# Thermal Hysteresis in Melting-Solidification of Nanoparticles
## 1 Introduction  
The melting of nanoparticles (NPs) has been widely studied both experimentally and theoretically [1; 2; 3; 4; 5; 6; 7; 8]. It has been well established that the melting temperatures of NPs decrease when their size decreases, and this fact has been explained through a number of models (see the review presented in [9]). Almost all of these models approximate the particle shape using an ideal sphere with a radius \(R\). The size-dependent melting temperature \(T_{m}\) can then be written in the generic form:  
\[T_{m}=T_{b}\cdot\left(1-\frac{X}{R}\right) \tag{1}\]  
Where \(T_{b}\) is the melting temperature of the bulk, and the quantity \(X\) depends on the model. The simplest model is based on the condition for thermodynamic equilibrium at melting temperature \(T_{m}\) for the particle in solid (S) and liquid (L) states, which implies phase coexistence. Formally, this entails an equivalence of the corresponding atomic (molecular) chemical potentials \(\mu_{S}=\mu_{L}\). The latter are determined from the corresponding free energies  
\[F_{S}=\frac{4\pi}{3}{R_{S}}^{3}n_{S}(-E_{S}-T\cdot S_{S})+4\pi R_{S}^{2}\gamma_ {S},\;\;F_{L}=\frac{4\pi}{3}R_{L}^{3}n_{L}(-E_{L}-T\cdot S_{L})+4\pi R_{L}^{2} \gamma_{L} \tag{2}\]  
where \(n_{S,L}\) are the atomic concentrations; \(E_{S,L}\) and \(S_{S,L}\) are the atomic cohesive energies and entropies, respectively; \(\gamma_{S,L}\) are the surface energies of the corresponding states; and \(T\) is the temperature. At \(T=T_{m}\), the condition \(\mu_{S}=\mu_{L}\) results in  
\[\Delta E-T_{m}\cdot\Delta S-\frac{2\Delta\gamma}{n_{L}R}=0\to T_{m}=\frac{ \Delta E}{\Delta S}\bigg{(}1-\frac{2\Delta\gamma}{\Delta En_{L}R}\bigg{)} \tag{3}\]  
where \(R\equiv R_{L}\), \(\Delta E=E_{S}-E_{L}\), \(\Delta S=S_{L}-S_{S}\), and \(\Delta\gamma_{T}=\gamma_{S}(n_{L}/n_{S})^{2/3}-\gamma_{L}\). Assuming that the factor \(\Delta E/\Delta S\) can be replaced with its asymptotic value at \(R\rightarrow\infty\), namely, \(\Delta E/\Delta S=T_{b}\), we obtain an expression similar to that given in Equation (1).  
In contrast to melting, the solidification of particles is a highly non-equilibrium process initiated from a solid nucleus formed inside the particle bulk due to some degree offluctuation. Once the nucleus exceeds the critical size, it can grow further into the rest of the particle. Such a process is characterized by a certain energy barrier and can be described according to classical nucleation theory (CNT) [10; 11]. The key parameter in the solidification process within the CNT is the nucleation probability  
\[P(R,T_{s})=N(R)M_{0}\cdot\exp\biggl{(}-\frac{\Delta F_{N}(R,T_{s})}{k_{B}T_{s}} \biggr{)}=\exp\biggl{(}-\frac{\Delta F_{N}(R,T_{s})}{k_{B}T_{s}}+\ln(N(R) \cdot M_{0})\biggr{)} \tag{4}\]  
where \(\Delta F_{N}(R,T_{s})\) is the nucleation barrier, i.e., the free energy barrier for the formation of the critical solid nucleus inside the liquid particle of radius \(R\), and \(k_{B}\) is the Boltzman factor. The factor \(N(R)=4/3\cdot\pi R^{3}/v\) is the number of nucleation sites, while \(v\) is the characteristic volume of the nucleation site. The factor \(M_{0}\) incorporates Zeldovich's factor and the attempt frequency to overcome the nucleation barrier and is assumed to be constant in the current context. The probabilistic nature of the solidification process results in a significant variation in the solidification temperature [12; 13; 14; 15; 16; 17] and significantly hampers its theoretical predictions [18; 19]. As a result, there is currently no clear picture of the size dependence of thermal hysteresis in nanoparticles.  
This study presents an attempt to qualitatively understand the behavior of thermal hysteresis in the melting-solidification of NPs as a function of particle radius.