The “critical limits for crystallinity” in nanoparticles of the elements: A combined thermodynamic and crystallographic critique

Abstract

The theme of the present work is the procedure for evaluating the minimum size for the stability of a crystalline particle with respect to the same group of atoms but in the amorphous state. A key goal of the study is the critical analysis of an extensively quoted paper by F.G. Shi [J. Mater. Res. 9 (1994) 1307–1313], who presented a criterion for evaluating a “crystallinity distance” (h) through its relation with the “critical diameter” (dC) of a particle, i.e., the diameter below which no particles with the crystalline structure are expected to exist at finite temperatures. Key assumptions of Shi's model are a direct proportionality relation between h and dC, and a prescription for estimating h from crystallographic information. In the present work the accuracy of the Shi model is assessed with particular reference to nanoparticles of the elements. To this end, an alternative way to obtain h, that better realizes Shi's idea of this quantity as “the height of a monolayer of atoms on the bulk crystal surface”, is explored. Moreover, a thermodynamic calculation of dC, which involves a description of the bulk- and the surface contributions to the crystalline/amorphous relative phase stability for nanoparticles, is performed. It is shown that the Shi equation does not account for the key features of the h vs. dC relation established in the current work. Consequently, it is concluded that the parameter h obtained only from information about the structure of the crystalline phase, does not provide an accurate route to estimate the quantity dC. In fact, a key result of the current study is that dC crucially depends on the relation between bulk- and surface contributions to the crystalline/amorphous relative thermodynamic stability.