alloys and phase diagrams

8/9/2019 Alloys and Phase Diagrams

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MetalsII. Phase diagrams of alloys

J. McCord

(mostly taken from Gottstein)


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Contents

some basic thermodynamics

Gibbs phase rule Binary alloys

Monotectic phase diagram

Continuous miscibility isomorphous alloy systems

Eutectic phase diagram

Peritectic phase diagram

Intermediate and intermetallic phases

Ternary alloys

Multicomponent phase diagrams

2.2


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Why alloying?

Purpose of alloying

Improve mechanical properties

Improve physical properties

Improve or impede chemical reactions (e.g. resistant to corrosion)

Reduce the overall material cost (simplify processing/reducing metal cost)

Steel & cast iron (Iron (Fe) & Carbon (C)

Brass (Copper (Cu) & Zinc (Zn))

Bronze (Copper (Cu) & Tin (Sn))

Sterling silver (92.5%Ag & 7.5%Cu)

2.3


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Formation of alloys

Alloys from a large number of elements

Binary and ternary phase diagrams

Alloying

Starting from homogeneous liquid solution

Changes in state, i.e. from liquid to solid, and the equilibrium between

the various phases present are illustrated using a phase diagram

Defining microstructure after solidification

2.4


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Thermodynamic considerations (brief) Ideal configurational entropy

Entropy of mixing SM(ideal solution)

W: Number of configurations

N= NA + NB: Number of atoms

2.5

( )

( ) ( )[ ]BBBB

AABB

BA

M

M

ccccNk

ccccNk

NN

NkS

WkS

+

+=

1ln1ln=

lnln

!!

!ln=

ln=

ions)concentrat(atomic=BA

ii

NNNc+

Fig. 2.1.Entropy of mixing for ideal solution.Note that for cB 0 and cB 1 approaches +/- infintiy. As aresult, it is impossible to produceperfectly pure crystals.

BM cS /

A cB B

Entropy

ofmixingSM


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Regular solution interaction of A-B

Free energy of mixing GM = HM - TSM

Enthalpy of mixing H

M

A-B interaction is neglected incalculating W

Pair interaction model

ij interaction energy

Zcoordination number

BAM cNZcH =

( )BBAAAB +2

1=

Fig. 2.2. Enthalpy of mixing HMfor regularsolutions.

< 0

> 0

A cB B

EnthalpyofmixingHM

2.6


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Composition dependence of G= H- TS

Fig. 2.3. Composition dependence of Gibbs free energy G and its component H and (-TS) at constanttemperature and pressure for (a) continuous miscibility ( < 0) and (b) miscibility gap at c1 < c < c2 ( > 0).Here, further alloying would increase G. Therefore, it is favorable to form to separate phases with

concentrations c1 and c2. A positive value of means that the A-A and B-B attraction is larger than the A-Battraction. Note the asymmetry of the diagrams despite the symmetry of SM and GM. This is due to HA HB.

SinceS/c for the pure components it is not possible to obtain perfectly pure materials.

TSHG -=

> 0

< 0

2.7


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Chemical potential i

Diffusion of A and B atoms between phases until establishment ofequilibrium condition.

Equilibrium condition for 2 components and two phases:

Tangent construction (see also Fig. 2.3(b))

Assumption: ncomponents & Pphases

Gibbs phase rule for number of degrees of freedom1

f = n - P + 2 . For p= const.: f = n - P + 1.

pTNi

i

ij

N

G

,,

=

Ni= ciN: number

of atoms of i

1There are (n 1)P concentration variables and the variables p and T. The equilibrium

conditions yield n(P 1) boundary conditions.

BB

AA

=

=

Fig. 2.4. Tangent construction.

2.8


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The Gibbs phase rule

f = number of degrees of freedomor number of externally controlled variables (e.g. temperature, pressure,composition)

n = number of components in the systemSmallest number of independent chemical constituents by means of which

the composition of every possible phase can be expressed

P = number of phases

2 = number of state variablesTemperature and pressure (usually incompressible at ordinary pressures,

pressure is considered fixed)

Gibbs phase rule can written as

f = n - P + 1

f = n - P + 2

2.9


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Equilibrium Phase Diagrams

provide information for the control of the micro- or phasestructure of an alloy system.

are for predicting phase transformations and theresulting microstructures.

represent the relationships between temperature andthe compositions and the quantities of phases atequilibrium.

are maps of the equilibrium phases associated withvarious combinations of temperature and composition.

2.10


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Equilibrium phases in two component systems

f = n - P + 1.

Fig. 2.5. At a given pressure a one component system (a) has a certain melting point. In a binarysystem, however, there is a melting range by liquidus and solidus lines. These can either fall (b) or rise

(c) with concentration. The tie line connects equilibrium concentration.2.11

tie line

liquidus line

solidus line

liquidus line

solidus line

S

L

S

L L+S

L+Sf = degrees of freedomn = number of components

P = number of phases

L

S

TM


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Monotectic phase diagram (Pb-Fe)

Complete miscibility gap in the solid and liquid phase.

Abb. 2.6. Monotectic phase diagram of lead iron PbFe.

GS 4.7

2.12

Pb-melt + Fe-melt

Pb-melt + Fe-crystal

Pb-crystal + Fe-crystal


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Monotectic phase diagram (Cu-Pb)

Abb. 2.7. The Copper-lead phase diagram.

2.13

Cu-solid + Pb-liquid


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Fig. 2.8. Composition dependence of Gibbs free energy of crystal or solid phase and melt L of abinary alloy with continuous miscibility for different temperatures T. The composition ranges of the

occurring phases correspond to an isothermal section through the phase diagram.

L

LL

LL

L L

LS

Continuous miscibility

2.14

L

L +


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Ni-Cu

2.15

Fig. 2.9. The copper-nickel isomorphousphase diagram

Point x

Alloy of 20%Cu 80% Ni at T =500C (solid)

Point y

Two-phase region (solid &liquid) at T = 1200C

Composition not defined bypoint y!

Point y


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The Lever rule

No equilibration in solid phase

c0 approached asymptotically

Strongly extended solidificationrange

Formation of layered crystals

Weight fraction mL of melt and m of solid

Conservation of atom numbers

1,120 =++= LL mmcmcmc

01

20

cc

cc

m

m

L

=

21

01

21

20

cc

ccm

cc

ccm

L

=

=

Lever rule at T= T1

Fig. 2.10. Illustration of the solidification of abinary alloy. The microstructure of the threestates, liquid, semi-solid, and solid areschematically illustrated.

system composition

L

Tie line

2.16


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Continuous miscibility (II)

Fig. 2.11. Relationship of the free-energy curves that

lead to a maximum in the phase diagram.2.17


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Continuous miscibility (III)

Fig. 2.12. Relationship of the free-energy curves that

lead to a minimum in the phase diagram.2.18


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Cu-Au phase diagram

Fig. 2.13. Cu-Au phase diagram with a minimum. In the solid state thereis a continuous solution. 2.19

weight %Au

atomic %Au

S L + S

L


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Solid solutions

Fig. 2.14. Different types of solid solution crystals. (a) Interstitial solid solution, (b) substitution solidsolution with random distribution, (c) ordered substitutional solid solution.

2.20

Substitutional or interstitial solid solution

Small solute atom

Interstitial solid solution

Substitution of the atoms in the solvent unit cell

Substitutional solid solution


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Fig. 2.15. Interstitial in the fcc and bcc lattice. (a) Octahedral interstice in the fcc, (b) tetrahedralinterstice in fcc, (c) octahedral interstice in bcc. The open circles indicate the different but equivalentinterstitial positions.

Interstitials in the fcc and bcc lattice

fcc

bcc


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Substitutional solid solutionHome-Rothery-Rules

1. Atomic size factor Difference in atomic radii between the two atom typesshould not exceed 15%.

2. Valences The number of valence electrons should not be very different.

3. Electronegativity The difference of electronegativity (chemical affinity)should be small. In case not, intermetallic compounds form instead of asubstitutional solid solution.

Crystal structure For appreciable solid solubility,the crystal structures for metals of both atom types

must be the same (FCC, BCC, ...) Melting point A metal of higher melting point

dissolves a metal of lower melting point to agreater extent than vice versa.

2.23


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Cu-Au phase diagram

Entropy contribution!!!

Fig. 2.16. Phase diagram of the system

CuAu that forms several solid stateordered phases at low temperatures.However, these turn to a disordered statewell below the melting point.

weight %Au

atomic %Au

2.24

S L + S

L


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Ordered solid solutions

Fig. 2.17. Ordered atom distributions of the AB-type can be realized in the (a) CsCl lattice (= B2structure); (b) AB3 in the Cu3Au lattice (=L12 structure). Both are consistent with a bcc or fcc solid

solution crystal. A ordered face centered cubic solution of the AB type (c) loses its cubic structure.Due to differences in the atomic radii a tetragonal crystal lattice is formed.

Cu atoms

Au atoms

Cu atoms

Au atoms

bcc

fcc

2.25


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Long range order parameter s(sublattices)

AB-alloy with bcc structure

Complete long range order p= 1, s= 1

Complete statistical disorder p= 0.5, s= 0

Wrong order p = 0, s= 1

Order-disorder transition at elevated T:mostly 2nd order transitions

x

xps

=

1

12 = ps

Fig. 2.18. The bcc lattice con-sists of two cubic simple sub-

lattices that are displaced alongthe -direction. With s= 1each of the sublattices isoccupied by only one sort ofatoms.

p: fraction of A-atoms in A-sublatticesx: fraction of A-atoms in the alloy

2.26


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Debye-Scherrer images of Cu3Au

Fig. 2.19. Occurrence of long-range ordered phases can be confirmed by superlattice lines in a Debye-Scherrer-image. Note the distinct lines for T


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Short range order parameter

No long range correlation needed

Short range order coefficients can be extended to more distant neighbors

Complete short range order = 1

No short range order = 0

Short-range order for < 0.

can be determined from the intensity of the diffuse scattering background. Derivations from ideal

crystal structure lead to diffuse x-ray scattering with intensity outside the Bragg-reflections.

dm

d

qq

qq

=

q: fraction of B-atoms as next neighbors of A-atomsqd: fraction of B-atoms as neighbors of A-atoms in

the disordered stateq

m: fraction of B-atoms as neighbors of A-atoms inthe completely ordered state

10

2.28


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Temperature dependence of order parameters

Fig. 2.20. Temperature dependence of the order parameters. Whereas the long range orderparameter s is zero above Tc, the short range order parameter exists also above Tc.

Temperature [K]


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Antiphase boundaries (Cu3Au)

(a)

(c)

(b)

Abb. 2.21. Change of electrical resistivity with theoccurrence of superlattices. (b) Sketch of a long-rangeordered Cu3Au structure divided into domains. Theregions are separated by antiphase boundaries. (c)

Visualization of antiphase boundaries by TEM.

quenched from 650 C

annealed at 200 C

atomic %Au

resistivity[10-6

cm]

2.30


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Formation of a solubility gap

Increasing energy of mixing GM> 0

Fig. 2.22. Schematic illustration of the evolution of the eutectic phase diagram resulting from awidening of the miscibility gap in the solid state. This is equivalent to an increasing energy of mixingGM> 0.

L

L

L

LL

LL

L

L

2.31

L


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Eutectic system

Fig. 2.23. Composition dependence of Gibbs free energy of crystal or solid phase and melt L of a

binary alloy with a miscibility gap for different temperatures T. The composition ranges of the occurringphases correspond to an isothermal section through the phase diagram.

L L

L

L

L L

SS

S

LL

L

SS

L

L

L

2.32

L

L


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Eutectic phase diagram

(solidification)

Gibbs phase rule

f = n - P + 1

Eutectic reaction

n= 2 (A, B) - components

P= 3 (L, ,) - phases

f= 0 (fixed point)

No degree of freedom

Simultaneous precipitation of and(mostly fine-lamellar microstructure).Fig. 2.24. Schematic eutectic phase diagram illustratingthe development of the microstructure of a sub eutecticalloy during solidification. At the eutectic composition cEthe alloy solidifies like a pure metal (no solidification

range).

weight %B

Invariant reaction L+

L

LL

2.33

hypoeutectic hypereutectic


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Copper-Silver phase diagram

Fig. 2.25. The eutectic Cu-Ag phase diagram.

2.34


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Fig. 2.26. Schematic representations of the equilibrium microstructures for a leadtin alloy of compositionC4 cooled from the liquid-phase region.

Example: PbSn lead-tin phase diagram

cE


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Eutectic microstructure

Fig. 2.27. Example of an eutectically solidified microstructure (c = cE)

in the AlZn system (95.16 weight% Zn). 2.36


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Peritectic phase diagram

Strongly different melting points

Increasing energy of mixing GM> 0 Peritectic phase diagram

Fig. 2.28. Schematic illustration of the evolution of the peritectic phase diagram resulting from awidening of the miscibility gap in the solid state accompanied by a strong difference of melting pointsof the componentsA and B. The first is equivalent to an increasing energy of mixing GM> 0.

L L L L

LLLL

L

2.37


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Example: Peritectic Fe-Ni phase diagram

Fig. 2.29. The FeNi phase diagram.

2.38


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Phase diagram with an intermetallic phase

GM, < 0 Tendency to form high melting-point AxBy compounds

In the intermediate and intermetallic phase the crystal structure is different from

the structures of the components A and B.

Fig. 2.30. Schematic illustration illustrating the formation of intermetallic phases due to a widening ofthe miscibility gap in the solid state.

LL

2.39


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Copper-Zinc phase diagram

Fig. 2.31. Copper zinc (brass) phase diagram. 2.40


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Summary of

equilibrium phasetransformations(binary alloys)

Abb. 2.32. Overview of differenttype of phase transformations.

2.41


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Multicomponent systems ternary systems

cA = 70 %cB = 10 %

cC = 20 %

cA = 70 %

cB = 10 %

cC = 20 %

Fig. 2.33. Representation of the composition in a ternary system.

2.42


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Eutectic ternary system

2.44

Fig. 2.35. (a) Perspective view of a ternary phase diagram. (b) Projection of liquidus plane on theconcentration valley including eutectic valleys and melt isotherms.(e1, e2, e3: eutectic points of binary alloys; E: Eutectic point of ternary alloy).

Combining three


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Combining threeeutectic binaries

Eutectic valleys

Fig. 2.36. A ternary phase diagramwith three eutectic binaries betweenA, B and C.

2.45

I h l i


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Isothermal sections

Fig. 2.37. Isothermal section for a ternary phase diagram. A few tie-lines are shown by dotted lines.The lines represent two-phase equilibriums and are also called conodes. For a given point on the tie-lines, the fractions of the two phases are determined by the lever rule. Here the temperature is below

the A-B eutectic but above the eutectic temperature of the ternaryA-C and B-C. 2.46

I th l ti


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Isothermal sections

Fig. 2.38. Isothermal section for a ternary phase diagram. A few tie-lines are shown by dotted lines.The lines represent two-phase equilibriums and are also called conodes. For a given point on the tie-lines, the fractions of the two phases are determined by the lever rule. Here the temperature is below

the binary eutectics but above the ternary eutectic temperature. 2.47

I th l ti


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Isothermal sections

Fig. 2.39. Isothermal section for a ternary phase diagram. A few tie-lines are shown by dotted lines.The lines represent two-phase equilibriums and are also called conodes. For a given point on the tie-lines, the fractions of the two phases are determined by the lever rule. Now the temperature is below

the ternary eutectic temperature. 2.48


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alloys and phase diagrams

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