On the role of surface energy and surface stress in phase-transforming nanoparticles

The role of surface energy and surface stress has been a topic of extensive discussion since the seminal work by Gibbs [Gibbs JW. The scientific papers of J. Willard Gibbs. Vol. I: Thermodynamics. New York and Bombay:Longmans, Green, and Co; 1906; Gibbs JW. Collected works. New Haven:Yale University Press; 1957]. Both quantities have the same value for liquids, but not for solids. The distinction between these terms is of special importance for phase transforming nanoparticles (precipitates, transforming or melting/solidifying single particles), since surface quantities scale as the inverse of the particle size relative to volume quantities. Continuum mechanics and, especially, the concept of configurational forces (stresses) provide a convenient framework for distinguishing between “surface energy”, “surface tension” and “surface stress”. Therefore, this progress report gives a rather detailed introduction into the continuum mechanics and thermodynamics of a moving surface. The transformation conditions for the cases where an entire nanoparticle transforms suddenly and when the transformation is interface-driven are discussed. A global transformation condition for a sudden phase-transforming nanoparticle is explained. For the interface-driven transformation, the concept of configurational forces is applied to derive a local transformation condition in a material point at the phase interface. Four examples of nanoparticles (growing precipitate, growing solid nucleus in liquid, melting particle, solidifying particle) are studied in detail. The surface energy and surface stress are shown to contribute to the thermodynamic driving force on the interface in different ways. These contributions are quantified and discussed with respect to the case of a sudden transformation of the nanoparticle.