Thermodynamics of induced vacancies

Experimental studies of the solid-state reduction of metal oxides showed that metallic particles may form within an oxide sample during its interaction with a reducing medium without melting or destruction of the crystal structure. Such phenomena are difficult to explain within the framework of the classical concepts of equilibrium point defects. In the present work, a new type of non-equilibrium point defect, referred to as induced vacancies, is introduced and thermodynamically substantiated. These defects arise in oxide crystals under the action of an external chemical agent that performs work associated with the removal of ions from lattice sites. Unlike equilibrium Frenkel and Schottky vacancies, induced vacancies are not generated by intrinsic fluctuations of the crystal lattice and therefore give rise to a fundamentally different non-equilibrium defect structure. Based on a simplified ionic model, analytical expressions are derived for the changes in internal energy, entropy, and Helmholtz free energy associated with the formation of induced vacancies, as well as for their chemical potentials. The paper considers two limiting models of electron redistribution ensuring charge neutrality after formation of an induced anion vacancy: redistribution of excess electrons among cations and formation of neutral atoms through the neutralization of a fraction of cations. Both mechanisms lead to identical thermodynamic consequences and yield the same expression for the bond energy of the crystal. The results demonstrate that accumulation of induced vacancies is accompanied by appearance of neutral metal atoms within the oxide matrix and creates thermodynamic conditions favorable for nucleation of metallic clusters. The proposed approach provides a thermodynamic basis for describing non-equilibrium defects during the solid-state reduction of oxides and may serve as a foundation for a more general theory of reduction and oxidation processes in metal-containing systems.