phase diagram

G. Selvaduray - SJSU

BINARY PHASE DIAGRAMS

Dr. Guna SelvadurayMaterials Engineering Program

San Jose State UniversitySan Jose, CA 95192-0086

Utility of Phase Diagrams

SolderingBrazingElectromigrationDiffusion ProblemsKirkendahl VoidingCorrosionElectrical Resistivity

Limitations to use of Phase Diagrams

Phase Diagrams are also known as Equilibrium DiagramsRate of Transformation is missingTTT (Time-Temperature-Transformation) diagrams are a complement to Phase Diagrams

Approach

Approach taken during this course will be phenomenologicalNo chemical thermodynamics will be used for derivations

Useful References1. M. Hansen & K. Anderko, Constitution of Binary Alloys, McGraw-

Hill, 19582. R.P. Elliot, Constitution of Binary Alloys, First Supplement,

McGraw-Hill, 19653. F.A. Shunk, Constitution of Binary Alloys, Second Supplement,

McGraw-Hill,19694. ASM International, ASM Handbook Volume 3: Alloy Phase

Diagrams, 19925. R. Hultgren, P.D. Desai, et al, Selected Values of the

Thermodynamic Properties of Binary Alloys, ASM International, 1973

Useful References (cont’d)6. E.M. Levine, C.R. Robbins & H.F. McMurdie, Phase Diagrams for

Ceramists, The American Ceramic Society, 19647. A. Reisman, Phase Equilibria-Basic Principles, Applications,

Experimental Techniques, Academic Press, 19708. A. Findlay, The Phase Rule and its Applications, Dover

Publications, 19519. G. Humpston & D.M. Jacobson, Principles of Soldering and Brazing,

ASM International, 1993

What is a “Phase”?

Sand and SaltCoffee and SugarOil and Vinegar

How many phases in each?

What is a “Phase”? (cont’d)A phase is a homogenous, physically distinct and mechanically separable portion of the material with a given chemical composition and structure.

For solids: Chemically and structurally distinct

For liquids: Miscibility

For gases: Always 1 phase

One Component Phase Diagram

The simplest case-WaterAlso known as a P-T diagramSign of [dP/dT] for:

Solid-LiquidLiquid-GasGas-Solid

equilibria

P-T Diagram for Water

Source: Barret, Nix & Tetelman, The Principles of EngineeringMaterials, 1973. p 118

One Component Phase DiagramRegion Number of Phases Degrees of Freedom

The Gibbs Phase Rule

P + F = C + 2

The Quasi-Chemical ApproachUnderstanding interactions on bond energies

Interaction between 2 species: A and BA-A and B-B bonds

Thermodynamic Parameter: Melting Point (T)

How does mixing of A-A and B-B bonds affect T?

The Ideal Case

(A-B) = x(A-A) + (1-x) (B-B)

Where x is the mole fraction of A in BTAlloy = TA + x ( TB - TA)

Examples:Copper – NickelSilicon – Germanium

Nickel-Copper Phase Diagram

Source: ??

Germanium-Silicon Phase Diagram

Hume Rothery Rules

1. Relative Size Ratio ±15%2. Crystal Structure-must be the same3. Electronegativity Difference – within

± 0.4 e.u.4. Valence must be the same

Eutectic Behavior

A-B < 0.5 (A-A + B-B)

TAlloy < TA , TB

Examples:Lead - TinGold - Silicon

Tin-Lead Phase Diagram

Gold-Silicon Phase Diagram

Hansen & Anderko, Constitution of BinaryAlloys, 1958. p. 232

Gold-Germanium Phase Diagram

Intermetallic Compound Formation

A-B > 0.5 (A-A + B-B)

TAlloy > TA , TB

Example:Gallium -Arsenic

Arsenic-Gallium Phase Diagram

Working with Phase DiagramsOverall CompositionSolidusLiquidusLimits of Solid SolubilityChemical Composition of Phases at any temperatureAmount of Phases at any temperatureInvariant ReactionsDevelopment of MicrostructureChemical Activity

Copper-Silver Phase Diagram

Source: Callister, Materials Science and Engineering:An Introduction, 2000. p. 256

Solidus and LiquidusSolidusTemperature at which alloy is completely

solid Temperature at which liquefaction beginsLiquidusTemperature at which alloy is completely

liquidTemperature at which solidification begins

Overall Composition

Concentration: Relative amounts of each constituentIt is the horizontal axis in all binary phase diagrams The scale can be in weight %, atomic % or mole %

Chemical Composition of Phases

It is the chemical composition of each phase in the systemIn a system having more than one phase, each phase will have a unique chemical composition which will be different from each other, and will also be different from the overall compositionNot to be confused with overall composition

Solid SolutionsWhat is a solid solution?

When foreign atoms are incorporated into a crystal structure, whether in substitutional or interstitial sites, the resulting phase is a solid solution of the matrix material (solvent) and the foreign atoms (solute)

Substitutional Solid Solution: Foreign (solute) atoms occupy “normal” lattice sites occupied by matrix (solvent) atoms, e.g. Cu-Ni;Ge-Si

Interstitial Solid Solutions: Foreign (solute) atoms occupy interstitial sites, e.g., Fe-C

G. Selvaduray - SJSUSource: Barret, Nix & Tetelman, The Principles of EngineeringMaterials, 1973. p 72

Types of Solid SolubilityUnlimited Solid Solubility: Solute and solvent are mutually soluble at all concentrations, e.g., Cu-Ni systemMeets the requirements of the Hume-Rothery Rules

Result is a “single phase alloy”

Limited or Partial Solid Solubility: There is a limit to how much of the solute can dissolve in the solvent before “saturation” is reached, e.g., Pb-Sn and most other systems

Does not meet the requirements of the Hume-Rothery Rules

Results in a “multi-phase alloy”

Amount of each phase

Dependent on the Overall Composition and Temperature

The (Inverse) Lever Rule

Tie-Lines

Lever Rule - 1

Source: Smith, Principles of Materials ScienceAnd Engineering, 1996, p.440

Cu-Ni Phase Diagram

Example Problem 1One kilogram of an alloy of 70% Pb and 30% Sn is

slowly cooled from 300ºC. Calculate the following:

a) Weight % of liquid and α at 250ºCb) Chemical composition of the liquid and α at 250ºCc) Weight % of the liquid and α just above the eutectic

temperatured) Chemical composition of the liquid and α at just above

the eutectic temperature

Pb-Sn Phase Diagram

Invariant Reactions

Eutectic: L = α (s) + β (s); e.g., Pb-Sn

Peritectic: α (s) + L = β (s); e.g., Pb-In

Monotectic: L1 = α (s) + L2; e.g., Cu-Pb

Syntectic: L1 + L2 = α (s); e.g., Na-Zn

Metatectic: β (s) + α (s) = L1 e.g., U-Mn

Pb-In Phase Diagram

Cu-Pb Phase Diagram

Microstructure Development

The microstructure developed depends on the overall composition and the cooling rate

Composition dependence of microstructure

Source: Askeland, The Science & EngineeringOf Materials, 1984, p 246

Example Problem 2For the 70% Pb and 30% Sn alloy, calculate:(a) The weight percent of alpha and beta phases

at 100ºC(b) The chemical composition of the α and β

phases at 100ºC(c) Amount of primary and secondary α(d) Amount of α formed during the eutectic reaction

Chemical ActivityWhat is activity?A measure of the “escaping tendency”Activity = 1 if species is in its standard

state (pure, most stable form, at temperature of interest)

What is the activity of a species in a solution?

Activity (a) =Activity Coefficient x Mole Fraction

Activity DeterminationsIDEAL CASE: Activity Coefficient = 1Therefore: Activity = Mole Fraction; e.g., Cu-Ni

NON-IDEAL CASE:

Positive Deviation: a>aid, i.e., activity coefficient>1e.g. Pb-Sn

Negative Deviation: a<aid, i.e., activity coefficient<1e.g. Ga-As

G. Selvaduray - SJSUSource: Gaskell, Introduction to ThermodynamicsOf Materials, 1973

Example Problem 3Draw an activity-composition diagram for the Cu-Ni system at 1200ºC

Draw an activity-composition diagram for the Ga-As system at 400ºC

SJSU-Selvaduray

Intermetallic Compounds

Line compoundsStoichiometric RatioStoichiometric Range

SJSU-Selvaduray

Au-Sn Phase Diagram

Ag-Sn Phase Diagram

Using Phase Diagrams to determine Heat TreatabilityHeat Treatment is based on “controlling” the solid state transformation rate

Heat treatment of steels: control of the eutectoid reactionAge hardening (precipitation strengthening) of aluminum alloys: control of precipitation reaction

Heat Treatment of Steels

The eutectoid reactionMartensiteAustenitePearliteTTT diagrams

Fe-C Phase Diagram

TTT Diagram

Source: Flinn & Trojan, Engineering Materialsand their Applications, 1986, p 239

Age Hardening/Precipitation Strengthening

Particularly relevant for aluminum alloys, e.g., aluminum lines on ICsPhase diagrams tell us if an alloy system is age-hardenable, and the composition range over which the alloy system is age-hardenableAl-Cu system

Age Hardening Al Alloys

Al-Cu Phase Diagram

Source: Hansen & Anderko, Constitutionof Binary Alloys, 1958. p. 85

Heat Treatment vs Strength

Source: ??

Heat Treatment vs Ductility

Source: ??

Coherent & Incoherent Precipitates

Source: ??

Effect of aging on Electromigration

Critical parameter: densityppt vs densitymatrix

If densityppt > densitymatrix

Region of compression is created around the ppt

Driving force is for migration of matrix atoms awayfrom ppt

If densityppt < densitymatrix

Region of tension is created around the ppt

Driving force of for migration of matrix atomstowards the ppt

Lead Frame Alloys

Alloy 42Copper alloy lead framesKovar

Lead Frame Alloy Compositions

Source: Electronic Materials Handbook Volume 1: Packaging, ASM International, 1989, p. 490

Fe-NiPhaseDiagram

Source: Hansen & Anderko,Constitution of BinaryAlloys, 1958. p. 85

Fe-CuPhaseDiagram

Cu-SnPhaseDiagram

Example Problem 4

Will the age hardening process characteristics affect the electrical resistivity (or conductivity) of lead frames?Will the conductivity increase or decrease with overaging?

Application of Phase Diagrams to Diffusion

Fick’s First Law: J = -D [dc/dx][dc/dx] is the concentration gradient and

driving force for diffusionIt this were true, multiphase alloys such

as Pb-Sn alloys must “self-homogenize”over time and transform into a single phase alloy

Activity: Driving Force for Diffusion

The driving force for diffusion to occur is the activity differenceIn the case of Pb-Sn alloys, the phases are:α (Pb rich) and β (Sn rich)

Diffusion of a species from one phase into another will not occur if:

aSn (beta) = aSn (alpha)aPb (beta) = aPb (alpha)

Relevance of solid solubility limits

Phase diagrams also tell us the maximum extent to which one species can diffuse into another

This is given by the solid solubility limits at the temperature of interest

The Cu-Ni example in standard textbooks is most often not applicable

Kirkendahl VoidingIf there is a major difference in solid

solubility limits, voiding can be expected to occur in the phase that permits less solid solubility

e.g., the Al-Au system

Interdiffusion does not necessarily occur at the same rate

Al-Au Phase Diagram

Effect of composition on properties

Mechanical PropertiesElectrical Resistivity

Composition vs Strength

Source: ??

Composition vs Resistivity

G. Selvaduray - SJSUSource: Askeland, The Science & EngineeringOf Materials, 1984, p 565

Determination of Phase Diagrams

Cooling CurvesDifferential Scanning CalorimetryThermomechanical AnalysisDifferential Thermal AnalysisMetallography/PetrographyEnergy Dispersive X-ray SpectroscopyElectron Microprobe AnalyzerX-ray DiffractionTransmission Electron Microscopy

Cooling Curves

Source: Smith, Principles of Materials ScienceAnd Engineering, 1996, p.441

Experimental measurement of ∆Hm from DSC

Mg-SiPhaseDiagram

Al-SiPhaseDiagram

Al-MgPhaseDiagram

Cr-Mo Phase Diagram

Cr-Ni Phase Diagram

Mo-NiPhaseDiagram

Au-Si Phase Diagram

Au-SnPhaseDiagram

Ternary Phase Diagrams

Three componentsOverall compositionNumber of phasesChemical composition of each phaseAmount of each phaseSolidification sequence

SJSU-Selvaduray

Example Problem 5

What is the maximum number of phases that can exist in a ternary system?

SJSU-Selvaduray

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