Phase transformations
Coefficient: 5; Credits: 3
Teachers: Prof. HAMANA Djamel
HAYOUNEAbdelali
Lecture:/ Exercises 3 Hour(s) per week x 14 weeks
About this Module
This module teaches about the basic knowledge of phase transformations that occur
in different materials. The different types of phase transformations, their
mechanisms, kinetics and effects on the physical and mechanical properties of
materials, will be introduced.
Module content
I. Phase diagrams of metallic alloys
I.1- General thermodynamic aspects
I.2- Concepts of alloys, phases, solid solutions and phase diagrams.
I.3- Phase diagrams for metallic alloys.
I.4- Study of a phase diagram.
I.5- Usefulness of phase diagrams in various fields.
II. Solidification
II.1- The driving force for solidification.
II.2- The microstructure of metallic materials after solidification.
II.3- Solidification of a pure metal and an alloy.
III. Solid state Phase transformations
III.1- General classification of solid-state phase transformations
III.2- The driving force for a solid-state transformation (general characteristics).
III.2- Kinetics of solid-state phase transformations (Avrami's Law)
III.3- Precipitation in metallic alloys
IV. Study of phase transformations in steels
IV.1- Fe-C phase diagrams
IV.2- Phase transformations in steels
IV.3 - TTT and CCT diagrams
V. Experimental methods for studying phase transformations
V.1- Choosing an experimental method
V.2- Different experimental methods
Laboratory Practice:
Lab 1: Metallography and microstructures of various metal alloys.
Lab 2: The Pb-Sn phase diagram.
Lab 3: Determination of the composition of a solid solution by X-ray Diffraction
Assessment method
- Final exam (50 %)
- Continuous assessment (2) (20 %)
- Laboratory practice (30 %)
Chapter0 : Phase diagrams : Part 1 : Constituants, phases and
structures
Definitions :
1. Alloys :
A metal alloy is a mixture of one metal and another metal or non-metal. Ceramics can also be
mixed to form alloys.

-
Exemples :
Copper (Cu) and zinc (Zn), when mixed, form an alloy called brass.
Magnesia (MgO) and alumina (Al2O3), when mixed in equal proportions, form spinel.
Iron (Fe) and carbon (C) mix to produce carbon steels.
2. Constituants :
Alloys are generally obtained by melting together and mixing their constituents  constituents are
the chemical elements that make up the alloy.

-
Exemples :
In brass, the constituents are Cu and Zn.
In carbon steel, the constituents are Fe and C.
In spinel, the constituents are Mg, Al and O.
A binary alloy contains two constituents, a ternary - three, a quartenary - four, and so on.
3. Symbols :
Constituents are designated by capital letters: A, B, C,... or by the element's chemical symbol: Al,
Fe, Cu, Zn, C,...
4. Concentration :
An alloy is described by specifying its constituents and composition, i.e. the concentration of each
constituent.
The mass (or weight) concentration - wt. % - of constituent A is:
Wt % A 
Weight of A
x100
Weignts
of
all
constituen
ts

The atomic concentration (at. %) andthe molar concentration (mol. %) of constituent
A is :
at %A
number of A atoms (or moles)
x100
 numbers of atoms (or moles) of all constutuents.
5. Structure :
Alloys are generally obtained by melting and mixing the constituents in the liquid state, although
they can also be made by depositing the constituents from their vapor, or by interdiffusing them in
the solid state. Whatever the production method, a binary alloy can take one of four forms:
a) a single solid solution,
b) two separate, practically pure components,
c) two separate solid solutions,
d) a chemical compound and a solid solution.
How can we tell which of these shapes we are dealing with?
By examining the alloy's microstructure. To do this :
-
-
the alloy is cut to present a flat surface which is then polished, first with emery
papers of increasingly fine grain size, then with diamond pastes (on rotating felt discs) until
it is mirror like.
Finally, the polished surface is etched, usually with a weakly acidic or basic reagent, to
reveal the microstructure, i.e., the arrangement of grains and phases: a brass door handle
often reveals this arrangement, under the effect of attack by salts from hand perspiration.
Grain boundaries stand out because the reagent attacks them preferentially. Light is reflected on
these planes, so that some grains appear light and others dark, depending on the intensity of light
reflected in the direction of observation.
Phases can also be distinguished, due to the attack of phase boundaries and because many
reagents are designed to attack one phase preferentially to others, causing a difference in contrast
between phases. Example: typical
Fig. 1: Micrograph Ni-50 % Cu alloy is typical.
The foundry alloy Al-11% wt. Si is typical of case b): the Si separates in the form of fine needles (=
1 µm diameter) of practically pure silicon in a matrix of pure Al (Fig. 2).
Fig. 2: Optical Micrograph of the Al-11% wt. Si alloy.
The Cd-60 % wt. Zn alloy illustrates case c): it consists of a zinc-rich phase containing 0.1% mass
Cd in solution, plus a cadmium-rich phase containing 0.8% mass. Zn in solution.
Fig. 3: Optical Micrograph of the Cd-60 % wt. Zn alloy.
Finally, the slowly cooled Al-4% wt. Cu alloy is typical of case d).
Fig. 4: Optical Micrograph of the Al-4 % wt. Cu alloy.
6. Phases :
All parts of an alloy with the same physical and chemical properties and composition are part of a
single phase.
Al-Si, Cd-Zn and Al-Cu alloys have two phases.
7. The constitution of an alloy:
The constitution of an alloy is described by:
a) the nature and number of the phases present;
b) the mass fraction of each phase (mass of the phase / total mass of the alloy);
c) the composition of each phase.
Alloy's properties (yield strength, toughness, oxidation resistance, etc.) depend on the critical
nature of its constitution and other structural features: the size (nm or µm or mm) and shape
(globular, rod-shaped, needle-shaped or plate-shaped) and distribution of the phases, which are
not described by the constitution. The constitution, size, shape and distribution of the phases
depend on the heat and mechanical treatments that the material has undergone.
Exemple:
Aluminum alloy with 4% copper by mass (duralumin, or dural for short) forms the basis of the 2000
series of aluminum-based alloys. Melting at around 650 °C, the Al solid solution can contain the
full 4% wt. of Cu in solution. At 20 °C, equilibrium solubility is only 0.1% Cu. If the material is
cooled slowly from 500 to 20 °C, (4% wt.- 0.1% wt.) = 3.9% wt. of copper separates from the
aluminum as massive inclusions of a new phase: not pure copper, but the compound Al2Cu. If, on
the other hand, the material is quenched (cooled very rapidly, usually by immersion in cold water)
from 500 °C to 20 °C, the copper atoms remain in solid solution and have not time to diffuse and
form the Al2Cu phase, and the remained solid solution; is a supersaturated solid solution.
At room temperature, diffusion is so slow that the alloy simply remains in this state, frozen in a
single phase. But if we heat it just a little to 160°C, for example, and keep it there ("artificial
ageing"), the copper begins to gather to form innumerable very small (nm-sized) platelet-like
particles, with a composition close to Al2Cu. When cooled to room temperature, this structure
becomes fixed once again. The yield strength and toughness of dural differ enormously under
these three conditions (slow-cooled, quenched and tempered and aged); the last confers the
highest yield strength and lowest toughness, as the small particles constitute a very effective
barrier to dislocations. It's important to be able to describe the constitution and structure of an alloy
quickly and accurately. So do the exercises below, even if they seem obvious to you.
8. Equilibrium conditions :
The Al - 4 %mass. Cu alloy in the previous example can exist at 20 °C in three different
states. Only one of the three - the slowly cooled state - is its equilibrium state, because after a
sufficiently long time, the others would evolve towards the same final state. At a given
temperature, therefore, there is an equilibrium constitution for an alloy, towards which it tends.
A sample is in its equilibrium constitution when, at a given and constant temperature T and
pressure P, it shows no tendency to change constitution over time. This is the stable constitution.
Alloys can exist in non-equilibrium states (for example, after rapid cooling in the previous case).
But it is always useful to know the equilibrium constitution. It is the basis for interpreting the
constitution of the actual alloy, and the most plausible non-equilibrium constitutions can frequently
be deduced from it.
9. State variable :
Ten different samples of identical composition, held at the same temperature and pressure, have
the same equilibrium constitution. Ten samples each with different compositions, or each
maintained at different pressures or temperatures, have ten different equilibrium constitutions.
The independent constitution variables, or state variables, are temperature T, pressure P and
composition c.
Example:
For the Al - Cu alloy described above:
Values of state variables
a) T = 500°C
P = 1 atm
Wt.Al = 96%
Wt.Cu = 4 %
Equilibrium constitution
Single-phase solid solution of copper in
aluminum
b) T = 20 °C
P = 1 atm
Wt.All = 96%
Wt. Cu= 4%.
Two phases: Al containing 0.1% Cu (solid
solution) and Al2Cu
These are equilibrium constitutions, as they are those reached after very slow cooling; slow
cooling allows time for equilibrium to be reached.
Some thermodynamic relationships exist between the state variables. In general, for a binary alloy,
we choose pressure P, temperature T and xB (% at. of constituent B) as independent state
variables. Although, in the following, we'll disregard P. The volume V and concentration x A( = 1 xB) can be deduced, so they are dependent variables. Of course, Wt. concentrations Wt. A and
Wt. B can also be used instead of atomic concentrations.
10. Equilibrium phase diagrams (or constitution diagrams):
The equilibrium constitution of an alloy can be determined experimentally by metallography and
other characterization methods such X-ray Diffraction and thermal analysis (described later). If
pressure is held constant at 1 atm, then the independent variables controlling the constitution of a
binary alloy are T and xB or Wt.B.
An equilibrium constitution diagram, or equilibrium diagram for short (or phase diagram), is a
diagram plotted along the T and xB (or Wt.B) axes. It shows the results of experiments to determine
the equilibrium constitution at each T and xB (or Wt.B).
Figure 5. shows the phase diagram of the lead (Pb) - tin (Sn) system (the range of alloys obtained
by mixing lead and tin, which includes tin solder alloys). The x-axis shows the composition Wight
of Sn (in wt%) at the bottom and xSn (in at. %) at the top. The vertical axis is the temperature axis
in °C. The diagram is divided into domains: regions where the number of phases is constant. In
non-hatched areas, the equilibrium constitution is single-phase: a liquid (top), or tin (Sn) with a
little lead in solid solution (right t), or lead (Pb) containing a little tin in solid solution (lef). In the
hatched areas, the equilibrium constitution comprises two phases: liquid plus solid (Sn), or liquid
plus solid (Pb), or solid (Pb) mixed with solid (Sn) (each containing a little of the other in solution).
Fig. 5: Phase diagram of the Pb - Sn system.
The phase diagram shows the equilibrium constitution of all binary alloys that can be prepared
from lead and tin, in all possible proportions, or, in abbreviated form, for the lead-tin system.
A binary system is a two-component system.
A ternary system is a three-component system.
11. The constitutive point :
The value of the state variables defines a point on the diagram: "the constitutive point" If this point
is given, then we can read the number of phases in equilibrium. We can also read their
composition and the quantity of each phase (this will be explained later). In this way, the diagram
provides us with the entire constitution of any given alloy at equilibrium. Refer to the definition of
constitution and check that this is the case.
12. One-component systems (unary phase diagram):
The equilibrium constitution of a one-component system is set by the variables P and T, so that
the equilibrium phases can be shown on a diagram with P and T axes. The diagram in figure 6
shows unary phase diagram for magnesium, showing the melting and boiling temperatures at one
atmosphere pressure.Iceand Ge, have similar unary phase diagram. Single-phase domains are
surfaces. Two phases coexist along lines. Three phases coexist at one point: the triple point.
Fig.6 : Schematic unary phase diagram for
Magnesium.
13. Cooling curve :
If a one-component system is cooled at constant
pressure and the temperature recorded as a
function of time, the resulting curve looks like the
one in figure below:
1 - steam;
2 - phase change from vapor to liquid (liquefaction or pouring);
3 - liquid;
4 - phase change from liquid to solid (solidification);
5 - solid.
The system is single-phase in regions 1, 3 and 5. Phase changes occur at temperatures
corresponding to regions 2 and 4. When a phase transformation occurs, the latent heat of
transformation is released (on cooling) or absorbed (on heating). For this reason, the temperature
remains almost constant during the transformation, producing plateaus 2 and 4; cooling continues
only when the transformation is achieved.
Phase transformations in the solid state (such as in iron) also have latent heats. These are
sometimes small, but with sensitive equipment for measuring cooling or heating curves, they can
be easily detected.
The plateaus of the cooling curve are called breakpoints. The two shown in the diagram are the
boiling point and the melting point of the material, at the given pressure.
14. Alloy composition
 General information :
- Dissolving properties of molten metals:
Metals in their liquid state have great dissolving power towards other metals in their solid state.
For example, mercury in its liquid state at room temperature can dissolve gold, which only melts at
1063 °C; aluminum, liquid at 660 °C, can dissolve copper, whose melting point of 1083 °C is much
higher.
The result is a more or less homogeneous liquid, known as a liquid alloy. This dissolving power
increases with temperature. Dissolving one metal into another is easier when both metals are in a
liquid state.
Dissolution may be accompanied by the release of heat (exothermic reaction), when the two
metals are chemically combined. In many cases, this heat release can be so great as to cause
dangerous liquid splashes during mixing.
There are three possible situations, when two metals are mixed in the liquid state:
a) Miscibility is complete: whatever the proportions of metals A and B mixed together, a single
homogeneous liquid is obtained. Ex: Pb-Sn, Fe-Ni, Cu-Au, Cu-Zn.
Fig. 7. Fe-Ni phase diagram.
b) Miscibility is zero: the two metals separate by density, producing two superimposed liquids.
Ex: Al-Pb.
Fig. 8. Al-Pb phase diagram.
c) Miscibility is partial: for certain proportions, e.g., a little B in a lot of A, we obtain a single liquid;
but for a proportion of B equivalent to that of A, there is saturation and we obtain two liquids
separated by density, one formed by A with a little B and the other formed by B with a little A. This
is known as a miscibility gap in the liquid state. Ex: Cu-Pb between around 50 % and 85% lead,
Cu-Cr between 38 and 92% chromium.
Fig. 9. Cu-Pb phase diagram.
Some of these alloys are used industrially.
To simplify the study of alloy solidification, we have only considered the first case, i.e. solidification
always takes place from a single homogeneous liquid, whatever the proportions of the two metals
present.
Three cases may arise during solidification:
a) The two metals, completely miscible in the liquid state, remain miscible in all proportions in
the solid state. Ex: Cu-Ni.
Fig. 10. Cu-Ni phase diagram.
b) Miscibility is zero in the solid state: Ex: Pb-Bi.
Fig. 11. Pb-Bi phase diagram.
c) Miscibility in the solid state is partial. This is the most frequent case.
- Notion of solid solution:
Depending on the attraction between dissimilar atoms, they may occupy statistically and arbitrary
ordered positions. When atoms occupy arbitrary positions, they form a solid solution (by analogy
with a liquid solution). One candistinguishtwo type of split solutions :
-
Primary solid solution:If the crystalline structure of the solvent metal is not modified by the
presence of foreign atoms,
Secondary splitsolution: If a different crystalline structure is formed.
The various types of solid solution can also be classified as: insertion solid, substitution solid
solution, superstructure, intermetallic combination of more or less variable composition.
- Insertion solid solution: is obtained when the dissolved atom occupies the interstices of the
solvent lattice. This interstitial solution can only exist when the atoms of the solute are very small
compared to those of the solvent.
According to Hume Rothery, the diameter of the dissolved atoms must be less than 0.59 times the
diameter of the base metal. Only elements such as hydrogen, nitrogen, boron, carbon and others,
whose atoms are very small, can form interstitial solid solutions.
Exemple :
Interstitial solid solution of C in CFC gamma iron: this is austenite, which can which can dissolve
up to 2% of C.
- Substitution solid solution:
This solution is formed when the atoms have approximately the same dimensions. Atomic
diameters must not differ by more than 14 % to obtain solid solutions of the first kind.
Atoms A and B have different diameters. As a result, the lattice of metal A is disturbed, and
stresses can develop, leading to variations in mechanical properties.
If the diameters of the atoms differ by more than 14%, deformations will be very significant and
saturation will occur (partial miscibility).
-
Superstructure :
In most substitutionalsolidsolutions, the distribution of atoms is disordered. However, in some
binary or ternary systems, dissimilar atoms may occupy symmetrical positions. In this later, we
have a superstructure or superlattice.
In these superstructures, the ratio of atomic concentrations of the constituents A and B is always
simple: AB or AB3.
Decreasingthe temperature favors the appearance of an ordered atomic grouping.Higher
temperatures, on the other hand, increase atomic mobility and promote disorder.
During these order-disorder transformations, mechanical, electrical and magnetic properties
change considerably.
Order-disorder transformations are reversible and progressive.
Solid solutions of appropriate atomic concentration may be in an ordered, partially ordered or
disordered state.
Ordered structure of the Cu3Au alloy
Ordered structure of the CuAu alloy
- Intermetallic compounds:
An intermetallic combination is a particular and original constituent of formula A mBn (m, n are
integers), with a well-defined m/n ratio. A distinction can be made between:
a) Defined compound of normal valence, where A and B atoms are covalently or electrovalently
bonded.
b) Abnormally valenced or non-stoichiometric compounds in which A and B atoms are bonded by
mixed metallic and covalent bonds.
The best known are the Hume Rothery compounds, which verify the following rule: the ratio of
peripheral electrons to the number of metal atoms has characteristic values. This ratio is called
electron concentration. The number of peripheral electrons is the element's valence.
Thermodynamic equilibrium: is described quantitatively by the free energy G = H - TS
H is enthalpy, which is a function of the internal energy of a system,
S is entropy, the disorder of atoms or molecules.
A system is at equilibrium if its free energy is at a minimum under a certain specified combination
of temperature, pressure and composition.:dG = 0.
Macroscopically: the characteristics of the system do not change over time but persist indefinitely;
in other words, the system is stable.
A change in temperature, pressure and/or composition for a system which is initially in equilibrium
will result in a change in the free energy  a spontaneous transition to another state for which the
free energy is lowered.
Phase diagrams:
Much of the information about controlling the phase structure of a system is given on a phase
diagram, also often called an equilibrium diagram.
There are three external parameters that can control the phase structure: temperature, pressure
and composition. Phase diagrams are constructed when various combinations of these
parameters are plotted against each other.
The simplest and easiest type of phase diagram to understand is the one-component diagram, in
which the composition is kept constant: pressure and temperature are the variables.
This one-component phase diagram (or unary phase diagram) is represented by a twodimensional graph P = f (T).
There are three different phases (regions): solid, liquid and vapor. These regions are delimited by
lines which are phase boundaries (marked aO, bO and cO). Each phase exists under equilibrium
conditions of temperature and pressure.
At any point on one of the curves marked aO, bO and cO, the two phases on either side of the
curve are in equilibrium (or coexisting) with each other. That is, the equilibrium between the solid
and vapor phases is along curve aO - similarly for the solid-liquid, curve bO, and the liquid-vapour,
curve cO.
When a boundary is crossed (as temperature and/or pressure is altered), one phase is
transformed into another.
Example: at atmospheric pressure, during heating, the solid phase transforms into the liquid
phase (i.e. melting occurs) at the point labelled 2 corresponding to a temperature of 0 ◦C.
The reverse transformation (liquid-solid, or solidification) takes place at the same point after
cooling.
Gibbs phase rule:
It describes the relationship between the number of components and the number of phases for a
given system and the conditions that may be allowed to change (e.g., temperature, pressure, etc.).
It has the general form: 2 + C = F + P (when temperature and pressure both can vary)
where: C: number of constituents; F: number of independent variables (T, P or composition); P:
number of phases in equilibrium.
Example:
Let’s consider the case of pure magnesium (Mg). Figure below shows a unary (C =1) phase
diagram.
Suppose we have a combination of pressure and temperature that put us at point A in the phase
diagram. At this point, all magnesium is liquid.
The number of phases is one (liquid)the phase rule tells us that there are two degrees of
freedom. 2 + C = F + P; therefore; 2 + 1 = F +1  F = 2
F = 2? We can change the pressure, the temperature, or both, and still be in an all-liquid portion of
the diagram. Another way, we must fix both the temperature and the pressure to know precisely
where we are in the liquid portion of the diagram.
Consider point B: the boundary between the solid and liquid portions of the diagram:
-
The number of components, C, is still one,
At point B, the solid and liquid coexist, or the number of phases P = 2.
From the phase rule Equation, 2 + C = F + P F = 1: there is only one degree of freedom.
If we change the temperature, the pressure must also be adjusted if we are to stay on the
boundary where the liquid and solid coexist. On the other hand, if we fix the pressure, the phase
diagram tells us the temperature that we must have if solid and liquid are to coexist.
At point X, solid, liquid, and vapor coexist: C = 1, P = 3  F = 0: we have no degrees of freedom;
all three phases coexist only if both the temperature and the pressure are fixed. A point on the
phase diagram at which the solid, liquid, and gaseous phases coexist under equilibrium conditions
is the triple point.
Binary phase diagrams:
Another extremely common type of phase diagram is one in which temperature and composition
are variable parameters, and pressure is held constant (1 bar).
Binary phase diagrams are maps that represent the relationships between temperature and
equilibrium phase compositions and quantities, which influence the microstructure of an alloy.
Many microstructures develop from phase transformations, the changes that occur when
temperature is altered (usually during cooling). This can involve the transition from one phase to
another, or the appearance or disappearance of a phase.
Binary phase diagrams are useful for predicting phase transformations and the resulting
microstructures (equilibrium or non-equilibrium).
Liquidus and Solidus Temperatures:
We define liquidus temperature as the temperature above which a material is completely liquid.
The solidus temperature for an alloy is the temperature below which the alloy is 100% solid.
The temperature difference between the liquidus and the solidus is the freezing range of the alloy
The simplest binary phase diagram to understand and interpret is the Cu-Ni type: The temperature
is plotted on the ordinate, and the abscissa represents the composition of the alloy (in mass or
atomic percentage). The composition ranges from 0 wt.% Ni (100 wt.% Cu) on the horizontal left to
100 wt.% Ni (0 wt.% Cu) on the right.
Example:
Determine the degrees of freedom and the
constitution of the Cu-40 % Ni alloy at (a) 1300 °C,
(b) 1250 °C, and (c) 1200 °C.
CHAPTER I
General thermodynamic aspects
I.1 Introduction:
The increasing use of metals and alloys in industry requires their detailed study in both liquid and
solid states. The most widely used method for studying metals and alloys is still thermodynamics,
which provides comprehensive information on their equilibrium properties.
The equilibrium properties of metals and alloys can be successfully described using statistical
thermodynamics. Statistical thermodynamics establishes the link between the macroscopic
properties of metals and their microscopic characteristics.
The statistical-thermodynamic approach to study metals and alloys makes it possible to guess
their equilibrium properties over a wide range of temperature and pressure variations.
Determining the equilibrium properties of metals and alloys on the basis of statistical
thermodynamics supposes the possibility of calculating the microscopic characteristics using the
quantum theory.
Thermodynamics: complete information on equilibrium properties:
- Statistical thermodynamics: link between microscopic characteristics and macroscopic
properties.
- The statistical-thermodynamic approach to the study of metals and alloys determines their
properties over a wide range of T and P variations.
- A metallic crystal is made up of a set of atoms arranged in space according to simple geometric
laws and which, under certain conditions, is in equilibrium.
- If these conditions are modified, this "configuration" can change  The crystal is then said to be
transformed.
Example, at room temperature, iron has a centred cubic structure (Fe - α). When it is heated
slightly above 910 °C (Fe - ). The transformation is reversible.
Suppose an iron bar is subjected to a very small temperature gradient in the vicinity of
910°C (Fig. 1.).
910
Fig. 1
The two parts α and  of the iron are separated by an interface.
- Raising the furnace temperature while maintaining the same thermal gradient  We can see that
the iron crystal  grows at the expense of the iron α and that the boundary between these two
phases moves to the left (Fig. 2).

910
Fe-
Fe- α
Fig. 2
After a certain time, the  crystal will have completely invaded the test tube. The transformation will
be total, and it is said to be "heterogeneous" because, while it is taking place, two phases are
present.
The choice of atoms between the α and  structures therefore only takes place at the interface. In
fact, only the atoms at the interface can have this choice.
I.2 The kinetics of a transformation:
Thermodynamics shows that if a system is in equilibrium at a given temperature T and pressure P,
its free energy F is minimal:
dFT,P = 0
If the system is not in equilibrium, under the imposed conditions, it tends to return to the state of
equilibrium  Its free energy decreases during the transformation: dFT,P< 0 and the particles that
make it up change configuration.
Example: The case of Fe (Fig. 3); at 910 °C the free energies of the α and  forms are equal. But
above this temperature: Fα> F which proves that Fe-α cannot exist above 910°C in the equilibrium
state because its transformation into Fe- leads to a decrease in the free energy.
Similarly, below 910°C, Fe- cannot exist in the equilibrium state because its transformation into
Fe-α leads to a decrease in the free energy.
However, when a system returns to a state of equilibrium, not all the particles undergo the
transformation simultaneously. Otherwise, at each instant, all the particles would be in an
intermediate configuration and neither the initial nor the final configuration could coexist during the
transformation, which is contrary to experiment since they can be detected.
Therefore, one can say that at any given moment, a small, unmeasurable fraction of the total
number of atoms is involved in the transformation, while the vast majority is already transformed or
untransformed. The atoms involved in the transformation, at a given moment, are those with a
high kinetic energy.
Fig.3: Variation of the free energy of Fe close to 910 °C.
1.3 Activated transient states:
When an atom evolves from an initial equilibrium state to a final one under isothermal conditions, it
passes through a series of intermediate states. As the free energies of the two configurations are,
by definition, minimal, the free energy of an atom or group of atoms first increases during the
evolution to a maximum, and then decreases towards its final value.
Thus, for a given temperature, Figure I.4 shows the variation of F as a function of the amplitude of
the transformation x. The atom that reaches the maximum free energy F *A is then in a transitional
state known as the "activated state", which is unstable because it can either return to the initial
state, or evolve towards the final state, with a much lower free energy.
Fig. 4: The variation of F as a function of the amplitude of the transformation x.
For an atom to undergo the transformation, its free energy must therefore reach at least a value
equal to F*A. the difference:
FA = F*A– Fi: is the "activation energy" of the transformation.
The driving force of the transformation is: F = Ff - Fiand is negative.
There is no connection between these two quantities. The existence of a driving force is necessary
for the transformation to take place, but it is not sufficient.
The additional free energy that enables the atom to overcome the thermodynamic barrier of
transformation will be provided by "thermal fluctuations". Let's be clear about the meaning of this
term: the distribution of thermal energy among a set of atoms is not uniform. Collisions resulting
from the random motion of particles produce large variations in the energy of individual atoms, and
fluctuations in the energy of a given particle over time. Those whose energy is greater than F* A
undergo the transformation. The others have to wait until they receive the necessary activation
energy.
We can therefore predict from the above that the speed of transformation will depend above all on
the value of the activation free energy.
1.4 Internal energy of activation:
This quantity is linked to the free energy of activation by the relation:FA = EA - TSA
Activation energy is the difference between the internal energy of an atom in the activated
transition state (F*A)and its initial internal energy (Fi). In the internal energy of an atom associated
with the other atoms in the crystal, a distinction must be made between the potential interaction
energy of the atoms, which depends on bonding forces, and the kinetic energy due to the atom's
thermal vibrations.
The equilibrium of an atom in its initial and final states forces it to occupy positions of minimal
potential energy. As a result, this magnitude must vary during the transformation according to a
curve similar to that shown in figure I. 4. But there is a crucial difference to note: E F can be lower
or higher than EI depending on whether the transformation is exothermic or endothermic, while F F
is always lower than FI.
It is currently impossible to calculate EA, Fortunately, the empirical activation energy can usually
be obtained from macroscopic experiments. In the case of the allotropic transformation of iron,
however, this is impossible, as the transformation is too rapid.
1.5 Kinetic energy distribution:
Statistics are used to determine the most probable distribution of energy of the particles of a
system. This is the distribution that corresponds to the greatest number of possible distributions,
since it is the one that can be obtained by the greatest number of possible means.
On the basis of this hypothesis, we demonstrate that the fraction of atoms with a thermal energy
equal to or greater than a given value EA is the temperature T:
n
EA
 exp( 
)
N
K BT
KB Boltzmann constant = 1.98 cal/mol/K
For an atom to pass from one phase to another, it must have a thermal or kinetic energy at least
equal to the activation energy EA.
We can then look for an expression for the speed of the single process that constitutes the
passage of an atom from the  structure to the  structure, i.e. the fraction of the total number of
atoms that reach the final configuration per unit of time dy/dt (not taking into account the
simultaneous reverse transformation).
This is proportional to :
- The frequency of vibration, which is assumed to be the same for all atoms;
- The probability of an atom having a thermal energy at each vibration at least equal toat least
equal to EA, i.e.: exp( 
EA
)
K BT
- The probability p that, when an atom has sufficient energy, it will satisfy thegeometrical
conditions required for the transformation. For example, in the case under consideration, p will
vary from 1/8 to 1/12, since in C.C and C.F.C systems the number of nearest-neighbor sites of a
given atom is equal to 8 and 12 respectively. Thus: dy / dt  pvexp(
EA )
K BT
Chapter II: Crystal growth and solidification
II.1 Introduction:
In principle, any substance can exist in three distinct physical states: solid, liquid or gas. It is
the balance between cohesive energy (bringing the atoms together) and thermal energy (tending
to separate them) that determines the physical state.
The thermal energy that results from the continuous movement of atoms is proportional to
the absolute temperature T (in Kelvin): Eth = kBT, kB being Boltzmann's constant (k = R/NA: perfect
gas constant R / Avogadro number NA), while the cohesive energy is independent of it to a first
approximation. This explains the transition of structures and states of matter with temperature and
Figure II.1 sketches this evolution. When the thermal energy is high relative to the cohesive
energy (high temperature), any structured or ordered state of the atoms is excluded.
Fig. II. 1: Sketch of the evolution of the degree of organization of atoms with increasing temperature
Matter exists in a completely disordered form in the gaseous state, the limit of which is
represented by the perfect gas (e.g. neon at normal pressure: 1 atmosphere at room temperature).
The liquid state is an intermediate state between the gaseous state and the crystalline solid. If a
gas is compressed to a temperature below its so-called critical temperature, it can be seen that at
a certain pressure the system becomes heterogeneous and the atoms or molecules, as the case
may be, organize themselves into much denser islands. During this transformation, known as
condensation, the atoms or molecules suddenly come closer together and contact is established
between them, forming a condensed state from the gas: the liquid. At the same time, the system's
kinetic energy and entropy decrease, and heat is released.
While the liquid state is therefore characterized by short-range order, the solid state is
characterized by long-range order.
Solidification occurs when any material changes from a liquid to a solid state, regardless of the
structure in which it is found. If it is crystalline, it is said to have crystallized.
Liquid-solid (solidification) and solid-liquid (fusion) phase transitions are certainly among the most
widespread in nature.
The properties of a solid (in particular it’s mechanical and thermal properties) depend very much
on the kinetics of solidification and more specifically on the size of the microstructures that form
during the phase transition.
However, it should be pointed out that we are still a long way from having satisfactory theories to
describe and explain the mechanisms involved in melting and solidification, even though they have
been known and used for a very long time.
II.2 Crystallization:
This is the transition from the liquid state to the crystalline state. It occurs when the substance
changes to a more stable thermodynamic state with a lower free energy (thermodynamic potential)
F.F = H - T S
Where H - Total energy of the system
T - The absolute temperature
S - Entropy.
Fig II.2: Variation in the free energy of a metal in the liquid (Fl) and solid (Fs) states, as a function of
temperature.
- At Tm (melting temperature):Fl = Fs (equilibrium between the liquid and solid phases).
- At this temperature the crystallization process has not yet begun.
Crystallization only occurs when the temperature of the metal is lower than T m, i.e. under
conditions of "supercooling" of the metal (actually called supercooling).
T = Tm -Tc is called the degree of supercooling.
Where TmandTcare the melting and crystallization temperatures respectively.
Figure II.3 shows the curves illustrating the crystallization of pure metals during cooling
Fig. II. 3: Cooling curve during the crystallization of a metal.
Crystallization takes place at lower temperatures than those corresponding to equilibrium
solidification(Tm -T). T depends on the nature and purity of the metal: the purer the liquid metal,
the more suitable it is for supercooling (undercooling).When very pure metals solidify, the level of
supercooling T can be very high.
Ex: For Sn we obtained T = 118°C.
and for SbT = 135°C.
However, in most cases the supercooling level does not exceed 10 to 30°C.
For a low cooling rate,T is low and crystallization occurs at a temperature close to that of the
equilibrium state (Tc close to Tm).At Tc the curve forms a plateau (a halt in the decrease in
temperature) which indicates the release of heat (called the latent heat of solidification).
The degree of supercoolingT increases as the cooling rate increases.
Crystallization begins with the formation of nuclei (crystallizationcenters) and continues with their
development and increase in number.
The figure below (Fig. II.4), shows that a liquid alloy whose supercooling has fallen below Tm has
many sectors in which stable crystalline nuclei, known as critical crystals, form and are capable of
growth.
- As long as the crystals in formation grow freely, their geometric shape is regular. As soon as they
come into contact with others, their shape is disturbed. Growth only continues in directions that
allow free access to the "feed" liquid, which leads to an irregular shape of the crystals, hence the
name crystallites or grains (Fig. II.4).
Fig. II.4: Diagram of the crystallisation of a metal.
II.3 Spontaneous formation of germination centers:
Crystallization gives rise to many complex phenomena. It is especially difficult to imagine
the primary stages of the process, when the first small crystal forms in a liquid and becomes the
crystallization center.
To understand the conditions under which this occurs, we need to have a clear idea of the
structure of the parent phase, i.e. the liquid metal. The models of the crystalline and liquid phases
already discussed above ( Fig. II.1):
- Liquid metal: the arrangement of the atoms is not as chaotic as in a gas, but their arrangement is
not as regular as that of a solid crystalline body, where the interatomic distances and orientation
(angle ratios) remain constant at large distances (long-range order).
In a liquid metal only the short-range order is perpetuated, and the atoms only retain their regular
arrangement over very short lengths.
Intense thermal agitation means that short-range order is dynamically unstable: once microvolumes with a regular arrangement of atoms have appeared, they may exist for a short time, then
dissociate and appear again in another elementary volume of the liquid.
As the temperature T decreases, short-range order increases and the size of these volumes
increases.
At a temperature T close to the melting temperature Tm, a liquid metal can be the site of small
groupings of atoms stacked in a similar way to crystals. Such groupings are called phase
fluctuations.
Crystallization produces nuclei of various sizes. The development of a nucleus is only possible if it
has reached a defined size from which its subsequent growth leads to a decrease in free energy,
which can be demonstrated as follows:
During crystallization, the free energy of a system decreases by VΔf (a certain volume of the liquid
metal changes to the solid state) and grows by S (formation of an interface boundary).
The general change in free energy can be translated by the followingexpression:
ΔF = - VΔf + S
Where
V - volume of the formed nuclei
S –It’s surface
 - the surface energy between the solid and liquid phases.
f = (Fl – Fs) where Fl et Fs are the free energies / unit of volume of liquid and solid phases
respectively. For a spherical nuclei :
ΔF = - 4/3 πr3 mΔf + 4 πr2 m 
Where r - is the radius of the spherical cluster
m - the number of the total nucleus
As the equation shows, the smaller the nucleus, the greater the ratio between its surface area and
its volume, and the greater the proportion of the total energy that corresponds to surface energy.
If we want to calculate the critical radius corresponding to the maximum of ΔF, we need to
cancel out the derivative of ΔF with respect to r:
d(ΔF)/dr/ (r =rc)= 0
So :
- 4 πrc2 mΔf + 8 πrc m  = 0
4 πrcm ( rcΔf – 2  ) = 0
Hence:rcΔf – 2  = 0
And: rc = 2  / Δf
* This expression is only true for values of Tc that are not very large.
AS can be seen in the figure II.5:
Fig II.5: Variation in the free energy of a metal during nucleation as a function of crystal size.
* If r<rc (critical radius) ΔF increases when r increases (surface energy increases Δf more than
volume free energy decreases it)  a nucleus of radius r<rc cannot therefore grow and will
dissolve in the liquid metal.
* If r>rc, the nucleus becomes stable and able to grow because ΔF decreases as r increases.
The formation of the critical nucleus corresponds to the maximum increase in free energy, which is
equal to one third of the work absorbed by the formation of the interface:
ΔFc = 1/3 Sc
This means that when atoms go from the liquid state to the solid state, the reduction in free energy
by volume is not enough to ensure the formation of a critical nucleus (it only compensates for 2/3
of the energy released by the formation of the nucleus joint).
So where does the energy needed to form a critical nucleus come from?
Its formation is favoured by the irregular distribution of energy between the atoms of matter.
At any given temperature T, the energy of most atoms is equal to a certain average value.
However, in small volumes of matter there are always a certain number of atoms whose
energy is higher or lower than this average: these fortuitous and temporary deviations from the
average value of the energy of isolated atoms or groups of atoms for a given temperature T are
called energy fluctuations. It is these fluctuations that provide the energy necessary for the
appearance of a critical nucleus.
Remarks:
- At a temperature T close to Tm, rc of a nucleus must be very large and the probability of its
formation very small. As the degree of supercooling ΔT increases, the value of ΔF increases.
- When ΔF increases (or Tc decreases), rc becomes smaller, as does the work required for its
formation.
II.4. Growth of nuclei:
The growth of nuclei is due to the passage of atoms from the "supercooled" liquid to the
crystalline state. A crystal grows in layers: the thickness of each layer being monoatomic.
Crystals grow according to one of the two basic processes below:
1.
Formation by fluctuation of a nucleus (i.e. a nucleus of monoatomic thickness) on the flat
faces of the crystal (Fig. II.6). The size of the two-dimensional nucleus must not be less
than the critical size. A smaller seed size compromises its stability, as the formation of
an additional interface increases the free energy of the system.
(a)
(b)
Fig.II.6: Atomically smooth solid/liquid interfaces with atoms represented by cubes. (a) Addition of a
single atom onto a Hat interface increases (b) Addition to a ledge (L).
2.
Growth of a two-dimensional nucleus by the addition of atoms from the supercooled
liquid. Once the two-dimensional nucleus has appeared on the flat surface, the further
development of the layer in formation becomes relatively easy, as convenient areas are
formed by the attachment of atoms from the liquid.
Fig. II.7: Spiral growth: a screw dislocation terminating in the solid/liquid interface showing the
associated ledge.
As can be seen in figure II.6, Addition of a single atom onto a Hat interface increases the number
of 'broken bonds' by fourthe interfacial energy, will be increased There is therefore little prob
ability of the atom remaining attached to the solid and it is likely to jump back into the liquid.
However,addition to a ledge (L) only increases the number of broken bonds by two, whereas at a
jog in a ledge (J) there is no increase The development of a crystal is made easier by the fact
that its faces are not perfect planes: the defects on the faces of the crystal are varied and in the
form of gradations.
A growing crystal always has dislocations. A screw dislocation, by opening out at the surface,
forms a step which makes it easy to attach atoms from the liquid (Fig.II.7).
Example: Growth of single crystals of Mg, Cd and Ag in the form of a spiral.
II.5. Number of crystallisation centres and crystal growth rate :
The volume of crystallised metal and the rate of its crystallisation are defined by :
1 - The germination power, i.e. the number of nuclei (NG) formed per unit of time and volume,
2 - The nuclei growth rate (VC), i.e. the increase in the linear dimensions of the crystal per unit of
time (mm/s).
The greater the germination
power and the rate of nucleus
growth,
the
faster
the
crystallisation
process.
At
equilibrium temperature (Te), the
number of nucleus and the
growth rate are zero, so
crystallisation does not take
place (Fig.II.8).
Fig. II.8:The Variation of Nucleation and Growth rates as function of temperature.
As the degree of supercooling increases, NG and VC increase to reach a maximum at a certain
supercooling value and then decrease. Under these conditions, N G reaches its maximum at lower
supercooling values than VC.
This variation in NG and VC as a function of the degree of supercooling can be explained as
follows:
As the degree of supercooling increases, the difference f between the free energies of the liquid
and solid increases, which contributes to an increase in the rate of crystallisation, i.e. the
germination power and the growth rate of the nuclei.
For highdegrees of supercooling (at low temperatures), there will be a decrease in the speed of
diffusion (of the diffusion coefficient D) which makes the germination and growth of seeds more
difficult. As a result, the number of nucleus and the speed at which they grow decreases.
At very low temperatures (higher degree of supercooling), the agitation of atoms due to diffusion is
so low that the great gain f in free energy by volume during crystallisation is insufficient to ensure
the germination and growth of nucleus (NG = 0; VC = 0): In this case, solidification leads to an
amorphous state.
Note:
The very rapid cooling of metal droplets gives the metal a glassy state characterised by special
physical and mechanical properties.
- Grain size :
The greater the number of seeds, the slower their growth rate and the smaller the crystal that has
grown from the seed (metal grain).The number of grains (S) and therefore their size as a function
of NG and VC can be calculated from the relationship :
S = 1,1 (VC/ NG)3/4
When the degree of supercooling is low (low cooling rate), the number of nuclei is small and the
grain will be large.
As the degree of supercooling increases, the number of nuclei increases and the grain size of the
solidified metal decreases.
The size of a metal grain has a significant effect on its mechanical properties, especially its
ductility and plasticity, which are more pronounced when the grain is small.
The degree of supercooling is not the only factor that determines grain size: there is also the
heating and casting temperature of the metal, its chemical composition and, above all, the
presence of impurities. The influence of all these factors is very strong.
II.6. Types of germination :
a) - Homogeneous germination: The formation of nucleus in a liquid metal which takes place in the
manner described above is called spontaneous germination. This spontaneous germination based
on phase and energy fluctuations can only occur in a very pure liquid metal. It can be considered
as homogeneous germination.
b) - Heterogeneous nucleation: Industrial metals always contain a large number of various
additions (oxides, non-metallic inclusions, etc.) which, under certain conditions, make nucleation
easier:
1. The addition must have a higher melting temperature than the base metal,
2. The surface tension at the interface between the addition and the nucleus (1) must be lower
than the surface tension at the interface between the nucleus being formed and the liquid metal
(2).
(1) can be less than(2)when the crystal lattices of the addition and the metal being crystallised
are the same, the difference between the crystalline parameter of these lattices being negligible (it
does not exceed 9%).
The more crystallisation centres there are, the finer the grain (and the finer the number of
additions). This type of germination is known as heterogeneous.
The crystallisation process usually starts from the walls of the mould, which play the same role as
the inclusions.
Remarks:
* Overheating a metal to a temperature significantly higher than Tm contributes to grain coarsening
(because additions can be dissolved).
* Modification is the addition of special modifiers introduced into a liquid metal to obtain a fine
grain following the mechanism described above:
- for aluminium alloys we uses Ti, V, Zr
- for steel Al, V, Ti
During melting, these additions form refractory combinations (carbides, nitrides, oxides) which first
solidify in the form of minute particles constituting the first crystalline nuclei serving as
heterogeneous nucleation sites.
Solidification of a fully miscible binary alloy in the solid state:
Qualitative study:
The crystals formed are mixed crystals of either metal.
Assume an alloy formed from two metals A and B, such that the melting point of A is higher than
that of B (Fig.II.14) metal A will solidify first , taking some atoms of metal B with it This will
deplete the liquid of metal A, and subsequent crystals will be richer in B. This process continues
until solidification is complete. Consequently, the compositions of the liquid and solid phases
therefore vary throughout solidification.
Fig.II. 14: Solidification curves of pure elements and an A-25 % B alloy.
Moreover, since metal’s atoms can diffuse into each other in the solid state, differences in
concentration diminish over time. The higher the solid-state temperature, the faster is the diffusion.
When a single phase solidifies, a number of small crystals form in the liquid, growing almost
exclusively in the direction of the axes of a cube. Following this primary crystallization, particular
points on the axes of the cube may be the origin of secondary crystallization, and so on (Fig.
II.15). The branched crystals formed mainly during solidification are called dendrites. Each
dendrite grows from a single crystallization center
Fig.II. 15: The development of thermal dendrites: (a) a spherical
nucleus; (b) the interface becomes unstable; (c) primary arms develop
in crystalJographic directions «(100) in cubic crystals); (d) secondary
and tertiary arms develop”.
Fig.II. 16: Dendrites in an aluminum alloy (X50).
Consider the equilibrium diagram for Cu-Ni alloys, and let's take a closer look at the 70 % Ni alloy,
at a temperature of 1350°C (Fig. II17) .
Horizontal rule:
At a given temperature, the compositions of the liquid and solid phases present are given by the
abscissas of the points of intersection of the horizontal corresponding to the given temperature,
with the liquidus and solidus. For example, for the 70% Ni alloy, at a temperature of 1350°C:
Fig. II.17: Cu-Ni phase diagram.
The liquid phase (point A) contains 56% nickel,and the solid phase (point B) contains 84% nickel.
N.B: at 1350 C°, the respective compositions of the liquid and solid phases will be the same for all
alloys with nickel contents between 56 and 84%.
Inverse segment rule:
The proportion by mass of the liquid phase, at a given temperature and for a given alloy, is
represented by the segment MB touching the solidus; the proportion by mass of the solid phase is
represented by the segment MA touching the liquidus (strong line).
Practical use of the diagrams:
Consider the equilibrium diagram for Cu-Ni alloys. When solidification begins, the first solid
nucleus to appear will have a composition of s1. During solidification, the composition of the solid
phase will change from s1 to c, and that of the liquid from c to L2. The composition of the last liquid
drop is L2(Fig. II18).
Fig. II.18: Evolution of phase’s composition during solidification.
If diffusion is relatively slow, there will be chemical heterogeneity between the initial crystalline
skeleton and the inter-dendritic spaces, which is detrimental to the alloy's mechanical properties.
Fig. II.19: Solidification intervals.
So far, we've assumed total diffusion, which is rarely the case, as industrial cooling rates are
generally too fast for normal equilibrium conditions.
In the case of zero diffusion (Fig.II. 20) , the solidus would be represented by curve3. The actual
solidus, curve 2, usually lies between the most unfavorable solidus (zero diffusion) and the
thermodynamic solidus, curve1.For steels, for example, the actual solidus can be as much as
200°C below the theoretical solidus.The heterogeneity resulting from this partial diffusion is called
minor segregation.
Fig.II. 20: Real Solidus.
Solidification of a partially miscible alloy in the solid state:
Two metals crystallizing from a homogeneous liquid can, in some cases, form a homogeneous
solid phase known as a solid solution. Very often, however, two solid phases are formed: a solid
solution of A in B, and another of B in A. This is known as partial miscibility.
Reciprocal solubilities are temperature-dependent, and we can represent the domains of existence
of the phases and  in a binary diagram (Fig. II. 21).
At temperature 1 and for a low concentration of B in metal A, correspondsto a homogeneous
phase . But above a concentration x1 of atoms B, the solid solution is no longer homogeneous,
and separates into two distinct phases:
- a sol sol (network of A with some B),
- a sol sol (network of B with some A).
As the temperature rises, the reciprocal solubilities of the two metals increase. At the limit, from
the critical temperature c onwards, there will be a continuous transition from sol sol to sol sol,
which presupposes a relative identity between the lattices of the metals in question. This is known
as a miscibility gap in the solid state. If this miscibility gap is entirely located below the solidus, it is
bounded upwards by a critical point. More often, however, binary diagrams show miscibility gaps
that reach the solidus (Fig.II.22).
Fig. II. 21: Miscibility gap.
Fig. II. 22: Different cases of the Miscibility gap.
Case of two solid solutions with eutectic point:
Consider the Cr-Ni binary phase diagram (Fig. II.23),the cooling curves vary according to the alloy.
They include a change in curvature if saturation is not reached (10% nickel alloy), a change in
curvature followed by a plateau if saturation is reached (40% nickel alloy), a plateau only if the
alloy has eutectic composition (50% nickel alloy).
Fig.II.23: Cr-Ni equilibrium diagram.
This is in accordance with the phase rule. In fact, on the eutectic plateau, we will find :
Two constituents, Ni and Cr, (C = 2),three phases, the liquid phase and theand(P = 3)
So F = 0
There are no more degrees of freedom and solidification takes place at a constant temperature or
eutectic temperature. The length of the plateau is proportional to the quantity of eutectic
formed.Equilibrium at point E, the eutectic point, can only exist for a given temperature and a
given composition.
Liq E
sol sol  ssolsol s
This is the eutectic reaction, which corresponds in a way to a co-solidification of the phases,
generally resulting in a very fine aggregate. In short, solidification takes place in two stages. Take,
- 40% Ni alloy.
for example, the Cr
Fig.II.24: Solidification and microstructure of the Cr-40 % Ni alloy.
From 1475 °C to 1345 °C, germination and growth of  nucleus which grow steadily and whose
average composition changes from 20 to 32 % nickel (point).
At 1345 °C, the residual liquid of composition E (50 % nickel) solidifies isothermally to give the
eutectic aggregate+ , which surrounds the already solidified  phase and cooling ends normally.
Application of the inverse segment rule:
Let's take the CrNi diagram and look more specifically at the 40% nickel alloy.
Fig. II.25: Parts of the Cr-Ni phase diagram.
% eutectic= M as  0,400,32 x100 44%
Eas 0.500,32
0,500,40
x10056%
% proeutectic = ME 
Eas 0,500,32
It is also interesting to know the proportions by mass of and  in the eutectic aggregate. These
proportions will be calculated as follows:
E s 0,550,50
% in the eutectic

x10022% 
ss 0,550,32
0,500,32
in the eutectic= Eas 
x10078%
s s 0,550,23
Vérification :
% Ni = Ni in : proeutectic+ eutectic+ eutectic= 100 [0,56x0,32 +0,44(0,78x0,55+0,22x0,32)]
= 40% (composition de l’alliage étudié).
Case of two solid solutions with a peritecticpoint :
The peritectic point P lies outside the segment ssrespective saturation limits of metals A and B.
Note, as in the previous paragraph, the existence of a peritectic plateau, a peritectic temperature,
etc.
a peritectic reaction,
Fig. II.26: Equilibrium diagram with peritectic point.
The length of the peritectic plateau will be maximum for an alloy with a composition s which can be
accurately determined by plotting the TAMMAN diagram.
Peritectic solidification takes place at constant temperature in accordance with the phase rule:
Two components A and B, n = 2, three phases, , and the liquid phase, P= 3  F = 0
Application of the inverse segment rule:
For an alloy such as I, solidification begins at the temperature  with the crystallisation of sol sol.
The respective compositions of the solid and liquid phases will tend during cooling towards sand
P. The portions of sol sol and liquid Ppresent at the peritectic temperature are given by the
inverse segment rule applied to the segment sMP:
s s
value of the equilibrium composition.
s P s P
% liqP = M as <
Hence, there is insufficient liquid P
sol solreacts with all the liquid P to give sol soland we finally obtain a mixture + . For an alloy
such as III, the rule of inverse segments applies to the segment sNP.
s s
% liq P = N as 
s P s P
There is an excess of liquid in relation to the fraction of sol sols. All the sol solseacts with only
part of the liquid; below the peritectic temperature, we therefore obtain a mixture of soil.sol and
liquid.
Fig. II.27: Peritectic Solidification.
Appearance of a metallic combination during solidification:
Metal compounds can be divided into two categories:
- Daltonides, which form a relatively stable combination at ambient temperature, of formula A mBn,
with a well-defined melting point or known decomposition temperature;
- Berthollides, whose composition varies within wide limits, their melting point being poorly defined.
On the diagrams, daltonides are represented by verticals, while berthollides, which can be
assimilated to solid solutions of the second kind, are represented by more or less extensive
domains.
When the melting point and decomposition temperature coincide, whether for daltonides or
berthollides, the intermetallic compound is said to be stable, or a congruent melting compound.
When decomposition begins well before the melting point, the compound is said to be unstable or
non-congruent melting.
Stable compound:
Any defined compound behaves as a true physico-chemical individual, endowed with
characteristic physical and chemical properties different from those of the metals A and B of which
it is composed.Such a compound must be considered as independent in the phase rule.
Exercise:
1. Explain briefly what is meant by the following terms: (a) a eutectic reaction, (b) a peritectic
reaction.
The phase diagram for the copper- antimony (Cu-Sb) system is shown below. The phase diagram
contains the intermetallic compound marked “X” on the diagram. Determine the chemical formula
of this compound. The atomic weights of copper and antimony are 63.54 and 121.75, respectively.
2. This phase diagram contains two eutectic reactions. For each reaction: (a) identify the phases
involved, (b) give the compositions of the phases, (c) give the temperature of the reaction.
3. A Cu-Sb alloy containing 95 wt% antimony is allowed to cool from 650 °C to room temperature.
Describe the different phase changes which take place as the alloy is cooled and make labeled
sketches of the microstructure to illustrate your answer.
4. Sketch a graph of temperature against time for the Cu-95 % wt. Sb alloy over the range 650 °C
- 500 °C and account for the shape of your plot.
C
u
S
b
p
h
a
s
e
d
i
a
g
r
a
m
CHAPTER III: General classification of phase transformations and problems of
germination of a new phase
1. Introduction
Phase transformations, in particular crystallo-atomic transformations of the structure, occur when
temperature, pressure and the type, number and ratio of the components of the system, i.e. the
chemical composition of the alloy, vary.
All these factors are used in modern technology to obtain metallic materials with specific
properties. Great importance is attached to phase transformations due to temperature variations:
polymorphism, variation in solubility, precipitation and order in solid solutions.
Phase transformation, whether complete or partial (stopped at an intermediate stage), ensures the
optimum properties of the material. These properties are determined not only by the composition
of the phases and their properties, but also by the alloy's microstructure, i.e. the size, morphology,
dispersion and distribution of the phases, as well as the destruction of the alloy's basic crystalloatomic structure or grain substructure.
The purpose of certain heat treatment operations is to form the appropriate substructure. In
general, the temperature of the final heat treatment operation is higher than the operating
temperature of the material, so the material retains its structure and properties for as long as
possible.
The problem of studying phase transformations is made difficult by the fact that the mechanism
and kinetics of a phase transformation of the same type (allotropic transformation, precipitation,
solid solution order, etc) can be very different.
The analysis of phase transformations in alloys requires the consideration of a combination of
various mechanisms that occur one after the other or simultaneously.
The theory of phase transformations in the solid state uses the same laws as those demonstrated
and well studied during the study of condensation and crystallisation processes; however, two
factors specific to transformations in a crystalline medium must still be taken into account:
a) The elastic energy factor:
During the formation of the new phase in a crystalline medium, in addition to the variation of the
chemical free energy in volume (F2 - F1) v and the energy of the interface boundaries (∑ i Si(v)),
the variation of the free energy F must necessarily include the energy of the elastic stress field
(dEel/dv)v:
So:  F = (F2 – F1) v + (∑ i Si(v)) + (dEel/dv)v
The elastic energy factor together with the energy of the interface boundaries determines the
morphology, orientation and distribution of the new phase particles during their formation and
growth.
b) The factor limiting the mobility by diffusion of the atoms in the crystal and the
cooperative movement of all the atoms:
Limiting the mobility of atoms makes it possible to obtain metastable and absolutely unstable
states using the usual means of modern technology (temperature and pressure).
A sudden variation in pressure over a wide range (from a few tens to a few hundreds of
atmospheres) at low temperature, or a high cooling rate under conditions of limited atomic
mobility, makes it possible to achieve very high supercooling (or supersaturation), which
completely changes the course of the phase transformation.
The cooperative nature of the displacement of atoms, in the absence (or at low levels) of
relaxation processes at low temperature, gives rise to a particular mechanism for transforming the
crystallographic structure, known as martensitic transformation). The kinetics of martensitic
transformations differ greatly from the kinetics of other phase transformations.
2. General classification of phase transformations:
The basis for the classification of phase transformations can be the comparison of the crystalloatomic structure and chemical composition of the phases in the initial state with the phases
produced by the transformation. In this respect, the transformation occurs through the formation of
the new phase (or several new phases), which differ(s) in :
- The crystalline structure, i.e. the coordination of the atoms in the lattice (e.g. allotropic
transformation in metals and alloys, ordering).
- The chemical composition when the coordination of the atoms in the lattice is preserved
(demixing of the solid solution).
- Structure and composition (precipitation of supersaturated solid solutions, eutectoid precipitation,
order-disorder transformation).
Note: This classification does not include transformations linked solely to electronic structure
(several cases of magnetic transitions and the superconducting state).
3. General characteristics of solid state transformations:
The previous chapter was devoted to the study of the solidification of metals and alloys, and
therefore concerned systems in which solid and liquid phases coexist in a more or less perfect
state of equilibrium.
However complete an analysis may be of the conditions under which solidification takes place, it is
only very rarely possible to determine the structure and properties of an alloy near its ordinary
temperature, as it is often the site of transformations that take place at temperatures below the
temperature at which solidification ends. The influence of these transformations in the solid state
on mechanical and physical properties in general is so great that the art of the metallurgist largely
lies in the means at his disposal to direct them in a direction favourable to his designs.
To be successful, the study of transformations in metals and solid alloys requires a very different
approach to that adopted for the study of solidification. In the latter case, knowledge of the
system's physico-chemical equilibrium conditions was considered essential. Determining these
conditions by studying the cooling curves experimentally, and justifying them by applying the
principles of thermodynamics, made it possible to interpret most of the phenomena observed. The
deviations resulting from the fact that the systems sometimes deviated from the state of
equilibrium during their evolution never seriously altered the term of this evolution. Moreover, an
appropriate choice of experimental conditions meant that, in almost all cases, the conditions of
reversibility could be approached as closely as possible.
In solid alloys, on the other hand, precise knowledge of the state of thermodynamic equilibrium of
the system under given conditions provides little information about what its actual constitution will
be following a heat treatment carried out under these conditions. The rate of transformation in the
solid state is in fact subject to the influence of specific factors whose intervention very often
thwarts the evolution of the system as might have been predicted from the study of the equilibrium
diagrams. For this reason, it is essential to precede the examination of the different types of
transformations in solid-state alloys with a study of the kinetics of these transformations, a study to
which this chapter is devoted.
The general characteristics of these kinetics can be summarised as follows:
a) It is difficult to initiate reactions, which results in significant delays in time or in the temperature
scale, depending on whether the reaction is carried out at constant or variable temperature,
b) The initiation of reactions is subject to the influence of defects in the crystal lattice and,
consequently, to anything that modifies the number and nature of these defects (work hardening,
irradiation by particles, etc).
c) The subsequent evolution of the reactions is most often dependent on the diffusion of the
reacting species in the matrix lattice, so that the influence of temperature on the reaction rate is in
fact the result of its influence on the diffusion coefficients.
4. Initiation of reactions by germination and growth:
Of all the transformations that can occur in metals or alloys in the solid state, one very important
category is represented by those in which a new phase comes into being within an initially singlephase homogeneous medium.
From a purely chemical point of view, this situation can result from very different types of reaction,
in particular:
- Precipitation of an intermetallic compound in a supersaturated solid solution, during a drop in
temperature4. A good example of such a reaction is the precipitation of the compound Al 3Mg2 by
slow cooling of a homogeneous aluminium-C% magnesium (Al-c% Mg) solid solution at high
temperature:
Solid Solution rich in Mg  Solid Solution depleted from Mg + Al3Mg2
Allotropic transformation of a pure metal, or a homogeneous alloy, by an abrupt change from the
stability range of one form to the stability range of the other form. An example is the cooling
transformation of iron-nickel alloys with a low nickel content:
Solid Solution(CFC)  Solid Solution (CC)
- Eutectoid transformation of a solid solution, this time giving rise to not one but two new phases.
The example of the formation of the eutectoid (Iron  + Fe3C) within the homogeneous solid
solution with 0.8% carbon in iron  below 720°C is well known:
Solid Solution  Solid Solution  + Fe3 C
If we compare these different transformations from a morphological point of view, we can see that
they begin in the same way.
In the alloy, initially heated to a high temperature where it exists as a single homogeneous phase,
then cooled to a temperature where this phase becomes unstable, small crystalline individuals of
the new phase appear after a certain time in a dispersed state.
These individuals grow at the expense of the matrix until they completely invade it in the case of
an allotropic transformation.
In the case of precipitation in a supersaturated solution, they stop growing once the solution has
reached a new equilibrium state.
It is customary to distinguish two successive stages in this process: germination or growth,
although this distinction is largely artificial; it has the advantage of facilitating study by
distinguishing two sets of fundamentally different factors, and this is why we will adopt it in what
follows.
5. Classical germination theory:
The theory to be developed is based on the evaluation of the stability and growth potential of a
germ of the new phase as a function of the size of this germ.
When a phase A, initially assumed to be homogeneous, is heated to a temperature where it
becomes unstable and tends to give rise to a new phase B, the latter only becomes stable when it
has reached a certain critical volume, which can be evaluated as follows.
The free energy of formation of the initial nucleus must be negative for the nucleus to be stable.
This has two components, one effectively negative representing the variation in free energy by
volume V F V associated with the formation of phase B from A, the other positive representing
the variation in surface free enthalpy F s associated with the creation of the interface between
the two phases. Assuming the nucleus is spherical and of radius r, we have:
4
F V  r 3 ( F B  F A)
3
FS = 4 r2 
FB and FA are the specific free energies by volume of B and A, and  is the specific surface free
energy of the A/B interface, the overall variation in free energy associated with the formation of the
seed is written as :
4
F  r 3 ( FB  FA )  4r 2
3
The curve showing the variation of F as a function of r is shown in the diagram. It shows a
maximum for a value r0 of r given by the relationship:
dF
 4r 2 ( F B  F A)  8r  0
dr
 r 0
 2
 2

F A F B F ( A  B)
The particular value r0 depends on the temperature through the term F (A  B ). It represents the
critical radius predicted by theory.
dF
> 0 , which means that for any nucleus whose radius is less
dr
than the critical radius, an increase in radius must lead to an increase in the free energy of the
system, which is a highly unlikely event.
If r < r0 
when r increases,
dF
< 0, which means that any increase in the
dr
volume of the nucleus with a radius greater than r0 leads to a decrease in the free energy of the
system, i.e. an increase in stability.
On the other hand, for r >r0  when r increases
In short, if we consider at a given temperature a population of nuclei of any radius, those whose
radius is less than r0 will tend to disappear, while only those whose radius is greater than r 0 will
develop.
In this way, we can explain why it is so difficult for a new phase to appear in an initially
homogeneous system, and justify the persistence of supersaturated states in conditions that are
sometimes very far from equilibrium.
Fig. III.1: Schematic variation of the free energy F accompanying the formation of a nucleus of a new phase as a
function of the radius r of this nucleus. FV and FS represent the two components of F considered in the text.
The strict application of this theory would make it impossible for a new phase to appear in a
homogeneous system, as no nucleus could reach the critical volume by a path that could be
justified by macroscopic thermodynamics.
But we must take into account the existence in any system of fluctuations in configuration
associated with fluctuations in energy, which cause fugitive embryos of phase B to appear within
phase A.
At every temperature, in a state of dynamic equilibrium, there is a distribution of such embryos
which, when they exceed the critical radius, become stable nuclei destined to develop. The
influence of temperature on the germination process can be introduced into this theory as follows:
the free energy of the transformation, assumed to take place at a temperature T, can be written as
follows:
F( AB )  ( H B  H A )  T (S B  S A )
If we bear in mind that at equilibrium temperature TE,
F( A B )  0, S B  S A 
HB  H A
TE
and assuming that : S and F are independent of temperature, which is acceptable as long as T
remains close to TE, we arrive at the relationship :
F ( A B )    
H (E  ) H .

E
TE
 is the deviation of the system temperature from the equilibrium temperature. If we include this value
of F in the expression found above for the critical radius, we get:
r 0
 2E
 2

F ( A  B) F
Assuming that  is T independent, the variation of r0 as a function of  is hyperbolic. One of the
branches of the hyperbola corresponds to the germination of phase B in phase A observed, for
example, on cooling, while the other branch corresponds to the reverse germination of phase A in
phase B on reheating.
It is easy to conclude from the above that the probability of the formation of stable germs
increases with the difference  characterising the extent of supersaturation.
The subsequent step of assessing the actual number of nuclei as a function of  for a given
system is based on a number of hazardous assumptions and will not be discussed here.
6. Spinodal decomposition theory:
The previous theory attributed the origin of the barrier responsible for the delay in germination to
the existence of a positive energy for the creation of interfaces. A different analysis of the
phenomenon has been very popular for a number of years, and we will describe it briefly.
We consider a solid solution that is initially homogeneous at the temperature and that may give
rise, within a certain range of composition, to a dismutation leading to two boundary solutions, one
of which is richer, and the other less rich, in one of the elements than the initial alloy.
The free energy curve of the homogeneous phase corresponds to a general shape given in the
figure III.2. Let's consider different alloys.
Fig. III.2: Variation of the free energy of a solid solution at temperature T, as a function of composition, in the region
where there is a miscibility gap.
Alloy of composition C1: the homogeneous single-phase state represented for an alloy of
composition C1 by point H is clearly less stable than any two-phase state such as that
represented, for example, by point I leading to a dismutation into two phases of respective
compositions I' and I''. Of all the possible two-phase states for this alloy, the only truly stable state
is that represented by point J, the barycentre of points J' and J'' representing the two component
phases. The transition from state H to the final state J via a succession of states such as I always
leads to a decrease in the overall free energy of the system, if we disregard the energy required to
create the interface. It does not therefore require and is therefore not subject to any restrictions
linked to the distribution of fluctuations.
- Alloy of composition C2: for such an alloy, on the other hand, the transition from the unstable
homogeneous state L to the theoretically most stable state represented by point N, the new
barycentre of phases J' and J'', requires the initial solid solution to undergo significant
concentration variations in certain regions in the direction of J''. This cannot happen, as in the
previous case, by means of an evolution with a regular lowering of , since the system would be
forced under these conditions to pass through two-phase intermediate states such as M = M' + M'',
which are less stable than L. The transition from L to N can therefore only occur through the
appearance of significant fluctuations in composition, i.e. with a low probability. Hence the
intervention in such an alloy of a significant energy barrier, generating delays in the formation of
new phases.
- Spinodal curve: the limits between the alloys corresponding to the two previous types
correspond to the two inflection points K' and K'' of the free energy curve, i.e. the particular
F
concentrations CK' and CK'' for which
 0 . These characteristic concentrations, which
C2
obviously vary as a function of temperature, can be plotted in an equilibrium diagram and form
what are known as spinodal curves (Fig.III.3). The dashed line in this figure shows the two
branches of the spinodal curve relating to the miscibility gap that affects the series of goldplatinum solid solutions at low temperatures, which is homogeneous at high temperatures, and
whose limits are shown in a solid line. If the free enthalpy diagram of this figure corresponds to the
isothermal section T of the figure, we find the characteristic concentrations C J', CJ'', CK', CK'' on the
abscissae of the latter. If we keep to the assumptions made above, in particular if we neglect the
contribution of interfacial energy, all the alloys whose representative points are located inside the
spinodal curves can pass from the single-phase state to the two-phase state without the
intervention of additional energy, whereas those whose representative points are located between
the spinodal curve and the miscibility limit require the intervention of such energy.
Fig. III.3. Solid-state miscibility gap in the Au-Pt system. The corresponding spinodal curve was established from the
evaluation of the free energy of gold-platinum alloys.
7. Shortcomings of previous theories:
In explaining the previous theories, we have been led to neglect several factors whose importance
should not be underestimated. This observation severely limits the validity of the quantitative
conclusions which a student, unfamiliar with the difficulties of reaction kinetics in the solid state,
might be tempted to draw from the foregoing. This is why it is particularly instructive to identify
these simplifications and to investigate how they may influence the mechanism of the
transformations in each particular case.
First of all, it should be noted that the two approaches are contradictory only in appearance. While
the first attributes decisive importance to interfacial energy, while the second neglects it in the
simplified form in which we have presented it, it is also possible to introduce this energy into the
spinodal theory by translating the spinodal curve in the figure, which becomes the point-and-stroke
curve below the previous one. The difference between these two curves, which can vary according
to temperature, represents the elastic deformation energy due to the differences in parameters
between enriched and depleted regions.
It should also be noted that although classical theory originally introduced the notion of interfacial
energy, it overlooks the fact that this energy can vary considerably, for the same pair of associated
phases, as a function of various factors, including the following:
-
morphology of the dispersed phase within the original matrix,
the orientation relationship between the two adjoining crystal lattices,
stresses due to changes in volume following the transformation.
The second of these factors seems, in particular, to allow considerable variations in the interfacial
energy, which can take on extremely low values, and consequently lead to extremely low
activation energies, in certain cases where there is perfect coherence between the two lattices.
The concept of interface coherence thus goes a long way towards overcoming the schematic
nature of previous concepts.
We shall see later, in studying certain transformations and in particular the phenomena of
precipitation in supersaturated solid solutions, how fruitful these considerations are in interpreting
the abundant experimental data available to the metallurgist.
8. Homogeneous and heterogeneous germination:
All the considerations that have been developed up to now on the initiation of reactions in the solid
state assume that at the moment when the system is placed in conditions of instability likely to
favour the appearance of a new reaction, the initial phase has a chemical composition and a
crystalline structure that are identical at all points.
In this hypothesis, the location of the nuclei is not predetermined and their distribution is assumed
to be linked to that of the fluctuations, obeying a law of chance.
These peculiarities characterise what is known as homogeneous germination, although the term
is ambiguous in that it applies to a system which, if it was indeed homogeneous before the
reaction, is obviously no longer homogeneous once the reaction has begun.
The above situation does not generally arise in a real solid in which germination tends to take
place, preferably and primarily, where there are imperfections. We are then in the presence of a
heterogeneous germination process. The various imperfections capable of initiating
transformations are, in particular, inter-crystalline joints, polygonisation surfaces, isolated
dislocations, stacking defects, and possibly point defects and their clusters.
The surface of the metal, by introducing a discontinuity in the crystal lattice, can also be the
starting point for transformations in the matrix.
9. Description of the global behaviour of phase transformations using Avrami theory:
The kinetics of liquid-solid and solid-solid phase transformations that proceed by a germination
mechanism generally obey a transformation law proposed by Avrami. Avrami's treatment gives an
equation that can be used to calculate the degree of advancement of the phase transformation as
a function of time.
The appearance and development of a new phase  within a pre-existing phase  can be
imagined as follows: initially, a germ appears within the parent phase . This germ constitutes an
element of the  phase. In a second stage, the germs grow at the expense of the  phase,
contributing to the progress of the transformation (fig. III.4).
Fig. III.4: Isothermal variation of the volume fraction f of the transformed phase as a function of the logarithm of time t
according to the law f = 1 - exp (- K tn). f corresponds to the transformed fraction and (1-f) is the untransformed
fraction of the of  phase.
The new phase does not necessarily grow at the same speed in all directions in space. In the
simplest cases, we can imagine uniform growth (spherulitic growth) in three directions in space. A
germ that appeared at time t = 0 will have reached a volume equal to :
V = (4/3)  (vt)3 = (4/3)  v3 t3 = (4/3)  (vt) v2 t2= (4/3)  r (v2 t2)
v being the speed of growth and (vt) = r the radius of the sphere at time t
A  grain that begins to develop after a time t =  will have reache, after a time t >  a volume of :
V’ = (4/3)  v3 (t – )3
The new phase grows freely during the initial stages of the transformation and this behaviour changes at a
certain conversion rate, when the growing phases come into contact with each other. By taking this
characteristic into account and basing ourselves on the laws of germination and growth, we obtain a
general equation called the Avrami equation, which gives the conversion rate (volume fraction f) as a
function of the transformation time:
f = 1 – exp (- K tn)
K is a global transformation rate constant which includes the various factors involved in the equations
describing germination and growth (which are highly dependent on temperature) and n varies according to
the type of transformation from 1 to 4. Thus, in the case of spherulitic growth initiated by sporadic
(dispersed) homogeneous germination at speed I, the exponent n = 4, and we calculate that :
K = ( / 3) I v3
Knowing the law of variation of K as a function of temperature, we can calculate the time required to reach
a given conversion rate at a given temperature (1, 50, 80%, etc.).
Chapter V : Carbon Steels
1. Introduction:
Iron is one of the oldest known metals. Methods of extracting and working it have been practised
for 3000 - 4000 years, although the large-scale production of carbon steels is a development of
the 19th century. From these carbon steels (which account for 90% of steel production), a range of
alloy steels has evolved: the low alloy steels (containing up to 6% of chromium, nickel, etc.); the
stainless steels (containing typically 18% chromium and 8% nickel), and the tool steels (alloyed
with chromium, molybdenum, tungsten, vanadium, and cobalt).
We already know something about the transformations that take place in steels and the
microstructures that they produce. In this chapter, we show how they are essential in determining
the mechanical properties of steels. We restrict ourselves to carbon steels.
Carbon is the cheapest and the most effective alloying element for hardening iron. Carbon is
added to iron in quantities ranging from 0.04 to 1.7 wt% to make low-, medium-, and high-carbon
steels. The mechanical properties are strongly dependent on both the carbon content and the type
of heat treatment. Steels can therefore be used in a wide range of applications.
2. The allotropic varieties of iron
Iron exists in two different allotropic varieties, i.e. with two crystalline forms: CC and CFC:
Ferrite: at low temperatures and up to 912°C (A3), its atoms are arranged in a CC lattice: It is then
called Iron α. The solubility of carbon in α-iron is very low: 0.02%C maximum at 723°C, less than
0.01%C at 300°C.
-Austenite: at temperatures above 912°C and up to 1394°C (A4) the crystal lattice is of the CFC
type: it is called γ-Iron. Iron γ easily dissolves carbon: 0.8 % C at 723°C, 2.14 % C at 1147°C.
δ-ferrite: above 1394°C and up to the melting point at 1538°C, iron regains the CC structure of αiron: it is then called δ-iron. It dissolves carbon slightly better than Iron α (0.07%C maximum at
1493°C).
The magnetic transition: Up to 768°C (A2) Curie point, iron is ferromagnetic, beyond which it
becomes paramagnetic. The ferromagnetic character is said of a substance that can take on a
strong magnetisation.
3. Fe-C diagrams
Carbon has various possible behaviours:
The formation of insertion solid solutions: either in iron α (ferrite) or in iron  (austenite). The solid
solution of carbon in iron  is called ferrite .
Carbon is more soluble in austenite (2.1 wt.%) than in ferrite α (0.02 wt.%. ) because of the
more favourable interstitial sites between the iron atoms:
The interstitial sites in austenite represents 25% of the volume of the CFC cell and the maximum
radius of the site is 0.052 nm, the radius of the carbon atom is 0.077nm  the insertion of a C
atom introduces a strong distortion of the crystal lattice  few interstitial sites will be occupied by
C atoms.
Ferrite is even less compact than austenite (void volume 32%)  C is even less soluble in it (the
maximum radius of the sites is 0.06 nm).
The formation of cementite: iron is a carburigenic element  formation of a Fe3C carbide known
as cementite. It is unstable and can decompose according to the reaction: Fe3C  3 Fe + C.
This decomposition requires appropriate thermal conditions  the iron - cementite phase diagram
is called the metastable diagram.
Formation of graphite: (carbon for hexagonal structure) can result from the decomposition of
cementite  iron - graphite diagram (stable diagram).
As conclusion: Fe-C alloys can undergo two types of evolution:
The first produces a carbon-rich phase with the formula Fe3C called cementite. The
corresponding diagram is called metastable or cementite.
The second forms a carbon-rich phase that remains in the pure graphite state and has zero
miscibility with iron. The corresponding diagram is called stable or graphite. To obtain it, the
cementite must be decomposed by cooling at a very slow rate and adding a catalyst with high
graphitising power, such as silicon (Figure 1).
Fig. 1: Fe-C diagrams for steels and cast irons. Dotted line: stable Fe-C diagram. Solid lines: metastable
Fe -Fe3C diagram.
4. Fe - C phase diagrams with cementite:
From Figure 2 below one can distinguish three types of alloys: (i) pure iron, (ii) steels (between
0.008%C and 2%C) and cast irons (from 2%C to 6.67%C).
Fig. 2: Metastable Fe -Fe3C diagram.
4.1.
The constituents of steel:
-ferrite: Insertion Solid solution of C in Iron α, with CC structure, whose properties are:
- Low yield strength (y = 150 MPa)
- Low mechanical strength (m280 MPa and HV hardness  80)
- High ductility (A = 35 %)
- Density ρ = 7.86 g.cm-3.
-ferrite: Insertion solution of a few C atoms in δ-Fe. Its structure is CC. It forms in the temperature
range 1394-1538 °C and contains about 0.11% C.
Austenite: Insertion Solid solution of C atom in Fe , with CFC structure, the amount of C reaches
≈2 % C at 1145°C. It is stable only at high temperatures. Austenite is very ductile.
Cementite (Fe3C): Cementite is a chemically defined compound CCD. Its chemical composition
equals 6.67wt% C, in a metastable state.Cementite occurs in the form of lamellae or globules in
perlite or needles in white cast iron. Its properties are : Yield strength greater than 2000 MPa
(hardness close to HV = 700), Density ρ = 7.82 g.cm-3.
Perlite: Is a two-phase mixture of ferrite (88.3 %) and cementite (11.7 %) containing 0.8 % C. It
has a microstructure constituted of alternating fine lamellae (lamellar perlite). Interesting
mechanical properties:
- Ductility from ferrite (A = 10%),
- Yield strength from cementite,
- Mechanical strength: m (MPa) = 180 + 3800 λ-1/2, with λ interlamellar distance in nm)
- It is more stable to corrosion.
4.2. Transformation lines:
Acm: indicates the end of the dissolution, after dissociation, of the cementite in the austenite where
this exists.
A0 (210°C): indicates the temperature of the magnetic transformation of the cementite.
A1: specifies the end of the transformation when the austenite cools. Austenite no longer exists
below this line.
A2: (Curie point ≈ 768°C) specifies the temperature of the loss of iron α magnetism.
A3: specifies the end of the transformation of ferrite into austenite on heating; ferrite no longer
exists above this line.
-A4: indicates the end of the transformation of the austenite into ferrite δ and/or liquid on heating.
Austenite no longer exists above this line.
Ac3, Ac4... can be found to indicate that the point is drawn during heating, or Ar3, Ar4 when
cooling is involved.
Note:
- The transformation lines were determined by dilatometry, magnetic measurements, thermal
analysis and resistivity measurements.
- The transformations take place with a hysteresis that is a function of the speed and direction of
the transformation: the transformations take place at a lower temperature when cooling than when
heating. The lower the cooling rate, the lower the hysteresis.
4.3.
Transformations : The iron-carbon diagram contains three isothermal reactions
characterised by stages:
Peritectic at 1147°C: marks the minimum temperature at which the liquid can exist.
Eutectoid at 723°C (A1): marks the end of the transformation of pearlite into austenite on heating.
Above 723°C, pearlite no longer exists.
Eutectic at 1487°C: of negligible importance from an industrial point of view.
5. Equilibrium microstructure:
Eutectoid steels: This particular alloy corresponds to the eutectoid point of the Fe-C diagram with
a carbon content of ~ 0.8% (variable content depending on the presence of other alloying
elements):
- At high T, the alloy contains only the austenite phase.
-On cooling, there is no change until the eutectoid T (~ 723 °C) is reached. Below this T, all the
austenite is precipitated as pearlite (Figure 3)
Fig. 3. Microstructure of a eutectoid alloy.
Hypo-eutectoid steel: (8.10-3 at 0.8 wt. % C)
-High T: the alloy contains only the austenite phase
-On cooling, two phases coexist: α + .
-Below eutectoid T, the entire  phase is transformed into pearlite, while the ferrite phase
undergoes little change. These steels are the most widely used industrially.
Fig. 4. Microstructure of a Hypo-eutectoid alloy.
Hyper-eutectoid steel: (0.8 to 2% by mass of C)
- High T: the alloy contains only the austenite phase .
- On cooling, cementite is formed.
-Below eutectoid T, the remaining austenite is transformed into pearlite. These steels are rarely
used industrially (very brittle).
Fig. 5. Microstructure of a Hyper-eutectoid alloy.
6. Influence of alloying elements
Putting alloying elements into solid solution in iron modifies the position of points A3 and A4.
These elements are classified according to their influence on the position of these points.
The effects depend on the nature of the element:
-
CFC structural elements (Ni, Al, Co, Cu) have a high solubility in austenite and ferrite (only
Pb remains free).
All the other elements, whatever the crystal lattice (CC: Cr, W, V, Mo, Nb, etc.) HCP (Ti),
quadratic (Mn, etc.) give carbides (M or M6C or complex formulae).
Cr, Mo and W can also form intermetallic compounds with iron.
Alpha- and gamma-forming elements: (Figure 6)
α-gen elements: any element that stabilises the CC phase, it raises the temperature of point A3
and lowers that of point A4.
-gen elements: any element that stabilises the CFC phase, it lowers the temperature of point A3
and raises that of point A4.
Fig. 6: α-gen and -gen elements.
7. Microstructures produced by slow cooling of carbon steels:
a. Pure iron: slow cooling of the  phase (T < 912 °C) α phase germinate at the  phase grain
boundaries and the structure transforms into α phase (Fig. 7). 
Fig. 7: Microstructures during the slow cooling of pure iron from the hot working temperature.
b. Eutectoid steel (0.8% C): below 723°C, austenite begins to transform into pearlite: pearlite
colonies germinate at the  GB and the microstructure becomes entirely pearlitic (Fig. 8). When
pearlite is cooled to room temperature, the concentration of carbon in the α decreases slightly. The
excess carbon reacts with iron at the α / Fe3C interfaces to form more Fe3C.
Fig. 8: Microstructures during the slow cooling of a eutectoid steel from the hot working temperature.
c. Hypoeutectoid steel (% C < 0.8%): As soon as the system enters the α +  range, the  phase
transforms into the primary α phase, which forms at thr GB of  and grows as the steel is cooled
from A3 to A1 (Fig. 9)
At A1, the  phase gives rise to pearlite  the resulting microstructure is made up of primary α
and pearlite.
Fig. 9 : Microstructures during the slow cooling of a hypoeutectoid steel from the hot working temperature.
d. Hypereutectoid steel (% C > 0.8%): the room-temperature structure consists of primary Fe3C
and pearlite (Fig. 10).
Fig. 10: Microstructures during the slow cooling of a hypereutectoid steel.
Mechanical properties of normalized steels:
Both the yield strength and tensile strength increase linearly with carbon content (Figure 11). This
is expected due to the increase of the Fe3C phase content which acts as a strengthening phase.
On the other hand, the ductility, falls rapidly as the carbon content goes up because the α / Fe3C
interfaces in pearlite are good at nucleating cracks.
Fig. 11: Mechanical properties of normalized carbon steels.
9. Diffusive FCC  BCC transformation in Iron :
We have already seen that the speed of a diffusive transformation depends strongly on
temperature. The diffusive FCC.--> BCC transformation in iron shows the same dependence, with
a maximum speed at around 700 °C (Fig. 12).
The rate of transformation is given by: Rate (volume %s-1) = interface area x interface speed
The nucleation rate is critically dependent on temperature. In iron, grain-boundary nucleation will
not occur unless it is cooled below 910 °C. Simple calculation of the number of atoms forming a
critical germ of the BCC  phase, starting from the FCC  phase, at 908 °C and 910 °C gives that
the critical germ for the germination is constituted of 100 atoms however that for 908 °C contains
only 30 atoms. The chances of assembling this small number of atoms are obviously far greater
than the chances of assembling 100 atoms and grain-boundary nucleation is thus much more
rapid at 908 °C than at 910 °C (Fig. 12).
Fig. 12: How the nucleation rate, during the FCC  BCC
transformation in Iron, depends on temperature.
The sequence of the nucleation and growth of α iron from  iron at a temperature of 908 °C is as
follows : After an incubation period, nuclei of α start to form at the most favorable sites (usually
grain corners). These nuclei then grow at the interface speed. With time, more nuclei form and
grow. The volume transforming per second increases rapidly as (a) more nuclei form and (b) the
total area of the α /  interface increases.
However, nucleation may well stop at 20 - 30% transformation owing to “site saturation. Further
growth then takes place only at existing interfaces. The total interface area continues to increase
for a while because of the increasing size of the α grains, so the volume rate increases as well. As
α grains start to impinge on one another, the interface area begins to decrease and the volume
rate decreases until all the  has transformed (when it becomes zero). This sequence of nucleation
and growth leads to the characteristic “S-” shaped curve shown in Figure 13
Fig. 13: Schematic sequence (a) to (d) of nucleation and growth of α from  phase.
10. Time-Temperature-Transformation or TTT diagram:
The rates of diffusive transformations can be plotes in the form of time-temperature-transformation
(TTT) diagrams or “C-curves” (Fig. 14). Semischematic only, the 1% and 99% curves represent
the start and end of the transformation.
Fig. 14: The diffusive   α transformation in
iron: the TTT diagram or “C-curve”.
Consider the 1 % transformed” curve on the diagram. The curve gives the time required for 1 % of
the  phase to transform to α at various temperatures. Because the transformation rate is zero at
both 910 and - 273 °C (Figure 12), the time required to give 1% transformation must be infinite at
these temperatures. And because the transformation rate is a maximum at 700 °C (Figure 12), the
time for 1% transformation must be a minimum at this temperature, which is why the 1% curve has
a “nose” there. The same arguments can be appliedbto the 25 %, 50 %, 75 %, and 99 % curves.
Exercise 1:
You have received samples of the following materials: (1) Pure iron, (2) 0.3% C steel; (3) 0.8% C
and 1.2% C steel.
Draw the microstructure you would see under the microscope, assuming that all the samples have
undergone slow cooling from 1100°C to room temperature.
Exercise 2: The respective densities of pure iron and cementite are 7.87 and 8.15 gcm -3
.Calculate the volume fraction of α and Fe3C in pearlite.
11. The displacive FCC  BCC transformation in iron:
The rapid cooling, at a rate of about 105°C s−1 of  iron (obtained by maintaining the temperature of
a iron sample at 914 °C for example) to room temperature, allows to miss the nose of the 1%
curve (Figure 15). The TTT diagram tells us that we would expect to end up with FCC  iron at
room temperature and that phase can survive for years at room temperature before the diffusive
transformation could take place.
Fig. 15: Effect of quenching FCC. iron from 914 °C to room temperature at a rate of about 105°C s−1 on the FCC 
BCC transformation.
FCC iron at room temperature would be undercooled by nearly 900 °C, and there would be a huge
driving force for the FCC  BCC transformation.
In reality, below 550 °C the driving force becomes so large that it cannot be contained; and the
iron transforms from FCC to BCC by the displacive mechanism. Small lens-shaped grains of BCC
nucleate at FCC grain boundaries and move across the grains at speeds approaching the speed
of sound in iron (Figure 16). The lenses stop growing when they hit the next grain boundary. Note
that, when a new phase in any material is produced by a displacive transformation it is always
referred to as “martensite”. Displacive transformations are often called “martensitic”
transformations as a result.
Fig. 16: The martensitic transformation in iron.
As the lenses grow, the lattice planes distort, and some of the driving force for the FCC to BCC
transformation is removed as strain energy. Hence at a given temperature, fewer lenses nucleate
and grow, and eventually the transformation stops and to get more martensite, we must cool the
iron down to a lower temperature.
12. Martensite transformation in steels
Cooling of a “eutectoid” steel (0.8 % C) with a relatively slow cooling rate it transforms by diffusion
into pearlite (the eutectoid mixture of α +Fe3C).
According to Fig. 17a showing the TTT diagram for an eutectoid steel (0.8 % C), it can be seen
that we will miss the nose of the 1% curve, if we quench the steel at a cooling rate of about 200 °C
s-1. Note that if the steel is quenched into cold water, then not all the  will transform to martensite.
The steel will contain some “retained”  which can only be turned into martensite if the steel is
cooled below the martensite finish (MF) temperature of - 50 °C.
(a)
(b)
Fig. 17 : (a) : The TTT diagram for a eutectoid steel, (b) : the structure of martensite in a carbon steel.
For this steel, above 723 °C, the carbon atoms are able to squeeze into the space between the
iron atoms to form a random interstitial solution in the FCC iron. When the steel is quenched, the
iron atoms transform displacively to martensite. It all happens so fast that the carbon atoms are
frozen in place and remain in their original positions leading to the stretching the of the BCC lattice
along one of the cube directions to make a body-centered tetragonal unit cell (Fig. 17 b).
TTT diagrams for steels :
The TTT diagrams are determined by quenching a specimen to a given temperature, holding it
there for a given time, and quenching to room temperature. The specimen is then sectioned,
polished, etched, and examined in the microscope. The percentage of Fe3C present in the
sectioned specimen allows one to find out how far the   α + Fe3C transformation has gone. The
TTT diagram is built up by doing a large number of experiments at different temperatures and for
different times. In order to get fast enough quenches, thin specimens are quenched into baths of
molten salt kept at the various hold temperatures (Fig 18).
Fig. 18: How TTT diagrams are built up using quench – hold - quench sequences.
A quicker alternative to quenching and sectioning is to follow the progress of the transformation
with a high-resolution dilatometer: both α and Fe3C are less dense than  and the extent of the
expansion observed after a given holding time tells us how far the transformation has gone.
- At a high temperature, with little undercooling, the formed pearlite in the steel is coarse: the
plates in any nodule are relatively large and widely spaced.
- At slightly lower temperatures we get fine pearlite.
- Below the nose of the C-curve, the transformation is too fast for the Fe3C to grow in the form of
plates (as in the pearlite). It grows instead as isolated stringers to give a structure called “upper
bainite”.
Fig.19 : The formation of Upper Bainite (UB).
-
At still lower temperatures the Fe3C grows as tiny rods and there is evidence that the α
forms by a displacive transformation called lower bainite (Fig.19).
Fig.20: The formation of Lower Bainite (LB).
Exercise:
The figure bellow shows the TTT diagram for a coarse-grained, carbon steel of eutectoid
composition. Samples of the steel are austenitized at 850 °C and then subjected to the quenching
treatments shown on the diagram. Describe the microstructure produced by each heat treatment.