Temperature dependence of the surface free energy and surface stress: An atomistic calculation for Cu(110)

We propose a method to deduce the free energies $\gamma$ and stresses $\tau$ of plane surfaces and solid-liquid interfaces in elemental systems from atomistic simulations involving nonhydrostatically stressed solid phases. The method is applied to compute the temperature dependencies of $\gamma$ and $\tau$ for the (110) Cu surface using Monte Carlo simulations with an embedded-atom potential. Both quantities decrease with temperature but remain different even near the bulk melting point despite extensive premelting of this surface. This difference is explained by the existence in the premelted surface structure of a solid-liquid interface with relatively small but finite values of $\gamma$ and $\tau$. Separate calculations of the (110) solid-liquid interface stress give a negative value, suggesting that this interface is in a state of compression. This study motivates future work on anisotropy of surface/interface free energies and stresses, and on the extension of this method to more complex systems.

Figure 1

Simulation blocks and positions of the interface and bulk regions employed in the calculations of thermodynamic properties of (a) solid surfaces and (b) solid-liquid interfaces. The quantities computed in different regions are indicated. The bracketed quantities refer to the surface/interface layer, whose bounds can be chosen arbitrary as long as they lie within the homogeneous bulk phases. The bulk regions can lie beyond the surface/interface layer, as in this figure, or inside it (not shown).

Figure 2

Typical Monte Carlo (MC) snapshot of (a) (110) solid film at 1320 K and (b) (110) solid-liquid coexistence system at 1327 K. The open circles mark instantaneous atomic positions projected on the $\left(1\overline{1}0\right)$ plane parallel to the page. The top and bottom surfaces of both systems are exposed to vacuum. The distances $h$, $d$, and $d^{\prime }$ are discussed in text.

Figure 3

Typical MC snapshot of the solid film at three temperatures. The open circles mark instantaneous atomic positions projected on the $\left(1\overline{1}0\right)$ plane parallel to the page. Note the perfectly ordered surface structure at low temperatures and premelting at high temperatures.

Figure 4

Thickness of the (110) surface layer as a function of temperature. The vertical dashed line indicates the bulk melting point.

Figure 5

Average stress $\tau$ and the surface structure factor ${\left|S\left(\mathbf{k}\right)\right|}_{\text{top}}$ of the (110) surface as a function of temperature. The vertical dashed line indicates the bulk melting point.

Figure 6

Temperature dependence of the average surface stress computed by MC simulations of systems with 1024, 2240, and 9856 atoms and by molecular-dynamics simulations of a 896 atom system. Note that the shape of the curve does not depend on the model size or the simulation method.

Figure 7

Temperature dependence of the surface-stress anisotropy ${\tau }_{22}-{\tau }_{11}$. The vertical dashed line indicates the bulk melting point.

Figure 8

Temperature dependence of the excess free energy and stress of the (110) surface. Three values of the free energy, ${\gamma }^l$, of the liquid-vacuum interface are shown for comparison. The vertical dashed line indicates the bulk melting point.

Figure 9

Temperature and bulk-strain dependencies of the surface stress obtained by MC simulations and by 0 K static calculations. In the latter case, the data are plotted against the temperature at which thermal expansion would give the corresponding strain. The close agreement below 800 K indicates the dominant role of the bond-stretching effect in the temperature dependence of surface stress of atomically ordered surfaces.

Figure 10

Temperature dependence of the excess entropy of the (110) Cu surface. The vertical dashed line indicates the bulk melting point.