GATE Metallurgical Engineering Physical Metallurgy Syllabus

Contrast diamond and graphite as covalent solids in terms of bonding and properties.

Diamond: $sp^{3}$ tetrahedral 3-D network, hard, electrically insulating. Graphite: $sp^{2}$ hexagonal layers with delocalized $\pi$ electrons within layers and weak van der Waals bonding between layers, making it soft, a lubricant, and electrically conducting along the layers.

Define a polymer and distinguish thermoplastics from thermosets.

A polymer is a long-chain macromolecule built from repeating monomer units linked by covalent bonds. Thermoplastics consist of linear/branched chains held by secondary bonds (soften on heating, remoldable); thermosets are cross-linked networks (rigid, decompose rather than melt on heating).

Define the degree of polymerization and number-average molecular weight $\bar{M}_n$.

The degree of polymerization $n = \dfrac{\bar{M}_n}{m_0}$ is the average number of monomer units per chain, where $m_0$ is the monomer molar mass. The number-average molecular weight is $\bar{M}_n = \sum x_i M_i$, with $x_i$ the number fraction of chains of mass $M_i$.

What is the glass transition temperature $T_g$ of a polymer?

$T_g$ is the temperature below which an amorphous (or amorphous regions of a semicrystalline) polymer becomes hard and glassy, and above which it becomes soft and rubbery due to the onset of large-scale chain segment motion.

State Bragg's law and define each symbol.

$$n\lambda = 2 d \sin\theta$$ where $n$ is the order of diffraction (integer), $\lambda$ the X-ray wavelength, $d$ the interplanar spacing, and $\theta$ the Bragg angle (angle between incident beam and the plane).

What physical condition must be satisfied for constructive interference in Bragg diffraction?

The path difference between rays reflected from adjacent parallel planes, equal to $2d\sin\theta$, must be an integer multiple of the wavelength $\lambda$, so the waves are in phase.

For a cubic crystal of lattice parameter $a$, write the interplanar spacing $d_{hkl}$ and combine it with Bragg's law.

$$d_{hkl} = \frac{a}{\sqrt{h^{2}+k^{2}+l^{2}}}$$ Substituting into Bragg's law gives $\sin^{2}\theta = \dfrac{\lambda^{2}}{4a^{2}}\left(h^{2}+k^{2}+l^{2}\right)$.

Which reflections ($hkl$) are allowed for FCC and for BCC crystals in X-ray diffraction?

FCC: reflections present only when $h,k,l$ are all even or all odd (unmixed). BCC: reflections present only when $h+k+l$ is even.

What is optical (light) metallography used for and what is the typical sample preparation sequence?

Optical metallography uses a reflected-light microscope to examine microstructure (grains, phases, inclusions). Preparation: sectioning, mounting, grinding (progressively finer abrasives), polishing to a mirror finish, and etching to reveal microstructural features.

Why is etching necessary in optical metallography and how does it reveal grain boundaries?

Etching is a controlled chemical attack that preferentially dissolves grain boundaries and different phases at different rates, creating surface relief and reflectivity differences; under reflected light, boundaries appear dark because they scatter light away from the objective.

What is the approximate maximum useful magnification and resolution limit of optical metallography, and what sets the limit?

Maximum useful magnification is about $\sim 1000$–$2000\times$ with a resolution limit of roughly $0.2\ \mu\mathrm{m}$, set by the wavelength of visible light through the Abbe diffraction limit $d \approx \dfrac{0.61\lambda}{\mathrm{NA}}$.

What is the basic working principle of the Scanning Electron Microscope (SEM)?

A focused beam of high-energy electrons is rastered across the specimen surface; signals (secondary electrons, backscattered electrons, X-rays) emitted from each point are detected and synchronized with the scan to build a magnified image.

Compare secondary electrons (SE) and backscattered electrons (BSE) as SEM imaging signals.

Secondary electrons are low-energy ($<50\ \mathrm{eV}$) electrons ejected from near the surface, giving high-resolution topographic contrast. Backscattered electrons are high-energy beam electrons elastically scattered back; their yield increases with atomic number $Z$, giving compositional (atomic-number) contrast.

Why does the SEM offer much greater depth of field and resolution than the optical microscope?

Electrons have a far shorter effective wavelength than visible light (giving higher resolution, $\sim$ few nm), and the small beam-convergence angle gives a large depth of field, so rough surfaces appear sharply in focus over a wide height range.

What is a point defect? Name the three main types in a crystal.

A point defect is a zero-dimensional disruption of the lattice at a single atomic site. The main types are vacancies (missing atoms), self-interstitials (extra atoms in interstitial sites), and substitutional/interstitial impurity atoms.

Write the equilibrium concentration of vacancies as a function of temperature.

$$\frac{n_v}{N} = \exp\left(-\frac{Q_v}{k_B T}\right)$$ where $n_v/N$ is the vacancy fraction, $Q_v$ the energy of vacancy formation, $k_B$ Boltzmann's constant, and $T$ the absolute temperature.

Distinguish Schottky and Frenkel defects in ionic crystals.

A Schottky defect is a paired cation–anion vacancy that preserves charge neutrality. A Frenkel defect is an ion displaced from its lattice site into an interstitial position, creating a vacancy–interstitial pair (usually the smaller cation).

What is a line defect (dislocation), and what are its two pure types?

A line defect, or dislocation, is a one-dimensional defect around which atoms are misaligned. The two pure types are the edge dislocation (an extra half-plane of atoms) and the screw dislocation (a helical lattice distortion).

Define the Burgers vector and state its orientation relative to the dislocation line for edge and screw dislocations.

The Burgers vector $\vec{b}$ measures the magnitude and direction of lattice distortion (slip) caused by a dislocation. It is perpendicular ($\perp$) to the line for an edge dislocation and parallel ($\parallel$) to the line for a screw dislocation.

How does dislocation motion (slip) relate to plastic deformation, and why does it lower the required stress?

Plastic deformation occurs by dislocations gliding on slip planes; only one row of bonds breaks and reforms at a time as the dislocation moves, so the theoretical shear strength is reduced by orders of magnitude compared with rigidly shearing the whole plane at once.

Define dislocation density and give its typical range for annealed vs heavily cold-worked metals.

Dislocation density is the total dislocation line length per unit volume (units $\mathrm{m/m^{3}} = \mathrm{m^{-2}}$). It is $\sim 10^{10}\ \mathrm{m^{-2}}$ in annealed metals and rises to $\sim 10^{15}$–$10^{16}\ \mathrm{m^{-2}}$ after heavy cold work.

What are surface (interfacial) defects? List the principal types.

Surface defects are two-dimensional boundaries separating regions of different crystallographic orientation or structure. Principal types: external (free) surfaces, grain boundaries, twin boundaries, stacking faults, and phase boundaries.

Distinguish low-angle from high-angle grain boundaries.

A low-angle (sub-) grain boundary has a misorientation $\lesssim 10$–$15^{\circ}$ and can be described as an array of dislocations (e.g. a tilt boundary of edge dislocations), with low boundary energy. A high-angle boundary has greater misorientation, a more disordered structure, and higher energy.

What is a twin boundary and how does a twin relate to the parent crystal?

A twin boundary is a special surface defect across which the lattice is a mirror image of the parent orientation. Twins form by annealing (annealing twins) or by shear (mechanical/deformation twins), and the boundary has very low energy when coherent.

Define a stacking fault and give an example in an FCC crystal.

A stacking fault is a localized error in the regular stacking sequence of close-packed planes. In FCC the ideal sequence is $\ldots ABCABC\ldots$; a fault such as $\ldots ABC\,AB\,ABC\ldots$ introduces a thin region of HCP-like stacking and is bounded by partial dislocations.

What is a coherent interface, and how does its energy compare with other interfaces?

A coherent interface is one where the two crystals' lattice planes are continuous across the boundary with one-to-one matching of atoms (perfect registry). It has the lowest interfacial energy ($\sim 1$–$200\ \mathrm{mJ/m^{2}}$) but may carry coherency strain energy if lattice parameters differ slightly.

What is coherency strain, and what happens to a coherent precipitate as it grows?

Coherency strain is the elastic strain stored when matched lattices have slightly different spacings, forcing them to distort to stay continuous. As a coherent precipitate grows, the total strain energy rises until it becomes energetically favorable to introduce misfit dislocations, transforming it to a semi-coherent interface.

Describe a semi-coherent interface and how the lattice misfit is accommodated.

A semi-coherent interface has partial atomic matching: regions of coherency are periodically interrupted by misfit (edge) dislocations that absorb the lattice mismatch. Its energy is intermediate between coherent and incoherent interfaces.

Define the lattice misfit $\delta$ and give the approximate spacing of misfit dislocations in a semi-coherent interface.

The misfit is $\delta = \dfrac{d_\beta - d_\alpha}{d_\alpha}$, where $d_\alpha$ and $d_\beta$ are the unstressed interplanar spacings of the two phases. Misfit dislocations are spaced approximately $D \approx \dfrac{b}{\delta}$ apart, where $b$ is the Burgers vector magnitude.

What is an incoherent interface and why does it have the highest interfacial energy?

An incoherent interface forms when the two adjoining crystals have very different structures or orientations so there is essentially no atomic matching across the boundary. The disordered, broken-bond structure gives the highest interfacial energy ($\sim 500$–$1000\ \mathrm{mJ/m^{2}}$), similar to a high-angle grain boundary.

State Fick's first law of diffusion and define its terms.

$$J = -D \frac{\partial C}{\partial x}$$ where $J$ is the diffusion flux (atoms per unit area per unit time), $D$ the diffusion coefficient, and $\dfrac{\partial C}{\partial x}$ the concentration gradient. The minus sign shows flux is down the gradient (steady-state diffusion).

State Fick's second law of diffusion (the diffusion equation) for non-steady-state diffusion.

$$\frac{\partial C}{\partial t} = D \frac{\partial^{2} C}{\partial x^{2}}$$ (for constant $D$), describing how concentration $C$ at position $x$ changes with time $t$ during non-steady-state diffusion.

Write the temperature dependence of the diffusion coefficient (Arrhenius relation) and define the terms.

$$D = D_0 \exp\left(-\frac{Q_d}{R T}\right)$$ where $D_0$ is the pre-exponential (frequency) factor, $Q_d$ the activation energy for diffusion, $R$ the gas constant, and $T$ the absolute temperature.

Give the error-function solution of Fick's second law for a semi-infinite solid with constant surface concentration (e.g. carburizing).

$$\frac{C_x - C_0}{C_s - C_0} = 1 - \operatorname{erf}\left(\frac{x}{2\sqrt{D t}}\right)$$ where $C_0$ is the initial concentration, $C_s$ the constant surface concentration, and $C_x$ the concentration at depth $x$ after time $t$.

What is the physical meaning of the diffusion length $\sqrt{Dt}$?

$\sqrt{Dt}$ is the characteristic distance over which significant diffusion occurs in time $t$; the depth of a diffusion profile scales as $x \propto \sqrt{Dt}$, so to keep the same penetration the product $Dt$ must be held constant.

Compare interstitial and substitutional (vacancy) diffusion mechanisms and their relative rates.

Interstitial diffusion proceeds by small atoms (e.g. C, N, H) jumping between interstitial sites and is fast, with low activation energy. Substitutional diffusion requires an adjacent vacancy for an atom to jump into (vacancy mechanism) and is much slower, with higher activation energy.