3.012 Fundamentals of Materials Science
Fall 2003
Lecture 19: 11.07.03 Binary phase diagrams
Today:
LAST TIME .............................................................................................................................................................................................. 2
Free energy and phase diagrams of ideal binary solutions ............................................................................................................... 2
BINARY SOLUTIONS WITH LIMITED MISCIBILITY IN THE SOLID STATE: EUTECTIC SYSTEMS .................................................................... 3
Free energy diagrams in eutectic systems ......................................................................................................................................... 3
THE PHASE RULE APPLIED TO BINARY PHASE DIAGRAMS ........................................................................................................................ 7
The phase rule and eutectic diagrams: invariant points .................................................................................................................... 7
REFERENCES ........................................................................................................................................................................................... 8
Reading:
Supplementary Reading:
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Lecture 19 – Binary phase diagrams
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3.012 Fundamentals of Materials Science
Fall 2003
Last time
Free energy and phase diagrams of ideal binary solutions

G (J/mole)
Last time, we examined the behavior of the free energy in an ideal binary solution as the temperature decreases,
shifting from a completely liquid system at all compositions to a system where the solid state is stable at all
compositions:
GS
GS
GL
GL
0
0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1
1
XBB
G (J/mole)
T > Tm,B > T m,A
GS
GS
GL
GL
0
0
0.2
0.2
0.4
0.6
0.6
Tm,B > T 1 > T m,A
XS
XL
0.8
0.8
T = Tm,B > T m,A
Tm,B > T 1 > T 2 > T m,A
1
1
XBB
Tm,B > T 2 > T 3 > T m,A
G (J/mole)
Tm,B > T = Tm,A
GS
GL
XS
XL
0
0.2
0.4
0.6
0.8
1
XB

The ideal solution is a model for material where the two components are completely miscible in the solid and
liquid states at all compositions- the only phase separation that occurs in the system is that between the solid and
liquid states.
o
However, many materials that form binary solutions are not miscible at all compositions and all
temperatures, particularly in the solid state. Materials that have incompatible crystal structures cannot
form homogeneous solid solutions- they are forced to phase separate into an A-rich solid phase and a Brich solid phase, to preserve their stable crystalline forms. Today we will examine the phase diagrams of
these and other common binary materials systems.
Lecture 19 – Binary phase diagrams
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3.012 Fundamentals of Materials Science
Fall 2003
Binary solutions with limited miscibility in the solid state: Eutectic 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  phase) and one that is
highly enriched in component B (the  phase). These binary systems, with unlimited liquid state miscibility and low
or negligible solid state miscibility, are referred to as eutectic systems.

Important examples of eutectic systems are X. show how structure dictates they must phase separate.
Component A
Bi
o
Solid-state crystal structures of several eutectic systems
Solid-state crystal
Component B
Solid-state crystal
structure A
structure B
Sn
Applications
Lead-free solders
in
microelectronics
The behavior just described, where the two components are completely miscible at high temperatures in
the liquid state and phase-separated into two solids at low temperatures would be represented by a
phase diagram as follows:
L
T
?

+ 

XB Ÿ
o

Now the question is, what happens in the region between high and low temperatures?
Let’s examine how the phase diagram of a eutectic system develops.
Free energy diagrams in eutectic systems

Suppose we have a binary solution of A and B. The molar free energy for each phase can be diagrammed for as
a function of composition, as we did for the simple ideal solution: 1
Lecture 19 – Binary phase diagrams
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3.012 Fundamentals of Materials Science
Fall 2003
(© W.C. Carter 2002, p. 201)
(© W.C. Carter 2002)
Lecture 19 – Binary phase diagrams
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Fall 2003
(© W.C. Carter 2002, p. 202)
(© W.C. Carter 2002, p. 203)
Lecture 19 – Binary phase diagrams
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Fall 2003
Bringing together the series of isothermal ‘slices’ for the system, we can construct the eutectic phase diagram:
(© W.C. Carter 2002, p. 203)
Lecture 19 – Binary phase diagrams
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The phase rule applied to binary phase diagrams
The phase rule and eutectic diagrams: invariant points

All phase diagrams must obey the phase rule. Does our eutectic diagram obey it?
(© W.C. Carter 2002)
o
What is the single degree of freedom in the 2-phase regions?1


If temperature is changed at fixed overall composition, then the change in volume fraction of
phases is determined. In other words, there is a relation between dT and dfsolid.
If the overall composition is changed with fixed phase fractions, then T is determined by the
change.
Lecture 19 – Binary phase diagrams
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Fall 2003
References
1.
2.
Carter, W. C. 3.00 Thermodynamics of Materials Lecture Notes http://pruffle.mit.edu/3.00/ (2002).
Bergeron, C. G. & Risbud, S. H. Introduction to Phase Equilibria in Ceramics (American Ceramic Society,
Westerville, OH, 1984) 158 pp.
Lecture 19 – Binary phase diagrams
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