Computational techniques for studying phase change behavior in nanoparticles and thin films have advanced significantly, enabling detailed investigations of size-dependent phenomena such as melting point depression and supercooling. These methods provide insights into nanoscale thermodynamics and kinetics, which differ markedly from bulk behavior due to high surface-to-volume ratios and quantum confinement effects. Key approaches include classical molecular dynamics (MD) with embedded atom potentials, phase-field modeling, and nucleation theory, each offering unique advantages for specific aspects of nanoscale phase transitions.

Classical molecular dynamics simulations are widely used to study melting and solidification in nanoparticles and thin films. Embedded atom method (EAM) potentials are particularly effective for metals, as they account for many-body interactions crucial for accurately describing phase transitions. MD simulations reveal that nanoparticles exhibit melting point depression, where the melting temperature decreases with particle size. For example, simulations of gold nanoparticles show a reduction in melting temperature from the bulk value of 1337 K to approximately 800 K for particles below 5 nm in diameter. This effect arises from the increased surface energy contribution, which destabilizes the solid phase. MD also captures supercooling, where nanoparticles remain liquid below their equilibrium freezing point due to kinetic barriers to nucleation. The extent of supercooling depends on the cooling rate and particle size, with smaller particles showing deeper supercooling. Thin films exhibit similar size-dependent behavior, with melting temperatures decreasing as film thickness approaches the nanometer scale. MD simulations can track atomic trajectories during phase transitions, providing detailed information on nucleation mechanisms, interface dynamics, and defect formation.

Phase-field modeling offers a mesoscale approach to simulate phase transitions in nanoparticles and thin films. This method treats interfaces as diffuse regions and solves coupled equations for the phase field and temperature or concentration fields. Phase-field models are particularly useful for studying microstructure evolution during solidification, including dendritic growth and pattern formation. The technique can incorporate anisotropic surface energies and kinetic effects, which are critical for understanding nanoscale phase behavior. For example, phase-field simulations of nanoparticle solidification reveal that the competition between surface energy minimization and crystallographic anisotropy leads to complex growth morphologies. In thin films, phase-field models predict the formation of metastable phases and grain structures influenced by substrate interactions and film thickness. The method's ability to handle large system sizes and long timescales makes it suitable for exploring polycrystalline films and multi-particle systems. However, phase-field models rely on phenomenological parameters, which must be calibrated using MD or experimental data to ensure accuracy.

Nucleation theory provides a theoretical framework for understanding the kinetics of phase transitions in nanoparticles and thin films. Classical nucleation theory (CNT) describes the formation of critical nuclei during melting or solidification, where the free energy barrier depends on the interfacial energy and volume free energy difference. CNT predicts that smaller nanoparticles have higher nucleation barriers due to their larger surface-to-volume ratios, leading to pronounced supercooling. Modified versions of CNT account for non-spherical nuclei and size-dependent interfacial energies, improving predictions for nanoscale systems. For instance, studies on aluminum nanoparticles show that CNT underestimates nucleation rates for particles below 10 nm, necessitating corrections for curvature effects and atomic-level disorder. In thin films, nucleation is influenced by substrate interactions, with heterogeneous nucleation often dominating due to template effects. Computational implementations of nucleation theory combine analytical models with MD or Monte Carlo simulations to predict phase transition temperatures and rates. These approaches are essential for designing nanomaterials with tailored thermal properties, such as phase-change memory devices.

Size-dependent melting point depression is a hallmark of nanoscale phase behavior, with computational studies providing quantitative relationships between particle size and melting temperature. The Gibbs-Thomson equation describes this effect, predicting a linear inverse relationship between melting temperature and particle radius. MD simulations validate this trend but also reveal deviations for very small nanoparticles due to non-equilibrium effects and surface premelting. For example, simulations of tin nanoparticles show that surface atoms begin disordered motion well below the core melting temperature, a phenomenon not captured by continuum models. Thin films exhibit similar behavior, with melting temperatures decreasing as film thickness approaches atomic dimensions. Phase-field and MD studies of metallic films demonstrate that substrate interactions can either suppress or enhance melting point depression, depending on interfacial bonding strength. These findings are critical for applications such as nanojoining and thermal barrier coatings, where precise control of phase stability is required.

Supercooling effects in nanoparticles and thin films are another area where computational techniques provide valuable insights. Supercooling occurs when a material remains liquid below its equilibrium freezing point due to the absence of nucleation sites. MD simulations show that the degree of supercooling increases with decreasing particle size, as smaller particles have higher nucleation barriers. For instance, simulations of nickel nanoparticles reveal supercooling ranges of up to 500 K for particles below 3 nm. Thin films also exhibit supercooling, but the effect is modulated by substrate interactions and film thickness. Phase-field models capture the competition between heterogeneous nucleation at substrates and homogeneous nucleation within the film, predicting transitions between these regimes as a function of cooling rate and interfacial energy. Nucleation theory provides analytical expressions for the critical cooling rate required to achieve deep supercooling, which is relevant for applications like glass formation and nanoscale casting.

Computational studies also address the role of defects and impurities in nanoscale phase transitions. MD simulations show that dislocations and grain boundaries can act as preferential nucleation sites, reducing supercooling in nanoparticles and thin films. For example, simulations of copper nanoparticles with twin boundaries demonstrate accelerated solidification compared to defect-free particles. Phase-field models incorporate defect energy contributions, predicting altered growth patterns in polycrystalline films. These findings highlight the importance of microstructure control in nanomaterial processing, where defect engineering can tailor phase transition behavior.

The integration of machine learning with traditional computational methods is an emerging trend in nanoscale phase change research. Machine learning potentials trained on MD data enable faster and more accurate simulations of large systems, while neural networks can predict phase diagrams and nucleation rates from structural descriptors. These approaches are particularly useful for high-throughput screening of nanomaterials for thermal management applications.

In summary, computational techniques such as classical MD, phase-field modeling, and nucleation theory provide powerful tools for studying phase change behavior in nanoparticles and thin films. These methods reveal the intricate interplay between size effects, interfacial phenomena, and kinetic barriers that govern nanoscale melting and solidification. The insights gained are essential for advancing applications in nanomanufacturing, energy storage, and electronic devices, where precise control of phase transitions is critical. Future developments in computational nanoscience will further refine these techniques, enabling predictive design of nanomaterials with optimized thermal properties.