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From Chapter 9 of Saunders and Miodownik ’ s CALPHAD book Computational methods

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##### From Chapter 9 of Saunders and Miodownik ’ s CALPHAD book Computational methods

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**From Chapter 9 of Saunders and Miodownik’s CALPHAD**bookComputational methods**for many years:**equil const used to express (the abundance of certain species) in terms of the amounts of other arbitrarily chosen species for one rxn → one equil const needed the # of rxn ↑ → many equil const needed but not available ∴ CALPHAD (CALculation of PHAse Diagram) method: considering the G energy of all phases → true equil calculation by of phase component → then, the # of unknowns considerably reduced in comparison to equil const approach**in the case of binary and ternary phase equil**→ analytic method is possible to calculate certain 2-phase equil ex) binary eutectic, negligible solubility in the terminal solid phase, ideal mixing in the liq, equil btw liq and a(A) : Fig. 9.1. Simple eutectic system with ideal mixing in the liquid and negl solid solubility in the terminal solid phases, a and b. Fig. 9.2. G/x diagram at 850K for eutectic system in Fig. 9.1.**no solubility**this is a special case not a good method**general method: equil btw a solu phase & stoichiometric**according to Hillert, how to calculate T ? general solution : a liq, liquidus of Φ (MTDATA used this methodology) Newton-Raphson method: for fixed xiΦ → first choosing arbitrary T → iteratively stepping T until ΔG below a small, defined accuracy limit**liq**a xa xo xl liq a xia • a more general case of both phases with a solubility range as xiΦ is not fixed, by fixing T and then finding xa and xliq from xo (To), however, it is converted to N in this section the alloy first assumed in single-phase a → an arbitrary amount of liq introduced → min G reached →fixingliq comp and changing a→ min G reached again → final minimization of G if the alloy chosen outside 2-phase equil (xia) → xiliq calculated to minimize G of a +l → by fixing xiliq and changing the amount of a → G ↓ → finally reaching equil btw a and liq**multi-component sys**calculation must be defined: # of degree of fredom reduced G of the sys calculated some iterative tech to minimize G how to reduce f (= v) : by defining a series of constraints such as balance in ionic sys composition range of each phase one of the earliest example of G minimization by White (1958) → chem equil in an ideal gas mixture of O, H, N with the species H, H2, H2O, N, N2, NH, NO, O, O2, OH # of atoms of element j (O, H, N only) in species (H, H2, H2O, NH, NO…)**→ presenting 2 methods of G minimization**the typical one is to use Lagrange’s method of multipliers: irrespective of # of phases and components here, finding NΦ(fraction or amount of each phase), xkΦ(mole fraction of k in the phase Φ) which minimize G this is the equil calculation Then, why this method ? with the constraints of ∴ by defining 0 0**# of p eq**# of c eq # of p x c eq # of p eq adding ∑0·mk + ∑0·lΦ to the original eq not changing its value → minimum condition of L ? thus, to solve 2p + c (p+1) non-linear eq by the Newton-Raphson method Germany SOLGASMIX ChemSage FactSage Canada FACT FactSage Sweden Thermo-Calc England MTDATA Germany, MPI Lukas program