L8_Binary_phase_diagrams.pdf

7/27/2019 L8_Binary_phase_diagrams.pdf

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MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Binary phase diagrams

Binary phase diagrams and Gibbs free energy curves

Binary solutions with unlimited solubility

Relative proportion of phases (tie lines and the lever principle)

Development of microstructure in isomorphous alloys

Binary eutectic systems (limited solid solubility)

Solid state reactions (eutectoid, peritectoid reactions)

Binary systems with intermediate phases/compoundsThe iron-carbon system (steel and cast iron)

Gibbs phase rule

Temperature dependence of solubility

Three-component (ternary) phase diagrams

Reading: Chapters 1.5.1 1.5.7 of Porter and Easterling,Chapter 10 of Gaskell


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Binary phase diagram and Gibbs free energy

BX 10

AG

A binary phase diagram is a temperature - composition mapwhich indicates the equilibrium phases present at a given

temperature and composition.The equilibrium state can be found from the Gibbs free energydependence on temperature and composition.

BG

G

We have also discussed thedependence of the Gibbs freeenergy from composition at agiven T:

We have discussed thedependence of G of a one-

component system on T:

G

T

ST

G

P

=

T

c

T

S

T

G P

PP

2

2

=

=

mixmixBBAA STHGXGXG ++=


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Binary solutions with unlimited solubility

BX 10

liquidAG

Lets construct a binary phase diagram for the simplest case: Aand B components are mutually soluble in any amounts in both

solid (isomorphous system) and liquid phases, and form idealsolutions.

We have 2 phases liquid and solid. Lets consider Gibbs freeenergy curves for the two phases at different T

liquidBG

solidG

T1 is above the equilibrium melting temperatures of both

pure components: T1 > Tm(A) > Tm(B) the liquid phasewill be the stable phase for any composition.

liquidG

1T

[ ]BBAABBAAid lnXXlnXXRTGXGXG +++=

solidBG

solid

AG


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Binary solutions with unlimited solubility (II)

B

X 10

solidBG

Decreasing the temperature below T1 will have two effects:

will increase more rapidly than

liquidBG

solidG

Eventually we will reach T2 melting point of pure

component A, where

liquidG

2T

liquidB

liquidA GandG

solidAG

solidBGand Why?

The curvature of the G(XB) curves will decrease. Why?

solidA

liquidA GG =

solid

A

liquid

A GG =


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Binary solutions with unlimited solubility (III)

solidBG

For even lower temperature T3 < T2 = Tm(A) the Gibbs freeenergy curves for the liquid and solid phases will cross.

liquid

BG

solidG

liquidG

3T

solidAG

As we discussed before, the common tangent construction can beused to show that for compositions near cross-over of Gsolid andGliquid, the total Gibbs free energy can be minimized byseparation into two phases.

BX 10

solidliquid

solid +liquid

1X 2X

liquidAG


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Binary solutions with unlimited solubility (IV)

liquidAG

solidB

liquidB GG =

solidG

liquidG

4T

solid

A

G

At T4 and below this temperature the Gibbs free energy of thesolid phase is lower than the G of the liquid phase in the wholerange of compositions the solid phase is the only stable phase.

BX 10

As temperature decreases below T3 continue

to increase more rapidly than

Therefore, the intersection of the Gibbs free energy curves, as

well as points X1 and X2 are shifting to the right, until, at T4= Tm(B) the curves will intersect at X1 = X2 = 1

liquidB

liquidA GandG

solidB

solidA GandG


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Binary solutions with unlimited solubility (V)

solidBG

Based on the Gibbs free energy curves we can now construct aphase diagram for a binary isomorphous systems

liquidBG

solidG

liquidG

3TsolidAG

BX 10

solidliquid

solid +liquid

liquidAG

2T

3T

4T

5T

1TT

4T

2T

1T


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Liquidus line separates liquid from liquid + solid

Solidus line separates solid from liquid + solid

Binary solutions with unlimited solubility (VI)

Example of isomorphous system: Cu-Ni (the complete solubilityoccurs because both Cu and Ni have the same crystal structure,FCC, similar radii, electronegativity and valence).

Liquid

Solid solution

Solidus lineLiquidus line


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In one-component system melting occurs at a well-definedmelting temperature.

In multi-component systems melting occurs over the range oftemperatures, between the solidus and liquidus lines. Solid andliquid phases are in equilibrium in this temperature range.

+ L

L liquid solution

liquid solution+

crystallites ofsolid solution

polycrystalsolid solution

Binary solutions with unlimited solubility (VII)

Liquidus

Solidus

A B20 40 60 80Composition, wt %

Temperature


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Interpretation of Phase Diagrams

For a given temperature and composition we can use phasediagram to determine:

1) The phases that are present

2) Compositions of the phases

3) The relative fractions of the phases

Finding the composition in a two phase region:

1. Locate composition and temperature in diagram

2. In two phase region draw the tie line or isotherm

3. Note intersection with phase boundaries. Read compositionsat the intersections.

The liquid and solid phases have these compositions.

BXsolidBX

liquidBX


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Finding the amounts of phases in a two phase region:

1. Locate composition and temperature in diagram

2. In two phase region draw the tie line or isotherm

3. Fraction of a phase is determined by taking the length of thetie line to the phase boundary for the other phase, anddividing by the total 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.

Interpretation of Phase Diagrams: the Lever Rule

1) All material must be in one phase or the other: W + W = 1

2) Mass of a component that is present in both phases equal tothe mass of the component in one phase + mass of thecomponent in the second phase: WC + WC = Co

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

W = (C0- C) / (C - C) and W = (C- C0) / (C - C)

Derivation of the lever rule:

W

+

W


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Composition/Concentration:

weight fraction vs. molar fraction

Composition can be expressed in

Molar fraction, XB, oratom percent (at %) that is useful whentrying to understand the material at the atomic level. Atom

percent (at %) is a number of moles (atoms) of a particularelement relative to the total number of moles (atoms) in alloy.For two-component system, concentration of element B in at. %is

Where nmA and nm

B are numbers of moles of elements A and B inthe system.

Weight percent (C, wt %) that is useful when making the

solution. Weight percent is the weight of a particular componentrelative to the total alloy weight. For two-component system,concentration of element B in wt. % is

100mm

mC

AB

Bwt +

=

100nn

nC A

mBm

Bmat

+=

where mA and mB are the weights of the components in the

system.

100XC Bat

=

B

BBm

A

mn = where AA and AB are atomic

weights of elements A and B.A

AAm

A

mn =


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Composition Conversions

Weight % to Atomic %:

Atomic % to Weight %:

100ACAC

ACC

BwtAA

wtB

AwtBat

B +=

100ACAC

ACC

BwtAA

wtB

BwtAat

A +=

WL = (Cwt

- Cwt

o) / (Cwt

- Cwt

L)

Of course the lever rule can be formulated for any specification

of composition:

ML = (XB - XB

0)/(XB - XB

L) = (Cat - Cat

o) / (Cat

- Cat

L)

M = (XB0 - XB

L)/(XB - XB

L) = (Cat0 - Cat

L) / (Cat

- Cat

L)

W = (Cwt

o- Cwt

L) / (Cwt

- Cwt

L)

100ACAC

ACC

AatAB

atB

BatBwt

B +=

100ACAC

ACC

AatAB

atB

AatAwt

A +=


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Phase compositions and amounts. An example.

Mass fractions: WL = S / (R+S) = (C - Co) / (C - CL) = 0.68

W = R / (R+S) = (Co- CL) / (C - CL) = 0.32

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


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Equilibrium (very slow) cooling


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Equilibrium (very slow) cooling

Solidification in the solid + liquid phase occursgradually upon cooling from the liquidus line.

The composition of the solid and the liquid changegradually during cooling (as can be determined by the

tie-line method.)

Nuclei of the solid phase form and they grow toconsume all the liquid at the solidus line.


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Non-equilibrium cooling


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Non-equilibrium cooling

Compositional changes require diffusion in solid and liquid

phases

Diffusion in the solid state is very slow. The new layersthat solidify on top of the existing grains have the equilibriumcomposition at that temperature but once they are solid theircomposition does not change. Formation of layered (cored)

grains and the invalidity of the tie-line method to determinethe composition of the solid phase.

The tie-line method still works for the liquid phase, wherediffusion is fast. Average Ni content of solid grains is higher. Application of the lever rule gives us a greater proportion

of liquid phase as compared to the one for equilibriumcooling at the same T. Solidus line is shifted to the right(higher Ni contents), solidification is complete at lower T, theouter part of the grains are richer in the low-meltingcomponent (Cu).

Upon heating grain boundaries will melt first. This can leadto premature mechanical failure.


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Binary solutions with a miscibility gap

Lets consider a system in which the liquid phase isapproximately ideal, but for the solid phase we have Hmix > 0

solidG

liquidG

1T

BX 10

solidG

liquidG12 TT

L8_Binary_phase_diagrams.pdf

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