Physics
Nanoparticles have a much greater surface to volume ratio than bulk materials. The increased surface to volume ratio means surface atoms have a much greater effect on chemical and physical properties of a nanoparticle. Surface atoms bind in the solid phase with less cohesive energy because they have fewer neighboring atoms in close proximity compared to atoms in the bulk of the solid. Each chemical bond an atom shares with a neighboring atom provides cohesive energy, so atoms with fewer bonds and neighboring atoms have lower cohesive energy. The average cohesive energy per atom of a nanoparticle has been theoretically calculated as a function of particle size according to Equation 1.
Where: D=nanoparticle size
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- d=atomic size
- Eb=cohesive energy of bulk
As Equation 1 shows, the effective cohesive energy of a nanoparticle approaches that of the bulk material as the material extends beyond atomic size range (D>>d).
Atoms located at or near the surface of the nanoparticle have reduced cohesive energy due to a reduced number of cohesive bonds. An atom experiences an attractive force with all nearby atoms according to the Lennard-Jones potential. The Lennard-Jones pair-potential shown in Figure 2 models the cohesive energy between atoms as a function of separation distance.
The cohesive energy of an atom is directly related to the thermal energy required to free the atom from the solid. According to Lindemann’s criterion, the melting temperature of a material is proportional to its cohesive energy, av (TM=Cav). Since atoms near the surface have fewer bonds and reduced cohesive energy, they require less energy to free from the solid phase. Melting point depression of high surface to volume ratio materials results from this effect. For the same reason, surfaces of bulk materials can melt at lower temperatures than the bulk material.
The theoretical size-dependent melting point of a material can be calculated through classical thermodynamic analysis. The result is the Gibbs–Thomson equation shown in Equation 2.
Where: TMB=Bulk Melting temperature
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- σsl=solid–liquid interface energy
- Hf=Bulk heat of fusion
- ρs=density of solid
- d=particle diameter
A normalized Gibbs–Thomson Equation for gold nanoparticles is plotted in Figure 1, and the shape of the curve is in general agreement with those obtained through experiment.
Read more about this topic: Melting-point Depression
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