Geoscience Reference
In-Depth Information
10
2
10
2
1/
φ
mixture model 1
10
1
10
1
mixture model 2
m
ix
tu
re
m
ode
l 1 (Sim
pson
et al
.)
10
0
10
0
mixture model 2 (Takei)
contiguity model
10
−
1
10
−
1
contiguity model
l=
33.7
l=
30
l=
25
10
−
2
10
−
2
0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
Melt fraction,
φ
Melt fraction,
φ
(a)
(b)
Fig. 3.10
(a) Shear viscosity
η
and (b) bulk viscosity
ξ
of the solid framework versus melt fraction
φ
.Both
η
and
ξ
are normalized to the shear viscosity of a melt-free system
0). Theoretical predictions from the contiguity
model (thick line) and the two mixture models (thin line, Simpson
et al
., 2010a,b; thin dotted line, Takei, 1998a)
differ. Slopes labeled
η
(
φ
=
λ
show the experimentally obtained dependence of
η
and
ξ
on
φ
.
slightly smaller and
ξ
significantly larger than
the singularity of
0, predicted by the
contiguity model, was actually observed in the
experimental studies. However, the observed
jumps (1-2 orders in Faul & Jackson, 2007, and
by a factor of 40 in McCarthy and Takei, 2011)
are much larger than the theoretical prediction (a
factor of 5). The difference may be explained by
considering the following ''chemical effect.'' In
the contiguity model, grain boundary diffusivity
is assumed to be the same in melt-free and
melt-bearing systems. However, in real material,
the melt-bearing and melt-free samples have
different chemical compositions, possibly having
a significant effect on grain boundary chemistry
and causing a significant increase in the grain
boundary diffusivity (Hiraga
et al
., 2004).
Figure 3.10b shows that the bulk viscosity
η
near
φ
=
η
(
φ
=
0), the dependence of
η
on
φ
is small, and
ξ/η
=
φ
−
1
and
.On
the other hand, the contiguity model predicts
ξ/η >>
1 at small values of
φ
η
and
ξ
to be much smaller than
η
(
φ
=
0), a large
dependence of
η
on
φ
, and similar behavior of
η
and
85 (Equation 3.20). Because
the contiguity model and the mixture model 2
are based on exactly the same microstructural
model, the differences between them come from
the different deformation mechanisms assumed
at the microscopic scale.
Also shown in Figure 3.10 are the slopes
ξ
given by
ξ/η
=
1
.
of
measured in the diffusion creep regime.
Figure 3.10a shows that the contiguity model can
explain the steep slope but that neither of the
mixture models can. The strong dependence of
η
λ
,
predicted by the contiguity model, is more than
2-3 orders of magnitude smaller than
ξ
on
φ
predicted by the contiguity model comes
2
(Equation 3.20), which was first
predicted by Cooper
et al
. (1989) (CK-model).
The contiguity model, which is an extension of
the CK-model, can reproduce
from
η
sk
∝
ϕ
predicted
by the mixture models. In numerical studies of
melt segregation dynamics, the value of
ξ
ξ
from a
2
as well as
η
sk
∝
ϕ
mixture model (
) has been used
(Sumita
et al
., 1996; Katz, 2008). Because the use
of
ξ/η
(
φ
=
0)
∼
1
/φ
predicting the bulk viscosity
ξ
, and the singular
behavior of
η
and
ξ
near
φ
=
0. As stated above,
ξ
from the contiguity model can significantly
