Application of a Solvation Shell Analysis to the Diffusivity of Interstitial Species in Binary Substitutional Solutions
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APPLICATION OF A SOLVATION SHELL ANALYSIS TO THE DIFFUSIVITY OF INTERSTITIAL SPECIES IN BINARY SUBSTITUTIONAL SOLUTIONS
D. Farkas Virginia Polytechnic Institute and State University, Engineering, Blacksburg, VA 24061
Dept.
of Materials
INTRODUCTION The problem of interstitial diffusion in substitutional lattices has been extensively studied due to the practical importance of carbon in ternary austenites, as well as H in alloys. The complete treatment of the problem is indeed a very complicated subject. A variety of models have been developed for the H trapping case some of which yield simple results [1-5]. These results have been applied successfully to limiting cases where the concentration of traps and diffusing species is very small. The problem of these models is that they do not account for chanqes in saddle-point energy. Some of the models recently developed [6] are based on the thermodynamic model developed by Wagner [7], known as the "cell model", "surrounded atom" or "coordination shell" approach. This has the advantage of relating thermodynamic measurements to diffusion behavior. The purpose of the present work is to discuss different models proposed for the effect of substitutional elements on the diffusivity of an interstitial species, accounting for the different saddle-point energies. The possibility of applying these models to the entire composition range is also discussed, as well as the anomalous cases of C in Fe-Si and Ni-Fe.
THERMODYNAMIC MODELS Several solution models have been published based on the solvation shell analysis developed by Wagner [7]for liquid alloys. In this analysis the interstitial atoms occupy coordination shells formed by j substitutional atoms and z-j solvent atoms, where z is the number of atoms in a coordination shell. Wagner assumed a random distribution of shells given by:
n.3
z) j
(Z-j)
v
e(J) u
(()
z where (j) is the number of ways in which substitutional atoms can be arranged in the solvation shell, and au is the concentration of substitutional solute u in the alloy. This distribution is not considered to change with the addition of the interstitial solute. For most practical cases these assumptions are generally valid. The random distribution will break down for cases where the interaction of u and v is very strong. The nonrandomness of the substitutional solution can be accounted for on the basis of regular solution theory and quasichemical calculations [8]. This is done as follows: If the energies Ej of the different types of sites are defined, following set of reactions can be considered site j + site I l
site j+
+ site 0
the
j = 1, 5
the set includes 5 reactions and the ecuilibrium constant for each reaction K. is
Mat. Res. Soc.Symp. Proc. Vol. 21 (1984) Q Elsevier Science Publishing Co.,
Inc.
488
-A-• K.
j
C CO Cj C1 ex
T
= e
-H
F L
~ jl
..... j 1
RT
-H
+ H 1
RT
0
where Cj denote the number of different arrangements of the j,
(2)
Z-j atoms.
The equilibrium may be described for the n.3 fractions of sites
n ij:-l nji
= K, n
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