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PHASE DIAGRAMS AND GLASS FORMATION

Marcello Baricco

Dipartimento di Chimica IFM-NISUniversità di Torino, TORINO (Italy)

OUTLINE

•Phase diagrams, free energy curves and the Calphad method -

•Metastable phase diagrams

•Thermodynamics of amorphisation

•Eutectic systems

•T0 - Driving forces for nucleation

•Multicomponent systems

•Conclusions

Phase diagrams and free energy

A.H.Cottrell, Theoretical Structural Metallurgy, Edward Arnold, London, 1955)

The CALPHAD approachThermodynamic models

•Regular solutions• Sublattice model

•Stoichiometric compounds•Magnetic contributions

•Lattice stabilities

Ab initiocalculations

Parameter description

Assessment(least square method)

Experimental techniques•Calorimetry

• Electromotive forces•XRD, SEM, etc.

Experimental data:•Thermodynamic properties

•Phase diagrams

),,(),(),,(),,( 0i

idii xPTGxTGxPTGxPTG ++=ϕ

The CALPHAD approachTemperature dependence:

G k k T k T T k TS k k T k T

H k k T k TCp k k T

= + + + + ... = - - [ + ] - + ...

= - - + ...= - - + ...

1 2 3 42

1 3 42

Composition dependence:

•Redlich-Kister polinomia•Stoichiometric compounds•Sub-lattice models•Associate model

0BA )( xxLxxG

nex −= ∑

forforBABA STHGyGxG yx ∆⋅−∆++=∆ βα 00

A truly metastable body is in internalequilibrium, though its free energy is above

that of the same mass in its stable state.

STABLE

UNSTABLE

METASTABLE

Arrangement of atoms

METASTABLE PHASE DIAGRAMS

T.B.Massalski, The 1988 Campbell Memorial Lecture ASM International, Metall. Trans. B, 20B (1989) 445

Thermodynamicsof glass-formation

viscosity

heat capacity

free energy

What we can learn from phasediagrams about glass formation?

•Empirical parameters

•Liquidus lines

•Phase selection and T0 curves

•Driving forces for nucleation

•Input data for kinetic modelling

Empirical parameters for bulk glass formation

lgrg TTT /=•Reduced glass transition temperature

•Stability of undercooled liquid

•Lu & Liu gamma

•Inoue’s magic rules

large negative enthalpy of mixing

atomic size ratios above 10%

several components (confusion principle)

gx TTT −=∆

)/( lgx TTT +=γ

A.Inoue, Bulk Amorphous Alloys, Mater. Sci. Found. 4 (1998) 1

Deep eutectics•Tg does not depend sharply on composition, so glass formation will be greatest near eutecticcompositions.

•The deeper the eutectic or steeper the adiacent liquidi, the greater is the expected tendency for glass formation.

•Deeper eutectics mean greater stability of the liquid and a relatively smaller degree of undercooling required for glass formation.

Cu-Mg phase diagram

0.0 0.2 0.4 0.6 0.8 1.0400

1400 Liang et al. CALPHAD 22 (1998) 527-544

Mole fraction Mg

Eutectic system

0.0 0.2 0.4 0.6 0.8 1.0

00.0 0.2 0.4 0.6 0.8 1.0

Mg molar fraction

hcp-Mg

CuMg2Cu2Mg

fcc-Cu

liquid

Free energy at 600 K

Free energy difference(solids-liquid) at 600 K

eutectic points

How deep is a eutectic?

Turnbull (M.Marcus and D.Turnbull, Mater. Sci. Eng. 23 (1976) 211)

negative deviation of the liquidus temperature from the ideal solution liquidus.

Donald and Davies (I.W.Donald, H.A.Davies, J. Non-cryst. Sol. 30 (1978) 77)

deviation of the liquidus temperature from the average melting temeparture of the pure components.

∆T = how deep is the eutectic

0.0 0.2 0.4 0.6 0.8 1.0

TavTid

B MOLAR FRACTION

Regular solution modelThe regular solution model considers interactions between similar atoms (A-A and B-B) and different atoms (A-B).

Regular solution model

∆ ∆G X G X G GA A B B mix = + +

∆ ∆ ∆ = .- G H T Smix mix mix

)](ln + )(ln [ - = BBAAmix XXXXRS∆

= BAmix XXH Ω∆

Regular solutionmodel

0.0 0.2 0.4 0.6 0.8 1.0250

1500 ΩL -20 -10 0 10

20 IDEAL

B MOLAR FRACTION

-1mol KJ 03 = ΩS

TmA = 800 K, Tm

B = 1200 K, ∆SmA = ∆Sm

B = 10 J mol-1 K-1

-20 -10 0 10 20

ΩL / KJ mol-1

-1mol KJ 03 = ΩS

A deep eutectic is the evidence of a strong interaction in the liquid.

Role of solid state solubility

0.0 0.2 0.4 0.6 0.8 1.0-1

liq ΩL=-10 sol ΩS=+10 sol ΩS=+20T = 500 K

B molar fraction

When a solubilty in the solid state isobserved, solidsolution isstabilised withrespect the pure component

Role of solid state solubilityHigher liquidusline and lowerliquidus slope

0.0 0.1 0.2400

800ΩS=

ΩL= -10 KJ mol-1 10 12 15 20

B molar fraction

Role of solidstate solubility

Alloy compositions where the melt normally nucleated in a stoichiometric intermetallic phase exhibit a greater tendency to glass formation.

Eutectics comprising a solid solution and an intermetallic phase should exhibit an asymmetry in the GFR, with a preference towards the side where the compound is the primary phase.

K.S.Dubey, P.Ramachandrarao, Int. J. Rap. Sol. 5 (1990) 127

Off-eutectic compositionsCu-Mg-Y

H.Ma , Q.Zheng, J.Xu, Y.Li, E.Ma, J. Mater. Res. 20 (2005) 2252H.Ma , L.L.Shi, J.Xu, Y.Li, E.Ma, APL 87

(2005) 181915

H.Tan, Y.Zhang, D.Ma, Y.P.Feng, Y.Li, Acta Mat. 51 (2003) 4551

Off-eutecticcompositions

Off-eutectic compositionsSkewed coupled zones

Cu-ZrW.J.Boettinger, MRS Symp. 8 (1982) 15

D.Wang, Y.Li, B.B.Sun, M.L.Sui, K.Lu, E.Ma, APL 84 (2004) 4029

H.Tan, Y.Zhang, D.Ma, Y.P.Feng, Y.Li, Acta Mat. 51 (2003) 4551

Thermodynamic description of metastable phases

liquid

crystal

Temperature

Experimental data and modelling for amorphous phase are necessary

Fe-B binarysystem

Undercooling experiments

1200 1300 1400 1500

Crystallisation of amorphous alloys

Fe83B17

1 W / g

Temperature / K

Primary γ-Fe

Eutectic γ-Fe+Fe3B

Fe3B ⇒ Fe2B

300 325 350 375 400 425 450 475 500

Temperature / °C

Fe85B15

20 K/min

Am ⇒ α-Fe+Am’

Am’ ⇒ Fe3B

Formation of Fe3B metastable phase

L.Battezzati, C.Antonione, M.Baricco, JAC 247 (1997) 164

Stable and metastable Fe-B phase diagram

0.0 0.2 0.4 0.6 0.8 1.0

1767 K

1452 K0.168

0.1811387 K

liquid

XBM.Palumbo, G.Cacciamani, E.Bosco, M.Baricco, Calphad 25 (2001) 625

Heat capacity of undercooled melts

I. -R. Lu, G. Wilde, G. P. Görler and R. WillneckerJ. Non-Cryst. Sol. 250-252, ( 1999) 577 R.Bush, W.Liu, W.L.Johnson, J. Appl. Phys. 83

(1998) 4134

Modelling of the liquid-amorphous phase and glass transition

H.-J. Fecht, W.L. Johnson / Materials Science and Engineering A 375–377 (2004) 2–8

Kubaschewski‘s approach

O. Kubaschewski et al., Materials Thermochemistry, Pergamon, NY,

23 −++= bTaTRCp

R -> gas constant

a, b → fitting parameters

• Suitable to describe the behaviour of the Cp on undercooling

• No glass transition (Kauzmann paradox)

Glass transition as a second order thermodynamic transition, analougous to Curie transition

Shao‘s approach

G. Shao, J. Appl. Phys. 88 (2000) 4443

)()1ln( τα fRTG amliq +−=∆ →

Hillert and Jarl polynomial for magnetic contribution to the Gibbs free energy τ=T/Tg

Composition dependence of both Tg and α

• Suitable for the description of glass transition

• Unable to correctly describe the behaviour of the Cp of liquid on undercooling

Combination of models

to describe glass transition as a second order thermodynamic transition

Shao‘s approach

Kubaschewski‘s approach

above Tg, Hillert and Jarl polynomial has been modified according to Kubaschewski‘s formula

...it is expected to correctly describe both glass transition and specific heat data...

Experimental and calculated Cp of liquid Cu25Mg65Y10

300 600 90020

Tg Exp. Busch Exp. this work no excess Cp Shao Kubaschewski this work

Temperature / K

M.Palumbo, M.Sarra, G.Cacciamani, M.Baricco, to appear on Mater. Trans.

T0 curvesThe T0 curve is the locus of the compositions and temperatures where the free energies of two phases are equal.

The T0 curve between the liquid and a solid phase determines the minimum undercooling of the liquid for the partition-less formation of the solid with the same composition.

T0 curves

Alloys with T0 curves which are only slightly depressed below the stable liquidus curves are good candidates for partitionlesssolidification in the entire composition range (dashed lines).

If T0 curves plunge to very low temperatures (continuous lines), single phase crystals cannot be

formed from the melt.

T0curves

T0 curves for Cu-Tistable metastable

L.Battezzati, M.Baricco, G.Riontino, I.Soletta, J. De Phys. 51 (1990) C4-79

T0 curves for Fe-B

0.00 0.05 0.10 0.15 0.20 0.25800

1800 with ∆Cp without ∆Cp

B molar fraction

M. Palumbo, E. Bosco, G. Cacciamani, M. Baricco, CALPHAD, 25, 4 (2001) 625

Driving force for solutions

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂=⎟⎟

⎞⎜⎜⎝

⎛∂

∂ LA

XAG µ−µ=∆Parallel tangent rule

A CN C0

Liquid(amorphous)

CrystallineSolution

µAXFr

Composition B

Driving forces for regular solutions

0.0 0.2 0.4 0.6 0.8 1.0600

XBl=0.1

ΩL=10 KJ/mol and ΩS=30 KJ/mol

Driving forces for regular solutions

Driving forces for glass-formingsystems

800 1000 12000

CuZr without ∆Cp CuZr with ∆Cp

Liquid 50% Cu

Temperature / K

Cu-MgCu-Zr

400 500 600 700 8000

hcp-Mg with ∆Cp CuMg2 with ∆Cp

hcp-Mg without ∆Cp CuMg2 fwithout ∆Cp

Temperature / K

M. Baricco, M. Palumbo, M. Satta, G.Cacciamani, ISMANAM 2006

Multicomponent systems

D. Wang, H. Tan and Y. Li, Acta Mater. 53 (2005) 2969

Cu-Zr-Al

Liquid-amorphous phase separation A phase separation in a ternary regular solution occurs when the following equation is fulfilled where L(0)

i;j < 0:

Cu-Mg-Y

Cu-Zr-Al

Conclusions

Thermodynamics is the science of the impossible. It enables us to tell with certainty what cannot happen, but is noncommittal about the things that are possible.

Thermodynamics is at its best when nothing more can happen, a condition called equilibrium.

The usefulness of metastable equilibrium diagrams lies in the fact that, like stable diagrams, there are rules for their construction which guide measurement and permit our experience to be organized.

J.W.Cahn. Thermodynamics of metastable equilibria, Proc. 2nd Conf. On Rapid SolidificationProcessing: Principles and Technology, March 1980

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