Alloy phase diagrams are
useful to metallurgists, materials engineers, and materials scientists in four
major areas:
In all
these areas, the use of phase diagrams allows research, development, and
production to be done more efficiently and cost effectively.
Several commonly used terms are described below.
Phases. All materials exist in gaseous,
liquid, or solid form (usually referred to as a phase), depending on the
conditions of state. State variables include compositions, temperature,
pressure, magnetic field, electrostatic field, gravitational field, and so on.
The term "phase" refers to that region of space occupied by a
physically homogeneous material. However, there are two uses of the term: the
strict sense normally used by scientists and the somewhat looser sense normally
used by materials engineers.
In the strictest sense, homogeneous means that the
physical properties throughout the region of space occupied by the phase are
absolutely identical, and any change in condition of state, no matter how
small, will result in a different phase. For example, a sample of solid metal
with an apparently homogeneous appearance is not truly a single-phase material,
because the pressure condition varies in the sample due to its own weight in
the gravitational field.
In a phase diagram, however, each single-phase field
(phase fields are discussed in a following section) is usually given a single
label, and engineers often find it convenient to use this label to refer to all
the materials lying within the field, regardless of how much the physical
properties of the materials continuously change from one part of the field to
another. This means that in engineering practice, the distinction between the
terms "phase" and "phase field" is seldom made, and all
materials having the same name are referred to as the same phase.
Equilibrium. There are
three types of equilibria: stable, metastable, and unstable. These three
conditions are illustrated in a mechanical sense in Fig. 1.
Fig. 1 Mechanical equilibria: (a) Stable. (b)
Metastable. (c) Unstable.
Stable equilibrium exists when the object is in its
lowest energy condition; metastable equilibrium exists when additional energy
must be introduced before the object can reach true stability; unstable
equilibrium exists when no additional energy is needed before reaching
metastability or stability. Although true stable equilibrium conditions seldom
exist, the study of equilibrium system is extremely valuable, because it
constitutes a limiting condition from which actual conditions can be estimated.
Polymorphism. The structure
of solid elements and compounds under stable equilibrium conditions is
crystalline, and the crystal structure of each is unique. Some elements and
compounds, however, are polymorphic (multishaped); that is, their structure
transforms from one crystal structure to another with changes in temperature.
The term allotropy (existing in another form) is usually used to describe
polymorphic changes in chemical elements.
Metastable Phases. Under some conditions, metastable crystal structures can form instead
of stable structure. Rapid freezing is a common method of producing metastable
structures, but some (such as Fe3C, or cementite) are produced at moderately
slow cooling rates. With extremely rapid freezing, even thermodynamically
unstable structures (such as amorphous metal glasses) can be produced.
Systems. A physical system consists
of a substance (or group of substances) that is isolated from its surroundings,
this concept is used to facilitate study of the effects of conditions of state.
"Isolated" means that there is no interchange of mass between the
substance and its surroundings. The substances in alloy systems, for example,
might be two metals, such as copper and zinc; a metal and a nonmetal, such as
iron and carbon, a metal and an intermetallic compound, such as iron and
cementite; or several metals such as aluminum, magnesium, and manganese. These
substances constitute the components bordering the system and should not be
confused with the various phases found with in the system. A system, however,
can also consist of a single component such as an element or compound.
Phase Diagram. In order to
record and visualize the result of studying the effect of stage variable on a
system, diagrams were introduced to show the relationships between the various
phases that appear within the system under equilibrium conditions, As such, the
diagrams are variously called constitutional diagrams, equilibrium
diagrams, or phase diagrams. A single-component phase diagram can be
simply a one- or two-dimensional plot showing the phase change in the substance
as temperature and/or pressure change. Most diagrams, however, are two-or
three-dimensional plots describing the phase relationships in systems made up
of two or more components, and these usually contain fields (areas) consisting
of mixed-phase fields, as well as single-phase fields.
Phase Rule. The phase rule, first announced by J. Willard Gibbs in 1876, relates
the physical state of a mixture to the number of constituents in the system and
to its conditions. It was also Gibbs who first called each homogeneous region
in a system by the term “phase.” When pressure and temperature are the state
variables, the rule can be written as:
f = c - p + 2 |
where f is the number of
independent variables (called degrees of freedom), c is the number of
components, and p is the number of stable phases in the system. |
If a system being considered is bordered
by two components, the system is called a binary system. Most metallurgical
problems are concerned only with a fixed pressure of 1 atm, and the phase
diagram is expressed by a two-dimensional plot of temperature and composition.
The Gibbs phase rule applies to
all states of matter (solid, liquid, and gaseous), but when the effect of
pressure is constant, the rule reduces to:
f = c - p +
1
The stable equilibria for binary
systems are summarized as:
Number of components |
Number of phases |
Degree of freedom |
Equilibrium |
2 |
3 |
0 |
Invariant |
2 |
2 |
1 |
Univariant |
2 |
1 |
2 |
Bivariant |
Miscible Solids. Many systems are bordered by components having the same crystal
structure, and the components of some of these systems are completely miscible
(completely soluble in each other) in the solid form, thus forming a continuous
solid solution. When this occurs in a binary system, the phase diagram usually
has the general appearance of that shown in Fig. 2.
Fig. 2 Schematic binary phase
diagram showing miscibility in both the liquid and solid states.
The diagram consists of two single-phase fields
separated by a two-phase field. The boundary between the liquid field and the
two-phase field in Fig. 2 is called the liquidus; that between the two-phase
field and the solid field is the solidus. In general, a liquidus is the locus
of points in a phase diagram representing the temperatures at which alloys of
the various compositions of the system begin to freeze on cooling or finish
melting on heating; a solidus is the locus of points representing the
temperatures at which the various alloys finish freezing on cooling or begin
melting on heating. The phases in equilibrium across the two-phase field in
Fig. 2 (the liquid and solid solutions) are called conjugate phases. If the
solidus and liquidus meet tangentially at some point, a maximum or minimum is
produced in the two-phase field, splitting it into two portions, as shown in
Fig. 3.
Fig. 3
Schematic binary phase diagrams with solid state miscibility where the liquidus
shows a maximum (a) and a minimum (b).
It also is possible to have a gap in miscibility in a
single-phase field; this is shown in Fig. 4. Point Tc, above which
phase α1 and α2 become indistinguishable, is a critical point. Lines a-Tc
and b-Tc, called solvus lines, indicate the limits of solubility of
component B in A and A in B, respectively. The configuration of these and all
other phase diagrams depends on the thermodynamics of the system, as discussed
later.
Fig. 4 Schematic binary phase
diagrams with a gap in miscibility in a single-phase field.
Eutectic Reactions. If the two-phase field in the solid region of Fig. 4 is expanded so
that it touches the solidus at some point, as shown in Fig. 5(a), complete
miscibility of the components is lost. Instead of a single solid phase, the
diagram now shows two separate solid terminal phases, which are in three-phase
equilibrium with the liquid at point P, an invariant point that occurred by
coincidence (Three-phase equilibrium is discussed in the next section.)
Fig. 5 Schematic binary phase
diagrams with invariant points. (a) Hypothetical diagram of the type shown in
Fig. 4 except that the miscibility gap in the solid touches the solidus curve
at invariant point P;an actual diagram of this type probably does not exist.
(b) and (c) Typical eutectic diagrams for components having the same crystal
structure (b) and components having different crystal structures (c). The eutectic
(invariant) points are labeled E. The dashed lines in (b) and (c) are
metastable extensions of the stable equilibria lines.
Then, if this two-phase field in the solid region is
even further widened so that the solvus lines no longer touch at the invariant
point, the diagram passes through a series of configurations, finally taking on
the more familiar shape shown in Fig. 5(b). The three-phase reaction that takes
place at the invariant point E, where the liquid phase freezes into a mixture
of two solid phases, is called a eutectic reaction (from the Greek word for
"easily melted"). The alloy that corresponds to the eutectic
composition is called a eutectic alloy. An alloy having a composition to the
left of the eutectic point is called a hypoeutectic alloy (from the Greek word
for "less than"); an alloy to the right is a hypereutectic alloy
(meaning "greater than"). In the eutectic system described
previously, the two components of the system have the same crystal structure.
This, and other factors, allows complete miscibility between them. Eutectic
systems, however, also can be formed by two components having different crystal
structures. When this occurs, the liquidus and solidus curves (and their
extensions into the two-phase field) for each of the terminal phases (see Fig.
5c) resemble those for the situation of complete miscibility between system
components shown in Fig. 2.
Three-Phase Equilibrium. Reactions involving three conjugate phases are not limited to the
eutectic reaction. For example, upon cooling, a single solid phase can change
into a mixture of two new solid phases or, conversely, two solid phases can
react to form a single new phase. These and the other various types of
invariant reactions observed in binary systems are illustrated in Fig. 6.
Fig. 6 This Schematic phase diagram
showing various types of invariant reactions observed in binary systems.
Intermediate Phases. In addition to the three solid terminal phases α, β, and ε, the
diagram in Fig. 6 displays five other solid phase fields, γ, δ, δ', η, and σ,
at intermediate compositions. Such phases are called intermediate phases. Many
intermediate phases, such as those illustrated in Fig. 6, have a fairly wide
range of homogeneity. However, many others have a very limited or no
significant homogeneity range.
When an intermediate phase of limited (or no)
homogeneity range is located at or near a specific ratio of component elements
that reflects the normal positioning of the component atoms in the crystal
structure of the phase, it is often called a compound (or line compound). When
the components of the system are metallic, such an intermediate phase is often
called an intermetallic compound. Three intermetallic compounds (with four
types of melting reactions) are shown in Fig. 7.
Fig. 7 Schematic phase diagram
showing three intermetallic line compounds and four melting reactions.
In the hypothetical diagram shown in Fig. 7, an alloy
of composition AB will freeze and melt isothermally, without the liquid or
solid phases undergoing changes in composition; such a phase change is called
congruent. All other reactions are incongruent; that is two phases are formed
from one phase on melting. Congruent and incongruent phase changes, however,
are not limited to line compounds: the terminal component B (pure phase ε) and
the highest-melting composition of intermediate phase δ' in Fig. 6, for
example, freeze and melt congruently, while δ' and ε freeze and melt
incongruently at other compositions.
Metastable Equilibrium. In Fig. 5(c), dashed lines indicate the portion of the liquidus and
solidus lines that disappear into the two-phase solid region. These dashed
lines represent valuable information, as they indicate a condition that would
exist under metastable equilibrium, such as might theoretically occur during
extremely rapid cooling.
When a
third component is added to a binary system, illustrating equilibrium
conditions in two dimensions becomes more complicated. One option is to add a
third composition dimension to the base, forming a solid diagram having binary
diagrams as its vertical sides. This can be represented as a modified isometric
projection, such as shown in Fig. 8. Here, boundaries of single-phase fields
(liquidus, solidus, and solvus lines in the binary diagrams) become surfaces;
single- and two-phase areas become volumes; three-phase lines become volumes;
and four-phase points, while not shown in Fig. 8, can exist as an invariant
plane. The composition of a binary eutectic liquid, which is a point in a
two-component system, becomes a line in a ternary diagram, as shown in Fig. 8.
Fig. 8 Ternary phase diagram
showing three-phase equilibrium.1
While three-dimensional projections can be helpful in
understanding the relationships in the diagram, reading values from them is
difficult. Ternary systems, therefore, are often represented by views of the
binary diagrams that comprise the faces and two-dimensional projections of the
liquidus and solidus surfaces, along with a series of two-dimensional
horizontal sections (isotherms) and vertical sections (isopleths) through the
solid diagram.
Vertical sections are often
taken through one corner (one component) and a congruently melting binary
compound that appears on the opposite face; when such a plot can be read like
any other true binary diagram, it is called a quasi-binary section. One
possibility of such a section is illustrated by line 1-2 in the isothermal
section shown in Fig. 9. A vertical section between a congruently melting
binary compound on one face and one on a different face might also form a
quasi-binary section (see line 2-3).
Fig. 9 Isothermal section of a
ternary diagram with phase boundaries deleted for simplification
All other vertical sections are not true binary
diagrams, and the term pseudobinary is applied to them. A common
pseudobinary section is one where the percentage of one of the components is
held constant (the section is parallel to one of the faces), as shown by line
4-5 in Fig. 9. Another is one where the ratio of two constituents is held
constant, and the amount of the third is varied from 0 to 100% (line 1-5).
Isothermal Sections. Composition values in the triangular isothermal sections are read
from a triangular grid consisting of three sets of lines parallel to the faces
and placed at regular composition intervals (see Fig. 10). Normally, the point
of the triangle is placed at the top of the illustration, component A is placed
at the bottom left, B at the bottom right, and C at the top. The amount of
constituent A is normally indicated from point C to point A, the amount of
constituent B from point A to point B, and the amount of constituent C from
point B to point C. This scale arrangement is often modified when only a corner
area of the diagram is shown.
Fig. 10 Triangular composition grid
for isothermal sections; X is the composition of each constituent in mole
fraction or percent
Projected Views. Liquidus, solidus, and solvus surfaces by their nature are not
isothermal. Therefore, equal-temperature (isothermal) contour lines are often
added to the projected views of these surfaces to indicate the shape of the
surfaces (see Fig. 11). In addition to (or instead of) contour lines, views
often show lines indicating the temperature troughs (also called
"valleys" or "grooves") formed at the intersections of two
surfaces. Arrowheads are often added to these lines to indicate the direction
of decreasing temperature in the trough.
Fig. 11 Liquidus projection of a
ternary phase diagram showing isothermal contour lines.1
The reactions between components,
the phases formed in a system, and the shape of the resulting phase diagram can
be explained and understood through knowledge of the principles, laws, and
terms of thermodynamics, and how they apply to the system.
Internal Energy. The sum of
the kinetic energy (energy of motion) and potential energy (stored energy) of a
system is called its internal energy, U. Internal energy is characterized
solely by the state of the system.
Closed System. A
thermodynamic system that undergoes no interchange of mass (material) with its
surroundings is called a closed system. A closed system, however, can
interchange energy with its surroundings.
First Law. The First
Law of Thermodynamics, as stated by Julius von Mayer, James Joule, and Hermann
von Helmholtz in the 1840s, states that energy can be neither created nor
destroyed. Therefore, it is called the Law of Conservation of Energy. This law
means that the total energy of an isolated system remains constant throughout
any operations that are carried out on it; that is, for any quantity of energy
in one form that disappears from the system, an equal quantity of another form
(or other forms) will appear. For example, consider a closed gaseous system to
which a quantity of heat energy, dQ,
is added and a quantity of work, dW,
is extracted. The First Law describes that change in the internal energy, dU, of the system as:
dU = dQ
- dW
In the vast majority of industrial
processes and material applications, the only work done by or on a system is
limited to pressure/volume terms. Any energy contributions from electric,
magnetic, or gravitational fields are neglected, except for electrowinning and
electrorefining processes such as those used in the production of copper,
aluminum, magnesium, the alkaline metals, and the alkaline earth metals. With
the neglect of field effects, the work done by a system can be measured by summing
the changes in volume, dV,
times each pressure causing a change. Therefore, when field effects are
neglected, the First Law can be written:
dU = dQ
- pdV
Enthalpy. Thermal energy changes under
constant pressure (again neglecting any field effects) are most conveniently
expressed in terms of the enthalpy, H,
of a system. Enthalpy, also called heat content, is defined by:
H = U + pV
Enthalpy, like internal energy, is a function of the state
of the system, as is the product pV.
Heat Capacity The heat capacity, C, of a substance is the
amount of heat required to raise its temperature one degree; that is:
C = dQ/dT
However, if the substance is kept at constant volume (dV = 0):
CV = (∂Q/∂T)V
= (∂U/∂T)V
If, instead, the substance is kept at constant pressure
(as in many metallurgical systems),
Cp
= (∂H/∂T)p
Second Law. While the
First Law establishes the relationship between the heat absorbed and the work
performed by a system, it places no restriction on the source of the heat or
its flow direction. This restriction, however, is set by the Second Law of
Thermodynamics, which was advanced by Rudolf Clausius and William Thomson (Lord
Kelvin). The Second Law states that the spontaneous flow of heat always is from
the higher temperature body to the lower temperature body. In other words, all
naturally occurring processes tend to take place spontaneously in the direction
that will lead to equilibrium.
Entropy. The Second Law is most conveniently
stated in terms of entropy, S, another property of state possessed by all
systems. Entropy represents the energy (per degree of absolute temperature, T) in a system that is not available
for work. In terms of entropy, the Second Law states that all natural processes
tend to occur only with an increase in entropy, and the direction of the
process is always such as to lead to an increase in entropy. For processes
taking place in a system in equilibrium with its surrounding, the change in
entropy is defined as:
ΔS
= ΔQ/T = ΔU + pΔV/T
Third Law. A principle
advanced by Theodore Richards, Walter Nernst, Max Planck, and others — often
called the Third Law of Thermodynamics,— states that the entropy of all
chemically homogeneous materials can be taken as zero at absolute zero
temperature (0 K). This principle allows calculation of the absolute values of
entropy of pure substances solely from heat capacity.
Gibbs Energy. Because both
S and V are difficult to control
experimentally, an additional term, Gibbs energy, G, is introduced, whereby:
G
= U + pV - TS
= H - TS
and
dG
= dU + pdV + Vdp - TdS
- SdT
However,
dU
= TdS + pdV
Therefore,
dG
= Vdp - SdT
Here, the change in Gibbs energy of a system undergoing a
process is expressed in terms of two independent variables, pressure and
absolute temperature, which are easily controlled experimentally. If the
process is carried out under conditions of constant pressure and temperature,
the change in Gibbs energy of a system at equilibrium with its surroundings (a
reversible process) is zero. For a spontaneous (irreversible) process, the
change in Gibbs energy is less than zero (negative); that is, the Gibbs energy
decreases during the process, and it reaches a minimum at equilibrium.
The areas (fields) in a phase diagram, and the position
and shapes of the points, lines, surfaces, and intersections in it, are
controlled by thermodynamic principles and the thermodynamic properties of all
of the phases that constitute the system.
Phase-Field Rule. The phase-field
rule specifies that at constant temperature and pressure, the number of phases
in adjacent fields in a multi-component diagram must differ by one.
Theorem of Le Chatelier.
The theorem of Henri Le Chatelier, which is based on thermodynamic principles,
states that if a system in equilibrium is subjected to a constraint by which
the equilibrium is altered, a reaction occurs that opposes the constraint, that
is, a reaction that partially nullifies the alteration.
Clausius-Clapeyron Equation.
The theorem of Le Chatelier was quantified by Benoit Clapeyron and Rudolf
Clausius to give:
dp/dT
= ΔH/TΔV
where dp/dT
is the slope of the univariant line in a p-T
diagram, ΔV is the difference
in molar volume of the two phases in the reaction, and ΔH is the difference in molar enthalpy
of the two phases (the heat of reaction).
Solutions. The shape of liquidus,
solidus, and solvus curves (or surfaces) in a phase diagram are determined by
the Gibbs energies of the relevant phases. In this instance, the Gibbs energy
must include not only the energy of the constituent components, but also the
energy of mixing of these components in the phase. Consider, for example, the
situation of complete miscibility shown in Fig. 2. The two phases, liquid and
solid, are in stable equilibrium in the two-phase field between the liquidus
and solidus lines. The Gibbs energies at various temperatures are calculated as
a function of composition for ideal liquid solutions and for ideal solid
solutions of the two components, A and B. The result is a series of plots
similar to those shown in Fig. 12(a) to (e).
At temperature T1, the liquid solution has the lower Gibbs
energy and, therefore, is the more stable phase. At T2, the melting temperature for component A,
the liquid and solid are equally stable only at a composition of pure A. At
temperature T3,
between the melting temperatures of components A and B, the Gibbs energy curves
cross. Temperature T4
is the melting temperature of component B, while T5 is below it.
Construction of the two-phase
liquid-plus-solid field of the phase diagram in Fig. 12(f) is as follows.
According to thermodynamic principles, the compositions of the two phases in
equilibrium with each other at temperature T3
can be determined by constructing a straight line that is tangential to both
curves in Fig. 12(c). The points of tangency, 1 and 2, are then transferred to
the phase diagram as points on the solidus and liquidus, respectively. This is
repeated at sufficient temperatures to determine the curves accurately. If, at
some temperature, the Gibbs energy curves for the liquid and the solid
tangentially touch at some point, the resulting phase diagram will be similar
to those shown in Fig. 3(a) and (b), where a maximum or minimum appears in the
liquidus and solidus curves.
Mixture. The two-phase field in Fig. 12(f)
consists of a mixture of liquid and solid phases. As stated above, the
compositions of the two phases in equilibrium at temperature T3 are C1 and C2.
The horizontal isothermal line connecting points 1 and 2, where these
compositions intersect temperature T3,
is called a tie line. Similar tie lines connect the coexisting phases
throughout all two-phase fields in binary systems.
Eutectic phase diagrams, a feature of which is a field
where there is a mixture of two solid phases, also can be constructed from
Gibbs energy curves. Consider the temperatures indicated on the phase diagram
in Fig. 13(f) and the Gibbs energy curves for these temperatures (Fig. 13a-e).
When the points of tangency on the energy curves are
transferred to the diagram, the typical shape of a eutectic system results.
Binary diagrams that have three-phase reactions other than the eutectic
reaction, as well as diagrams with multiple three-phase reactions, also can be
constructed from appropriate Gibbs energy curves.
Fig. 12 Use of Gibbs energy curves to construct a
binary phase diagram that shows miscibility in both the liquid and the solid.
Fig. 13 Use of Gibbs energy curves to construct a
binary phase diagram of the eutectic type.
Curves and Intersections.
Thermodynamic principles also limit the shape of the various boundary curves
(or surfaces) and their intersections. For example, see the PT diagram shown in Fig. 14. The
Clausius-Clapeyron equation requires that at the intersection of the triple
curves in such a diagram, the angle between adjacent curves should never exceed
180°, or alternatively, the extension of each triple curve between two phases
must lie within the field of third phase.
Fig. 14 Pressure-temperature phase diagram.
The angle at which the boundaries of two-phase
fields meet also is limited by thermodynamics. That is, the angle must be such
that the extension of each beyond the point of intersection projects into a
two-phase field, rather than a one-phase field. An example of correct
intersections can be seen in Fig. 15(b), where both the solidus and solvus
lines are concave. However, the curvature of both boundaries need not be
concave.
Fig. 15 Binary phase diagrams with invariant points.
(a) Hypothetical diagram in which the miscibility gap in the solid touches the
solidus curve at invariant point P; an actual diagram of this type probably
does not exist. (b) and (c) Typical eutectic diagrams for (b) components having
the same crystal structure, and (c) components having different crystal
structures; the eutectic (invariant) points are labeled E. The dashed lines in
(b) and (c) are metastable extensions of the stable-equilibria lines.
Congruent Transformations.
The congruent point on a
phase diagram is where different phases of same composition are in equilibrium.
The Gibbs-Konovalov Rule for
congruent points, which was developed by Dmitry Konovalov from a thermodynamic
expression given by J. Willard Gibbs, states that the slope of phase boundaries
at congruent transformations must be zero (horizontal). Examples of correct
slope at the maximum and minimum points on liquidus and solidus curves can be
seen in Fig. 16.
Fig. 16 Binary phase diagrams with solid-state
miscibility where the liquidus shows (a) a maximum and (b) a minimum.
Higher-Order Transitions.
First-order transitions are
those involving distinct phases having different lattice parameters,
enthalpies, entropies, densities, and so forth. Transitions not involving
discontinuities in composition, enthalpy, entropy, or molar volume are called higher-order transitions and occur less
frequently. The change in the magnetic quality of iron from ferromagnetic to
paramagnetic as the temperature is raised above 771 °C (1420 °F) is an example
of a second-order transition: no phase change is involved and the Gibbs phase
rule does not come into play in the transition.
Another example of a higher-order transition
is the continuous change from a random arrangement of the various kinds of
atoms in a multicomponent crystal structure (a disordered structure) to an arrangement where there is
some degree of crystal ordering
of the atoms (an ordered structure,
or superlattice), or the
reverse reaction.
Composition Scales.
Phase diagrams to be used by scientists are usually plotted in atomic
percentage (or mole fraction), while those to be used by engineers are usually
plotted in weight percentage.
Lines and Labels. Magnetic
transitions (Curie temperature and Néel temperature) and uncertain or
speculative boundaries are usually shown in phase diagrams as nonsolid lines of
various types.
The components of metallic systems, which usually are pure
elements, are identified in phase diagrams by their symbols. Allotropes of
polymorphic elements are distinguished by small (lower-case) Greek letter
prefixes.
Terminal solid phases are normally designated by the
symbol (in parentheses) for the allotrope of the component element, such as
(Cr) or (αTi). Continuous solid solutions are designated by the names of both
elements, such as (Cu,Pd) or (βTi, βY).
Intermediate phases in phase diagrams are normally labeled
with small (lower-case) Greek letters. However, certain Greek letters are
conventionally used for certain phases, particularly disordered solutions: for
example, β for disordered body-centered cubic (bcc), ζ or ε for disordered close-packed
hexagonal (cph), γ for the γ-brass-type structure, and δ for the δ CrFe-type
structure.
For line compounds, a stoichiometric phase name is used in
preference to a Greek letter (for example, A2B3 rather
than δ). Greek letter prefixes are used to indicate high- and low-temperature
forms of the compound (for example, αA2B3 for the
low-temperature form and βA2B3 for the high-temperature
form).
Lever Rule. A tie line is an
imaginary horizontal line drawn in a two-phase field connecting two points that
represent two coexisting phases in equilibrium at the temperature indicated by
the line. Tie lines can be used to determine the fractional amounts of the
phases in equilibrium by employing the lever rule. The lever rule is a mathematical expression
derived by the principle of conservation of matter in which the phase amounts
can be calculated from the bulk composition of the alloy and compositions of
the conjugate phases, as shown in Fig. 17(a).
Fig. 17 Portion of a binary phase diagram containing
a two-phase liquid-plus-solid field illustrating (a) application of the lever
rule to (b) equilibrium freezing, (c) nonequilibrium freezing, and (d) heating
of a homogenized sample.1
At the left end of the line between α1 and L1, the bulk composition is Y% component B and 100 - Y% component A, and consists of 100% a
solid solution. As the percentage of component B in the bulk composition moves
to the right, some liquid appears along with the solid. With further increases
in the amount of B in the alloy, more of the mixture consists of liquid, until
the material becomes entirely liquid at the right end of the tie line. At bulk
composition X, which is less
than halfway to point L1,
there is more solid present than liquid. The lever rule says that the
percentages of the two phases present can be calculated as follows:
It should be remembered that the calculated amounts of the
phases present are either in weight or atomic percentages, and, as shown in
Table 1, do not directly indicate the area or volume percentages of the phases
observed in microstructures.
In order to relate the weight fraction of a phase
present in an alloy specimen as determined from a phase diagram to its
two-dimensional appearance as observed in a micrograph, it is necessary to be
able to convert between weight-fraction values and area-fraction values, both
in decimal fractions. This conversion can be developed: |
The weight fraction of the phase is determined
from the phase diagram, using the lever rule. |
Volume portion of the phase = (Weight fraction of
the phase)/(Phase density) |
Total volume of all phases present = Sum of the
volume portions of each phase. |
Volume fraction of the phase = (Weight fraction
of the phase)/(Phase density × total volume) |
It has been shown by stereology and quantitative
metallography that areal fraction is equal to volume fraction.2 (Areal
fraction of a phase is the sum of areas of the phase intercepted by a
microscopic traverse of the observed region of the specimen divided by the
total area of the observed region.) Therefore: |
Areal fraction of the phase = (Weight fraction of
the phase)/(Phase density × total volume) |
The phase density value for the preceding
equation can be obtained by measurements or calculation. The densities of
chemical elements, and some line compounds, can be found in the literature.
Alternatively, the density of a unit cell of a phase comprising one or more
elements can be calculated from information about its crystal structure and
the atomic weights of the elements comprising it as follows: |
Weight of each element = number of atoms ×
[(Atomic weight)/(Avogadro's number)] |
Total cell weight = Sum of weights of each
element |
Density = Total cell weight/cell volume |
For example, the calculated density of pure
copper, which has a face-centered cubic (fcc) structure and a lattice
parameter of 0.36146 nm, is: |
|
This compares favorably with the published value
of 8.93. |
Phase-Fraction Lines.
Reading phase relationships in many ternary diagram sections (and other types
of sections) can often be difficult due to the great many lines and areas
present. Phase-fraction lines
are used by some to simplify this task. In this approach, the sets of often
nonparallel tie lines in the two-phase fields of isothermal sections (see Fig.
18a) are replaced with sets of curving lines of equal phase fraction (Fig.
18b). Note that the phase-fraction lines extend through the three-phase region
where they appear as a triangular network. As with tie lines, the number of
phase-fraction lines used is up to the individual using the diagram. While this
approach to reading diagrams may not seem helpful for such a simple diagram, it
can be a useful aid in more complicated systems.3,4
Fig. 18 Alternative systems for showing phase
relationships in multiphase regions of ternary-diagram isothermal sections. (a)
Tie lines. (b) Phase-fraction lines.3
Solidification. Tie lines and the
lever rule can be used to understand the freezing of a solid-solution alloy.
Consider the series of tie lines at different temperature shown in Fig. 17(b),
all of which intersect the bulk composition X. The first crystals to freeze
have the composition α1. As the temperature is reduced to T2 and the solid crystals
grow, more A atoms are removed from the liquid than B atoms, thus shifting the
composition of the remaining liquid to composition L2. Therefore, during freezing, the
compositions of both the layer of solid freezing out on the crystals and the
remaining liquid continuously shift to higher B contents and become leaner in
A. Therefore, for equilibrium to be maintained, the solid crystals must absorb
B atoms from the liquid and B atoms must migrate (diffuse) from the previously
frozen material into subsequently deposited layers. When this happens, the
average composition of the solid material follows the solidus line to
temperature T4
where it equals the bulk composition of the alloy.
Coring. If cooling takes place too
rapidly for maintenance of equilibrium, the successive layers deposited on the
crystals will have a range of local compositions from their centers to their
edges (a condition known as coring). Development of this condition is
illustrated in Fig. 17(c). Without diffusion of B atoms from the material that
solidified at temperature T1
into the material freezing at T2,
the average composition of the solid formed up to that point will not follow
the solidus line. Instead it will remain to the left of the solidus, following
compositions α'1 through α'3. Note that final freezing does not occur until
temperature T5,
which means that nonequilibrium solidification takes place over a greater temperature
range than equilibrium freezing. Because most metals freeze by the formation
and growth of "treelike" crystals, called dendrites, coring is sometimes called dendritic segregation. An example of
cored dendrites is shown in Fig. 19.
Fig. 19 Copper alloy 71500 (Cu-30Ni) ingot.
Dendritic structure shows coring: light areas are nickel-rich; dark areas are
low in nickel. 20×.2
Liquation. Because the lowest
freezing material in a cored microstructure is segregated to the edges of the
solidifying crystals (the grain boundaries), this material can remelt when the
alloy sample is heated to temperatures below the equilibrium solidus line. If grain-boundary
melting (called liquation or
"burning") occurs while the sample is also under stress, such as
during hot forming, the liquefied grain boundaries will rupture and the sample
will lose its ductility and be characterized as hot short.
Liquation also can have a deleterious effect on the
mechanical properties (and microstructure) of the sample after it returns to
room temperature. This is illustrated in Fig. 17(d) for a homogenized sample.
If homogenized alloy X is
heated into the liquid-plus-solid region for some reason (inadvertently or
during welding, etc.), it will begin to melt when it reaches temperature T2; the first liquid to
appear will have the composition L2.
When the sample is heated at normal rates to temperature T1, the liquid formed so far
will have a composition L1,
but the solid will not have time to reach the equilibrium composition α1.
The average composition will instead lie at some intermediate value such as α'1.
According to the lever rule, this means that less than the equilibrium amount
of liquid will form at this temperature. If the sample is then rapidly cooled
from temperature T1,
solidification will occur in the normal manner, with a layer of material having
composition a1 deposited on existing solid grains. This is followed by layers
of increasing B content up to composition α3 at temperature T3, where all of the liquid
is converted to solid. This produces coring in the previously melted regions
along the grain boundaries and sometimes even voids that decrease the strength
of the sample. Homogenization heat treatment will eliminate the coring, but not
the voids.
Eutectic Microstructures.
When an alloy of eutectic composition is cooled from the liquid state, the
eutectic reaction occurs at the eutectic temperature, where the two distinct
liquidus curves meet. At this temperature, both α and β solid phases must
deposit on the grain nuclei until all of the liquid is converted to solid. This
simultaneous deposition results in microstructures made up of distinctively
shaped particles of one phase in a matrix of the other phase, or alternate
layers of the two phases. Examples of characteristic eutectic microstructures
include spheroidal, nodular, or globular; acicular (needles) or rod; and
lamellar (platelets, Chinese script or dendritic, or filigreed). Each eutectic
alloy has its own characteristic microstructure, when slowly cooled (see Fig.
20). Cooling more rapidly, however, can affect the microstructure obtained (see
Fig. 21). Care must be taken in characterizing eutectic structures because
elongated particles can appear nodular and flat platelets can appear elongated
or needlelike when viewed in cross section.
Fig. 20 Examples of characteristic eutectic
microstructures in slowly cooled alloys. (a) 40Sn-50In alloy showing globules
of tin-rich intermetallic phase (light) in a matrix of dark indium-rich
intermetallic phase. 150×. (b) Al-13Si alloy showing an acicular structure
consisting of short, angular particles of silicon (dark) in a matrix of
aluminum. 200×. (c) Al-33Cu alloy showing a lamellar structure consisting of
dark platelets of CuAl2 and light platelets of aluminum solid solution. 180×.
(d) Mg-37Sn alloy showing a lamellar structure consisting of Mg2Sn
"Chinese-script" (dark) in a matrix of magnesium solid solution.
250×.2
Fig. 21 Effect of cooling rate on the microstructure
of Sn-37Pb alloy (eutectic soft solder). (a) Slowly cooled sample shows a
lamellar structure consisting of dark platelets of lead-rich solid solution and
light platelets of tin. 375×. (b) More rapidly cooled sample shows globules of
lead-rich solid solution, some of which exhibit a slightly dendritic structure,
in a matrix of tin. 375×.2
If the alloy has a composition different than the eutectic
composition, the alloy will begin to solidify before the eutectic temperature
is reached. If the alloy is hypoeutectic, some dendrites of α will form in the
liquid before the remaining liquid solidifies at the eutectic temperature. If
the alloy is hypereutectic, the first (primary) material to solidify will be
dendrites of β. The microstructure produced by slow cooling of a hypoeutectic
and hypereutectic alloy will consist of relatively large particles of primary constituent, consisting of the
phase that begins to freeze first surrounded by relatively fine eutectic
structure. In many instances, the shape of the particles will show a
relationship to their dendritic origin (see Fig. 22a). In other instances, the
initial dendrites will have filled out somewhat into idiomorphic particles (particles having
their own characteristic shape) that reflect the crystal structure of the phase
(see Fig. 22b).
Fig. 22 Examples of primary-particle shape. (a)
Sn-30Pb hypoeutectic alloy showing dendritic particles of tin-rich solid
solution in a matrix of tin-lead eutectic. 500×. (b) Al-19Si hypereutectic
alloy, phosphorus-modified, showing idiomorphic particles of silicon in a
matrix of aluminum-silicon eutectic. 100×2
As stated earlier, cooling at a rate that does not allow sufficient
time to reach equilibrium conditions will affect the resulting microstructure.
For example, it is possible for an alloy in a eutectic system to obtain some
eutectic structure in an alloy outside the normal composition range for such a
structure. This is illustrated in Fig. 23. With relatively rapid cooling of
alloy X, the composition of
the solid material that forms will follow line α1-α'4
rather than solidus line to α4. As a result, the last liquid to
solidify will have the eutectic composition L4
rather than L3 and
will form some eutectic structure in the microstructure. The question of what
takes place when the temperature reaches T5
is discussed later.
Fig. 23 Binary phase diagram, illustrating the
effect of cooling rate on an alloy lying outside the equilibrium
eutectic-transformation line. Rapid solidification into a terminal phase field
can result in some eutectic structure being formed; homogenization at
temperatures in the single-phase field will eliminate the eutectic structure; β
phase will precipitate out of solution upon slow cooling into the a + β field.
Adapted from Ref 1.
Eutectoid Microstructures.
Because the diffusion rates of atoms are so much lower in solids than liquids,
nonequilibrium transformation is even more important in solid/solid reactions
(such as the eutectoid reaction) than in liquid/solid reactions (such as the
eutectic reaction). With slow cooling through the eutectoid temperature, most
alloys of eutectoid composition such as alloy 2 in Fig. 24 transform from a
single-phase microstructure to a lamellar structure consisting of alternate
platelets of α and β arranged in groups (or "colonies"). The
appearance of this structure is very similar to lamellar eutectic structure
(see Fig. 25). When found in cast irons and steels, this structure is called
"pearlite" because of its shiny mother-of-pearl-like appearance under
the microscope (especially under oblique illumination); when similar eutectoid
structure is found in nonferrous alloys, it often is called
"pearlite-like" or "pearlitic."
Fig. 24 Binary phase diagram of a eutectoid system.
Adapted from Ref 1.
Fig. 25 Fe-0.8C alloy showing a typical pearlite
eutectoid structure of alternate layers of light ferrite and dark cementite.
500×.2
The terms, hypoeutectoid and hypereutectoid
have the same relationship to the eutectoid composition as hypoeutectic and
hypereutectic do in a eutectic system; alloy 1 in Fig. 24 is a hypoeutectoid
alloy, while alloy 3 is hypereutectoid. The solid-state transformation of such
alloys takes place in two steps, much like freezing of hypoeutectic and
hypereutectic alloys except that the microconstituents that form before the
eutectoid temperature is reached are referred to as proeutectoid constituents
rather than “primary.”
Microstructures of Other Invariant Reactions.
Phase diagrams can be used in a manner similar to that used in the discussion
of eutectic and eutectoid reactions to determine the microstructures expected
to result from cooling an alloy through any of the other six types of reactions
listed in Table 2.
Solid-State Precipitation.
If alloy X in Fig. 23 is
homogenized at a temperature between T3
and T5, it will
reach equilibrium condition; that is, the β portion of the eutectic constituent
will dissolve and the microstructure will consist solely of α grains. Upon
cooling below temperature T5,
this microstructure will no longer represent equilibrium conditions, but
instead will be supersaturated with B atoms. In order for the sample to return
to equilibrium, some of the B atoms will tend to congregate in various regions
of the sample to form colonies of new β material. The B atoms in some of these
colonies, called Guinier-Preston zones,
will drift apart, while other colonies will grow large enough to form
incipient, but not distinct, particles. The difference in crystal structures
and lattice parameters between the α and β phases causes lattice strain at the
boundary between the two materials, thereby raising the total energy level of the
sample and hardening and strengthening it. At this stage, the incipient
particles are difficult to distinguish in the microstructure. Instead, there
usually is only a general darkening of the structure. If sufficient time is
allowed, the β regions will break away from their host grains of α and
precipitate as distinct particles, thereby relieving the lattice strain and
returning the hardness and strength to the former levels. While this process is
illustrated for a simple eutectic system, it can occur wherever similar
conditions exist in a phase diagram; that is, there is a range of alloy
compositions in the system for which there is a transition on cooling from a
single-solid region to a region that also contains a second solid phase, and
where the boundary between the regions slopes away from the composition line as
cooling continues. Several examples of such systems are shown schematically in
Fig. 26.
Fig. 26 Examples of binary phase diagrams that give
rise to precipitation reactions.2
Although this entire process is called precipitation hardening, the term normally refers only to
the portion before much actual precipitation takes place. Because the process
takes a while to be accomplished, the term age
hardening is often used instead. The rate at which aging occurs
depends on the level of supersaturation (how far from equilibrium), the amount
of lattice strain originally developed (amount of lattice mismatch), the
fraction left to be relieved (how far along the process has progressed), and
the aging temperature (the mobility of the atoms to migrate). The β precipitate
usually takes the form of small idiomorphic particles situated along the grain
boundaries and within the grains of α phase. In most instances, the particles
are more or less uniform in size and oriented in a systematic fashion.
Impossible Diagrams.
Thermodynamic principles also limit the shape of the various boundary curves
and their intersections. If a phase boundary of a phase diagram violates such
thermodynamic requirements, the diagram is thermodynamically impossible at
least in the related segment. Various impossible phase relationships often
found in published phase diagrams are summarized in Fig. 27.
Fig. 27 Summary of impossible phase relationships in
published phase diagrams
These problems are:
A:
The liquidus and solidus must meet at the melting point of the pure element.
B:
Two liquidus curves must meet at one composition at a eutectic temperature.
C:
A tie line must terminate at a phase boundary.
D:
Two solvus boundaries (or two liquidus, or two solidus, or a solidus and a
solvus) of the same phase must intersect at one composition at an invariant
temperature.
E:
A phase boundary must extrapolate into a two-phase field after crossing an
invariant point.
F:
A two-phase field cannot be extended to a pure element end.
G:
Two boundaries of g must not be continuous at the invariant temperature. They
must cross one another.
H:
An invariant temperature line should involve equilibrium among three phases.
I:
There should be a two-phase field between two single phase fields.
J:
When two phase boundaries touch at a point, they should touch at an extremity
of temperature.
K:
A touching liquidus and solidus (or any two touching boundaries) must have a
horizontal common tangent at the congruent point. In this case, the slope of
the solidus appears to be discontinuous at the melting point.
L:
A local minimum point in the lower part of a single-phase field cannot be drawn
without an additional boundary in contact with it (minimum congruent point or
monotectic reaction in this case).
M:
A local maximum point in the lower part of a single-phase field cannot be drawn
without a monotectic, monotectoid, syntactic, and syntectoid reaction occurring
at a lower temperature. Alternatively, a solidus curve must be drawn to touch
the liquidus at point M. (If the maximum is not local, as in a miscibility gap,
this is not a phase rule violation.)
N:
The temperature of an invariant reaction must be constant. (The reaction line
must be horizontal.)
O:
A phase boundary cannot terminate within a phase field (except the case when
the boundary is unknown beyond this point).
P:
The liquidus should not have a discontinuous sharp peak at the melting point of
a compound. (See exceptions below.)
Q:
The compositions of all three phases at an invariant reaction must be
different.
R:
Temperatures of liquidus and solidus (or any two boundaries) must either
increase or decrease together from one point on the pure element line as the
content of a second element increases.
S:
A four-phase equilibrium is not allowed in a binary system. (See exceptions
below.)
T:
Two separate phase boundaries that create a two-phase field between two phases
in equilibrium should not cross one another.
Although phase rules are not violated, three additional
unusual situations (X, Y, and Z) are also included in Fig. 27. These unlikely
situations are discussed in the next section.
An additional problem, not shown in Fig. 27:
For example, a fcc phase and a bcc phase cannot
form a continuous phase. There must be a two-phase field between them.
Exceptions
Improbable Diagrams.
Some diagrams involve errors that are generally acceptable from the viewpoint
of phase rule, but the proposed phase boundaries have atypical or abnormal
forms, or have been forced to have abnormal forms in order to satisfy the phase
rule, or uncertain experimental data. First, it must be noted that an abrupt
change of slope of a phase boundary, as shown X, Y, and Z in Fig. 27, is
thermodynamically unlikely. An abrupt change of slope can occur only if the
thermodynamic property of either one of the two phases in equilibrium suddenly
changes at the corresponding temperature or composition. Because the
thermodynamic properties are expected to change gradually in one phase field,
the phase boundary slope is also expected to change gradually. If an abrupt
change of slope is real, it must be related to a unique situation affecting a
phase associated with this phase field, such as the onset of an order-disorder
transformation, or a magnetic transition. Figure 28 shows various types of
improbable phase boundaries.
Fig. 28 Various types of improbable phase boundaries
a:
G + L two-phase field is too narrow. The opening angle of G + L at 0 at.% must
be much larger because the heat of vaporization of an element is usually much
greater than the heat of fusion.
b:
Extrapolation of the liquidus should not cross the 0 at.% line. Otherwise,
problem F of Fig. 27 occurs.
c:
The liquidus of δat point c is too flat in comparison with the liquidus of δat
point e. Problems c, d, and e are related. Because entropy of fusion of
elements and compounds cannot differ much, curvatures of liquidus curves for
compounds in a binary system must be similar. A phase with a sharper liquidus
tends to decompose into two neighboring phases at low temperatures.
d:
A compound with a flat liquidus is stable and will not decompose at low
temperatures.
e:
Liquidus at point e is too sharp in comparison with the liquidus at point c.
f:
Extrapolation of the liquidus of λ2 must have a peak at the
composition of λ2. Otherwise, problem P of Fig. 27 occurs.
g:
Change of liquidus slope associated with an allotropic transformation must be
small.
h:
Two compounds having similar compositions cannot be stable over a wide
temperature range.
i:
A phase field of a compound cannot extend over a neighboring phase. Problem T
of Fig. 27 occurs.
j:
The congruent melting point of AmBn compound is too far away from its
stoichiometric composition.
k:
The liquidus is too asymmetric. According to the author’s criterion, a liquidus
is already too asymmetric if the liquidus width ratio to the left and right of
a compound exceeds 2 to 3.
l:
The transformation temperature of ε to β2 should be higher than the
melting point of ε. Otherwise, the β2 phase is stable above point j.
m:
Extrapolation of two boundaries of L + β2 should not cross. Problem
T of Fig. 27 occurs.
n:
A two-phase field must be narrower at higher temperatures.
o:
The slope is too flat to have a maximum point at the composition of φ.
p:
The liquid miscibility gap is too close to the edge of a phase diagram.
q:
The liquidus slope is too steep. The initial slope of a liquidus must conform
to the van’t Hoff relationship. If no solubility can be assumed for the solid
phase, extrapolation of the initial liquidus should go through the horizontal
axis at 0 K near approximately 110 at.%.
r:
Extrapolation of two boundaries of L + β3 should cross at the 100
at.% line, not at some composition exceeding 100 at.%. Problem A of Fig. 27.
s:
Two phase boundaries should have different initial slopes.
t:
The slopes of two phase boundaries are too far apart.
There are many other improbable phase relationships that
cannot be generalized in Fig. 28. Please refer to the related articles: H.
Okamoto and T.B. Massalski, J. Phase
Equilibria, Vol 12, 1991, p 148–168; H. Okamoto, J. Phase Equilibria, Vol 12, 1991, p
623–643; H. Okamoto and T.B. Massalski, J.
Phase Equilibria, Vol 14, 1993, p 316–335; and H. Okamoto and T.B.
Massalski, J. Phase Equilibria,
Vol 15, 1994, p 500–521.
The information in this article
was largely taken from: