Thermodynamics of interstitial systems

Scripta METALLURGICA

Vol. 22, pp. I05S-I056, 1988 Printed in the U.S.A.

Pergamon Press plc All rights reserved

THERMODYNAMICS OF INTERSTITIALSYSTEMS Rex B. McLellan Department of Mechanical Engineering and Materials Science William Marsh Rice University, Houston, Texas 77251 [Received March 18, 1988) {Revised A p r i l 11, 1988) The thermodynamics of austenitic solid solutions has long been an object of intensive study. V i r t u a l l y all of the effort has been devoted to binary N or C austenite and ternary solutions containing a substitutional (U) solute in addition to the i n t e r s t i t i a l component ( i ) and the major component (V) ( i . e . Fe). The fact that in such systems the (V,U) and i species occupy d i s t i n c t sublattices in the solution crystal gives rise to geometrical constraints enabling f a i r l y simple s t a t i s t i c a l methods to be used in analyzing thermodynamic data. These various models have been discussed and reviewed for the binary Fe-C system by many authors [1,2,3,4]. Even though the assumptions inherent in f i r s t and second order calculations are simple, the equations representing the thermodynamic functions in a closed form are most cumbersome. The present author is aware of no thermodynamic work on d i - i n t e r s t i t i a l austenites. Because of the strong C-C and N-N repulsive interactions in the C and N austenite solutions and the large areas of the T.B~ (temperature-interstitial s o l u b i l i t y ) fields in both the binary systems, i t is clear tha't the ternary Fe-C-N system is of great interest in the development of solution thermodynamics. Experimental work on this system is now underway in the author's laboratory and, although not yet completed, raises the question of s t a t i s t i c a l models in the treatment of d i - i n t e r s t i t i a l systems. Both f i r s t - o r d e r [5] and cumulant expansion techniques [6] (second-order) have been formulated for V-U-i ternary solutions, but the treatments, especially the l a t t e r [ 6 ] , are excessively complex. The same problems exist in the treatment of d i - i n t e r s t i t i a l ( V - i - j ) solutions and thus i t becomes pertinent to pose the question as to how accurate the simple (zeroth order) formulations are in comparison to the more complex f i r s t and second order models. This is the motivation behind the present communication. The most comprehensive thermodynamic measurements on the Fe-C system are due to Ban-ya, E l l i o t , and Chipman (BEC) [7,8] and span the f i e l d T = 1173-1673 K, e~ = atom l a t i o C/Fe = 0 - 0.07. The resulting p_artia]: configurational entropy 2~ and enthal~y H. - H. are given in Fig. ( I ) . The value of H. - H. is almost independent of ~(B~ is B: at ihfini~e d i l u t i o n of C) and Si is that of an ideal ~ n t e r s t i t i a l system. Now the i~act t6at S. is that of an ideal solution makes i t clear that, for C-austenite, departures.from regular ~olution behavior are small. The f i r s t order model [7,8] gives a value of H. shown by the upper dashed line - - ( ' ~ -- -) in Fig. (1) which corresponds to a C-C repulsivelinteraction (aci) of 8.25 kJ/mol and is v i r t u a l l y independent of temperature in the range 900-1400 K. The S. curve is shown by the uppermost (- - -) curve. This degree of compatibility between the f i r s t - o r d e r model and experiment is to be expected for a high-temperature system. Let us now investigate the zeroth order model ( i . e . the entropy is i d e n t i c a l l y that of an ideal system). The thermodynamic functions of entropy-dominated i n t e r s t i t i a l systems [4] can be written in terms of a~i and a series of functions ¢,~6in theac form, -

-

®

(-i)

~

Hi - Hi and

i~

: - kT s ~-_- ~.l "kT-" ¢6 £=I xs = ¢6 AEi 6 0i ~i = k 6=1Z £--F.(k-T-) [ i - (-1)66] - k In (_.i-z-~i)

(I) (2)

The functions ¢o are derivatives of the semi-invar~nt cumulants [9] and have been given up to 6=4 previousl~y [4]. The " b e s t - f i t " values of 2 and H. obtained by f i t t i n g eqns (1) and (2) to the BEC data at a mean temperature I of 14~5 K is shown by the dashed

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INTERSTITIAL SYSTEMS

Vol.

22, No.

7

l i n e ( ÷++% R~) and the uppermost curve of Fig. (1) ( - - - ), which is identical to the f i r s t - o r d e r cuFve for Si/k. These curves are labeled z=4 to denote that all the cumulants were considered. Now i f we take only the i n i t i a l term ~=1 in eqns (1) and (2) we obtain the results, 1

si

1

1

= - k In(ei/(Z

(3)

1

(4)

- el))

where ¢1 has been taken from previous work [4] and z is the number of nearest-neighbor metal atoms surrounding an octahedral site. These relations are the z~roth order results [10]. Thus in this approximation S. is identical to the experimental result and H showslthe same linear behavior with respect I to o. and temperature-independence exhibited by the BEC ~ata. The value of H. for the zeroth approximation using the aei-value d~duced from eqns (I) and (2) ( i . e . 7.53 kJ/mol at 1423 K) is shown in Fig. (1) (labeled ~ = I , O-order) ( -a-~-l- ). This simple conclusion is that the zeroth order model is a useful approximation for high-temperature i n t e r s t i t i a l systems and has the advantage of great simplicity. This conclusion is not meant to indicate that the more sophisticated methods should not be used f o r ~ solutions (indeed they should), but that the zeroth order approximation should be an adequate starting point in the study of ( v - j - i ) d i - i n t e r s t i t i a l solutions where f i r s t and second order treatments are excessively complex. I t is encouraging to note that in respect to binary Fe-N austenites, the f i r s t - o r d e r treatment [11,12] yields much smaller values of ae~ (3.97 kJ/mol) so that the zeroth order method should be an even better approximation. The calculated value of H. for the zeroth approximation (~ = 1) is contrasted with the result for (~ = 4) using AEi = 3.97 kj/mol at T = 1423 K in the two lowest plots (labeled N) in Fig. (1). The small energy effects are i l l u s t r a t e d by 0.01 eV (0.96 kJ/mol) marker. The energy required to transport a C-atom from a rest-state in a vacuum into the fcc Fe l a t t i c e is 6.76 eV (650 kJ/mol) [13].

4,0

3,0

2,0 ~n-

1,-t, Olor~r. - / i

32 ..o-

,y~,t

_..t-

Io.o,.r o~W v I 0 0.02

I 0,04

I 0,06

I 0,08

O, lq

Measuredand Calculated Partial Enthalpies and Entropies Acknowledgement The author is grateful for support provided by the Robert A. Welch Foundation. References

[1] [Z] [3]

FIG. 1.

M. Yiwen and T. Y. Hsu, Acta metall. 34, 325 (1986). G. J. S h i f l e t , J. R. Bradley, and H.-I~'. Aaronson, Metall. Trans. A9 999 (1978). H. I. Aaronson, H. A. Domian, and G. M. Pound, Trans. metall--Soc. AIME, 236, 768 (1966).

[4] R. B. McLellan and C. Ko, Acta metall. 35, 2151 (1987). [5] K. Alex and R. B. McLellan, J. Phys. Ch~. Solids, 32, 449 (1971). [6] J. C. Langeberg and R. B. McLellan, J. Phys. Chem. ~ i d s 35, 999 (1974). [7] R. B. McLellan and W. W. Dunn, J. Phys. Chem. Solids 30, 2~1 (1969). [8] R. B. McLellan and W. W. Dunn, Scripta metall. 4, 321--~1970). [9] J. G. Kirkwood, J. Chem. Phys. 6, 70 (1938). [10] K. Alex and R. B. McLellan, J.-Fhys. Chem. Solids 31, 2751 (1970). [11] R. B. McLellan and K. Alex, Scripta metall. 4, 967--(-1970). [12] D. Atkinson and C. Bodsworth, J . I . S . I . 32, 5~7 (1970). [13] R. B. McLellan, Trans. metall. Soc. A . I ~ . E . 233, 1664 (1965).