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Binary phase diagrams


Binary phase diagrams The Gibbs phase rule (I) Let s consider a simple one-component system. In the areas where only one phase is stable both pressure and ... – PowerPoint PPT presentation

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Title: Binary phase diagrams

Binary phase diagrams
Binary phase diagrams and Gibbs free energy curves
Binary solutions with unlimited
solubility Relative proportion of phases (tie
lines and the lever principle) Development of
microstructure in isomorphous alloys Binary
eutectic systems (limited solid solubility) Solid
state reactions (eutectoid, peritectoid
reactions) Binary systems with intermediate
phases/compounds The iron-carbon system (steel
and cast iron) Gibbs phase rule Temperature
dependence of solubility Three-component
(ternary) phase diagrams
Reading Chapters 1.5.1 1.5.7 of Porter and
Easterling, Chapter 10 of Gaskell
Binary phase diagram and Gibbs free energy
A binary phase diagram is a temperature -
composition map which indicates the equilibrium
phases present at a given temperature and
The equilibrium state can be found from the Gibbs
free energy dependence on temperature and
We have discussed the dependence of G of a one
component system on T
We have also discussed the dependence of the
Gibbs free energy from composition at a given T
G XAGA XBGB Hmix -T?Smix
Binary solutions with unlimited solubility
Lets construct a binary phase diagram for the
simplest case A and B components are mutually
soluble in any amounts in both solid (isomorphous
system) and liquid phases, and form ideal
We have 2 phases liquid and solid. Lets
consider Gibbs free energy curves for the two
phases at different T
T1 is above the equilibrium melting temperatures
of both pure components T1 gt Tm(A) gt Tm(B) ? the
liquid phase will be the stable phase for any
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Decreasing the temperature below T1 will have two
  1. GAliquid dan GBliquid will decrease more rapidly
    than GAsolid and GBsolid . WHY
  1. The curvature of the G(XB) curves will decrease.
    WHY ?

Eventually we will reach T2 the melting point
pure component A, where GAliquid GAsolid
For even lower temperature T3 lt T2 Tm(A) the
Gibbs free energy curves for the liquid and solid
phases will cross.
As we discussed before, the common tangent
construction can be used to show that for
compositions near cross-over of Gsolid and
Gliquid, the total Gibbs free energy can be
minimized by separation into two phases.
As temperature decreases below T3, GAliquid and
GBliquid continue to increase more rapidly than
GAsolid and GBsolid
Therefore, the intersection of the Gibbs free
energy curves, as well as points X1 and X2 are
shifting to the right, until, at T4 Tm(B) the
curves will intersect at X1 X2 1
At T4 and below this temperature the Gibbs free
energy of the solid phase is lower than the G of
the liquid phase in the whole range of
compositions the solid phase is the only stable
Based on the Gibbs free energy curves we can now
construct a phase diagram for a binary
isomorphous systems
Example of isomorphous system Cu-Ni (the
complete solubility occurs because both Cu and Ni
have the same crystal structure, FCC, similar
radii, electronegativity and valence).
Liquidus line separates liquid from liquid
solid Solidus line separates solid from liquid
In one-component system melting occurs at a
well-defined melting temperature.
In multi-component systems melting occurs over
the range of temperatures, between the solidus
and liquidus lines. Solid and liquid phases are
in equilibrium in this temperature range.
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Interpretation of Phase Diagrams
For a given temperature and composition we can
use phase diagram to determine
  1. The phases that are present
  2. Compositions of the phases
  3. The relative fractions of the phases

Finding the composition in a two phase region
1. Locate composition and temperature in
diagram 2. In two phase region draw the tie line
or isotherm 3. Note intersection with phase
boundaries. Read compositions at the
intersections. The liquid and solid phases have
these compositions.
Interpretation of Phase Diagrams the Lever Rule
Finding the amounts of phases in a two phase
1. Locate composition and temperature in
diagram 2. In two phase region draw the tie line
or isotherm 3. Fraction of a phase is determined
by taking the length of the tie line to the phase
boundary for the other phase, and dividing by the
total length of tie line
The lever rule is a mechanical analogy to the
mass balance calculation. The tie line in the two
phase region is analogous to a lever balanced on
a fulcrum.
Derivation of the lever rule
  • All material must be in one phase or the other
    Wa WL 1
  • 2) Mass of a component that is present in both
    phases equal to the mass of the component in one
    phase mass of the component in the second
    phase WaCa WßCß Co
  • 3) Solution of these equations gives us the
    Lever rule.

Wß (C0 - Ca) / (Cß- Ca) and Wa (Cß?- C0) /
(Cß - Ca)
Composition/Concentration weight fraction vs.
molar fraction
Composition can be expressed in
Molar fraction, XB, or atom percent (at) that is
useful when trying to understand the material at
the atomic level. Atom percent (at ) is a
number of moles (atoms) of a particular element
relative to the total number of moles (atoms) in
alloy. For two -component system, concentration
of element B in at. is
Where nmA and nmB are numbers of moles of
elements A and B in the system.
Weight percent (C, wt ) that is useful when
making the solution. Weight percent is the weight
of a particular component relative to the total
alloy weight. For two component system,
concentration of element B in wt. is
where mA and mB are the weights of the components
in the system.
where AA and AB are atomic weights of elements A
and B.
Composition Conversions
Weight to Atomic
Atomic to Weight
Of course the lever rule can be formulated for
any specification of composition
ML (XBa - XB0)/(XBa - XBL) (Cata - Cato) /
(Cata- CatL)
Ma (XB0 - XBL)/(XBa - XBL) (Cat0 - CatL) /
(Cata- CatL)
WL (Cwta - Cwto) / (Cwta- CwtL)
WL (Cwto - CwtL) / (Cwta- CwtL)
Phase compositions and amounts. An example.
Co 35 wt. , CL 31.5 wt. , Ca 42.5 wt.
Mass fractions WL S / (RS) (Ca - Co) / (Ca-
CL) 0.68
Wa R / (RS) (Co - CL) / (Ca- CL) 0.32
Development of microstructure in isomorphous
alloys Equilibrium (very slow) cooling
Development of microstructure in isomorphous
alloys Equilibrium (very slow) cooling
  • Solidification in the solid liquid phase occurs
    gradually upon cooling from the liquidus line.
  • The composition of the solid and the liquid
    change gradually during cooling (as can be
    determined by the tie-line method.)
  • Nuclei of the solid phase form and they grow to
    consume all the liquid at the solidus line.

Development of microstructure in isomorphous
alloys Non-equilibrium cooling
Development of microstructure in isomorphous
alloys Non-equilibrium cooling
  • Compositional changes require diffusion in solid
    and liquid phases
  • Diffusion in the solid state is very slow. ? The
    new layers that solidify on top of the existing
    grains have the equilibrium composition at that
    temperature but once they are solid their
    composition does not change. ? Formation of
    layered (cored) grains and the invalidity of the
    tie-line method to determine the composition of
    the solid phase.
  • The tie-line method still works for the liquid
    phase, where diffusion is fast. Average Ni
    content of solid grains is higher. ? Application
    of the lever rule gives us a greater proportion
    of liquid phase as compared to the one for
    equilibrium cooling at the same T. ? Solidus line
    is shifted to the right (higher Ni contents),
    solidification is complete at lower T, the outer
    part of the grains are richer in the low-melting
    component (Cu).
  • Upon heating grain boundaries will melt first.
    This can lead to premature mechanical failure.

Binary solutions with a miscibility gap
Lets consider a system in which the liquid phase
is approximately ideal, but for the solid phase
we have ?Hmix gt 0 ( A and B atoms dislike each
At low temperatures, there is a region where the
solid solution is most stable as a mixture of two
phases a1 and a2 with compositions X1 and X2.
This region is called a miscibility gap.
Eutectic phase diagram
For an even larger ?Hmix the miscibility gap can
extend into the liquid phase region. In this case
we have eutectic phase diagram.
Eutectic phase diagram with different crystal
structures of pure phases
A similar eutectic phase diagram can result if
pure A and B have different crystal structures.
Eutectic systems - alloys with limited solubility
Three single phase regions (a - solid solution of
Ag in Cu matrix, ß solid solution of Cu in Ag
matrix, L - liquid)
Three two-phase regions (a L, ß L, a ß)
Solvus line separates one solid solution from a
mixture of solid solutions. Solvus line shows
limit of solubility
Eutectic systems - alloys with limited solubility
Eutectic or invariant point - Liquid and two
solid phases coexist in equilibrium at the
eutectic composition CE and the eutectic
temperature TE.
Eutectic isotherm - the horizontal solidus line
at TE.
Eutectic reaction transition between liquid and
mixture of two solid phases, a ß at eutectic
concentration CE.
The melting point of the eutectic alloy is lower
than that of the components (eutectic easy to
melt in Greek).
Eutectic systems - alloys with limited solubility
Compositions and relative amounts of phases are
determined from the same tie lines and lever
rule, as for isomorphous alloys
For points A, B, and C calculate the compositions
(wt. ) and relative amounts (mass fractions) of
phases present.
Development of microstructure in eutectic alloys
Several different types of microstructure can be
formed in slow cooling an different compositions.
Lets consider cooling of liquid lead tin
system as an example.
In the case of lead-rich alloy (0-2 wt. of tin)
solidification proceeds in the same manner as for
isomorphous alloys (e.g. Cu-Ni) that we discussed
L ? aL ? a
Development of microstructure in eutectic alloys
At compositions between the room temperature
solubility limit and the maximum solid solubility
at the eutectic temperature, ß phase nucleates as
the a solid solubility is exceeded upon crossing
the solvus line.
Development of microstructure in eutectic alloys
Solidification at the eutectic composition
No changes above the eutectic temperature TE. At
TE all the liquid transforms to a and ß phases
(eutectic reaction).
Development of microstructure in eutectic alloys
Solidification at the eutectic composition
Compositions of a and ß phases are very different
? eutectic reaction involves redistribution of Pb
and Sn atoms by atomic diffusion This
simultaneous formation of a and ß phases result
in a layered (lamellar) microstructure that is
called eutectic structure.
Formation of the eutectic structure in the
lead-tin system. In the micrograph, the dark
layers are lead reach a phase, the light layers
are the tin-reach ß phase.
Development of microstructure in eutectic alloys
Compositions other than eutectic but within the
range of the eutectic isotherm
Primary a phase is formed in the a L region,
and the eutectic structure that includes layers
of a and ß phases (called eutectic a and eutectic
ß phases) is formed upon crossing the
eutectic isotherm.
Development of microstructure in eutectic alloys
Microconstituent element of the microstructure
having a distinctive structure. In the case
described in the previous page, microstructure
consists of two microconstituents, primary
a phase and the eutectic structure.
Although the eutectic structure consists of two
phases, it is a microconstituent with distinct
lamellar structure and fixed ratioof the two
How to calculate relative amounts of
Eutectic microconstituent forms from liquid
having eutectic composition (61.9 wt Sn)
We can treat the eutectic as a separate phase and
apply the lever rule to find the relative
fractions of primary a phase (18.3 wt Sn) and
the eutectic structure (61.9 wt Sn)
We P / (PQ) (eutectic)
Wa Q / (PQ) (primary)
How to calculate the total amount of a phase
(both eutectic and primary)?
Fraction of a phase determined by application of
the lever rule across the entire a ß phase
Wa (QR) / (PQR) (a phase)
Wß P / (PQR) (ß phase)
Binary solutions with ?Hmix lt 0 - ordering
If ?Hmix lt 0 bonding becomes stronger upon mixing
? melting point of the mixture will be higher
than the ones of the pure components.
For the solid phase strong interaction
between unlike atoms can lead to (partial)
ordering ? ?Hmix canbecome larger than ?XAXB
and the Gibbs free energy curve for the solid
phase can become steeper than the one for liquid.
At low temperatures, strong attraction between
unlike atoms can lead to the formation ofordered
phase a.
Binary solutions with ?Hmix lt 0 - intermediate
If attraction between unlike atoms is very
strong, the ordered phase may extend up to the
In simple eutectic systems, discussed above,
there are only two solid phases (a and ß) that
exist near the ends of phase diagrams.
Phases that are separated from the composition
extremes (0 and 100) are called intermediate
phases. They can have crystal structure different
from structures of components A and B.
?Hmixlt0 - tendency to form high-melting point
intermediate phase
Phase diagrams with intermediate phases example
Example of intermediate solid solution phases in
Cu-Zn, a and ? are terminal solid solutions, ß,
ß, ?, d, e are intermediate solid solutions.
lPhase diagrams for systems containing compounds
For some sytems, instead of an intermediate
phase, an intermetallic compound of specific
composition forms. Compound is represented on the
phase diagram as a vertical line, since the
composition is a specific value.
When using the lever rule, compound is treated
like any other phase, except they appear not as a
wide region but as a vertical line
This diagram can be thought of as two joined
eutectic diagrams, for Mg-Mg2Pb and Mg2Pb-Pb. In
this case compound Mg2Pb(19wt Mg and 81wt Pb)
can be considered as a component.
A sharp drop in the Gibbs free energy at the
compound composition should be added to Gibbs
free energy curves for the existing phases in the
Eutectoid Reactions
The eutectoid (eutectic-like in Greek) reaction
is similar to the eutectic reaction but occurs
from one solid phase to two new solid phases.
Eutectoid structures are similar to eutectic
structures but are much finer in scale (diffusion
is much slower in the solid state).
Upon cooling, a solid phase transforms into two
other solid phases (d ? ? e in the example
Looks as V on top of a horizontal tie line
(eutectoid isotherm) in the phase diagram.
Eutectic and Eutectoid Reactions
The above phase diagram contains both an eutectic
reaction and its solid-state analog, an eutectoid
Peritectic Reactions
A peritectic reaction - solid phase and liquid
phase will together form a second solid phase at
a particular temperature and composition upon
cooling - e.g. L a ? ß
These reactions are rather slow as the product
phase will form at the boundary between the two
reacting phases thus separating them, and slowing
down any further reaction.
Peritectoid is a three-phase reaction similar to
peritectic but occurs from two solid phases to
one new solid phase (a ß ?).
Example The IronIron Carbide (FeFe3C) Phase
In their simplest form, steels are alloys of Iron
(Fe) and Carbon (C). The Fe-C phase diagram is a
fairly complex one, but we will only consider the
steel part of the diagram, up to around 7 Carbon.
Phases in FeFe3C Phase Diagram
  • a-ferrite - solid solution of C in BCC Fe
  • Stable form of iron at room
  • The maximum solubility of C is 0.022
  • Transforms to FCC ?-austenite at 912
  • ?-austenite - solid solution of C in FCC Fe
  • The maximum solubility of C is 2.14
    wt .
  • Transforms to BCC d-ferrite at 1395
  • Is not stable below the eutectic
    temperature (727oC unless cooled rapidly
  • d-ferrite solid solution of C in BCC Fe
  • The same structure as a-ferrite
  • Stable only at high T, above 1394
  • Melts at 1538 C

  • Fe3C (iron carbide or cementite)
  • This intermetallic compound is
    metastable, it remains
  • as a compound indefinitely at room
    T, but decomposes
  • (very slowly, within several years)
    into a-Fe and C
  • (graphite) at 650 - 700 C
  • Fe-C liquid solution

A few comments on FeFe3C system
C is an interstitial impurity in Fe. It forms a
solid solution with a, ?, d phases of iron
Maximum solubility in BCC a-ferrite is limited
(max. 0.022 wt at 727 C) - BCC has relatively
small interstitial positions
Maximum solubility in FCC austenite is 2.14 wt
at 1147 C - FCC has larger interstitial positions
Mechanical properties Cementite is very hard and
brittle - can strengthen steels. Mechanical
properties also depend on the microstructure,
that is, how ferrite and cementite are mixed.
Magnetic properties a -ferrite is magnetic below
768 C, austenite is non-magnetic
Classification. Three types of ferrous alloys
Iron less than 0.008 wt C in a-ferrite at
room T steels 0.008 - 2.14 wt C (usually lt
1 wt ) a-ferrite Fe3C at room T Cast
iron 2.14 - 6.7 wt (usually lt 4.5 wt )
Eutectic and eutectoid reactions in FeFe3C
Eutectic 4.30 wt C, 1147 C
Eutectic and eutectoid reactions are very
important in heat treatment of steels
Development of Microstructure in Iron - Carbon
Microstructure depends on composition (carbon
content) and heat treatment. In the discussion
below we consider slow cooling in which
equilibrium is maintained.
Microstructure of eutectoid steel (II)
When alloy of eutectoid composition (0.76 wt C)
is cooled slowly it forms pearlite, a lamellar or
layered structure of two phases a-ferrite and
cementite (Fe3C)
The layers of alternating phases in pearlite are
formed for the same reason as layered structure
of eutectic structures redistribution C atoms
between ferrite (0.022 wt) and cementite (6.7
wt) by atomic diffusion.
Mechanically, pearlite has properties
intermediate to soft, ductile ferrite and hard,
brittle cementite.
In the micrograph, the dark areas areFe3C layers,
the light phase is a-ferrite
Microstructure of hypoeutectoid steel (I)
Compositions to the left of eutectoid (0.022 -
0.76 wt C) hypoeutectoid (less than eutectoid
-Greek) alloys.
Microstructure of hypoeutectoid steel (II)
Hypoeutectoid alloys contain proeutectoid ferrite
(formed above the eutectoid temperature) plus the
eutectoid perlite that contain eutectoid ferrite
and cementite.
Microstructure of hypereutectoid steel (I)
Compositions to the right of eutectoid (0.76 -
2.14 wt C) hypereutectoid (more than eutectoid
-Greek) alloys.
Microstructure of hypereutectoid steel
Microstructure of hypereutectoid steel (II)
Hypereutectoid alloys contain proeutectoid
cementite (formed above the eutectoid
temperature) plus pearlite that contain eutectoid
ferrite and cementite.
How to calculate the relative amounts of
proeutectoid phase (a or Fe3C) and pearlite?
Application of the lever rule with tie line that
extends from the eutectoid composition (0.75 wt
C) to a (a Fe3C) boundary (0.022 wt C) for
hypoeutectoid alloys and to (a Fe3C) Fe3C
boundary (6.7 wt C) for hypereutectoid alloys.
Fraction of a phase is determined by application
of the lever rule across the entire (a Fe3C)
phase field
Example for hypereutectoid alloy with composition
Fraction of pearlite
WP X / (VX) (6.7 C1) / (6.7 0.76)
Fraction of proeutectoid cementite
WFe3C V / (VX) (C1 0.76) / (6.7 0.76)
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The Gibbs phase rule (I)
Lets consider a simple one-component system.
In the areas where only one phase is stable both
pressure and temperature can be
independently varied without upsetting
the equilibrium ? there are 2 degrees of freedom.
Along the lines where two phases coexist in
equilibrium, only one variable can be
independently varied without upsetting
the two-phase equilibrium (P and T are related by
the Clapeyron equation) ? there is only one
degree of freedom.
At the triple point, where solid liquid and vapor
coexist any change in P or T would upset the
three-phase equilibrium ? there are no degrees of
In general, the number of degrees of freedom, F,
in a system that contains C components and can
have Ph phases is given by the Gibbs phase rule
F C - Ph 2
Lets now consider a multi-component system
containing C components and having Ph phases
A thermodynamic state of each phase can be
described by pressure P, temperature T, and C - 1
composition variables. The state of the system
can be then described by Ph(C-12) variables
But how many of them are independent? The
condition for Ph phases to be at equilibrium are
T? T? T? .. Ph -1 equations
P? P? P? .. Ph -1 equations
??A ??A ??A .. Ph -1 equations
C sets of equation
??B ??B ??B .. Ph -1 equations
Therefore we have (Ph 1)(C 2) equations that
connect the variables in the system.
The number of degrees of freedom is the
difference between the total number of variables
in the system and the minimum number of equations
among these variables that have to be satisfied
in order to maintain the equilibrium.
F Ph(C1) (Ph-1)(C2) C Ph2
F C Ph2
Gibbs Phase Rule
The Gibbs phase rule example (an eutectic
F C Ph 2
P constant
F C Ph 1
C 2
F 3 Ph
  • In one-phase regions of the phase diagram T and
    XB can be changed independently.
  • In two-phase regions, F 1. If the temperature
    is chosen independently, the compositions of both
    phases are fixed.
  • Three phases (L, a, ß) may be in equilibrium only
    at a few points along the eutectic isotherm (F

Temperature dependence of solubility
Lets consider a regular binary solution with
limited solubilities of A in B and B in A.
The closer is the minimum of the Gibbs free
energy curve Ga(XB) to the axes XB 0, the
smaller is the maximum possible concentration of
B in phase a. Therefore, to discuss the
temperature dependence of solubility lets find
the minimum of Ga(XB).
minimum of G(XB)
Greg (1-XB)GA XBGB ?(1-XB)XB RT(1-XB)
ln(1-XB) XBlnXB
Solid solubility of B in a increases
exponentially with temperature
Multicomponent systems (I)
The approach used for analysis of binary systems
can be extended to multi component systems.
Representation of the composition in a ternary
system (the Gibbs triangle). The total length of
the red lines is 100
The Gibbs free energy surfaces (instead of curves
for a binary system) can be plotted for all the
possible phases and for different temperatures.
The chemical potentials of A, B, and C of any
phase in thissystem are given by the points where
the tangential plane to the free energy surfaces
intersects the A, B, and C axis.
A three-phase equilibrium in the ternary system
for a given temperature can be derived by means
of the tangential planeconstruction.
For two phases to be in equilibrium, the chemical
potentials should be equal, that is the
compositions of the two phases in equilibrium
must be given by points connected by a common
tangential plane (e.g. l and m).
The relative amounts of phases are given by the
lever rule (e.g.using tie-line l-m).
A three phase triangle can result from a common
tangential plane simultaneously touching the
Gibbs free energies of three phases e.g. points
x, y, and z).
Eutectic point four-phase equilibrium between a,
ß, ?, and liquid
An example of ternary system
The ternary diagram of Ni-Cr-Fe. It includes
Stainless Steel (wt. of Cr gt 11.5 , wt. of Fe
gt 50 ) and Inconeltm (Nickel based super alloys).
Inconel have very good corrosion resistance, but
are more expensive and therefore used in
corrosive environments where Stainless Steels are
not sufficient (Piping onNuclear Reactors or
Steam Generators).
Another example of ternary phase diagramoil
water surfactant system
Surfactants are surface-active molecules that can
form interfaces between immiscible fluids (such
as oil and water).
A large number of structurally different phases
can be formed, such as droplet, rod-like, and
bicontinuous microemulsions, along with
hexagonal, lamellar, and cubic liquid crystalline
Make sure you understand language and concepts
  • ?? Common tangent construction
  • ?? Separation into 2 phases
  • ?? Eutectic structure
  • ?? Composition of phases
  • ?? Weight and atom percent
  • ?? Miscibility gap
  • ?? Solubility dependence on T
  • ?? Intermediate solid solution
  • ?? Compound
  • ?? Isomorphous
  • ?? Tie line, Lever rule
  • ?? Liquidus Solidus lines
  • ?? Microconstituent
  • ?? Primary phase
  • ?? Solvus line, Solubility limit
  • ?? Austenite, Cementite, Ferrite
  • ?? Pearlite
  • ?? Hypereutectoid alloy
  • ?? Hypoeutectoid alloy
  • ?? Ternery alloys
  • ?? Gibbs phase rule

Elements of phase diagrams