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Chapter Outline: Phase Diagrams. Microstructure + Phase Transformations in Multicomponent Systems. Definitions and basic concepts Phases and microstructure Binary isomorphous systems (complete solid solubility) Binary eutectic systems (limited solid solubility)

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Chapter Outline: Phase Diagrams

Microstructure + Phase Transformations

in Multicomponent Systems

  • Definitions and basic concepts
  • Phases and microstructure
  • Binary isomorphous systems (complete solid solubility)
  • Binary eutectic systems (limited solid solubility)
  • Binary systems with intermediate phases/compounds
  • The iron-carbon system (steel and cast iron)
  • Not tested: The Gibbs Phase Rule

Components and Phases

Component - chemical species

(Fe + C in steel; H2O + NaCl in salt water).

Binary alloy 2 two components,

Ternary alloy – 3, etc.

Phase – a portion with distinct, uniform physical or chemical characteristics

Single-phase system: Homogeneous.

Two or more phases

Mixture or Heterogeneous system.

e.g. water + ice, separated by a phase boundary


Solubility Limit

Solvent - host or major component

Solute - minor component (Chapter 4).

Solubility Limit = maximum amountthat can be dissolved in a phase

(e.g. alcohol has unlimited solubility in water, sugar has a limited solubility, oil is insoluble).

Same concepts for solids: Cu and Ni are mutually soluble in any amount (unlimited solid solubility), while C has a limited solubility in Fe.



Properties of an alloy depend on proportions of the phases and on how they are arranged at the microscopic level.

Microstructure: number of phases, their proportions, and their arrangements

Microstructure of cast Iron

Alloy of Fe with 4 wt.% C. There are several phases. The long gray regions are flakes of graphite. The matrix is a fine mixture of BCC Fe and Fe3C compound.

Phase diagrams help understand and predict microstructures


Equilibrium and Metastable States

Equilibrium: at constant temperature, pressure and composition system is stable

(Equilibrium is achieved given sufficient time, but that may be very long. )

Metastable:System appears to be stable

Equilibrium  minimum in the free energy.

  • Under conditions of constant temperature, pressure and composition, change is toward lower free energy.


  • Stable equilibrium is state with minimum free energy.
  • Metastable state is a local minimum of free energy.

Free Energy



Phase diagram

Phase diagram - combinations of temperature, pressure or composition for which specific phases exist at equilibrium

H2O: diagram shows temperature and pressure at which ice (solid),water (liquid) and steam (gas) exist.


Phase diagram

Show what phases exist at equilibrium and what transformations we can expect when we change T, P, or composition

Consider binary alloys only

Pressure constant at one atmosphere.


Binary Isomorphous System (I)

Assume Complete Solubility


 + L

Three phases :

Liquid (L) , solid + liquid (+L), solid ()

Liquidusline separates liquid from liquid + solid

Solidus line separates solid from liquid + solid


Binary Isomorphous Systems (II)


Complete solubility occurs because Cu and Ni have the same crystal structure (FCC), similar radii, electronegativity and valence


Binary Isomorphous System (III)

One-component: melting occurs at a well-defined temperature.

Multi-component: melting occurs over range of temperatures between solidus and liquidus lines.

Solid and liquid phases are in equilibrium in this temperature range.


Liquid solution

 + L

Liquid solution


Crystallites of

Solid solution


Solid solution


Interpretation of Phase Diagrams


temperature + composition 


1) Phases present

2) Compositions of phases

3) Relative fractions of phases

  • Composition in a two phase region:
  • 1. Locate composition and temperature
  • 2. Draw tie line or isotherm
  • Note intersection with phase boundaries
  • Read compositions at the intersections
  • Liquid and solid phases have these compositions

The Lever Rule

Amounts of each phase in two phase region

Locate composition and temperature

Draw tie line or isotherm

Fraction of a phase =length of tie line to other phase boundary divided by the length of tie line

The lever rule is a mechanical analogy to the mass balance calculation. The tie line in the two-phase region is analogous to a lever balanced on a fulcrum.


The Lever Rule

Mass fractions: WL = S / (R+S) = (C- Co) / (C- CL)

W = R / (R+S) = (Co- CL) / (C- CL)


Derivation of the lever rule

Wand WL are fractions of  and L phases

1) All material is in one phase or the other:

W+ WL = 1

2) Composition equal composition in one phase + composition second phase

at given T:

Co = WC + WLCL

3) Solution gives Lever rule.

WL = (C- Co) / (C- CL)

W = (Co- CL) / (C- CL)


Phase compositions and amounts. An example.

Co = 35 wt. %, CL = 31.5 wt. %, C = 42.5 wt. %

Mass fractions: WL = (C- Co) / (C- CL) = 0.68

W = (Co- CL) / (C- CL) = 0.32


Microstructure in isomorphous alloys

Equilibrium (very slow) cooling