Imperfections in Solids and Diffusion

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Imperfections in Solids and Diffusion
References used in preparation Callister, 2000
and Smith, 2006
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OUTLINE OF THE LECTURE

• Imperfections
• Point
• Linear
• Interfacial
• Microscopic Examination of Dislocations
• Diffusion
• Diffusion mechanisms
• Steady-state diffusion
• Non-steady state diffusion
• Factor affecting diffusion

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• A crystalline defect is lattice irregularity and
  its classification is made based on the geometry
  or dimensionality of the defect. Impurities in
  solids may be considered as defects for pure
  materials.
• The properties of the materials may change by the
  presence of imperfections.
• For example Pure silver (Ag) softer than
  alloyed Ag (sterling silver 92.5 Ag and 7.5
  Cu), which is harder and stronger. In this
  example Cu atoms changes the perfection of the Ag
  metal.
• There are small and localized regions of
  impurities in semiconducting materials used in
  integrated circuit microelectronic devices in
  computers, calculators and home appliances.
• Imperfections may exist as a result of
  solidification of the material. During
  solidification, the material may cease to
  replicate the unit cells, for instance, formation
  of grain boundaries in metallic materials.
• On the other hand imperfections can be introduced
  to the material on purpose to change the property
  (like in Ag, semiconducting materials.)

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Stages in Solidification
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Types of Imperfections in Materials

• Point Defects
• Vacancies and self-interstitials
• Impurities in solids
• 2. Linear Defects
• Dislocations
• 3. Interfacial Defects
• Grain boundaries

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Point Defects

• Vacancies and Self-Interstitials
• Vacancy is observed when an atom is missing. This
  type of defect is commonly observed. The
  existence of vacancies is closely related with
  the continuous increase in entropy (randomness)
  since it decreases organization in a crystalline
  structure (2nd law of thermodynamics).

Self-interstitial
Vacancy
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Energy required for the formation of vacancy
Equilibrium number of vacancies for a given
quantity
Total number of atomic sites
Absolute temperature (K)
kBoltzmanns constant1.38x10-23 J/atom-K or
8.62x10-5 eV/atom-K. Use the appropriate constant
depending on the unit of Qv.
Based on this equation, it is clear that as T
increases the number of vacancies increases
exponentially.
For most of the metals just below melting point,
the fraction of vacancies (N/Nv) is on the order
of 10-4. This means one lattice out of 10,000 is
empty.
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• Self-Interstitials- these are the atoms crowding
  into interstitial sites (void spaces, which are
  not occupied under ordinary circumstances). In
  metals, this type of defect causes large
  distortions in the surrounding lattice because
  the interstitial atoms usually larger than the
  void space.

Example Nv? Given Cu Qv0.9 eV/atom, atomic
weight 63.5 g/mol, ?8.4 g/mol Use the
equations
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• Point defects in Ceramics vacancies and
  interstitials are possible. These are illustrated
  below

Cation interstitial
Cation vacancy
Anion vacancy
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• In ceramics, conditions of electroneutrality must
  be maintained even in the defect structures. As a
  result, defects in ceramics do not occur alone.
  For example, one type of defect has a cation
  vacancy and cation interstitial pair, which is
  called Frenkel defect. Another type involves a
  pair vacancies of a cation and anion, which is
  called Schottky defect.

Schottky
Frenkel
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• The ratio of cation to anion is not affected from
  the defect type, and therefore these are called
  stoichiometric ceramics. But there may be
  nonstoichiometric ceramics composed of ions with
  more than one states.

For every two Fe3 formation, there is a single
Fe2 vacancy. This maintains the the
electroneutrality, but not the stoichiometry.
Therefore the chemical formula of The FeO is
usually shown as Fe1-xO (x is a small fraction).
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• Impurities in solids Impurity or foreign atoms
  are always present, that is, it is difficult to
  find a pure material. Even if the purity is
  99.9999, then 1022 to 1023 impurity atoms will
  be present in 1 m3 of the metal.
• Alloys are materials in which impurity is added
  intentionaly to alter the characteristics (
  mechanical strength, corrosion resistance) of the
  materials composing them.
• The addition of impurity atoms to a metal forms a
  solid solution.
• Solute is the element or compound present in
  minor concentration.
• Solvent is the host element with high
  concentration.
• Solid solutions are compositionally homogeneous,
  and the solute atoms are randomly dispersed
  within the solid.
• Impurity point defects can be found as,
• 1) Substitutional 2) Interstitial

Substitutional
Interstitial
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• Substitutional solutions Here are some important
  features for this type of solid solution
  (Hume-Rothery rules),
• Atomic size difference should be less than 15
  b/w the two atoms. Otherwise the solute atoms
  will create substantial lattice distortions and a
  new phase will form.
• Crystal structure should be the same.
• Electronegativities should be similar to prevent
  the ionic bonding.
• Other factors being equal, a metal has a higher
  tendency to dissolve another metal of higher
  valency.
• For example Cu-Ni solid solution. The type is
  substitutional. WHY?
• Because
• Radii for Cu and Ni are 0.128 and 0.125 nm
  respectively.
• They both have FCC structure.
• Their electronegativities are 1.9 and 1.8 for Cu
  and Ni respectively.
• The most common valence is 1 for Cu and 2 for
  Ni.

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• Interstitial solutions Impurity atoms fill the
  voids or interstices among the host atoms. For
  metals with a high APF, the interstitial
  positions are small in size and therefore the
  diameter of the impurity atoms should be smaller.
  In general, the concentration of impurity atoms
  is low (lt10 ). The formation of interstitial
  solutions introduce lattice strains on the
  adjacent host atoms. For example C forms an
  interstitial solid solution when it is added to
  Fe.
• 2 is the maximum solubility of C (r0.071 nm)
  in Fe (r0.124 nm).
• Impurities in Ceramics Substantial and
  interstitial solutions are possible.

Substitional cation
Interstitial
Ionic sizes and charges should be similar for
high solubility.
Substitional anion
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• There are two ways of specifying the composition
  of solid solutions
• Weight percent and atom percent.
• Weight percent

m1 and m2 are the masses of the solute and
solvent.
Atom percent
nm1 is the number of moles of an element (or
number of atoms). m1 is the mass and A1 is the
atomic weight. C1 is the atom percentage.
Conversions from weight percent to atom percent
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• To convert concentration from weight percent to
  mass/volume

For density in units g/cm3, these expressions
yield C1 and C2 in kg/m3.
To calculate the density and atomic weight of a
binary alloy
Note that the equations on this page are deriven
by assuming the total alloy volume is exactly
equal to the sum of the volumes of the
individual elements. This assumption may be true
for dilute solutions, but brings in some error
to the calculations.
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Linear Defects

• It is a group of point defects forming a linear
  one dimensional defect in the structure of the
  material. Dislocation is the most commonly
  observed type of linear defect.

1. edge dislocation
Extra portion of a plane of atoms terminating
within the lattice structure causing localized
lattice distortion. The magnitude and direction
of the distortion is expressed in terms of a
Burgers vector, b, which is perpendicular to edge
dislocation line.
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• 2) screw dislocation
• It may be thought of as being formed by a shear
  stress that is applied to produce this type of
  dislocation.

The upper part is shifted one atomic distance to
the right relative to the bottom portion.
dislocation line
Dislocation line is linear.
Burgers vector is parallel to dislocation line.
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• Mixed dislocations very common

Edge dislocation
Screw dislocation
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• All crystalline materials contain dislocations,
  which may have introduced during solidification,
  plastic deformation, and as a result of thermal
  stresses that a rapidly cooled material
  experiences.
• Interfacial Defects
• Interfacial defects are boundaries that have two
  dimensions and separate regions of different
  crystal structures or crystallographic
  orientations.
• For example
• 1)External surface is the boundary of material
  at which the structure terminates. Atoms at the
  surface have higher energy state than the atoms
  at the inner parts of the structure. This energy
  is called surface energy (J/m2 or erg/cm2). To
  minimize this energy, materials minimizes the
  surface area.
• 2)Grain boundary Crystalline solids composed of
  a collection of many small crystals or grains are
  called polycrystalline. The growth of grains
  happens during solidification. Grain boundary is
  the line separating two small grains or crystals
  having different crystallographic orientations.

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• Solidification

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• Grain boundary is a type of interfacial defect
  and usually is several atom distances wide.
• Various degrees of crystallographic misalignment
  between adjacent grains are possible.

The atoms are bonded less regularly along the
grain boundary and there is an interfacial or
grain boundary energy similar to the surface
energy. The magnitude of the energy depends on
the degree of misorientation. It is higher for
higher angle grain boundary. Grains boundaries
are important parts of the structure since they
are reactive chemically and impurity atoms
prefer grain boundaries. Fine grained materials
have higher grain boundary energy. Discuss WHY?

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• Twin Boundary is a special type of grain
  boundary across which there is a mirror lattice
  symmetry.

observed due to atomic displacements produced
as a result of shear force application and during
annealing heat treatments following deformation.
Twinning is observed on a specific plane and
direction. Annealing twin is commonly observed in
materials with FCC structures. Mechanical twins
are observed in BCC and HCP metals.
Other interfacial defects are stacking faults
(FCC), phase boundaries, ferromagnetic domain
walls.
Cracks, pores, foreign inclusions are examples
for bulk or volume defects. Atomic vibrations
can be thought as imperfections at any time fixed.
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• Examination of dislocations can be performed
  using microscopic techniques.
  

relatively large grains
microscopic dislocations
Microscopic techniques involves a photographic
equipment in conjunction with the microscope.
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• Optical Microscopy there is a pretreatment
  necessary for the sample as grounding and
  polishing the surface to a smooth and mirrorlike
  finish. The microstructure is revealed by using a
  chemical reagent (etching). The chemical
  reactivity of the grains depends on
  crsytallographic orientation.

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• Grain boundaries dissolve more than grains and
  using a chemical reagent can make the grooves
  deeper so that they can be examined using a light
  microscope.

The upper limit of magnification is 2000
diameter.
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• Electron Microscopy
• Higher magnifications than optical is possible.
• Beams of electrons are used instead of light
  radiation.
• Transmission electron microscopy (TEM) a
  specimen is prepared as a thin foil to ensure the
  transmission through the specimen. Magnifications
  up to 1,000,000x is possible.
• Scanning electron microscopy (SEM) Surface
  features of the specimen are examined. Surface
  may or may not be polished or etched, but it must
  be electrically conductive. If the material is
  nonconductive, then it is coated by a conductive
  material.
• Scanning probe microscopy (SPM) This technique
  creates a topographical map of the surface on an
  atomic scale.
• -Examination in nanoscale (109X) is possible.
• -Resolution is higher.
• -Three dimensional images can be produced.
• -They may operate in a variety of environments.

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• Grain Size Distribution
• Grain size may be estimated using an intercept
  method.
• On a photomicrograph, draw straight lines all the
  same in length and count the grains intersecting
  each line, and then divide the length of the line
  by an average number of grains intersected. The
  average grain diameter can then be calculated by
  using the magnification of the photomicrograph.
• Another technique is ASTM technique Use a 100x
  magnified photograph of the material. ASTM has
  prepared standard comparison charts corresponding
  to different grain sizes numbered from 1 to 10
  (grain size number, n). The larger the number,
  the smaller the grains. Determine the grain size
  number by comparison and use the equation below
  to find out the number of grains per square inch
  (N)

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• DIFFUSION
• Heat treatment of the materials causes atomic
  diffusion, which is usually desired to improve
  the properties of the materials. For example the
  steel gear, which case hardened.

Hardness of the material and resistance to
failure have been improved by diffusing excess C
or N into the outer surface layer.
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• Diffusion is material transport by atomic motion.
  Diffusion can be explained using a diffusion
  couple example

after heating
before heating
This process is also called interdiffusion or
impurity diffusion.
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• Diffusion Mechanisms
• Diffusion is a stepwise migration of the atoms
  fro lattice site to another lattice site. For
  this to happen
• there must be an empty adjacent site
• atom that is moving should have enough energy to
  break bonds with the neighboring atoms and cause
  some distortion. The energy is vibrational in
  nature.
• Vacancy diffusion the interchange of an atom
  from a normal lattice position to an adjacent
  vacant lattice resulting in a motion of vacancies
  in opposite direction.

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• Interstitial Diffusion Atoms move from an
  interstitial position to another one closeby.
  Migrating atoms should be small in size such as
  N, C, H and O. This type of diffusion has a
  faster rate than vacancy diffusion.

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• Steady State diffusion Diffusion is a function
  of time when it is necessary to find out how fast
  the diffusion is. This rate is explained by
  diffusion flux (J)

In differential form
mass
time
crosssectional area
If the diffusion flux is the same regardless of
time, then a steady state condition exists.
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• The steady state diffusion in a single direction
  is simple

Ficks first law
D is diffusion coefficient (m2/sec). The negative
sign indicates that the direction of diffusion is
down the concentration gradient. Concentration
gradient is the driving force in diffusion
reactions. One example for steady state diffusion
is the purification of hydrogen gas using
palladium sheet.
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• Nonsteady state diffusion The diffusion flux and
  concentration gradient at a selected point vary
  with time causing a net accumulation or
  depletion.

For nonsteady state diffusion Ficks first law is
not valid. Therefore, we use partial differential
equation as
Ficks second law
Depending on the selected boundary conditions
there may be different solutions for
this equation. For example semi-finite solid
in which the surface concentration is held
constant. Assume a) before diffusion, diffusing
solute concentration is uniform, Co. b) the value
of x is zero at the surface and increases with
distance into the solid. c) the time is zero at
the instant before the diffusion begins.
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Gaussian error function
concentration at the depth x after time t
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• Factors affecting diffusion
• 1) Diffusing species- different materials have
  different diffusion coefficient, which is also
  the indication of the diffusion rate.

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• 2) Temperature
• Temperature dependence can be expressed as
  follows

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