MCEN90014 · Materials Engineering
CalPhaD & Thermo-Calc
CalPhaD & Thermo-Calc is the Week-5 core of MCEN90014 Materials Engineering at the University of Melbourne, where it links the thermodynamics of Gibbs free energy to the phase diagrams the subject is built on. It shows how the equilibrium between two phases is fixed by the common-tangent construction on their Gibbs-energy-versus-composition curves, which is the same as saying each component has an equal chemical potential in every phase. This is the theory the 20% computational assignment automates in Thermo-Calc when it calculates a phase diagram.
What this chapter covers
- 01Write the molar Gibbs energy of a binary solution G(X,T) and identify the ideal entropy-of-mixing term RT(X_A ln X_A + X_B ln X_B)
- 02Read the chemical potential of a component as the intercept of the tangent to the G-X curve on each pure-component axis
- 03Derive the equal-chemical-potential equilibrium condition mu_i^alpha = mu_i^beta from minimising the total Gibbs energy (dG = 0)
- 04Carry out the common-tangent construction and explain why one line touching both curves means equal chemical potentials
- 05Locate the two touch compositions as the ends of a tie-line and identify the single-phase and two-phase composition bands
- 06Build a T-X phase diagram by repeating the common tangent as temperature falls, tracing the liquidus and solidus
- 07Compute the molar Gibbs energy of mixing for ideal and regular solutions and see how a positive interaction parameter can open a miscibility gap
- 08Convert tie-line compositions into phase fractions with the lever rule (opposite-arm rule)
- 09Describe the CalPhaD method and how Thermo-Calc finds the minimum-Gibbs-energy state for the Fe-C-Cr assignment
Common tangent to phase fractions
- +1Because one common tangent touches both curves, its axis intercepts are shared, so the equilibrium condition mu_A(alpha) = mu_A(L) and mu_B(alpha) = mu_B(L) holds. The two touch compositions are the ends of the tie-line.
- +1The overall composition C_0 = 0.30 lies between C_alpha = 0.15 and C_L = 0.50, so the alloy is in the two-phase alpha + L region and the lever rule applies.
- +1Lever rule: the fraction of a phase equals the opposite tie-line arm over the whole tie-line, W_L = (C_0 - C_alpha)/(C_L - C_alpha) and W_alpha = (C_L - C_0)/(C_L - C_alpha).
- +1Liquid fraction: W_L = (0.30 - 0.15)/(0.50 - 0.15) = 0.15/0.35 = 0.429.
- +1Solid fraction: W_alpha = (0.50 - 0.30)/(0.50 - 0.15) = 0.20/0.35 = 0.571.
- +1Check: W_L + W_alpha = 0.429 + 0.571 = 1.000, as required.
Key terms
- Molar Gibbs free energy G
- The Gibbs energy per mole of a phase, G = H - TS (J/mol), plotted against composition as a G-X curve. At constant temperature and pressure the phase with the lowest G at a given composition is the stable one.
- Chemical potential
- The Gibbs energy per mole of a component in a solution (J/mol). Graphically it is the intercept of the tangent to the G-X curve on that component's pure axis: mu_A = G - X_B(dG/dX_B) and mu_B = G + (1 - X_B)(dG/dX_B).
- Common-tangent construction
- The single straight line tangent to two phases' G-X curves at once. Its two touch points give the equilibrium compositions of the coexisting phases, and its shared intercepts set their chemical potentials equal.
- Equal-chemical-potential condition
- The equilibrium requirement that every component has the same chemical potential in all coexisting phases, mu_i(alpha) = mu_i(beta). It follows from minimising the total Gibbs energy (dG = 0) and is the algebraic form of the common tangent.
- Tie-line and lever rule
- A horizontal line across a two-phase field joining the equilibrium phase compositions; the fraction of each phase is the length of the opposite arm divided by the whole tie-line.
- Regular solution
- A solution model that adds an enthalpy of mixing, Delta H_mix = Omega X_A X_B, to the ideal entropy term. A positive interaction parameter Omega (J/mol) raises the Gibbs energy and, if large enough or the temperature low enough, splits the curve into two wells - a miscibility gap.
- CalPhaD method
- CALculation of PHAse Diagrams: assessed Gibbs-energy models G(X,T) are fitted to experimental data for every phase, then the state of lowest total Gibbs energy (the common-tangent state) is computed numerically at each temperature and composition.
- Thermo-Calc
- The CalPhaD software used in the subject. Given the elements, composition and a database, it returns the equilibrium phases present, their fractions and their compositions - the automated version of the common-tangent construction, and the tool for the computational assignment.
CalPhaD & Thermo-Calc FAQ
Is the common-tangent construction the same thing as equal chemical potentials?
Yes. They are one statement in two languages. A single straight line tangent to both Gibbs-energy curves has one intercept on each pure-component axis, so the chemical potential of A read from the alpha touch point equals the one read from the beta touch point, and likewise for B. That is exactly mu_i(alpha) = mu_i(beta), the condition you get by minimising the total Gibbs energy. If an exam question asks you to justify the construction, state both the geometric version (common tangent) and the algebraic version (equal chemical potentials).
What is the difference between CalPhaD and Thermo-Calc?
CalPhaD (CALculation of PHAse Diagrams) is the method: store an assessed Gibbs-energy model for each phase, fitted to reproduce known data, then find the lowest-total-Gibbs-energy state at each temperature and composition. Thermo-Calc is one piece of software that implements that method, and it is the package used for the computational assignment. So CalPhaD is the theory and Thermo-Calc is the tool that runs it.
Can AI help me with CalPhaD and Thermo-Calc in MCEN90014?
For understanding, yes; for the assessed report, be careful. An AI tutor like Sia can explain the common-tangent construction step by step, walk through why chemical potentials are tangent intercepts, or check your reasoning on a lever-rule practice question. It cannot sit the invigilated final exam or guarantee a mark. The computational assignment is individually assessed and the University of Melbourne engineering GenAI policy is strict - AI-generated content is not permitted in submitted work - so your Thermo-Calc analysis and report must be your own. Always confirm what is allowed on Canvas.
Studying with AI? Sia — free AI mechanical engineering tutor works through MCEN90014 step by step.
Exam move
Treat this chapter as the reason phase diagrams exist rather than a set of facts to memorise. Learn the pictures first: one tangent to a single Gibbs-energy curve gives the two chemical potentials as axis intercepts, and one common tangent to two curves is what phase equilibrium looks like. Then be able to say why - minimising the total Gibbs energy forces mu_i to be equal in every phase - and to turn tie-line compositions into phase fractions with the lever rule (remembering the opposite-arm rule and the check that the fractions sum to one). Because the final exam is worth 50%, is a hurdle you must pass, and provides a formula sheet, and because all ten questions carry equal weight, budget about one-tenth of the exam per question and keep a few minutes to check signs and units; confirm the exam duration on the timetable in Canvas. The same theory underpins the 20% Thermo-Calc assignment, so practising the hand construction is the most efficient way to prepare for both.
