Experimental observation of oscillatory cellular patterns in three-dimensional directional solidification

We present a detailed analysis of oscillatory modes during three-dimensional cellular growth in a diffusive transport regime. We ground our analysis primarily on in situ observations of directional solidification experiments of a transparent succinonitrile $0.24\mathrm{wt}\%$ camphor alloy performed in microgravity conditions onboard the International Space Station. This study completes our previous reports [Bergeon et al., Phys. Rev. Lett. 110, 226102 (2013); Tourret  et al., Phys. Rev. E 92, 042401 (2015)] from an experimental perspective, and results are supported by additional phase-field simulations. We analyze the influence of growth parameters, crystal orientation, and sample history on promoting oscillations, and on their spatiotemporal characteristics. Cellular patterns display a remarkably uniform oscillation period throughout the entire array, despite a high array disorder and a wide distribution of primary spacing. Oscillation inhibition may be associated to crystalline disorientation, which stems from polygonization and is manifested as pattern drifting. We determine a drifting velocity threshold above which oscillations are inhibited, thereby demonstrating that inhibition is due to cell drifting and not directly to disorientation, and also explaining the suppression of oscillations when the pulling velocity history favors drifting. Furthermore, we show that the array disorder prevents long-range coherence of oscillations, but not short-range coherence in localized ordered regions. For regions of a few cells exhibiting hexagonal (square) ordering, three (two) subarrays oscillate with a phase shift of approximately $\pm {120}^{\circ }$ (${180}^{\circ }$), with square ordering occurring preferentially near subgrain boundaries.

Figure 1

(a) Example of an initial raw image $(3.2\times 3.2\mathrm{m}{\mathrm{m}}^2)$. (b) Binary image evidencing some defects resulting from dark cells (DC), tip-splitting events (TS), or groove bottoms (GB). (c) Binary image after processing with “opening function”. (d) Contours of apparent cell area on corrected binary image superimposed on the initial raw image. (e) Tagged binary image ($V=1\mu \mathrm{m}/\mathrm{s}, G_1=19\mathrm{K}/\mathrm{cm}$).

Figure 2

(a) Evolution of the primary spacing $\left(\lambda \right)$ as a function of the pulled length $(L=Vt)$ at $G_1=19\mathrm{K}/\mathrm{cm}$. Primary spacings for the same pulling velocity are compared ($V=1\mu \mathrm{m}/\mathrm{s}$) for different experimental situations. $•$: long solidification at constant pulling rate; $\mathrm{\Delta }$: pulling rate jump from $V_1=1\mu \mathrm{m}/\mathrm{s}$ to $V_2=8\mu \mathrm{m}/\mathrm{s}$ after 30 mm; □: pulling rate jump from $V_1=8\mu \mathrm{m}/\mathrm{s}$ to $V_2=1\mu \mathrm{m}/\mathrm{s}$ after 30 mm. For experiments with pulling rate jumps, only the data corresponding to the part at $V=1\mu \mathrm{m}/\mathrm{s}$ are represented. (b) Distribution of the number of nearest neighbors at different times (or pulled lengths) for the long solidification at constant pulling rate $V=1\mu \mathrm{m}/\mathrm{s}, G_1=19\mathrm{K}/\mathrm{cm}$ [symbol $•$ in (a)].

Figure 3

(a) Global view of the interface at $t=13.9\mathrm{h}$ ($L=50\mathrm{mm}$), for $V=1\mu \mathrm{m}/\mathrm{s}$ and $G_1=19\mathrm{K}/\mathrm{cm}$, with the corresponding FFT image highlighting the high spatial array disorder. (b) Enlarged view of first neighbors in the area marked with a white square in (a) at different times: 1.5 h (5.4 mm), 6.7 h (24.1 mm), and 14.1 h (50.8 mm). (c) Enlarged view of area marked with a black square in (a) at different times $t_0=14.3\mathrm{h}, t_0+\tau /4=14.6\mathrm{h}, t_0+\tau /4=14.8\mathrm{h}$, and $t_0+\tau =15.1\mathrm{h}$ (oscillation period $\tau =48min$).

Figure 4

Oscillation period τ as a function of pulling velocity $V$ ($G_1=19\mathrm{K}/\mathrm{cm}$): ■, experimental; - -, fit by the power law $t=2.8\times {10}^3{V}^{-1.5}$, with $\tau$ in s and V in μm/s.

Figure 5

Different oscillation behaviors observed experimentally for $V=1\mu \mathrm{m}/\mathrm{s}$ and $G_1=19\mathrm{K}/\mathrm{cm}$ started from rest. (a) Standard oscillation. (b) Oscillation with tip-splitting events: the first tip splitting of the parent cell ($•$) around 7 h leads to two cells ($\blacksquare$ and $\square$) among which only one survives ($\blacksquare$); this cell is also subject to a tip splitting after about 10 h, and the two cells survive, oscillate in phase for some time before $•$ shifts out of phase and is eventually eliminated. (c) Oscillation with phase shift: the cell marked with $♦$ is initially in phase with $•$ before shifting its phase to synchronize with $▴$.

Figure 6

Examples of characteristic tip-splitting events: (a)–(d) tip splitting with overgrowth of one of the “daughter” cells by its “sister”; (e),(f) tip splitting where the two “daughters” survive and grow ($t_0=6.89\mathrm{h}, V=1\mu \mathrm{m}/\mathrm{s}$ and $G_1=19\mathrm{K}/\mathrm{cm}$).

Figure 7

(a) Enlargement of the square area with white border of Fig. 3. Some cells are tagged to identify their instantaneous phases on the unit circle presented in (b); the same color of tag denotes that phases are close ($L=56\mathrm{mm}, t=15\mathrm{h}, V=1\mu \mathrm{m}/\mathrm{s}, G_1=19\mathrm{K}/\mathrm{cm}$).

Figure 8

(a) Interface area presenting a locally hexagonal order formed by cells 1–7. (b) Apparent areas of cells of the three subarrays distinguished in the hexagonal structure formed by cells 1–7 ($•$, cell 3; $▴$, cell 4; and $\blacksquare$, cell 7) ($L=56\mathrm{mm}, t=16\mathrm{h}, V=1\mu \mathrm{m}/\mathrm{s}, G_1=19\mathrm{K}/\mathrm{cm}$).

Figure 9

Instantaneous phases of the cells tagged in Fig. 8 reported on the unit circle ($t=16\mathrm{h}, V=1\mu \mathrm{m}/\mathrm{s}, G_1=19\mathrm{K}/\mathrm{cm}$). The same tag color indicates that cells oscillate in phase.

Figure 10

(a) Oscillation in a regular square structure for $V=1\mu \mathrm{m}/\mathrm{s}$ and $G_1=19\mathrm{K}/\mathrm{cm}$. (b) Cells 43, 48, 55, and 56 oscillate in phase; as do cells 10, 54, and 57; and the two groups oscillate in phase opposition as illustrated by the apparent surface areas as a function of time for one cell of each subarray ($•$, cell 54; $▴$, cell 43). (c) Top view of the interface at $t=2.3\mathrm{h}$ (8.3 mm) where the subboundaries are visible with cells arranged perpendicular to them.

Figure 11

Stability maps for a quarter of cell in a hexagonal pattern for a pulling velocity $V=1\mu \mathrm{m}/\mathrm{s}$ (a) and $1.5\mu \mathrm{m}/\mathrm{s}$ (b), and different temperature gradients (shown with different colors and symbols, with steps of 2 K/cm, and 1 K/cm in the vicinity of the stability gap), plot as tip undercooling $\mathrm{\Delta }$ of the stable cells vs primary spacing $\lambda$. The thick yellow line is a schematic illustration of the stability limit in the $(\mathrm{\Lambda },\mathrm{\Delta })$ plane. Zoomed-in insets mark configurations explored for oscillatory breathing mode oscillations (Fig. 12).

Figure 12

Breathing mode oscillations amplitude (a) and period (b) vs temperature gradient $G$ for different cases represented in the insets of Fig. 11. Gray background zones in (a) represent noisy oscillations exhibiting no neighbor synchronization for low amplitudes $(\le 0.1)$, and configurations unstable with respect to elimination and/or tip splitting for high oscillation amplitudes $(\ge 0.8)$.

Figure 13

Distribution of normalized oscillation amplitudes vs individual primary spacing for (a) experimental normalized amplitudes reported for 250 cells (measured between 13.5 and 16.5 $\mathrm{h}$ of solidification, $V=1\mu \mathrm{m}/\mathrm{s}, G_1=19\mathrm{K}/\mathrm{cm}$); (b) 3D phase field simulation for 51 cells between 2 and 10 h of solidification (spatially extended simulation with $k=0.21, {\varepsilon }_4=0.007, V=1\mu \mathrm{m}/\mathrm{s}, G=28\mathrm{K}/\mathrm{cm}$ [36]).

Figure 14

Long time evolution of the primary spacing observed in the experiment at $V=0.75\mu \mathrm{m}/\mathrm{s}$, used to extend the analysis performed in Fig. 13: (a) the experimental normalized oscillation amplitudes as a function of primary spacing are reported for $\approx 50$ cells, for four different time intervals, namely $\blacksquare$: 4.1–8.5 h; $▵$: 8.5–13.7 h; $♦$: 13.7–16.3 h; ○: 16.3–21.9 h. In (b), all data points are averaged by classes of primary spacing of 10 μm width to evidence the weak dependence of the oscillation amplitude upon primary spacing ($V=0.75\mu \mathrm{m}/\mathrm{s}, G_1=19\mathrm{K}/\mathrm{cm}$).

Figure 15

(a) Top view of the interface during steady-state growth ($t=15\mathrm{h}$; $L=40.5\mathrm{mm}$) for $V=0.75\mu \mathrm{m}/\mathrm{s}$ and $G_1=19\mathrm{K}/\mathrm{cm}$. (b) Evolution of the primary spacing as a function of the pulled length ($L$) for the two regions identified in (a): stable region $\left(\mathrm{\Delta }\right)$; oscillating region (♦). (c) Trajectories of the cells in the boxed areas of (a) between 8.5 h (23 mm) and 16 h (43 mm).

Figure 16

Top view of the interface at $t=2.3\mathrm{h}$ ($L=12\mathrm{mm}$) (a) and 10.8 h (58 mm) (b), with $V=1.5\mu \mathrm{m}/\mathrm{s}$ and $G_1=19\mathrm{K}/\mathrm{cm}$. In (b) the arrows represent the trajectories of the cells followed from 6 to 11 h of solidification (32–59 mm), and three different zones are zoomed 2.5 times to evidence the difference of morphology between the cells of the left side, which are clearly tilted, and the cells of the right side.

Figure 17

Ratio of the characteristic times of drift ${\tau }_{\mathrm{d}}$ and oscillation period τ as function of growth rate ($V_i$), for the different data sets given in Table 2: the filled triangles ($▴$) correspond to oscillating conditions whereas the empty triangles ($▵$) correspond to oscillation inhibition. Diamonds ($♦$) correspond to nonoscillating conditions, using the oscillating period measured in the same area before the oscillation stops. The cross (x) corresponds to the oscillation stop at $V=1.5\mu \mathrm{m}/\mathrm{s}$ in the right side of the interface, explained by the analysis of the stability gap and its relation to oscillation (cf. Fig. 11) ($G_1=19\mathrm{K}/\mathrm{cm}$).

Figure 18

Analysis of the characteristics of the cells during the growth stage at $V=1\mu \mathrm{m}/\mathrm{s}$ after a jump of pulling rate from $V=8\mu \mathrm{m}/\mathrm{s}$. Maps of (a) drift direction and (b) drift velocity of each cell, and (c) primary spacing, with dashed lines indicating frontiers between regions of different drift directions, indicated by arrows (length of arrows not proportional to drift velocity). $V=1\mu \mathrm{m}/\mathrm{s}, G_1=19\mathrm{K}/\mathrm{cm}, t=4.3\mathrm{h}, L=41.8\mathrm{mm}$ (30 mm at $8\mu \mathrm{m}/\mathrm{s}+11.8\mathrm{mm}$ at $1\mu \mathrm{m}/\mathrm{s}$).