Application of the maximal entropy production principle to rapid solidification: A multi-phase-field model

Wang, K.; Liu, Feng; Ehlen, Georg; Herlach, D.M.

doi:10.1016/j.actamat.2013.01.041

Cited by 33 publications

(23 citation statements)

References 45 publications

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“…In recent works [40][41][42], similar variational formulations were carried out within the framework of two-phase approach by utilizing the Lagrange multiplier, while the purposes, details of the procedures, and thus the final forms of the time evolution equations differ from those discussed in this paper. The mobility for φ given by Eq.…”

Section: ) Wherementioning

confidence: 99%

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Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

2016

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We present the variational formulation of a quantitative phase-field model for isothermal low-speed solidification in a binary dilute alloy with diffusion in the solid. In the present formulation, cross-coupling terms between the phase field and composition field, including the so-called antitrapping current, naturally arise in the time evolution equations. One of the essential ingredients in the present formulation is the utilization of tensor diffusivity instead of scalar diffusivity. In an asymptotic analysis, it is shown that the correct mapping between the present variational model and a free-boundary problem for alloy solidification with an arbitrary value of solid diffusivity is successfully achieved in the thin-interface limit due to the cross-coupling terms and tensor diffusivity. Furthermore, we investigate the numerical performance of the variational model and also its nonvariational versions by carrying out two-dimensional simulations of free dendritic growth. The nonvariational model with tensor diffusivity shows excellent convergence of results with respect to the interface thickness.

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Section: ) Wherementioning

confidence: 99%

“…The relation given by Eq. (2.4) is taken into account by employing the Lagrange multiplier [39][40][41][42]. To be more specific, the time derivative of the following functional is considered:…”

Section: B Time Evolution Equationsmentioning

confidence: 99%

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

2016

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“…The phase-field variable φ takes values of +1 in a solid and −1 in a liquid and it continuously changes from +1 to −1 inside the solid-liquid interface. In this formulation, we employ the two-phase approach, in which the thermodynamic quantities in the solid and liquid are separately defined and the interface is described as a mixture of two phases [10,32,34,35]. As briefly mentioned in the previous section, one of the important points in the present formulation is that the diffusion fluxes for the conserved variables in both the solid and liquid are first formulated and then they are related to each other by introducing the local equilibrium conditions.…”

Section: A Entropy Functional and Time Evolution Equationsmentioning

confidence: 99%

“…(21) is always positive or equal to zero. This can be achieved by introducing phenomenological transport coefficients [5,7,35]. Care must be taken to ensure the following points [31].…”

Section: A Entropy Functional and Time Evolution Equationsmentioning

confidence: 99%

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Variational formulation of a quantitative phase-field model for nonisothermal solidification in a multicomponent alloy

2017

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A variational formulation of a quantitative phase-field model is presented for nonisothermal solidification in a multicomponent alloy with two-sided asymmetric diffusion. The essential ingredient of this formulation is that the diffusion fluxes for conserved variables in both the liquid and solid are separately derived from functional derivatives of the total entropy and then these fluxes are related to each other on the basis of the local equilibrium conditions. In the present formulation, the cross-coupling terms between the phase-field and conserved variables naturally arise in the phase-field equation and diffusion equations, one of which corresponds to the antitrapping current, the phenomenological correction term in early nonvariational models. In addition, this formulation results in diffusivities of tensor form inside the interface. Asymptotic analysis demonstrates that this model can exactly reproduce the free-boundary problem in the thin-interface limit. The present model is widely applicable because approximations and simplifications are not formally introduced into the bulk's free energy densities and because off-diagonal elements of the diffusivity matrix are explicitly taken into account. Furthermore, we propose a nonvariational form of the present model to achieve high numerical performance. A numerical test of the nonvariational model is carried out for nonisothermal solidification in a binary alloy. It shows fast convergence of the results with decreasing interface thickness.

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The Hodograph Equation

Galenko

2024

Phase Field Theory in Materials Physics

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Application of the maximal entropy production principle to rapid solidification: A multi-phase-field model

Cited by 33 publications

References 45 publications

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

Variational formulation of a quantitative phase-field model for nonisothermal solidification in a multicomponent alloy

The Hodograph Equation

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