# Thermal Hysteresis in Melting-Solidification of Nanoparticles
## 2 Results and Discussion
### Thermal Hysteresis  
The value of the parameter \(\delta_{0}\) can be defined as equal to that for the bulk; for most simple metals, it is close to 0.17 [12; 13; 23; 24]. Notably, decreasing \(\delta_{0}\) to 0.1 (which is typical for a SnAgCu alloy [25]) does not change the qualitative behavior of \(\delta\) as a function of \(R\). To qualitatively understand the role of the parameters \(\alpha_{0}\) and \(\beta_{0}\) in the present context of the relative thermal hysteresis, Equation (10) was solved numerically for \(n_{S}\approx n_{L}=40\)\(nm^{-3}\), defining the reference particle radius as \(R_{0}=50\)\(\mu\). The results calculated for a number of \(\alpha_{0}\) and \(\beta_{0}\) values are presented in Figure 1. According to the figure, the value of relative thermal hysteresis slightly increases when the particle radius decreases in the range of 50-0.5 \(\mu\). A similar trend was observed for the SnAgCu alloy when the particle size  
In contrast, the behavior of \(\delta(R)\) for the Bi particles shows a noticeable increase with a decreasing \(R\) in the range of \(R>30\) nm and a steep decrease for smaller radii. Interestingly, the NP melting temperatures of the corresponding materials show a noticeable decrease in the same ranges of \(R\) values as those of \(\delta\). The available experimental data for thermal hysteresis in NPs are rather ambiguous, in some cases showing a significant decrease at low \(R\) values, which is in agreement with our numerical results [29]. A more quantitative verification of the proposed analytical approach can be performed by comparing its predictions with the results of numerical simulations, as illustrated in the insert in Figure 2 wherein Palladium is used as an example. As one can see, the relative hysteresis calculated using the simulated melting-solidification temperatures [28] (symbols) is quite closely approximated by the analytical curve obtained by solving Equation (10). Direct evidence of the decreased nucleation barrier, and hence thermal hysteresis for small NPs, can be found by observing the phase fluctuation in the Bi NPs under continuous electron beam illumination [16].  
The presented analysis is based on several assumptions, which may restrict its applicability. The main assumption is that solidification can be described within a CNT for all particles in the size range of interest. An additional assumption about the nucleation probability is that the nucleation site volume \(v\) is independent of the particle size, which is true if the latter is large enough compared to \(v\). Very little is known about the nucleation site volume in real systems. It is most likely that its size is of the order of a few interatomic distances, in which case the above assumptions seem to be realistic for nanoparticles with a radius over 1 nm. This can be applicable to particles made of pure metals and alloys provided that the alloy composition is homogeneous throughout the particles. Thus, we may conclude that the predicted size dependence of thermal hysteresis can be observed for particles made of these materials.