Grain Boundary Properties: Energy, Mobility

1
Grain Boundary PropertiesEnergy, Mobility

• 27-750, Spring 2003
• A.D. Rollett

2
References

• Interfaces in Crystalline Materials, Sutton
  Balluffi, Oxford U.P., 1998. Very complete
  compendium on interfaces.
• Interfaces in Materials, J. Howe, Wiley, 1999.
  Useful general text at the upper
  undergraduate/graduate level.
• Grain Boundary Migration in Metals, G. Gottstein
  and L. Shvindlerman, CRC Press, 1999. The most
  complete review on grain boundary migration and
  mobility.
• Materials Interfaces Atomic-Level Structure
  Properties, D. Wolf S. Yip, Chapman Hall,
  1992.

3
Why learn about grain boundary properties?

• Many aspects of materials behavior and
  performance affected by g.b. properties.
• Examples include- stress corrosion cracking in
  Pb battery electrodes, Ni-alloy nuclear fuel
  containment, steam generator tubes- creep
  strength in high temp. alloys- weld cracking
  (under investigation)- electromigration
  resistance (interconnects)
• Precipitation of second phases at grain
  boundaries depends on interface energy (
  structure).

4
Properties, phenomena of interest

• 1. Energy (excess free energy ? wetting,
  precipitation)
• 2. Mobility (normal motion ? grain growth,
  recrystallization)
• 3. Sliding (tangential motion ? creep)
• 4. Cracking resistance (intergranular fracture)
• 5. Segregation of impurities (embrittlement,
  formation of second phases)

5
Grain Boundary Diffusion

• Especially for high symmetry boundaries, there is
  a very strong anisotropy of diffusion
  coefficients as a function of boundary type. This
  example is for Zn diffusing in a series of lt110gt
  symmetric tilts in copper.

6
Grain Boundary Sliding
640C

• Grain boundary sliding should be very structure
  dependent. No surprise therefore that Biscondis
  results show that the rate at which boundaries
  slide is highly dependent on misorientation in
  fact there is a threshold effect with no sliding
  below a certain misorientation at a given
  temperature.

600C
500C
Biscondi, M. and C. Goux (1968). "Fluage
intergranulaire de bicristaux orientés
d'aluminium." Mémoires Scientifiques Revue de
Métallurgie 55(2) 167-179.
7
Grain boundary energy current status?

• Limited information available
• Deep cusps exist for a few lt110gt CSL types in fcc
  (S3, S11), based on both experiments and
  simulation.
• Extensive simulation results Wolf et al.
  indicate that interfacial free volume is good
  predictor. No simple rules available, however, to
  predict free volume.
• Wetting results in iron Takashima, Wynblatt
  suggest that a broken bond approach (with free
  volume and twist angle) provides a reasonable
  5-parameter model.
• If binding energy is neglected, an average of
  the surface energies is a good predictor of grain
  boundary energy in MgO Saylor, Rohrer. This
  may be useful in oxide structures but is not
  likely to be successful in metallic systems.
• Minimum dislocation density structures Frank
  provide a good model of g.b. energy in MgO, and
  may provide a good model of low angle grain
  boundary mobility.

8
Experimental Impact of Energy

• Wetting by liquids is sensitive to grain boundary
  energy.
• Example copper wets boundaries in iron at high
  temperatures.
• Wet versus unwetted condition found to be
  sensitive to grain boundary energy in FeCu
  system Takashima et al., ICOTOM-12 (1999).

9
Wetting comparison

• High energy (light) boundaries should be wet
  (W).
• Low energy boundaries (dark) should be dry (U).
• Example of Cu wetting boundaries in Fe with (311)
  on one side.
• Takashima, M., P. Wynblatt, and B.L. Adams,
  Correlation of grain boundary character with
  wetting behavior. Interface Science, 2000. 8 p.
  351-361.

10
1. Grain Boundary Energy

• First categorization of boundary type is into
  low-angle versus high-angle boundaries. Typical
  value in cubic materials is 15 for the
  misorientation angle.
• Typical values of g.b. energies vary from 0.32
  J.m-2 for Al to 0.87 for Ni J.m-2.
• Read-Shockley model describes the energy
  variation with angle for low-angle boundaries
  successfully in many experimental cases, based on
  a dislocation structure.

11
LAGB to HAGB Transition

• LAGB steep risewith angle.HAGB plateau

Disordered Structure
Dislocation Structure
Gjostein, N. A. and F. N. Rhines (1959).
"Absolute interfacial energies of 001 tilt and
twist grain boundaries in copper." Acta
metallurgica 7 319-330.
12
1.1 Read-Shockley model

• Start with a symmetric tilt boundary composed of
  a wall of infinitely straight, parallel edge
  dislocations (e.g. based on a 100, 111 or 110
  rotation axis with the planes symmetrically
  disposed).
• Dislocation density (L-1) given by1/D
  2sin(q/2)/b ? q/b for small angles.

13
1.1 Tilt boundary
D
14
1.1 Read-Shockley contd.

• For an infinite array of edge dislocations the
  long-range stress field depends on the spacing.
  Therefore given the dislocation density and the
  core energy of the dislocations, the energy of
  the wall (boundary) is estimated (r0 sets the
  core energy of the dislocation) ggb E0 q(A0
  - lnq), whereE0 µb/4p(1-n) A0 1
  ln(b/2pr0)

15
1.1 LAGB experimental results

• Experimental results on copper.

Gjostein Rhines, Acta metall. 7, 319 (1959)
16
1.1 Read-Shockley contd.

• If the non-linear form for the dislocation
  spacing is used, we obtain a sine-law variation
  (Ucore core energy) ggb sinq Ucore/b -
  µb2/4p(1-n) ln(sinq)
• Note this form of energy variation may also be
  applied to CSL-vicinal boundaries.

17
1.1 Low-angle g.b. properties

• Recently, the properties of low angle grain
  boundaries have been measured by the MIMP at CMU.
• The results confirm the Read-Shockley
  relationship.
• A variation of energy with misorientation axis
  was also found boundaries with lt111gt
  misorientation axes had the lowest energies
  whereas those with lt100gt axes had the highest.
  The variation was only over a range of /- 10,
  however.

18
1.1 Low Angle Grain Boundary Energy, Yang et al.
High
117
105
113
205
215
335
203
Low
8411
323
727
"Measuring relative grain boundary energies and
mobilities in an aluminum foil from triple
junction geometry", C.-C. Yang, W. W. Mullins and
A. D. Rollett, Scripta Materiala 44 2735-2740
(2001).
? vs.
19
1.2 Energy of High Angle Boundaries

• No universal theory exists to describe the energy
  of HAGBs.
• Based on a disordered atomic structure for
  general high angle boundaries, we expect that the
  g.b. energy should be at a maximum and
  approximately constant.
• Abundant experimental evidence for special
  boundaries at (a small number) of certain
  orientations for which the atomic fit is better
  than in general high angle g.bs.
• Each special point (in misorientation space)
  expected to have a cusp in energy, similar to
  zero-boundary case but with non-zero energy at
  the bottom of the cusp.
• Atomistic simulations suggest that g.b. energy is
  (positively) correlated with free volume at the
  interface.

20
1.2 Exptl. vs. Computed Egb
lt100gtTilts
S11
lt110gtTilts
S3, 111 plane CoherentTwin
Hasson Goux
21
Dislocation models of HAGBs

• Boundaries near CSL points expected to exhibit
  dislocation networks, which is observed.

lt100gt twists
22
1.2 Atomistic modeling

• Extensive atomistic modeling has been conducted
  using (mostly) embedded atom potentials and an
  energy-relaxation method to locate the minimum
  energy configuration of a (finite) bicrystal.
  See Wolf Yip, Materials Interfaces
  Atomic-Level Structure Properties, Chapman
  Hall, 1992.
• Grain boundaries in fcc metals Cu, Au

23
Atomistic models results
g.b. plane

• Results of atomistic modeling confirm the
  importance of the more symmetric boundaries.
• Example of symmetric tilt boundary energy for
  embedded-atom-method calculations using either
  Lennard-Jones, copper or gold interatomic
  potentials.

Wolf Yip
24
Coordination Number
Reasonable correlation for energy versus the
coordination number for atoms at the boundary
suggests that broken bond model may be
applicable, as it is for solid/vapor surfaces.
Wolf Yip
25
Surface Energies vs. Grain Boundary Energy

• A recently revived, but still controversial idea,
  is that the grain boundary energy is largely
  determined by the energy of the two surfaces that
  make up the boundary (and that the twist angle is
  not significant).
• This is has been demonstrated to be highly
  accurate in the case of MgO, which is an ionic
  ceramic with a rock-salt structure. In this
  case, 100 has the lowest surface energy, so
  boundaries with a 100 plane are expected to be
  low energy.
• The next slide, taken from the PhD thesis work of
  David Saylor, shows a comparison of the g.b.
  energy computed as the average of the two surface
  energies, compared to the frequency of boundaries
  of the corresponding type. As predicted, the
  frequency is lowest for the highest energy
  boundaries, and vice versa.

26
Hypothetical Boundary Energy
Population
gs1gs2
0
60
110
27
Comparison of CSL and boundary plane models for
lt100gt misorientation axes
Studies of MgO smoke show a high fraction of
high coincidence boundaries (low S
CSL). Chaudhari and Matthews, J. Appl. Phys. 42
(1971) 3063.
28
G.B. frequencies in MgO
Data from Saylors work on MgO suggests that
the boundary plane is more important to energy
than is CSL structure. The S5 symmetric tilt
boundary has two 210 planes, for example
boundaries with (310) and (100) are far more
frequent, however.
100
(MRD)
(MRD)
S5 100,36.86
S9 110,38.94
29
G.B. Energy Metals Summary

• For low angle boundaries, use the Read-Shockley
  model with a logarithmic dependence well
  established both experimentally and
  theoretically.
• For high angle boundaries, use a constant value
  unless near a CSL structure with high fraction of
  coincident sites and plane suitable for good
  atomic fit.
• In ionic solids, the grain boundary energy may be
  simply the average of the two surface energies
  (modified for low angle boundaries). This is
  less likely to be valid for metals.

30
Grain Boundary Mobility

• The mobility of grain boundaries is dominated by
  solute. Solutes tend to segregate to any
  interface and lower the free energy of the system
  (because of size misfit, for example). Therefore
  for a boundary to move away from the segregated
  solute requires energy to be supplied.
• Mobility is also strongly sensitive to boundary
  type (i.e. atomic structure). High mobilities
  tend to be associated with CSL structures.
• Mechanistic explanation is still lacking (and
  controversial!). Even a detailed understanding
  of mobility in pure systems is not available.

31
Mobility general characteristics

• Low angle boundaries with dislocation structures
  are, in general, much less mobile than high
  boundaries. In broad terms, the diffusion
  distances are much larger in the case of LAGBs
  (diffusion between dislocations).
• With only a few high mobility boundary types, the
  general picture is of very low mobility for
  LAGBs, moderate mobility for general boundaries,
  and the occasional peak for high angle boundaries.

32
Grain Boundary Energy, Mobility

• Contrast the sharply peaked properties derived
  from (2D) simulations (Upmanyu) with the broad
  mobility maximum from recrystallization
  experiments.

33
2.1 Low Angle G.B. Mobility

• Mobility of low angle boundaries dominated by
  climb of the dislocations making up the boundary.
• Even in a symmetrical tilt boundary the
  dislocations must move non-conservatively in
  order to maintain the correct spacing as the
  boundary moves.

34
LA-gtHAGB Transition
High Angle Boundaries
Transfer of atoms from the shrinking grain to the
growing grain by atomic bulk diffusion mechanism
1
0
.
9
0
.
8
0
.
7
0
.
6
Relative Boundary Mobility
0
.
5
0
.
4
0
.
3
0
.
2
0
.
1
0
0
1
0
2
0
3
0
M
i
s
o
r
i
e
n
t
a
t
i
o
n

A
n
g
l
e

(
d
e
g
r
e
e
s
)
Low Angle Boundaries
Transfer of vacancies between two adjacent sets
of dislocations by grain boundary diffusion
mechanism
35
Tilt Boundary Motion
Burgers vectors inclined with respect to the
boundary plane in proportion to the
misorientation angle.
climb
glide
(Bauer and Lanxner, Proc. JIMIS-4 (1986) 411)
36
Low Angle GB Mobility

• Huang and Humphreys (2000) coarsening kinetics
  of subgrain structures in deformed Al single
  crystals. Dependence of the mobility on
  misorientation was fitted with a power-law
  relationship, Mkqc, with c5.2 and k3.10-6
  m4(Js)-1.
• Yang, et al. mobility (and energy) of LAGBs in
  aluminum strong dependence of mobility on
  misorientation boundaries based on 001
  rotation axes had much lower mobilities than
  either 110 or 111 axes.

37
LAGB Mobility in Al, experimental
001
Low
117
105
113
205
215
335
203
8411
111
High
101
323
727
M vs.
38
LAGB Axis Dependence

• We can explain the (strong) variation in LAGB
  mobility from lt111gt axes to lt100gt axes, based on
  the simple tilt model lt111gt tilt boundaries have
  dislocations with Burgers vectors nearly perp. to
  the plane. lt100gt boundaries, however, have
  Burgers vectors near 45 to the plane. Therefore
  latter require more climb for a given
  displacement of the boundary.

39
Symmetrical lt001gt 11.4o grain boundarygt nearly
45o alignment of dislocations with respect to
the boundary normal gt ? 45o ?/2
Symmetrical lt111gt 12.4o grain boundarygt
dislocations are nearly parallel to the boundary
normal gt ? ?/2
40
2.1 Low Angle GB Mobility, contd.

• Winning et al. Measured mobilities of low angle
  grain lt112gt and lt111gt tilt boundaries under a
  shear stress driving force. A sharp transition
  in activation enthalpy from high to low with
  increasing misorientation (at 13).

41
Dislocation Modelsfor Low Angle G.B.s
Sutton and Balluffi (1995). Interfaces in
Crystalline Materials. Clarendon Press, Oxford,
UK.
42
LAGB to HAGB Transitions

• Read-Shockley forenergy of low angleboundaries
• Exponentialfunction for transitionfrom low- to
  high-angle boundaries

43
HAGB mobility theory

• The standard theory for HAGB mobility is due to
  Burke Turnbull, based on thermally activated
  atomic transfer across the interface.
• For the low driving forces typical in grain
  growth, recrystallization etc., it gives a linear
  relation between force and velocity (as typically
  assumed).

44
Burke-Turnbull

• Given a difference in free energy for an atom
  attached to one side of the boundary versus the
  other, ?P, the rate at which the boundary moves
  is

Given similar attack frequencies and activation
energies in both directions,
45
Velocity Linear in Driving Force

• Then, for small driving forces compared to the
  activation energy for migration, ?Pb3kT, which
  allows us to linearize the exponential term.

Mobility
46
Linearity of migration rate against driving force

• At what point is the linear relation no longer
  reasonable?
• The criterion is the ratio of the driving force
  to the thermal activation energy, kT.
• For a radius of curvature of 10nm, for example,
  and a g.b. energy of 1 J.m-2, ?P108 Pa. In
  terms of energy per atom in, say, Al, we multiply
  the volumetric pressure by the volume per atom,
  16 Å3, to obtain ?P 1.6.10-21 J. Compare with
  kT 1.2 10-20 at 600C. Clearly for very small
  scale microstructures (and low temperatures) we
  may expect the linearity to break down.

47
HAGB Mobility

• The basic Burke-Turnbull theory ignores details
  of g.b. structure
• The terrace-ledge-kink model may be useful the
  density of sites for detachment and attachment of
  atoms can modify the pre-factor.
• Atomistic modeling is starting to play a role
  see work by Upmanyu Srolovitz M. Upmanyu, D.
  Srolovitz and R. Smith, Int. Sci., 6, (1998)
  41..
• Much room for research!

graduate
48
HAGB Mobility the U-bicrystal

• The curvature of the end of the interior grain is
  constant (unless anisotropy causes a change in
  shape) and the curvature on the sides is zero.
• Migration of the boundary does not change the
  driving force
• Simulation and experiment

Dunn, Shvindlerman, Gottstein,...
49
HAGB M Boundary velocity
Simulation Experiment
Steady-state migration initial and final
transients
50
HAGB M 2D simulation results

• Extract boundary energy from total energy vs.
  half-loop height (assume constant entropy)
• MM/g

Note misorientation angle shown in plots is 1/2
of total angle.
51
HAGB M Activation energy
simulation
S19
S7
S13
specialboundary
Q (e)
experiment
S7
Lattice diffusion between dislocations
Q (eV)
Simulations exhibit much smaller activation
energies than experiments, possibly because
solutes affect experimental results.
52
3D simulations reduced mobility (M) vs.
Misorientation
(m4/Js)
(deg)
Zhang, Upmanyu, Srolovitz
53
Mobility and Energy vs. Misorientation
(J/m2)
(m4/Js)
(deg)
(deg)
Zhang, Upmanyu, Srolovitz
54
Mobility vs. Misorientation
(m4/Js)
(deg)
Zhang, Upmanyu, Srolovitz
55
Reduced Mobility, M

• In many experiments on g.b. mobility, only the
  migration rate can be measured and the boundary
  curvature. If the energy of the boundary is not
  known (or must be assumed to be constant) then
  one can only derive the reduced mobility, M.
  Where M and E are the mobility and energy defined
  in the standard way, M M E.

56
High Angle GB Mobility

• Large variations known in HAGB mobility.
• Classic example is the high mobility of
  boundaries close to 40lt111gt (which is near the
  S7 CSL type).
• Note broad maximum.

Gottstein Shvindlerman grain boundary
migration in metals
57
Mobility of HAGBs with stored energy driving force

• Huang Humphreys, The effect of solute elements
  on grain boundary mobility during
  recrystallization of single-phase aluminum
  alloys, Proc. Conf. Rex Gr.Gr., Aachen, vol.1
  409 (2001).
• As previously observed, broad peak in mobility
  observed centered on 40 lt111gt misorientation
  with 10 FWHM w.r.t. misorientation angle.
  Similar decrease with deviation from lt111gt axis.

58
Simulation Results Misorientation Axis Dependence
T 0.7Tm
lt111gt tilt misorientations fastest moving
boundaries Dramatic decrease in mobility with
deviation in tilt axis
Unpublished work by Upmanyu
59
HAGB M Issues dirt

• Solutes play a major role in g.b. mobility by
  reducing absolute mobilities at very low levels.
• Simulations typically have no impurities
  included therefore they model ultra-pure
  material.

60
HAGB Impurity effects

• Impurities known to affect g.b. mobility
  strongly, depending on segregation and mobility.
• CSL structures with good atomic fit less affected
  by solutes
• Example Pb bicrystals

special
general
61
HAGB M impurity effect on recrystallization
kinetics
increasing Cu content
V (cm.s-1)
decreasing Fe content
1/T
F. R. Boutin, J. Physique, C4, (1975) C4.355.
R. Vandermeer and P. Gordon, Proc. Symposium on
the Recovery and Recrystallization of Metals, New
York, TMS AIME, (1962) p. 211.
62
Impurity (solute) effect on mobility, contd.

• Example of Ga additions to Al (LHS) show that at
  low levels, certain solutes can increase
  mobility. Adding 10ppm Ga to 99.999 Al
  increases the mobility whereas adding 410ppm Ga
  to Al decreases the mobility (as expected).
• Note the low levels of solute that have
  measurable effects on mobility. Example (RHS) of
  adding Cu to Al shows an effect at 0.0002 a/o,
  i.e. at the ppm level.

Gottstein Shvindlerman grain boundary mobility
in Al.
Gordon Vandermeer impurities in aluminum
63
GB Mobility Summary

• The properties of low angle grain boundaries are
  dictated by their discrete dislocation structure
  energy logarithmic with angle mobility
  exponential with angle.
• The kinetic properties of high angle boundaries
  are (approx.) plateau dictated by local atomic
  transfer. Special boundary types have low energy
  and high/low mobility.

64
Theory Diffusion

• Atom flux, J, between the dislocations
  iswhere DL is the atom diffusivity (vacancy
  mechanism) in the latticem is the chemical
  potentialkT is the thermal energyand W is an
  atomic volume.

65
Driving Force

• A stress t that tends to move dislocations with
  Burgers vectors perpendicular to the boundary
  plane, produces a chemical potential gradient
  between adjacent dislocations associated with the
  non-perpendicular component of the Burgers
  vector where d is the distance between
  dislocations in the tilt boundary.

66
Atom Flux

• The atom flux between the dislocations (per
  length of boundary in direction parallel to the
  tilt axis) passes through some area of the matrix
  between the dislocations which is very roughly
  Ad/2. The total current of atoms between the
  two adjacent dislocations (per length of
  boundary) I is SB.

67
Dislocation Velocity

• Assuming that the rate of boundary migration is
  controlled by how fast the dislocations climb,
  the boundary velocity can be written as the
  current of atoms to the dislocations (per length
  of boundary in the direction parallel to the tilt
  axis) times the distance advanced per dislocation
  for each atom that arrives times the unit length
  of the boundary.

68
Mobility (Lattice Diffusion only)

• The driving force or pressure on the boundary is
  the product of the Peach-Koehler force on each
  dislocation times the number of dislocations per
  unit length, (since db/v2q).
• Hence, the boundary mobility is SBSee
  also Furu and Nes (1995), Subgrain growth in
  heavily deformed aluminium - experimental
  investigation and modelling treatment. Acta
  metall. mater., 43, 2209-2232.

69
Theory Addition of a Pipe Diffusion Model

• Consider a grain boundary containing two arrays
  of dislocations, one parallel to the tilt axis
  and one perpendicular to it. Dislocations
  parallel to the tilt axis must undergo
  diffusional climb, while the orthogonal set of
  dislocations requires no climb. The flux along
  the dislocation lines is

70
LatticePipe Diffusion

• The total current of atoms from one dislocation
  parallel to the tilt axis to the next (per length
  of boundary) is where d is the radius of the
  fast diffusion pipe at the dislocation core and
  d1 and d2 are the spacing between the
  dislocations that run parallel and perpendicular
  to the tilt axis, respectively.

71
Boundary Velocity

• The boundary velocity is related to the
  diffusional current as above but with
  contributions from both lattice and pipe
  diffusion

72
Mobility (Lattice and Pipe Diffusion)

• The mobility Mv/(tq) is now simplyThis
  expression suggests that the mobility increases
  as the spacing between dislocations perpendicular
  to the tilt axis decreases.

73
Effect of twist angle

• If the density of dislocations running
  perpendicular to the tilt axis is associated with
  a twist component, thenwhere f is the twist
  misorientation. On the other hand, a network of
  dislocations with line directions running both
  parallel and perpendicular to the tilt axis may
  be present even in a pure tilt boundary assuming
  that dislocation reactions occur.

74
Effect of Misorientation

• If the density of the perpendicular dislocations
  is proportional to the density of parallel ones,
  then the mobility iswhere a is a
  proportionality factor. Note the combination of
  mobility increasing and decreasing with
  misorientation.

75
Results Ni Mobility

• Nickel QL2.86 eV, Q?0.6QL, D0LD0?10-4 m2/s,
  b3x10-10 m, Wb3, db, a1, k8.6171x10-5 eV/K.

M (10-10 m4/J s)
T (K)
q ()
76
Theory Reduced Mobility

• Product of the two quantities MMg that is
  typically determined when g.b. energy not
  measured. Using the Read-Shockley expression for
  the grain boundary energy, we can write the
  reduced mobility as

77
Results Ni Reduced Mobility

• g01 J/m2 and q25, corresponding to a maximum
  in the boundary mobility at 9.2.

log10M (10-11m2/s)
q ()
T (K)
78
Results AluminumMobility vs. T and q
The vertical axis is Log10 M.
log10M (µm4/s MPa)
g0 324 mJ/m2, q 15, DL(T) 1.76.10-5
exp-126153 J/mol/RT m2/s, D?(T) 2.8.10-6
exp-81855 J/mol/RT m2/s, db, b 0.286 nm, W
16.5.10-30 m3 b3/v2, a 1.
q ()
T (K)
79
Comparison with Expt. Mobility vs. Angle at
873K
Log10M (µm4/s MPa)
0 -1-2 -3 -4 -5
Log10M (µm4/s MPa)
q ()
M. Winning, G. Gottstein L.S. Shvindlerman,
Grain Boundary Dynamics under the Influence of
MechanicalStresses, Risø-21 Recrystallization,
p.645, 2000.
80
Comparison with Expt. Mobility vs. Angle at
473K
Log10M (µm4/s MPa)
Log10M (µm4/s MPa)
4 32 1
q ()
81
Discussion on LAGB mobility

• The experimental data shows high and low angle
  plateaus the theoretical results are much more
  continuous.
• The low T minimum is quite sharp compared with
  experiment.
• Simple assumptions about the boundary structure
  do not capture the real situation.

82
2.1 LAGB mobility conclusion

• Agreement between calculated (reduced) mobility
  and experimental results is remarkably good.
  Only one (structure sensitive) adjustable
  parameter (a 1), which determines the position
  of the minimum.
• Better models of g.b. structure will permit
  prediction of low angle g.b. mobilities for all
  crystallographic types.