G–D Map Method for Analyzing Effects of Elements on Size Control of In Situ Formed Mg₂Si in Mg Melts

Abstract

An approach for selecting alloying elements to control the size of an in situ formed phase is proposed based on a G–D map, which is derived from basic thermodynamic and kinetic principles. The G–D map is divided into four zones. Elements in Zone III are potential refiners whereas those in Zone II should be avoided in processing. When applied to the Mg₂Si/Mg system, we find that Sr, P, Sb and Sn refine Mg₂Si by increasing the nucleation rate and inhibiting the growth rate. Conversely, Ti, Mn, Zr and V contribute to coarse-shaped Mg₂Si by postponing nucleation. The content of alloying elements should be matched with the Si content, and the required amount can be pre-estimated from the G–D map. Mg₂Si formed in Mg-Al and Mg-Zn alloy is smaller than that in Mg-Si and Mg-Mn alloy, because Al and Zn suppress the diffusion of Si, particularly at high Si contents. The predictions of this approach are consistent with experimental observations. Hence, the G–D map can provide useful guidance for quantitative phase refinement design.

Appendix A

1.1 Calculation of the Self-diffusion Coefficient $D_{i}^{*}$

The self-diffusion coefficient for binary liquid alloys based on hard sphere theory is given by Reference 36:

$$ D_{i}^{*} = \frac{{l_{i} }}{2}\left( {\frac{\pi RT}{{M_{i} }}} \right)^{1/2} \frac{C\left( \eta \right)}{{Z_{x} - 1}}, $$

(A1)

where $l_{i}$ and $M_{i}$ are the atom radius and molar mass of component i, $Z_{x}$ is the compressibility factor, and $C\left( \eta \right)$ is the correction factor.

$$ l_{i} = \left( {\frac{{3\eta_{i} M_{i} }}{{4\pi N_{o} \rho_{i} }}} \right)^{1/3} , $$

(A2)

$$ C\left( \eta \right) = \frac{{0.73\eta_{m} }}{\eta }, $$

(A3)

where $\rho_{i}$ is the density of component i, which can be obtained in Reference 46 and is shown in Table A2. $N_{o}$ is the Avogadro number, and $\eta_{i}$ is the packing fraction, hence:

$$ \eta_{i} = - 3.57 \times 10^{ - 2} + 1.367 \times 10^{ - 1} Z_{i} - 1.747 \times 10^{ - 2} Z_{i}^{2} + 1.225 \times 10^{ - 3} Z_{i}^{3} - 3.508 \times 10^{ - 5} Z_{i}^{4} . $$

(A4)

$$ Z_{x} = \frac{{9.385\rho_{x} }}{{M_{x} T}}\left( {\sqrt {\frac{{M_{i} T_{i} }}{{\rho_{{m_{i} }} }}} x_{i} + \sqrt {\frac{{M_{j} T_{j} }}{{\rho_{{m_{j} }} }}} \left( {1 - x_{i} } \right)} \right), $$

(A5)

where T and $T_{i}$ are the temperature of the liquid melt and the melting temperature of component i;$\rho_{{m_{i} }}$ is the density of component i at the melting temperature; $M_{x}$ and $\rho_{x}$ are respectively the molar mass and the density of the liquid:

$$ M_{x} = x_{i} M_{i} + \left( {1 - x_{i} } \right)M_{j} , $$

(A6)

$$ \rho_{x} = \frac{{M_{x} }}{{x_{i} M_{i} /\rho_{i} + (1 - x_{i} )M_{j} /\rho_{j} }}, $$

(A7)

From Eqs. [A2] through [A7], the self-diffusion coefficient of $D_{\text{Si}}^{*}$, $D_{\text{Mg}}^{*}$ and $D_{X}^{*}$ can be calculated.

1.2 Calculation of the Activity $a$

According to the ternary activity model introduced by Fan et al.[37,38] in i − j − k ternary systems, we extended it into quaternary system i − j − h − k:

$$ lna_{i} = 1 + lnx_{i} - \ln \left( {1 - \mathop \sum \limits_{j} x_{j} A_{ji} } \right) - \mathop \sum \limits_{j} \left( {\frac{{x_{j} \left( {1 - A_{ij} } \right)}}{{1 - \mathop \sum \nolimits_{k} x_{k} A_{kj} }}} \right), $$

(A8)

The pair of adjustable parameters, $A_{ij}$ and $A_{ji}$, were solved by:

$$ \frac{{a_{ij} f_{ij} \left( {1 + u_{i} \left( {\psi_{i} - \psi_{j} } \right)} \right)}}{{RTV_{j}^{2/3} }} = A_{ij} - { \ln }( 1- A_{ij} ), $$

(A9)

where $a_{ij} , f_{ij}$ is denoted as:

$$ a_{ij} = 1 - 0.1T\left( {\frac{1}{{T_{i} }} + \frac{1}{{T_{j} }}} \right), $$

(A10)

$$ f_{ij} = \frac{{2pV_{i}^{2/3} V_{j}^{2/3} \{ 9.4\left( {n_{i}^{{\frac{1}{3}}} - n_{j}^{{\frac{1}{3}}} } \right)^{2} - \left( {\psi_{i} - \psi_{j} } \right)^{2} - 0.73\left( {r/p} \right)}}{{(n_{i}^{{\frac{1}{3}}} )^{ - 1} - (n_{j}^{{\frac{1}{3}}} )^{ - 1} }}, $$

(A11)

where $\psi$ is the electron density, V is the molar volume, and $n$ is the Wigner-Seitz electron density at the cell boundary; r, b and $p$ are model parameters in the Miedema model, and their values can be found in References 47, 48, which are listed in Table A2.

See Tables A1 and A2.

Table A1 Standard Gibbs Free Energy Change[32]

Table A2 Values of Parameters Used in the Calculations[46,47,48]