3.012 Fundamentals of Materials Science
Fall 2005
Lecture 19: 11.23.05 Binary phase diagrams Today: LAST TIME .........................................................................................................................................................................................2
Eutectic Binary Systems...............................................................................................................................................................2
Analyzing phase equilibria on eutectic phase diagrams ...........................................................................................................3
Example eutectic systems.............................................................................................................................................................4
INVARIANT POINTS IN BINARY SYSTEMS ..........................................................................................................................................5
The phase rule and eutectic diagrams: eutectic invariant points .............................................................................................5
Other types of invariant points....................................................................................................................................................6
Congruent phase transitions........................................................................................................................................................7
INTERMEDIATE COMPOUNDS IN PHASE DIAGRAMS3 ........................................................................................................................8
EXAMPLE BINARY PHASE DIAGRAMS ...............................................................................................................................................9
DELINEATING STABLE AND METASTABLE PHASE BOUNDARIES: SPINODALS AND MISCIBILITY GAPS ........................................ 11
Conditions for stability as a function of composition ............................................................................................................. 11
SUPPLEMENTARY INFORMATION (NOT TO BE TESTED): TERNARY SOLUTION PHASE DIAGRAMS ............................................... 14
REFERENCES................................................................................................................................................................................... 16
Reading:
W.D. Callister, Jr., “Fundamentals of Materials Science and Engineering,” Ch. 10S Phase Diagrams, pp. S67-S84.
Supplementary Reading:
Ternary phase diagrams (at end of today’s lecture notes)
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Last time Eutectic Binary Systems •
It is commonly found that many materials are highly miscible in the liquid state, but have very limited mutual miscibility in the solid state. Thus much of the phase diagram at low temperatures is dominated by a 2-phase field of two different solid structures- one that is highly enriched in component A (the α phase) and one that is highly enriched in component B (the β phase). These binary systems, with unlimited liquid state miscibility and low or negligible solid state miscibility, are referred to as eutectic systems.
P = constant T = 800
L
T
G
�
�
!+L X
�
X
L
L
X X
�
!
"+L "
!+"
XB
XB ! Figure by MIT OCW.
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Analyzing phase equilibria on eutectic phase diagrams Next term, you will learn how these thermodynamic phase equilibria intersect with the development of microstructure in materials:
L (C4 wt% Sn)
Z j
300
Temperature (0C)
�+L
200
L L
500
k
� (18.3 wt% Sn) �
�+L
I
m L (61.9 wt% Sn)
100
600
�
�
Eutectic structure
400
300
Primary � (18.3 wt% Sn)
�+�
200 � (97.8 wt% Sn) Z'
0
0 (Pb)
Eutectic � (18.3 wt% Sn)
20
60 C4 (40)
100
80
100 (Sn)
Composition (wt% Sn)
Schematic representations of the equilibrium microstructures for a lead-tin alloy of composition C4 as it is cooled from the liquid-phase region.
Figure by MIT OCW.
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Temperature (0F)
•
3.012 Fundamentals of Materials Science
Fall 2005
Example eutectic systems Solid-state crystal structures of several eutectic systems Component A Solid-state crystal Component B Solid-state crystal structure A structure B Bi rhombohedral
Sn tetragonal Ag FCC
Cu FCC (lattice parameter:
(lattice parameter: 4.09 Å at 298K) 3.61 Å at 298K) Pb FCC Sn tetragonal
2800
Liquid
2600
1200 Ag
1000
Cu
T(0K)
1100
900
Temperature (0C)
1300
2400
L
MgO ss
CaO ss
2200 MgO ss + CaO ss
2000
800
CaO ss + L
MgO ss + L
1800
700 600 Ag
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Xcm
0.8
0.9
Eutectic phase diagram for a silver-copper system. Figure by MIT OCW.
Cu
1600 MgO
20
40
60
80
Wt %
Eutetic phase diagram for MgO-CaO system.
Figure by MIT OCW.
•
Eutectic phase diagrams are also obtained when the solid state of a solution has regular solution behavior (can you show this?)
Lecture 19 – Binary phase diagrams 11/23/05
100
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CaO
3.012 Fundamentals of Materials Science
Fall 2005
Invariant points in binary systems The phase rule and eutectic diagrams: eutectic invariant points •
All phase diagrams must obey the phase rule. Does our eutectic diagram obey it?
T
L !+L !
"+L !+"
"
XB !
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Other types of invariant points •
Other transformations that occur in binary systems at a fixed composition and temperature (for constant pressure) are given titles as well: o Two 2-phase regions join into one 2-phase region:
Eutectic:
L (α + β)
(upper two region is liquid)
Eutectoid:
γ (α + β)
(upper region is solid)
o One 2-phase region splits into two 2-phase regions:
Peritectic:
(α + L) β
(upper two-phase region is solid + liquid)
Peritectoid:
(α + γ) β
(upper two-phase region is solid + solid)
β+L 700
β+�
�
β
600
L 5980C
P �+α
5600C E β+α
500
60
70
1200 Temperature (0F)
Temperature (0C)
�+L
1000 α
80
α+L
90
Composition (wt % Zn)
A region of the copper-zinc phase diagram that has been enlarged to show eutectoid and peritectic invariant points , labeled E (5600C, 74 wt% Zn) and P (5980C, 78.6 wt% Zn), respectively.
Figure by MIT OCW.
Note that each single-phase field is separated from other single-phase fields by a twophase field.
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Congruent phase transitions •
Congruent phase transition: complete transformation from one phase to another at a fixed composition
LS G
T SL
XB
XB !
G
XB
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Intermediate Compounds in phase diagrams3 •
Stable compounds often form between the two extremes of pure component A and pure component B in binary systems- these are referred to as intermediate compounds.
AB2 + L L B+L
A+L T
AB2 + B
A + AB2 AB2
A
B
Mol%
Figure by MIT OCW.
•
When the intermediate compound melts to a liquid of the same composition as the solid, it is termed a congruently melting compound. Congruently melting intermediates subdivide the binary system into smaller binary systems with all the characteristics of typical binary systems.
•
Intermediate compounds are especially common in ceramics, as the pure components may form unique molecules at intermediate ratios. Shown below is the example of the system MnO-Al2O3: 20500 2000 X
1900
18500
1800 1700 1600
23 MnO + L 15200
8
23
10
23
MnO
MnO + MA
20
17700
MA +L
1500 1400
Al2O3 + L
MA +L
17850
MnO + Al2O3
Temperature (0C)
L
40
60
MA + Al2O3
80
WI%
Al2O3
Figure by MIT OCW.
System MnO + Al2O3 (MA = MnO + Al2O3)
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Example binary phase diagrams •
Phase diagrams in this section are © W.C. Carter, 2002: Graphics were produced by Angela Tong, MIT. 1500 1400 1300
T(Temperature), oC
1200 1100 1000 900
(Au,Ni)
800 700 600 500 400 300
0
10
20
Au
30
40
50
60
70
80
90
100
Ni
Atomic percent nickel
o
Phase diagram for Gold-Nickel showing complete solid solubility above about 800 C and below o about 950 C. The miscibility gap at low temperatures can be understood with a regular solution model.
Liquid
Li2Al
LiAl
800
Figure by MIT OCW.
T (temperature) oC
700 600 Alpha Alpha 500 Beta
400 300 200 100
Zintl crystal Structure of LiAl (�
phase)4
0
Al
10
20
30
Beta 40 50 60 70 Atomic Percent Lithium
80
90
100 Ll
Phase diagram for light metals Aluminum-Lithium. There are five phases illustrated; however the BCC Lithium end-member phase shows such limited Aluminum solubility that it hardly appears on this plot. There are two eutectics and one peritectoid reactions. The LiAl(�) and Li2Al intermetallic phases are ordered compounds where the atoms order on sublattices, thus changing the symmetry of the material. The ordered phases show limited solubility where, for instance, the extra Li will occupy sites where an Al usually sits, or occupies an interstitial position.
Lecture 19 – Binary phase diagrams 11/23/05
Figure by MIT OCW.
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1100 1000 900
T (temperature) oC
800 700 AuSn
600
AuSn2
500 AuSn4
400 300 200 100
0
10
20
Au
30
40
50
60
70
80
90
Atomic Percent Tin
100 Sn
Phase diagram of Gold-Tin has seven distinct phases, three peritectics, two eutectics, and one eutectoid reactions. Figure by MIT OCW.
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Delineating stable and metastable phase boundaries: spinodals and miscibility gaps •
We saw last time that a homogeneous solution can spontaneously decompose into phase-separated mixtures when interactions between the molecules are unfavorable. The example we started with was the regular solution model for the free energy, where the enthalpy of mixing two components might be positive (i.e., in terms of total free energy, unfavorable).
Conditions for stability as a function of composition •
For closed systems at constant temperature and pressure, the Gibbs free energy is minimized with respect to fluctuations in its other extensive variables. Thus, if we allow the composition of a binary system to vary, the system will move toward the minimum in the free energy vs. XB curve:
•
If the system is at the minimum in G, then we can write:
•
We can perform a Taylor expansion for a fluctuation in Gibbs free energy, assuming the only variable that can vary is composition (XB):
o If we examine the consequence of a fluctuation in composition near the extremum point of the free energy curve, the first derivative of G is zero. If we assume the third-order (and higher) terms are negligible, then the condition for stable equilibrium is controlled by the value of the second derivative.
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G
XB ! INSIDE SPINODAL: SYSTEM UNSTABLE TO SMALL COMPOSITION FLUCTUATIONS
OUTSIDE SPINODAL: SYSTEM STABLE TO SMALL COMPOSITION FLUCTUATIONS
A
B
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Figure by MIT OCW.
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Supplementary Information (not to be tested): Ternary solution phase diagrams •
A 3-component analog to the binary phase diagram is also commonly encountered in materials science & engineering problems. For a 3 component system, a triangular 2D phase equilibrium map can be used to represent the stable phases as a function of composition in the 3-component ternary solution:
327
271
Bi-Pb Peritectic
232
tc/0C
Bi-Sn Eutectic
183 Pb-Sn Eutectic
125
Bi-Pb Eutectic Triple Eutectic at 960C c ti s F ra Mas
Bi
139
f Bi on o
Pb
Mass Fraction of Sn
Mass
F r a ct
io n o
f Pb
Sn
3-Dimensional Depiction of Temperature-Composition Phase Diagram of Bismuth, Tin, and Lead at 1atm. The diagram has been simplified by omission of the regions of solid solubility. Each face of the triangular prism is a two-component temperature-composition phase diagram with a eutectic. There is also a peritectic point in the Bi-Pb phase diagram.
Figure by MIT OCW.
•
3D depictions are necessary to show all 3 composition variables and temperature at fixed pressure (temperature is shown in the vertical axis)- in this arrangement, each face of the triangular column is the rd equivalent binary phase diagram (2 components present while 3 component is not present).
•
A single horizontal slice from the 3D ternary construction provides the phase equilibria as a function of composition for a fixed value of temperature and pressure:
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A
X
B
=0
3.012 Fundamentals of Materials Science
=1
!
X
B
"
#
B
C
XB = 0
XB = 1 •
Similar to binary phase diagrams, tie lines are used to identify the compositions and phase fractions in multi-phase regions of the ternary diagram:
XR �
�+ε
�+� �+�+ε
ε
ε+�
XG
�
XB
Figure by MIT OCW. o The phase rule allows 3-phase equilibria to lie within fields of the ternary diagram, unlike the binary systems, where 3-phase equilibria are confined to invariant points. Lecture 19 – Binary phase diagrams 11/23/05
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References 1. 2. 3. 4. 5. 6.
McCallister, W. D. (John Wiley & Sons, Inc., New York, 2003). Lupis, C. H. P. (Prentice-Hall, New York, 1983). Bergeron, C. G. & Risbud, S. H. (American Ceramic Society, Westerville, OH, 1984). Lindgren, B. & Ellis, D. E. Molecular Cluster Studies of Lial with Different Vacancy Structures. 1471-1481 (1983). Carter, W. C. (2002). Mortimer, R. G. Physical Chemistry (Academic Press, New York, 2000). Materi als Scienc e and Engineering: An I ntroducti on
C hemic al T hermodynamics of Materials
Lecture 19 – Binary phase diagrams 11/23/05
I ntroducti on to Phase Equilibria i n C eramics
Journal of Physics F-Met al Phy sics
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