Models for diffusion with trapping effects

Scripta METALLURGICA

Vol. iS, pp. 909-912, 1981 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

MODELS FOR DIFFUSION WITH TRAPPING EFFECTS R. R. de A v i l l e z , * R. J. Lauf+ and C. J. Altstetter Department of Metallurgy and Mining Engineering and the Materials Research Laboratory University of I l l i n o i s at Urbana-Champaign Urbana, IL 61801

(Received May 19, 1981) (Revised June 2, 1981) Many models have been proposed to explain the effect of traps on i n t e r s t i t i a l diffusion. Some models are very general but either require extensive use of a computer (I-3), or the expressions for the diffusion coefficient are too involved (4-6). These models, therefore can not be easily compared with experimental data. Simple models may have conceptual limitations, however, they can sometimes provide a consistent description of the experimental data within their v a l i d i t y . The models proposed by Oriani (7), by McLellan (8) and by Perkins and Padgett (9) are discussed and compared using experimental data for oxygen diffusion in niobium-based alloys (9,10). Both Oriani's and McLellan's models assume no change of the saddle point energy for the activated migration of an i n t e r s t i t i a l atom, hence the atom has the same probablility for a l l jumps until i t is trapped. Further, they also assume one species of trap site, and equilibrium between the i n t e r s t i t i a l population in trapped and normal sites. We have analyzed the Oriani model without the restriction of small trap population, and i t was shown to lead to the major characteristics of diffusion with traps (11). Its general expression may be written as D

~--

1

=

n

T

1+ ~

F1

Li--~--~n_~

(1}

2

ex p -

where Dn and D are the normal d i f f u s i v i t y and the d i f f u s i v i t y with trapping effects, respectively; and the fractional occupancy of trap sites, @t' and normal sites, Bn, are related by: @t 1_0t

:

~

@n

exp - ~

(2>

with the free energy, Ag = Ah - TAS, where Ah is the partial molar enthalpy difference, and As is the nonconfigurational entropy difference between a normal and a trap site. T is the ratio of trap sites to normal sites and is given by T = ) Xs,

(3}

In the present work substitutional solute atoms are assumed to be responsible for trapping. Thus @is the number of trap sites associated with a substitutional solute atom, 8 is the number of i n t e r s t i t i a l sites per l a t t i c e site, and X. is the atom fraction of substitutional solute, k and T have their usual meanings. Equation ( I ) , is, however, used mostly at the *currently at the Instituto Nacional de Technologia, Rio de Janeiro, Brazil +currently at the Oak Ridge National Laboratory, Oak Ridge, TN

909 0036-97481811080909-04502.00/0 Copyright (c) 1981 Pergamon Press Ltd.

910

DIFFUSION WITH TRAPPING EFFECTS

Vol.

high temperature l i m i t , ag/kT +0, or for very dilute systems, B~, B_ << 1, d i f f u s i v i t y is independent of the fractional occupancy of trap and norm~al s'Ites: D ~n

I

:

I +~

.

1 5 , No.

8

where the

(4)

exp " ~ T

The McLellan model was derived from reaction rate theory and was intended to account for the different chemical environments for i n t e r s t i t i a l solute atoms in a perfect and in a defect alloy crystal. Its general derivation leads to a rather involved expression, which, however, yields the following relation when the fractional occupancy of trap sites is much smaller than unity. D : ~nn

i - T + T exp -

(5)

The s t a t i s t i c s of jumping are greatly complicated in the formulation of Koiwa (1), when the activation energy for diffusion is changed (by AE) due to the substitutional atom responsible for trapping. The equation Koiwa derives for the l i m i t i n g case of AE + 0 is equivalent to Eq. 4. Perkins and Padgett (9) obtained essentially the same result when AE was a r b i t r a r i l y set equal to one-half the binding energy. Although the assumption of an energy barrier change, AE, is physically reasonable, i t complicates the calculation of trap capture and escape probabilities and introduces yet another parameter to f i t the model to the experimental data. Equations (4) and (5) are rather suitable for direct comparisons of model to experiment, and the oxygen d i f f u s i v i t y data of Perkins and Padgett (9) and Lauf and A l t s t e t t e r (10) for Nb-based alloys were used for this purpose. A nonlinear least-squares program was employed in the analysis. The values of the binding energy that give the best f i t to the data are displayed in Table 1. Both models were f i t t e d assuming octahedral i n t e r s t i t i a l site occupancy in a BCC crystal (B = 3) and a substitutional solute atom trap for which all adjacent sites are equivalent (¢ = 6), so that ~ = 2X . The numbers in the parentheses are the errors for a 95% confidence level. The results o~ the Oriani model, Eq. (4), are in good agreement with the previously reported values (10) while the errors are even smaller. The binding energies calculated from the McLellan model are about 25% smaller than those from Oriani's model. Table I Binding Energies (kJ/mole) Between a Substitutional Solute Atom and an Oxygen I n t e r s t i t i a l Atom In Niobium Models for Diffusion with Traps~ . . . .

Alloy +

McLellan Model ¢= 6

Oriani Model ¢= 6

Nb-.9Ti Nb-.95Zr Nb-I.4V Nb-2.7V Nb-5,1V Nb-4.1Ta

50.7 (5.1%) 51.4 (2.0%) 42.0 (3,8%) 33.9 (2.8%) 36.5 (2.5%) 25.8 (10.2%)

67.3 67.6 54.7 45.7 52.3 33.8

e

(5.0%) (2.4%) (4.0%) (3.2%) (3.6%)

(5.8%)

Percentage in parentheses refer to twice the standard error of f i t t i n g . ¢ is defined as the number of trap sites per substitutional solute atom + D i f f u s i v i t y data of Lauf and A l t s t e t t e r (10).

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8

DIFFUSION WITH TRAPPING

EFFECTS

911

The f i t of the model to the data is equally good for the two models. For example, experimental points are compared to calculated lines for the two models, Figs. 1 and 2 for oxygen diffusion in a Nb-O.95 at% Zr alloy (10). I t was impossible to discern the most appropriate model solely from this analysis. I t appears, in the present case, that the refinement incorporated in the McLellan model does not improve the agreement between theory and experiment. More experimental points at low temperatures and use of the more general expressions might be necessary to allow a more d e f i n i t i v e comparison. Perkins and Padgett (9) measured oxygen d i f f u s i v i t i e s for three Nb-Zr alloys containing up to 1 at% Zr. They found that the temperature dependence of the oxygen d i f f u s i v i t y was essentially the same as for pure Nb, though the d i f f u s i v i t i e s were lower in the ternary alloys. Their result for the I at% Zr alloy is shown as a dashed line in Fig. 1. By their own model or by those of Oriani and McLellan their data indicate no binding of 0 to Zr in a niobium matrix, and the depression of the d i f f u s i v i t y in the alloy must be explained solely on the basis of entropic considerations. The main limitation of the models proposed by Oriani and McLellan is the assumption of local equilibrium between the atomic populations in normal and trap sites throughout the diffusion process. Indeed, s t a t i s t i c a l calculation (1) showed that the assumption of local equilibrium may be inappropriate i f there is a change in the saddle point energy. Further, the expressions for the fractional occupancy, % and 0., depend on the total concentration of i n t e r s t i t i a l s (8,11), which is usually a functibn of ~me. Hence, the general relations for the diffusion coefficient (e.g., Eq. (I) and equations in Ref. 8 and 11) can not be used except, perhaps, in a tracer diffusion technique where the total concentration of i n t e r s t i t i a l solute is kept constant. TEMPERATURE (*C) 1000 800

1200 l

l

-24,

I

I

I

~

TEMPERATURE (eC) I

600

1200

I

l

Nb - 0.95 Zr

X

~

1000 l

-2&

!

800 t

~

I

Nb-

0.95

Zr Model

~ ~ellon

Oriani Model

600 i

l

10-" E

E -26

:> 10-t2 ~ -z8 N

B

10"12 ~.

-2s

•\. \ •\.

\

°-

\

I 0.6

I 0.8

I

I

10-13

-30

~I

1.0

i

1.2

REC~PflOCAL TEMPERATURE (*W 1) 1=10"31

Fig. 1 Arrhenius plot for oxygen diffusion in Nb-O.95Zr alloy (lO). D i f f u s i v i t i e s were f i t t e d according to Oriani's model, 4=6. The upper line is for diffusion of oxygen in pure niobium. The dashed line is Perkins and Padgett's result for a l% Zr alloy (9).

10-13

~

0.6

l

I

~

0.8

I

I

1,0

1.2

RECIPROCAL TEMPERATURE [el("1) IxlO"3}

Fig. 2

Arrhenius p l o t f o r oxygen d i f f u s i o n in Nb-O.95Zr alloy (I0). D i f f u s i v i t i e s were f i t t e d according to McLellan's model, 4=6. The upper l i n e is f o r diffusion of oxygen in pure niobium.

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DIFFUSION WITH TRAPPING EFFECTS

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1S, No. 8

The above restrictions may be avoided altogether i f one uses the general model of diffusion with traps proposed by McNabb and Foster (12). Although their mathematical analysis of trapping may be employed with most boundary and i n i t i a l conditions, i t does not have a general analytic solution. Caskey and Pillinger (13) developed an approximate solution to these equations by the f i n i t e difference method. Their solution allows the determination of the concentration of diffusing species as a function of time and position. Therefore, the fractional occupancies, B+ and en, may be calculated from the experimental data; and the study of the fractional occupancies a~ a function of time and temperature w i l l provide information on the assumption of local equilibrium ( i . e . , whether Eq. (2) is satisfied) and the trapping energy. Calculations using the Caskey and Pillinger approach have been made and w i l l be reported in detail elsewhere (14). Acknowledgement This research was performed in the Materials Research Laboratory of the University of I l l i n o i s at Urbana-Champaign under the sponsorship of the U.S. Department of Energy, Contract DOE-EY76-C-02-1198. Dr. de Avillez is pleased to acknowledge the Comissao Nacional de Energia Nuclear, Rio de Janeiro, Brazil for fellowship support. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

M. Koiwa, Acta Met. 22, 1259 (1974). K.W. Kehr, D. Richter and R.H. Swendser, J. Phys. F: Metal Phys. 8, 433 (1978). G.E. Murch, Phil. Mag. A41, 157 (1980). J.P. Stark, "Solid State Diffusion," Chap. 2, John Wiley, New York (1976). K.E. Blazek, Trans. JIM 19, 253 (1978). A. Barbu, Acta Met. 28, 499 (1980). R.A. Oriani, Acta Met. 18, 147 (1970). R.B. McLellan, Acta Met. 27, 1655 (1979). R.A. Perkins and R.A. Padgett, Jr., Acta Met. 25, 1221 (1977). R.J. Lauf and C.J. Altstetter, Acta Met. 27, 1157 (1979). R.R. de Avillez, Ph.D. thesis, University of I l l i n o i s , Urbana, IL (1981). A. McNabb and P.K. Foster, Trans. AIME 227, 618 (1963). G.R. Caskey, Jr. and W.L. Pillinger, Met. Trans. 6A, 467 (1975). R.J. Lauf, R.C. Frank and C.J. Altstetter, to be published.