**Glossary**

### CALPHAD Method

One of the fundamental goals in metallurgical engineering is to be able to control the final physical, mechanical, and/or chemical properties of an alloy. To achieve this goal, one must understand the relationship between the alloy's chemical composition, the processing conditions, the resulting microstructure, and the alloy's final properties. During processing, and often also during use, most alloys undergo one or more phase transformations, which are best understood through the use of phase diagrams. Experimentally determined phase diagrams, however, are usually available for binary systems only (e.g., Co-Cr, Fe-C, Fe-Co, Fe-Cr, Fe-Mn, Fe-Mo, Fe-Ni, Fe-Ti, Fe-V, Ni-Cr, Ti-Al, Ti-B, Ti-N, etc), to some extent for ternary systems (e.g., Fe-Cr-C, W-Co-C, W-Fe-C, W-Ni-C, etc), and very rarely for higher-order systems. This is where the CALPHAD (**CAL**culation of **PHA**se **D**iagrams) method comes in.

In the early seventies computational thermodynamics started with the pioneering work of Dr. Larry Kaufman, evolving to what is known today as the CALPHAD method. The method is based on the concept of deriving the thermodynamic functions of a system from all available experimental data. The thermodynamic functions are expressed as polynomials of temperature and chemical composition. The numerical values of the polynomial coefficients are obtained using numerical optimization techniques.

The CALPHAD method is based on the fact that a phase diagram is a representation of the thermodynamic properties of a system. Thus, if the thermodynamic properties are known, it would be possible to calculate the multi-component phase diagrams. Thermodynamic descriptions of lower-order systems (e.g., the Gibbs energy of each phase) are combined to extrapolate higher-order systems.

The Gibbs energy of a phase is described by a model that contains a relatively small number of experimentally optimized variable coefficients. Examples of experimental information used include melting and other transformation temperatures, solubilities, as well as thermodynamic properties such as heat capacities, enthalpies of formation, and chemical potentials.

For **pure elements** and **stoichiometric compounds**, the following model is most commonly used:

G_{m} - H_{m}^{SER} = a + b·T +c·T·ln(T) + Σd_{i}·T^{i}

where

G_{m} - H_{m}^{SER} is the Gibbs energy relative to a standard element reference state (SER),

H_{m}^{SER} is the enthalpy of the element in its stable state at the temperature of 298.15 Kelvin and the pressure of 10^{5} Pascal (1 bar), and

a, b, c, and d_{i} are the model parameters.

For **multi-component solution phases**, the following expression for the Gibbs energy is used:

G = G^{°} + ^{id}G_{mix} + ^{xs}G_{mix}

where

G^{°} is the Gibbs energy due to the mechanical mixing of the constituents of the phase,

^{id}G_{mix} is the ideal mixing contribution, and^{xs}G_{mix} is the excess Gibbs energy of mixing (the non-ideal mixing contribution).

If the **sub-lattice model** is used to describe solution phases, then the G^{°}, ^{id}G_{mix}, and ^{xs}G_{mix} of an A-B binary system with two sub-lattices, (A,B)_{p}(A,B)_{q}, can be expressed as follows:

G^{°} = y^{I}_{A}·y^{II}_{A}·G^{°}_{A:A} + y^{I}_{A}·y^{II}_{B}·G^{°}_{A:B} + y^{I}_{B}·y^{II}_{A}·G^{°}_{B:A} + y^{I}_{B}·y^{II}_{B}·G^{°}_{B:B}^{id}G_{mix} = p·R·T·[y^{I}_{A}·ln(y^{I}_{A}) + y^{I}_{B}·ln(y^{I}_{B})] + q·R·T·[y^{II}_{A}·ln(y^{II}_{A}) + y^{II}_{B}·ln(y^{II}_{B})]

^{xs}G_{mix} = y^{I}_{A}·y^{I}_{B}·[y^{II}_{A}·Σ_{k=0} L^{k}_{A,B:A}·(y^{I}_{A} - y^{I}_{B})^{k} + y^{II}_{B}·Σ_{k=0} L^{k}_{A,B:B}·(y^{I}_{A} - y^{I}_{B})^{k}] +

+ y^{II}_{A}·y^{II}_{B}·[y^{I}_{A}·Σ_{k=0} L^{k}_{A:A,B}·(y^{II}_{A} - y^{II}_{B})^{k} + y^{I}_{B}·Σ_{k=0} L^{k}_{B:A,B}·(y^{II}_{A} - y^{II}_{B})^{k}]

where

y^{I} and y^{II} are the site fractions of components A and B in the first and second sub-lattices, respectively,

G^{°}_{A:A}, G^{°}_{A:B}, G^{°}_{B:A}, and G^{°}_{B:B} is the Gibbs energy of A_{p}A_{q}, A_{p}B_{q}, B_{p}A_{q}, B_{p}B_{q} compounds, respectively,

L^{k}_{A,B:A} and L^{k}_{A,B:B} are the k^{th}-order interaction parameter between component A and B in the first sub-lattice, and

L^{k}_{A:A,B} and L^{k}_{B:A,B} are the k^{th}-order interaction parameter between component A and B in the second sub-lattice.

In the expressions for interaction parameters, a comma separates interacting components in the same sub-lattice, and a colon separates components that occupy different sub-lattices. The equations for G^{°}, ^{id}G_{mix}, and ^{xs}G_{mix} can be generalized for multi-component and multi-sub-lattice phases.

If a phase in a multi-component solution is described with a single sub-lattice model, then the G^{°}, ^{id}G_{mix}, and ^{xs}G_{mix} contributions to the Gibbs energy can be expressed as follows:

G^{°} = Σ_{i} c_{i}·G^{°}_{i}^{id}G_{mix} = R·T·Σ_{i} c_{i}·ln(c_{i})^{xs}G_{mix} = Σ_{i} Σ_{j>i} c_{i}·c_{j}·Σ_{k} L^{k}_{i,j}·(c_{i} - c_{j})^{k}

where

c_{i} and c_{j} is the mole fraction of species i and j, respectively, and

L^{k}_{i,j} is a binary interaction parameter between species i and j.

The binary interaction parameter L^{k}_{i,j} is dependant on the value of k. When the value of k is equal to zero or one, the equation for ^{xs}G_{mix} becomes regular or sub-regular, respectively.

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