Effect of size on the surface energy of noble metal nanoparticles from analytical and numerical approaches

Surface energy is a key quantity that controls many physical properties of materials, yet determining its value at the nanoscale remains challenging. By using $N$-body interatomic potentials and performing analytical calculations, we develop a robust approach to determine the surface energy of metallic nanoparticles as a function of particle size and temperature. A strong increase in the surface energy is obtained when the size of the nanoparticle decreases in both the solid and liquid states. However, we show how the use of the classical spherical approximation to characterize the surface area of a nanoparticle leads to an almost constant surface energy with size as usually done to characterize the thermodynamic and kinetic properties of NPs in many works. We then propose a correction of the spherical approximation that is particularly useful for small size nanoparticles to improve the different models developed in the literature so far.

Figure 1

Energy profiles along the radius of NPs containing 147, 923, and 5083 atoms (from top to bottom). Left: Solid NPs ($T=5$ K); right: Liquid NPs ($T=1500$ K). The gray and blue areas represent the core and surface atoms, respectively.

Figure 2

Calculated surface energies for cuboctahedron Cu NPs as a function of size based on an (a) exact surface $S\left(k\right)$ and (b) a spherical (sa) approximation as well as corrected spherical approximation (csa). Both figures present analytical and MC results.

Figure 3

Slice views showing the presence of core and surface populations for NPs containing 923 atoms in solid state where the contribution of subsurface atoms can be identified. Energies are in eV.

Figure 4

(a) Analysis of the local energies in a histogram form for Cu solid NPs containing 147, 923, and 5083 atoms at 5K. For the sake of clarity, the values for the number of atoms in case of the ${\mathrm{NP}}_{147}$ have been increased by a factor of 10. (b) Analysis of the local energies on surface sites of solid ${\mathrm{Cu}}_{923}$ NP where different populations can be identified according to the following hierarchy: $E_f<E_e<E_v$. Note that the distinction between the different facets ($\left\{111\right\}$ and $\left\{100\right\}$) is not visible at this scale. Energies are in eV/at.

Figure 5

Spherical approximation to determine the effective surface area from TEM images of Cu NPs in (a) solid (facetted NPs at 573 K) and (b) liquid states (nonfaceted NPs with a spherical shape at 1073 K). On the right: Comparison of the sphere of radius $R$ obtained from the atomic density (${\rho }_{\infty }$) with configurations from MC simulations for different sizes and states of NPs containing 55 atoms, 561 atoms, and 2869 atoms. For the liquid cases, it is obvious that a snapshot is not so representative of the situation since the particle is strongly deformed during the simulation.

Figure 6

Relative difference between the surface area calculated according to the spherical approximation and the exact one of the cuboctahedron as a function of size.

Figure 7

(a) Calculated surface energies for liquid Cu NPs as a function of size within a spherical approximation and corrected spherical approximation. (b) Analysis of the local energies on surface of liquid NPs containing 561 and 5083 atoms. Energies are in eV.