Atomistic simulations of crystal-melt interfaces in a model binary alloy: Interfacial free energies, adsorption coefficients, and excess entropy

Monte Carlo and molecular-dynamics simulations are employed in a study of the equilibrium structural and thermodynamic properties of crystal-melt interfaces in a model binary alloy system described by Lennard-Jones interatomic interactions with zero size mismatch, a ratio of interaction strengths equal to 0.75, and interspecies interactions given by Lorentz-Berthelot mixing rules. This alloy system features a simple lens-type solid-liquid phase diagram at zero pressure, with nearly ideal solution thermodynamics in the solid and liquid solution phases. Equilibrium density profiles are computed for (100)-oriented crystal-melt interfaces and are used to derive the magnitudes of the relative adsorption coefficients $\left({\Gamma }_i^{\left(j\right)}\right)$ at six temperatures along the solidus/liquidus boundary. The values for ${\Gamma }_1^{\left(2\right)}$, the relative adsorption of the lower melting-point species (1) with respect to the higher melting point species (2), are found to vary monotonically with temperature, with values that are positive and in the range of a few atomic percent per interface site. By contrast, values of ${\Gamma }_2^{\left(1\right)}$ display a much more complex temperature dependence with a large peak in the magnitude of the relative adsorption more than ten times larger than those found for ${\Gamma }_1^{\left(2\right)}$. The capillary fluctuation method is used to compute the temperature dependence of the magnitudes and anisotropies of the crystal-melt interfacial free energy $\left(\gamma \right)$. At all temperatures we obtain the ordering ${\gamma }_{100}>{\gamma }_{110}>{\gamma }_{111}$ for the high-symmetry (100), (110), and (111) interface orientations. The values of $\gamma$ monotonically decrease with decreasing temperature (i.e., increasing concentration of the lower melting-point species). Using the calculated temperature-dependent values of $\gamma$ and ${\Gamma }_1^{\left(2\right)}$ in the Gibbs adsorption theorem, we estimate that roughly 25% of the temperature dependence of $\gamma$ for the alloys can be attributed to interface adsorption, while the remaining contribution arises from the relative excess entropy $S_{xs}^{\left(2\right)}$.

Figure 1

(a) Activities as a function of composition for the solid and liquid phases for the alloy with $\beta =0.75$ and $\alpha =1.0$. Solid and dashed lines correspond to solid and liquid phases, respectively, while the dotted line represents ideal-solution behavior. While both solid and liquid are plotted for $T=0.472$ and $\text{0.603}{ϵ}_{22}/k_B$, they are so close in value as to not be readily distinguishable. (b) Mixing free energy with common tangent illustrated for $T=0.5025$ ${ϵ}_{22}/k_B$. (c) Zero-pressure phase diagram calculated for this system.

Figure 2

Density profiles for each of the temperatures considered in this study. The solid line represents the total density, where the solid is located on the right and the liquid regions are shown on the left in each panel. The dashed line shows the density of species 1, while the dashed-dotted line is the density for species 2. Units on temperature $T$ are ${ϵ}_{22}/k_B$.

Figure 3

Density (${\rho }_1$ and ${\rho }_2$) and composition ($x_1$ and $x_2$) profiles at $T=0.5025{ϵ}_{22}/k_B$. A small density peak at the interface is visible in the profile for ${\rho }_1$.

Figure 4

${\Gamma }_1^{\left(2\right)}$ and ${\Gamma }_2^{\left(1\right)}$ (atoms/site) versus temperature for $\beta =0.75$, $\alpha =1.0$. Note the different scales of ${\Gamma }_1^{\left(2\right)}$ and ${\Gamma }_2^{\left(1\right)}$. Error bars represent 95% confidence based on the adsorption subaverages as described in the text.

Figure 5

(Color online) Sample interface height profile for an alloy with $\beta =0.75$ and $\alpha =1.0$. The interface heights are determined for each column in a $10\times 10$ grid using a weighted mean value of the solid and liquid order parameters.

Figure 6

Stiffnesses versus $T$ for the (100) and (110) orientations. Error bars represent 95% confidence levels as described in the text. Units on temperature $T$ are ${ϵ}_{22}/k_B$.

Figure 7

(Color online) Interfacial free energy for each orientation, with the average orientation ${\gamma }_0$ also shown. Error bars represent 95% confidence levels as described in the text.

Figure 8

Relative contributions to the interfacial free energy from the adsorption and relative excess entropy. Error bars on the interfacial free energies calculated from MD data represent 95% confidence levels as described in the text.

Figure 9

(Color online) Calculated vs predicted stiffnesses for the (100) orientation of the $\beta =0.75$, $\alpha =1.0$ system with $T=0.6185$ and $T=0.539{ϵ}_{22}/k_B$.

Figure 10

(Color online) Calculated vs predicted stiffnesses for the (110) orientation of the $\beta =0.75$, $\alpha =1.0$ system with $T=0.6185$ and $T=0.539{ϵ}_{22}/k_B$.