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)
Lecture 19 – Binary phase diagrams
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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.
Lecture 19 – Binary phase diagrams
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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
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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
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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.
Lecture 19 – Binary phase diagrams
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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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Fall 2005
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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Fall 2005
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:
Lecture 19 – Binary phase diagrams
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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
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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
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I ntroducti on to Phase Equilibria i n C eramics
Journal of Physics F-Met al Phy sics
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