F L Mota

K Ji

L Strutzenberg Littles

R Trivedi

A Karma

N Bergeon

Influence of macroscopic interface curvature on dendritic patterns during directional solidification of bulk samples: Experimental and phase-field studies

Keywords: directional solidification, microgravity experiments, phase-field simulation, pattern formation, interface curvature

The envelope of the solidification front naturally develops a macroscopic curvature in a multitude of solidification processes. However, its effect on dendritic microstructure formation remains poorly understood. Here we exploit a microgravity environment where convection is suppressed to investigate quantitatively the effect of curvature on dendritic array growth during directional solidification of a transparent succinonitrile 0.46 wt% camphor alloy. In addition, we interpret the results using both theoretical analyses and phase-field simulations. In situ observations reveal that even a weak macroscopic interface curvature can have a major effect on both the array pattern evolution and grain structure. First, convex and concave interfaces lead to a continuous increase and decrease in time of the average primary spacing, respectively, which only attains a stationary value for a flat interface. We show that this results from the formation of drifting velocity gradients along the interface under the combined effect of the crystal misorientation and the interface misorientation. These grains form at sample boundaries and overgrow pre-existing interior grains of smaller misorientation to the detriment of well-oriented crystal growth. In addition, they induce a change of array structure when the directional normal to the sample boundary is parallel to one of <100> axis of secondary branches. These results provide new insights for controlling and optimizing directionally solidified alloy microstructures.

boundary is parallel to one of <100> axis of secondary branches. These results insights for controlling and optimizing directionally addition, they induce a change of array structure when the directional normal to the sample boundary is parallel to one of <100> axis of secondary branches. These results insights for controlling and optimizing directionally misorientation. These grains form at sample boundaries and overgrow pre grains of smaller misorientation to the detriment of well addition, they induce a change of array structure when the directional normal to the sample misorientation. These grains form at sample boundaries and overgrow pre grains of smaller misorientation to the detriment of well under the combined effect of the crystal misorientation an misorientation. These grains form at sample boundaries and overgrow pre this results from the formation of drifting velocity gradients under the combined effect of the crystal misorientation an First, convex and concave interfaces lead to a continuous increase and decrease in he average primary spacing, respectively, which only attains a stationary value for a results from the formation of drifting velocity gradients observations reveal that even a weak macroscopic interface curvature can have a major effect on both the array pattern evolution and grain First, convex and concave interfaces lead to a continuous increase and decrease in he average primary spacing, respectively, which only attains a stationary value for a camphor alloy. In addition, we interpret the results using both theoretical analyses and observations reveal that even a weak macroscopic interface curvature can have a major effect on both the array pattern evolution and grain array growth during directional solidification of a transparent succinonitrile camphor alloy. In addition, we interpret the results using both theoretical analyses and observations reveal that even a weak macroscopic interface igate quantitatively the effect of curvature on dendritic array growth during directional solidification of a transparent succinonitrile we exploit a microgravity environment where igate quantitatively the effect of curvature on dendritic array growth during directional solidification of a transparent succinonitrile naturally develops a macroscopic curvature in a multitude of solidification processes. However, its effect on dendritic microstructure we exploit a microgravity environment where naturally develops a macroscopic curvature in a multitude of solidification processes. However, its effect on dendritic microstructure naturally develops a macroscopic curvature in a

Introduction

During alloy solidification, a variety of interface patterns develop when the growth conditions exceed the threshold for the planar interface stability. Pattern selection occurs under dynamic conditions of growth in which the unstable pattern goes through the process of reorganization into a rather periodic array [START_REF] Cross | Pattern formation outside of equilibrium[END_REF] of cells or dendrites. Since the characteristics of patterns strongly influence the properties of the material in solidification processing, it is mandatory to clarify the specific dependence of the solid-liquid interface pattern on processing conditions. Numerous studies have been directed at predicting the morphological characteristics, and especially the primary spacing, of growing patterns as a function of growth control parameters [START_REF] Hunt | Solidification and casting of metals[END_REF][START_REF] Kurz | Dendrite Growth at the Limit of Stability -Tip Radius and Spacing[END_REF][START_REF] Trivedi | Interdendritic Spacing .2. A Comparison of Theory and Experiment[END_REF][START_REF] Han | Primary Spacing Selection in Directionally Solidified Alloys[END_REF][START_REF] Somboonsuk | Interdendritic Spacing .1. Experimental Studies[END_REF][START_REF] Somboonsuk | Dynamical studies of dendritic growth[END_REF][START_REF] Trivedi | Pattern-Formation during the Directional Solidification of Binary-Systems[END_REF][START_REF] Eshelman | Cellular spacings I. Steady-state growth[END_REF][START_REF] Seetharaman | Cellular spacings II. Dynamical studies[END_REF][11]. Most of those studies assume planar interfaces on a large scale. However, solidification processes encountered in nature and application relevant processing usually involve a slight curvature due to large-scale perturbation or instability, or a closed interface geometry. For example, most metallic alloys exhibit macroscopically convex interfaces during solidification as a result of higher thermal conductivity in the solid phase [12].

The effects of solid-liquid interface curvature have been investigated in different fields of material processing. For example, during unidirectional ice-templating of aqueous titanium oxide suspensions [13,14], a technique to obtain sophisticated structures exhibiting enhanced functionalities, the authors found a disparity in the mean pore width between the outer edge of the sample and its center due to the interface curvature. The pore width, by association with the microstructural characteristics, plays an important role in the material properties. Another example can be found in solidification of binary eutectic alloys. Both experiments and simulations demonstrated the crucial influence of a slight curvature of the isotherms on the long-time dynamics of bulk rod-like eutectic patterns, providing a small symmetry-breaking perturbation in eutectic growth [START_REF] Perrut | Directional and single-crystal solidification of Ni-base superalloys: Part I. The role of curved isotherms on grain selection[END_REF]. In the presence of a transverse temperature gradient (or, equivalently, a tilt of the isotherms), the pattern drifts and one of the crucible border becomes a source for new structures. Lamellae formed on this boundary invade the whole interface, leading to the emergence of a regular lamellar array parallel to the transverse temperature gradient direction. The directional solidification of single crystals of Ni-based superalloys which is of major industrial interest for the production of turbine blades observed that the larger the deviation of the local solid-liquid interface from planarity, the larger will be the range of orientations that can occur in the grain texture, thus degrading the quality product [16].

The use of transparent organic analogs, that freeze like metals but are transparent to visible light, is a precious tool for in situ observation of the solid-liquid interface [17]. Weiss et al. [18] studied solidification of an organic alloy in a large cylindrical crucible where the solidliquid interface is curved due to the evacuation of latent heat generated upon solidification through the crucible wall. In such a bulk system, the combination of curvature and convection in the fluid leads to solute macrosegregation and radial variation of microstructure [START_REF] Jamgotchian | Localized microstructures induced by fluid flow in directional solidification[END_REF] and in cellular arrays, the local interface slope induces pattern advection in which new cells are permanently born, while moving towards the center where they are eventually eliminated [18]. Bottin-Rousseau and Pocheau [START_REF] Bottin-Rousseau | Self-organized dynamics on a curved growth interface[END_REF] experimentally investigated the spatiotemporal implications of an artificially curved growth interface in directional solidification of a based superalloys which is of major industrial interest for the production of turbine the deviation of th orientations that can occur in the grain texture, thus degrading the quality product based superalloys which is of major industrial interest for the production of turbine invade the whole interface, leading to the emergence of a regular lamellar array parallel to the transverse temperature gradient direction.

based superalloys which is of major industrial interest for the production of turbine becomes invade the whole interface, leading to the emergence of a regular lamellar array parallel to the transverse temperature gradient direction. temperature gradient (or, equivalently, a tilt of the isotherms), the patt becomes a source for new structures. Lamellae formed on invade the whole interface, leading to the emergence of a regular lamellar array parallel to time dynamics of breaking perturbation in eutectic growth temperature gradient (or, equivalently, a tilt of the isotherms), the patt experiments and simulations demonstrated the crucial influence of a slight curvature of the time dynamics of breaking perturbation in eutectic growth association with the microstructural characteristics, plays can be found in experiments and simulations demonstrated the crucial influence of a slight curvature of the due to the interface curvature association with the microstructural characteristics, plays can be found in found a disparity in the mean pore width between the due to the interface curvature association with the microstructural characteristics, plays obtain found a disparity in the mean pore width between the during unidirectional ice obtain found a disparity in the mean pore width between the been investigated during unidirectional ice conductivit been investigated scale perturbation or instability, or a closed interfa metallic alloys exhibit macroscopically convex interfaces conductivity in the solid phase and application relevant process scale perturbation or instability, or a closed interfa metallic alloys exhibit macroscopically convex interfaces transparent alloy in thin sample geometry. Cell advection along the interface slope, associated to stretching, is observed. It induces repetitive cell nucleation by tip-splitting. This phenomenon first occurs in a disordered manner but the cells eventually become ordered over long times and the cell nucleations spontaneously organize themselves in avalanches [START_REF] Pocheau | Curvature induced periodic attractor on growth interface[END_REF].

The main problem with studying bulk samples comes from the presence of significant convection in the melt. Fluid flow modifies the structure of the solute boundary layer by sweeping, thus causing non-uniform morphological instability with the formation of a nonuniform microstructure and non-uniform primary spacing [START_REF] Jamgotchian | Localized microstructures induced by fluid flow in directional solidification[END_REF][START_REF] Drevet | Solidification of aluminium lithium alloys near the cell/dendrite transition-influence of solutal convection[END_REF][START_REF] Trivedi | Microstructure development in the directionally solidified Al-4.0 wt% Cu alloy system[END_REF][START_REF] Mota | Convection Effects During Bulk Transparent Alloy Solidification in DECLIC-DSI and Phase-Field Simulations in Diffusive Conditions[END_REF][START_REF] Witusiewicz | Effect of convection on directional solidification in transparent succinonitrile 2.2wt.%(d)camphor alloy[END_REF][START_REF] Reinhart | Influence of natural convection during upward directional solidification: A comparison between in situ X-ray radiography and direct simulation of the grain structure[END_REF]. The easiest way to avoid or drastically reduce convection is to reduce the size of the samples. However, to study bulk samples, the reduced-gravity environment of space is required to eliminate fluid flow. The studies presented here were conducted using the Directional Solidification Insert (DSI) dedicated to in situ and real time characterization of the dynamical selection of the solidliquid interface morphology on bulk samples of transparent materials. The DSI is part of the DECLIC instrument (DEvice for the study of Critical LIquids and Crystallization) and it is installed within the International Space Station (ISS) to benefit from a microgravity environment. It is jointly operated by the French Space Agency (CNES) and NASA. A first series of experiments was performed in 2010-2011, mainly dedicated to cellular growth, which led to unprecedented observations and improvement of numerical models of solidification [START_REF] Bergeon | Spatiotemporal Dynamics of Oscillatory Cellular Patterns in Three-Dimensional Directional Solidification[END_REF][28][START_REF] Tourret | Oscillatory cellular patterns in three-dimensional directional solidification[END_REF][30][31][32]. A second series of experiments was performed in 2017-2018 on a sample of higher solute concentration, dedicated to dendritic growth. This article reports new experimental results and related phase-field (PF) simulations that quantitatively explore the effects of interface curvature on dendritic patterns. This paper is organized as follows. In the experimental part (Section 2), we first briefly describe the experimental device and methods before describing key characteristics of the growing interfaces, such as macroscopic shapes and the crystallographic orientation of the single crystal. Section 3, dedicated to results, is split into three subsections. The effects of interface curvature in terms of microstructure characteristics are first described, before focusing on its effects on primary spacing. We will provide evidence that the spatial variation of the drifting velocity is at the origin of spatio-temporal structuration of primary spacing. The last subsection will then analyze how the pattern drifting velocity profile is built along the interface. In Section 4, we first describe a quantitative PF model for solidification of dilute binary alloys and the implementation of curved isotherms that induce the macroscopic curvature. Then we present the modeling results and investigate the effects of convex and concave curvatures on the microstructure evolution. Finally, conclusions are summarized in Section 5.

Experiments

Device DECLIC-DSI and experiments

The DECLIC-DSI includes a Bridgman type furnace and the experimental cartridge [33,34]. Solidification is performed by pulling the experimental cartridge containing the alloy inside the thermal gradient G (between 10 and 30 K/cm) at a pulling rate V ranging between 0.1 In Section 4, binary alloys and the implementation of curved isotherms that induce the macroscopic curvature. Then we present the modeling results and investigate the effects of convex will then In Section 4, on primary spacing drifting velocity is at the origin of spatio will then analyze curvature in terms of microstructure characteristics are first described, before focusing on on primary spacing drifting velocity is at the origin of spatio ection 3, dedicated to results, curvature in terms of microstructure characteristics are first described, before focusing on on primary spacing experimental device and methods interfaces, such as macroscopic shapes and the crystallograp ection 3, dedicated to results, the study of Critical LIquids and Crystallization) and it is and 30 µm/s. The different diagnostics, with illustrations of their specific use, have been previously described in numerous papers, for example in ref. [30,32,[START_REF] Pereda | Experimental characterization and theoretical analysis of cell tip oscillations in directional solidification[END_REF]. In the current article, only images obtained by the direct axial observation, providing top-view interface images, and transverse observations, providing side-view interface images, are analyzed.

The organic transparent alloy used is a succinonitrile (SCN) -camphor, with a nominal concentration C0 = 0.46 wt% camphor. A single crystalline solid seed with a direction <100> parallel to the pulling axial direction is kept unmelted during all experiments. Further details about the experimental procedure can be found in previous works [30,34,[START_REF] Weiss | Cellular pattern dynamics on a concave interface in three-dimensional alloy solidification[END_REF].

In this paper, we will focus on a series of experiments performed with a thermal gradient of G = 12.5 K/cm, with a pulling velocity V varying from 1.5 to 12 µm/s. The in situ and real time videos corresponding to the solidification experiments are provided as supplementary material [37]. For each experiment, a length of 60 mm is solidified at a constant pulling rate. Whatever the pulling rate is, the formation of the pattern follows the same classical stages described in Figure 1 for V = 3 µm/s, corresponding to snapshots of video2 of supplementary material [37]. Selected top-views of the solid-liquid interface at different times are given in Figure 1. The corresponding side-views illustrate the recoil of the interface within the furnace, associated to the curvature change, similar to previous studies [30]. We start from rest (V = 0, t = 0), with a smooth and convex interface (Fig. 1a). After triggering of the pulling, we first observe a rather complex array of more or less linear ridges along sub-boundaries followed by the development of a more or less uniform corrugation within the sub-grains (Fig. 1b). The interface dynamics is very fast and interface modulations amplifies to eventually lead to a clear but highly disordered dendritic pattern (Fig. 1c). The dynamics then slows down, and a stage of continuous pattern evolution starts (Fig. 1d). In this paper, we will concentrate on this last stage. The dendritic patterns after 45 mm of pulling are illustrated in Figure 2a for the different velocities (only 20 mm for V = 12 µm/s). They obviously differ in primary spacing that becomes finer when the pulling velocity increases. Additionally, it is worth noting that parasite sub-grains coming from the crucible border progressively invade the well-oriented central grain for pulling rates higher than 3 µm/s. The parasite sub-grains are characterized by a higher misorientation of their dendrite growth direction; they are then more tilted so that the light is more diffused and sub-grains appear darker than the wellorientated central grain (see the videos of [37] to clearly visualize this phenomenon). The higher the pulling rate is, the faster the invasion is: at V = 12 µm/s after 37 mm of solidification the central grain has almost vanished.

The pattern is characterized in terms of primary spacing and order/disorder level. Each structure of an image is first identified and labeled before applying a Voronoi tessellation to identify the first neighbors. Primary spacing corresponds to the average of the center-tocenter distances of a structure with its first-neighbors. Each dendrite is tracked from one image to the following, so that its possible motion (drift in the interface plane) can be analyzed (trajectory, velocity). More details on the methods of data treatment can be found elsewhere [START_REF] Pereda | Primary spacing evolution during microstructure formation in 3D directional solidification: methods of data treatment of microgravity experiments conducted in the DECLIC-DSI[END_REF].

On the videos of supplementary material [37], the most striking observation is that, seen from the top, dendrites are drifting in the interface plane, with characteristics that depend on pulling velocity as stated in Figure 2b. Drifting velocity maps are given for each velocity higher the pulling rate is, the faster the invasion is: at V = 12 µm/s after 37 mm of solidification the central grain has almost vanished. orientated central grain (see the videos of higher the pulling rate is, the faster the invasion is: at V = 12 µm/s after 37 mm of are characterized by a higher misorientation of their dendrite growth direction more tilted so that the light is more diffused and sub orientated central grain (see the videos of oriented central grain for pulling rates higher than 3 µm/s. are characterized by a higher misorientation of their dendrite growth direction more tilted so that the light is more diffused and sub worth noting that parasite sub oriented central grain for pulling rates higher than 3 µm/s. are characterized by a higher misorientation of their dendrite growth direction for the different velocities (only 20 mm for V = 12 µm/s) primary spacing that becomes finer when the pulling velocity increases.

-grains coming from the crucible border progressively invade

The dendritic patterns for the different velocities (only 20 mm for V = 12 µm/s) primary spacing that becomes finer when the pulling velocity increases. eventually lead to a clear but highly disordered dendritic pattern (Fig. 1c).

ontinuous pattern evolution starts he dendritic patterns he interface dynamics is very fast and interface eventually lead to a clear but highly disordered dendritic pattern (Fig. 1c).

ontinuous pattern evolution starts the development of a more or less uniform corrugation within the sub he interface dynamics is very fast and interface eventually lead to a clear but highly disordered dendritic pattern (Fig. 1c). we first observe a rather complex array of more or less linear ridges along sub ess uniform corrugation within the sub rest (V = 0, t = 0), with a smooth and convex interface (Fig. 1a). we first observe a rather complex array of more or less linear ridges along sub ess uniform corrugation within the sub ustrate the recoil of the interface within the furnace, associated to the curvature change, similar to previous studies rest (V = 0, t = 0), with a smooth and convex interface (Fig. 1a). liquid interface at different times ustrate the recoil of the interface within the furnace, associated to the curvature change, similar to previous studies , corresponding to snapshots of video2 of supplementary liquid interface at different times For each experiment, a length of 60 mm is solidified at Whatever the pulling rate is, the formation of the pattern follows the , corresponding to snapshots of video2 of supplementary after 30 mm of pulling, where the individual drift velocity of each dendrite defines its color. The corresponding drifting direction maps were also obtained, and the arrows superposed to Figure 2b correspond to drift velocity vectors. At 3 µm/s the drift is roughly homogeneous in value and direction on the whole interface. For the other pulling rates, the main direction of drift remains the same from top left to bottom right but its spatial homogeneity is modulated differently with pulling rate. At V = 1.5 µm/s, dendrites tend to move towards the border and the drifting velocity increases towards the bottom and the right; at V = 6 and 12 µm/s, dendrites tend to move towards the center and the drifting velocity decreases towards the bottom and the right. The origin of the drifting characteristics and its consequence on pattern are analyzed in §3.

Macroscopic interface shape

Macroscopic shape and motion of the interface are investigated by side view observation. The macroscopic interface shape is a consequence of the isotherms shape that is usually curved in 3D configurations. to the liquidus isotherm at rest, and to the cell/dendrite tips isotherm while pulling. Seen from the side, the interface appears as a curved line that can generally be fitted by a parabola. The interface curvature can be quantified by the curvature radius of the parabo corresponds to the difference between the interface positions measured in the center and on the border of the crucible. With this second method of quantification, positive and negative values respectively correspond to convex and concave interfaces. Side view images of the solid-liquid interface are given in Figure 3a, at rest and during pulling, for different velocities, where one can easily distinguish the liquid and solid part of the sample.

The origin of curvature and the analysis of its evolution while pulling has been largely discussed in a previous paper [30] and present experiments are consistent with the key points already reported. In Figure 3a, the interface at rest is convex. While pulling, the interface recoils and its shape becomes more and more concave as pulling velocity increases. So, we observe that: at V = 1.5 µm/s, the interface is still convex but less than at rest; it is macroscopically flat at 3 µm/s; and it becomes slightly concave at 6 µm/s. For concave interfaces, the central part of the interface (at a lower position) is hidden from the side camera view by the dendrites at the crucible wall (at a higher position). Consequently, the interface shape is not clearly visible, as shown in Figure 3a for V = 6 µm/s. At V = 12 µm/s, the interface is out of the field of view.

The software package CrysMAS® designed for the global modeling of solidification processes in complex furnaces is used here to compute the thermal field and study the interface position and shape, as detailed in [30]. Figure 3b gives the interface shapes in the adiabatic area calculated with CrysMAS® for the experimental conditions. At V = 1.5 µm/s, the experimental side view of the interface at steady-state allows to determine the interface shape, which is additionally represented in Figure 3b. The experimental shape superposes well to the calculated one, thus validating the numerically determined shapes. Experimental shapes will be used when available, as for V = 1.5 and 3 µm/s; shapes extracted from CrysMAS simulations will be used for other cases, at V = 6 and 12 µm/s. interface shape is not clearly visible, as the interface is out of by the interface shape is not clearly visible, as pically flat at interfaces, the central part of the interface (at dendrites at the crucible wall V pically flat at 3 interfaces, the central part of the interface (at its shape V = 1.5 µm/s µm/s discussed in a previous paper [30] points already reported. In Figure becomes and the analysis of its evolution while pulling has been largely Seen from the side, fitted by a parabola. The interface curvature can be macroscopic interface shape is a consequence of the isotherms the liquidu Seen from the side, acroscopic shape and motion of the interface are investigated by side view observation. macroscopic interface shape is a consequence of the isotherms shape that is usually curved the liquidu

Crystallographic orientation

A single crystal seed is used, with a <100> direction aligned with the pulling axis as closely as possible. The experiments described in this work were performed sequentially and it was clearly observed that, in the early stages of solidification, the -sub-grain dominates. Stray sub-grains may progressively invade the central area at large velocities, but we will focus on the dendrite behavior in the central well-oriented grain.

The growth direction of a cell/dendrite rotates from the thermal gradient direction at a low velocity to the closest <100> direction as the pulling rate increases [START_REF] Akamatsu | Similarity law for the tilt angle of dendrites in directional solidification of non-axially-oriented crystals[END_REF][START_REF] Deschamps | Growth directions of microstructures in directional solidification of crystalline materials[END_REF]. A sub-grain is characterized by a collective drift of the structures, of specific direction and amplitude, caused by its particular misalignment to the pulling/thermal axis. To determine the orientation of the central grain, we measure the drifting velocity for a flat interface (at V = 3 µm/s). The average amplitude of drift velocity is used to evaluate the tilt angle between the growth velocity and the vertical interface velocity . For a flat interface, the thermal gradient is parallel to the interface velocity as shown in the configuration of Figure 4a, and the tilt angle is simply determined by the relation:

(1)
The misorientation of a sub-grain is defined by the angle between the preferred <100> growth direction (noted as in Figure 4) and . The cell/dendrite growth direction varies relatively to and this variation depends on the Péclet number (where is the primary spacing and D is the solute diffusion coefficient in the liquid) and is generally described by a phenomenological equation [START_REF] Akamatsu | Similarity law for the tilt angle of dendrites in directional solidification of non-axially-oriented crystals[END_REF][START_REF] Ghmadh | Directional solidification of inclined structures in thin samples[END_REF][42][START_REF] Song | Propagative selection of tilted array patterns in directional solidification[END_REF]: [START_REF] Hunt | Solidification and casting of metals[END_REF] where is the angle between the preferred <100> direction and the growth direction, f and g are two fitting constants. Equation ( 2) was previously used to describe the drifting dynamics under a flat interface with the local interface slope , for which . Here, we also use it to describe the drifting dynamics under a curved interface with varying at different positions and assume that f and g are independent of curvature.

As illustrated in Figure 4a, the growth direction always lies between the thermal gradient and the preferred <100> direction for different interface slopes. For a given value of Pe, the ratio is constant, so that when increases due to the variation of , also increases and the drifting velocity decreases, as described in Figure 4a.

We use the 3D PF simulation and the method described in Refs. [START_REF] Tourret | Oscillatory cellular patterns in three-dimensional directional solidification[END_REF]31,32] to determine f and g values associated to the SCN-0.46 wt% camphor alloy, with growth conditions and -12 . In the PF simulation, we construct a regular hexagonal array with a flat interface and impose a constant crystallographic tilt angle (f and g are independent of according to [44]). The dendritic array drifting due to misorientation is fitted by Eq. ( 2 Based on the experimentally measured average primary spacing and the average drifting velocity , together with the PF-determined f and g, the crystal misorientation = 2.7°was obtained for V = 3 µm/s when the interface is flat. For the other pulling velocities when interfaces are curved, we use Eqs. ( 1)-( 2) in the central region where the interface is locally flat to calculate . The same value = 2.7° was obtained for V = 1.5 and 6 µm/s.

Results

Microstructure characteristics

The question here is to determine how the pattern organization is affected by the drifting and curvature.

First, let us focus on the central well-oriented grain. Whatever the pulling rate is, the maps of numbers of first neighbors are similar. They reveal a hexagonal basis, with numerous defects (mainly pairs of 5 7 neighbors) and high disorder. Such disordered hexagonal arrays are typical of solidified cellular patterns [28,[START_REF] Billia | Statistical-Analysis of the Disorder of 2-Dimensional Cellular Arrays in Directional Solidification[END_REF], and they also appeared dominant in large scale 3D PF simulations [46]. It is experimentally observed that in spite of the cubic symmetry which is a determinant for the secondary formation, the dendritic pattern organization does not obviously reflect the crystal anisotropy and the spatial ordering seems similar to the natural hexagonal structure observed for cellular structures.

Regarding the grain structure, these experiments reveal a very striking consequence of concave curvature. At V = 1.5 µm/s and 3 µm/s, convex and flat interfaces respectively, no major evolution of the grain structure is observed: only one major grain is observed without stray grains invasion. However, when the interface is concave, at V = 6 and 12 µm/s, the central region containing the well-oriented grain is invaded by sub-grains coming from the crucible border. These invading sub-grains, or stray grains, are characterized by various drifting directions as well as a large range of drifting velocities (Figure 5a andb), generally higher than the average drifting velocity of the central well-oriented sub-grain thus witnessing of a higher misorientation. It means that the concavity favors the formation and propagation of stray grains characterized by off-axis crystallographic orientations. At V = 6 µm/s (Figure 5a), all stray grains originate upstream of the central grain (from the top and left image sides). They drift in roughly the same direction but their higher velocities define convergent sub-boundaries with the central grain. At V = 12 µm/s (Figure 5b), stray grains also invade from the bottom side of the image: with a drift direction roughly opposite to the central grain so that convergent sub-boundaries are also formed. In both cases, it appears that the central grain is overgrown by these stray grains, in spite of its lowest misorientation. This is in contradiction with the classical conjecture that the best oriented grain cannot be overgrown and the grain boundary follows the growth direction of the most favorably oriented grain. This conjecture has been formalized and justified by Walton and directions as well as higher than the average ing of a higher misorientation These i directions as well as higher than the average region containing the well These invading sub grain However, when region containing the well concave curvature. At V = 1.5 µm/s and 3 µm/s, grain structure is However, when

Regarding the grain structure, these concave curvature. At V = 1.5 µm/s and 3 µm/s, structure is similar to the natural hexagonal structur

Regarding the grain structure, these determinant for the secondary reflect the crystal anisotropy and the spatial ordering seems similar to the natural hexagonal structur experimentally determinant for the secondary reflect the crystal anisotropy and the spatial ordering seems [28,[START_REF] Billia | Statistical-Analysis of the Disorder of 2-Dimensional Cellular Arrays in Directional Solidification[END_REF], and experimentally observe determinant for the secondary ) and high disorder. and reveal a hexagonal basis, with numerous defects ) and high disorder. Such disordered they also appeared dominant in large scale Whatever hexagonal basis, with numerous defects Whatever the pattern organization is affected by the drifting and the pattern organization is affected by the drifting and Chalmers [47] on the basis of the lower undercooling of a grain with lower misorientation with respect to the thermal gradient compared to neighboring grains with higher misorientation. Unusual overgrowth of favorably oriented dendrites at converging grain boundary has already been reported [48][49][50][START_REF] Takaki | Two-dimensional phase-field simulations of dendrite competitive growth during the directional solidification of a binary alloy bicrystal[END_REF], but Tourret et al. [START_REF] Tourret | Grain growth competition during thin-sample directional solidification of dendritic microstructures: A phase-field study[END_REF] have found that it is rare and limited -for low misorientations similar to our case -to some convergent boundaries with opposite drifting directions. Our observations indicate that curvature favors the occurrence of this unusual phenomenon and extends the conditions of its appearance to other types of converging boundaries: in the case of curvature, it even occurs for convergent boundaries with the same drifting directions.

At 12 µm/s, the final grain texture resulting from the complete overgrowth of the initial central grain is formed by several off-axis grains. Such an observation has been reported previously by [16] in the case of directional growth of Ni-based superalloys that present a concave interface: post-mortem texture analyses evidence that the larger the concavity is, the wider the range of orientations in the final texture. A possible interpretation, proposed by the authors, is that the orientation of the local gradient varies in a curved interface so that the most favored orientation varies all along the interface. Our observations invalidate this proposition as it cannot explain the continuous propagation of off-axis grains. They keep propagating and overgrowing the central well-oriented grain during their whole drift that starts at the border and ends in the center while the thermal gradient orientation varies continuously during this motion. The final texture formed by off-axis grains results from a general unusual overgrowth phenomenon favored by concavity.

The last point to discuss regarding the microstructure is the appearance of an unusual pattern organization at V = 6 and 12 µm/s in some areas of the stray grains. These areas are noted from 1 to 4 in Figure 5b and are enlarged in Figure 5c. The classical organization of a tilted dendritic pattern is visible, for example, in area 0 of Figure 5b andc. The effect of tilt is mainly to modify the 4 th order symmetry appearance in top-view observation by hiding half of the dendrite. 3D phase-field simulations of misoriented dendritic patterns lead to similar microstructures as for example in Fig. 1 of Takaki et al. [START_REF] Takaki | Phase-field study on an array of tilted columnar dendrites during the directional solidification of a binary alloy[END_REF]. In area 0, no specific ordering appears. On the other hand, in areas 1 to 4, dendrite primary trunks are aligned along the directions of the secondary branches, indicated by the yellow and blue dashed lines. It is worth noting that these aligned dendrites were not present and visible at the beginning of the pattern formation: they probably were formed after the initial transient along the border (out of visibility) then propagated towards the center and became visible. The location of the different areas with respect to the crucible border is schematically given in Figure 5d, together with the two directions of secondary branches in yellow and blue dashed lines. The secondary branches of the aligned structures in areas 1 to 4 are normal to the crucible border. At these locations, a tilted dendrite that drifts away from the border will be associated to a periodic source via a tertiary branching process, with a quite similar mechanism observed in thin samples of tilted patterns by Song et al. [START_REF] Song | Propagative selection of tilted array patterns in directional solidification[END_REF]. The aligned structures of areas 1 to 4 result from the propagation of these periodically generated dendrites. In area 0, the tilted dendrites that drift far from the border are also sources. But as the orientation of secondary branches regarding the crucible border is not specific, the tertiary branching is neither locked on the secondary branches nor periodic. As will be discussed in Section 4.2.3, the aligned structure caused by the invasion of dendrites from the source at the border can also be observed in the pattern formation: they (out of visibility) different areas with respect to the crucible border is schematically given in worth noting that these aligned dendrites were not present and visible at the beginning of pattern formation: they . On the other hand, in areas 1 to 4, dendrite directions of the secondary branches, worth noting that these aligned dendrites were not present and visible at the beginning of microstructures as for example in . On the other hand, in areas 1 to 4, dendrite directions of the secondary branches, field simulations of misoriented dendritic patterns lead to similar microstructures as for example in . On the other hand, in areas 1 to 4, dendrite dendritic pattern is visible order symmetry appearance in top field simulations of misoriented dendritic patterns lead to similar and are enlarged in for example order symmetry appearance in top

The last point to discuss regarding the microstructure is the appearance of an unusual 12 µm/s in and are enlarged in

The last point to discuss regarding the microstructure is the appearance of an unusual 12 µm/s in al unusual overgrowth phenomenon favored by concavity.

The last point to discuss regarding the microstructure is the appearance of an unusual

The final texture formed by off the central well that starts at the border and ends in the center while the thermal gradient orientation

The final texture formed by off al unusual overgrowth phenomenon favored by concavity.

as it cannot explain the continuous propagation of well that starts at the border and ends in the center while the thermal gradient orientation most favored orientation varies all along the interface.

as it cannot explain the continuous propagation of oriented concavity is, the wider the range of orientations in the final texture ocal gradient most favored orientation varies all along the interface. mortem texture analyses evidence that the larger the concavity is, the wider the range of orientations in the final texture. A possible interpretation, ocal gradient 3D PF simulation. It is worth noting that we observed such arrangements in all dendritic experiments, whatever the experimental parameters are (concentration, thermal gradient, pulling velocity) as soon as the interface is concave and the drifting velocity is high enough to induce the formation of several dendrites by tertiary-branching at the border. We believe that this arrangement is very difficult to detect by post-mortem analysis, thus explaining that it has never been reported.

Primary spacing evolution

Figure 6 shows the time evolution of primary spacing for the different pulling velocities.

Those measurements are limited to the well-oriented grain and they start after the initial transient that leads to the formation of the well-developed pattern; during this earlier stage, the primary spacing increases. For V = 12 µm/s, the well-oriented grain disappears after 36.8 mm of growth, thus explaining the early stop of the curve. As usual in a solidification pattern, a band of primary spacing is observed; minimum and maximum values of spacing are also represented on the graphs of Figure 6. It is well-known that a wide band of stable spacings exists, limited by elimination at its lowest value and tertiary branching for dendritic array at its largest value [START_REF] Han | Primary Spacing Selection in Directionally Solidified Alloys[END_REF][START_REF] Somboonsuk | Dynamical studies of dendritic growth[END_REF][START_REF] Seetharaman | Cellular spacings II. Dynamical studies[END_REF][START_REF] Warren | Prediction of Dendritic Spacings in a Directional-Solidification Experiment[END_REF][START_REF] Hunt | Numerical modeling of cellular/dendritic array growth: spacing and structure predictions[END_REF][START_REF] Ding | On primary dendritic spacing during unidirectional solidification[END_REF][START_REF] Chen | History dependence of primary dendrite spacing during directional solidification of binary metallic alloys and interdendritic convection[END_REF][START_REF] Gurevich | Phase-field study of spacing evolution during transient growth[END_REF][START_REF] Gurevich | Phase-field study of three-dimensional steady-state growth shapes in directional solidification[END_REF]. However, in our measurements, the minimal and maximal spacing values do not correspond to limits of the stable spacing band as elimination or tertiary branching do not occur in the central well-oriented grain, but may sometimes be observed in the frame of competition with invading grains at the sub-boundary.

Regarding the average spacing curve for each velocity (Figure 6), a striking difference appears. At V = 3 µm/s, the average spacing stabilizes after the initial transient stage whereas for the other pulling velocities, it keeps evolving during the whole solidification. At V = 1.5 µm/s, the spacing increases with time; at V = 6 and 12 µm/s, it decreases. This difference in evolution can be linked to the macroscopic interface curvature inversion from convex at V = 1.5 µm/s to concave when V is higher than 3 µm/s. For convex interfaces, Bottin-Rousseau et al. [START_REF] Bottin-Rousseau | Self-organized dynamics on a curved growth interface[END_REF] evidenced a progressive stretching of cells while they drift from center to border that eventually led to tip-splitting when the upper stable limit is reached. Inversely for concave interfaces, still for cellular patterns, Weiss et al. [START_REF] Weiss | Cellular pattern dynamics on a concave interface in three-dimensional alloy solidification[END_REF] evidenced and analyzed the size decrease of cells while they drift from the border toward the center. It appears here that similar behaviors occur in the dendritic patterns.

To better understand the origin of this variation, we analyzed the spatial homogeneity of the spacing with time for V = 1.5 µm/s. This experiment is chosen as the spacing evolution is large and due to convexity, no invading sub-grain enters the studied area. So, rather than consider the whole interface as in Figure 6, nine small areas of interest Ai are defined as described in Figure 7a. In each area Ai, measurements of spacing and drifting velocity are performed on all dendrites of the area (average of 20 dendrites per area) and repeated every 170 s (step of 250 µm of growth). The average spacing evolution during the second third of the solidification (from L = 20 to 40 mm) was measured in each area and the variation in % is given in Figure 7b. We note that even if the spacing increases everywhere, its increase is larger in central and bottom right areas (A5-A6-A8-A9).

The origin of cell stretching/reduction is linked to a variation of the drifting velocity along the interface [START_REF] Bottin-Rousseau | Self-organized dynamics on a curved growth interface[END_REF][START_REF] Weiss | Cellular pattern dynamics on a concave interface in three-dimensional alloy solidification[END_REF]. In the case of a misoriented dendritic pattern on a flat interface, at V = To better understand the origin of this variation, we analyzed the spatial homogeneity of the time similar behaviors occur in To better understand the origin of this variation, we analyzed the spatial homogeneity of the interfaces, still for cellular patterns, Weiss et al. decrease of cells while they drift similar behaviors occur in splitting when the upper stable limit is reached. Inversely for concave interfaces, still for cellular patterns, Weiss et al. decrease of cells while they drift evidenced a progressive stretchin splitting when the upper stable limit is reached. Inversely for concave interfaces, still for cellular patterns, Weiss et al. evolution can be linked to the macroscopic interface curvature 1.5 µm/s to concave when V is higher than 3 µm/s evidenced a progressive stretchin with time evolution can be linked to the macroscopic interface curvature 1.5 µm/s to concave when V is higher than 3 µm/s 3 µm/s, the average spacing stabilizes after the initial transient stage whereas her pulling velocities, it keeps evolving during the whole solidification. At with time; at V = Regarding the average spacing curve for each velocity 3 µm/s, the average spacing stabilizes after the initial transient stage whereas her pulling velocities, it keeps evolving during the whole solidification. At me of competition with invading grains at the Regarding the average spacing curve for each velocity me of competition with invading grains at the do not correspond to limits of the stable spacing band as elimination or in the central well oriented grain, me of competition with invading grains at the . However, in our measurements, the minimal and maximal do not correspond to limits of the stable spacing band as elimination or oriented grain, exists, limited by elimination at its lowest value and tertiary branching for dendritic array at . However, in our measurements, the minimal and maximal ; minimum and maximum values known that a wide band of stable spacings exists, limited by elimination at its lowest value and tertiary branching for dendritic array at early stop of the curve. As usual in ; minimum and maximum values known that a wide band of stable spacings 3 µm/s, the primary spacing remains stable and the analysis of drifting velocity reveals a homogeneity of drifting velocity all along the interface (Figure 2b). However, at V = 1.5 µm/s, on a convex interface, the drifting velocity map reveals large variations along the interface. The average drifting velocity in each area Ai was measured and compared to the average considering the whole interface. Figure 7c represents the respective differences in each area, where the arrows indicate the drifting direction and their length is proportional to the velocity. A very clear and strong drifting velocity gradient appears along the diagonal A1-A5-A9. This gradient is roughly parallel to the drifting direction. Still along drifting direction, we notice strong gradients along A1-A4-A7, A4 & A5 to A8 and A2 to A6. The downward motion along the drifting direction, associated to gradients of drifting velocity, induces the dendrite stretching and so the large increase of spacing in A4 to A9 (Figure 7b).

As demonstrated here, the spatial variation of drifting velocity is at the origin of spatiotemporal structuration of primary spacing. It is then worth studying the formation of the drifting velocity profile along the interface.

Local curvature and drift

The effect of curvature on cellular patterns has been previously described respectively for convex [START_REF] Bottin-Rousseau | Self-organized dynamics on a curved growth interface[END_REF] and concave [START_REF] Weiss | Cellular pattern dynamics on a concave interface in three-dimensional alloy solidification[END_REF] interfaces. A phenomenon of advection appears that drives cells to drift along the interface slope. This advection is linked to the existence, due to the curvature, of an angle between the imposed pulling rate and the thermal gradient which is normal to the interface. Cells grow perpendicular to the solid liquid interface as long as the growth direction is not tilted by anisotropy. Thus, the cell growth velocity has a radial component which in top view makes them drift down the interface slope.

In the present case, dendritic patterns grow with misorientation on a curved interface. To roughly visualize the combined effects of curvature and of crystalline misorientation in terms of pattern drift, we can first suppose that each effect induces its own drifting contribution and the resulting drift is just their addition. Even if this approach is not completely correct, it highlights the main characteristics of drifting velocity fields. The different contributions are drawn in Figure 8 for convex and concave interfaces. Misorientation induces a translational drift which is constant all along the interface (green arrows) on a macroscopically flat interface. On a curved interface, advection induces an axisymmetry (blue arrows), towards the periphery for a convex interface and towards the center for a concave one, and its component increases when distancing from the center. The resulting drifting velocity is represented by the red arrows. Comparing with the flat interface case where the drifting velocity field is constant all over the interface, the interface curvature induces a divergence of the drifting velocity field for a convex interface and a convergence for a concave one. Also, gradients of drifting velocity are built due to curvature. This structuration of the drifting velocity field is consistent with our observations, given in Figure 2b.

This approach should be corrected as the variation of the growth velocity direction with respect to the orientation of thermal gradient and misorientation -described by Eq. ( 2) and Figure 4 -is not considered. A better model can be done in the plane resulting from the transverse interface cutting along the drifting direction, i.e. along the orange line (AB) of Figure 8. We then obtain interface profiles schematically drawn in Figure 4b. For a convex n a curved interface the periphery for a convex interface and towards the center for a concave one, ncreases when distancing from the center.

drift which is consta n a curved interface the periphery for a convex interface and towards the center for a concave one, characteristics for convex and concave interfaces. drift which is constant all along the interface and the resulting drift is just their addition. characteristics for convex and concave interfaces.

of pattern drift, we can first suppose that each effect induce and the resulting drift is just their addition. characteristics visualize the combined effects of curvature and of crystalline misorientation in terms of pattern drift, we can first suppose that each effect induce In the present case, dendritic patterns grow with misorientation on a curved i visualize the combined effects of curvature and of crystalline misorientation in terms by anisotropy. Thus, the cell growth velocity has a radial makes them

In the present case, dendritic patterns grow with misorientation on a curved i normal to the interface. Cells grow perpendicular to the solid by anisotropy. Thus, the cell growth velocity has a radial drift between the imposed pulling normal to the interface. Cells grow perpendicular to the solid This advection is linked to the existence, due to the pulling

The effect of curvature on cellular patterns has been previously described respectively for phenomenon of advection appears This advection is linked to the existence, due to the rate and the

The effect of curvature on cellular patterns has been previously described respectively for phenomenon of advection appears

The effect of curvature on cellular patterns has been previously described respectively for temporal structuration of primary spacing. It is then worth studying the format interface (V = 1.5 µm/s), if one goes from A to B, it means going from configurations to to of Figure 4a, and the resulting drifting velocity simultaneously increases. For a slightly concave interface (V = 6 µm/s), going from A to B means going from configurations to to , and the resulting drifting velocity simultaneously decreases. For a highly concave interface (V = 12 µm/s), we can reach configuration which is associated to an inversion of the drifting velocity. This was experimentally observed with the drift velocity of the stray grains around point B for V = 12 µm/s (Figure 5b).

Equation ( 2) can be applied to the experimental interface shape to get the theoretical variation of drifting velocity along the interface, from A to B. The interface shapes given in Figure 3 can be fitted by z(r) = ar² + b, where z is the height, r the distance from the center and, a and b fitting constants. The local interface slope (r) can be determined by the z(r) derivative. The local misorientation is linked to : , with the misorientation for a flat interface ( = 2.7° as mentioned in §2.3). Moreover, with the relation and Eq. ( 2), we obtain:

(3)

Equation ( 1) can also be adjusted to calculate for a curved interface as follows:

,

where can be calculated by Eq. ( 3) with a given set of f and g values.

We compare the results of such calculations to experimental measurements of drift velocities at V = 1.5 and 6 µm/s in Figure 9. The dendrites included in the measurements are marked in red in Figure 9a andd; they are chosen along the interface diagonal to represent an interface cut. The experimental data of drift velocities (Figure 9c andf) and primary spacing (Figure 9b ande) come from five close images to reduce the dispersion which is unavoidable as only a few dendrites are located in the interest area, and curves are drawn using a sliding average calculated on eight successive points. The theoretical profiles of the drifting velocity in these figures have been calculated using Eq. ( 4) with local Peclet number calculated using experimental primary spacing curves of Figure 9b ande.

For V = 1.5 µm/s, the experimental and calculated curves superpose almost perfectly, thus validating the chosen approach to combine misorientation and curvature and evaluate drifting velocity. The experimental data of d come from at V = 1.5 and 6 µm/s in Figure 9.

in Figure 9a andd; The experimental data of d such at V = 1.5 and 6 µm/s in Figure 9.

in Figure 9a andd; they are such calculations to experimental measurements of at V = 1.5 and 6 µm/s in Figure 9. can be calculated by Eq. ( 3) with a given set of can be calculated by Eq. ( 3

Model, simulations, and parameters

Model equations

We use a well-established quantitative PF model [START_REF] Karma | Phase-field formulation for quantitative modeling of alloy solidification[END_REF][START_REF] Echebarria | Quantitative phase-field model of alloy solidification[END_REF] for solidification of dilute binary alloys that is applicable to different thermal conditions. The model equations are [START_REF] Han | Primary Spacing Selection in Directionally Solidified Alloys[END_REF] , and

, where and are two functions depending on thermal conditions. Here is the PF tha ies smoothly across the diffuse interface, and is a nonlinear preconditioning PF defined as . A dimensionless supersaturation field is coupled to the PF, with the definition alloys that is applicable to different thermal conditions. The model equations are established quantitative PF model alloys that is applicable to different thermal conditions. The model equations are established quantitative PF model

Model, simulations, and parameters Model, simulations, and parameters

spacing evolution (Section 4.2.1), the drifting dynamics of (Section 4.2.2), and the pattern organization (Section 4.2.3).

spacing evolution (Section 4.2.1), the drifting dynamics of the numerical implementation ing the effects of curvature on the primary spacing evolution (Section 4.2.1), the drifting dynamics of (Section 4.2.2), and the pattern organization (Section 4.2.3).

then we introduce the temperature field with in Section 4.1.3. ing the effects of curvature on the primary for solidification of dilute binary alloys that is ap then we introduce the temperature field with simulations and describe applicable to different thermal PF model, we simulations and , [START_REF] Somboonsuk | Dynamical studies of dendritic growth[END_REF] where is the interface solute partition coefficient, is the solute concentration in molar fraction, is the liquidus concentration at the reference temperature , with the melting temperature of pure solvent and the liquidus slope. The length and time are scaled by the diffuse interface thickness and the relaxation time [START_REF] Karma | Phase-field formulation for quantitative modeling of alloy solidification[END_REF][START_REF] Echebarria | Quantitative phase-field model of alloy solidification[END_REF], respectively. The dimensionless diffusivity in Eq. ( 6) is given by , where and . The capillary length is defined at the reference temperature by , with the Gibbs-Thomson coefficient of the solid-liquid interface. The coupling factor is . We consider a standard fourfold anisotropy of the interface free-energy,

, where is the average interface free energy in the plane, is a measure of the strength of the anisotropy, and is the angle between the local surface normal vector and a fixed crystalline axis. For a crystal orientation angle with respect to the reference axis, the anisotropy function in the PF model is . We impose the misorientation angle in PF simulations using the rotation matrix method with details given in [START_REF] Tourret | Grain growth competition during thin-sample directional solidification of dendritic microstructures: A phase-field study[END_REF].

Temperature field

We consider a 2D temperature field for a thin-sample geometry, which can also represent the thermal condition for the cross-section of a 3D cylindrical sample that is parallel to the pulling velocity. With the assumption that the temperature field does not vary with time, can be described by , [START_REF] Trivedi | Pattern-Formation during the Directional Solidification of Binary-Systems[END_REF] where is a constant temperature gradient directed along the vertical -direction, is a function that describes the radial temperature variation along the -direction, is a reference temperature at and . When , Eq. ( 8) reduces to a classical frozen temperature approximation in one dimension. Here, we consider a parabolic shape of the isotherm with extrema at the sample center , where is an arbitrary reference position, is a measure of the amplitude of the macroscopic curvature, is the sample size in the y-direction, and . Then, we can obtain , where is proportional to the maximum variation of the radial temperature at a given sample height. In PF simulations, we use a value for a convex interface at obtained from the direct experimental observations of the interface shape. However, as explained in section 2.2, the interface shape of a concave interface cannot be directly observed from the side view in the DECLIC experiment because the lower center is obscured by dendrites near the crucible boundary. Thus, we use a value obtained from the CrysMAS® simulation for a concave interface with DECLIC experimental conditions at .

To incorporate the temperature field into the PF model, we use the thermal conditions in Eq.

( with respect to the refe in PF simulations using the rotation matrix method with details is the angle between the local surface normal vector and a fixed with respect to the refe

Results of simulations 4.2.1 Primary spacing evolution

We investigate the evolution of the primary spacing under the influence of the macroscopic curvature in PF simulations where a well-oriented single crystal grows in the thin-sample geometry. Figure 10a shows a macroscopically convex interface in the PF simulation at . The locations of cell tips (dots) in the central region follow well the isotherm shape (curves) described by Eq. ( 8) with , which indicates that cell tips still grow at the same temperature under a weak macroscopic curvature. For a few cells near the simulation boundaries, their tips are located slightly below the isotherm that passes through most cell tips in the central region, and those cells at the boundaries are eliminated due to the wetting boundary conditions applied at and . Under the influence of the convex interface curvature, cells drift laterally towards the boundaries, and the evolution of the primary spacing can be significantly affected by the curvature over a long period of time. Figure 10b shows the spatiotemporal evolution of in the simulation of Figure 10a, where we only measure cell tips that are at least 200 away from the simulation boundaries in the -direction. At an early time , the value is unevenly distributed. Then, increases over time across the entire simulation domain, and its spatial distribution becomes smoother. At a time , has lower values near the simulation boundaries and a maximum value at the sample center. As shown in Figure 11a, the average primary spacing increases as cells are eliminated at the boundaries. Meanwhile, both the maximum spacing found at the sample center, and the minimum spacing found near the boundaries increase over time, which is in good qualitative agreement with the experimental observation in Figure 5 for .

Figure 10c shows a macroscopically concave interface in the PF simulation at . In this case, the locations of cell tips (dots) follow well the isotherm shape (curves) described by Eq. ( 8) with , and the effects of the simulation boundaries are small. Figure 10d shows the spatiotemporal evolution of in the simulation of Figure 10c, where we only measure cell tips that are at least 150 away from the simulation boundaries in thedirection. At an early time , the primary spacing is unevenly distributed and has a local maximum near the sample center. Under the influence of the concave interface curvature, cells drift laterally towards the sample center, and near the sample center decreases. Meanwhile, near the right boundary slowly increases over time. At a later time , has the minimum value near the sample center and higher values near simulation boundaries. As shown in Figure 11b, the average primary spacing does not change significantly, while both and slightly decrease over time. This observation partially agrees with the experimental observation in Figure 6 for , where in the latter decreases visibly over time. One possible explanation is that the value obtained from the CrysMAS® simulation for an interface at rest underestimated the magnitude of the concave curvature at . Thus, the curvature effects are potentially smaller in the PF simulation than the experiments.

Drifting dynamics

We investigate the drifting dynamics that usually results from the combined effects of macroscopic curvature and crystal misorientation in experiments. Here, we consider a macroscopically curved interface for a tilted array in a thin-sample geometry. Figure 12a shows the distribution of cell drifting velocity in a PF simulation at , where the interface is macroscopically convex and the crystalline misorientation angle is . Since the crystalline misorientation has a larger effect than the convex curvature, all cells drift towards the right side, i.e., in the same direction of the misorientation. The measured profile (solid line in blue) increases from left to right, and a velocity gradient that prevents a stationary pattern can be clearly observed. This velocity gradient is caused by the fact that the cell drifting is suppressed by the convex curvature on the left side of the simulation domain, while the drifting is favored by the convex curvature on the right side as analyzed in section 3.3. The measured profile is compared to the calculated profile using Eq. (4) (dashed line in red), where the local primary spacing and the constants fitted for a flat interface ( and for ) were used to determine the local drifting velocity.

Figure 12b shows the distribution of cell drifting velocity in a PF simulation at , where the interface is macroscopically concave with a misorientation angle . All cells drift in the same direction of the misorientation, and the measured profile (solid line in blue) decreases from left to right. This is because the cell drifting is favored by the concave curvature on the left side of the simulation domain, while the drifting is suppressed by the concave curvature on the right side. The measured profile is compared to the calculated profile using Eq. ( 4) with and for .

For both convex and concave interfaces, a reasonably good overall agreement is found between the profiles measured in PF simulations and calculated using Eq. ( 4). In both cases, however, there is a notable quantitative difference between measured and predicted values that increases in magnitude with position from the left to the right sample boundary and changes sign in between. One possible explanation for this discrepancy is that the constants and , which were fitted for a flat interface with a small misorientation angle, may depend on curvature. The fact that the sign of the difference between calculated and measured values near each boundary depends on the sign of the macroscopic curvature, which has opposite signs for convex and concave interfaces, supports this interpretation. Solutal interactions of cells with the sample boundaries, which act as a source (left boundary) and sink (right boundary) of cells could further contribute to the difference between calculated and measured values. However, the latter alone cannot easily explain how the sign of this difference changes with the sign of macroscopic curvature.

Pattern organization

We also perform a benchmark PF simulation in 3D to investigate the pattern organization observed in experiments at higher velocities ( and ). Figure 13 shows the top view of a PF simulation with growth conditions and , where the interface is macroscopically concave with a curvature amplitude . Here we only focus on the range from the left border to the sample center, i.e., . The values which has opposite signs for convex and concave interfaces, olutal interactions of cells with may depend on curvature. The fact that the sign of the values near each boundary which has opposite signs for convex and concave interfaces, and changes sign in between , which were fitted for a flat interface with a small misorientation angle, may depend on curvature. The fact that the sign of the in magnitude and changes sign in between , which were fitted for a flat interface with a small misorientation angle, re is a notable in magnitude and changes sign in between profiles measured in PF simulations re is a notable quantitative difference between measured and predicted The initial condition is a disordered hexagonal array that was selected in a 3D PF simulation with a flat interface and no misorientation. We impose a misorientation angle at . As shown in Figure 13a at , the disordered hexagonal array drifts to the right, and new dendrites are created at the source on the left border. Here, the dendritic array drifts in the direction that is parallel to the preferred growth direction of secondary branches, and those secondary branches are normal to the simulation border. This simulation configuration can be compared to the experimental configuration of areas 1 to 4 of Figure 5, where the secondary branches are normal to the crucible border and the drifting velocity direction drives the pattern away from the border. The newly created dendrites form an aligned structure where dendrite primary trunks are aligned along the directions of the secondary branches. They progressively invade the disordered hexagonal array and form an invasion front (orange line) that shifts over time in the direction of the misorientation, as shown in Figures 13b-c. The aligned structure in the simulation is caused by the first type of source, i.e., the simulation boundary that represents a crucible border, and this observation qualitatively agrees with experimental observations in Figure 5.

Conclusion

In this paper, we have presented the analysis of a series of directional solidification experiments carried out in the DECLIC-DSI onboard the International Space Station on a transparent organic alloy (succinonitrile-0.46wt% camphor). Dendritic patterns formation and evolution have been studied for a large range of pulling velocity. Changing this velocity macroscopic curvatures, i.e. curvatures of the envelope of the solidification front on a sample scale much larger than the dendritic array spacing, are obtained depending on the pulling velocity. Curvature naturally arises in a multitude of solidification processes from casting to additive manufacturing. Yet, the effect of curvature is largely unknown.

Our paper reports for the first time a quantitative study of the effect of the macroscopic curvature of the solidification front on the formation of dendritic microstructures, supported by both experiments and 3D phase-field simulations. Quantitative phase-field simulations with the implementation of curved isotherm shapes corresponding to the experiments were performed at experimental growth conditions. We demonstrated, both experimentally and numerically, that even a weak curvature can have a major effect on microstructure evolution and the striking results are summarized: -A solid-liquid interface macroscopic curvature prevents the array from reaching a stationary state, which results in a primary spacing that does not stabilize: it increases continuously if the interface is convex and, inversely, decreases if the interface is concave. -Dendrite stretching or reduction is caused by a radially symmetrical curvature-induced pattern drift (away from or towards this central region for a convex or concave interface, manufacturing. Yet, the effect of curvature is largely unknown.

Our paper reports for the first time a quantitative study of the effect of the macroscopic curvature of the solidification front on the formation of dendritic microstructur Curvature naturally arises in a multitude of solidification processes from manufacturing. Yet, the effect of curvature is largely unknown.

Our paper reports for the first time a quantitative study of the effect of the macroscopic vatures, i.e. curvatures of the envelope of the solidification front on a sample scale much larger than the dendritic array spacing, Curvature naturally arises in a multitude of solidification processes from vatures, i.e. curvatures of the envelope of the solidification front on a sample scale much larger than the dendritic array spacing, The newly created dendrites respectively). This curvature-induced drift interacts with the unidirectional pattern drift induced by a small misorientation resulting in a macroscopic spatial gradient of drift velocity across the entire sample. Under the combined effect of curvature and misorientation, the spacing increases or decreases in the average drift direction induced by misorientation for a convex or concave interface, respectively. -Interface concavity favors the stray grains propagation, which can even lead to the complete overgrowth of the well-oriented grain by off-axis grains. Overgrowth of the favorably oriented grain by off-axis subgrains of worse orientation appears to be generalized to all convergent boundaries in case of concavity. This generalization leads to a final grain texture formed by several off-axis grains. -In the case of concavity, unusual arrangements of dendrites form along the directions of the secondary branches. These alignments result from the formation, due to drifting, of a periodic dendritic source by tertiary branching along the crucible border when secondary branches are roughly normal to the crucible border. Experiments performed in the low gravity environment of space provided a unique opportunity to observe the formation and evolution of directionally solidified 3D extended dendritic patterns in diffusive transport conditions. The results presented here highlight the fundamental role of the macroscopic interface curvature on the characteristics and structuration of the pattern. Interface curvature is unavoidable in solidification processes and therefore it is important to be able to identify its effects. From this point of view, the results obtained here complement previous works elucidating the role of the complex and non-fixed thermal field in 3D configuration [30,31], or of the influence of subgrains [32]. These characteristics, often dismissed as instrumental or experimental biases, are nevertheless critical in the real-world formation of solidification microstructures and must therefore be integrated in process relevant analyses and solidification models. ent previous works fixed thermal field in 3D configuration [30,31] as instrumental or experimental biases, are and therefore it is important to be able to identify its effects. From this point of view, the ent previous works structuration of the pattern. Interface curvature is unavoidable in solidification processes and therefore it is important to be able to identify its effects. From this point of view, the elucidating

The results presented here highlight the curvature on the characteristics and structuration of the pattern. Interface curvature is unavoidable in solidification processes opportunity to observe the formation and evolution of directionally solidified

The results presented here highlight the curvature on the characteristics and Experiments performed in the low gravity environment o opportunity to observe the formation and evolution of directionally solidified

The results presented here highlight the a periodic dendritic source by tertiary branching along the crucible border secondary branches are roughly normal to the crucible border. Experiments performed in the low gravity environment of space

[ 30 ]

 30 points already reported. In Figure are given in Figure where one can easily distinguish the liquid and solid part of the sample and the analysis of its evolution while pulling has been largely 3a where one can easily distinguish the liquid and solid part of the sample the border of the crucible. With this second method of quantification, convex and concave at rest the border of the crucible. With this second method of quantification, quantified by the curvature radius of the parabo corresponds to the difference between the interface positions measured in the center and on the border of the crucible. With this second method of quantification, convex and concave fitted by a parabola. The interface curvature can be corresponds to the difference between the interface positions measured in the center and on

  ), which gives the f and g values at different pulling velocities. As a result, we e also use position s illustrated in Figure 4a e also use it to describe the drifting dynamics positions constants. dynamics under a flat interface with the local interface slope to describe the drifting dynamics is the angle between the preferred <100> direction and the growth direction, constants. dynamics under a flat interface with the local interface slope is the angle between the preferred <100> direction and the growth direction, constants. phenomenological equation and this variation depends is the solute diffusion coefficient in the liquid phenomenological equation ) and this variation depends on the Péclet number is the solute diffusion coefficient in the liquid grain is defined by the angle and on the Péclet number grain is defined by the angle grain is defined by the angle is simply determined by the relation (1) as shown is simply determined by the relation as shown is used to evaluate the tilt angle obtain f = 1.13 and g = 1.38 for V = 1.5 ; f = 0.62 and g = 1.80 for V = 3 ; f = 0.42 and g = 1.76 for V = 6 ; f = 0.60 and g = 1.35 for V = 12 .

  For V = 6 µm/s, the experimental variation is quite well reproduced by calculation in the central area, for r                                               as only a few dendrites are located in the interest area average calculated on eight successive points. figures have been calculated (Figure 9b and e) come from as only a few dendrites are located in the interest area average calculated on eight successive points.

  temperature approximation in one dimension. Here, w reference position, reference temperature at frozen temperature approximation in one dimension. Here, w function that describes the radial temperature variation along the frozen temperature approximation in one dimension. Here, w is a constant temperature gradient directed along the vertical function that describes the radial temperature variation along the is a constant temperature gradient directed along the vertical in PF simulations using the rotation matrix method with details

  near cells drift laterally near the right minimum value near the sample center and higher values near simulation boundaries. As direction. At an early time the sample center. Under the influence of the concave interface curvature, cells drift laterally towards the sample center, and measure cell tips that are at least 150 direction. At an early time the sample center. Under the influence of the concave interface curvature, shows the spatiotemporal evolution of measure cell tips that are at least 150 this case, the locations of cell tips (dots) follow well the , and the effects of the simulation boundaries are shows the spatiotemporal evolution of shows a macroscopically concave interface in the PF simulation at this case, the locations of cell tips (dots) follow well the , and the effects of the simulation boundaries are shows a macroscopically concave interface in the PF simulation at the sample center boundaries increase over time, which is in good qualitative agreement with the experimental increases as cells are eliminated at the boundaries. the sample center, boundaries increase over time, which is in good qualitative agreement with the experimental As shown in Fig increases as cells are eliminated at the boundaries. and the minimum spacing As shown in Fig increases over time across the entire simulation domain has lower values near the simulation boundaries and As shown in Fig only measure cell tips that are at least 200 the increases over time across the entire simulation domain shows the spatiotemporal evolution of only measure cell tips that are at least 200 value is unevenly distributed. Then, can be significantly affected by the curvature over a long period of in the simulation of ls drift laterally towards the boundaries, can be significantly affected by the curvature over a long period of

For

  both convex and concave interfaces, a reasonably good overall agreement is found profiles measured in PF simulations concave curvature on the right side. The measured and For both convex and concave interfaces, a reasonably good overall agreement is found curvature on the left side of the simulation domain The measured This is because the cell drifting is favored by the concave curvature on the left side of the simulation domain in the same direction of the misorientation, and the measured the cell drifting is favored by the concave the distribution of cell drifting velocity where the interface is macroscopically concave with a misorientation angle he measured the cell drifting is favored by the concave the distribution of cell drifting velocity where the interface is macroscopically concave with a misorientation angle in a PF simulation at the local primary spacing and the constants were used to determine the local drifting curvature on the right side o the calculated the constants wetting boundary condition is applied at to represent the sample border, and the noflux (symmetric) boundary condition is applied at to represent the sample center, where the local curvature is zero. The periodic boundary conditions are applied at and . The simulation domain size is , and the total simulated time is 4 h.

  and evolution have been studied for a large range of pulling velocity. Changing this velocity vatures, i.e. curvatures of the envelope of the solidification front on a sample scale much alloy (succinonitrile and evolution have been studied for a large range of pulling velocity. Changing this velocity In this paper, we have presented the analysis of a series of carried out in the DECLIC alloy (succinonitrile In this paper, we have presented the analysis of a series of carried out in the DECLIC experimental observations in Fig experimental observations in Fig The aligned structure in the simulation is caused by the first type of source, i.e., the simulation boundary that represents a crucible border, and this observation experimental observations in Fig over time in the direction of the misorientation The aligned structure in the simulation is caused by the first type of progressively invade the disordered hexagonal array and form an over time in the direction of the misorientation The newly created dendrites aligned structure where dendrite primary trunks are aligned along the directions of the progressively invade the disordered hexagonal array and form an configuration of areas 1 to 4 of Figure where the secondary branches are normal to the crucible border

  Aeronautics and Space Administration . It is based on observations with the DECLIC DEvice for the study of Critical LIquids and Crystallization) work was supported by CNES (French Space Agency) and by Aeronautics and Space Administration work was supported by CNES (French Space Agency) and by process relevant dismissed world formation of process relevant fixed thermal field in 3D configuration dismissed as instrumental or experimental biases, are formation of

Figure 2 .

 2 Figure 2. a) Top-view images of the interface for the different pulling velocities, V = 1.5, 3 , 6 and 12 µm/s, after pulling for respectively 8.3 h, 4.2 h, 2.1 h and 28 min. b) Maps of drift velocity (scale in nm/s) for the different pulling velocities for a grown length of 30 mm. The arrows represent the drift directions.

Figure 3 .

 3 Figure 3. a) Experimental side-views of the solid-liquid interface and b) numerical calculations of the interface shape at rest and at steady-sate for the different pulling velocities.

Figure 4 .

 4 Figure 4. a) Scheme of the basic cases of drifting velocity for various local interface slope . b) Occurrence of these basic cases along the interface for different types and amplitudes of curvature. Directions: , orientation of the preferred <001>; , thermal gradient; , interface velocity; , dendrite growth velocity; , dendrite drift velocity. Angles: , interface slope; between and ; between and ; between and (equal to for a local flat interface); between and .

Figure 5 .

 5 Figure 5. Study of invading sub-grains at V = a) 6 and b) 12 µm/s. Top view images of the interface after respectively 115 min and 61 min of growth. Rough limits of different straygrains are drawn in a) to help visualization. For the different sub-grains, drifting velocity vectors of some dendrites are drawn and the corresponding module given. On b), straygrains where dendrite tips are aligned with secondary branches appear : areas 1 to 4, enlarged in c). In areas 1 to 4, examples of alignments are given with markers positioned on dendrite tips. Such alignments are not observed in area 0 roughly located at 45° of secondary branches as illustrated on the scheme of d).

Figure 6 .

 6 Figure 6. Time evolution of primary spacing for the different pulling velocities. In red : average values; maximum and minimum values are respectively in blue and yellow. For V = 1.5, 3 and 6 µm/s, measurements were performed during the whole solidification, starting just after the initial transient and ending after 60 mm of pulling. At V = 12 µm/s, measurements end after 36.8 mm of pulling with the disappearance of the well-oriented grain.

Figure 7 .

 7 Figure 7. Spatial variation of primary spacing (b) and drifting velocity (c) for V = 1.5 µm/s. Measurements are performed in small regions defined in a). The evolution of spacing is characterized between 20 and 40 mm of pulling and compared to the local initial spacing for each area. The drifting velocity in each area is compared to the average value.

Figure 8 .

 8 Figure 8. Scheme of building of the drifting velocity field for convex and concave interfaces. Misorientation induces a translational drift which is constant all along the interface (green arrows). Advection on a curved interface (blue arrows) is axisymmetric and its component increases when distancing from the center. The resulting drifting velocity is represented by the red arrows.

Figure 9 .

 9 Figure 9. Analysis of drifting velocity profile along diagonal. All dendrites tagged in red are included in measurements for a) V = 1.5 and d) 6 µm/s. b) and e) Spatial primary spacing variation of the dendrites respectively tagged in a) and d). c) and f) Experimental and theoretically calculated profiles of the drifting velocity of the dendrites respectively tagged in a) and d). The r direction is schematically shown in a).

Figure 10 .

 10 Figure 10. (a) Phase-field simulations of directional solidification of a SCN-0.46 wt% camphor alloy in a thin-sample geometry with a convex interface at V = 1.5 m/s. (b) The evolution of the primary spacing in phase-field simulations of (a), where t1 = 2.8, t2 = 4.6, t3 = 6.4, t4 = 8.2, and t5 = 10.0 h are the time after the beginning of the phase-field simulation from a planar interface. (c) Simulations with a concave interface at V = 6 m/s. (d) The evolution of (c), where t1 = 1.6, t2 = 3.4, t3 = 5.2, t4 = 7.0, and t5 = 8.8 h. The curves in (a) and (c) represent isotherms, and dots represent cell tips.

Figure 11 .

 11 Figure 11. The evolution of the average primary spacing ave, the maximum primary spacing max, and the minimum primary spacing min in phase-field simulations of directional solidification of an SCN-0.46 wt% camphor alloy in a thin-sample geometry with (a) a convex interface at V = 1.5 m/s and (b) a concave interface at V = 6 m/s.

Figure 12 .

 12 Figure 12. Analysis of drifting velocity profiles in phase-field simulations with an SCN-0.46 wt% camphor alloy performed for G = 12 K/cm, a) V = 1.5 m/s, and b) V = 6 m /s. The solid lines in blue represent the measured drifting velocity profiles, and the dashed lines in red represent the theoretically calculated drifting velocity profiles.

Figure 13 .

 13 Figure 13. The invasion of dendrites formed at the left boundary (a source) in 3D phase-field simulations of directional solidification of an SCN-0.46 wt% camphor alloy with G = 12 K/cm and V = 6 m/s, where the time t = (a) 0.12 h, (b) 2 h, and (c) 4 h. The interface is concave, and the misorientation angle is 0 = 2.7°. A dendrite pattern close to the hexagonal structure is gradually replaced by an aligned pattern. The orange lines represent the invasion front.

  International Space Station (ISS) to benefit from a microgravity It is jointly operated by the French Space Agency (CNES) and NASA., mainly dedicated to cellular growth liquid interface morphology on bulk samples of transparent materials. the study of Critical LIquids and Crystallization) and it is installed within the International Space Station (ISS) to benefit from a microgravity and real time characterization of the dynamical selection of the solid liquid interface morphology on bulk samples of transparent materials.

	installed within the
	It is jointly operated by the French Space Agency (CNES) and NASA.
	2011, mainly dedicated to cellular growth 2011, mainly dedicated to cellular growth
	and improvement of numerical models and improvement of numerical models
	second series of experiments second series of experiments
	dedicated to dendritic growth. dedicated to dendritic growth.
	This article reports new experimental results and This article reports new experimental results and This article reports new experimental results and
	effects of interface curvature on dendritic pat effects of interface curvature on dendritic pat
	In the experimental part ( In the experimental part (
	experimental device and methods
	interfaces, such as macroscopic shapes and the crystallograp

field modeling with macroscopic interface curvature

  Computational modeling of the DECLIC experiments for directional solidification of a transparent dilute alloy to date has focused exclusively on observations obtained in a velocity range where the solidification front is flat[START_REF] Reinhart | Influence of natural convection during upward directional solidification: A comparison between in situ X-ray radiography and direct simulation of the grain structure[END_REF] 28,[START_REF] Tourret | Oscillatory cellular patterns in three-dimensional directional solidification[END_REF] 31, 32]. However, solidification fronts with a macroscopic curvature are frequently observed[30]. In this section, by implementing a radial temperature gradient into the quantitative PF model, we can initiate a macroscopically convex or concave interface during PF simulations and investigate its longtime effects on the microstructure evolution. We first describe a quantitative phase-field model for solidification of dilute binary alloys that is applicable to different thermal conditions in Section 4.1.1, then we introduce the temperature field with a radial gradient in Section 4.1.2 and the numerical implementation in Section 4.1.3. The results of PF

	  
	              
	
	
	
	4 Phase-
		r
		can be
	= 2.7° as mentioned in = 2.7° as mentioned in
	to calculate	for a curved interface
	) with a given set of

simulations are discussed in Section 4.2, including the effects of curvature on the primary spacing evolution (Section 4.2.1), the drifting dynamics of misoriented cells under a curvature (Section 4.2.2), and the pattern organization (Section 4.2.3).

Acknowledgments

This work was supported by CNES (French Space Agency) and by NASA (National Aeronautics and Space Administration) Grants No. NNX12AK54G, 80NSSC19K0135, and 80NSSC22K0664. It is based on observations with the DECLIC-DSI (Directional Solidification Insert of the DEvice for the study of Critical LIquids and Crystallization) embarked on the ISS (International Space Station).

, [START_REF] Eshelman | Cellular spacings I. Steady-state growth[END_REF] and , [START_REF] Seetharaman | Cellular spacings II. Dynamical studies[END_REF] where , , and are scaled by the diffuse interface thickness , and is scaled by the relaxation time . The dimensionless pulling velocity and thermal length in Eqs. ( 9)- [START_REF] Seetharaman | Cellular spacings II. Dynamical studies[END_REF] are defined by and , where is the dimensional thermal length.

Implementation

We solve Eqs. ( 5)-( 6) on a cubic lattice of grid spacing using a finite-difference implementation of spatial derivatives and an Euler explicit time stepping scheme with a constant time step . The spatial discretization utilizes a simple seven-point stencil that contains six nearest-neighbor points in 100 directions on a cubic lattice [START_REF] Ji | Isotropic finite-difference approximations for phase-field simulations of polycrystalline alloy solidification[END_REF]. The PF model is solved on massively parallel graphic processing units (GPUs) with the computer unified device architecture (CUDA) programming language. In addition, we use the multi-GPU technique to improve the efficiency of large-scale simulations [32].

We perform numerical simulations for directional solidification of a SCN-0.46 wt% camphor alloy with the DECLIC experimental conditions, including the temperature gradient directed along the vertical -direction and the pulling velocity and 6

. The alloy parameters include the partition coefficient the liquidus slope [START_REF] Mota | Quantitative determination of the solidus line in the dilute limit of succinonitrile-camphor alloys[END_REF], the diffusion coefficient , and the crystalline anisotropy . The numerical parameters are chosen to be the interface thickness , the grid spacing , and the explicit time step for simulations at ; , , and for simulations at . In the following sections, the simulation domain size of a thin-sample geometry is for simulations at , and for simulations at , where is the sample thickness in the -direction. We consider wetting boundary conditions where a layer of liquid is systematically wetting the sample walls at , , , and . Since the triple line where the solid-liquid interface contacts with the sample wall is located deep within the mushy zone, we choose a simplified description of the wetting boundary condition by forcing a contact angle of the interface with the walls on the nearest layer of grid points next to the sample walls, with details given in [START_REF] Tourret | Grain growth competition during thin-sample directional solidification of dendritic microstructures: A phase-field study[END_REF]. Initially, we create a curved interface with a parabolic shape at the liquidus temperature, which also matches the isotherm shape. The simulated time is set to 10 hours, which takes approximately 10 and 20 days of computation time with two Nvidia V100 GPUs for simulations at and 6 , respectively. We also perform PF simulations in a 3D bulk sample, as will be discussed in Section 4. The spatial discretization utilizes directions on a cubic lattice on massively parallel graphic processing units (GPUs) with the computer unified Euler explicit time stepping scheme with a The spatial discretization utilizes a simple seven directions on a cubic lattice on a cubic lattice of grid spacing Euler explicit time stepping scheme with a a simple seven using a finite

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: relationships that could have appeared to influence the work reported declare that they have relationships that could have appeared to influence the work reported declare that they have s Quantitative determination of the solidus line in the dilute limit of succinonitrile