Mech 473 Lectures
Professor Rodney Herring
Phase Diagrams
A phase is a state of matter with the following characteristics:
•
It has the same structure or atomic arrangement throughout
•
It has roughly the same composition and properties
throughout.
•
There exists a definite interface between it and its
surroundings or adjoining phases.
Phase Diagrams
A phase diagram is a graphical representation of the phases
that are present in a material at various temperatures and
pressures and compositions.
• It usually describes the equilibrium conditions
• Sometimes non-equilibrium conditions are also shown
when well known.
Phase Diagram
• It indicates the melting/solidification temperatures of the
constituents
• It indicates the compositions of alloys where solidification
begins and the temperature range over which it occurs.
For a pure substance, the Pressure-Temperature phase diagram
simply tells which forms (solid, liquid, gas) of the material exist
under different P-T conditions.
Phase diagram for water.
Phase
diagram for
magnesium,
showing the
melting and
boiling
temperatures
at one
atmosphere
pressure.
General Types of Phase Diagrams
There are two general types of alloys having phase diagrams.
• Substitutional alloys
• Interstitial alloys
Subtitutional alloys have elements, which are incorporated into
regular lattice positions within the unit cell.
An example is Tin and Zinc alloying additions to Copper to
form bronze and brass, respectively
Interstitial alloys have elements, which are incorporated into the
interstitial sites of the unit cell.
An example is carbon in iron to form steel.
Gibb’s Phase Rule
Gibb’s phase rule describes the thermodynamic state of a material.
This famous rule is used to determine the number of phases that can
coexist in equilibrium in a given system.
It has the general form: F = C – P + 2
C is the number of components, usually elements or compounds, in
the system.
F is the number of degrees of freedom, or number of variables, such
as temperature, pressure, or composition that are allowed to
change independently without changing the number of phases in
equilibrium.
P is the number of phases present
The constant “2” in the equation implies that both temperature and
pressure are allowed to change.
Gibb’s Phase Rule
For the triple point of water:
• One component, i.e., water.
• 3 phases present, i.e. vapor, liquid, and solid.
• F = 1 – 3 + 2 = 0, so this is an invariant point on the diagram
Most binary phase diagrams used in materials science are
temperature and composition diagrams at a constant 1
atmosphere of pressure.
The constant pressure will reduce the degrees of freedom from
“2” in Gibb’s equation to “1” for a binary phase diagram
Thus, F = C – P + 1.
Liquidus/Solidus Temperatures
The liquidus temperature is the temperature above which a material is
completely liquid.
The solidus temperature is the temperature which the alloy is 100% solid.
The freezing range of the alloy is the temperature difference between the
liquidus and solidus where the two phases exists, ie., the liquid and solid.
The cooling curve for an
isomorphous alloy during
solidification. The changes
in slope of the cooling
curve indicate the liquidus
and solidus temperatures.
This is the Mech 285 Solidification Lab.
Tie Line
A binary phase diagram between
two elements A and B. When an
alloy is present in a two phase
region, a tie line at the
temperature of interest fixes the
composition of the two phases.
This is a consequence of the
Gibbs phase rule, which provides
for only one degree of freedom.
Lever Rule
The Lever Rule is used to calculate the weight % of the phase in any two-phase
region of the Phase diagram (and only the two phase region!)
In general:
• Phase percent = opposite arm of lever
x 100
total length of the tie line
For example,
weight fraction, %X s 
wo  wl
x 100
ws  wl
(of solid phase)
ws  wo
weight fraction, %X l 
x 100
ws  wl
(of liquid phase)
Lever Rule
When a material solidifies it
does not have a constant
concentration throughout the
material but there will be
concentration gradients,
which will significantly alter
the properties of the material.
This is an important concept.
In the example of Cu and Ni, the
concentration of Ni that freezes
at 1270 oC is 50 wt%, at 1250 oC
is 45 wt% and 1200 oC is 40
wt%.
Lever Rule
Calculate the amount of a phase and L phase present in a Cu 40% Ni alloy at 1250 C
In general:
• Percent a phase = (% Ni in alloy) – (% Ni in L)
% Ni in L - % Ni in a
weight fraction, %X s 
40  32
x 100  62 %
45  32
(of solid a phase)
weight fraction, % L  38 %
(of liquid phase)
x 100
Solidification of a Solid-Solution Alloy
The change in structure
and composition of a Cu40% Ni alloy during
equilibrium solidification
showing that the liquid
contains 40% Ni and the
first solid contains Cu-52%
Ni. At 1250 C,
solidification has advanced
and the phase diagram tells
us that the liquid contains
32% Ni and the solid
contains 45% Ni, which
continues until just below
the solidus, all of the solid
contains 40% Ni, which is
achieved through diffusion.
Nonequilibrium Solidification and Segregation
When cooling is too
fast for atoms to
diffuse and produce
equilibrium
conditions,
nonequilibrium
concentrations are
produced. The first
solid formed contains
52% Ni and the last
solid only 25% Ni
with the last liquid
containing only 17%
Ni. The average
composition of Ni is
40% but it is not
uniform.
Microsegregation and Homogenization
The nonuniform composition produced by nonequilibrium
solidification is known as segregation.
Microsegregation, also known as interdendritic segregation and
coring, occurs over short distances on the micron length scale.
Microsegregation can cause hot shortness which is the melting of
the material below the melting point of the equilibrium solidus.
Homogenization, which involves heating the material just below
the non-equilibrium solidus and holding it there for a few
hours, reduces the microsegregation by enabling diffusion to
bring the composition back to equilibrium.
Microsegregation and Homogenization
Macrosegregation can also exist where there exist a large
composition difference between the surface and the center of a
casting, which cannot be affected by diffusion as the distance is
too large.
Hot working breaks down the cast macrostructure enabling
the composition to be evened out.
Phase Diagrams with Intermediate
Phases and Compounds
Many combinations of two elements produce more complicated
phase diagrams than the isomorphous systems and the simple
eutectic systems.
• Many equilibrium diagrams often show intermediate phases and
compounds when either incomplete solubility or compound
formation occurs.
• These new phases are distinguished by the labels “terminal
phases” and “intermediate phases”.
• Their phase diagrams look complex.
Phase Diagrams with Intermediate
Phases and Compounds
• The terminal solid-solution phases occur at the ends of the phase
diagrams, bordering on the pure components, e.g., the alpha
phase and the beta phase in the Pb-Sn phase diagram.
• Intermediate phases commonly have new compounds and are
called intermediate compounds or intermetallic compounds.
– An intermediate compound is made up of two or more elements that
produce a new phase with its own composition, crystal structure, and
properties.
– Intermediate compounds are almost always very hard and brittle.
– An example is Fe3C in steels.
Intermetallic Compounds
Non-stoichiometric intermetallic compounds
• These have a range of compositions and are often called
intermediate solid solutions.
• An example is the iron-chrome alloy.
• The 50/50 composition orders at temperatures below ~900 oC to
form the s –phase, which causes embrittlement in stainless steels.
Intermetallic Compounds
Stoichiometric intermetallic compounds
• These have a fixed composition such as iron carbide (Fe3C).
• On the phase diagram they are represented by a straight line.
The aluminum-antimony
(AlSb) phase diagram
includes a stoichiometric
intermetallic compound, g.
Phase Diagrams Containing Three-Phase Reactions
In more complex phase diagrams, the type of melting is sometimes used to
describe the type of intermediate compound that occurs along with a
particular type of solid state reaction.
• Congruently melting compounds are those that maintain their specific
composition right up to the melting point.
This appears as a localized “dome” in the liquidus region of the phase
diagram.
• Incongruent melting compounds do not occur directly from the liquidus,
but are formed by some form of solid-state reaction.
The five most important three-phase reactions that occur in phase diagrams
are:
• Eutectic – a liquid transforms into two solids upon cooling
• Eutectoid – a solid transforms into two new solids
• Peritectic – a liquid plus a solid transforms into a new solid
• Peritectoid – two solids transforms into a new solid
• Monotectic – a liquid transforms into a new liquid and a solid.
Three phase reaction type, reaction equation and
appearance on a phase diagram.
Rules of Three Phase Reactions
• Locate a horizontal line on the phase diagram. The horizontal line, which
indicates the presence of a three-phase reaction, represents the temperature at
which the reaction occurs under equilibrium conditions.
Rules of Three Phase Reactions
• Locate three distinct points on the horizontal line: the two end points plus a
third point, often near the center of the horizontal line. The center point
represents the composition at which the three-phase reaction occurs.
Rules of Three Phase Reactions
• Write in the reaction from the phase(s) above the center point transforming to
the phase(s) below the point. In most cases the reaction will be a eutectic,
eutectoid, peritectic, etc.
11-1
Ternary Phase Diagrams
• Many alloy systems are based on three or more elements.
• To describe the changes in structure with temperature, a threedimensional phase diagram is required, which is somewhat
complicated.
• In the plot below the shaded area is the temperature at which
freezing begins.
Hypothetical ternary
phase diagrams where
binary phase diagrams
are present at the three
faces.
A
Ternary Phase Diagrams
• To reduce the complexity of the plot, we can hold the
temperature constant to make a isothermal plot or hold the
phase transformation constant such as the liquidus and show its
temperature dependence.
Hypothetical ternary
phase diagrams where
binary phase diagrams
are present at the three
faces.
A
Ternary Phase Diagrams
• In a liquidus plot, the freezing/melting temperatures for each
composition are plotted onto a triangular diagram.
• This presentation is helpful in predicting the melting/freezing
temperature of the alloys.
• And, the plot includes the primary phases that form during
solidification for any given composition.
A liquidus plot for the
hypothetical ternary
phase diagram.
Ternary Phase Diagrams
• The isothermal plot of a ternary phase diagram is usually
constructed using an equilateral triangle with the composition of
each pure element (or compound) at each corner of the triangle
to predict the phases at a particular temperature.
An isothermal plot at
room temperature for
the hypothetical ternary
phase diagram.
Summary of Vocabulary
Nomenclature:
• Phase
• Component
• Phase diagram
• Liquidus
• Solidus
• Solvus
Basic Information on a Phase Diagram
• Equilibrium phase(s) as a function of temperature and composition
• Equilibrium composition of each phase
• Relative amount of each phase in a two phase region
• Reactions during heating and cooling
Summary of Vocabulary
Important Rules
• Phase rule: F + P = C + 2
• Lever rule: XA = (Wo – WB)/(WA – WB)
The End
(Any questions or comments?)
Isomorphous Phase Diagrams
A phase diagram shows the phases and their compositions at any combination
of temperature and alloy composition
When only two elements or two compounds are present in a material a “binary
phase diagram” can be constructed.
In isomorphous binary phase diagrams only one solid phase forms as the two
components in the system display complete solid solubility.
Examples include the Cu-Ni and NiO-MgO systems.
Note that the
concentrations
can be expressed
in wt% or mole
%.
Rapidly Solidified Powders
Many complex metal alloys are made by rapidly solidifying a
spray of fine droplets of material, usually consisting of
complex compositions, in a quenching gas such as argon,
nitrogen or water.
Examples are nickel- and cobalt-based super alloys and some
stainless steels.
This process minimizes microsegregation, macrosegregation and
porosity since the process happens so rapidly that there is no
time for segregation or diffusion.
The fine particles are then processed into shapes using sintering,
hot pressing and hot isostatic pressing (HIP).
Intermetallic Compounds
Properties and Applications of Intermetallics
• Intermetallics such as Ti3Al and Ni3Al have an ordered crystal structure where
the Ti and Al atoms occupy specific locations in the crystal rather than random
locations as in most solid solutions.
• In TiAl the Ti atoms are located at the corner and the top and bottom faces of
the unit cell whereas Al atoms are only at the other four faces of the unit cell.
• This ordered structure makes it difficult for dislocations to move, which results
in poor ductility at low temperatures, which increases at high temperatures.
• TiAl also has a high activation energy for diffusion, giving good creep
resistance at elevated temperatures.
The unit cell of two
intermetallic
compounds: a) TiAl has
an ordered tetragonal
structure and b) Ni3Al
has an ordered cubic
structure.
Eutectic Phase Diagrams
• Many alloy systems are based on only two elements.
• A good example is the lead-tin system, which is used for soldering but because
of the toxicity of Pb, it is now being replaced with other Sn alloys.
Solid Solution Alloys
• A single phase solid solution forms during solidification.
• Examples include Pb-2 wt% Sn.
• These alloys strengthen by solid-solution strengthening, by strain hardening
and by controlling the solidification process to refine the grain structure.
Solidification and
microstructure of
Pb-Sn alloy
showing singlephase solid
solution at 2 wt%.
Eutectic Phase Diagrams
Alloys that exceed the solubility limit
• Pb-Sn alloys between 2% - 29% Sn also solidify to produce a single solid
solution, however, as the solid-state reaction continues, a second solid phase,
b, precipitates from the a phase.
• The solubility of Sn in solid Pb at any temperature is given by the solvus
curve. We’ll see later how this is important for age hardening materials.
• Any alloy containing between 2% and 19% Sn that cools past the solvus
exceeds the solubility resulting in the precipitation of the b phase.
Solidificaton, precipitation
and microstructure of Pb10%Sn alloy. Some
dispersion strengthening
occurs as the b solid
precipitates.
Eutectic Phase Diagrams
Alloys that exceed the solubility limit
• The Pb – 61.9% Sn alloy has the eutectic composition.
• The eutectic composition has the lowest melting temperature.
• The eutectic composition has no freezing range as solidification occurs at
one temperature (183 C in the Pb-Sn alloy).
• The Pb-Sn eutectic reaction forms two solid solutions and is given by:
L61.9% Sn a19% Sn + b97.5% Sn
• The compositions are given by the ends of the eutectic line.
Solidificaton and microstructure
of eutectic alloy of Pb-61.9%Sn.
Often eutectic alloys have a
special microstructure as shown.
Eutectic Phase Diagrams
Cooling curve for a eutectic
alloy is a simple thermal
arrest, since eutectics freeze or
melt at a single temperature.
a) Atom redistribution during
lamellar growth of a Pb-Sn eutectic.
Sn atoms from the liquid
preferentially diffuse to the b plates,
and Pb atoms diffuse to the a plates.
b) photograph of the Pb-Sn eutectic.
Review example 11-4 in Askeland and Phule, which shows you how
to calculate the amount and composition of eutectic phases.
Eutectic Phase Diagrams
Hypoeutectic alloy
• This is an alloy whose composition will be between the left-hand-side of the
end of the tie line and the eutectic composition.
• For the Pb-Sn alloy, it is between 19% and 61.9% Sn.
• In the hypoeutectic alloy, the liquid solidifies at the liquidus temperature
producing solid, a and is completed by going through the eutectic reaction.
Solidificaton and microstructure of
a hypoeutectic alloy of Pb-30%Sn.
Eutectic Phase Diagrams
Hypereutectic alloy
• This is an alloy whose composition will be between the right-hand-side of the
end of the tie line and the eutectic composition.
• For the Pb-Sn alloy, it is between 61.9% and 97.5% Sn.
• The primary or proeutectic solid that forms the bphase before the eutectic
phase is different from the eutectic solid and leads to a variation in
microstructure.
a) A hypereutectic alloy
of Pb-Sn and b) a
hypoeutectic alloy of PbSn where the dark
constituent is the Pb-rich
a phase and the light
constituent is the Sn-rich
b phase and the fine
plate structure is the
eutectic..
Strength of Eutectic Alloys
Each phase in the eutectic can be solid-solution strengthened since increasing the
alloying addition to the phase increases its strength.
Some eutectics can be strengthened by cold working.
Adding grain refiners, or inoculants, during solidification can decrease grain size.
The amount and microstructure of the eutectic can also be controlled.
• Each eutectic colony can nucleate and grow independently having the
orientation of the lamellae being identical.
• The lamellae orientation changes on crossing from one colony boundary to
another.
• By refining the colony size by inoculation, the strength can be improved.
• The eutectic is strengthened by decreasing the interlamellar spacing.
Colonies in the Pb-Sn
eutectic and the effect of
growth rate, R, on the
interlamellar spacing, l, in
the eutectic, which follows
the relationship:
Growth rate (cm/s)
l  cR 1/ 2
Strength of Eutectic Alloys
Interlamellar spacing
This is the distance between the center of one alamella to the
center of the next alamella.
A small interlamellar spacing indicates that the amount of ab
interface area is large.
A small interlamellar spacing therefore increases the strength of
the eutectic.
The interlamellar spacing in a
eutectic microstructure.
Microstructure of Eutectic Alloys
Not all eutectics give a lamellar structure.
The morphology of the two phases depends on the cooling rate, presence of
impurities, and the nature of the alloy.
An example is a Al-Si alloy where the Si portion of the eutectic grows as thin
platelets (much have a high surface interface energy) growing as thin, flat
platelets, which concentrate stresses leading to reduced ductility and
toughness.
Modification causes the Si phase to grow as thin, interconnected rods between
dendrites of Al, which increases strength and elongation.
Typical eutectic
microstructures of Al-Si
where a) shows needlelike plates and b) shows
a modified structure of
rounded rods, which are
difficult to see in this
micrograph.
Non-equilibrium Solidification of Eutectic Alloys
If the alloys cools too quickly, a non-equilibrium solidus curve is produced.
Example, for a Pb-15%Sn alloy, the a phase should freeze at 230 C, which is
well above the eutectic temperature of 183 C.
As the a phase continues to grow until, just above 183 C, the remaining
non-equilibrium liquid contains 61.9 %Sn, the eutectic composition.
This liquid then transforms to the eutectic microconstituent, surrounding
the primary alpha phase. For the conditions shown in the figure below, the
amount of eutectic is:
% eutectic 
15  10
100  9.6%
61.9  10
At near equilibrium conditions,
100% a phase should form.
For non-equilibrium solidification
a microstructure of a phase and a
eutectic microconstituent form if
the solidification is too rapid.