Journal of Applied Mechanics Vol.11, pp.517-523
Aggregation
of Unequal-sized
(August 2008)
and Oppositely
JSCE
Charged
Colloidal Particles
in a Shear
Flow
勇断流 中におけ る異径 ・異符号帯電 コロイ ド粒子の凝集
Motoyoshi
Kobayashi*
小林幹佳
* Member of JSCE, Dr. (Agr. Sci.), Faculty of Agriculture, Iwate University (Ueda 3-18-8, Morioka, Iwate 020-8550, Japan)
Understanding
the rateof aggregationof colloidalparticlesis veryimportantto considerthetransportof substancesin
waterenvironments.
Theoreticalinvestigations
on aggregation
kineticshavebeenpexfonnedbyusingtrajectoryanalyses
to elucidatethe influenceof electrochemical
and hydrodynamicinteractionson aggregation.So far, however,this
analysishas not been fullyappliedto the aggregationbetweenunequal-sizedparticleswith oppositesign of surface
electricchargein a shear flow. In this study, the rate of shear hetemaggregation
between oppositelycharged and
unequal-sizedparticleswas evaluatedby using the trajectoryanalysis.The calculatedresultsindicatethat the rate of
heteroaggregation
increaseswithdecreasingsaltconcentrationand thatthe effectof attractiveelectricdoublelayer force
is more significantat highershearrates.Atlowsaltconcentrations,
in addition,thepresentevaluationrevealsthatlarger
valuesof scaledcaptureefficiencywerefound not for thehomoaggegationbetweenequal-sizedparticlesbut for the
hetemaggregation
betweenunequal-sizedparticles.The calculationalso suggestsan experimentalplan,that is, pure
heteroaggregation
withoutelectricdoublelayereffectcan be obtainedbyusingthebinarymixtureof oppositelycharged
and unequal-sized
particlesin 1-10mMaqueoussolutions.
KeyWords:coagulation,
flocculation,captureefficiency,
trajectoryanalysis
Smoluchowski4,9,10), who
1. Introduction
derived
a
theoretical
equation
describing the number of colliding spheres with a radius of R1
toward a sphere with a radius R2 per unit time JSmol:
Colloidal particles such as clay minerals and natural organic
matter are ubiquitous in water environment These particles play
an essential role in the fate and transport of substances sorbed
with the particles1-4).
One of typical natures of colloidal particles
is their aggregation, coagulation, or flocculation. Through
aggregation, the size of transport unit increases. And then,
increased particle size enhances the sedimentation rate of
particles and causes drastic changes of particle transport.
Therefore, understanding aggregation process is of vital
importance for controlling and predicting transport properties of
substances in hydro-environments. Flocculation is
also
(1)
where G is velocitygradientof flow and N1 is the number
concentrationof particleswitha radius R1. The Smoluchowski
expressionneglectsall the interactionsactingbetweencolliding
particles. In reality, however, various electrochemicaland
hydrodynamicinteractionsact betweencollidingparticlesand
thus alterthe collidingtrajectorybetweenparticles.As a result,
an introductionof correctionfactorcalledcaptureor aggregation
efficiencya is needed to modifythe expressionof aggregation
rate J4,10-18):
important in water purification4,5).For these reasons, many
experimental and theoretical works on aggregation or
(2)
flocculationhave been carried out5-8).
In
order
The rate of aggregation is a key to understand the whole
aggregation process. Physicochemical and hydrodynamic
analysis
conditions of colloidal suspensions affect the aggregation rate of
colloidal particles4,7).
A theoretical treatment on the aggregation
colliding
kinetics
in
a
simple
shear
flow
was
started
by
―517―
solve
as
to
has
evaluate
been
trajectory
spheres
equations
under
physicochemical
interaction
the
used11-16).
is usually
capture
In
efficiency ƒ¿,
doing
describing
this
the
the
influence
of
interactions 11-16,19).
given
by
the
sum
a trajectory
analysis,
relative
one
hydrodynamic
The
has
movement
as
to
of
well
physicochemical
of electric
double
layer
(EDL)
and
forces
the van
is
der Waals
referred
to
Verwey-Overbeek
dependent
the
solved
flow
of
in
the
a rapid
aggregation,
presence
given
of only
sign
of
where
the
and
vdW
de
EDL
in
force
the capture
is
a shear
colliding
Their
equation
and
capture
the
potential.
is
and
Ven
the
force;
correlation
force,
force
surface
evaluated
EDL
surface
the
and
EDL
Van
charged suspendedparticlesin a shear flow has never been
studied.In the naturalwater,aggregationof colloidalparticles
of the two
spheres ƒ¿11
repulsive
by a simple
The
medium.
(R1=R2=R)
same
sum
at particle's
equations
of
be expressed
The
Derjaguin-Landau
suspension
trajectory
presence
have
cannot
the
potential
equal-sized
the
spheres
of
forces.
force4,20,21).
the electrostatic
salt concentration
efficiency
as
(DLVO)
on
Mason12)
(vdW)
results
except
absent
for
In
the
efficiency ƒ¿11,0
is
by
(3)
with a dimensionless parameter
proceedsthroughthe collisionbetweenunequal-sizedparticles
and/or aggregates3,6,7)
under the influenceof flow fields, and
both positivelyand negativelycharged particlesexist28,29).
In
addition,heteroaggregation
of small colloidalparticles,which
are thecauseof turbidity,withoppositelychargedlargeparticles,
whichhavelargesedimentation
velocity,can be used to increase
the rate of solid-liquidseparation(Fig. 1). Therefore,to study
the shear aggregationbetween unequal-sizedand oppositely
chargedparticlesis usefulfromscientificand engineeringpoints
of view,becauseit givesbetterknowledgeon aggregationto us.
In this context,this paper describes the results of trajectory
analysis for the evaluation of capture efficiency of
heteroaggregation
with attractiveEDL force in a simple shear
flow.
(4)
where ƒÊ
dynamic
constant
viscosity
reflecting
of
the magnitude
medium,
A12
of vdW
the
Hamaker
force, c1=0.79,
0.87,
Mixing
and
0.95
the
rate
for
R=2,
of
shear
heteroaggregation
Lawler5)
written
0.5 ƒÊm.
Later,
aggregation
between
obtained
for the shear
1, and
were
unequal-sized
correlation
equations
heteroaggregation
trajectory
analyses
performed
spheres13-15).
of the capture
in the absence
for
the
Han
and
defined
small
and
large
particles
force ƒ¿12,0
as
where ă=R2/R1
are
charged
efficiency
of EDL
Heteroaggregation
(5)
d
oppositely
on
constants
(R1>R2)
is the ratio
depending
on
of particle
a dimensionless
radius,
in flow fields
a, b, c, and
parameter
HA
by
(6)
Both Eqs. (4) and (6) asses the relative importance of van der
Fig.1
Large
settling
Rapid
solid-liquid
Schematic
Waals attractive force to hydrodynamic force. Experimental
application;
studies on aggregation in flow fields have been carried out for
separation.
velocity
illustration
separation
of heteroaggregation
heteroaggregates
enhance
and its
solid-liquid
the homoaggregation of equal-sized spheres with repulsive EDL
or without EDL force10,12,17,18,22).
These investigations have
2. Method
qualitatively supported the validity of the prediction by
trajectory analyses.
Recently, the effects of attractive EDL force acting between
positively charged and negatively (oppositely) charged particles
on aggregation and deposition have been tested4,23-27).
Previous
researches studying the effect of attractive EDL on the
aggregation rate by Brownian diffusion23,24)
and the deposition
kinetics in porous media4,25)reported that decreasing salt
concentration enhance the rates of particle deposition and
unequal-sized and oppositely
―518―
to evaluate
is given
Consider
shown
sphere
sphere
in a simple
in Fig.
Evaluation
trajectory
spheres,
and
potential ƒÕ2,
capture
efficiency ƒ¿
from
trajectory
below.
two
potential
rectangular
aggregation.
As far as the author knows, however, the rate of
heteroaggregation between
A method
analysis
2, the
of
capture
with
shear
relative
coordinates
equations
2
1 with
a
flow
a radius
radius
with
position
of
R2
efficiency
a
the
can
and
a surface
a
surface
a shear
rate
sphere
2 is given
(x, y, z) or by spherical
describing
R1 and
of G.
coordinates
be
done
relative
by
As
by
(r, ƒÆ,ƒÓ).
solving
motion
of
is expressed by the sum of Fvdw and Fel. The vdW force is
described by13,14)
(16)
for
p<1
and
(17)
for
Fig.2Coordinatesfor trajectoryanalysis.
1•…p,
where
Hamaker
aggregatingspheres.The relativemotion of collidingsphereis
governedby the followingtrajectoryequations13,34):
p=2ƒÎh/ƒÉL
The
h
distance
constant,
using
p
the London
EDL
force
as KC1 and
NaC1
between
particles'
surfaces,
A12
distance
defined
by
dimensionless
wave
in a simple
is given
length ăL
(typically
salt solution
100nm).
of 1:1 electrolyte
such
by4,21)
(18)
(7)
with
(19)
(8)
(9)
where
t time,r*=r/R1
dimensionless
centers,t*=Gt
distance
dimensionless
time,
between
A,
B,
where
kB
the
and
C
interaction
functions4,19)
that
depend
ratio ă=R2/R1
(R1>R2).
At
the
large
on r* and
separation
particles,
the functions
A, B, and
permittivity,
C are shown
NA
the
The
measure
of
length.
the
particles
are oppositely
are written
in one
(i) and
expressions,
and
h*=r*-ă-1
between
not
0.513,14).
is
surfaces.
was
shown
here,
are available
referred
part
of
as
EDL
the
different
sat
EDL
sign,
the
For
expressions
force
that
stopped
Eqs
with
layer
(EDL)
respectively.
force
The
the
(7)-(9)
hydrodynamic
were
numerically
Calculations
(x*=x/R1,
when
In
the
above
dimensionless
were
started
z*=z/R1).
Each
the position
separation,
by Wang14)
are
separation,
mean
that
spheres
of
sphere
2 resulted
The
of
or (iii) y*=10.
2 is captured
case
on
and
was
boundary
(i) r*-ƒÉ-1<ƒÂ*=ƒÂ/R1,
(ii) ƒÓ>ƒÎ/2,
sphere
work, ƒÂ=0. 1nm
shape
fates;
used
of
the
no
by
the
to avoid
The
sphere
other
aggregation
where ƒÂ
hand,
numerical
cross-section
calculating
trajectories
der Waals
are
introduced
electrochemical
(vdW)
In the
divergence.
y*=-20
can
starting
points
be
found
by
xc*(z*)
in Eq.
force
trial
points
(x*,
distance
(Fig.
2).
All
calculations
started
from
the boundary
result
in aggregation.
From
the shape
in the present
and
(7)
known
at
from
interpolated
adopted
force
is,
indicates
happens.
capture
cases
1, that
equations ă ,
separation
intermediate
three
2 is separated
from
van
is
is,
cross-section,
the capture
efficiency
a was
evaluated
study.
attractive
a
around
decreasing
that
trajectories
and trial
that
the capture
The
to
regarded
with
metho30).
occurs.
present
for
(ii)
z*) within
approximate
is thus
1/Kis
(18)
have
interactions,
aggregation
The
0.2,
and
diffuse
Eq.
colliding
of the following
is the minimum
(15)
=0.1,
M
as follows13,14):
(14)
Similar
in
hydrodynamic
(13)
for ă=1.
of
charged.
to calculate
y*=y/R1=-20
calculation
functions
e
concentration
concentration
length
increases
from
by the Runge-Kutta
from
length
the
and ƒÕ2
physicochemical
solved
1/Kis
and
surfaces
and
hand,
relative
permittivity,
number
Cs salt
Debye
of
It is clear
when ƒÕ1
In order
(12)
other
and
of
The
thickness
attractive
(11)
the
the
number
dimension
Debye
concentration.
on
dielectric
n=1000NACs
Avogadro's
colloidal
distances,
length, ƒÃr.
•@
as13,14
(10)
small
temperature,
distance
as
very
Debye
vacuum
charge,
(=mol/L).
At
reciprocal
absolute
the
salt,
between
T
are
elementary
radius
constant,
the
particles'
dielectric
hydrodynamic
Boltzmann
electric
as
Fvdw
as DLVO
the following
equation13,34):
double
and
Fel,
(20)
force
―519―
of
In
this
study,
compute
function
force
part
and
above
Evaluated
force
EDL Ą=KR1,
in
applied
the ratio
to
as a
of the vdW
of the EDL
the relative
order
to
are plotted
the ratio
, and
of
was
results
CA (or HA),
forces •@
diffuse
to
thickness
make
the
results
meaningful.
Results
3.1.
described
a
parameters8,11,12);
to hydrodynamic
universal
3.
procedure
efficiency
of dimensionless
the vdW
of
the
capture
and
Discussion
Capture
Efficiency
without
Electric
Double
Layer
Force
Values
of
equal-sized
(EDL)
capture
particles
force ƒ¿11,0
CA in Fig.
by
Eq.
Eq.
(3) obtained
4
are
seen
by
this
This
by
van
evaluated
from
the
agreement
Hamaker
between
constant
10-21
J
for
that
experimental
useful
fields.
the
the
evaluation
in
be
probably
previous
1•~10-21
due
adjacent
are
the
3•~10-21
present
Fig.4 Capture efficiency of heteroaggregation without
EDL force: Symbols and lines are evaluated
to obtain
of
j and
the
figure
•~
describes
is
found
to
be
kinetics
lower
than
to
in
that
the
by
concluded
values
that
A12
of A12 in the experiment
roughness
or
by this work and by Eq. (5), respectively
1
The
aggregation
which
EDL
values
comparable
reduction
surface
efficiency
without
with
A12 are
studies10,22),
J. The
to
of
of
work.
in order
analysis
(5).
evaluated
reasonably
trajectory
they
flow
respectively.
analysis
values
predictions,
suggested
to be
made
by Eq.
calculations.
experiment,
1.4ƒÊm,
the
for
While
theoretical
3.2.
and
were
of capture
plotting,
assumed
capture
particles
by the present
compared
the
and
trajectory
behavior,
practically
should
A12 are
the
previous
a shear
are
In
in
4 are drawn
dat18)
in
theory
R=1ƒÊm
demonstrates
flow
symbols,
as lines.
by
calculations
with
without
work
symbols
of ƒ¿11,0 and ƒ¿12,0
experimental
by
fit
The
values
(R1=R2=R)
shown
best
the
for
this
unequal-sized
in Fig.
of homoaggregation
EDL force.
layer
is drawn
The
study
the calculation
5, previous
denoted
this
agreement
confirms
calculations
curves
good
homoaggregation
the
by
Mason12).
efficiency
quantity
by
in the figure
and
force ƒ¿12,0.
graphs,
show
double
the dimensionless
between
of HA. The
work
force,
Ven
values
electrical
line
Fig.3 Capture
between
are calculations
The
de
EDL
values
In Fig.
of
respectively.
without
three
As
against
triangles
of heteroaggregation
calculated
for
and
homoaggregation
without
are plotted
(5),
efficiency
for
calculated
3. Circles
and
Fig.
efficiency
accumulation
is
of
salt
to surface22).
Capture
Efficiency
Electric
In
Double
Figs.
6
in
Layer
and
7,
the
Presence
of
Attractive
Force
values
of
capture
efficiency
between
Fig.5
unequal-sized
of radius
spheres
ratio ă.
shear
rates
these
figures,
Values
of flow,
the
are plotted
were
capture
against ĄKR1
of HA, reflecting
written
in Figs.
efficiency
the
6 and
with
for
four
relative
effect
7, respectively.
attractive
Capture
EDL ƒ¿12
(R1=R2=R)
without
(symbols)
are
by
that
see the effect
without
of EDL
EDL ƒ¿12,0.
force
By
using
this
homoaggregation
EDL
force.
compared
with
Experiments
calculations
In
(Lines).
is
To
1
expression,
respectively.
we can
of
of
and
normalized
efficiency
values
clearly.
―520―
are
obtain
assumed
better
fits,
for
A12
R=1
( 10-21J)
and
1.4 ƒÊm,
=3
retardation
force,
prevents
with
and
the
van
effectively
and
of the
radius
from
,0 against
means
more
The
prevention
force
is more
between
is clear
(Fig.
particles
in the
3). The
and
calculation
attractive
EDL
retardation
force
the
trend
that
the
concentration)
attractive
for
significant
for
i.e.,
for
in Figs.
reduction
for
the
smaller ƒÉ. ƒ¿12/ƒ¿12
heteroaggregation.
retardation
of unequal-sized
particles32).
4). Increasing
6
of
aggregation
the hydrodynamic
heteroaggregation
of equal-sized
of ƒ¿12,0 (Fig.
found
enhances
smaller ă,
by
are
sharper
is obtained
EDL
of aggregation
that
on ƒ¿12/ƒ¿12,0
figures
This
range
is also
and
indicated
magnitude
of
heteroaggregation
unequal-sized
attractive
ratio ă
effectively
than
by the
of
effect
force
colliding
the hydrodynamic
the
r(salt
that
rate
spheres
efficiency
This
Waals
overcomes
7. It is clear
This
Capture
der
away
gains ƒ¿12/ƒ¿12,0.
Effects
Fig.6
pushes
aggregation31,32).
only
force
which
spheres
attractive
EDL
captured
particles
force
and
by
compared
with
the system
decreasing
r increases
the
number
of
with
therefore
raises
the
capture
efficiency
EDL force.
where
only
the
van
der
Waals
force
works.
Fig.7
Capture
efficiency
between
EDL
of
unequal-sized
heteroaggregation
spheres
with
attractive
Fig.8
force.
Capture
efficiency
between
These
figures
decreasing Ą,
tendency
namely,
is similar
deposition
1/K
the range
and
increased
range
capture
of
disappears;
the
physicochemical
are not
By
is,
Decreasing
particles
At
and
of
higher
der
Waals
the
value
of
with
HA
by
EDL
become
force.
large.
becomes
the
increase
of
case,
length
enhance
in the
effect
with
EDL
of
force
dominant
the
in
aggregation
force.
Fig. 7, we
capture
rates,
result
In this
EDL
aggregation
force
spheres
EDL force.
The
the Debye
force
the
forces.
6 with
of
from
force
(ƒ¿12/ƒ¿12,0 are found
shear
rate
unequal-sized
attractive
heteroaggregation
of particle
of attractive
thus
with
concentration.
of EDL
higher Ą,
by attractive
Fig.
the
of EDL
magnitude
the normalized
values
for
and
,0 increases
the kinetics
as deduced
interaction
raised
comparing
HA on
higher
and
van
on
the presence
the magnitude
rates.
salt
studies
media4,25)
colliding
aggregation
rates
decreasing
salt concentrations,
The
rates
with
diffusion23,24)in
lower
that ƒ¿12/ƒ¿12
to previous
in porous
Brownian
At
demonstrate
of
see the influence
efficiency ƒ¿12/ƒ¿12,0.
for
lower
the
increases
values
same
radius
the
of shear
That
is,
Fig.9
of HA, that
ratio ă.
Capture
between
hydrodynamic
attractive
―521―
efficiency
of
unequal-sized
EDL force.
heteroaggregation
spheres
with
Figures
8
and
9
examine
the
), ie., the magnitude
on ƒ¿12/ƒ¿12 ,0. The
increasing
for
the degree
effect
can
say
the
that
the range
twice
of EDL
the magnitude
force,
of EDL
increases
orders
Acknowledgement
This study was supported by the MEXT KAKENHI
As
minor
of magnitude
(18688013) and the Maeda Engineering Foundation.
in ƒ¿12/ƒ¿12 ,0. Thus,
reflected
force
REFERENCES
by Ą is more
to enhance
the
rate
1) Buffle, J., The key role of environmental
colloids/nanoparticles
for the sustainabilityof life,Environ.
Chem.,3, pp.155-158,2006.
2) Kretzschmar, R., Borkovec, M., Grolimund, D., and
Elimelech,M., Mobilesubsurfacecolloidsand their role in
contaminanttransport,Adv. Agronomy,66, pp.121-193,
1999.
3) Li, X.Y., Zhang,J.J., and Lee, J.H.W, Modellingparticle
size distributionin marine waters, Water Research, 38,
of aggregation.
3.3.
Suggestion
for
Equation
capture
(5),
efficiency
used
to
which
in the
compute
no
validate
with
as fractal
decreasing ă
the same
to raise
mM
salt
sign
causes
homo
data
The
present
increasing
by
r
Cs=1-10
and
mM
coincides
because
Thus,
from
the
binary
particles
use
4.
the
doing
of
mM
solutions.
binary
mixture
experiments
capture
oppositely
evaluated
the
effects
can
be realized
charged
and
finding
(ƒ¿12
is
that
pure
by
using
unequal-sized
of this
oppositely
rate
and
of
work
charged
allows
particles
of ƒ¿12,0 in future.
on the
efficiency
capture
for
effect
heteroaggrgation
results
also
heterocoagulation
using
the
binary
heteroaggregation
unequal-sized
particles
analysis.
heteroaggregation
The
calculated
of
by the trajectory
concentration.
by
conclude
The
in
where
force17,18).
can
of
namely
EDL
we
to test the behavior
efficiency
charged
was
pure
EDL
with
homoaggregation
of repulsive
oppositely
of
weak
particles,
solutions,
results,
to
and
Conclusion
The
and
mM
rise
enhancement
becomes
sized
presence
without
gives
around Ą=200-500,
1-10
present
mixture
in 1-10
to
In
need
100-1000
particles17,18),
the
force
micrometer
of the
heteroaggregation
when
for
to
difficult
that
EDL
disappears
solution
with ƒ¿12,0.
negligible
of
pp.1305-1317,2004.
4) Elimelech,M., Gregory,G., Xia, X, and Williams, R.A.,
Particle deposition and aggregation, paperback ed.
Butterworth-Heinemann,
1995.
5) Han, M.,and Lawler,D.F, The (relative)insignificance
of G
in flocculation,J. American Water WorksAssoc., 84(10),
of
the
particles
we generally
medium
concentration
indicates
attractive
to
on
smaller
of heteroaggregation
calculation
heterocoagulation
out
of ƒ¿12,0
study
and
heteroaggregation
interpretation
carried
to the difficulty
charge,
high
been
So far,
decrease
due
suspension
Such
and
(5);
larger
electric
of
waters3).
experimental
between
has
of
aggregates,
been
Eq.
is probably
of surface
effect ƒ¿12.0,
has
by
values
between
in marine
perform
aggregation.
simultaneous
the
reason
concentration
to induce
objects,
is, if we
heteroaggregation
with
of EDL
kinetics
prediction
The
That
of calculated
experiment
numerical
experiment.
rapid
absence
systematic
the
Approach
equation
aggregation
are regarded
however,
us
Experimental
the correlation
The
increases
of attractive
is more
between
suggests
without
mixture
effect
flow
indicate
that
decreasing
salt
double
layer
force
at higher
shear
rates
unequal-sized
of
in a shear
with
an experimental
EDL
between
results
electric
profound
mM.
with
force.
of Nel has
increase
unequal-sized particles in solutions with concentrations of 1-10
(
potential, •@
EDL
increment
of Ą, two
in about
Nel
surface
that ƒ¿12/ƒ¿12,0
to the influence
than
of
of the attractive
of enhancement,
of Nel results
important
of electric
demonstrate
to the increase
compared
increase
we
figures
Nel due
effects
particles.
approach,
The
that
is,
can be accomplished
oppositely
charged
and
―522―
pp.79-91,1992.
6) Maggi, F., Mieta, F., and Winterwerp,J.C., Effect of
variable fractal dimensionon the floc size distributionof
suspendedcohesivesediment,J. Hydrology,343,pp.43-55,
2007.
7) McAnally,W.H., and Mehta,A.J., Aggregationrateof fine
sediment,J. HydraulicEng., 126,pp.883-892,2000.
8) Masliyah,J.H. and Bhattachrjee,S., Electrokineticand
colloidtransportphenomena,Wiley,2006.
9) von Smoluchowski,M., Versuch einer mathematischen
thorie der koagulationkinetic
kolloider loesungen,Z. Phy.
Chem.,92, pp. 129-168,1917.
10)Adachi,Y.,Dynamicaspectsof coagulationand flocculation,
Adv.Colloidand InterfaceSci.,56, pp.1-31, 1995.
11)Zeichner,G.R., and Schowalter,W.R., Use of trajectory
analysisto study stabilityof colloidal dispersionsin flow
fields,AlChEJ., 23, pp.243-54, 1977.
12)Van de Ven, T.G.M.,
and Mason, S.G., The
microrheologyof colloidal dispersionsVII. Orthokinetic
doublet formationof spheres, ColloidPolymer Sci., 255,
pp.468-479,1977.
13)Vanni, M., and Baldi, J.B., Coagulationefficiencyof
colloidal particles in shear flow, Avd. Colloid Interface
Sci.,.97,pp.151-177,2002.
14)Wang, Q., A study on shear coagulation and
heterocoagulation,
J. ColloidInterfaceSci.,150,pp.418-427,
1992.
15)Higashitani,K., Ogawa,R., Hosokawa,G., and Matsuno,Y.,
Kinetictheoryof shearcoagulationfor particlesin a viscous
fluid,J. Chem.Eng.Japan, 15, pp.299-304,1982.
16)Adler, P. M., Heterocoagulationin shear flow, J. Colloid
InterfaceSci.,83, pp.106-115,1981.
17)Kobayashi,M., Maekita, T., Adachi, Y., and Sasaki, H.,
Colloid stabilityand coagulationrate of polystyrenelatex
particles in a turbulent flow, International J. Mineral
Processing,73, pp.177-181,2004.
18)Sato,D., Kobayashi,M., and Adachi,Y., Captureefficiency
and coagulationrate of polystyrene latex particles in a
laminarshear flow:Effectsof ionic strengthand shear rate,
Colloidsand SurfacesA,266, pp. 150-154,2005.
19)Batchelor, G.K., and Green, J. T., The hydrodynamic
interactionof two small freely-movingspheresin a linear
flowfield,J. FluidMech.,56, pp.375-400,1972.
20)Hiemenz,P. C., and Rajagopalan,R., Principlesof colloid
and surfacechemistry,3rd ed., MarcelDekker,1997.
21)Ohshima,H., and Furusawa,K. ed., Electricalphenomena
at interfaces: Fundamentals, measurements, and
applications,2nded., MarcelDekker,1998.
22)Sato, D., Kobayashi,M., and Adachi, Y., Effect of floc
structureon therate of shearcoagulation,J. ColloidInterface
Sci.,272, pp.345-351,2004.
23)Lin,W., Kobayashi,M., Skarba,M., Mu, C., Galletto,P. and
Borkovec M.: Heteroaggregationin binary mixtures of
oppositelychargedparticles,Langmuir,22, pp.1038-1047,
2006.
24) Yu, W. L, and Borkovec, M., Distinguish
heteroaggregationfrom homoaggregationin mixed binary
particlesuspensionsby multianglestatic and dynamiclight
scattering,J. Phys.Chem.B, 106,pp.13106-13110,2002.
―523―
25)Elimelech,M., and Song, L.: Theoreticalinvestigationof
colloid separation from dilute aqueous suspensionsby
oppositelychargedgranularmedia,SeparationsTechnol.,2,
pp.2-12,1992.
26)Kobayashi, M., Ifugou ni taidenshita biryuushikan no
sendangyoushuusokudo:
kidoukaiseki,Riron ouyou rikigku
kouenkai kouen ronbunshu, 56, pp.497-498, 2007, in
Japanese.
27)Kobayashi,M., Kineticsof shearcoagulationof oppositely
charged particles: A trajectory analysis,Theoretical and
AppliedMechanicsJapan, 56, pp. 267-272,2007.
28)Hunter, K. A. and Liss, P. S., The surface charge of
suspendedparticlesin estuarineand coastalwaters,Nature,
282, pp.823-825,1979.
29)Newton,P. P. and Liss,P. S.:, Positivelychargedsuspended
particles:Studiesin an iron-richriverand its estuary,Limnol.
Oceanogr.,32, pp1267-1276,1987.
30)e.g., Kawamura, T., Suuchi keisan nyuumon, 1st ed.,
Saiensusha,2006, inJapanese.
31)Higashitani,K., Yamauchi,K., Matsuno,Y., and Hosokawa,
G.: Turbulentcoagulationof particlesdispersedin a viscous
fluid,J. Chem.Eng.Japan, 16,pp.299-304,1983.
32)Adler, P. M., Interaction of unequal spheres I.
Hydrodynamicinteraction: Colloidal forces, J. Colloid
InterfaceSci.,84, pp.461-474,1981.
(Received:April 14, 2008)