Professor: Syed Takwi TAs: Azeen (PhD candidate, chemical engineering, nanocomposites research) and Karel (MASc, chemical engineering; BSc materials science and engineering, University of Toronto) Term: Winter 2021 (1211) Lectures: Mondays 9:30 AM; Tuesdays and Thursdays 8:30 AM Tutorials: Mondays, Wednesdays, and Fridays (one section per student)

Introduction — What Is Materials Science?

The Central Paradigm of Materials Science

The opening lecture framed the entire course around a single unifying idea sometimes called the central paradigm (or classical paradigm) of materials science. It states that processing determines structure, structure determines properties, and properties determine performance (the application). These four elements are tightly coupled:

A useful extension — the modified paradigm — adds a fifth vertex: reusability/recyclability/environment. Engineers today must consider whether a material can be recovered and reused at the end of its service life. The refrigerant CFC illustrates this: it performed its cooling function flawlessly but was discarded because of its environmental toxicity.

In a more complete picture the four (or five) elements form a tetrahedron rather than a line, because processing, structure, and properties each influence the others simultaneously.

The Six Categories of Material Properties

Properties are always defined as the response of a material to an applied stimulus. Professor Takwi organized them into six families:

Mechanical properties arise when a load is applied. They include strength (resistance to fracture), stiffness (tendency to return to the original shape after the load is removed — related to the elastic modulus), ductility (ability to be drawn into wires without fracture), and hardness.

Electrical properties describe how a material responds to an applied electric field. Conductivity indicates how freely charge carriers move; capacitance reflects the ability to store charge.

Thermal properties relate to how a material responds to a change in temperature — thermal conductivity, heat capacity, and thermal expansion.

Magnetic properties describe the response to an applied magnetic field, including ferromagnetism and paramagnetism.

Optical properties govern the interaction with light, the most visible example being transparency. Glass is transparent because of its non-crystalline atomic structure; metals are opaque because their free electrons absorb and re-emit photons.

Deteriorative (chemical) properties characterize reactivity with the environment — corrosion resistance and chemical stability in acidic or basic media.

A critical observation: all properties depend on structure, and structure is controlled by processing. This is why materials science is inseparable from engineering.

The Four Primary Classes of Solid Materials

Advanced Materials

Beyond the four primary classes, two families are especially relevant to nanotechnology:

Smart materials sense a change in their environment and respond in a programmed way. A piezoelectric actuator, for example, changes shape under an applied electric field; this is why a smartphone camera can autofocus.

Nanomaterials are defined not by chemistry but by scale: at least one dimension in the range 1–100 nm. At the nanoscale, materials can exhibit properties completely absent in the bulk form. Graphene — a single atomic layer of graphite — is 200 times stronger than steel, an excellent electrical conductor, and nearly transparent, properties that bulk graphite does not replicate.

Material Selection Criteria

Selecting a material for a given application involves at minimum three considerations. First, identify the critical property: for a smartphone screen, transparency and fracture resistance are paramount. Second, assess property retention: a material that is initially transparent but yellows under UV exposure (as certain polymers do) is unsuitable for prolonged outdoor use. Third, evaluate cost: ceramic coatings can be extraordinarily transparent and hard, but at a price per unit area that makes them impractical for a device replaced every two years. Cost is almost always the dominant decision-making factor once technical requirements have been met.

Atomic Structure

Why Atomic Structure Matters

The classic example is carbon. Both graphite and diamond are pure carbon, yet they are completely different materials. Diamond is the hardest natural substance and an electrical insulator; graphite is soft enough to mark paper and is a good electrical conductor. The difference is entirely structural: in diamond, each carbon atom forms four covalent bonds in a tetrahedral (sp³) geometry, producing a rigid three-dimensional lattice. In graphite, carbon atoms bond in sp² hybridization forming flat hexagonal rings; the layers stack above each other and can slide easily, giving graphite its lubricating quality. The conducting electrons in graphite are delocalized within the planes, enabling conductivity.

Fundamentals of Atomic Notation

An element in the periodic table is described by two numbers. The atomic number Z is the number of protons in the nucleus — it uniquely identifies the element. The atomic weight (also called atomic mass) is the weighted average of the atomic masses of the element’s naturally occurring isotopes, reported in atomic mass units (AMU). One AMU per atom equals one gram per mole.

Key constants: the charge on an electron (or proton) is 1.602 × 10⁻¹⁹ C; the mass of a proton or neutron is ~1.67 × 10⁻²⁷ kg; the mass of an electron is ~9.11 × 10⁻³¹ kg; Avogadro’s number Nₐ = 6.022 × 10²³ atoms/mol.

Atomic Models

Classical mechanics fails at the atomic scale. In the 19th century, experimental observations — blackbody radiation, the photoelectric effect, atomic emission spectra — could not be explained by Newtonian physics. This motivated the development of quantum mechanics.

Bohr’s atomic model (early 20th century) proposed that electrons orbit the nucleus in discrete shells, each associated with a fixed energy. For hydrogen, the ground state energy is −13.6 eV. Bohr could predict the energies of spectroscopic lines for hydrogen, but could not explain multi-electron atoms or determine the spatial probability of finding an electron.

The wave mechanical model improved on Bohr by treating the electron as both a particle and a wave (wave-particle duality). Instead of a fixed orbit, the electron is described by a probability distribution or electron cloud — the wavefunction squared gives the probability of finding the electron in a given region. This model introduces four quantum numbers to completely characterize each electron in an atom.

Quantum Numbers

Every electron in an atom is characterized by four quantum numbers:

Pauli Exclusion Principle: No two electrons in the same atom can have the same set of four quantum numbers. Equivalently, each orbital holds at most two electrons, and they must have opposite spin.

Orbital capacities: s holds 2 electrons (1 orbital); p holds 6 electrons (3 orbitals × 2); d holds 10 electrons (5 orbitals × 2); f holds 14 electrons (7 orbitals × 2).

Electron Configuration

The ground state of an atom is the lowest-energy electron configuration — all electrons in the lowest available orbitals. To write it, fill orbitals in order of increasing energy.

Example — Oxygen (Z = 8): 1s² 2s² 2p⁴. The outermost (valence) shell contains 2 + 4 = 6 electrons.

Valence electrons are those in the outermost shell (or, more precisely, those outside the nearest noble-gas core configuration). For iron (Z = 26), the noble-gas core is argon (1s² 2s² 2p⁶ 3s² 3p⁶). Iron adds 3d⁶ 4s², giving 8 valence electrons.

Electronegativity and Electropositivity

Electropositive elements readily give up electrons (metals, left side of periodic table). Electronegative elements readily accept electrons (non-metals, right side). Electronegativity increases going up a group and going right across a period; the most electronegative element is fluorine. In a mixture, the more electronegative element tends to acquire electrons from the less electronegative one.

Interatomic Bonding

Forces Between Atoms

When two atoms are brought together from infinite separation, they experience both attractive and repulsive forces. At large distances, both are negligible. As separation r decreases, attraction dominates first; at very small r, repulsion from overlapping electron clouds dominates. Equilibrium interatomic distance r₀ (~0.3 nm) is where the net force equals zero. The bonding energy E₀ is the energy required to separate the atoms from r₀ to infinity.

Energy and force are related by E = ∫F dr (energy is the integral of force with respect to separation). Equivalently, force is the derivative of energy with respect to r. In convention: attractive potential energy is negative; repulsive potential energy is positive. Attractive force is positive (pulls atoms together); repulsive force is negative.

Primary (Chemical) Bonds

Ionic Bonding

The attractive energy is: Eₐ = −A/r, where A = (1/4πε₀)|Z₁||Z₂|e², Z₁ and Z₂ are the valences (absolute values), and e = 1.602 × 10⁻¹⁹ C. The repulsive energy is Eᵣ = B/rⁿ, where B and n are determined experimentally for each ionic system.

Ionic bonds are non-directional: the Coulombic force acts equally in all directions from an ion, so each positive ion attracts all neighbouring negative ions symmetrically. Example: NaCl, where Na⁺ and Cl⁻ ions each have six nearest neighbours.

Covalent Bonding

Covalent bonds are directional: they point along the line connecting the two bonded atoms (or along specific geometric directions in hybridized systems). In H₂, the single electron from each hydrogen shares the 1s orbital, satisfying the Pauli principle with opposite spins. Carbon in diamond forms four sp³ hybrid bonds at tetrahedral angles; carbon in graphite forms three sp² bonds in a plane with the fourth electron delocalized into a π bond — which is why graphite conducts electricity.

Hybridization: When an atom forms bonds, its atomic orbitals can mix to form hybrid orbitals with different geometry. sp³ hybridization (tetrahedral, 109.5°) is found in diamond and methane CH₄; sp² hybridization (trigonal planar, 120°) is found in graphite and benzene.

Percent ionic character: The bond between two atoms is rarely purely covalent or purely ionic. The percent ionic character is: %IC = {1 − exp[−0.25(Xₐ − X_b)²]} × 100, where Xₐ and X_b are the electronegativities of the two atoms. Carbon–hydrogen bonds have ~3.9% ionic character (essentially covalent); sodium–chlorine has a high ionic character.

Metallic Bonding

Metallic bonds are non-directional: the electron sea surrounds ions equally in all directions. The freely mobile electrons explain why metals are good electrical and thermal conductors — electrical conductivity is simply the flow of these electrons in response to an electric field.

Secondary (Physical / Van der Waals) Bonds

Secondary bonds are weaker than primary bonds and arise from electric dipoles — charge imbalances within a molecule or atom. They exist between all molecules but are usually masked by stronger primary bonds. They become significant in noble gases (which form no primary bonds) and between molecules already joined by primary bonds.

There are three types of dipole interaction:

• Induced–induced (London dispersion forces): Temporary dipoles arise from random electron fluctuations and induce dipoles in neighbours. Example: liquefaction of Cl₂.
• Induced–permanent: A permanently polar molecule induces a dipole in a nearby symmetric molecule.
• Permanent–permanent: Two permanently polar molecules attract each other.

The hydrogen bond is the strongest secondary bond. It forms specifically between a hydrogen atom covalently bonded to F, O, or N and another electronegative atom on a neighbouring molecule. It is responsible for the unusually high boiling point of water and for the structure of proteins and DNA. In HF⋯HF chains, the bond between the F of one molecule and the H of the next is the hydrogen bond.

Applications of Van der Waals forces include adhesives, emulsifiers (soaps, detergents), and desiccants.

Crystal Structure of Metals

Crystalline vs. Non-Crystalline Solids

The crystal structure of a material refers to the way its atoms (or ions or molecules) are arranged in space.

The Unit Cell

Because a crystal repeats indefinitely, its entire structure can be described by a single unit cell — the smallest repeating unit that retains the full geometry of the crystal. The unit cell is characterized by three edge lengths (a, b, c) and three angles (α, β, γ) between them. The atom positions at the corners and interior of the unit cell are called lattice points.

There are 7 crystal systems (cubic, tetragonal, orthorhombic, rhombohedral, hexagonal, monoclinic, triclinic) and 14 Bravais lattices within them. For most engineering metals, only the cubic systems need to be mastered in detail.

The Three Metallic Crystal Structures

Most elemental metals crystallize in one of three structures at room temperature:

Face-Centred Cubic (FCC)

In the FCC structure, atoms sit at each corner of a cube and at the centre of each face. There are no atoms in the interior. Using the standard counting formula:

Number of atoms per unit cell = N_interior + N_face/2 + N_corner/8 = 0 + 6/2 + 8/8 = 4 atoms per unit cell

The relationship between the lattice parameter a and the atomic radius R is obtained by noting that atoms touch along a face diagonal: 4R = a√2, so a = 2R√2.

The atomic packing factor (APF) is the fraction of the unit cell volume occupied by atoms: APF = (volume of atoms)/(volume of unit cell) = (4 × (4/3)πR³)/a³ = 0.74. This is the maximum possible for equal spheres.

The coordination number (number of nearest neighbours touching a reference atom) in FCC is 12.

Metals with FCC structure at room temperature: γ-iron, aluminum, copper, nickel, silver, gold, platinum.

Body-Centred Cubic (BCC)

In the BCC structure, atoms sit at each corner and one complete atom at the centre of the cube. Atoms touch along the body diagonal: 4R = a√3, so a = 4R/√3.

Number of atoms per unit cell = 0/2 + 0 + 8/8 + 1 = 2 atoms per unit cell APF = 0.68 (less dense than FCC) Coordination number = 8

Metals with BCC structure: α-iron, vanadium, chromium, molybdenum, tungsten (wolfram).

Hexagonal Close-Packed (HCP)

In the HCP structure, atoms are arranged in a hexagonal prism. The base-plane lattice parameter a = 2R. The unit cell contains 6 atoms. The APF = 0.74 (same as FCC) and the coordination number = 12 (also same as FCC).

Despite identical APF and coordination number, FCC and HCP differ in their stacking sequence: FCC stacks as ABCABC… while HCP stacks as ABABAB…. This difference is visible in the crystallographic directions perpendicular to the close-packed planes.

Metals with HCP structure: magnesium, α-titanium, zinc, zirconium, beryllium.

Theoretical Density

Knowing the crystal structure allows calculation of theoretical density:

ρ = (n × A) / (Vunit cell × Nₐ)

where n = number of atoms per unit cell, A = atomic weight (g/mol), V = volume of unit cell (cm³), and Nₐ = 6.022 × 10²³/mol.

Example — copper (FCC): R = 0.128 nm, A = 63.5 g/mol, n = 4. Since a = 2R√2, V = a³ = (2R√2)³ = 16R³√2. Calculated ρ = 8.89 g/cm³; literature value = 8.94 g/cm³. The small discrepancy reflects impurities and defects in real copper.

Polymorphism and Allotropy

Carbon is the most famous example: graphite (sp², hexagonal layers), diamond (sp³, tetrahedral), and graphene (a single graphite layer) are all allotropes. Iron is another: α-iron (BCC, stable at room temperature) and γ-iron (FCC, stable above 912 °C). Tin transforms between white tin (body-centred tetragonal) and gray tin (diamond cubic) at 13.2 °C.

Crystallographic Directions, Planes, and X-Ray Diffraction

Crystallographic Point Coordinates

Any point within a unit cell is specified by three fractional coordinates (p, q, r) multiplying the unit cell edge lengths a, b, c respectively. The origin (0, 0, 0) is at one corner; the body-centre is (0.5, 0.5, 0.5); a face centre on the z = 1 plane is (0.5, 0.5, 1).

Crystallographic Directions

A crystallographic direction is defined by a vector from one lattice point to another, expressed as [uvw] (square brackets). The procedure:

1. Set up the x, y, z coordinate system along the unit cell edges a, b, c.
2. Identify two lattice points. Subtract the tail coordinates from the head coordinates.
3. Normalize each component by the corresponding unit cell length.
4. Clear fractions by multiplying all components by the smallest integer that makes them whole numbers.
5. Negative indices are written with a bar above the number (e.g., [1̄10]).

Directions that are equivalent by symmetry (e.g., [100], [010], [001], [1̄00]…) belong to the same family of directions, written with angle brackets: ⟨100⟩.

For hexagonal crystals, a four-index Miller–Bravais notation [uvtw] is used (t = −(u + v)), because three axes at 120° are needed to describe the hexagonal base plane.

Crystallographic Planes

Planes are indexed using Miller indices {hkl} (curly braces for a family, parentheses for a specific plane). The procedure:

1. If the plane passes through the chosen origin, translate the origin by one unit cell along any axis.
2. Find where the plane intersects each of the three axes and express those intersections as multiples of the lattice parameters (∞ if parallel).
3. Take reciprocals.
4. Clear fractions to obtain the smallest integers h, k, l.
5. Enclose in parentheses: (hkl).

Linear density = number of atoms centred on the direction vector / length of that vector. Planar density = number of atoms centred on a plane / area of that plane. Dense crystallographic planes are important in deformation because slip (plastic deformation) occurs preferentially along close-packed planes and directions.

Single Crystals, Polycrystalline Materials, and Anisotropy

A single crystal has a perfectly uniform lattice orientation throughout. Silicon wafers for microelectronics are grown as single crystals.

Most engineering materials are polycrystalline: they consist of many randomly oriented crystalline grains separated by grain boundaries (regions of crystallographic mismatch). During solidification, many nuclei form simultaneously and grow until they impinge.

Anisotropy is the dependence of a property on crystallographic direction. The elastic modulus of iron is 125 GPa along ⟨100⟩ and 272 GPa along ⟨111⟩ — a factor of ~2. Isotropic materials (e.g., tungsten, polycrystalline aggregates with random grain orientations) exhibit properties independent of direction.

X-Ray Diffraction (XRD)

X-rays have wavelengths comparable to interatomic spacings (~0.1 nm), so crystals act as diffraction gratings for X-rays. When an X-ray beam encounters a crystal, it is elastically scattered by the electrons of each atom. Along specific angles, scattered beams from different planes interfere constructively, producing diffraction peaks.

For a cubic crystal, the spacing between {hkl} planes is: d_hkl = a / √(h² + k² + l²). For a hexagonal crystal, a more complex formula applies.

Procedure to find unit cell parameter from an XRD pattern:

1. Read 2θ from the diffractometer spectrum; divide by 2 to obtain θ.
2. Apply Bragg’s law: d = nλ / (2 sin θ).
3. Use the appropriate formula to find a from d and the {hkl} indices assigned to that peak.

XRD is used to identify unknown materials, determine lattice parameters, study phase transformations, and characterize thin films and powders.

Ceramic Crystal Structures

Ceramics are ionic or mixed ionic–covalent compounds. Their structures differ from simple metallic structures because there are two types of ions (with different radii) and the electrostatic requirement of charge neutrality.

The ionic packing factor (analogous to APF) = sum of volumes of all ions in the unit cell / volume of the unit cell.

Classic examples include: NaCl (rock salt structure — built on an FCC framework, with Na⁺ ions occupying octahedral interstitial sites); CsCl (similar to BCC but with two ion types); SiO₂ (silica — tetrahedral coordination); graphite/graphene (classified as ceramic because carbon is non-metallic). Carbon nanotubes (CNTs) are rolled graphene sheets; fullerenes (C₆₀, “buckyballs”) are closed-cage carbon allotropes.

Polymer Crystallinity

Polymers have long chain-like molecules that fold back and forth on themselves, forming lamellae — folded-chain crystalline regions. Most commercial polymers are semi-crystalline (partially crystalline, partially amorphous). Their degree of non-crystallinity is the main reason many plastics are translucent or transparent.

Imperfections in Solids

Why Defects Matter

No real material is a perfect crystal. Defects — departures from the ideal periodic arrangement — profoundly influence mechanical strength, electrical conductivity, diffusion rates, and optical properties. Crucially, vacancies are thermodynamically necessary: a perfectly defect-free crystal would have lower entropy, but entropy favours disorder.

Chemical Imperfections: Solid Solutions and Alloys

Example — Sterling silver: 92.5 wt% silver (solvent) + 7.5 wt% copper (solute).

There are two types of solid solution imperfections:

Substitutional Defects

A foreign atom replaces a host atom at its lattice site. The Hume-Rothery rules determine whether significant substitutional solubility is possible:

1. Atomic radius: The size difference must be less than 15%.
2. Crystal structure: Both elements must have the same crystal structure.
3. Electronegativity: They must have similar electronegativities; otherwise they prefer to form an intermetallic compound.
4. Valence: The greater the valence difference, the lower the solubility.

Nickel and copper satisfy all four rules (similar radius, both FCC, similar electronegativity, both +2 preferred valence) and are completely miscible in all proportions.

Interstitial Defects

A foreign atom is small enough to fit into the spaces (voids) between host atoms without displacing them. The maximum concentration of interstitial impurity is ~10%, because the voids are small. Interstitials can occupy tetrahedral sites (coordination 4) or octahedral sites (coordination 6). Carbon dissolved in iron (forming steel) is a classic example of interstitial solid solution.

Composition Calculations

Weight percent (wt%): mass of component 1 / total mass × 100.

Atom percent (at%): number of moles of component 1 / total moles × 100.

Conversion formula (wt% → at%): For a two-component system with atomic weights A₁ and A₂: C₁’ = (C₁A₂) / (C₁A₂ + C₂A₁) × 100

where C₁ and C₂ are weight percentages and C₁’ is the atom percent of component 1. Example: 97 wt% Al + 3 wt% Cu gives 98.7 at% Al.

Structural Imperfections

Structural defects are classified by their dimensionality:

Point Defects (0D)

Example calculation: For copper at 1000 °C, given Qᵥ = 0.9 eV/atom, A = 63.5 g/mol, and ρ = 8.4 g/cm³:

N = Nₐ × ρ / A = 6.022 × 10²³ × 8.4 × 10⁶ / 63.5 ≈ 8.0 × 10²⁸ atoms/m³

Nᵥ = 8.0 × 10²⁸ × exp(−0.9 / (8.62 × 10⁻⁵ × 1273)) ≈ 2.2 × 10²⁵ vacancies/m³

Linear Defects (1D) — Dislocations

Mixed dislocations contain both edge and screw character simultaneously. Dislocations are the primary mechanism of plastic deformation in metals — slip occurs when dislocations move through the lattice. This topic will be revisited in the mechanical properties section.

Interfacial / Planar Defects (2D)

External surfaces: Atoms at a surface have fewer bonds than interior atoms; they behave differently, giving rise to surface energy and surface tension.

Grain boundaries: The boundary between two grains of different crystallographic orientation. Grain boundaries impede dislocation motion and therefore increase strength (grain boundary strengthening / Hall–Petch effect).

Phase boundaries: The interface between two regions of different phases (e.g., solid and liquid coexisting).

Twin boundaries: A special grain boundary across which the lattice is a mirror image. Twinning is important in some HCP metals.

Stacking faults: An interruption in the normal ABCABC… or ABABAB… stacking sequence.

Ferromagnetic domain walls: Boundaries separating regions of different magnetic-dipole orientation within the same material.

Bulk / Volume Defects (3D)

These include pores (voids within the material), cracks (planar discontinuities), foreign inclusions (large foreign particles), and corrosion products. They are visible under an optical or electron microscope and have the greatest effect on fracture behaviour.

Atomic vibrations can also be considered a form of defect: at any temperature above 0 K, atoms vibrate about their equilibrium positions. At sufficiently high amplitude (high temperature), these vibrations rupture bonds and cause melting.

Microscopic Examination

Why Microscopy?

Microstructure — the arrangement of grains, grain boundaries, phases, and defects visible at the scale of 100 nm to several mm — profoundly governs material properties. Microscopy lets engineers directly observe and quantify grain size, grain shape, and the distribution of phases.

Optical Microscopy

Optical microscopes use visible light. They can achieve magnifications up to ~2000×. Before examination, the sample surface must be prepared: polished to a smooth finish and then etched with a chemical reagent. Etching preferentially attacks grain boundaries and surface irregularities, revealing the microstructure. Different crystallographic orientations reflect light differently, so polished-and-etched surfaces show grain boundaries as dark lines under reflected light.

Electron Microscopy

For features finer than the resolution of light (~200 nm), electron beams (with much shorter effective wavelengths) are needed.

Transmission Electron Microscope (TEM): A very thin foil of the material is illuminated by an electron beam. The transmitted (not reflected) electrons form an image on a screen or detector. Magnification up to ~1,000,000×. Capable of imaging individual dislocations, grain boundaries, and nanoscale precipitates.

Scanning Electron Microscope (SEM): An electron beam rasters over the sample surface. Reflected (secondary and backscattered) electrons are detected and displayed on a cathode-ray screen. Magnification from 10× to ~50,000×. The surface must be electrically conductive; non-conducting samples are coated with a thin gold or carbon film. SEM gives detailed surface topography and can be coupled with energy-dispersive X-ray spectroscopy (EDS) for elemental analysis.

Scanning Probe Microscopy (SPM)

SPM uses a nanometer-sharp tip that physically scans the sample surface. As the tip moves, the interaction force (in atomic force microscopy, AFM) or the tunnelling current (in scanning tunnelling microscopy, STM) is measured and used to map surface topography. SPM can achieve sub-nanometre resolution, can image individual atoms, and can operate in air, liquid, or vacuum without extensive sample preparation. AFM can also measure forces between atoms and characterize soft biological materials (DNA, cell membranes).

The different microscopy techniques span complementary resolution ranges: optical (~0.2 µm limit), SEM (~1 nm), TEM (~0.1 nm), SPM (~0.01 nm).

Diffusion in Solids

What Is Diffusion?

Diffusion is the transport of atoms (or molecules) through a material by atomic motion. In solids, it is slow but measurable and is the basis for many manufacturing processes: carburizing steel, forming semiconductor junctions by dopant diffusion, and bonding dissimilar materials.

The driving force for diffusion is the concentration gradient: atoms move from regions of high concentration to regions of low concentration, tending toward a uniform distribution.

Mechanisms of Diffusion

Vacancy diffusion: An atom jumps into an adjacent vacant lattice site. The atom’s original site becomes the new vacancy. This mechanism operates for both host atoms (self-diffusion) and substitutional impurities. A significant vacancy concentration is required.

Interstitial diffusion: A small foreign atom jumps from one interstitial site to an adjacent one. Because interstitial sites are more numerous than vacancies and small atoms cause little lattice distortion when moving, interstitial diffusion is generally faster than vacancy diffusion. Carbon in iron and hydrogen in metals are common examples.

Fick’s First Law (Steady-State Diffusion)

Under steady state, the concentration profile does not change with time — the flux entering one face of a membrane equals the flux leaving the other. Fick’s first law relates the diffusion flux to the concentration gradient:

J = −D (dC/dx)

where J = diffusion flux (kg m⁻² s⁻¹ or atoms m⁻² s⁻¹), D = diffusion coefficient (or diffusivity, m² s⁻¹), C = concentration (kg m⁻³ or atoms m⁻³), and x = position (m). The negative sign indicates flow from high to low concentration.

Example: A 2.5 mm thick iron plate separates a carbon-rich atmosphere (C = 1.2 kg/m³) from a carbon-deficient atmosphere (C = 0.8 kg/m³). Given D = 3 × 10⁻¹¹ m²/s:

J = −(3 × 10⁻¹¹) × (0.8 − 1.2) / (10 × 10⁻³ − 5 × 10⁻³) = 2.4 × 10⁻⁹ kg m⁻² s⁻¹

Fick’s Second Law (Non-Steady-State Diffusion)

Under non-steady state, the concentration profile changes with time — there is accumulation within the material. Fick’s second law:

∂C/∂t = D (∂²C/∂x²)

For a semi-infinite solid with a constant surface concentration Cs imposed at time t = 0 on a solid with initial uniform concentration C₀:

(Cx − C₀) / (Cs − C₀) = 1 − erf(x / 2√(Dt))

where Cx = concentration at depth x after time t, and erf is the Gaussian error function (values are tabulated; use linear interpolation between table entries when the argument is not listed exactly).

Example: Steel initially contains 0.25 wt% C. The surface is exposed to a carbon-rich atmosphere maintaining Cs = 1.20 wt% C. Find the time to achieve Cx = 0.80 wt% C at x = 0.5 mm depth, given D = 1.6 × 10⁻¹¹ m² s⁻¹.

(0.80 − 0.25)/(1.20 − 0.25) = 0.5789/0.95 = 0.610 → erf(z) = 1 − 0.610 = 0.390 → z ≈ 0.392 (from table + interpolation)

z = x / 2√(Dt) → t = x² / (4z²D) = (5 × 10⁻⁴)² / (4 × 0.392² × 1.6 × 10⁻¹¹) ≈ 25,400 s ≈ 7.1 hours

Factors Affecting the Diffusion Coefficient

Diffusing species: Interstitial atoms (small, many available sites) diffuse faster than substitutional atoms (need vacancies, larger atoms). In FCC iron, carbon diffuses faster than nickel.

Temperature: The diffusion coefficient follows an Arrhenius relation:

D = D₀ exp(−Qd / RT)

where D₀ = temperature-independent pre-exponential, Qd = activation energy for diffusion (J/mol or eV/atom), R = gas constant (8.314 J/mol·K or 8.62 × 10⁻⁵ eV/K), and T = absolute temperature (K). Increasing temperature dramatically increases diffusivity. A plot of ln D vs. 1/T is linear with slope −Qd/R.

Short-circuit diffusion paths: Diffusion is faster along grain boundaries and external surfaces than through the bulk, because these regions are less ordered. However, their contribution to the overall flux is small because their cross-sectional area is tiny compared with the bulk material.

Mechanical Properties

The Importance of Mechanical Properties

Engineers design structures and devices that must withstand loads — bridges, aircraft, biomedical implants, consumer electronics. To do so, they must understand how materials respond to applied forces. As introduced in Lecture 1, a material property is any characteristic exhibited in response to a stimulus; mechanical properties are the responses to an applied load.

Key mechanical properties include strength (resistance to fracture or permanent deformation), stiffness (resistance to elastic deformation), ductility (ability to deform plastically before fracture), hardness (resistance to surface penetration), and toughness (energy absorbed before fracture).

Mechanical Testing Methods

Four broad test types:

Tensile test: A specimen is pulled in uniaxial tension. The most common mechanical test. Produces the stress–strain curve from which Young’s modulus, yield strength, tensile strength, and ductility are extracted.

Compression test: A specimen is compressed between platens. Used for brittle materials (ceramics, concrete) that cannot be gripped for tensile tests.

Shear test: A shear force is applied parallel to the cross-section. Relevant to adhesive joints, rivets, and bolts.

Torsional test: A specimen is twisted. Relevant to shafts, drills, and fasteners.

Stress and Strain

For the tensile test:

For shear:

• Shear stress τ = F/A₀ (force parallel to the area)
• Shear strain γ = tan θ ≈ θ for small angles, where θ is the angle of deformation

For torsion:

• Shear stress varies across the cross-section; maximum at the outer surface
• Shear strain similarly varies radially

Elastic Deformation and Young’s Modulus

At small stresses, deformation is elastic — the material returns to its original shape when the load is removed. In the elastic regime, stress and strain are proportional:

σ = Eε (Hooke’s Law)

For shear: τ = Gγ, where G is the shear modulus.

For hydrostatic pressure: Δp = −KΔV/V, where K is the bulk modulus.

These three moduli are related through Poisson’s ratio ν:

G = E / [2(1 + ν)] and K = E / [3(1 − 2ν)]

Young’s moduli of selected materials (approximate): diamond ~1000 GPa; steel ~200 GPa; aluminum ~70 GPa; glass ~70 GPa; PTFE (Teflon) ~0.5 GPa. The enormous range reflects differences in bonding — covalent/ionic bonds are stronger and stiffer than metallic bonds, which are in turn much stiffer than Van der Waals forces.

The elastic modulus is an example of an anisotropic property — it depends on crystallographic direction (as noted in the crystallography lectures). Tungsten is an exception, being essentially isotropic.

Summary of Key Formulas

Closing Thoughts

NE 125 establishes the foundational vocabulary of materials science that underpins all subsequent nanotechnology engineering courses. The central thread is simple but profound: structure at every scale — from the spin of a single electron to the orientation of a millimetre-scale grain — determines the macroscopic properties we observe and engineer. Understanding this chain of causation, from bonding to crystal structure to defects to diffusion to mechanical response, equips the nanotechnology engineer to ask the right questions when designing and selecting materials for nanoscale applications.

The emergence of graphene as a one-atom-thick material with extraordinary properties that graphite does not exhibit illustrates this point perfectly: reducing dimensionality to the ultimate limit transforms a mundane lubricant into a material platform that Nobel laureates have devoted careers to exploring. That is the promise of nanotechnology — and it is built on exactly the materials science principles covered in this course.