1. Microstructure-Properties: I Lecture 6: The Effect of Grain Size?
27-301
18th September, 2002
Prof. A. D. Rollett

2. Bibliography
Electroceramics, A.J. Moulson & J.M. Herbert, Chapman & Hall, ISBN , /M92e
Physical Ceramics (1997), Y.-T. Chiang, D.P. Birnie III, W.D. Kingery, Wiley, New York,
Mechanical Behavior of Materials (1966), F. McClintock and A. S. Argon, Addison Wesley.
Electronic Materials (1990), edited N. Braithwaite & G. Weaver, (The Open University) Butterworths.
Mechanical Behavior of Materials, T.H. Courtney, McGraw-Hill, ISBN , ,C86M
Microstructure and Properties of Materials, J.C.M. Li, editor, World Scientific, ISBN

3. Objective
This lecture is concerned with the effects of grain size on properties.
Two examples will be given:
(1) The effect of grain size on resistance in ceramics used for varistors (e.g. in surge protectors).
(2) The effect of grain size on mechanical properties (Hall-Petch effect, Nabarro-Herring creep).
If time permits, the discussion will be extended to magnetic hardness also.

4. Varistors
Varistor = variable resistor, i.e. a circuit element whose resistance varies with the voltage applied.
As typically fabricated, they have highly non-linear response and are useful as voltage limiters.
They operate by retaining high resistance to some voltage, above which their resistance drops rapidly.
For short times they can pass large currents thereby preventing the voltage from rising much above the breakdown voltage.
Varistors can therefore function as self-reseting circuit breakers (actually shunts, not breakers!).

5. Examples of Varistor Circuit Components
Electroceramics

6. Macrostructure of a Surge Arrester
The size and structure of the device depends on the application, e.g. at what voltage it is designed to limit to, and how much current it must be able to pass in a given surge.
Electroceramics

7. Current-voltage characteristic
At low voltages, the response is ohmic, i.e. the current is proportional to the voltage. At higher voltages the response is power-law, with a large exponent (compare this to the power-law relationship for plastic flow!). The better the device, the larger the exponent. The typical breakdown voltage ranges from tens to hundreds of volts.
Electroceramics

8. Varistor application
A varistor (“VDR” in the figure) is typically included in parallel with the load so that the latter never sees anything above some maximum voltage.
Electroceramics

9. Material, microstructure
Varistors can be made from a range of semiconducting ceramics: SiC, ZnO, TiO2 and SrTiO3.
ZnO with Bi dopant and other oxides (Co, Sb, Fe) is standard material.
Critical feature is the segregation of the dopant to the grain boundaries.

10. Varistor microstructure
The real microstructure contains a range of grain sizes and shapes (left). For the purposes of understanding varistor behavior, one can idealize the microstructure as a “brick” structure, i.e. a regular lattice of cubical grains.
Electroceramics

11. ZnO
ZnO has a 3.2eV band gap and so the presence of electron donor additions such as Co, Sb, Fe to make it an n-type extrinsic semiconductor are vital. The presence of the donor sites makes the grain interiors conductive.
The Bi segregates strongly to grain boundaries (and other interfaces) where it provides acceptor states. The presence of the acceptor states locally depresses the Fermi level in the grain boundary.

12. ZnO, contd.
A typical ZnO compact has grain size 10-50µm, with an intergranular phase of thickness nm.
The high Bi-content intergranular phase has high resistance, ~ 106 m.
Heating to high temperatures (typical = 1250°C) drives off oxygen, leaving vacancies on the oxygen sub-lattice (wurtzite structure). Thermal activation can ionize these vacancies, thereby releasing electrons into the conduction band (giving n-type conduction).
Typical compositions include ~1mol% dopants: 96.5ZnO-0.5Bi2O3-1.0CoO-0.5MnO-1.0Sb2O3-0.5Cr2O3.

13. p-n diode junctions (silicon)
It is useful to go back to basics and consider how to form a p-n diode in terms of doped semiconductors.
Consider a block of Si with two (adjacent) regions of doping - one p-type and one n-type.
p-type means that conduction is hole-dominated (acceptor dopant atoms). n-type means electron-dominated conduction (donor dopant atoms).
p-type
n-type

14. Fermi levels
For acceptor dopants (e.g. boron), the Fermi level is low in the gap. For donor dopants (e.g. phosphorus, arsenic) the Fermi level is high in the gap.
Electron Energy
Electron Energy Ee Ee
Conduction band
Band Gap
Band Gap
valence band
p-type
n-type

15. Electron energies at junction
When we join the p-type to the n-type, the rule is that the Fermi level is constant throughout the material (otherwise there would be a net flow of electrons in the material). The result is a bending of the energy levels in the junction region.
Junction of p- & n-types Ee
Electron Energy
Band Gap
p-type
n-type

16. Potential vs. electron energy
Electric potential (voltage) is the opposite of electron energy (from the change in sign).
Holes move down gradients in electric potential: electrons move down gradients in electron energy.
By equilibrating Fermi levels, no net electron (or hole) flow will occur between the p- and n-type regions. + -
Potential (V)
~0.8V Ee
p-type
n-type

17. Junction Region
In addition to the gradients in electron energy and potential, there is some flow of electrons from the n-type into the p-type region with recombination of the carriers.
This depletes the concentrations of holes and electrons on either side of the junction.
Carrier depletion obviously decreases conductivity.
Conduction: the conductivity depends (linearly) on the carrier concentration, n, mobility, µ, and charge,e;  = n e µ

18. Conductivity in a semiconductor
Typical values for n-type doped silicon (subscript “n” denotes quantity in n-type): majority carrier concentration, nn, = 1022 electrons.m-3 mobility, µn, = 0.35 m2V-1s-1 and charge,e, = C. minority carrier concentration, pn, = holes.m-3 mobility, µh, = m2V-1s-1 and charge,e, = C.
Remember: electric field = -1*gradient of potential; E = -dV/dx

19. Junction region p-type n-type
The local electric field repels electrons on the n-type side, and repels holes on the p-type side.
Only minority carriers on either side of the junction are available to carry current.
[Electronic Materials]

20. Biasing a p-n junction - +
Now we consider what happens when we apply an external voltage (electric potential) to the system and require a current to flow through the junction.
p-type
n-type
Forward bias - +
Forward bias
Reverse bias

21. Biasing, contd.
Forward bias = lowers the potential (voltage) on the n-type side, and raises it on the p-type side. This tends to diminish the depletion zone (from both sides).
Reverse bias = as expected, this raises the potential (voltage) on the n-type side, and lowers it on the p-type side. This tends to widen the depletion zone (from both sides).

22. Biasing: minority carrier conc.
Bias voltage changes the density of minority carriers at the edge of the depletion zone and thus the current that can be carried across the zone.
Increasing forward bias increases the number of majority carriers (holes) in the p-type side which flow into the n-type side, raise the (minority carrier) level on that side and increase current capacity. The density is proportional to the exponential of the voltage across the junction.
[Electronic Materials]

23. Grain Boundary electric double layer
The electronic structure at a grain boundary in a ceramic is understood as having acceptor states (not well understood!) that cause a local increase in the electron energy. This constitutes a barrier to electron motion through the material.
Electroceramics

24. Band Structure at a Grain Boundary
Equilibration of the chemical potential of electrons throughout the solid equalizes the Fermi levels inside and outside the boundaries. Charge redistribution occurs. Conduction electrons are depleted from the boundary vicinity (and go into the acceptor states in the boundary).
A potential energy barrier at the boundary is created.
Applying a voltage across the material tilts the energy levels until breakdown occurs.
Electroceramics

25. Grain Boundary control
As a consequence,the electrical properties depend on (a) the doping of the grain boundaries and (b) the microstructure through the number and arrangement of the boundaries.
Chiang gives an example of estimating the breakdown voltage based on a 3V breakdown for an individual boundary. For a 1mm thick device with a 10µm grain size, one expects about 100 boundaries through the thickness, which predicts a breakdown voltage of ~300V.

26. Voltage-Current Characteristic
This is the characteristic that one can observe across a polycrystal, i.e. a breakdown voltage of about 300V. The inverse slope, a, is a measure of varistor quality.
Electroceramics

27. Relation to Diodes
Each boundary can be regarded as a pair of back-to-back Schottky barriers, i.e. metal-semiconductor junctions.
Chemistry of the boundaries is not well understood. Bi3+ is an electron donor solute, so it is not clear how it functions as an acceptor in the boundary!
The oxidation state is important: quenched samples of ZnO exhibit little or no breakdown. Apparently, oxidation of the grain boundaries during post-sintering cool-down is important for development of the critical properties.

28. Typical Varistor Application
Electroceramics

29. Hall-Petch Effect
The Hall-Petch effect is remarkably simple to express but still difficult to explain in fundamental terms.
At ambient conditions (no creep), yield strength rises as the grain size decreases.
The variation in strength can be described by a power-law relationship: sy = s0 + kd-1/2
The Hall-Petch effect is named for E.O. Hall and N.J. Petch from their papers of the early 1950’s, e.g. “The Cleavage Strength of Crystals” N.J. Petch, J. Iron & Steel Inst., 174,

30. Dislocation Pile-ups
The central idea is that dislocations are forced to pile up at grain boundaries, either because there is a barrier to crossing over into the next grain, or because a source must be activated in the next grain.
[Courtney]

31. Pile-up at a Boundary
The classical explanation for the Hall-Petch effect is that some stress concentration in a given grain is required to initiate slip in its neighboring grain. That stress concentration is most plausibly obtained through a dislocation pile-up, see figure The essence of the argument is that stress is higher as the number of dislocations increases. Thus the larger the grain size, the more quickly (in terms of macroscopic strain) is the critical stress reached at which slip is initiated in the neighboring grain. The form of the equation describing the pile-up stress contains a term in √(d/r) where d is the grain diameter and r is the (average) distance to the source in the neighboring grain from the boundary.

32. Material Dependence
The Hall-Petch constant, k in the equation, varies considerably amongst materials. This in itself raises some questions about the mechanism(s) underlying the effect. The explanation given is purely geometrical and although the material dependence could be explained through the ratio d/r, it is not clear why this should be so!
Solutes tend to enhance the magnitude of the Hall- Petch effect.
[Courtney]

33. Grain Size and Fracture
Grain size also has a marked effect on fracture, which was, in fact, part of Petch’s original contribution.
McClintock & Argon

34. Nanocrystalline materials
All this suggests that remarkably strong materials can be generated if very small grain sizes can be achieved. This, of course, is one aim of nanocrystalline materials in which grain sizes are obtained that are well less than one micron. The processing (in metals) relies on either compaction of fine powders (which requires second phase particles in order to maintain the small grain sizes at sintering temperatures) or heavy deformations allied with recrystallization. This is an exciting area and is a lively area of research and development.
How to make nanocrystalline material? Powders, ball milling, equal-angle channel extrusion, thin film deposition (chemical vapor deposition, physical vapor deposition, laser ablation).

35. Nanocrystalline materials
Limitations of nanocrystalline materials?

36. Stress concentration
The square root term is akin to the stress concentration at the tip of a penny-shaped crack (in fracture mechanics). Thus, (applied - 0) √(d/4r) = *, where * is the critical stress for dislocation source activation, 0 is the resistance to dislocation motion in each grain, and applied is the applied shear stress. Again, the larger the diameter, d, the more dislocations in the pile-up for a given applied stress (minus the resistance). Rearranging, we get {Courtney - Eq. 5.8}. applied = 0 + 2 * √r d-1/2 = 0 + kd-1/2.

37. Creep An important property of materials is their resistance to creep.
Creep is irreversible (plastic) flow at low rates under low stresses.
We will return to this issue in later lectures because of its importance.
Creep is highly sensitive to temperature because thermal activation makes the largest contribution to plastic flow when the stress is too small to overcome mechanical barriers to dislocation motion.

38. Diffusion
Creep is therefore a phenomenon associated with high temperatures.
High temperature is a relative term: one contribution that thermal activation makes is by increasing diffusion rates.
Diffusion coefficients (D = D0exp-{Q/RT}) are strongly (exponentially!) dependent on temperature.
The activation energy (enthalpy, strictly speaking) is approximately proportional to the melting point of the material.
At the same temperature, a higher melting point material will exhibit slower diffusion than a lower melting point material.

39. Diffusivity, Activation Energy
Porter & Easterling: diffusivity at the melting point is constant for a given class of material. Similarly, the activation energy normalized by RTm is constant for a given class of material.

40. Homologous Temperature
Therefore it is common to use homologous temperature as a measure of relative temperature: T’ = T/Tmelt
Therefore we expect materials tested at the same homologous temperature to show similar behavior.
Materials will tend to creep at high homologous temperatures because diffusion allows changes in shape.

41. Creep Mechanisms: diffusion
For the purposes of this lecture, we will consider just one creep mechanism: self-diffusion between grain boundaries.
Assumption: grain boundaries are perfect sources and sinks of vacancies.
Therefore a tensile stress (for example) on a polycrystalline body sets up a driving force for vacancy motion.
[Courtney 7.5]

42. Nabarro-Herring Creep
The creep mechanism involving diffusion to/from grain boundaries through the bulk lattice is known as Nabarro-Herring creep for the scientists who identified it.
The reason for the grain size dependence is simple: the diffusion path length is proportional to the grain size: - since the vacancy concentration at the boundaries is fixed by the stress and the path length is proportional to the grain size, the concentration gradient is inversely proportional to the grain size. - the creep rate (i.e. the strain rate) is proportional to the vacancy flux and is thus inversely proportional to the grain size.

43. Other Creep Mechanisms
Dislocation Glide. This is self-explanatory: dislocations move (conservatively) in response to shear stresses.
Nabarro-Herring Creep. Creep can occur by mass transport, i.e. diffusion of atoms from regions of lower (algebraically) stress to regions of higher (more tensile) stress. This is equally effective in amorphous materials as in crystalline.
Coble Creep. Mass transport can occur either in the bulk (leading to N-H Creep) or along interfaces such as grain boundaries. In the latter case it is known as Coble creep. Both of these mechanisms result in a significant grain size dependence.
Solute Drag Creep. For dislocations gliding at high T, not only do the solute atoms interact with the dislocations but they can also move sufficiently rapidly for the drag effect to be significant.
Dislocation Climb-Glide Creep. In between the (low) temperatures at which only dislocation glide is important, and the (high) temperatures at which diffusion dominates (at low stresses), a combination of glide and climb controls creep. That is to say, dislocation motion carries most of the strain but the dislocations circumvent obstacles by climb.
Grain Boundary Sliding accommodated by diffusional flow. In superplasticity especially, sliding of one grain relative to another is very important.
Grain Boundary Sliding accommodated by Dislocation Flow. This is the same mechanism of g.b. sliding but the accommodation is achieved by dislocation glide. Clearly one expects this to dominate over diffusion at lower temperatures.

44. Nabarro-Herring Creep: grain size dependence
The creep rate in Nabarro-Herring creep is inversely proportional to the square of the grain size.
The quadratic dependence of creep rate on grain size arises from distributing the vacancy flux over the (average) area of a grain facet.
Bottom line: small grain size lowers creep resistance, and large grain size increases creep resistance.
Ideal microstructure (w.r.t. grain structure) is a single crystal.

45. N-H Creep: derivation: 1
Difference in vacancy concentration, Nv, (which provides the driving force). The activation energies are modified from the unstressed values by exp{+/- stress*atomic volume}=exp-{sW/kT}. Compression decreases the concentration (slightly) and tension raises it.
Given small stresses (1-50 MPa), W~10-29m3, kT~ J, sW/kT~0.02 «1. This permits us to linearize the driving force.

46. N-H Creep: derivation: 2
Given small stresses (1-50 MPa), W~10-29m3, kT~ J, sW/kT~0.02 «1. This permits us to linearize the driving force. ∆NV ~ sW/kT exp-{Qvacancy/kT}.
The vacancy flux is given by Fick’s first law: JV = DV(dNV/dx)
The distance over which the diffusion occurs is approximated by the grain diameter, d: JV = DV (sW/kT)(1/d) exp-{Qm/kT}.

47. N-H Creep: derivation: 3
If we multiply the flux by the area over which diffusion takes place, which we approximate by the area of a grain boundary facet, d2, we obtain the rate of change of volume. We can also include the diffusion coefficient in the expression, where Qm is the activation energy for vacancy motion; DV=D0Vexp-{Qm/kT} dV/dt = JV d2 dV/dt = d2D0Vexp-{Qm+QVacancy/kT} (sW/kT)(1/d).

48. N-H Creep: derivation: 4
Rate of change in length = volume rate (dV/dt) / area; area ~ d2. dd/dt = D0Vexp-{Qm+QVacancy/kT} (sW/kT)(1/d).
Strain = Change in length / length = dd/d. dd/dt = D0Vexp-{Qm+QVacancy/kT} (sW/kT)(1/d2).
Collecting terms:
Note the grain size dependence!

49. Creep: general characteristics
Low temperature deformation is characterized by work hardening: high temperature by a short transient hardening, followed by steady-state flow.
Similarly, constant load leads to steady-state flow at high T, but cessation of flow at low T.

50. Deformation Mechanism Maps
Deformation maps provide a convenient graphical view of the different regimes of deformation behavior as a function of temperature and stress.

51. Grain Size and NH Creep
Smaller grain sizes expand the range over which Nabarro-Herring (and Coble) creep are observed.
Alternatively, the map can be drawn in the space of stress (normalized by modulus) and grain size (normalized by Burgers vector). In this map, the temperature is fixed.

52. Creep Resistance: Superalloys
Nickel-based “superalloys” originated with the Ni-Cr alloys used for heating elements in furnaces. In these, and the subsequent superalloys, their oxidation resistance is critical. Very few materials possess ductility, oxidation resistance and strength at high temperatures.
The term superalloy refers - loosely - to the use of this alloy class at unusually high homologous temperatures.
They are based on the Ni-Cr-Al ternary system but have many other alloy additions.
The key to their success is the presence of a second phase, close to Ni3Al (“gamma-prime”) that is coherent with the matrix (more on this in 302!) and whose strength increases with temperature.

53. Refractory Materials (Engines)
Tmelt
Carbon oxidizes Tungsten oxidizes MgO brittle (room T) SiC brittle (room T) Mo oxidizes Nb oxidizes Al2O brittle (room T) Cr brittle Zr expensive (nuclear fuel elements) Pt expensive (crucibles) Fe oxidizes Ni √

54. Superalloys: Temperature of Use
Microstructure and Properties of Materials: Stoloff
Why is temperature important? The efficiency of the engine is very sensitive to hot zone temperature. Therefore even small increases in the temperature to which turbine blades can be exposed produces significant gains in engine efficiency

55. Grain Size effect on creep rate
[Microstructure and Properties of Materials: Stoloff]

56. Ni-Al phase diagram
Note the use of high Al contents in order to obtain very high volume fractions >50%) of the Ni3Al phase (gamma-prime) that exhibits INCREASING hardness/strength with increasing temperature (alloy d, right). The ordered crystal structure of the g’) provides some of the enhanced strength.
Microstructure and Properties of Materials: Stoloff

57. Superalloy strengthening
The coherent interface between the g’ and the matrix is important because it means that the precipitate does not coarsen (again, more on this in 302).
The increasing strength with temperature (of the g’) is critical: most materials soften with temperature. This effect is not completely understood and is likely the result of dislocation-dislocation interactions.
The phase relationships allow a very large volume fraction of cuboidal g’ to be precipitated in the matrix: this is a very effective barrier to dislocation creep.
Microstructure and Properties of Materials: Stoloff

58. Contemporary Issues
An example of a contemporary issue is the effort being made by the manufacturers of gas turbine engines to transfer the technology for single crystal turbine blades from aircraft engines over to land-based gas turbines (for power generation).
What’s the issue? Hot stage turbine blades for an aircraft engine are comparatively small - a few centimeters long. The equivalent component for a land-based turbine is an order of magnitude large cm long! Making these depends on control of directional solidification: faults give rise to new orientations and grain boundaries which weaken the material.
Solution? Either much improved furnaces for directional solidification (very expensive and perhaps not feasible) or fabricate the single crystal from smaller pieces.

59. Bonded “Single” Crystals
The approach adopted by one manufacturer is to grow thin single crystal pieces (slabs) of the nickel superalloy using standard directional solidification. Control of the crystal soldification direction is still feasible because of the thinness. Then two pieces, chosen for similar orientation, are bonded together.
The bonding process relies on a boron-rich bonding alloy (braze) that produces transient melting.
Xtal A + bond + Xtal B Bonded Xtal

60. Summary
Grain size is a critically important aspect of polycrystalline materials.
In the case of varistors, a special electronic structure in the grain boundary layer produces a back-to-back diode that has a well-defined breakdown voltage. The electrical characteristics of the device are directly related to the electrical properties of the boundary and the grain size.
In the case of the Hall-Petch effect, in most materials, both the strength and the toughness increase as the grain size is reduced. This effect can be explained by the resistance of the boundaries to plastic flow (in the case of strength) and/or the decreased microcrack size in the case of fracture.
Grain size can play a major role in controlling creep resistance. Larger grain size increases creep resistance - hence the use of single crystals where feasible, especially for superalloys.

61. Example Problem for Varistors
How much Bismuth oxide must I add to ZnO (proportions by weight) in order to dope the grain boundaries to the desired level?
Can estimate a minimum amount by assuming that we need, say, 2 layers of Bi atoms along every boundary in order to accomplish the required doping.
From here on, it is college chemistry to make the estimate.
Suppose that the grain size in the ZnO is 12µm (as in the exam question). That is, 3V grain boundary breakdown voltage, 250V device breakdown, 1mm thick.

62. Bi doping levels in ZnO
Step 1: grain boundary area per unit volume, AV, = 1/d = m2/m3.
Step 2: atomic area, Aatom ~ m2
Step 3: no. of atoms per unit volume = AV/Aatom = 8.3/9 104/10-20 ~ 1024 atoms/m3.
Step 4: moles(Bi)/m3 = 1024/Nav = 1.7 moles/m3.
Step 5: molecular weight of Bi2O3=464 gms
Step 6: weight(Bi2O3) per m3 = 770 gms
Step 7: density of ZnO = 5.6 Mg/m3.
Step 8: weight proportions are 1:7200, Bi2O3:ZnO.

63. 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).

64. 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 <angle brackets> indicates average quantity

65. Euler’s equations
3D: simple polyhedra (no re-entrant shapes) V + F = E [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

66. 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 E V E E V V E E E E V V E E E

67. 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, E = 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 _ _

68. 2D sections split each edge
6-sided grain = unit cell; each vertex has 1/3 in each unit cell; each boundary has 1/2 in each cell
divide each vertex by 3

69. 3D Topology: polyhedra

70. 2D Topology: polygons

71. Typical section
[Underwood]
Correction terms (Eb, C1’,C2’) allow finite sections to be interpreted.
C1’:=no. incomplete corners against 1 polygon; C2’:= same for 2 polygons

72. Grain size measurement: area based
Grain count method: <A>=1/NA
Number of whole grains= 20 Number of edge grains= 21 Effective total=Nwhole+Nedge/ = 30.5 Total area= 0.5 mm2 Thus, NA= 61 mm-2; <A>=16.4 µm2
Assume spherical (?!) grains, <A> mean intercept area= 2/3πr2  d = 2√(3<A>/2π)= 5.6 µm.
[Underwood]

73. 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)

74. S:V and Mean Intercept Length
Mean intercept length from intercepts of particles of alpha phase: <L3> = 1/N Si (L3)i (2.33)
Can also be obtained as: <L3> = LL/NL (2.34)
Substituting: <L3> = 4VV/SV, (2.35) where fractions refer to alpha phase only.

75. S:V example: sphere
For a sphere, the volume:surface ratio is D[iameter]/6.
Thus <L3>sphere = 2D/3.
In general we can invert the relationship to obtain the surface:volume ratio, if we know (measure) the mean intercept: <S/V>alpha = 4/<L3> (2.38)

76. Table 2.2 <L3>:= mean intercept length, 3D objects
<V>:= 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]

77. Grain size measurement: intercepts
From Table 2.2 [Underwood], column (a), illustrates how to make a measurement of the mean intercept length, based on the number of grains per unit length of test line. <L3> = 1/NL
Important: use many test lines that are randomly oriented w.r.t. the structure.
Assuming spherical (?!) grains, <L3> = 4r/3, [Underwood, table 4.1], if LT= 25µm, LTNL= 5  d = 6<L3>/4 = 6/NL4 = 6*5/4 = 7.5µm.

78. Particles and Grains (4.7)
“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 as number of interceptions with the matrix. Thus lineal fraction of occupied by matrix is lNL, equal to the volume fraction, 1-VV-alpha.

79. 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, <L3>: <L3> = s - l [mm] (4.8)

80. Particle Relationships
Application: particle coarsening in a 2-phase material; strengthening of solid against dislocation flow.
Eqs , with LA=πPL/2=πNL= πSV/4
dimension: length units (e.g.): mm
Use l to calculate critical resolved shear stresses.
Alternate notation for 3rd eq.: l = d (1-f) / f

81. 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 / √PA (4.18a)
3D: ∆3 = (PV)-1/3 (4.18)
Based on l~1/NL, ∆3  (πr2 l)1/3 for small VV, ∆2  (π/2 rl)1/2

82. Application of ∆2 Hull & Bacon; fig 10.17
Percolation of dislocation lines through arrays of 2D point obstacles.
Caution! “Spacing” has many interpretations: select the correct one!
Hull & Bacon; fig 10.17

83. 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.

84. Sections through dispersions of spherical objects
Even mono-disperse spheres exhibit a variety of diameters in cross section. Only if you know that the second phase is monodisperse may you measure diameter from maximum cross-section!