#### Imperfection in Solids, Physics tutorial

Crystalline Defects:

Crystalline defect is the lattice irregularity having one or more of its dimensions on order of atomic dimension. There are five main categories of crystalline defects:

• Zero dimensional: Point defects
• One dimensional: Linear defects (dislocations)
• Two dimensional: Planar (surface) defects
• Three dimensional: Volume (bulk) defects
• Vibrations.

Point Defects:

There are two main kinds of geometrical defects in the crystal. There are those which are much localized and are of atomic dimensions. These are known as point defects. Example of this is impurity atom that can be either substitutional or interstitial impurity.

i) When some atoms are not accurately in their right place, lattice is said to have imperfections or defects.

ii) Several properties of solids e.g. electrical resistance and mechanical strength are run by presence of certain kinds of defects in lattice.

Point defects permit for diffusion to happen in solid state. There are different categories of point defects:

a) Vacancy:

These are created by thermal vibrations of crystal lattice and/or from need to maintain charge neutrality.

b) Self-interstitial or interstitialcy:

A position on crystal lattice which is not usually occupied is known as interstitial site. If it is occupied by same atomic species it is self-interstitial or self-interstitialcy. It produces large distortions in crystal lattice and so concentrations are small.

c) Impurities:

• solid solutions-homogeneous single phase materials which are completely analogous to liquid solution (alcohol and water)
• solute/solvent/solubility - miscibility
• random solid solution vs. ordered solid solution
• substitutional solid solution (Cu/Ni) vs. interstitial solid solution
• Iron has 0.025% C and γ iron has 2.08% C)
• nonstoichiometric compounds (Fe1-xO)
• second phases - totally analogous to liquid mixture (oil and water)
• Hume-Rothery Rules - utilized to forecast solubility (miscibility):

i) Less than 15% difference in atomic radii

ii) Same crystal structure

iii) Similar electronegativities

iv) Same valence.

d) Schottky defect: a pair of oppositely charged ion vacancies.

e) Frenkel defect: a vacancy-interstitialcy combination.

Linear Defects (Dislocations):

Unlike point defects, these are kinds of disorder that extend beyond volume of one or two atoms. It is line defect that can extend right through crystal or it can form closed loops.

Line defect is lattice distortion created about line formed by solidification process, plastic deformation, vacancy condensation or atomic mismatch in solid solutions. Dislocations describe observation of plastic deformation at lower stress than would be needed in perfect lattice. They also describe phenomenon of work hardening. There are two basic kinds of dislocations, edge dislocation and screw dislocation.

Edge dislocation:

These are caused by termination of the plane of atoms in middle of crystal. In such a case, adjacent planes are not straight, but instead bend around edge of terminating plane so that crystal structure is completely ordered on either side. Analogy with stack of paper is appropriate: if half a piece of paper is put in the stack of paper, defect in the stack is only visible at the edge of half sheet.

Screw dislocation:

This is more hard to visualize, but basically includes structure in which helical path is traced around linear defect (dislocation line) by atomic planes of atoms in crystal lattice.

Planar (Interfacial) Defects:

External surfaces:

Atoms at any surface are not in their perfect crystal positions as they will have decrease number neighbors than atoms inside volume of material. Therefore, the atoms will be at higher energies. During solidification, materials attempt to minimize this.

Grain boundaries:

Grains are produced during solidification procedure. The grain boundary is area of mismatch between volumes of material which have a common orientation of crystallographic axes. Atoms at grain boundaries are not in their perfect crystal positions and therefore, grain boundary is less dense. These atoms are at higher energies than atoms inside the volume of grain. Thickness is of order of 2-5 atoms wide.

There are different categories of grain boundaries:

• Low angle tilt boundary - a few isolated edge dislocations
• High angle tilt boundary - more complex.

At ambient temperatures, grain boundaries give strength to the material. So in general, fine grained materials are stronger than coarse grained ones as they have more grain boundaries per unit volume. Though, at higher temperatures, grain boundaries act to weaken the material because of corrosion and other factors. Grain size can be measured by:

• Grain size number
• Average grain diameter
• Grain density.

The size and shape of grains are estimated by number of factors during solidification:

• Lots of nucleation sites → fine grains
• Fewer nucleation sites → coarse grains
• Equal growth in all directions → equiaxed grains
• Thermal gradients → elongated or columnar grains
• Slow growth → coarse grains
• Rapid growth → fine grains.

Many materials are polycrystalline (or polygranular). There are some applications where expense and time to generate single crystal materials (and therefore no grain boundaries) is justified:

• Silicon wafer chips.
• Twin boundaries: a special kind of grain boundary across which there exist mirror image of crystal lattice. It is generated by mechanical shear stresses and/or annealing some materials.
• Stacking faults: interruption of stacking sequence of close packed planes.
• Phase boundaries: surface area between grains in multiphase materials.
• Ferromagnetic domain walls: boundary between regions which have different orientation of magnetic dipoles.

Volume (Bulk) Defects:

These are introduced during processing and fabrication.

• Pores
• Cracks
• Foreign inclusions

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