Phase Equilibrium in Nanoscale Alloys
Jörg Weissmüller, Peter Bunzel, Gerhard Wilde, Christian Lemier
Institut für Nanotechnologie, Forschungszentrum Karlsruhe
Technische Physik, Universität des Saarlandes
When the size of a particle is reduced then the excess free energy due
to the surface and to internal interfaces, gA (g - interfacial free
energy; A - area), diminishes more slowly than the free energies of
the bulk phases; the interfacial excess will therefore increasingly
affect the free energy balance of phase transformations. While some
related phenomena have been studied since the 19th century, there is
an ongoing interest in reversible phase transformations in
nanoparticles, specifically, size-dependent melting.
Phenomenologically, the origin of the size-dependence of the
temperature of fusion T[f] in elemental solids is a change Dg in the
interfacial free energy upon melting, in other words a term of the
form A[ ]Dg. It is straightforward to derive the size-dependence of
T[f] for a given value of Dg, but atomistic theories for interfacial
energies are often not accurate enough to allow the prediction of the
numerical value of Dg, or even of its sign. By contrast, meaningful
predictions are obtained by analysis of the equilibrium in nanoscale
alloys in terms of phenomenological approaches. This includes the
effects of grain boundary segregation on the position of phase
coexistence lines in alloy phase diagrams,1 and on the stability of
nanocrystalline alloys against grain growth,2 as well as the effect of
grain boundary induced stress on the miscibility gap in solid
solutions.3 While segregation and stress are of relevance in processes
subject to the constraint of constant interfacial area, the two-phase
equilibrium in matrix-isolated alloy particles involves reversible
changes of the interfacial area as a function of the phase fraction,
in other words terms of the form g DA. As we shall show, even with the
simplest constitutive equations this term leads to substantial shifts
in the phase diagram and even to qualitative changes in the nature of
the associated phase transformations.
1: J. Weissmüller, H. Ehrhardt, Phys. Rev. Lett. 81 (1998), 1114.
2: J. Weissmüller, NanoStruct. Mater. 3 (1993), 261.
3: J. Weissmüller, C. Lemier, Phys. Rev. Lett. 82 (1999), 213.