Grain Boundaries

1
Grain Boundaries
2

• In the last four lectures, we dealt with point
  defects (e.g. vacancy, interstitials, etc.) and
  line defects (dislocations).
• There is another class of defects called
  interfacial or planar defects
• They occupy an area or surface and are therefore
  bidimensional.
• They are of great importance in mechanical
  metallurgy.
• Examples of these form of defects include
• grain boundaries
• twin boundaries
• anti-phase boundaries
• free surface of materials
• Of all these, the grain boundaries are the most
  important from the mechanical properties point of
  view.

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• Crystalline solids (most materials) generally
  consist of millions of individual grains
  separated by boundaries.
• Each grain (or subgrain) is a single crystal.
• Within each individual grain there is a
  systematic packing of atoms. Therefore each
  grain has different orientation (see Figure 16-1)
  and is separated from the neighboring grain by
  grain boundary.
• When the misorientation between two grains is
  small, the grain boundary can be described by a
  relatively simple configuration of dislocations
  (e.g., an edge dislocation wall) and is,
  fittingly, called a low-angle boundary.

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Figure 16.1. Grains in a metal or ceramic the
cube depicted in each grain indicates the
crystallographic orientation of the grain in
schematic fashion
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• When the misorientation is large (high-angle
  grain boundary), more complicated structures are
  involved (as in a configuration of soap bubbles
  simulating the atomic planes in crystal
  lattices).
• The grain boundaries are therefore
• where grains meet in a solid.
• transition regions between the neighboring
  crystals.
• Where there is a disturbance in the atomic
  packing, as shown in Figure 16-2.
• These transition regions (grain boundaries) may
  consist of various kinds of dislocation
  arrangements.

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Figure 16.2. At the grain boundary, there is a
disturbance in the atomic packing.
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• In general, a grain boundary has five degrees of
  freedom.
• We need three degrees to specify the orientation
  of one grain with respect to the other, and
• We need the other two degrees to specify the
  orientation of the boundary with respect to one
  of the grains.
• Grain structure is usually specified by giving
  the average diameter or using a procedure due to
  ASTM according to which grains size is specified
  by a number n in the expression N 2n-1, where N
  is the number of grains per square inch when the
  sample is examined at 100x.

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Tilt and Twist Boundaries

• The simplest grain boundary consists of a
  configuration of edge dislocations between two
  grains.
• The misfit in the orientation of the two grains
  (one on each side of the boundary) is
  accommodated by a perturbation of the regular
  arrangement of crystals in the boundary region.
• Figure 16.3 shows some vertical atomic planes
  termination in the boundary and each termination
  is represented by an edge dislocation.

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• Figure 16.3. Low-angle tile boundary.

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Figure 16-3(b). Diagram of low-angle grain
boundary. (a) Two grains having a common 001
axis and angular difference in orientation of
(b) two grains joined together to form a
low-angle grain boundary made up of an array of
edge dislocations.
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• The misorientation at the boundary is related to
  spacing between dislocations, D, by the following
  relation
• where b is the Burgers vector.
• As the misorientation q increases, the spacing
  between dislocations is reduced, until, at large
  angles, the description of the boundary in terms
  of simple dislocation arrangements does not make
  sense.

(for q very small) (16-1)
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• For such a case, ? becomes so large that the
  dislocations are separated by one or two atomic
  spacing
• the dislocation core energy becomes important and
  the linear elasticity does not hold.
• Therefore, the grain boundary becomes a region
  of severe localized disorder.
• Boundaries consisting entirely of edge
  dislocations are called tilt boundaries, because
  the misorientation, as can be seen in Figure
  16.3, can be described in terms of a rotation
  about an axis normal to the plane of the paper
  and contained in the plane of dislocations.

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• The example shown in figure 16.3 is called the
  symmetrical tilt wall as the two grains are
  symmetrically located with respect to the
  boundary.

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• A boundary consisting entirely of screw
  dislocations is called twist boundary, because
  the misorientation can be described by a relative
  rotation of two grains about an axis.
• Figure 16.4 shows a twist boundary consisting of
  two groups of screw dislocations.
• It is possible to produce misorientations between
  grains by combined tilt and twist boundaries. In
  such a case, the grain boundary structure will
  consist of a network of edge and screw
  dislocations.

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Figure 16.4. Low-angle twist boundary.
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Calculation of the Energy of a Grain Boundary

• The dislocation model of grain boundary can be
  used to compute the energy of low-angle
  boundaries (qlt 10o).
• For such boundaries the distance between
  dislocations in the boundary is more than a few
  interatomic spaces, as
• (16-2)

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• Consider a tilt boundary consisting of edge
  dislocations with spacing D. Let us isolate a
  small portion of dimension D, as in Figure 16.5,
  with a dislocation at its center.
• The energy associated with such a portion, E,
  includes contributions from the regions marked I,
  II, and III in figure 16.5.

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Figure 16.5. Model for the computation of grain
boundary energy.
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• EI is the energy due to the material inside the
  dislocation core of radius rI.
• EII is the energy contribution of the region
  outside the radius and inside the radius R KD gt
  b, where K is constant less than unity.
• In this region II, the elastic strain energy
  contributed by other dislocations in the boundary
  is very small.
• EII is mainly due to the plastic strain energy
  strain energy associated with the dislocation in
  the center of this portion.

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• EIII, the rest of the energy in this portion,
  depends on the combined effects of all
  dislocations.
• The total strain energy per dislocation in the
  boundary is, then,
• Consider now a small decrease, , in the
  boundary misorientation. The corresponding
  variation in the strain energy is

(16-3)
(16-4)
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• and
• The new dimensions of this crystal portion are
  shown in Fig. 16-6.
• The region immediately around the dislocation,
  contributing an energy EI , does not change.
• This region does not change because EI , the
  localized energy of atomic misfit in the
  dislocation core, is independent of the
  disposition of other dislocations.

(16-5)
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Figure 16-6. New dimensions of a portion of
crystal after a decrease in the boundary
misorientation.
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• Thus, dEI 0. EII increases by a quantity dEII,
  corresponding to an increase in R by dR.
• EIII, however, does not change with an increase
  in D, because although the volume of region III
  increases, the number of dislocations
  contributing to the strain energy of this region
  decreases.

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Role of Grain Boundaries

• Grain boundaries have very important role in
  plastic deformation of polycrystalline materials.
  
• We outline below the important aspects of the
  role of grain boundaries.
• 1. At low temperature (Tlt0.5Tm, where Tm is the
  melting point in K), the grain boundaries act as
  strong obstacles to dislocation motion. Mobile
  dislocations can pile up against the grain
  boundaries and thus give rise to stress
  concentrations that can be relaxed by initiating
  locally multiple slip.

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• 2. There exists a condition of compatibility
  among the neighboring grains during the
  deformation of polycrystals that is, if the
  development of voids or cracks is not permitted,
  the deformation in each grain must be
  accommodated by its neighbors.
• This accommodation is realized by multiple slip
  in the vicinity of the boundaries which leads to
  a high strain hardening rate.
• It can be shown, following von Mises, that for
  each grain to stay in contiguity with others
  during deformation, there must be operating at
  least five independent slip systems - Taylors
  Theorem.

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• This condition of strain compatibility leads a
  polycrystalline sample to have multiple slip in
  the vicinity of grain boundaries.
• The smaller the grain size, the larger will be
  the total boundary surface area per unit volume.
• In other words, for a given deformation in the
  beginning of the stress-strain curve, the total
  volume occupied by the work-hardened material
  increases with the decreasing grain size.
• This implies a greater hardening due to
  dislocation interactions induced by multiple slip.

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• 3. At high temperatures the grain boundaries
  function as sites of weakness.
• Grain boundary sliding may occur, leading to
  plastic flow and/or opening up of voids along the
  boundaries.
• 4. Grain boundaries can act as sources and
  sinks for vacancies at high temperatures, leading
  to diffusion currents as, for example, in the
  Nabarro Herring creep mechanism.
• 5. In polycrystalline materials, the individual
  grains usually have a random orientation with
  respect to one another.

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• The term polycrystalline refers to any material
  which is composed of many individual grains.
• However, some materials are actually used in
  their single crystal state silicon for
  integrated circuits and nickel alloys for
  aircraft engine turbine blades are two examples.
• The sizes of individual grains vary from
  submicrometer (for nanocrystalline structures) to
  millimeters and even centimeters (for materials
  especially processed for high-temperature creep
  resistance).
• Figure 16.7 shows typical equiaxed grain
  configurations for polycrystalline tantalum and
  titanium carbide.

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Figure 16.7. Micrographs showing polycrystalline
Tantalum
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• One example of a material property that is
  dependent on grain size is the strength of a
  material as grain size is increased the material
  becomes weaker (see Fig.16.8). Note that
• strength is expressed in units of stress (MN/m2)
• grain size of a material can be altered
  (increased) by annealing
• Hardness measurement (e.g., by vickers indenter)
  can provide a measure of the strength of the
  material.

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Figure 16.8 The dependence of strength on grain
size for a number of metals and alloys.
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Grain Size Measurements

• Grain structure is usually specified by giving
  the average diameter. Grain size can be measured
  by two methods.
• (a) Lineal Intercept Technique This is very
  easy and may be the preferred method for
  measuring grain size.
• (b) ASTM Procedure This method of measuring
  grain size is common in engineering applications.
  

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• Lineal Intercept Technique
• In this technique, lines are drawn in the
  photomicrograph, and the number of grain-boundary
  intercepts, Nl, along a line is counted.
• The mean lineal intercept is then given as
• where L is the length of the line and M is the
  magnification in the photomicrograph of the
  material.

10-1
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Figure 16.9. Micrographs showing polycrystalline
TiC
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• In Figure 16.9 a line is drawn for purposes of
  illustration.
• The length of the line is 6.5 cm. The number of
  intersections, Nl, is equal to 7, and the
• magnification M 1,300. Thus,
• Several lines should be drawn to obtain a
  statistically significant result.

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• The mean lineal intercept l does not really
  provide the grain size, but is related to a
  fundamental size parameter, the grain-boundary
  area per unit volume, Sv, by the equation
• The most correct way to express the grain size
  (D) from lineal intercept measurements is
• Therefore, the grain size (D) of the material of
  Figure 10.4 is

10-2
10-3
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• ASTM Procedure
• With the ASTM method, the grain size is specified
  by the number n in the expression N 2 n-1,
  where N is the number of grains per square inch
  (in an area of 1 in2), when the sample is
  examined at 100 power micrograph.
• Example
• In a grain size measurement of an aluminum
  sample, it was found that there were 56 full
  grains in the area, and 48 grains were cut by the
  circumference of the circle of area 1 in2.
  Calculate ASTM grain size number n for this
  sample.

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• Solution
• The grains cut by the circumference of the
  circle are taken as one-half the number.
  Therefore,