11.3 Planar defects and grain boundaries

Planar defects are two-dimensional imperfections in crystal structures that significantly impact material properties. These include grain boundaries, stacking faults, twin boundaries, and antiphase boundaries, each with unique characteristics and effects on material behavior.

Understanding planar defects is crucial for manipulating material properties. Grain boundaries, for instance, affect strength through the Hall-Petch effect, while twin boundaries can enhance ductility. Characterization techniques like electron microscopy and X-ray diffraction help engineers optimize materials for specific applications.

Types of planar defects

• Planar defects are two-dimensional imperfections in the crystal structure that extend over a significant area within the material
• They can have a profound impact on the mechanical, electrical, and chemical properties of materials
• Different types of planar defects include grain boundaries, stacking faults, twin boundaries, and antiphase boundaries

Grain boundaries vs stacking faults

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• Grain boundaries form when two grains with different crystallographic orientations meet, resulting in a region of atomic mismatch and disorder
  • They can be classified as low-angle (misorientation angle < 15°) or high-angle (misorientation angle > 15°) boundaries
• Stacking faults occur when there is an error in the regular stacking sequence of atomic planes, often due to the insertion or removal of a partial plane
  • Examples include intrinsic stacking faults (removal of a plane) and extrinsic stacking faults (insertion of an extra plane)

Twin boundaries

• Twin boundaries form when two crystals share a common crystallographic plane, but one is a mirror image of the other
• They can be classified as mechanical twins (formed by deformation) or annealing twins (formed during heat treatment)
• Twin boundaries have a specific misorientation angle that depends on the crystal structure (60° for FCC, 86.3° for BCC)
• Examples of twinning include deformation twinning in magnesium alloys and annealing twins in copper

Antiphase boundaries

• Antiphase boundaries (APBs) occur in ordered alloys when the ordered structure is disrupted by a shift in the atomic positions
• They separate two domains that have the same crystal structure but are out of phase with each other
• APBs can form during ordering transformations or as a result of plastic deformation
• Examples include APBs in ordered intermetallic compounds like Ni3Al and Fe3Al

Structure of grain boundaries

Low-angle vs high-angle boundaries

• Low-angle grain boundaries (misorientation angle < 15°) consist of an array of dislocations that accommodate the small misorientation between grains
  • The dislocation spacing decreases with increasing misorientation angle
• High-angle grain boundaries (misorientation angle > 15°) have a more complex structure with a higher degree of atomic disorder
  • They cannot be described by simple dislocation arrays and require more advanced models

Coincidence site lattice (CSL) model

• The CSL model describes special high-angle grain boundaries with a high degree of atomic matching across the boundary
• CSL boundaries are characterized by a Σ value, which represents the reciprocal density of coincidence sites
  • Examples include Σ3 (twin boundary), Σ5, and Σ11 boundaries
• CSL boundaries often have lower energy and better properties compared to general high-angle boundaries

Microscopic degrees of freedom

• Grain boundaries have five microscopic degrees of freedom: three for the relative orientation of the grains and two for the boundary plane orientation
• These degrees of freedom determine the atomic structure and properties of the grain boundary
• Grain boundary engineering aims to control these degrees of freedom to optimize the material's properties

Energy of grain boundaries

Elastic strain energy

• Grain boundaries introduce elastic strain in the surrounding lattice due to the atomic mismatch and dislocations present
• The elastic strain energy increases with increasing misorientation angle up to about 15°, after which it remains relatively constant
• The strain energy contributes to the overall energy of the grain boundary

Chemical energy

• The chemical energy of a grain boundary arises from the difference in bonding and coordination of atoms at the boundary compared to the bulk
• It depends on the local atomic structure and composition of the boundary
• Segregation of solute atoms to the grain boundary can lower the chemical energy and stabilize the boundary

Relationship between energy and misorientation angle

• The grain boundary energy generally increases with increasing misorientation angle up to about 15°, after which it remains relatively constant
• Special low-energy boundaries, such as CSL boundaries, can have significantly lower energies than general high-angle boundaries
• The energy of a grain boundary also depends on the boundary plane orientation, with certain orientations being more favorable than others

Mechanical properties of grain boundaries

Strengthening mechanisms

• Grain boundaries act as obstacles to dislocation motion, leading to strengthening of the material
• The two main strengthening mechanisms are dislocation pile-up at grain boundaries and the Hall-Petch effect
• Dislocation pile-up occurs when dislocations accumulate at a grain boundary, creating a stress concentration that can trigger slip in the adjacent grain
• The Hall-Petch effect relates the yield strength of a material to its grain size, with smaller grains leading to higher strength

Hall-Petch relationship

• The Hall-Petch relationship is given by: $\sigma_y = \sigma_0 + k_y d^{-1/2}$, where $\sigma_y$ is the yield strength, $\sigma_0$ is the friction stress, $k_y$ is the Hall-Petch coefficient, and $d$ is the grain size
• The relationship arises from the increased difficulty of dislocation motion in fine-grained materials due to the higher density of grain boundaries
• The Hall-Petch effect is most pronounced in materials with grain sizes larger than ~20 nm, below which other deformation mechanisms may dominate

Grain boundary sliding and creep

• Grain boundary sliding is a deformation mechanism where grains slide past each other along their boundaries, accommodated by diffusion or dislocation motion
• It becomes more prominent at high temperatures (typically > 0.4 Tm, where Tm is the melting temperature) and low strain rates
• Grain boundary sliding contributes to creep deformation, where materials deform slowly under constant load at elevated temperatures
• Fine-grained materials are more susceptible to grain boundary sliding and creep due to the higher density of grain boundaries

Kinetic properties of grain boundaries

Grain boundary diffusion

• Grain boundary diffusion is the enhanced diffusion of atoms along grain boundaries compared to bulk diffusion
• It occurs due to the higher atomic disorder and free volume present at grain boundaries
• Grain boundary diffusion plays a key role in various kinetic processes, such as sintering, creep, and superplasticity
• The activation energy for grain boundary diffusion is typically 0.4-0.7 times that of bulk diffusion

Grain boundary migration and grain growth

• Grain boundary migration is the movement of grain boundaries to reduce the overall grain boundary area and energy
• It is driven by the curvature of the grain boundary, with convex boundaries moving towards their center of curvature
• Grain growth occurs when larger grains grow at the expense of smaller grains through grain boundary migration
• The average grain size increases with time according to the equation: $D^n - D_0^n = kt$, where $D$ is the average grain size at time $t$, $D_0$ is the initial grain size, $n$ is the grain growth exponent (typically 2-4), and $k$ is a temperature-dependent constant

Grain boundary segregation

• Grain boundary segregation is the enrichment of solute atoms at grain boundaries compared to the bulk
• It occurs due to the lower energy of solute atoms at grain boundaries and the strain relief provided by their segregation
• Segregation can have both positive and negative effects on material properties, depending on the solute and matrix combination
• Examples include the segregation of bismuth in copper, which causes embrittlement, and the segregation of boron in nickel, which enhances grain boundary cohesion

Characterization techniques for grain boundaries

Electron microscopy (TEM, SEM, EBSD)

• Transmission electron microscopy (TEM) provides high-resolution imaging of grain boundaries and their atomic structure
  • It can reveal the presence of dislocations, segregation, and other features at the boundary
• Scanning electron microscopy (SEM) allows for the visualization of grain boundaries on the surface of a sample
  • It can be used to study grain size, morphology, and distribution
• Electron backscatter diffraction (EBSD) is a technique that provides information on the crystallographic orientation of grains and the misorientation between them
  • It can be used to construct grain boundary maps and study the distribution of special boundaries (e.g., CSL boundaries)

X-ray diffraction

• X-ray diffraction (XRD) can be used to study the average grain size and microstrain in a material
• The broadening of XRD peaks is related to the grain size through the Scherrer equation: $D = K\lambda / (\beta \cos\theta)$, where $D$ is the grain size, $K$ is a shape factor, $\lambda$ is the X-ray wavelength, $\beta$ is the peak width, and $\theta$ is the diffraction angle
• XRD can also be used to study the texture (preferred orientation) of grains in a material

Atomic force microscopy (AFM)

• Atomic force microscopy (AFM) is a high-resolution scanning probe technique that can image grain boundaries on the surface of a sample
• It can provide information on the topography and mechanical properties (e.g., stiffness, adhesion) of grain boundaries
• AFM can be used to study the early stages of grain growth and the interaction of grain boundaries with other defects (e.g., dislocations, precipitates)

Technological applications of grain boundaries

Nanocrystalline materials

• Nanocrystalline materials have grain sizes less than 100 nm and a high volume fraction of grain boundaries
• They exhibit unique mechanical, electrical, and magnetic properties due to the increased influence of grain boundaries
• Examples include nanocrystalline metals with high strength and hardness, and nanocrystalline ceramics with enhanced ductility and toughness
• The properties of nanocrystalline materials can be tailored by controlling the grain size and grain boundary structure

Grain boundary engineering

• Grain boundary engineering is the process of controlling the distribution of grain boundaries in a material to optimize its properties
• It involves the selective introduction of special boundaries (e.g., CSL boundaries) through thermomechanical processing
• Grain boundary engineered materials can have improved strength, ductility, corrosion resistance, and resistance to intergranular cracking
• Examples include grain boundary engineered nickel-based superalloys for aerospace applications and grain boundary engineered stainless steels for nuclear reactors

Role in corrosion and stress corrosion cracking

• Grain boundaries are often preferential sites for corrosion and stress corrosion cracking (SCC) due to their higher energy and diffusivity compared to the bulk
• Intergranular corrosion occurs when grain boundaries are preferentially attacked by a corrosive environment
• SCC involves the combined action of stress and a corrosive environment, leading to the propagation of cracks along grain boundaries
• The susceptibility of a material to corrosion and SCC can be reduced by grain boundary engineering, which introduces more resistant boundaries (e.g., CSL boundaries)
• Other strategies include the use of corrosion-resistant coatings, the control of grain size and shape, and the addition of alloying elements that promote grain boundary cohesion