Chapter 5
Less Common Stereological Measures
This chapter deals with manual stereological measurements that provide
access to geometric properties that are related to the curvature of the objects of
analysis. It begins with the concept of the spherical image of surfaces and its relation to the number and connectivity of the features the surfaces enclose. The probe
for this analysis is a plane that is visualized to sweep through the three dimensional
structure. This probe is implemented through the disector, or more generally by
serial sectioning. The concept of integral mean curvature is then developed with some
insight into the meaning of this rather abstract notion. Plane probes yield a feature
count in the plane which measures integral mean curvature. The same information
may be obtained in a more general way through the area tangent count. Polyhedral
features have edges and corners which contribute to the spherical image of the
feature and its integral mean curvature. The latter property provides access to the
mean dihedral angle at edges through a combination of counting measurements.
Three-Dimensional Features: Topological Properties and the Volume
Tangent Count
The topological properties, number NV and connectivity CV, provide rudimentary information about three dimensional feature sets. A fundamental property
of surfaces that is related to these topological properties derives from the concept
of the spherical image, W of the surface.
If a surface is smoothly curved, every point P on it has a tangent plane, as
shown in Figure 5.1a. (If the surface is not smoothly curved, i.e., has edges and
corners, then they also contribute to the spherical image.) A tangent plane at P also
has a vector, pointing outward from the surface, called the local surface normal.
This vector represents the local orientation of the patch of surface at P. This orientation can be mapped in longitude and colatitude as a point P¢ on the sphere of
orientation, as shown in Figure 5.1b. The point P¢ on the unit sphere is called the
spherical image of the point P on the surface. It represents the local direction of the
surface. For a small patch on a curved surface the collection of its normal directions map as a small patch on the unit sphere, as shown in Figure 5.1c,d. This is the
spherical image of the patch.
Now think about a convex surface (no bumps, dimples or saddles), as shown
in Figure 5.2a. The spherical images of the collection of normal directions for such
a surface cover the sphere of orientation exactly once. For every point that represents a direction on the sphere there is a point on the surface that has a normal
in that direction. No normal direction is represented more than once. Thus, the
79
80
Chapter 5
Figure 5.1. The spherical image of a point P on a surface (a) is a point P¢ on the unit
sphere (b). The spherical image of a patch of surface dS (c) is a patch dW on the unit
sphere (d). (For color representation see the attached CD-ROM.)
spherical image of any convex body covers the unit sphere exactly once, as shown
in Figure 5.2a, and may be evaluated as the area of the unit sphere, which is 4p
steradians1. This result is the same for every convex body, no matter what its shape
or size. The spherical image thus has the character of a topological property, at least
for convex bodies, i.e., it has a value that is independent of the metric properties of
the body. For a collection of N convex bodies, the spherical image is 4pN.
Particles with smooth bounding surfaces that are not convex, as shown in
Figure 5.2b and c, may have three different kinds of surface patches: convex,
concave, and saddle surface, as shown in the Figure. At any point on a convex patch
of surface, the tangent plane is outside the surface. At points on a concave patch,
the tangent plane is inside the surface. At any point on a saddle surface patch the
tangent plane lies partly inside the surface, partly outside. Saddle surface patches
(curved outward in one direction and inward in another like a potato chip) are
required to smoothly transition between convex and concave elements, or to surround holes in the particle, as in the inside of a doughnut, as shown in Figure 5.2c.
Patches of each kind of surface have their own collection of normal directions which
may be mapped on the unit sphere.
1
A steradian is a unit of solid angle represented by an area on the sphere of orientations analogous to the unit, radian, which applies to angles in a plane. The circle in a plane subtends
2p radians (the circumference of a circle with unit radius). The sphere in three dimensions
subtends a solid angle of 4p steradians.
Less Common Stereological Measure
81
Figure 5.2. Convex bodies (a) are composed of convex surface elements. Non-convex
(b,c) bodies have convex and saddle surfaces, and may also have concave surface
elements. Holes in multiply connected closed surfaces (d) contribute a net of -4p spherical image for each hole. (For color representation see the attached CD-ROM.)
It can be shown, and this appears remarkable at first sight, that if the parts
of area of the unit sphere that are covered by the convex and concave patch images
are counted as positive and the parts covered by saddle images are counted as negative, then the result, called the net spherical image,
W net = Wconvex + Wconcave - W saddle = 4p (1 - C )
(5.1)
where C is the connectivity of the particle. For simply connected features (no holes),
C = 0, and Wnet = 4p. This is a generalization of the result for convex bodies described
above. Where the shape departs from a simple convex closed surface, the extra
convex and concave spherical image is exactly balanced by the (negative) spherical
image of saddle surfaces on the feature, so that the unit sphere again is covered
(algebraically speaking) exactly once. If the feature has holes in it, every hole
requires a net of (-4p) steradians of saddle surface, which thus gives rise to equation (5.1). For a collection of N particles in a structure, with a collective connectivity of C, the net spherical image of the collection of features is
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Chapter 5
W net = 4p ( N - C )
(5.2)
The difference (N - C) is called the Euler characteristic of the particles in
the structure.
The probe used to measure these properties is the sweeping tangent plane.
Figure 5.3 shows a set of particles in a specimen. Imagine that the plane shown at
the top of the figure is swept through the structure from top to bottom. Such a probe
can be realized in practice by confocal microscopy in which a plane of focus is translated through a transparent three dimensional specimen. For opaque specimens
where this is not possible, the sweeping plane probe may be visualized by producing a set of closely spaced plane sections (serial sections) through the structure and
examining the changes that occur between successive sections. The disector probe
(Sterio, 1984) is the minimum serial sectioning experiment, employing information
from two adjacent planes.
The events of interest to the measurement of spherical image are the formation of tangents with the particle surfaces as the plane sweeps through the particles in the structure. Three kinds of tangent events may occur: at a convex surface
element (++); at a concave element (--), or at an element of saddle surface (+ -).
These three situations are pictured in Figure 5.3. These events are marked and
counted separately. The result is normalized by dividing by the volume swept
through by the sweeping probe. The spherical image per unit volume of each class
of surface in the structure is given by
WV ++ = 2pTV ++
WV - - = 2pTV - WV +- = 2pTV +-
(5.3)
Combining equations (5.1), (5.2) and (5.3) leads to a simple relation between
the Euler characteristic of a collection of particles and the volume tangent counts
NV - CV =
1
1
(WV ++ + WV -- - WV +- ) = (2pTV ++ + 2pTV -- - 2pTV +- )
4p
4p
(5.4)
which further simplifies to
1
NV - CV = TVnet (TV ++ + TV -- - TV +- )
2
(5.5)
For a collection of convex bodies CV = 0, as are TV - - and TV +-. Every particle has exactly two convex tangents, and counting tangents is evidently equivalent
to counting particles. More generally, for a collection of simply connected (not necessarily convex) particles, no matter what their shape, CV = 0, and the net tangent
count gives the number of particles in the structure. At the other extreme, for a fully
connected network (one particle) NV << CV, and the net tangent count (which will
be mostly saddle tangents) gives (minus) the connectivity. The case where both
number and connectivity are about the same order of magnitude is fortunately rare
in microstructures. In this case the separation of number and connectivity is a challenge. A sufficient volume of the sample must be examined, either in a confocal
microscope or by serial sections, to encompass many whole particles so that
a
b
Figure 5.3. The sweeping plane probe forms tangents with convex (++), concave
(- -) and saddle (+ -) surface elements. (For color representation see the attached CDROM.)
Probe population:
Sweeping plane in three dimensional space
This sample:
The specimen
Calibration:
None required
Event:
Plane forms tangents with convex, concave and saddle surface
Measurement:
Count each class of tangent formed
This count:
T++ = 16; T- - = 1; T+ - = 11
Relationship:
N - C = (1/2) Tnet = (1/2) (T++ + T- - - T+ -)
Geometric property:
N - C = (6 - 3) = (1/2) · (16 + 1 - 11) = 6/2 = 3
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Chapter 5
a
Figure 5.4. The simple disector analysis measures the number density, NV, of convex
features in three dimensional space. (For color representation see the attached CDROM.)
Probe population:
Set of disector planes in three dimensional space
Calibration:
A0 = 64 mm2; h = 0.5 mm, V0 = 32 mm3
Event:
Feature in counted unbiased frame on the reference plane
does not appear on the look-up plane
Measurement:
Count these events
This count:
N=6
Relationship:
NV = N/V0
Geometric property:
NV = 6/32 mm3 = 0.19 mm-3 = 1.9 · 1011 per cm3
beginnings and ends of specific particles may be identified and counted to determine NV. This information may then be combined with the measurement of the
Euler characteristic to estimate the connectivity.
The disector is a minimum serial sectioning experiment composed of two
closely spaced planes. The distance between the planes, h, must be measured so that,
together with area of the field analyzed on the plane, the volume contained in the disector sample is known. The spacing h must be small enough so that tangent events that
occur in the volume between the planes can be inferred unambiguously. Usually this
means that the spacing must be a fraction (perhaps 1/4 or 1/5) of the size of the smallest features being analyzed. Figure 5.4 shows the two planes of the disector viewed
Less Common Stereological Measure
85
b
Figure 5.4. Continued
side by side for a simple structure composed of convex bodies. An unbiased
counting frame2 with area, A0, is delineated on the first plane, called the reference plane.
The second plane is called the look-up plane. Each feature on the reference plane is
viewed in conjunction with the corresponding region on the look up plane. If the particle still persists on the look-up plane it is not marked. If there is no particle section
observed in the region then it is inferred that the particle came to an end between the
planes. The particle is marked on the reference plane to indicate that a convex tangent
has occurred within the volume of the disector. Counting the marks is equivalent to
counting particle ends. Since in this simple case of convex particles each particle has
only one end, this is equivalent to counting particles. If N particle ends are counted,
then N/(h · A0) is an estimate of NV. If the disector is chosen to sample the population of disector volumes uniformly then this provides an unbiased estimate of NV.
2
A potential bias in counting particles in a plane probe frame arises because some particles
will intersect the boundary of the frame. Further, large particles are more likely to intersect
the frame boundary than small ones. The bias can be eliminated by assigning a count of 1/2
to particles that cross the frame. A more general strategy involving the concept of an unbiased frame is described in detail in a later section of this chapter.
86
Chapter 5
a
Figure 5.5. The sweeping tangent analysis analysis using the disector measures the
Euler characteristic of features in three dimensional space. The events are most easily
seen when the two images are overlaid (Figure 5.5c), and are marked with arrows (red
= T++, green = T- -, blue = T+ -). Note that features are not counted if any part of them
touches that touch the exclusion line in either image. (For color representation see the
attached CD-ROM.)
Probe population:
Set of disector planes in three dimensional space
Calibration:
A0 = (18 mm)2 = 324 mm2; h = 0.5 mm, V0 = 162 mm3
Event:
Features in counted unbiased frame on the each plane
indicate various tangent events in the volume
Measurement:
Count these events
This count:
T+ + = 3 (red)
T- - = 3 (green)
T+ - = 5 (blue)
Relationship:
NV - CV = 1/2 · (TV+ + + TV- - - TV+ -)
Geometric property:
NV - CV = 1/2 · (3 + 3 - 5)/162 mm3 = 0.0031 mm-3
= 3.1 · 109 cm-3
If the look-up and reference planes are now reversed, particles on the second
plane that do not appear on the first imply that a convex tangent has formed within
the volume of the disector. On the average, the number of counts in the second case
will be equal to that in the first, but this will not be true for individual disector fields.
Figure 5.5 shows a more complex structure with particles that have concave
as well as convex boundary segments on a section. The presence of concave
b
c
Figure 5.5. Continued
88
Chapter 5
boundary elements on the plane probes results from sections through saddle surface
elements and/or concave elements in the three dimensional structure. The sweeping
plane probe is expected to form tangents of all three types of surfaces. A tangent
is formed with a convex surface element (T++) if an isolated particle on the reference
plane does not appear on the look-up plane, and also if an isolated particle on the
look-up plane does not appear on the reference plane. A tangent with a concave
surface element (T--) is indicated if a hole on the reference plane disappears on the
look-up plane, and also if a hole seen on the look-up plane disappears on the reference plane. If one feature on the reference plane becomes two on the look-up
plane, or two on the reference plane becomes one on the look-up plane, then a
tangent with an element of saddle surface (T+-) is inferred. The three classes of
tangent planes inferred from the disector sample in Figure 5.5 are indicated with
red (++), green (--) and blue (+-) markers. The analysis of the disector is presented
in the figure caption. The Euler characteristic estimated from this single observation is 0.0031 mm-3. If it is assumed that these particles are simply connected (CV =
0), the number of particles can be estimated to be 0.0031 mm-3. Relative amounts of
each of the three classes of spherical image may also be estimated from the tangent
counts by applying equations (5.3).
Three Dimensional Features: the Mean Caliper Diameter
The notion of the diameter has a straightforward meaning for a sphere. For
more general convex bodies, the “diameter” of a particle is different in different directions. Figure 5.6 illustrates the concept of the caliper diameter, D, of a convex
Figure 5.6. The mean caliper diameter, D, is the distance between parallel tangent
planes in any direction on a convex particle. This distance is sensed by plane probes
in three-dimensional space. (For color representation see the attached CD-ROM.)
Less Common Stereological Measure
89
particle. Choose a direction and visualize the sweeping plane probe moving along
that direction. The probe will form two tangents with the particle. The perpendicular distance between these two tangent planes is the caliper diameter of the particle
in that direction. This diameter has a value for each direction in the population of
orientations. If the value of the diameter is averaged over the hemisphere of orientation (the other hemisphere gives redundant information), the mean caliper diameter of the particle is obtained. This concept may be applied to convex bodies of any
shape. It provides one measure of the particle size of features in the structure.
An estimate of the mean caliper diameter may be obtained using a plane
probe through the three dimensional structure. Consider the subpopulation of plane
probes that all have the same normal direction. Individual planes in this subpopulation are located by their position along the direction vector. In order for a plane
probe to produce an intersection with a particle in the structure, its position must
lie between the planes that are tangent to the ends of the particle in that orientation. The event of interest is that the plane probe intersects the particle. This event
will produce a two dimensional feature, a section through the particle, observed on
the plane probe. The number of these events, i.e., the number of these two dimensional features that appear on the probe, is counted. The count is normalized by
dividing by the area of the probe sample to give NA, the number of features per unit
area of probe scanned. The governing stereological relationship is,
N A = NV D
(5.6)
The number of particles per unit volume can be estimated separately using
the disector probe as described in the last section. With this information, the mean
caliper diameter of particles in the structure can be evaluated from the expected
value of a feature count on plane probes. This result is limited to convex bodies, but
the shapes of the particles are not otherwise constrained. In order to obtain the
mean caliper of particles averaged over orientation it will be necessary to sample
the population of plane orientations in three dimensions. The difficulties in obtaining an unbiased sample of orientations of plane probes have already been discussed.
For a single orientation of plane probes, this count provides a measure of
the mean caliper diameter of particles in the direction perpendicular to the probe
planes. This result can be visualized as an application of equation (4.14) in Chapter
4, which is the stereological basis for measuring the length of lineal features in a
three dimensional structure. Visualize a subpopulation of parallel planes that share
some normal direction given by coordinates (q, f) on the orientation sphere. Replace
each convex particle with a stick that is the same length as the caliper diameter in
the direction perpendicular to those planes D(q, f). Then each particle section on
the plane probe may be visualized as resulting from an intersection of a plane probe
with one of these sticks. NA for the particle sections is the same as PA for the collection of sticks for this structure. The feature count thus measures the collective
length of these sticks, or more precisely, the collective lengths of the caliper diameters of the particles in the structure, in the direction that is perpendicular to the
plane probes.
N A (q , f ) = NV D (q , f )
(5.7)
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Chapter 5
For a collection of particles that show a tendency to be elongated or flattened in a particular direction in space measurement of NA on planes with different orientations provides one measure of the anisotropy of the particle shapes.
Ratios of NA counts on planes oriented in different directions directly yield ratios
of the mean caliper diameters in those directions:
N A (q1 , f1 )
N D (q1 , f1 ) D (q1 , f1 )
= V
=
N A (q 2 , f2 )
NV D (q 2 , f2 ) D (q 2 , f2 )
(5.8)
Note that these caliper diameters are measured in the direction that is parallel to the normal to the plane probe.
Thus a feature count on a section, when combined with a disector measurement of NV gives access to the mean caliper diameter of convex particles, one convenient measure of particle “size”. If the particle shapes are anisotropically
arranged in space, then a feature count on an oriented plane probe provides a
measure of the mean particle diameter in the direction perpendicular to the probe.
Mean Surface Curvature and Its Integral
This section introduces curvature related geometric properties that are significantly more abstract than the straightforward geometric properties defined and
measured in previous sections. These hard-to-visualize properties are introduced
here because they have geometric meaning that is general for structures of arbitrary
geometry, and they can be measured (they are what is measured) by the feature count
when the features are not limited to being convex, as they were in the previous
section. For some special classes of features (plates or muralia, or rods or tubules)
these geometric properties have a meaning that can be visualized and put to conceptual use. For more general kinds of three dimensional features, although the
meaning is abstract, at the very least the measurement provides an additional
descriptor of the geometry of the system.
The central concept, the mean curvature at a point on a smooth surface, is
the geometric property of surfaces through which capillarity effects operate. If
surface energy or surface tension plays a role in the problem being investigated,
access to a measure of the mean curvature provides geometric information that has
direct thermodynamic meaning (DeHoff, 1993).
In order to present the concept of curvature at a point on a surface, it is first
useful to recall the concept of curvature of a plane curve in two dimensional space.
Figure 5.7 shows a curve in the xy plane. To define the curvature at a point P place
two other points, A and B, on the curve. Construct a circle through these three
points. This circle has a center, O, and a radius r. Now let A and B approach P. The
point O moves and the radius changes. In the limit as A and B arrive at P, the center
approaches the center of curvature for the point P, and the radius becomes the
radius of curvature. This limiting circle, which just kisses the curve, is called the
osculating circle. The curvature of the curve at P is the reciprocal of this radius. It
can also be shown that the curvature is the rate of rotation of the tangent to the
curve as the point P moves along the arc length, s:
Less Common Stereological Measure
91
Figure 5.7. Curvature at a point P on a curve in two dimensional space is the reciprocal of the radius of a circle that “passes through three adjacent points” on the curve
at P.
k=
1 dq
=
r ds
(5.9)
This two dimensional concept is the basis for defining the curvature at a point
on a surface in three dimensional space.
Figure 5.8a shows a smooth surface. Focus on the point P on this surface.
There is a tangent plane at P, and a local surface normal, perpendicular to that
plane. Imagine a plane that contains this normal direction and intersects this surface
at P. The intersection of this “normal plane” with the surface is a plane curve that
passes through P. The radius of curvature, r, and the curvature, k, of that curve
may be defined at P using the concepts developed in the previous paragraph.
Next visualize another normal plane that makes some arbitrary angle with
the first plane. The curve of intersection of this plane with the surface also has a
value of curvature at P. If the piece of surface is a piece of a sphere, these two curvatures will be the same. However, for an arbitrarily shaped surface, the curvature
values on two different normal planes will be different. As the normal plane is
rotated, the curvature of the intersecting curve at P changes smoothly. As the
normal plane is rotated through half a circle, there will be some direction for which
the curvature has a maximum value, and another direction for which its value is a
minimum. In differential geometry, which deals with the geometry of entities in
the vicinity of a point on the entity, it is shown that the directions for which the
maximum and minimum values occur are 90° apart. These two directions in
the tangent plane are called the principle directions. The curvatures in these two
directions, k1 and k2, are called the principle normal curvatures at the point P,
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Chapter 5
Figure 5.8. Curvature at a point P on a surface is reported by two principle normal curvatures k1 and k2 measured on two orthogonal normal planes through P. Of all the possible planes through P that contain the normal direction (a), the two with the minimum
and maximum radii (b) give the principle normal curvatures.
corresponding to radii r1 and r2. In differential geometry it is shown that these two
curvature values completely describe the local geometry near a point on a smooth
surface. The local configuration for an element of surface around a point P is shown
in Figure 5.8b.
For a spherical surface with radius R, k1 = k2 = 1/R at every point on the
surface. For a right circular cylinder of radius r, the maximum curvature occurs on
a plane perpendicular to the axis: k1 = 1/r. The minimum curvature occurs on a plane
containing the cylinder axis, which intersects the surface in a straight line so that
k2 = 0. The values of k1 = 1/r and k2 = 0 are the same at all points on the cylindrical surface. For smooth surfaces with arbitrary geometry, k1 and k2 vary smoothly
from point to point and have some distribution of values over the surface area.
For a surface that encloses a three dimensional volume, i.e., for a closed
surface, these curvatures may be assigned a sign. If the curvature vector points
toward the inside of the surface it is defined to be positive; if it points outward, it
is negative. The possible combinations of signs of the curvature give rise to the three
classes of surface elements described earlier in this chapter:
Convex surface (++)—if both curvatures point inward and are thus positive;
Concave surface (- -)—if both point outward and are thus negative;
Saddle surface (+-)—if one points inward and the other points outward.
These three classes of surfaces were illustrated in Figure 5.2. In general, if
a surface has any departure from a simple convex shape, extra convex lumps, or
Less Common Stereological Measure
93
concave dimples, it must possess patches of saddle surface that connect these
smoothly. At the boundaries of saddle surface one of the curvatures changes sign
(passes from {+-} to {++}, or from {+-} to {--}), so that one of the principle curvatures passes through zero. Such points occur only along the lines bounding saddle
patches (or on the surface of a cylinder), and are called parabolic points.
Two combinations of the principle normal curvatures find widespread application in geometry, topology, physics and stereology:
1
The mean curvature, defined by H = –2 (k1 + k2), the algebraic average of the
two curvatures, and
The Gaussian curvature, defined by K = k1 · k2.
Both have a value at each point on a surface and thus vary smoothly over
the surface.
The mean curvature, H, is important for two distinct reasons. Capillarity
effects play a crucial role in the formation and performance of most microstructures. These effects arise from the physical action of curved surfaces on their surroundings. The geometric property of the surfaces that determines the nature of
these capillarity phenomena is shown in thermodynamics to be the local mean curvature, H. Thus an important component of the behavior of surfaces involves the
local mean curvature, its distribution and its average value over the surfaces
involved.
The second compelling reason for interest in this geometric property has to
do with its integrated value for the collection of surfaces in a microstructure, called
the integral mean curvature. This property has a single value for a closed surface. It
may also be evaluated for a collection of surfaces that make up a microstructure.
Visualization of its meaning is elusive. Consider an element (a small patch) of
surface with incremental area dS; suppose H is the value of mean curvature at that
element. Compute the product HdS for the element. Then add these values together
for all of the surface elements that bound the particle. The resulting quantity,
defined mathematically by
M ∫ Ú Ú HdS
(5.10)
S
has units of length. H has units of length-1 and dS has units of length2. (The double
integral signs ÚÚ signifies that the local incremental value HdS is summed over the
whole area of the surface, S.) This abstract geometric property has geometric
meaning that can be visualized for particular classes of three dimensional features.
As a simple example, apply the concept to a spherical particle of radius R.
The principle curvatures are both equal to (1/R), and have the same value at every
point on the sphere. Thus, for every point on a sphere the mean curvature is simply
1/R:
H=
1Ê1 1ˆ 1Ê 1 1ˆ 1
+
=
+
=
2 Ë r1 r2 ¯ 2 Ë R R ¯ R
Put this value into the definition of M:
(5.11)
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Chapter 5
1
1
1
1
dS = Ú Ú dS = Ssphere = 4pR 2 = 4pR
R
R
R
R
S
S
M sphere = Ú Ú HdS = Ú Ú
S
(5.12)
This result generalizes through the Minkowski formula; for general convex
bodies,
Mconvex = 2pD
(5.13)
where D is the mean caliper diameter defined in the previous section.
We have seen that for a collection of convex bodies the feature count on a
plane probe, NA, reports the product of the number density and the mean caliper
diameter. For non-convex bodies, i.e., for the general case, a generalization of the
feature count reports the integral mean curvature, M. In the discussions up to now,
the classes of features considered produced sections that are free of holes on the
plane probe. However, in the general case of arbitrarily shaped particles in three
dimensional space, some plane probes will produce features with holes in them. In
this most general kind of structure, the feature count generalizes to the Euler characteristic of the particle sections, which is defined to be the number of particles
minus the number of holes (N - C). It is useful to think of this as a count of the
net number of closed loops bounding particles in the structure, where loops enclosing particles are counted as positive and those enclosing holes are negative. The fundamental stereological relationship is
NA
net
= NA - CA =
1
MV
2p
(5.14)
where MV is the integral mean curvature of the surfaces bounding the particles
in the three dimensional structure divided by the volume of the structure. Thus,
no matter how complex the features in three dimensions are, the net feature count
on a plane probe provides an unbiased estimate of this abstract property, the
integral mean curvature. It is the three dimensional property that the feature count
measures.
Applying Minkowski’s formula for convex bodies, equation (5.13), to equation (5.14):
NA
convex
=
1
1
1
MVconvex =
NV MVconvex =
NV 2pD = NV D
2p
2p
2p
(5.15)
recovers equation (5.6), now seen to be a special case of equation (5.14).
Figure 5.9 reproduces the microstructure in the left side of the disector in
Figure 5.4 so that a feature count may be made and illustrated. In this case, the
value of the mean curvature MV estimated to be 4.45 mm/mm3 can be used to estimate the mean caliper diameter. The value of NV obtained from the disector analysis in Figure 5.4 is 0.19 mm-3. Apply the Minkowski formula, equation (5.13), to
compute:
D=
MV
4.45
=
= 3.72 mm
2pNV 2 ◊ p ◊ 0.19
(5.16)
Less Common Stereological Measure
95
Figure 5.9. A feature count measures the integral mean curvature, MV, of the bounding surfaces in three dimensional space. (For color representation see the attached CDROM.)
Probe population:
Planes in three dimensional space
This sample:
The unbiased frame in the field
Calibration:
L0 = 6.9 mm; A0 = (6.9 mm)2 = 47.9 mm2
Event:
Feature lies “within” the unbiased frame
Measurement:
Count the features
This count:
N = 34
Relationship:
·NAÒ = (1/2p) · MV
Normalized count:
NA = 34 counts/47.9 mm2 = 0.71 counts/mm2
Geometric property:
MV = (2p) · 0.71 = 4.45 mm/mm3
Figure 5.10 reproduces the microstructure on the left side of the disector in
Figure 5.5. A net area loop count estimates the integral mean curvature for this
structure of general geometry.
The integral mean curvature is a key geometric property of a collection of
plate shaped particles, or, more generally muralia, i.e., “thick surfaces”, features that
are small in one dimension and extensive in the other two, as shown in Figure 5.11a.
For such features it can be shown that M measures the length of the perimeter of
the edge of the the plate or muralia, L (DeHoff, 1977):
96
Chapter 5
Figure 5.10. Net feature count for non-convex features in the area measures the integral mean curvature, MV, of the surfaces bounding the features in the three dimensional
microstructure. (For color representation see the attached CD-ROM.)
Probe population:
Planes in three dimensional space
This sample:
The unbiased frame in the field
Calibration:
L0 = 17.7 mm; A0 = (17.7 mm)2 = 313 mm2
Event:
Feature (particle or hole) lies “within” the unbiased frame
Measurement:
Count the features
This count:
8 particles; 2 holes
Relationship:
·NAÒnet = ·NAÒ+ - ·NAÒ- = (1/2p) · MV
Normalized count:
NAnet = 8 - 2 counts/313 mm2 = 0.019 counts/mm2
Geometric property:
MV = (2p) · 0.019 = 0.120 mm/mm3
M muralia =
p
L
2
(5.16)
At the other extreme of feature shapes, the integral mean curvature reports
visualizable information for rod or more generally for tubule features. A feature satisfies the definition of a tubule if it is small in two of its dimensions and extensive
in the third, as shown in Figure 5.11b. For this class of features,
Less Common Stereological Measure
97
Figure 5.11. Muralia are “thick surfaces; a flat muralium is a plate. Tubules are “thick
space curves”; a straight tubule is a rod.
Mtubules = pL
(5.17)
where L is the length of the tubule.
Figure 5.12 shows a plane probe section through a microstructure consisting of muralia. The vast majority of sections will appear as curved strips in the
microstructure. Orientations and positions of the probe that produce fat sections
are not very likely. For muralia equation (5.16) yields,
NA
muralia
=
1
LVmuralia
4
(5.18)
where LVmuralia is the total perimeter of plates or length of edges of muralia in unit
volume of structure. If it is assumed that Figure 5.12 is an unbiased sample of the
population of positions and orientations of plane probes in the structure, the feature
count, NA = 0.074 (1/mm2) gives, for the total length of feature edges in the three
dimensional structure = 0.295 (mm/mm3).
Most sections through tubules are small and equiaxed, as shown in Figure
5.13. Orientations and positions which produce a very long section only occur for
planes that are nearly parallel to the local axis of the tubule, and are close to it in
position. Such sections will be relatively rare, particularly if the radius is small. From
equation (5.17) the expected value of the feature count gives
NA
tubules
=
1
LVtubules
2
(5.19)
where LVtubules is the collective length of the rods or tubules in unit volume. For the
example shown in Figure 5.13, NA = 0.29 (1/mm2) which estimates a total tubule
length of 0.58 (mm/mm3).
98
Chapter 5
Figure 5.12. A feature count on a section through muralia measures the integral mean
curvature, MV, or the muralia surfaces in three dimensional space. In this case, MV is
simply related to the totallength of edges of the muralia, LVmuralia (For color representation see the attached CD-ROM.)
Probe population:
Planes in three dimensional space
This sample:
Area within the unbiased frame
Calibration:
A0 = 122 mm2
Event:
Feature lies “within” the unbiased frame
Measurement:
Count the features
This count:
N=9
Relationship:
·NAÒ = (1/2p) · MVmuralia = 1/4 LVmuralia
Normalized count:
NA = 9 counts/122 mm2 = 0.074 counts/mm2
Geometric property:
LVmuralia = 4 NA = 4 · 0.074 = 0.295 mm/mm3
A plane probe intersects particles in three dimensional space to produce a
collection of two dimensional features on the sectioning plane. The number of such
features, normalized by dividing by the area of the probe, is proportional to the integral mean curvature of the surfaces bounding the three dimensional particles in the
structure divided by the volume of the specimen (equation 5.14). Integral mean curvature has simple geometric meaning for convex bodies, for which it reports the
mean caliper diameter, platelets or muralia, for which it reports the total length of
edge, and tubules, for which it reports the total tubule length. These results assume
the set of plane probes in the sample are drawn uniformly from the population of
Less Common Stereological Measure
99
Figure 5.13. A feature count is used to estimate MV, the integral mean curvature. For
tubules MV is proportional to the LVtubules length of the tubules. (For color representation
see the attached CD-ROM.)
Probe population:
Planes in three dimensional space
This sample:
Area within the unbiased frame
Calibration:
L0 = 6.9 mm; A0 = (6.9 mm)2 = 47.9 mm2
Event:
Feature lies “within” the unbiased frame
Measurement:
Count the features
This count:
N = 14
Relationship:
·NAÒ = (1/2p) · MVtubules = 1/2 LVtubules
Normalized count:
NA = 14 counts/47.9 mm2 = 0.29 counts/mm2
Geometric property:
MVtubules = 2pNA = 1.84 mm/mm3
LVmuralia= 2 · NA = 0.58 mm/mm3
positions and orientations of plane probes in three dimensional space. Oriented
plane probes give information about these lengths in the direction that is perpendicular to the probes.
The Sweeping Line Probe in Two Dimensions
Figure 5.14 shows a feature in two dimensional space. Any point P on the
boundary has a tangent direction and, perpendicular to the tangent and pointing
100
Chapter 5
a
b
Figure 5.14. The circular image of a convex feature in a plane is equivalent to the unit
circle. Every point on the feature periphery has a normal direction that corresponds to
one point on the circle, as shown by the colored vectors in Figure 14a. A segment dS
on the feature maps to a segment of arc dq on the circle. A non-convex feature has
regions of negative curvature as shown in Figure 14b. The angular range which these
cover (shown in green) exactly cancels the extra positive curvature (shown in blue and
orange), again leaving a net circular image of 2p. (For color representation see the
attached CD-ROM.)
Less Common Stereological Measure
101
outward, a normal direction. Mapping these directions onto a unit circle creates the
circular image of the points, and a small arc of the boundary ds has a small range
of normal directions that map as a segment of arc on the unit circle, dq, as shown
in Figure 5.14a. The circular image of a convex feature maps point for point on the
unit circle exactly once. Thus, no matter what the shape of the particle boundary,
so long as it is convex, its circular image is exactly 2p radians, the circumference of
the unit circle.
If the boundary shape departs from convex, as shown in Figure 5.14b, then
part of it will consist of convex arc segments and part will be made up of concave
arc segments. Mapping the rotations of the boundary normal on the unit circle as
the point P moves aroung the perimeter of the particle will produce ranges of
overlap as shown in Figure 5.14d. However, if concave segments are defined to contribute a negative circular image, then the net rotation of the normal vector around
the perimeter remains exactly 2p radians because the point chosen to begin and end
the map is the same point. Thus, like the spherical image of a surface discussed
earlier, the circular image of the boundary of a two dimensional feature also has
the character of a topological property. It is equal to 2p radians, no matter what is
the size and shape of the particle enclosed by the boundary.
In a two dimensional structure particles of the b phase are said to be “multiply connected” if they have holes in them. The connectivity of a two dimensional
particle, C, is equal to the number of holes in the particle. The boundary of each
of these holes, no matter what their size or shape, contributes a net circular image
of (-2p) because overall, a hole contributes a net of 2p of concave arc. Thus, the
net circular image of a two dimensional particle with C holes in it is
q net = q + - q - = 2p (1 - C )
(5.20)
For a collection of N features with a total of C holes in them,
q net = 2p ( N - C )
(5.21)
The difference in the topological properties (N - C) is called the Euler characteristic of the two dimensional collection of particles. These concepts for a two
dimensional structure mirror those presented earlier for the spherical image and the
topological properties of three dimensional closed surfaces.
It was shown earlier that a volume tangent count, based on a sweeping plane
probe through a three dimensional structure, measured the spherical image of the
surface bounding particles in the volume. An analogous probe and measurement
applies to the boundaries of two dimensional features in two dimensions. Figure
5.15 illustrates the concept of the sweeping line probe in two dimensions. The line
in the figure is swept across the field. In the process the moving line forms tangents
with elements of the ab boundary. Tangents formed with elements of the boundary that are convex (T+) with respect to the b phase and elements that are concave
(T-) may be marked separately and subsequently counted. Dividing these counts by
the area of the field included in the count gives the area tangent count. This is the
two dimensional analogue of the volume tangent count described in a previous
section.
102
Chapter 5
Figure 5.15. An area tangent count is used to estimate the Euler characteristic and
integral mean curvature. A horizontal line is swept down across the image and the locations of tangents with convex and concave boundaries are counted. (For color representation see the attached CD-ROM.)
Probe population:
Sweeping line in two dimensional space
This sample:
Area within the frame
Calibration:
L0 = 20 mm; A0 = (20 mm)2 = 400 mm2
Event:
Line forms tangents with convex and concave boundaries
Measurement:
Count each class of tangent formed
This count:
T+ = 45 (red)
T- = 19 (green)
Relationship:
NA - CA = (1/2) · (T+ - T-) = (1/2p) · MV
Normalized count:
NA - CA = (1/2) · (45 - 19) /400 mm2 = 0.0325 counts/mm2
Geometric property:
MV = 0.204 mm/mm3
The area tangent count applied to boundary elements of a collection of features measures the circular image of those features:
q A+ = pTA+ ;
q A- = pTAq Anet = p (TA+ - TA- ) = pTAnet
(5.22)
Combine this result with equation (5.21) to show that the Euler characteristic of a collection of two dimensional features with holes is simply related to the
net area tangent count:
Less Common Stereological Measure
1
1
N A - C A = TAnet = (TA+ - TA- )
2
2
103
(5.23)
For a collection of convex features CA = 0 (there are no holes) and TA- = 0,
so that NA = 1/2 · TA+; the result that every particle has two tangents is self evident
in this case. If the features are simply connected, i.e., have no holes, then every
bounding loop has two terminal convex tangents and, for each concave tangent there
is a balancing extra convex tangent. The net tangent count still gives two per particle. If there are holes, equation 5.23 gives the Euler characteristic of the collection
of features.
If the two dimensional structure in this discussion results from probing a
three dimensional microstructure with a plane then the expected value of the Euler
characteristic on the section is an estimator of the integral mean curvature of the
corresponding collection of ab surfaces in the volume, according to equation (5.14).
Thus, the net tangent count provides an unbiased estimate of the integral mean curvature:
TA net = 2 N A
net
=
1
MV
p
(5.24)
It may appear that the tangent count merely provides the same information
that a feature and hole count could provide, since in a two dimensional structure
the features are visible in the field and can be separately marked and counted. The
tangent count would appear to give redundant information. However, there are
some valid arguments for considering replacing the feature count with the tangent
count:
1. Tangents occur at a point; a point is either inside the boundary of the field,
or outside it. Bias due to particles intersecting the boundary of the field is
not a factor in the tangent count;
2. In some structures, e.g., lamellar structures, or structures in which both
phases occupy about the same area fraction, features of, say, the b phase may
wander in and out of the boundary of the field several times so that it is not
possible to make the feature count.
3. In a three phase structure (a + b + e) part of the boundary of b particles is
ab boundary, and part may be be boundary. In this case it is possible make
separate area tangent counts of these two kinds of interface and assess the
circular image (and the integral mean curvature) of each.
4. If the boundary on a section has vertices, separate application of the tangent
count to smooth segments versus vertices provides a measure of the dihedral
angle at edges in the three dimensional structure the section samples. This is
described in more detail in an example below.
5. Since separate counts are obtained for convex and concave elements of
boundary the circular images of convex and concave arc in the plane probe
structure may be separately estimated. While this information does not have
104
Chapter 5
Figure 5.16. An edge is formed by two surfaces meeting to form a space curve. The
angle between the surface normals at a point P on an edge, c, is called the dihedral angle.
(For color representation see the attached CD-ROM.)
a simple relation to the geometry of the parent three dimensional microstructure (DeHoff, 1978) it may be useful information in some applications.
Thus, the net tangent provides an unbiased estimator of MV in all of its
applications.
Edges in Three Dimensional Microstructures
Because microstructures are space filling, triple line structures, such as those
described in the qualitative microstructural state discussion in Chapter 3, are
common. A triple line results when three “cells”, some or all of which may be the
same phase, are incident in three dimensional space. Three surfaces also meet at a
triple line, namely the three pairwise incidences of the cells involved. From the viewpoint of any particular cell, the triple lines it touches are edges of that polyhedral
shaped body. More generally an edge is a geometric feature which results from the
intersection of two surfaces that bound a feature to form a space curve, as shown
in Figure 5.16. In addition to the properties of space curves (length, curvature and
torsion) an element of edge has a dihedral angle. The two surfaces that meet at an
element of edge each have a local normal vector. The dihedral angle, c in Figure
5.16, is the angle of rotation between these two surface normals.3 In general this
angle varies from point to point along the edge.
3
Like surfaces, edges may be convex (ridge with a maximum); concave (valley with a minimum)
or saddle (ridge with a minimum or valley with a maximum) in character. Valley edges by
definition have a negative value of c.
Less Common Stereological Measure
105
The dihedral angle also plays an important role in the physical behavior
of structures in which surface tension plays a role. In the thermodynamics of surfaces that meet at triple lines it is shown that the dihedral angles at the three
edges are determined by the relative surface energies of the three surfaces that
meet there. For example, the tendency of one phase to spread over a surface
between two other phases, i.e., to “wet” the surface, is measured by this dihedral
angle. This property plays an important role in soldering, adhesion, liquid phase
penetration, welding, powder processing and the shape distribution of phases at
interfaces.
An element of length of an edge may be thought of as a limiting case of an
element of surface for which one of the radii of curvature goes to zero, i.e., sharpens to an angle. With this point of view it can be shown that for a polyhedral particle the edges make their own contribution to the integral mean curvature of the
boundary of a particle:
Medges =
1
c ◊ dL
2 ÚL
(5.24)
where the integration is over the length of edge in the structure. To lend
some credence to this assertion that this property is in fact the contribution to M
due to edges it may be shown that this result may be used in conjunction with
Minkowski’s formula for convex bodies, equation (5.13), to compute the mean
caliper diameter of polyhedral shapes. Consider the cube with edge length e shown
in Figure 5.17a. The dihedral angle at all points on the twelve edges is p/2. The total
length of edge is 12e. The surfaces (faces) all have zero curvature, so that the contribution of the surfaces to M is zero. Combine equation 5.24 with the Minkowski
formula:
Mcube = M faces + Medges = 0 +
Dcube =
3
e
2
p
1p
L = 12 e = 3pe = 2pDcube
22
4
(5.25)
This is the same result that is obtained by evaluating the caliper diameter of
a cube as a function of orientation and averaging over orientation.
Similar analysis of other shapes provides an efficient way to calculate D averaged over orientation. This is generally far easier than the modelling approach using
Monte-Carlo sampling introduced in Chapter 11, but the latter method can also
produce the distribution of intercept lengths or areas which are needed to unfold
particle size distributions.
Figure 5.17b shows a cylinder with radius r and length l. In this case there
are contributions to M from both the curved surface and the edges. On the curved
face of the cylinder the curvature in the axial direction is zero, and the mean curvature everywhere has the value (1/2r). The dihedral angle at every point on the
edges is p/2. Evaluate M for a cylinder:
106
Chapter 5
Figure 5.17. The integral mean curvature for features with edges contains a contribution from edges. For the cube (or any flat faced polyhedron), all of the integral mean
curvature resides at the edges. A feature with curved faces and edges has both
contributions. (For color representation see the attached CD-ROM.)
Mcyl = M faces + Medges = Ú Ú H ◊ dS +
S
1
H ◊ dL
2 ÚL
1
1 p
1
p
2prl + 2 ◊ 2pr
= Ú Ú dS + Ú dL =
2r
2L2
2r
4
S
(5.26)
Mcyl = pl + p 2 r = 2pDcyl
1
p
Dcyl - l + r
2
2
Again, this is the same result that is obtained if D is computed as a function
of orientation and averaged over the hemisphere. Note that for the limiting cases of
p
1
a plate (r >> l), D = –2 r, while for a rod (l >> r), D = –2 l.
Edges can also be treated as a limiting case of a curved surface in the sense
that the integral mean curvature of an edge can be measured by the tangent count,
equation (5.23):
TA
=
edges
1
11
c ◊ dL
MVedges =
p
p 2 LÚV
(5.27)
where c is the dihedral angle measured at any point along the edge. An average dihedral angle may be defined,
c =
Ú c ◊ dL
LV
Ú
dL
=
2 MVedges
LV
(5.28)
LV
MVedges can be measured with the tangent count at edges, using equation
(5.27), and LV can be measured by counting points of intersection of the edge line
Less Common Stereological Measure
107
Figure 5.18. The mean dihedral angle at edges, ·cÒ on the b (colored) phase of this
structure can be measured with a triple point count combined with a tangent count. (For
color representation see the attached CD-ROM.)
Probe population:
Planes in three dimensional space and sweeping line probes
in two dimensions
Calibration:
L0 = 14.7 mm; A0 = (14.7 mm)2 = 215 mm2
Event:
Plane intersects aab triple line;
Sweeping line forms tangent at aab triple line
Measurement:
Count the triple points; count the tangents
This count:
49 triple points, 32 tangents
Relationship:
·cÒ = p TA/PA
Normalized count:
PA = 45 counts/215 mm2 = 0.23 counts/mm2
TA = 32 counts/215 mm2 = 0.15 counts/mm2
Geometric property:
·cÒ = p · 0.15/0.23 = 2.05 radians = 117 degrees
with the plane probe. Thus, the average dihedral angle for any specific type of edge
can be estimated from the ratio
c =
2p ◊ TAedges pTAedges
=
2 PA
PA
(5.29)
Figure 5.18 shows a two phase structure with aab triple lines. A sweeping
line probe is visualized to move from the top to the bottom of the field. Tangents
with the aab edge are marked and counted. The total number of aab triple points
108
Chapter 5
is also counted. With the usual assumptions about the sample plane, these properties are normalized and used to estimate MVedge and LVaab. These computations are
indicated in the caption to Figure 5.18. The ratio is then used to estimate the average
dihedral angle on aab edges of the b particles. The result, 2.05 radians or 117
degrees, is the average value of the angle between the surface normals at the edge.
This result may be used, for example, in the assessment of the relative surface energies of the ab and aa interfaces in this system.
Summary
The disector probe is required to obtain information about the topological
properties, number (NV) and connectivity (CV) of surfaces bounding three dimensional features. This is achieved by comparing the structures on a reference plane
and a look-up plane in order to infer the occurrence of tangents with the surfaces
formed by a plane that is visualized to sweep the volume between the planes of the
disector. Counts of the three kinds of tangents measure the spherical image of
convex, concave and saddle surfaces in the structure. The Euler characteristic,
(NV - CV), may be estimated from these tangent counts in the disector probe
NV - CV
1
1
TVnet = [TV++ + TV-- - TV+- ]
2
2
For most structures the connectivity is small in comparison with the number
of disconnected parts of the structure, and the disector probe provides the primary
strategy for estimating number density in microstructures.
If the three dimensional particles are convex, then a feature count on a plane
probe estimates the product of the number density and mean caliper diameter:
NV = NV D
Feature counts on oriented plane probes measure the caliper diameter in the
direction of the plane probe normal.
N A (q , f ) = NV D (q , f )
For more general structures, the (net) feature count (or equivalently, the
net area tangent count) reports the integral mean curvature of the surface in the
structure:
NA
net
= NA - CA =
1
1
1
TAnet = TA+ - TA- =
M
2
2
2p
Particles of arbitrary shape in the three dimensional structure may, for some
positions and orientations of a plane probe, produce sections with holes in them.
In this general case, the feature count is interpreted as the number of features in the
section minus the number of holes. This net number of features can also be determined from the area tangent count, which visualizes a line that sweeps across the
field of view and forms tangents with convex and concave elements of the boundaries of particle sections. In the general case, either of these counts reports the value
Less Common Stereological Measure
109
of integral of the mean surface curvature of the boundaries of particles in the three
dimensional microstructure. For convex bodies, M is proportional to the mean
caliper diameter. For muralia (plates) it reports the length of perimeter of the features in three dimensions. For tubules (rods), M reports their total length in the
volume.
If the features in the three dimensional collection of particles have edges,
then these edges contribute to the integral mean curvature of the boundaries of the
particles so that
MVtotal = MVsurfaces + MVedges = Ú
1
Ú H ◊ dS + 2 Ú c ◊ dL
SV
LV
where c is the dihedral angle between the surface normals at the each point on the
edge. This result may be used to estimate the average dihedral angle along a collection of edges in the structure.
c =
pTAedges
PA