General Phase Diagram Sections

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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.
2
CaO-MgO phase diagram
3
Diagram CaO-Al2O3-SiO2
4
C-Fe-W System at 5 wt W
5
A Phase Diagram for the Fe-Cr-V-C System
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Fe-Cr-V-C system at 1.5 wt Cr and 0.1 wt V
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PT diagram for Al2TiO5
8
Fe-O2 System
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System Fe-Cr-O2
10
SO2-O2-Cu System at 1000 K
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Fe2O3-MgO-SiO2-O2 SystemSiO2/(Fe2O3 MgO SiO2)
0.20 in air
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Fe-Cr-S-O System
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A Phase Diagram for the Fe-Cr-V-C System
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The Law of Adjoining Phase Regions

• As a phase boundary line is crossed, one and
  only one phase appears or disappears.

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(a1 a2 an)
(a1 a2 an)
(a1 a2 an)
b
g
(a1 a2 an)
b g

• A node in a general true phase diagram section.

16
CaO-MgO phase diagram
17

• An isobaric binary temperature-composition
  diagram with the eutectic opened-up to show
  that this is an infinitely narrow 3-phase region.

18
SO2-O2-Cu System at 1000 K
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• A potential-potential phase diagram opened-up
  to show that the lines are infinitely narrow
  2-phase regions.

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• Potential Variables
• (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)

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• 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.

22
Zero Phase Fraction (ZPF) Lines
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Zero Phase Fraction (ZPF) Lines
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Fe-S-O Predominance diagram (ZPF lines)
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Choice of variables to always give a true phase
diagram (single-valued) everywhere
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N-Component System (A-B-C--N)
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Choice of variables

• N-component system
• (1) Choose n potentials f1, f2, , fn
• (2) From the non-corresponding extensive
  variables (qn1, qn2, ), form (N1-n)
  independent ratios (Qn1, Qn2, , QN1).
• Example
• f1, f2, , fn Qn1, Qn2, , QN1 are then
  the (N1) variables of which 2 are chosen as axes
  and the remainder are held constant.

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MgO-CaO Binary System

• S T
• V -P
• nMgO mMgO
• nCaO mCaO

• f1 T
• f2 -P

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CaO-MgO phase diagram
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Fe-Cr-S-O System

• S T
• V -P
• nFe mFe
• nCr mCr

• f1 T
• f2 -P

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Fe-Cr-S-O System
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Pressure vs. Volume diagram for H2O
This is NOT a true phase diagram.
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Fe-Cr-C System

• S T
• V -P
• nC mC
• nFe mFe
• nCr mCr

• f1 T
• f2 -P
• f3 mC
• (NOT OK)
• (OK)

Requirement
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Fe-Cr-C system, T 1300 K, XCr
nCr/(nFenCrnC) vs. (carbon activity)
This is NOT a true phase diagram.
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Corresponding phase diagrams of the Fe-O system
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Other Sets of Extensive Variables and
Corresponding Potentials
Gibbs-Duhem
Gibbs-Duhem
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Mg-Si binary phase diagram
38
Mg-Si, DH vs XSi phase diagram