Microstructure: Stereology

1
Microstructure Stereology

• A. D. Rollett
• 27-765
• Spring 2001

2
Objectives

• To lay out methods of measuring characteristics
  of microstructure grain size, shape,
  orientation phase structure grain boundary
  length, curvature etc. stereology.
• To illustrate the principles used in extracting
  grain boundary properties from geometrycrystallog
  raphy of grain boundaries microstructural
  analysis.

3
Objectives, contd.

• To understand how Herrings equations lead to a
  method of obtaining (relative) grain boundary
  (and surface) energies as a function of boundary
  type.
• To understand how curvature-driven grain boundary
  migration leads to a method of obtaining
  (relative) mobilities.

4
Stereology

• The inference of 3D (internal) structure from 2D
  sections through bodies.
• Based on Quantitative Stereology, E.E.
  Underwood, Addison-Wesley, 1970.- equation
  numbers given where appropriate.
• Also useful M.G. Kendall P.A.P.Moran,
  Geometrical Probability, Griffin (1963).

5
Motivation grain size

• Secondary recrystall. in Fe-3Si at 1100C
• Grain size becomes heterogeneous, anisotropic
  how to measure?

6
Motivation precipitate sizes, frequency, shape,
alignment

• Gamma-prime precipitates in Al-4a/oAg.
• Ppts aligned on 111 planes, elongated.

7
Topology - why study it?

• The behavior of networks of interfaces is largely
  driven by their topology. The connectivity of
  the interfaces matters more than dimensions.
• Example in a 2D boundary network, whether a
  grain shrinks or grows depends on the number of
  sides (von Neumann-Mullins), not its dimensions
  (although there is a size-no._of_sides
  relationship).

8
Shrink vs. Grow (Topology)
9
Topology of Networks

• Consider the body as a polycrystal in which only
  the grain boundaries are of interest.
• Each grain is a polyhedron with facets, edges
  (triple lines) and vertices (corners).
• Typical structure has three facets meeting at an
  edge (triple line/junction or TJ) Why 3-fold
  junctions? Because higher order junctions are
  unstable to be proved.
• Four edges (TJs) meet at a vertex (corner).

10
Definitions

• G ? B Grain polyhedral object polyhedron
  body
• F Facet face grain boundary
• E Edge triple line triple junction TJ
• C ? V Corner Vertex points
• n number of edges around a facet
• overbar or ltangle bracketsgt indicates average
  quantity

11
Eulers equations

• 3D simple polyhedra (no re-entrant shapes) V
  F E 2 G 1
• 3D connected polyhedra (grain networks) V F
  E G 1
• 2D connected polygons V F E 1
• Proof see What is Mathematics? by Courant
  Robbins (1956) O.U.P., pp 235-240.

12
Grain Networks

• A consequence of the characteristic that three
  grain boundaries meet at each edge to form a
  triple junction is this 3V 2E

E
E
V
V
E
E
E
E
V
V
E
E
E
V
V
E
E
E
13
2D sections

• In a network of 2D grains, each grain boundary
  has two vertices at each end, each of which is
  shared with two other grain boundaries
  (edges) 2/3 E V, or, 2E 3V ? E
  1.5V
• Each grain has an average of 6 boundaries and
  each boundary is shared n 6 E n/2 G
  3G, or, V 2/3 E 2/3 3G 2G

_
_
14
2D sections
split each edge
6-sided grain unit celleach vertex has 1/3
in eachunit celleach boundaryhas 1/2 in each
cell
divide each vertex by 3
15
3D Topology polyhedra
16
2D Topology polygons
17
Typical section
Underwood

• Correction terms (Eb, C1,C2) allow finite
  sections to be interpreted.

C1no. incomplete corners against 1 polygon
C2 same for 2 polygons
18
Grain size measurement area based

• Grain count method ltAgt1/NA
• Number of whole grains 20Number of edge grains
  21Effective totalNwholeNedge/2
  30.5Total area 0.5 mm2Thus, NA 61 mm-2
  ltAgt16.4 µm2
• Assume spherical (?!) grains, ltAgt mean intercept
  area 2/3pr2? d 2v(3ltAgt/2p) 5.6 µm.

Underwood
19
3D vs 2D polygonal faces

• Average no. of edges on polygonal faces is less
  than 6 for typical 3D grains/cells.
• Typical ltngt5.14
• In 2D, ltngt6.

20
Measurement of Volume fractions

• Typical method of measurement is to identify
  phases by contrast (gray level, color) and either
  use pixel counting (point counting) or line
  intercepts.
• Volume fractions, surface area (per unit volume),
  diameters and curvatures are readily obtained.

21
Definitions
Subscripts P per test point L per unit
of line A per unit area V per unit
volume T totaloverbar averageltxgt
average of x
22
Point Counting

• Issues- Objects that lie partially in the test
  area should be counted with a factor of 0.5.-
  Systematic point counts give the lowest
  coefficients of deviation (errors) coefficient
  of deviation/variation standard deviation
  divided by the mean, CVs(x)/ltxgt.

23
Relationships between Quantities

• VV AA LL PP mm0
• SV (4/p)LA 2PL mm1
• LV 2PA mm2
• PV 0.5LVSV 2PAPL mm3 (2.1-4).
• These are exact relationships, provided that
  measurements are made with statistical uniformity
  (randomly).

24
Dimensions
25
Delesses Principle Measuring volume fractions
of a second phase

• The French geologist Delesse pointed out (1848)
  that AAVV (2.11).
• Rosiwal pointed out (1898) the equivalence of
  point and area fractions, PP AA (2.25).
• Relationship for the surface area per unit volume
  derived from considering lines piercing a body
  by averaging over all inclinations of the line

26
Derivation Delesses formula
27
Surface Area (per unit volume)

• SV 2PL (2.2).
• Derivation based on random intersection of lines
  with (internal) surfaces. Probability of
  intersection depends on inclination angle, q.
  Averaging q gives factor of 2.

28
SV 2PL

• Derivation based on uniform distributionof
  elementary areas.
• Consider the dA to bedistributed over the
  surface of a sphere.
• Projected area dA cosq.
• Point intersection proportional to projected area
  on the plane.

29
SV 2PL
30
Length of Line per Unit Area, LA compared with
the Intersection Points Density, PL

• Set up the problem with a set of test lines
  (vertical, arbitrarily) and a line to be sampled.
  The sample line can lie at any angle what will
  we measure?

ref p38/39 in Underwood
31
LA p/2 PL, contd.
?x

• The number of points of intersection with the
  test grid depends on the angle between the sample
  line and the grid.

l
l cos q
q
l sin q
The projected length l sin q PL ?x
Line length in area l LA.
32
LA p/2 PL, contd.

• Probability of intersection with test line given
  by average over all values of q

q
Density of intersection points, PL,to Line
Density per unit area, LA, is given by this
probability.
33
Line length per unit volume vs. Points per unit
area

• Equation 2.3 states that LV 2PA.
• Practical application estimating dislocation
  density from intersections with a plane.
• Derivation based on similar argument to that for
  surfacevolume ratio. Probability of
  intersection of a line with a plane depends on
  the inclination of the line w.r.t plane
  therefore we average a term in cos(theta).

34
Oriented structures 2D

• For highly oriented structures, it is sensible to
  define specific directions (axes) aligned with
  the preferred directions (e.g. twinned
  structures) and measure LA w.r.t. the axes.
• For less highly oriented structures, orientation
  distributions should be used (just as for pole
  figures!)

35
Oriented structures 3D

• Again, for less highly oriented structures,
  orientation distributions should be used (just as
  for pole figures!) note the incorporation of the
  normalization factor on the RHS of (Eq. 3.32).

See also Ch. 12 of Bunges book
36
SV and 2nd phase particles

• Convex particles any two points on particle
  surface can be connected by a wholly internal
  line.
• Sometimes it is easier to count the number of
  particles intercepted along a line, NL then the
  number of surface points is double the particle
  number. Also applies to non-convex particles if
  interceptions counted. Sv
  4NL (2.32)

37
SV and Mean Intercept Length

• Mean intercept length from intercepts of
  particles of alpha phase ltL3gt 1/N Si
  (L3)i (2.33)
• Can also be obtained as ltL3gt LL/NL (2.34)
• Substituting ltL3gt 4VV/SV, (2.35)where
  fractions refer to alpha phase only.

38
SV example sphere

• For a sphere, the volumesurface ratio is
  Diameter/6.
• Thus ltL3gtsphere 2D/3.
• In general we can invert the relationship to
  obtain the surfacevolume ratio, if we know
  (measure) the mean intercept ltS/Vgtalpha
  4/ltL3gt (2.38)

39
Table 2.2
ltL3gt mean intercept length, 3D objects ltVgt
mean volume l length (constant) of test lines
superimposed on structure p number of (end)
points of l-lines in phase of interest LT test
line length
Underwood
40
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. ltL3gt 1/NL
• Important use many test lines that are randomly
  oriented w.r.t. the structure.
• Assuming spherical (?!) grains, ltL3gt 4r/3,
  Underwood, table 4.1, if LT 25µm, LTNL 5?
  d 6ltL3gt/4 6/NL4 65/4 7.5µm.

41
Particles and Grains

• Where the rubber meets the road, in stereology,
  that is!
• Mean free distance, l uninterrupted
  interparticle distance through the matrix
  averaged over all pairs of particles (in contrast
  to interparticle distance for nearest neighbors
  only).

(4.7)
Number of interceptions with particles is same
asnumber of interceptions with the matrix. Thus
linealfraction of occupied by matrix is lNL,
equal to thevolume fraction, 1-VV-alpha.
42
Mean Random Spacing

• The number of interceptions with particles per
  unit test length NL PL/2. Reciprocal of this
  quantity is the mean random spacing, s, which is
  the mean uninterrupted center-to-center length
  between all possible pairs of particles.
  Thus,the particle mean intercept length, ltL3gt
  ltL3gt s - l mm (4.8)

43
Particle Relationships

• Application particle coarsening in a 2-phase
  material strengthening of solid against
  dislocation flow.
• Eqs. 4.9-4.11, with LApPL/2pNL pSV/4
• dimension lengthunits (e.g.) mm

44
Nearest-Neighbor Distances

• Also useful are distances between nearest
  neighbors S. Chandrasekhar, Stochastic problems
  in physics and astronomy, Rev. Mod. Physics, 15,
  83 (1943).
• 2D ?2 0.5 / vPA (4.18a)
• 3D ?3 0.554 (PV)-1/3 (4.18)
• Based on l1/NL, ?3 ? 0.554 (pr2 l)1/3for small
  VV, ?2 ? 0.500 (p/2 rl)1/2

45
Application of ?2

• Percolation of dislocation lines through arrays
  of 2D point obstacles.
• Caution! Spacing has many interpretations
  select the correct one!

Hull Baconfig 10.17
46
Measurement of Regularly Shaped Particles

• Purpose how can we relate measurements in plane
  sections to what we know of the geometry of
  regularly shaped objects with a distribution of
  sizes?
• In general, the mean intercept length is not
  equal to the grain diameter, for example! Also,
  the proportionality factors depend on the
  (assumed) shape.

47
Sections through dispersions of spherical objects
Even mono-disperse spheresexhibit a variety of
diametersin cross section.Only if you know that
the second phase is monodispersemay you measure
diameterfrom maximum cross-section!
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50
Application regular shapes

• 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
51
True dimension(s) from measurements examples

• Spheres radius, r 8NL/3pNA.
• Truncated octahedron, or tetrakaidcahedron edge
  length, a, L3/1.69 0.945 NL/NA.Volume of
  truncated octahedron 11.314a3 9.548
  (NL/NA)3.Equivalent spherical radius, based on
  Vsphere 4p/3 r3 and equating volumes
  rsphere 1.316 NL/NA.

52
Measurements on Sections
Areas are convenient if automated pixel
counting available Chords are convenient for
use of random test lines nL number of chords
per unit length
53
Size distributions from measurement

• Distribution of cross sections very different
  from 3D size distribution.
• Measurement of chord lengths is most reliable,
  i.e. experimental frequency of nL(l) versus l.
• Lord Willis Cahn Fullman
• ltDgt mean diameter s(D) standard devNV
  number of particles (grains) per unit volume.

54
Number per unit volume

• Lord Willis
• ?l size intervalaj median of class
  intervals (can use average of the size,l, in the
  jth interval)
• ASTM Bulletin 177 (1951) 56.

55
Number per unit volumeCahn Fullman

• Cahn FullmanTrans AIME 206 (1956) 610.D
  diameter lnumerical differentiation of nL(l)
  required.

56
Projections of Lines Spektor

• Z v(D/22 - l/22)
• Consider a cylindrical volume of length L, and
  radius Z centered on the test line. Volume is
  pZ2L and the intercepted chord lengths vary
  between l and D.

57
Projections of Lines, contd.

• Number of chords per unit length of line nL
  pZ2NV p/4 (D2 - l2)NV.where NV is the no. of
  spheres per unit vol.
• For a dispersion of spheres, sum up

58
Projections of Lines, contd.

• The terms on the RHS can be related to the total
  surface area, SV, and the total no of particles
  per unit volume, NV, respectively

Differentiating this expression gives
59
Projections of Lines, contd.

• The first two terms cancel out also we note that
  d(nL)lDmax - d(nL)0l, so that we obtain

60
Projections of Lines, contd.

• In order to relate a distribution of the number
  of spheres per unit volume to the distribution of
  chord lengths, we can take differences nL is a
  number of chords over an interval of lengths, ?l
  is the length interval (essentially the
  LordWillis result).