E-Journal of Advanced Maintenance Vol.7-2 (2015) 160-165
Japan Society of Maintenology

Numerical Evaluation of Material’s Degradation under Various Irradiation Conditions
Toshiki NAKASUJI1,*, Yasunori YAMAMOTO1 and Kazunori MORISHITA2
1
2

Graduate School of Energy Science, Kyoto University, Kyoto 611-0011, Japan
Institute of Advanced Energy, Kyoto University, Kyoto 611-0011, Japan

ABSTRACT
Structural materials during a use in a fission reactor are subjected to irradiation by high energy neutrons, andtherefore the properties, functions, and performances of materials become gradually degraded. Since keeping
material’s integrity is a key to success towards a stable operation of reactors, materials degradation due toirradiation should be taken into account when reactor design and maintenance are considered. One of the issues
to be solved for realization of a long term operation is how the integrity of materials in use is ensured for a longtime beyond an actually-experienced operation period. To do this, the future ageing behavior of materials should
be understood in advance. Usually, an attempt is made to understand the behavior using the existing irradiation
facilities with accelerated irradiation environments. In many cases, however, material’s behavior depends muchon acceleration coefficients and its dependency is not so simple. To overcome such a difficulty, a numericalsimulation study is employed. In the present study, reaction rate analysis is performed to investigate material
microstructure changes under various irradiation conditions. Our results have shown that volume swelling due toirradiation is very much different depending on irradiation conditions provided by such irradiation facilities as
HFIR, JOYO, KUR, 14MeV fusion neutrons. Based on the results, irradiation correlation methodology is
discussed to predict material’s degradation under arbitrary irradiation conditions using the actual existing dataobtained with accelerated irradiation conditions.
KEYWORDS

ARTICLE INFORMATION

Radiation damage, Rate theory, Defect cluster, Neutron spectrum,
Irradiation correlation

Article history:
Received 5 December 2014
Accepted 4 June 2015

1.

Introduction

In fission reactors, component materials are exposed to irradiation by high energy neutrons,
where athermal point defects (vacancies and self-interstitial atoms (SIAs)) are produced by atomic
displacements. Those defects may diffuse freely in the material depending on temperatures, and form
such clusters as voids (vacancy clusters) and dislocation loops (SIA clusters), which will cause
microstructural changes and accompanying macroscopic property changes (e.g., volume swelling,
ductility loss, and embrittlement). To keep the integrity of reactors, material degradation due toirradiation should be considered when reactor design and maintenance are considered.
In Japan, the operational period of fission reactors is restricted primarily less than 40 years by thesafety regulation revised in 2012. The operational period can be extended until 60 years provided thatreactor integrity is validated by performing so-called the special inspection. Since the operational
experiences of Japanese commercial reactors are around 40 years at the longest, actual data ofmaterial degradation do not exist for longer irradiations than 40 years. Material’s behavior due to suchan unexperienced, longer term irradiation must be obtained by extrapolating the data obtained at otheraccelerated irradiation facilities.
Dpa (displacement per atom) is one of the most important parameters to characterize irradiation
conditions. Dpa indicates an irradiation dose that is obtained using incident energy spectra, incident
flux, and cross section of reactions between incident particles and target materials. Dose rate (dpa/s) is
usually employed as an acceleration coefficient in the research field of materials irradiation.
*

Corresponding author, E-mail: t-nakasuji@iae.kyoto-u.ac.jp

ISSN-1883-9894/10 © 2010 – JSM and the authors. All rights reserved.

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T. Nakasuji, et al./ Numerical Evaluation of Material’s Degradation
under Various Irradiation Conditions

Unfortunately, however, dpa is not linearly correlated with actual materials degradation due toirradiation, which cannot always be a good index for describing materials degradation. Namely, evenwhen dpa values are identical, microstructural and mechanical property changes are different
depending on irradiation facilities employed. In the present study, a relationship between dpa (ordpa/s) and material’s microstructure in ferritic steel is theoretically evaluated, where defect production
and cluster formation processes are simulated using the reaction rate theory equations.

2.

Quantification of irradiation condition

2.1. Numerical method
Various irradiation facilities were quantified by the SPECTER code [1], which can calculate thedpa rate, PKA (primary knock-on atom) energy spectra, gas production rate, and total damage energy(Kerma) for 41 material elements, using the input data of incident neutron energy spectra ranging
from 10-10 to 20 MeV. All nuclear reaction cross-section data was obtained from ENDF/B-V
distributed by Brookhaven National Laboratory. The target element for the evaluation is iron, which isa major element of steel.
2.2. dpa rate of various irradiation facilities
The dpa rates were calculated by the SPECTER code for such existing irradiation facilities as
HFIR (The High Flux Isotope Reactor at ORNL), JOYO (Experimental fast reactor at JAEA), KUR(Kyoto University Reactor), JMTR (Japan Materials Testing Reactor), and 14 MeV fusion neutrons.
Figure 1 shows the dpa rate as a function of neutron flux for various irradiation facilities. As
shown in the figure, neutron flux and the corresponding dpa rate are quite different depending on
irradiation facilities. When the data of the core region of reactors is however focused, the dpa rate is in
proportion to the neutron flux. By means of the least squares method, a relationship between them isobtained as follows;P  7.9  1026 ,where P is the dpa rate in dpa/s,  is the neutron flux in n/m2/s. In the following section, theinfluence of dpa rate on microstructural changes in steel is discussed based on the rate theory analysis.

DPA rate, P [dpa/s]

10-4
10-5
10-6
10

Monochromatic neutron 10MeV
1MeV
SPECTER Code
0.1MeV
Material : Fe

-7

Fusion(14MeV)

HFIR
JMTR

10-8
10-9
10

KUR
JOYO
(fuel area)

-10

10-11
10-12
10-13

Approx. line
-26
JOYO P=7.9×10 Φ(outer wall)

10-14 11101013 1015 1017 1019 1021
Neutron flux (E>0.1MeV), Φ [n/m2/s]

Fig. 1

3.
3.1.

DPA rate (dpa/s) as a function of neutron flux for various irradiation facilities

Simulation of defect cluster formation process
Reaction rate theory model of defect accumulation processes161

T. Nakasuji, et al./ Numerical Evaluation of Material’s Degradation
under Various Irradiation Conditions

We have developed the defect cluster formation model on a basis of the reaction rate theory,
where defect energetics obtained by the molecular dynamics study [2] was employed. With theenergetics thus employed, the nucleation process of defect clusters is appropriately described andhence, for example, the critical size for nucleation, an important parameter in the nucleation theory,can be defined dynamically during calculations and appropriately incorporated into the model,
although the conventional nucleation model [3] was described with only a simple, inflexible
assumption. Model assumptions employed here are as follows.






Only Frenkel pairs are produced by atomic displacements.
Only single vacancies and self-interstitial atoms (SIAs) are mobile.
An SIA cluster is immobile.
A vacancy and an SIA are mutually annihilated by recombination reaction.
An interstitial-type dislocation loop (an SIA cluster) has disk shape, and a void (vacancy
cluster) has spherical shape.
 Dislocation lines play a role as a permanent sink for mobile point defects.
 The dislocation bias effect is considered in which a dislocation line prefers absorbing SIAs toabsorbing vacancies. Dislocation loops created during irradiation have also the bias effect.
Using the assumptions above, reactions of point defect diffusion, defect clustering,recombination reaction, and absorption to sinks were considered. Rate equations were simultaneously
solved using the Gear method [4] to obtain time evolution of defect clusters. The defect clusters
created during irradiation were followed for every cluster size of up to a 500-member cluster. The
clusters greater than the 500-member cluster are all classified into one group category, where only theaverage concentration and average size are defied for the group. Note that, in order to understand theeffect of dpa rate clearly, so-called the effect of cascade clusters on microstructural evolution was
regarded beyond the scope of the present study. The cascade cluster effect is under investigation andpublished in the near future. In addition, the effect of 1D-motion of SIA clusters is not also consideredhere because our target is ferritic steel that contains a lot of impurity atoms which may play a role as
an obstacle for the movement [5].
3.2.

Time evolution of defect concentrations

Figure 2 shows the time evolution of defect concentrations when the dpa rate is fixed at 1 × 10-8
dpa/s and temperature is 573 K. In the early stage of irradiation (around 10-5 s), the concentrations ofDose [dpa]Dose [dpa]10-1910-1710-1510-1310-1110-9 10-7 10-5 10-3 10-1 101

10-1910-1710-1510-1310-1110-9 10-7 10-5 10-3 10-1 101
10-2

10-2

10-8
10-10
10-12
10-14
10-16
10-18
-20

10

V1

1
2
10
100

Defect concentration

Defect concentration

Dose rate: 10-8 dpa/s

10-4 Temperature: 473 K
10-6 Cv Ci Cluster size

V2
V10

V100

I1
I2
I10

I100

10-6
10-8

Dose rate: 10-8 dpa/s
Cv

Ci

V1

Temp.
673 K
573 K
473 K

10-10
10-12
10-14
10-16

I1

10-18
10-20
10-22 -11 -9 -7 -5 -3 -1 1 3 5 7 9
10 10 10 10 10 10 10 10 10 10 10

10-22 -11 -9 -7 -5 -3 -1 1 3 5 7 9
10 10 10 10 10 10 10 10 10 10 10Time [s]Time [s]Fig. 2

10-4Time evolution of defect concentrationfor 1×10-8dpa/s and 573 K

Fig. 3 Time evolution of point defectconcentration at the temperature of 473 K,
573 K, and 673 K.

162

T. Nakasuji, et al./ Numerical Evaluation of Material’s Degradation
under Various Irradiation Conditions

point defects increase by atomic displacements. And then, SIAs are absorbed by dislocation lines anddislocation loops start to form at 10-5 - 101 s. Finally, after voids are formed at 10-1 - 107 s, defectconcentrations are saturated.
Figure 3 shows the time evolution of point defect concentrations at temperatures of 473 K, 573 K,
and 673 K. The concentrations vary depending on temperature, even when the dpa rate is fixed at thesame value. Vacancy concentration increases with decreasing irradiation temperature at the late stageof irradiation. This is because defect annihilation at a permanent sink takes less frequently due to lessmobility of point defects under lower temperatures.

3.3.

Vacancy supersaturation and void critical size

In order to understand defect accumulation processes in more detail, defect diffusion flux in thematerial matrix was defined by the product of diffusion coefficient (D) and defect concentration (C),where the diffusion coefficient is expressed by E D  exp   m  .
 kT 
Here, D is the diffusion coefficient, Em is the migration energy, T is temperature, k is the Boltzmann
constant. Using the defect diffusion flux defined here, vacancy supersaturation was defined by adifference between the fluxes of vacancies and SIAs in the material matrix, as follows;
SV 

DVCV  DICI ,DVCV eq

where CV eq is the equilibrium vacancy concentration. The subscript V and I denotes a vacancy and an
SIA, respectively. When the value of vacancy supersaturation is positive, the vacancy flux is greaterthan the SIA flux, in which voids grow easily.
Figure 4 shows the time evolution of vacancy supersaturation. In the early stage of irradiation,
the vacancy supersaturation is negative, because the mobility of an SIA is higher than that of avacancy. And then, the vacancy supersaturation turns to positive by the effect of dislocation bias, inwhich voids start to nucleate and grow up stably. The figure shows that it takes longer time to reachpositive values when the mobility of point defects is lower at lower temperatures. Figure 5 shows thetemperature dependence of vacancy supersaturation as a
Dose [dpa]function of dpa rate. The dose is fixed at 1 dpa. At low
10 10 10 10 10 10 10 10 10 10 10
temperatures the vacancy supersaturation takes a negative
15
Dose rate: 10 dpa/s
value. In contrast, the vacancy supersaturation is positive10
at high temperatures, where the values slightly decrease
Temperature
473 K
5
with increasing temperature.
573 K
673K
The vacancy supersaturation is only an index that0presents a balance between the influxes of vacancies and-5
SIAs to a void. In order to understand void nucleation-10
process precisely, the thermal stability of voids should
also be considered. When those considerations are made,-15
the critical size of void nucleation is evaluated as a-20function of temperature and dpa rate, as shown in Figure
10 10 10 10 10 10 10 10 10 10 10Time [s]6. The critical size is an index showing how easy void
nucleation takes place. When temperature is larger, theFig. 4 Time evolution of vacancy
critical size becomes larger, indicating that a void issupersaturationdifficult to nucleate.
-19

-17

-15

-13

-11

-9

-7

-5

-3

-1

1

-3

-1

1

3

5

7

9

Sv
sgn(Sv)×log(|Sv|)

-8

-11

163

-9

-7

-5

T. Nakasuji, et al./ Numerical Evaluation of Material’s Degradation
under Various Irradiation Conditions
20
-13

Sv
sgn(Sv)×log(|Sv|)

10

dpa/s

20

at 1 dpa
10-10 dpa/s
10-7 dpa/s

Critical size of void30

10-4 dpa/s

10
0
-10
-20
-30
300

400

500

600

at 1 dpa
15

1010-13 dpa/s

510-7 dpa/s
10-4 dpa/s

0
300

700

400

500

600

700

Temperature [K]Temperature [K]Fig. 5 Temperature dependence of vacancy
supersaturation as a function of dpa rate

3.4.

10-10 dpa/s

Fig. 6 Temperature dependence of void criticalsize for nucleation as a function of dpa rate

Average diameter and number density of voids, and volume swelling

Figure 7 shows temperature dependence of void number density as a function of dpa rate. As
shown in the figure, the number density of voids has a peak against temperatures. At the lower
temperature side, a void hardly nucleates due to low mobility of vacancies. While, at the higher
temperature side, a void is thermally unstable in which the critical size is larger. Figure 8 shows thetemperature dependence of void average diameter as a function of dpa rate. The diameter becomeslarger when temperature increases. At a given temperature, the diameter is larger when the dpa rate
decreases. The results may indicate that a small number of large voids are created under hightemperatures, while a large number of small voids are formed to be distributed in the material at low
temperatures. This behavior is qualitatively consistent with experiments [6].
The swelling was calculated using the following equation,S  i  void ,where S is the calculated swelling, i is the average size of voids,  void is the number density ofvoids. Notice that, a void in consideration here is restricted for a relatively large void with diameter
greater than 1 nm.
Figure 9 shows the calculated void swelling as a function of temperature for various dpa rates.
Again, the swelling has a peak against temperature. When temperature is low enough, defectdiffusivity is too small, and hence, a void cannot form. With increasing temperatures, the diffusivity
gradually increases and a void becomes easy to be created. However, when temperature becomes too
high, the thermal stability of voids becomesdecreased, which results in less formation of10
at 1 dpa
voids. Thus, the peak is formed by a balance
10 dpa/s
10 dpa/s
between the defect diffusivity and void thermal
10
10 dpa/s
stability. Interestingly, moreover, the peak
10 dpa/s
10
temperatures depend on the dpa rate as shown in
the figure. The peak shifts to the higher
10
temperature side when the dpa rate increases.
This is reflected by the fact that above balance is
10changed depending on the dpa rate. Since thedpa rate represents an acceleration coefficient
10
300
400
500
600
700
that is a characteristic parameter of irradiation
Temperature[K]
facilities, the peak shift shown here should beconsidered when accelerated irradiation is Fig. 7 Temperature and DPA rate dependence of
number density of voids
performed.
27

Void density [m-3]

-13

26

-7

25

24

23

22

164

-10

-4

120
110 at 1 dpa
100
90
80
70
60
50
40
30
20
10
0
300
400

0.20
10-13 dpa/s
-13

10

dpa/s

10-10 dpa/s-7

10 dpa/s

0.10

4.

10-10 dpa/s
10-7 dpa/s

0.05

10-4 dpa/s

10-4 dpa/s

500

600

0.00
300

700

Temperature [K]Fig. 8at 1 dpa

0.15

Swelling [%]Void diameter [nm]T. Nakasuji, et al./ Numerical Evaluation of Material’s Degradation
under Various Irradiation Conditions

Temperature and dose rate dependence of the void average diameters

Fig. 9400

500

Temperature [K]

600

700

Void swelling as a function of temperature
for various DPA rate ranging from 10-13
dpa/s to 10-4 dpa/s.

Conclusion

Numerical simulations have been performed to clarify a relationship between irradiation
conditions and material degradation behavior. The results are as follows;1.

We developed numerical models for microstructural evolution due to irradiation, whichcontain the detail process of defect cluster nucleation.
2. In the early stage of irradiation, the vacancy supersaturation takes a negative value. And then,the vacancy supersaturation becomes positive due to dislocation bias effect, when voids start
to nucleate and grow.
3. The critical size of void nucleation increases with increasing temperature. When temperature
is fixed, the critical size increases with decreasing the dpa rate. The critical size is determinedby a balance between the vacancy supersaturation and the thermal stability of voids.
4. The void swelling has a peak against irradiation temperature, which shifts to the higher
temperature side when the dpa rate increases.

From these understandings obtained here, the following suggestions should be considered whenreactor design and maintenance are performed. Namely, when material degradation is evaluated byaccelerated irradiation facilities, it is necessary to consider a difference in peak temperatures for void
swelling.

Acknowledgement
This work was supported by JSPS KAKENHI Grant Number 25420907.

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[3] T. Yoshiie, Q. Xu, K. Sato, Nuclear Instruments and Methods in Physics Research Section B:, 303 (15),(2013), pp. 37-41.
[4] S. D. Cohen and A. C. Hindmarsh, Computers in Physics, 10 (2), (1996), pp. 138-143.
[5] A. Barbu and E. Clouet, Solid State Phenomena, 129, (2007), pp. 51-58.
[6] Jia, X., Dai, Y., Journal of Nuclear Materials, 356 (1-3), (2006), pp. 105-111.

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