[Paper Review] Melting of sodium clusters

Juan A. Reyes Nava, Ignacio L. Garzón+2 more|ArXiv.org|Jul 17, 2002

nanoparticles nucleation surface interactions2 references16 citations

TL;DR

This study investigates the melting transition of sodium clusters (Naₙ, N = 13–147) using microcanonical molecular dynamics simulations with a many-body Gupta potential to model metallic bonding. It reveals size-dependent melting mechanisms, irregular melting temperature trends, and a 40% underestimation of the Na₅₅ melting point compared to experiment, highlighting limitations of phenomenological potentials in quantitatively capturing cluster melting despite qualitative agreement with experimental trends.

ABSTRACT

Thermal stability properties and the melting-like transition of Na_n, n=13-147, clusters are studied through microcanonical molecular dynamics simulations. The metallic bonding in the sodium clusters is mimicked by a many-body Gupta potential based on the second moment approximation of a tight-binding Hamiltonian. The characteristics of the solid-to-liquid transition in the sodium clusters are analyzed by calculating physical quantities like caloric curves, heat capacities, and root-mean-square bond length fluctuations using simulation times of several nanoseconds. Distinct melting mechanisms are obtained for the sodium clusters in the size range investigated. The calculated melting temperatures show an irregular variation with the cluster size, in qualitative agreement with recent experimental results. However, the calculated melting point for the Na_55 cluster is about 40 % lower than the experimental value.

Motivation & Objective

• To investigate the solid-to-liquid transition in sodium clusters using long-timescale microcanonical molecular dynamics simulations.
• To test the performance of the many-body Gupta potential in describing melting transitions with extended simulation times (up to 50 ns).
• To compare melting temperatures and mechanisms obtained via MD with those from Monte Carlo simulations and first-principles methods.
• To analyze the role of surface melting and structural evolution during the melting process in Naₙ clusters.
• To assess the convergence of physical observables like caloric curves, heat capacity, and root-mean-square bond-length fluctuations over long simulation times.

Proposed method

• Microcanonical molecular dynamics simulations were performed on Naₙ clusters (N = 13, 20, 55, 135, 142, 147) using a many-body Gupta potential based on the second moment approximation of a tight-binding Hamiltonian.
• The potential energy function is defined as Vi = A∑j≠i e^(-p(r_ij/r₀ - 1)) - ξ(∑j≠i e^(-2q(r_ij/r₀ - 1)))^(1/2), modeling metallic bonding via effective two- and three-body interactions.
• Physical observables such as caloric curves, heat capacity, and root-mean-square (rms) bond-length fluctuations were computed over simulation times of ~50 nanoseconds.
• Melting temperatures were determined using two criteria: the peak of the heat capacity and the Lindemann criterion (δ ≈ 0.15).
• Structural evolution during heating was analyzed by tracking atomic trajectories and identifying stages such as surface melting and complete diffusive motion.
• Simulations were initialized from the lowest-energy isomers, including icosahedral structures and their incomplete variants (e.g., Na₁₃₅, Na₁₄₂ derived from Na₁₄₇).

Experimental results

Research questions

• RQ1 How does the melting temperature of sodium clusters vary with size, and does this variation align with experimental observations?
• RQ2 What are the dominant melting mechanisms in Naₙ clusters, and how do they evolve from surface to bulk melting?
• RQ3 How do long simulation times (up to 50 ns) affect the convergence of melting indicators like heat capacity and Lindemann parameter?
• RQ4 Why is the calculated melting temperature of the Na₅₅ cluster significantly lower than the experimental value, and what does this imply about the accuracy of the Gupta potential?
• RQ5 How do the results from microcanonical MD simulations compare quantitatively and qualitatively with those from Monte Carlo simulations and first-principles methods?

Key findings

• The melting temperature of the Na₅₅ cluster is calculated to be approximately 40% lower than the experimental value, indicating a significant quantitative discrepancy.
• Melting temperatures show an irregular dependence on cluster size, with the highest value observed for Na₁₄₂ (190 K via heat capacity peak), consistent with experimental trends.
• The Lindemann criterion yields lower melting temperatures than the heat capacity maximum, indicating that structural softening precedes the onset of full diffusive motion.
• Melting initiates at the cluster surface, with increased atomic mobility propagating inward, confirming surface melting as a precursor to bulk melting.
• The caloric curves, heat capacities, and rms bond-length fluctuations for Na₁₃₅, Na₁₄₂, and Na₁₄₇ are nearly identical whether starting from complete or incomplete icosahedral structures, indicating structural robustness.
• Longer simulation times (up to 50 ns) lead to lower melting temperatures compared to shorter simulations, suggesting improved convergence of time-averaged observables.