Materials Science and Engineering

UNIT 1
Materials Science and Engineering (MSE): An Overview

• MSE is an interdisciplinary eld focused on understanding and improving materials by exploring the relationships
between composition, structure, processing, and properties to develop advanced materials for practical applications.

Core Concepts:

• 🔬 Composition – Chemical makeup of a material.

• 🧱 Structure – Arrangement of atoms from atomic to microstructural level.

• ⚗ Synthesis – How materials are created from raw or synthetic sources.

• 🛠 Processing – How materials are shaped into functional components.

• 📈 Properties – Mechanical, electrical, thermal, and chemical behaviors.

•
• 🚀 Performance – How materials behave under real-world conditions.
fi
Classi cation of Materials

• There are di erent ways of classifying materials. One way is to describe ve

groups

• Metals and alloys;

• Ceramics, glasses, and glass-ceramics;
• Polymers (plastics);
• Semiconductors; and
• Composite materials
fi
ff
fi
 Category Examples Key Properties Common Applications

Steel, Al, Cu, Ti, Ni, Cast Structures, engines, aircraft, automotive
Metals & Alloys High strength, ductile, conductive, shock-resistant
Iron parts

Alumina, Zirconia, Bricks, Turbine blades, sensors, coatings, space

Ceramics Hard, brittle, high melting point, good insulators
Tiles shuttle tiles

Glasses & Glass- Amorphous, optical clarity, brittle but strong when
Silica glass, Zerodur™ Windows, ber optics, telescope mirrors
Ceramics tempered

Nylon, PVC, Rubber, Lightweight, corrosion-resistant, insulating, Plastics, packaging, clothing, electronics,
Polymers
Te on exible or rigid insulation

Semiconductors Silicon, Germanium, GaAs Moderate conductivity, tunable via doping Microchips, solar cells, transistors
fl
fl
fi
 Crystalline Materials

Materials in which atoms or molecules are arranged in a regular, repeating pattern over long distances (long-range order).

Characteristics:
• Highly ordered atomic arrangement.

• Well-de ned unit cells repeated throughout the material.

• Exhibit sharp melting points.
• Easier to study due to symmetry.

Examples:

• Metals (Iron, Copper, Aluminum)

• Alloys (Steel, Brass)

• Minerals (Quartz, Salt)

•
fi
Non-Crystalline (Amorphous) Materials

Materials with no long-range order in their atomic arrangement. Atoms are randomly distributed, although short-range order may exist.

Characteristics:
• Irregular atomic structure.
• No true unit cell or periodicity.

• Gradual softening rather than sharp melting.

• Often transparent due to lack of crystal boundaries.

Examples:

• Glasses (Silica glass, Borosilicate)

• Polymers (Some plastics like polystyrene)

• Amorphous metals (metallic glasses)

•
 Atomic Arrangements in Materials

Type of Order Description Examples

- Atoms or ions are randomly arranged with no xed

No Order Inert gases (e.g., Argon), plasma in uorescent tubes
pattern.

- Atoms have a speci c arrangement with only nearest

Short-Range Order Steam (H₂O vapor), amorphous silicon, silicate glass, some
neighbors.
(SRO) polymers
- No periodicity beyond basic units.

- Repeating 3D arrangement of atoms/ions over large

Long-Range Order Metals, alloys, ceramics, semiconductors, single crystals,
distances (>100 nm).
(LRO) polycrystalline materials
- Forms crystalline structures.
fi
fl
fi
Mixed Structures (Partially Crystalline)
Some materials contain both crystalline and amorphous regions within the same structure.

Characteristics:
• Intermediate properties between crystalline and amorphous.

• Often formed by rapid cooling or controlled processing.

• Properties can be tailored (e.g., strength, transparency).

Examples:

• Polymers like polyethylene (semi-crystalline)

• Silica: Exists in both crystalline (quartz) and amorphous (glass) forms

•
• Some ceramics and glasses used in electronics

Structure Type Atomic Order Examples Notes

**Crystalline** Long-range order Metals, alloys, minerals Sharp melting point, high strength
**Non-Crystalline** Short-range only Glasses, amorphous plastics No clear melting point
**Mixed** Both orders present Silica (quartz + glass), polymers Tailored properties possible
Crystal structure

A crystal structure describes the ordered,

three-dimensional arrangement of atoms,
ions, or molecules in a crystalline
material. A crystal structure is made of atoms.

It's characterized by a lattice, which A crystal lattice is made of points.

provides the framework, and a basis, A crystal system is a set of axes. In other words, the structure is an ordered
array of atoms, ions or molecules.
which is the group of atoms or molecules
attached to each lattice point. This
ordered arrangement results in the
unique properties and symmetry of
crystals.
https://youtu.be/SpanlpL_SYI
Key Words

• Lattice:
A crystal lattice is an abstract framework of points in space that repeats periodically. It de nes the overall shape and
symmetry of the crystal structure.

• Basis:
The basis is a group of atoms, ions, or molecules that is associated with each lattice point. The basis, combined with the
lattice, de nes the complete crystal structure.

• Unit Cell:
A unit cell is the smallest repeating unit of the crystal structure that still retains all the symmetry and characteristics of the
entire structure.

• Types of Crystal Structures:

There are various crystal systems (like cubic, tetragonal, hexagonal, etc.) and lattice types (like simple cubic, body-
centered cubic, face-centered cubic, etc.) that de ne the di erent possible crystal structures.

• Importance:
Understanding crystal structures is crucial for understanding the physical properties of materials, such as their strength,
electrical conductivity, and optical properties. Techniques like X-ray di raction are used to determine crystal structures.
fi
fi
ff
ff
fi
Crystal Systems
What is a Crystal System?
• A crystal system classifies crystals based on:
• - Crystallographic axes
• - Axis lengths and angles
• - Internal atomic symmetry

• Crystals form repeating 3D patterns called lattices.

7 Crystal Systems
 System Axes (Length) Angles Between Axes Symmetry Examples

Triclinic a≠b≠c α ≠ β ≠ γ ≠ 90° Very low Kyanite, Turquoise

Monoclinic a≠b≠c Two 90°, one ≠ 90° Low Gypsum, Howlite

Orthorhombic a≠b≠c All 90° Medium Topaz, Zoisite

Trigonal a=b≠c 120°, 90° mixed Medium-High Quartz, Calcite

Hexagonal a=b≠c 120° (basal), c ⟂ plane High Beryl, Apatite

Tetragonal a=b≠c All 90° High Perovskite, Zircon

Cubic a=b=c All 90° Very high Diamond, Gold, Silver

 Triclinic Crystal System
• Axes: a ≠ b ≠ c, Angles: α ≠ β ≠ γ ≠ 90°
• - Least symmetrical
• - Irregular, inclined crystal faces
• - Examples: Labradorite, Amazonite, Kyanite,
Turquoise
 Monoclinic Crystal System
• Axes: a ≠ b ≠ c, Angles: α = γ = 90°, β ≠ 90°
• - One inclined axis
• - Prisms with inclined faces
• - Examples: Gypsum, Kunzite, Howlite,
Vivianite
 Orthorhombic Crystal System
• Axes: a ≠ b ≠ c, Angles: all 90°
• - Rhombic prisms, double pyramids
• - Moderate symmetry
• - Examples: Topaz, Tanzanite, Zoisite,
Danburite
 Trigonal Crystal System
• Axes: a = b ≠ c, 3-fold symmetry
• - Base: triangle (120°)
• - Forms: Rhombohedra, 3-sided pyramids
• - Examples: Quartz, Calcite, Ruby, Agate
 Hexagonal Crystal System
• Axes: 4 axes (a1 = a2 = a3 ≠ c), angles: 120°, 90°
• - 6-fold symmetry
• - Forms: 6-sided prisms, double pyramids
• - Examples: Beryl, Apatite, Sugilite
 Tetragonal Crystal System
• Axes: a = b ≠ c, Angles: all 90°
• - Forms: 4-sided prisms, trapezohedrons
• - Symmetry: 4-fold
• - Examples: Zircon, Cassiterite, Perovskite
 Cubic Crystal System
• Axes: a = b = c, Angles: all 90°
• - Most symmetrical
• - Forms: Cubes, octahedrons, dodecahedrons
• - Examples: Diamond, Gold, Silver, Garnet
 Introduction to Unit Cells
• A unit cell is the smallest repeating unit in a
crystal lattice.
• It defines the symmetry and structure of the
entire crystal.
• Atoms or ions occupy lattice points.
 Classification of Unit Cells
• i. Primitive Cubic (Simple Cubic)
• a. Body-Centered Cubic (BCC)
• b. Face-Centered Cubic (FCC)
• c. Hexagonal Close-Packed (HCP)
 Primitive Cubic Unit Cell
• Atoms at 8 corners
• Each corner atom shared among 8 unit cells
• Total atoms per unit cell = 1
• Packing efficiency: ~52%
• Example: Polonium (rare)
Body-Centered Cubic (BCC) Unit Cell
• Atoms at 8 corners + 1 atom at center
• Total atoms per unit cell = 2
• Packing efficiency: ~68%
• Examples: Chromium (Cr), Iron (Fe),
Molybdenum (Mo), Tantalum (Ta), Tungsten
(W)
Face-Centered Cubic (FCC) Unit Cell
• Atoms at 8 corners + centers of 6 faces
• Total atoms per unit cell = 4
• Packing efficiency: ~74%
• Examples: Aluminum (Al), Copper (Cu), Gold
(Au), Lead (Pb), Silver (Ag), Nickel (Ni)
Hexagonal Close-Packed (HCP) Unit Cell
• Hexagonal prism shape
• Coordination number = 12
• Total atoms per unit cell = 6
• Packing efficiency: ~74%
• Examples: Magnesium (Mg), Titanium (Ti), Zinc
(Zn)
Bravais lattices.
🔶 14 Bravais Lattices (Grouped into 7 Crystal Systems)

Crystal System Lattice Types

Cubic Simple, Body-Centered, Face-Centered

Tetragonal Simple, Body-Centered

Orthorhombic Simple, Body-Centered, Face-Centered, Base-Centered

Monoclinic Simple, Base-Centered

Triclinic Simple

Hexagonal Simple

Rhombohedral Simple
What is Atomic Packing Factor?

• The Atomic Packing Factor (APF) is the fraction of space

occupied by atoms in a unit cell.
• APF = Volume of atoms in unit cell / Volume of unit cell
• Helps determine:
• - Packing density
• - Void fraction
• - Material behavior (strength, ductility)
FCC (Face-Centered Cubic) Structure

• Atoms in Unit Cell: 8×1/8 + 6×1/2 = 4

• Atomic Contact Along: Face diagonal √2a = 4R R = √2a/4

• Volume of Atoms: V = 4 × (4/3)πR³

• APF_FCC = [4 × (4/3)πR³] / a³ = 0.74
• Metals: Al, Cu, Au (Highly ductile)
BCC (Body-Centered Cubic)
Structure

• Atoms in Unit Cell: 8×1/8 + 1 = 2

• Atomic Contact Along: Body diagonal √3a = 4R R = √3a/4

• Volume of Atoms: V = 2 × (4/3)πR³

• APF_BCC = [2 × (4/3)πR³] / a³ = 0.68
• Metals: Fe, Cr, Mo (Strong, less ductile)
Crystal Structure Atoms per Unit Cell APF Packing E ciency (%)

Simple Cubic (SC) 1 0.52 52%

Body-Centered Cubic (BCC) 2 0.68 68%

Face-Centered Cubic (FCC) 4 0.74 74%

Hexagonal Close-Packed (HCP) 6 (theoretical unit) 0.74 74%

ffi
HCP (Hexagonal Close-Packed)
Structure

• Atoms in Unit Cell: 12×1/6 + 2×1/2 + 3 = 6

• Atomic Contact in Basal Plane: a = 2R, c/a ≈ 1.633
• Volume of Unit Cell: V = (3√3/2)a²c
• APF_HCP = [6 × (4/3)πR³] / [(3√3/2)a²c] ≈ 0.74
• Metals: Zn, Ti, Mg (Strong, limited slip systems)
 Imperfections in Crystals

• Perfect crystals are theoretical models.

• All real crystals contain imperfections or defects.

• These imperfections in uence a material’s:
◦ Mechanical strength
◦ Electrical and optical properties
◦ Processing behavior
• Examples:
◦ A awless diamond is more valuable.
◦ Silicon wafers require near-perfect crystals for IC chips.
fl
fl
Why Imperfections Occur

• Imperfections arise due to:

◦ Solidi cation errors
◦ Processing methods (e.g., rolling, forging)
◦ Alloying to improve properties
• Even during ideal growth, the lattice may not replicate perfectly.

• Some defects are bene cial (like alloying for strength).

fi
fi
• Crystal imperfections are broadly classi ed into four major classes as below:
• 1)Point defects (Zero dimensional)
• 2)Line defects (One dimensional)
• 3)Planar or surface defects (Two dimensional)
• 4)Volume defects
fi
 Point Defects

• Occur at atomic scale

• Types:
◦ Vacancy: Atom missing from lattice site
◦ Schottky Defect: Missing a pair of ions (cation + anion)
◦ Interstitialcy: Extra atom squeezed into space
◦ Frenkel Defect: Atom displaced from normal position to interstitial site

• E ect: Di usion, conductivity, strength

ff
ff
Point defects: (a) vacancy, (b) ion-pair vacancy, (c) interstitialcy, and (d)
displaced ion.
Line Defects – Dislocations

• Dislocation: One-dimensional defect, a misalignment in lattice

• Important in explaining plastic deformation in metals

Two major types:

1. Edge Dislocation – Extra half-plane of atoms

2. Screw Dislocation – Helical mis t caused by shear

• Enable slip; without them, metals wouldn’t deform easily
fi
Line defects: (a) edge dislocation and (b) screw dislocation.
 Surface Defects

• Extend in two dimensions

• Types:
◦ Grain Boundaries: Between crystals of di erent orientations
◦ Twin Boundaries: Mirror symmetry interface
◦ Stacking Faults: Incorrect atomic stacking sequence
◦ External Surfaces: Termination of lattice

• E ect: In uence strength, corrosion resistance, di usion

ff
fl
ff
ff
• Volume Defects
• Volume defects such as stacking faults may arise when there is only small
dissimilarity between the stacking sequence of close packed planes in FCC
and HCP metals.

• Stacking faults are of two types called as intrinsic and extrinsic Intrinsic fault
results in one break whereas extrinsic fault results in two breaks in the
sequence

• The volume defects may a ect their mechanical, electrical and optical
properties.
ff
• Solid Solution
• Solid Solution or an Alloy is a phase, where two or more elements are
completely soluble in each other.

• Solid solutions have important commercial and industrial applications, as

such mixtures often have superior properties to pure materials.

• Many metal alloys are solid solutions.

• Ex: Cu-Ni, Au-Ag etc.
• In a solid solution, the metal in the major proportion is called the solvent (host
or parent or matrix) and the metal in the minor proportion is called the solute.
• There are two types of Solid Solutions:
• i. Substitutional Solid Solution
• ii. Interstitial Solid Solution
• Substitutional Solid Solution

• In this type of solid solution, the solute atoms substitute the atoms
of solvent in the crystal structure of the solvent.
• The substitutional solid solution are generally ordered at lower
temperatures and at higher disordered temperatures.
• Temperature is the deciding factor.
Cont..

• There are two types of substitutional solid solutions:

• i. Ordered Substitutional Solid Solution (OSSS)
• ii. Disordered Substitutional Solid Solution (DSSS)
• OSSS: In this type, the solute atoms substitute the solvent atoms in an orderly
manner, taking up xed positions of symmetry in lattice. This solid solution has
uniform distribution of solute and solvent atoms.

• DSSS: In this type, the solute atoms do not occupy any xed positions but are
distributed at random in the lattice structure of solvent. The concentration of
solute atoms vary considerably through out lattice structure.
fi
fi
Hume-Rothery Rules for Solid Solution Formation
These rules determine whether a substitutional solid solution will form between two
metallic elements.

1. Crystal Structure Factor

• The solute and solvent must have the same crystal structure.

• ✔ Promotes atomic compatibility in the lattice.

• Example: Au (Gold) and Ag (Silver) both have FCC structures.

2. Relative Atomic Size Factor

• The di erence in atomic radii should be less than 15%.

• ✔ Minimizes lattice distortion.

• ❌ Greater di erences reduce solubility.

ff
ff
Hume-Rothery Rules for Solid Solution Formation
3. Valency Factor

• Elements with the same valence (number of outer electrons) mix more readily.

• ✔ A metal tends to dissolve another metal of the same or lower valency more easily.

• ❌ A metal with higher valency dissolves one with lower valency less easily.

4. Electronegativity Factor

• The electronegativity di erence between solute and solvent should be small.

• ✔ Large di erences lead to compound formation instead of solid solutions.

✅ Conclusion:

All four rules must be reasonably satis ed for complete solid solubility.
Partial satisfaction may still result in limited solubility or intermetallic compounds.
•
ff
ff
fi
Interstitial Solid Solution
An interstitial solid solution forms when small atoms (solute) occupy the interstitial
spaces in the crystal lattice of a host metal (solvent) without replacing its atoms.

Characteristics:

• Solute atoms are signi cantly smaller than solvent atoms.

• They t into voids between host atoms.

• Common in metals with open structures like BCC and FCC.

• Examples include carbon in iron, hydrogen in palladium, nitrogen in iron, etc.

Conditions for Formation:

• The atomic radius of the solute must be less than 60% of the solvent atom.

• Typically involves non-metals as solutes and metals as solvents.

fi
fi
 Interstitial Solid Solution
E ects:
• Increases hardness and strength.

• Reduces ductility.
• In uences electrical and thermal conductivity.

Example:

• Carbon steel is a classic interstitial solution where carbon atoms occupy

interstitial spaces in the iron lattice, greatly enhancing strength.
ff
fl
 Intermediate Phases
• De nition: Intermediate phases are phases with chemical compositions between two pure
metals, typically possessing crystal structures di erent from those of the base (parent)
metals.
• Occurrence in Alloys:

◦ Alloys may consist solely of a solid solution phase or may include intermediate phases.

◦ An intermediate phase often behaves like a compound, formed from two or more
elements, at least one being a metal.
• Compound Formation:

◦ Compounds are chemical combinations of elements with positive and negative valence,
represented by chemical formulas (e.g., H₂O, NaCl).
◦ Once formed, compounds lose the individual properties of constituent elements and
•
exhibit distinct physical, mechanical, and chemical properties.
fi
ff
Types of Intermediate Alloy Phases
1. Intermetallic or Valence Compounds

• Formed exclusively from metal-metal combinations.

• Created based on chemical valence rules.

• Properties:
◦ Poor ductility
◦ Poor electrical conductivity
◦ Complex crystal structures

• Examples:
◦ Mg₂Pb, Mg₂Sn, CaSe, Cu₂Se

•
2. Interstitial Compounds

• Resemble interstitial solid solutions but have a xed stoichiometry.

• Formed when small non-metal atoms occupy interstitial spaces in a

metal lattice.
• Properties:
◦ Metallic nature
◦ High melting point
◦ Extremely hard
• Example:
◦ Fe₃C (Iron carbide)
fi
. Electron Compounds
• Exhibit variable compositions and do not follow valence laws.
• Characterized by a speci c electron-to-atom ratio.
• Example:
◦ Cu₉Al₄
▪ 9 Cu atoms (1 valence e⁻ each) + 4 Al atoms (3 valence e⁻ each) = 21 electrons

▪ Total atoms = 13 → e⁻/atom = 21:13

• Properties:
◦ Similar to solid solutions
◦ Wide composition range
◦ High ductility
◦ Low hardness

•
fi
 PHASE DIAGRAMS
De nition: Phase diagrams (also called equilibrium diagrams or constitutional diagrams) represent the
relationships between phases in a system as a function of temperature, pressure, and composition.

Equilibrium Condition: These diagrams typically re ect equilibrium conditions—the state of minimum free
energy of the system.
Graphical Maps: They are graphical tools used to predict the phase stability and transformation behavior of
materials.

Use in Alloy Systems: Widely used in alloy systems to understand solidi cation, melting, and other phase
changes.

Metastable Phases: Sometimes, metastable phases (non-equilibrium phases) are shown, especially in
engineering alloys where such phases are technologically important.

Metastable Equilibrium: When shown, metastable phases indicate a metastable equilibrium—a temporary
but stable condition due to kinetic hindrance.

Application: Essential for materials selection, heat treatment, and processing of metals, ceramics, and
other materials.
•
fi
fl
fi
 Components
• De nition: Components are the independent chemical species that constitute an
alloy or a system.
• Example – Plain Carbon Steel:
◦ Chemical species: Iron (Fe) and Carbon (C).
◦ However, in steels, carbon usually exists as cementite (Fe₃C) rather than elemental
graphite.
◦ Although graphite is more stable thermodynamically, cementite (Fe₃C) is the
commonly observed metastable phase.
• Metastable Diagram Usage:
◦ Phase diagrams of steel typically consider Fe and Fe₃C as the components.
◦ Even though Fe, C, and Fe₃C are all present, only Fe and C are independent
components.
◦ Fe₃C is a compound formed from Fe and C, not an independent component.
fi
• THE PHASE RULE

• The Gibbs phase rule can be stated as follows :

• F=C-P+2

• where F is the degrees of freedom,

• C is the number of components, and
• P is the number of phases in the system.
Phases in Materials

Phases refer to di erent states of aggregation of matter—solid, liquid, and gas—each representing a physically
distinct and mechanically separable portion of a system.
• Gaseous Phase: Always forms a single phase, regardless of the components present. This is because atoms or
molecules in gases are completely mixed at the atomic level, leading to uniform properties throughout.
• Liquid Phase:
◦ A homogeneous liquid solution, such as salt dissolved in water, is a single phase. Here, sodium ions (Na⁺),
chloride ions (Cl⁻), and water molecules are thoroughly mixed at the molecular level.
◦ A heterogeneous liquid mixture, such as water and oil, forms two distinct liquid phases, since the
components do not mix uniformly and separate due to di erences in polarity and density.
• Solid Phase:
◦ A solid solution—such as an alloy where di erent types of atoms occupy positions within the same crystal
lattice—is a single solid phase. The atomic mixing occurs at the unit cell level, resulting in uniform structure
and properties.
◦ If the solid contains regions of di erent crystal structures or compositions, it is considered to have multiple solid
•
phases.
ff
ff
ff
ff
Degrees of Freedom – De nition

Degrees of Freedom (F):

• The number of independent variables that can be changed without altering
the number of phases in equilibrium.

Variables include:
• Temperature
• Pressure
• Composition of each phase
fi
 Binary Phase Diagrams
• A binary phase diagram represents equilibrium
relationships between two components in a
system as a function of temperature and
composition.

• It shows the phases present and their

compositions under equilibrium conditions.

🔹 Key Features:

• Axes:

◦ Y-axis → Temperature (°C or K)

◦ X-axis → Composition (usually weight % or

atomic % of component B)
• Pressure is usually held constant (typically 1 atm),
•
and the vapor phase is ignored.

https://learnmetallurgy.com/study/physical/topic/binary-phase-diagrams.php
Types of Binary Diagrams:

1. Isomorphous system – Complete solubility in solid and liquid states

(e.g., Cu–Ni).

2. Eutectic system – Limited solubility; includes eutectic point.

3. Peritectic system

4. Eutectoid and Peritectoid systems

 Isomorphous System

• De nition: Complete solubility in both liquid and solid states.

• Example: Copper–Nickel (Cu–Ni) system.
• Features:
◦ Only one type of solid solution (α-phase).
◦ Single-phase regions: Liquid (L), Solid (α), and L + α.
◦ Smooth transition during cooling from liquid to solid solution.
fi
Eutectic Systems
In metallurgy and materials science, phase diagrams are used to understand how di erent phases form and coexist in alloys at various temperatures and
compositions. One of the most important types is the eutectic phase diagram, which arises due to limited solid solubility between two components.

🔸 Solid Solubility and Hume-Rothery Rules

• Not all metallic systems exhibit complete solid solubility.

• The Hume-Rothery rules provide guidelines for extensive solid solubility, such as:
◦ Similar atomic sizes
◦ Same crystal structure
◦ Comparable electronegativities
◦ Similar valency
• However, many alloy systems do not meet these criteria perfectly, leading to partial or limited solubility.

As a result, two terminal solid solutions are formed—each based on one of the pure elements.

🔸 Two Key Types of Binary Diagrams Due to Limited Solubility

1. Eutectic Diagram: Formed when components have similar melting points.

•
2. Peritectic Diagram: Formed when one component has a much higher melting point than the other.
ff
Eutectic Point
• Eutectic composition (Ce): 28.1 wt%
Cu
• Eutectic temperature (Te): 780°C
• At this exact composition and
temperature:
◦ The liquid phase transforms
isothermally into two solid phases:
α (Ag-rich FCC solid solution) and β
(Cu-rich FCC solid solution).

• This is known as the eutectic

reaction:

• L⟶α+β(at T=780∘C)
 Peritectic Phase Diagram

When two elements have signi cantly di erent melting points and
limited mutual solubility, their binary phase diagram often exhibits a
peritectic reaction rather than a eutectic.

🔹 Example: Ni–Re (Nickel–Rhenium) System

• Nickel (Ni): Melting point ≈ 1455°C

• Rhenium (Re): Melting point ≈ 3186°C
• Due to this large di erence in melting points, the Ni–Re system
forms a peritectic diagram.

📌 Peritectic Reaction (at Peritectic Temperature Tp = 1622°C):

L+β→α(on cooling)
• A liquid phase (L) and a solid phase (β) react upon cooling to
form a new solid phase (α).
• This reaction is invariant — three phases (L, β, α) are in
equilibrium at a xed temperature and composition, with zero
•
degrees of freedom (F = 0) by Gibbs Phase Rule.
fi
ff
fi
ff
The Lever Rule: Determining Phase Fractions

The lever rule is a graphical method used in binary phase diagrams to

determine the relative amounts (fractions) of two coexisting phases (such as
liquid and solid) at equilibrium for a given overall alloy composition and
temperature.

It is applicable in any two-phase region (e.g., solid + liquid, α + β, etc.).

•
Basic Concept LR
• Imagine a horizontal tie-line
drawn across the two-phase
region at the speci ed
temperature.

• The ends of the tie-line represent

the compositions of the two
coexisting phases.

• The overall composition (C₀) lies

somewhere along this line (called
the fulcrum point).

• Think of this tie-line as a lever with

the fulcrum at C₀.

📌 Opposite arm rule: Each phase

fraction is given by the length of the
lever arm on the opposite sideof the
fulcrum.

•
fi
Peritectoid System

• Solid-state analog of peritectic system.

• Reaction: α + β → γ

• Occurs entirely in the solid state.

Invariant Reactions in Binary Systems

An invariant reaction is a phase transformation that occurs at a xed

temperature and composition in a binary system, involving the equilibrium of
three phases.

✅ According to the Gibbs Phase Rule for a binary system (C = 2):

F=C−P+1=2−3+1=0

So, F = 0, meaning there are zero degrees of freedom (neither temperature nor
composition can change).

fi
Key Points:
• All occur isothermally at a xed composition and temperature (F = 0).
• Each involves three phases in equilibrium.
• Used to analyze microstructure evolution and heat treatment behavior of
alloys.
fi
S.
Reaction Reaction Phases in
No Phase State Description
Name Equation Equilibrium
.
Liquid → A single liquid transforms into two distinct solid phases upon cooling.
L→α+
1 Eutectic Solid + L, α, β Common in systems with limited solubility. Produces ne, layered
β
Solid microstructures.
Solid →
Eutectoi A single solid phase decomposes into two di erent solid phases
2 γ→α+β Solid + γ, α, β
d isothermally. Found in steel (austenite → ferrite + cementite).
Solid
Liquid +
Peritecti L + β → A liquid and one solid phase react to form a new solid phase. Occurs at a
3 Solid → L, β, α
c α temperature between the melting points of the pure components.
Solid
Solid +
Peritecto Two solid phases combine to form a new solid phase. Occurs entirely in
4 γ+β→α Solid → γ, β, α
id the solid state.
Solid
Liquid →
Monotec L₁ → L₂ + A single liquid splits into another liquid and a solid phase. Happens in
5 Liquid + L₁, L₂, α
tic α immiscible liquid systems.
Solid
Liquid +
Syntecti L₁ + L₂ → Two immiscible liquids combine to form a single solid phase. Rare in
6 Liquid → L₁, L₂, α
c α practical systems
Solid
ff
fi
Thank you