Tilt and Twist Grain Boundaries

Low angle grain boundary is an array of aligned edge dislocations. This type of grain boundary is called tilt boundary (consider joint of two wedges).

Transmission electron microscope image of a small angle tilt boundary in Si. The red lines mark the edge dislocations, the blue lines indicate the tilt angle.

Twist boundary - the boundary region consisting of arrays of screw dislocations (consider joint of two halves of a cube and twist an angle around the cross section normal).

Low-energy twin boundaries with mirrored atomic positions across boundary may be produced by deformation of materials. This gives rise to shape memory metals, which can recover their original shape if heated to a high temperature. Shape-memory alloys are twinned and when deformed they untwin. At high temperature the alloy returns back to the original twin configuration and restore the original shape.

Electron Microscopy

Dislocations in Nickel (the dark lines and loops), transmission electron microscopy image, Manchester Materials Science Center.

High-resolution Transmission Electron Microscope image of a tilt grain boundary in aluminum, Sandia National Lab.

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Interfacial Defects

External Surfaces:

Surface atoms have have unsatisfied atomic bonds, and higher energies than the bulk atoms ⇒ Surface energy, γ (J/m2)

• Surface areas tend to minimize (e.g. liquid drop)

• Solid surfaces can “reconstruct” to satisfy atomic bonds at surfaces.

Grain Boundaries:

Polycrystalline material comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries. Surfaces and interfaces are reactive and impurities tend to segregate there. Since energy is associated with interfaces, grains tend to grow in size at the expense of smaller grains to minimize energy. This occurs by diffusion (Chapter 5), which is accelerated at high temperatures.

High and Low Angle Grain Boundaries:

Depending on misalignments of atomic planes between adjacent grains we can distinguish between the low and high angle grain boundaries.

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Edge and screw dislocations

Dislocations shown in previous slide are edge dislocations, have Burgers vector directed perpendicular to the dislocation line.

There is a second basic type of dislocation, called screw dislocation. The screw dislocation is parallel to the direction in which the crystal is being displaced (Burgers vector is parallel to the dislocation line).

Mixed/partial dislocations (not tested)

The exact structure of dislocations in real crystals is usually more complicated than the ones shown in this pages. Edge and screw dislocations are just extreme forms of the possible dislocation structures. Most dislocations have mixed edge/screw character.

To add to the complexity of real defect structures, dislocation are often split in "partial" dislocations that have their cores spread out over a larger area.

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Description of Dislocations—Burgers Vector

To describe the size and the direction of the main lattice distortion caused by a dislocation we should introduce socalled Burgers vector b. To find the Burgers vector, we should make a circuit from from atom to atom counting the same number of atomic distances in all directions. If the circuit encloses a dislocation it will not close. The vector that closes the loop is the Burgers vector b.

Dislocations shown above have Burgers vector directed perpendicular to the dislocation line. These dislocations are called edge dislocations.

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Composition Conversions

Weight % to Atomic %:

Atomic % to Weight %:

Dislocations are linear defects: the interatomic bonds are significantly distorted only in the immediate vicinity of the dislocation line. This area is called the dislocation core. Dislocations also create small elastic deformations of the lattice at large distances.

Dislocations are very important in mechanical properties of material. Introduction/discovery of dislocations in 1934 by Taylor, Orowan and Polyani marked the beginning of our understanding of mechanical properties of materials.

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Interstitial Solid Solutions

Interstitial solid solution of C in α-Fe. The C atom is small enough to fit, after introducing some strain into the BCC lattice.

Factors for high solubility:

• For fcc, bcc, hcp structures the voids (or interstices) between the host atoms are relatively small ⇒ atomic radius of solute should be significantly less than solvent Normally, max. solute concentration ≤ 10%, (2% for C-Fe).

Composition / Concentration:

Composition can be expressed in

• weight percent, useful when making the solution
• atom percent, useful when trying to understand the material at the atomic level

Weight percent (wt %): weight of a particular element relative to the total alloy weight. For two component system, concentration of element 1 in wt. % is

Atom percent (at %): number of moles (atoms) of a particular element relative to the total number of moles (atoms) in alloy. For two-component system, concentration of element 1 in at. % is

where nm1 = m'1/A1 m'1 is weight in grams of element 1, A1 is atomic weight of element 1)

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Substitutional Solid Solutions

Factors for high solubility:

• Atomic size factor - atoms need to “fit” ⇒ solute and solvent atomic radii should be within ~ 15%

• Crystal structures of solute and solvent should be the same

• Electronegativities of solute and solvent should be comparable (otherwise new inter-metallic phases are encouraged)

• Generally more solute goes into solution when it has higher valency than solvent

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Other point defects: self-interstitials, impurities

Schematic representationof different point defects:

• vacancy(1);
• self-interstitial(2);
• interstitial impurity(3);
• substitutional impurities(4,5)

The arrows show the local stresses introduced by the point defects.

Self-interstitials:

Self-interstitials in metals introduce large distortions in thesurrounding lattice ⇒ the energy of self-interstitial

formation is ~ 3 times larger as compared to vacancies (Qi ~ 3×Qv) ⇒ equilibrium concentration of self-interstitials is very low (less than one self-interstitial per cm3 at room T).

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How many vacancies are there?

The equilibrium number of vacancies formed as a result of thermal vibrations may be calculated from thermodynamics:

where Ns is the number of regular lattice sites, kB is the Boltzmann constant, Qv is the energy needed to form a vacant lattice site in a perfect crystal, and T the temperature in Kelvin (note, not in oC or oF).

Using this equation we can estimate that at room temperature in copper there is one vacancy per 1015 lattice atoms, whereas at high temperature, just below the melting point there is one vacancy for every 10,000 atoms. Note, that the above equation gives the lower end estimation of the number of vacancies, a large numbers of additional (nonequilibrium) vacancies can be introduced in a growth process or as a result of further treatment (plastic deformation, quenching from high temperature to the ambient one, etc.)

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