PHASE DIAGRAMSPhase – a chemically and structurally homogenous region of a material. Region of uniform physical and chemical characteristics. Phase boundaries separate two distinct phases. A single phase system is called homogeneous. A system with two or more phases is called heterogeneous.Phase Diagram – a graphic representation showing the phase or phases present for a given composition, temperature and pressure.Component – the chemical elements which make up the alloy.

Solvent atoms: primary atomic species. Host atoms Solute atoms: the impurities. Normally the minor component

Solubility Limit - Maximum concentration of solute atoms that may dissolve in the solvent to form a solid solution. The excess of solute forms another phase of different composition. Example: water-sugar

Phase Diagrams of Pure Substances•Predicts the stable phase as a function of Ptotal and T. Example: water can exist in solid, liquid and vapor phases, depending on the conditions of temperature and pressure.•Characteristic shape punctuated by unique points.

– Phase equilibrium lines– Triple Point (three different phases of water in equilibrium)– Critical Point

Example: In the pressure-temperature (PT) phase diagram of water there exists a triple point at low pressure (4.579 torr) and low temperature (0.0098oC) where solid, liquid and vapor phases of water coexists.Vaporization Line – Liquid and vapor coexistsFreezing Line – Liquid and solid coexist.Sublimation Line – Solid and vapor coexist

Gibbs Phase RuleFrom thermodynamic considerations, J.W. Gibbs (1839-1903 American physicist –University of Yale) derived the following equation:

P + F = C + 2Where: P = number of phases which coexists in a given system; F = degrees of freedom; C = number of components in the system ; 2 = one can vary temperature and pressure

F = 0 zero degrees of freedom. Neither P or T can be change (a point–invariant point)F = 1 one degree of freedom. One variable (P or T) can be changed independently (a line)F = 2 two degrees of freedom. Two variables (P or T) can be changed independently (an area).Example- For pure substance where P and T can be changed

P + F = C + 2 = 1 + 2 = 3Pure substance in a triple point, then C = 1 (one component) and P = 3 (number of phases that coexist)The value of F is zero (zero degrees of freedom) the three phases coexist in a point.

- For pure substance where P and T can be changed P + F = 1 + 2 = 3

Pure substance in a freezing line, then C = 1 (one component) and P = 2 (number of phases that coexist)The value of F is one (one degree of freedom) the two phases (solid and liquid) coexist in a line.

PHASE•Homogeneous portion of the system with uniform physical and chemical characteristics

– Sugar – water syrup (H20 and C12 H22 O11)– Solid sugar (C12 H22 O11)

•A difference in either physical or chemical properties constitutes a phase– Water and ice– FCC and BCC polymorphic forms of an element

MicrostructureThe structure observed under a microscope

Al Brake – more than one phase

Iron-chromium alloy – one phase (solid solution)

Phase Equilibria• Free energy: a function of the internal energy of a system• Equilibrium: a system is at equilibrium if its free energy is at a minimum• Phase equilibrium: for a system which has more than one phase• Phase Diagram is a diagram with T and Composition as axes. They define the

stability of the phases that can occur in an alloy system at constant pressure (P). The plots consist of temperature (vertical) axis and compositional (horizontal) axis.

• Constitution: is described by(a) the phases present(b) the composition of each phase(c) the weight fraction of each phase

• Binary alloy: A mixture of two metals is called a binary alloy and constitute a two-component system.

• Each metallic element in an alloy is called a separate component. [Sometimes a compound is considered a component, (e.g., iron carbide)]

• Isomorphous System: In some metallic systems, the two elements are completely soluble in each other in both the liquid and solid states. In these systems only a single type of crystal structure exists for all compositions of the components (alloy) and therefore it is called isomorphous system.

Binary isomorphous systems

T

(a) Phases PresentPoint A: at T=1100oC60wt% Ni – 40wt% CuOnly α phase is presentPoint B: at T= 1250oC35wt%Ni – 65wt% Cu Both α & liquid phases are present

at equilibrium

(b) Composition of each phaseSingle phase:Point A:

60wt%Ni – 40%Cu alloy at 1100oC

Two-phase region:Tie line: across the two-phase region at the temperature of the alloyPoint B: T=1250oCComposition of Liquid phase: CL=31.5wt%Ni – 68.5%CuComposition of α phase:Cα=42.5wt%Ni- 57.5wt%Cu

(c) Weight fraction of each phaseSingle phase: 100% Ex: Point A: 100% α phase

Two-phase region: Ex: Point B

LEVER RULE (Inverse Lever Rule)

L

oL

L

CCCCW

SRSW

−−

=

+=

α

α

L

Lo

CCCC

SRRW

−−

=+

=α

α

Example: Point B: C0 = 35wt%NiCα = 42.5%, CL = 31.5% 68% or ...

.

32% or ...

.

680531542

35542

320531542

53135

=−−

=−−

=

=−

−=

−−

=

Ls

osL

Ls

Lo

ccccW

ccccWα

Volume fractionFor an alloy consisting of α and β phases, the volume fraction of the α phase is

defined as

ββαα

βββ

ββαα

ααα

βαβα

αα

ρρρ

ρρρ

vvv

Wvv

vW

VVvv

vV

+=

+=

=++

=

;

, 1 Then, the weight fractions are

Where να and νβ are the volumes of α and β

β

β

α

α

α

α

α

ρρ

ρWW

W

V+

=

β

β

α

α

β

β

β

ρρ

ρWW

W

V+

=

Derivation of the lever rule1) All material must be in one phase or the other:

2) Mass of a component that is present in both phases equal to the mass of the component in one phase + mass of the component in the second phase:

3) Solution of these equations gives us the Lever rule.

L

oL cc

ccW−−

=α

α

L

Lo

ccccW

−−

=α

α

1=+ LWWα

oLL ccWcW =+αα

Equilibrium Cooling - Development of Microstructure in Isomorphous Alloys

Example:

35wt%Cu-65wt%Ni system – Slow coolingfrom point a to point e

a: 1300oC: complete liquid with 35wt%Cu-65wt%Ni

b: ~1260oC: first solid begin to form

(α-46wt%Ni)

c: ~1250oC: α-43wt%Ni, L-32wt%Ni

d:~1220oC: last liquid to solidify

e: 35wt%Cu – 65wt%Ni solid phase

Nonequilibrium Cooling - Development of Microstructure in Isomorphous Alloys

Fast cooling

Compositional changes require diffusion

•Diffusion in the solid state is very slow. ⇒ The new layers that solidify on top of the existing grains have the equilibriumcomposition at that temperature ⇒ Formation of layered (cored) grains. Tie-line method to determine the composition of the solid phase is invalid.•The tie-line method works for the liquid phase, where diffusion is fast. •Solidus line is shifted to the right (higher Ni contents), solidification is complete at lower T, the outer part of the grains are richer in the low-melting component (Cu).•Upon heating grain boundaries will melt first. This can lead to premature mechanical failure.

Complete solidification occurs at lower temperature and higher Nickel concentration than equilibriumSolid can’t freeze fast enough: solidus line effectively shifted to higher Ni concentrations. Shift increases with faster cooling rates, slower diffusion

Mechanical properties of isomorphous alloys

Solid solution strengthening

Invariant Points in Binary Systems•Binary alloys – two components at ambient pressure. Gibbs rule states that

P + F = 2 + 1= 3.•If three phases coexists (P = 3), they coexist at a point (zero degrees of freedom – the invariant point, at a specific temperature and chemical composition•Types of invariant points:

eutectic, eutectoid, peritecticperitectoid, monotectic etc.

Some Important Invariant Points

EutecticCoolingHeating

EutectoidCoolingHeating

L+α

α+β

L L+β

Peritectic CoolingHeating

α+γ

α+β

γ β+γ

δ+γ γ

δ +L

γ +L

eutectic: Liquid/solid reactioneutectoid: solid/solid reaction

βα +→Lβα +←L

βαγ +→βαγ +←

γδ →+ Lγδ ←+ L