ture at the base of the grain-boundary groove.
δsDsγ Ω
B=
(3)
The stress along the grain boundary is easily de-
kT
rived with that end condition and requires that
KY equal sin(A/2) at the base of the groove when
where δ s = the surface diffusive width,
equilibrium is established and the flux disappears.
Ds = the coefficient of surface diffusion
Thus, the theory requires that equilibrium can
γ = the surface free energy for the solid
only be achieved when the curvature is the same
vapor surface
everywhere. This does ignore crystallographic dif-
Ω = the atomic volume
ferences between the two grains, differences that
k = Boltzmann's constant.
clearly arise when two ice grains are observed to
sinter, but for now we will have to accept this
The surface flux is then taken as
limitation.
When material is removed from the grain
δsDsγ
K.
Js =
boundary by grain-boundary diffusion and moves
(4)
kT
s
onto the free surfaces of the two grains by surface
Zhang and Schneibel (1995) assumed a straight
diffusion, the fluxes are in balance when, at the
grain boundary from which parallel layers of mat-
base of the groove,
ter are removed during sintering. These molecules
J b = 2 Js .
(6)
diffuse away by flux along the grain boundary, a
flux that arises from the stress gradient along the
Zhang and Schneibel (1995) used this and two
boundary. The flux is given by
other conditions to describe the changing geom-
etry as two particles grow in size and move closer.
δ D γ dσ
Jb = b b
They do so until the two grains achieve the final
(5)
kT
dy
condition, capped spheres, as shown in Figure 10.
where the subscript b refers to the grain bound-
The two other conditions are that the dihedral
ary, σ is the normal stress acting across the grain
angle remain constant and that the base of the
boundary, and y is the radial distance across the
groove move outward, as shown in Figure 10.
boundary. The normal stress at the base of the
In their numerical calculations they used the
grain-boundary groove due to the pull of the
ratio of the grain boundary and surface diffusiv-
vaporice surfaces is γ K0, where K0 is the curva-
ities, Γ, where
Dihedral Angle
Early
Shape
Shage
Final
Figure 10. Two idealized particles
Shape
shown at an early and the final
stages of sintering. The dihedral
angle is maintained at a constant
value throughout sintering in this
example. However, it appears that
the dihedral angle changes during
sintering of ice grains. (After Zhang
and Schneibel 1995.)
8
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