Molecular dynamics study of equilibrium concentration profiles and the gradient energy coefficient in Cu-Pb nanodroplets

Atomistic simulations were used to equilibrate the concentration profiles in $5.3\mathrm{nm}$ liquid droplets for an embedded atom method model of Cu-Pb. The strong tendency of Pb to surface segregate establishes a nonuniform profile, which decays over a length comparable to the particle radius. With the free energy vs composition determined from separate Monte Carlo simulations, the composition profile can be analyzed in terms of the Cahn-Hilliard diffuse interface model and the gradient energy coefficient, $\kappa$, can be obtained. Results from three different temperatures indicate that $\kappa$ lies in the range $1.0–1.4\times {10}^{-10}\mathrm{J}∕\mathrm{m}$. In addition to the gradient energy coefficient, various aspects of phase equilibria at the nanoscale have been investigated. The critical point of the liquid-liquid miscibility was found to decrease by $\approx 50\mathrm{K}$ for the $5.3\mathrm{nm}$ particle size, the melting point of pure Cu is suppressed by $\approx 110\mathrm{K}$, and the liquidus slope is shallower than that of the bulk system.

Figure 1

The chemical potential difference vs composition in Cu-Pb at $T=1000\mathrm{K}$. The edge dimension of the cubic periodic cells is given in the legend.

Figure 2

The free energy of mixing vs composition for Cu-Pb at $T=1000\mathrm{K}$. The top curve, showing the double well shape and the common tangent construction, are the results for a pressure of zero. The distance between the $P=0$ free energy curve and the common tangent represents the quantity $\Delta f$. The bottom curve was computed at a pressure of $280\mathrm{MPa}$, the approximate pressure of the nanosize droplets.

Figure 3

The Pb composition profile across the two-phase interface in the bulk liquid Cu-Pb system at $T=1000\mathrm{K}$. The gradient energy coefficient is found to be $2.18\times {10}^{-10}\mathrm{J}∕\mathrm{m}$.

Figure 4

Pb concentration as a function of radial distance in nanosize liquid droplets at $T=1000\mathrm{K}$. Solid line is a fit to the Cahn-Hilliard model leading to a prediction of the gradient energy coefficient.

Figure 5

(a) Pb concentration as a function of radial distance in nanosize liquid droplets at $T=1050\mathrm{K}$. The fit shown by the solid line yields $\kappa =1.04\times {10}^{-19}\mathrm{J}∕\mathrm{m}$. (b) Results for $1100\mathrm{K}$ where $\kappa =1.42\times {10}^{-10}\mathrm{J}∕\mathrm{m}$.

Figure 6

The gradient energy coefficient as a function temperature.

Figure 7

A slice through the solid-liquid two-phase Cu-Pb nanosphere. The dark atoms represent Pb and the white atoms are Cu. The average composition of the sphere is $c=0.18$ and the temperature is $T=1000\mathrm{K}$.

Figure 8

The phase diagram for the bulk Cu-Pb system (the solid lines and points) compared to the liquidus (dashed lines/open points) for the $5.3\mathrm{nm}$ particle size.