Oscillatory cellular patterns in three-dimensional directional solidification

We present a phase-field study of oscillatory breathing modes observed during the solidification of three-dimensional cellular arrays in microgravity. Directional solidification experiments conducted onboard the International Space Station have allowed us to observe spatially extended homogeneous arrays of cells and dendrites while minimizing the amount of gravity-induced convection in the liquid. In situ observations of transparent alloys have revealed the existence, over a narrow range of control parameters, of oscillations in cellular arrays with a period ranging from about 25 to 125 min. Cellular patterns are spatially disordered, and the oscillations of individual cells are spatiotemporally uncorrelated at long distance. However, in regions displaying short-range spatial ordering, groups of cells can synchronize into oscillatory breathing modes. Quantitative phase-field simulations show that the oscillatory behavior of cells in this regime is linked to a stability limit of the spacing in hexagonal cellular array structures. For relatively high cellular front undercooling (i.e., low growth velocity or high thermal gradient), a gap appears in the otherwise continuous range of stable array spacings. Close to this gap, a sustained oscillatory regime appears with a period that compares quantitatively well with experiment. For control parameters where this gap exists, oscillations typically occur for spacings at the edge of the gap. However, after a change of growth conditions, oscillations can also occur for nearby values of control parameters where this gap just closes and a continuous range of spacings exists. In addition, sustained oscillations at to the opening of this stable gap exhibit a slow periodic modulation of the phase-shift among cells with a slower period of several hours. While long-range coherence of breathing modes can be achieved in simulations for a perfect spatial arrangement of cells as initial condition, global disorder is observed in both three-dimensional experiments and simulations from realistic noisy initial conditions. In the latter case, erratic tip-splitting events promoted by large-amplitude oscillations contribute to maintaining the long-range array disorder, unlike in thin-sample experiments where long-range coherence of oscillations is experimentally observable.

Figure 1

Image processing steps for the experimental videos: (a) Initial raw image ($3.2\times 3.2{\mathrm{mm}}^2$); (b) contours of apparent surface of cell caps on corrected binary image superimposed to the initial image; (c) tagged binary image. Cells A and B correspond to the cells analyzed in Fig. 2. The raw video (a) is provided in the Supplemental Material [32].

Figure 2

Time evolution of the areas of cells A and B in Fig. 1 (symbols), fitted to sinusoidal functions (lines).

Figure 3

Schematics of the domains for the simulation of hexagonal arrays of cells. A quarter of cell in such an array of spacing $\mathrm{\Lambda }$ can be obtained by simulating a domain of size $\mathrm{\Lambda }/2\times \mathrm{\Lambda }\sqrt{3}/4$ in $x\times y$ (red box) with no-flux boundary conditions in $x$ and at the low $y$ boundary and an “antisymmetric” boundary condition on the top $y$ boundary (see text). A hexagonal array of cells containing three groups of cells A, B, and C can be simulated with 1.5 cells within a domain of size $3\mathrm{\Lambda }/2\times \mathrm{\Lambda }\sqrt{3}/2$ in $x\times y$ (blue box) with no-flux boundary conditions in $y$ and “helical” boundary conditions on $x$ (see text).

Figure 4

Postprocessing steps of the phase-field simulations: (a) Solid-liquid interface, i.e., contour $\psi (x,y,z)=0$ seen from the liquid; (b) isothermal intersect (blue lines) of the interface; (c) binary image of the areas of cells; and (d) final processed result with numbered cells and Delaunay triangulation from the centers of cells. The three cells labeled A, B, and C in (c) correspond to the cells illustrated in Figs. 5 and 8. Movies of the entire simulation represented as in (a) and (d) are presented in the Supplemental Material [32].

Figure 5

Time evolution of the areas of cells A, B, and C of Fig. 4. The areas of cells (symbols) are fitted to a sinusoidal function (lines) in order to extract their oscillation period $\tau$.

Figure 6

Trajectories of the normalized areas of cells A, B, and C of Fig. 4 in the $\left[A\right(t),A(t+\tau /4\left)\right]$ plane (a) and phases of all cells at a given time $t$ on the unit circle (b), where cells A, B, and C appear as larger full symbols.

Figure 7

Oscillating cellular pattern observed from the liquid side at $V=1\mu \mathrm{m}$/s in the experiment at $G\approx 19\mathrm{K}$/cm (a) and in the phase-field simulation at $G=28\mathrm{K}$/cm (b). The high spatial disorder of the array is highlighted by both the ring-shaped fast Fourier transform of the image [“FFT” inset in (a)] and the large number of array defects (the number of nearest neighbors of each cell is indicated on the right-hand side). Tip-splitting events are at the origin of the creation and subsequent elimination of new cells in the array, hence preventing the array from stabilizing as a perfect hexagonal arrangement. All cells oscillate with nearly the same period but different phases. However, as illustrated in Fig. 8, in locally ordered regions, regular hexagonal or square patterns exhibit coherent breathing-mode oscillations.

Figure 8

Oscillation phases $\theta$ plotted on the unit circle at 2 times half a period ($\tau /2$) apart with different colors and symbols for the experiment (a) and the simulation (b) from Fig. 7. The large scatter of phases indicates the absence of global coherence of oscillations. However, in locally ordered regions, temporary synchronization between first-neighbor cells appears in both experiment and simulation, either with phase opposition (a) or with a $\pm 2\pi /3$ phase shift (b), which correspond to the basic breathing modes.

Figure 9

Oscillation dynamics of penta-hepta defects in the phase-field simulation of Fig. 7. For two different penta-hepta pairs of cells $(A,B)$ in (a) and (b): The top panel shows the time evolution of the number of neighboring cells to cell $A$ and cell $B$ and the bottom graph shows the time evolution of the area of cells A ($\circ$), B ($\diamond$), as well as each surrounding cell C through J (gray symbols). Each cell area is normalized with respect to its time-averaged value ${\int }_0^{\tau }A\left(t^{\prime }\right)d{t}^{\prime }/\tau$, its minimum, and its maximum value within the represented time range. Panels on the right-hand side illustrate the solid-liquid interface with labeled cells A through J at three different times respectively marked (${\mathrm{a}}_{\mathrm{k}}$) and (${\mathrm{b}}_{\mathrm{k}}$) on the left-hand side graphs. The time origin in (a) and (b) is, respectively, $t_a\approx 342\min$ and $t_b\approx 162\min$.

Figure 10

Phase-field predictions of stable states for hexagonal arrays of cells for different temperature gradients at a velocity $V=1\mu \mathrm{m}$/s (a), and for different velocities at a temperature gradient $G=12\mathrm{K}$/cm (b). The vertical axis $\mathrm{\Delta }$ is the cell tip undercooling below the liquidus temperature normalized by the freezing range $m{c}_{\infty }(1-1/k)$. Calculations in (a) were performed with ${\epsilon }_4=0.007$, while calculations in (b) are with ${\epsilon }_4=0.011$. Both sets of parameters exhibit the opening of a stable spacing gap in the vicinity of $G=12\mathrm{K}$/cm and $V=1\mu \mathrm{m}$/s, yielding two stable branches when increasing $G$ or decreasing $V$. The gray line in (a) and (b) illustrates schematically the limit of stability where cells exhibit oscillations of their position (i.e., undercooling) in time within a narrow range of spacings. This is exemplified for $V=1\mu \mathrm{m}$/s and $G=16\mathrm{K}$/cm at $\mathrm{\Lambda }\approx 188$, 268, at $282\mu \mathrm{m}$ in Fig. 11. The green background zone for $V=1.1$ to $1.5\mu \mathrm{m}$/s in (b) illustrates the area investigated in Fig. 14, where breathing-mode oscillations can be induced by a change of growth conditions (see Sec. 4d), as exemplified at $V=1.1\mu \mathrm{m}$/s in Figs. 14 and 14, respectively, for $\mathrm{\Lambda }\approx 193$ and $206\mu \mathrm{m}$.

Figure 11

Phase-field prediction of the evolution of cell tip undercooling (a) and cell tip area (b) with time at $G=16\mathrm{K}$/cm and $V=1\mu \mathrm{m}$/s for stable cells at $\mathrm{\Lambda }\approx 188$ ($\circ$) and $282\mu \mathrm{m}$ ($\square$) and an unstable cell at $\mathrm{\Lambda }\approx 268\mu \mathrm{m}$ ($\diamond$), corresponding to the points marked with similar black symbols in Fig. 10.

Figure 12

Short-range correlation of hexagonal patterns oscillations at $V=1\mu \mathrm{m}$/s. Inside the “Hexagon” region of Fig. 7, three groups of cells oscillate coherently with a mutual phase difference of $\pm 2\pi /3$ (a). The phase-field simulation at $G=28\mathrm{K}$/cm and $\mathrm{\Lambda }=165\mu \mathrm{m}$ in (b) reproduces this coherent oscillation with a similar period.

Figure 13

Sustained oscillations of cell areas for a hexagonal array at $G=13\mathrm{K}$/cm, $V=1\mu \mathrm{m}$/s, and $\mathrm{\Lambda }\approx 215\mu \mathrm{m}$ [see Fig. 10]. The $2\pi /3$ breathing-mode oscillations appearing at the time scale of a few of hours in (a) are sustained in time, as shown over a 40 h time period in (b). This long-time scale dynamics exhibits an evolution of the oscillation amplitude of cell areas in (b), which also follows a $2\pi /3$ breathing mode with a period of approximately 7 h. This additional oscillation also appears in the evolution of phase difference between the three group of cells in (c), each oscillating around $2\pi /3$, i.e., ${120}^{\circ }$.

Figure 14

Presence of oscillatory breathing modes in continuous stable branches of spacing for $G=12\mathrm{K}$/cm and ${\epsilon }_4=0.011$ [green background zone in Fig. 10]. In (a), full symbols show the average normalized cell area oscillation amplitude for $2\pi /3$ breathing modes; open symbols represent simulations with elimination of one cell, resulting in $\pi$ breathing oscillations of the two remaining cells. For $2\pi /3$ modes, the oscillation periods are given in (b). The area evolution with time for the three cells at $V=1.1\mu \mathrm{m}$/s for $\mathrm{\Lambda }=193\mu \mathrm{m}$ and $\mathrm{\Lambda }=206\mu \mathrm{m}$—tagged in (a), (b), and Fig. 10—appear in (c) and (d), respectively.

Figure 15

Oscillation period $\tau$ vs growth velocity $V$ for ($\diamond$) experiments, ($\circ$) simulations with ${\epsilon }_4=0.007$ and $G=28\mathrm{K}$/cm, and ($\square$) simulations with ${\epsilon }_4=0.011$ and $G=12\mathrm{K}$/cm (see Sec. 3c). Average oscillations periods from simulations of a spatially extended domain (full symbol at $V=1\mu \mathrm{m}$/s) are similar to those for an imposed hexagonal pattern at the high spacing end of the leftmost stable spacing branch (open symbols). Data points are fitted to power laws $\tau \sim V^{\alpha }$ (lines). Simulations with $G=28\mathrm{K}$/cm yield an exponent $\alpha \approx -2.67$. In contrast, simulations with a lower critical velocity $V_c\approx 0.26\mu \mathrm{m}$/s at $G=12\mathrm{K}$/cm yield $\alpha \approx -1.91$, in better agreement with the experimental data with $\alpha \approx -1.51$.