Imperfection of Solids: Point Defects

• Materials Science and Engineering is the subject being studied in the first semester of the academic year 2024-2025.

Imperfection of Solids

• Point defects are one type of imperfection in solids.

Point Defects

• Vacancies are the simplest type of point defect where an atom is missing from its normal lattice site.
• Vacant lattice site means that the location is normally occupied.
• Self-interstitials occur when an atom from the crystal is crowded into an interstitial site, a small void space not normally occupied.
• Self-interstitials introduce large distortions in the surrounding lattice due to the atom's size relative to the interstitial position.
• The equilibrium number of vacancies (Nv) depends on temperature, increasing as temperature increases.
• Nv is calculated using the formula Nv = N * exp(-Qv / kT),
• N is the total number of atomic sites (m3), Qv is the energy required for vacancy formation (J/mol or eV/atom), T is absolute temperature in Kelvin, k is Boltzmann's constant (1.38 x 10-23 J/atom K).

Self-Interstitial Formation

• Self-interstitial formation is not highly probable, resulting in very low concentrations compared to vacancies.

Number-of-Vacancies Computation Example

• To find the equilibrium number of vacancies per cubic meter for copper at 1000°C the energy vacancy formation is 0.9 eV/atom.
• The atomic weight and density at 1000°C for copper being 63.5 g/mol and 8.4 g/cm³, respectively.
• First, Calculate the total number of atomic sites (N) using Avogadro's number (6.022 x 10^23 atoms/mol), density, and atomic weight.
• Then, use the formula Nv = N * exp(-Qv / kT) to find the number of vacancies (Nv).
• The number of vacancies for copper at 1000°C is approximately 2.2 x 10^25 vacancies/m³.

Impurities in Solids

• Adding impurity atoms to a metal results in a solid solution and/or a new second phase.
• This depends on the type of impurity, its concentration, and the alloy's temperature.
• Impurity atoms can either be substitutional or interstitial.

Solid Solutions

• Solid solutions form when solute atoms are added without changing the crystal structure or forming new structures.
• Impurity atoms are randomly and uniformly dispersed within the solid, making the solution compositionally homogeneous.
• Impurity point defects in solid solutions are either substitutional or interstitial.

Hume-Rothery Rules for Solid Solutions

• Atomic Size Factor: Appreciable solute quantities are accommodated only if the atomic radii difference between the two atom types is less than about 15%. Otherwise, solute atoms cause substantial lattice distortions and a new phase forms.
• Electronegativity Factor: The greater the electronegativity difference between elements, the higher the likelihood of forming an intermetallic compound over a substitutional solid solution.
• Crystal Structure: For appreciable solid solubility, both atom types' metals must have the same crystal structures.
• Valences: A metal tends to dissolve another metal of higher valency more readily than one of lower valency, all other factors being equal.

Interstitial Sites in Crystal Structures

• FCC and BCC crystal structures have two types of interstitial sites: tetrahedral and octahedral.
• These sites are distinguished by the coordination number, which is the number of nearest neighbor host atoms.

Radius Computation of BCC Interstitial Site

• The octahedral interstitial site for BCC is located at the center of a unit cell edge.
• The radius (r) of an impurity atom fitting into a BCC octahedral site without causing lattice strains is related to the host atom's atomic radius (R) by the equation r = ((2 / √3) - 1) * R = 0.155R.

Specification of Composition

• For an alloy containing two hypothetical atoms (1 and 2), the concentration of 1 in weight percent (C1) is defined by C1 = (m1 / (m1 + m2)) * 100, where m1 and m2 are the weights of elements 1 and 2, respectively.
• The number of moles (nm1) of a hypothetical element 1 is computed as nm1 = m'1 / A1, where m'1 is the mass in grams and A1 is the atomic weight of element 1.
• The concentration of element 1 in atom percent (C'1) in an alloy is defined by C'1 = (nm1 / (nm1 + nm2)) * 100.

Composition Conversions

• Converting from weight percent (C1, C2) to atom percent (C'1, C'2) are calculated using the atomic weights (A1, A2) using: C'1 = (C1A2 / (C1A2 + C2A1)) * 100 and C'2 = (C2A1 / (C1A2 + C2A1)) * 100.
• Converting from weight percent to mass of one component per unit volume of material: C'1' = (C1 / (ρ1) / (C1 / ρ1 + C2 / ρ2)) * 10^3 and C'2' = (C2 / (ρ2) / (C1 / ρ1 + C2 / ρ2)) * 10^3.
• Density (ρave) and atomic weight (Aave) of a binary alloy: ρave= 100 / (C1/ρ1 + C2/ρ2) and Aave= (C'1A1 + C'2A'2) /100

Composition Conversion Example

• To determine the composition in atom percent of an alloy with 97 wt% aluminum and 3 wt% copper:
• Use the formula C'Al = (CAl * ACu) / (CAl * ACu + CCu * AAl) * 100,
• Substituting the atomic weights of aluminum (26.98 g/mol) and copper (63.55 g/mol):
• C'Al = (97 * 63.55) / (97 * 63.55 + 3 * 26.98) * 100 = 98.7 at%.

Dislocations - Linear Defects

• A dislocation is a linear or one-dimensional defect around which atoms are misaligned.
• An edge dislocation is a linear defect centered on the line defined along the end of an extra half-plane of atoms.
• The dislocation line is perpendicular to the plane of the page in the case of an edge dislocation.
• The density of dislocations in a crystal is measured by counting etch-pit points at random cross-sections.

Screw Dislocation

• A screw dislocation is formed through a shear stress that has caused distortion.
• Atomic distortion for a screw dislocation is linear and along a dislocation line.

Mixed Dislocations

• Most dislocations are neither pure edge nor pure screw but exhibit components of both types.
• The magnitude and direction of the lattice distortion is expressed in terms of a Burgers vector, denoted by b.

Interfacial Defects

• Interfacial defects are boundaries with two dimensions that separate regions of materials with different crystal structures or crystallographic orientations.

Types of Imperfections Include:

• External surfaces
• Grain boundaries
• Phase boundaries
• Twin boundaries
• Stacking faults.

External Surfaces

• Surface atoms are not bonded to the maximum number of nearest neighbors, thus existing in a higher energy state compared to atoms at interior positions.

Grain Boundaries

• Within the boundary region of grain boundaries, there is some atomic mismatch transitioning from the crystalline orientation of one grain to another.
• The misalignment between adjacent grains can have various degrees of crystallographic variance.
• A tilt boundary is a small-angle grain boundary formed when edge dislocations align.
• A twist boundary occurs when the angle of misorientation is parallel to the boundary, described by an array of screw dislocations.

Phase Boundaries

• Phase boundaries exist in multiphase materials, where a different phase exists on each boundary side.
• Phase boundaries influence the mechanical characteristics of metal alloys.
• The region of material between these boundaries is termed a twin.

Twin Boundaries

• Special type of grain boundary across which there is a specific mirror lattice symmetry.
• Atoms on one side of the boundary are in mirror image positions to the atoms on the other side.

Bulk or Volume Defects

• Volume defects are three dimensional, including pores, cracks, foreign inclusions, and other phases.
• Dispersion hardening is a procedure where foreign particles act as obstacles to dislocation movement, facilitating plastic deformation.

Atomic Vibrations

• Every atom in a solid vibrates rapidly around its lattice position within the crystal.
• Rising temperature increases average energy, with temperature measuring the average vibrational activity.
• At room temperature, typical vibrational frequency is 10^13 vibrations per second, and amplitude is a few thousandths of a nanometer.

Microscopic Techniques

• Optical microscopy uses a light microscope to study the microstructure, along with optical and illumination systems.
• Metallography uses microscopic techniques to examine metals.
• Small grooves form along grain boundaries due to etching, as atoms in these regions are chemically active and dissolve faster.

Electron Microscopy

• Transmission Electron Microscopy: An electron beam passes through the specimen, revealing internal microstructural features through beam scattering differences. Can reach magnifications approaching 1,000,000, and is frequently used to study dislocations.
• Scanning Electron Microscopy: The specimen's surface is scanned with an electron beam, and the reflected beam is collected and displayed. Magnifications range from 10 to over 50,000, with great depths of field.

Scanning Probe Microscopy

• Scanning Probe Microscopy generates a topographical map, on an atomic scale with a representation of surface features.
• Examination is done on the nanometer scale with magnifications as high as 10^9 are possible; much better resolutions are attainable than with other microscopic techniques
• It generates three-dimensional magnified images, providing topographical information about features of interest.
• These microscopes may be operated in vacuum, air, and liquid, in its most suitable environment.

Grain-Size Determination

• Grain size is often determined when considering the properties of polycrystalline and single-phase materials.
• There are generally two common grain-size determination techniques: Linear intercept method or comparison method.

Linear Intercept Method:

• Number of grain boundary intersections by straight test lines is counted to determine grade size.

Comparison Method:

• Grain Structures are compared with standardized charts based on grade areas.
• Linear intercept method involves drawing random lines through photomicrographs and counting the grain boundaries intersected.