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Phase Field Modeling for Multiphase Problems

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A special issue of Materials (ISSN 1996-1944). This special issue belongs to the section "Materials Simulation and Design".

Deadline for manuscript submissions: closed (20 August 2022) | Viewed by 14719

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Prof. Luisa Silva

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Guest Editor

Institut du Calcul Intensif, Ecole Centrale de Nantes, Nantes, France
Interests: advanced numerical techniques used in High Performance Computing; stable/stabilized immersed finite element methods; interface capturing through modified level-set set; phase-field methods; anisotropic mesh adaptation; developments in a massively parallel context

Special Issue Information

Dear Colleagues,

A phase field model is a mathematical model for solving interfacial problems. Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification.

The diffuse-interface approach allows us to study the evolution of arbitrary complex grain morphologies without any presumption on their shape or mutual distribution. It is also easy to account for different thermodynamic driving forces for microstructure evolution, such as bulk and interfacial energy, elastic energy and electric or magnetic energy, and the effect of different transport processes, such as mass diffusion, heat conduction, and convection. The phase-field method has already proven its usefulness to simulate microstructural evolution for several applications, e.g., during solidification, solid-state phase transformations, fracture, etc.

This Special Issue will explore the following: the concept of diffuse interfaces, phase-field variables, thermodynamic driving forces for microstructure evolution and the kinetic phase-field equations, common techniques for parameter determination and numerical solutions for the equations, and possibilities to solve the equations describing microstructural evolution.

This Special Issue will collect scientific works that solve multiphase problems by using phase-field modeling. Research articles, communications, and reviews are all welcome.

Prof. Luisa Silva
Guest Editor

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Keywords

phase field model
microstructure evolution
phase-field equations
solidification
solid-state phase transformations
fracture

Published Papers (7 papers)

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Research

18 pages, 3223 KiB

Open AccessArticle

Study of Microstructural Morphology of Ti-6Al-4V Alloy by Crystallographic Analysis and Phase Field Simulation

by Hao Xiang, Wim Van Paepegem and Leo A. I. Kestens

Materials 2022, 15(15), 5325; https://0-doi-org.brum.beds.ac.uk/10.3390/ma15155325 - 02 Aug 2022

Cited by 1 | Viewed by 1677

Abstract

Formation of a habit plane during martensitic transformation is related to an invariant plane strain transformation, which involves dislocation glide and twins. In the current work, the Phenomenological Theory of Martensitic Transformation (PTMT) is employed to study the crystallographic features while the phase field simulation is used to study the microstructure evolution for martensitic transformation of Ti-6Al-4V alloy. Results show that mechanical constraints play a key role in the microstructure evolution. It is shown that a twinned structure with very small twinned variants is geometrically difficult to form due to the lattice parameters of Ti-6Al-4V alloy. It is concluded that the predicted habit plane from the PTMT is consistent with results of the micro-elastic theory. The formation of a triangular morphology is favored geometrically and elastically. Full article

(This article belongs to the Special Issue Phase Field Modeling for Multiphase Problems)

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21 pages, 3870 KiB

Open AccessArticle

Rotating Directional Solidification of Ternary Eutectic Microstructures in Bi-In-Sn: A Phase-Field Study

by Kaveh Dargahi Noubary, Michael Kellner and Britta Nestler

Materials 2022, 15(3), 1160; https://0-doi-org.brum.beds.ac.uk/10.3390/ma15031160 - 02 Feb 2022

Cited by 7 | Viewed by 1642

Abstract

For the first time, the experimental processing condition of a rotating directional solidification is simulated in this work, by means of a grand-potential-based phase-field model. To simulate the rotating directional solidification, a new simulation setup with a rotating temperature field is introduced. The newly developed configuration can be beneficent for a more precise study of the ongoing adjustment mechanisms during temperature gradient controlled solidification processes. Ad hoc, the solidification of the ternary eutectic system Bi-In-Sn with three distinct solid phases

α, β, δ

is studied in this paper. For this system, accurate in situ observations of both directional and rotating directional solidification experiments exist, which makes the system favorable for the investigation. The two-dimensional simulation studies are performed for both solidification processes, considering the reported 2D patterns in the steady state growth of the bulk samples. The desired

α β α δ

phase ordering repeat unit is obtained within both simulation types. By considering anisotropy of the interfacial energies, experimentally reported tilted lamellae with respect to normal vectors of the solidification front, as well as predominant role of

α β

anisotropy in tilting phenomenon, are observed. The results are validated by using the Jackson–Hunt analysis and by comparing with the existing experimental data. The convincing agreements indicate the applicability of the introduced method. Full article

(This article belongs to the Special Issue Phase Field Modeling for Multiphase Problems)

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10 pages, 2241 KiB

Open AccessArticle

Lattice Phase Field Model for Nanomaterials

by Pingping Wu and Yongfeng Liang

Materials 2021, 14(23), 7317; https://0-doi-org.brum.beds.ac.uk/10.3390/ma14237317 - 29 Nov 2021

Cited by 6 | Viewed by 1453

Abstract

The lattice phase field model is developed to simulate microstructures of nanoscale materials. The grid spacing in simulation is rescaled and restricted to the lattice parameter of real materials. Two possible approaches are used to solve the phase field equations at the length scale of lattice parameter. Examples for lattice phase field modeling of complex nanostructures are presented to demonstrate the potential and capability of this model, including ferroelectric superlattice structure, ferromagnetic composites, and the grain growth process under stress. Advantages, disadvantages, and future directions with this phase field model are discussed briefly. Full article

(This article belongs to the Special Issue Phase Field Modeling for Multiphase Problems)

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14 pages, 3747 KiB

Open AccessArticle

The Morphology and Solute Segregation of Dendrite Growth in Ti-4.5% Al Alloy: A Phase-Field Study

by Yongmei Zhang, Xiaona Wang, Shuai Yang, Weipeng Chen and Hua Hou

Materials 2021, 14(23), 7257; https://0-doi-org.brum.beds.ac.uk/10.3390/ma14237257 - 27 Nov 2021

Cited by 2 | Viewed by 1413

Abstract

Ti-Al alloys have excellent high-temperature performance and are often used in the manufacture of high-pressure compressors and low-pressure turbine blades for military aircraft engines. However, solute segregation is easy to occur in the solidification process of Ti-Al alloys, which will affect their properties. In this study, we used the quantitative phase-field model developed by Karma to study the equiaxed dendrite growth of Ti-4.5% Al alloy. The effects of supersaturation, undercooling and thermal disturbance on the dendrite morphology and solute segregation were studied. The results showed that the increase of supersaturation and undercooling will promote the growth of secondary dendrite arms and aggravate the solute segregation. When the undercooling is large, the solute in the root of the primary dendrite arms is seriously enriched, and when the supersaturation is large, the time for the dendrite tips to reach a steady-state will be shortened. The thermal disturbance mainly affects the morphology and distribution of the secondary dendrite arms but has almost no effect on the steady-state of the primary dendrite tips. This is helpful to understand the cause of solute segregation in Ti-Al alloy theoretically. Full article

(This article belongs to the Special Issue Phase Field Modeling for Multiphase Problems)

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10 pages, 4394 KiB

Open AccessArticle

Controlling Equilibrium Morphologies of Bimetallic Nanostructures Using Thermal Dewetting via Phase-Field Modeling

by Taejin Kwak and Dongchoul Kim

Materials 2021, 14(21), 6697; https://0-doi-org.brum.beds.ac.uk/10.3390/ma14216697 - 07 Nov 2021

Cited by 2 | Viewed by 1627

Abstract

Herein, we report a computational model for the morphological evolution of bimetallic nanostructures in a thermal dewetting process, with a phase-field framework and superior optical, physical, and chemical properties compared to those of conventional nanostructures. The quantitative analysis of the simulation results revealed nano-cap, nano-ring, and nano-island equilibrium morphologies of the deposited material in thermal dewetting, and the morphologies depended on the gap between the spherical patterns on the substrate, size of the substrate, and deposition thickness. We studied the variations in the equilibrium morphologies of the nanostructures with the changes in the shape of the substrate pattern and the thickness of the deposited material. The method described herein can be used to control the properties of bimetallic nanostructures by altering their equilibrium morphologies using thermal dewetting. Full article

(This article belongs to the Special Issue Phase Field Modeling for Multiphase Problems)

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30 pages, 5641 KiB

Open AccessArticle

Phase-Field Model for the Simulation of Brittle-Anisotropic and Ductile Crack Propagation in Composite Materials

by Christoph Herrmann, Daniel Schneider, Ephraim Schoof, Felix Schwab and Britta Nestler

Materials 2021, 14(17), 4956; https://0-doi-org.brum.beds.ac.uk/10.3390/ma14174956 - 30 Aug 2021

Cited by 2 | Viewed by 1813

Abstract

In this work, a small-strain phase-field model is presented, which is able to predict crack propagation in systems with anisotropic brittle and ductile constituents. To model the anisotropic brittle crack propagation, an anisotropic critical energy release rate is used. The brittle constituents behave linear-elastically in a transversely isotropic manner. Ductile crack growth is realised by a special crack degradation function, depending on the accumulated plastic strain, which is calculated by following the

J_{2}

-plasticity theory. The mechanical jump conditions are applied in solid-solid phase transition regions. The influence of the relevant model parameters on a crack propagating through a planar brittle-ductile interface, and furthermore a crack developing in a domain with a single anisotropic brittle ellipsoid, embedded in a ductile matrix, is investigated. We demonstrate that important properties concerning the mechanical behaviour of grey cast iron, such as the favoured growth of cracks along the graphite lamellae and the tension–compression load asymmetry of the stress–strain response, are covered by the model. The behaviour is analysed on the basis of a simulation domain consisting of three differently oriented elliptical inclusions, embedded in a ductile matrix, which is subjected to tensile and compressive load. The material parameters used correspond to graphite lamellae and pearlite. Full article

(This article belongs to the Special Issue Phase Field Modeling for Multiphase Problems)

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19 pages, 8176 KiB

Open AccessArticle

3D Phase Field Modeling of Multi-Dendrites Evolution in Solidification and Validation by Synchrotron X-ray Tomography

by Shuo Wang, Zhipeng Guo, Jinwu Kang, Meishuai Zou, Xiaodong Li, Ang Zhang, Wenjia Du, Wei Zhang, Tung Lik Lee, Shoumei Xiong and Jiawei Mi

Materials 2021, 14(3), 520; https://0-doi-org.brum.beds.ac.uk/10.3390/ma14030520 - 21 Jan 2021

Cited by 3 | Viewed by 3388

Abstract

In this paper, the dynamics of multi-dendrite concurrent growth and coarsening of an Al-15 wt.% Cu alloy was studied using a highly computationally efficient 3D phase field model and real-time synchrotron X-ray micro-tomography. High fidelity multi-dendrite simulations were achieved and the results were compared directly with the time-evolved tomography datasets to quantify the relative importance of multi-dendritic growth and coarsening. Coarsening mechanisms under different solidification conditions were further elucidated. The dominant coarsening mechanisms change from small arm melting and interdendritic groove advancement to coalescence when the solid volume fraction approaches ~0.70. Both tomography experiments and phase field simulations indicated that multi-dendrite coarsening obeys the classical Lifshitz–Slyozov–Wagner theory

R^{n} - R_{0}^{n} {= k}_{c} ({t - t}_{0})

, but with a higher constant of n = 4.3. Full article

(This article belongs to the Special Issue Phase Field Modeling for Multiphase Problems)

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