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Chapter 9. Phase Diagrams. Phase Diagram Vocabulary. Unary Phase Diagrams – H 2 O. 1 atmosphere. Unary Phase Diagram – Pure Fe. Gibbs Phase Rule (Section 9.17). Tells us how many phases can exist under a given set of circumstances. P+F=C+2 P = number of phases

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Chapter 9

Chapter 9

Phase Diagrams

Gibbs phase rule section 9 17
Gibbs Phase Rule (Section 9.17)

  • Tells us how many phases can exist under a given set of circumstances.


    • P = number of phases

    • F = number of degrees of freedom – number of variables that can be changed independently of all other variables in the system

    • C=number of components

    • The number two indicates the ability to change temperature and pressure; these are non-compositional variables that affect the phases.

  • Modified Gibbs phase rule

    • Most engineering systems function at a pressure of 1 atmosphere, i.e. we have picked the pressure as one of our degrees of freedom. Therefore,

      P+F = C+1

  • Binary isomorphous system
    Binary Isomorphous System

    • Two components are completely soluble in each other in both solid and liquid phases

    • Hume-Rothery’s Rules (Section 4.3 text 7th edition)

      • Atomic size difference not greater than 15%

      • Crystal structure is the same for both components

      • Similar electronegativity (i.e. no ionic bonding)

      • Elements have a similar valance

    • Example: Cu-Ni System

      • rCu = 0.128 nm rNi = 0.125 nm

      • Both have a face centered cubic (fcc) structure

      • Electronegativity Cu = 0.19; Ni = 0.18

      • Valance – Cu+ and Cu++; Ni++

    Cooling curves during solidification
    Cooling Curves during Solidification

    Solidification occurs at constant temperature while latent heat of fusion is released

    Cooling curves for a binary isomorphous alloy
    Cooling curves for a binary isomorphous alloy

    • Features:

    • Solidus – locus of temperatures below which all compositions are solid

      • Start of solidification during cooling

    • Liquidus – locus of temperatures above which all compositions are liquid

      • Start of melting during heating

    Modified gibbs phase rule
    Modified Gibbs Phase Rule

    • In the liquid or solid phase:

      • P=1, C=2

      • P+F=C+1

      • F=2

      • Both composition and temperature can be varied while remaining in the liquid or solid phase

    • In the L+a region

      • P=2, C=2

      • P+F=C+1

      • F=1

      • If we pick a temperature, then compositions of L and a are fixed

      • If we pick a composition, liquidus and solidus temperatures are fixed



    Tie line and lever rule
    Tie Line and Lever Rule

    • At point B both liquid and a are present

    • WL×R = WS×S





    • Non-equilibrium cooling results in

      • Cored structure

      • Composition variations in the solid phase as layers of decreasing Ni concentration are deposited on previously formed a phase

      • Solidification point is depressed

      • Melting point on reheat is lowered

    • Homogenization or reheating for extended times at temperature below e’

    Effect on mechanical properties
    Effect on Mechanical Properties

    Due to solid solution strengthening, alloys tend to be stronger and less ductile than the pure components.

    Binary eutectic system

    Eutectic temperature

    α (solid solution) + β (solid solution)



    61.9% Sn


    18.3% Sn

    97.8% Sn

    Binary Eutectic System

    • The two components have limited solid solubility in each other

    • Solubility varies with temperature

    • For an alloy with the Eutectic composition the liquid solidifies into two solid phases

    Binary eutectic system1
    Binary Eutectic System

    • Apply Modified Gibbs Phase Rule

      • Phases present: L, a and b (P=3)

      • Components: Pb and Sn (C=2)

      • P+F=C+1

      • F=0  no degrees of freedom

      • Therefore, three phases can coexist in a binary system only at a unique temperature and for unique compositions of the three phases

      • Upon cooling, there is a temperature arrest during the solidification process (eutectic reaction)

    Microstructures in the eutectic system
    Microstructures in the Eutectic System

    • Depending on the system, eutectic solidification can result in:

      • Lamellar structure – alternating plates

      • Rod-like

      • Particulate

    Amounts of phases at different temperatures
    Amounts of Phases at different temperatures

    • At Teutectic + DT

    • At Teutectic - DT

    Other reactions in the binary system
    Other Reactions in the Binary System

    • Upon Cooling the following reactions are also possible

      • Peritectic L + a b

      • Monotectic L1 L2 + a

      • Eutectoid a b + g

      • Peritectoid a + b g

    Copper zinc system
    Copper-Zinc System

    • Terminal phases

    • Intermediate phases

    • Several peritectics

    • Eutectoid

    • Two phase regions between any two single phase regions

    Mg pb system
    Mg-Pb System

    • Intermediate Compound Mg2Pb

    • Congruently melting

      Mg2Pb  L


    Portion of the ni ti system
    Portion of the Ni-Ti System

    • Congruently melting intermediate phase g

      g  L


    Iron carbon system
    Iron-Carbon System

    • Reactions on cooling

    • Peritectic

      L + d  g

    • Eutectic

      L  g + Fe3C

    • Eutectoid

      g  a + Fe3C

    Cast Iron


    Iron carbon or iron fe 3 c
    Iron-Carbon or Iron-Fe3C

    • In principle, the components of the phase diagram should be iron (Fe) and carbon/graphite (C).

      • Fe and C form an intermediate compound Fe3C, which is very stable

      • There isn’t anything of interest at carbon contents greater than 25 at.% or 6.7 wt.% C.

      • Fe3C is considered to be a component, and the binary phase diagram is drawn using Fe and Fe3C.

    • Names of phases:

      • Ferrite - a iron – bcc structure

      • Austenite – g iron – fcc structure

      • High temperature d iron – bcc structure

      • Cementite – Fe3C

    • Steels have carbon contents <2%, usually <1.2%

    • Cast irons have carbon contents >2%

    Phase transformations in steels
    Phase Transformations in Steels

    Eutectoid Composition – 0.76wt% C


    Alternating plates (lamellae) of Fe and Fe3C

    Austenite  Ferrite + Cementite (at 727ºC upon cooling)

    0.76wt.%C 0.022wt.%C 6.7wt.% C

    Phase transformations in steels1
    Phase Transformations in Steels

    • Hypoeutectoid composition <0.76 wt% C

    • Proeutectoid ferrite nucleates and spreads along austenite grain boundaries at T>727ºC

    • Remaining austenite converts to pearlite during eutectoid transformation

    Phase transformations in steels2
    Phase Transformations in Steels

    • Hypereutectoid composition >0.76 wt% C

    • Proeutectoid cementite nucleates and spreads along austenite grain boundaries at T>727ºC

    • Remaining austenite converts to pearlite during eutectoid transformation

    Phase transformations in steels3
    Phase Transformations in Steels



    Proeutectoid ferrite


    Proeutectoid cementite

    Effect of alloying elements
    Effect of Alloying Elements

    • Addition of an alloying element increases the number of components in Gibbs Phase Rule.

    • The additional degree of freedom allows changes in the eutectoid temperature or eutectoid Carbon concentration