Origin of the quantum shape effect

The quantum size and shape effects are often considered difficult to distinguish from each other because of their coexistence. Essentially, it is possible to separate them and focus solely on the shape effect by considering a size-invariant shape transformation, which changes the discrete energy spectra of strongly confined systems and causes the quantum shape effects. The size-invariant shape transformation is a geometric technique of transforming shapes by preserving the boundary curvature, topology, and the Lebesgue measure of a bounded domain. The quantum shape effect is a quite different phenomenon from quantum size effects, as it can have the opposite influence on the physical properties of nanoscale systems. While quantum size effects can usually be obtained via bounded continuum approximation, the quantum shape effect is a direct consequence of the energy quantization in specifically designed confined geometries. Here, we explore the origin of the quantum shape effect by theoretically investigating the simplest system that can produce the same physics: quantum particles in a one-dimensional box separated by a moving partition. The partition moves quasistatically from one end of the box to the other, allowing the system to remain in equilibrium with a reservoir throughout the process. The partition and the boundaries are impenetrable by particles, forming two effectively interconnected regions. The position of the partition becomes the shape variable. We investigate the quantum shape effect on the thermodynamic properties of confined particles considering their discrete spectrum. In addition, we applied an analytical model based on dimensional transitions to predict thermodynamic properties under the quantum shape effect accurately. A fundamental understanding of quantum shape effects could pave the way for employing them to engineer physical properties and design better materials at the nanoscale.

Thermodynamic insight into the growth of calcia inclusions at the nanoscale: the case of Fe-O-Ca melt

A thermodynamic model was developed to investigate the relationship between the thermodynamics of nano-CaO as a deoxidation reaction product and their size in an Fe–O–Ca melt. The results of thermodynamic model coupling with DFT (density functional theory) calculation prediction showed that the solubility product of calcium and oxygen for nanoscale CaO decreased with the increase of calcia product size in an Fe–O–Ca melt. The existing experimental data about the Ca-deoxidation equilibrium in liquid iron are covered by the region between the bulk-calcia equilibrium curve and the nano-CaO of 2 nm size curve. This result indicates that the partial product in most of the Ca-deoxidation experiments could be nanoscale CaO particles. Most of the Ca-deoxidation experimental equilibrium states are not reaching the equilibrium state between bulk calcia and liquid iron but a multi-equilibria between bulk- and nano-CaO and liquid iron.

Thermodynamics of the formation of nano-CaO in liquid iron is important to explore the relationship between deoxidation reaction and size of inclusions, and is very useful in the size controlling of inclusions.

Size, dimensionality and composition effects on the Debye temperature of nanocrystals

As an important property for reflecting the binding forces between atoms, the Debye temperature of nanocrystals can be tuned by size, dimensionality and composition. In order to understand how these factors influence the Debye temperature, in this contribution, a new nanothermodynamic model without any adjustable parameter was established by considering the surface stress and bond number simultaneously. As expected, the Debye temperature decreases with a decrease in size if the dimensionality is given, while the size effect on nanowires is stronger than that on thin films and weaker than that on nanoparticles. It is also found that the Debye temperature of nanoalloys decreases with the increase of the component with smaller cohesive energy for the same size and dimensionality. The validity of the model is proved by the good consistency between the model predictions and experimental and computer simulation results.

Size-dependent Raman shift of semiconductor nanomaterials determined using bond number and strength

Significant variations in Raman shifts with decreasing material size, D, have been detected in Raman spectroscopy. In this study, we propose a simple and unified model to determine and explain the size-dependent Raman shift, ω(D), of low-dimensional semiconductor nanomaterials. ω(D) was found to be a function of bond number in a system, with an obvious decline in Raman shift observed when size dropped to the nanoscale. This arose from a decrease in coordination number, Z(D), and increase in single bond strength, ε(D). The predicted results show good agreement with experimental data for a series of semiconductor nanomaterials, showing that bond number can be used to calculate Raman shifts of nanomaterials. Moreover, this theoretical model was successfully applied to both single crystals and some binary semiconductor nanomaterials. Furthermore, bond number, which is directly related to the nanomaterial shape and size, becomes the only parameter required to determine ω(D) in this model, as both Z(D) and ε(D) can be determined from the bond number. This indicates that the established model has the potential to determine Raman shifts of nanomaterials with different shapes and sizes.

Finite size and length effects on bandgap of CdSe nanorods

E _g(D, L) function of CdSe nanorods versus L and D.

Modeling the melting temperature of nanoscaled bimetallic alloys

T _m(x,D,d) of Pd_0.5Pt_0.5 nanocrystals versus D with different dimensions.