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
of G
Assessment(least square method)
Experimental techniques•Calorimetry
• Electromotive forces•XRD, SEM, etc.
Experimental data:•Thermodynamic properties
•Phase diagrams
),,(),(),,(),,( 0i
exi
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
2 3 4
1 3 42
3 4
1 2
2
lnln
Composition dependence:
•Redlich-Kister polinomia•Stoichiometric compounds•Sub-lattice models•Associate model
νBA
0BA )( xxLxxG
nex −= ∑
=ν
ν
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.
∆G
Eatt
STABLE
UNSTABLE
METASTABLE
dG=0
dG=0
Gib
bs fr
ee e
nerg
y
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
600
800
1000
1200
1400 Liang et al. CALPHAD 22 (1998) 527-544
Te
mpe
ratu
re /
K
Mole fraction Mg
Eutectic system
GFR
Eutectic system
0.0 0.2 0.4 0.6 0.8 1.0
-5
00.0 0.2 0.4 0.6 0.8 1.0
-35
-30
-25
-20
-15Fr
ee E
nerg
y / K
J m
ol-1
Mg molar fraction
hcp-Mg
CuMg2Cu2Mg
fcc-Cu
liquid
Free
Ene
rgy
/ KJ
mol
-1
GFR
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
500
750
1000
1250
TavTid
∆T
BA
TEM
PE
RA
TUR
E /
K
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
500
750
1000
1250
1500 ΩL -20 -10 0 10
20 IDEAL
BA
TEM
UR
E /
K
B MOLAR FRACTION
-1mol KJ 03 = ΩS
TmA = 800 K, Tm
B = 1200 K, ∆SmA = ∆Sm
B = 10 J mol-1 K-1
PE
RA
T
-20 -10 0 10 20
0
100
200
300
400
500
600
700
∆T
/ K
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
0
1
2
3
XL
liq ΩL=-10 sol ΩS=+10 sol ΩS=+20T = 500 K
Free
ene
rgy
/ KJ
mol
-1
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
500
600
700
800ΩS=
ΩL= -10 KJ mol-1 10 12 15 20
A
Tem
pera
ture
/ K
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
∆Hm
∆Hx
glass
liquid
crystal
Tg Tm
Enth
alpy
Temperature
Experimental data and modelling for amorphous phase are necessary
Fe-B binarysystem
Undercooling experiments
1200 1300 1400 1500
Crystallisation of amorphous alloys
b
a
Fe83B17
1 W / g
Hea
t flo
w
Temperature / K
Primary γ-Fe
Eutectic γ-Fe+Fe3B
Fe3B ⇒ Fe2B
300 325 350 375 400 425 450 475 500
Hea
t Flo
w
Temperature / °C
EXO
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
1000
1500
2000
2500
1767 K
1452 K0.168
0.63
β
fcc
bcc
bcc
0.1811387 K
liquid
FeB
Fe2B
Fe3B
Tem
pera
ture
/ K
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,
1993
23 −++= bTaTRCp
R -> gas constant
a, b → fitting parameters
• Suitable to describe the behaviour of the Cp on undercooling
• No glass transition (Kauzmann paradox)
Modelling of the liquid-amorphous phase and glass transition
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
)(τf
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
Modelling of the liquid-amorphous phase and glass transition
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
30
40
50
60
70
Tm
Tg Exp. Busch Exp. this work no excess Cp Shao Kubaschewski this work
Hea
t cap
acity
/ J
mol
-1 K
-1
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
1000
1200
1400
1600
1800 with ∆Cp without ∆Cp
Tem
pera
ture
/ K
B molar fraction
M. Palumbo, E. Bosco, G. Cacciamani, M. Baricco, CALPHAD, 25, 4 (2001) 625
Driving force for solutions
0N C
L
C
X
CG
CG
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂=⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂ LA
XAG µ−µ=∆Parallel tangent rule
A CN C0
Liquid(amorphous)
CrystallineSolution
µAL
µAXFr
ee e
nerg
y
Composition B
Driving forces for regular solutions
0.0 0.2 0.4 0.6 0.8 1.0600
800
1000
1200
BA
TA
TL
Tem
pera
tura
(K)
xB
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
1
2
3
4
5
CuZr without ∆Cp CuZr with ∆Cp
Liquid 50% Cu
Driv
ing
forc
e / K
J m
ol-1
Temperature / K
Cu-MgCu-Zr
400 500 600 700 8000
2
4
6
hcp-Mg with ∆Cp CuMg2 with ∆Cp
hcp-Mg without ∆Cp CuMg2 fwithout ∆Cp
Driv
ing
forc
e / K
J m
ol-1
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