Solidification

• Solidification is the result of casting molten material and involves two steps:

  • Nuclei form.

  • Nuclei grow to form crystals, resulting in a grain structure.

• The process starts with a molten material that is entirely liquid.

• Crystals grow until they meet each other, forming the grain structure.

Grain Boundaries

• Grain boundaries are regions between crystals.

• They represent the transition from the lattice of one region to that of another.

• Grain boundaries are slightly disordered.

• They have low density, resulting in:

  • High mobility

  • High diffusivity

  • High chemical reactivity

• Polycrystalline material contains many grain boundaries.

Imperfections in Solids

• No crystal is perfect; imperfections always exist.

• Imperfections are important because many material properties are due to their presence.

Types of Imperfections

• Point Defects

  • Vacancy atoms

  • Interstitial atoms

  • Substitutional atoms

• Line Defects

  • Dislocations

• Area Defects

  • Grain Boundaries

Point Defects: Vacancies

• The simplest point defect is a vacancy, which is a missing atom at a lattice site.

• Vacancies increase the randomness (entropy) of the crystal, leading to a lack of order.

• The equilibrium number of vacancies Nv for a given material depends on temperature and can be calculated using the following equation: Nv = N \exp \left( -\frac{Q_v}{kT} \right)

  • Where:

    • N = total number of atomic sites

    • T = Temperature in Kelvins

    • k = Boltzmann’s constant (1.38 × 10^{-23} J/atom-K or 8.62 × 10^{-5} eV/atom-K)

    • Q_v = Energy required for vacancy formation

• Absolute temperature in Kelvins (K) is given by \degree C + 273.

• Boltzmann’s constant per mole of atoms becomes the gas constant R = 8.31 J/mol-K.

• For most metals, the fraction of vacancies \frac{N_v}{N} just below the melting temperature is approximately 10^{-4}, meaning one lattice site out of 10,000 will be empty.

Point Defects: Self-Interstitial

• A self-interstitial is an atom from the crystal that occupies a small void space (interstitial site) that is normally unoccupied.

• Self-interstitials introduce relatively large distortions in the surrounding lattice.

Calculating Number of Vacancies

• Example: Calculate the equilibrium number of vacancies per cubic meter for copper at 1000°C.

  • Given:

    • Energy for vacancy formation, Q_v = 0.9 eV/atom

    • Atomic weight of copper, A_{Cu} = 63.5 g/mol

    • Density of copper at 1000°C, \rho = 8.4 g/cm^3

Solution

• First, find the total number of atomic sites N using the equation: N = NA \frac{\rho}{A{Cu}}

  • Where: N_A = 6.023 \times 10^{23} atoms/mol
    N = \frac{(6.023 \times 10^{23} atoms/mol) \times (8.4 g/cm^3)}{63.5 g/mol} = 8.0 \times 10^{28} atoms/m^3

• Convert 1000°C to Kelvin:
  T = 1000 \degree C + 273 = 1273 K

• The number of vacancies at 1000°C is:
  Nv = N \exp \left( -\frac{Qv}{kT} \right)
  N_v = (8.0 \times 10^{28} atoms/m^3) \exp \left( -\frac{0.9 eV}{(8.62 \times 10^{-5} eV/K) \times (1273 K)} \right) = 2.2 \times 10^{25} vacancies/m^3

Impurities in Solids

• It is impossible to have a metal with only one type of atom.

• Foreign atoms (impurities) will always be present and exist as crystalline point defects.

• Even at 99.999% purity, there are approximately 10^{22} to 10^{23} impurity atoms per cubic meter of material.

• This is why alloys are created by intentionally adding impurity atoms to impart specific characteristics to the material.

• Adding impurity atoms to metals results in solid solutions and/or a new second phase.

Solid Solution

• Solute and solvent are terms used to describe a solution.

  • Solvent: The element or compound in a large quantity.

  • Solute: The element or compound in a minor concentration.

• In a solid solution, the crystal structure is maintained, and no new structures are formed.

Types of Solid Solutions

• Impurity point defects in solid solutions are of two types:

  • Substitutional solution

  • Interstitial solution

• Substitutional solution: Impurity atoms replace or substitute host atoms.

• Interstitial solution: Impurity atoms fill the voids or interstices among host atoms.

Factors Determining Solid Solubility

• The degree to which a solvent dissolves a solute depends on the following factors:

  • Atomic Size Factor: The difference in atomic radii of the two atoms must be less than \pm 15\%.

  • Crystal Structure: Both atom types must have the same crystal structure.

  • Electronegativity: If one atom is more electronegative and the other more electropositive, they may form an intermetallic compound instead of a substitutional solid solution.

  • Valences: One atom is more likely to dissolve in another when it has a higher valence.

Example of Substitutional Solid Solution

• Copper and nickel form a substitutional solid solution.

  • Atomic radii: Copper (0.128 nm), Nickel (0.125 nm)

  • Crystal structure: Both have FCC crystal structure

  • Electronegativity: Copper (1.9), Nickel (1.8)

  • Valences: Copper (+1), Nickel (+2)

Dislocations – Linear Defects

• Virtually all crystalline materials contain dislocations.

• Dislocations can be introduced during:

  • Solidification

  • Plastic deformation

  • Thermal stresses due to rapid cooling

• Dislocations can be found in metals, ceramics, and polymers.

• Dislocations can be observed in crystalline materials using electron microscopes.

Observation of Dislocations

• Mainly by transmission electron microscopy (TEM) with image formed by:

  • 'Misfit' dislocations at interface

  • Dark-field (Ni3Al) either diffraction contrast (conventional TEM) or phase contrast (HREM - atomic resolution)

What is a Dislocation?

• A dislocation is a linear or one-dimensional defect around which some of the atoms are misaligned.

• Dislocations can be classified as:

  • Edge dislocation

  • Screw dislocation

  • Mixed dislocation

• Edge dislocation: A linear defect that centers around a line defined along the end of the extra half-plane of atoms.

Shear Stress and Dislocation Motion

• When a shear stress is applied to a dislocation, atoms are displaced, causing the dislocation to move one Burgers vector in the slip direction.

• Continued movement of the dislocation creates a step, and the crystal is deformed.

• The motion of a caterpillar (or a fold in a rug) is analogous to the motion of a dislocation.

• The slip direction is always in the direction of the Burgers vector of the dislocation.

Screw Dislocation

• Screw dislocation is a linear defect formed by shear stress that produces distortion.

• One region of the crystal is shifted one atomic distance relative to the other.

Mixed Dislocation

• Most dislocations found in crystalline materials are neither pure edge nor pure screw dislocations.

• A combination of the three dislocations exists simultaneously and are called mixed dislocations.

Burgers Vector

• Burgers vector expresses the magnitude and direction of the lattice distortion.

• The nature of a dislocation is defined by the relative orientations of the dislocation line and Burgers vector:

  • Edge dislocation: Perpendicular

  • Screw dislocation: Parallel

  • Mixed dislocation: Neither perpendicular nor parallel

Interfacial Defects

• Interfacial defects are boundaries that have two dimensions and normally separate regions of materials that have different crystal structures and/or crystallographic orientations.

• Interfacial defects can fall into the following categories:

  • External surfaces

  • Grain boundaries

  • Twin boundaries

  • Stacking faults

  • Phase boundaries

External Surfaces

• Surface atoms of a crystal that are not bonded to the maximum number of nearest neighbors are not satisfied and give rise to surface energy (J/m² or erg/cm²).

• To reduce this energy, materials tend to minimize the total surface energy if possible.

Grain Boundaries

• A grain boundary is a boundary separating two small grains or crystals having different crystallographic orientations in polycrystalline materials.

• Various degrees of crystallographic misalignment between adjacent grains are possible.

Twin Boundaries

• A twin boundary is a special type of grain boundary across which there is a specific mirror lattice symmetry.

• Atoms on one side of the boundary are located in mirror-image positions of the atoms on the other side.

• The region of material between these boundaries is termed a twin.