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Grain Boundaries

NANO54

Foothill College

Overview


What are grain boundaries?


What is grain boundary engineering?


Applications with metals


Applications with ceramics


Characterization tools

Grain Boundaries

A
grain boundary

is the interface between two
grains
, or
crystallites, in a polycrystalline material. Grain boundaries
are defects in the
crystal

structure, and tend to decrease the
electrical

and
thermal conductivity

of the material. The high
interfacial energy and relatively weak bonding in most grain
boundaries often makes them preferred sites for the onset of
corrosion and for the
precipitation

of new phases from the
solid. They are also important to many of the mechanisms of
creep
. On the other hand, grain boundaries disrupt the
motion of
dislocations

through a material, so reducing
crystallite size is a common way to improve strength, as
described by the
Hall

Petch

relationship.

http://en.wikipedia.org/wiki/Grain_boundary

Grain Boundaries

High resolution STEM image from a
grain boundary in gold at the atomic
level, imaged on an FEI Titan STEM
80
-
300.

http://www.fei.com/resources/image
-
gallery/grain
-
boundary
-
gold
-
1234.aspx

Grain Boundary
Deformations

A beautiful molecular dynamics
simulation of copper nanocrystals
undergoing
...

http://virtualexplorer.com.au/special/meansvolume/contribs/jessell/lectures/lec1.html

Lattice Point
Defects

They can speed up the process of
crystal growth by orders of magnitude
(geometry). The distorted crystal lattice
around defects provides rapid diffusion
pathways within crystals (geometry).
They are intimately involved in several
deformation mechanisms (kinematics).
Provide a driving force for many
deformation processes (dynamics).
Can weaken the strength of a crystal
by several orders of magnitude
(dynamics). The movement of
dislocations can lead to the formation
of crystallographic preferred
orientations. Defects in crystals:


Point

0
-
D known as
point defects

vacancies, interstitials,
discrete 2nd phase


http://virtualexplorer.com.au/special/meansvolume/contribs/jessell/lectures/lec1.html

Grain Boundary Inhibition

Both low
-

and high
-
angle boundaries are retarded by the presence of particles via
the so
-
called
Zener pinning

effect. This effect is often exploited in commercial alloys
to minimize or prevent re
-
crystallization or
grain growth

during
heat
-
treatment
.


http://en.wikipedia.org/wiki/Grain_boundary

Why do we study
microstructures?

To establish the link between process, environment & microstructure via general constitutive equation eg
Dorn 1957 The (Micro)structure is a function of the competing processes in a rock: (PROCESS RATE =
DRIVING FORCE*KINETICS) as they act on an initial (Micro)structure. By understanding this relationship
we can interpret microstructures in terms of the history of temperature (
T
) , pressure (
s
), the CO2 fluid
pressure (
ƒCO
2
) and other boundary conditions that control both the driving force and the kinetics. To
provide evidence of deformation processes, which in turn provides:


evidence of rheology during deformation, which is important for geodynamic interpretation


interpretation of deformation history


evidence of metamorphic environment


interpretation of seismic anisotropy and microstructural geochemistry

http://virtualexplorer.com.au/special/meansvolume/contribs/jessell/lectures/lec1.html

Grain Boundary Formation

When molten metal is cooled down, the metal atoms settle out into a crystal lattice.
Given enough time and ideal conditions, the crystal lattice can grow to be very large,
with a perfect internal crystalline structure. Ideal conditions are seldom found, and
the reality is that almost every solid metal exists as jumble of crystals of varying
sizes. Each individual crystal in the body is called a grain. These grains are
crystalline structures that have lots of imperfections, which distort the crystal lattice.

Grain Boundary Engineering

Grain
-
boundary strengthening

(or
Hall

Petch
strengthening
) is a method of
strengthening

materials by
changing their average
crystallite

(grain) size. It is based on
the observation that
grain boundaries

impede dislocation
movement and that the number of
dislocations

within a
grain have an effect on how easily dislocations can traverse
grain boundaries and travel from grain to grain. So, by
changing grain size one can influence dislocation
movement and
yield strength
. For example,
heat treatment

after plastic deformation and changing the rate of
solidification are ways to alter grain size.

http://en.wikipedia.org/wiki/Grain_boundary_strengthening

Precipitation Hardening

The islands of B embedded in grains of A act as barriers to the movement of dislocations and this increases the yield
stress. The aging process has to be carried out very carefully as the size and spacing of the islands increases with
longer aging and beyond an ideal point the strength decreases and the component becomes prone to work hardening.

Hall
-
Petch Strengthening

In grain
-
boundary strengthening
the
grain boundaries

act as
pinning points

impeding further
dislocation propagation. Since
the lattice structure of adjacent
grains differs in orientation, it
requires more energy for a
dislocation to change directions
and move into the adjacent
grain. The grain boundary is
also much more disordered than
inside the grain, which also
prevents the dislocations from
moving in a continuous slip
plane. Impeding this dislocation
movement will hinder the onset
of plasticity and hence increase
the yield strength of the material

http://en.wikipedia.org/wiki/Grain_boundary_strengthening

Grain Size Effects

This is a schematic roughly
illustrating the concept of
dislocation pile up and how it
effects the strength of the
material. A material with larger
grain size is able to have more
dislocation to pile up leading to
a bigger driving force for
dislocations to move from one
grain to another. Thus you will
have to apply less force to
move a dislocation from a
larger than from a smaller
grain, leading materials with
smaller grains to exhibit higher
yield stress.

http://en.wikipedia.org/wiki/Grain_boundary_strengthening

Grain Boundaries Metals

Most of the metallic materials we are widely using in technology are poly
-
crystals which
consist of a large number of grains having different crystal orientation. It is empirically
known that various properties of polycrystalline metals can be improved by grain refinement.

Bulk Nanostructured Metals (BNM)

The BNMs, which are considered as the materials "full of grain boundaries", expected to exhibit peculiar properties that have

ne
ver
seen in conventional metallic materials. Usually, slip movement of dislocations, which is one of the lattice defect, results
in
the
deformation of metallic materials. High density of grain boundaries in BNMs changes the energy of each dislocation, leading t
o
some unique behaviors, especially mechanical properties that are beyond our expectation. For example, BNMs exhibit strength
four times higher than the same materials with conventional grain size. As a result, aluminum can be as strong as steel.

Grain Boundary
Strengthening

A technique used to improve the
yield stress of a metal. The surface
area of a grain is proportional to the
square of its linear size whilst the
number of dislocations is proportional
to the third power. By reducing the
grain size (up to a critical point) the
number of dislocations that ca pile up
at the grain boundary is reduced.
This leads to a greater stress being
required to force the dislocations
through the grain boundary in order
to create more dislocations on the
other side. This results in a greater
yield stress. In many alloys ‘grain
refiners’ are introduced that provide
nucleation sites for the cooling melt.
This results in many small grains
being produced, thereby producing a
stronger alloy.

Representation of the grain boundary between three separate grains. (all the atoms are
the same species and the color is only used to separate the individual grains)

http://spaceflight.esa.int/impress/text/education/Glossary/Glossary_G.html

Grain Boundaries
Al
2
O
3

and SWNT

There has been growing interest in
incorporating single
-
wall carbon nanotubes
(SWNTs) as toughening agents in brittle
ceramics. Here we have prepared dense
Al
2
O
3
/SWNT composites using the spark
-
plasma sintering (SPS) method. Vickers
(sharp) and Hertzian (blunt) indentation
tests reveal that these composites are
highly contact
-
damage resistant, as shown
by the lack of crack formation. However,
direct toughness measurements, using the
single
-
edge V
-
notch beam method, show
that these composites are as brittle as
dense Al
2
O
3

(having a toughness of 3.22
MPa m
0.5
). This type of unusual mechanical
behaviour was also observed in SPS
-
processed, dense Al
2
O
3
/graphite
composites. We argue that the highly
shear
-
deformable SWNTs or graphite
heterogeneities in the composites help
redistribute the stress field under
indentation, imparting the composites with
contact
-
damage resistance. These
composites may find use in engineering
and biomedical applications where contact
loading is important.

http://www.nature.com/nmat/journal/v3/n8/full/nmat1161.html

Grain Boundary
Precipitation

TEM image showing grain boundaries and precipitation within the grains.
http://www.steeluniversity.org/content/html/eng/1340
-
0060.htm


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

= 2
n
-
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:






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


very small) (16
-
1)


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.

Grain Boundaries and
Dislocations

However a dislocation cannot carry on
moving forever. Eventually it will run into
a grain boundary, the jumbled mess of
atoms at the edge of a grain. It is stuck
here unless it can jump the boundary into
the next grain. Other dislocations pilling
up behind the first one can have the
effect of increasing the surrounding
stresses to the point that the dislocation
is forced through the boundary in to the
neighboring grain. This applies to all
grains. If the grain size is decreased
though there will be an increase in area
of the grain boundary (proportional to the
square of the grain size) compared to the
number dislocations in the grain
(proportional to the cube of the grain
size). This decreases the number of
dislocations that can pile up in order to
force another dislocation across the grain
boundary. This in turn increases the yield
stress which in turn increases the
hardness of the metal.


http://spaceflight.esa.int/impress/text/education/Mechanical Properties/Dislocations_02.html

Solid Solutions

This is one of the things that can happen when alloys such as steel, brass and
bronze are made. Individual atoms of the alloying element can be dissolved into the
crystal structure of the main material. If the atom takes the place of a normal atom
then it is called a ‘substitutional defect’; smaller atoms put the crystal lattice in to
tension and larger atoms put it into compression. Atoms that are much smaller than
the majority atoms can sit in between the lattice points and are called interstitial
defects. Either way, the stresses that these defects create in the crystal lattice are
‘pinning points’ that restrict the motion of dislocations and so strengthen the material.


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 (

<

10
o
).



For such boundaries the distance between dislocations in
the boundary is more than a few interatomic spaces, as:



(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.


E
I

is the energy due to the material inside the dislocation
core of radius
r
I
.




E
II
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.



E
II
is mainly due to the plastic strain energy strain energy
associated with the dislocation in the center of this
portion.


E
III
, 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)


and





The new dimensions of this crystal portion are shown in
Fig. 16
-
6.



The region immediately around the dislocation, contributing
an energy E
I

, does not change.



This region does not change because E
I

, the localized
energy of atomic misfit in the dislocation core, is
independent of the disposition of other dislocations.

(16
-
5)

Figure 16
-
6. New dimensions of a portion of crystal after a

decrease in the boundary misorientation.


Thus, dE
I

=0. E
II

increases by a quantity dE
II
,
corresponding to an increase in R by dR.



E
III
, 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<0.5T
m
, where T
m

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 high
-
temperature 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/m
2
)


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
,
N
l
, 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

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,
N
l
,
is equal to 7, and the


magnification
M

= 1,300. Thus,






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,
S
v
, 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


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 in
2
), 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 in
2
.
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,

Summary


Grain boundaries


Grain boundary engineering


Important for metals, alloys, ceramics


Powder metallurgy


Characterization tools


References


Grain Boundaries
Mechanical Metallurgy EMA
4225 Fall 2001 Eric Kalu (used with permission)


Finite Element Modeling of the Deformation of
3D Polycrystals Including the Effect of Grain Size


Grain Boundary, Surface Energies,
Measurement
-

27
-
750, Spring 2006 A.D. Rollett


http://www.chemguide.co.uk/atoms/structures/metals.html