class27/1
•Polycrystals composed of many small single crystals (called grains) with different orientations
joined at interfaces called grain boundaries.
•Microstructures have both geometric and crystallographic characteristics that influence their
properties.
•Among geometric are the distribution of grain sizes (larger vs. small) & shape/aspect ratios
(equiaxed vs. acicular).
•Among crystallographic are the hkl orientation of the crystallites with respect to external sample
reference frame (macroscopic surface of the sample), and the hkl orientation of the crystallites with
respect to one another (the misorientation, e.g. low vs. high angle tilt and grain boundaries).
•Traditionally looked at in 2-D but more recently in 3-D with advancements in characterization tools.
Fig. Contact atomic force
microscope (AFM) image
of thermally etched Al2O3
polycrystal (from Rohrer
book)
grain
grain
boundary
Polycrystallography
Grains tend to be randomly
orientated → different
crystallographic (hkl)
orientations
Polycrystallography (continued)
•The crystallographic orientation of the grains with respect to external sample surface is typically
shown on special stereographic projection known as a pole figure.
Fig. (001) pole figure
shows the positions of
poles of each Al2O3
polycrystalline grain.
•The grains’ numerical label is used to designate the location of its pole.
•The center of the circle represents the surface normal [001] ┴ (001).
•This information can be constructed from XRD pole figure (shown above right), SEM/EBSD
(electron backscatter diffraction-next slide), or TEM/SAED (selected area electron diffraction).
class27/2
Grains tend to be
randomly orientated
(from Rohrer book)
Recall Stereographic projections (Case study of how structure determines properties)
Figure above. Pole figure for all
Berkovich nanoindentations
(technique to measure hardness
and elastic modulus). Black spots
on the stereographic triangle
represent various indentations.
•Hardness and Elastic Modulus
vary from grain to grain which
exhibit different crystallographic
orientation
Microcrystalline Fe
sample taken with UNT’s
Environmental-SEM (Quanta)
with electron backscatter
diffraction (EBSD) detector
one nanoindent
class27/3
Preferred Orientation (Texture)
•Eq. (3) from last class becomes invalid when you have (a) preferred orientation (texture), i.e. the
crystals making up the specimen are not randomly orientated in space.
•Preferred orientation of crystal grains cause large disagreements between calculated and
observed intensities.
•The reason for this is that each peak in the pattern is caused by diffraction from a different subset
of the particles (or grains) in the material.
•If the particles are distributed in a truly random fashion, then the number of particles in each
orientation should be identical.
•However, if the particles have a shape anisotropy, this might not be true. Eq. (3) assumed
particles of approximately the same size/shape.
•Assume particles are hexagonal platelets (below Figure). In a packed powder sample, plate-like
particles are most likely to lie with their basal plane, (0001), parallel to the reference plane.
•Thus the (0001) diffraction peak will originate from a greater number of particles than other peaks,
such as the (1010).
•As a result of this preferred orientation, the (000l)-type reflections will be intensified relative to
(hki0)-type reflections.
•In general anything that changes the assumed random distribution of particles will affect the
distribution of relative intensities.
•Also, a few large particles, in an otherwise
fine powder pressed sample, can affect the
distribution of intensities (Figure):•Bulk materials produced by sintering, casting, directional
solidification & deformation frequently causes some texture.
Figure shows texture in a powder diffraction sample. Highly
anisotropic particles are likely to have similar alignments.
class27/4
•Deformation texturing is due to the tendency of
the grains in a polycrystalline material to rotate
during plastic deformation:
•Each grain undergoes slip and rotation in a
complex way that is determined by the imposed
forces and by the slip and rotation of adjoining
grains, the result is a preferred (nonrandom)
orientation.
•It can occur in metals, ceramics, rocks and in
both natural and artificial polymeric fibers and
sheets.
•Macroscopic properties of materials are
influenced by texture due to anisotropy
Preferred Orientation (Texture)(continued)
class27/5
(Mechanical Metallurgy)
•Pole figures usually give a statistical
distribution of poles from a very large
number of grains.
•Example on right is for Al alloy (6111)
sheet (cubic texture) that has been rolled
and recrystallized causing the <100> axes
to be preferentially aligned along rolling,
normal and transverse directions.RD=rolling direction
TD=transverse direction
3 poles with respect to sample normal.
Pole figures can be prepared for any
set of planes
Preferred Orientation (Texture)(continued)
Grains tend to be
preferentially
orientated or
textured along
<100> axes:
Recall the simplified stereographic
(001) projection for cubic crystal.class27/6
[001]
XRD pole
figure
maps for
6111 Al
alloy.
[010]
[100]
If the material had randomly oriented grains, the
pole figure would not have strong intensity spots
Thin Film Growth often results in
Preferred Orientation (Texture)•Chemical vapor deposition of ZnO (Zincite) thin film → Wurtzite crystal structure.
•Columnar growth of the ZnO grains normal to Si substrate which exhibit high (002) texture:
Cross-sectional
BFTEM
XRD (002)
Pole figure:showing
strong out of
plane fibrous
(002) texture
(z-axis ┴ to
the substrate). class27/7
20 30 40 50 60 70
0
500
1000
1500
2000
2500
3000
ZrO2
ZrO2
ZrO2
ZrO2
(202)
(211)(200)
(110)
(0002)
ZnO
Inte
nsit
y (
co
un
ts)
2 (deg)
ZrO2
(101)
ZnO
(1011)
ZnO
(1012)
ZnO
(1013)
X-ray
diffraction
(XRD) scan
50 nm
From H. Mohseni, P.C. Collins, and T. W. Scharf, “Nanocrystalline
Orientation and Phase Mapping of Textured Coatings Revealed by Precession
Electron Diffraction,” Nanomaterials and Energy, 1(6), 318-323 (2012).
TD
ND
GD
Precession electron
diffraction in the
TEM:
class27/8
• Using the ZnO XRD measured intensities (previous slide) and standard intensities
(PDF on next slide), these ZnO (002)-orientated grains are clearly textured, since the
TC was calculated to be 3 (a TC > 1 indicates preferential orientation/texture):
1. Harris, G.B. Quantitative measurement of preferred orientation in rolled uranium bars, Phil. Mag. 1952, 424, 113-123.
2. Barrett, C.S., Massalski, T.B. Structure of Metals, Pergamon, Oxford, 1980, p. 204.
Texture Coefficient (TC) Calculation
• TC of the previous ZnO film can be computed via the Harris method [1,2] considering
a Wurtzite ZnO standard (PDF# 01-071-3830 shown on next slide).
• TC measures the relative degree of preferred orientation (PO) among crystal planes
and is expressed as
where I(hkl) and Io(hkl) are the measured intensity and standard integrated intensity (from
PDF) for (hkl) reflection, respectively, and N is the number of reflections observed.
03
625
402
919
69
100
424
240
2915
4
1
2402915002
103
103
102
102
101
101
002
002
002002.
...
./)(TC
)(
)(
)(
)(
)(
)(
)(
)(
)()(=
+++
=
Preferred Orientation (Texture)(continued)
class27/10
Randomly
oriented
PE chains
Semi-oriented
PE chains after
1200%
deformation
Highly oriented
PE chains after
3600%
deformation
•Eq. (3) also becomes invalid when you have (b) extinction, i.e. crystal imperfection (mosaic
structure).
•The crystal does not have its atoms arranged on a perfectly regular lattice extending from one side
of the crystal to other; instead, the lattice is broken up into a number of tiny blocks, each slightly
disorientated from one another. Size is on order of 100 nm.
•TEM proved this in 1960’s, showed whether single crystals or individual grains in a polycrystalline
aggregate had a substructure defined by the dislocations present.
•The density of these dislocations tend to group themselves into walls (sub-grain boundaries)
surrounding small volumes having a low dislocation density (sub-grains or cells).
•This structure will increase the integrated intensity of the reflected beam relative to that
theoretically calculated for an ideally perfect crystal.
•We should not regard these as deviations from the Bragg law.
•A single atom scatters an incident beam of x-rays in all directions in space, but a large number of
atoms arranged in a perfectly periodic array in 3-D to form a crystal diffracts x-rays in relatively few
directions. It does so precisely because the periodic arrangement of atoms causes destructive
interference of the diffracted x-rays in all directions except those predicted by the Bragg law, and in
these directions constructive interference occurs.
•It is not surprising therefore that measurable diffraction occurs at non-Bragg angles whenever any
crystal imperfection results in the partial absence of one or more of the necessary conditions for
perfect destructive interference at these angles.
•These imperfections are generally slight compared to the overall regularity of the lattice, with the
result that diffracted x-ray beams are confined to very narrow angular ranges centered on the
angles predicted by the Bragg law for ideal conditions.
Extinction
class27/11
•This relation between destructive interference and structural
periodicity can be further illustrated by a comparison of x-ray
scattering by solids, liquids and gases.
•For crystalline solid, the curve is almost zero everywhere except at
diffracted beam angles.
•Both amorphous solids or liquids have structures characterized by
an almost complete lack of periodicity and tend to “order” only in the
sense that the atoms are fairly tightly packed together and show a
statistical preference for a particular interatomic distance.
•The result is an x-ray scattering curve showing nothing more than
one or two broad maxima.
•For monoatomic gases, which have no structural periodicity
whatsoever, in such gases, the atoms are arranged perfectly at
random and their relative positions change constantly with time.
•No maxima, just a regular decrease of intensity with increase in
scattering angle. The curve would be entirely featureless, horizontal,
if it were not for the fact that isolated atoms scatter x-rays more
intensely at low 2 angles than at high, i.e. atomic scattering factor
(f) plot we saw before.
•Note that x-ray radiation can cause damage/imperfections in your
material especially during long measurements→
Extinction (continued)
Compare measurements made in beginning vs. endclass27/12
Determination of Grain
size/particle size (t)Hypothetical
case
of diffraction
occurring
only at the
exact Bragg
angle
•The width of the diffraction curve on left increases as
the thickness of crystal decreases because the angular
range (21-22) increases as m (=t/d) decreases. m is
mth plane, d is interplanar distance between 1st and 2nd
plane….mth plane and t is total thickness.
•The width B is usually measured in radians at FWHM….
class27/13Debye-Scherrer formula
t=?
Example:
from B.D. Cullity “Elements of XRD”
XRD peak shifts + broadening
class27/14
Useful to know in mechanical properties
and thermal expansion
Two causes of line broadening are small particle
size (D-S formula) and nonuniform strain:
Differentiating Bragg’s law:
tan22d
db −==
where b is extra line broadening due to fractional
variation in strain, d/d. Thus we can measure strain
in material.