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Enhanced diffusion under irradiation
262
Journal of Nuclear Materials 108 & 109 (1982) 262-266 North-Holland Publishing Company
ENHANCED DIFFUSION UNDER IRRADIATION Tomoo KIRIHARA Dept. of Nuclear Engineering
Nagoya University, Furo-cho, Chikusa-kr
Nagoya 464, Japan
Received 8 January 1982; accepted 1 February 1982.
Diffusion and creep under irradiation have been generally treated by kinetic theory. Thermodynamic concepts on those phenomena are described here. The work ( W) done to crystalline solids is expressed by the equation Q=TS*=-
W+ H*,
where H* and S* are the changes of enthalpy and entropy under irradiation. The thermodynamic functions depend on the energy, the dose rate and the dose of incident flux. The enhanced diffusion coefficient (D*) becomes, InD*-lnD=W/kT=-S/k+H*/kT.
For high energy particles such as fission fragments, H*%H+,
therefore, lnD*=(S+
-S*)/k.
For neutron damage, InD*=(S+-S*)/k-(H+-H*)/kT,
where the superscript+means activation parameter without irradiation. The above relations were applied to the diffusion of ‘95Auin Au, that of U in UO,, and the creep of 321 stainless steel under irradiation.
1. InMon
2. Theoretical
Theoretical consideration of enhanced or induced diffusion phenomena has been developed with kinetic rate theory since Damask and Dienes [I] proposed their theory. Under irradiation the thermal spike model proposed by Seitz and Koehler [2] has mainly been extended to irradiated fuel materials [3,4,5]. Thermodynamic consideration, however, have been disregarded. In a preceding paper [6] the irradiation creep of nuclear fuels was investigated based on thermodynamic concepts. The activation free energy, the activation enthalpy, and the activation entropy under irradiation were explained. The self-diffusion of ‘9sAu under neutron irradiation by Acker et al. (71, uranium diffusion in UO, under fission damage by Mat&e [8], and the creep of 321 stainless steel under 4 MeV proton irradiation by Hudson et al. [9] were investigated with above concepts.
Incident particles do thermodynamic work on a system (a crystalline solid) except heat generation. The particles produce knock-on atoms and leave vacancies. The knocked-out atoms become interstitials in the lattice matrix, clusters, dislocation loops and dislocations. These defects, including vacancies, vary the configurational entropy of the system. Such entropy change is denoted by S,*. Interstitials in the lattice matrix also increase the lattice parameter, and then the vibrational frequencies are decreased. Therefore, the vibrational entropy is decreased by an amount ST. Since the lattice is relaxed and may vacancies are produced, the enthalpy of vacancy migration should be changed by Hz. Since a
0022-3115/82/0000-0000/$02.75
0 1982 North-Holland
number of vacancies are produced with irradiation, the formation enthalpy decreases by the amount HF. When
the changes of the entropy and the enthalpy under irradiation are denoted by S* and H*, then the follow-
T. Kirihara/
Enhanceddiffusionunderirradiation
263
ing relations are obtained: ** s*=s~+s:
(1)
and H*=H;+H,*.
(2)
According to the first and the second laws of thermodynamics, TS*=
-W+H*.
(3)
Therefore, W=
-TS*+H*.
(4)
The free energy of diffusion in non-irradiated can be generally described as, G+ = -TS+
+H+,
system (5)
where G+, S+ and H+ are the activation free energy, the activation entropy and the activation enthalpy of diffusion ***. From eq. 4 the activation free energy of diffusion under irradiation (G+ *) can be expressed as: G+*=G+-W = -T(S+
-S*)
+ (H+
-H*).
(6)
Accordingly, the diffusion coefficient under irradiation (D*) becomes
Fig. 1. ,Schematicpresent+ion of the enhanced diffusion coefficient (D*) for high and low energy damage: H*c If+ and H*= H+.
and H*
lnD*=ln(va2)+(S+-S*)/k-(H+-H*)/kT,(7) where Y and a are the vibrational frequency of atoms and the nearest neighbor distance of atoms for nonirradiated state, respectively. In eq. (7) the changes of YU~and of the correlation factor are included in S*: lnD* - 1nD = W/kT
H*/k : inclination of the line WlkT
(8)
/
and W/kT
/
= - S*/k
+ H*/kT.
(9)
As a consequence, W/kT, S*/k and H*/kT are obtained from experimental values of D* and D. At a constant dose rate and dose of incident particles, when diffusion takes place by the same mechanism, e.g. vacancy mechanism, in a certain temperature range in which there is no recovery under irradiation, the values of S* and H* become constant. In such a case the relations between In D* and In D for H* = H+
/’
1’
/’ /’ ,/’ E /’ r,/
/ /” I’ i /’ fs”,k
l * For simplicity the delta (A) is omitted. *** If the mechanism of diffusion under irradiation is different from that of the non-irradiated state, such as interstitial diffusion, the same procedure can be applied in the case where H* and S* are constant because the energy relation for the non-irradiated state is used only as a reference state.
l/T
Fig. 2. Schematic presentation of W/kT vs. l/T, entropy changes P/k, enthalpy change H* and critical temperature T, under irradiation are indicated.
T. Kirihara /
264
Enhanced diffusion under irradiation 694 II 765.7
3. Appliitions to experimental data !i
3.1. Enhanced diffusion of t9’Au in Au
vi-
S/k=
S* = 1.62 X 10e3 eV/K (156 J/K - mol)
O
and H* = 1.24eV (119.5 KJ/mol). i
= 1nD + W/kT
694
613
llTV04
+ (-0.5
1
Fig. 4. W/&T vs. l/T calculated froth self-diffusion coefficients of Au under neutron irradiation [7].
eV/kT). 451
533
I 25
I;a78
1.74eV/KT,
the equation for D* is obtained referring to eq. (7) as:
= -21.95
I 20
,
Since thermal diffusion data are expressed as,
lnD*(cm2s-‘)
K
18.8
I' ,' 0 8' Tc l ' J IIQ I "15 , I
Therefore,
= -3.17-
457 1
Tc .766K
= - 18.78 + 1.239 eV/kT.
lnD(cm2s-‘)
533 I
5
Self-diffusions of ‘95Au under neutron damage was investigated from 417 to 694 K by Acker et al. [7). In their experiment the dose rate was constant (6 X 10 16/m2 s) and the doses were nearly constant (1.3 - 1.7 X 1023/m2) for six specimens at various temperatures. Thermal diffusion data were also indicated. These results are illustrated in fig. 3. The calculated values of W/kT vs. l/T are obtained from eq. (9) with four data points below 533 K, and are represented in fig.4. The equation for W/kT is expressed as: W/kT
613 I
In fig. 4 the values of W/kT
at higher temperature (613 and 694 K) show lower values than the above equation. These results indicate that there is a certain recovery step of defects between 533 and 613 K.
K
3.2. Enhanced diffurion of U in UO, Enhanced diffusion of U in UO, and Pu in (U, Pu)O, under fission damage was extensively studied by Mat&e and his colleagues [8]. An almost constant value of the diffusion coefficient of uranium in UO, was obtained for a normalized fission rate of 5 X lo’* fission/m3s below 1000°C. The average value of the diffusion coefficient was found to be 5.7 X 10-l’ cm2/s from his figure. Since his data at 800 and 130°C were located close to the average line and, in addition, these two specimens were irradiated to almost the same dose (- 1.5 X 1O22 fission/&) [lo], the values of W/kT below 800°C (1073 K) were obtained using the thermal diffusion coefficient recommended by Matzke [ 111: L
I
15
I
20
I
25
lITa110
Fig. 3. Self-diffusion coefficient of ‘95Auin Au [7].
L
lnD(cm2s-‘)
= In 0.26 - 5.61 eV/kT.
The calculated values of W/kT,
S/k,
H* and TC are
T. Kirihara / lm 40-
1lOo
therefore corresponds to the thermal diffusion coefficient at T,, but the activation entropy is quite different from the thermal one.
K
600
265
Enhanced diffusion under irrodiotion
1606
30-
3.3. Irradiation protons
Hudson et al, [9] reported an irradiation creep of 32 1 stainless steel with 4 MeV protons. They obtained an activation energy above 520°C. which was similar to the energy of self-diffusion of Ni, and indicated a slight temperature dependence between 400 and 500°C for the irradiated 321 stainless steel (60% cold-worked) with 100 MN applied stress. They also showed a thermal creep in their text. From those data, thermal creep was obtained;
10 Tc ,I’ I,, ,
0. I
10 -
I’
/I Tc ~1606
//’ 20-
30-
,‘I
#’
K
d/k=36.0 <.
#’
541.3KJlmol (5.6levlatom
creep of 321 stainless steel with 4 MeV
1
’
Ini (h-l)
::36.03
= 1nA + S+/k = -31.1
40-
If it could be assumed that the irradiation creep was expressed in a similar manner to the above relation, W/kT and the other thermodynamic function were obtained as;
50-
I
I
1
5
lo
15
- 3.06 eV/kT
- 3.06 eV/kT.
l/Tr
Fig. 5. W/kTvs. l/Tcalculated from enhanced diffusion of U in UO, under fission damage [8].
H+ = 2.8 eV (295 KJ/mol),
S* = 3.35 X 10e3 eV/K (323.45 J/K.
mol),
and T, = 838 K.
in fig. 5. The values of H+ are identical to as expected from the above procedure. The value of S* was 3.10 X 10m3 eV/K (299.5 J/K. mol). Actually, the observed values of the diffusion coefficient of U in UO, at 130 and 800°C irradiated under the above conditions were almost the same. Therefore, the relation H* = H+ is satisfied. Considering eq. (2):
illustrated
H+,
H;-H;=O,
(10)
and Hz-Hz=O,
(11)
where H: and Hz are the activation enthalpies (energies) of vacancy formation and of migration. The vacancy concentration estimated as the “UO” vacancy from the measurements of the lattice parameter and the density change was about 0.2 at.% at I X 102* fission/m’ at a temperature between 50 and 200°C [ 121. Accordingly, it is reasonable that the formation energy of vacancies under the above irradiation conditions is zero, even if the temperature is 13O’C. The migration enthalpy (energy) is very low and nearly equal to zero. The diffusion coefficicient of the above conditions
Even if any mechanisms were operated under irradiation, the values of the above thermodynamic functions were correct.
4. Conclusion For enhanced diffusion under irradiation a thermodynamic consideration was proposed. More experimental works, in which the dose rate and the dose are constant for various temperature, are needed in order to clarify the changes in the thermodynamic functions (H* and S*) due to irradiation.
Acknowledgement The author is indebted to Dr. Hj. Matzke (Institute for Transuranium Elements, EUR) for his continuous interest and valuable discussion in connection with this study. Thanks are also due to Drs. H. Blank, C. Ron& and Prof. J. Takamura for their valuable discussions. This work was done as a part of the International Joint
266
T. Kirihara /
Enhanced diffurion under irradiation
Research sponsored by the Japan Society for the Promotion of science.
151H. Blank, EUR 6600 EN (1979) p. 307. 161T. Kirihara, N. Obata and N. Nakae, J. Nucl. Mater. 90 (1980) 148.
[71 D. Acker, M. Bexeler, G. Brebec and M. Spin, in: J. References [I] A.C. Damask and G.J. Dienes, in: Point Defects in Metals (1963) (Gordon and Breach, 1963). [Z] F. Seitx and J.S. Koehler, in: Solid State Physics, Eds. F. Seitx and D. Tumbull, vol. 2 (1956) p. 305. [3] D. Bmcklacher and W. Dienst, J. Nucl. Mater. 42 (1977) 285. [4] A. Hbh and Hj. Mat&e, Nucl. Mater. 48 (1973) 157.
Gittus, Irradiation Effects in Crystalline Solids (Applied Science Publishers, London) p. 313. V’l Hj. Mat&e, EUR 6600 EN (1979) p. 289. [91 J.A. Hudson, R.S. Nelson and R.J. McEloy, J. Nucl. Mater. 65 (1977) 279. 1101Hj. Matxke, Private communication. 1111Hj. Matxke, in: Pu 1975 and Other A&tides (1976) p. 801. [12] N. Nakae, Y Iwata and T. Kirihara, J. Nucl. Mater. 80 (1979) 314.
Journal of Nuclear Materials 108 & 109 (1982) 262-266 North-Holland Publishing Company
ENHANCED DIFFUSION UNDER IRRADIATION Tomoo KIRIHARA Dept. of Nuclear Engineering
Nagoya University, Furo-cho, Chikusa-kr
Nagoya 464, Japan
Received 8 January 1982; accepted 1 February 1982.
Diffusion and creep under irradiation have been generally treated by kinetic theory. Thermodynamic concepts on those phenomena are described here. The work ( W) done to crystalline solids is expressed by the equation Q=TS*=-
W+ H*,
where H* and S* are the changes of enthalpy and entropy under irradiation. The thermodynamic functions depend on the energy, the dose rate and the dose of incident flux. The enhanced diffusion coefficient (D*) becomes, InD*-lnD=W/kT=-S/k+H*/kT.
For high energy particles such as fission fragments, H*%H+,
therefore, lnD*=(S+
-S*)/k.
For neutron damage, InD*=(S+-S*)/k-(H+-H*)/kT,
where the superscript+means activation parameter without irradiation. The above relations were applied to the diffusion of ‘95Auin Au, that of U in UO,, and the creep of 321 stainless steel under irradiation.
1. InMon
2. Theoretical
Theoretical consideration of enhanced or induced diffusion phenomena has been developed with kinetic rate theory since Damask and Dienes [I] proposed their theory. Under irradiation the thermal spike model proposed by Seitz and Koehler [2] has mainly been extended to irradiated fuel materials [3,4,5]. Thermodynamic consideration, however, have been disregarded. In a preceding paper [6] the irradiation creep of nuclear fuels was investigated based on thermodynamic concepts. The activation free energy, the activation enthalpy, and the activation entropy under irradiation were explained. The self-diffusion of ‘9sAu under neutron irradiation by Acker et al. (71, uranium diffusion in UO, under fission damage by Mat&e [8], and the creep of 321 stainless steel under 4 MeV proton irradiation by Hudson et al. [9] were investigated with above concepts.
Incident particles do thermodynamic work on a system (a crystalline solid) except heat generation. The particles produce knock-on atoms and leave vacancies. The knocked-out atoms become interstitials in the lattice matrix, clusters, dislocation loops and dislocations. These defects, including vacancies, vary the configurational entropy of the system. Such entropy change is denoted by S,*. Interstitials in the lattice matrix also increase the lattice parameter, and then the vibrational frequencies are decreased. Therefore, the vibrational entropy is decreased by an amount ST. Since the lattice is relaxed and may vacancies are produced, the enthalpy of vacancy migration should be changed by Hz. Since a
0022-3115/82/0000-0000/$02.75
0 1982 North-Holland
number of vacancies are produced with irradiation, the formation enthalpy decreases by the amount HF. When
the changes of the entropy and the enthalpy under irradiation are denoted by S* and H*, then the follow-
T. Kirihara/
Enhanceddiffusionunderirradiation
263
ing relations are obtained: ** s*=s~+s:
(1)
and H*=H;+H,*.
(2)
According to the first and the second laws of thermodynamics, TS*=
-W+H*.
(3)
Therefore, W=
-TS*+H*.
(4)
The free energy of diffusion in non-irradiated can be generally described as, G+ = -TS+
+H+,
system (5)
where G+, S+ and H+ are the activation free energy, the activation entropy and the activation enthalpy of diffusion ***. From eq. 4 the activation free energy of diffusion under irradiation (G+ *) can be expressed as: G+*=G+-W = -T(S+
-S*)
+ (H+
-H*).
(6)
Accordingly, the diffusion coefficient under irradiation (D*) becomes
Fig. 1. ,Schematicpresent+ion of the enhanced diffusion coefficient (D*) for high and low energy damage: H*c If+ and H*= H+.
and H*
lnD*=ln(va2)+(S+-S*)/k-(H+-H*)/kT,(7) where Y and a are the vibrational frequency of atoms and the nearest neighbor distance of atoms for nonirradiated state, respectively. In eq. (7) the changes of YU~and of the correlation factor are included in S*: lnD* - 1nD = W/kT
H*/k : inclination of the line WlkT
(8)
/
and W/kT
/
= - S*/k
+ H*/kT.
(9)
As a consequence, W/kT, S*/k and H*/kT are obtained from experimental values of D* and D. At a constant dose rate and dose of incident particles, when diffusion takes place by the same mechanism, e.g. vacancy mechanism, in a certain temperature range in which there is no recovery under irradiation, the values of S* and H* become constant. In such a case the relations between In D* and In D for H* = H+
/’
1’
/’ /’ ,/’ E /’ r,/
/ /” I’ i /’ fs”,k
l * For simplicity the delta (A) is omitted. *** If the mechanism of diffusion under irradiation is different from that of the non-irradiated state, such as interstitial diffusion, the same procedure can be applied in the case where H* and S* are constant because the energy relation for the non-irradiated state is used only as a reference state.
l/T
Fig. 2. Schematic presentation of W/kT vs. l/T, entropy changes P/k, enthalpy change H* and critical temperature T, under irradiation are indicated.
T. Kirihara /
264
Enhanced diffusion under irradiation 694 II 765.7
3. Appliitions to experimental data !i
3.1. Enhanced diffusion of t9’Au in Au
vi-
S/k=
S* = 1.62 X 10e3 eV/K (156 J/K - mol)
O
and H* = 1.24eV (119.5 KJ/mol). i
= 1nD + W/kT
694
613
llTV04
+ (-0.5
1
Fig. 4. W/&T vs. l/T calculated froth self-diffusion coefficients of Au under neutron irradiation [7].
eV/kT). 451
533
I 25
I;a78
1.74eV/KT,
the equation for D* is obtained referring to eq. (7) as:
= -21.95
I 20
,
Since thermal diffusion data are expressed as,
lnD*(cm2s-‘)
K
18.8
I' ,' 0 8' Tc l ' J IIQ I "15 , I
Therefore,
= -3.17-
457 1
Tc .766K
= - 18.78 + 1.239 eV/kT.
lnD(cm2s-‘)
533 I
5
Self-diffusions of ‘95Au under neutron damage was investigated from 417 to 694 K by Acker et al. [7). In their experiment the dose rate was constant (6 X 10 16/m2 s) and the doses were nearly constant (1.3 - 1.7 X 1023/m2) for six specimens at various temperatures. Thermal diffusion data were also indicated. These results are illustrated in fig. 3. The calculated values of W/kT vs. l/T are obtained from eq. (9) with four data points below 533 K, and are represented in fig.4. The equation for W/kT is expressed as: W/kT
613 I
In fig. 4 the values of W/kT
at higher temperature (613 and 694 K) show lower values than the above equation. These results indicate that there is a certain recovery step of defects between 533 and 613 K.
K
3.2. Enhanced diffurion of U in UO, Enhanced diffusion of U in UO, and Pu in (U, Pu)O, under fission damage was extensively studied by Mat&e and his colleagues [8]. An almost constant value of the diffusion coefficient of uranium in UO, was obtained for a normalized fission rate of 5 X lo’* fission/m3s below 1000°C. The average value of the diffusion coefficient was found to be 5.7 X 10-l’ cm2/s from his figure. Since his data at 800 and 130°C were located close to the average line and, in addition, these two specimens were irradiated to almost the same dose (- 1.5 X 1O22 fission/&) [lo], the values of W/kT below 800°C (1073 K) were obtained using the thermal diffusion coefficient recommended by Matzke [ 111: L
I
15
I
20
I
25
lITa110
Fig. 3. Self-diffusion coefficient of ‘95Auin Au [7].
L
lnD(cm2s-‘)
= In 0.26 - 5.61 eV/kT.
The calculated values of W/kT,
S/k,
H* and TC are
T. Kirihara / lm 40-
1lOo
therefore corresponds to the thermal diffusion coefficient at T,, but the activation entropy is quite different from the thermal one.
K
600
265
Enhanced diffusion under irrodiotion
1606
30-
3.3. Irradiation protons
Hudson et al, [9] reported an irradiation creep of 32 1 stainless steel with 4 MeV protons. They obtained an activation energy above 520°C. which was similar to the energy of self-diffusion of Ni, and indicated a slight temperature dependence between 400 and 500°C for the irradiated 321 stainless steel (60% cold-worked) with 100 MN applied stress. They also showed a thermal creep in their text. From those data, thermal creep was obtained;
10 Tc ,I’ I,, ,
0. I
10 -
I’
/I Tc ~1606
//’ 20-
30-
,‘I
#’
K
d/k=36.0 <.
#’
541.3KJlmol (5.6levlatom
creep of 321 stainless steel with 4 MeV
1
’
Ini (h-l)
::36.03
= 1nA + S+/k = -31.1
40-
If it could be assumed that the irradiation creep was expressed in a similar manner to the above relation, W/kT and the other thermodynamic function were obtained as;
50-
I
I
1
5
lo
15
- 3.06 eV/kT
- 3.06 eV/kT.
l/Tr
Fig. 5. W/kTvs. l/Tcalculated from enhanced diffusion of U in UO, under fission damage [8].
H+ = 2.8 eV (295 KJ/mol),
S* = 3.35 X 10e3 eV/K (323.45 J/K.
mol),
and T, = 838 K.
in fig. 5. The values of H+ are identical to as expected from the above procedure. The value of S* was 3.10 X 10m3 eV/K (299.5 J/K. mol). Actually, the observed values of the diffusion coefficient of U in UO, at 130 and 800°C irradiated under the above conditions were almost the same. Therefore, the relation H* = H+ is satisfied. Considering eq. (2):
illustrated
H+,
H;-H;=O,
(10)
and Hz-Hz=O,
(11)
where H: and Hz are the activation enthalpies (energies) of vacancy formation and of migration. The vacancy concentration estimated as the “UO” vacancy from the measurements of the lattice parameter and the density change was about 0.2 at.% at I X 102* fission/m’ at a temperature between 50 and 200°C [ 121. Accordingly, it is reasonable that the formation energy of vacancies under the above irradiation conditions is zero, even if the temperature is 13O’C. The migration enthalpy (energy) is very low and nearly equal to zero. The diffusion coefficicient of the above conditions
Even if any mechanisms were operated under irradiation, the values of the above thermodynamic functions were correct.
4. Conclusion For enhanced diffusion under irradiation a thermodynamic consideration was proposed. More experimental works, in which the dose rate and the dose are constant for various temperature, are needed in order to clarify the changes in the thermodynamic functions (H* and S*) due to irradiation.
Acknowledgement The author is indebted to Dr. Hj. Matzke (Institute for Transuranium Elements, EUR) for his continuous interest and valuable discussion in connection with this study. Thanks are also due to Drs. H. Blank, C. Ron& and Prof. J. Takamura for their valuable discussions. This work was done as a part of the International Joint
266
T. Kirihara /
Enhanced diffurion under irradiation
Research sponsored by the Japan Society for the Promotion of science.
151H. Blank, EUR 6600 EN (1979) p. 307. 161T. Kirihara, N. Obata and N. Nakae, J. Nucl. Mater. 90 (1980) 148.
[71 D. Acker, M. Bexeler, G. Brebec and M. Spin, in: J. References [I] A.C. Damask and G.J. Dienes, in: Point Defects in Metals (1963) (Gordon and Breach, 1963). [Z] F. Seitx and J.S. Koehler, in: Solid State Physics, Eds. F. Seitx and D. Tumbull, vol. 2 (1956) p. 305. [3] D. Bmcklacher and W. Dienst, J. Nucl. Mater. 42 (1977) 285. [4] A. Hbh and Hj. Mat&e, Nucl. Mater. 48 (1973) 157.
Gittus, Irradiation Effects in Crystalline Solids (Applied Science Publishers, London) p. 313. V’l Hj. Mat&e, EUR 6600 EN (1979) p. 289. [91 J.A. Hudson, R.S. Nelson and R.J. McEloy, J. Nucl. Mater. 65 (1977) 279. 1101Hj. Matxke, Private communication. 1111Hj. Matxke, in: Pu 1975 and Other A&tides (1976) p. 801. [12] N. Nakae, Y Iwata and T. Kirihara, J. Nucl. Mater. 80 (1979) 314.