J O U R NA L O F M AT E R I A L S S C I E N C E L E T T E R S 1 6 ( 1 9 9 7 ) 1 1 9 1 – 1 1 9 4

Irradiation-enhanced solute diffusion in alloys

R. G. FAULKNER, SHENHUA SONGInstitute of Polymer Technology and Materials Engineering, Loughborough, Leicestershire LE11 3TU, UK

P. E. J. FLEWITT, S. B. FISHERBerkeley Technology Centre, Magnox Electric plc, Berkeley, Gloucestershire GL13 9PB, UK

Irradiation-enhanced diffusion of solute atoms playsan important part in the kinetic process of irradiationeffects in materials, such as irradiation-inducedsegregation and precipitation [1, 2]. As a conse-quence, it is necessary to evaluate this diffusionenhancement.

In our previous work [2], we proposed anapproach to evaluating irradiation-enhanced solutediffusion. The approach is based on the followingidea. Irradiation-created vacancies affect the vacancyformation energy term in the diffusion activationenergy but do not affect the vacancy migrationenergy term. We may calculate the relative concen-trations of freely migrating point defects under boththermal equilibrium and irradiation-induced non-equilibrium conditions. The ratio of these twoconcentrations will determine the extent to whichthe activation energy for solute diffusion is reducedby lowering the effective vacancy formation energy.Hence the irradiation-reduced activation energy forsolute diffusion, E�i , is given by [2]

E�i � Ei ÿC r

ν

Ceν � C r

νEν

f (1)

where Ei is the activation energy for thermaldiffusion of solute atoms, Eν

f is the vacancyformation energy, and Ce

ν and Crν are the thermal

equilibrium and irradiation-created vacancy concen-trations, respectively. It may be seen that Equation 1has no clear physical basis. Consequently, a newapproach that has a clear physical basis needs to bedeveloped. In this work, the new approach has beenestablished on the basis of the vacancy diffusionmechanism.

For diffusion of substitutional solute atoms in acrystal matrix, the diffusion coefficient, Di, can begiven by [3]

Di � θω pν (2)

where θ is a material constant, ω is the probabilitythat a solute atom jumps into a vacant nearest-neighbour lattice site, and pν is the probability thatany given nearest-neighbour lattice site is vacant. pνis approximately equal to the vacancy concentrationCν (fraction of vacant lattice sites).

ω may be obtained from

ω � ν exp ÿ

Gm

kT

� �(3)

where ν is the vibrational frequency of the soluteatom, k is the Boltzmann constant, T is the absolutetemperature. Gm is the free energy required for asolute atom to migrate from an equilibrium positionto a nearest neighbour site and is a function oftemperature T.

As a consequence, Equation 2 can be rewritten as

Di � θνCν exp ÿ

Gm

kT

� �

� δ(T)Cν

(4)

where δ(T ) � θν exp (ÿGm=kT ).Under irradiation, the vacancy concentration, Cν,

is given by

Cν � Ceν � C r

ν (5)

As a result, Equation 4 can be rewritten as

D�

i � δ(T )(Ceν � C r

ν) (6)

where D�

i is the irradiation-enhanced solute diffusioncoefficient.

In the absence of irradiation, D�

i is equal to thethermal diffusion coefficient DT

i . Consequently, δ(T)is given by

δ(T ) � DTi =Ce

ν (7)

Thus the irradiation-enhanced solute diffusioncoefficient, D�

i , can be acquired from

D�

i � DTi

Ceν � C r

ν

Ceν

� �(8)

The thermal diffusion coefficient, DTi , is given by

DTi � Doi exp ÿ

Ei

kT

� �(9)

where Ei and Doi are the activation energy and thepre-exponential constant for solute diffusion in thematrix, respectively. The thermal equilibrium va-cancy concentration, Ce

ν, is given by

Ceν � Aν exp ÿ

Eνf

kT

� �(10)

where Aν is a constant correlating with the vibra-tional entropy of atoms around the vacancy, and Eν

fis the vacancy formation energy.

The irradiation-created vacancy concentration maybe evaluated by the following approach. Duringsteady-state irradiation [4]:

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BG ÿ λC rνC r

I ÿ k2dν DνC r

ν � 0 (11a)

BG ÿ λC rνC r

I ÿ k2dI DI C r

I � 0 (11b)

where G is the point defect production rate or doserate; Dν and DI are the diffusion coefficients ofvacancies and interstitials in the matrix, respectively;Cr

I and Crν are the irradiation-generated interstitial

and vacancy concentrations, respectively; B is thedose rate correction factor, i.e. the fraction of freelymigrating defects escaping from the cascade; λ is thelong-range recombination coefficient of freely mi-grating point defects outside the cascade, given by[5]

λ �2lDI

b2(12)

where b is the jump distance of interstitials; k2dν and

k2dI are the sink strengths for the vacancy and the

interstitial, respectively, given approximately by [6]

k2dν � r

1=2 6R� r

1=2

� �(13a)

k2dI � (Z Ir)1=2 6

R� (Z Ir)1=2

� �(13b)

where r is the dislocation density, R is the grainsize, and ZI is the bias parameter defining thepreferential interaction between interstitials anddislocations as compared with that between vacan-cies and dislocations.

The dislocation density, r, is dependent on theheat treatment temperature and time, and increaseswith increasing neutron irradiation dose. After thesteady state point defect concentration is achieved,the dislocation density will no longer vary withincreasing dose, but will be dependent upon tem-perature. In this scenario, r may be approximatelygiven by [2]

r � ro expEd

kT

� �(14)

where ro is the dislocation density constant whichdepends on the pre-irradiation dislocation density,the dose rate, etc. and Ed is the activation energy fordislocation recovery processes.

From Equations 11a and 11b, one may obtain

C rν �

BGF(η)

Dν k2dν

(15a)

where

F(η) �2η

[(1� η)1=2ÿ 1] (15b)

η �4λBG

k2dν k2

dI Dν DI(15c)

The diffusion coefficients of vacancies and inter-stitials, Dν and DI, are respectively given by

Dν � Doν exp ÿ

Eνm

kT

� �(16a)

DI � DoI exp ÿ

EIm

kT

� �(16b)

where Doν and DoI are the pre-exponential constantsfor diffusion of vacancies and interstitials, respec-tively and Eν

m and EIm are the migration energies of

the vacancies and the interstitials, respectively.It is evident that the model described above has

clearer physical bases compared to the earlier one.This new model has been applied to the evaluationof neutron irradiation-enhanced phosphorus diffusionin the α-Fe matrix. Data used in the theoreticalcalculations are listed in Table I.

Fig. 1 shows the phosphorus diffusion coefficientsas a function of temperature for thermal diffusionand irradiation-enhanced diffusion, at a neutron doserate of 10ÿ8 dpa sÿ1 and a dislocation density con-stant (ro) of 1014 mÿ2. Above ,350 8C, the irradia-tion-enhanced diffusion coefficient is the same as thethermal diffusion coefficient. This is because, above350 8C, the irradiation-created vacancy concentrationis so low due to the higher mobility of vacancies thatthere are no apparent irradiation-enhanced diffusioneffects. Below ,350 8C, there are apparent irradia-tion-enhanced diffusion effects in that the differencebetween the irradiation-enhanced and thermal diffu-sion coefficients increases with decreasing tempera-ture.

The temperature dependences of the phosphorusdiffusion coefficient for irradiation-enhanced diffu-sion at different neutron dose rates or differentdislocation density constants are shown in Figs 2 and3. It is easy to see that either the neutron dose rate

1E 2 16

1E 2 19

1E 2 22

1E 2 25

1E 2 28

1E 2 31

1E 2 34

1E 2 37

1E 2 40

1E 2 430 100 200 300 400 500 600 700

Temperature, (°C)

Diff

usio

n co

effic

ient

, (m

2s2

1 )

Figure 1 Temperature dependences of phosphorus diffusion coeffi-cient in the α-Fe matrix for thermal diffusion (D1-j) and irradiation-enhanced diffusion (D2-m), at a dose rate of 10ÿ8 dpa sÿ1 and adislocation density constant (ro) of 1014 mÿ2.

TABLE I Data used in the theoretical calculation

Aν 1 [3] Ed (eV) 0.1 [12]Eν

f (eV) 1.4 [7] Cg (at %) 0.072Ei (eV) 2.68 [2] B 0.01 [13]Eν

m (eV) 1.24 [8] R (µm) 20EI

m 0.3 [9] b (nm) 0.143 [14]Doi (m2 sÿ1) 7.12 3 10ÿ3 [2] ZI 1.1 [15]Doν (m2 sÿ1) 5 3 10ÿ5 [10]DoI (m2 sÿ1) 5 3 10ÿ6 [11]

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or the dislocation density has an obvious effect onthe irradiation-enhanced diffusion coefficient. As aconsequence, choosing appropriate parameters isquite important in order accurately to evaluateirradiation-enhanced diffusion. Fig. 4 represents theirradiation-enhanced phosphorus diffusion coeffi-

cients as a function of irradiation temperature inthe α-Fe matrix, determined by both the presentmodel and the earlier model. It is clear that theextent to which the phosphorus diffusion is enhancedfor the present model is much smaller than that forthe earlier model.

Temperature dependences of equilibrium phos-phorus grain boundary segregation during neutronirradiation in α-Fe, predicted by our combinedequilibrium and irradiation-induced non-equilibriumgrain boundary segregation model [2], are shown inFig. 5. It is clear that the equilibrium segregationkinetics of phosphorus are enhanced by the irradia-tion-enhanced phosphorus diffusion and, neverthe-less, the extent to which the segregation kinetics areenhanced is much smaller for the present irradiation-enhanced diffusion model than for the earlier model.

AcknowledgementsThis work was financially supported by MagnoxElectric plc.

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1E 2 17

0 100 200 300 400 500 600 700Temperature, (°C)

Diff

usio

n co

effic

ient

, (m

2s2

1 )1E 2 19

1E 2 21

1E 2 23

1E 2 25

1E 2 27

Figure 2 Temperature dependences of phosphorus diffusion coeffi-cient in the α-Fe matrix for irradiation-enhanced diffusion at dose ratesof (j) 1 3 10ÿ8 dpa sÿ1 and (m) 1 3 10ÿ10 dpa sÿ1 (dislocationdensity constant (ro) � 1014 mÿ2).

1E 2 17

0 100 200 300 400 500 600 700Temperature, (°C)

Diff

usio

n co

effic

ient

, (m

2s2

1 )

1E 2 19

1E 2 21

1E 2 23

1E 2 25

1E 2 27

Figure 3 Temperature dependences of phosphorus diffusion coeffi-cient in the α-Fe matrix for irradiation-enhanced diffusion atdislocation density constants of (j) 1 3 1014 mÿ2 and (m)1 3 1016 mÿ2 (dose � 10ÿ8 dpa sÿ1).

1E 2 12

0 100 200 300 400 500 600 700Temperature, (°C)

Diff

usio

n co

effic

ient

, (m

2s2

1 )

1E 2 14

1E 2 18

1E 2 20

1E 2 24

1E 2 26

1E 2 16

1E 2 22

Figure 4 Irradiation-enhanced phosphorus diffusion coefficient in theα-Fe matrix as a function of irradiation temperature, determined byboth the present model (D1-j) and the earlier model (D2-m), at a doserate of 10ÿ8 dpa sÿ1 and a dislocation density constant (ro) of1014 mÿ2.

10000

1000

100

10

150 150 250 350 450 550 650

Temperature, (°C)

Cb/

Cg

Figure 5 Temperature dependences of thermal (C1-r) and irradiation-enhanced (C2-j) is associated with the present irradiation-enhanceddiffusion model; and C3 (m) is associated with the earlier irradiation-enhanced diffusion model) equilibrium phosphorus segregation tograin boundaries in α-Fe (dose rate � 10ÿ8 dpa sÿ1, dose � 1 dpa, anddislocation density constant (ro) � 1014 mÿ2).

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Received 30 January 1997and accepted 21 March 1997

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