27-301
Microstructure-Properties
Lecture 1:
Microstructure Measurement
Profs. A. D. Rollett, M. De Graef
Updated 30th August, 2015
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Course Objectives
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The main objective of this course is to introduce you to the concept of
microstructure-property relationships and the power of being able to control
material properties through microstructure.
Previous courses have given you basic knowledge about microstructure and
shown you how to use basic ideas in crystal structure (201), thermodynamics
(215), defects (202) and phase relationships (217) in order to understand and
predict microstructure.
Now we will continue on to show you how the properties of materials depend
on their microstructure, which allows you to change the properties via
controlling microstructure.
301 (this required course) focuses mainly on single-phase materials but with
an introduction to composites.
367 (required course) teaches you about materials selection and builds upon
all the previous courses.
The final stages of the MSE program, e.g. 401, show you how to relate
properties (thus microstructure & processing) to design.
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Q&A
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Name a pioneer in studying microstructure and
explain what they did. Sorby, for example, cut and
polished specimens (esp. of steels) and examined
them using optical microscopy.
Name a method for measuring grain size and explain
it. Reduce an image to lines representing grain
boundaries; draw test lines across the image; count
the number of intersections per unit length; compute
the mean intercept length.
Name a second method for measuring grain size and
explain it. Reduce an image to lines representing
grain boundaries; count the area belonging to each
grain in the image (e.g. by counting pixels), noting
whether the grain is complete or touches an edge;
compensate for the finite thickness of the boundary
lines (if necessary); compute the average area per
grain as total area divided by the number of grains
(considering edge grains as contributing each ½);
compute, if needed, the circle-equivalent radius.
Explain the difference between a low angle grain
boundary (LAGB) and a high angle grain boundary
(HAGB); describe the variation in energy and mobility.
A LAGB can be constructed as an array of dislocations;
it has low energy and mobility compared to the HAGB.
A HAGB has a disordered atomic structure (and
cannot be represented by dislocations because the
cores would overlap) with high (constant) energy and
mobility. The energy of LAGBs can be described by
the Read-Shockley model; for mobility we can use an
exponential function.
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Explain the difference between an analytical
property and a complex property. Allowing for this
distinction being particular to this course, an
analytical property is one that can be described
with a linear relationship, and is sometimes
calculable from first principles, e.g. elastic modulus.
A complex property is one that is generally nonlinear and dependent on the behavior of large
populations of defects, e.g. work hardening, e.g.
fracture.
Explain what is meant by (crystallographic) texture.
There is a certain relationship between the crystal
axes of each grain and a frame attached to the
sample. Any bias in the alignment of crystal axes is
called texture. A consequence of texture is that any
anisotropy present in the single crystal properties is
then present in the polycrystalline material. We use
transformation of axes to convert properties known
in the crystal frame, for example, to the sample
frame. The relationship between single crystal and
polycrystal properties may, however, be complex.
Explain pole figures and inverse pole figures. A pole
figure is a map of a chosen plane normal (pole) with
respect to the specimen frame; typically the
individual points are binned, smoothed and
graphed as a contour plot. An inverse pole figure is
the same plot but a chosen sample direction (e.g.
the rolling direction) is depicted with respect to the
crystal frame.
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Microstructural Parameters, Properties
Properties
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Strength
Toughness
Formability
Conductivity
Corrosion Resistance
Piezoelectric strain
Dielectric constant
Magnetic Permeability
Microstructural
Parameters
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Grain size
Grain shape
Phase structure
Composite structure
Chemical composition
(alloying)
• Crystal structure
• Defect structure (e.g.
porosity)
Think of this course as learning how to connect
quantities in the two columns quantitatively
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Analytical vs. Complex Properties
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“Analytical” Properties
Piezoelectricity
Dielectric constant
Piezomagnetism
Pyromagnetism
Magnetostriction
Thermal Conductivity
Thermoelectricity
Elastic Modulus
Density
Specific Heat
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“Complex” Properties
Strength
Toughness
Fatigue Resistance
Corrosion Resistance
Formability
Creep Resistance
Magnetic Permeability
Weldability
Wear Resistance
Castability
“Analytical” properties are generally linear, albeit anisotropic, that can often be calculated from first principles.
“Complex” properties are rarely linear, sometimes based on a threshold value, and require constitutive relations
to quantify them.
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Optimization of Multiple Properties
Strength
Weldability
Thermal Conductivity
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Toughness
Corrosion
Resistance
It is generally the case that any given material must meet minimum requirements
for multiple properties.
Therefore it is not feasible to optimize one property at the expense of all others.
On the “spider diagram” example, one can visualize this requirement by seeing
that the polygon must have vertices at some distance from the origin along every
axis.
This will be discussed further in Lab 2 and in the Capstone course (401).
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Topics/ Lecture List
• What is microstructure? How do we measure microstructure?
• What is a material property? How does crystal symmetry
affect material properties? How do we represent material
properties in a mathematical framework?
• Strength-toughness-microstructure relationships.
• Recrystallization for grain size control.
• How grain size affects properties such as strength, creep,
electrical conductivity (varistors).
• Toughness-microstructure relationships.
• Fatigue-microstructure relationships.
• Composites - basic structure, properties
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Details
• All relevant materials for the course are published on the Blackboard
website.
• CAs for Labs in 301: see the posted Syllabus.
• Homework each week, posted on website.
• Quiz each week; mostly Wednesdays.
• 3 Exams: see the posted Schedule.
• Two Labs: Lab 1 on recrystallization+ measurement of
microstructure (written report); Lab 2 on mechanical behavior +
optimization of properties (oral presentations plus copies of slides
with full supplemental information).
• Labs will run concurrently (lack of equipment), so consult the Lab
groups, schedule.
• Computer skills: Excel, Matlab (for homeworks) and ImageJ (for
image analysis, Labs).
• Extensive in-class interaction; reading of lecture notes expected so
that we can discuss concepts in class.
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Microstructure = internal structure
• Biology was revolutionized when
Leeuwenhoek and others started to use
microscopes to look at the internal
structure of plants. They were able to
relate many characteristics of plants to
their cell structure, for example.
*http://www.ucmp.berkeley.edu/history/leeuwenhoek.html
• Similarly, Sorby† was one of the first
to make cross-sections of materials
such as iron and examine them in
the microscope, so that he could
relate properties to structure.
† http://www.shu.ac.uk/sorby/hcsorby.shtml
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What is microstructure?
• Microstructure originally meant the structure inside a material that could
be observed with the aid of a microscope.
• In contrast to the crystals that make up materials, which can be
approximated as collections of atoms in specific packing arrangements
(crystal structure), microstructure is the collection of defects in the material.
• What defects are we interested in? Interfaces (both grain boundaries and
interphase boundaries), dislocations (and other line defects), and point
defects.
• Since the invention of prefixes for units, the micrometer (1 µm) happens to
correspond to the wavelength of light. Light, obviously is used to form
images in a light/optical microscope. Thus microstructure has come to be
accepted as those elements of structure with length scale of order 1 µm.
• Since we commonly examine materials in the microscope, we generally
observe grains as crystallites in polycrystals, separated by grain boundaries.
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Grain Orientation
Polycrystal with no texture,
i.e. no crystallographic preferred orientation
Think of each crystallite as a separate
lattice orientation, represented by a cube
102
110
100
221
111
103
123
112
122
100
102
221
110 111
100
Colors are often used in images to denote the
orientation: in “inverse pole figure maps”, the color is
based on the crystal direction that is parallel to a
sample direction (often the surface normal); in the
absence of texture, all colors are present
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Grain Orientation Texture
Polycrystal with orientation texture
[111]
Axial texture: in this example (often observed in rolled sheets of
bcc metals), most grains have a <111> // sample normal
Grain colors correlate with orientation; here
the strong color indicates a strong texture
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Grain Orientation Distribution
Axial texture is easily represented on an inverse pole figure
Strong axial <110> texture
MRD
8.0
4.0
2.0
1.0
Data from a Si-Fe alloy that has a strong Goss texture component, which means
crystals aligned with <100>//RD and <011>//ND
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Tensor Transformation Rule
• There are quantities known as tensors. In order for a quantity to “qualify”
as a tensor it has to obey the axis transformation rule.
• The idea is that a tensor quantity represents something physical that is the
same regardless of which frame of reference (i.e. set of axes) one uses to
describe it. Examples are electric current (vector), stress (2nd rank tensor),
elastic modulus (4th rank tensor).
• The transformation rule defines relationships between transformed
(quantities with a prime) and untransformed tensors (no prime) of various
ranks.
• The Einstein convention applies, i.e. a repeated index represents a
summation over that index. In the 1st example below, “V” means a vector
and the index “j” is repeated (on the RHS) and therefore one performs a
sum on that index.
Vector:
2nd rank
3rd rank
4th rank
V’i = aijVj
T’ij = aikajlTkl
T’ijk = ailajmaknTlmn
T’ijkl = aimajnakoalpTmnop
• This rule is a critical piece of information, which you must know how to
use.
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Axis Transformations for Texture
• We now define how we use axis transformation to deal with problems
involving texture (crystal orientation).
• The standard definition of crystal orientation is with an orientation matrix,
g, which is exactly equivalent to the “a” on the previous slide. We associate
a unit vector, ê, with each axis in both the crystal and the sample reference
frames. The orientation matrix is then calculated as follows.
gij = êcrystal • êsample
• Given any quantity expressed in sample coordinates, it can be transformed
(converted, if you like) to crystal coordinates by using the version of the
tensor transformation rule that is appropriate to the rank of the tensor
concerned (previous slide).
• To transform in the opposite direction (crystal to sample), one uses the
same transformation but reverses the sense of rotation. This sounds
complicated but it’s actually simple because one uses the transpose, gT.
That we can do this is because of the properties of rotations, which can be
described by orthogonal matrices.
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Axis Transformation: Examples
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As a practical illustration, consider where the color in the inverse pole figure map comes from.
What we want is to know which crystal axis is parallel to the surface normal of a sample. By
convention, we assume a perfectly flat surface and set the sample z-axis perpendicular to the
surface (with the x- and y-axes in the plane).
This means that we need to transform a vector, V=[0,0,1], where the numbers are the x, y
& z coefficients of the vector. (Vectors are 1st rank tensors).
Using a Matlab routine called EulerToMatrix_ADR.m, we can obtain an orientation matrix from
a set of three Euler angles (to be explained at a later date), specified as [0.,54.5,45.].
Orientation Matrix: =
--1-->
--2-->
--3-->
--1-->
0.70711
0.41062
0.57567
--2-->
-0.70711
0.41062
0.57567
--3-->
0
-0.81412
0.58070
By performing matrix multiplication of this matrix with the vector (treat it as a column vector),
V = [0, 0, 1], we obtain the new, transformed vector,
V’ = [0.576, 0.576, 0.581] .
In relation to the color maps already shown, this transformed vector is very close to [111],
which means that we would color that grain blue.
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Texture Measurement
• The best method of quantifying texture is to
measure the crystallographic orientation of a
statistically representative set of grains in the
material.
• Historically, this was a tedious exercise with Laue
X-ray diffractograms that was only possible on
single crystals or very coarse polycrystals.
• The standard characterization method for texture
(crystallographic preferred orientation) in an
average sense is to measure x-ray pole figures.
• A more modern technique is that of Orientation
Imaging Microscopy (OIM), which is readily
available in the SEM (and much used here at
CMU). This Electron Back Scatter Diffraction
(EBSD) technique produces a map of orientation
measured on a regular grid of points (pixels).
• Now synchrotron x-rays can measure 3D
orientation maps non-destructively
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What is a Grain Boundary?
• Grain boundaries control the process of recrystallization. This
suggests that it is worth knowing something about the
structure and properties of boundaries.
• Regular atomic packing disrupted at the boundary by the
change in lattice directions.
• In most crystalline solids, a grain boundary is very thin
(one/two atoms).
• Disorder (broken bonds) unavoidable for geometrical reasons;
therefore large excess free energy. This interfacial energy is
analogous to the surface tension in a soap bubble (and many
investigations on grain growth have been made with soap
froths).
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Crystal orientations at a g.b.
TJACB
gB
∆g=gBgA-1
gD
TJABC
gC
gA
Here, “g” represents a crystal orientation, e.g. as a rotation matrix.
Also, “TJ” means triple junction; ∆g represents misorientation i.e. difference in orientation
across a grain boundary (also a rotation).
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Grain Boundary Structure
• High angle boundaries: can be thought of as two
crystallographic planes joined together (with or w/o a twist of
the lattices).
• Low angle boundaries are arrays (walls) of dislocations: this is
particularly simple to understand for pure tilt boundaries [to be
explained]. From this comes the Read-Shockley model for grain
boundary energy (as a function of misorientation).
• Grain boundary energy increases monotonically with
misorientation as a consequence of increasing dislocation
density.
• Transition: in the range 10-15°, the dislocation structure
changes to a high angle boundary structure.
• The grain boundary mobility increases abruptly at this
transition in structure.
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LAGB to HAGB Transitions
• The distinction between low
and high angle grain boundaries
(LAGB versus HAGB) is
important in recrystallization
(L2). Only the HAGB are able to
move under the driving force of
energy stored in the dislocation
structure.
• The Read-Shockley equation
describes the energy of low
angle boundaries.
• An exponential
function is useful for describing
the sharp transition in mobility
from low- to high-angle
boundaries
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Example: tilt boundary = array of
parallel edge dislocations
b
• Low angle boundaries are arrays of parallel edge dislocations if the
rotation between the lattices is small and the rotation axis lies in
the boundary plane. In this example, the rotation axis between the
two crystals is perpendicular to the plane of the picture. Again the
energy of a dislocation array, which is a low angle grain boundary is
described by the Read-Shockley model.
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Grain Boundary Misorientation Texture
Polycrystal with orientation and
misorientation texture
Red grain boundaries have the same misorientation
We will need special methods to represent Orientation Texture and
Grain Boundary Texture – see 27-750
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Types of Defect
Porter & Easterling
• Interfaces - phase boundaries
– Micrograph shows theta-prime
precipitates (Al2Cu) in Al
• Interfaces - grain boundaries
– Micrograph actually shows a soap froth
but the image is representative of a grain
boundary network in a polycrystal.
• Dislocations (line defects)
Reed-Hill
– This messy image illustrates the difficulty
of counting dislocations: generally one
avoids counting individual dislocations!
www.feic.com/support/tem/stdisloc.htm
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slides
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Quantification of Microstructure
• It is essential to quantify microstructure in order to
be able to predict properties quantitatively.
• What you quantify depends on the property, i.e.
what question you ask of the material.
• Examples of quantitative microstructural
parameters:
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Grain size
Void fraction
Second phase particle size
Aspect ratio of second phase particles or grains
Average distance between particles
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Grain Size Measurement
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How to measure with linear intercepts
How to measure with area counts
Grain size number (the E112 ASTM standard)
The problem of plane sections (stereology)
The problem of grain shape
See:
http://www.metallography.com/grain.htm
• Useful references: : Quantitative Stereology, E.E.
Underwood, Addison-Wesley, 1970; Practical
Stereology by John C. Russ.
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Grain size Measurement
Linear Intercept Methods
• In the case of ‘space-filling’ grains, Stereology provides us with a simple relationship:
<L> = 1/NL = 1/PL.
• Definitions:
<L>:= mean intercept length
NL := number of intercepts per unit length
PL := number of points per unit length
• Test lines can be straight or circles depending on method (next page highlights
comparison)
• Grain shape determines whether the mean intercept length, <L>, is a 2D or 3D
parameter.
Equiaxed  <L3>
Columnar  <L2>
Why?
Area-based Method (Planimetric Method)
• Another simple measure of grain size in 2D sections is based on area. Count the number
of grains, NA, and also measure the area in the image (an edge grain is counted as 0.5).
Divide the area by the number to obtain <A>. One may then calculate the areaequivalent diameter (or circle-equivalent diameter),
<D> = 2√(<A>/π).
• This approach can only be applied to equiaxed morphologies! Slight deviations from
equiaxed lead to significant errors in the estimated grain size.
This size, <D>, is not the same as the mean intercept length, <L>!
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Line Intercept Technique
• One can either count the number of
intercepts per unit length along a
straight line (which is sensitive to the
orientation of the line)
• Or, one can count intercepts around a
circle (eliminates any anisotropy in the
microstructure) and divide by the
perimeter length of the circle to
obtain PL.
• Grain size = <L3> = PL-1
• Note some elementary image analysis:
increasing the contrast on the original
images made it much easier to
perceive the two separate phases.
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public use of these slides
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Grain size measurement: intercepts
• From Table 2.2 [Underwood], column (a),
illustrates how to make a measurement of the
mean intercept length, based on the number of
grains per unit length of test line.
<L3> = 1/NL
• Important: use many test lines that are
randomly oriented with respect to the structure.
• Assuming spherical† grains, <L3> = 4r/3,
[Underwood, Table 4.1], there are 7
intersections and if we take the total (red) test
line length, LT= 798µm, then LTNL= 7, so NL=
1/114 µm-1
 d = 2*3*<L3>/4 = 6/(4*NL) = 6*114/4
= 171 µm.
† Ask yourself what a better assumption about grain shape
might be!
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Grain size measurement: area based
[Underwood]
Fig. 7.12
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Grain count method:
<A>=1/NA
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Number of whole grains= 20
Number of edge grains= 21
Effective total=Nwhole+Nedge/2
= 30.5
Total area= 0.5 mm2
Thus, NA= 61 mm-2; <A>=16,400 µm2
2D size: d = 2√(<A>/π) = 177 µm.
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For 3D, assume spherical grains, <A> mean intercept area= 2/3πr2
 d = 2√(3<A>/2π)= 144 µm.
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Neither of these values is equal to the value determined on the
previous slide, using the linear intercept method.
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Alternative Representation:
ASTM Grain Size Number
• ASTM has defined a standard, E112, for grain size
measurement.
• ASTM has a grain size parameter, G, which can be
calculated based on either area or linear measurements.
• This ASTM grain size number, G, is commonly employed
within industry and earlier research efforts (before
computer technologies were available).
• Higher grain size number means smaller grain size.
American Standards and Test Methods, Designation E112, (1996).
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Analysis with ImageJ*
• Read an image into ImageJ
(Mac OS X, in this case),
threshold the image so as to
eliminate background pixels;
“close” (or use erode/dilate)
to fill in pixels within
particles; “analyze particles”
to identify and measure each
particle in the selected area.
* The URL for ImageJ is: http://rsb.info.nih.gov/ij/ and the version in 2015 is 1.49
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Grain Size
analysis with
ImageJ
A (traced) grain image
has to be
(1) read in;
(2) skeletonized;
(3) dilated;
(4) inverted.
Then each grain can be
analyzed as if it were a
particle. This is point
counting.
You must apply a
correction to each grain
area to account for the
finite width of the
boundaries.
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Recommended
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Books, References
Porter, D.A. & K.E. Easterling, Phase transformations in metals and alloys, Chapman &
Hall (669.94 P84P2)
Newnham, R.E., Properties of Materials: Anisotropy, Symmetry, Structure. Oxford
University Press, 2004.
Nye, J.F., Physical Properties of Crystals. Clarendon Press, 1957 (548.8 N99PA); this book
has been reprinted fairly recently.
Underwood, E.E., Quantitative Stereology, 1971, Addison-Wesley
Humphreys, F.J. and M. Hatherly (1996). Recrystallization and related annealing
phenomena. Oxford, UK, Elsevier Science, Oxford, UK
Russ, J.C. and R.T. Dehoff (1999). Practical Stereology, New York, Plenum Press.
Reed-Hill, R.E. (1973). Physical Metallurgy Principles, New York, Van Nostrand.
http://www.metallography.com/grain.htm - grain size measurement
http://www.stereologysociety.org/ - resources for stereology
http://www.doitpoms.ac.uk/ - a nice library of micrographs and microstructures at the
Univ. Cambridge (UK)
Howard, C.V. and M.G. Reed (1998). Unbiased Stereology, New York, Springer-Verlag.
Martin, J., R.D. Doherty, et al. (1997). Stability of microstructure in metallic systems.
Cambridge, England, Cambridge University Press.
Nutting & Baker, The Microstructure of Metals, 1965, The Institute of Metals (UK).
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Summary
• Objectives of the course have been described.
• Examples of ‘typical’ microstructures
described.
• Methods for measuring grain size described:
first method is based on linear intercepts.
Second method is based on average area on
the cross section.
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Supplemental Slides
• The following slides provide supplemental
information, e.g. to explain what stereology is.
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Plane sectioning
• General problem: how can one estimate the true 3D
distribution of sizes based on (2D) plane sections through
the objects? This challenge is the essence of stereology.
• Imagine an array of spheres randomly arranged in space:
cut through the space with a plane: examine the spheres.
What do you see?
• Answer: a array of circles of variable size, of which the
maximum possible diameter (size) is the sphere diameter.
• How to measure? The easiest is to draw random straight
lines and measure intercepts.
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Stereology
• Stereology provides us with relationships between
quantities that are measurable on a plane section, and 3D
quantities that we may need.
• Two quantities that are (relatively) easily measured are
the number of points per unit length along a line, PL, and
the number of points per unit area, PA. This nomenclature
can be extended to volumes (V) and surfaces (S).
• Quantities per unit volume have to be calculated (table).
• In this course, we will describe a few key results from
stereology, mainly motivated by the need to be able to
measure grain size. No derivations of the equations will
be carried out.
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Definitions of Stereological Parameters
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VV, or, Vf or f := volume fraction
AA := area fraction
LL := line length fraction
PP := point fraction
SV := surface area per volume
LA := line length per area
PL := points per length
NL := points per length; usually points of intersection with
interfaces/boundaries
<L>:= mean intercept length; with subscript 2 or 3 for length in 2 or
3 dimensions
PA := points per area
LV := line length per volume
PV := points per volume
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Example: volume fractions
• In two-phase structures, it is important to quantify how
much of the second phase is present in the material.
• The standard measure here is volume fraction, often
written as vf.
• Fortunately, stereology shows us that the volume fraction
is simply equal to the area fraction on the section plane
(which is also equal to the point/pixel fraction):
vf = AA = LL = PP (= VV).
• This allows us to counts pixels in a digitized image to
estimate area and volume fractions.
• Similar approach used for particle size.
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Point Fraction Counts
Example from “Unbiased Stereology” by
Howard and Reed
• 2-phase microstructure: phase-1 is dark gray, phase-2 is light gray.
• Point counting is extremely simple: from the total of grid points
that fall within the image, count the number that fall within phase1 and the number that fall within phase-2. Divide by the total
number of points (within the image) to obtain the point fractions
of the two phases, which are equal to the area fractions, which are
equal to the volume fractions.
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Basic relationships in stereology
Underwood, Quantitative Stereology, 1971, Addison-Wesley
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Quantitative Relationships
• On the left hand column,
PP = LL = AA = Vv
• For the right hand triangle, more complex relationships
exist:
SV = 2PL = (4/π)LA
LV = 2PA
PV = 1/2 LvSV = 2PAPL
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Quantitative Relationships: SV  NL
Space-filling structures
Dispersed Phases
N L  PL
2 N L  PL
SV 
4

L A  2PL
SV  4N L
SV  2N L
L
2SV  L
1
NL
4SV  L
When we analyze the grain characteristics in typical metal alloys, we will use the lefthand relationships; for particle statistics (VV<<1), the right-hand equation is valid.
It is apparent that a factor of 2 is the difference between the two approaches, which can
be attributed to the sharing of grain boundary area between 2 grains.
J.C. Russ, Practical Stereology, Plenum Press, New York (1986).
E.E. Underwood, Quantitative Stereology, Addison-Wesley,
MA (1970).
45
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46
Measurements on sections
(of spherical particles)
• Diameters require great care for accuracy.
• Areas are convenient if automated pixel
counting available (to give PA).
• Chords are convenient for use of random
test lines: nL := number of chords per unit length
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47
Mean free path, l, versus
Nearest neighbor spacing, ∆
• It is useful (and therefore important) to keep the
difference between mean free path and nearest
neighbor spacing separate and distinct.
• Mean free path is how far, on average, you travel from
one particle until you encounter another one.
• Nearest neighbor spacing is how far apart, on average,
two nearest neighbors are from each other.
• They appear at first glance to be the same thing but
they are not!
• They are related to one another, as we shall see in the
next few slides.
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48
Nearest-Neighbor Distances
• Also useful are distances between nearest
neighbors: S. Chandrasekhar, “Stochastic problems in
physics and astronomy”, Rev. Mod. Physics, 15, 83
(1943).
• r := particle radius
• 2D: ∆2 = 0.5 / √PA
(4.18a)
• 3D: ∆3 = 0.554 (PV)-1/3
(4.18)
• Based on l~1/NL, ∆3  0.554 (πr2 l)1/3
for small VV,
∆2  0.500 (π/2 rl)1/2
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49
Application of ∆2
• Percolation of dislocation
lines through arrays of 2D
point obstacles.
• Caution! “Spacing” has
many interpretations: select
the correct one!
•
In general, if the obstacles are weak
and the dislocations are nearly
straight then the relevant spacing is
the mean free path. Conversely, if the
obstacles are strong and the
dislocations bend then the relevant
spacing is ∆2.
Hull & Bacon;
fig. 10.17
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50
Particle Pinning - Summary
• Strong obstacles + flexible
entities: nearest neighbor
spacing, ∆2, applies.
• Weak obstacles + inflexible
entities: mean free path, l,
applies.
• This applies to dislocations or
grain boundaries or domain
walls.
• Note the similar dependence on
particle size, r, but very different
dependence on volume fraction,
f!
r
l»
f
(a )
f º VV
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51
Grain Shape in 3D
•
•
•
•
•
•
•
Extracting sizes of grains in 3 dimensions is not straightforward!
For spheres, we can write that V=4/3πr3=4.189r3. A=4πr2=12.57r2,
L3:=mean intercept length =1.33r. V=1.78L33, mean intercept area =
2/3πr2.
Unfortunately, grains in real materials are not spheres!
Therefore it is not obvious what the relationships between volume, size,
surface area and number of sides might be.
Real grains can be approximated by the Kelvin tetrakaidecahdron, or
truncated octahedron.
Edge length:=a, V=11.314a3, A=26.785a2, L3=1.69a. V=2.34L33, mean
intercept area = 3.77a2.
Read more about shapes in 3D versus cross-sectional measurements in
Underwood’s book on Stereology.
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52
3D Shape Example
To confirm a material has an equiaxed
microstructure, two orthogonal sections
must be sampled.
Truncated Octahedron
a= <L3>/1.69 = 36mm/1.69 = 21.3mm
300mm
<L>=38mm ± 2.5mm
300mm
<L>=34mm ± 3.5mm
It is evident that the material has an equiaxed morphology and the mean intercept
length from either plane is representative of the entire microstructure.
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53
Grain shapes, contd.
• For grains in
polycrystalline
solids, the shapes
are approximated
by
tetrakaidecahedra:
a-ttkd to b-ttkd.
(a) soap froth; (b) plant pith cells; (c) grains in Al-Sn
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54
Polymer Composite
AFM Images
•
•
These images are taken from
“The application of atomic force
microscope of the
characterization of industrial
polymer materials” by G.K. Bar
and G.F. Meyers, MRS Bulletin,
July 2004, p 464.
How would you go about
measuring the volume fractions
in these materials? How would
you estimate the true 3D
distribution of particle sizes?
What would you do about the
areas of dark phase included
inside the other light phase in
the 40/60 composite?
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