A generalized field method for multiphase transformations using interface fields

Steinbach, Ingo; Pezzolla, F.

doi:10.1016/s0167-2789(99)00129-3

Cited by 492 publications

(346 citation statements)

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“…Within the multi-phase field model, [13,14] grain growth in polycrystalline materials is described by the temporal evolution of phase field, φ i (r, t), which represents a probability of finding a grain with an orientation, i, at given spatial point, r, and time, t. The temporal evolution of φ i is described by the following equation, [14] ( ) . In the present study, σ is set to σ = 0.79 J m -2 [15] and W is given to be W = 6·Δx with the square grid spacing,…”

Section: Computational Detailsmentioning

confidence: 99%

“…The FCG cannot grow along its short axis direction because of the pinning effect of liquid phase which also prevents the FCRB from moving. [11] In the light of these facts, we omitted the initial formation process of FCG and we dealt with only the grain growth starting from already formed FCG structure during cooling as detailed below.Within the multi-phase field model, [13,14] grain growth in polycrystalline materials is described by the temporal evolution of phase field, φ i (r, t), which represents a probability of finding a grain with an orientation, i, at given spatial point, r, and time, t. The temporal evolution of φ i is described by the following equation, [14] ( ) . In the present study, σ is set to σ = 0.79 J m -2 [15] and W is given to be W = 6·Δx with the square grid spacing,…”

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confidence: 99%

See 1 more Smart Citation

Austenite Grain Growth in Peritectic Solidified Carbon Steels Analyzed by Phase-Field Simulation

Ohno

Tsuchiya

Matsuura

2012

Metall Mater Trans A

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Formation of coarse columnar grains (CCG) in as-cast austenite structure of peritectic carbon steels is one of the serious problems in continuous casting processes. It was recently elucidated that the formation of CCG is ascribed to a discontinuous grain growth.Furthermore, the critical condition for the discontinuous growth to occur was elicited on the basis of phase field simulations and a theory of grain growth. In this study, by means of the phase field simulations, the detailed investigation is carried out for grain coarsening of as-cast austenite structure. It is demonstrated in the two-dimensional simulations that the coarsest grain structure emerges by the discontinuous growth in the vicinity of the critical condition. In addition, a model for predicting the upper limit of grain size during the discontinuous growth is proposed. The model successfully describes the experimental result with reasonable accuracy. 2 I. INTRODUCIONAs-cast γ-austenite structure in continuously cast slabs of peritectic carbon steels consists of coarse columnar grains (CCG) in the vicinity of the slab surfaces. Since the size of CCG is closely related to the occurrence of surface cracking and low ductility of the slabs, [1][2][3][4] the precise description and prediction of the as-cast γ grain size are quite important issues in the field of casting of steels.A number of attempts have been made for the description and prediction of size of the CCG during continuous casting processes. [5][6][7][8][9][10] In the early works, the CCG structure has been supposed to originate from continuous grain growth which occurs during cooling below a temperature for completion of transformation to γ single phase, T γ . [5][6][7][8] However, our recent study by means of rapid unidirectional solidification experiment revealed that the formation of CCG structure is actually ascribed to a discontinuous grain growth. [11] Figure 1 shows the as-cast γ grain structure observed in a 0.2 mass% carbon steel quenched during the unidirectional solidification. The CCG develop from the mold side to the upper part.Importantly, fine columnar γ grains (FCG) exist ahead of the CCG region. The boundary between FCG and CCG regions, which in this paper is termed the FCG/CCG region boundary (FCRB), is identified in Figure 1. From temperature measurements, it was found that the temperature at FCRB is always T γ during the solidification and, hence, the FCG region corresponds to liquid + γ two phase field, while the CCG region is γ single phase field. 3Although the FCG always form ahead of the growing CCG during the solidification, the FCG near FCRB shrink with the growth of CCG. More specifically, coarse γ grains, which formed in the vicinity of the mold wall, preferentially develop along the temperature gradient below T γ at the expense of FCG, finally growing to CCG. This process is characterized by vertical motion of FCRB in Figure 1 and is classified as a discontinuous grain growth. Since the liquid phase in FCG region prevents the migration of FCRB, the FC...

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Section: Computational Detailsmentioning

confidence: 99%

mentioning

confidence: 99%

Austenite Grain Growth in Peritectic Solidified Carbon Steels Analyzed by Phase-Field Simulation

Ohno

Tsuchiya

Matsuura

2012

Metall Mater Trans A

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“…In this work, each individual g grain is assigned its own phase-field variable, according to the multiphase field formalism. [26][27][28]32 The spatial distribution of the liquid solder, the substrate, and the nucleated grains in the system is mathematically expressed by using N arrays of phase fields / i (x, t) (i = 1, …, N), which can be expressed with multiphase field variables. The phase-field variables in this model can be defined as / 1 for the solid substrate, / 2, …, NÀ1 for the nucleating grains, and / N for the liquid solder.…”

Section: Multiphase Field and Diffusion Equationsmentioning

confidence: 99%

“…IMC growth has been modeled previously [24][25][26] by using phasefield approaches. 27,28 These mathematical models have proved to be a very useful tool to investigate the microstructural evolution of IMCs as a function of factors such as solid/liquid interfacial energies, grain boundary energies, relative mass transport enhancement by grain boundaries, and so forth. [24][25][26] So far, these simulation efforts have focused on the evolution of IMC layers at late stages.…”

Section: Introductionmentioning

confidence: 99%

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Formation and Growth of Intermetallic Compound Cu6Sn5 at Early Stages in Lead-Free Soldering

Park

Arróyave

2010

Journal of Elec Materi

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In this work, the early stages of the formation and growth of the intermetallic compound Cu 6 Sn 5 during soldering reactions between a Cu substrate and liquid Sn are examined through phase-field simulations. The liquid Sn-based solder (L phase) and the copper substrate (a phase) are considered to be under metastable equilibrium conditions that eventually lead to nucleation of the Cu 6 Sn 5 intermetallic compound (IMC) (g phase) at the solid/liquid interface. Nucleation is incorporated into the model through a classical treatment considering that individual nucleation events follow a Poisson distribution function. The driving forces for the nucleation and phase transformations are obtained by coupling the phase-field simulations to CALPHAD models. In the phase-field simulations, physical properties such as liquid surface as well as IMC interfacial energies are treated parametrically to probe the behavior of the system under various growth conditions. The simulations are compared with previous works and are shown to have good (qualitative) agreement with recent detailed studies on the early stages of the interaction between Cu and liquid Sn.

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Application of the Phase‐Field Model to Four‐Phase Reactions in Ternary Alloys

Kundin¹,

Emmerich²

2013

TMS2013 Supplemental Proceedings

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A generalized field method for multiphase transformations using interface fields

Cited by 492 publications

References 4 publications

Austenite Grain Growth in Peritectic Solidified Carbon Steels Analyzed by Phase-Field Simulation

Austenite Grain Growth in Peritectic Solidified Carbon Steels Analyzed by Phase-Field Simulation

Formation and Growth of Intermetallic Compound Cu6Sn5 at Early Stages in Lead-Free Soldering

Application of the Phase‐Field Model to Four‐Phase Reactions in Ternary Alloys

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