General Phase Diagram Sections Arthur D. Pelton Centre de Recherche en Calcul Thermochimique École Polytechnique de Montréal Montréal, Québec, CANADA General rules of construction of all true phase diagram sections; Proper choice of variables and constants to give a “true” phase diagram section (with a unique equilibrium state at each point); A general algorithm for calculating all true phase diagram sections. 1

CaO-MgO: phase diagram 2

Diagram: CaO-Al2O3-SiO2 3

C-Fe-W System at 5 wt% W 4

A Phase Diagram for the Fe-Cr-V-C System 5

Fe-Cr-V-C system at 1.5 wt% Cr and 0.1 wt% V 6

P–T diagram for Al2TiO5 7

Fe-O2 System 8

System Fe-Cr-O2 9

SO2-O2-Cu System at 1000 K 10

Fe2O3-MgO-SiO2-O2 System SiO2/(Fe2O3+ MgO + SiO2) = 0.20 in air 11

Fe-Cr-S-O System 12

A Phase Diagram for the Fe-Cr-V-C System 13

The Law of Adjoining Phase Regions «As a phase boundary line is crossed, one and only one phase appears or disappears.» 14

(a1 + a2 + … + an) (a1 + a2 + … + an) (a1 + a2 + … + an) + b + g A node in a general true phase diagram section. 15

CaO-MgO: phase diagram 16

T XB B A L b + L a + b + L a + L a b a + b An isobaric binary temperature-composition diagram with the eutectic “opened-up” to show that this is an infinitely narrow 3-phase region. 17

SO2-O2-Cu System at 1000 K 18

f1 a + b f2 b + g a + g a + b + g a b g A “potential-potential” phase diagram “opened-up” to show that the lines are infinitely narrow 2-phase regions. 19

Potential Variables Extensive Variables T (temperature) P (pressure) (the same for all phases at equilibrium) T (temperature) P (pressure) mi (chemical potential) Extensive Variables (not the same for all phases at equilibrium) Xi (composition) V (volume) 20

If only potential variables are held constant, then all tie-lines lie in the plane of the section. In this case, the compositions of the individual phases at equilibrium can be read from the diagram and the lever rule applies. 21

Zero Phase Fraction (ZPF) Lines MC fcc bcc M7C3 M23C6 22

Zero Phase Fraction (ZPF) Lines LIQUID LIQUID + a L+b SOLID a SOLID b 2 SOLIDS (a + b) Solidus Liquidus Solvus line a LIQUID b 23

Fe-S-O Predominance diagram (ZPF lines) 24

Choice of variables to always give a true phase diagram (single-valued) everywhere 25

N-Component System (A-B-C-…-N) Gibbs-Duhem: Phase Diagram 26

Choice of variables N-component system (1) Choose n potentials: f1, f2, … , fn (2) From the non-corresponding extensive variables (qn+1, qn+2, … ), form (N+1-n) independent ratios (Qn+1, Qn+2, …, QN+1). Example: [f1, f2, … , fn; Qn+1, Qn+2, …, QN+1] are then the (N+1) variables of which 2 are chosen as axes and the remainder are held constant. Phase Diagram 27

MgO-CaO Binary System S T V -P nMgO mMgO nCaO mCaO f1 = T f2 = -P Phase Diagram 28

CaO-MgO: phase diagram 29

Fe-Cr-S-O System S T V -P nFe mFe nCr mCr f1 = T f2 = -P Phase Diagram 30

Fe-Cr-S-O System 31

Pressure vs. Volume diagram for H2O S+L L+G S+G P V This is NOT a true phase diagram. Phase Diagram 32

Fe-Cr-C System S T V -P nC mC nFe mFe nCr mCr Requirement: f1 = T f2 = -P f3 = mC (NOT OK) (OK) Requirement: Phase Diagram 33

This is NOT a true phase diagram. Fe-Cr-C system, T = 1300 K, XCr = nCr/(nFe+nCr+nC) vs. (carbon activity) M23C6 M7C3 bcc fcc cementite log(ac) Mole fraction of Cr 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -3 -2 -1 1 2 This is NOT a true phase diagram. Phase Diagram 34

“Corresponding” phase diagrams of the Fe-O system a-Iron g-Iron d-Iron Liquid iron Liquid oxide Wustite Magnetite Hematite -500 -400 -300 -200 -100 RT ln PO2 (kJ) + a-Iron + Magnetite Oxygen Liquid iron Liquid iron + liquid oxide 700 900 1100 1300 1500 1700 1900 0.50 0.54 0.58 0.62 Mole fraction XO Temperature (K) 35

Other Sets of Extensive Variables and Corresponding Potentials Gibbs-Duhem: Gibbs-Duhem: 36

Mg-Si binary phase diagram 37

Mg-Si, DH vs XSi phase diagram 38