Materials &'ience and Engineering, A 171 ( 1993 ) 241 -248 241

Radiation-induced grain boundary segregation in dilute alloys

R. G. Faulkner institttte of Polymer Technology and Materials Engineering, Loughborough Lhtiversi O, of Technolo,q~', l.oughborough, Leics LEI 1 3TU (UK)

N. C. Waite Argonne National Laboratories, 97(X) South ('ass A venue, Argonne, IL (~')439 (USA)

E. A. Little Materials' attd Chernist~. Division, Harwell Laboratory, Didcot, ()xfordshire OXI 10RA (UK)

T. S. Morgan Institute of Polymer 7ethnology and Materials Engineering, Loughborough University of Technology, Loughborough, l.ei~w LEI 1 3TU (UK)

i Received January 19, 1993; in revised form April 13, 1993)

Abstract

An innovative analytical model of radiation-induced segregation (RIS) in dilute alloys is presented. The approach is based on non-equilibrium segregation theory and can be extended to cover a multiplicity of sink types in a wide variety of metallic matrices. The model circumvents the requirement for complex mathematical procedures inherent in current rate theory analyses of RIS. Calculations are made for nickel segregation to grain boundaries in ¢~-l-'e and ~-Fc and for zinc segregation in silver. Adequate fits between the limited available experimental surface segregation data and the model predictions arc obtained. The analysis is used to deduce trends in grain boundary segregation in austcnitic and ferritic steels to provide insight into materials selection for nuclear plants to meet requirements for minimal RIS.

1. Introduction

Radiation-induced segregation (R1S) is now recog- nized as an important non-equilibrium phenomenon that can affect microstructural evolution in alloys at elevated temperatures in an irradiation environment. Experimental studies confirm the effect in a number of metallic systems and RIS has been detected at both free surfaces [ 1-5] and, more recently, at internal interfaces, such as grain boundaries [6-9].

The mechanism of RIS is essentially the coupled transport of solute atoms by fluxes of vacancy and interstitial point defects generated during irradiation. The coupling arises from binding energy interactions between the point defects and the solute species; the solute species then migrate as bound complexes, either towards or away from point defect sinks--the direction of solute flow depending on the magnitude of the binding energy [1, 2]. In general, the undersize solutes bind strongly to interstitials in a mixed dumb-bell configuration, leading to marked enrichment at the sink. In contrast, oversize solutes exhibit weak binding

to vacancies; this gives rise to solute depletion at the sink and a corresponding enrichment in the matrix, because the preferential exchange of solute atoms with vacancies moving towards the sink results in solute drift in the opposite direction [101. Not all solutes conform to this pattern; some oversize solutes, such as germanium in nickel, migrate towards sinks, implying strong vacancy-solute binding. However, such excep- tions can be rationalized to some extent by redefining the size misfit term [ 1 1 ].

RIS is essentially a non-equilibrium process, because, in the absence of continued irradiation, the solute build-up is eliminated by diffusion [4]. Such non- equilibrium processes rely on steady state or transient point defect fluxes being established as a result of spa- tial variations in point defect concentrations. These variations can also arise during quenching (for the case of vacancies) as well as irradiation. In both cases, regions in close proximity to sinks become depleted in point defects relative to the matrix and the point defects--alone or complexed with solute atoms--then diffuse down the concentration gradient to the sink.

0921-5093/93/$6.00 © 1993 - Elsevier Sequoia. All rights reserved

242 R. G. Faulkner et al. / Radiation-induced grain boundary segregation in dilute alloys

For dilute alloys, the concept of a point defect-solute atom complex, as detailed above, is thus useful in describing the RIS mechanism. The situation is more complicated when solute concentrations exceed 1%-2%, because the individual nature of a complex is then lost.

To a first approximation, dominant RIS towards sinks can be effectively modelled in terms of intersti- tial-solute point defect complex formation, as described above, and this approach is adopted in this paper. The critical parameters required for such modelling are the point defect-solute binding energy and the mechanism of solute transport, as defined by the migration energy of the complex; these aspects have been treated for the case of interstitial complexes by Dederichs et al. [12] and Robrock [13]. It is noted that corresponding parameters for vacancy complexes have been evaluated by Faulkner [14] as applicable to non-equilibrium segregation during quenching [15].

Conventional modelling of RIS is based on rate theory concepts [1, 2, 10]; in this approach, the fluxes of all atomic species are defined by a hierarchy of partial differential equations which are solved simul- taneously to give elemental distributions. However, an analytical model has distinct advantages in terms o f mathematical simplicity and is hence derived in this' paper. The analysis is based on an extension of a vacancy-solute complex model for dilute alloys, as presented previously by Faulkner [16]. The model is then applied to predict the segregation behaviour of nickel to grain boundaries in neutron-irradiated a-Fe and y-Fe and of zinc migration in irradiated silver, for comparison with experimental data where available.

tration of solute on the boundary relative to the bulk level within the grains, is obtained.

The concentration of impurity-point defect com- plexes is given by [17]

cc = k~caci exp ~ (1)

where cj is the point defect concentration, q is the solute concentration, k~ is a geometrical constant, E h is the point defect-solute binding energy, k is Boltz- mann's constant and T is the absolute temperature.

The magnitude of this concentration of complexes can be obtained by substituting the appropriate values of the defect concentration into the above expression. At the interface at which segregation is occurring, it can be assumed that the defects are present in their equilib- rium concentrations. This is a good approximation in the case of surface segregation and is thought to be a reasonable approximation for grain boundary segrega- tion. This gives the equilibrium point defect concen- tration as

where Ef is the defect formation energy and A is a geometrical constant.

Hence, from eqn. ( 1 ), the ratio cc/q at the boundary is given by

ci = k c A e x p / ~ - - ] (3)

2. Theory

The first step in the analysis is to define the increased concentration of solute created on the grain boundary or sink plane during the non-equilibrium segregation process. This is achieved by assuming a fixed steady state concentration of point defects created by the neutron irradiation. The concentration gradient around the sink is defined in terms of the concentration differential set up between the enhanced, irradiation-induced concentration away from the sink (i.e. within the grains) and the equilibrium thermal concentration present on the sink plane. If there is mutual attraction between the point defects and solute atoms, then these will be dragged with the point defects down the concentration gradient towards the sink plane. Using the law of mass action, the con- centration of solute atoms can be inversely related to the point defect concentration. Thus, an expression for the segregation ratio cb/Cg, representing the concen-

In the matrix, the defect concentration is increased by the addition of irradiation-induced point defects. The total steady state concentration of defects is thus given by [18]

cd=A exp (~TEf) + (~k02) (4)

where G is the defect production or dose rate, D is the defect diffusion coefficient and kd 2 is the sink strength of the defect.

From eqn. (1), the value of cc/c i within the bulk of the grains is

(c / ~ = k c A e x p ~ - - - ~ ] ~k~d2exp - ~ (5)

Dividing eqn. (5) by eqn. (3) gives

(cJc,)s=l+ G (E,) (c¢/ci~o ~ exp ~ (6)

R. G. Faulkner et al. / Radiation-induced grain boundary segregation in dilute alloys 243

The left-hand side of this equation can be rearranged to give

(Ci/C~)b = 1 + - - ~ exp (7) ( ci/c~)~ A Dkd-

From previous work by Faulkner [17] the ratio of solute concentration on the boundaries (Cb) to that within the grains (Cg), as predicted by eqn. (7), needs to be modified to account for the absolute concentrations of the complexes, which is controlled by the binding energy E b. Thus, we have

c,._ E. ( c,/q). (8) q E, (cdc,,)g

Combining this with eqn. (7) then gives

c g - E , + A - D ~ 2exp k-T (9)

Equation (9) represents the level of non-equilibrium segregation that will take place if no other processes are acting. However, this expression does not give any indication of the time dependence of such segregation. To obtain such information, it is necessary to substitute this expression into a diffusion equation for the point defect-solute complexes, such as

cob } ,10, Cg -- C b

where c is the concentration of impurity at a distance x from the boundary, t is the time, De is the complex diffusion coefficient and erf is the gaussian error function. (To determine the magnitude of the boundary segregation, x is given the value l x 10-9m in the present studies.)

Equation (10) can then be rearranged to give

. . . . + 1 - e (11) Cg Cg

where c/cg is the time-dependent segregation ratio and cb/q is given by eqn. (9).

The sink strength kd 2 is a function of the microstruc- ture and is primarily dependent on the grain size d, dislocation density p and the interstitial bias Z,. A typi- cal definition used here is given by Bullough et al. [ 18] a s

(12)

A further refinement to the model is included by introducing the concept of a critical time t¢, at which steady state is achieved. At this critical time, the supply of solute from the grain centres becomes exhausted. After this time, the reverse flow of solute atoms caused by the non-equilibrium concentration gradients created during segregation will equal that of the forward flow of the complexes. The factors controlling this critical time are the diffusion characteristics of the complexes (D~.) and that for the isolated solute atoms in the matrix (Di), and the grain size d [17]. t c is derived as [17]

d26 In(De/D,) t,, = (13)

4(D~ -D~)

where 6 is a dimensionless constant describing the effective diffusion time, with a value taken as 0.05 [ 17].

This concept allows for irradiation-induced solute depletion around boundaries to be predicted. If the diffusion rate for the complexes is less than that for the isolated solute atoms, then a negative critical time will be predicted and all the segregation parameters can be inverted; the model can then be applied in exactly the same manner to predict depletion as for segregation.

The diffusion coefficients indicated above are given by the following relationships:

D =D 0 exp( -Em/kT )

1)~ = D 0 exp( - Eml/kT)

D i = Di0 exp( - Emi/kT)

(14)

(15)

(16)

where D 0 is the pre-exponential constant for diffusion of point defects and interstitial-impurity complexes (usually taken as that for self-diffusion), D~0 is the pre- exponential constant for solute diffusion in the matrix, E m is the point defect migration energy, Emi is the migration energy of the complex (usually taken as the impurity activation energy for diffusion) and Emi is the activation energy for solute diffusion.

3. Experimental and theoretical results

The model derived above has been applied to cal- culate the RIS of nickel in both austenitic and ferritic steels, and of zinc in silver. The cases of nickel in austenitic steel and zinc in silver are included to validate the model, because experimental and theoreti- cal data are available for radiation-induced surface segregation in these systems.

As previously implied, the model assumes that the dominant transport mechanism is via interstitial- rather than vacancy-based complexes. The justification for this is that (a) the binding energy of undersize solutes to interstitials is much greater than that to vacancies; (b)

244 R. G. Faulkner et al. / Radiation-induced grain boundary segregation in dilute alloys

the ratio of the binding to formation energies of inter- stitial complexes is greater than that for vacancy complexes; and (c) the activation energy for migration of interstitial complexes is less than that for vacancy complexes. Thus, for the cases under consideration here, the binding energy for vacancy-nickel complexes in iron is about 0.03 eV [14], whereas the value is about 1.0 eV for interstitial-nickel complexes [19]. Similarly, the vacancy-zinc and interstitial-zinc binding energies in silver are 0.05 eV and 0.2 eV respectively [1]. However, it should be noted that, while nickel is undersized in y-Fe, it is slightly oversized in a-Fe; nevertheless, all indications are that nickel exhibits strong positive RIS in both systems.

The values of the fixed input parameters for the above equations required for the calculations are given in Table 1.

The predictions of the RIS model for a range of radiation-materials conditions, together with com- parisons with published experimental and rate theory data, are illustrated in Figs. 1-9 for nickel segregation in a-Fe or y-Fe and in Figs. 10-12 for zinc segregation in silver. The input variables examined include the dose (or time), dose rate, irradiation temperature, point defect formation energy and grain size, while the outputs from the analyses are the segregation ratio C/Cg and the saturation time t c. Comparisons are carried out for fixed base-line values of the dose and dose rate of 1 dpa and 1 x 10 -6 dpa s-~, respectively, and grain sizes as listed in Table 1, except where sensitivity to any one of these parameters is examined. It should be noted here that the number of displaced atoms (i.e. vacancy-interstitial pairs) produced by electron, ion or neutron irradiation is conveniently given in units of dpa

TABLE 1. Values of input parameters used to calculate RIS of nickel in iron and of zinc in silver

(displacements per atom), calculated according to the internationally accepted NRT standard [26]. It should also be noted that "grain sizes" as low as 0.1 pm are considered for the a-Fe matrix. This microstructural condition essentially simulates the interlath spacing of the ferritic-martensitic steels favoured for nuclear plants; recent experimental data have demonstrated that the martensite lath boundaries in these alloys act as potent sites for RIS [25]. Furthermore, the relatively high dislocation density (1016 m-2) selected for model- ling represents a cold-worked condition for an austen- itic steel (i.e. y-Fe) and a martensitic condition for the a-Fe, again in accordance with current nuclear applica- tions for these materials. The irradiation temperature ranges covered are 180-680 °C for iron and 0-450 °C for silver. The temperature range for iron and the dose rates examined are thus also broadly relevant to nuclear plant conditions, to enable the technological significance to be assessed. For example, typical dose rates for current nuclear power plant are as follows: fast reactor core, 1 x 10 -6 dpa s-~; thermal reactor core, 1 × 10 -s dpa s-~; inner wall of pressur~vessel (pressurized water reactor), 5 x 10-11 dpa s-~.

Figure 1 compares the segregation behaviour of nickel in a-Fe and y-Fe as a function of the irradiation temperature. The effects of the dose rate on segrega- tion as a function of the irradiation temperature in a- Fe and y-Fe are shown in Figs. 2 and 3, respectively, while the effect of the dose on the segregation tempera- ture dependences are shown in Figs. 4 and 5. The increase in segregation as a function of the dose at a temperature of about 270 °C, i.e. close to the peak segregation temperature in a-Fe, is shown in Fig. 6. Figure 7 illustrates the effect of the grain size on segre- gation in a-Fe, while the saturation or critical time at which steady state segregation occurs in a-Fe as a function of the grain size is given in Fig. 8. Finally, Fig. 9 compares the predicted nickel grain boundary segre.

Parameter Nickel in Nickel in Zinc in y-Fe a-Fe silver

a 1.0 [20] 1.0 [201 1.0 [20] Eb(eV) 1.01191 1.0119] 0.211] E,(eV) 3.0112] 2.5 [12[" 1.6112] a Em(eV ) 0.3121] h 0.25 [211 b 0.1 [1] Era, (eV) 3.3 [22] 2.53 [23] 1.56 [24] Do (m 2 s- ') 1.0 [20] 1.0 [20] 1.0 [20] Z, 1.1 1.1 1.1 p(m -2) 1016 10 '6 1016 D~0(m-'s -~) 5x10-~[22] 3.2x10-'[23] 0.46x10-a[24] Era, (eV) 3.3 [22] 2.53 [231 1.56 [24] d (~m) 10.0 0.1 1.0

'Assumes lower values for less close-packed and lower melting point structures. bAssumes that slightly lower values are applicable for less close- packed structures.

10 ~'

103 i ~-Fe I02 I 10

,L I

°~;o 26o ~oo' ,-oo' ~oo' ~ao ~oo TEMPERATURE (°C)

Fig. I. Model predictions comparing RIS of nickel in ct-Fc and y-Fe as a function of irradiation temperature: l dpa; 1 x 10 -6 dpa s- ~.

R. G. Faulkner et al. / Radiation-induced grain boundar>' segregation in dilute alloys 245

10'il . i0-~ dpo.s I' l x 10 Sdpo.s.~

1 C : L ~ ~ ' lx

t3" t0?' ] 10 "3 dpaS

' 0 ' ~

1 ,-.

01 . . . . . i _~-_. 1 ~oo 2oo 300 400 5()o 600 ~-oo

TEMPERATURE (°C)

Fig. 2. Model predictions of RIS of nickel in a-Fe as a function of irradiation temperature and dose rate: l dpa.

i 0 5 r . . . . . . . 10 dpo

lOaF

lo~1 1"0dpo

t-~ ~ 102 .

10~

1. _ : = : . . . . . . . . . . . . . . . . . . , . . . . . . .

0~[- . ~ l.__. .L~_ _.J 100 200 300 400 500 600 700

T E M P E R A T U R E ( ° C )

Fig. 5. Model predictions of RIS of nickel in a-Fe as a function of irradiation temperature and dose: I x 10 ~ dpa s- ~.

r . . . . . .

102 i 1 • 10 .6 dpo s '

,S 10[- 1 , lO-adpo s -I .3 dpe.s

I o ~ . . . . . . . . ~ _ _ __,_ . J

100 200 300 400 500 600 700

TEMPERATURE {°C )

Fig. 3. Model predictions of RIS of nickel in 7-Fe as a function of irradiation temperature and dose rate: 1 dpa.

~7

10cF . . . . .

I +"

102 I

10

i

01+ J OO1 . . . . 0"1 . . . . . I - 10

DOSE ( dpo )

Fig. 6. Model predictions of dose dependence of RIS of nickel in a-Fe near the peak segregation temperature of 270 °C: I x 10-" dpa s- t.

T0+[

I

10,

i ,3

1-0 dpo

1~0 .1 dpo

I i : - - - " " ~ : - - - / - - ~ ' - ~

°~oo 260- 300 400 ~o +oo= 7--m T E M P E R A T U R E ( ° C )

Fig. 4. Model predictions of RIS of nickel in 7-Fe as a function of irradiation tcmperature and dose: 1 x 10-~ dpa s- ~.

gation in 7 - F e with the rate- theory modelling results in concentrated alloys, as given by English et al. [5].

Turning next to the Z n - A g system, Fig. 10 compares the calculated saturation time with the experimental results of Johnson and Lam [1]. The experimental results of Johnson and Lam for the tempera ture dependence of zinc segregation to thin foil surfaces in electron-irradiated silver are shown in Fig. 11 and are compared with the segregation profile predicted for the same conditions f rom the present model. Finally, Fig.

10~[ .... d=--OOprn . . . .

j l O 2-

10 ~

1 I-

01 . . . . . ~ _ . t . . . . +00 200 300 400 +50 +,+0 7-00

T E M P E R A T U R E (oC)

Fig. 7. Model predictions of RIS of nickel in a-Fe as a function of irradiation temperature and grain size: l dpa; l x lO-" dpa s- ).

12 illustrates the effect of varying the interstitial forma- tion energy in silver on the magnitude of the predicted grain boundary zinc segregation under irradiation.

4. D i s c u s s i o n

The results presented above theoretically predict RIS of nickel in austenitic and ferritic iron and zinc

246 R. G. Faulkner et al. / Radiation-induced grain boundary segregation in dilute alloys

I0 e

~°z~ I ~~ 10-210<t10zl r d = 0-.llIJ

100 200 300 400 500 600 700 TEMPERATURE { °C)

Fig. 8. Model predictions of saturation times for RIS of nickel in a-Fe as a function of irradiation temperature and grain size: 1 dpa; 1 x 10-6 dpa s - I .

10'[

10 ~-

'3 10 ~ T ~

1 -50 50 150 250 350

TEMPERATURE (°C)

Fig. 11. Comparison of model predictions for temperature dependence of RIS of zinc in silver with published experimental data in thin foils: 1 dpa; 1 x 10-~ dpa s- L

6" i 10

i f O0 100 200 300 400 500 600 700

TEMPERATURE [°C )

Fig. 9. Comparison of model predictions for temperature dependence of RIS of nickel in y-Fe with published rate-theory data: 1 dpa; 1 x 10 -6 dpa s-1.

I0'2F---- 1

I#° I 108'- = .

J 10 s-

"~ I0~i

102tEt : 1,6e v

-I00 10(3 20O 30O z.O0 500 TEMPERATURE [°C)

Fig. ] 2. Mode] predictions of RJS of zinc in silver as a function of irradiation temperature and interstitial formation energy: 1 dpa; 1 x 10-6dpas-L

_ 108!

< 10 ~ w >-

w I04 ~E

102

~ 16" • 16, i

-100

y

Data ~ Ja~son and Lore I)

t 0 100 200 300 400 500

TEMPERATURE I°C }

Fig. 10. Comparison of model predictions for temperature dependence of saturation times for RIS of zinc in silver with published experimental data in thin foils: 1 dpa; 1 x 10-rdpa s-1.

segregation in silver. While the magnitude and kinetics of segregation of other elements in these matrices may also be significant, the results may be taken as essen- tially representative of the general behaviour expected in the three matrixes described.

Though the body of experimental data on RIS is quite sparse, the limited validation possible suggests that the analytical framework developed here repre-

sents a reasonable basis for modelling the phenom- enon. Figures 10 and 11 show that the Johnson and Lam [1] data for zinc segregation in silver are broadly comparable with the model predictions. However, the spread of the segregation is over a wider range of temperature in the work of Johnson and Lam, and the model estimates of the saturation time are very high at low temperatures. Against this, it is noted that the experimental data are from surface segregation in electron irradiations of thin foils 100 nm thick. These conditions are somewhat far removed from grain boundary segregation in bulk material during neutron irradiation. There is also a broad fit between the model predictions and the rate theory modelling of concen- trated nickel segregation in 7-Fe [5]; however, there are similar reservations about the temperature spread (Fig. 9).

Regarding the sensitivity of the model in predicting the temperature of peak R/S, the main parameters governing this feature are the diffusion coefficients for the impurities and complexes. Table 1 and Fig. 1 can be viewed together to indicate that a change in the activation energy for diffusion from 2.53 to 3.3 eV

R. G. Faulkner et al. / Radiation-induced grain boundary segregation in dilute alloys 247

produces a shift of about 150°C when comparing nickel RIS in a-Fe and 7-Fe (the pre-exponential constants for impurity diffusion only vary by a factor of about 1.5 and the pre-exponential constants for com- plex diffusion are the same). Usually, well-established diffusion constants are accurate to within +0.1 eV, so the confidence limits to be placed on the peak RIS temperature prediction are about + 20 °C.

The predictions from the model suggest that the magnitude of the interfacial segregation (a) increases significantly with increasing dose, leading to saturation (Figs. 5 and 6) and (b) increases moderately with decreasing dose rate (Fig. 2); furthermore, (c) the peak segregation is shifted to lower temperatures with decreasing dose rate and (d) the segregation profiles are characterized by a relatively sharp low temperature cut-off. It is interesting to note that trends (c) and (d) are in general agreement with those deduced from the earlier rate-theory model of Johnson and Lam [ 1 ].

The model predictions also highlight the importance of microstructure. Therefore, while the grain size only slightly affects the magnitude of the segregation (Fig. 7), it has a larger effect on the saturation time (Fig. 8). This has implications for fine-grained materials, because these will have very short saturation times t c and, thus, it is unlikely that large contributions to grain boundary segregation of solutes will accumulate. The saturation or critical time concept given in eqn. (13) also shows promise for predicting depletion effects where the saturation time is negative. This will happen when the diffusion of the complex is slower than the solute diffusion rate. It is expected that this condition will apply for oversize solutes diffusing by the split dumb-bell mechanism. There is some experimental evidence to support this contention given by Johnson and Lam [1 ].

One limitation of the analysis is that segregation levels greater than 100% are predicted at the peak in the segregation-temperature curves. Figure 12 illus- trates that this maximum is strongly dependent on the formation energy of the interstitial (El). Data on inter- stitial formation energies are very limited and usually indicate values in excess of 3 eV [12]. However, it might be reasonable to expect that, in some circum- stances, E r could drop below 2 eV, especially in low melting point matrices. Other factors which affect the magnitude of the maximum segregation parameter are the entropy terms used in the diffusion equations (Do) and the pre-exponential factor in the defect concentra- tion equation (A). Lam et al. [20] stated that the entropy of formation term Sf is zero for most metallics and, hence, A would be unity. Dederichs et al. [12] stated that most entropy terms for vacancy diffusion in metallic matrices are between 0.5k and 3k; hence, D~ = exp(Sf,k) will be close to unity. It is assumed here

that, to a good approximation, the same reasoning can be applied to the interstitial case.

A further point that can be made about the high maximum segregation values is that the values refer to C/Cg. The absolute values of c at the boundary are usually a factor of 100 lower if dilute alloys are being considered. Also, the concentration represents that found at a point 1 nm from the boundary plane. If the concentration is integrated over a region, for example, 10 nm wide on either side of the boundary as segrega- tion proceeds, then the segregation parameter will be further reduced as follows:

' 2c(D~t)l 2 t17) c - 1 x 1 0 - ~ '

These two factors can reduce C/Cg by as much as a factor of 104 in most cases considered in these studies.

An additional aspect which needs to be considered arises from the consequences of the spatially hetero- geneous distribution of radiation damage created by fast ion or neutron irradiation. It is well established that such damage is concentrated initially in collision cascades comprising a central core of vacancies with the interstitials distributed at the periphery. This con- figuration results in significant athermal recombination (i.e. annihilation) and agglomeration into defect clusters, leaving a significantly reduced population of freely migrating defects available to participate in long- range solute transport processes. Recent results suggest that this can amount to as little as 1.5% of the total cal- culated displacement production [27]. At present, quantification of these effects is very difficult but it is clear that the process will lead to a further reduction in the magnitude of the predicted solute segregation.

The potential differences in RIS response between austenitic (i .e.) ,-Fe) and ferritic (i.e. a-Fe) steels high- lighted by the model are of significance for nuclear reactor core and pressure vessel applications. For example, Fig. 1 demonstrates that the peak segregation temperature in ferritics is shifted downwards com- pared with that for austenitics; this shift is about 150 °C for the input conditions selected. The reasons for this are the faster diffusion rates in, and smaller grain sizes of, ferritic compared with austenitic matrices. This is particularly true where martensitic or ferritic-marten- sitic steels are considered [25, 28]. There are consider- able data to suggest that the levels of interracial solute-impurity segregation in steels are likely to affect their microstructural stability and their anticipated degradation of mechanical properties during service [29]. Thus, the analytical trends imply that there is considerable advantage to be gained by judicious selec- tion of either ferritic or austenitic steels for nuclear applications, depending on the thermal conditions

248 R. G. Faulkner et aL / Radiation-induced grain boundary segregation in dilute alloys

experienced by the different components, provided other nuclear requirements are satisfied.

Finally, Fig. 6 can be used in conjunction with Fig. 2 to determine the level of segregation expected for given combinations of the dose and dose rate. Conceptually, this should enable estimation of the segregational state of components in various neutron flux environments in currently operating nuclear plant. The present model clearly represents an initial attempt to provide an analytical f ramework for these strategies. Additional refinements and applications to other segregants and radiation conditions are thus appropriate to explore fully the technological applications.

5. Conclusions

(1) A model has been developed for non-equilib- rium solute segregation to interfacial sinks in dilute alloys, based on coupled interstitial point defect and solute fluxes. The analysis has been applied to the phenomenon of RIS and trends deduced for the effects of the dose, dose rate, irradiation temperature and grain size on the level of segregation.

(2) The results indicate similar trends to those predicted by other models based on mathematically more complicated rate theory. Reasonable fits with limited experimental surface irradiation-induced segre- gation results for nickel in austenitic iron and zinc in silver are obtained.

(3) The model predictions suggest that segregation to grain boundaries will increase with (a) lower dose rates, (b) higher doses, (c) higher interstitial formation energy and (d) larger grain size.

(4) Furthermore, the maximum predicted RIS is shifted towards lower temperatures under conditions of (a) lower dose rates, (b) higher doses and (c) ferritic rather than austenitic steel matrices.

(5) The results presented are illustrative of nickel segregation in iron and zinc in silver under irradiation. The kinetics of segregation of other elements in these matrices may vary but the trends are deemed to be representative of the general behaviour expected for RIS in the two matrices described.

Acknowledgment

T he studies were undertaken as part of a joint

research programme supported by the Argonne National Laboratories, USA.

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