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Akamatsu S, Bottin-Rousseau S, Witusiewicz VT, Hecht U, Plapp M, Ludwig A, Mogeritsch J, Şerefoğlu M, Bergeon N, Mota FL, Sturz L, Zimmermann G, McFadden S, Sillekens W. Microgravity studies of solidification patterns in model transparent alloys onboard the International Space Station. NPJ Microgravity 2023; 9:83. [PMID: 37852993 PMCID: PMC10584907 DOI: 10.1038/s41526-023-00326-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2022] [Accepted: 09/17/2023] [Indexed: 10/20/2023] Open
Abstract
We review recent in situ solidification experiments using nonfaceted model transparent alloys in science-in-microgravity facilities onboard the International Space Station (ISS), namely the Transparent Alloys (TA) apparatus and the Directional Solidification Insert of the DEvice for the study of Critical Liquids and Crystallization (DECLIC-DSI). These directional-solidification devices use innovative optical videomicroscopy imaging techniques to observe the spatiotemporal dynamics of solidification patterns in real time in large samples. In contrast to laboratory conditions on ground, microgravity guarantees the absence or a reduction of convective motion in the liquid, thus ensuring a purely diffusion-controlled growth of the crystalline solid(s). This makes it possible to perform a direct theoretical analysis of the formation process of solidification microstructures with comparisons to quantitative numerical simulations. Important questions that concern multiphase growth patterns in eutectic and peritectic alloys on the one hand and single-phased, cellular and dendritic structures on the other hand have been addressed, and unprecedented results have been obtained. Complex self-organizing phenomena during steady-state and transient coupled growth in eutectics and peritectics, interfacial-anisotropy effects in cellular arrays, and promising insights into the columnar-to-equiaxed transition are highlighted.
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Affiliation(s)
- S Akamatsu
- Sorbonne Université, CNRS-UMR 7588, Institut des NanoSciences de Paris, case courrier 840, 4 place Jussieu, 75252, Paris, Cedex 05, France.
| | - S Bottin-Rousseau
- Sorbonne Université, CNRS-UMR 7588, Institut des NanoSciences de Paris, case courrier 840, 4 place Jussieu, 75252, Paris, Cedex 05, France
| | | | - U Hecht
- Access e.V., Intzestr. 5, 52072, Aachen, Germany
| | - M Plapp
- Laboratoire de Physique de la Matière Condensée, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120, Palaiseau, France
| | - A Ludwig
- Department Metallurgy, University of Leoben, Franz-Josef-Str. 18, 8700, Leoben, Austria
| | - J Mogeritsch
- Department Metallurgy, University of Leoben, Franz-Josef-Str. 18, 8700, Leoben, Austria
| | - M Şerefoğlu
- Department of Metallurgical and Materials Engineering, Marmara University, 34854 Maltepe, Istanbul, Turkey
| | - N Bergeon
- Aix Marseille Univ, Université de Toulon, CNRS, IM2NP, Marseille, France
| | - F L Mota
- Aix Marseille Univ, Université de Toulon, CNRS, IM2NP, Marseille, France
| | - L Sturz
- Access e.V., Intzestr. 5, 52072, Aachen, Germany
| | - G Zimmermann
- Access e.V., Intzestr. 5, 52072, Aachen, Germany
| | - S McFadden
- School of Computing, Engineering, and Intelligent Systems, Ulster University, Northland Road, Derry/Londonderry, Northern Ireland, BT48 7JL, UK
| | - W Sillekens
- European Space Agency, ESTEC - Research and Utilisation Group, Directorate of Human and Robotic Exploration Programmes, Keplerlaan 1, 2201 AZ, Noordwijk, Netherlands
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Zanella R, Tellier RL, Plapp M, Tegze G, Henry H. Three-dimensional numerical simulation of droplet formation by Rayleigh–Taylor instability in multiphase corium. Nuclear Engineering and Design 2021. [DOI: 10.1016/j.nucengdes.2021.111177] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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Philippe T, Henry H, Plapp M. A regularized phase-field model for faceting in a kinetically controlled crystal growth. Proc Math Phys Eng Sci 2020; 476:20200227. [PMID: 33071578 DOI: 10.1098/rspa.2020.0227] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2020] [Accepted: 09/01/2020] [Indexed: 11/12/2022] Open
Abstract
At equilibrium, the shape of a strongly anisotropic crystal exhibits corners when for some orientations the surface stiffness is negative. In the sharp-interface problem, the surface free energy is traditionally augmented with a curvature-dependent term in order to round the corners and regularize the dynamic equations that describe the motion of such interfaces. In this paper, we adopt a diffuse interface description and present a phase-field model for strongly anisotropic crystals that is regularized using an approximation of the Willmore energy. The Allen-Cahn equation is employed to model kinetically controlled crystal growth. Using the method of matched asymptotic expansions, it is shown that the model converges to the sharp-interface theory proposed by Herring. Then, the stress tensor is used to derive the force acting on the diffuse interface and to examine the properties of a corner at equilibrium. Finally, the coarsening dynamics of the faceting instability during growth is investigated. Phase-field simulations reveal the existence of a parabolic regime, with the mean facet length evolving in t , with t the time, as predicted by the sharp-interface theory. A specific coarsening mechanism is observed: a hill disappears as the two neighbouring valleys merge.
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Affiliation(s)
- T Philippe
- Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique, CNRS, IP Paris, 91128 Palaiseau, France
| | - H Henry
- Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique, CNRS, IP Paris, 91128 Palaiseau, France
| | - M Plapp
- Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique, CNRS, IP Paris, 91128 Palaiseau, France
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Claudin P, Jarry H, Vignoles G, Plapp M, Andreotti B. Physical processes causing the formation of penitentes. Phys Rev E Stat Nonlin Soft Matter Phys 2015; 92:033015. [PMID: 26465564 DOI: 10.1103/physreve.92.033015] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2015] [Indexed: 05/05/2023]
Abstract
Snow penitentes form in sublimation conditions by differential ablation. Here we investigate the physical processes at the initial stage of penitente growth and perform the linear stability analysis of a flat surface submitted to the solar heat flux. We show that these patterns do not simply result from the self-illumination of the surface-a scale-free process-but are primarily controlled by vapor diffusion and heat conduction. The wavelength at which snow penitentes emerge is derived and discussed. We found that it is controlled by aerodynamic mixing of vapor above the ice surface.
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Affiliation(s)
- P Claudin
- Laboratoire de Physique et Mécanique des Milieux Hétérogènes, PMMH UMR 7636 ESPCI, CNRS, University Paris Diderot, University Pierre et Marie Curie, 10 rue Vauquelin, 75005, Paris, France
| | - H Jarry
- Laboratoire de Physique et Mécanique des Milieux Hétérogènes, PMMH UMR 7636 ESPCI, CNRS, University Paris Diderot, University Pierre et Marie Curie, 10 rue Vauquelin, 75005, Paris, France
| | - G Vignoles
- Laboratoire des Composites ThermoStructuraux, LCTS UMR 5801 CNRS, UB1, CEA-Safran, 3 allée La Boetie, 33600 Pessac, France
| | - M Plapp
- Condensed Matter Physics, Ecole Polytechnique, CNRS, 91128 Palaiseau, France
| | - B Andreotti
- Laboratoire de Physique et Mécanique des Milieux Hétérogènes, PMMH UMR 7636 ESPCI, CNRS, University Paris Diderot, University Pierre et Marie Curie, 10 rue Vauquelin, 75005, Paris, France
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Abstract
Local structural arrest in random packings of colloidal or granular spheres is quantified by a caging number, defined as the average minimum number of randomly placed spheres on a single sphere that immobilize all its translations. We present an analytic solution for the caging number for two-dimensional hard disks immobilized by neighbor disks which are placed at random positions under the constraint of a nonoverlap condition. Immobilization of a disk with radius r = 1 by arbitrary larger neighbor disks with radius r > or = 1 is solved analytically, whereas for contacting neighbors with radius 0 < r < 1, the caging number can be evaluated accurately with an approximate excluded volume model that also applies to spheres in higher Euclidean dimension. Comparison of our exact two-dimensional caging number with studies on random disk packing indicates that it relates to the average coordination number of random loose packing, whereas the parking number is more indicative for coordination in random dense packing of disks.
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Affiliation(s)
- A Wouterse
- Van't Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute, Utrecht University, Padualaan 8, 3584 CH, Utrecht, The Netherlands
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Abstract
A phase-field model that allows for quantitative simulations of low-speed eutectic and peritectic solidification under typical experimental conditions is developed. Its cornerstone is a smooth free-energy functional, specifically designed so that the stable solutions that connect any two phases are completely free of the third phase. For the simplest choice for this functional, the equations of motion for each of the two solid-liquid interfaces can be mapped to the standard phase-field model of single-phase solidification with its quartic double-well potential. By applying the thin-interface asymptotics and by extending the antitrapping current previously developed for this model, all spurious corrections to the dynamics of the solid-liquid interfaces linear in the interface thickness W can be eliminated. This means that, for small enough values of W, simulation results become independent of it. As a consequence, accurate results can be obtained using values of W much larger than the physical interface thickness, which yields a tremendous gain in computational power and makes simulations for realistic experimental parameters feasible. Convergence of the simulation outcome with decreasing W is explicitly demonstrated. Furthermore, the results are compared to a boundary-integral formulation of the corresponding free-boundary problem. Excellent agreement is found, except in the immediate vicinity of bifurcation points, a very sensitive situation where noticeable differences arise. These differences reveal that, in contrast to the standard assumptions of the free-boundary problem, out of equilibrium the diffuse trijunction region of the phase-field model can (i) slightly deviate from Young's law for the contact angles, and (ii) advance in a direction that forms a finite angle with the solid-solid interface at each instant. While the deviation (i) extrapolates to zero in the limit of vanishing interface thickness, the small angle in (ii) remains roughly constant, which indicates that it might be a genuine physical effect, present even for an atomic-scale interface thickness.
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Affiliation(s)
- R Folch
- Laboratoire de Physique de la Matière Condensée, CNRS/Ecole Polytechnique, Palaiseau, France
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Folch R, Plapp M. Towards a quantitative phase-field model of two-phase solidification. Phys Rev E Stat Nonlin Soft Matter Phys 2003; 68:010602. [PMID: 12935120 DOI: 10.1103/physreve.68.010602] [Citation(s) in RCA: 57] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2003] [Indexed: 11/07/2022]
Abstract
We construct a diffuse-interface model of two-phase solidification that quantitatively reproduces the classic free boundary problem on solid-liquid interfaces in the thin-interface limit. Convergence tests and comparisons with boundary integral simulations of eutectic growth show good accuracy for steady-state lamellae, but the results for limit cycles depend on the interface thickness through the trijunction behavior. This raises the fundamental issue of diffuse multiple-junction dynamics.
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Affiliation(s)
- R Folch
- Laboratoire de Physique de la Matière Condensée, CNRS/Ecole Polytechnique, 91128 Palaiseau, France
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Akamatsu S, Plapp M, Faivre G, Karma A. Pattern stability and trijunction motion in eutectic solidification. Phys Rev E Stat Nonlin Soft Matter Phys 2002; 66:030501. [PMID: 12366091 DOI: 10.1103/physreve.66.030501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2002] [Indexed: 05/23/2023]
Abstract
We demonstrate by both experiments and phase-field simulations that lamellar eutectic growth can be stable for a wide range of spacings below the point of minimum undercooling at low velocity, contrary to what is predicted by existing stability analyses. This overstabilization can be explained by relaxing Cahn's assumption that lamellae grow locally normal to the eutectic interface.
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Affiliation(s)
- S Akamatsu
- Groupe de Physique des Solides, CNRS UMR 7588, Universités Denis-Diderot et Pierre-et-Marie-Curie, Tour 23, 2 place Jussieu, 75251 Paris Cedex 05, France
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Lo TS, Karma A, Plapp M. Phase-field modeling of microstructural pattern formation during directional solidification of peritectic alloys without morphological instability. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 63:031504. [PMID: 11308654 DOI: 10.1103/physreve.63.031504] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2000] [Indexed: 05/23/2023]
Abstract
During the directional solidification of peritectic alloys, two stable solid phases (parent and peritectic) grow competitively into a metastable liquid phase of larger impurity content than either solid phase. When the parent or both solid phases are morphologically unstable, i.e., for a small temperature gradient/growth rate ratio (G/v(p)), one solid phase usually outgrows and covers the other phase, leading to a cellular-dendritic array structure closely analogous to the one formed during monophase solidification of a dilute binary alloy. In contrast, when G/v(p) is large enough for both phases to be morphologically stable, the formation of the microstructure becomes controlled by a subtle interplay between the nucleation and growth of the two solid phases. The structures that have been observed in this regime (in small samples where convection effects are suppressed) include alternate layers (bands) of the parent and peritectic phases perpendicular to the growth direction, which are formed by alternate nucleation and lateral spreading of one phase onto the other as proposed in a recent model [R. Trivedi, Metall. Mater. Trans. A 26, 1 (1995)], as well as partially filled bands (islands), where the peritectic phase does not fully cover the parent phase which grows continuously. We develop a phase-field model of peritectic solidification that incorporates nucleation processes in order to explore the formation of these structures. Simulations of this model shed light on the morphology transition from islands to bands, the dynamics of spreading of the peritectic phase on the parent phase following nucleation, which turns out to be characterized by a remarkably constant acceleration, and the types of growth morphology that one might expect to observe in large samples under purely diffusive growth conditions.
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Affiliation(s)
- T S Lo
- Physics Department and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Massachusetts 02115, USA
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Karma A, Lee YH, Plapp M. Three-dimensional dendrite-tip morphology at low undercooling. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 2000; 61:3996-4006. [PMID: 11088190 DOI: 10.1103/physreve.61.3996] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/1999] [Indexed: 05/23/2023]
Abstract
We investigate the three-dimensional morphology of the dendrite tip using the phase-field method. We find that, for low undercoolings, this morphology is ostensibly independent of anisotropy strength except for a localized shape distortion near the tip that only affects the value of the tip radius rho [which is crudely approximated by rho approximately (1-alpha)rho(Iv) where rho(Iv) is the Ivantsov tip radius of an isothermal paraboloid with the same tip velocity and alpha is the stiffness anisotropy]. The universal tip shape, which excludes this distortion, is well fitted by the form z=-r(2)/2+A(4)r(4) cos 4straight phi where |z| is the distance from the tip and all lengths are scaled by rho(Iv). This fit yields A4 in the range 0.004-0.005 in good quantitative agreement with the existing tip morphology measurements in succinonitrile [LaCombe et al., Phys. Rev. E 52, 2778 (1995)], which are reanalyzed here and found to be consistent with a single cos 4straight phi mode nonaxisymmetric deviation from a paraboloid. Moreover, the fin shape away from the tip is well fitted by the power law z=-a|x|(5/3) with a approximately 0.68. Finally, the characterization of the operating state of the dendrite tip is revisited in the light of these results.
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Affiliation(s)
- A Karma
- Physics Department and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Massachusetts 02115, USA
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Abstract
We present a novel computational method to simulate accurately a wide range of interfacial patterns whose growth is limited by a large-scale diffusion field. To illustrate the computational power of this method, we demonstrate that it can be used to simulate three-dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance.
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Affiliation(s)
- M Plapp
- Physics Department and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Massachusetts 02115, USA
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Plapp M, Karma A. Eutectic colony formation: a stability analysis. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1999; 60:6865-89. [PMID: 11970626 DOI: 10.1103/physreve.60.6865] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/1998] [Revised: 04/09/1999] [Indexed: 04/18/2023]
Abstract
Experiments have widely shown that a steady-state lamellar eutectic solidification front is destabilized on a scale much larger than the lamellar spacing by the rejection of a dilute ternary impurity and forms two-phase cells commonly referred to as "eutectic colonies." We extend the stability analysis of Datye and Langer [V. Datye and J. S. Langer, Phys. Rev. B 24, 4155 (1981)] for a binary eutectic to include the effect of a ternary impurity. We find that the expressions for the critical onset velocity and morphological instability wavelength are analogous to those for the classic Mullins-Sekerka instability of a monophase planar interface, albeit with an effective surface tension that depends on the geometry of the lamellar interface and, nontrivially, on interlamellar diffusion. A qualitatively new aspect of this instability is the occurrence of oscillatory modes due to the interplay between the destabilizing effect of the ternary impurity and the dynamical feedback of the local change in lamellar spacing on the front motion. In a transient regime, these modes lead to the formation of large scale oscillatory microstructures for which there is recent experimental evidence in a transparent organic system. Moreover, it is shown that the eutectic front dynamics on a scale larger than the lamellar spacing can be formulated as an effective monophase interface free boundary problem with a modified Gibbs-Thomson condition that is coupled to a slow evolution equation for the lamellar spacing. This formulation provides additional physical insights into the nature of the instability and a simple means to calculate an approximate stability spectrum. Finally, we investigate the influence of the ternary impurity on a short wavelength oscillatory instability that is already present at off-eutectic compositions in binary eutectics.
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Affiliation(s)
- M Plapp
- Physics Department and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Massachusetts 02115, USA
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