore capture more interstitials than vacancies. Dislocations generally act as suitable biased sinks but it is important for the quantitiative assessment of swelling that the magnitude of the bias is known. There have been several theoretical estimates of the dislocation bias, E, which is normally defined as:
-L=
ar at
( Zi - Z,) in the study of swelling.
Y+F(~I)[Zi-Zvl&i 4ar,z[k:+k,2][k:+k:]
(1)
’
+ cl h_ W(V)[Zi- Z”l[YPh -= at f&q+k:][k;+k:]
’
(2)
where
and B=(Zi-Z,)/Zi, where Zi and Z, are dimensionless parameters usually referred to as the sink strengths for interstitials and vacancies respectively [l]. The theoretical values of B range from 1% to 30% (eg. ref. [2]) while experimental determinations, rather fewer in number, generally indicate a value in the lower half of this range [3]. The present work was undertaken in order to establish a more reliable bias measurement, using a technique which gives as a by-product a value of the vacancy migration energy, E;, in a metal where this is already well known. Agreement with accepted values of Ez could then be used to establish the validity of the technique. The technique used to study the defect bias in nickel in this work follows directly that used by Fisher et al. [3] in their analysis of copper. Eqs. (1) and (2), which describe the rate of growth of void and dislocation loop radii [3], demonstrate the importance of the magnitude 0022-3115/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
and the symbols are defined at the end of this article (Nomenclature). Unfortunately neither the magnitudes of Zi and Z, themselves nor of the bias are well established. The expressions used for Zi and Z, were derived from calculation of the dislocation-point defect interactions involving the stress field of the dislocation and the field due to the point defect relaxation volumes AV, and AVv [4]. Various workers have calculated the vacancy formation volume, AV., in nickel and have arrived at similar results [5] which suggest a value of approximately 0.19 L?. With regard to the interstitial relaxation volume the situation is less clear, since calculated values vary by a factor of two [5]. In the current simulation we therefore use a AV, value of 0.1952 but vary the value of AVi in order to fit the experimental data. B.V.
Thus the determination of ( Z, - Z, ) and EA, is the major requirement of any experiment designed to measure a defect bias. Eq. (2) suggests that the results of a dislocation loop growth experiment at one temperature would yield a pair of possible values for (B. Ei) where B represents the interstitial bias. Repeating the experiment at different temperatures and plotting the values of bias against those values of EG which give the observed growth rate at each temperature should yield a pair of values of bias and Ei which satisfy results at all temperatures. If (Z, - Z,) and Ek can be determined. then a comparison of loop and void growth in the early stages of irradiation will yield information on the void sink strength. Using eqs. (1) and (2). the following relationship between the void and loop growth rates can be obtained: i-,_ -= 4
[ Yp, + kz]47rr;! byp,
= ki4vr,’ byp,
The HVEM is the most suitable instrument to carry out these measurements as temperature can be varied accurately and easily and both void and loop growth data can be obtained in a single experiment.
2. Experimental
Nickel was received in the form of 50 pm sheets of 99.995% purity. Discs of 3 mm diameter were punched from this and annealed for one hour at 760°C. The discs were then thinned for examination and irradiation in a 1 MeV electron microscope. In eq. (2) the observable parameters are the void and loop radii, the void and loop densities, the dislocation density and the foil thickness. The dislocation line density has been measured using the method of Ham [6] and the foil thickness has been measured by counting the number of grain-boundary fringes. Stereomicroscopy has been used to measure the void and loop densities and allowance has been made for loop invisibility and for the denuded layers adjacent to the foil surfaces. Dislocation loop radii were measured as half the largest apparent distance between opposite edges of the loop [7,8]. The void dimensions were obtained by assuming that the voids were spherical and measuring the radius of the smallest sphere which would enclose the void [9]. All high voltage micrographs were taken at magnifications of 20000 x or 30 000 X . Focussing at very high magnifications using thick foils is difficult and thus the magnifications used were as high as possible.
3. Analysis Quantitative analysis of the experimental data was initially performed using the simulation program of Fisher et al. [3] in order to find values for E,:, and JC;. In the program these parameters are varied and the pairs of values which give the best match between the theoretical and the observed void growth rates are recorded. The simulation program is based on the rate theorv continuum model and allows for the effects of recombination on the sink strength calculations. Growth rates are calculated for a small time increment and the sink strengths are then updated at the end of this increment. New defect concentrations are calculated which are used to calculate new growth rates for the next time increment. Any discussion of the nucleation problem ix avoided in the program by starting with the observed density of dislocation loops at an arbitrary small size and introducing the voids at the dose and size at which they are first observed. In test calculations, Fisher et al. found that the loop growth rate was insensitive to starting size and chose an initial loop radius of 5 nm. It is reasonabie to introduce the observed loop density immediately as the loop nucleation time is effectively zero. If the loop nucleation time was measurably nonzero then the calculations could be started at this later time. From the loop growth rate and measured loop density a saturation radius and saturation time can be determined at which loop interaction begins. At times greater than this the dislocation network density was introduced into the program. Fig. 1 shows a typical irradiation sequence with loops interacting to form a network and then voids appearing and subsequently growing. This irradiation took place at a temperature of 485°C and will be used as an example of evaluation of the experimental data. 3.1. Derioation of data to be input to the progrum The electron flux was measured with a Faraday cage and used to calculate the dose and damage rate. The displacement rate was calculated to be about 2 X 10 ’ cross-section of 40 barns dpa SK’, using a displacement 1101. The mean loop radius was determined from the radii of about one hundred loops in each of the micrographs. The loop density was calculated by counting the number of loops occurring in a given volume of the specimen allowing for invisibility criteria. The dislocation density was measured using the method of Ham [6]. In the dislocation loop regime the dislocation line density,
B. Cochrane et al. / Loop - growrh and void-swelling in nickel
a
b
e Fig. 1. The void growth sequence in nickel at &ET. Dislocation loops ace visible in (a) and hegin to interact in (bf. Voids are present after 2 dpa and then grow; (a) 0.27 dpa, (b) 0.60 dpa, (c) 2.10 dpa, (d) 3.56 dpa, (e) 4.05 dpa, (f) 6.46 dpa.
B. Cochrane et al.
n Void
I
Flose
/
d.p.a.
Fig. 2. Mean void and loop radii nickel irradiated at 485’C.
pd. was calculated
400
as a function
of dose for
using the formula:
p, = 27rr,N,.
(5)
The mean void radius was obtained by measuring one hundred voids in each micrograph. The void density was calculated in a similar way to the dislocation loop density. The loop interaction radius is given by: RI = ( $TN~)~“~
(6)
The void radius and density data are plotted against dose in figs. 2 and 3, together with the dislocation loop radii and dislocation line densities. The magnitude of the error in the dislocation line density depends on the error in the estimate of the total foil thickness. This error has been minimised by counting the number of thickness fringes near the s = 0 position. Using this method foil thickness can be measured to an accuracy of *5% [ll]. In a 1 MeV microscope it is almost impossible to operate with only two beams as strong systematic reflections always arise. Thus there will be an error in the magnitude of the extinction distance used to.calculate
Fig. 3. Void and dislocation irradiated at 485°C.
line density
vs dose for nickel
the foil thickness. The total error in the measurement of foil thickness has been estimated to be about 20% which is translated directly to be the error in the dislocation line density. This is-in accordance with the estimate h) Ham [6]. The estimate of void and dislocation loop densities depends on the accuracy obtained using stereomicroscopy. 11 has been pointed out by Hirsch et al. [II] that the i58 error claimed by Nankivell [12] can only be achieved with careful attention to the microscope tilt holder. By matching a large number of voids in the stereo pair the error in estimating the defect layer width has been minimised but will still be of the order of 10%. Measuring the void dimension by taking the radius of the smallest sphere which would fit over it would give a large error in calculating the extent of volume-swelling. This experiment however requires only information on the increase in the void size and so this is a convenient parameter to measure. The void radii can be measured to an accuracy of +0.05 mm on the micrographs which are printed to 60K magnification. Thus the error in the void size measurements is of the order of 2 nm. Loop radius measurements are less accurate since the
83
B. Cochrane et al. / Loop -growth and void -swelling in nickel Table 1 Void and loop growth
data
Irradiation temperature (“C) Loop density (1014/cm3) Interaction radius (nm) Saturation time(s) Growth rate (nm/s) Max. dislocation density (10” cm/cm3) Min. dislocation density (lo9 cm/cm-‘) Foil thickness (nm) Dose rate (1O-3 dpa/s) Dose at which voids appear (dpa) Radius with which voids appear (nm)
440 2.9 93 660 0.136
485 1.76
525 0.40
110
182
300 0.37
413 0.44 0.45
1.7
1.2
4.0 490
2.0 430
2.5 500
2.0
2.0
2.0
2.1
3.6
I
of loop growth
1.4
1.2 EL leV
5.6
9.5
width of the dark band from which measurements were made is variable. Fig. la shows some loops with no dark band along their longer axis and some with a dark band. Measurements of loop diameter were always made to the inside of this dark band, if it was visible. The loop diameter could be measured to an accuracy of +0.25 mm on the plate which corresponds to 54 nm on the specimen. The data to be input to the program is summarised, for three temperatures, in table 1. The experiment carried out at 440°C was not continued into the void regime. Table 1 does not include the estimated errors as it shows the specific values which were used to simulate the microstructural development. 3.2. Analysis
I
1.0
rates
The output of the simulation routine is searched to find, for each value of vacancy migration energy, a calculated saturation time close to the observed saturation time. The interstitial relaxation volume and vacancy migration energies for which this condition holds are noted. These are plotted for the three temperatures in fig. 4. The quoted values of A5 are kO.02552. Each curve in this figure represents pairs of values of AVi and Ez which fit the loop growth data at a particular temperature. The values of AVi and Ez where the curves cross satisfy the loop growth rate at all three temperatures. Thus a value for the bias, B, and Ei can be found
Fig. 4. Interstitial relaxation volume (AV,) vs vacancy tion energy (Ez) using the Fisher model [3].
migra-
of the precise variation of B and Ei with temperature. From fig. 4 it can be seen that the curves cross over with EG between 1.0 eV and 1.2 eV. The corresponding value of AK is about 0.25 ( f 0.025)8. Thus Ei = 1.1 (fO.l) eV and AK = 0.2552. The bias corresponding to these values is 5.5%. The value of Ei is in good agreement with that of 1.2(+0.2) eV found by Dlubek et al. [13] using a positron annihilation method. It also agrees with the values of 1.2 eV deduced by Yoo and Stiegler [7] and Kiritani and Takata [14]. Yoo and Stiegler found the interstitial bias to be 5% for nickel which (again) agrees well with the value obtained in this work and that of Harbottle and Silvent [15]. The model used to deduce the results, although simple, assumes a continuum surface sink strength which in some circumstances can lead to appreciable errors in the sink strengths [16,17]. However the parameter crucial to a bias measurement is the difference between the vacancy and interstitial sink strengths. A better model for interstitial loop growth, which directly accounts for point-defect loss to the surfaces, has been developed by Miller et al. ((181, hereafter referred to as MFW) who applied it to experimental results from copper. They found that the results could still be interpreted in terms of a pair of values of (AVi, EG) which can be used to establish the dislocation bias. They deduced that B is about 2% for copper. We have applied the MFW analyirrespective
ble pair of values at one temperature, although not the most acceptable at all temperatures. What is still needed is a set of experiments accurately determining loop growth rate as a function of depth at several temperatures.
Ed, and AP’, having been determined from the loop growth data. the subsequent void growth may be analysed in detail. This has been carried out for the irradiations at 485°C and 525’C. Fig. 6 shows the experimental results at the two temperatures and the computed results assuming a simple diffusion-controlled mechanism for void growth. The agreement between the computed and experimental curves is reasonable, espe-
1.0
I.4
1.2 EL leV
Fig. 5. Interstitial relaxation volume (Ay ) vs vacancy tion energy (Ez) using the MFW analysis [lS].
migra-
sis to the results reported here for nickel and obtain the curves shown in fig. 5. This alternative analysis of the experimental data leads to the similar conclusion that a consistent pair of values for (Al/;, Eg) is 0.22G and 1.0 eV, which corresponds to a bias of 4%. An encouraging feature of the MFW analysis is that, for both copper (ref. [IS], fig. 3) and nickel (this work, fig. 5), the data for the three temperatures studied vary regularly with temperature, which is not the case when the simpler analysis (fig. 4) is employed. Although the results presented here show very reasonable agreement with measured values of EG for nickel [13,19,20] there is still a need for more careful experiments. The MFW analysis predicts that loops at different depths in the foil will grow at different rates. Miller [21] has recently shown that the analysis can be used to interpret the detailed results presented by YOO and Stiegler on loop growth in nickel [7]. However. in this case the best fit to the data is for Avj = 0.4D and EJ, = 1.4 eV, which implies a bias value of 24%. The discrepancy between this result and the results in the present paper probably arises because the Yoo and Stiegler data were collected at a single temperature (45O”C), at a dose rate 100 times lower than that in our experiments, and include a great deal of scatter. It is worth pointing out that the best fitting set of (AC’,, Eg). i.e., 0.49, 1.4 eV, Iies close to our curve for 440°C (fig. 4) and therefore is according to our results an accepta-
60. I‘;
i
Fig. 6. Experimental void growth data at two temperatures compared with the theoretical behaviour assuming simple diffusion controlled growth.
B. Cochrane et ai. / Loop-growth
cially at low doses. At higher doses however the computed curve rises above the experimental one. Fisher et al found their calculated curve to be higher than the experimental in their analysis of void growth in copper [3]. They suggested increasing the value of the void interface radius, r*, to lower the calculated growth rate. This implies that the growth is being controlled by an interfacial reaction. Fig. 7 shows the experimental data points at 485°C along with those computed with various values of r*. The agreement between the calculated and experimental growth curves is best with r* = 10 nm. This is of the same order as found by Fisher et al. [3]. Assuming an interface-controlled mechanism may not be the best explanation. This factor enters eqs. (1) and (2) as: Y=47rr,2/(rv+r*).
and void-swelling
85
in nickel Avx = 0.25fi iz;=
l.lSaV
IZO.OC
100.00 0 Q
80.00 2 ;; e 77 60.00 D >
+
40.00
Av,
= 0.25~1
+ exparrmenrai f’m=
l.lSeV
0
20.00
120.00
Void for
bias
of
0.5%
intersritlals
*vet- vaca"c,*s.
0.00
100.00
Dose
Id. p, a.
Fig. 8. Void growth behaviour at 485“C assuming simple diffusion control and a small void bias for interstitials.
80.00 5 0 : -O 60.00 ;; jr
9
* ewperlmental 4Q.00
20.00
0.00
J-0 0.0
I.00
2.00
3.00
Dose
I. 00
d.
Tc= 2.5h
.
r’=
524
0
r*=
lOOA
B
r*=
ISOA
D
r*=
200h
6.00
7.00
5.00
8.00
/d.p.a.
void growth data at 485’C compared with theoretical behaviour assuming an interface controlled mechanism with various interface radii. Fig. 7. Experimental
Thus as the void radius increases, the r, term becomes dominant. This means that this model will not affect the calculated growth rate in such a way as to lower the growth rate at large radii more than at small radii. It should also be noted that this model involves a thin spherical shell surrounding a void which modifies the capture efficiency of the void for point defects. As large voids are strongly faceted this model may not be applicable. An alternative approach wouid be to assert that voids may, as do dislocations, have a preference for interstitials over vacancies. This preference for interstitials could, as with dislocations, be due to the stronger interaction with an interstitial than with a vacancy due to either size effect interactions or stress field interactions. The void bias is likely to be less than the dislocation bias and thus more vacancies than interstitials would still arrive at void surfaces because, although the
86
B. Cochrane et al. / Loop -growth and cwrd - swlling
interstitials are more strongly attracted to the void than vacancies, there would be fewer interstitials available as they are even more strongly attracted to the dislocations. This possibility has been investigated with the result shown in fig. 8. Void bias values between 0 and 5% were tried and the best agreement between the experimental and calculated growth curves was obtained with a void bias of 0.5% for interstitials. Mansur and Wolfer [22] investigated the effect of solute segregation to voids by calculating the change in the capture efficiencies for point defects of voids with, and without. a surrounding shell. During the course of this work they suggest that a void with no surrounding,,shell has a large preference for interstitials over vacancies. For voids of radius greater than 4 nm they show a value of the order of 5% to 10% for this bias. Our experimental value thus reinforces their qualitative argument but is in better agreement with the observed occurence of swelling.
in nickel
to surfaces (m ’ ). rmfp for the loss of vacancies to all internal sinks (i.e. dislocations and voids) (mm ’ ). rmpf for the loss of interstitials to all internal sinks (m-l). defect density (mm “), mean radius (m), effective increase of void radius due to interfacial effects (m). void sink strength (m). dimensionless parameter related to the sink strength of a dislocation for the defect indicated by the subscript. Analytical approximations have been given for example by [1X] ( - ). recombination coefficient (s- ’ ). point defect relaxation volume (m? ). displacement rate (s-l). dislocation density (m ’ ), void density (m ’ ). atomic volume (m’). loss of defects
k,
k, N I
r* I’
Z
a
AC’ P P
4. Conclusions Subscripts
Our main conclusion is that the dislocation bias can be measured using the techniques of Fisher et al. [3] and Miller et al. [18]. The value obtained for the bias in nickel is in the range 4-6%. which is consistent with an Ez of 1.1 + 0.1 eV. The success of the methods in arriving at both the accepted value of Ez and also at a sensible value of the bias means that the technique may be applied to other materials in which the parameters are not so well known. This method also allows the void sink strength to be separately considered in the analysis. from which it is apparent that voids also have a small bias for interstitials.
d
h i L s V
dislocation hole ( = void) interstitial loop surface vacancv
References PI S.B. Fisher and R.J. White. Rad. Eff. 30 (1976) 17. F.A. Nichols,
Rad. Eff. 39 (1978) 169.
;Si S.B. Fisher, R.J. White and K.M. Miller. Phil. Mag. A40 Acknowledgements We would like to thank CEGB Berkeley Nuclear Laboratories for supporting this work. BC thanks the Nothern Ireland Dept. of Education for a research studentship.
Nomenclature B b D
ElYI k,
dislocation bias (- ), Burgers vector (m). diffusivity (m*/s). vacancy migration energy (eV), reciprocal mean free path (rmfp) for the surface sink, such that kf is a factor accounting for the
(1979) 239. [41 P.T. Heald and K.M. Miller. J. Nucl. Mater. 66 (1976) 107. 151 K.M. Miller and P.D. Bristowe, Phys. Stat. Sol.(b) X6 (1978) 93. [61 R.K. Ham, Phil. Mag. 6 (1961) 1183. [71 M.H. Yoo and J.O. Stiegler, Phil. Mag. 36 (1977) 1305. PI D.I.R. Norris. Phil. Mag. 22 (1970) 1273. 2 (1969) 89. 191 U.E. Wolff, Metallography IlO1 D.I.R. Norris. J. Nucl. Mater. 40 (1971) 66. PII P.B. Hirsch. A. Howie, R.B. Nicholson, D.W. Pashley and M.J. Whelan. Electron Microscopy of Thin Crystals (Butterworths. London, 1965) p. 243. WI J.F. Nankivell, Brit. J. Appl. Phys. 13 (1962) 126. P31 G. Dlubek, 0. Brumonev and P. Sickert. Phys. Stat. Sol (A). 39 (1978) 169. (141 M. Kiritani and H. Takata, J. Nucl. Mater 69170 (1978) 277. and A. Silvent, Proc. Int. Conf. on Irradia1151 J.E.Harbottle
B. Cochrane et al. / Loop -growth and void -swelling in nickel tion Behaviour of Metallic Materials for Fast Reactor Core Materials, Eds. J. Poirier and J.M. Dupouy (CEA. Paris, 1979) p. 391. [16] M.H. Wood, B. Jones and S.M. Pierce, Res. Mechanica 3 (1981) 283. [17] SM. Pierce, R. Bullough and M.H. Wood, Res. Mechanica 4 (1982) 191. [18] K.M. Miller, S.B. Fisher and R.J. White, J. Nucl. Mater. 110 (1982) 265.
87
[19] W. Wycsik and M. Feller-Kniepmeyer. Phys. Stat. Sol. (a) 37 (1976) 183. [20] S.K. Khanna and K. Sonnenberg, Rad. Eff. 59 (1981) 91. [21] K.M. Miller, J. Nucl. Mater. 115 (1983) 216. [22] L.K. Mansur and W.G. Wolfer, J. Nucl. Mater 69&70 (1978) 825.
79
120 (1984) 79-87
THE QUANTITATIVE
ANALYSIS
OF LOOP-GROWTH
AND VOID-SWELLING
IN NICKEL
B. COCHRANE a, S.B. FISHER b, KM. MILLER b and P.J. GOODHEW B ’ Dept. of Metallurgv and Materials Technology, Uniuersity of Surrey, Guildford GlJ2 ZXH, UK ’ Cenlral Electricity Generating Board, Berkeley Nuclear Laboratories, Berkelqv, Gloucestershire, GLl3 9PB, UK Received
17 January
1983; accepted
10 September
1983
Dislocation loop and void growth experiments at several temperatures in the HVEM have been used to investigate point defect behaviour in nickel. It has been found that the dislocation bias is about 5% for interstitials and that voids also have a
preference for interstitial defects. The value of the void bias is much less than the dislocation bias, being about 0.576, so that swelling still occurs.
The vacancy
migration
energy,
Ez, has been confirmed
to be 1.1 +O.l eV.
1. Introduction
of the function
In the absence of an applied stress, irradiation-induced void growth can only occur in the presence of point-defect sinks which are biased and which therefore capture more interstitials than vacancies. Dislocations generally act as suitable biased sinks but it is important for the quantitiative assessment of swelling that the magnitude of the bias is known. There have been several theoretical estimates of the dislocation bias, E, which is normally defined as:
-L=
ar at
( Zi - Z,) in the study of swelling.
Y+F(~I)[Zi-Zvl&i 4ar,z[k:+k,2][k:+k:]
(1)
’
+ cl h_ W(V)[Zi- Z”l[YPh -= at f&q+k:][k;+k:]
’
(2)
where
and B=(Zi-Z,)/Zi, where Zi and Z, are dimensionless parameters usually referred to as the sink strengths for interstitials and vacancies respectively [l]. The theoretical values of B range from 1% to 30% (eg. ref. [2]) while experimental determinations, rather fewer in number, generally indicate a value in the lower half of this range [3]. The present work was undertaken in order to establish a more reliable bias measurement, using a technique which gives as a by-product a value of the vacancy migration energy, E;, in a metal where this is already well known. Agreement with accepted values of Ez could then be used to establish the validity of the technique. The technique used to study the defect bias in nickel in this work follows directly that used by Fisher et al. [3] in their analysis of copper. Eqs. (1) and (2), which describe the rate of growth of void and dislocation loop radii [3], demonstrate the importance of the magnitude 0022-3115/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
and the symbols are defined at the end of this article (Nomenclature). Unfortunately neither the magnitudes of Zi and Z, themselves nor of the bias are well established. The expressions used for Zi and Z, were derived from calculation of the dislocation-point defect interactions involving the stress field of the dislocation and the field due to the point defect relaxation volumes AV, and AVv [4]. Various workers have calculated the vacancy formation volume, AV., in nickel and have arrived at similar results [5] which suggest a value of approximately 0.19 L?. With regard to the interstitial relaxation volume the situation is less clear, since calculated values vary by a factor of two [5]. In the current simulation we therefore use a AV, value of 0.1952 but vary the value of AVi in order to fit the experimental data. B.V.
Thus the determination of ( Z, - Z, ) and EA, is the major requirement of any experiment designed to measure a defect bias. Eq. (2) suggests that the results of a dislocation loop growth experiment at one temperature would yield a pair of possible values for (B. Ei) where B represents the interstitial bias. Repeating the experiment at different temperatures and plotting the values of bias against those values of EG which give the observed growth rate at each temperature should yield a pair of values of bias and Ei which satisfy results at all temperatures. If (Z, - Z,) and Ek can be determined. then a comparison of loop and void growth in the early stages of irradiation will yield information on the void sink strength. Using eqs. (1) and (2). the following relationship between the void and loop growth rates can be obtained: i-,_ -= 4
[ Yp, + kz]47rr;! byp,
= ki4vr,’ byp,
The HVEM is the most suitable instrument to carry out these measurements as temperature can be varied accurately and easily and both void and loop growth data can be obtained in a single experiment.
2. Experimental
Nickel was received in the form of 50 pm sheets of 99.995% purity. Discs of 3 mm diameter were punched from this and annealed for one hour at 760°C. The discs were then thinned for examination and irradiation in a 1 MeV electron microscope. In eq. (2) the observable parameters are the void and loop radii, the void and loop densities, the dislocation density and the foil thickness. The dislocation line density has been measured using the method of Ham [6] and the foil thickness has been measured by counting the number of grain-boundary fringes. Stereomicroscopy has been used to measure the void and loop densities and allowance has been made for loop invisibility and for the denuded layers adjacent to the foil surfaces. Dislocation loop radii were measured as half the largest apparent distance between opposite edges of the loop [7,8]. The void dimensions were obtained by assuming that the voids were spherical and measuring the radius of the smallest sphere which would enclose the void [9]. All high voltage micrographs were taken at magnifications of 20000 x or 30 000 X . Focussing at very high magnifications using thick foils is difficult and thus the magnifications used were as high as possible.
3. Analysis Quantitative analysis of the experimental data was initially performed using the simulation program of Fisher et al. [3] in order to find values for E,:, and JC;. In the program these parameters are varied and the pairs of values which give the best match between the theoretical and the observed void growth rates are recorded. The simulation program is based on the rate theorv continuum model and allows for the effects of recombination on the sink strength calculations. Growth rates are calculated for a small time increment and the sink strengths are then updated at the end of this increment. New defect concentrations are calculated which are used to calculate new growth rates for the next time increment. Any discussion of the nucleation problem ix avoided in the program by starting with the observed density of dislocation loops at an arbitrary small size and introducing the voids at the dose and size at which they are first observed. In test calculations, Fisher et al. found that the loop growth rate was insensitive to starting size and chose an initial loop radius of 5 nm. It is reasonabie to introduce the observed loop density immediately as the loop nucleation time is effectively zero. If the loop nucleation time was measurably nonzero then the calculations could be started at this later time. From the loop growth rate and measured loop density a saturation radius and saturation time can be determined at which loop interaction begins. At times greater than this the dislocation network density was introduced into the program. Fig. 1 shows a typical irradiation sequence with loops interacting to form a network and then voids appearing and subsequently growing. This irradiation took place at a temperature of 485°C and will be used as an example of evaluation of the experimental data. 3.1. Derioation of data to be input to the progrum The electron flux was measured with a Faraday cage and used to calculate the dose and damage rate. The displacement rate was calculated to be about 2 X 10 ’ cross-section of 40 barns dpa SK’, using a displacement 1101. The mean loop radius was determined from the radii of about one hundred loops in each of the micrographs. The loop density was calculated by counting the number of loops occurring in a given volume of the specimen allowing for invisibility criteria. The dislocation density was measured using the method of Ham [6]. In the dislocation loop regime the dislocation line density,
B. Cochrane et al. / Loop - growrh and void-swelling in nickel
a
b
e Fig. 1. The void growth sequence in nickel at &ET. Dislocation loops ace visible in (a) and hegin to interact in (bf. Voids are present after 2 dpa and then grow; (a) 0.27 dpa, (b) 0.60 dpa, (c) 2.10 dpa, (d) 3.56 dpa, (e) 4.05 dpa, (f) 6.46 dpa.
B. Cochrane et al.
n Void
I
Flose
/
d.p.a.
Fig. 2. Mean void and loop radii nickel irradiated at 485’C.
pd. was calculated
400
as a function
of dose for
using the formula:
p, = 27rr,N,.
(5)
The mean void radius was obtained by measuring one hundred voids in each micrograph. The void density was calculated in a similar way to the dislocation loop density. The loop interaction radius is given by: RI = ( $TN~)~“~
(6)
The void radius and density data are plotted against dose in figs. 2 and 3, together with the dislocation loop radii and dislocation line densities. The magnitude of the error in the dislocation line density depends on the error in the estimate of the total foil thickness. This error has been minimised by counting the number of thickness fringes near the s = 0 position. Using this method foil thickness can be measured to an accuracy of *5% [ll]. In a 1 MeV microscope it is almost impossible to operate with only two beams as strong systematic reflections always arise. Thus there will be an error in the magnitude of the extinction distance used to.calculate
Fig. 3. Void and dislocation irradiated at 485°C.
line density
vs dose for nickel
the foil thickness. The total error in the measurement of foil thickness has been estimated to be about 20% which is translated directly to be the error in the dislocation line density. This is-in accordance with the estimate h) Ham [6]. The estimate of void and dislocation loop densities depends on the accuracy obtained using stereomicroscopy. 11 has been pointed out by Hirsch et al. [II] that the i58 error claimed by Nankivell [12] can only be achieved with careful attention to the microscope tilt holder. By matching a large number of voids in the stereo pair the error in estimating the defect layer width has been minimised but will still be of the order of 10%. Measuring the void dimension by taking the radius of the smallest sphere which would fit over it would give a large error in calculating the extent of volume-swelling. This experiment however requires only information on the increase in the void size and so this is a convenient parameter to measure. The void radii can be measured to an accuracy of +0.05 mm on the micrographs which are printed to 60K magnification. Thus the error in the void size measurements is of the order of 2 nm. Loop radius measurements are less accurate since the
83
B. Cochrane et al. / Loop -growth and void -swelling in nickel Table 1 Void and loop growth
data
Irradiation temperature (“C) Loop density (1014/cm3) Interaction radius (nm) Saturation time(s) Growth rate (nm/s) Max. dislocation density (10” cm/cm3) Min. dislocation density (lo9 cm/cm-‘) Foil thickness (nm) Dose rate (1O-3 dpa/s) Dose at which voids appear (dpa) Radius with which voids appear (nm)
440 2.9 93 660 0.136
485 1.76
525 0.40
110
182
300 0.37
413 0.44 0.45
1.7
1.2
4.0 490
2.0 430
2.5 500
2.0
2.0
2.0
2.1
3.6
I
of loop growth
1.4
1.2 EL leV
5.6
9.5
width of the dark band from which measurements were made is variable. Fig. la shows some loops with no dark band along their longer axis and some with a dark band. Measurements of loop diameter were always made to the inside of this dark band, if it was visible. The loop diameter could be measured to an accuracy of +0.25 mm on the plate which corresponds to 54 nm on the specimen. The data to be input to the program is summarised, for three temperatures, in table 1. The experiment carried out at 440°C was not continued into the void regime. Table 1 does not include the estimated errors as it shows the specific values which were used to simulate the microstructural development. 3.2. Analysis
I
1.0
rates
The output of the simulation routine is searched to find, for each value of vacancy migration energy, a calculated saturation time close to the observed saturation time. The interstitial relaxation volume and vacancy migration energies for which this condition holds are noted. These are plotted for the three temperatures in fig. 4. The quoted values of A5 are kO.02552. Each curve in this figure represents pairs of values of AVi and Ez which fit the loop growth data at a particular temperature. The values of AVi and Ez where the curves cross satisfy the loop growth rate at all three temperatures. Thus a value for the bias, B, and Ei can be found
Fig. 4. Interstitial relaxation volume (AV,) vs vacancy tion energy (Ez) using the Fisher model [3].
migra-
of the precise variation of B and Ei with temperature. From fig. 4 it can be seen that the curves cross over with EG between 1.0 eV and 1.2 eV. The corresponding value of AK is about 0.25 ( f 0.025)8. Thus Ei = 1.1 (fO.l) eV and AK = 0.2552. The bias corresponding to these values is 5.5%. The value of Ei is in good agreement with that of 1.2(+0.2) eV found by Dlubek et al. [13] using a positron annihilation method. It also agrees with the values of 1.2 eV deduced by Yoo and Stiegler [7] and Kiritani and Takata [14]. Yoo and Stiegler found the interstitial bias to be 5% for nickel which (again) agrees well with the value obtained in this work and that of Harbottle and Silvent [15]. The model used to deduce the results, although simple, assumes a continuum surface sink strength which in some circumstances can lead to appreciable errors in the sink strengths [16,17]. However the parameter crucial to a bias measurement is the difference between the vacancy and interstitial sink strengths. A better model for interstitial loop growth, which directly accounts for point-defect loss to the surfaces, has been developed by Miller et al. ((181, hereafter referred to as MFW) who applied it to experimental results from copper. They found that the results could still be interpreted in terms of a pair of values of (AVi, EG) which can be used to establish the dislocation bias. They deduced that B is about 2% for copper. We have applied the MFW analyirrespective
ble pair of values at one temperature, although not the most acceptable at all temperatures. What is still needed is a set of experiments accurately determining loop growth rate as a function of depth at several temperatures.
Ed, and AP’, having been determined from the loop growth data. the subsequent void growth may be analysed in detail. This has been carried out for the irradiations at 485°C and 525’C. Fig. 6 shows the experimental results at the two temperatures and the computed results assuming a simple diffusion-controlled mechanism for void growth. The agreement between the computed and experimental curves is reasonable, espe-
1.0
I.4
1.2 EL leV
Fig. 5. Interstitial relaxation volume (Ay ) vs vacancy tion energy (Ez) using the MFW analysis [lS].
migra-
sis to the results reported here for nickel and obtain the curves shown in fig. 5. This alternative analysis of the experimental data leads to the similar conclusion that a consistent pair of values for (Al/;, Eg) is 0.22G and 1.0 eV, which corresponds to a bias of 4%. An encouraging feature of the MFW analysis is that, for both copper (ref. [IS], fig. 3) and nickel (this work, fig. 5), the data for the three temperatures studied vary regularly with temperature, which is not the case when the simpler analysis (fig. 4) is employed. Although the results presented here show very reasonable agreement with measured values of EG for nickel [13,19,20] there is still a need for more careful experiments. The MFW analysis predicts that loops at different depths in the foil will grow at different rates. Miller [21] has recently shown that the analysis can be used to interpret the detailed results presented by YOO and Stiegler on loop growth in nickel [7]. However. in this case the best fit to the data is for Avj = 0.4D and EJ, = 1.4 eV, which implies a bias value of 24%. The discrepancy between this result and the results in the present paper probably arises because the Yoo and Stiegler data were collected at a single temperature (45O”C), at a dose rate 100 times lower than that in our experiments, and include a great deal of scatter. It is worth pointing out that the best fitting set of (AC’,, Eg). i.e., 0.49, 1.4 eV, Iies close to our curve for 440°C (fig. 4) and therefore is according to our results an accepta-
60. I‘;
i
Fig. 6. Experimental void growth data at two temperatures compared with the theoretical behaviour assuming simple diffusion controlled growth.
B. Cochrane et ai. / Loop-growth
cially at low doses. At higher doses however the computed curve rises above the experimental one. Fisher et al found their calculated curve to be higher than the experimental in their analysis of void growth in copper [3]. They suggested increasing the value of the void interface radius, r*, to lower the calculated growth rate. This implies that the growth is being controlled by an interfacial reaction. Fig. 7 shows the experimental data points at 485°C along with those computed with various values of r*. The agreement between the calculated and experimental growth curves is best with r* = 10 nm. This is of the same order as found by Fisher et al. [3]. Assuming an interface-controlled mechanism may not be the best explanation. This factor enters eqs. (1) and (2) as: Y=47rr,2/(rv+r*).
and void-swelling
85
in nickel Avx = 0.25fi iz;=
l.lSaV
IZO.OC
100.00 0 Q
80.00 2 ;; e 77 60.00 D >
+
40.00
Av,
= 0.25~1
+ exparrmenrai f’m=
l.lSeV
0
20.00
120.00
Void for
bias
of
0.5%
intersritlals
*vet- vaca"c,*s.
0.00
100.00
Dose
Id. p, a.
Fig. 8. Void growth behaviour at 485“C assuming simple diffusion control and a small void bias for interstitials.
80.00 5 0 : -O 60.00 ;; jr
9
* ewperlmental 4Q.00
20.00
0.00
J-0 0.0
I.00
2.00
3.00
Dose
I. 00
d.
Tc= 2.5h
.
r’=
524
0
r*=
lOOA
B
r*=
ISOA
D
r*=
200h
6.00
7.00
5.00
8.00
/d.p.a.
void growth data at 485’C compared with theoretical behaviour assuming an interface controlled mechanism with various interface radii. Fig. 7. Experimental
Thus as the void radius increases, the r, term becomes dominant. This means that this model will not affect the calculated growth rate in such a way as to lower the growth rate at large radii more than at small radii. It should also be noted that this model involves a thin spherical shell surrounding a void which modifies the capture efficiency of the void for point defects. As large voids are strongly faceted this model may not be applicable. An alternative approach wouid be to assert that voids may, as do dislocations, have a preference for interstitials over vacancies. This preference for interstitials could, as with dislocations, be due to the stronger interaction with an interstitial than with a vacancy due to either size effect interactions or stress field interactions. The void bias is likely to be less than the dislocation bias and thus more vacancies than interstitials would still arrive at void surfaces because, although the
86
B. Cochrane et al. / Loop -growth and cwrd - swlling
interstitials are more strongly attracted to the void than vacancies, there would be fewer interstitials available as they are even more strongly attracted to the dislocations. This possibility has been investigated with the result shown in fig. 8. Void bias values between 0 and 5% were tried and the best agreement between the experimental and calculated growth curves was obtained with a void bias of 0.5% for interstitials. Mansur and Wolfer [22] investigated the effect of solute segregation to voids by calculating the change in the capture efficiencies for point defects of voids with, and without. a surrounding shell. During the course of this work they suggest that a void with no surrounding,,shell has a large preference for interstitials over vacancies. For voids of radius greater than 4 nm they show a value of the order of 5% to 10% for this bias. Our experimental value thus reinforces their qualitative argument but is in better agreement with the observed occurence of swelling.
in nickel
to surfaces (m ’ ). rmfp for the loss of vacancies to all internal sinks (i.e. dislocations and voids) (mm ’ ). rmpf for the loss of interstitials to all internal sinks (m-l). defect density (mm “), mean radius (m), effective increase of void radius due to interfacial effects (m). void sink strength (m). dimensionless parameter related to the sink strength of a dislocation for the defect indicated by the subscript. Analytical approximations have been given for example by [1X] ( - ). recombination coefficient (s- ’ ). point defect relaxation volume (m? ). displacement rate (s-l). dislocation density (m ’ ), void density (m ’ ). atomic volume (m’). loss of defects
k,
k, N I
r* I’
Z
a
AC’ P P
4. Conclusions Subscripts
Our main conclusion is that the dislocation bias can be measured using the techniques of Fisher et al. [3] and Miller et al. [18]. The value obtained for the bias in nickel is in the range 4-6%. which is consistent with an Ez of 1.1 + 0.1 eV. The success of the methods in arriving at both the accepted value of Ez and also at a sensible value of the bias means that the technique may be applied to other materials in which the parameters are not so well known. This method also allows the void sink strength to be separately considered in the analysis. from which it is apparent that voids also have a small bias for interstitials.
d
h i L s V
dislocation hole ( = void) interstitial loop surface vacancv
References PI S.B. Fisher and R.J. White. Rad. Eff. 30 (1976) 17. F.A. Nichols,
Rad. Eff. 39 (1978) 169.
;Si S.B. Fisher, R.J. White and K.M. Miller. Phil. Mag. A40 Acknowledgements We would like to thank CEGB Berkeley Nuclear Laboratories for supporting this work. BC thanks the Nothern Ireland Dept. of Education for a research studentship.
Nomenclature B b D
ElYI k,
dislocation bias (- ), Burgers vector (m). diffusivity (m*/s). vacancy migration energy (eV), reciprocal mean free path (rmfp) for the surface sink, such that kf is a factor accounting for the
(1979) 239. [41 P.T. Heald and K.M. Miller. J. Nucl. Mater. 66 (1976) 107. 151 K.M. Miller and P.D. Bristowe, Phys. Stat. Sol.(b) X6 (1978) 93. [61 R.K. Ham, Phil. Mag. 6 (1961) 1183. [71 M.H. Yoo and J.O. Stiegler, Phil. Mag. 36 (1977) 1305. PI D.I.R. Norris. Phil. Mag. 22 (1970) 1273. 2 (1969) 89. 191 U.E. Wolff, Metallography IlO1 D.I.R. Norris. J. Nucl. Mater. 40 (1971) 66. PII P.B. Hirsch. A. Howie, R.B. Nicholson, D.W. Pashley and M.J. Whelan. Electron Microscopy of Thin Crystals (Butterworths. London, 1965) p. 243. WI J.F. Nankivell, Brit. J. Appl. Phys. 13 (1962) 126. P31 G. Dlubek, 0. Brumonev and P. Sickert. Phys. Stat. Sol (A). 39 (1978) 169. (141 M. Kiritani and H. Takata, J. Nucl. Mater 69170 (1978) 277. and A. Silvent, Proc. Int. Conf. on Irradia1151 J.E.Harbottle
B. Cochrane et al. / Loop -growth and void -swelling in nickel tion Behaviour of Metallic Materials for Fast Reactor Core Materials, Eds. J. Poirier and J.M. Dupouy (CEA. Paris, 1979) p. 391. [16] M.H. Wood, B. Jones and S.M. Pierce, Res. Mechanica 3 (1981) 283. [17] SM. Pierce, R. Bullough and M.H. Wood, Res. Mechanica 4 (1982) 191. [18] K.M. Miller, S.B. Fisher and R.J. White, J. Nucl. Mater. 110 (1982) 265.
87
[19] W. Wycsik and M. Feller-Kniepmeyer. Phys. Stat. Sol. (a) 37 (1976) 183. [20] S.K. Khanna and K. Sonnenberg, Rad. Eff. 59 (1981) 91. [21] K.M. Miller, J. Nucl. Mater. 115 (1983) 216. [22] L.K. Mansur and W.G. Wolfer, J. Nucl. Mater 69&70 (1978) 825.