Grain Boundary, Surface Energies, Measurement

1
Grain Boundary, Surface Energies, Measurement

• 27-750, Spring 2003
• A.D. Rollett

2
Interfacial Energies

• Practical Applications Rain-X for windshields.
  Alters the water/glassglass/vapor ratio so that
  the contact angle is increased.

streaky clear
3
Impact on Materials

• Surface grooving where grain boundaries intersect
  free surfaces leads to surface roughness,
  possibly break-up of thin films.
• Excess free energy of interfaces (virtually all
  circumstances) implies a driving force for
  reduction in total surface area, e.g. grain
  growth (but not recrystallization).
• Interfacial Excess Free Energy g

4
Herring Equations

• We can demonstrate the effect of interfacial
  energies at the (triple) junctions of boundaries.
• Equal g.b. energies on 3 g.b.s

1
g1g2g3
2
3
120
5
Definition of Dihedral Angle

• Dihedral angle, c angle between the tangents to
  an adjacent pair of boundaries (unsigned). In a
  triple junction, the dihedral angle is assigned
  to the opposing boundary.

1
g1g2g3
2
3
c1 dihedralangle for g.b.1
120
6
Isotropic Material

• An material with uniform grain boundary energy
  should have dihedral angles equal to 120.
• Likely in real materials? No! Low angle
  boundaries (crystalline materials) always have a
  dislocation structure and therefore a monotonic
  increase in energy with misorientation angle
  (Read-Shockley model).

7
Unequal energies

• If the interfacial energies are not equal, then
  the dihedral angles change. A low g.b. energy on
  boundary 1 increases the corresponding dihedral
  angle.

1
g1ltg2g3
2
3
c1gt120
8
Unequal Energies, contd.

• A high g.b. energy on boundary 1 decreases the
  corresponding dihedral angle.
• Note that the dihedral angles depend on all the
  energies.

1
g1gtg2g3
3
2
c1lt 120
9
Wetting

• For a large enough ratio, wetting can occur, i.e.
  replacement of one boundary by the other two at
  the TJ.

g1gtg2g3Balance vertical forces ? g1
2g2cos(c1/2) Wetting ? g1 ? 2 g2
g1
1
g2cosc1/2
g3cosc1/2
3
2
c1lt 120
10
Experimental Methods for g.b. energy measurement
G. Gottstein L. Shvindlerman, Grain Boundary
Migration in Metals, CRC (1999)
11
Triple Junction Quantities
12
Triple Junction Quantities

• Grain boundary tangent (at a TJ) b
• Grain boundary normal (at a TJ) n
• Grain boundary inclination, measured
  anti-clockwise with respect to a(n arbitrarily
  chosen) reference direction (at a TJ) f
• Grain boundary dihedral angle c
• Grain orientationg

13
Force Balance Equations/ Herring Equations

• The Herring equations(1951). Surface tension as
  a motivation for sintering. The Physics of Powder
  Metallurgy. New York, McGraw-Hill Book Co.
  143-179 are force balance equations at a TJ.
  They rely on a local equilibrium in terms of free
  energy.
• A virtual displacement, dr, of the TJ (L in the
  figure) results in no change in free energy.

14
Derivation of Herring Equs.
A virtual displacement, dr, of the TJ results in
no change in free energy.
15
Force Balance

• Consider only interfacial energy vector sum of
  the forces must be zero to satisfy equilibrium.
• These equations can be rearranged to give the
  Young equations (sine law)

16
Inclination Dependence

• Interfacial energy can depend on inclination,
  i.e. which crystallographic plane is involved.
• Example? The coherent twin boundary is obviously
  low energy as compared to the incoherent twin
  boundary (e.g. Cu, Ag). The misorientation (60
  about lt111gt) is the same, so inclination is the
  only difference.

17
Twin coherent vs. incoherent

• Porter Easterling fig. 3.12/p123

18
The torque term
Change in inclination causes a change in its
energy,tending to twist it (either back or
forwards)
df
1
19
Inclination Dependence, contd.

• For local equilibrium at a TJ, what matters is
  the rate of change of energy with inclination,
  i.e. the torque on the boundary.
• Recall that the virtual displacement twists each
  boundary, i.e. changes its inclination.
• Re-express the force balance as (s?g)

torque terms
surfacetensionterms
20
Herrings Relations
21
Torque effects

• The effect of inclination seems esoteric should
  one be concerned about it?
• Yes! Twin boundaries are only one example where
  inclination has an obvious effect. Other types
  of grain boundary (to be explored later) also
  have low energies at unique misorientations.
• Torque effects can result in inequalities
  instead of equalities for dihedral angles.

B.L. Adams, et al. (1999). Extracting Grain
Boundary and Surface Energy from Measurement of
Triple Junction Geometry. Interface Science 7
321-337.
22
Aluminum foil, cross section
surface

• Torque term literally twists the boundary away
  from being perpendicular to the surface

23
Why Triple Junctions?

• For isotropic g.b. energy, 4-fold junctions split
  into two 3-fold junctions with a reduction in
  free energy

90
120
24
The n-6 Rule

• The n-6 rule is the rule previously shown
  pictorially that predicts the growth or shrinkage
  of grains (in 2D only) based solely on their
  number of sides/edges. For ngt6, grain grows for
  nlt6, grain shrinks.
• Originally derived for gas bubbles by von Neumann
  (1948) and written up as a discussion on a paper
  by Cyril Stanley Smith (W.W. Mullins advisor).

25
Curvature and Sides on a Grain

• Shrinkage/growth depends on which way the grain
  boundaries migrate, which in turn depends on
  their curvature.
• velocity mobility driving force driving
  force g.b. stiffness curvature v Mf M
  (g g) k
• We can integrate the curvature around the
  perimeter of a grain in order to obtain the net
  change in area of the grain.

26
Integrating inclination angle to obtain curvature

• Curvature rate of change of tangent with arc
  length, s k df/ds
• Integrate around the perimeter (isolated grain
  with no triple junctions), k M g

27
Effect of TJs on curvature

• Each TJ in effect subtracts a finite angle from
  the total turning angle to complete the perimeter
  of a grain

3
1
f1-f3
2
28
Isotropic Case

• In the isotropic case, the turning angle (change
  in inclination angle) is 60.
• For the average grain with ltngt6, the sum of the
  turning angles 6n 360.
• Therefore all the change in direction of the
  perimeter of an n6 grain is accommodated by the
  dihedral angles at the TJs, which means no change
  in area.

29
Isotropy, nlt6, ngt6

• If the number of TJs is less than 6, then not all
  the change in angle is accommodated by the TJs
  and the g.b.s linking the TJs must be curved such
  that their centers of curvature lie inside the
  grain, i.e. shrinkage
• If ngt6, converse occurs and centers of curvature
  lie outside the grain, i.e. growth.
• Final result dA/dt pk/6(n-6)
• Known as the von Neumann-Mullins Law.

30
Test of the n-6 Rule

• Grain growth experiments in a thin film of 2D
  polycrystalline succinonitrile (bcc organic, much
  used for solidification studies) were analyzed by
  Palmer et al.
• Averaging the rate of change of area in each size
  class produced an excellent fit to the (n-6)
  rule.
• Scripta metall. 30, 633-637 (1994).

Note the scatter in dA/dt within each size class
31
Grain Growth

• One interesting feature of grain growth is that,
  in a given material subjected to annealing at the
  same temperature, the only difference between the
  various microstructures is the average grain
  size. Or, expressed another way, the
  microstructures (limited to the description of
  the boundary network) are self-similar and cannot
  be distinguished from one another unless the
  magnification is known. This characteristic of
  grain growth has been shown by Mullins (1986) to
  be related to the kinetics of grain growth. The
  kinetics of grain growth can be deduced in a very
  simple manner based on the available driving
  force.
• Curvature is present in essentially all grain
  boundary networks and statistical self-similarity
  in structure is observed both in experiment and
  simulation. This latter observation is extremely
  useful because it permits an assumption to be
  made that the average curvature in a network is
  inversely proportional to the grain size. In
  other words, provided that self-similarity and
  isotropy hold, the driving force for grain
  boundary migration is inversely proportional to
  grain size.

32
Grain Growth Kinetics

• The rate of change of the mean size, dltrgt/dt,
  must be related to the migration rate of
  boundaries in the system. Thus we have a
  mechanism for grain coarsening (grain growth) and
  a quantitative relationship to a single measure
  of the microstructure. This allows us to write
  the following equations. v ??M ? / r
  dltrgt/dtOne can then integrate and obtain
• ltrgt2 - ltrt0gt2 ??M ?? t
• In this, the constant ? is geometrical factor of
  order unity (to be discussed later). In
  Hillerts theory, ? 0.25. From simulations, ?
  0.40.

33
Experimental grain growth data

• Data from Grey Higgins (1973) for
  zone-refined Pb with Sn additions, showing
  deviations from the ideal grain growth law
  (nlt0.5).
• In general, the grain growth exponent (in terms
  of radius) is often appreciably less than the
  theoretical value of 0.5

34
Grain Growth Theory

• The main objective in grain growth theory is to
  be able to describe both the coarsening rate and
  the grain size distribution with (mathematical)
  functions.
• What is the answer? Unfortunately only a partial
  answer exists and it is not obvious that a unique
  answer is available, especially if realistic
  (anisotropic) boundary properties are included.
• Hillert (1965) adapted particle coarsening theory
  by Lifshitz-Slyozov and Wagner Scripta metall.
  13, 227-238.

35
Hillert Normal Grain Growth Theory

• Coarsening rate ltrgt2 - ltrt0gt2 0.25 k t
  0.25 Mg t
• Grain size distribution (2D), fHere, r
  r/ltrgt.

36
Grain Size Distributions

• (a) Comparison of theoretical distributions due
  to Hillert (dotted line), Louat (dashed) and the
  log-normal (solid) distribution. The histogram
  is taken from the 2D computer simulations of
  Anderson, Srolovitz et al. (b) Histogram showing
  the same computer simulation results compared
  with experimental distributions for Al (solid
  line) by Beck and MgO (dashed) by Aboav and
  Langdon.

37
Development of Hillert Theory

• Where does the solution come from?
• The most basic aspect of any particle coarsening
  theory is that it must satisfy the continuity
  requirement, which simply says that the (time)
  rate of change of the number of particles of a
  given size is the difference between the numbers
  leaving and entering that size class.
• The number entering is the number fraction
  (density), f, in the class below times the rate
  of increase, v. Similarly for the size class
  above. ?f/?t ?/?r(fv)

38
Grain Growth Theory (1)

• Expanding the continuity requirement gives the
  following
• Assuming that a time-invariant (quasi-stationary)
  solution is possible, and transforming the
  equation into terms of the relative size, r
• Clearly, all that is needed is an equation for
  the distribution, f, and the velocity of grains,
  v.

39
Grain Growth Theory (2)

• General theories also must satisfy volume
  conservation
• In this case, the assumption of self-similarity
  allows us to assume a solution for the
  distribution function in terms of r only (and not
  time).

40
Grain Growth Theory (3)

• A critical part of the Hillert theory is the link
  between the n-6 rule and the assumed relationship
  between the rate of change, vdr/dt.
• N-6 rule dr/dt Mg(p/3r)(n-6)
• Hillert dr/dt Mg /21/ltrgt-1/r Mg
  /2ltrgt r - 1
• Note that Hillerts (critical) assumption means
  that there is a linear relationship between size
  and the number of sides n 61 0.5 (r/ltrgt -
  1) 3 1 r

41
Anisotropic grain boundary energy

• If the energies are not isotropic, the dihedral
  angles vary with the nature of the g.b.s making
  up each TJ.
• Changes in dihedral angle affect the turning
  angle.
• See Rollett and Mullins (1996). On the growth
  of abnormal grains. Scripta metall. et mater.
  36(9) 975-980. An explanation of this theory is
  given in the second section of this set of slides.

42
v Mf, revisited

• If the g.b. energy is inclination dependent, then
  equation is modified g.b. energy term includes
  the second derivative. Derivative evaluated
  along directions of principal curvature.

Care required curvatures have sign sign
ofvelocity depends on convention for normal.
43
Sign of Curvature
Porter Easterling, fig. 3.20, p130

• (a) singly curved (b) zero curvature, zero
  force (c) equal principal curvatures, opposite
  signs, zero (net) force.

44
Example of importance of interface stiffness

• The Monte Carlo model is commonly used for
  simulating grain growth and recrystallization.
• It is based on a discrete lattice of points in
  which a boundary is the dividing line between
  points of differing orientation. In effect,
  boundary energy is a broken bond model.
• This means that certain orientations
  (inclinations) of boundaries will have low
  energies because fewer broken bonds per unit
  length are needed.
• This has been analyzed by Karma, Srolovitz and
  others.

45
Broken bond model, 2D
10

• We can estimate the boundary energy by counting
  the lengths of steps and ledges.

46
Interface stiffness

• At the singular point, the second derivative goes
  strongly positive, thereby compensating for the
  low density of defects at that orientation that
  otherwise controls the mobility!

47
How to Measure Dihedral Angles and Curvatures
2D microstructures
Image Processing
(1)
(2) Fit conic sections to each grain boundary
Q(x,y)Ax2 Bxy Cy2 Dx EyF 0
Assume a quadratic curve is adequate to describe
the shape of a grain boundary.
48
Measuring Dihedral Angles and Curvatures
(3) Calculate the tangent angle and curvature at
a triple junction from the fitted conic
function, Q(x,y)
Q(x,y)Ax2 Bxy Cy2 Dx EyF0
49
Application to G.B. Properties

• In principle, one can measure many different
  triple junctions to characterize crystallography,
  dihedral angles and curvature.
• From these measurements one can extract the
  relative properties of the grain boundaries.

50
Energy Extraction
Measurements atmany TJs bin thedihedral angles
by g.b. type average the sinceach TJ gives a
pair of equations
D. Kinderlehrer, et al. , Proc. of the Twelfth
International Conference on Textures of
Materials, Montréal, Canada, (1999) 1643.
51
Mobility Extraction
(?1?1sin?1)m1 (?2?2sin?2)m2 (?3?3sin?3)m3 0
m1 m2 m3 ? mn
?1?1sin?1 ?2?2sin?2 ?3?3sin?3 0 0 0
0
0 ...0 0
0 ...0 ?
? ? ? ? ?
0 0
0
0
52
Summary (1)

• Force balance at triple junctions leads to the
  Herring equations. These include both surface
  tension and torque terms.
• If the interfacial energy does not depend on
  inclination, the torque terms are zero and
  Herring equations reduce to the Young equations,
  also known as the sine law.
• In 2D, the curvature of a grain boundary can be
  integrated to obtain the n-6 rule that predicts
  the growth (shrinkage) of a grain.
• Normal grain growth is associated with
  self-similarity of the evolving structures which
  in turn requires the area to be linear in time.
• Hillert extended particle coarsening theory to
  predict a stable grain size distribution and
  coarsening rate.

53
Extensions of Herring Equations and Grain Growth
Theory

• The next two sections explain the use of the
  capillarity vector and an extension of grain
  growth theory to the situation of anisotropic
  grain boundary properties.

54
Capillarity Vector

• The capillarity vector is a convenient quantity
  to use in force balances at junctions of
  surfaces.
• It is derived from the variation in (excess free)
  energy of a surface.
• In effect, the capillarity vector combines both
  the surface tension (so-called) and the torque
  terms into a single quantity

55
Equilibrium at TJ

• The utility of the capillarity vector, x, can be
  illustrated by re-writing Herrings equations as
  follows, where l123 is the triple line (tangent)
  vector. (x1 x2 x3) x l123 0
• Note that the cross product with the TJ tangent
  implies resolution of forces perpendicular to the
  TJ.

56
Capillarity vector definition

• Following Hoffman Cahn 1972, A vector
  thermodynamics for anisotropic surfaces. I.
  Fundamentals and application to plane surface
  junctions. Surface Science 31 368-388., define
  a unit surface normal vector to the surface,
  , and a scalar field, rg( ), where r is a
  radius from the origin. Typically, the normal is
  defined w.r.t. crystal axes.

57

• Definition x grad(rg)
• From which, d(rg) grad(rg) dr (1)
• Giving, d(rg) x(rd dr)
• Compare with the rule for products d(rg)
  rdg gdrgives x g (2), and,
  xd dg (3)
• Combining total derivative of (2) and (3) dg-
  dg xd dx - xd 0 dx
  (4)

58

• The physical consequence of Eq (2) is that the
  component of x that is normal to the associated
  surface, xn, is equal to the surface energy,
  g. xn g
• Can also define a tangential component of the
  vector, xt, that is parallel to the surface
  xt x - g (?g/?q)maxwhere the tangent
  vector is associated with the maximum rate of
  change of energy.

59
Young Equns, with Torques

• Contrast the capillarity vector expression with
  the expanded Young eqns.

60
Expanded Young Equations

• Project the force balance along each grain
  boundary normal in turn, so as to eliminate one
  tangent term at a time

61
Abnormal Grain Growth
62
Objective

• The objective of this section of the lecture is
  to introduce the student to some theory
  concerning abnormal grain growth.
• The theory describes the circumstances under
  which one can expect abnormal grain growth to
  occur.
• It is applicable mainly to metals and other
  single-phase materials (metals).
• Many technological materials have a liquid phase
  present at grain growth (sintering) temperatures.

63
Assumptions

• The theory makes a number of simplifying
  assumptions.
• Only 2D grain growth is treated.
• Grain boundary properties are assumed to be
  constant (uniform) everywhere except for the
  boundaries between certain special grains (type
  A) and grains in the matrix (type B).
• The special grains share the same energy and
  mobility on their boundaries.
• The special grains are isolated (no A-A
  boundaries).

64
Philosophy of Approach

• This development is based on Rollett, A. D. and
  W. W. Mullins (1996). On the growth of abnormal
  grains. Scripta metall. et mater. 36(9)
  975-980.
• Analysis confined to 2D because it is possible to
  integrate the curvature around the perimeter of a
  grain and relate it to the rate of change of
  area not possible in 3D.

65
Microstructure
turning anglez

• Each A grain is surrounded by B grains.
• We are interested in whether the A grain grows
  faster than the B grains during growth.
• 4-sided example should shrink.

Figure 1.
66
Parameters

• GgAB/gBB in which the g's are the boundary
  energies on the AB and BB boundaries,
  respectively.
• µ MAB/MBB in which the Ms are the boundary
  mobilities.
• A area of a grain n number of sides of a
  grain.
• S arc length bangle made by directed tangent
  to line relative to a fixed reference direction.
• Define a function a, of the energy ratio

67
Turning angle at a vertex

• Fig. 1 shows an irregular 4-sided A grain
  surrounded by B grains. The vertices are assumed
  to be in equilibrium. If torque terms are
  neglected, the turning angle z of the tangent to
  the AB boundary at a vertex is then given by
• Note the limiting case of wetting of the matrix
  grain boundaries by the A grain occurs at G1/2
  or zp.

68
dA/dt vs. curvature

• We now derive a relationship between the rate of
  change of area of the A grain, its number of
  sides, and the equilibrium angles at each vertex.
  The curvature rule for a two dimensional grain
  may be writtenwhere v is the outward
  velocity in the normal direction, and db/ds is
  the curvature.

69
Integrate around the grain

• If Eq. 2 is integrated around the A grain with
  n sides of fixed MAB and gAB, the left hand side
  just gives the rate of change of area A of the
  grain and on the right, the integral of db/ds
  gives 2p-nz, since there is a discontinuity of
  the AB tangent angle of z at each of the n
  vertices as illustrated in Fig. 1.

70
The n-6 rule

• The previous equation shows that the rate of area
  change is constant if all boundaries are
  equivalent, as they are for a B grain
  surrounded by B grains in the matrix, then G1,
  a1 and the equation reduces to the n-6 rule
  for the B matrix grains

71
Grow or Shrink?

• The equation shows that the A grain will grow if
  ngtn6/a and shrink if the reverse is true. Fig. 2
  shows a plot of v as a function of G.

72
Grow or Shrink? Contd.

• If ngtn the A grain grows and if nltn it shrinks.
  Note if Glt1/v3, the A grain grows for any number
  of sides n3, which corresponds to the wetting
  criterion.
• We now turn to the comparison of the ultimate
  size of the A grain to that of the B grains. For
  this purpose, following Thompson et al. and
  Rollett et al., we study the time dependence of
  the ratio rRA/ltRBgt where the R's are
  area-equivalent radii.

Thompson, C., H. Frost, et al. (1987). The
relative rates of secondary and normal grain
growth. Acta metallurgica 35 887-890.
Rollett, A. D., D. J. Srolovitz, et al. (1989).
Simulation and Theory of Abnormal Grain Growth-
Variable Grain Boundary Energies and
Mobilities. Acta Metall. 37 2127.
73
Growth of A grains

• Using the rule for differentiating a quotient we
  have where simply means that
  is averaged over (isolated) A grains of the same
  RA but otherwise possibly of different shape and
  n, and similarly for .

74
Growth of A grains, contd.

• To evaluate the first derivative in the brackets
  of the previous equation, we combine Eq. 3 and
  the expression ApRA2 to get For ltnRAgt, the
  average number of sides of A grains with a fixed
  RA, we use the expression ltnRAgt 3r
  3which is linear in the size of the A grain and
  gives the limit 3 as RA becomes very small it
  approximates the number of circular B grains that
  would fit on the circumference of the A grain

75
Growth of A grains, contd.

• Combining these equations
• Now we need something to predict the coarsening
  rate in the matrix (the B grains), for which we
  turn to Hillerts classical grain growth theory.

76
Hillerts theory

• Assume that the presence of A grains does perturb
  the B matrix the theory may be obtained by
  combining two ingredients the n-6 rule,
  expressed in area equivalent radius form and
  averaged over all B grains of radius RBand
  the linear relation sides(size)

77
Hillerts theory, contd.

• Abbruzzese et al. found that this relationship
  is experimentally supported. It also satisfies
  the Euler requirement that ltngt6. Combining Eqs.
  11 and 12 with standard coarsening theory, one
  obtains the Hillert result

Abbruzzese, G., I. Heckelmann, et al. (1991).
Topological foundation and kinetics of texture
controlled grain growth. Textures and
Microstructures 14-18 659-666. Courant, R.
and H. E. Robbins (1941). What is Mathematics,
Oxford University Press.
78
Abnormal Growth main result

• Combining equations gives the following

The sign of G is the same as that of
79
Finding a limiting (abnormal) size

• Think of the result as a quadratic equation and
  consider which way the relative size is changing
  for there to be a fixed relative size (r), there
  must be a restoring force.
• Thus, for a stable r to exist, corresponding to a
  fixed ratio of the A to B grain size as growth
  proceeds, there must be a positive root r of the
  quadratic equation G0 at which ?G/?rlt0, so that
  the sign of
• The mathematical condition for real roots of G0
  to exist is

80
The positive root max. size

• Physically m must also be positive. If condition
  15 is not satisfied, Glt0 for all r (the Eq. above
  shows Glt0 for large r) so that under these
  conditions the A grain would shrink and
  disappear. If the condition above is satisfied,
  it is easily shown that the upper root, given by
  is then the stable one. If 1/2ltG1/v3 (3gta2),
  then Ggt0 for any (positive) rltr so that any A
  grain in this range will grow toward the stable
  r.

81
Minimum size for abnormal growth

• If, however, Ggt1/v3 (alt2), the initial value of r
  must lie in the range r- ltr lt r for growth
  toward r to occur, where r- is given byif
  rltr-, Glt0 so shrinkage and disappearance of the A
  grain would occur. In the special case that the
  condition above is an equality, which represents
  a double-root curve in the (m, G) plane (e.g.
  mGa1 as discussed by Thompson et al.), then
  . In this
  case G is never positive so grain A would again
  ultimately disappear.

82
Details

• The case of alt2 (Ggt1/v3) is illustrated in the
  following figure which shows a plot of G as a
  function of r for m2 and several values of G.
  There are two positive roots of G0. The effect
  of decreasing G is to expand the range over which
  G is positive. The meaning of Ggt0 or
  gt0 is that the average growth rate of the A grain
  will exceed that of the mean size of the matrix
  so that the expected value of r will increase
  until it reaches the stable upper root of G0.
  Note that this analysis addresses only the
  average value of . If the range of r over
  which Ggt0 is small then fluctuations in the
  growth rate, due for example, to fluctuations in
  the number of neighbors, may cause to become
  negative for long enough to cause A to disappear.

83
Asymptotes

• Note also that one only expects the relative size
  to approach the asymptote rather slowly under
  experimental conditions. If the range of r over
  which Ggt0 is small then fluctuations in the
  growth rate, due for example, to fluctuations in
  the number of neighbors, may cause to become
  negative for long enough to cause A to disappear.
  Note also that one only expects the relative size
  to approach the asymptote rather slowly under
  experimental conditions.

84
Relative growth rate - plot
Figure 3. Plot of relative growth rate versus
relative size for µ2 with four different values
for the ratio of grain boundary energies, G0.7,
1.0 (solid line), 1.1 and 1.3. Note that only
the curve for G1.3 shows both upper and lower
roots (r0) but that the other three cases also
have upper roots that increase with decreasing G.
The results predict abnormal grain growth (Ggt0)
over a range of relative size that decreases with
increasing boundary energy ratio.
85
Contours in the (µ,G) plane

• The next figure shows the (m,G) plane with
  regions of r (stable root) delineated no real
  roots exist in the upper left triangle whereas
  two roots exist in the lower right triangle
  (Ggt1/v3). Over the range 0.5ltG1/v3, only one
  positive root exists (r- is negative), and for
  G0.5, wetting occurs. Several contours of
  constant relative size are shown these were
  calculated by setting G0 in Eq. 14a, and solving
  for µ with fixed values of r. Also shown is the
  curve (thick line) along which the upper and
  lower roots are equal this was calculated from
  Eq. 16 with an equality. Each contour of
  constant r touches this line at a point for which
  rr- for µ and G less than this point, the
  contour describes the branch for r conversely,
  for µ and G larger than this point, the contour
  describes the branch for r-. In the region of two
  positive roots, each µ,G pair is intersected by
  two contours the difference between r and r-
  defines the range of relative size over which
  abnormal growth is likely to occur. Eq. numbers
  refer to the Scripta metall. Paper

86
Contours in the (µ,G) plane plot
Figure 4. The m, G plane with regions of stable r
delineated. For Glt0.5, wetting occurs for
0.5G1/v3, only one positive root exists below
the double root curve (dotted line) and for
Ggt1/v3 two positive roots exist. Contours of
constant r (solid curves) and r- (dashed curves)
have been drawn the difference between the r
and r- values at any point in the two root region
defines the range of relative sizes over which
abnormal growth can occur.
87
Abn. Gr. Gr. Conclusions

• Abnormal growth is promoted by high mobility of
  the abnormal grain relative to the matrix.
• Abnormal growth is constrained by high energy of
  the abnormal grain.
• A combination of high mobility and high energy
  could occur in a sub-grain structure with a few
  misoriented subgrains.
• Complex dependence of abnormal growth revealed by
  contours in the (µ,G) plane.

88
Summary (2)

• The capillarity vector allows the force balance
  at a triple junction to be expressed more
  compactly and elegantly.
• It is important to remember that the Herring
  equations become inequalities if the inclination
  dependence (torque terms) are too strong.