Computer simulations of two-dimensional and three-dimensional ideal grain growth

We developed an efficient computation scheme for the phase-field simulation of grain growth, which allows unlimited number of the orientation variables and high computational efficiency independent of them. Large-scale phase-field simulations of the ideal grain growth in two-dimensions (2D) and three-dimensions (3D) were carried out with holding the coalescence-free condition, where a few tens of thousands grains evolved into a few thousand grains. By checking the validity of the von Neumann-Mullins law for individual grains, it could be shown that the present simulations were correctly carried out under the conditions of the ideal grain growth. The steady-state grain size distribution in 2D appeared as a symmetrical shape with a plateau slightly inclined to the small grain side, which was quite different from the Hillert 2D distribution. The existence of the plateau stems from the wide separation of the peaks in the size distributions of the grains with five, six, and seven sides. The steady-state grain size distribution in 3D simulation of the ideal grain growth appeared to be very close to the Hillert 3D distribution, independent of the initial average grain size and size distribution. The mean-field assumption, the Lifshitz-Slyozov stability condition, and all resulting predictions in the Hillert 3D theory were in excellent agreement with the present 3D simulation. Thus the Hillert theory can be regarded as an accurate description for the 3D ideal grain growth. The dependence of the growth rate in 3D simulations on the grain topology were discussed. The large-scale phase-field simulation confirms the 3D growth law obtained from the Surface Evolver simulations in smaller scales.

Figure 1

(Color online) The effect of $N_p$ (the number of the phase-fields allowed at a given grid point) on the microstructural evolution. (a) Computation with $N_p=10$ to $t=100\mathrm{\Delta }t$ and further computations with (b) $N_p=4$, (c) $N_p=6$, and (d) $N_p=10$ to $t=1500\mathrm{\Delta }t$. The computations were carried out in a $150\times 150\times 150$ grid system. As can be seen in the regions marked by the white circles, the microstructure computed with $N_p=6$ is almost identical to that with $N_p=10$, whereas there exists clearly discernable difference between microstructures with $N_p=4$ and $N_p=6$.

Figure 2

(Color online) The effect of $N_p$ on the changes of the average grain size with time in a $150\times 150\times 150$ grid system. The data were taken from the simulations of Fig. 1. With increasing $N_p$, the evolution curves with time converge to the case with $N_p=10$. Three curves with $N_p\ge 6$ were almost perfectly overlapped.

Figure 3

(Color online) Time-evolutions of the $n$-sided regular polygonal grains. (a) Changes of grain area with time. The circles indicate the simulation results and the straight lines depict the predictions by the von Neumann-Mullins law. (b) An example of the time-evolution of a regular pentagonal grain, where four successive microstructures were taken at $t=17$, $170$, $239$, and $241$, respectively.

Figure 4

(Color online) (a) Area change rates $dA∕dt$ of 4331 individual grains plotted with the relative grain size at $t=205$ in the 2D grain growth. The side numbers of individual grains were discriminated by different symbols. The arrow marks denote the predictions by the von Neumann-Mullins law. (b) Average area change rates $⟨dA∕dt⟩$ and their standard deviations plotted with the side numbers of grains. The straight line depicts the prediction by the von Neumann-Mullins law.

Figure 5

(Color online) Grain size distributions in the 2D grain growth in a $2400\times 2400$ grid system. The numbers of grains decreased from $12618$ at $t=1500\mathrm{\Delta }t$ to 3555 at $t=6000\mathrm{\Delta }t$.

Figure 6

(Color online) Effect of the initial condition on the steady-state grain size distribution in the 2D grain growth in a $2400\times 2400$ grid system; (a) three different distributions at the early stage $(t=40\mathrm{\Delta }t)$ and (b) the distributions averaged over duration $t=(1500–6000)\mathrm{\Delta }t$ after reaching the steady state.

Figure 7

(Color online) A typical microstructure evolution during the 3D ideal grain growth in a $420\times 420\times 420$ grid system; (a) $28181$ grains at $t=100\mathrm{\Delta }t$ and (b) 2012 grains at $t=4000\mathrm{\Delta }t$.

Figure 8

(Color online) Evolution of the grain size distribution during the 3D ideal grain growth in a $420\times 420\times 420$ grid system. (a) Distribution profiles at different time steps of $50\mathrm{\Delta }t$, $100\mathrm{\Delta }t$, $200\mathrm{\Delta }t$, and $400\mathrm{\Delta }t$ move toward the steady state and (b) after $t=400\mathrm{\Delta }t$ the grain size distribution remains unchanged at the steady state which corresponds to the Hillert distribution. The initial number of grains was $29320$, which decreased to $18445$ at $t=400\mathrm{\Delta }t$ and 2630 at $t=3200\mathrm{\Delta }t$. All curves show the instantaneous distributions at the times indicated.

Figure 9

(Color online) Effect of initial conditions on grain size distribution in a 3D grain growth in a $420\times 420\times 420$ grid system. (a) Three different distributions at an early stage $(t=100\mathrm{\Delta }t)$, where the numbers of grains were $38349$, $28181$, and $21911$ for the distributions denoted by squares, triangles, and inverted triangles, respectively. (b)–(d) Time-averaged distributions after the system reached the steady state, which evolved from three distributions shown in (a).

Figure 10

(Color online) Simulation test of mean-field approximation [Eq. (14)] in the 3D ideal grain growth. 1400 grains randomly sampled from about 2837 grains at $t=3000\mathrm{\Delta }t$ were shown as dots in a $rdr∕dt$ vs $r∕⟨r⟩$ plane. The $dr∕dt$ values were measured from the changes during a single time step. The grains with 9–20 faces were discriminated by symbols, where the numbers written count the faces of the grain. The thick straight line is the linear least-square fitting over all the points in the figure. Note that $r=r_c=⟨r⟩$ $9∕8$ when $dr∕dt$ on this line, as predicted in the Hillert 3D theory.

Figure 11

(Color online) Test of LS stability condition in the 3D grain growth. All the grains at $t=3000\mathrm{\Delta }t$ were shown as circles in a $d\rho ∕d\tau$ vs $\rho$ plane and the curve depicts the stability function $-(\rho -2{)}^2∕(2\rho )$ in the Hillert theory.

Figure 12

(Color online) (a) Time evolution of the average grain size in the 3D grain growth. The circles and solid curve, respectively, indicate the simulation result and a nonlinear least-squares fitting with Eq. (16) over the time interval $400\mathrm{\Delta }t<t<4000\mathrm{\Delta }t$ $(\mathrm{\Delta }t=0.0308)$ with a fixed exponent $n=2$. The $k$ value from the fitting is 2.38. (b) The variation of the mean-field constant $M$ in Eq. (16) with time. After reaching the steady state at about $t=400\mathrm{\Delta }t$, $M$ varies slightly in a range of $5.9\pm 0.4$ with time.

Figure 13

(Color online) The distributions obtained from the deterministic models. The squares are from the vertex model [15], triangles from the Surface Evolver program [18], and diamonds from the present phase-field model, respectively. The distribution from the Surface Evolver program is similar with that from this study, except for a clear evidence of the plateau. The simulations based on five different vertex models always showed the existence of the plateau, of which only one was shown in this figure.

Figure 14

(Color online) Size distribution of each topological class. The squares are the overall size distribution contributed by all grains and the other symbols are the size distributions for the topological classes denoted by indicated numbers. Note the peak heights and peak positions of the size distributions with $n=5$, $6$, and $7$. Such distribution profiles of the major topological classes are combined to form a plateau with small undulations in overall size distribution for all grains.

Figure 15

(Color online) The steady-state grain size distributions from various simulations of the 3D ideal grain growth; the phase-field simulation by Krill and Chen [31] (squares), Surface Evolver program by Wakai et al. [20] (circles), vertex simulation by Weygand and Brechet [17] (triangles), and the present phase-field simulation (stars). The thick curve depicts the 3D distribution predicted by the Hillert theory.

Figure 16

(Color online) Distributions of grains with the number of faces per grain in the 3D simulations. Triangles denote the results from the vertex simulation [17], circles from the Surface Evolver program [20], squares from Krill and Chen [31], and stars from the present simulation, respectively.

Figure 17

(Color online) Change of $V^{-1∕3}dV∕dt∕\left(2m\sigma \right)$ as a function of the number of faces per grain. The solid, dashed, and dotted curves denote the Mullins equation [57], Hilgenfeldt’s equation [59], and its improved form [60], respectively. Filled circles are the present simulation results, which were obtained from the volume change rates of 2837 grains at $t=3000\mathrm{\Delta }t$ during a single computation time step.