Sub T 1.2 Thernodynamics for phase diagrams ...

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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