APPLICATION OF STATISTICAL MECHANICS TO SOLID SOLUTIONS OF INTERSTITIAL IMPURITIES IN BINARY ALLOYS A. BISCARINI{, B. COLUZZI and F. M. MAZZOLAI Dipartimento di Fisica, UniversitaÁ di Perugia, via A. Pascoli 5, 06100 Perugia, Italy (Received 8 June 1999; accepted 14 June 1999) AbstractÐA statistical mechanical model has been developed for ternary interstitial alloys in the frame- work of the canonical ensemble. Short-range repulsive interactions among interstitials have been taken into account assuming a hard-sphere pair interaction potential. Sites of dierent geometrical kind (octahedral, tetrahedral, etc.) have been considered and further distinguished on the basis of either the chemical compo- sition of their ®rst shell of neighbouring metal atoms or of the distribution of the two metallic species within this shell. Analytical expressions for the thermodynamic functions of the system and for the occu- pancy probabilities of the dierent interstitial sites have been derived for a given alloy and assigned site energies. A general procedure for the evaluation of the blocking factors and the interstitial jump probabil- ities is proposed and applied to random f.c.c. and b.c.c. binary alloys containing H. The results are dis- cussed in relation to diusivity and anelastic relaxations associated with the interstitial impurity. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Theory and modelling; Thermodynamics; Interstitial lattice defects; Alloys 1. INTRODUCTION In recent years interest in ternary interstitial alloys has greatly increased due to their potential use in technological applications [1, 2]. The large variety of available interstitial sites (IS) with dierent neighbouring atomic environments introduces com- plicated features in the statistical thermodynamic modelling of these systems. To simplify the problem interstitial impurity±metal atom interactions are fre- quently dealt with in the limit of the ``local environ- ment model'' [3, 4]. According to this model the energy of an isolated interstitial atom (IA) located in a given site (``site energy'') is determined only by the chemical composition of and the atomic distri- bution within the ®rst coordination shell of metal atoms surrounding the site. At ®nite dilutions, short-range repulsive interactions among interstitials are generally taken into account by a pair inter- action potential of hard-core type, while long-range interactions are usually accounted for in terms of site energy changes associated with lattice expansion due to IA. Even under these simplifying approximations a calculation of con®gurational contributions to ther- modynamic functions appears not to be accessible [5]. Thus, additional assumptions have been intro- duced such as absence of interactions among inter- stitials [6±8], non-overlapping hard spheres [9], interactions among interstitials located within sub-lattices of sites with identical local atomic environment [10], occupancy of IS of only one kind (octahedral, tetrahedral, etc.) [10]. In the present work an attempt is made to calcu- late the thermodynamic functions of ternary inter- stitial alloys within the framework of the local environment and hard-sphere potential models by assuming that a certain number of shells of sites around an IA are unavailable for occupancy by a second interstitial (blocked) and, moreover, that a given site may be simultaneously blocked by more than one IA (blocking overlapping). A further aim of the work is the derivation of analytical ex- pressions for the occupation probabilities of the dierent IS as deduced by minimization of the Helmholtz free energy, in order to justify the modi- ®ed Fermi±Dirac distributions intuitively proposed in our previous papers dealing with H in random f.c.c. [11] and b.c.c. [12] binary alloys. Finally, it was hoped that the calculation of the jump prob- ability factors between pairs of neighbouring sites might help to derive the diusion coecient of interstitials and to predict their associated anelastic relaxations. 2. ASSUMPTIONS The present calculations will be made under the following assumptions: Acta mater. Vol. 47, No. 12, pp. 3447±3455, 1999 # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Printed in Great Britain 1359-6454/99 $20.00 + 0.00 PII: S1359-6454(99)00197-4 {To whom all correspondence should be addressed. 3447