Nanocrystalline surface energy studied within the Gibbs thermodynamic framework

In the equilibrium thermodynamics framework, expressions were obtained that determine the dependences of the specific surface energy $\ensuremath{\sigma}$ and surface pressure ${P}_{Sf}$ on the size ($N$) and shape of a freestanding nanocrystal at different pressure $P$ and temperature $T$. Based on these expressions, the behavior of the $\ensuremath{\sigma}(P,T,N)$ and ${P}_{Sf}(P,T,N)$ functions for fcc Au have been studied. The calculations performed for the macrocrystal showed good agreement with the experimental data. Calculations for a nanocrystal have shown that at $P=0$, the ${P}_{Sf}(N)$ function lies in the negative region, i.e., the nanocrystal is stretched by surface pressure the more the higher of temperature, or the more the nanocrystal shape is deviated from an energy-optimal shape. With a decrease in $N$ at $P=0$, the function $\ensuremath{\sigma}(N)$ decreases more noticeably the higher of temperature, or the more the nanocrystal shape deviates from an energy-optimal shape. Based on these results, it was shown that the increase in the $\ensuremath{\sigma}(N)$ function obtained in some articles with an isomorphic-isothermal decrease in $N$ does not correspond to the physical properties of nanoparticles. In these articles, the nanoparticle was compressed by surface pressure, which increased with an isomorphic-isothermal decrease of the $N$ value. This compression led to the corresponding growth of the $\ensuremath{\sigma}(N)$ function both with an isomorphic-isothermal decrease in size and with an isomeric (i.e., at $N=\mathrm{const}$) increase in the nanoparticle temperature.