A Mechanistic Model for Mud Flocculati
o
n in Freshwater Rivers
J
.
A.
Nghiem
1
, W. W. Fischer
1
,
G.
K. Li
1
†
, M. P. Lamb
1
1
Division of Geological and Planetary Sciences, California Institute of Technology
Correspond
ing
author:
Justin
A.
Nghiem
(
jnghiem@caltech.edu)
†
Current address: Department of Earth Science
, University of Cal
ifornia, Santa Barbara
Key Points:
Turbulence, sediment concentration and mineralogy, water chemistry, and organics affect
river floc
size
and settling velocity.
Controlling variabl
es are incorporated into a
model for mud floc diameter
and settling
velocity in rivers
calibrated on a global dataset
.
Mud flocs in rivers can respond to river hydraulics and biogeochemistry with
implications for the carbon cycle and fluvial morphodynamics.
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. Please cite this article as
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.
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Abstract
The transport and deposition of mud in riv
ers are key processes in fluvial geomorphology and
biogeochemical
cycles
. Recent work indicate
s
that flocculation might regulate fluvial mud
transport by increasing mud settling velocities, but we lack a calibrated mechanistic model for
flocculation in fre
shwater rivers. Here, we developed
and calibrated
a semi
-
empirical model for
floc
diameter
and settling velocity in rivers. We compiled a global dataset of
river
suspended
sediment concentration
-
depth profiles
and
invert
ed
them
for
in situ
settling velocity using the
Rouse
-
Vanoni equation
.
On average,
clay
and silt
(
diameter
s
< 39
μ
m) a
re
flocculated with
settling velocit
y
of
1.
8
mm s
-
1
and floc
diameter
of
13
0
μm
.
Among
model variables
,
Kolmogorov microscale has the strongest positive correlation with floc
diameter
, supporting the
idea that turbulent shear limits floc size. Sediment Al/Si
(
a mineralogy proxy
)
has the strongest
negative correlation with floc diameter and settling velocity,
indicating the importance of clay
abundance and composition
for
flocculation. Floc
settling velocity
increases with greater mud
and organic matter concentrations
,
consistent with flocculation driven by particle collisions and
binding by organic matter
whi
ch is often concentrated in mud
. Relative charge density (a salinity
proxy) correlates with smaller floc settling velocities,
a finding that
m
ight reflect the primary
particle size distribution and
physical
hosting of organic matter
.
T
he calibrated model e
xplains
river floc settling velocity data
within a factor of about
two
. Results highlight that flocculation
can impact the fate of
mud and
particulate
organic carbon
, holding implications for global
biogeochemical cyc
l
es
.
Plain Language Summary
The fate of fine sediment in rivers is important
for
understanding contaminant
dispersal
, organic
carbon
burial
, and
the
construction of
river floodplains
and deltas
.
I
ndividual grains of silt and
cl
ay dispersed in water settle under the pull of gravity at
extremely slow rates. However, in
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natural rivers, these
mud
particles can aggregate together
into
larger structures called
flocs
,
resulting in far faster settling rates. Here, w
e built on prior work from estuar
i
e
s to develop a
settling velocity
model for
flocculated mud
in freshwater rivers. Our results demonstrate that
mud
settling velocity
increase
s
in rivers with
less
vigorous turbulence
because turbulence can
break flocs apart
.
Mud s
ettling velocity also increase
s
with
greater concentrations of
mud and
particulate
organic matter
, which promote particle collisions and binding.
Cou
n
terintuitively,
settling velocity decrease
s
with greater clay abundance and greater river water salinity, possibly
due to how the
y
affect
organic matter in binding
mud particle
s into
flocs.
Our results improve
understanding of floc
behavior in rivers
and indicate potential links between the routing of mud
and organic matter, river geomorphology, and global climate
.
1 Introduction
Mud (grain diameter,
D
< 62.5 μm) dominates the sediment load carried by rivers globally (e.g.,
Baronas et al., 2020; Lupker et al., 2011) and its fate is important for
our understanding of
fluvial
geomorphology and biogeochemical cycling. For example, mud
-
rich fluvial deposits
are a major
component of the rock record (Aller, 1998; McMahon and Davies, 2018; Zeichner et al., 2021).
Mud cohesion increases bank strength in alluvial rivers
,
affect
ing
river morphodynamics (
e.g,
Dunne and Jerolmack, 2020;
Kleinhans et al., 2018;
Lapôt
re et al., 2019
;
Millar and Quick,
1998
).
M
ud
i
s
also
a primary carrier of organic carbon and pollutants
because of its high specific
surface area
(
e.g.,
France
-
Lanord and Derry, 1997; Galy et al., 2015; Pizzuto et al., 2014).
Despite its importance, we la
ck well
-
tested mechanistic models for mud transport in rivers.
M
ud
in rivers
has traditionally been treated as washload
,
or
sediment
that is
too fine to
regularly
settle to and interact with the
river
bed (Church, 2006; Garcia, 2008
).
In contrast, recent
work suggest
s
that flocculation
—
the
aggregation of particles
into composite structures called
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flocs
—
can enhance
mud
settling velocities
and
drastically
affect mud
transp
ort dynamics
in
rivers
(Bouchez et al., 2011b;
Lamb et al., 2020;
Zeichner et al., 2021
).
S
imilar to sand,
flocculated
mud might be in a dynamic interchange between the
flow and
bed mat
erial
(Lamb et
al., 2020)
.
Mud flocculation has been well
-
studied in estuarine and marine systems where flocs
form in part because
salinity
promotes van der W
aals attraction
between particles
(
e.g.,
Hill et
al., 2000;
Mehta and Partheniades, 1975
;
Winterwerp, 2002;
Fig. 1
). In addition, flow turbulence,
sediment concentration, organic matter concentration, and clay mineralogy are important for
estuarine and
marine flocculation (
e.g.,
Kranck and Milligan, 1980; Meade, 1972; Verney et al.,
2009). In contrast to the wealth of studies on flocculation in saline environments,
knowledge
on
flocculation in freshwater rivers
is
relatively
limited
(
e.g.,
Bungartz and W
anner, 2004; Droppo
and Ongley, 1994; Droppo et al., 1997).
S
tudies in rivers identified flow characteristics, organic matter concentration, and
suspended sediment concentration as
potential controls on
floc size, settling velocity, and
strength
(Fig. 1)
.
Through microscopy of samples from Canadian rivers
,
Droppo and Ongley
(1994)
observed organic matri
c
es
b
inding
together mineral sediment
into flocs
.
They observed
correlations
between
floc size and suspended sediment concentration, attached bacteria count,
and particulate organic carbon concentration. Bungartz et al. (2006) characterized floc setting
velocities at three transects along a lake outlet and found faster
-
settling flocs at h
igh
er
discharge
,
a result
t
hey
attributed
to
faster
floc
gro
wth at higher flow turbulence
. They also showed that
settling patterns of suspended sediment and particulate organic carbon were similar, supporting
the idea that flocculation controlled transport
of both mineral sediment and organic carbon.
Gerbersdorf et al. (2008) examined bed material
composition
in the Neckar River, Germany, and
identified rich networks of microbe
-
derived extracellular polymeric substances (EPS). They
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found positive correlatio
ns between concentrations of EPS moieties and the critical shear stress
for erosion, indicating that EPS
can
help
stabilize bed sediment.
Lamb et al. (2020) use
d
a
field
data
compilation
to
infer
the presence of
widespread
mud flocculation in rivers. They
showed
that
in situ
particle settling velocity can be inferred by fitting the Rouse
-
Vanoni equation to grain
size
-
specific suspended sediment concentration
-
depth profiles. However,
they
did not explain the
order
-
of
-
magnitude variation in the inferred floc
settling velocities.
Experiments have also
supported
organic matter
, dissolved species, and sediment
concentration
as important control
s
on freshwater flocculation
(Fig. 1)
. Chase (1979) showed
that
the presence of
organics
increased floc settling velocity
,
a result
attributed to the interaction
of sediment surface coatings, organic chemistry, and dissolved solutes.
Subsequent experiments
showed that s
ediment concentration positively correlated with floc size while fluid shear rate
affected floc size and se
ttling velocity differently (
e.g.,
Burban et al., 1990; Tsai et al., 1987).
More recent e
xperiments
examining the role of organics on flocculation
in freshwater
highlight
ed
the
importance
of nutrient
s, biomass,
and organic matter
composition
on floc size
and settling velocity (
Furukawa et al., 2014; Lee et al., 2017;
Lee et al., 2019; Tang and Maggi,
2016
; Zeichner et al., 2021
)
.
For instance,
Zeichner et al. (2021) showed in experiments
modeled
after
rivers
that organic matter increased clay
floc settling velocities by up to three orders of
magnitude,
depending
on organic matter type and clay mineralogy.
Process
-
based flocculation theory is required to
link
field studies and experiments
into a
coherent framework.
F
loc population balance model
s
use
particle aggregation and breakage
kernels
,
and have been successful at reproducing floc size distributions (e.g., Lick and Lick,
1988
;
Spicer and Pratsinis, 1996
; Xu et al., 2008
). These studies
show
ed
that sediment
concentration
and fluid shear
enhance floc aggregation by increasing particle collision
frequency
,
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but
greater shear
cause
s
floc breakage
(Fig. 1)
. Winterwerp (1998)
introduced a simplified
model
(hereafter, the Winterwerp model)
tracking
a characteristic floc
diameter
(e.g.,
the
media
n)
,
making it more easily
coupled to hydrodynamic models
(e.g., Maggi, 2008; Son and Hsu, 2011;
Winterwerp, 2002).
The Winterwerp model includes the effects of fluid shear and sediment
concentration,
but
subsumes other factors into coefficients
of the
aggregation and breakage rates.
T
he model
d
escribe
s
well the equilibrium s
ize
of flocculated estuarine mud
(Winterwerp, 1998)
and flocs in
saline laboratory experiments (e.g., Kuprenas et al., 2018; Maggi, 2009; Son and
Hsu, 2008).
However, t
hese models ha
ve yet to be compared or adapted to freshwater rivers.
Here, we built on the
Winterwerp
approach
to develop a semi
-
empirical process
-
based
model for
mud
flocculation in
freshwater
rivers.
First, w
e proposed new forms for flocculation
efficiency coefficient
s to explicitly cast floc diameter and settling velocity as functions of
physicochemical variables that prior work
has
show
n
are important
for flocculation
in
freshwater
: turbulence, sediment concentration, sediment mineralogy, organic matter
concentration
, and dissolved ion concentration (Fig. 1).
Next
, we calibrated
the new
model
against field data. W
e compiled a global dataset of river grain size
-
specific suspended sediment
concentration
-
depth profiles and inverted them for
in situ
settling velocity
using the Rouse
-
Vanoni equation
(
Lamb et al., 2020
).
T
ogether with a river
geo
chemistry data compilation, we
fitted
the
model
to help explain the variance in floc settling velocities
. Finally, the results
are
discussed
in the conte
xt of
fluvial
geomorphol
ogy, organic carbon, tectonics, and climate.
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Figure 1.
Schematic
of a
cross
-
section
through a river water column
illustrating
p
hysicochemical
processes
operating
at different scales
that could
be
important
for mud
flocculation
in rivers
.
Key
variables
are
turbulence (
Ko
lmo
gorov microscale,
η
)
,
volumetric
mud
concentration
(
C
m
)
, sediment mineralogy
(
molar
Al/Si
of river suspended sediment
)
,
organic matter concentration (
fraction of sediment surface covered by organic matter,
θ
), and
dissolved species concentrations (
relative charge density of river water
,
Φ
)
.
These variables
affect the diameter,
D
f
, and settling velocity,
w
s
,floc
, of flocs composed of primary particles
with diameter
D
p
.
2 Model Development
2.1 Winterwerp Model
Winterwerp (1998) proposed a flocculation model in which fluid shear drives particle collision
s
and floc aggregation and
breakage. The model casts the time rate of change of floc diameter,
D
f
,
(or median
D
f
for a floc size distribution) as the difference
of
floc aggregation and breakage
rates:
d
퐷
푓
d
푡
=
푘
퐴
푛
푓
휂
2
휈퐶
퐷
푓
(
퐷
푓
퐷
푝
)
3
−
푛
푓
−
푘
퐵
푛
푓
휂
2
휈
퐷
푓
(
퐷
푓
−
퐷
푝
퐷
푝
)
3
−
푛
푓
(
휏
푡
휏
푦
)
푗
(
1
)
On the right
-
hand side of E
quation (1), t
he first term is the floc aggregation rate
, scaled by
the
aggregation efficienc
y
,
k
A
(dimensionless)
,
and the second term is the floc breakage rate
, scaled
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by the breakage efficiency
,
k
B
(dimensionless)
. The shear rate,
G
(s
-
1
),
quantifies
fluid mixing
and rela
t
e
s
to the smallest turbulence length scale
—
the Kolmogorov microscale,
휂
=
√
휈
퐺
⁄
(m), where
ν
is the fluid kinematic viscosity (m
2
s
-
1
) (Tennekes and Lumley, 1972).
Greater f
luid
mixing
and volumetric sediment concentration,
C
(volume sediment/total volume;
dimensionless),
d
rive
more frequent collisions of
primary particles wit
h diameter
D
p
(m)
and
thereby
increase aggregation rate (Fig. 1).
Flocs break up if fluid shear is too high relative to floc strength, an effect that
Winterwerp (1998) expressed in
E
quation (1) using the ratio of fluid stress on the floc,
휏
푡
=
휌
(
휈
휂
⁄
)
2
(Pa), and floc strength,
휏
푦
=
퐹
푦
퐷
푓
2
⁄
(Pa), where
ρ
is fluid density (kg m
-
3
)
.
F
y
is floc
yield strength (in terms of force)
and has been
estimated
to be of order 10
-
10
N (Matsuo and
Unno, 1981).
F
loc fractal dimension,
푛
푓
∈
[
1
,
3
]
(dimensionless),
descri
bes floc structure
assum
ing it
is approximately self
-
similar (Kranenburg, 1994). Floc structure
can
var
y
from a
linear string of particles (
푛
푓
=
1
) to a solid, compact particle (
푛
푓
=
3
). An average
푛
푓
=
2
is
typical
for natural flocs (e.g., Tambo and
Watanabe, 1979; Winterwerp, 1998).
In practice,
n
f
describes the relationship between floc diameter and floc density by
푅
푓
푅
푠
⁄
=
(
퐷
푓
퐷
푝
⁄
)
푛
푓
−
3
where
R
f
is the floc submerged specific gravity (dimensionless) and
R
s
is the submerged specific
gravity of the
primary particle sediment (dimensionless
) (
Kranenburg, 1994).
Although the
parameter
j
in
E
quation (1)
is
an empirical constant, Winterwerp (1998)
used
푗
=
1
/
2
to ensure
that floc settling velocity, floc diameter, and sediment concentration are linearly re
lated to each
other based on estuarine floc data. We retained
j
as a fit parameter to maintain generality.
2.2 Modifications to the Winterwerp
M
odel for river flocs
We proposed changes to floc strength, and floc aggregation and breakage
efficiencies
to
adapt
the Winterwerp model to rivers
.
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2.2.1 Floc strength
Experiments in freshwater have shown that, for constant
D
f
, floc settling velocity,
w
s
,floc,
increases with larger mixing rate due to an increase in floc density (Burban et al., 1990). This
behavior
suggests that flow conditions during floc formation can affect floc strength, where more
porous and lighter flocs are weaker because they have fewer
inter
particle contacts and
vice
versa
.
Bache (2004) proposed that floc strength,
휏
푦
, is
a balance of loc
al turbulent kinetic energy per
unit volume acting on the floc and the energy per unit volume required to rupture the floc:
휏
푦
=
휌
30
(
휈
휂
)
2
(
퐷
푓
휂
)
2
(
2
)
T
he power
-
law
form
of Equation (2)
holds in general but the
numerical
constants
apply for small
퐷
푓
휂
⁄
(Bache, 2004).
2.2.2 Floc aggregation and breakage efficiencies
In
the Winterwerp
model
,
all co
ntributions to flocculation outside of
fluid shear and sediment
concentration are captured in the
constant
floc aggregation and breakage
efficiency terms,
k
A
and
k
B
, respectively
.
We investigated whether
k
A
and
k
B
in rivers
depend on organic matter
concentration, sediment mineralogy, and dissolved ion concentration, as functions rather than fit
constants.
Organic matter can adsorb onto sediment surfaces and
form connective “bridges”
between grains (Ruehrwein and Ward, 1952; Smellie and La Mer, 1958; Molski, 1989; Fig. 1).
In rive
rs, biogenic molecules like EPS
can act as sticky media for bridging flocculation (Droppo
and Ongley, 1994; Gerbersdorf et al., 2008
; Larsen et al., 2009; Lee et al., 2019). Smellie and La
Mer (1958) proposed a functional form of bridging flocculation efficiency,
푘
퐴
,
푘
퐵
−
1
∝
휃
(
1
−
휃
)
(
3
)
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in which
θ
is the fraction of the sediment surface covered by
a polymeric substance. We used
E
quati
on (3) and calculated
θ
for organic matter
(
Section 3.3
)
.
We accounted for sediment mineralogy using the molar elemental ratio Al/Si as a proxy
variable (Fig. 1). More intensely weathered
rocks
typically
generate
sediment with larger Al/Si
because chemical
weathering produces
Al
-
rich
clay minerals (e.g., Ito and Wagai, 2017; Jackson
et al., 1948; Lupker et al., 2012).
M
ineralogy can affect
flocculation
because
it
determines the
range of potential chemical interactions between
particles through cation
exchange capacity
(CEC)
and therefore the ability to attract cations in solution (Mehta and McAnally, 2008).
Furthermore, c
ations
can affect the ability of organic matter to adsorb to
particle
surfaces and the
physical orientation of
adsorbed
organic matte
r (Galy et al., 2008; Mehta and McAnally, 2008).
We used a simple power law model as a starting point,
푘
퐴
∝
(
Al/Si
)
퐴
1
(
4
)
푘
퐵
∝
(
Al/Si
)
퐵
1
(
5
)
where
A
1
and
B
1
are
dimensionless
fit constants.
Dissolved ions in river water
might
promote flocculation throu
gh the same mechanism as
salinity
by
boost
ing
the effectiveness of van der Waals attraction
between particles
(
e.g.,
Seiphoori et al., 2021; Fig. 1).
To express ionic effects, w
e used a dimensionless parameter, Φ,
to
quantify the relative densities of
charges in solution and on the sediment (Rommelfanger et al.,
2020):
Φ
=
휆퐼
CEC
휌
푠
퐿
2
⁄
(
6
)
in which the Debye length,
λ
(m), is the average length from the particle in which an electrostatic
effect from the charged surface is sustained,
I
is the solution
ionic strength ([number ions] m
-
3
),
CEC is the sediment cation exchange capacity ([number ions] kg
-
1
)
,
ρ
s
is sediment density
(kg m
-
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3
)
, and
L
(m) is a grain length scale that is nominally the face length of a plate
-
shaped clay
particle, which we set to
퐷
푝
.
Physically, Φ
quantifies the
ionic
strength of river water
relative to
the ionic
strength in
a
volume surrounding
primary particles.
A
s
Φ
increases
, the positive charge
in the water
within the Debye length overcomes the negative charge on the sediment
surface and
causes attraction between nearby sediment grains
(Rommelfanger et al., 2020)
. We proposed
power
-
law relations as starting points
to relate Φ and the flocculation efficiencies
:
푘
퐴
∝
Φ
퐴
2
(
7
)
푘
퐵
∝
Φ
퐵
2
(
8
)
where
A
2
and
B
2
are
dimensionless
fit constants.
2.3
River floc model
We substituted
Equations (2)
–
(8) into E
quation (1) to derive a modified semi
-
empirical model
for
floc diameter,
D
f
:
d
퐷
푓
d
푡
=
푘
퐴
′
휃
(
1
−
휃
)
(
Al/Si
)
퐴
1
Φ
퐴
2
푛
푓
휂
2
휈퐶
퐷
푓
(
퐷
푓
퐷
푝
)
3
−
푛
푓
−
푘
퐵
′
(
Al/Si
)
퐵
1
Φ
퐵
2
휃
(
1
−
휃
)
푛
푓
휂
2
휈
퐷
푓
(
퐷
푓
−
퐷
푝
퐷
푝
)
3
−
푛
푓
(
휂
퐷
푓
)
2
푗
(
9
)
in which
푘
퐴
′
and
푘
퐵
′
are new dimensionless constants that absorb all constant dimensionless
parameters related to floc aggregation and breakage, respectively. At dynamic equilibrium, the
time derivative o
f
D
f
vanishes
,
resulting in
퐷
푓
=
푘휂
(
퐶
휃
2
(
1
−
휃
)
2
)
푞
(
Al/Si
)
푟
Φ
푠
(
1
−
퐷
푝
퐷
푓
)
−
푞
(
3
−
푛
푓
)
(
10
)
in which
푘
=
(
푘
퐵
′
푘
퐴
′
⁄
)
1
(
2
푗
)
⁄
,
푞
=
−
1
(
2
푗
)
⁄
,
푟
=
(
퐵
1
−
퐴
1
)
(
2
푗
)
⁄
, and
푠
=
(
퐵
2
−
퐴
2
)
(
2
푗
)
⁄
. We
consolidated the unknown dimensionless coefficients and variables into the coefficient
k
and
Accepted
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exponents
q
,
r
,
and
s
.
D
f
appears on both sides of E
quation
(10)
,
so
w
e simplified the equation by
assuming that
퐷
푓
≫
퐷
푝
:
퐷
푓
=
푘휂
(
퐶
휃
2
(
1
−
휃
)
2
)
푞
(
Al/Si
)
푟
Φ
푠
(
11
)
T
he assumption
퐷
푓
≫
퐷
푝
makes
D
f
independent of
D
p
in E
quation (11) and implies a model
domain of validity
of intermediate fluid shear such that
D
f
does not converge to
D
p
.
We
validated
the assumption through analysis of our field data compilation
(
Section 4.1
)
.
The equilibrium
D
f
model is
plausible in rivers
because experiments and field studies have shown the time scale for
unsteady floc behavior to reach equilibrium
in river conditions
is typically on the order of tens of
minutes to hours, and most dynamic river processes (e.g., floods) have longer time scal
es
(e.g.,
Bungartz et al., 2006;
Garcia
-
Aragon et al., 2011).
Floc settling velocity
,
w
s
,floc
,
relates to
D
f
using an adaptation of the Stokes settling law
for flocs
(
Strom and Keyvani, 2011;
Winterwerp, 1998)
as
푤
푠
,
floc
=
푅
푠
푔
퐷
푝
2
푐
1
휈
(
퐷
푓
퐷
푝
)
푛
푓
−
1
(
12
)
Substituting Equation (11) into Equation (12) yields a model for
w
s
,floc
푤
푠
,
floc
=
푅
푠
푔
퐷
푝
푐
1
휈
[
푘
휂
퐷
푝
(
퐶
휃
2
(
1
−
휃
)
2
)
푞
(
Al/Si
)
푟
Φ
푠
]
푛
푓
−
1
(
13
)
F
locs have irregular shapes and variable porosity
which
complicate the relationship between floc
diameter and settling velocity
(
v
an Leussen
, 1988).
In E
quation (1
3
),
the effects of floc
s
hape
and porosity
on
w
s
,floc
are captured in the
dimensionless
parameters
c
1
and
n
f
.
W
e held
them
constant
at
푐
1
=
20
(Strom and Keyvani, 2011; Winterwerp, 1998) and
푛
푓
=
2
(Kranenburg,
1994; Tambo and Watanabe, 1979).
Combining
t
he
se
assumptions
with
Equation (13)
yields
푤
푠
,
floc
=
푅
푠
푔
퐷
푝
20
휈
푘휂
(
퐶
휃
2
(
1
−
휃
)
2
)
푞
(
Al/Si
)
푟
Φ
푠
(
14
)