Precipitate growth kinetics under inhomogeneous concentration fields using a phase-field model

We investigate precipitation dynamics in the presence of a local solute gradient using phase-field simulations. During the homogenization heat treatment of the solidified Inconel 718 alloy, high Nb concentration within the Laves phases or at the core of the secondary arms results in Nb diffusion into the $\gamma$ matrix. The volume fraction and spatial distribution of precipitation during subsequent annealing can be controlled by tailoring the Nb concentration gradient in the matrix during homogenization. We use a surrogate Ni-Fe-Nb alloy for Inconel 718 to explore the growth dynamics of $\delta$ precipitates related to the local Nb concentration levels. The simulations indicate that in the presence of a Nb concentration gradient the growth rate of $\delta$ precipitates is higher than in a matrix of uniform average Nb concentration. The higher growth rate is a result of the higher local thermodynamic driving force at the interface between the solute-rich matrix and the $\delta$ interface. We propose a phenomenological model to describe the diffusion-controlled growth kinetics of the $\delta$ phase under a solute concentration gradient.

Figure 1

Initial Nb profiles along the $z$ axis. Three different profiles (red, green, and blue lines for cases 1, 2, and 3, respectively) are used as the initial conditions for PF simulations in Fig. 2. The simulation domain size along the $z$ axis is 210 nm ($N_z$ = 210 and $\mathrm{\Delta }x$ = 1 nm).

Figure 2

Microstructures from phase-field simulations and concentration maps at a mid-2D section. Simulation results in (a), (b), and (c) used initial Nb profiles of cases 1, 2, and 3 in Fig. 1, respectively. Left microstructures show $\delta$ precipitates (silver) after 200 s at $T= 850{}^{\circ }\mathrm{C}$. Right color maps are the Nb fields at the midsection of the left microstructures. (d) $\delta$ fractions along the $z$ axis for cases 1–3. In all simulations, $\delta$ seeds are distributed initially with fractions according to the black dashed line shown in (d).

Figure 3

Radius and thickness of a $\delta$ precipitate as a function of time under initially uniform solute fields. The left image in (a) shows a radius $r$ of the $\delta$ (silver) within the blue (111) plane. On the right, the $\varphi$ profile along the white arrow across the center of the $\delta$ on the green ($10\overline{1}$) plane is used to measure the thickness. (b) Shows $r^2-r_0{\left(t_0\right)}^2$ of a $\delta$ precipitate, where $r$ is the average radius and $r_0$ is the radius at a reference time $t_0$. We use three different $r_0\left(t_0\right)$ [nm] = 1.7 for $t_0=0$ s (black line), 29.2 for $t_0$ = 150 s (blue line), and 52.4 for $t_0$ = 300 s (red line). (c) The square of the half thickness as a function of time. In both plots, solid and dashed lines are for the simulation results and linear fits, respectively. The insets show the log-log plots.

Figure 4

Growth of a $\delta$ precipitate under different Nb gradients. (a) Initial Nb profiles along the $z$ axis at the center using different gradient slopes. The $\delta$ nucleus is located at the center with a high $X^{\text{Nb}}$ = 0.24. Black solid line is for the uniform $X^{\text{Nb}}$ = 0.1 field within the matrix. Red, green, and blue dashed lines are for the gradients of $\pm 0.002, \pm 0.004$, and $\pm 0.006$, respectively. (b) The volume fractions at different times. The inset is for the log-log plot.

Figure 5

$\delta$ precipitates for the volume fraction of $V_f=0.015$. The volume fraction is reached around $t=295.21$ s for the uniform (a), 154.02 s for the shallow (b), 190.97 s for the moderate (c), and 209.51 s for the deep gradient cases (d). Blue plane indicates the (111) plane.

Figure 6

Volume fraction as a function of time from the simulations (solid lines) and prediction (dashed lines). Dashed lines are the predicted growth rates using Eq. (10). For the uniform case, the equation becomes Eq. (9) due to $F_R\left(t\right)=1$. The inset is for the log-log scale.