Grain Boundaries
• 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.
• 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.
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
• 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.
Figure 16.2. At the grain boundary, there is a disturbance in the atomic
packing.
• 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.
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.
Figure 16.3. Low-angle tile boundary.
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.
• The misorientation at the boundary is related to spacing
between dislocations, D, by the following relation:
D
b
b

   (for  very small)

2 sin  
2
(16-1)
where b is the Burgers vector.
• As the misorientation  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 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.
• 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.
• 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.
Figure 16.4. Low-angle twist boundary.
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 (< 10o).
• For such boundaries the distance between dislocations in
the boundary is more than a few interatomic spaces, as:
b
1
o
   10  rad
D
6
or
D  6b
(16-2)
• 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.
Figure 16.5. Model for the computation of grain boundary
energy.
• 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 > 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.
• 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,
E  EI  EII  EIII
(16-3)
• Consider now a small decrease, d , in the boundary
misorientation. The corresponding variation in the strain
energy is
dE  dE I  dE II  dE III
(16-4)
and
d
dD dR



(as R  KD )

D
R
(16-5)
• 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.
Figure 16-6. New dimensions of a portion of crystal after a
decrease d in the boundary misorientation.
• 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.
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 (T<0.5Tm, 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.
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.
• 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.
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.
• 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 hightemperature creep resistance).
• Figure 16.7 shows typical equiaxed grain configurations for
polycrystalline tantalum and titanium carbide.
Figure 16.7. Micrographs showing polycrystalline Tantalum
• 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.
Figure 16.8 The dependence of strength on grain size for a
number of metals and alloys.
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.
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:

L
l 
Nl M
10-1
where L is the length of the line and M is the magnification
in the photomicrograph of the material.
Figure 16.9. Micrographs showing polycrystalline TiC
• 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,
100 X 103
 11m
l 
7 X 1300

• Several lines should be drawn to obtain a statistically
significant result.
• 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

2
l 
Sv
10-2
• The most correct way to express the grain size (D) from
lineal intercept measurements is:
3
D  l
2
10-3
• Therefore, the grain size (D) of the material of Figure 10.4
is:
3
D 
2
X 11  16.5
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.
Solution
The grains cut by the circumference of the circle are taken as
one-half the number. Therefore,
N  56  48
2
 56  24  80  2n 1
  1
But n   ln N
ln 2 

  1
 n   ln 80
ln 2 

 4.38
 1  7.35
0.69

