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

)(

)(

)(

)(

)(

)(

)(

)(

)()(=

+++

=

class27/9

Powder Diffraction File for

Zincite-ZnO (Wurtzite)

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.

class27/15

Determine its approximate % crystallinity….

XRD pattern for a Semi-

crystalline Polymer