Solid-liquid surface tensions of critical nuclei and nucleation barriers from a phase-field-crystal study of a model binary alloy using finite system sizes

Phase-field-crystal (PFC) modeling has emerged as a computationally efficient tool to address crystal growth phenomena on atomistic length and diffusive time scales. We use a two-dimensional phase-field-crystal model for a binary system based on Elder et al. [Phys. Rev. B 75, 064107 (2007)] to study critical nuclei and their liquid-solid phase boundaries, in particular the nucleus size dependence of the liquid-solid interface tension as well as of the nucleation barrier. Critical nuclei are stabilized in finite systems of various sizes, however, the extracted interface tension as function of the nucleus radius $r$ is independent of system size. We suggest a phenomenological expression to describe the dependence of the extracted interface tension on the nucleus radius $r$ for the liquid-solid system. Moreover, the numerical PFC results show that this dependency can not be fully described by the nonclassical Tolman formula.

Figure 1

The phase diagram of the model in the one-mode approximation, using $n_0=0$ and the parameters given in Table 1. The dotted red line at $\Delta B_{0,c}\approx 0.0282$ shows the location of the eutectic temperature, above there is possible liquid-solid coexistence regime, below solid-solid coexistence. The dashed lines indicate a metastable solid-solid transition. The full line at $\Delta B_0=0.05$ indicates where nucleation is investigated below.

Figure 2

The isothermal section of the equilibrium phase diagram at $\Delta B_0=0.05$.

Figure 3

(left) Planar solid slab in the numerical box. Across the box the chemical potentials for the two species are constant and are at the coexistence values ${\mu }_{n\left[c\right],\mathrm{slab}}^{\mathrm{coex}}$. The densities $n$ and $c$ for the solid ($s$) and liquid ($l$) phases correspond to their bulk coexistence values $n_{l\left[s\right],\mathrm{slab}}^{\mathrm{coex}}$ and $c_{l\left[s\right],\mathrm{slab}}^{\mathrm{coex}}$. (right) Solid nucleus with radius $r$ stabilized in the finite numerical box. Across the box the chemical potentials for the two species are constant but different from the slab coexistence values, ${\mu }_{n\left[c\right],\mathrm{drop}}^{\mathrm{coex}}\ne {\mu }_{n\left[c\right],\mathrm{slab}}^{\mathrm{coex}}$. The densities $n$ and $c$ for the solid ($s$) and liquid ($l$) phase are at their coexistence values $n_{l\left[s\right],\mathrm{drop}}^{\mathrm{coex}}$ and $c_{l\left[s\right],\mathrm{drop}}^{\mathrm{coex}}$. Note that the coexisting densities for the solid (liquid) phase for a slab (left) or a drop (right) $n_{l\left[s\right],\mathrm{drop}\left[\mathrm{slab}\right]}^{\mathrm{coex}}$ and $c_{l\left[s\right],\mathrm{drop}\left[\mathrm{slab}\right]}^{\mathrm{coex}}$ are defined as the bulk densities in the solid (liquid) phase having the chemical potentials ${\mu }_{n,\mathrm{drop}\left[\mathrm{slab}\right]}^{\mathrm{coex}}$ and ${\mu }_{c,\mathrm{drop}\left[\mathrm{slab}\right]}^{\mathrm{coex}}$ using a solid (liquid) unit cell. For larger nuclei one expects that the average densities extracted from the center of the nucleus agree with the pure bulk densities. For smaller nuclei, this is not necessarily the case.

Figure 4

Typical initial states for simulations performed with system size $L=30$ unit cells and (a) $P_s=0.1$ with a circular solid cluster surrounded by liquid phase, (b) $P_s=0.9$ with a circular liquid droplet surrounded by solid phase, (c) $P_s=0.5$ with a circular solid cluster surrounded by liquid phase, and (d) $P_s=0.5$ with a solid slab surrounded by liquid phase. The x axis represents $L_x/\Delta x$ whereas the y axis represents $L_y/\Delta y$.

Figure 5

A stable solid crystal coexists with the surrounding liquid phase in a system with $P_s=0.2$ and $L=30$ unit cells; the atomic density field (left) and the concentration field (right) are shown. The x axis represents $L_x/\Delta x$ whereas the y axis represents $L_y/\Delta y$.

Figure 6

A stable liquid droplet coexists with the surrounding solid phase in a system with $P_s=0.8$ and $L=30$ unit cells; the atomic density field (left) and the concentration field (right) are shown. The x axis represents $L_x/\Delta x$ whereas the y axis represents $L_y/\Delta y$.

Figure 7

Equimolar excess energy density ($F_{I,\mathrm{equim}}$) as a function of the solid fraction ($P_{s,\mathrm{equim}}$) for various system sizes.

Figure 8

Equimolar interfacial line tension ($f_{I,\mathrm{equim}}$) as a function of the droplet equimolar radius ($r_{\mathrm{equim}}$) for various system sizes. The dotted straight line at $f_{I,\mathrm{equim}}=0.00217978$ is the interface tension for the slab configuration.

Figure 9

The difference in grand potential (nucleation barrier) $\Delta \Omega$ as a function of $r_{\mathrm{equim}}$ calculated from the numerical simulations for various system sizes and the predictions based on the classical nucleation theory ($\Delta {\Omega }_{\mathrm{CNT}}$).