Thermodynamics-Based Computational Design of Al-Mg-Sc-Zr Alloys

Abstract

Alloying additions of Sc and Zr raise the yield strength of Al-Mg alloys significantly. We have studied the effects of Sc and Zr on the grain refinement and recrystallization resistance of Al-Mg alloys with the aid of computational alloy thermodynamics. The grain refinement potential has been assessed by Scheil–Gulliver simulations of solidification paths, while the recrystallization resistance (Zener drag) has been assessed by calculation of the precipitation driving forces of the Al₃Sc and Al₃Zr intermetallics. Microstructural performance indices have been derived, used to rank several alloy composition variants, and finally select the variant with the best combination of grain refinement and recrystallization resistance. The method can be used, with certain limitations, for a thermodynamics-based design of Al-Mg and other alloy compositions.

Appendix

Introduction of Sc in the COST 507 light-alloy thermodynamic database

The present appendix presents the procedure for the introduction of Sc in the COST 507 thermodynamic database. This was necessary in order to carry out thermodynamic calculations involving Sc as an alloying element in Al-Mg alloys with Thermo-Calc software.

The fundamental thermodynamic properties of elemental Sc, i.e., a stable crystal structure, atomic mass, and standard element reference (SER) enthalpy and entropy are given in Table IV. The following descriptions of the molar Gibbs free energies of the intermetallic phases were retrieved from the literature:[22]

$$ G^{{{\text{Al}}_{ 3} {\text{Sc}}}} = 0.75G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} + 0.25G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} - 42000 + 7.1932T $$

(A1)

$$ G^{{{\text{Al}}_{ 2} {\text{Sc}}}} = 0.667G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} + 0.333G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} - 48000 + 7.5T $$

(A2)

$$ G^{\text{AlSc}} = 0.50G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} + 0.50G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} - 46000 + 6.9371T $$

(A3)

$$ G^{{{\text{AlSc}}_{ 2} }} = 0.333G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} + 0.667G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} - 37000 + 3.7333T $$

(A4)

Table IV Fundamental Thermodynamic Properties of Elemental Sc

In Eqs. [A1] through [A4], G ^φ represents the molar Gibbs free energy of intermetallic phase φ, $G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} (T)$ and $G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} (T)$ represent the Gibbs free energy of pure Al and pure Sc, respectively, at their stable crystal structure as a function of temperature, while T is the absolute temperature. The Gibbs free energies of pure Al and pure Sc as functions of temperature are given by the following equations:

• 1. Al in the fcc phase

for 298 K ≤ T ≤ 700 K (25 °C ≤ T ≤ 427 °C):

$$ G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} = -7976.15 + 137.093038T - 24.3671976T\ln (T) - 0.001884662T^{2} - 8.77664 \times 10T^{3} + 74092T^{ - 1} $$

(A5)

for 700 K ≤ T ≤ 933 K (427 °C ≤ T ≤ 660 °C):

$$ G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} = - 11276.24 + 223.048446T - 38.5844296T\ln (T) + 0.018531982T^{2} - 5.764227 \times 10^{ - 6} T^{3} + 74092T^{ - 1} $$

(A6)

for T ≥ 933 K (T ≥ 660 °C):

$$G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} = - 11278.378 + 188.684153T - 31.748192T\ln (T) - 1.230524 \times 10^{28} T^{ - 9} $$

(A7)

• 2. Sc in the hcp phase

for 298 K ≤ T ≤ 800 K (25 °C ≤ T ≤ 527 °C):

$$ G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} = - 8689.547 + 153.48097T - 28.1882T\ln (T) + 0.00321892T^{2} - 1.64531 \times 10^{ - 6} T^{3} + 72177T^{ - 1} $$

(A8)

for 800 K ≤ T ≤ 1608 K (527 °C ≤ T ≤ 1335 °C):

$$ G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} = - 7511.295 + 132.759582T - 24.9132T\ln (T) - 5.73295 \times 10^{ - 4} T^{2} - 8.59345 \times 10^{ - 7} T^{3} $$

(A9)

for T ≥ 1608 K (T ≥ 1335 °C):

$$ G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} = + 261143.04 - 1817.92245T + 241.441051T\ln (T) - 0.117529396T^{2} + 8.7398 \times 10^{ - 6} T^{3} - 50607159T^{ - 1} $$

(A10)

The corresponding mathematical description of the Gibbs free energy of solid-solution phases in the CALPHAD method is implemented by the Redlich–Kister–Mugiannu formalism.[23] According to this formalism, the molar Gibbs free energy of a nonstoichiometric phase φ is given by a relation of the following form:

$$ G^{\phi } = \sum\limits_{i} {x_{i} } G_{i}^{^\circ ,\phi } + RT\sum\limits_{i} {x_{i} \ln x_{i} } + \sum\limits_{i} {\sum\limits_{i > j} {x_{i} x_{j} } } \sum\limits_{u} {L_{i,j}^{u,\phi } } (x_{i} - x_{j} )^{u} $$

(A11)

In Eq. [A11], x _i and x _j represent the mole fractions of components i and j, respectively, $G_{i}^{^\circ ,\phi }$ is the Gibbs free energy of pure component i, when existing in the crystal structure of phase φ (e.g., $G_{\text{Al}}^{{^\circ ,{\text{fcc}}}}$, $G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}}$, etc.), while $L_{i,j}^{u,\phi }$ are the interaction parameters of order u, through which the excess energy that arises from the fact that the solid-solution phase φ is not ideal is taken into account. The available interaction parameters for the Al-Sc binary system are listed here. Interaction parameters not contained in the list are taken to be equal to zero.

$$ L_{\text{Al,Sc}}^{{0,{\text{fcc}}}} = - 107664.4 + 8.3T $$

(A12)

$$ L_{\text{Al,Sc}}^{{0,{\text{bcc}}}} = - 1 0 6 5 0 6. 3- 2. 7 7 4 6T $$

(A13)

$$ L_{\text{Al,Sc}}^{{1,{\text{bcc}}}} = - 1 9 9 0 8. 5 $$

(A14)

$$ L_{\text{Al,Sc}}^{{0,{\text{hcp}}}} = - 106375.1 + 15.3157T $$

(A15)

$$ L_{\text{Al,Sc}}^{{0,{\text{liquid}}}} = - 126270.4 + 17.6241T $$

(A16)

Finally, the relevant $G_{i}^{^\circ ,\phi }$ parameters of Eq. [A11] are given here. It should be noted that parameters $G_{\text{Al}}^{{^\circ ,{\text{fcc}}}}$ and $G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}}$ have already been described in Eqs. [A5] through [A10] and are not repeated here.

$$ G_{\text{Al}}^{{^\circ ,{\text{bcc}}}} = 10083 - 4.813T + G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} $$

(A17)

$$ G_{\text{Al}}^{{^\circ ,{\text{hcp}}}} = 5481 - 1.8T + G_{\text{Al}}^{{^\circ ,{\text{hcp}}}} $$

(A18)

• 3. Al in the liquid phase

for 298 K ≤ T ≤ 934 K (25 °C ≤ T ≤ 661 °C):

$$ G_{\text{Al}}^{{^\circ ,{\text{liquid}}}} = 11005.029 - 11.841867T + 7.934 \times 10^{ - 20} T^{7} + G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} $$

(A19)

for T ≥ 934 K (T ≥ 661 °C):

$$ G_{\text{Al}}^{{^\circ ,{\text{liquid}}}} = 10482.382 - 11.253974T + 1.231 \times 10^{28} T^{ - 9} + G_{\text{Al}}^{{^\circ ,{\text{fcc}}}} $$

(A20)

$$ G_{\text{Sc}}^{{^\circ ,{\text{fcc}}}} = 5 0 0 0 + G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} $$

(A21)

• 4. Sc in the bcc phase

for 298 K ≤ T ≤ 1000 K (25 °C ≤ T ≤ 727 °C):

$$ G_{\text{Sc}}^{{^\circ ,{\text{bcc}}}} = 1979.728 - 1.024135T + G_{\text{Sc}}^{{^\circ ,{\text{hcp}}}} $$

(A22)

for 1000 K ≤ T ≤ 1608 K (727 °C ≤ T ≤ 1335 °C):

$$ G_{\text{Sc}}^{{^\circ ,{\text{bcc}}}} = 230161.408 - 2004.05469T + 276.76664T\ln (T) - 0.167120107T^{2} + 1.5637371 \times 10^{ - 5} T^{3} - 33783257T^{ - 1} $$

(A23)

for T ≥ 1608 K (T ≥ 1335 °C):

$$ G_{\text{Sc}}^{{^\circ ,{\text{bcc}}}} = - 25928.011 + 283.642312T - 44.2249T\ln (T) $$

(A24)

• 5. Sc in the liquid phase

for 298 K ≤ T ≤ 1608 K (25 °C ≤ T ≤ 1335 °C):

$$ G_{\text{Sc}}^{{^\circ ,{\text{liquid}}}} = 6478.66 + 45.427539T - 10.7967803T\ln (T) - 0.020636524T^{2} + 2.13106 \times 10^{ - 6} T^{3} - 158106T^{ - 1} $$

(A25)

for T ≥ 1608 K (T ≥ 1335 °C):

$$ G_{\text{Sc}}^{{^\circ ,{\text{liquid}}}} = - 11832.111 + 275.871695T - 44.2249T\ln (T) $$

(A26)

After retrieving the necessary thermodynamic data for the Al-Sc system, the data were incorporated in the COST 507 database. This modified COST 507+Sc database was then used in conjunction with Thermo-Calc software to calculate the binary Al-Sc system. The calculation was performed up to 50 at. pct Sc and is shown in Figure A1. The comparison with the published experimental binary Al-Sc[22] shows that the calculated diagram is in good agreement with the experimental one and, therefore, the modified COST 507+Sc database can be used for the thermodynamic calculations in this work.

Fig. A1

Calculated binary Al-Sc phase diagram with modified COST 507+Sc database