Growth and coarsening of eutectic M₇C₃ carbide in Fe–Cr–C alloy during isothermal aging: phase field simulation and experimental research

Abstract

Combined with phase field simulation and experimental research, the growth and coarsening of M₇C₃ carbide precipitated from supersaturated austenite matrix were studied. Adopting the CALculation of PHAse Diagram (CALPHAD) method and thermodynamic database, the real Gibbs free energy of the austenite and M₇C₃ carbide phases in the Fe–Cr–C ternary alloy was established. The growth and coarsening of M₇C₃ carbide were simulated by phase field method. The microstructure and elements distribution of M₇C₃ carbide in Fe–Cr–C alloy were observed by field emission scanning electron microscope (FESEM) and energy-dispersive spectrometer (EDS) after various isothermal aging times (0 h, 0.5 h, 2 h) at 1273 K, respectively. Results show that the growth and coarsening of the precipitated M₇C₃ carbide phase undergo the following process during isothermal aging: first, a single precipitated phase nucleus is precipitated from the matrix; afterward, the single precipitated phase continues to coarsen and grow to form a plate-like structure, or two precipitated phases contact to form a single bar-like structure; then, several precipitated phase nucleuses contact to form an elongated bar shape, and finally, the elongated bar-shaped structure contacts and grows to form a typical lamellar eutectic carbide. In addition, through the analysis of the concentration field, the precipitated M₇C₃ carbide is a shell structure. The outer layer is a carbon-rich zone, and the inner part is a chromium-rich zone. By comparing the simulated morphology evolution with the experimental microstructure characterization and elements distribution via EDS, the correctness of the proposed phase field model has been verified.

Appendix

The molar Gibbs energy for the FCC phase was described using a two-sublattice model ${\text{(Fe,Cr)(C,Va)}}$ (Table

Table 3 Thermodynamics parameters for the Fe–Cr–C system [34] (in SI units and R = 8.31448)

3).

$$\begin{aligned} G_{{\text{m}}}^{{{\text{fcc}}}} & { = }y_{{{\text{Fe}}}} y_{{\text{C}}} {}^{{0}}G_{{\text{Fe:C}}}^{{{\text{fcc}}}} { + }y_{{{\text{Fe}}}} y_{{{\text{Va}}}} {}^{{0}}G_{{\text{Fe:Va}}}^{{{\text{fcc}}}} { + }y_{{{\text{Cr}}}} y_{{\text{C}}} {}^{{0}}G_{{\text{Cr:C}}}^{{{\text{fcc}}}} { + }y_{{{\text{Cr}}}} y_{{{\text{Va}}}} {}^{{0}}G_{{\text{Cr:Va}}}^{{{\text{fcc}}}} \\ & \quad { + }\,{\text{RT(}}y_{{{\text{Fe}}}} {\text{ln}}y_{{{\text{Fe}}}} { + }y_{{{\text{Cr}}}} {\text{ln}}y_{{{\text{Cr}}}} {\text{) + RT(}}y_{{{\text{Va}}}} {\text{ln}}y_{{{\text{Va}}}} { + }y_{{\text{C}}} {\text{ln}}y_{{\text{C}}} {) + }{}^{{\text{E}}}G_{{\text{m}}}^{{{\text{fcc}}}} \\ \end{aligned}$$

(A.1)

where

$${}^{{\text{E}}}G_{{\text{m}}}^{{{\text{fcc}}}} { = }y_{{{\text{Cr}}}} y_{{{\text{Fe}}}} {(}y_{{\text{C}}} L_{{\text{Cr,Fe:C}}}^{{{\text{fcc}}}} { + }y_{{{\text{Va}}}} L_{{\text{Cr,Fe:Va}}}^{{{\text{fcc}}}} {) + }y_{{{\text{Va}}}} y_{{\text{C}}} {(}y_{{{\text{Cr}}}} L_{{\text{Cr:Va,C}}}^{{{\text{fcc}}}} { + }y_{{{\text{Fe}}}} L_{{\text{Fe:Va,C}}}^{{{\text{fcc}}}} {)}$$

(A.2)

where the site fraction of component $i$ is marked as $y_{i}$ and the relationship between site fraction and mole fraction can be expressed as:

$$y_{{{\text{Fe}}}} { = }x_{{{\text{Fe}}}} {/(1 - }x_{{\text{C}}} {),}\quad \, y_{{{\text{Fe}}}} { + }y_{{{\text{Cr}}}} { = 1}$$

(A.3)

$$y_{{\text{C}}} { = }x_{{\text{C}}} {/(1 - }x_{{\text{C}}} {),}\quad \, y_{{\text{C}}} { + }y_{{{\text{Va}}}} { = 1}$$

(A.4)

All the ${}^{0}G$ values are given relative to the enthalpy of selected reference states for the elements at 298.15 K. The comma separates components that interact in the same sublattice and colon is used to separate between different sublattices. $R$ and $T$ are gas constant and temperature. The $L_{i,j}^{\phi }$(binary interaction parameters) can be concentration dependent according to a Redlich–Kister polynomial,

$$L_{i,j}^{\phi } = {}^{0}L_{i,j}^{\phi } + {}^{1}L_{i,j}^{\phi } (x_{i} - x_{j} ) + {}^{2}L_{i,j}^{\phi } (x_{i} - x_{j} )^{2} + \cdots {}^{n}L_{i,j}^{\phi } (x_{i} - x_{j} )^{n}$$

(A.5)

where

$${}^{v}L_{i,j}^{\phi } = a + bT(v = 0,1,2)...$$

(A.6)

The M₇C₃ carbide was treated with the two-sublattice model ${\text{(Fe,Cr)}}_{{7}} {\text{C}}_{{3}}$ and its molar free energy was described as

$$G_{{\text{m}}}^{{{\text{M}}_{{7}} {\text{C}}_{{3}} }} { = }y_{{{\text{Fe}}}} {}^{{0}}G_{{\text{Fe:C}}}^{{{\text{M}}_{{7}} {\text{C}}_{{3}} }} { + }y_{{{\text{Cr}}}} {}^{{0}}G_{{\text{Cr:C}}}^{{{\text{M}}_{{7}} {\text{C}}_{{3}} }} { + 7}RT{(}y_{{{\text{Fe}}}} {\text{ln}}y_{{{\text{Fe}}}} { + }y_{{{\text{Cr}}}} {\text{ln}}y_{{{\text{Cr}}}} {) + }y_{{{\text{Fe}}}} y_{{{\text{Cr}}}} L_{{\text{Cr,Fe:C}}}^{{{\text{M}}_{{7}} {\text{C}}_{{3}} }}$$

(A.7)

where ${}^{0}G_{i:{\text{C}}}^{{{\text{M}}_{7} {\text{C}}_{3} }}$ is the Gibbs energy of a pure binary carbide, the binary interaction parameter $L_{{\text{Cr,Fe:C}}}^{{{\text{M}}_{{7}} {\text{C}}_{{3}} }}$ in Eq. A.7 is

$$L_{{\text{Cr,Fe:C}}}^{{{\text{M}}_{7} {\text{C}}_{3} }} = {}^{0}L_{{\text{Cr,Fe:C}}}^{{{\text{M}}_{{7}} {\text{C}}_{{3}} }} + {}^{1}L_{{\text{Cr,Fe:C}}}^{{{\text{M}}_{{7}} {\text{C}}_{{3}} }} (x_{{\text{Cr,Fe}}} - x_{{\text{C}}} ) + {}^{2}L_{{\text{Cr,Fe:C}}}^{{{\text{M}}_{{7}} {\text{C}}_{{3}} }} (x_{{\text{Cr,Fe}}} - x_{{\text{C}}} )^{2}$$

(A.8)