Crystal growth and melting in NiZr alloy: Linking phase-field modeling to molecular dynamics simulations

We compare results from molecular dynamics simulations with those from phase-field modeling concerning the solidification and melting kinetics of a planar ${\left[{\text{Ni}}_c{\text{Zr}}_{1-c}\right]}_{\text{liquid}}{\text{-Zr}}_{\text{crystal}}$ interface. Our study is an illustration that both approaches may predict the same quantitative physical description when the key parameters calculated within the atomistic molecular dynamics approach are used to construct the mesoscopic phase-field model. We show in this way that a thermodynamic consistent phase-field model can be applied down to the range of atomic structure. At the same time, molecular dynamics simulation seems to be capable to treat correctly relaxation dynamics, driven by thermodynamic forces, in a nonequilibrium state of solidification and melting. We discuss, in particular, how the free energy from atomistic calculations is used to design the phase dependent free-energy density in the phase-field model. Bridging the gap between both simulation approaches contributes to a better understanding of the thermodynamic and kinetic processes underlying the solidification and melting processes in alloys out of chemical equilibrium. The effective thermodynamic enhancement of the diffusivity through the strong negative enthalpy of mixing in the NiZr solution is discussed.

Figure 1

(Color online) Upper panel: two-phase crystal-melt system equilibrated at $T=1900\text{K}$. Lower panel: the same sample after cooling it down to $T=1700\text{K}$ and relaxing it for 20 ns. The hell and dark atoms represent Ni and Zr, respectively. Clearly visible is an expansion of the crystalline fraction on account of the molten part, which implies an increase in the melt concentration $c_l\left(T\right)$ under cooling conditions.

Figure 2

Zr-rich part of the Ni-Zr phase diagram with discrete MD data points (solid dots) and a second degree polynomial interpolation function.

Figure 3

(Color online) Crosses: MD-calculated FE density for Zr-rich ${\text{Ni}}_c{\text{Zr}}_{1-c}$ melts at $T=1700\text{K}$, the solid line is a third-order polynomial interpolation function. Squares: MD-calculated FE density for crystalline Zr at low Ni concentrations $\left(<2\mathrm{\%}\right)$, the dashed line extrapolates these values to higher concentrations according to the regular solution model, see text for details. The dashed-dotted line is the double tangent.

Figure 4

(Color online) Diffusion profiles for Ni and Zr around the crystal-liquid interface at $T=1700\text{K}$, recorded 40 ns after quenching the system down from $T=1900\text{K}$. The corresponding concentration profile is also shown.

Figure 5

(Color online) Diffusion profiles for Ni and Zr around the crystal-liquid interface at $T=1300\text{K}$, recorded 40 ns after quenching the system down from $T=1900\text{K}$.

Figure 6

Evolution of the concentration profile during MD isothermal solidification simulation at $T=1500\text{K}$ after undercooling from $T=1900\text{K}$. The horizontal dotted lines represent the equilibrium concentrations at the temperatures $T=1500\text{K}$ (upper line) and $T=1900\text{K}$ (lower line).

Figure 7

(Color online) Interface position as a function of time during a solidification process at $T=1700\text{K}$, starting from a two-phase system equilibrated at $T=1900\text{K}$ (see snapshot of Fig. 1). The diagram compares results from MD, PF, and sharp-interface models.

Figure 8

(Color online) Ni-concentration profiles after 20 ns of a solidification process, from the same MD and PF simulations described in Fig. 7. The dashed line indicates the initial liquid equilibrium concentration $c_l^0=c_l\left(1900\text{K}\right)$. The thin solid line corresponds to the diffuse PF profile.

Figure 9

(Color online) Sensitivity of the PF growth velocity to variation in $\sigma$ and $\epsilon$ with the same undercooling conditions as in Fig. 7. The results of each parameter modulation is compared with the reference simulation of Fig. 7.

Figure 10

Front position $x_f$ after 40 ns of solidification as a function of undercooling $\Delta T$ for MD and PF simulations. The initial temperature is $T=1900\text{K}$ for all simulations.

Figure 11

(Color online) Interface position as a function of time for a melting process at $T=2100\text{K}$ starting from the same two-phase samples as in Fig. 7.

Figure 12

(Color online) Ni-concentration profiles after 20 ns of a melting process from the same MD and PF simulations as in Fig. 11. The dashed line indicates the initial liquid equilibrium concentration $c_l^0=c_l\left(1900\text{K}\right)$ and the thin solid line illustrates the profile of the PF variable $\varphi$.