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Effects of defect trapping and solute segregation on defect recombination rates and void swelling in irradiated alloys
Journal of Nuclear Materials 69 & 70 (1978) 821-824 0 North-Holland Publishing Company
EFFECTS OF DEFECT TRAPPING AND SOLUTE SEGREGATION RATES AND VOID SWELLING IN IRRADIATED
ON DEFECT RECOMBINATION
ALLOYS *
P.R. OKAMOTO, N.Q. LAM, H. WIEDERSICH and R.A. JOHNSON ** Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439,
1. Introduction
USA
with bound complexes. Since the resulting differential equations which describe the diffusion profiles of individual defects, i.e., free vacancies (V), free interstitials (I), free solutes (i), and bounds complexes and the details of the calculation procedures are given elsewhere in these Proceedings [S], only those aspects relevant to void swelling are discussed here.
Recent studies [l-5] of void formation in irradiated metals and alloys have shown that minor solute additions of substitutional elements which contract the lattice parameter of the host material tend to segregate to sinks such as voids, dislocation loops, and external surfaces. The authors have pointed out [l] that this radiation-induced segregation effect may be a consequence of trapping of interstitials by undersized solutes. Swanson and co-workers [6] have shown that stable mixed (100) dumbbell interstitials are formed in a number of alloys. The migration of mixed dumbbells to sinks will not only lead to the observed segregation effect but also to the depletion of the matrix of the strong trapping element. The present calculations were undertaken to investigate synergistic effects of trapping and of segregation on the temperature dependence of void swelling in irradiated alloys.
3. Effects of defect trapping The differential equations in ref. [8] can be used to investigate the effects of defect trapping alone, i.e., in the absence of segregation effects, by assuming all bound complexes to be immobile and by replacing the diffusion terms by sink loss terms linear in the defect concentrations. The differential equations then reduce to rate equations for the spatially averaged total defect concentrations (= free plus bound defects), for the I’s (CT) and V’s (C$), i.e.,
ac;r -at _ -K.
2. The model The present calculations are based on a kinetic model of radiation-induced solute segregation [7] which takes into account the effects of vacancy and interstitial binding to substitutional solutes on the diffusion of free defects, free solutes and bound solute-defect complexes to extended sinks. Also included are reaction terms for the formation and dissociation of bound complexes, the intrinsic recombination rates of free defects and the extrinsic recombination rates of free defects
ac$
at
=
- K&C:C$
,
-PI&F
K. - K&C’C;rC; - PVv$ [C; - (C;)EQ] , (1)
where
(C$yQ=C;Q[ 1 + 12Cf exy(H&/kT)] ) , T
_
aI -
* Work supported by the U.S. Energy Research and Develop ment Administration. ** Visiting scientist from the Materials Science Department, University of Virginia, Charlottesville, Virginia 22901.
aIv + 12aviCf exp(I!f&/k~) 1 + 12p exp(H&/k7)
T _ aIv + 42aEaC’F exp(H&/kT) av1 + 42Cf exp(HiJkT) 821
’
’
(2)
822
P.R. Okamoto et al. /Effects
vv = vv/[ 1 + 12Cie exp(HQJk~]
of
defect trapping
0
200
400
in irradiated alloys
.
Here Ke is the defect-production rate; aIV, LZV~, and Bria are geometrical factors related to the recombination volumes of the free defects, bound V’s and bound I’s, respectively; vI, vv and PI, Pv, respectively, are the jump frequencies and sink factors (= total sink annihilation probability per defect jump) for the free I’s and V’s; H& and Hti are the solute-interstitial and solute-vacancy binding energies; Cf is the initial free solute concentration and C’tQ is the thermal equilibrium free vacancy concentration. Eq. (1) is identical in form to the basic rate equations governing void growth in pure metals [9] except that the free defect recombination factors, jump freEQ have been replaced by the effective quencies, and Cv values given by eq. (2). The steady-state solution to eq. (1) leads to a general expression for the void swelling rate of the form d(AV/V)/dt = 6 Ke Pff, where 8 is a function of the sink factors and defect bias parameters and Seff = 1 - (K&C~C~/Ke) is the fraction of radiationproduced defects annihilating at sinks. The dominant temperature dependence of d(AV/ V)/dt is contained in Seff which, under appropriate conditions, reduces to the following low and high temperature approximations.
0.0
of solute segregation
600
800
(3) Eqs. (2) and (3) show that defect trapping promotes recombination and, hence, reduces Seff primarily as a consequance of two effects: (i) an apparent decrease in the defect jump frequancy and (ii) an apparent increase in the total thermal vacancy concentration. For V-trapping, i.e., when HE, = 0, and a$$ >>a&v$, effect (i) reduces Seff at low temperatures but not at high temperatures. Thus, effect (i) alone would tend to shift the swelling peak to higher temperatures as illustrated by the dashed curves in fig. la. The apparent increase in the thermal vacancy concentration, effect (ii), reduces Seff at high temperatures but not at low temperatures and thus, in the absence of effect (i) would cause the swelling peak to shift to lower temperatures as illustrated by the dotted curve in fig. la. For strong I-trapping, i.e., when HOi = 0 and arvr << aqv$, the apparent decrease in the interstitial mobility is responsible for reducing Seff at both low and high temperatures. Although the physical reasons for reducing‘Seff at high temperatures differ for the two trapping mechanisms the net effect is qualitatively the same, a reduction in
1000
Fig. 1. Temperaturedependenceof the swellingparameterSeff showingthe absenceof a temperatureshift in the swellingpeak. (a) for va~cy-taping and (b) for inte~titi~-~app~g. The dashed curve in (a} shows that the peak temperature shifts to higher temperature when only the decrease in vacancy mobility, or to lower temperature (dotted curve) when only the increase in the total equilibrium vacancy concentration is taken into account; both curves are for ff$ = 0.5 eV.
P.R. Oknmoto
et al. / Effects of defect trappingof sohte segregationin irradii?Eed a&vu
HTio (eVf
H& (eV1
to-51 0
0.34
0.58
0.4
0.8
823
0.82
1.06
1.2
1.6
0134
0.58
0.82
1.06
2,o
Hita (sVI Fig, 2, Steady-statesolute concentrative at the foil center as a function of solute-interstitia1 binding energy for several temperatures and defect production rates. (#a = IiF f zH&, L = 1W5 cm, Hv = 0.84 eV, 17; = 1.0 eV, and HP = 0.1 eV.)
Seff with little change in the peak swelling temperature, T, = SOO”C, as shown by the sohd curves in figs. la and lb. The main difference $ that a si~i~~ant reduction in Seff OCCUTS for V-trapping when H&r exceeds a few tenths of an eV whereas, for I-trapping, Jf& must exceed the difference, (I@ - HIM), in the free defect migration energies. The latter follows from the condition aTvT << a$~$, i.e., the recombination rate of moving vacancies with trapped interstitials must become competitive with the intrinisic recombination rate.
4. Effects of segregation By causing solute depletion of the matrix, solute segregation to sinks will reduce the effectiveness of defect trapping in promoting recombination. To investigate the effect of radiation-induced segregation on swelling, the differential equations of ref. [8] describing the segregation process were solved numerically for a thin foil geometry. The loss of free defects, free solutes and mobile complexes to the foil surfaces are described by diffusion currents rather than sink loss terms linear in the
defect concentrations. To simulate solute segregation to internal sinks in an infinite medium, the foil thickness was chosen so that the effective sink efficiency of the foil surfaces alone was approximately equivalent to the sink factors (PI = PV = 10W4) used to obtain fig. 1. Fig. 2 shows the steady-state solute concentration at the center of the foil in units of the initial solute concentration for the case of solute-interstitial binding. For these calculations the migration energy of the bound solute-interstitial complex was taken as Hpa = HP + 2~;~. The main points can be summarized as follows: * ( a ) maximum depletion occurs when &@a*I?: = 0.84 eV. Hence, maximum depletion occurs for solutes that trap interstitials up to stage III; (b) the temperature for maximum depletion occurs on the low temperature side of the peak swelling temperature Tp of the pure metal (Tp = 500°C for Ke = 1Oe3 dpa/s and Tt, = 325°C for Ke = 10M6 dpa/s); and (c) the extent of solute depletion is greater at low (reactor ir* A detailed interpretation of these effects is given in refs. 17’1 and [ 81 where it is also shown that little segregation OCCURS as a result of solute-vacancy
binding.
P.R. Okamoto et al. /Effects of defect trapping of solute segregation in irradiated alloys
824
“IO0
200
300
400
!TiOO 600
700
Tf”CI
tb) 7ol-
’
1
I
I
0 STEP-HEIGHT DATA
--I
fig. 3a for Ke = 10v3 dpafs, HRa = 1.5 eV for C$= 10W3 and lo-‘. The shift in the swelling peak to lower temperatures is a consequance of(b), i.e., maximum depletion occurs on the Iow temperature side of Tp’ Therefore, the pure metal swelhng peak is less suppressed at the low temperature side than at the high temperature side resulting in a peak shift to lower temperature. Such shifts have recently been observed by Johnston and co. workers [lo] in Si doped Fe-Cr-Ni ternary alloys during heavy-ion irradiations (see fig. 3b). Si has been shown to segregate to sinks in Fe-Cr-Ni alloys when irradiated with heavy-ions or with high energy electrons in an HVEM [ 1,2]. Johnston’s results are in qualitative agreen~ent with the present calculations (based on a combination of trapping and segregation effects) with regard to temperature, and compositional dependence of the swelling behavior of alloys which depend on minor solute additions to suppress void formation. An implication of the present calculations is that segregation, and hence, the suppression of void swelling by defect trapping is sensitive to the defect production rate. Thus, a doped alloy which is resistent to swelling during ion bombardment or irradiation in the HVEM, may be significantly less resistant to swelling under in-reactor service conditions.
References
[II P.R. Okamoto and H. Wiedersich. J. Nucl. Mat. 53 (1974) 336.
121P.R. Okamoto and H. Wiedersich, The Physics of Irradia575
625 TE~~A~RE
675
725
c”Ct
Fig. 3. (a) Seff at the center of the foil as a function of temperature for several solute-~nterstit~~ binding energies and initial sohrte concentrations. (Other parameters as in fig. 2.) (b) The effect of minor Si additions on the temperature dependence of swelling in Ni-ion irradiated Fe-Cr-Ni alloys. (After Johnston et al., ref. [lo]).
radiation) than at high displacement rates (heavy ion or HVEM irradiation). The effect of solute depletion on the swelling parameter Sef’ evaluated at the center of the foil is shown in
tion Produced Voids, R.S. Nelson, ed., AERE-R7934 (1975) p. 231. 131 A. Barbu and A.J. ArdeU, Scripta Met. 9 (197.5) 1233. f41 P.R. Okamoto, A. Taylor and H. Wiidersicb, Fundamental Aspects of Radiation Damage in Metals, M.T. Robinson and F.W. Young, Jr. eds, ERDA CONF-?5I~U6-F~, Vol. II (1976) p. 1188. 151 E.A. Kenik, Scripta Met. (in press). [61 M.L. Swanson, L.M. Howe and A.F. Quennevihe, Fundamental Aspects of Radiation Damage in Metals, MT. Robinson and F.W. Young, Jr., eds. ERDA CONF-751006Pl, Vol. I(lP76) p. 316. [71 R.A. Johnson and N.Q. Lam, Phys. Rev. 13 (1976) 4264. I81 R.A. Johnson and N.Q. Lam, this Conference. 191H. Wiedersich, Rad. Effects 12 (1972) 111. 1101W.G. Johnston, T. Lam&en, J.H. Rosolowski and A.M. Turkalo, 3. of Metals, 28 (1976) 19.
EFFECTS OF DEFECT TRAPPING AND SOLUTE SEGREGATION RATES AND VOID SWELLING IN IRRADIATED
ON DEFECT RECOMBINATION
ALLOYS *
P.R. OKAMOTO, N.Q. LAM, H. WIEDERSICH and R.A. JOHNSON ** Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439,
1. Introduction
USA
with bound complexes. Since the resulting differential equations which describe the diffusion profiles of individual defects, i.e., free vacancies (V), free interstitials (I), free solutes (i), and bounds complexes and the details of the calculation procedures are given elsewhere in these Proceedings [S], only those aspects relevant to void swelling are discussed here.
Recent studies [l-5] of void formation in irradiated metals and alloys have shown that minor solute additions of substitutional elements which contract the lattice parameter of the host material tend to segregate to sinks such as voids, dislocation loops, and external surfaces. The authors have pointed out [l] that this radiation-induced segregation effect may be a consequence of trapping of interstitials by undersized solutes. Swanson and co-workers [6] have shown that stable mixed (100) dumbbell interstitials are formed in a number of alloys. The migration of mixed dumbbells to sinks will not only lead to the observed segregation effect but also to the depletion of the matrix of the strong trapping element. The present calculations were undertaken to investigate synergistic effects of trapping and of segregation on the temperature dependence of void swelling in irradiated alloys.
3. Effects of defect trapping The differential equations in ref. [8] can be used to investigate the effects of defect trapping alone, i.e., in the absence of segregation effects, by assuming all bound complexes to be immobile and by replacing the diffusion terms by sink loss terms linear in the defect concentrations. The differential equations then reduce to rate equations for the spatially averaged total defect concentrations (= free plus bound defects), for the I’s (CT) and V’s (C$), i.e.,
ac;r -at _ -K.
2. The model The present calculations are based on a kinetic model of radiation-induced solute segregation [7] which takes into account the effects of vacancy and interstitial binding to substitutional solutes on the diffusion of free defects, free solutes and bound solute-defect complexes to extended sinks. Also included are reaction terms for the formation and dissociation of bound complexes, the intrinsic recombination rates of free defects and the extrinsic recombination rates of free defects
ac$
at
=
- K&C:C$
,
-PI&F
K. - K&C’C;rC; - PVv$ [C; - (C;)EQ] , (1)
where
(C$yQ=C;Q[ 1 + 12Cf exy(H&/kT)] ) , T
_
aI -
* Work supported by the U.S. Energy Research and Develop ment Administration. ** Visiting scientist from the Materials Science Department, University of Virginia, Charlottesville, Virginia 22901.
aIv + 12aviCf exp(I!f&/k~) 1 + 12p exp(H&/k7)
T _ aIv + 42aEaC’F exp(H&/kT) av1 + 42Cf exp(HiJkT) 821
’
’
(2)
822
P.R. Okamoto et al. /Effects
vv = vv/[ 1 + 12Cie exp(HQJk~]
of
defect trapping
0
200
400
in irradiated alloys
.
Here Ke is the defect-production rate; aIV, LZV~, and Bria are geometrical factors related to the recombination volumes of the free defects, bound V’s and bound I’s, respectively; vI, vv and PI, Pv, respectively, are the jump frequencies and sink factors (= total sink annihilation probability per defect jump) for the free I’s and V’s; H& and Hti are the solute-interstitial and solute-vacancy binding energies; Cf is the initial free solute concentration and C’tQ is the thermal equilibrium free vacancy concentration. Eq. (1) is identical in form to the basic rate equations governing void growth in pure metals [9] except that the free defect recombination factors, jump freEQ have been replaced by the effective quencies, and Cv values given by eq. (2). The steady-state solution to eq. (1) leads to a general expression for the void swelling rate of the form d(AV/V)/dt = 6 Ke Pff, where 8 is a function of the sink factors and defect bias parameters and Seff = 1 - (K&C~C~/Ke) is the fraction of radiationproduced defects annihilating at sinks. The dominant temperature dependence of d(AV/ V)/dt is contained in Seff which, under appropriate conditions, reduces to the following low and high temperature approximations.
0.0
of solute segregation
600
800
(3) Eqs. (2) and (3) show that defect trapping promotes recombination and, hence, reduces Seff primarily as a consequance of two effects: (i) an apparent decrease in the defect jump frequancy and (ii) an apparent increase in the total thermal vacancy concentration. For V-trapping, i.e., when HE, = 0, and a$$ >>a&v$, effect (i) reduces Seff at low temperatures but not at high temperatures. Thus, effect (i) alone would tend to shift the swelling peak to higher temperatures as illustrated by the dashed curves in fig. la. The apparent increase in the thermal vacancy concentration, effect (ii), reduces Seff at high temperatures but not at low temperatures and thus, in the absence of effect (i) would cause the swelling peak to shift to lower temperatures as illustrated by the dotted curve in fig. la. For strong I-trapping, i.e., when HOi = 0 and arvr << aqv$, the apparent decrease in the interstitial mobility is responsible for reducing Seff at both low and high temperatures. Although the physical reasons for reducing‘Seff at high temperatures differ for the two trapping mechanisms the net effect is qualitatively the same, a reduction in
1000
Fig. 1. Temperaturedependenceof the swellingparameterSeff showingthe absenceof a temperatureshift in the swellingpeak. (a) for va~cy-taping and (b) for inte~titi~-~app~g. The dashed curve in (a} shows that the peak temperature shifts to higher temperature when only the decrease in vacancy mobility, or to lower temperature (dotted curve) when only the increase in the total equilibrium vacancy concentration is taken into account; both curves are for ff$ = 0.5 eV.
P.R. Oknmoto
et al. / Effects of defect trappingof sohte segregationin irradii?Eed a&vu
HTio (eVf
H& (eV1
to-51 0
0.34
0.58
0.4
0.8
823
0.82
1.06
1.2
1.6
0134
0.58
0.82
1.06
2,o
Hita (sVI Fig, 2, Steady-statesolute concentrative at the foil center as a function of solute-interstitia1 binding energy for several temperatures and defect production rates. (#a = IiF f zH&, L = 1W5 cm, Hv = 0.84 eV, 17; = 1.0 eV, and HP = 0.1 eV.)
Seff with little change in the peak swelling temperature, T, = SOO”C, as shown by the sohd curves in figs. la and lb. The main difference $ that a si~i~~ant reduction in Seff OCCUTS for V-trapping when H&r exceeds a few tenths of an eV whereas, for I-trapping, Jf& must exceed the difference, (I@ - HIM), in the free defect migration energies. The latter follows from the condition aTvT << a$~$, i.e., the recombination rate of moving vacancies with trapped interstitials must become competitive with the intrinisic recombination rate.
4. Effects of segregation By causing solute depletion of the matrix, solute segregation to sinks will reduce the effectiveness of defect trapping in promoting recombination. To investigate the effect of radiation-induced segregation on swelling, the differential equations of ref. [8] describing the segregation process were solved numerically for a thin foil geometry. The loss of free defects, free solutes and mobile complexes to the foil surfaces are described by diffusion currents rather than sink loss terms linear in the
defect concentrations. To simulate solute segregation to internal sinks in an infinite medium, the foil thickness was chosen so that the effective sink efficiency of the foil surfaces alone was approximately equivalent to the sink factors (PI = PV = 10W4) used to obtain fig. 1. Fig. 2 shows the steady-state solute concentration at the center of the foil in units of the initial solute concentration for the case of solute-interstitial binding. For these calculations the migration energy of the bound solute-interstitial complex was taken as Hpa = HP + 2~;~. The main points can be summarized as follows: * ( a ) maximum depletion occurs when &@a*I?: = 0.84 eV. Hence, maximum depletion occurs for solutes that trap interstitials up to stage III; (b) the temperature for maximum depletion occurs on the low temperature side of the peak swelling temperature Tp of the pure metal (Tp = 500°C for Ke = 1Oe3 dpa/s and Tt, = 325°C for Ke = 10M6 dpa/s); and (c) the extent of solute depletion is greater at low (reactor ir* A detailed interpretation of these effects is given in refs. 17’1 and [ 81 where it is also shown that little segregation OCCURS as a result of solute-vacancy
binding.
P.R. Okamoto et al. /Effects of defect trapping of solute segregation in irradiated alloys
824
“IO0
200
300
400
!TiOO 600
700
Tf”CI
tb) 7ol-
’
1
I
I
0 STEP-HEIGHT DATA
--I
fig. 3a for Ke = 10v3 dpafs, HRa = 1.5 eV for C$= 10W3 and lo-‘. The shift in the swelling peak to lower temperatures is a consequance of(b), i.e., maximum depletion occurs on the Iow temperature side of Tp’ Therefore, the pure metal swelhng peak is less suppressed at the low temperature side than at the high temperature side resulting in a peak shift to lower temperature. Such shifts have recently been observed by Johnston and co. workers [lo] in Si doped Fe-Cr-Ni ternary alloys during heavy-ion irradiations (see fig. 3b). Si has been shown to segregate to sinks in Fe-Cr-Ni alloys when irradiated with heavy-ions or with high energy electrons in an HVEM [ 1,2]. Johnston’s results are in qualitative agreen~ent with the present calculations (based on a combination of trapping and segregation effects) with regard to temperature, and compositional dependence of the swelling behavior of alloys which depend on minor solute additions to suppress void formation. An implication of the present calculations is that segregation, and hence, the suppression of void swelling by defect trapping is sensitive to the defect production rate. Thus, a doped alloy which is resistent to swelling during ion bombardment or irradiation in the HVEM, may be significantly less resistant to swelling under in-reactor service conditions.
References
[II P.R. Okamoto and H. Wiedersich. J. Nucl. Mat. 53 (1974) 336.
121P.R. Okamoto and H. Wiedersich, The Physics of Irradia575
625 TE~~A~RE
675
725
c”Ct
Fig. 3. (a) Seff at the center of the foil as a function of temperature for several solute-~nterstit~~ binding energies and initial sohrte concentrations. (Other parameters as in fig. 2.) (b) The effect of minor Si additions on the temperature dependence of swelling in Ni-ion irradiated Fe-Cr-Ni alloys. (After Johnston et al., ref. [lo]).
radiation) than at high displacement rates (heavy ion or HVEM irradiation). The effect of solute depletion on the swelling parameter Sef’ evaluated at the center of the foil is shown in
tion Produced Voids, R.S. Nelson, ed., AERE-R7934 (1975) p. 231. 131 A. Barbu and A.J. ArdeU, Scripta Met. 9 (197.5) 1233. f41 P.R. Okamoto, A. Taylor and H. Wiidersicb, Fundamental Aspects of Radiation Damage in Metals, M.T. Robinson and F.W. Young, Jr. eds, ERDA CONF-?5I~U6-F~, Vol. II (1976) p. 1188. 151 E.A. Kenik, Scripta Met. (in press). [61 M.L. Swanson, L.M. Howe and A.F. Quennevihe, Fundamental Aspects of Radiation Damage in Metals, MT. Robinson and F.W. Young, Jr., eds. ERDA CONF-751006Pl, Vol. I(lP76) p. 316. [71 R.A. Johnson and N.Q. Lam, Phys. Rev. 13 (1976) 4264. I81 R.A. Johnson and N.Q. Lam, this Conference. 191H. Wiedersich, Rad. Effects 12 (1972) 111. 1101W.G. Johnston, T. Lam&en, J.H. Rosolowski and A.M. Turkalo, 3. of Metals, 28 (1976) 19.