3.012 Fundamentals of Materials Science Fall 2005
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 
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.
XB !
T
L
! "
! + "
! + L " + L


X

X

X
L
X
L
XB
G
P = constant
T = 800
Figure by MIT OCW.
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3.012 Fundamentals of Materials Science Fall 2005
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:
Z
Z'
 + L
 + 
 + L

L
L


L
(C4 wt% Sn)
 (18.3
wt% Sn)
j
k
I
m
Eutectic structure
Primary 
(18.3 wt% Sn)
 (97.8 wt% Sn)
Eutectic 
(18.3 wt% Sn)
300
200
100
0
0 20 60 80
600
500
400
300
200
100
100
0C)
0F)
C4 (40)
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.
(Pb) (Sn)
L (61.9 wt% Sn)
Temperature(
Temperature(
Figure by MIT OCW.
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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

Ag FCC

(lattice parameter:

Sn tetragonal
Cu FCC
(lattice parameter:
4.09  at 298K) 3.61  at 298K)
Pb FCC Sn tetragonal
Figure by MIT OCW.
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?)
1300
1200
1000
900
800
700
600
Ag 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Cu
Cu
Xcm
T(0K)
Liquid
Ag
1100
Eutectic phase diagram for a silver-copper system.
2800
2600
2400
2200
2000
1800
1600
MgO CaO
20 40 60 80 100
0C)
L
MgO ss + L
MgO ss
CaO ss + L
CaO ss
MgO ss + CaO ss
Wt %
Eutetic phase diagram for MgO-CaO system.
Temperature(
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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?
XB !
T
L
! "
! + "
! + L " + L
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3.012 Fundamentals of Materials Science Fall 2005
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 <-> ( + ) (upper two region is liquid)
 Eutectoid:  <-> ( + ) (upper region is solid)
o One 2-phase region splits into two 2-phase regions:
 Peritectic: ( + L) <->  (upper two-phase region is solid + liquid)
 Peritectoid: ( + ) <->  (upper two-phase region is solid + solid)
700
600
500
60 70 80 90
1200
1000
Composition (wt % Zn)
 + L
 + 

 + L
P 5980C
L
 + 

  + L
 + 5600C
E
0C)
0F)
labeled E (5600 0
Temperature(
Temperature(
A region of the copper-zinc phase diagram that has been enlarged to show eutectoid and peritectic invariant points ,
C, 74 wt% Zn) and P (598 C, 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
XB !
T
S
L
XB
G
XB
G
L
S
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3.012 Fundamentals of Materials Science Fall 2005
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.
2
L
AB2 + B
A AB2 B
Mol%
T
B + L
AB2 + L
A + AB
A + L
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:
2000
1900
1800
1600
1400
1700
1500
15200
17850
18500
17700
20500
X
Al2O3 + L
2O3
2O3
MnO + MA
MnO + L
L
MnO Al2O3
20 40 60 80
23
23
2310
WI%
0C)
2O3 2O3)
8
MA + Al
MnO+Al
MA +L
MA +L
Temperature(
System MnO + Al (MA = MnO + Al Figure by MIT OCW.
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3.012 Fundamentals of Materials Science Fall 2005
Example binary phase diagrams
* Phase diagrams in this section are  W.C. Carter, 2002: Graphics were produced by Angela Tong, MIT.
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
0 10 20 30 40 50 60 70 80 90 100
(Au,Ni)
T(Temperature),oC
Au Ni
Atomic percent nickel
Phase diagram for Gold-Nickel showing complete solid solubility above about 800
o
C and below
about 950
o
C. The miscibility gap at low temperatures can be understood with a regular solution
model. Figure by MIT OCW.
800
700
600
500
400
300
200
100
0 10 20 30 40 50 60 70 80 90 100
Alpha
Liquid
Beta
Atomic Percent Lithium
T(temperature)o
C
Al Ll
Alpha
Beta
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.
LiAl
Li2Al
Zintl crystal Structure of LiAl ( phase)4
Lecture 19 ­ Binary phase diagrams Figure by MIT OCW. 9 of 16
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3.012 Fundamentals of Materials Science Fall 2005
800
900
1000
700
600
500
400
300
200
100
0 10 20 30 40 50 60 70 80 90 100
o
C
Au Sn
and one eutectoid reactions.
AuSn
AuSn4
AuSn2
1100
Atomic Percent Tin
T(temperature)
Phase diagram of Gold-Tin has seven distinct phases, three peritectics, two eutectics,
Figure by MIT OCW.
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3.012 Fundamentals of Materials Science Fall 2005
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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3.012 Fundamentals of Materials Science Fall 2005
XB !
G
A
INSIDE SPINODAL: SYSTEM UNSTABLE
TO SMALL COMPOSITION
FLUCTUATIONS
B
OUTSIDE SPINODAL: SYSTEM STABLE
TO SMALL COMPOSITION
FLUCTUATIONS
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3.012 Fundamentals of Materials Science Fall 2005
Figure by MIT OCW.
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3.012 Fundamentals of Materials Science 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
125
232
183
139
Bi-Pb
Peritectic
Bi-Pb
Eutectic
Bi-Sn
Eutectic
Pb-Sn
Eutectic
at 960C
Mass Fraction of Bi
Mass Fraction of Pb
Bi SnMass Fraction of Sn
tc/0C
prism is a two-component temperature-composition phase diagram with
Pb
Triple Eutectic
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
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
equivalent binary phase diagram (2 components present while 3
rd
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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3.012 Fundamentals of Materials Science Fall 2005
A
B C
XB = 0XB = 1
X
B=0
X
B=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:
Figure by MIT OCW.

+
  + 

 + 
 +  + 
XG
XB
XR
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.
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References
1. McCallister, W. D. Materi als Science and Engineering: An Introducti on (John Wiley & Sons, Inc., New York, 2003).
2. Lupis, C. H. P. C hemical Thermodynamics of Materials (Prentice-Hall, New York, 1983).
3. Bergeron, C. G. & Risbud, S. H. Introducti on to Phase Equilibria i n C eramics (American Ceramic Society, Westerville, OH, 1984).
4. Lindgren, B. & Ellis, D. E. Molecular Cluster Studies of Lial with Different Vacancy Structures. Journal of Physics F-Metal Physics 13,
1471-1481 (1983).
5. Carter, W. C. (2002).
6. Mortimer, R. G. Physical Chemistry (Academic Press, New York, 2000).
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