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Atomic Transport

&

Phase Transformations

Lecture 8

PD Dr. Nikolay Zotov

[email protected]

Lecture I-8 Outline

Binary phase diagrams with limited solubility in the liquid state

Classification of (intermediate) intermetallic compounds

Formation of intermetallic compounds

Gibbs energy of intermediate phases

Examples of phase diagrams with intermediate phases

Calculation of phase diagrams with intermediate phases

Binary Phase DiagramsClassification

WL = 0

WS = 0

Unlimited Solubilityin Liquid and Solid

Limited Solubilty

in Solid in Liquid in Liquidand in Solid

WS > 0 WL > 0

WL > 0

WS > 0with intermetallicphases WS < 0

Binary Phase Diagrams

Limited solubility in the Liquid State (WL > 0)

Regular solution

ln(gAL) = WL(1 – XA

L)2 > 0 gAL > 1

ln(gBL) = WL(1 – XB

L)2 > 0 gBL > 1 Tendency for phase separation in the liquid state

If WL> 0 a miscibility gap will form in the liquid state!



DeHoff (2006)

WL = 0 WL = 10000 J/mol WL = 20000 J/mol



# very different melting points

# inflection point in the liquidus curve

# retrograde solubility of Tl in Ag above 300 oC;

# very low solubility of Ag in Tl;

# Allotropic phase transition in Tl at 234 oC;

a(Tl) hexagonal P 63/mmc

ß(Tl) cubic Im-3m



# Maximum of the liquid missibility gap at 1071 K

# WL = 2R TgmL ~ 17800 J/mol

# Eutectic point TE ~ 591 K

# below 591 K two phase mixture Pb + Zn

Zn hexagonal; Pb cubic

L1 + L2

L

L1 + Zn

Pb + Zn

TgmL

E



●

●

●

●

●

●

A

B

C

D

F

G

A T > 1071 K homogeneous liquid L with XZn = 0.4

B L starts to segregate

XZn(L1) ~ 0.4, XZn(L2) ~ 0.92

fraction(L1) ~ 100%

C Two liquids mixture

XZn(L1) ~ 0.11, XZn(L2) ~ 0.98 (almost pure Zn melt)

fraction (L1) ~ 56 %

D The Zn-rich liquid disappears (Zn crystallizes)

Mixture of Pb-rich liquid (L1) + Zn

XZn(L1) ~ 0.06, XZn(Zn) ~ 0.999

fraction(Zn) = 38%

F Eutectic Tie-line; L1 + Pb + Zn in equilibrium

XZn(Pb) ~ 0.024, XZn(Zn) ~ 0.99

G Two phase mixture of solid Pb and solid Zn



# Temperature of the maximum of theSolubility gap Tgm

L ~ 879 K at XPb ~0.38

# WL = 2RTgmL ~ 14600 J/mol

# B Two – liquid mixture

XPb(L1) ~ 0.13, XPb(L2) ~ 0.76

# Monotectic point M

(MM‘ – monotectic tie-line)

XPb(L1) ~ 0.016, XPb(L2) ~ 0.97 XPb(fcc) ~ 0.998

# Monotectic reaction

L2 ↔ L1 + Pb

# Below 302 K two phase mixture of Ga and Pb

Ga orthorhombic oC8 Cmca

Pb cubic cF4 F m-3m

TgmL

L1L1 + L2

L2

L

L1 + Pb

Ga + Pb

MM‘

●

B



M

# TgmL ~ 3162 K

WL ~ 52490 J/mol

# Monotectic point M

Monotectic T = 2124 K;

# Solubility of Ag in bcc V

increases with increasing T

# Solubility of V in Ag –

negligible:

V bcc structure I m-3 m

(5+, 4+, 3+, 2+)

Ag fcc structure F m -3 m

TgmL



# Maximum of the liquid immisibility

gap unknown →

WL probably very high

# Very high Monotectic point 3240 oC

# practically no mutual solubility

below 2000 oC


Monotetctic Reaction - DG curves


Monotectic Reactions

# temperature of the maximum of

the liquid miscibility gap 849 K

at XTl = 0.32;

# WL ~ 14100 J/mol;

# Monotectic Temperature 559 K;

# Allotropic phase transition

L1 + ß → L1 + a Tl at 234 oC;

# 2 phase mixture Ga + aTl below 30 oC.


Invariant reactions - Summary

Limited Reaction Description

Solubility

Solid Eutectic l ↔ a + ß

Eutectoid g ↔ a + ß

Peritectic l + a ↔ ß

Metatectic ß ↔ l + a

Liquid Monotectic l1 ↔ l2 + a


Impossible eutectic phase diagrams

Gibbs phase rule:

The invariant tie -line means that the there isno degree of freedom: the composiiton and thetemperature are fixed

Inclined tie-line in an eutectic phase diagram means thatT can be varied as a function of composition, which is notpossible.


Impossible eutectic phase diagrams

Prince (1966)

Complete misciblity is not possible

Addition of B into A (or B in A) will increase the Entropy of mixing and reduce the Gibbs eneryg of the solution

Binary Phase DiagramsClassification

WL = 0

WS = 0

Unlimited Solubilityin Liquid and Solid

Limited Solubilty

in Solid in Liquid in Liquidand in Solid

WS > 0 WL > 0

WL > 0

WS > 0with intermetallicphases WS < 0

Intermetallic Compounds

Classification

Entropy

ordered disordered

Variations in chemical composition

Stoichiometric Non-stoichiometricDefect compounds

Electronic configuration

Normal ValenceCompounds

Hume-Rotheryphases

Binary Phase DiagramsIntermediate phases - Formation

# Enthalpy stabilisation with respect to the termianl solid solutions

Strong preference for the formation of bonds between unlike atoms in the solid

De = eAB – ½ (eAA + eBB ) < 0 → WS < 0

# usually (very) different structures from the terminal solid solutions

Binary Phase DiagramsFormation of intermediate phases

WS < 0

WL = 0

# WS < 0 and WL = 0 Increase of the melting temperatures

of the solid solutions with respect to the pure components

is observed;

# congruent melting at XBCM

# Congruent melting (freezing) – the melt freezes in a solid phase

with the same composition

# Incongruent melting (freezing) – the solid and the melt do not have

the same composition

XBCM


Extention of the Regular solution model:

DH mix = XB(1 – XB)

WS(XB) = - a[b(1-2XB)4 + c/(d – XB)2]

The second term describes a strong tendency for

unlike bond formation

-ac/(d – XB)2 ; XB → d deep minimum

0,0 0,2 0,4 0,6 0,8 1,0

-4000

-2000

0

DH

mix

(J/m

ol)

XB

a

a+ß a+ß a

Appearance of an

intermediate phase ß

Relatively wide

compositional range

Prince (1966)

ß

WS(XB); WS(XB) < 0


Stoichiometric (line) compounds – compounds with

‚Infinitely‘ large curvature!!!


The curvature of the DG curve the ß phase determines the stability range

Prince (1966)

Binary Phase DiagramsPhase diagrams with Intermediate Phases

FeCr

s – FeCr phase

Tetragonal, P 42/mnm; 30 atoms in the unit cell

Compositional range 45 – 49 wt% Cr

Fe DHM = 13.8 kJ/mol

Cr DHM = 21.0 kJ/mol


Very large compositional range

35 – 65 wt% V!!!

Fe V

WS << 0


# Intermediate phase ß with relatively wide compositional range

# congruent melting of the ß phase for the composition X*.

# The phase diagram could be regarded as two eutectic phase

diagrams: A – X and X – B;

∂XBb/∂T ~ DHsol

ab/ T(XBß – XB

a) ∂2Gmb/∂(XB

b)2;

∂T/ ∂XBb ~ T ∂2Gm

b/∂(XBb)2 (X* – XB

a) / DHsolab ;

# very large curvature of the Gibbs free energy of the

ß phase ↔ vertical phase boundary

# very small solubility of the a phase in ß ↔

vertical phase boundary

X*


# Congruent melting at 1065 oC;

# extremely narrow stability range of Mg2Sn(Stoichiometric compound);

# Mg2Sn cubic, antifluoride structure, F m -3 m

# Extremely low solubility of Mg in Sn and Sn in Mg

Different structure

Compared to termianl phases

Mg hexagonal

P 63/mmc

Sn tetragonal

I 41/


# Congruent melting at 1065 oC

# extremely narrow stability range

(Stoichiometric compound);

# Mg2Si cubic, antifluoride structure, F m -3 m

# Extremely low solubility of Mg in Si and

Si in Mg

Mg hexagonal

P 63/mmc

Si cubic

F d -3 m


●A

●B

●

●

A Liquid with composition XSi(L) = 0.4

B L + Mg2Si

XSi(Mg2Si) = 0.333, XSi(L) = 0.45

fraction (Mg2Si) ~ 50%

C Mg2Si + Si solid state 2-phase mixture

fraction fraction (Mg2Si) ~ 89%

C


Multiple intermetallic compounds

# Ti5Si3 is congruently melting;

Larger range of stability at high temperatures

Hex P 63/mcm

# Ti3Si, Ti5Si4 and TiSi are incongruently

melting, stoichiometric compounds

# TiSi2 congruently melting

TiSi2

Binary Phase DiagramsCalculation of Phase diagrams with Intermediate Phases

Thermodynamic description of Compounds

Compound Energy Formalism

Stoichiometric Compounds – each atom occupies a unique sub-lattice, no substitutional disorder

G Comp = S xirefGi + GComp

Form (T); xi site fraction of the unique sublatticerefGi(T) – Gibbs energy of element occupying sub-lattice i in its

reference state

GCompForm (T) – Gibbs energy of formation of the compound.

G(T) = A + BT + (CTlnT + DT2 +)

Solid solutions (liquid/Solid) – Regular solid solutions + Redlich-Kisler expansion

G sol = XTirefGTi + XSi

refGSi + RT[XTiln(XTi) + XSiln(XSi)] + XTiXSiW

W = oL(T) + 1L(T)(XSi – XTi) + 2L(T)(XSi – XTi) Fiore et al (2016)


Description of intermediate compounds

Ti3SiSi occupies 1 sub-lattice (XSi= ¼)

Ti atoms fill 3 sub-lattices with the same sitesymmetry, each contributes XTi = ¼

Ti

Si

G Comp = S xirefGi +



From the SGTE Data Base

Element S 298

(J/mol.K)

Ti 30.72

Si 18.8

Fiore et al (2016)

GTi3Si = ¼(3217 – 298x18.8) + ¾(4824 – 298x30.72) - 20000 + 3.2T

GTiSi = ½ (3217 – 298x18.8) + ½(4824 – 298x30.72) - 155062 + 7.6T

GTi5S4 = 4/9(3217 – 298x18.8) + 5/9(4824 – 298x30.72) - 711000 + 23.4T

GTiSi2 = 2/3 (3217 – 298x18.8) + 1/3 (4824 – 298x30.72) - 17538 + 4.5T

refG = refDH – 298 refS



Fiore et al (2016)

Solid solutions



The eutectic tie-lines are given in red

Non-vertical phasebaoundaries often relatedto non-stoichiometric compounds

Fiore et al (2016)


CuZn

# five peritectic reactions

l + a ↔ ß

l + ß ↔ g

l + g ↔ d

l + d ↔ e

l + e ↔ h

# only the e phase is stoichiometric and

has a vertical phase boundary

# the other intermediate phases (solid

Solutions) have inclined phase boundaries,

because the curvatures of the Gibbs energy

curves are not so steep