Younggil Song

Silvère Akamatsu

Sabine Bottin-Rousseau

Alain Karma

Propagative selection of tilted array patterns in directional solidification

Keywords: numbers: 68.08.-p, 05.70.Ln, 64.70.D, 81.30.Fb

We investigate the dynamics of tilted cellular/dendritic array patterns that form during directional solidification of a binary alloy when a preferred-growth crystal axis is misoriented with respect to the temperature gradient. In situ experimental observations and phase-field simulations in thin samples reveal the existence of a propagative source-sink mechanism of array spacing selection that operates on larger space and time scales than the competitive growth at play during the initial solidification transient. For tilted arrays, tertiary branching at the diverging edge of the sample acts as a source of new cells with a spacing that can be significantly larger than the initial average spacing. A spatial domain of large spacing then invades the sample propagatively. It thus yields a uniform spacing everywhere, selected independently of the initial conditions, except in a small region near the converging edge of the sample, which acts as a sink of cells. We propose a discrete geometrical model that describes the large-scale evolution of the spatial spacing profile based on the local dependence of the cell drift velocity on the spacing. We also derive a nonlinear advection equation that predicts the invasion velocity of the large-spacing domain, and sheds light on the fundamental nature of this process. The models also account for more complex spacing modulations produced by an irregular dynamics at the source, in good quantitative agreement with both phasefield simulations and experiments. This basic knowledge provides a theoretical basis to improve the processing of single crystals or textured polycrystals for advanced materials.

I. INTRODUCTION

The directional solidification of nonfaceted dilute alloys leads to the freezing of columnar-like microstructures in the bulk solid, which are known to largely determine the properties in use of as-cast materials in metallurgy [START_REF] Flemings | Solidification Processing[END_REF][START_REF] Kurz | Fundamentals of Solidification[END_REF][START_REF] Dantzig | Solidification[END_REF]. These microstructures are formed during growth as a trace left behind by self-organized solidification front patterns. Their main morphological features are primarily determined by the redistribution of chemical species by diffusion in the liquid and capillary effects at the propagating solid-liquid interface, which can be considered in local thermodynamic equilibrium at slow growth rate [START_REF] Mullins | [END_REF][5][6][7][START_REF] Hunt | Solidification and Casting of Metals[END_REF][START_REF] Somboonsuk | [END_REF][10][11][12][13][14].

During directional solidification (DS), melt crystallization is performed while a sample is pulled with a velocity V in a uniaxial temperature gradient G [6][7][START_REF] Hunt | Solidification and Casting of Metals[END_REF][START_REF] Somboonsuk | [END_REF][10][11][12][13]. When the sample is pulled from rest, the solute concentration builds up in front of the recoling planar interface that becomes morphologically unstable [START_REF] Mullins | [END_REF]15,[START_REF] Losert | Proc. Natl. Acad. Sci[END_REF], thereby forming shallow cells that become progressively deeper. During this initial transient, cells grow competitively due to solutal interactions between neighbors and become eliminated. Surviving cells then reach a steady-state, forming a periodic array of finger-like shapes -"cells" and "dendrites"-with a primary spacing λ (typically, a few tens of micrometers), which can vary within a * a.karma@northeastern.edu large interval [START_REF] Hunt | Solidification and Casting of Metals[END_REF][START_REF] Somboonsuk | [END_REF][10][11][12][13][14][15][START_REF] Losert | Proc. Natl. Acad. Sci[END_REF][START_REF] Hunt | [END_REF][18][19][20][21][22][23][24]. In particular, phase-field (PF) simulation studies have shown that steady-state arrays of deep cells (smooth fingers) or dendrites (branched fingers) can be stable for a large range of primary spacings λ min < λ < λ max [19][20][21]. Moreover, the largest stable spacing λ max is typically several fold the smallest stable spacing λ min in the velocity range where arrays of deep cells or dendrites form, and V is much larger than the onset velocity V c of morphological instability [19][20][21].

Pattern selection has been studied extensively both theoretically and experimentally in a thin-sample geometry [7][START_REF] Hunt | Solidification and Casting of Metals[END_REF][START_REF] Somboonsuk | [END_REF][10][11][12][13][14]. To date, however, those studies have primarily focused on understanding what spacing is dynamically selected by the transient growth competition between neighboring cells for a well-oriented single crystal [15,[START_REF] Losert | Proc. Natl. Acad. Sci[END_REF][20][21][22]25]. Competitive growth continuously increases the average spacing until a stable spacing is reached. Therefore, it typically selects a spacing slightly larger than λ min due to the fact that the cell elimination stops when a stable spacing is reached. In practice, the spatial λ distribution along the array is rarely uniform and can be strongly influenced by the orientation of the growing crystal with respect to the temperature gradient axis both during, and well after, the initial cell elimination process. However, the long-time evolution of the array pattern towards steady-state in a large crystal as a function of its orientation remains largely unexplored.

The crystal-orientation dependency of DS front patterns originates from the sensitivity of growth shapes to the surface-tension anisotropy of the solid-liquid interface, which is well-known to select both the orienta- tion and tip operating state of freely growing dendrites [26,27]. In a DS setting, however, the growth orientation is also influenced by the temperature gradient. As a result, cells/dendrites grow at an angle α with respect to the temperature gradient axis (x axis), which differs from the crystal misorientation angle α 0 (between the x axis and one of the principal crystal axes). This difference is illustrated in Fig. 1, which shows experimental and PF simulation images of tilted array patterns from the present investigation. The relationship between α and α 0 has been extensively studied previously both experimentally [28,29] and numerically [30][31][32][33]. Those studies have shown that α/α 0 is typically a monotonously increasing function of the dimensionless Péclet number Pe = λ/l d that is largely insensitive to G, where D is the solute diffusion coefficient in the liquid and l d = D/V is the solutal diffusion length. The ratio α/α 0 varies from a value much smaller than unity for Pe 1, where the growth direction of cells is primarily determined by diffusive interactions with neighboring cells, to unity in a dendritic regime (Pe 1) where the growth direction is primarily determined by crystalline anisotropy.

This current knowledge of tilted finger patterns remains limited to a steady-state growth regime where λ is spatially uniform along the array. In this paper, we present an experimental and numerical study of the largescale dynamics of those patterns in the generic case where λ is spatially non-uniform. Non-uniformity is expected to be generic and its origin in the setting of the growth of a single misoriented crystal is illustrated in Fig. 1. Due to the lateral drift, there is a qualitatively different operating dynamics on the two edges of the sample, each of them acting as a virtually immobile grain boundary (GB). Tilted cells travel away from the right edge in Fig. 1, referred to as the divergent GB, which acts as a source of new cells with a spacing significantly larger than λ min , towards the left edge referred to as a convergent GB, which acts as a sink of cells that are eliminated when their spacing falls below λ min . To investigate the pattern dynamics inside the sample resulting from the source and sink at the edges, we used a model transparent alloy, namely the well-characterized succinonitrile-dcamphor (SCN-DC) alloy in a dilute-concentration range, in semi-thin samples. In such a confined-3D geometry, deep cells with a three-dimensional (3D) shape arrange within a single row. This permits real-time observation of the evolution of the solid-liquid interface with a standard optical microscope. Quantitative numerical simulations using the physical parameters of dilute SCN-DC alloys were carried out using a well-developed quantitative PF model of dilute binary alloy DS [23,24,34,35]. Systematic numerical results were obtained by varying V in a deep-cell regime, and the misorientation angle α 0 between 5 • and 20 • .

Both experiments and numerical simulations show that, after a long transient, the cell source at the divergent GB operates regularly. As a result, the total number of cells in the sample remains constant and the spacing reaches a steady-state spatial distribution where λ is uniform and much larger than λ min in most of the sample, and drops abruptly to a small value ∼ λ min inside a boundary layer near the convergent GB. This steadystate distribution is established by the lateral propagation of a front separating regions of larger and smaller spacings from the divergent GB towards the convergent GB. Importantly, this propagation takes place on a much longer time scale than the competitive growth transient at the start of an experiment during which cell elimination establishes the initial spacing distribution inside the sample with a spacing slightly larger than λ min [15,[START_REF] Losert | Proc. Natl. Acad. Sci[END_REF][20][21][22]25]. Consequently, propagative spacing selection can be investigated separately from the establishment of the initial spacing distribution. This initial distribution is not studied in detail here since, on the long time scale of the present experiments and simulations, it is completely "erased" by the propagative source-sink selection mechanism that establishes an entirely different spatial distribution of larger λ.

We propose a simplified geometrical model that can be numerically studied to predict the long-time spatiotemporal evolution of the array spacing. In addition, we derive a nonlinear advection equation that describes the evolution of the spacing distribution in a continuum limit and can be used to predict analytically the invasion velocity of the large spacing region inside the sample using only the relationship between local growth orientation and local spacing (α/α 0 vs. Pe) as input into the theory. Those models yield predictions in excellent quantitative agreement with numerical and experimental results. Moreover, they show that the spacing modulations, which are initially present or possibly provoked during the course of a long-time DS run by any experimental imperfection, are advected in the same direction as the cell drifts at a rate that is noticeably smaller than V d ≡ V tan α, and eventually eliminated at the convergent GB.

The numerical and experimental methods are described in Section II A. We propose an overview of the main results in Section III A, and some useful details about the relationship between the growth orientation of tilted cells and the local spacing in reference to previous works in Section III B. Simple models for analyzing the (primary) spacing evolution are presented in Section III C. Numerical and experimental results are presented, analyzed and discussed in Sections III D and III E, respectively. Conclusions are presented in the last section.

II. METHODS

A. Phase-field model

We used a recent 3D phase-field (PF) model that describes the solidification dynamics of a single-phase alloy in a one-sided limit (no diffusion in the solid) [19,20,32,[36][37][38]. This model includes the so-called antitrapping current that has been previously introduced in PF models to quantitatively predict the interface dynamics during solidification experiments of dilute binary alloys [37,38]. We also replaced the classical phase field ϕ (ϕ = +1 in the solid and -1 in the liquid) with the preconditioned phase field ψ defined by ϕ ≡ tanh (ψ/ √ 2), thus permitting to enhance the numerical stability of the PF simulations with large grid spacings [39]. We used a GPU (Graphic Processing Unit) architecture to improve the computational performance [21,23,30,40].

In order to produce a tilted-cell array in the 3D PF model, the misorientation angle α 0 is introduced into the surface tension anisotropy as described in Refs. [30,40]. Then, the angles α 0 and α (Fig. 1) are measured from the x axis that is parallel to the temperature gradient G. The detailed equations for the ψ field including a misorientation angle and dimensionless solute concentration U can be found elsewhere [30,40]. We implemented the thermophysical data of a SCN-0.24wt% camphor alloy [23,24,35,[41][42][43], which are listed in Table I along with numerical parameters for the PF model. This model used a frozen-temperature approximation, and the temperature gradient was set to G = 12 K cm -1 .

We performed PF simulations using three different values of the solidification velocity, V = 4, 12, and 20 µm s -1 for different interface thicknesses W/d 0 ≈ 47.56, 28.26, and 17.23, respectively, where

d 0 = Γ/(|m|c ∞ (1/k - 1)
) is the chemical capillary length with the Gibbs-Thomson coefficient Γ and the nominal composition c ∞ = 0.24wt%. Then, the grid spacing ∆x and the Euler explicit time step ∆t are chosen as ∆x ≈ 3.00, 1.78, and 1.09 µm, and ∆t ≈ 5.00 × 10 -3 , 1.75 × 10 -3 , and 6.55 × 10 -4 s in order of increasing V (see Table I).

For a thin-sample geometry, sample sizes were set to L x × L y × L z = 1290 × 2490 × 114 µm 3 for V = 4 µm s -1 , 709 × 2563 × 82 µm 3 for V = 12 µm s -1 , and 485 × 1998 × 50 µm 3 for V = 20 µm s -1 , where L x , L y , and L z correspond to the domain size along the x, y, and z axes, respectively. In all situations, the sample width L y was large enough to form more than ten cells in steady-state (with the exception of the periodic-array simulations presented at the beginning of Section III B). Thermal fluctuations were introduced by adding a random noise with a strength F ψ = 0.01 onto the ψ field [23,30,32,35,40]. No-flux (or reflection) boundary conditions were imposed at the lower/upper limits of the simulation domain (x = 0 and x = L x ), the immobile GBs (y = 0 and y = L y ), and the walls that delimit the sample along the z axis (z = 0 and z = L z ). We moreover imposed a finite-wetting condition at the z = 0 and z = L z boundaries with a slope dψ/dz| z=0 = -dψ/dz| z=H = +1 [21,32,40]. The long duration time of the simulations (20000 s for V = 4 µm s -1 ; 2000 s for higher velocities) allowed us to simulate the dynamics of tilted-cell arrays over several centimeters. The cell-tip positions within a single row of cells were measured at the middle of the sample, i.e. in a x-y plane located at z = H/2, where H = L z is the sample thickness, as illustrated in Fig. 2a. These measurements in turn were used to calculate the local spacing λ, the drift velocity V d , and the cell-tilt angle α.

For calibration purposes, we also carried out PF simulations of a single cell in a channel with periodic boundary conditions in the y direction. This was used to measure V d in a perfectly ordered array with a fixed spacing λ = L y . For those simulations, we used an initial guess made of a pre-calculated axial cell (α 0 = 0 • ) with λ = 114, 82, and 50 µm for V = 4, 12, and 20 µm s -1 , respectively, in a squared-section simulation box (i.e. L y = L z ). Then, we re-started a simulation using the axial cell by implementing a finite misorientation angle α 0 . The λ value was changed by adding (or subtracting) a gridspacing unit ∆x to the width L y of the sample, and using a bilinear interpolation from the previous steady-state simulation.

B. Experiments

A SCN-0.3 ± 0.05mol% DC alloy was prepared with purified SCN (distillation) and DC (sublimation). Alloy samples were enclosed in thin containers made of two flat glass plates glued to each other, and separated by polymer spacers of thickness H = 100 µm. The sample length (along the x axis) was of about 64 mm, and the width L y (along the y axis) of 2.8 ± 0.2 mm (this typically corresponds to 20 cells). Solidification experiments were carried out over a total length of 30 to 40 mm, beyond which substantial macrosegregation effects were observed. Previous to DS, the samples were filled by capillarity with the melted alloy under a low-pressure argon atmosphere, and then rapidly cooled down to room temperature. During this process, multiple nucleation sites were activated, thus resulting in the formation of a polycrystal.

The DS apparatus was made of two temperatureregulated copper blocks, separated from each other by a distance of 5 mm (for details, see Ref. [44] and references therein). A fixed temperature gradient G = 75 ± 5 K cm -1 established by heat conduction in the sample along the x axis. Solidification was performed by translating the sample along the x axis at a velocity V = 4 µm s -1 with a DC motor. The whole setup was installed on the stage of a standard optical microscope. The solid-liquid interface was visualized (transmitted light) in real time over the entire width of the sample with a numerical camera connected to a PC for image storage and analysis.

At the beginning of the experiment, the alloy was partially melted directionally down to a crystal selector placed on the cold end of the sample [45]. A single, randomly oriented crystal was then grown and expanded along a funnel-shaped region until it occupied the whole width of the sample. The orientation of the crystal was measured, within a 1 • error margin, by growing fully developed dendrites at a large velocity (V > 200 µm s -1 ). The operation was repeated until an "in-plane" crystal such that a [100] axis (the preferred dendrite growth axis in SCN-based alloys at low growth rate) belonged to the sample x-y plane was selected.

III. RESULTS AND DISCUSSION

A. Overview

In both PF simulations and experiments, regular solidification runs were started (time t = 0 s) after a planar solid-liquid interface was equilibrated at rest (V = 0 µm s -1 ). Small cells were observed to appear during the initial solute-redistribution transient after the velocity of the planar interface becomes larger than the threshold velocity V c [START_REF] Mullins | [END_REF]15,[START_REF] Losert | Proc. Natl. Acad. Sci[END_REF]. Their spacing was systematically and substantially smaller than the lower stability limit of steady periodic cells at the imposed velocity. This entailed an initial coarsening dynamics by progressive elimination of a large proportion (about one out of two) of the initial cells, which lasted until a one-row pattern of confined-3D cells with a stable average spacing was formed. Three key features must be noted at this stage: (i) after the very last cell-elimination event in the middle of the sample, the cellular pattern presented a nonuniform spatial spacing distribution λ(y); (ii) tilted cells drift away from the divergent GB, which thus acts as a source of new cells via a tertiary branching process; (iii) cells are eliminated in the vicinity of the convergent GB. The aim of the present study is to characterize and model the long-time behavior of tilted-cell patterns by taking those three elements -nonuniform initial spacing distribution, cell source at the divergent GB, and cell sink at the convergent GB-into account.

Let us go into more quantitative details. We note the time t 0 at which the initial coarsening process was completed (this time has no need to be defined with a better accuracy than, say, ±λ/2V ). In typical PF simulations, we measured t 0 ≈ 4040 s for α 0 = 5 • (but 2040 s for higher α 0 values) at V = 4 µm s -1 , 900 s for V = 12 µm s -1 , and 220 s for V = 20 µm s -1 . This represents a long-duration process as compared to the reference diffusion time

τ d = D/V 2 .
Those simulations were performed over t max = 2000 s and 20000 s for V = 4 µm s -1 and the higher V , respectively. Then, the overall duration was systematically in the range of 2t 0 and 10t 0 (t max was close to 2t 0 in the experiments).

Therefore, the long-time dynamics after t 0 was fully disconnected from both the initial solute redistribution transient, and cell-spacing rearrangement processes.

In the simplified spatiotemporal diagram of Fig. 2a (PF simulation), the drifting motion of the cells (circles) is made clearly visible. Different colors correspond to different cells, and those cells on a same horizontal position are from one microstructure at a time step. While they grow vertically (along the x axis), they drift laterally from the divergent GB (on the right-hand side) to the convergent GB (on the left-hand side). Changing a color of a circle at the boundaries corresponds to an appearance of a cell by tertiary branching at the divergent GB and cell elimination at the convergent GB. As illustrated in Fig. 2b, those operations at the boundaries occur quite regularly. A similar sink-and-source dynamics is observed in the experiments.

We measured the spacing distribution λ(y) at different 3) (also see Table II), respectively. Blue empty circles: modulated arrays (see text).

time steps (Fig. 2c). In the middle of the initial pattern at t 0 , the spacing was slightly modulated about an average value of, here, about 130 µm and exhibited a steep variation close to the convergent and divergent GBs. During the solidification run, the spacing value λ M delivered from the source propagated across the sample. At the end of the process, the spacing distribution reached a steady-state characterized by a large plateau, signalling the extension of a periodic pattern from the divergent GB over the major part of the sample width, and a steep drop in a region close to the convergent GB. The value of λ M was larger than the average spacing at time t 0 . The latter feature was systematically observed in our PF simulations. It is also compatible with previous reports of a progressive increase of the average primary spacing over long timescales during DS experiments [14,[48][49][50][START_REF] Esaka | Columnar Dendrite Growth in Succinonitrile-Acetone Alloys[END_REF][START_REF] Esaka | Solidification Processing[END_REF]. This qualitative description of the tilted-cell pattern dynamics opens to further questions, among which that of the rate of the advection (vs. smoothing out) of incipient (or accidental) modulations.

B. Dependence of the cell-tilt angle on the local spacing in modulated patterns

For reference purposes, we measured the dependence of the cell-tilt angle α on the Péclet number in periodic arrays for the SCN-DC alloy under consideration. We performed PF simulations with periodic boundary con-ditions, for various values of V (= 4, 12, and 20 µm s -1 ), α 0 (= 5, 10, 15 and 20 • ), and L y (= λ). We let λ increase to the upper limit λ max at which a tertiary branching instability occurs [20], and decrease to the lower limit λ low (Table II). Below the lower limit of a spacing, a cell has a planar interface perpendicular to the temperature gradient axis like a 2D ribbon shape [19].

When a cell travels laterally, its drift velocity V d was calculated as

V d = dy t dt , (1)
where y t was the position of the cell tip along the y axis, and t was the time. Then, the tilt angle α was calculated by using the geometrical relation

tan α = V d /V . (2)
The α/α 0 data are shown as red full squares in the graphs of Fig. 3 and can be used to define the function α = α 0 F (Pe) in Eq. ( 3), which is related to the misorientation angle α 0 and the local adimensionalized spacing Pe = λV /D. We also measured local V d and λ values in the modulated tilted-cell patterns observed during large-scale PF simulations. Each data set cumulates measurements performed approximately every 63 s, 8 s, and 3 s for V = 4, 12, and 20 µm s -1 , respectively, starting from the above defined time t 0 to the end of the simulations t max .

We systematically discarded the sidebranching and sinking cells located close to the GBs. The obtained α/α 0 data are reported as blue empty circles in the graphs of Fig. 3 along with the periodic-array measurements. The two kinds of data are obviously close to each other. The modulated-pattern data exhibit a substantial dispersion, but it is worth noting that the most departing points were measured in regions with steep λ(y) variations near the GBs.

We finally recall that the following law:

α α 0 ≡ F (Pe) = 1 - 1 1 + f Pe g , (3)
where f and g are alloy-dependent constants, has been previously proposed for a general, yet empirical description of the F (Pe) function [28][29][30][31][32][33]. Best-fit adjustments of that law with our periodic-array data points are shown in the graphs of Figs. 3a-f. The corresponding constants f and g are listed in Table II. Two comments are in order. First, the parameters f and g at a fixed α 0 (see the data for α 0 = 15 • in Table II) vary when V is changed. In other words, PF simulations of confined-3D patterns show that the parameters can change when V > 15V c , where the critical velocity V c = DG/∆T 0 with ∆T 0 = mc ∞ (1/k-1), as observed in the 2D patterns [33].

On the other hand, the α/α 0 = F (Pe) function exhibits the same "universality" for 2D arrays at V < 15V c [28].

Second, the best-fitting curves agree relatively well with the PF simulations, but not perfectly. Considering that not only the F (Pe) function, but also the derivative of that function will be used later on in this paper for a quantitative analysis of the spatiotemporal dynamics of tilted-cell arrays, we think that using the (interpolated) results of the PF simulations in the models presented in the next section is more reliable than using the approximate law of Eq. (3).

TABLE II. Control parameters (pulling velocity V and misorientation angle α0) of PF simulations using periodic tilted-cell arrays. λ low (λmax): lower (upper) limit of the spacing interval. f and g: fitting parameters in Eq. (3). The fitting curves are shown in Fig. 3 as black lines. We insert here the definition of the two models that we use for the analysis of our numerical and experimental data presented in the next section.

V [µms -1 ] α0 [ • ] λ low [µm] λmax [µm] f [-] g [-] 4

Geometrical model

We propose a simple "geometrical" model based on a discrete treatment of the lateral motion of individual cells to describe the spatiotemporal evolution of the spacing of tilted-cell patterns. We make the reasonable assumption that the growth orientation of a tilted cell within an array with a slowly varying spatial modulation of spacing is described locally by the α/α 0 = F (Pe) function determined for a periodic array (red squares in Fig. 3). This assumption yields the evolution equation

dy i dt = -V d = -V tan α 0 F y i -y i+1 D/V , (4)
for the lateral coordinate y i of the tip of the i th cell shown schematically in Fig. 4, where the local spacing between two cell tips is taken equal to y i -y i+1 (with the positive y axis pointing to the right and the cell index i increasing from right to left). We note that, in the framework of this simple phenomenological model, the local spacing influencing the growth orientation of the i th could be chosen as well as y i-1 -y i or any weighted sum p(y i -y i+1 ) + (1 -p)(y i-1 -y i ), with p varying from 0 to 1, of the spacings defined with the two neighboring cell tips. We investigated different values of p and empirically found that p = 1 gave the best predictions. This finding suggests, as previously announced by J. Deschamps [START_REF] Deschamps | [END_REF], that the lateral velocity of a cell in the array is more strongly influenced by the spacing with its front neighbor, towards which it is traveling, than its rear neighbor. This assertion is moreover supported by the finding in Ref. [33] that the asymmetry of the shape of a tilted cell is typically strong on the side of the preceding neighbor cell (here, cell i+1), whereas it is much less pronounced on the side of the following cell (here, cell i-1). In all numerical solutions of the geometrical model presented here, we will therefore use the value p = 1. This geometrical model quantitatively predicts the spacing evolution of tilted cells away from the convergent and divergent GBs observed in experiments [START_REF] Deschamps | [END_REF].

Incoming

In order to account for the source-sink dynamics at the two GBs, we create a new cell with a spacing λ M when the distance of the tip of the right-most cell closest to the divergent GB (right edge of the sample) exceeds λ M (source), and we remove the left-most cell when its distance to the convergent GB (left edge of the sample) becomes smaller than λ e . Eq. ( 4) defines a set of coupled ordinary different equations for the time-evolution of cell tips that are solved using an Euler explicit time stepping scheme with a time step ∆t = 0.1 s for the experiments and simulations. The above conditions for cell creation and removal are checked at each time step.

Nonlinear advection equation

To complement the discrete geometrical model, we derive an equation for the slow spatiotemporal evolution of λ treated as a continuous function of the lateral position y. This equation can be readily obtained from the conservation of the total number of cells in the array away from the edges where cells can be created or eliminated. For this, we define the cell density (number of cell per unit length of the array) ρ ≡ λ -1 . The cell density obeys the continuity equation

∂ t ρ -∂ y [V d • ρ] = 0 . ( 5
)
This equation is simply the statement that, in the absence of cell creation or elimination, the change of the total number cells in a segment of the array away from the sample edges (Fig. 4), which is much larger than the cell spacing but much smaller than the characteristic scale of variation of λ, must be equal to the difference of flux of cells entering or leaving through the left and right boundaries of this segment. Note that the minus sign of the second term in the continuity equation originates from the fact that the local drift velocity of cells towards the left is defined positively (V d ≥ 0) while the positive y axis points towards the right. Substituting ρ = 1/λ in Eq. ( 5), we obtain the advection equation for λ

∂ t λ = V λ ∂ y λ , (6)
where we have defined

V λ ≡ V d -λ dV d dλ . ( 7
)
This equation is nonlinear because, as before, V d is a nonlinear function of λ defined by

V d = V tan α = V tan(α 0 F (Pe))
where the local growth orientation α = α 0 F (Pe) and hence the drift velocity V d is assumed to be a function of the local spacing.

In addition, V λ using Eq. ( 3) can be negative at a low spacing, however non-uniform arrays in both the simulations and the experiments select a spacing larger than the one at V λ = 0 µm s -1 . Hence, we do not consider such conditions in the current article.

Note that, using the standard relations between a frequency ω ≡ V d k ω and a wave number k ω = 2π/λ of traveling waves, V λ defined by Eq. ( 7) can also be written in form of a group velocity V λ = dω/dk ω .

Eq. ( 6) is reminiscent of Burgers' equation [START_REF] Burgers | The Nonlinear Diffusion Equation[END_REF], which is well-known to form shocks. In contrast, in the simulations shown in Fig. 2c, the front separating regions of larger and smaller spacing propagates at a nearly constant velocity without becoming steeper, as would be expected if shock formation was present. The shock-less behavior can be related the fact that the drift velocity is predominantly a linear function of λ in the front region. Hence, the drift velocity can be expanded in the form

V d (λ) = V d (λ p ) + [dV d /dλ] λ=λp (λ -λ p ) + . . . (8)
up to small O((λ -λ p ) 2 ) corrections, where λ p is defined as the "median spacing", which is the characteristic value of λ in the propagating front region. It is arbitrarily defined here as the value of λ that is half way between the smaller and larger values of λ in regions ahead and behind the front, respectively, where λ is spatially uniform. Indeed, values of α (blue circles in Fig. 3), and hence the drift velocity V d = V tan α ≈ V α for small α, are to a good approximation of a linear function using the dimensionless spacing. Substituting the approximation Eq. ( 8) into Eq. ( 7) yields the prediction

V λ ≈ V d (λ p ) -[dV d /dλ] λ=λp λ p . (9)
There are two implications of this result. First, the advection velocity is approximately constant in the entire front separating large and small spacings regions, such that the front simply propagates in a shape-preserving manner with a velocity V λ without forming shocks. Second, since the slope of the drift velocity-spacing relation is positive ([dV d /dλ] λ=λp > 0), the front propagation velocity V λ is typically several fold smaller than the local drift velocity V d (λ p ) in the front region. In fact V λ defined by Eq. ( 9) is the λ = 0 intercept of a straight line fit of the V d (λ) curve and this intercept must always be smaller than V d (λ p ). We use Eq. ( 9) to predict the front propagation velocity in simulations and experiments. For the simulations, we compute V d and its slope using the growth orientation vs. spacing relation α/α 0 = F (Pe) determined from simulations with periodic arrays of different spacing (red full squares in Fig. 3) and λ p extracted from simulations. We have also computed V λ as the λ = 0 intercept of a straight line fit of the α/α 0 vs. Pe values measured on the propagating front during simulations (blue empty circles in Fig. 3), and found that the two methods predict values of V λ within 16% of each other. We only report in this paper values of V λ computed with the first method for the simulations. For the experiments, the slope dV d /dλ is estimated by a best-fit line of Eq. ( 3) that is fit to the experimental measurements (empty blue diamonds in Fig. 10b), then we compute V λ at λ p . The λ = 0 intercept of a straight line fit of the α/α 0 vs. Pe values measured on the propagating front during experiments is also close to the computed V λ . The nonlinear advection equation, i.e. Eq. ( 6), neglects the effect of "phase diffusion", which generically relaxes spatially modulated nonequilibrium patterns towards a spatially uniform spacing [START_REF] Cross | [END_REF]. This effect could be incorporated by the addition of a diffusion term D λ ∂ 2 λ/∂y 2 on the right-hand-side of Eq. ( 6). While "phase diffusion" has been shown to strongly influence the stability of lamellar eutectic growth patterns [56,57], its effect on array patterns of deep cells and dendrites in the presence of crystalline anisotropy turns out to be much weaker. We explicitly computed D λ by PF simulations of welloriented (α 0 = 0) arrays in which we analyzed the decay of slowly varying spatial modulations of the spacing and found D λ to have a very small value (D λ ≈ 20.1 µm 2 s -1 ), which would have a negligible effect on front propagation in the present PF simulations and experiments. The details to estimate D λ are shown in Appendix A.

D. Comparison of phase-field simulations with theoretical models

We performed systematic PF simulations of confined-3D arrays of tilted cells by varying the misorientation angle α 0 and the solidification velocity V (also see Section III A and Fig. 2). Let us first focus on the effect of the misorientation angle. We considered four different α 0 values, namely, 5, 10, 15, and 20 • , and kept the velocity V = 4 µm s -1 . The spatiotemporal evolution of the corresponding DS patterns is shown in the graphs of Fig. 5a. In each of those graphs, the measured λ(y) (symbols) are plotted at successive simulation times from t 0 to t max . For all considered α 0 values, the propagation of the spacing λ M delivered from the source towards the inner-grain region is clearly visible. In the simulations, the selected spacings at the divergent GB are typically λ M ≈ 180, 192, 201, and 218 µm for α 0 = 5, 10, 15, and 20 • , respectively. As mentioned above, λ M was systematically larger than the initial average spacing at t 0 . Note that most simulations reached a steady-state regime, with a uniform λ M plateau and a steep drop near the sink, except for the smaller misorientation angle α 0 = 5 • , that is obviously for a very slow lateral drift velocity.

We implemented the geometrical model (Section III C 1) by using, for each pair of V and α 0 data, the spacing profile at t 0 as an initial condition (red continuous line). We let cells move with an instantaneous local velocity V d (Pe) that was linearly interpolated from the ordered array simulations of Fig. 3. We used the steadystate values of λ M for the cell branching criterion close to the divergent GB, and manually set the cell elimination limit to λ e = 54 µm at the convergent GB for the V = 4 µm s -1 simulations (gray shaded area on the right and left side, respectively, in the graphs of Fig. 5a). The calculated spacing profiles are shown in the form of continuous lines in Fig. 5a. They superimpose remarkably well with the simulation results, in consideration of the simplicity of the geometrical model, and bring a clear evidence to the general dynamical scheme that can be summarized as follows: advection and elimination of the initial modulations, formation of a propagation front of a λ = λ M plateau, and convergence towards a steady-state profile.

There are slight discrepancies between the geometrical model and the PF simulations, which can be essentially classified according to two separate features. First, the shape of the steep-drop region close to the convergent GB is not fully reproduced. This is in agreement with our observation (Section III B) that the local cell-drift dynamics may severely depart from the V d (Pe) law of an ordered-array if the spacing gradients are large. Second, the prediction by the geometrical model is apparently less reliable for α 0 = 5 • . We can advance that a phase-diffusion process can predominate for a vanishing misorientation and induce spreading of the propagation front that cannot be fairly reproduced by the geometrical model.

We tested the prediction using a group velocity, i.e. V λ in Eq. ( 7) (see Section III C 2). For each PF simulation, we defined a median spacing λ p between the initial average spacing at t 0 (close to the lower plateau of a λ distribution) and λ M (Table III). The position of the spacing λ p propagates from an initial position y 0 at t 0 towards the sink, as schematically indicated by the black arrows in Fig. 5a. We followed this propagation as a function of time by linearly interpolating the positions y p that correspond to λ p in the λ(y) profiles at different times. We reported the quantity Y = |y p -y 0 | as a function of t -t 0 in the graphs of Fig. 5b. It appears clearly that the propagation is essentially linear, and that the TABLE III. Calculated V d and V λ at the median spacings λp in confined-3D array simulations. (second row), • (third row), and 20 • (fourth row) at V = 4 µm s -1 (PF simulations). In the graphs of (a), we reported the spacing profiles measured from the simulations (symbols), and calculated (solid lines) using the geometrical model (Section III C 1), with the PF distribution at time t0 as a initial condition. Different symbols (and color online) are corresponded to different times. Right (left) gray shaded areas: location of the source (sink) in the model. Arrows are used to schematically trace the position yp of the median spacing λp. In the graphs of (b), we reported the abscissa yp corresponding to λ = λp in the corresponding graphs of (a) as a function of time t -t0, using, for convenience, Y = |yp -y0| = 0 for t = t0 (circles). Slopes of the straight lines correspond to V λ (solid lines) and V d (dashed lines). See text for definitions.

V [µm s -1 ] α0 [ • ] λp [µm] V d [µm s -1 ] V λ [µm s -1 ]
spacing propagation velocity, i.e. the slope of a linear fit of the Y(t -t 0 ) data (black empty circles), compares well with the group velocity V λ calculated with Eq. ( 7), i.e. the slope of the red continuous lines. It can also be seen that V d (the slope of the blue dashed lines) is much larger than both V λ and the Y(t -t 0 ) slope. The calculated V d and V λ at a given spacing λ p for α 0 are listed in Table III. It is worth mentioning that the calculated V λ in Table III is always close to the intercept of a linearized V d (λ) (Section III C 2). We also performed PF simulations at higher V (12 and 20 µm s -1 ), and kept α 0 = 15 • as a constant (Fig. 6a). As in the above cases, λ M is substantially larger than the initial average spacing, and it propagates from the source. In addition, a steady-state spacing profile is reached with a large uniform λ M plateau. The region of the spacing drop near the sink is narrower as V increases. The quantitative agreement between the geometrical model and the PF simulations also seems all the better as V increases.

We again measured the position of the median spacing λ p (105 µm for V = 12 µm s -1 ; 85 µm for V = 20 µm s -1 ; see black arrows in Fig. 6a -and not by V d (Table III).

In practice, the value of V λ (for the chosen λ p values) and the ratio V λ /V d increases as V increases (Fig. 7). If one refers to the relation F (Pe) for α 0 = 15 • (Fig. 3), it should be noticed that α ≈ α 0 is reached for lower Pe values when V increases. Then, the derivative term in Eq. ( 7) becomes negligible (because dV d /dλ ∼ dα/dλ vanishes), which leads to V λ → V d . Accordingly, it is indeed expected that the propagation rate of the spacing approaches V d (V λ /V d → 1) when V increases.

In most PF simulations of tilted-cell patterns, a longlived steady-state was observed after the λ M plateau had formed. Overall, their main features were negligibly sensitive to the small-scale dynamics of the tertiary branching (amplification of and competition between sidebranches). However, an anomalous event at the source can entail a marked modification of the tiltedcell pattern on a large scale. In Fig. 8a, one can see the growth and the propagation of two cells with a spacing significantly smaller than λ M . The initial pattern (bot-tom of Fig. 8a) was in a steady-state situation (similar to that corresponding to the final time in the graph of Fig. 6a for α 0 = 15 • and V = 20 µm s -1 ). The simultaneous growth of two incipient sidebranches close to the divergent GB, and their transformation into two individual small-spacing cells is clearly seen (Fig. 8a). This induces a persistent depression in the spacing profile, which both travels laterally and smoothes out (Fig. 8b). This anomalous event did not occur additionally during the simulation, and a steady-state configuration was recovered. Again, the geometrical model predicts this complex behavior quite well as illustrated in Fig. 8b. We also followed the propagation of the depression in the spacing profile (with λ p = 105 µm). As shown in Fig. 8c, the corresponding Y data from t 1 = 1358 s closely align with the linear law of slope V λ (see Table III). In the PF simulations, this complex behavior was only occurred at this condition, i.e. relatively large α 0 and V .

E. Comparison of theoretical models and experiment

Fig. 9a shows a time series of snapshots of a tilted-cell array (α 0 = 15.5 ± 0.5 • ) during a DS experiment of the SCN-0.3mol% DC alloy. The image at t 0 ≈ 4000 s is taken just after the last event of a series of cell eliminations (horizontal red line on the bottom of the spatiotemporal diagram of Fig. 9b), which occurred during the early stages of the experiment. On average, the primary spacing λ evolved from about 130 µm (before cell eliminations) to 180 µm (plateau region). The cell-tilt angle was falling between about 8 and 12 • , and was thus substantially lower than α 0 , as expected. simplified spatiotemporal diagram corresponding to this experiment (such an image is built by recording a line normal to the growth direction at a fixed position behind the cell tips at regular time intervals, and piled up along an axis representing the time). It reveals that the cellular array is entirely "replaced" by cells created near the source more than twice over the solidification time. As in the PF simulations, new cells are regularly created by tertiary branching (see, e.g., the images at times t 5 and t 6 in Fig. 9a) at the divergent GB source, and eliminated at the convergent GB sink -one can note some occurrences of a surface dendrite along the sample edge at the convergent GB.

The evolution of the spacing profile from time t 0 to t 6 is shown in Fig. 10a. At t 0 , the λ distribution exhibits steep modulations, which are a trace of the cell eliminations during the initial transient. At this stage, the average spacing was relatively close to the value λ M delivered at the source -this feature contrasts with PF simulations, but is of minor importance as concerns the rest of the dynamics. As in the simulations, the initial modulations of the spacing profile simultaneously smooth out, and are advected laterally towards the convergent GB. Moreover, in the end of the experiment, the spacing profile presents a wide plateau with an essentially uniform spacing λ M , starting from the source and extending over the major part of the sample, and a narrow region close to the sink, within which the spacing drops.

For further analysis, we measured the dependence of F (Pe) in Eq. ( 3), as it could be extracted from the modulated patterns during the experiment of Fig. 9. If one refers to the PF simulations (Fig. 3), those data can be expected to be quite close to a F (Pe) law that would characterize perfectly periodic patterns, in spite of their large dispersion (note that producing periodic tilted-cell arrays with various, controlled values of λ is out of reach experimentally). We then used a best-fit procedure for adjusting F (Pe) in Eq. ( 3) with D/V = 67.5 µm to the experimental data. We obtained f = 0.558 and g = 1.17. The red fitting curve with these parameters in Fig. 10b can describe the experimental measurements (blue diamonds).

We followed the same procedure as above for implementing the geometrical model. We used the spacing profile at t 0 (Fig. 10a) as an initial condition, and set λ M = 175 µm and λ e = 54 µm. The agreement between the experimental data and the geometrical model is good, in spite of the significantly large uncertainties on the F (Pe) law, and the large amplitude of the initial λ modulations. The profile at the end of the experiment (green diamonds) is quite faithfully reproduced by the geometrical model. We also analysed the spacing propagation by following the abscissa Y corresponding to λ p = 163 µm (the black arrow in Fig. 10a). The results are shown in Fig. 10c. Here again, the measured Y(t-t 0 ) data (black diamonds) essentially follow a linear variation. In the present case, the slope of this variation is essentially close to the group velocity V λ = 0.3568 µm s -1 (the slope of red continuous line), within experimental accuracy. It is much lower than V d = 0.6664 µm s -1 (the slope of blue dashed line), as it could be expected from the analysis of the PF simulations.

IV. CONCLUSIONS

We have used thin-sample phase-field (PF) simulations and experiments to investigate the effect of a finite crystal misorientation on the selection of finger-shaped cellular/dendritic array patterns that form during directional solidification of binary alloys. Previous studies have characterized the relationship between the tilt angle α of the pattern and the spacing λ in a situation where λ is spatially uniform [28][29][30][31][32][33]. The present study focused on the more general situation where the array spacing is spatially non-uniform. For a misoriented monocrystal, nonuniformity originates from the creation or elimination of cells at the two lateral edges of the sample and can give rise to different spatial non-uniformities of spacing depending on whether tertiary branching at the divergent boundary is regular or irregular. Our main finding is that the spatial modulation of spacing propagates later-ally at a velocity that is generally slower than the lateral drift velocity V d = V tan α of cell/dendrite tips when λ is spatially uniform.

We have shown that this nontrivial behavior follows from the requirement that cells must move so as to conserve their total number away from boundaries. This conservation law yields a continuity equation for the linear density of cells along the array, which transforms into a nonlinear advection equation for the spacing with an advection velocity V d -λ dV d /dλ. The latter is generally less than V d since the tilt angles, and hence V d , are monotonously increasing functions of λ. The difference between the two velocities is more pronounced for cells with strong diffusive interactions (Pe = λV /D of order unity) than for well-developed dendrites (Pe 1) since the slope dV d /dλ is a decreasing function of Pe.

We have further found that, to a good approximation, V d depends linearly on λ over the limited range of variation of λ when the spacing is non-uniform. This implies that the advection velocity is nearly uniform even when the spacing is non-uniform, and thus that the spatial modulation of spacing is advected in a shape-preserving manner without forming shocks, which generically form when the advection velocity is a strongly varying function of the underlying field as in Burgers' equation [START_REF] Burgers | The Nonlinear Diffusion Equation[END_REF]. This is best exemplified by the propagation of a front separating regions of large and small uniform spacings when tertiary branching at the divergent boundary is regular. The front propagation thus selects a spacing that is closer to the largest stable spacing λ max , independently of the initial conditions. The selected spacing is generally significantly larger than the initial spacing selected by the transient growth competition between neighboring cells, which falls closer to the smallest stable spacing λ min [15,[START_REF] Losert | Proc. Natl. Acad. Sci[END_REF][20][21][22]25].

Front propagation is a relatively slow process, especially for small misorientation since V tan α 0 is an upper bound of the advection velocity. In the present simulations and experiments the time to establish a larger uniform spacing propagatively is at least an order of magnitude larger than the duration of the cell elimination transient leading to the establishment of the initial nonuniform spacing. Therefore, in the light of the present results, the spacing distribution is expected to strongly depend on the duration of the experiment even in the simplest case where tertiary branching at the divergent boundary creates cells with a constant spacing. When tertiary branching at the divergent boundary is irregular so as to create cells with highly variable spacings, the spacing variation can become undulatory. Undulations are advected towards the convergent boundary but also sporadically created at the divergent boundary, such that a steady-state uniform distribution of spacing may never be reached even at arbitrarily large time.

While the present study focused on a misoriented monocrystal, we expect the insights to be relevant for interpreting experimental observations in polycrystals with multiple grains terminated by divergent or convergent grain boundaries. Similarly to the sample edges, those boundaries can act as sources and sinks of cells, respectively. While tertiary branching has been found to depend in a non-trivial way on grain-boundary bicrystallography [30,40], propagation of spacing non-uniformities inside each grain should be governed by the same geometrical model and advection equation developed here for a misoriented monocrystal. The extension of those models to spatially extended three-dimensional patterns is an interesting future prospect. Considering a linear-response regime, the relaxation of a smoothly modulated, well-oriented cellular pattern can be described by a diffusion-like equation:

∂ t λ = D λ ∂ yy λ .
(A2)

Our fitting procedure above consists of solving this equation considering the lowest Fourier mode with a wave vector π/L. Accordingly, the time-dependent amplitude A λ (t) is given by:

A λ (t) = A 0 λ exp - π 2 D λ L 2 t , (A3)
where A 0 λ = 40.9 µm is the amplitude at t = 0 s. Then, D λ can be estimated using a best-fit adjustment to the A λ (t) data (black circles in Fig. A1c). This yields here D λ = 20.1 µm 2 s -1 . The effective diffusion time L 2 /D λ = 310000 s (L = 2.5 mm) is then at least one order of magnitude larger than the values of the time L/V λ over which a spacing modulation is transported from one edge to the other in tilted-cell patterns, even for small misorientation angles (see Table III). It may be noted that the phase diffusion coefficient in a tilted-cell array may differ from the present value of D λ , however a quantitative difference is not expected to be as large as questioning substantially our conclusion. In brief, we bring a clear evidence that phase diffusion is essentially negligible as far as the propagative dynamics of tilted-cell patterns is concerned.

FIG. 1 .

 1 FIG. 1. Tilted-cell arrays in thin-sample directional solidification of a misoriented crystal (misorientation angle α0) in a dilute succinonitrile-d,camphor alloy of concentration c∞. (a) Experimental observation (c∞ = 0.3 mol%; V = 4 µm s -1 ; G = 75 K cm -1 ; α0 ≈ 15.5 • ) of a horizontal dimension Ly = 2780 µm. (b) Phase-field numerical simulation (c∞ = 0.24 wt%; V = 4 µm s -1 ; G = 12 K cm -1 ; α0 = 15 • ) of Ly = 2490 µm. In these images, as well as in the following ones, the field of view covers the entire lateral width of the sample; the temperature gradient axis x is vertical, and the liquid is located on the top. The arrow in (a) describes drifting motion of the pattern along the lateral axis y. α and λ are for a cell-tilt angle and a primary spacing, respectively.

FIG. 2 .

 2 FIG. 2. Numerical simulation of a tilted-cell array in a SCN-0.24wt% camphor alloy (α0 = 15 • ; V = 4 µm s -1 ; G = 12 K cm -1 ). (a) Simplified spatiotemporal diagram: cell tip positions (circles) at regular time intervals (≈ 63 s) from t0 to the end of the simulation at tmax = 20000 s. Dashed horizontal lines and numbers are related to the solidified length V (t -t0). Bottom: profile of the cross-section of the solid-liquid interface in the median plane of the sample (z = H/2) at time t0. (b) Tilted-cell arrays at times t = 2072 s, 6032 s, and 10043 s (bottom to top) of the same simulation as in (a). (c) Spatial distribution of the primary spacing λ(y) for 9 successive times corresponded to different symbols. Dotted lines are guides to the eye (color online).

FIG. 3 .

 3 FIG.3. Ratio α/α0 between the cell-tilt angle an the misorientation angle as a function of the Péclet number Pe, measured from PF simulations for different α0 and V values. Red full squares and black solid lines: periodic arrays, and best-fitting curves using Eq. (3) (also see TableII), respectively. Blue empty circles: modulated arrays (see text).

1 FIG. 4 .

 14 FIG.[START_REF] Mullins | [END_REF]. Schematic tilted-cell array far away from any GB. λ: primary spacing. V d : lateral drift velocity. Red circles: cell tips. Within such a region of the pattern, the cell density is conserved.

FIG. 5 .

 5 FIG.5. Spatiotemporal evolution of the primary spacing λ and spacing-propagation dynamics for α0 = 5 • (first row), 10 • (second row), • (third row), and 20 • (fourth row) at V = 4 µm s -1 (PF simulations). In the graphs of (a), we reported the spacing profiles measured from the simulations (symbols), and calculated (solid lines) using the geometrical model (Section III C 1), with the PF distribution at time t0 as a initial condition. Different symbols (and color online) are corresponded to different times. Right (left) gray shaded areas: location of the source (sink) in the model. Arrows are used to schematically trace the position yp of the median spacing λp. In the graphs of (b), we reported the abscissa yp corresponding to λ = λp in the corresponding graphs of (a) as a function of time t -t0, using, for convenience, Y = |yp -y0| = 0 for t = t0 (circles). Slopes of the straight lines correspond to V λ (solid lines) and V d (dashed lines). See text for definitions.

FIG. 6 .FIG. 7 .

 67 FIG.6. Spatiotemporal evolution of the primary spacing λ (a) and spacing-propagation dynamics (b) for V = 12 µm s -1 (first row) and 20 µm s -1 (second row) α0 = 15 • . Same representations as in Fig.5. The spacing limits for the geometrical model are (λM , λe) = (140 µm, 27 µm) for V = 12 µm s -1 and (117 µm, 19 µm) for V = 20 µm s -1 . Color online.

Fig

  Fig.9ashows a time series of snapshots of a tilted-cell array (α 0 = 15.5 ± 0.5 • ) during a DS experiment of the SCN-0.3mol% DC alloy. The image at t 0 ≈ 4000 s is taken just after the last event of a series of cell eliminations (horizontal red line on the bottom of the spatiotemporal diagram of Fig.9b), which occurred during the early stages of the experiment. On average, the primary spacing λ evolved from about 130 µm (before cell eliminations) to 180 µm (plateau region). The cell-tilt angle was falling between about 8 and 12 • , and was thus substantially lower than α 0 , as expected. Fig.9b shows a

FIG. 9 .

 9 FIG. 9. Tilted-cell array during a directional-solidification experiment in a 100 µm thick sample of a SCN-0.3mol% DC alloy (V = 4 µm s -1 ; G = 75 ± 5 K cm -1 ). (a) Snapshots at times t0 ≈ 4000 s, t1 = t0 + 600 s, t2 = t0 + 1350 s, t3 = t0 + 2100 s, t4 = t0 + 2850 s, t5 = t0 + 3600 s, and t6 = t0 + 4500 s. (b) Simplified spatiotemporal diagram. Same horizontal dimension as in the snapshots of (a). Vertical dimension: ≈ 6300 s (25.2 mm) of solidification time (length). Horizontal line: time t0.

  FIG. 10. (a) Symbols: spacing distribution at times t0 to t6 during the experiment of Fig. 9. Solid lines: geometrical model. Gray shaded areas: same as in Fig. 5. (b) Symbols: ratio α/α0 (with α0 = 15.5 • ) as a function of Pe (measured during the same experiment). Solid line: best-fitting curve using Eq. (3) with f = 0.558 and g = 1.17. (c) Symbols: abscissa Y corresponding λp = 163 µm as a function of time t -t0. Solid (dashed) lines: slope V λ (V d ).

  FIG. A1. Relaxation of a non-uniform spacing array. We performed a PF simulation at V = 4 µm s -1 and G = 19 K cm -1 using a well-oriented grain (the upper image in a) to estimate a phase diffusivity D λ . Initial λ relax slowly towards the mean spacing λm = 155 µm over 8000 s (the lower image in a). The simulation measurements (symbols in b) show a cosine distribution Eq. (A1) (continuous lines) away from the boundaries (gray regions in b). We measure the time-dependent spacing amplitude A λ (t) every 1000 s (black circles in a), and the fitting curve (red line in c) of Eq. (A3) with D λ = 20.1 µm 2 s -1 agrees well with the simulation measurements.

TABLE I .

 I Material, control, and numerical parameters. Thermophysical parameters of SCN alloys are taken from Refs.[23, 35, 41-44, 46, 47]. For the PF simulations, we used a fixed W/∆x = 1.2 and other numerical parameters depend on V . * Experimental parameters (V = 4 µm s -1 ); see Section III E.

	Parameter	Symbol	Value	Unit
	Solute concentration	c∞	0.24	wt% camphor
			0.3 ± 0.05 *	mol% d-camphor
	Temperature gradient	G	12	K cm -1
			75 ± 5 *
	misorientation angle	α0	5, 10, 15, 20	degrees ( • )
			15.5 ± 0.5 *
	Pulling velocity	V	4, 12, 20	µm s -1
	Diffusion coefficient	D	270	µm 2 s -1
	Liquidus slope	m	-1.365	K wt% -1
	Partition coefficient	k	0.21	-
	Gibbs-Thomson coefficient	Γ	6.478 × 10 -8	K m
	Anisotropy strength	ε4	0.011	-
	Interface thickness	W	47.56, 28.26, 17.23	d0
	Grid spacing	∆x	3.00, 1.78, 1.09	µm
	Time step	∆t	5.00 × 10

-3

, 1.75 × 10 -3 , 6.55 × 10 -4 s

ACKNOWLEDGMENTS

We thank G. Faivre for fruitful discussions, and P. Ott for her help in the preparation of the experiments. This research was supported by National Aeronautics and Space Administration (NASA) grant NNX16AB54G and by the French Center for Spatial Research (CNES).

APPENDIX A: PHASE DIFFUSION

We performed an independent PF simulation using a well-oriented grain (the upper image of Fig. A1a) in order to obtain an estimate of the phase diffusion coefficient D λ , independently of any drifitng motion of the cells. For this simulation, we used the same material and numerical parameters as above for a directional-solidification run of a SCN-0.24wt% camphor alloy at V = 4 µm s -1 and G = 19 K cm -1 (see Table I). First, we carried out a PF simulation with α 0 = 10 • over t = 8000 s. As we observed previously in Fig. 5a, an array at t = 8000 s was spatially non-uniform on the scale of the simulation box. Using this misoriented and non-uniform array, we abruptly changed the crystal angle to α 0 = 0 • . Then, the entire cells in the misoriented array becomes welloriented approximately after 2000 s. Hence, we measured the spacing distribution from this point t = 0 s (the upper image of Fig. A1a) to 8000 s (the lower image of Fig. A1a). As illustrated in Fig. A1a-b, the spacing distribution slowly relaxed towards uniformity, keeping the mean spacing λ m constant.

Fig. A1b shows the spacing distribution at t = 0 s (blue squares) and t = 8000 s (red circles). Except close to the edges, that is, for a system size L = 2200 µm comprised between y 0 = 100 µm and y 1 = 2300 µm (meaning that the gray regions in Fig. A1b are excluded from the analysis), the spacing distribution can be well fitted by a cosine function of the form:

where A λ (t) is a time-dependent spacing amplitude, while λ m , y 0 , and a phase shift φ 0 remain fixed over time.

We estimated λ m = 155 µm and φ 0 = 3.4. This being defined, we computed A λ (t) every 1000 s from t = 0 s to t = 8000 s (black circles in Fig. A1c).


Sabine Bottin-Rousseau

Oriane Senninger

Gabriel Faivre

Silvère Akamatsu

Special interphase orientation relationships and locked lamellar growth patterns in thin In-In 2 Bi eutectics

Keywords:

Lamellar eutectic alloys have two main modes of growth (floating or locked) depending on the orientation of the two eutectic phases. According to recent theoretical studies [Ghosh et al, Phys. Rev. E 91, 022407 (2015)], the key factor is the existence or otherwise of a sufficiently deep minimum in the surface energy plot of the interphase boundaries. We present an experimental study aiming at testing this prediction. We used a near-eutectic In-In2Bi alloy that solidifies into a tetragonal phase (In-11.5 at%Bi) and a hexagonal phase δ (In2Bi). Thin samples of this alloy were solidified using a rotating directional solidification stage and analyzed for crystallographic texture by X-ray diffraction. A series of samples with low (1 to 3 o ) mosaic spreads were found to contain either of the following coincidence relationships: A = {2 11 0} δ //{111} or E = {1 102} δ //{111} . The A-type samples exhibited a locked mode of growth, and the E-type samples a floating mode of growth. Given that, according to standard crystallographic criteria, A-type interphase boundaries have a much smaller surface energy than E-type ones, these findings are in agreement with the theoretical predictions.

I. INTRODUCTION

Directionally solidified lamellar eutectic alloys often grow with special interphase orientation relationships, the term special meaning that a coincidence relationship between some low-index lattice planes of the two eutectic phases is achieved [1][2][3][4][5]. The consequences of this fact on the growth dynamics of lamellar eutectics are rarely mentioned. Yet, an interphase boundary lying parallel to two coincident low-index lattice planes must have a particularly low energy [START_REF] Sutton | Interfaces in Crystalline Materials[END_REF]. Consequently, the surfaceenergy plot of the interphase boundary (γ B -plot) in a eutectic grain (EG) -i.e. a region of uniform orientation of the two eutectic phases -with a special interphase orientation relationship must have a deep minimum in the direction of the coincident lattice planes and, more globally, be strongly anisotropic.

There has been much recent progress on the issue of the growth dynamics of lamellar eutectics with strongly anisotropic γ B -plots, at least in thin samples. It was shown experimentally that thin eutectic grains should be classified into three main categories according to the nature (floating, locked or nearly locked) of the growth dynamics they display [8]. A floating EG is recognizable by two main features: eutectic lamellae grow parallel to the thermal gradient; the interlamellar spacing undergoes a continuous process of homogenization called spacing-diffusion [START_REF] Akamatsu | [END_REF]. Conversely, in a locked EG, eutectic lamellae run parallel to some fixed crystallographic direction even if it is tilted to a large angle from the thermal gradient axis, and the interlamellar spacing distribution does not change appreciably over time even if it is strongly non-uniform [8]. The intermediate nearly locked category comprises all the EGs, in which eutec-tic lamellae are tilted to a noticeable angle with respect to the thermal gradient but nevertheless undergo an efficient spacing-diffusion process. Theoretical and numerical studies showed that these different dynamics are related to different degrees of anisotropy of the interphase boundary -the anisotropy of the solid-liquid interfaces being assumed to be negligible in lamellar eutectics [9,10]. In brief, according to these studies, floating EGs have nearly isotropic γ B -plots, locked EGs have γ Bplots with deep minima in directions that corresponds to locked lamellar planes, and nearly locked EGs have γ Bplots with relatively shallow minima that attract, but do not lock, the lamellar planes. Floating and nearly locked EGs will be jointly referred to as "floating" henceforth.

These theoretical views are self-consistent but have not yet been tested by means of a simultaneous study of interphase orientation relationships and lamellar growth dynamics. In this article, we present a real-time observation study of lamellar patterns in thin-sample directional solidification of a near-eutectic In-In 2 Bi alloy coupled with crystal orientation measurements by X-ray diffraction. The In-In 2 Bi alloy is used here as a convenient and wellcharacterized low-melting point model eutectic alloy [START_REF] Akamatsu | [END_REF]. The use of a rotating directional solidification stage allows a grain-by-grain study of the γ B -plot. The results demonstrate a strong correlation between special interphase orientation relationships and anisotropy effects on lamellar eutectic growth, in agreement with the theoretical predictions. They also provide information about the preferred interphase orientation relationships of the In-In 2 Bi alloy and their mechanism of formation.

II. METHODS

A. Alloy and samples

Details about the alloy and the experimental methods used during this study can be found elsewhere [START_REF] Akamatsu | [END_REF]8]. We will limit ourselves to a few points. The In-Bi phase diagram has a eutectic point at 72.7 o and 22.5 at%Bi [START_REF] Massalsky | Binary Alloys Phase Diagram[END_REF]. The eutectic solid phases are a tetragonal solid solution (In-11.5 at%Bi) and the hexagonal intermetallic compound δ (In 2 Bi). The phase decomposes eutectoidally into δ and a tetragonal terminal solid solution (In-8at%Bi) at about 49 o .

The and (In) phases have most often been described as body centered tetragonal with two atoms per unit cell, but may also be described as face centered tetragonal (fct) with four atoms per unit cell [START_REF] Becker | [END_REF]13]. The latter option, which allows and (In) to be viewed as slightly deformed face centered cubic crystals, is adopted here. The δ phase has a hexagonal Ni 2 In structure with six atoms per unit cell. The Miller notation of its lattice directions is adopted here. A slightly hypereutectic In-In 2 Bi alloy was used. This alloy forms regular -δ lamellar patterns with a volume phase fraction of about 0.5 during directional solidification. Figure 1a describes the directionalsolidification samples and coordinate system used here. The alloy is enclosed between two 0.3 mm-thick glass plates separated by a 13 µm-thick funnel-shaped spacer delimiting a 0.1 mm-wide grain selector at one end of the sample and a 30 mm-wide space outside the selector. The molten alloy is introduced into this container by a vacuum suction method and then solidified by rapid cooling to room temperature (primary solidification). The sample is then inserted into the directional solidification setup and placed under a undirectional thermal gradient (0.9 Kmm -1 ) in such a position that the 72.7 • C isotherm, and thus the solid-liquid interface, are located inside the selector. The part of the sample that remains solid serves as seed for the forthcoming solidification process. During this process, the growth front and the solid behind it are observed through the upper glass lid of the sample using reflected-light microscopy. A scanning of the growth front over the whole sample width is performed at regular intervals. Images are recorded with a monochrome digital camera, and transferred to a computer for further analysis.

B. Grain selection process

The first and critical stage of a solidification run consists of growing a single eutectic grain that exhibits a well-characterized (floating or locked) eutectic growth dynamics. A description of the method used for this purpose can be found elsewhere [START_REF] Akamatsu | [END_REF]8,14]. We limit ourselves to the following points. The grain selection process is conducted in such a way that one EG overgrows the seed and fills the whole sample width. The monocrystallinity of the emerging EG as well as the nature of its growth dynamics can be qualitatively assessed in real time, allowing one to decide immediately on the continuation or reset of the solidification run. A quantitative assessment of crystal quality performed by XRD diffraction at the end of the solidification run leads to the eventual recruitment or rejection of the sample. Five locked samples (1 to 5) and four floating ones (6 to 9) were used during this study. Unexpectedly, some samples (1 to 3) contained two interrelated EGs instead of one, with interesting consequences. We will return to this point later on. The nine selected samples turned out to contain only two types of special EGs, as will be seen below. This does not mean that no other types of EGs were present in the seed, but only that, if there were, they were eliminated during the grain selection process.

C. Rotating directional solidification

The solidification setup used during this study allows one to switch between standard directional solidifica-tion (DS) and rotating directional solidification (RDS) -i.e. between translation and rotation with respect to the thermal gradient (Fig. 1). RDS-runs are usually pursued over several complete revolutions of the sample. Although the grown microstructure is destroyed by melting after a half turn, an uninterrupted recording of the growth front allows one to perform retroactively a reconstruction, or space-time map, of the whole solidification process. It has been shown that the trajectories of the interphase boundaries generated during RDS are approximately homothetic to the 2D Wulff form of the interphase boundary [9,10]. In particular, any straight segment of a RDS lamellar trajectory -or, equivalently, any locked lamellar plane -corresponds to a facet of the 2D Wulff form and thus to a deep minimum in the γ Bplot.

During this study, apart from the initial eutectic-grain selection process, solidification runs were performed first by RDS, and then by DS. These two stages were separated by a short rest stage, during which the majority phase (δ) invaded the growth front. The space-time maps are therefore composed of two main parts: a lower part representing a π-RDS map; an upper part representing a conventional DS recording. An example is given in Section III A.

D. X-ray diffraction analysis

The orientation of the crystals in the xyz coordinate system was determined by X-ray diffraction analysis in situ, i.e. without removing the glass lids from the sample and without letting the sample cool below 60 o . The pole figure method for texture analysis was used in order to get information on crystal mosaicity across the whole solidified area. To allow the use of conventional X-ray diffractometers, the upper glass lid was thinned with fluorhydric acid, before solidification, over an area covering the region of interest. X-ray measurements were performed in a Rigaku Smartlab 9kW diffractometer equipped with a home-made thermostatic chamber set at 60 o . Sample transfer from the solidification bench to this thermostatic chamber were without any cooling of the region of interest below 60 o . Constraints linked to the geometry of the samples prevented the detection of XRD peaks with an angle to the y axis (or polar angle) larger than about 63 o , without negative consequences. The scanning pitch of the pole figures was of 1 o for both the polar angle and the azimuth. The orientations of the different crystals present in a sample were calculated from the poles figure by standard methods. Details about the uncertainties resulting from the finite width of the XRD peaks are given in Section III A.

E. Identification of special interphase orientation relationships

A special interphase orientation relationship (OR) is defined by two mutually parallel low-index lattice planes -called coincident planes (CPs) -and two mutually parallel low-index lattice directions (CDs) belonging to the CPs. This set of CPs and CDs also defines an epitaxial interphase boundary, whose surface energy γ EB characterizes the OR. As is common practice, we evaluated γ EB using two criteria, namely, the atomic density and the misfit of the CPs. The misfit can be defined as the relative difference in atomic density of the CPs or, in a more refined way, as the deformation required to form a dense coincidence site lattice (CSL) along the interphase boundary [START_REF] Sutton | Interfaces in Crystalline Materials[END_REF]. It should be noted that low-energy heterophase boundaries with misfits as high as 15% were observed in epitaxial and eutectic growth [17,18].

The atomic density of a lattice direction can be defined as the inverse of the shortest distance between neighbor atoms along this direction. The shortest vectors between neighbors are 1 2 < 101 > and 1 2 < 110 > in the phase, and 1 2 [001] δ , 1 3 [210] δ and 1 12 < 843 > δ in the δ phase. The atomic densities of these lattice directions are close to one another. The atomic density of a lattice plane can be defined as ρ = n 2D d/Ω, where n 2D is the number of atoms per unit cell of the 2D lattice of the plane, d is the interplanar spacing and Ω is the volume of the 3D unit cell. For those lattice planes which contain several layers with different densities, the densest layer should be chosen. The densest lattice planes of are {111} and those of δ are {2 11 0} δ . For convenience, we shall express the atomic densities of the different lattice planes in units of the atomic density of {2 11 0} δ (ρ m ≈ 9.57nm -2 ).

The method for the comparison of experimental data and theoretical ORs uses lists of equivalent lattice planes and directions of δ and sorted in decreasing order of atomic density (Table II). The method includes the following steps: (i) determination of the orientations of the crystals composing the EG under study; (ii) calculation of the misorientation angles between all the possible combinations of distinct δ and lattice planes in this EG; (iii) elimination of the combinations, for which the misorientation angle is larger than a predefined limit value (generally 2 o ); (iv) application of the same procedure to the lattice directions; (v) determination of a list of OR candidates combining the selected CP candidates with the CD candidates that belong to them; (vi) determination of the best OR candidate according to the misorientation angles of the CPs and CDs and the evaluation of γ EB ; (vii) characterization of the global orientation of the EG with respect to the sample by the angle ψ CP between the CPs of the best OR candidate and the yz plane and the angle θ δ between (0001) δ and the xz plane. The misorientation angles of the CPs are denoted by ∆ CP and those of the CDs by ∆ CD hereafter.

F. Nomenclature of special orientation relationships

We have built a nomenclature of the ORs of In-In 2 Bi on the model of the nomenclature currently used for Al-Al 2 Cu [5,[START_REF] Bonnet | Conference on in situ Composites; P209[END_REF][START_REF] Kokotin | [END_REF]. The following rules will apply:

(i) Sets of equivalent CPs are designated by capital letters, as in the following examples:

A = {2 11 0} δ //{111} , (1)
B = {2 11 0} δ //{100} . (2)
(ii) Sets of equivalent CDs are designated by numbers. Sets of equivalent ORs are thus denoted by dual symbols, as in the following examples:

A1 = {2 11 0} δ //{111} [001] δ // < 101 > , (3)
A2 = {2 11 0} δ //{111} [001] δ // < 1 10 > . (4)
(iii) Subscripts are used to distinguish between distinct ORs belonging to the same set of equivalent ORs.

The following examples will be used below:

A1 1 = (2 11 0) δ //(111) [001] δ //[01 1] ; (5)
A1 2 = ( 12
A3 1 = {2 11 0} δ //{111} [483] δ //[10 1] . (6)
A few additional points of terminology are: (i) A1 1 has two variants related to each other by a πrotation of the crystal around the axis normal to the CPs (π-variants). (ii) One can switch from A1 and A2 trough a rotation by ≈58.5 o of the crystal around the normal to the CPs ("in-plane" rotation). This remark can be generalized to any set of ORs sharing the same CPs. (iii) Because of the similarity in local structure of the two phases, there exist sets of interrelated ORs at small angular distances from each other ("neighbor" ORs).

III. RESULTS

A. Grain structure of the samples

Figure 2 shows the space-time map of the solidification process undergone by Sample 1. This map reveals a fully locked growth dynamics, which will be studied in Section III D. Figure 3a shows the corresponding XRD pole figures. They reveal that Sample 1 contained a single δ crystal and two crystals ( 1 and 2 ). In other words, it contained two eutectic grains, one (EG 1 ) formed by δ and 1 and the other (EG 2 ) by δ and 2 . The boundary between EG 1 and EG 2 that is shown in Fig. 2 was traced in real time during solidification. The correspondence between EGs and XRD peaks was determined using Xray-opaque masks. It is also apparent from Fig. 3 that the orientation of the crystals with respect to the sample is a singular one; however, this feature -as well as the presence of two EGs instead of one -is specific for Samples 1, 2 and 3, and will not be commented on until Section III E.

Figure 4 shows the substructures of two representative XRD peaks from Sample 1. These substructures contain a narrow high-intensity central peak and a broader low-intensity "base". The half widths of the two components are of about 1 o and 2 o , respectively. The same type of bimodal substructure was observed in all the samples studied. Five good-quality samples had the same peak and base widths as Sample 1. In the other four samples, described as medium quality herefater, the half width of

B. A-type orientation relationships

The main features of the five samples that exhibited a locked growth dynamics are displayed in Table III. It is striking that all the EGs of these samples had either A1 or A2 as best OR candidate. Let us explain this point in detail in the example of Sample 1 (Fig. 5). A1 1 , as defined in Eq. ( 5), obviously was a good OR A1 or A2. Coincident site lattices were obtained by superposing a deformed (111) plane and an undeformed (2 11 0) δ plane. For A1 interface boundaries, a CSL with a coinciding-site density of about 1/2 (in ρ m units) was obtained by means of a deformation of principal components e I ≈ 0.073 and e II ≈ -0.018 (Fig. 6b). For A2 interface boundaries, a CSL with a coinciding-site density of about 1/3 was obtained through a deformation of principal components e I ≈ 0.006 and e II ≈ -0.055 (Fig. 6c). Thus, A1 and A2 both have a low surface energy, according to generally accepted crystallographic criteria. They are jointly referred to as A-type ORs henceforth.

Sample EG OR ∆ CP ∆ CD θ δ ψ CP [deg] [deg] [deg] [deg] 1 1 A1 0.
The data of Table III can then be summed up as follows: all the locked EGs were A-type and all the A-type EGs were locked.

C. E-type orientation relationships

In all the samples that exhibited a floating growth dynamics, the best CP candidates were

E = {1 102} δ //{111} , (8)
but the best CD candidates varied from sample to sample (Table V). In each sample, the selected E-type OR was part of a set of neighbor ORs. The selection was performed using the same criteria (misorientation angles, surface energy) as above. A representative example (Sample 9) is given in Fig. 8 and Table VI. The relevance of E regarding the surface energy criterion is not immediately obvious since {1 102} δ has a low atomic density compared to {111} . However, the 2D lattice of {1 102} δ contains a pseudo-hexagonal motif very similar to the motif formed by the second-nearest neiborghs in {111} , in accordance with the fact that the density ratio of these planes is close to 1/3 (Fig. 7a). As shown in this figure, the {1 102} δ planes contain layers of composition InBi, which may contribute to lowering the energy of E-type interphase boundaries. In the example of Sample 9, a moderate (e I ≈ 0.08 and e II ≈ -0.10) deformation of the 2-D lattice of (111) would suffice to produce a CSL including all the sites of the (1 102) δ plane for the E3 1 interphase boundary (Fig. 7b). Incidentally, this is also true of the F 1 1 interphase boundary, since the density ratio of the CPs of this boundary is close to 1/2.

Sample OR ∆ CP CDs ∆ CD θ δ ψ CP [deg] [deg] [deg] [deg] 6 E1 0.5 < 021 > δ // <

D. Locked lamellar planes

E-type samples showed quasi-circular or oval RDS lamellar trajectories, indicative of a near-isotropic or mildly anisotropic floating growth dynamics [8]. By con-Black: !. White: ". VI).

trast, A-type samples displayed polygonal-shaped RDS lamellar trajectories characteristic of a strongly locked growth dynamics (Fig. 2). Each rectilinear segment of an RDS trajectory is the intersection of a particular locked lamellar plane (LP) and the sample plane. Knowing that LPs may not be perpendicular to the sample plane [8], we characterized their orientation by two angles, namely, the tilt angles with respect to z of their trace on the xz plane (longitudinal tilt angles) and the yz plane (transverse tilt angles). Longitudinal tilt angles were measured in spacetime maps with a margin of error of ±1 o . Transverse tilt angles were measured in metallographic transverse crosssections in some samples and were found to be smaller than the experimental uncertainty (≈5 o ), except in Sample 5. Longitudinal tilt angles will be simply called tilt angles henceforth.

As shown in Fig. 9, each of the two EGs of Sample 1 contained three different LPs called LP 1 , LP 2 and LP 3 , which lay a few degrees from (2 11 0) δ , (11 20) δ and ( 12 10) δ -and thus from the CPs of A1 1 , B1 1 and A1 2 , respectively (see Table IV). The angle between LP 1 and (2 11 0) δ was of about 3 o in EG 1 and 6 o in EG 2 . In each EG, let us take the trace of (2 11 0) δ instead of z as origin of the tilt angles. As shown in Figure 10, the tilt angles thus measured were close to 0 o (LP 1 ), 60 o (LP 2 ) or 120 o (LP 3 ) in Samples 1 to 4. In Sample 5, the following facts were found: only two locked planes (LP 2 and LP 3 ) were observed; these planes had large transverse tilt angles and lay close to the [120] δ and [1 10] δ zones, respectively; the floating lamellae observed in the vicinity of 0 o also had a relatively large transverse tilt angle (Fig. 11a). Given that all the A-type EGs had the same γ B -plot except that the global orientation of it changed from EG to EG, the above observations suggest that the γ B -plot of A-type EGs contains three deep valleys, which intersect the (0001) δ plane a few degrees from the {2 11 0} δ poles and change from deep to shallow at some distance from this plane (Fig. 11b).

E. Twin crystals and the -(In) transition in

A-type samples

Let us consider the orientation relationships between the two crystals of the two-grained samples, and between these crystals and the coordinate system of the sample. Within experimental error, the main findings are the following:

(i) The two crystals of Sample 1 were twin crystals related to each other by a π-rotation about a < 011 > axis, which was nearly perpendicular to the sample plane (Fig. 5b). The ORs of the two EGs were thus π-variants of A1 1 ;

(ii) Sample 2 had the same two-grained structure as Sample 1 with only a slightly different orientation of the δ crystal with respect to the sample plane (Table IV).

(iii) In Sample 3, EG 1 had a A1 OR on a {2 11 0} δ plane; EG 2 had a A2 OR on another {2 11 0} δ plane; 1 and 2 were twin crystals related to each other by a πrotation about a < 211 > axis (Fig. 12). The twin axis was nearly perpendicular to the sample plane.

These observations strongly suggest that A-type ORs were created in the nucleation stage by epitaxial nucleation of one phase onto the other. Since, after their formation by nucleation, the crystals were cooled to room temperature and then reheated up to the eutectic temperature, a preliminary requirement for this assumption Black: !. White: " 1 . to be justified is that the orientation of the crystals was preserved through the eutectoid transformation at 49 o . To investigate this matter, we cycled some directionally solidified samples between 60 o and 30 o in the RX diffractometer and collected XRD pole figures at these temperatures. The results obtained with Sample 2 are shown in Figure 13. It can be seen that the -(In) transition consisted of a deformation of the unit cell through which the 4-fold axis of the phase transformed into a 2-fold axis of the (In) phase and vice versa, while the orientation of the axes was approximately preserved. As there are two 2-fold axes and a single 4-fold axis per crystal, one crystal might give rise, upon cooling, to two (In) variants, which in turn might give rise to three variants upon reheating. This was not observed in A-type samples. In fact, no changes at all were observed in the pole figures of the phases between the beginning and end of a thermal cycle, even at the sub-structure level (Fig. 13c). The orientation and substructure of the (single) δ crystal contained in A-type samples were also perfectly preserved during the eutectoid transition.

IV. DISCUSSION

This study has clearly demonstrated a strong correlation between special interphase orientation relationship and locked lamellar growth, in agreement with theoretical predictions. However, it has also revealed several unexpected facts, which need to be discussed. A first point is the relatively large (3 to 6 o ) angular deviation found between the locked lamellar plane LP 1 and the CPs in A-type EGs. According to recent theoretical and numerical studies, the normal to a locked lamellar plane points is unlikely to lie at more than 1 o from the direction of a minimum of the γ B -plot [9,10]. It must therefore be admitted that the minimum of the γ B -plot corresponding to LP 1 lies several degrees from the normal to the CPs in A-type EGs. In fact, a similar assumption about the other two locked lamellar planes of A-type EGs has been made above: LP 3 , for example, was assumed to lie somewhere between a ( 12 10) δ and a ( 111) plane making an angle of about 8 o with each other. Speculations on the structure of these interphase boundaries on the atomic scale would be useless. We can simply note that the CPs of A-type EGs are not parallel to a plane of symmetry of the γ B -plot and that, therefore, there is no obvious reason for them to coincide with a minimum of the γ B -plot.

Another interesting question raised by our findings is the formation mechanism of A-type ORs. We first note that the small (1 to 3 o ) orientation scatter found in these samples discards any hypothesis based on a slow rearrangement of the crystal orientations during growth [3]. In our case, the best assumption seems to be that A-type ORs were created in the nucleation stage and preserved without substantial change during the ensuing growth process. As shown above, this would be compatible with the -(In) transitions undergone by the crystals before the onset of directional solidification. Concerning the nucleation process itself, two scenarios are admissible, given the experimental uncertainties: either δ nucleated first from the liquid and the two crystals nucleated simultaneously on the same (in Sample 1 and 2) or different (in Sample 3) {2 11 0} planes of the δ nucleus, or nucleated first in the form of a twin crystal. In any case, an interaction between the nucleus and the sample walls must have taken place since the twin axis of the crystal were found to be nearly perpendicular to the sample plane.

V. CONCLUSION

Two different types of special interphase orientation relationships, namely,

A = {2 11 0} δ //{111} (9)
and

E = {1 102} δ //{111} , (10)
have been identified in thin In-In 2 Bi samples. All the Atype eutectic grains exhibited a strongly locked lamellar growth, while all the E-type ones displayed a more or less anisotropic floating lamellar growth. As shown above, the surface energy of A-type interphase boundaries is substantially smaller than that of E-type ones, according to well-established crystallographic criteria. Thus the results of this study unequivocally validate the assumption of a link between locked lamellar growth and epitaxial interphase boundaries that underlies current theories of anisotropy effects in lamellar eutectic growth [9,10]. As a final remark, we wish to stress that the good agreement between experimental and theoretical results is at present limited to 2-D lamellar eutectics. This study has indeed shown that even a thin lamellar eutectics can cease to be a 2-D system under the influence of a strong anisotropy of the surface energy of the interphase boundary. Future studies should therefore focus on anisotropy effects related to interphase boundaries in bulk lamellar eutectics.

FIG. 1 .

 1 FIG. 1. Solidification processes. The thermal gradient is directed along the z axis. The sample plane lies parallel to the xz plane. (a) Standard directional solidification. The sample is being pulled in the -z direction. g.s.p.: grain selection process. (b) Rotating directional solidification. The sample is being rotated anticlockwise about the y axis. Arrows: directions of motion with respect to the liquid.

  10) δ //( 111) [001] δ //[01 1] ;

FIG. 2 .

 2 FIG. 2. Sample 1.Lower portion of the space-time map of the solidification process. The darker phase is δ (In2Bi) and the lighter one is . The alloy is slightly hypereutectic. RDS: rotating directional solidification at 2.5 × 10 -4 rad s -1 . DS: standard directional solidification at 0.5 µm s -1 . Only a portion of the total DS length (≈20mm) is represented. EG1, EG2: eutectic grains. Dash line: boundary between EG1 and EG2. Numbers: tilt angles of the lamellar trajectories with respect to z in various locked areas. CR: region near the center of rotation.

2 -FIG. 4 .

 24 FIG. 4. Sample 1. Zoomed-in views of the XRD peaks marked by arrows in Fig 3. The sub-peaks are represented by circles of diameters proportional to their intensity.

FIG. 6 . 2 -

 62 FIG. 6. 2-D lattices of (2 11 0) δ and (111) superimposed. Cross: origin of coordinates. White arrow: 1 2 [ 110] . Black arrow: 1 12 [483] δ . Full vs open black circles : In vs Bi atoms. (a) A2 orientation relationship. Dotted line: unit cell of (2 11 0) δ . (b) and (c) A2 and A1 orientation relationships with a slight deformation of the (111) plane (see the text). Dotted lines: unit cell of the CSLs.

FIG. 8 .

 8 FIG. 7. 2-D lattices of (1 102) δ and (111) superimposed. Cross: origin of coordinates. White arrow: 1 2 [101] . Black arrow: [110] δ . Full vs open black circles : In vs Bi atoms. (a) Symmetrical orientation relationship (not observed). (b) E3 orientation relationship with a slight deformation of the (111) plane (see the text). Dotted line: unit cells of the CSL.

FIG. 9 .

 9 FIG. 9. Sample 1. Stereographic projections showing the coincident lattice planes of A11, B11 and A12 (see TableIV) in in EG1 (top) and EG2 (bottom). LP1, LP2 and LP3: locked lamellar planes.

3 FIG. 10 .

 310 FIG. 9. Sample 1. Stereographic projections showing the coincident lattice planes of A11, B11 and A12 (see TableIV) in in EG1 (top) and EG2 (bottom). LP1, LP2 and LP3: locked lamellar planes.

FIG. 11 .

 11 FIG. 11. Stereographic projections. (a) Sample 5. Coincident lattice planes and directions, and locked lamellar planes (LP1 and LP2). FL: floating lamellar planes. (b) Sketch of the valleys of the γB-plot of A1 EGs in the coordinate system of the δ phase. Continuous lines: deep valleys. Dotted lines: shallow valleys.

FIG. 12 .

 12 FIG. 12. Sample 3. Stereographic projections showing the existing orientation relationships between the δ and crystals in EG1 (top) and EG2 (middle), and between the two crystals (bottom).

  Table I gives the lattice parameters of the different phases as measured from our eutectic alloy by X-ray powder diffraction.

	Phase	a	c	c/a
		[ Å] [ Å]
	δ	5.496 6.585 1.198
		4.914 4.490 0.914
	(In) 4.600 5.019 1.091

TABLE I .

 I Lattice parameters of the eutectic and eutectoid phases (this study). A fct unit cell is used for and (In).

TABLE II .

 II First terms of the lists of equivalent lattice planes of the eutectic phases sorted in decreasing order of atomic density. ρ: atomic density in units of the atomic density of {2 11 0} δ .

	δ phase n2D ρ
	{2 11 0} 6	1
	{0001} 2 0.799
	{10 10} 2 0.577
	{2 11 4} 6 0.514
	{2 11 2} 4 0.512
	{30 34} 6 0.416
	{30 32} 4 0.347
	{10 12} 2 0.328
	phase n2D ρ
	{111} 4 1.059
	{100} 2 0.947
	{001} 2 0.865
	{110} 2 0.670
	{101} 2 0.639
	{311} 4 0.566
	{113} 4 0.530
	{331} 4 0.432
	{210} 2 0.424
	{313} 4 0.415

  A1 1 by an in-plane rotation of ≈0.3 o and need not be distinguished from it. The data of Table IV are clearly in favor of A1 1 . Moreover, Figure6ashows that the

			z
			(0002) "
					x
			δ-phase
		White: (0002) δ . Black: {2020} δ"
			z
			(13 1)
		€			x
			ε-phase
			White: {200} ε . Black: {311} ε"
	FIG. 3. Sample 1. Pole figures. Open circles : location of
	non-detected peaks (polar angle > 63 o ).
							6 1.4 3.0	3
							2 A1 1.4 0.9	"	"
		OR	CPs	ρ δ misfit ∆ CP 1	∆ CP 2	2	1 A1 0.5 2.0 8.0	7
		D11 ( 1010) δ //(311) 0.58 0.020 0.5 1.3		2 A1 0.3 1.8	"	"
		A11 (2 11 0) δ //(111)	1 0.057 0.6 1.4	3 *	1 A1 1.0 4.8 25	3
		B11 (11 20) δ //(100)	1 0.054 4.2 2.9		2 A2 3.6 3.5	"	30
		C11 ( 1100) δ //(011) 0.58 0.101 4.3 2.8	4 *	1 A2 2.1 2.7 27	27
		D12 (01 10) δ //( 311) 0.58 0.020 8.0 6.5	5 *	1 A1 2.6 3.3 41	36
		A12 ( 12 10) δ //( 111)	1 0.057 8.2 6.7
						TABLE III. Samples exhibiting a locked growth dynamics.
	TABLE IV. Sample 1. Neighbor ORs belonging to the [001] δ //	Asterisk: medium-quality sample. EG: eutectic grain(s). OR:
	[101] zone. ρ δ : atomic density of lattice planes of the δ phase.	best OR candidate.
	∆ CP 1	and ∆ CP 2	: misorientation angles of the CPs in EG1 and
	EG2, respectively.

candidate for both the EGs of this sample, but the calculations yielded a large number of other potentially significant neighbor ORs. Those of them which belong to the [001] δ //[101] zone are shown in Table

IV

. Incidentally, A3 1 , as defined in Eq. (

7

), is only separated from

TABLE V

 V

					z
				[110]
				[4 8 3]
					[011] [001]	x
		!	(111) (21 1 0)	[101] [483]
	!		Black: !. White: " 1 . !
					z
				[101]
				[4 8 3]
					[011] [001]	x
		!	(111) (21 1 0)	[110] [483]
	!				!
					z
					[011]
		[010] [001]	[01 1 ]	[00 1 ] [010]	x
			!	[100] ! [100]
			Black: " 1 . White: " 2 .
	FIG. 5. Sample 1. Stereographic projections showing the ex-
	isting orientation relationships between the δ and crystals in
	EG1 (top) and EG2 (middle), and between the two crystals
	(bottom).
	OR	CPs	ρ δ /ρ ∆ CP	CDs	∆ CD
	[deg] E31 (1 102) δ //(111) 0.31 1.0 [ 201] δ //[0 11] 1.8 [deg] F 11 (10 12) δ //(011) 0.51 1.9 [ 201] δ //[0 11] 1.8 (1 210) δ //(31 1) 1.8 2.1 [210] δ //[1 30] 3.5 ( 11 24)	7 8 9 *	2 31 > 5.0 E2 0.4 < 021 > δ // < 21 3 > 0.2 20 5 E1 0.9 < 021 > δ // < 2 31 > 0.9 40 E3 1.0 < 021 > δ // < 0 11 > 1.8 53	59 55 25 5

. Samples exhibiting a floating growth dynamics. Asterisk: medium-quality sample. OR: best OR candidate. Black: !. White: " 2 . δ //(3 13) 1.2 3.6 [ 110] δ //[ 23 1] 5.4 TABLE VI. Sample 9. Main neighbor ORs. ρ δ and ρ : atomic densities of lattice planes in the δ and phases, respectively.

Acknowledgments

We thank Richard Soucek for his ongoing technical help. We are greatly indebted to Gwenaëlle Rousse for her help in performing and analyzing X-ray powder diffraction patterns and Sarah Hidki for performing the X-ray pole figure measurements. We warmly thank Ulrike Hecht for many enlightening discussions and for performing EBSD measurements. This work was financially supported by ANPHASES M-ERA.NET project (ANR-14-MERA-004).