1
Objective
Microstructure-Properties: I
Lecture 6:
The Effect of Grain Size?
Grain
Size
Varistors
HallPetch
Creep
27-301
18th September, 2002
Prof. A. D. Rollett
2
Bibliography
• Electroceramics, A.J. Moulson & J.M. Herbert, Chapman &
Hall, ISBN 0-412-29490-7, 621.381/M92e
• Physical Ceramics (1997), Y.-T. Chiang, D.P. Birnie III, W.D.
Kingery, Wiley, New York, 0-471-59873-9.
Objective • Mechanical Behavior of Materials (1966), F. McClintock and A.
S. Argon, Addison Wesley.
Grain
Size
• Electronic Materials (1990), edited N. Braithwaite & G.
Weaver, (The Open University) Butterworths.
Varistors
• Mechanical Behavior of Materials, T.H. Courtney, McGraw-Hill,
HallISBN 0-07-013265-8, 620.11292,C86M
Petch
• Microstructure and Properties of Materials, J.C.M. Li, editor,
Creep
World Scientific, ISBN 981-02-2403-6
3
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Grain
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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
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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
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Examples of
Varistor
Circuit
Components
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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.
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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 powerlaw relationship for plastic flow!). The better the device, the
larger the exponent. The typical breakdown voltage ranges
Objective
from tens to hundreds of volts.
Grain
Size
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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.
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HallPetch
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Electroceramics
9
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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
Objective
Grain
Size
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.
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Electroceramics
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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
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ZnO, contd.
• A typical ZnO compact has grain size 10-50µm,
with an intergranular phase of thickness 1-1000nm.
• 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-1.0Sb2O30.5Cr2O3.
13
Objective
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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 electrondominated conduction (donor dopant atoms).
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p-type
n-type
Fermi levels
14
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Size
• 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.
Ee
Electron
Energy
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Ee
Electron
Energy
Conduction band
Band
Gap
Band
Gap
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valence band
p-type
n-type
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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
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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)
Creep
-
+
~0.8V
Ee
p-type
n-type
17
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Junction Region
• In addition to the gradients in electron energy and
potential, there is some flow of electrons from the ntype 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;
=neµ
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Conductivity in a semiconductor
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• 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, = -1.6.10-19C.
minority carrier concentration, pn, = 2.3.1010
holes.m-3
mobility, µh, = 0.044 m2V-1s-1
and charge,e, = +1.6.10-19C.
• Remember: electric field = -1*gradient of potential;
E = -dV/dx
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Junction region
p-type
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• 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]
n-type
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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.
Reverse bias
p-type
n-type
+
-
Forward bias
Forward bias
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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).
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Biasing: minority
carrier conc.
•
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•
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]
Grain Boundary electric double layer
23
Objective
• 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.
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Size
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Electroceramics
Band Structure at a Grain Boundary
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Objective
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• 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
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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
Objective
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.
Grain
Size
Varistors
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Electroceramics
27
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Relation to Diodes
• Each boundary can be regarded as a pair of backto-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 postsintering cool-down is important for development of
the critical properties.
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Typical Varistor Application
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Grain
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HallPetch
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Electroceramics
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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:
y = 0 + 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, 25-28.
Dislocation Pile-ups
30
Objective
Grain
Size
• 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.
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[Courtney]
31
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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 5.5. 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.
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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 HallPetch 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.
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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
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powders (which requires second phase particles in order to
Grain
maintain the small grain sizes at sintering temperatures) or
Size
heavy deformations allied with recrystallization. This is an
Varistors
exciting area and is a lively area of research and
development.
HallPetch
• How to make nanocrystalline material? Powders, ball milling,
equal-angle channel extrusion, thin film deposition (chemical
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vapor deposition, physical vapor deposition, laser ablation).
35
Nanocrystalline materials
• Limitations of nanocrystalline materials?
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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) = *,
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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
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Creep
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
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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
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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:
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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
Objective
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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.
Creep
[Courtney 7.5]
42
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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.
Other Creep Mechanisms
43
•
•
•
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•
•
Creep
•
•
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
Objective
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Size
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Creep
• 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
Objective
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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-{/kT}. Compression
decreases the concentration (slightly) and tension
raises it.
Q
   
Nvacancy  exp  vacancy exp 
 kT  kT 
Qvacancy 
 
compression
exp  
Nvacancy  exp 
 kT 
 kT 
tension
• Given small stresses (1-50 MPa), ~10-29m3,
kT~1.4.10-20J, /kT~0.02 «1. This permits us to
linearize the driving force.
46
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N-H Creep: derivation: 2
• Given small stresses (1-50 MPa), ~10-29m3,
kT~1.4.10-20J, /kT~0.02 «1. This permits us to
linearize the driving force.
∆NV ~ /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 (/kT)(1/d) exp-{Qm/kT}.
47
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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} (/kT)(1/d).
48
N-H Creep: derivation: 4
• Rate of change in length = volume rate (dV/dt) /
area; area ~ d2.
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dd/dt = D0Vexp-{Qm+QVacancy/kT} (/kT)(1/d).
• Strain = Change in length / length = dd/d.
dd/dt = D0Vexp-{Qm+QVacancy/kT} (/kT)(1/d2).
• Collecting terms:
Dbulk  
Ý ANH  2  

 d kT 
Note the grain size dependence!
49
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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
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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
Objective
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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.
Objective
• The term superalloy refers - loosely - to the use of this alloy
Grain
class at unusually high homologous temperatures.
Size
• They are based on the Ni-Cr-Al ternary system but have many
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other alloy additions.
Hall• The key to their success is the presence of a second phase,
Petch
close to Ni3Al (“gamma-prime”) that is coherent with the matrix
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(more on this in 302!) and whose strength increases with
temperature.
53
Objective
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Refractory Materials (Engines)
• Carbon
Tungsten
MgO
SiC
Mo
Nb
Al2O3
Cr
Zr
Tmelt
3800
3650
3100
3000
2880
2740
2290
2160
2125
Creep
Pt
Fe
Ni
2042
1808
1726
oxidizes
oxidizes
brittle (room T)
brittle (room T)
oxidizes
oxidizes
brittle (room T)
brittle
expensive
(nuclear fuel elements)
expensive (crucibles)
oxidizes
√
54
Superalloys: Temperature of Use
Microstructure and
Properties of Materials:
Stoloff
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Creep
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
Objective
Grain
Size
Varistors
HallPetch
Creep
[Microstructure and Properties of Materials: Stoloff]
56
Objective
Grain
Size
Varistors
HallPetch
Creep
Ni-Al phase
diagram
Note the use of high Al
contents in order to
obtain very high volume
fractions >50%) of the
Ni3Al phase (gammaprime) 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
Superalloy
strengthening
57
• The coherent
interface
between
the g’ and the
Objective
matrix is
Grain
important
Microstructure and Properties of Materials: Stoloff
Size
because it
means that the
Varistors
precipitate does not coarsen (again, more on this in 302).
Hall• The increasing strength with temperature (of the g’)is
Petch
critical: most materials soften with temperature. This effect
Creep
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.
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).
Objective • What’s the issue? Hot stage turbine blades for an aircraft
engine are comparatively small - a few centimeters long. The
Grain
equivalent component for a land-based turbine is an order of
Size
magnitude large - 30-70 cm long! Making these depends on
Varistors
control of directional solidification: faults give rise to new
Hallorientations and grain boundaries which weaken the material.
Petch
• Solution? Either much improved furnaces for directional
solidification (very expensive and perhaps not feasible) or
Creep
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
Objective
together.
Grain
• The bonding process relies on a boron-rich bonding alloy
Size
(braze) that produces transient melting.
Varistors
HallPetch
Creep
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
Objective
characteristics of the device are directly related to the
Grain
electrical properties of the boundary and the grain size.
Size
• In the case of the Hall-Petch effect, in most materials, both the
Varistors
strength and the toughness increase as the grain size is
Hallreduced. This effect can be explained by the resistance of the
Petch
boundaries to plastic flow (in the case of strength) and/or the
Creep
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
Objective
Grain
Size
Varistors
HallPetch
Creep
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
Objective
Grain
Size
Varistors
HallPetch
Creep
Bi doping levels in ZnO
• Step 1: grain boundary area per unit volume, AV, =
1/d = 8.3 104 m2/m3.
• Step 2: atomic area, Aatom ~ 0.32 10-18 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
Objective
Grain
Size
Varistors
HallPetch
Creep
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
Objective
Grain
Size
Varistors
HallPetch
Creep
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
Objective
Grain
Size
Varistors
HallPetch
Creep
Euler’s 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.
66
Objective
Grain Networks
• A consequence of the characteristic that three grain
boundaries meet at each edge to form a triple
junction is this:
3V = 2E
Grain
Size
E
Varistors
HallPetch
Creep
E
E
V EV
E
E
V
V
E
E
V EV
E
E
E
67
Objective
Grain
Size
Varistors
HallPetch
Creep
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
_
_
68
Objective
Grain
Size
Varistors
HallPetch
Creep
2D sections
6-sided grain
= unit cell;
each vertex
has 1/3 in each
unit cell;
each boundary
has 1/2 in
each cell
split each edge
divide each vertex by 3
69
Objective
Grain
Size
Varistors
HallPetch
Creep
3D Topology: polyhedra
70
Objective
Grain
Size
Varistors
HallPetch
Creep
2D Topology: polygons
71
Typical section
[Underwood]
Objective
Grain
Size
Varistors
HallPetch
Creep
• Correction terms (Eb, C1’,C2’) allow finite sections to
be interpreted.
C1’:=no. incomplete corners against 1 polygon;
C2’:= same for 2 polygons
Grain size measurement: area based
72
Objective
Grain
Size
Varistors
HallPetch
Creep
• Grain count method:
<A>=1/NA
• Number of whole grains= 20
Number of edge grains= 21
Effective total=Nwhole+Nedge/2
= 30.5
Total area= 0.5 mm2
Thus, NA= 61 mm-2; <A>=16.4 µm2
• Assume spherical (?!) grains, <A> mean intercept area=
2/3πr2
 d = 2√(3<A>/2π)= 5.6 µm.
[Underwood]
73
Objective
Grain
Size
Varistors
HallPetch
Creep
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
Objective
Grain
Size
Varistors
HallPetch
Creep
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
Objective
Grain
Size
Varistors
HallPetch
Creep
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
Objective
Grain
Size
Varistors
HallPetch
Creep
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
Objective
Grain
Size
Varistors
HallPetch
Creep
• 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
Objective
Grain
Size
Varistors
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).
HallPetch
Creep
l
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.
(
a
)
1  VV

NL
(4.7)
79
Objective
Grain
Size
Varistors
HallPetch
Creep
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, , which
is the mean uninterrupted center-to-center
length between all possible pairs of particles.
Thus,
the particle mean intercept length, <L3>:
<L3> =  l [mm] (4.8)
80
Particle Relationships
Objective
Grain
Size
Varistors
HallPetch
Creep
• Application: particle
coarsening in a 2-phase
material; strengthening of
solid against dislocation
flow.
• Eqs. 4.9-4.11, 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
VV(a )
L3  4 (a )
SV
l
l
l
(a )
1 VV
 4 (a )
SV
(
a
)
1  VV
 L3
(
a
)
VV
(a)
1  VV
  (a)
LA
81
Objective
Grain
Size
Varistors
HallPetch
Creep
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 = 0.554 (PV)-1/3
(4.18)
• Based on l~1/NL, ∆3  0.554 (πr2 l)1/3
for small VV,
∆2  0.500 (π/2 rl)1/2
82
Objective
Grain
Size
Varistors
HallPetch
Creep
Application of ∆2
• 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
Objective
Grain
Size
Varistors
HallPetch
Creep
• 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
Objective
Grain
Size
Varistors
HallPetch
Creep
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 crosssection!