CALPHAD Method

The CALPHAD method is fundamentally based on constructing and optimizing thermodynamic models of each phase in a material system. This involves integrating diverse sources of information into self-consistent expressions for Gibbs energy, which are then used to predict phase equilibria and properties. The core methodology consists of four key steps:

3.1 Data Assessment

The starting point in any CALPHAD modeling effort is a rigorous evaluation of available experimental and theoretical data. This includes:

• Phase diagrams from differential thermal analysis (DTA), X-ray diffraction (XRD), or microscopy,
• Thermochemical measurements such as enthalpies (from calorimetry), heat capacities, activities (from EMF), and vapor pressures,
• Ab initio calculations (e.g., DFT-based enthalpies of formation),
• Estimated or extrapolated data using empirical rules.

Critically assessing and selecting consistent and reliable data is essential. The quality of the resulting CALPHAD model depends strongly on the validity and coverage of this foundational dataset.

3.2 Thermodynamic Modelling

Each phase in the system is described using an analytical expression for its molar Gibbs free energy as a function of temperature, pressure, and composition. The expression typically includes:

• A reference term based on pure elements,
• An ideal mixing term (entropy),
• An excess term to describe interactions,
• Additional terms for magnetic, elastic, or electronic contributions when relevant.

The Compound Energy Formalism (CEF) is widely used to handle ordered phases, stoichiometric compounds, interstitial solutions, and ionic materials. In CEF, atoms or species are distributed across multiple sublattices, capturing order-disorder transitions, defects, and site preferences. This flexibility makes CEF essential for modeling real complex phases.

3.3 Optimization and Extrapolation

Thermodynamic model parameters are optimized by fitting to experimental or simulated data. The goal is to reproduce known phase equilibria and thermodynamic properties as accurately and consistently as possible. This optimization is typically performed using nonlinear least-squares minimization, as implemented in Thermo-Calc PARROT module or the Pandat Optimizer.

A key component of this modeling process is the use of Redlich-Kister polynomials to describe the excess Gibbs energy of mixing in solution phases:

$$ G^{ex} = x_A x_B \sum_{i=0}^{n} L_i (x_A – x_B)^i $$ 

Here

• \( x_A \) and \( x_B \) are the mole fractions of components A and B, respectively.
• \(L_i \) are interaction parameters that may themselves be temperature-dependent.

The expression captures non-ideal interactions and can be extended to ternary and multicomponent systems via generalized Redlich-Kister expansions.

These polynomials are especially useful because they:

• Allow flexible modeling of asymmetric phase diagrams,
• Have well-understood thermodynamic behavior and convergence properties,
• Are easily differentiable, aiding Gibbs energy minimization.

Once binary and ternary systems are assessed, extrapolation to multicomponent systems becomes possible using geometric schemes that combine lower-order data in a thermodynamically consistent way. The most commonly used models include Muggianu, Kohler, and Toop, each designed to handle extrapolation differently depending on the symmetry or asymmetry of the system:

3.3.1 Muggianu Model

Assumes symmetric behavior and averages binary interaction parameters across all components. It ensures smooth extrapolation but is best suited for systems with similar atomic sizes and behaviors.

$$ G^{ex}_{ABC} = x_A x_B L_{AB}^{ABC} + x_B x_C L_{BC}^{ABC} + x_C x_A L_{CA}^{ABC} $$

This is the simplified form of the Muggianu extrapolation expression, where:

• \( L_{AB}^{ABC} \), \( L_{BC}^{ABC} \), and \( L_{CA}^{ABC} \) are the composition-dependent interaction parameters in the ternary system,
• \( x_A \), \( x_B \), and \( x_C \) are the mole fractions of components A, B, and C, respectively.

3.3.2 Kohler Model

Also symmetric but maintains binary interaction behavior in the ternary by weighting deviations in a way that respects pure component influence. Works well when the system is relatively ideal. Often used when composition is evenly distributed.

3.3.3 Toop Model

Designed for asymmetric systems, where one component (element) dominates. Toop model weights binary excess terms based on proximity to the main element, making it especially suitable for systems like dilute solutions or those with strong composition bias (e.g., A-rich systems).

These three models allow the CALPHAD-based predictions to scale efficiently from binary assessments to practical multicomponent alloy predictions, using existing data without needing full higher-order experiments.

3.4 Equilibrium Calculations

With optimized Gibbs energy models stored in databases, thermodynamic software can compute equilibrium phase diagrams and property diagrams by minimizing the total Gibbs energy of the system. Calculations can be performed under constraints such as:

• Constant temperature and pressure,
• Fixed composition,
• Specified chemical potential,
• Fixed volume fraction (e.g., when calculating the liquidus and solidus temperatures) or other variables.

Results include stable and metastable phase diagrams, phase fractions, phase transformation temperatures, activities, thermodynamic potentials, and transformation paths. Some of these outputs can also serve as inputs to microstructure simulations (e.g. TC-PRISMA and MICRESS), kinetic models (e.g., DICTRA), and finite element tools.