Non-Equilibrium Interface Dynamics Theory

doi:10.11900/0412.1961.2024.00306

Acta Metall Sin

2025, Vol. 61

Issue (1): 29-42 DOI: 10.11900/0412.1961.2024.00306

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Non-Equilibrium Interface Dynamics Theory

WANG Haifeng(

), PU Zhenxin, ZHANG Jianbao

Advanced Lubrication and Sealing Materials Research Center, State Key Laboratory of Solidification Technology, Northwestern Polytechnical University, Xi'an 710072, China

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WANG Haifeng, PU Zhenxin, ZHANG Jianbao. Non-Equilibrium Interface Dynamics Theory. Acta Metall Sin, 2025, 61(1): 29-42.

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Abstract

Recently, the rapid advancement of extreme non-equilibrium material processing and fabrication techniques, such as 3D printing and rapid die-casting, has led to the continuous development of new materials with exceptional properties. However, current non-equilibrium processing technologies face technical challenges, such as the lack of clear guidelines for process optimization, which considerably limits the advancement and application of advanced materials. The solidification and solid phase transformations involved in materials prepared through non-equilibrium processing pertain to a non-equilibrium dissipative system and manifest throughout the entire dynamic process of material fabrication. By investigating key scientific issues such as non-equilibrium phase transformation dynamics, non-equilibrium solute diffusion, and solute-drag effects, developing a theoretical framework for the entire non-equilibrium material processing, from solidification to solid phase transformation is possible. This not only provides theoretical support for the design and fabrication of non-equilibrium materials but also introduces novel concepts for optimizing process parameters in non-equilibrium processing technologies. This review is crucial for advancing non-equilibrium phase transformation theory and deepening our understanding of fundamental theoretical research. Interfaces play a critical role in microstructure control during material processing, thereby making an accurate theoretical description of their kinetics is especially important. This review focuses on the common characteristics of liquid/solid interfaces during melting, solid/liquid interfaces during solidification, and solid/solid interfaces during solid state phase transformations and summarizes and analyzes the history and current state of sharp-interface models for interface kinetics. Using the solidification of binary alloys as an example, the review first introduces interface kinetic theories under local non-equilibrium conditions, covering descriptions of interface kinetic processes and interface kinetic models for steady-state and non-steady-state conditions. The physical nature of one-step and two-step trans-interface diffusion is demonstrated. Next, the review describes interface kinetic theories under full non-equilibrium conditions by comparing the applications of the kinetic energy method and the effective mobility method for non-equilibrium solute diffusion in bulk phases. Thereafter, it introduces interface kinetic theories incorporating the partial solute drag effect present and discusses limitations in current methods for addressing partial solute drag. This study aims to enhance understanding of interface kinetics, offering insights into microstructure control. Finally, an outlook on the future of non-equilibrium interface kinetic theories is provided, which outlines directions for future research.

Key words: interface kinetics non-equilibrium interface non-equilibrium solute diffusion solute-drag effect

Received: 03 September 2024

ZTFLH:

TG111

Fund: National Natural Science Foundation of China(51975474);Fundamental Research Funds for the Central Universities(3102019JC001)

Corresponding Authors: WANG Haifeng, professor, Tel: (029)88460311, E-mail: haifengw81@nwpu.edu.cn

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https://www.ams.org.cn/EN/10.11900/0412.1961.2024.00306 OR https://www.ams.org.cn/EN/Y2025/V61/I1/29

Fig.1 Schematics for the interface kinetic processes in different phase transformation types (L—liquid, S—solid, I—interface, A—solvent atom A, B—solute atom B, α—solid phase α, β—solid phase β, the same below)
(a) liquid/solid interface during melting
(b) solid/liquid interface during solidification
(c) solid/solid (α/β) interface during solid-state phase-transformation

Fig.2 Schematic for the solid/liquid interface kinetic processes when the solid with the aimed composition is solidified directly from the liquid

Fig.3 Mole Gibbs free energy diagram for the solid/liquid interface kinetic processes under a steady-state condition (Green and red curves represent the Gibbs free energy curve of the solid phase and liquid phase, respectively, the same below. Total Gibbs free energy dissipated by the interface after solidification of 1 mol liquid is $Δ G t o t a l$ = $(1 - C S *) Δ μ A * + C S * Δ μ B *$ . By translating the tangent of Gibbs free energy curve of solid at $C S *$ to the Gibbs free energy curve of liquid at $C L *$ , $Δ G t o t a l$ is divided into two parts: The upper one for trans-interface diffusion is $Δ G D$ = $(C L * - C S *) (Δ μ A * - Δ μ B *)$ and latter part for interface migration is $Δ G m$ = $(1 - C L *) Δ μ A * + C L * Δ μ B *$ . $Δ G m$ —Gibbs free energy of interface migration, $Δ G D$ —Gibbs free energy of trans-interface diffusion, $C S *$ —the solid composition at the interface, $C L *$ —the liquid composition at the interface, $Δ μ A *$ —the chemical potential of A across the interface, $Δ μ B *$ —the chemical potential of B across the interface, $μ A L *$ and $μ A S *$ —the chemical potential at the interface for liquid and solid phase B, $μ B L *$ and $μ B S *$ —the chemical potential at the interface for liquid and solid phase B, respectively, $g m S$ —the mole Gibbs free energy of the solid, $g m L$ —the mole Gibbs free energy of the liquid)

Fig.4 Mole Gibbs free energy diagram for the solid/liquid interface kinetic processes under a non-steady-state condition (Total Gibbs free energy dissipated by the interface after solidification of 1 mol liquid is $Δ G t o t a l$ = $(1 - C t r a n s) Δ μ A * + C t r a n s Δ μ B *$ . By translating the tangent of Gibbs free energy curve of solid at $C S *$ to the Gibbs free energy curve of liquid at $C L *$ , $Δ$ G_total is divided into two parts: The upper one for trans-interface diffusion is $Δ G D$ = $(C L * - C t r a n s) (Δ μ A * - Δ μ B *)$ and latter part for interface migration is $Δ G m$ = $(1 - C L *) Δ μ A * + C L * Δ μ B *$ . The difference in $Δ$ G_total between the steady-state condition and the non-steady-state condition is $Δ G t r a n s ‐ S$ = $(C t r a n s - C S *) (Δ μ A * - Δ μ B *)$ , which is the Gibbs free energy dissipated to adjust the actual composition transferred across the interface from $C S *$ to C_trans ( $C t r a n s$ —the solute component that finally crosses the interface and enters the solid phase)

Fig.5 Schematics for solute trans-interface diffusion in one step (a) and two steps (b) (For the former, there is only one flux for solute trans-interface diffusion, i.e., $J D$ = $J B L *$ = $(C L * - C S *) V / V m$ , whereas for the latter, there are two fluxes for solute trans-interface diffusion, i.e., $J D S → I$ = $(C t r a n s * - C S *) V / V m$ = $- J B S *$ and $J D I → L$ = $(C L * - C t r a n s *) V / V m$ = $J B L *$ . J_D—the flux of solute trans-interface diffusion, $J D S → I$ —the flux of trans-interface diffusion from solid phase to interface, $J D I → L$ —the flux of trans-interface diffusion from interface to liquid phase, $C t r a n s *$ —the solute component at the interface that finally crosses the interface and enters the solid phase, V—the velocity of the migrating interface, V_m—mole volume)

Fig.6 Mole Gibbs free energy diagram for the solid/liquid interface kinetic processes with a partial solute-drag effect under a non-steady-state condition (The total Gibbs free energy dissipated by the interface after solidification of 1 mol liquid is $Δ G t o t a l$ = $(1 - C t r a n s) Δ μ A * + C t r a n s Δ μ B *$ . By translating the tangent of Gibbs free energy curve of solid at $C S *$ to the Gibbs free energy curve of liquid at C_eff, $Δ$ G_total is divided into two parts. The upper one for trans-interface diffusion is $Δ G D$ = $(C e f f - C t r a n s) (Δ μ A * - Δ μ B *)$ and latter part for interface migration is $Δ$ G_m = $(1 - C e f f) Δ μ A * + C e f f Δ μ B *$ . The difference in $Δ$ G_D between the steady-state condition and the non-steady-state condition is $Δ G t r a n s ‐ S$ = $(C t r a n s - C S *) (Δ μ A * - Δ μ B *)$ , which is the Gibbs free energy dissipated to adjust the actual composition transferred across the interface from $C S *$ to C_trans for trans-interface diffusion. The difference in $Δ$ G_m between the two cases with a full solute-drag effect and with a partial solute-drag effect is $Δ G L ‐ e f f$ = $(C L * - C e f f) (Δ μ A * - Δ μ B *)$ , which is the Gibbs free energy dissipated to adjust the liquid composition ahead of the solid/liquid interface from $C L *$ to C_eff for interface migration. It should be pointed out that Hareland et al. ^[30] confused the concepts of C_trans under non-steady-state and C_eff. Furthermore, it is also queried that the definition of C_eff can be stilled be used for a non-steady state condition. For example, there will be a problem when $C S *$ < C_eff < C_trans (C_eff—effective concentration)

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