A model describing neutron irradiation-induced segregation to grain boundaries in dilute alloys

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, NOVEMBER 1996—3381

A Model Describing Neutron Irradiation-Induced Segregationto Grain Boundaries in Dilute Alloys

R.G. FAULKNER, SHENHUA SONG, and P.E.J. FLEWITT

A model describing neutron irradiation-induced grain boundary segregation at a given temperatureis established for dilute alloys based on a complex diffusion mechanism and combined with Mc-Lean’s equilibrium segregation model. In the model, irradiation-enhanced solute diffusion is takeninto consideration. The diffusion equations are more rigorously solved than in earlier models, so thatan accurate definition of the grain boundary solute concentration is given as a function of time. Theeffect of the temperature dependence of dislocation density is accommodated and the estimationmethod for complex diffusion is reappraised. Theoretical predictions are made for segregation ofphosphorus in neutron-irradiated a-Fe. There exists a transition temperature below which combinedirradiation-induced nonequilibrium and irradiation-enhanced equilibrium segregation is dominant andabove which thermal equilibrium segregation is dominant. The peaks in the temperature dependenceof segregation shift to lower temperatures with decreasing neutron dose rate and/or increasing neutrondose. The combined radiation-induced nonequilibrium and radiation-enhanced equilibrium peak seg-regation temperature is about 150 7C for P grain boundary segregation in neutron-irradiated a-Fe atdose rate 5 1026 dpa/s and dose 5 1 dpa. The thermal equilibrium segregation peak is around 5507C for the same conditions. Comparison of some experimental and predicted results shows that thepredictions are generally consistent with the observations.

I. INTRODUCTION

GRAIN boundaries are regions of discontinuity in thecrystal structure of a material and, as a consequence, theyare important in controlling the overall physical and me-chanical properties. Some engineering problems can be en-countered and associated with the chemical composition ofthese specific regions of the microstructure of the material.The local chemical composition of a grain boundary canresult from segregation of alloying or impurity elementssteming from either equilibrium or nonequilibrium pro-cesses.[1,2,3] It is these changes in composition that can havea profound effect on fracture processes. Induced changes infracture mode from transgranular cleavage to intergranularbrittle fracture have been identified in some ferritic steelsand other materials as a result of neutron irradiation-in-duced segregation to grain boundaries.[4–7] During the past20 years, a great deal of experimental and theoretical re-search has been directed towards a basic understanding ofneutron irradiation-induced segregation in a series of alloys.Therefore, a good understanding of irradiation-induced seg-regation of alloying and impurity elements is of consider-able importance to understanding the fracture processes inengineering components and structures operating in a nu-clear environment.[8]

Early experimental techniques made it difficult to rigor-ously evaluate the chemical composition of grain bounda-ries. However, advances in high resolution techniques, such

R.G. FAULKNER, Professor of Physical Metallurgy, and SHENHUASONG, Research Associate, are with the Institute of Polymer Technologyand Materials Engineering, Loughborough University of Technology,Loughborough, Leicestershire LE11 3TU, United Kingdom. P.E.J.FLEWITT, Structural Integrity Manager, is with Berkeley TechnologyCentre, Nuclear Electric plc, Berkeley, Gloucestershire GL13 9PB, UnitedKingdom.

Manuscript submitted February 6, 1996.

as Auger electron spectroscopy (AES) and scanning trans-mission electron microscopy (STEM) fitted with energydispersive and electron energy loss spectrometers, enablethe segregation to be examined in detail.[9] In particular, thededicated STEM instrument fitted with a high-brightnessfield emission electron source (FEGSTEM) provides abeam of ;1-nm-diameter incident on the foil specimen,thereby providing the capability to examine and quantifygrain boundary segregation. Irradiation-induced segregationto grain boundaries is more difficult to detect experimen-tally due to the active nature of the specimens. However,progress has been made using both AES and STEM tech-niques by Mahon et al.,[10] Norris et al.,[11] Kameda andBevolo,[12] Morgan et al.,[13] and Carter et al.[14] Mechanismsfor irradiation-induced segregation can be classified into ei-ther inverse Kirkendall effects or solute-point defect com-plex effects.[15] Irradiation-induced segregation theory hasreceived detailed attention from Johnson and Lam,[16] Lamet al.,[17] Okamoto and Wiedersich,[18] Murphy and Perks,[19]

and Grandjean et al.[20] These rate theory models, alsoknown as Kirkendall models, are based upon the conceptthat components of an alloy diffuse via vacancies at differ-ent rates so that a composition gradient may induce a netflux of vacancies across a lattice plane, even if the vacancydistribution is initially uniform. During neutron irradiation,the inverse situation occurs near sinks where gradients inthe vacancy and interstitial concentrations may cause a netflux of solute and solvent atoms across the lattice plane. Inmulticomponent systems, relative diffusion rates of variouscomponents determine their enrichment or depletion. Theabove researchers have supported their rate theory modelswith experimental observations of segregation to free sur-faces in irradiated nickel alloys and austenitic stainlesssteels. These rate theory models have input parameters thatrequire a knowledge of the diffusion of point defects, im-purity-point defect binding energies and irradiation-en-

3382—VOLUME 27A, NOVEMBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 1—The basis of the overall model that describes solute segregationto grain boundaries under neutron irradiation at a given temperature (GB5 grain boundary).

hanced solute diffusion. Most of these data are not readilyavailable. Moreover, the models require the use of curvefitting procedures to match the experimental observations,and they do not take into account grain size, dislocationdensity, or other microstructure features.

For solutes that have strong interactions with point de-fects, for example, phosphorus and silicon in steels andnickel alloys,[21,22] solute-point defect complexes play animportant part in solute segregation to grain boundaries.[15]

Usually, in this scenario, the solutes are undersized and thecomplexes are solute-interstitial pairs. This is because (1)the migration of solute-vacancy complexes is much moredifficult than that of solute-interstitial complexes,[16,23] and(2) the interaction of the undersized solute atom with theinterstitial is much stronger than that with the vacancy,while there is either no or very weak interaction betweenthe oversized solute atom and the interstitial.[21,22] As a con-sequence, it is certain[16,21,22] that for undersized solutes suchas phosphorus and silicon in steels and nickel alloys, thediffusion of solute-interstitial complexes should be domi-nant during neutron irradiation-induced segregation. Thesolute-defect complex mechanism developed by Faulkneret al.[24] describes the solute drag effects caused by the flowof point defects to grain boundaries when nonequilibriumconcentrations of these defects are formed in the matrixremote from the boundary. The derived relation predicts themaximum segregation level during irradiation as a functionof neutron irradiation conditions.

In the present work, we have developed a kinetic modelbased upon the basic model of Faulkner et al.[24] In SectionII, we describe the model for irradiation-induced segrega-tion of solute or impurity atoms to grain boundaries in apolycrystalline dilute alloy at a given temperature. As aconsequence, this model considers contributions for bothsolute-interstitial complexes leading to nonequilibrium seg-regation and solute atoms leading to thermal equilibriumsegregation at the temperature of neutron irradiation. InSection III, we present the results of this model for P im-

purity segregation in a-Fe at a neutron dose rate of 1026

dpa/s and at a neutron dose of 1 dpa. Comparison withexperimental results demonstrates that the predicted levelof grain boundary composition is consistent with the ob-servations.

II. MODEL

A. Background

Any model that describes the segregation of solute orimpurity atoms to grain boundaries in a dilute alloy sub-jected to neutron irradiation must also take into account thetemperature at which the irradiation has been undertaken.As shown in Figure 1, the present model considers the si-multaneous diffusion of solute-interstitial complexes (re-sulting in nonequilibrium segregation) and solute atoms(resulting in thermal equilibrium segregation) to grainboundaries at a given temperature. In addition, it is neces-sary to consider contributions from the enhancement of sol-ute diffusion arising from the neutron irradiation.

B. Neutron Irradiation-Induced Segregation

For undersized solutes in a crystalline matrix that havestrong interactions with interstitials, the flux of solute-in-terstitial complexes to a grain boundary sink will be dom-inant in the course of irradiation-induced segregation. Inthis work, we will only consider this case. As detailed inReferences 15, 25, and 26, the segregation mechanism re-lies on the formation of sufficient quantities of interstitial-solute complexes. Solute atoms, interstitials, and theircomplexes are in equilibrium with each other at a giventemperature. Neutron irradiation produces the cascade ofdefects and leads to the formation of Frenkel pairs and, asa result, the interstitial concentration exceeds the thermalequilibrium concentration in the matrix. However, at sinks,such as grain boundaries and free surfaces, the interstitialconcentration approaches the thermal equilibrium concen-tration. The decrease in interstitial concentration results inthe dissociation of the complexes into interstitials and sol-ute atoms. This, in turn, leads to a decrease in complexconcentration in the proximity of grain boundaries. Mean-while, in regions away from the grain boundary, the steady-state concentration of the complexes always remains. As aresult, a complex concentration gradient appears betweenthe grain boundary and the adjacent grains. The concentra-tion gradient of complexes causes their migration, leadingto an excess solute concentration in the vicinity of grainboundaries. It is obvious that the larger the supersaturationlevel of interstitials induced by neutron irradiation, thelarger the segregation level of solute atoms resulting at theboundary.

The basic model of Faulkner et al.[24] is utilized to de-scribe the process. This model originally predicted the max-imum magnitude of segregation expected in dilute alloyson the basis of a thermodynamic argument using the equi-librium concentration of interstitials expected at grainboundaries ratioed to the quasi-equilibrium concentrationexpected within grains during neutron irradiation. The max-imum concentration of segregation during irradiation, ,mCbr

is given by[24]

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, NOVEMBER 1996—3383

iI IE BG Eb fmC 5 C [1 1 exp ( )] [1]br g I 2E A D k kTf I I dI

where Cg is the solute concentration in the matrix, is theiIEb

solute-interstitial binding energy, is the interstitial for-IEf

mation energy, DI is the interstitial diffusion coefficient inthe matrix, AI is a constant showing the vibrational entropyof atoms around the interstitial, G is the point defect pro-duction rate or neutron dose rate, k is Boltzmann’s constant,T is the absolute temperature, B is the dose rate correctionfactor, i.e., the fraction of freely migrating defects, and 2kdI

is the grain interior sink strength for the interstitial, givenby[24]

1 162 2 2k 5 (Z r ) [ 1 (Z r ) ] [2]d I I IR

where ZI is the bias parameter defining the preferential in-teraction between interstitials and dislocations comparedwith that between vacancies and dislocations. It is evidentthat the maximum segregation concentration, , is mainlymCbr

dependent on the irradiation temperature and is independentof the irradiation time.

The first improvement on the original model[24] relates tothe neutron dose rate correction factor, B. As discussed inReference 27, an additional feature which acts to reduce theabsolute magnitude of the solute segregation arises becauseonly a small proportion of point defects created by irradi-ation may undergo long-range migration. Recent studies byRehn[28] and Naundorf et al.[29] have shown that the ratio offreely migrating point defects which can escape from thecascade, produced by charged particle or neutron irradia-tion, to those initially created amounts in the cascade, isonly about 1 pct. More recent studies[30] indicate that thevalue of B is in the range of 0.01 to 0.05. Therefore, it isnecessary to assume that the effective dose rate is only avery small part of the NRT (Norgett–Robinson–Torrens[31])calculated value, i.e., a dose rate correction factor, B,should be applied to the term containing the dose rate.Moreover, as discussed in Reference 32, long-range recom-bination of freely migrating vacancies and interstitials out-side the cascade should be taken into account for highirradiation dose rates associated with ion or electron irra-diation, whereas it may be neglected for low irradiationdose rates associated with neutron irradiation. Therefore,long-range effects may be negligible in the case of thiswork.

The second improvement on the original model[24] thatwe have introduced concerns modeling the composition dis-tribution on the grain boundary plane, which is detailedsubsequently. The interstitial concentration in the matrix,CI, at the steady state of neutron irradiation, may be ap-proximately given by[24]

I2E BGfC 5 A exp ( ) 1 [3]I I 2kT D kI d I

Recently, several studies have assumed that the defect con-centration is uniform up to the boundary.[33,34] In this sce-nario, the complex concentration is approximately givenby[24]

iIE bC 5 k C C (x, t) exp ( ) [4]c c I i kT

where kc is a geometrical constant and Ci(x,t ) is the soluteconcentration.

We have considered a very small thickness for the grainboundary of the order of three atom distances as comparedwith the grain size, and assumed the segregation to theboundary to be a linear flow in a semi-infinite mass. In lightof Eq. [4], the relevant diffusion equation describing thisprocess may be given by

2] C (x, t) ]C (x, t )i iD 5 [5]c 2]x ] t

where Dc is the diffusion coefficient of complexes in thematrix.

Owing to the small thickness of the boundary regionwhere the concentration gradient may be neglected, we mayenvisage, for convenience, an interface between grainboundary and interior located exactly at x 5 0 where thesolute concentration, C, is C 5 Cbr(t )/a, where Cbr(t) is thesolute concentration at the concentrated layer when irradi-ation time is equal to t and changes with irradiation timeat a given irradiation temperature, and a 5 /Cg. In termsmCbr

of Fick’s first law and the principle of mass conservation,the interface should satisfy the following boundary condi-tions,

C 5 C (t) /abr [6]]C d ]C (t) ad ]CbrD ( ) 5 5 ( )c ]x x 5 0 2 ] t 2 ] t x 5 0

where d is the thickness of the concentrated layer, and thefactor 1/2 is associated with the fact that the complexesdiffuse to the boundary from both adjacent grains. In termsof the preceding boundary conditions, the solution of Eq.[5] is given by[25]

=2 D tcC (t) 2 C 4D tbr g c5 1 2 exp ( ) erfc ( ) [7]m 2 2C 2 C a d adbr g

Eq. [7] is an isothermal kinetic relationship for irradiation-induced grain boundary segregation. It describes the grainboundary segregation level as a function of irradiation timeat a given irradiation temperature.

The time required for reaching the steady-state segrega-tion during irradiation may be determined by a critical time.At this critical time, the net supply of solute atoms fromthe grain centers becomes exhausted. After this, the reverseflow of solute atoms created by the nonequilibrium segre-gation concentration gradient is equal to the forward oneof the complexes. Critical time, tc, is given by[26]

2dR ln (D /D*)c it 5 [8]c 4(D 2 D*)c i

where d is a numerical constant (quoted by Faulkner[26] as0.05), R is the grain size, and is the irradiation-enhanced*Di

diffusion coefficient of solute atoms in the matrix.Third, we now describe a new method for estimating

irradiation-enhanced diffusion coefficience of solute atoms,. It is well known that vacancy formation plays an im-*Di

portant part in the solute diffusion. We can calculate therelative concentrations of freely migrating defects underboth thermal equilibrium and irradiation-induced nonequi-librium conditions. The ratio of these two concentrations

3384—VOLUME 27A, NOVEMBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A

will determine the extent to which the activation energy forsolute diffusion is reduced by lowering the effective va-cancy formation energy. Hence the irradiation-reduced ac-tivation energy for solute diffusion, , is given by*Ei

rCn nE*5 E 2 E [9]i i fe rC 1 Cn n

where Ei is the activation energy for diffusion of soluteatoms in the presence of the thermal equilibrium vacancyconcentration, is the vacancy formation energy, andv eE Cf v

and are the thermal equilibrium and irradiation-generatedrCv

vacancy concentrations, respectively.Eq. [9] is reasonable because in the absence of irradia-

tion-generated vacancies the diffusion activation energy be-comes the thermal diffusion activation energy. In modeling,the radiation-enhanced diffusion coefficient has been*Di

employed. The increased concentration of vacancies pro-duced by irradiation enhances the solute diffusion rate butdoes not affect the equilibrium segregation level.

The thermal equilibrium vacancy concentration, , iseCv

given by

nEfeC 5 A exp (2 ) [10]n n kT

where Av is a constant showing the vibrational entropy ofatoms around the vacancy and is the vacancy formationvEf

energy. The irradiation-generated vacancy concentrationmay be approximately given by[24]

BGrC 5 [11]n 2D kn dn

where G is the point defect production rate or dose rate, Dv

is the diffusion coefficient of vacancies in the matrix, B isthe dose rate correction factor, i.e., the fraction of freelymigrating defects, and is the grain interior sink strength2kdv

for the vacancy, given approximately by[24]

1 162 2 2k 5 r ( 1 r ) [12]dn R

where r is the dislocation density and R is the grain size.During steady-state neutron irradiation at a given tem-

perature, the irradiation-generated vacancy concentration isinvariant in the matrix, but in regions local to a grainboundary, it would be less than that indicated by Eq. [11].At present, it is difficult to evaluate vacancy distributionsnear a grain boundary, but this effect could be shown bymultiplying Eq. [11] by a factor ε (,1).

The fourth improvement on the original model[24] in-volves combining the neutron irradiation-induced nonequi-librium and thermal equilibrium segregation predictions(Section II–C). Nonequilibrium segregation is a kinetic pro-cess while equilibrium segregation is a thermodynamic pro-cess. Therefore, it can be envisaged that these two processesare independent of one other. In calculations, the total seg-regation level is taken to be the sum of the nonequilibriumand equilibrium segregation levels minus the bulk concen-tration of the solute.

The fifth new feature in the modeling of neutron irradi-ation-induced segregation is the introduction of a temper-ature dependence of dislocation density. The dislocationdensity, r, is dependent upon the heat treatment temperature

and time and increases with increasing neutron irradiationdose. In general, r is relatively high in irradiated materialsbecause the collapse of vacancy plates created by irradia-tion can produce a large number of dislocation loops. Inaddition, the interstitial loop component of the damagestructure can also produce a large number of dislocationnetworks. In general, at lower irradiation temperatures dis-location climb is difficult, and thus the higher dislocationdensity may be maintained. However, at higher irradiationtemperatures, recovery processes may result in an overallnet reduction in dislocation density by mechanisms, suchas climb of dislocations into grain and subgrain bounda-ries[35] and dislocation recombination. Consequently, thedislocation density is dependent on the pre-irradiation heattreatment, neutron dose rate, dose, and irradiation temper-ature. Norris et al.[36] have suggested that the dislocationdensity always increases with increasing neutron dose foraustenitic stainless steels and is independent of the activa-tion energy for dislocation recovery processes. This seemsto be unreasonable because recently the experimental re-sults of Loomis and Smith[37] have demonstrated that thedislocation density first increases during irradiation, but itreaches a stable value after a certain dose. Moreover, thework of Hashimoto and Shigenaka[38] has shown that thetemperature dependence of dislocation density duringsteady-state irradiation obeys an exponential function. As aresult, it is assumed here that after the steady-state concen-tration of point defects is reached, the dislocation densitywill no longer vary with increasing dose, but will dependon temperature. In this case, r may be approximately givenby

Edr 5 r exp ( ) [13]o kT

where ro is the dislocation density constant which dependson the preirradiation dislocation density, the dose rate, etc.,and Ed is the activation energy for dislocation recovery pro-cesses.

The sixth improvement on the original model concernsthe estimation of the migration energy of the complexes,which is necessary for estimation of Dc for Eqs. [7] and[8]. Data on migration energies of solute-interstitial com-plexes are sparse. We have introduced a new approach toestimating these data. In accordance with the discussion inReference 39, there are two mechanisms to lead to long-range migration of mixed dumbbells (impurity-interstitialcomplexes): one of which is dissociation together withreformation, as well as cage motion of the mixed dumbbell;the other is rotation together with cage motion. The cage-motion activation energy of the mixed dumbbell is muchsmaller than the migration energy of the self-interstitial. Forthe ideal dumbbell, the mixed dumbbell rotation energy is,in the light of computer simulation, about 3 to 4 times asmuch as the migration energy of the self-interstitial, and itis approximately 0.9 to 1.2 eV for the a-Fe matrix. Thedissociation of the mixed dumbbell requires the sum of theself-interstitial migration energy and the mixed dumbbellbinding energy. In terms of our calculated value of thephosphorus-interstitial binding energy (0.57 eV[21]) plus themigration energy of 0.3 eV for the a-Fe matrix,[40] this en-ergy is about 0.87 eV. Druce et al.[34] have adopted therotation mechanism for migration of impurity-interstitial

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, NOVEMBER 1996—3385

Table I. Data Used in the Theoretical Calculation

AI 1[41]

Av 1[33]

E vf (eV) 1.4[42]

E If (eV) 3.0

EiIb (eV) 0.57

Ei (eV) 2.68[33]

Ecm (eV) 0.87

EIm (eV) 0.3[40]

Evm (eV) 1.24[43]

Doi (m2 s21) 7.12 3 1023 [33]

Doc (m2 s21) 8 3 1027 [16]

DoI (m2 s21) 5 3 1026 1[16]

Dov (m2 s21) 5 3 1025 [33,44]

ZI 1.1[45]

Ed (eV) 0.1[46]

Cg (at. pct) 0.072B 0.01ε 0.1b 0.775[47]

Q (eV) 0.397[47]

d (nm) 1.0

(mixed dumbbell) complexes to predict phosphorus segre-gation in reactor pressure vessel steels, and employed therotation energy of mixed dumbbells as the migration en-ergy, but the value (;0.42 eV) is much smaller than 0.9eV. There is some evidence[12] that the mobility of P-inter-stitial complexes with weaker binding appears to be higherthan that of S-interstitial complexes with stronger bindingin Fe-base ferritic alloys, and the experimental results onradiation-induced solute segregation to grain boundaries aresatisfactorily interpreted by the dissociation mechanism formigration of impurity-interstitial (mixed dumbbell) com-plexes. In the present work, we have utilized the dissocia-tion mechanism because it requires less energy than therotation mechanism, and the complex migration energy of0.87 eV has, therefore, been employed.

C. Thermal Equilibrium Segregation

McLean[1] states that, for a solute atom with a bindingenergy to the lattice, Q, at any temperature, T, there willbe an increased concentration of that solute on boundariesor interfaces, C`(T). The driving force for this is the re-duction of energy, Q, of the solute on placing it in a strain-free region at the grain boundary, C`(T), is given by

bC exp (Q /kT)gC (T) 5 [14]` 1 1 bC exp (Q /kT)g

where Cg is the solute concentration in the matrix, b is aconstant representing the vibrational entropy of the grainboundary region, and k is Boltzmann’s constant. McLeanrefined these ideas by accounting for time, realizing cor-rectly that the finite time is required to reach equilibriumand this is controlled by the diffusivity of the solute in thematrix, Di.

The equilibrium segregation kinetics during irradiation,Cbq(t), derived by means of Eq. [10], are given by

=2 D tiC (t ) 2 C (0) 4D tbq b i5 1 2 exp ( ) erfc ( ) [15]2 2C (T ) 2 C (0) a d a d` b i i

where Cb(0) is the solute concentration at the grain bound-ary at irradiation time 5 0, i.e., the segregation level duringthe preirradiation heattreatment of the material that was as-sumed in this work to be Cg, and ai is the equilibriumenrichment ratio, given by

a 5 C (T)/C [16]i ` g

It should be, furthermore, noted here that although Eq. [7]is the same as Eq. [15] in form, they are much different innature. Eq. [15] depicts the equilibrium grain boundary seg-regation level induced by the solute equilibration at theboundary, whereas Eq. [7] describes the nonequilibriumgrain boundary segregation level induced by the complexdiffusion to the boundary.

D. Summary

The model described previously provides a realistic pic-ture of the mechanisms which lead to the segregation ofsolute or impurity atoms to grain boundaries of materialssubjected to neutron irradiation at a given temperature. Theimprovements on the earlier model can be listed as follows:(1) a dose rate correction factor has been applied to accountfor the reduced freely migrating defect population, com-pared to that predicted by the NRT model; (2) the analyticalequations to describe the solute concentration at the grainboundary plane as a function of irradiation conditions havebeen solved reasonably and satisfactorily; (3) irradiation-enhanced solute diffusion, which is shown to be importantat low temperatures, is accommodated; (4) the thermalequilibrium segregation model and the radiation-inducedsegregation model have been combined into a unifiedmodel to depict grain boundary segregation in irradiatedmaterials; (5) a proper description of the effect of disloca-tion density has been incorporated; and (6) the activationenergy for diffusion of complexes has been reappraised andbetter predictions are now possible.

III. RESULTS AND DISCUSSION

We will now apply the model described in Section II tothe segregation of phosphorus in an a-Fe matrix subjectedto neutron irradiation at given temperature. An importantparameter is the impurity-interstitial binding energy. Thisvalue is generally unavailable. For this reason, we use aphosphorus-interstitial binding energy in the a-Fe matrix,calculated in Reference 21, of 0.57 eV. Other data used inthe theoretical calculation are listed in Table I.

The diffusion coefficients are given by the following re-lations:

cEmD 5 D exp (2 ) [17a]c oc kTnEmD 5 D exp (2 ) [17b]n on kTIEmD 5 D exp (2 ) [17c]I oI kT

E*iD*5 D exp (2 ) [17d]i oi kT

3386—VOLUME 27A, NOVEMBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 2—Temperature dependences of irradiation-induced segregation(Rad.), equilibrium segregation (ES), and total segregation (Total) ofphosphorus in a-Fe (dose rate 5 1026 dpa/s, dose 5 1 dpa, r0 5 1018

m22, and R 5 20 mm).

(a)

(b)

(c)

Fig. 3—Temperature dependences of (a ) irradiation-induced, (b )equilibrium, and (c ) total phosphorus segregation in a-Fe at differentneutron doses (dose rate 5 1026 dpa/s, r0 5 1018 m22, and R 5 20 mm).

where Dc, Dv, and DI are the diffusion coefficients of solute-interstitial complexes, vacancies, and interstitials in the ma-trix, respectively; is the newly introduced radiation-*Di

enhanced diffusion coefficient of solute atoms determinedin Section II–B; Doc, Dov, DoI, and Doi are the pre-exponen-tial constants for diffusion of solute-interstitial complexes,vacancies, interstitials, and solute atoms, respectively; ,cEm

and are the migration energies for diffusion of solute-v IE Em m

interstitial complexes, vacancies, and interstitials in the ma-trix, respectively; is the irradiation-reduced activation*Ei

energy for diffusion of solute atoms; T is the absolute tem-perature; and k is Boltzmann’s constant.

In practice, a-Fe is not used for practical engineeringapplications, although simple C-Mn and low-alloy steelsare. Consequently, compared with experimental data, it isnecessary to take these differences into consideration.When evaluating the thermal diffusion coefficient of soluteatoms, it is necessary to take into account the fact that mo-lybdenum significantly influences phosphorus diffusivity.Some low-alloy steels employed for nuclear power me-chanical engineering applications, such as 2.25Cr1Mo steelare characterized by molybdenum contents in the range of0.4 to 1.0 wt pct. The effect of other alloying elements,such as Mn, Cr, and Ni on the phosphorus diffusivity, isconsiderably weaker than that of molybdenum and may beneglected. Hence, the determination of the phosphorus dif-fusion coefficient in an a-Fe matrix alloyed with Mo needsto be modified, as compared with that in an Mo-free a-Fematrix. The modified value[33] has been applied to this workbecause an Mo-containing steel is employed in the exper-imental section of this article.

Some of the results, predicted by the model, are shownin Figures 2 through 6. Figure 2 shows temperature de-pendences of P segregation to grain boundaries in a-Fe forradiation-induced segregation, thermal equilibrium segre-gation, and total combined segregation. These are in thetemperature range of 60 7C to 600 7C for a neutron doserate of 1026 dpa/s, a neutron dose of 1 dpa for a-Fe witha grain size of 20 mm, and a dislocation density parameterof 1018 m22. For equilibrium segregation, there exist twosegregation peaks, one of which is caused mainly by irra-diation-enhanced equilibrium segregation, the other is

brought about mainly by thermal equilibrium segregation,and moreover, there is a segregation transition temperature,below which neutron irradiation-enhanced equilibrium seg-regation is dominant and above which thermal equilibriumsegregation is dominant. As a result, the temperature de-pendence of total segregation has two segregation peaks inthe range of 60 7C to 600 7C, one of which is caused mainlyby combined irradiation-induced nonequilibrium and irra-diation-enhanced equilibrium segregation, the other iscaused mainly by thermal equilibrium segregation.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, NOVEMBER 1996—3387

(a)

(b)

(c)

Fig. 4—Temperature dependences of (a ) irradiation-induced, (b )equilibrium, and (c ) total phosphorus segregation in a-Fe at differentneutron dose rates (dose 5 1 dpa, r0 5 1018 m22, and R 5 20 mm).

Fig. 5—Temperature dependences of total phosphorus segregation in a-Fe at different dislocation density constants (m22) (dose rate 5 1026 dpa/s,dose 5 1 dpa, and R 5 20 mm).

Fig. 6—Temperature dependences of total phosphorus segregation in a-Fe at different grain sizes (mm) (dose rate 5 1026 dpa/s, dose 5 1 dpa,and r0 5 1018 m22).

Effects of neutron dose in the range of 0.01 to 1.00 dpaon the irradiation-induced segregation, equilibrium segre-gation, and combined segregation are illustrated in Figures3(a) through (c), respectively. They indicate that the seg-regation peaks shift to lower temperatures with increasingneutron dose at the same neutron dose rate.

Figures 4(a) through (c) show effects of neutron dose ratein the range of 1026 to 1028 dpa/s on the irradiation-inducedsegregation, equilibrium segregation, and combined segre-gation respectively. Clearly, the phosphorus segregationpeaks all shift to lower temperatures with decreasing neu-

tron dose rate at the same neutron dose. Influences of dis-location density on the total combined segregation in thetemperature are illustrated in Figure 5. Clearly, the dislo-cation density has an effect on both radiation-induced seg-regation and equilibrium segregation. The influence ofdislocation density is reflected in the kinetics of equilibriumsegregation (Eqs. [9], [12], and [15]) and the quasi-ther-modynamics of irradiation-induced segregation (Eq. [1]), aswell as critical time (Eq. [8]). Grain size, when changingin the range of 1 to 100 mm, has a considerable effect onirradiation-induced segregation (Figure 6). Grain size effecton irradiation-induced segregation is caused mainly by itsinfluence on the critical time (Eq. [8]).

The grain size effect shown in Figure 6 is useful in ex-

3388—VOLUME 27A, NOVEMBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A

(a)

(b)

Fig. 7—(a ) Irradiation temperature as a function of irradiation time, usedin Ref. 50. (b ) The predicted temperature dependence of phosphorussegregation in the irradiated a-Fe for neutron dose rate 5 1.05 3 1028

dpa/s and neutron dose 5 0.042 dpa (B 5 0.01, ε 5 0.1, R 5 1 mm, andr0 5 5 3 1018 m22).

plaining why some of the predictions for the enrichmentsuggest that the boundary concentration will be greater than100 pct. It is necessary to take careful account of the overallmicrostructure of the particular material concerned. For ex-ample, certain low-alloy steels, such as 2.25Cr1Mo steel,will have a microstructure of tempered martensite or bain-ite, where there are vast numbers of lath and lath packetboundaries present. There is some evidence to suggest[13]

that these boundaries act as good sinks for point defects.Although the prior austenite grain size is ;100 mm, theappropriate grain size is that corresponding to these laths,and is about 1 mm. In this case, the predicted phosphorusboundary concentrations will be much less than 100 pct,and the higher enrichments predicted for the larger grainsizes appropriate to changing the prior austenite grain sizeare somewhat irrelevant to such a microstructure.

It is seen from Figure 2 that the combined irradiation-induced nonequilibrium and irradiation-enhanced equilib-rium peak segregation temperature and the thermalequilibrium peak segregation temperature are about 150 7Cand 550 7C, respectively, for P grain boundary segregationin a-Fe at dose rate 5 1026 dpa/s and dose 5 1 dpa. There-fore, when analyzing irradiation-induced embrittlement oflow-alloy ferritic steels, one should simultaneously think ofequilibrium and radiation-induced nonequilibrium grainboundary segregation of embrittling elements, such asphosphorus. If we assume that irradiation-induced nonhar-dening embrittlement is brought about by P grain boundarysegregation during irradiation, we may predict from thiswork that, at dose rate 5 1026 dpa/s and dose 5 1 dpa, theferritic steels will exhibit considerable phosphorus segre-gation if they are irradiated in the vicinity of 150 7C and550 7C. In practice, when low-alloy ferritic steels are tem-pered in the range from 400 7C to 550 7C, they will exhibittemper embrittlement associated with grain boundary seg-regation of phosphorus,[48,49] which leads to intergranularfracture. Similar intergranular fracture could be predictedto be associated with combined irradiation-induced none-quilibrium and equilibrium segregation of phosphorus inthe lower temperature regime.

A recent experimental study[50] found that after a 0.077wt pct P-doped 2.25Cr1Mo steel, water quenched, was ir-radiated to a neutron dose of 0.042 dpa at a neutron doserate of 1.05 3 1028 dpa/s (neutron energy .1 MeV) in therange from 250 7C to 290 7C in a thermal, light-water re-search reactor, named SAPHIR, operated by the Swiss Fed-eral Institute for Reactor Research (Figure 7(a)),FEGSTEM microanalysis of the grain boundary demon-strated no apparent neutron irradiation-induced segregationof phosphorus. The segregation level of phosphorus wasdetermined as ;4 at pct. This measured value compareswith that of ;3.6 at pct produced during quenching becauseof thermal nonequilibrium segregation and predicted by thethermal nonequilibrium segregation model established byXu and Song.[25] Under the previously mentioned irradiationconditions, the predicted temperature dependence of phos-phorus segregation in a-Fe is shown in Figure 7(b), whichdemonstrates that there is no apparent radiation effect inthe range of 250 7C to 300 7C, which is in agreement withthe present modeling.

The predicted results have been compared, also, withsome observed data for a pressure vessel C-Mn weld metal,as shown in Table II. The observed grain boundary com-positions were obtained using either the FEGSTEM orAES.[51] Table II indicates that the predictions are generallyconsistent with the experimental values.

IV. CONCLUSIONS

1. A combined equilibrium and radiation-induced none-quilibrium grain boundary solute segregation model has beenestablished for dilute alloys based upon the solute-point de-fect complex mechanism. Provision is made for relating thedamage rate to an equilibrium level of freely migrating de-fects. Irradiation-enhanced solute diffusion is taken into ac-count. The diffusion equations provide an accurate definitionof the grain boundary solute concentration. The effect of thetemperature dependence of dislocation density has been ac-

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, NOVEMBER 1996—3389

Table II. Comparison of the Experimental and Predicted Results for C-Mn Steel

MaterialIrradiation

Temperature (7C)Bulk P Content

(Wt Pct)IrradiationCondition

Observed Value(At. Pct)

Predicted Value(At. Pct)

1 198 0.039 ;2 3 10212 dpa/s9.4 3 1024 dpa

20 22

1 260 0.039 ;2 3 1029 dpa/s9.2 3 1023 dpa

37 35

2* 292 0.040 ;2 3 1029 dpa/s9.5 3 1023 dpa

42 38

* AES results; all the others obtained by FEGSTEM.Dislocation density constant; r0 5 5.5 3 1016 m22 for material 1 and 3.0 3 1015 m22 for material 2.Dose rate correction factor: B 5 0.01 for 198 7C irradiation and B 5 0.02 for 260 7C and 292 7C irradiation.Vacancy concentration correction factor: ε 5 0.1.Interstitial formation energy: EI

f 5 3.0 eV.Grain size: R 5 20 mm.It is assumed in the calculation that the bulk P concentration is 0.072 at. pct.

commodated and the estimation method for quantifying com-plex diffusion has been reappraised. The analysis has beenapplied to predictions of phosphorus grain boundary segre-gation in the a-Fe matrix under neutron irradiation.

2. The predicted results demonstrate that (a) there existsa segregation transition temperature below which combinedirradiation-induced nonequilibrium and irradiation-en-hanced equilibrium segregation is dominant, and abovewhich thermal equilibrium segregation is dominant; (b)peaks in the temperature dependence of segregation shift tolower temperatures with decreasing neutron dose rateand/or increasing neutron dose; and (c) the combined irra-diation-induced nonequilibrium and irradiation-enhancedequilibrium peak segregation temperature and thermal equi-librium peak segregation temperature are about 150 7C and550 7C, respectively, for phosphorus grain boundary seg-regation in a-Fe at a neutron dose rate of 1026 dpa/s and aneutron dose of 1 dpa. Comparison of the predictions withsome existing experimental observations indicates that themodel works satisfactorily.

ACKNOWLEDGMENTS

This work was supported by the Nuclear Electric plc.

LIST OF SYMBOLS

AI a constant showing the vibrational energy ofatoms around the interstitial

Av a constant showing the vibrational energy ofatoms around the vacancy

B the dose rate correction factor, i.e., the fractionof freely migrating defects

Cbq(t) the equilibrium segregation level as a functionof irradiation time t at a given temperature

Cbr(t ) the irradiation-induced nonequilibriumsegregation level as a function of irradiationtime t at a given temperature

mCbr the maximum concentration of irradiation-induced segregation at a given temperature

Cc the complex concentrationCg the solute concentration in the matrixCI the interstitial concentration in the matrix at the

steady state of neutron irradiation

cCv the equilibrium vacancy concentrationrCv the irradiation-generated vacancy concentration

C`(T) the maximum equilibrium segregation level attemperature T

d the grain boundary enriched thicknessDc the diffusion coefficient of complexes in the

matrixDi the diffusion coefficient of solute atoms in the

matrix*Di the irradiation-enhanced diffusion coefficient of

solute atoms in the matrixDI the diffusion coefficient of interstitials in the

matrixDv the diffusion coefficient of vacancies in the

matrixDoc the pre-exponential constant for diffusion of

complexesDoi the pre-exponential constant for diffusion of

solute atomsDoI the pre-exponential constant for diffusion of

interstitialsDov the pre-exponential constant for diffusion of

vacanciesiIEb the solute-interstitial binding energyIEf the interstitial formation energyvEf the vacancy formation energy

Ei the activation energy for solute diffusion in thematrix

*Ei the irradiation-reduced activation energy forsolute diffusion in the matrix

cEm the migration energy of complexesIEm the migration energy of interstitialsvEm the migration energy of vacancies

G the defect production rate, i.e., neutron dose ratek Boltzmann’s constantkc a geometrical constant

2kdI the grain interior sink strength for theinterstitial

2kdv the grain interior sink strength for the vacancyQ the equilibrium segregation energyR the grain sizet the timetc the critical timeT the absolute temperature

3390—VOLUME 27A, NOVEMBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A

x the distanceZI the bias parameter defining the preferential

interaction between interstitials anddislocations compared with that betweenvacancies and dislocations

a the nonequilibrium enrichment ratioai the equilibrium enrichment ratiob a constant showing the vibrational entropy of

the grain boundary regiond a numerical constant, also known as the critical

time constantε a factor characterizing the vacancy

concentration in regions near the grainboundary

r the dislocation density

REFERENCES

1. D. McLean: Grain Boundaries in Metals, Oxford University Press,London, 1957, p. 118.

2. E.D. Hondros and D. McLean: in Grain Boundary Structure andProperties, G.A. Chadwick and D.A. Smith, eds., Academic Press,London, 1976, pp. 353-83.

3. M.P. Seah: Acta Metall., 1980, vol. 28, pp. 955-62.4. D.R. Harries and A.D. Markwich: Phil. Trans. R. Soc., 1980, vol.

A295, pp. 197-207.5. J. Kameda and A.J. Bevolo: Scripta Metall., 1987, vol. 21, pp. 1499-

1503.6. J. Kameda, X. Mao, and A.J. Bevolo: J. Nucl. Mater., 1991, vols.

179–181, pp. 1037-37.7. W.J. Phythian and C.A. English: J. Nucl. Mater., 1993, vol. 205, pp.

162-77.8. P.E.J. Flewitt, G.H. Williams, and M.B. Wright: Mater. Sci. Technol.,

1993, vol. 9, pp. 75-82.9. P.E.J. Flewitt and R.K. Wild: The Physical Characterisation of

Materials, IOP Publishing, Bristol, United Kingdom, 1994.10. G.J. Mahon, A.W. Nicholls, I.P. Jones, C.A. English, and T.M.

Williams: in Radiation-Induced Sensitisation of Stainless Steels,D.I.R. Norris, ed., Central Electricity Generating Board, London,1986, pp. 99-115.

11. D.I.R. Norris, C. Baker, and J.M. Titchmarsh: in Materials forNuclear Reactor Core Applications, British Nuclear Energy Society,London, 1987, pp. 277-83.

12. J. Kameda and A.J. Bevolo: Acta Metall., 1989, vol. 37, pp. 3283-96.

13. T.S. Morgan, E.A. Little, R.G. Faulkner, and J.M. Titchmarsh: inEffects of Radiation on Materials: 15th Int. Symp., ASTM STP 1125,R.E. Stoller, A.S. Kumar, and D.S. Gelles, eds., ASTM, Philadelphia,PA, 1992, pp. 633-44.

14. R.D. Carter, D.L. Damcott, M. Atzmon, G.S. Was, and E.A. Kenik:J. Nucl. Mater., 1993, vol. 205, pp. 361-73.

15. P.R. Okamoto and L.E. Rehn: J. Nucl. Mater., 1979, vol. 83, pp. 2-23.

16. R.A. Johnson and N.Q. Lam: Phys. Rev., 1976, vol. B13, pp. 4364-75.

17. N.Q. Lam, P.R. Okamoto, H. Wiedersich, and A. Taylor: Metall.Trans. A, 1978, vol. 9A, pp. 1707-14.

18. P.R. Okamoto and H. Wiedersich: J. Nucl. Mater., 1974, vol. 53, pp.336-45.

19. S.M. Murphy and J.M. Perks: J. Nucl. Mater., 1990, vol. 171, pp.360-72.

20. Yves Grandjean, Pascal Bellon, and Georges Martin: Phys. Rev.,1994, vol. B50, pp. 4228-31.

21. R.G. Faulkner, S. Song, and P.E.J. Flewitt: Mater. Sci. Technol., inpress.

22. L.E. Rehn, P.R. Okamoto, D.I. Potter, and H. Wiedersich: J. Nucl.Mater. 1978, vol. 74, pp. 242-51.

23. N.Q. Lam and G.K. Leaf: J. Mater. Res., 1986, vol. 1, pp. 251-67.24. R.G. Faulkner, N.C. Waite, E.A. Little, and T.S. Morgan: Mater. Sci.

Eng., 1993, vol. A171, pp. 241-48.25. T. Xu and S. Song: Acta Metall., 1989, vol. 37, pp. 2499-2506.26. R.G. Faulkner: J. Mater. Sci., 1981, vol. 16, pp. 373-83.27. C.A. English, S.M. Murphy, and J.M. Perks: J. Chem. Soc., Faraday

Trans., 1990, vol. 86, pp. 1263-71.28. L.E. Rehn: J. Nucl. Mater., 1990, vol. 174, pp. 144-50.29. V. Naundorf, M.-P. Macht, and H. Wollenberger: J. Nucl. Mater.,

1992, vol. 186, pp. 227-36.30. H. Trinkaus, V. Naundorf, B.N. Singh, and C.H. Woo: J. Nucl.

Mater., 1994, vol. 210, pp. 244-53.31. M.J. Norgett, M.T. Robinson, and I.M. Torrens: Nucl. Eng. Des.,

1975, vol. 33, pp. 50-54.32. C.H. Woo and B.N. Singh: Phil. Mag., 1992, vol. A65, pp. 889-912.33. A.V. Nikolaeva, Yu. A. Nikolaev, and A.M. Kryukov: J. Nucl.

Mater., 1994, vol. 218, pp. 85-93.34. S.G. Druce, C.A. English, A.J.E. Foreman, R.J. McElroy, and I.A.

Vatter: Report No. AEA-RS-2126, Harwell Laboratory, Oxfordshire,United Kingdom, Nov. 1991.

35. E.A. Little and L.P. Stoter: in Effects of Radiation on Materials: 11thInt. Conf., ASTM STA 782, H.R. Brader and J.S. Perrin, eds., ASTM,Philadelphia, PA, 1982, pp. 207-33.

36. D.I.R. Norris, C. Baker, C. Taylor, and J.M. Titchmarsh: in Effectsof Radiation on Materials: 15th Int. Symp., ASTM STP 1125, R.E.Stoller, A.S. Kumar, and D.S. Gelles, eds., ASTM, Philadelphia, PA,1992, pp. 603-20.

37. B.A. Loomis and D.L. Smith: in Effects of Radiation on Materials:15th Int. Symp., ASTM STP 1125, R.E. Stoller, A.S. Kumar, and D.S.Gelles, eds., ASTM, Philadelphia, PA, 1992, pp. 885-96.

38. T. Hashimoto and N. Shigenaka: J. Nucl. Mater., 1992, vol. 189, pp.161-69.

39. P.H. Dederichs, C. Lehmann, H.R. Schober, A. Scholz, and R. Zeller:J. Nucl. Mater., 1978, vols. 69–70, pp. 176-99.

40. Physical Metallurgy, 3rd ed., R.W. Cahn and P. Haasen, eds., North-Holland, Amsterdam, 1983, p. 1189.

41. N.Q. Lam, A. Kumar, and H. Wiedersich: in Effects of Radiation onMaterials: 11th Int. Conf., ASTM STP 782, H.R. Brager and J.S.Perrin, eds., ASTM, Philadelphia, PA, 1982, pp. 985-1007.

42. S.M. Kim and W.J.L. Buyers: J. Phys. F: Met. Phys., 1978, vol. F8,pp. L103-L108.

43. M. Kiritani, H. Takata, K. Moriyama, and F. Euchi Fujita: Phil. Mag.,1979, vol. A40, pp. 779-802.

44. T.M. Williams, A.M. Stoneham, and D.R. Harries: Met. Sci., 1976,vol. 10, pp. 14-19.

45. R. Bullough, M.R. Hayns, and M.H. Wood: J. Nucl. Mater., 1980,vol. 90, pp. 44-59.

46. D. Hull and D.J. Bacon: Introduction to Dislocations, 3rd ed.,Pergamon, Oxford, 1984, p. 79.

47. T. Qgura: Trans. Jpn. Inst. Met., 1981, vol. 22, pp. 109-17.48. R.A. Mulford, C.J. McMahon, Jr., D.P. Pope, and H.C. Feng: Metall.

Trans. A 1976, vol. 7A, pp. 1183-95.49. Yuan Zhexi, Song Shenhua, R.G. Faulkner, and Xu Tingdong: Acta

Metall. Mater., 1994, vol. 42, pp. 127-32.50. S. Song: Ph.D. Thesis, Loughborough University, Loughborough,

Leicestershire, United Kingdom, 1995.51. P.E.J. Flewitt: Nuclear Electric Technology Centre, Berkeley, United

Kingdom, unpublished research, 1995.