Comparative analysis of different numerical schemes in solute trapping simulations by using the phase-field model with finite interface dissipation

Abstract

Two different numerical schemes, the standard explicit scheme and the time-elimination relaxation one, in the framework of phase-field model with finite interface dissipation were employed to investigate the solute trapping effect in a Si-4.5 at.% As alloy during rapid solidification. With the equivalent input, a unique solute distribution under the steady state can be obtained by using the two schemes without restriction to numerical length scale and interface velocity. By adjusting interface width and interface permeability, the experimental solute segregation coefficients can be well reproduced. The comparative analysis of advantages and disadvantages in the two numerical schemes indicates that the time-elimination relaxation scheme is preferable in one-dimensional phase-field simulation, while the standard explicit scheme seems to be the only choice for two- or three dimensional phase-field simulation. Furthermore, the kinetic phase diagrams in the Si-As system were predicted by using the phase-field simulation with the time-elimination relaxation scheme.

Phase field crystal simulation of the structure evolution between the hexagonal and square phases at elevated pressures

“…Among the available models, the traditional phase-field method has recently emerged as a powerful numerical method for studying the evolution of microstructure. For example, Yang et al predicted the kinetic phase diagrams in the Si-As system by using the phase-field simulation with the time-elimination relaxation scheme [3]. Uehara [4] proposed a phase field model for predicting deformation behavior under an applied stress.…”

Kinetic Phase Diagrams of Ternary Al-Cu-Li System during Rapid Solidification: A Phase-Field Study

“…During the phase transition process, a local redistribution flux through the phase boundary is considered at the interface, and this solute exchange process between phases is described by a kinetic equation. The model has been applied to simulate the solute trapping in different binary alloys during rapid solidification [ 22 , 24 , 25 ] and later to predict the binary kinetic phase diagram [ 29 ].…”

“…Generally, this standard explicit scheme is easily coded but inefficient to predict interface moving velocity dependent kinetic phase diagram. Thus, a different numerical scheme is proposed to eliminate the temporal variable in the evolution equations of phase field and concentration by introducing a reference of moving frame, z = x − Vt , which moves with a constant velocity V at the center of the interfacial region given by ϕ = 1/2 at z = 0 [ 20 , 21 , 29 ]. With the input of interface velocity and initial alloy composition, the steady-state concentration profile and temperature at the interface can be predicted via the relaxation resolution.…”

“…Thus, with the input of the fixed interface velocity and varied initial alloy compositions, the corresponding interface velocity-dependent kinetic phase diagram can be obtained. Moreover, in the work of [ 20 , 21 ], the interface temperature was obtained by fixing the interface at origin point in the moving reference frame via a temperature relaxation equation, while in [ 29 ], the interface temperature is calculated via the Gibbs-Thomson equation directly and such numerical scheme is named as “the time-elimination relaxation scheme”. Additionally, it has been shown that with the equivalent input the time-elimination relaxation scheme and the standard explicit scheme can reach the unique solute distribution and temperature-velocity relationship in the length scale from nanometer to micrometer [ 29 ].…”

“…Moreover, in the work of [ 20 , 21 ], the interface temperature was obtained by fixing the interface at origin point in the moving reference frame via a temperature relaxation equation, while in [ 29 ], the interface temperature is calculated via the Gibbs-Thomson equation directly and such numerical scheme is named as “the time-elimination relaxation scheme”. Additionally, it has been shown that with the equivalent input the time-elimination relaxation scheme and the standard explicit scheme can reach the unique solute distribution and temperature-velocity relationship in the length scale from nanometer to micrometer [ 29 ]. However, all the reported interface velocity-dependent kinetic phase diagrams in the literature are only limited to several simple binary systems [ 9 , 16 , 21 , 29 ].…”