Phase-field modelling of directional melting of lamellar and rod eutectic structures

Tamás Pusztai1, László Rátkai1, Levente Horváth, László Gránásy1,2

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

The melting of 2D lamellar and 3D rod eutectic structures was studied by multi-phase-field simulations. A binary model eutectic system was investigated in a standard directional solidification setup. Switching from solidification to melting was realised by inverting the pulling speed of the sample. Steady state melting profiles were obtained as the long-time solutions of the evolution equations solved in the minimal representative domains of the periodic structures. Series of simulations were performed using different volume fractions of the initial solid phases, different values of the temperature gradient, and different widths of the simulation domain. It was found that melting occurs with a nearly flat interface if the average composition of the initial solid structure is equal to the eutectic composition. If the volume fraction of the solid phases is changed and the average composition becomes off-eutectic, then the melting positions of the phases decouple: the phase of sub-eutectic amount will melt near the eutectic temperature, while the phase of super-eutectic amount will melt near its liquidus temperature corresponding to the off-eutectic mean composition. The lamellar/rod spacing imposed by the width of the simulation domain has only minor effect on the interface temperature. Besides the steady states, we could observe two kinds of instabilities in the regime of non-planar melting both in 2D and 3D. By increasing the lamellar spacing, oscillations may appear around the trijunction and along the phase boundaries. By decreasing the lamellar spacing and increasing the pulling speed, the lamellae/rods of the phase that protrude deeper in the melt become thinner and may eventually break up to a series of small spherical particles before melting completely.

Three-dimensional numerical simulation of droplet formation by Rayleigh–Taylor instability in multiphase corium

R. Zanella, R. Le Tellier, Mathis Plapp1, György Tegze2, Hervé Henry1

1Laboratoire Physique de la Matière Condensée, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
2Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary

During a severe accident in a nuclear reactor, the melting of the core may lead to the formation of a multiphase liquid pool (corium) in the vessel lower head. The heat transfer at the boundary with the vessel is affected by diffusive and convective mass fluxes. In particular, the development of Rayleigh–Taylor instabilities influence the thickness of the top metallic layer and therefore the “focusing effect” of the heat flux, which is the main risk for the vessel integrity. We use a Cahn–Hilliard pseudo-binary model to describe the uranium/oxygen/zirconium/iron mixture. The diffusion and the convection are governed by the Cahn–Hilliard equation and the Navier–Stokes equations under the Boussinesq approximation. In this work, the model is isothermal and the buoyancy force is only due to the gradient of chemical composition. The model is solved in three dimensions with a pseudo-spectral code. The initial configuration consists of a light layer of iron-rich fluid above a heavy layer of uranium/oxygen/zirconium mixture. A thin layer of heavier metallic phase lays at the interface and eventually triggers a Rayleigh–Taylor instability. The metallic phase forms a plume which falls downward and then breaks up into droplets due to the Rayleigh-Plateau instability. The phenomenon is alimented by diffusion which generates the heavy metallic phase at the interface. The droplet formation observed in an experiment of corium stratification transient from the literature is qualitatively captured. The mobility, the viscosity and the surface tension are shown to have an influence on the mass transfer.

Phase-Field Modeling of Biomineralization in Mollusks and Corals: Microstructure vs Formation Mechanism

László Gránásy1,2, László Rátkai1, Gyula Tóth3, Pupa Gilbert, Igor Zlotnikov4, Tamás Pusztai1

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
3Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
4B CUBE - Center for Molecular Bioengineering, Technische Universität Dresden, Germany

While biological crystallization processes have been studied on the microscale extensively, there is a general lack of models addressing the mesoscale aspects of such phenomena. In this work, we investigate whether the phase-field theory developed in materials’ science for describing complex polycrystalline structures on the mesoscale can be meaningfully adapted to model crystallization in biological systems. We demonstrate the abilities of the phase-field technique by modeling a range of microstructures observed in mollusk shells and coral skeletons, including granular, prismatic, sheet/columnar nacre, and sprinkled spherulitic structures. We also compare two possible micromechanisms of calcification: the classical route, via ion-by-ion addition from a fluid state, and a nonclassical route, crystallization of an amorphous precursor deposited at the solidification front. We show that with an appropriate choice of the model parameters, microstructures similar to those found in biomineralized systems can be obtained along both routes, though the time-scale of the nonclassical route appears to be more realistic. The resemblance of the simulated and natural biominerals suggests that, underneath the immense biological complexity observed in living organisms, the underlying design principles for biological structures may be understood with simple math and simulated by phase-field theory.

Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification

Frigyes Podmaniczky1, László Gránásy1,2

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

Two-step nucleation and subsequent growth processes were investigated in the framework of the single mode phase-field crystal model combined with diffusive dynamics (corresponding to colloid suspensions) and hydrodynamical density relaxation (simple liquids). It is found that independently of dynamics, nucleation starts with the formation of solid precursor clusters that consist of domains with noncrystalline ordering (ringlike projections are seen from certain angles), and regions that have amorphous structure. Using the average bond order parameter q¯6, we distinguished amorphous, medium range crystallike order (MRCO), and crystalline local orders. We show that crystallization to the stable body-centered cubic phase is preceded by the formation of a mixture of amorphous and MRCO structures. We have determined the time dependence of the phase composition of the forming solid state. We also investigated the time/size dependence of the growth rate for solidification. The bond order analysis indicates similar structural transitions during solidification in the case of diffusive and hydrodynamic density relaxation.

Topics: Phase field crystal

Crystal nucleation and growth of spherulites demonstrated by coral skeletons and phase-field simulations

Chang-Yu Sun, László Gránásy1,2, Cayla Stifler, Tal Zaquin, Rajesh Chopdekar, Nobumichi Tamura, James Weaver, Jun Zhang, Stefano Goffredo, Giuseppe Falini, Matthew Marcus, Tamás Pusztai1, Vanessa Schoeppler3, Tali Mass, Pupa Gilbert

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
3B CUBE - Center for Molecular Bioengineering, Technische Universität Dresden, Germany

Spherulites are radial distributions of acicular crystals, common in biogenic, geologic, and synthetic systems, yet exactly how spherulitic crystals nucleate and grow is still poorly understood. To investigate these processes in more detail, we chose scleractinian corals as a model system, because they are well known to form their skeletons from aragonite (CaCO3) spherulites, and because a comparative study of crystal structures across coral species has not been performed previously. We observed that all 12 diverse coral species analyzed here exhibit plumose spherulites in their skeletons, with well-defined centers of calcification (CoCs), and crystalline fibers radiating from them. In 7 of the 12 species, we observed a skeletal structural motif not observed previously: randomly oriented, equant crystals, which we termed “sprinkles”. In Acropora pharaonis, these sprinkles are localized at the CoCs, while in 6 other species, sprinkles are either layered at the growth front (GF) of the spherulites, or randomly distributed. At the nano- and micro-scale, coral skeletons fill space as much as single crystals of aragonite. Based on these observations, we tentatively propose a spherulite formation mechanism in which growth front nucleation (GFN) of randomly oriented sprinkles, competition for space, and coarsening produce spherulites, rather than the previously assumed slightly misoriented nucleations termed “non-crystallographic branching”. Phase-field simulations support this mechanism, and, using a minimal set of thermodynamic parameters, are able to reproduce all of the microstructural variation observed experimentally in all of the investigated coral skeletons. Beyond coral skeletons, other spherulitic systems, from aspirin to semicrystalline polymers and chocolate, may also form according to the mechanism for spherulite formation proposed here.

Phase field benchmark problems for nucleation

W. Wu, D. Montiel, J.E. Guyer, P.W. Voorhees, Warren JA, D. Wheeler, L Gránásy, Pusztai T, O.G. Heinonen

We present nucleation phase field model benchmark problems, expanding on our previous benchmark problems on diffusion, precipitation, dendritic growth, linear elasticity, fluid flow and electrochemistry. Nucleation is the process in which either a new thermodynamic phase or a new structure is created, such as solidification from the melt, or self-assembly of particulates. Based on where the nucleation occurs, it can be divided into two main categories: homogeneous nucleation and heterogeneous nucleation. In the first nucleation benchmark problem, we focus on homogeneous nucleation for both single seed under different initial conditions and multiple seeds. The second nucleation benchmark problem focuses on athermal heterogeneous nucleation and nucleation behavior near the free growth limit with different undercooling driving forces.

Topics: Heterogeneous nucleation

Two- and three-dimensional simulations of Rayleigh–Taylor instabilities using a coupled Cahn–Hilliard/Navier–Stokes model

R. Zanella, G. Tegze, R. Le Tellier, H. Henry

We report on two- and three-dimensional numerical simulations of Rayleigh–Taylor instabilities in immiscible fluids. A diffuse-interface model that combines the Cahn–Hilliard equation, governing the evolution of the volume fraction of one fluid, and the Navier–Stokes equations, governing the bulk velocity and pressure, is used. The study is limited to low Atwood numbers owing to the use of the Boussinesq approximation. The code is based on a pseudo-spectral method. A linear analysis is first performed in a two-dimensional case of Rayleigh–Taylor instability to confirm that the model very well captures this phenomenon in the case of inviscid or viscid fluids. One key aspect of this work is that the influence of the thermodynamic parameters related to the Cahn–Hilliard equation (interface thickness and mobility) is quantitively studied. Three-dimensional results of Rayleigh–Taylor instabilities in viscous fluids are then presented to show the possibilities of this modeling. We observe the effect of the viscosity and the wavelength of an initial single-mode perturbation on the mass transport during the nonlinear regime.

Ultrafine Fe-Fe2Ti eutectics by directed energy deposition: Insights into microstructure formation based on experimental techniques and phase field modelling

G. Requena, K. Bugelnig, F. Sket, S. Milenkovic, G. Rödler, A. Weisheit, J. Gussone, J. Haubrich, P. Barriobero-Vila, Tamás Pusztai1, Gránásy László, A. Theofilatos, J.C. da Silva, U. Hecht

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary

We investigated the Fe-Fe2Ti eutectic microstructure obtained by Directed Energy Deposition (DED) with a hypereutectic composition of Fe-17.6 at.% Ti. Ultrafine lamellar spacings as low as 200 nm were achieved, features which otherwise can only be obtained in thin specimens, e.g. by suction casting. However, at interlayer boundaries (ILBs) a globular morphology of the primary Fe2Ti phase is observed with halos of the Fe phase. For the given DED conditions the crystalline structure is thus discontinuous across the ILBs. Both 2D and 3D analysis methods were used to quantify the microstructure, including high resolution synchrotron holographic X-ray computed tomography (HXCT). The generic behaviour of eutectic systems under conditions that qualitatively correspond to those of laser additive manufacturing was explored by phase-field modelling for selected nucleation scenarios and alloy compositions spanning from eutectic to hyper-eutectic. While providing valuable insights into microstructure formation, the simulations point out the need to further deepen our understanding about melting under additive manufacturing conditions in order to implement suitable nucleation and / or free growth models. The simulations also show that globular ILBs can be prevented when using exactly eutectic alloy compositions.