Fluctuations, bistability and hysteresis connected to phase transformations of nanoparticles

Based on the definition of fluctuation in connection to phase transformations, structural fluctuations are spontaneous transitions from an equilibrium phase to a non-equilibrium phase, followed by a back-transformation into the equilibrium phase, criteria for the fluctuation of isolated single particles and ensembles of nanoparticles are developed. It is important to realize that, in case of ensembles, the probability for fluctuation depends on the number of transformed particles. Especially the latter criteria are deduced from a statistical model describing fluctuation processes. Furthermore, this statistical model leads to the conclusion that the equilibrium state of fluctuation processes is characterized by a minimum of the free enthalpy. Interestingly, this equilibrium state is independent of the character of the nanoparticles either being conventional particles or ones characterized by indistinguishability. A detailed thermodynamic analysis, studying isothermal and adiabatic processes, of the behavior of a single isolated particle and an ensemble under isothermal and adiabatic conditions allows formulating a set of seven theorems. In the adiabatic case, the calculations indicate the existence of bistability or hysteresis in the temperature range of transformation. Experimentally, these phenomena are well documented, however, in most cases, attributed to activation phenomena. As a result of this study, at least connected to nanoparticles, the interpretation of these experiments needs thorough examination. Furthermore, a complete or partial adiabatic enclosure of the specimen, which is in experimental reality unavoidable, causes a shift of the transformation temperatures. This result enforces a new view on phase diagrams, especially on those for nanoparticles.