E2 Surface-Volume Ratio and Surface Energy

Topic

The surface-to-volume ratio of this nanostructure—a parameter that defines a specific category within this class of materials based on dimensionality and characteristic size—directly increases the surface energy per unit volume of the nanostructure itself. Each atom located on the surface has fewer neighbors than an equivalent atom within the interior volume, placing it in a higher energy state and thereby contributing to the total surface energy. A higher surface-to-volume ratio results in greater surface energy per unit volume; at the nanoscale, this surface energy comes to completely dominate the nanostructure's thermodynamic behavior—an effect that would be entirely negligible in a bulk material.

In addition to contributing this surface energy, the nanoparticle's surface is curved; according to the Laplace equation, this curvature generates additional internal pressure. The Laplace equation describes how any curved surface subject to surface tension compresses its enclosed volume with an additional pressure—a pressure that increases as the curvature becomes more pronounced. Consequently, a smaller nanoparticle with more pronounced curvature experiences higher additional internal pressure, as dictated by the Laplace equation.

This elevated surface energy, combined with the additional internal pressure generated by the nanoparticle's curvature, leads to a depression of the melting point—well below that of the bulk material—in accordance with the Gibbs-Thomson effect (which also influences the sintering temperature of such nanostructures). Melting the nanoparticle reduces this elevated surface energy along with the system's overall free energy, making the process thermodynamically favorable at a lower temperature than that required to melt the equivalent bulk material. The fact that this surface energy is higher in a smaller nanoparticle means that—in accordance with the Gibbs-Thomson effect—the resulting melting point depression is greater.

Consequently, this model—characterized by the surface energy and melting point depression mentioned at the outset—underlies two consequences described by other models in this set: on one hand, the high surface-to-volume ratio (which determines the surface energy here) accelerates the nanostructure's oxidation compared to that of the equivalent bulk material, as described by the first model; on the other hand, this same high surface energy—which lowers the nanoparticle's melting point—also lowers the nanostructure's sintering temperature well below that of the bulk material (via the Gibbs-Thomson effect, as described by the second model), leading to premature thermal coalescence. Since both depressions—the melting point depression described by this model and the sintering temperature depression described by the other—are distinct manifestations of the same Gibbs-Thomson effect, they ultimately share the same underlying cause: the high surface energy with which this model began.

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