Sub T 1.2 Thernodynamics for phase diagrams
THERMODYNAMICS is a branch of physics
and chemistry that covers a wide field, from
the atomic to macroscopic scale. In materials
science, thermodynamics is a powerful tool for
understanding
and solving problems. The part
of thermodynamics that concerns the physical
change of state of a chemical system following
the laws of thermodynamics is called chemical
thermodynamics. The thermodynamic properties
of individual phases can be used for eval
uating
their relative stability and heat evolution
during phase transformations or reactions. Traditionally,
the most common application of
thermodynamics is in the form of phase diagrams.
Phase diagrams are visual representations
of the state of a materia
l, generally
equilibrium, as a function of temperature, pressure,
and, in multicomponent systems, the concentrations
of the constituent components. They
are therefore frequently hailed as basic blueprints
or roadmaps for alloy design, development,
processi
ng, and basic understanding.
The importance of phase diagrams is also
reflected by the publication of many handbooks
(Ref 1). Thermodynamic quantities of pure substances
are tabulated in a number of compilations
(Ref 2), while only a few compilations
of th
ermochemical data for solution phases
exist (Ref 3, 4). Most of these compilations provide
the data in tabular form, although some
provide them in the form of analytical functions
or in electronic form. The Scientific Group
Thermodata
Europe compilation of binary systems
(Ref 4) provides selected thermochemical
values in tabulated and graphical form. Thiscompilation also includes phase diagrams calculated
from the same thermodynamic functions
as the tabulated thermochemical values.
It
is the first compilation that makes use of the
correlations between thermodynamics and
phase diagrams established by Gibbs.
Thermodynamics
The thermodynamic quantities that are most
frequently used in materials science are the
enthalpy, in the form of the
heat content of a
phase, the heat of formation of a phase, or the
latent heat of a phase transformation; the heat
capacity, which is the change of heat content
with temperature; and the chemical potential
or chemical activity, which describes the effect
of
composition change in a solution phase on its
energy. All of these thermodynamic quantities
are part of the energy content of a system and
are governed by the three laws of thermodynamics.
Detailed physical derivations of the
relationships of these quanti
ties can be found
in many textbooks (Ref 5, 6).A system is considered to be in equilibrium
when its energy is at a minimum. Under constant
pressure conditions for a closed system,
this energy is the Gibbs energy,
G
:
where
H
is the heat content or enthalpy,
T
is the
temperature, and
S
is the entropy. The entropy
is a rather abstract entity and can be envisioned
as a measure of the disorder of atoms in the system.
The temperature dependence of the Gibbs
energy of a pure substan
ce is usually expressed
as a power series of
T
:
where
a
. . .
h
are coefficients (Ref 7). Pressure
and magnetic states result in additional contributions
to the Gibbs energy. However, pressure
dependence for condensed systems at nearatmospheric
pressure
s is usually ignored.
For multicomponent systems, it has proven
useful to distinguish three contributions from
the concentration dependence to the Gibbs
energy of a phase,
The first term,
G
0
, corresponds to the Gibbs
energy of a mechanical mixture of
the constituents
of the phase
where
x
i
is the mole fraction of each of the
components, and
G
i
(
T
) is the Gibbs energy
of the pure component. The second term in
Eq 3,
G
ideal
, is the contribution from the
configurational entropy from atoms substituting
for
each other on equivalent sites in a random
mixture. In substitutional solutions, such as liquid
or disordered solid solutions, it is given as:
The third term in Eq 3, G
xs
, is the so

called
excess term. The excess Gibbs energy or Gibbs
energy
of mixing of a binary regular solution is
given as:
In a regular solution, the parameter
O
is a
measure of the enthalpy of mixing of the phase
of interest and represents the deviation of the
Gibbs energy from ideal behavior. It is negative
if A

B neigh
bors are favored in the solution and
positive if A

A and B

B are favored in the solution.
The solution is called subregular when
is
a function of the composition and quasi

regular
when
O
is also a function of temperature.
In general, these solutions are
referred to as
regular

type solutions. The formalisms needed
for the description of
G
xs
of ordered phases
with extended homogeneity ranges or liquid
phases with short

range order are generally
more complex, since an internal order parameter
must be taken i
nto account. The order
parameter is zero for the completely disordered
state and is at maximum for the perfectly
ordered state. Under equilibrium conditions,
the Gibbs energy must also have a minimum
with respect to this internal parameter.
For a given tem
perature and concentration,the equilibrium (minimum Gibbs energy) may
be single phase or a mixture of phases. The
Gibbs energy of a mixture of phases is the
weighted sum of Gibbs energies of the individual
phases:
where
f
j
is the phase fractions of the
individual
phases. The requirement of minimum Gibbs
energy for a system in equilibrium requires that
the chemical potential,
m
j
i
, of component
i
in
phase
j
must be equal in all phases:
The chemical potentials of the two elements ina
binary regular solution phase are (Ref 6):
The Gibbs energy for a given composition can
also be calculated from the chemical potentials:
The chemical activity is related to the chemical
potential by:
where
G
0
i
is the reference state of the pure
component.
In an ideal solution, the activity of a
component is equal to its molar fraction,
a
i
=
x
i
. This relation is known as Raoult’s law. Real
solutions, such as regular

type solutions, deviate
from Raoult’s law. The deviation of the
activity from the
molar fraction of the component
is described by the activity coefficient,
g
i
,
a
i
=
g
i
x
i
; this relation is known as Henry’s
law for dilute solutions.
The activity,
a
i
, of component
i
is defined as:
The vapor pressure of a component in solution,
P
i
, is
reduced in comparison to that of
the pure component,
P
0
i
. The vapor pressure
of an individual component is the same over
all phases in equilibrium
The requirement that the chemical potential of
a component must be the same for all phases in
equilibrium
directly links the Gibbs energy to
the phase diagram, as illustrated in Fig. 1. The
Gibbs energy functions of three phases, liquid
(l),
a
, and
b
, are plotted as functions of the concentration,
x
B
(
x
A
+
x
B
= 1 for a binary system)
for
various fixed temperatures. The minimum of
the Gibbs energy is given by the lowest of the individual
Gibbs energy curves and tangents to the
Gibbs energy curves. The equilibrium is a single
phase if the minimum Gibbs energy is given by
the curve of this p
hase. The equilibrium is a twophase
mixture if the minimum Gibbs energy is
given by the common tangent of two curves
(representing the line of equal chemical potential
for all components) and the equilibrium compositions
of the two phases are given by the
tangent
points. The line in the phase diagram connecting
these two points is called a tie line. The relative
position of the Gibbs energy curves varies with
temperature and causes the concentrations of the
tangent points to change as the temperature
changes. The phase boundaries plotted in a phase
diagram are the locii of these tangent points as a
function of temperature. A special temperature
can exist where three phases may occur in equilibrium,
for example,
T
3
in Fig. 1. Although a phase
diagram is
usually determined from direct measurement
of concentrations and temperature, it is
Metastable Phase Diagrams
Metastable equilibria occur when a phase is
missing, for example, if the phase fails to nucleate.
The failure of an equilibrium phase to form
may result in the formation of a metastable
phase. In the Gibbs energy diagram, the Gibbs
energy curve of a metastable phase always is
above the minimum Gibbs energy envelope of
the equilibrium phases. The most well

known
example
for such a case is the iron

carbon system,
shown in Fig. 3. Metastable cementite,
Fe
3
C, results if graphite, or carbon, fails to form.
The phase diagram shown in Fig. 3 is actually the
superimposition of two phase diagrams, the stable
equilibrium iron

car
bon diagram (solid lines)
and the metastable Fe

Fe
3
C diagram (dashed
lines). In general, the absence of an equilibrium
phase results in extended solubility in the
remaining equilibrium phases. The liquidus of
the metastable phase is always below the liquid
us
of the equilibrium phase that is absent.
Phase diagrams provide useful information
for understanding the solidification of alloys.
In addition to the liquidus and final freezing
temperatures, important quantities for the mathematical
treatment of solidi
fication processes
are obtained from the phase diagram. The ratio
of the concentrations of the liquidus,
x
L
i
, and
the solidus,
x
S
i
, at the same temperature is
described by the partition coefficient or distribution
coefficient,
k
i
:
Phase diagram
information can also be used
in two simple models to describe the limiting
cases of solidification behavior. For solidification
obeying the lever rule at each temperature
during cooling, complete diffusion is assumed
in
the solid as well as in the liquid. Thus, all
phases are assumed to be in thermodynamic
equilibrium at each temperature during solidification.
For the solidification of a single solid
phase in a binary alloy, rewriting Eq 17 using
f
S
= 1
_
f
L
and assumi
ng a constant partition
coefficient gives:
In comparison, for solidification following the
Scheil

Gulliver path (for convenience, here
called the Scheil path), no diffusion in the solid
and complete diffusion in the liquid are
assumed. This case, where t
hermodynamic
equilibrium exists only as local equilibrium at
the liquid/solid interface, produces microsegregation
with the lowest possible final freezing
temperature and therefore represents the most
extreme case of solidification with microsegregation.
An alloy that solidified following the
Scheil path is not in equilibrium after
solidification.
An expression similar to Eq 20 can be
obtained for the Scheil path by equating the solute
rejected during the formation of a small
amount
of solid, resulting in an increase of
solid in the liquid (Ref 10):
Integration from
x
S
B
¼
k
B
x
0
B
at
f
S
= 0 gives the
so

called Scheil

Gulliver equation (Ref 10):
While the Scheil

Gulliver equation can also be
derived
for ternary and higher

component
alloys, it requires knowledge of the tie lines.
An alloy that is not in equilibrium after solidification
can be the result of a phase failing to
nucleate, incomplete diffusion, or solute
trapping. The description of such s
olidification
processes with equilibrium phase diagram
information alone is not possible. However,
for most alloys where the solidifying phase is
a substitutional solid solution, predictions of
the Scheil

Gulliver model are closer to reality
than those of
the lever model, while for
interstitial solid solutions where the interstitial
element diffuses rapidly, predictions of the
lever model are closer to reality. Modeling of
real solidification behavior requires knowledge
of the temperature dependence of the
partition
coefficients and liquidus slopes and a kinetic
analysis of microsegregation and back diffusion
at each temperature.
The solidification of a binary alloy is schematically
shown in Fig. 4 (the sketches are oversimplified
to
emphasize the concentration
gradients). The solidification of the alloy labeled
with “
x
” begins at
T
L
. In the case of lever solidification,
the composition of the growing
a
phase
changes homogeneously during cooling according
to the tie lines (horizontal
lines) until the temperature
T
P
, the peritectic formation of the
b
phase, is reached. At this temperature,
b
phase
is formed from the
a
phase and the remaining liquid
phase, and solidification is complete. On further
cooling, the compositions and phase
fra
ctions of the
a
and
b
phases continue to
change according to the lever rule.
In the case of Scheil solidification, only the
composition of newly growing
a
phase changes
according to the tie lines during cooling; the
composition of
a
phase formed at higher
temperatures
remains unchanged because of the
absence of diffusion in the solid phase, and
the remaining liquid becomes enriched in solute.
On reaching the peritectic temperature,
T
P
, solidification of the
b
phase begins. The
amount of
a
becomes frozen at
this point. No
formation of
b
phase from
a
phase reacting
with the liquid phase occurs because of the
absence of diffusion. Solidification of the
b
phase continues until all liquid is consumed or
when the eutectic temperature,
T
E
, is reached,
where
the remaining liquid solidifies in a eutectic
reaction. An A

rich alloy that freezes as single
a
phase in lever solidification may show the
formation of
b
phase in Scheil solidification.
A B

rich alloy that freezes as two phases,
b
+
g
, in lever
solidification may actually have all
three solid phases,
a
+
b
+
g
, in Scheil solidification
if its composition is on the A

rich side of
the peritectic point. In some alloys, annealing at
elevated temperatures but below the solidus
temperature can be used
to remove nonequilibrium
phases and concentration gradients from
microsegregations.
Lever solidification in a ternary alloy is, in
general, similar to that of a binary alloy, but,
due to the added degree of freedom, a threedimensional
representation is nec
essary. The
phase boundaries that are crossed during cooling
and the sequence of phase formation can
be read from a temperature

concentration section
(e.g., an isopleth, a vertical section through
Fig. 5a or b). However, such a section provides
only very l
imited information on the compositions
of the individual phases or phase fractions,
since tie lines are rarely in the same
plane as the section. A temperature

concentration
section is ill suited to obtain information
for Scheil
solidification; more information can
be obtained from the liquidus projection, which
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