Chapter 3
Geometry of Microstructures
The generic term that is used to describe the physical elements that make up
a microstructure in this text applies the concept of a phase borrowed from classical
thermodynamics. Each phase in a microstructure is a set of three dimensional features. To belong to the same phase the features must have the same internal physical properties. Usually this means features of one phase have the same chemical
makeup and the same atomic, molecular, crystal or biological structure. The collection of parts of the structure that belong to the same phase is one example of a
feature set.
Microstructures are space filling, not-regular, not-random arrangements of
the feature sets of phases in three dimensional space. A microstructure may be a
single phase tessellation1 consisting of four feature sets, as shown in Figure 3.1: polyhedral cells that have faces, edges and vertices arranged to fill the three dimensional
space. Alternatively a microstructure may consist of two phases—two distinguishable feature sets—labeled, for instance, a and b, as shown in Figure 3.2, either or
both of which may be cell structures, that fit together with precision to fill up the
space occupied by the structure. Microstructures frequently consist of several distinguishable phases each contributing to the total collection of feature sets in the
system, as shown in Figure 3.3. In some cases, voids or porosity may be present and
is also treated as a measurable phase.
In the description of a microstructure it may be useful to focus on the properties of a particular feature set such as the b phase and its boundaries, edges
and vertices if it has them. Each member of the feature set has its own collection
of geometric properties. For example, particles of the b phase each have their own
volume, surface area and lineal dimensions such as tangent diameter. Properties of
the whole collection of features in the set are called global properties. Examples
include the total volume, or boundary surface area, or number of particles in the
set. A more complete description of the structure might incorporate information
about the spatial distribution of the b phase in the context of other feature sets in
the structure.
To facilitate the organization of an exercise in microstructural characterization, this chapter introduces three levels of characterization of microstructures:
1. The qualitative microstructural state;
2. The quantitative microstructural state; and
3. The topographic microstructural state.
1
A tesselation is a subdivision of space into cells which then fill that space.
29
30
Chapter 3
a
b
Figure 3.1. Idealized illustration of a space filling tesselation with a cell removed to
show faces (red), edges (triple lines, blue) and vertices (quadruple points, green). (For
color representation see the attached CD-ROM.)
Geometry of Microstructures
31
c
Figure 3.1. Continued
The first of these levels of description is a list of the classes of feature sets
that exist in the structure. The second level makes the description quantitative by
assigning numerical values to geometric properties that are appropriate to each
feature set. The third level of description deals with nonuniformities in the spatial
distribution that may exist in the structure.
The Qualitative Microstructural State
Microstructures consist, not only of the three dimensional features that make
up the phases in the structure, but also of two dimensional surfaces, one dimensional lines and zero dimensional points associated with these features. The qualitative microstructural state is a list of all of the classes of feature sets that are found
in the structure. Surfaces, edges and points arise from the incidence of three dimensional particles or cells. For example the incidence of two cells in space forms a
surface. The kind of surface is made explicit by reporting the class of the two cells
that form it. If both belong to the a feature set, the surface is an aa surface; if one
is from the a set and the other is a member of the collection of b cells, the interface is an ab interface. And so on. Triple lines are formed by the incidence of three
cells, (e.g., an abb triple line results from the incidence of an a cell with two b cells
a
b
c
Figure 3.2. Particles of a second phase may distribute themselves at vertices (a), along
triple lies (b), on the faces (c), or within the cell volume (not shown) in a space filling
structure. The white phase is a, the colored phase is b. (For color representation see
the attached CD-ROM.)
Geometry of Microstructures
33
Figure 3.3. Microstructures may consist of a number of phases. In this example, based
on a superalloy metal, there are five: a matrix phase (white background), a precipitate
(red) distributed through the matrix, and three different phases (blue, yellow, and green)
distributed along the undulating cell boundary.
in space). Quadruple points require the incidence of four cells, and are so labeled,
(e.g., aabb).
The qualitative microstructural state can be assessed or inferred by inspection of a sufficient number of fields to represent the structure. In many cases, one
field will be enough for this qualitative purpose. In making this assessment keep in
mind that the process of sectioning the structure reduces the dimensions of the features by one, as shown in Figure 3.4. Sections through three dimensional cells of
volume features appear as two dimensional areas on the section. Sections through
two dimensional surfaces appear as one dimensional lines or curves on the section.
Lineal features in space appear only as points of intersection with the sectioning
plane; triple lines in a cell structure intersect as triple points. Point feature sets in
three dimensions, such as quadruple points, are in general not observable on a sectioning plane.
Exhaustive lists of possible feature classes for single phase and two phase
structures are given in Table 3.1. The features that may exist in a three phase structure are listed in Table 3.2. Examples of each feature class are presented on the sections shown in Figure 3.4.
Characterization of any given microstructure should begin with an
explicit list of the feature sets it contains. This exercise is trivial for single
phase structures. For systems with two or more phases this exercise has three
purposes:
34
Chapter 3
a
b
c
Figure 3.4. The dimension of each feature is reduced by one when a three dimensional
structure is sectioned by a plane. In the illustration, phases a (an array of transparent
grains or cells) and b (colored), surfaces aa (grain or cell boundaries) and ab, and triple
lines aaa and aab are sectioned so that volumes are represented by areas, surfaces
by lines, and lines by points. Points in the original volume are not observable in the
plane. (For color representation see the attached CD-ROM.)
1. To force you to think in terms of the three dimensional geometry of the
structure;
2. To identify in the list of possible features those that are absent;
3. To provide an explicit basis for eliminating from characterization those
feature sets that may not be of interest.
Geometry of Microstructures
35
Table 3.1. Feature classes that may exist in one and two phase microstructures
Feature Class
Volumes
Surfaces
Triple Lines
Quadruple Points
Total Number of Classes
Single Phase (a)
a
aa
aaa
aaaa
4
Two Phase (a + b)
a, b
aa, ab, bb
aaa, aab, abb, bbb
aaaa, aaab, aabb, abbb, bbbb
14
For example, if the b phase is porosity, then it contains no internal
boundaries. This means that bb surface features are absent, and any triple lines
and quadruple points that contain two or more b’s in their designation are
also absent. The list of feature sets contained in such a structure is: a, b; aa, ab;
aaa, aab; aaaa, aaab. There are no features described by bb; abb, bbb; aabb,
abbb, bbbb.
The description of the qualitative microstructural state may be further
expanded by ascribing qualitative aspects of shape, scale, topology or topography.
Shape is frequently conveyed qualitatively by comparing structural features with
familiar objects: alveoli in the lung are “bunches of grapes”; solidifying grains are
“dendritic” or tree like, as are neural cells; a structure may be equiaxed, plate-like
or rod-like.
The Quantitative Microstructural State
Associated with each of the classes of feature sets listed in the last section is
one or more geometric properties. Use of stereology to estimate values for these
properties constitutes specification of the quantitative microstructural state. Properties that are stereologically accessible have the useful attribute that they have
unambiguous meaning for feature sets of arbitrary shape or complexity. These geometric properties may be associated with individual features in the feature set, or as
global properties of the whole feature set.
Metric Properties
As an example, consider a dispersion of particles of b in an a matrix,
as shown in Figure 3.5. In three dimensions individual particles of b each have a
value for their volume. Over the collection of all of the b particles there is some
Table 3.2. Feature classes that may exist in a three phase microstructure
Feature Class
Volumes
Surfaces
Triple Lines
Quadruple Points
Total Number of Classes
Three Phases (a + b + e)
a, b, e
aa, ab, ae, bb, be, ee
aaa, aab, aae, abb, aee, bbb, bbe, bee, eee, abe
aaaa, aaab, aaae, aabb, aaee, abbb, aeee, bbbb, bbbe,
bbee, beee, eeee, aabe, abbe, abee
34
36
Chapter 3
Figure 3.5. A simple two-phase microstructure provides a sample of the volumes and
surfaces that exist in the three-dimensional structure.
distribution of values of volume. Further, taken together, the entire collection of b
particles has a global property which is its total volume. Stereological measures are
generally related to these global properties of the full set of features. They are
usually reported in normalized units as the value of the property per unit volume
of structure. Thus, the total volume of the set of b particles divided by the total
volume of the structure that contains them is reported as the volume fraction,
written VV, of the b phase. The volume fraction occupied by a feature set may be
quantitatively estimated through the point count, the most used relationship in stereology, see Chapters 2 and 4.
Surfaces that exist as two dimensional feature sets in the three dimensional
microstructure possess the property area. Each feature in a surface or interfacial
feature set in the structure has a value of its area. The feature set as a whole has a
global value of its surface area. The concept of the area of a surface has unambiguous meaning for feature sets of arbitrary size, shape, size distribution or complexity. Cell faces in a tessellation have an area. The collection of ab interfaces on
the set of b particles in an a matrix has an area. The surface that separates particles from the gas phase in a stack of powder has an area. The normalized global
property measured stereologically is the surface area per unit volume, written SV,
sometimes called the surface area density of the two dimensional feature set. The
surface area may be estimated quantitatively with the line intercept count described
in Chapters 2 and 4.
Geometry of Microstructures
37
Lines or space curves that exist in the three dimensional microstructure, such
as the aaa or aab lines in the structure in Figure 3.4, possess a length. Individual
line segments, such as edges in a cell network, each have a length. The full feature
set has a global value of its length, reported as LV, the length per unit volume in
stereological measurements. Many feature sets, such as fibers in composite materials, axons in neurons, capillary blood vessels, or plant roots, approximate lineal features. Length density of each of these feature sets may be estimated stereologically
by applying the area point count on plane probes through the structure as presented
in Chapters 2 and 4.
Topological Properties
Line length, surface area and volume are called metric properties because
they depend explicitly on the dimensions of the features under examination. Geometric properties that do not depend upon shape, size, or size distribution are called
the topological properties. Most familiar of these properties is the number of disconnected parts of a feature set. The b phase particles in Figure 3.6 may be counted
in principle to report their number. The surfaces bounding these particles also may
be counted. In this case the number of disconnected parts in the collection of ab
surfaces is the same as the number of particles in the three dimensional b feature
set. (This will not be true if the particles are hollow spheres, for example. Then each
particle is bounded by two surfaces and the number of surfaces is twice the number
of particles.) The number of edges in the network of triple lines in a tessellation is
countable. The normalized value of this generic topological property is the number
density, NV.
A less familiar topological property is the connectivity of a feature set. Connectivity reports the number of extra connections that features in the set have with
themselves. To visualize the connectivity of a three dimensional feature determine
how many times the feature could be sliced with a knife without dividing it into two
parts, as shown in Figure 3.6. The sphere, the potato or the blob in the top row in
Figure 3.6 all have connectivity zero, since any cut will divide any of these features
into two parts. Features that may be deformed into a sphere without tearing or
making new joints are “topologically equivalent to a sphere” and are said to be
“simply connected”.
The features in the second row in Figure 3.6 can all be cut once without
separating the feature into two parts. These features all have connectivity equal
to one and are topologically equivalent to a torus. Connectivities of the remaining
features are given on the figure. A microstructure that consists of a network,
such as a powder stack, or the capillaries in an organ, may have a very large
value of connectivity. The normalized property is the connectivity per unit volume,
CV.
A third topological property of a feature set, called the Euler (pronounced
“oiler”) characteristic, c, is the difference between the number of connected parts
and the global connectivity of the feature set:
c=N-C
(3.1)
38
Chapter 3
Figure 3.6. Connectivity, C, is the maximum number of cuts that can be made through
a three-dimensional feature without separating it into two parts.
This combination is a useful measure of the topological properties that combines both rudimentary concepts. It is more directly accessible stereologically, as will
be shown in Chapter 5. Measurements of the topological properties require a sample
of the three dimensional volume of the structure either by serial sectioning and
reconstructing the three dimensional feature set or by applying the disector, see
Chapters 2 and 5.
Table 3.3 reviews the primary geometric properties of three dimensional
microstructures that can be estimated with stereology. Values for each may be estimated and appropriately assigned among the list of feature sets contained in the
qualitative microstructural state to provide the quantitative microstructural state for
the microstructure.
Some additional geometric properties may be defined that involve the
concept of the local curvature at a point on a surface, or at a point on a line.
Table 3.3. Geometric properties of feature sets
Features
a, b . . .
a, b . . .
a, b . . .
aa, ab . . .
aaa, aab . . .
Geometric Property
Topological Properties
Number Density
Connectivity Density
Metric Properties
Volume Fraction
Surface Area Density
Length Density
Symbol/Units
NV m-3
CV m-3
VV m3/m3
SV m2/m3
LV m/m3
Geometry of Microstructures
39
These concepts require more detailed development. They are unfamiliar and
not as widely used as those listed in Table 3.3. For the interested reader they are
developed in detail in Chapter 5, along with the stereological methods that permit
their estimation.
Ratios of Global Properties
Ratios of selected global properties also provide some useful measures of
microstructural geometry. A representative measure of the scale (i.e., size) of the
features in the feature set is the mean lineal intercept, ·lÒ. This property is the
average length of lines that intersect the features in the set. It is the mean surfaceto-surface distance through the three dimensional features in the structure. It is given
by
l =
4V 4VV
=
S
SV
(3.2)
This property must be applied knowledgeably in interpreting its meaning.
For a collection of same size spheres it is equal to 2/3 the sphere diameter.
4
4Ê pR 3 ˆ
Ë3
¯ 4
4V
=
l =
= R
3
S
4pR 2
(3.3)
If the spheres have a size distribution, this property is related to a ratio of
the third to the second moment of the distribution function; it is not the mean particle radius (Underwood, 1970a). For a collection of rod shapes ·lÒ reports the diameter and provides no information about their length (·lÒ = 3/2D). For plates, it is
primarily determined by the plate thickness (·lÒ = 2t) and provides no information
about dimensions in the plane of the plate.
Other accessible ratios of global properties report the mean cross sectional
area of particles, ·AÒ, and the average mean surface curvature, ·HÒ, which may be a
useful quantity when capillarity or surface tension effects play a role in shaping the
microstructure. A more detailed presentation of the measurement and meaning of
these ratios is given in Chapter 5.
The Topographic Microstructural State
Many if not most microstructures exhibit nonuniformities in the distribution
and disposition of the feature sets that make them up. Quantitative measures of
these nonuniformities lead to the specification of the topographical microstructural
state.
Gradients: Variation with Position
If a metal rod is heated and then dropped into a water bath the outside
quenches while the inside cools more slowly. The change in microstructure that
40
Chapter 3
accompanies this heat treatment will yield a structure that varies from the surface
to the axis of the rod because different cooling rates produce different microstructures. Figure 3.7 shows an axial section through a cylindrical sample which has a
radial gradient of the second phase with the amount decreasing from the surface
toward the axis of the cylinder. Other properties, like the number density or surface
density will also vary along the radius of the rod. Variations of geometric properties with position in the structure are called gradients.
Such spatial variations are common in biological structures because the function of an organ usually requires the structure to vary from the surface to the core.
If the positional variations may be captured in a plane, then global values of the
properties may be obtained on a single properly chosen sample plane which averages these variations appropriately. If the spatial variation is more complex, estimation of global properties requires a carefully designed sample of sections and
fields for observation, see Chapter 6. As an alternative you may wish to characterize the gradient, i.e., to quantify the variation of, say, the volume fraction of the b
phase with distance from the surface. In this case the sample design requires estimation of VV at each of a selected set of positions along the radius by viewing a
number of fields that lie along a plane that is parallel to the surface. In designing
such a sampling strategy it is necessary that you have a clear concept of the geometry of the spatial distribution of the feature set.
Figure 3.7. An axial section through a cylindrical sample with a gradient in the amount
of the b phase decreasing radially from the surface to the center. (For color representation see the attached CD-ROM.)
Geometry of Microstructures
41
Anisotropies: Variations with Orientation
Muscle fibers tend to be aligned along the macroscopic axis of the muscle.
Similarly, if a polycrystalline metal bar is deformed directionally as in rolling or
extruding, the grains are stretched out in the deformation direction and flattened
normal to that direction, as shown in Figure 3.8. Wool fibers in felt tend to lie in
the same plane, although they are (ideally) uniformly distributed in orientation
within that plane. This tendency for features to be more or less aligned with one or
more preferred directions in space is given the generic term of anisotropy.
The anisotropy of a collection of particles resides in the surfaces that bound
the particles. To visualize this attribute of structures in three dimensions imagine
that the surface bounding the particles is made up of a large number of small
patches of equal area. The direction associated with a patch is described by the
vector that is perpendicular to the tangent plane at the patch, i.e., locally perpendicular to the surface, called the local surface normal. If these vectors are distributed uniformly over the sphere that describes all possible orientations in three
dimensional space then the feature set is said to be isotropic. If the distribution of
vectors is not uniform on the sphere, but tends to cluster about certain directions,
the feature set exhibits anisotropy. This nonuniformity with respect to orientation
in space can be quantified using appropriate sampling strategies that involve
making line intercept counts on oriented line probes, as discussed in Chapter 4. Such
Figure 3.8. Grains in a polycrystal that has been directionally deformed are elongated
in the direction of deformation.
42
Chapter 3
nonuniformities may also be appropriately averaged by using a sampling strategy
based upon the method of vertical sections (Baddeley et al., 1986), also discussed
in Chapter 4.
Collections of space curves or other linear features in space may also exhibit
anisotropy. This anisotropy may also be quantified by making area point counts on
oriented plane probes (Gokhale, 1992), see Chapter 4.
Associations
In a truly uniform structure the spatial distribution of any one feature set is
not influenced by other feature sets that may exist in the system. In real microstructures there is always some tendency for relations between the various constituents
in the system. There are associations among elements of different feature sets. Mitochondria may tend to be in contact with cell walls. Particles of the b phase may tend
to reside at grain boundaries in the matrix, or at grain edges or corners, as shown
in Figures 3.2 and 3.9a. In a three phase structure g particles (nucleoli) may be completely contained within b particles (nuclei) and have no contact with the a matrix,
as shown in Figure 3.9b.
Quantitative measures of tendencies for positive or negative associations of
elements in a microstructure are based on comparisons of measured values of properties that report the interactions with values computed from a model that assumes
the distributions are in fact independent. For example, the b particles in Figure 3.9a
form triple lines of the type aab when they lie on the aa grain boundary. An area
point count (PA) measures the total length of these triple lines per unit volume of
structure. A line intercept count (PL) reports the area of aa grain boundary and,
separately, the area of ab boundary. The length of aab triple line that would result
if the interactions between the two interfacial feature sets were random can be computed. Comparison of the measured length of aab triple line with that expected for
the random structure gives a quantitative measure of the tendency for b particles
to be associated with aa grain boundaries.
Inclusions or other small particles in the structure may appear as clustered,
random or ordered (uniformly distributed). A variety of measures of this tendency
have been proposed; most are based on comparisons between the observed properties and those of some model spatial distribution of points that can be easily
visualized.
Summary
There are at three levels of characterization in the description of the geometric state of a microstructure. The qualitative microstructural state is simply a list
of the three, two, one and zero dimensional features that exist in the structure at
hand.
Each of the feature sets in the list has geometric properties that have
unambiguous meaning for features of arbitrary complexity. These properties may
be visualized for individual features in the microstructure or as global properties for
Geometry of Microstructures
43
a
b
Figure 3.9. Illustration of association tendencies in microstructures: (a) all of the b
particles (orange) are on aaa triple points, and thus on the grain edges in three
dimensions (Figure 3.2b); (b) all of the blue nucleoli (g) are contained within the yellow
nuclei (b). (For color representation see the attached CD-ROM.)
44
Chapter 3
the whole feature set. There are two classes of geometric properties associated with
these sets:
Topological properties including number density NV, connectivity density
CV, and Euler characteristic cV;
Metric properties including volume fraction VV, surface area density
SV, length density LV, and curvature measures to be developed in
Chapter 5.
Evaluation of one or more of these geometric properties constitutes a step
in the assessment of the quantitative microstructural state.
Real microstructures exhibit variations of some properties with macroscopic
position in the structure. Some feature sets may display variation with orientation
in the macroscopic specimen. Proper sample design may provide appropriate averages of global properties. Alternatively these gradients and anisotropies may be
assessed quantitatively. Comparison of appropriate combinations of these properties with predictions from random or uniform models for the structure provide measures of tendencies for feature sets to be positively or negatively associated with each
other.