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Collision efficiencies of diffusing spherical particles: hydrodynamic, van der waals and electrostatic forces
1
Advances in Colloid and Interface Science. 20 (1984) l-20 Ekevier Science Publishers B-V.. Amsterdam - printed in The Netherlands
COLLISION
EFFICIENCIES
DER WAALS
AND
IRAKLIS
A.
VALIOULIS
and
Institute
of
California
SPHERICAL
PARTICLES:
HYDRODYNAMIC,
Pasadena,
California,
USA
OF DIFFUSING
ELECTROSTATIC
V%
FORCES
E.
JOHN
LIST
Technology,
CONTENTS I.
ABSTRACT
II.
INTRODUCTION
III.
HYDRODYNAMIC
IV-
VAN
V.
COLLISION
VI.
DOUBLE
..~...~~.........._..~~.~.~~.~~~~~~.~~~~.~~~~..~.~.~~~.~.
INTERACTIONS
DER WAALS
FORCES
LAYER
CONCLUSIONS
ACKNOWLEDGEMENTS
3 5
_--.__._____--______~~.--~---~-..~~~...._.~
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..-.-.---_-*.---12
FORCES
VII,
1
.~.__.~~_~_~~~__~~~_~-~~~~~.~~~~.~~~...~
~~~_~~~~~__~~~~~_~~~~~~~~~~~~~~.~~~~~~~~~.~~~
EFFICIENCIES
VIII.
_I..~~-~~~~.~_.~~-~_~..~~~..-~~..~~....~...~...~~.....
17
~-~~~~_~~..~_~.~~_~~..-_..-___~~~~~~~~__~_____.______19
ABSTRACT
1.
A practical mate
for
1 imitation
the
Brownian
collision
motion
particles
in
causes
in
surround of
1
~..~~...~...I.~....~.....~~..~~..~~~~.~~~~~~~~~.~~~~~
between
the
of
forces
der
collision
the
can
van
Waals
are
energy
of
this
computed
various
the of
the
for
which or
they
corrections
such
forces for
ionic
in suspended
particle
ions
Results and
convenient
For
forces)
when
esti-
particles
paper
sizes.
attraction
a way
cloud
layer
In
classical
forces.
disturbance
the
(double
forces).
of
in
the
from
‘s
spherical
interparticle
from
particle
probability
presented
diffusing of
arise
Waals
particles
der
are
two
Smoluchowski
of
forces),
charged (van
approaching
electrolyte
ication
of
(hydrodynamic
origin
two
values
such
fluid
Smoluchowski’s
appl
non-consideration
electrically
molecular
the
probabiiity the
water
the an
is
‘of
can
operate several
strength
particle
be
to
of
collision
modeling,
INTRODUCTION
II.
Suspended
which
particles
They
flows.
the
of
the
to
predict
affect
are both
suspension
individual the
0001-8686/84~$06_00
is
0
bulk
contact.
of
the
and
in
most
properties
in
particles behavior
ubiquitous the
the
environmental of
the
Information properties
suspension_
on of
The
or
fluid the
the
industrial the
physical
flow
knowledge
1984 l3lsevier Science Publishers B-V_
and
is of
surfaces
with
characteristics required
the
in
fluid-particle
order
2 interactions, which collision mass
the
important
in
For
is
in
given
rate, the
and
collision
are
much
equal
t
r 2 with
1 radius
r_,,
surface than
the
is
B(r,,r2)
=$y k
fluid
dynamic
is
move
interparticle
causes an
in
in
flows
in
of
suspended
thus
become
in
which
it
is
particle
respective
particles
number
probabi
is
that
encounters
of
fluid,
diffusion
assumed
1 ity)
suspension
occur,
then
with
radii
r,
N,
and
concentrations B(rl,
r2),
the
the
and
r2
N2
representing
the
process:
(1)
r 2)
is
time
the (ref.
function
of
3) the
asymptotic
by
two
flux
D,i-02, The
a
molecular
computed
pair
sphere
viscous
equal in
to
the
volume
mean
free-path
solving
a
to
the Di
relaxation
of
surface is
a
t =
fixed
the for
sphere
diffusivity
of
is
infinity.
time,
the
fluid
equation
function
at
When
.the
specifically,
of
the
two
the
fluid.
Ptore
distribution unity
of
diffusion
particles.
where
and
common volume
unit
held
Then 2r’/9v,
at
for
the
of
particle zero
times collision
(r, + r212 r r i
2 constant,
T
is
the
absolute
temperature
and
u
is
the
viscosity. ignores
on
particle
straight
fluid
interactions,
paths,
modify
their
arise
from
can
However, relative the
(hydrodynamic
cha-ged
molecular
origin
collision
enhancement) collision
Brownian
If
of
unit
than
Particle
forces the
binary
6(r1. per
fixed
forces
electrically
1 inear’
distribution
matter
flows of
N,N2
i=1,2.
Boltzmann’s
such
r2)
the
the
expression
particles
only
Brownian
diffusivity
for of
function
where
to
2)_
volume
the
larger is
is
r
This
of
process
interactions
O_lum-1Oum
(co1 1 ision of
distribution
radius
larger
the
the
suspended
(ref.
their
sweep
8fr , , r-2)
the
of
B(rT,
function
pair
unit
probability
particles
water
per
=
particles
at
modifies
of
range
that
function
rate
collision
so
dynamics
diffusing
with
fare
size
product
co1 1 ision
co11 ision
the
modeling
Particle-particle
mechanism
dilute
by
geometry
The
the
successful Coagulation,
particles,
space,
the
coagulation
collision
a
for
other.
occurs.
sufficiently
and
sufficient each
of
size
quantifying
particles
dominant
the
not with
coalescence
particle
coagulation
is
interact
and
in
is
however,
particles
(van
rate
particle der
Waals
E(rl,r2) (as
given
(double forces).
by
that
motion.
then Eqs.
from layer
the the
1 and
assumes
suspended
presence cloud
forces))
which 2)
it
approach
For
A correction defined
is.
particles
disturbance
forces),
is
as
and
of
of or
ions
they
(collision multiplies incorporates
that
each
other,
particles the
which can
in
particle
be
surround of
efficiency the
or
‘recti-
the
influence
3 of
the
inter-particle
In
this
disperse
forces
paper
we
spherical
particles double
and
Hamaker
constant,
presented
in
to of
A,
a way
Smoluchowski applies
are
The
motions
first
sizes
in
flow
over
relative
for
Spielman
(ref.
efficiencies
Stokes’
Results the
by
collision
of
4) two
accounting
a wide
size
particle
of
range
the
of
for
mono-
van
values
particles,
co1 1 ision
for
diffusing
of
der the
‘2/r,,
are
modeling.
INTERACTIONS 1)
(ref.
extremely
considered.
Brownian
of
process.
used
the
forces. and
collision
a method
compute
convenient
‘s to
the
various
layer
HYDRODYNAMIC
III.
tion
extend
suspensions
Waals
on
classical dilute
two
model systems
particles
independently
are of
for
Brownian
where
only
treated
each
as
other
motion
binary rigid
with
induc_ed
coagula-
particle’encounters
spheres
a constant
describing relative
diffusion
coefficient
=
%2
D,
f
where
the
D1 =
kTb,,
are
D2
,
single
this
particle
D2=
functions
Stokes’
(3)
the
For
formulation
a
particle
motions generates
as
as
location
of
the
first
as
the
such In
an
a
velocity
unbounded at
particles
2
advective
flux
conservative
of
particles
point
in
space,
relative JF. force
to The
particle latter
F derivable
tend
s-=
causes induces Thus,
relative
extending
system
any
is
at a
are
determined
b =
1/(6nur).
to
correlate
The
motion
distance
particle
Eq.
of
3 becomes
However, the
of
s
in
located
a velocity
by
order
one the that
at s-’
at
the
increasingly
decreases.
the by
which
order
5).
separation modified
interactions
assumed:
(ref.
which
decreases.
of
which
b2
mobility
forces
gradient
dipole
and
the
separation
particle
4)
r
gradient
force
particle
(ref.
particle
particle
This
act
b,
radius
hydrodynamic
the
fluid. to
Spielman
mobilities of
a velocity
distance
invalid
particle particle
ignores
particle surrounding
coefficients
kTb2
of
law.
diffusion
a the
1 due
to from
a
coefficient (ref.6)
hypothetical mean
arises from
diffusion Einstein’s
dynamic
radial
number
Brownian the
potential
V and
of
account
flux is
an
acting
is JD of
balanced
arbitrary between
for
argument-
equilibrium
density
diffusion
action
to
ingenious
by
an
steady the
particles:
JD f
JF=
0
(5)
4 JD = -D,2 (dN2/dr), where
N2 is
velocity
to
where
b is
Under
the
N2 = N;
N” is 2
relative
the
=
flux
JF = -N2
conservative
relative force
radial
F
mobility the
which
number
is
a function
density
of
of
particles
separation. 2 must be
(8) of
relative
particles
particle
2 at
diffusion
infinite flux
inter-particle
is
(D,2/(kT))(dV/dr)N2
induced
by the
(9)
conservative
force
F
b(dV/dr)
(10)
The hypothetical 3 and
equilibrium
10 the
situation
relative
particle
(Eq.
5)
is
invoked
then
to deduce
from
diffusivity
= bkT
(11)
which
is
it
now assumed
is
is only become exact
a function
of that
justified
if
superposable solution
centers
obtained
irrelevant
in Stokes’ equations
unphysical
behavior
order
of
the
which
diverge
of
to
for
the
ignored
so
mobi 1 ity
two spheres
Einstein
force that
F is the
b can moving
(ref,
7).
Both
of
centers
of
considered,
two approaching
becomes by the
molecular particle
Following
when the
since
6) This
two fluxes
be computed
from
the
along
line
of
the the
(ref.
removed.
their
rotational
motion
particles all
are
motions
are
4).
between
explained
line are
motion
fluid at
are relative
particles
force
even
and Jeffery
flow
is
The
equations
by Sthnson
when spherical
separation. valid
effects
_ 5) _
perpendicular
The hydrodynamic
forces
11 is
inertial
Stokes’
linearised the
interparticle Eq.
(ref
of
the motion
independent
of
particle
number density
JD = -D12(dN2/dr)
and
2 and u the
by the
,
the
D12
particles
(7)
conditions
Then
Eqs-
of
particles
F = -dV/dr
exp(-V/(kT))
the
(6)
distributed
distance.
and
the
equilibrium
Boltzmann
where
number density
imparted
u = bF,
Here
the
JF = N2 - u
singular breakdown
mean free
contact
particles at
can
path.
zero of
determined
separation.
continuum Van der
be considered
flow
Waals
from
the
This at short
to overcome
distances range this
5
difficulty
in
the
collision
VAN DER WAALS
IV. The
attractive
dipoles
by
of
formula
for
vA Here
r
the
is
Schenkel
-
the
Waals
arise
in
electron
assumed and
(ref.
(
from
additivity and
the
synchronized
clouds of
energy
the
of
the
inter-
pairwise
derived
his
VA between
inter-
well-known
spherical
particles.
r2
(12)
‘1 -r2)
particle 3)
the
the
molecules
2rl r’-
between
recommended
forces
interaction
2rl r2 (r,+r2)zt
Kitchener
and
8) atoms
der
distance
and
formula
van
Waals
charges
(ref.
constituent
A 6kT
--
iz=
the
der
fluctuating
Hamaker
bodies.
actions
FORCES
London-van
created
acting
problem.
centers
incorporated
best-fit
and
A
is
the
retardation
approximation
Hamaker
effects
to
their
constant.
in
Hamaker’s
numerical
integrations
vA kT=
A -- 6kT
vA kT=
‘2
w
1
1 f
1.77~
’
--
where the
%&
p =
2nh/a
and
particles,
a
than
induced waves
and
10). it
The
render
separation been
then
that
The
the 11).
(ref.
in
A
is
only
12)
involving
mation
action
can
(ref.
used
ments
a
due
interparticle
the
the
to
Eq.
13
account Hamaker
the
1OOnm is so
of
the
when
finite
Eq.
from kinetics
Smoluchowski
of
of
for a
A, 13,
the
for
because
of of
Hamaker
the
its
the
of
effects. particle
simple
a
polydisperse
diffusing 4)
form, from
particles.
13 may
(ref.
the
electromagnetic
constant
Eq. in
i-s
of
is by
characteristic
retardation
function
coagulation
distance emitted
single
population
foGc&
between
length;
efficiencies
propagation
experiment,
equation
conservative
collision
a
‘effective’
an
distance wave
radiation
only
monodisperse such
London
interparticle
the
qualitatively
an
the
the
time
Lconstant’,-
minimum the
particles.
of
incorporates
calculate
p 2, 0.57
’
problem, size
occurs
(13)
dimensionless
k =
wavelength
coagulating
modeling
the
Nevertheless, to
obtained
generalized of
is
Since
wavelength latter
in
characteristic
dipoles
is
absolute
retardation
the
(ref.
the
1
- %$r
and
length
of
Electromagretic larger
h
(r-r2-r,)/rT
another
function
X/r,. -
cx =
h =
X introduces become
0 < p < 0.57
be
a
particles
has
experiProvided
good
approxi-
suspension. under
the
6 DN2
-divJ,2
at= with
N2
boundary
= D
The
and
steady
r2
D
[(
12
aN A+N2bx ar
dVA
,
(14)
)I
conditions
and
N2 = N;
a --I t-2 ar
=
VA = -m
when
r = r +r 1 2
VA = D
when
r = m
state
particles
2 into
-4arZJ
=
(19
equation
solution
of
this
a sphere
of
radius
gives
the
diffusive
flux
J12 of
r,+r2
4n D_ NT (r,+r2)
(16)
D12 is
where
the
interparticle rate
depends
A collision
E-‘(r
as of
relative
forces
particle
and
on the
s the
integral
efficiency
of
can
the
in the
coefficient
separation
particle
s=
interactions
absence
of
any
The collision r/r 1’ over all separations.
be defined
(17)
l”2)
the
enhancenent
any
of
interactions
(ref,
13)
V,
COLLISION The
co11 ision
between
stability
relative
the
the
obtained
by Stimson
and Jeffery
the
convergence to converge n th- partial
performed separations
particles.
coefficients,
by sumning
assumed
over
the
co1 1 ision
E(r,,r2)
is
were
determined
rate
the
in
inverse
the of
absence. Fuchs’
EFFICIENCIES diffusion
separation
A single
rate
factor.
particle
is
diffusion
dimensionless
criterion when the sum of
to a precision h c 0.001,
D,2. series
(ref.
7)
c=O.OOOl condition
a series. of
the
the
thirteen
asymptotic
solution (as
corrected
was
used
I(S,,T All
the
significant formula
as
to Stokes1 for
a function
by Spielman, each
-S,)/Sn] numerical figures.
series,
c c was
ref. which
4). were
fulfilled;
calculations For
of
equations
were
dimensionless
Sn
7 developed the
by Brenner
series The
the
larger
the
results.
of
of
the
particles
of
13)
equal
son between
the
retardat
parameter
less
ion
than
retarded as
rl
(or
as
as
computed very the
of
Q= 0.1
collision
forces with
with
relative
for
a compari-
values
of
the
separations
s
The curve
the
unretarded
for
the
potential
to
the
1,
(Figs.
5,
A/(kT)
examined
and a=1
the
as
results
illustrated
to hydrodynamic
of
For
forces
Waals
operate.
energy
to dominate
the
of
collision
range.
potential
for
this
both
equal
provided
the
in Fig. for
(with
X= 100nm. to
size
a check
for
at
more between
7 where different
the
A/(kT)
relative
particle
size
lower
and Burtscher of
pairs
at
than of
in
to a
all
and
of the
the
and are (ref.
15).
particles values
about
similar
the
of
than
interacting
a= 0.1)
particles
reduction
3
The enhancement
retardation,
less
co11 ision
corresponds
are
include
= lum or
on the
shown in Figs. 0.1
efficiencies not
efficiency (r,
particles
a=
and the
by Schmidt-Ott
collision
is
rI=O.lrrm)_
absolute
who did
particles
reduced
particles
considered
the
obtained
large
forces
interacting are
The computed
smaller
is
the
forces
15),
reduce
and for
efficiency
tend shorter
results;
corresponds
decrease.
forces
6 and 7)-
van der
2
H represent
and hydrodynamic
unretarded
respectively
by Twomey (ref_ with
the
W in Fig.
marked with
the
forces
Spielman’s
size
decreases
particles
Hydrodynamic
as
become of
van der Waals
r, =lum rate
calculated
well
and e=
radius
is
and
calculations.
the
when only
in agreement
der Waals
The hydrodynamic
van der
of
interacting
van
Waals
validity
effect
12)
electromagnetic
marked with
curves
importance
increases.
The effect
due
different
for
(Eq.
u -f = corresponds
curves
the
the agreed
collision
limit
by the
ignored;
increasing
The efficiencies
efficzency
are
when both
assumes
particles
ones
the
curve
r is 2 alter
not
1 is
dimensionless
significantly.
the
The
efficiencies
Fig.
distances
potential
represented
efficiencies
attraction
the
interparticle
more
where
did
unretarded
collision
for
for
formula.
the
500,
range
A/(W).
potential
1 approaches
interactions
Retardation
particle
larger
of
collapse
-z r/r-2<
the
compute
for
separation.
lo-’
both to
Simpson’s
account
integration
values
retarded
Q increases);
calculations
collision
and 4 for
used
various
attractive
since
case.
In the
process
the
retardation, were
and
calculations
using to
particle
separation
The curves
in Fig.
decreases
decreasing
extending
for
for the
potential
hydrodynamic
the
and
0.001;
reduces
unretarded
with
of
unretarded
about
retardation
numerically
potential
0~.
up the
was used
step
a dimensionless
size
speeds
integration
significance
(Eq.
this
separations.
17 was performed
two particles;
the
retarded
small
integrand
over
the
To assess the
in Eq.
ranged
was used;
at
decreasing
variation
integration
14)
slowly
integration
A successively rapid
(ref,
converges
20,
size.
collision
at various
of
the This
efficiency values
COMPARISON OF POTENTIALS ------
0
02
_ 0.4
UNRETARDED Q=1 0 = O-3 Q = 0.1
0.6
0.8
1
DlMENStONLESS SEPARATION h
Fiq
_ 1, Dimensionless tion
for
van der Waals potential vs, various values of the retardation
dimensionless parameter
EFFECT OF RETARDATION q
-
1 “‘““I
’ “““1
’ ’ ““‘I
’ “F
UNRETARDED
--Qal QXO.3 ----.--Q~O_,
I
___& cI I
t
o-:
5
104
IO-’
go-2
lo-’
IO0
IO’
HAMAKER GROUP A/(kT) Fig.
2.
separa-
,./*1 a
i
)
-
particle QI
Effect of retardation on the collision cles for various values of A/(kT).
efficiency
102
of
equal
size
parti-
9
VAN CER WAALS’ FORCES
-
r&r,
r
=1
--r&r, =3 cl=0 .I ---Q/r* =5 ----r&r, =I0 --rz /r, = 20 --_.m- r&r, =I00
If= 16.
IO-=
/
IO-'
10-l
IO"
IO'
4/ IO2
HAMAKER GROUP A/(kT) Fig.
3_
Coilision efficiencies for various values of (a= 0.1).
VAN
DER
of particies with various relative sizes and A/(kT) when only van der Waals forces operate
WAALS'
FORCES
1.6
IO* HAMAKER Fig.
4.
Collision efficiencies for various values of (a= 1).
IO+ lop GROUP A/(kT)
Id
Id
of particles with various relative sizes and A/(kT) when only van der Waals forces operate
10 of A/(kT) dynamic
is shown. and
van
: ER stands
der Waals
HYDRODYNAMIC
AND
VAN
for
forces
DER
the
collision
operate;
WAALS
----
A/(kT)
= 1O-3
-----
A/[kT)
= IO-+
_-.__-
A,(kT)
= I‘,-’
----.-
A/( kT) = IO
EW
efficiency
is the
when
collision
both
hydro-
efficiency
when
FORCES
-..--.- ,?,/(kT]= 10”
0
25
50 r,‘=
75
100
I
Fig,
5_ Collision efficiencies of particles with various relative sizes and various values of A/(kT) when van der Waals and hydrodynamic forces operate (a=O.l).
only
van
particle
der Waals
forces
attractive
act,
energy
theoretically
impossible
force
the particles
between
The
curves
decreases. since
shown
In the
in Stokes ' flow
grows
without
approach
limit
zero
as
the hydrodynamic
bound
as
the
A -+ 0 collisions
the
for
interare
repulsive
particle
separation
decreases_ For
small
A/(M)
particles
the effect
of
(r,=O.lum the van
particularly
strong
(see
efficiencies
as the
ratio
of
the hydrodynamic 6:
the collision
with
the
rest
Reported equal
size
of
Fig.
4).
the
efficiency
This
on
reverses
size
Witness
of equal
large
collision
the
trend
r22> r,) decreases
force.
at
the
the
particles
toward
caused
upper
of is
smaller
by
two
is very
values
efficiency
the action
curves
large
in
compared
curve.
experimental particles
of equal
forces
r../r, [where
interparticle
Fig.
or a=l)
der Waals
collision
in the
size
efficiencies range
O.lum
range to lum
from
(see
0.35
ref.
to 0.7
12 for
for
a recent
I1 survey),
a
Hamaker
result
which
constant
of
according
about
HYDRODYNAMIC
to
2 - 10-l’
Fig.
5
Joules
AND VAN DER WAALS’
-
AI
= C4
---
A/(kT)
= lO-3
---
A/CkT)
= IO-”
--.-
A/(kT1=
----.-
A/(kT)
= IO
-------
A/CkTl
= 50
--.-
A/( kT)
= IO2
implies
(at
a maximum
300°K)
for
value
the
for
retarded
the potential-
FORCES
IO-’
Ct=l
J
0
25
50
75
100
‘2/‘1 Fig.
Collision various operate
6.
According colloids
to in
efficiencies of particles w h en van values of A/(kT) (a = 1) _
tyklema
ranges
to
Hamaker
corresponding
the
groups
respectively
(for
efficiencies
determined
Estimation
tric
IS)
ionic
complicates
indirect tions-
the
species the
in
of on
determination
experimental Experimental
of a
A
about
25
collision
can
the
of
A
and E(O)
hydrophobic
2.
and
lO_”
Joules
0-06,
efficiency which
of
carried
respectively. of
are
the
and
out
dielectric
be the
more
in
about the
0.75
range
of
not
medium.
well
known
radiation data
constant
Lifshitz’s
w dependent
dispersive
the
promising
Hamaker
by
frequency
the
electromagnetic
from may
be of
particies
on
determination
most
about
for
experimentally-
function
measurement
relative size and hydrodynamic forces
of
to
potential),
knowledge
the
the
solution
to
retarded
constant
requires
E(W)
information
the
constant Joules
300°K)
correspond
Hamaker
This
permittivities
incomplete the
of
method.
lo-”
(at
to
(ref.
these
Hamaker
about
and
collision
5,
from
According 0.35,
Fig.
17).
(ref.
water
with various der Uaals and
(ref.
and
suggests
for
practical can
be
di elecHowever,
effect
of
19) that
its
appiica-
carried
out
indirectly
-from
wetting,
the
,_mr---
r2/r,
-*1**..’
’
10-S
coagulation layer
means of
the
reliable
process
are
of
then
sizes
can
of
the
value
on surface
film
thinning
tension (ref.
IO'
A/(kTl
monodisperse and
the
dispersion the
* .
since
be obtained
in
---I
IO2
the
systems
(or
the
readily
from
is
Figs.
5 and 6) Hodeling
5 or
Fig.
Waals
that
determined
by
efficiency can give
of
Dynamic
efficiencies
Fig.
assumed
A collision
General
collision
is
rate
12).
constant.
via
it
coagulation
(ref.
Hamaker
suspension
then,
and
19).
. * -m
*.‘.***’
calculations
of
in a polydisperse
the
is
a
coagulation (ref.
20)
between
Equation
particles
of
charge.
Since
6.
DOUBLE LAYER FORCES_ Dispersed
dispersion
particles is
opposite
cally
excess
forces
due
extend
tion
of
to
is
formed
oppositely to
the
in ‘the outer to
the
outwards
counter
thermal from
ions
waters
neutral,
Close
ions of
According
established
in natural
electrically
sign.
adsorbed
the
med i urn.
to
. *-‘**-*’
GROUP
numerical
can be accomplished
of
16'
negligible
half-life
reasonably
of
data
of
7
* **‘**J
experiments
and
VI.
from
kinetics
= loo
HGAKER
forces
determined
unlike
the
Evaluation of the relative importance of hydrodynamic and van der forces in modifying the co11 ision rate of t-wo spherical particles.
7.
rapid
coagulation,
and
r2/r,
= 5
01 . . **..A
double
of
adhesion
EFFECT OF HYDRODYNWIC FORCES . . ‘1”“, . * . L”“, . . . .“-‘, . . * 8”.‘, . * =l“V ra/r, = 1 ra/r,= 20 Q=t r&r, = 50 -r&,=3
125
Fig.
kinetics
adsorption,
the
the
(Stern
layer). ions
the
the
particle
diminishing
with
The outer
surface distance.
ions)
(ref.
between ions.
carries
a compact
(counter
model
layer of
phase
surface
Gouy-Chapman (diffuse)
an electric
aqueous
particle
charged
motion
carry
21)
charge
specifi-
(Gouy)
layer
dispersing
the
an equilibrium
causes
the
of
of
electrostatic
This into
an equal layer
forces the
solution,
diffuse the
the
consists is and layer concentra-
13 This
local
induces ficant
distribution
double
layer
of
charges
interaction
simplifications
are
in an electrically
forces
needed
between
in order
neutral
approaching
to describe
solution
particles.
Signi-
quantitatively
the
repulsion
J’
y$&= AV: energy
Fig.
8.
Schematic illustration surface separation.
interparticle charged are
and Van Leeuwen of
natural
a thin
equilibrium
suggests
that
for
particles
the during
outlined
a method
particles
to
the
the
be presented
diffuse
layer
the
the
of
which
density
scale
of the
between
size (ref.
the
potential His
spheres
same
range
O.lFrm
22).
Lykl ema
the
interaction of
electro-
size
of
the
potential
remains
Valioulis
(ref.
between
expressions
are
superposition obtained
negatively
analysis
with
surface
particle
encounters the
Brownian
interaction.
linear
carry
Their
compared
than
of
restoration
particles.
electrostatic
the
the
the
the
of
particle
but
size
of
for
Brownian
charge.
in
scale
thin
rather
He invokes
interaction
the
system
sizes
their
scale
are
the
same negative here.
time
time
a function
binary
particles with
as
dilute only
different
compared
surfaces
charge time
that
suspended
layers
to compute
carrying
not the
surface
so
have
compared
on the double
constant
can
with
energy
A sufficiently
layer
23)
potential
assumed
waters
particles
chemical
will
is
double
(ref.
two colloidal
the
forces.
The particles
In most
lOurn have
layer
particles
considered.
charge. to
double
spherical
of
barrier
by Bell
two spherical lengthy
and
approximation et
al.
24)
(ref.
.25)
for
Derjaguin’s The s ion
be
computed
the
v =
VA
The
salient
features
electrostatic
of
behavior
separations
the
subject
Waals
and
interaction
at
Brownian
van
of
der
constant
diffu-
layer
energy
attractive VE
to double
forces
the
Waals
two
potential
surface
VA
charge
curve 8.
outweighs
two
the
the
the
interaction
smai!
and
This
particles
and
can 17)
(Eq. barrier
coliision
is
At
intermediate
even
prevent
efficiency;
an
the
rest
der
Waals
in
from
colliding.
the
potential
coagulation
exponential
Since
factor
factor
of
the
the
significant
the
separations
reduces
them
V as
most
separations
a maximum
barrier
involves
V against
particle
creating
energy
the
energy
large
repulsion.
predominates
barrier).
energy
of At
the
governing
curve
in
the
Fig.
8
is
of
importance. 3 shows
the
$I= 0.5,
on
the
is
non-zero
of
density
of
particles
The
(or
the
Practically the energy
separation,
can
retarded barrier rr - r
is
is
(where
range is
of
the
the
(no
as
abrupt,
so
a
density
in
of
1=0.05
the
ionic
which
the
‘rapid’
to
interactions.
quantitative
in
the
unstable
strength
strength efficiency
coagulation
change
time)
and
surface
electrostatic
ionic
the
a = 0.67-10-6Cb/cm2. of
significant
the
strength
effect
regime
observation
on
undisturbed
A/(6kT)
by
attraction
of
number
state
the
criterion
of
the
solution of
coagula-
exist.
change
potential for
the
of
ionic
charge
influenced
A/(6kT)
presented no
surface
the
not
during
computations there
illustrate
stable
smaller
transition
stability)
a
This
is
kinetically
to
to
overlap.
The
dimensionless
11
For
energy
pairs.
(negative)
10 and
behavior
shifts
increases.
For
curves
from the
dispersion
5,
van
interacting
same
corresponding
particle
transition
the
efficiency.
the
the
the
the
Figs.
collision
of
of
have
sequence
where
effect
efficiency
particles
potential
the
energy
the
collision
when
the Fig.
efficiency
of
Fig.
tion
der
potential
in
(energy
between
The
sum*
of
shown
collision
The
the
repulsion
curve
1 i ttle
is
of
small
particles van
17 where now the
electrostatic
are
the
height
spherical
hydrodynamic,
Eq.
For
used.
(19)
der Waals
energy
is
v”R
i-
van
both
from
repulsive
separation
rate
for
particles
and
distances.
approximation of
accounting
approaching
the
interparticle
26)
collisionefficiency
and
can
.large (ref.
(Eq.
in
is
unretarded
potential
(Eq.
the
transition
from
to
13)
coagulation tc”
the
is is
the
used.
typically
double
layer
This at
slow
agreement
12) rapid occurs
a dimensionless
thickness),
*There is no firm basis for the superposition of the van double layer force. The interrelation of the two forces, understood (see discussion’in r&f. 19, p.'54).
of
is
used.
coagulation because particle
order
der Waals however,
1 where and the is not well
15
I
I
L
,,‘,I
1
L
I=O.OS
,
L,,,,
20
0.8 S 9 G 0.6 Liz B 5 0.4 Gi 3 9 0.2 -
Fig.
9.
Collision
I
efficiencies
L
,
of
I,,,,
0.8
*______--_--
-
in Brownian
L
1=0.25 _
particles
L
L
diffusion
(T=o.Os).
I,,,,
20___-.^
-
A
6kT Fig.
10.
Collision
efficiencies
of
particles
in Brownian
diffusion
(I=
0.25).
16 retardation The
effects
rapid
energy
of
are
not
variation
of
attraction
important.
the
occurs
collision in
the
efficiency
‘slow’
with
coagulation
the
van
regime.
der
Waals
Accord
i ng
to
I=O.5 >0.8
-
z w EO.6
-
B E =0.4i S uo.2-
Ii.
Fig_ Figs. of
Collision
9,
efficiencies
10 and
particle
11
size_
theoretical
the
transition
This
is
calculations
here,
so
the
appl
ications,
initially
is
Brownian
motion
is
from
stable
(ref.
slow
the
t:,2
is
to
in
reduced
Brownian rapid
diffusion
coagulation
experimental
Collision
for
time
system
in
with
28) _
half-life
monodisperse
particles
consistent
(ref.
dispersion The
of
are
time
states
of
which
the
number
to
is
results
efficiencies
one-half
the
most
(I=
0.5).
independent
(ref.
27)
very
small
and
practical
fi of
particles
original
value
in
1)
s
(20)
t1/2
= T&i
here
any
only
co11 isions
particle
interactions between
collision
efficiency
E(rT.r2)
=
where
in
between Eq.
an
by
ignored.
particles
defined
in
Eq.
17
(Eq.
is approximate
20
of
radius
is
equivalent
r
are
since
considered)
_
(21)
hydrodynamic, particles
21 reduces
to
The
to
S t1/2 t1/2
t,,2 the
as
are
primary
are
(ref.
van
der
considered_
i
21)
Waals
and
electrostatic
For water
at ambient
interactions
temperature
(20°C)
17
V2
= E(r,
,r2)
N is
the
where
2.10’1 7 number of
The number of 10’
cm-a
have
(22)
corresponding
particle
number densities
efficiency
smaller
Consequently,
by the
and Hull electrolyte of
with
is
actions
occur
equally of
valid
slow
10 and
11
the
Their
polydisperse
particles
(ref.
is
systems
population
of
practical
Honig
the
critical
in a monodisperse of
energy
and
waters
coagulation,
interest.
criterion
strength
24)
all
rapid of
order
A co1 1 ision
and computed
coagulation
ionic
for to
of
Natural
29).
dispersion
van der Waals
a polydisperse
the
rate
van der Waais The
work described
given
tend
to
collision
rate
applications concentration forces
coagulation
particles
both
on the
sediment
layer
of
for
the
the
system
onset
i t can
particles
the
Their
be used
for
of
electrolyte,
attraction.
and when hydrodynamic
thus
to
inter-
evaluate
any combination
can
of
of
coagulating
serve
this
in Table
particles
results
particles
of
of
the
unlike
purpose3.
treatment, onset
determined
solely
for
is
experiments
with
account
into
long-range the
abrupt,
the
particles.
hydrodynamic
interacting
dependence
particles.
purification,
rapid
Brohnian
hydrodynamic,
functional
is
relative
mobility
and
the
of
important.
Once co1 1 isions
forces)
of
For
prediction
coagulation
coagulation. from
the
for
two approaching
water
hydrodynamic
coagulation
take the
only
of
improve theory
and the
to
of
occur, the
Hamaker
so a quantitative
criterion
of
exist.
From well-controlled From the
of
discharges)
the
to account
and rate
sizes
been
classical
between
potential
collision
relative
determine
The onset
stability
the
has
1)
interactions
in ocean
is
paper (ref.
efficiencies
(wastewater
rate
(modified
constant.
layer
van der Waals
affect
practical Double
co1 1 ision
and double
short-range
forces
in this
by Smoluchowski’s
The computed
diffusion.
of
for
of
example,
(ref.
from
surfaces_
of
the
cme3
interactions
onset
for
= E - 55 hrs.
ONS
CONCLUSI
collision
of
a stable
in seconds.
is,
t,,2 10’
9,
is
u and A.
The aim of
the
the
charge
and
of
transition
in Figs.
a function
between
stability
VII,
the
is
t l/2 sludge
lOs-
hydrodynamic at
charge
time order
the
curves
constant
surface
analysis
1,
the
cm3 and sewage
imp1 ies
only
ignored
coagulation
particle
the
in
30)
of
0.001
concentration
particles
rapid
than
bend
(ref,
per
in primary
to a half-life
purposes. given
parti cl es
particles
an
‘effective’
calculations sizes
can
The curves
Interpolation
can
an
initially
monodisperse
Hamaker constant presented
be estimated. in Figs.
be used
here
the
Tables
5 and 6 are for
intermediate
can
suspension
be determined.
collision
efficiencies
1 and 2 and Figs. given
in parametric
values
of
the
5 and 6 form
Hamaker
18 The
constant. sizes.
Dynamic which
of
Equation involve
TABLE
E(rl
to
and
obtain
coagulating
is
.r,)
interest
so
thris it
realistic
obtained
can
be
results
particles,
(ref-
for
all
efficiencies
der
Uaals
in
modeling
for
Brownian
diffusion;
5
3
1
lo-* 10-3
I_0040 I_0042 I_0053 1_0098 1.0248 1.0691 I-1983
;::I: IO0 10 102 der
Slaals
and
‘2”1
retardation
1.0027 I_0028 l-0035 1.0064 1.0157 1.0435 1.1254
10
1.0022 l-0023 l.0025 1.0037 1.0075 1.0189 l-0540
:-x8;: 1 :a030 1.0040 1.0116 l.0251 1.0905
hydrodynamic
20
50
100
1_0021 l.0022 l-0023 l.0029 l.0049 1.0120 1.0300
1.0021 1.0021 1_0021 1.0024 1.0032 1.0049 1.0142
1.0020 1.0020 1 .ooto 1.0022 -1-0026 l-0040 1_0082
forces
5
3
1
10-4
0.2409
lC-’
10-z lOi=
0.2791 O-3286 0.3867
:: 102
5.5546 O-5477 0-7194
TABLE
10
20
50
100
O-4810
0_36lS 0,-k;
O-2971 0.3401 Cl.3931 0.4512
0:5207 0.6841 O-7430 O-8266
0.6562 0.5806 O-7700
0.5150 0.5981 O-7335
0.8763 o-8936
0.6198 0.6620 0.7060
0-8796
0.8659 0.8862 0.9070 O-9341
0.9101 O-9237
2
Collision den-
efficiencies waa1s
‘2~‘l
for
Brarnirn
diffusion;
retardation
parameter
==l
forces 1
3.
5
10
20
50
1_0021 1.0021 1.0024
1.0011 1.0011 1_0012 1_0019
100
A/&T) 10-b 10-x IO-’ 10-1 100 10 10’ der
Uaals
r2/r1 A/
1.0041
1.0027
l-0043
1_oots
1.0061 1.0159 1.0568 l-1866 IS546
1.0041 I-0108 l-0375 1.1198 l-354?
and
hydrodynamic
1
3
l-0024 1.0026 1.0035 l-0083 : xz5 1:2572
1.0022 1.0023 1.0028 l-0056 1.0169 1.0520 l-1537
l-0040 1.0101 1.0296
l-0045 l-0127
l-0866
l-0379
IO
20
50
O-6276 0.6630 O-7112
l_OOll 1.0011 l_OOll I_0015 I.0028 1.0071 1.0201
forces 5
100
CkT> IO-' 10-3 10-z 10-x 10’ :x2
General
parametera=O.l
A/&T)
van
particle
the
31)-
A/CkT)
“a”
of
into of
forces
‘2”l
van
pairs
incorporated
’
Coiiision van
function
(r1,r21,
O-2410 0.2797 O-3331
O-4078
0.5181 O-7025 l.0955
o-2996
0.3647
0.4862
O-3411 o-3992
0.4090 o-4694
0.5298 o-5879
O-4755
O-5450
O-6553
0.5775
0.6384
~- _-
0.7362
0.8217 0.9682
X-zz: o:a441 0.8740 0.9019
X-t;;: 0:slos o.Yz8l 0.9437 0.9603 0.9844
physical
processes
19 TABLE
3
Apprmtimations hydrodynamic E(r,,r2)
=
for collision efficiencies and van der UaaIs forces a +
bx
*
Retardation
parameter
A/(kT)
a
10-2
Retardation
bx
0.25950
*0.88410 2 C r2/rl
IO-’
*valid
for
VIII.
ACKNOWLEDGEMENTS authors
NA82RAD00004 Quality
accounting
for
‘2/r,
10-e
dx
1O-6
2.6953 2.9043 3.0409
1.3845
1 cx
3.0439 3.0405 3.0423 Z-8985 2.4525 1.4595 O-34764
0.31685 0.39643 0.51062 *Ct.69014
Assessment
cx
-4.9962 -5.3031 -5-4760 -5.4055 -5.0576 -4-3310 -2.4753
o=
0.21770
x =
Brorrnian diffusion for lc r2frl< 100)
o-1
2.9593 3.0338 3.0339 2.9251 2-6954
a
The
dx’.
lO-2
parameter
A/(M)
*
Q= bx
0.21811 0.25878 0.3r151 0.37254 0.44285
10-1 1 10 102
10-p 10-s 70-z 10-1 1 10 10’
cx’
in (valid
10-u
dx
1O-6
Z-8666 2.9227 3.0742 3 -0677
-5.2527 -5.3293 -5.5222 -5-4343 -4-6867 -2.7279 -0_5300
c 100
gratefully
Division and
Laboratory
acknowledge
- National
the Mellon at
Ocean
Foundation
the
financial
Service through
- NOAA
support
of
through
Grant
a grant
to the
the Ocean No.
Environmental
Caitech.
REFERENCES 1 H, Smoiuchowski, Physik Z., 17(1316)557. W.U. Keck Laboratory, California Institute 2 J.R. Hunt, Report No. AC-5-80, of Technology, Pasadena, California, 1980. 3 J. Chandrasekhar, Rev. Mod. Phys., 15(1943)1. 4 A.L, Spielman, J. Colloid Interface Sci., 33(1970)5625 G.K. Batchelor, J. Fluid Mech., 74(1376)1. 6 A. Einstein, The Theory of the Brownian Movement, Dover, New York, 1926. 7 M. Stimson and G-B_ Jeffrey, Proc. Roy. Sot. London Ser. A, 111(1926)110. 8 H.C. Hamaker, Physica, 4(1937)1058. 3 J.H. Schenkel and J-A. Kitchener, Trans. Faraday Sot., 56(1960)161. 10 H.B.G. Casimir and D. Polder. Phys. Rev., 73(1948)360. 11 J. Mahanty and B.W. Ninham, Dispersion Forces, Academic Press, New York, 1976. 12 G.R. Zeichner
13 14 15 16 17 18 13
and W-R, Schowalter, J. Colloid Interface Sci., 71(1979)237. of Aerosols, Pergaman Press, New York, 1364. N.A. Fuchs, The Mechanics H. Brenner, Advan. Chem. Eng.. 6(1966)328S. Twomey, Atmospheric Aeroscis, Elsevier, Amsterdam/New York, 1377. A. Schmidt-Ott and H, Burtscher, J. Colloid Interface Sci., 89(1982)353J. Lyklema, Advan. Colloid Sci., 2(1968)84. E.M. Lifshitz, Sov. Phys, JETP, i(1356)73. B. Vincent and S.G. Whittington, in E. Hatijevic (Ed,), Surface and Colloid Science, Plenum,Press, New York, 1982.
20 20 F. Gelbard. Y. Tamhour and J.H. Seinfeld, J. Colloid Interface Sci., 76(1980)54121 E.J_W. Verwey and J.T.G. Overbeek, Theory of the Stability of Lyophobic Co1 loids, Elsevier, Amsterdam, 1948. 22 W, Stumm and J-J. Morgan, Aquatic Chemistry, Wiley-Interscience, 1981. 23 J. Lyklema and H.P. Van Leeuwen, Adv. Colloid interface Sci., 16(1982)227. 24 1-A. Valioulis, Ph.D. Thesis, California institute of Technology, Pasadena, California, 1983. S. Levine and L.N. McCartney, J. Colloid Interface Sci., 25 G.M. Bell, 33(1970)335. 26 B.V. Derjaguin, Discuss. Faraday Sot., 18(1994)85. 27 R-H, Ottewill and J-N, Show, Disc, Faraday Sot,, 42(1966)194, 28 E.P. Honig. G-J. Roeberson and D.H. Wiersema, J. Colloid interface Sci., 36(1971)97. 29 C.R. O’Helia, Environ, Sci, Tech., 14, 9(198o)lo52. 30 E-P. Honig and P-H. Mull, J, Colloid Interface Sci., 36(1971)258. 31 1-A. Valiouiis anti E.J. List, Environ. Sci. Tech. (in press).
Advances in Colloid and Interface Science. 20 (1984) l-20 Ekevier Science Publishers B-V.. Amsterdam - printed in The Netherlands
COLLISION
EFFICIENCIES
DER WAALS
AND
IRAKLIS
A.
VALIOULIS
and
Institute
of
California
SPHERICAL
PARTICLES:
HYDRODYNAMIC,
Pasadena,
California,
USA
OF DIFFUSING
ELECTROSTATIC
V%
FORCES
E.
JOHN
LIST
Technology,
CONTENTS I.
ABSTRACT
II.
INTRODUCTION
III.
HYDRODYNAMIC
IV-
VAN
V.
COLLISION
VI.
DOUBLE
..~...~~.........._..~~.~.~~.~~~~~~.~~~~.~~~~..~.~.~~~.~.
INTERACTIONS
DER WAALS
FORCES
LAYER
CONCLUSIONS
ACKNOWLEDGEMENTS
3 5
_--.__._____--______~~.--~---~-..~~~...._.~
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..-.-.---_-*.---12
FORCES
VII,
1
.~.__.~~_~_~~~__~~~_~-~~~~~.~~~~.~~~...~
~~~_~~~~~__~~~~~_~~~~~~~~~~~~~~.~~~~~~~~~.~~~
EFFICIENCIES
VIII.
_I..~~-~~~~.~_.~~-~_~..~~~..-~~..~~....~...~...~~.....
17
~-~~~~_~~..~_~.~~_~~..-_..-___~~~~~~~~__~_____.______19
ABSTRACT
1.
A practical mate
for
1 imitation
the
Brownian
collision
motion
particles
in
causes
in
surround of
1
~..~~...~...I.~....~.....~~..~~..~~~~.~~~~~~~~~.~~~~~
between
the
of
forces
der
collision
the
can
van
Waals
are
energy
of
this
computed
various
the of
the
for
which or
they
corrections
such
forces for
ionic
in suspended
particle
ions
Results and
convenient
For
forces)
when
esti-
particles
paper
sizes.
attraction
a way
cloud
layer
In
classical
forces.
disturbance
the
(double
forces).
of
in
the
from
‘s
spherical
interparticle
from
particle
probability
presented
diffusing of
arise
Waals
particles
der
are
two
Smoluchowski
of
forces),
charged (van
approaching
electrolyte
ication
of
(hydrodynamic
origin
two
values
such
fluid
Smoluchowski’s
appl
non-consideration
electrically
molecular
the
probabiiity the
water
the an
is
‘of
can
operate several
strength
particle
be
to
of
collision
modeling,
INTRODUCTION
II.
Suspended
which
particles
They
flows.
the
of
the
to
predict
affect
are both
suspension
individual the
0001-8686/84~$06_00
is
0
bulk
contact.
of
the
and
in
most
properties
in
particles behavior
ubiquitous the
the
environmental of
the
Information properties
suspension_
on of
The
or
fluid the
the
industrial the
physical
flow
knowledge
1984 l3lsevier Science Publishers B-V_
and
is of
surfaces
with
characteristics required
the
in
fluid-particle
order
2 interactions, which collision mass
the
important
in
For
is
in
given
rate, the
and
collision
are
much
equal
t
r 2 with
1 radius
r_,,
surface than
the
is
B(r,,r2)
=$y k
fluid
dynamic
is
move
interparticle
causes an
in
in
flows
in
of
suspended
thus
become
in
which
it
is
particle
respective
particles
number
probabi
is
that
encounters
of
fluid,
diffusion
assumed
1 ity)
suspension
occur,
then
with
radii
r,
N,
and
concentrations B(rl,
r2),
the
the
and
r2
N2
representing
the
process:
(1)
r 2)
is
time
the (ref.
function
of
3) the
asymptotic
by
two
flux
D,i-02, The
a
molecular
computed
pair
sphere
viscous
equal in
to
the
volume
mean
free-path
solving
a
to
the Di
relaxation
of
surface is
a
t =
fixed
the for
sphere
diffusivity
of
is
infinity.
time,
the
fluid
equation
function
at
When
.the
specifically,
of
the
two
the
fluid.
Ptore
distribution unity
of
diffusion
particles.
where
and
common volume
unit
held
Then 2r’/9v,
at
for
the
of
particle zero
times collision
(r, + r212 r r i
2 constant,
T
is
the
absolute
temperature
and
u
is
the
viscosity. ignores
on
particle
straight
fluid
interactions,
paths,
modify
their
arise
from
can
However, relative the
(hydrodynamic
cha-ged
molecular
origin
collision
enhancement) collision
Brownian
If
of
unit
than
Particle
forces the
binary
6(r1. per
fixed
forces
electrically
1 inear’
distribution
matter
flows of
N,N2
i=1,2.
Boltzmann’s
such
r2)
the
the
expression
particles
only
Brownian
diffusivity
for of
function
where
to
2)_
volume
the
larger is
is
r
This
of
process
interactions
O_lum-1Oum
(co1 1 ision of
distribution
radius
larger
the
the
suspended
(ref.
their
sweep
8fr , , r-2)
the
of
B(rT,
function
pair
unit
probability
particles
water
per
=
particles
at
modifies
of
range
that
function
rate
collision
so
dynamics
diffusing
with
fare
size
product
co1 1 ision
co11 ision
the
modeling
Particle-particle
mechanism
dilute
by
geometry
The
the
successful Coagulation,
particles,
space,
the
coagulation
collision
a
for
other.
occurs.
sufficiently
and
sufficient each
of
size
quantifying
particles
dominant
the
not with
coalescence
particle
coagulation
is
interact
and
in
is
however,
particles
(van
rate
particle der
Waals
E(rl,r2) (as
given
(double forces).
by
that
motion.
then Eqs.
from layer
the the
1 and
assumes
suspended
presence cloud
forces))
which 2)
it
approach
For
A correction defined
is.
particles
disturbance
forces),
is
as
and
of
of or
ions
they
(collision multiplies incorporates
that
each
other,
particles the
which can
in
particle
be
surround of
efficiency the
or
‘recti-
the
influence
3 of
the
inter-particle
In
this
disperse
forces
paper
we
spherical
particles double
and
Hamaker
constant,
presented
in
to of
A,
a way
Smoluchowski applies
are
The
motions
first
sizes
in
flow
over
relative
for
Spielman
(ref.
efficiencies
Stokes’
Results the
by
collision
of
4) two
accounting
a wide
size
particle
of
range
the
of
for
mono-
van
values
particles,
co1 1 ision
for
diffusing
of
der the
‘2/r,,
are
modeling.
INTERACTIONS 1)
(ref.
extremely
considered.
Brownian
of
process.
used
the
forces. and
collision
a method
compute
convenient
‘s to
the
various
layer
HYDRODYNAMIC
III.
tion
extend
suspensions
Waals
on
classical dilute
two
model systems
particles
independently
are of
for
Brownian
where
only
treated
each
as
other
motion
binary rigid
with
induc_ed
coagula-
particle’encounters
spheres
a constant
describing relative
diffusion
coefficient
=
%2
D,
f
where
the
D1 =
kTb,,
are
D2
,
single
this
particle
D2=
functions
Stokes’
(3)
the
For
formulation
a
particle
motions generates
as
as
location
of
the
first
as
the
such In
an
a
velocity
unbounded at
particles
2
advective
flux
conservative
of
particles
point
in
space,
relative JF. force
to The
particle latter
F derivable
tend
s-=
causes induces Thus,
relative
extending
system
any
is
at a
are
determined
b =
1/(6nur).
to
correlate
The
motion
distance
particle
Eq.
of
3 becomes
However, the
of
s
in
located
a velocity
by
order
one the that
at s-’
at
the
increasingly
decreases.
the by
which
order
5).
separation modified
interactions
assumed:
(ref.
which
decreases.
of
which
b2
mobility
forces
gradient
dipole
and
the
separation
particle
4)
r
gradient
force
particle
(ref.
particle
particle
This
act
b,
radius
hydrodynamic
the
fluid. to
Spielman
mobilities of
a velocity
distance
invalid
particle particle
ignores
particle surrounding
coefficients
kTb2
of
law.
diffusion
a the
1 due
to from
a
coefficient (ref.6)
hypothetical mean
arises from
diffusion Einstein’s
dynamic
radial
number
Brownian the
potential
V and
of
account
flux is
an
acting
is JD of
balanced
arbitrary between
for
argument-
equilibrium
density
diffusion
action
to
ingenious
by
an
steady the
particles:
JD f
JF=
0
(5)
4 JD = -D,2 (dN2/dr), where
N2 is
velocity
to
where
b is
Under
the
N2 = N;
N” is 2
relative
the
=
flux
JF = -N2
conservative
relative force
radial
F
mobility the
which
number
is
a function
density
of
of
particles
separation. 2 must be
(8) of
relative
particles
particle
2 at
diffusion
infinite flux
inter-particle
is
(D,2/(kT))(dV/dr)N2
induced
by the
(9)
conservative
force
F
b(dV/dr)
(10)
The hypothetical 3 and
equilibrium
10 the
situation
relative
particle
(Eq.
5)
is
invoked
then
to deduce
from
diffusivity
= bkT
(11)
which
is
it
now assumed
is
is only become exact
a function
of that
justified
if
superposable solution
centers
obtained
irrelevant
in Stokes’ equations
unphysical
behavior
order
of
the
which
diverge
of
to
for
the
ignored
so
mobi 1 ity
two spheres
Einstein
force that
F is the
b can moving
(ref,
7).
Both
of
centers
of
considered,
two approaching
becomes by the
molecular particle
Following
when the
since
6) This
two fluxes
be computed
from
the
along
line
of
the the
(ref.
removed.
their
rotational
motion
particles all
are
motions
are
4).
between
explained
line are
motion
fluid at
are relative
particles
force
even
and Jeffery
flow
is
The
equations
by Sthnson
when spherical
separation. valid
effects
_ 5) _
perpendicular
The hydrodynamic
forces
11 is
inertial
Stokes’
linearised the
interparticle Eq.
(ref
of
the motion
independent
of
particle
number density
JD = -D12(dN2/dr)
and
2 and u the
by the
,
the
D12
particles
(7)
conditions
Then
Eqs-
of
particles
F = -dV/dr
exp(-V/(kT))
the
(6)
distributed
distance.
and
the
equilibrium
Boltzmann
where
number density
imparted
u = bF,
Here
the
JF = N2 - u
singular breakdown
mean free
contact
particles at
can
path.
zero of
determined
separation.
continuum Van der
be considered
flow
Waals
from
the
This at short
to overcome
distances range this
5
difficulty
in
the
collision
VAN DER WAALS
IV. The
attractive
dipoles
by
of
formula
for
vA Here
r
the
is
Schenkel
-
the
Waals
arise
in
electron
assumed and
(ref.
(
from
additivity and
the
synchronized
clouds of
energy
the
of
the
inter-
pairwise
derived
his
VA between
inter-
well-known
spherical
particles.
r2
(12)
‘1 -r2)
particle 3)
the
the
molecules
2rl r’-
between
recommended
forces
interaction
2rl r2 (r,+r2)zt
Kitchener
and
8) atoms
der
distance
and
formula
van
Waals
charges
(ref.
constituent
A 6kT
--
iz=
the
der
fluctuating
Hamaker
bodies.
actions
FORCES
London-van
created
acting
problem.
centers
incorporated
best-fit
and
A
is
the
retardation
approximation
Hamaker
effects
to
their
constant.
in
Hamaker’s
numerical
integrations
vA kT=
A -- 6kT
vA kT=
‘2
w
1
1 f
1.77~
’
--
where the
%&
p =
2nh/a
and
particles,
a
than
induced waves
and
10). it
The
render
separation been
then
that
The
the 11).
(ref.
in
A
is
only
12)
involving
mation
action
can
(ref.
used
ments
a
due
interparticle
the
the
to
Eq.
13
account Hamaker
the
1OOnm is so
of
the
when
finite
Eq.
from kinetics
Smoluchowski
of
of
for a
A, 13,
the
for
because
of of
Hamaker
the
its
the
of
effects. particle
simple
a
polydisperse
diffusing 4)
form, from
particles.
13 may
(ref.
the
electromagnetic
constant
Eq. in
i-s
of
is by
characteristic
retardation
function
coagulation
distance emitted
single
population
foGc&
between
length;
efficiencies
propagation
experiment,
equation
conservative
collision
a
‘effective’
an
distance wave
radiation
only
monodisperse such
London
interparticle
the
qualitatively
an
the
the
time
Lconstant’,-
minimum the
particles.
of
incorporates
calculate
p 2, 0.57
’
problem, size
occurs
(13)
dimensionless
k =
wavelength
coagulating
modeling
the
Nevertheless, to
obtained
generalized of
is
Since
wavelength latter
in
characteristic
dipoles
is
absolute
retardation
the
(ref.
the
1
- %$r
and
length
of
Electromagretic larger
h
(r-r2-r,)/rT
another
function
X/r,. -
cx =
h =
X introduces become
0 < p < 0.57
be
a
particles
has
experiProvided
good
approxi-
suspension. under
the
6 DN2
-divJ,2
at= with
N2
boundary
= D
The
and
steady
r2
D
[(
12
aN A+N2bx ar
dVA
,
(14)
)I
conditions
and
N2 = N;
a --I t-2 ar
=
VA = -m
when
r = r +r 1 2
VA = D
when
r = m
state
particles
2 into
-4arZJ
=
(19
equation
solution
of
this
a sphere
of
radius
gives
the
diffusive
flux
J12 of
r,+r2
4n D_ NT (r,+r2)
(16)
D12 is
where
the
interparticle rate
depends
A collision
E-‘(r
as of
relative
forces
particle
and
on the
s the
integral
efficiency
of
can
the
in the
coefficient
separation
particle
s=
interactions
absence
of
any
The collision r/r 1’ over all separations.
be defined
(17)
l”2)
the
enhancenent
any
of
interactions
(ref,
13)
V,
COLLISION The
co11 ision
between
stability
relative
the
the
obtained
by Stimson
and Jeffery
the
convergence to converge n th- partial
performed separations
particles.
coefficients,
by sumning
assumed
over
the
co1 1 ision
E(r,,r2)
is
were
determined
rate
the
in
inverse
the of
absence. Fuchs’
EFFICIENCIES diffusion
separation
A single
rate
factor.
particle
is
diffusion
dimensionless
criterion when the sum of
to a precision h c 0.001,
D,2. series
(ref.
7)
c=O.OOOl condition
a series. of
the
the
thirteen
asymptotic
solution (as
corrected
was
used
I(S,,T All
the
significant formula
as
to Stokes1 for
a function
by Spielman, each
-S,)/Sn] numerical figures.
series,
c c was
ref. which
4). were
fulfilled;
calculations For
of
equations
were
dimensionless
Sn
7 developed the
by Brenner
series The
the
larger
the
results.
of
of
the
particles
of
13)
equal
son between
the
retardat
parameter
less
ion
than
retarded as
rl
(or
as
as
computed very the
of
Q= 0.1
collision
forces with
with
relative
for
a compari-
values
of
the
separations
s
The curve
the
unretarded
for
the
potential
to
the
1,
(Figs.
5,
A/(kT)
examined
and a=1
the
as
results
illustrated
to hydrodynamic
of
For
forces
Waals
operate.
energy
to dominate
the
of
collision
range.
potential
for
this
both
equal
provided
the
in Fig. for
(with
X= 100nm. to
size
a check
for
at
more between
7 where different
the
A/(kT)
relative
particle
size
lower
and Burtscher of
pairs
at
than of
in
to a
all
and
of the
the
and are (ref.
15).
particles values
about
similar
the
of
than
interacting
a= 0.1)
particles
reduction
3
The enhancement
retardation,
less
co11 ision
corresponds
are
include
= lum or
on the
shown in Figs. 0.1
efficiencies not
efficiency (r,
particles
a=
and the
by Schmidt-Ott
collision
is
rI=O.lrrm)_
absolute
who did
particles
reduced
particles
considered
the
obtained
large
forces
interacting are
The computed
smaller
is
the
forces
15),
reduce
and for
efficiency
tend shorter
results;
corresponds
decrease.
forces
6 and 7)-
van der
2
H represent
and hydrodynamic
unretarded
respectively
by Twomey (ref_ with
the
W in Fig.
marked with
the
forces
Spielman’s
size
decreases
particles
Hydrodynamic
as
become of
van der Waals
r, =lum rate
calculated
well
and e=
radius
is
and
calculations.
the
when only
in agreement
der Waals
The hydrodynamic
van der
of
interacting
van
Waals
validity
effect
12)
electromagnetic
marked with
curves
importance
increases.
The effect
due
different
for
(Eq.
u -f = corresponds
curves
the
the agreed
collision
limit
by the
ignored;
increasing
The efficiencies
efficzency
are
when both
assumes
particles
ones
the
curve
r is 2 alter
not
1 is
dimensionless
significantly.
the
The
efficiencies
Fig.
distances
potential
represented
efficiencies
attraction
the
interparticle
more
where
did
unretarded
collision
for
for
formula.
the
500,
range
A/(W).
potential
1 approaches
interactions
Retardation
particle
larger
of
collapse
-z r/r-2<
the
compute
for
separation.
lo-’
both to
Simpson’s
account
integration
values
retarded
Q increases);
calculations
collision
and 4 for
used
various
attractive
since
case.
In the
process
the
retardation, were
and
calculations
using to
particle
separation
The curves
in Fig.
decreases
decreasing
extending
for
for the
potential
hydrodynamic
the
and
0.001;
reduces
unretarded
with
of
unretarded
about
retardation
numerically
potential
0~.
up the
was used
step
a dimensionless
size
speeds
integration
significance
(Eq.
this
separations.
17 was performed
two particles;
the
retarded
small
integrand
over
the
To assess the
in Eq.
ranged
was used;
at
decreasing
variation
integration
14)
slowly
integration
A successively rapid
(ref,
converges
20,
size.
collision
at various
of
the This
efficiency values
COMPARISON OF POTENTIALS ------
0
02
_ 0.4
UNRETARDED Q=1 0 = O-3 Q = 0.1
0.6
0.8
1
DlMENStONLESS SEPARATION h
Fiq
_ 1, Dimensionless tion
for
van der Waals potential vs, various values of the retardation
dimensionless parameter
EFFECT OF RETARDATION q
-
1 “‘““I
’ “““1
’ ’ ““‘I
’ “F
UNRETARDED
--Qal QXO.3 ----.--Q~O_,
I
___& cI I
t
o-:
5
104
IO-’
go-2
lo-’
IO0
IO’
HAMAKER GROUP A/(kT) Fig.
2.
separa-
,./*1 a
i
)
-
particle QI
Effect of retardation on the collision cles for various values of A/(kT).
efficiency
102
of
equal
size
parti-
9
VAN CER WAALS’ FORCES
-
r&r,
r
=1
--r&r, =3 cl=0 .I ---Q/r* =5 ----r&r, =I0 --rz /r, = 20 --_.m- r&r, =I00
If= 16.
IO-=
/
IO-'
10-l
IO"
IO'
4/ IO2
HAMAKER GROUP A/(kT) Fig.
3_
Coilision efficiencies for various values of (a= 0.1).
VAN
DER
of particies with various relative sizes and A/(kT) when only van der Waals forces operate
WAALS'
FORCES
1.6
IO* HAMAKER Fig.
4.
Collision efficiencies for various values of (a= 1).
IO+ lop GROUP A/(kT)
Id
Id
of particles with various relative sizes and A/(kT) when only van der Waals forces operate
10 of A/(kT) dynamic
is shown. and
van
: ER stands
der Waals
HYDRODYNAMIC
AND
VAN
for
forces
DER
the
collision
operate;
WAALS
----
A/(kT)
= 1O-3
-----
A/[kT)
= IO-+
_-.__-
A,(kT)
= I‘,-’
----.-
A/( kT) = IO
EW
efficiency
is the
when
collision
both
hydro-
efficiency
when
FORCES
-..--.- ,?,/(kT]= 10”
0
25
50 r,‘=
75
100
I
Fig,
5_ Collision efficiencies of particles with various relative sizes and various values of A/(kT) when van der Waals and hydrodynamic forces operate (a=O.l).
only
van
particle
der Waals
forces
attractive
act,
energy
theoretically
impossible
force
the particles
between
The
curves
decreases. since
shown
In the
in Stokes ' flow
grows
without
approach
limit
zero
as
the hydrodynamic
bound
as
the
A -+ 0 collisions
the
for
interare
repulsive
particle
separation
decreases_ For
small
A/(M)
particles
the effect
of
(r,=O.lum the van
particularly
strong
(see
efficiencies
as the
ratio
of
the hydrodynamic 6:
the collision
with
the
rest
Reported equal
size
of
Fig.
4).
the
efficiency
This
on
reverses
size
Witness
of equal
large
collision
the
trend
r22> r,) decreases
force.
at
the
the
particles
toward
caused
upper
of is
smaller
by
two
is very
values
efficiency
the action
curves
large
in
compared
curve.
experimental particles
of equal
forces
r../r, [where
interparticle
Fig.
or a=l)
der Waals
collision
in the
size
efficiencies range
O.lum
range to lum
from
(see
0.35
ref.
to 0.7
12 for
for
a recent
I1 survey),
a
Hamaker
result
which
constant
of
according
about
HYDRODYNAMIC
to
2 - 10-l’
Fig.
5
Joules
AND VAN DER WAALS’
-
AI
= C4
---
A/(kT)
= lO-3
---
A/CkT)
= IO-”
--.-
A/(kT1=
----.-
A/(kT)
= IO
-------
A/CkTl
= 50
--.-
A/( kT)
= IO2
implies
(at
a maximum
300°K)
for
value
the
for
retarded
the potential-
FORCES
IO-’
Ct=l
J
0
25
50
75
100
‘2/‘1 Fig.
Collision various operate
6.
According colloids
to in
efficiencies of particles w h en van values of A/(kT) (a = 1) _
tyklema
ranges
to
Hamaker
corresponding
the
groups
respectively
(for
efficiencies
determined
Estimation
tric
IS)
ionic
complicates
indirect tions-
the
species the
in
of on
determination
experimental Experimental
of a
A
about
25
collision
can
the
of
A
and E(O)
hydrophobic
2.
and
lO_”
Joules
0-06,
efficiency which
of
carried
respectively. of
are
the
and
out
dielectric
be the
more
in
about the
0.75
range
of
not
medium.
well
known
radiation data
constant
Lifshitz’s
w dependent
dispersive
the
promising
Hamaker
by
frequency
the
electromagnetic
from may
be of
particies
on
determination
most
about
for
experimentally-
function
measurement
relative size and hydrodynamic forces
of
to
potential),
knowledge
the
the
solution
to
retarded
constant
requires
E(W)
information
the
constant Joules
300°K)
correspond
Hamaker
This
permittivities
incomplete the
of
method.
lo-”
(at
to
(ref.
these
Hamaker
about
and
collision
5,
from
According 0.35,
Fig.
17).
(ref.
water
with various der Uaals and
(ref.
and
suggests
for
practical can
be
di elecHowever,
effect
of
19) that
its
appiica-
carried
out
indirectly
-from
wetting,
the
,_mr---
r2/r,
-*1**..’
’
10-S
coagulation layer
means of
the
reliable
process
are
of
then
sizes
can
of
the
value
on surface
film
thinning
tension (ref.
IO'
A/(kTl
monodisperse and
the
dispersion the
* .
since
be obtained
in
---I
IO2
the
systems
(or
the
readily
from
is
Figs.
5 and 6) Hodeling
5 or
Fig.
Waals
that
determined
by
efficiency can give
of
Dynamic
efficiencies
Fig.
assumed
A collision
General
collision
is
rate
12).
constant.
via
it
coagulation
(ref.
Hamaker
suspension
then,
and
19).
. * -m
*.‘.***’
calculations
of
in a polydisperse
the
is
a
coagulation (ref.
20)
between
Equation
particles
of
charge.
Since
6.
DOUBLE LAYER FORCES_ Dispersed
dispersion
particles is
opposite
cally
excess
forces
due
extend
tion
of
to
is
formed
oppositely to
the
in ‘the outer to
the
outwards
counter
thermal from
ions
waters
neutral,
Close
ions of
According
established
in natural
electrically
sign.
adsorbed
the
med i urn.
to
. *-‘**-*’
GROUP
numerical
can be accomplished
of
16'
negligible
half-life
reasonably
of
data
of
7
* **‘**J
experiments
and
VI.
from
kinetics
= loo
HGAKER
forces
determined
unlike
the
Evaluation of the relative importance of hydrodynamic and van der forces in modifying the co11 ision rate of t-wo spherical particles.
7.
rapid
coagulation,
and
r2/r,
= 5
01 . . **..A
double
of
adhesion
EFFECT OF HYDRODYNWIC FORCES . . ‘1”“, . * . L”“, . . . .“-‘, . . * 8”.‘, . * =l“V ra/r, = 1 ra/r,= 20 Q=t r&r, = 50 -r&,=3
125
Fig.
kinetics
adsorption,
the
the
(Stern
layer). ions
the
the
particle
diminishing
with
The outer
surface distance.
ions)
(ref.
between ions.
carries
a compact
(counter
model
layer of
phase
surface
Gouy-Chapman (diffuse)
an electric
aqueous
particle
charged
motion
carry
21)
charge
specifi-
(Gouy)
layer
dispersing
the
an equilibrium
causes
the
of
of
electrostatic
This into
an equal layer
forces the
solution,
diffuse the
the
consists is and layer concentra-
13 This
local
induces ficant
distribution
double
layer
of
charges
interaction
simplifications
are
in an electrically
forces
needed
between
in order
neutral
approaching
to describe
solution
particles.
Signi-
quantitatively
the
repulsion
J’
y$&= AV: energy
Fig.
8.
Schematic illustration surface separation.
interparticle charged are
and Van Leeuwen of
natural
a thin
equilibrium
suggests
that
for
particles
the during
outlined
a method
particles
to
the
the
be presented
diffuse
layer
the
the
of
which
density
scale
of the
between
size (ref.
the
potential His
spheres
same
range
O.lFrm
22).
Lykl ema
the
interaction of
electro-
size
of
the
potential
remains
Valioulis
(ref.
between
expressions
are
superposition obtained
negatively
analysis
with
surface
particle
encounters the
Brownian
interaction.
linear
carry
Their
compared
than
of
restoration
particles.
electrostatic
the
the
the
the
of
particle
but
size
of
for
Brownian
charge.
in
scale
thin
rather
He invokes
interaction
the
system
sizes
their
scale
are
the
same negative here.
time
time
a function
binary
particles with
as
dilute only
different
compared
surfaces
charge time
that
suspended
layers
to compute
carrying
not the
surface
so
have
compared
on the double
constant
can
with
energy
A sufficiently
layer
23)
potential
assumed
waters
particles
chemical
will
is
double
(ref.
two colloidal
the
forces.
The particles
In most
lOurn have
layer
particles
considered.
charge. to
double
spherical
of
barrier
by Bell
two spherical lengthy
and
approximation et
al.
24)
(ref.
.25)
for
Derjaguin’s The s ion
be
computed
the
v =
VA
The
salient
features
electrostatic
of
behavior
separations
the
subject
Waals
and
interaction
at
Brownian
van
of
der
constant
diffu-
layer
energy
attractive VE
to double
forces
the
Waals
two
potential
surface
VA
charge
curve 8.
outweighs
two
the
the
the
interaction
smai!
and
This
particles
and
can 17)
(Eq. barrier
coliision
is
At
intermediate
even
prevent
efficiency;
an
the
rest
der
Waals
in
from
colliding.
the
potential
coagulation
exponential
Since
factor
factor
of
the
the
significant
the
separations
reduces
them
V as
most
separations
a maximum
barrier
involves
V against
particle
creating
energy
the
energy
large
repulsion.
predominates
barrier).
energy
of At
the
governing
curve
in
the
Fig.
8
is
of
importance. 3 shows
the
$I= 0.5,
on
the
is
non-zero
of
density
of
particles
The
(or
the
Practically the energy
separation,
can
retarded barrier rr - r
is
is
(where
range is
of
the
the
(no
as
abrupt,
so
a
density
in
of
1=0.05
the
ionic
which
the
‘rapid’
to
interactions.
quantitative
in
the
unstable
strength
strength efficiency
coagulation
change
time)
and
surface
electrostatic
ionic
the
a = 0.67-10-6Cb/cm2. of
significant
the
strength
effect
regime
observation
on
undisturbed
A/(6kT)
by
attraction
of
number
state
the
criterion
of
the
solution of
coagula-
exist.
change
potential for
the
of
ionic
charge
influenced
A/(6kT)
presented no
surface
the
not
during
computations there
illustrate
stable
smaller
transition
stability)
a
This
is
kinetically
to
to
overlap.
The
dimensionless
11
For
energy
pairs.
(negative)
10 and
behavior
shifts
increases.
For
curves
from the
dispersion
5,
van
interacting
same
corresponding
particle
transition
the
efficiency.
the
the
the
the
Figs.
collision
of
of
have
sequence
where
effect
efficiency
particles
potential
the
energy
the
collision
when
the Fig.
efficiency
of
Fig.
tion
der
potential
in
(energy
between
The
sum*
of
shown
collision
The
the
repulsion
curve
1 i ttle
is
of
small
particles van
17 where now the
electrostatic
are
the
height
spherical
hydrodynamic,
Eq.
For
used.
(19)
der Waals
energy
is
v”R
i-
van
both
from
repulsive
separation
rate
for
particles
and
distances.
approximation of
accounting
approaching
the
interparticle
26)
collisionefficiency
and
can
.large (ref.
(Eq.
in
is
unretarded
potential
(Eq.
the
transition
from
to
13)
coagulation tc”
the
is is
the
used.
typically
double
layer
This at
slow
agreement
12) rapid occurs
a dimensionless
thickness),
*There is no firm basis for the superposition of the van double layer force. The interrelation of the two forces, understood (see discussion’in r&f. 19, p.'54).
of
is
used.
coagulation because particle
order
der Waals however,
1 where and the is not well
15
I
I
L
,,‘,I
1
L
I=O.OS
,
L,,,,
20
0.8 S 9 G 0.6 Liz B 5 0.4 Gi 3 9 0.2 -
Fig.
9.
Collision
I
efficiencies
L
,
of
I,,,,
0.8
*______--_--
-
in Brownian
L
1=0.25 _
particles
L
L
diffusion
(T=o.Os).
I,,,,
20___-.^
-
A
6kT Fig.
10.
Collision
efficiencies
of
particles
in Brownian
diffusion
(I=
0.25).
16 retardation The
effects
rapid
energy
of
are
not
variation
of
attraction
important.
the
occurs
collision in
the
efficiency
‘slow’
with
coagulation
the
van
regime.
der
Waals
Accord
i ng
to
I=O.5 >0.8
-
z w EO.6
-
B E =0.4i S uo.2-
Ii.
Fig_ Figs. of
Collision
9,
efficiencies
10 and
particle
11
size_
theoretical
the
transition
This
is
calculations
here,
so
the
appl
ications,
initially
is
Brownian
motion
is
from
stable
(ref.
slow
the
t:,2
is
to
in
reduced
Brownian rapid
diffusion
coagulation
experimental
Collision
for
time
system
in
with
28) _
half-life
monodisperse
particles
consistent
(ref.
dispersion The
of
are
time
states
of
which
the
number
to
is
results
efficiencies
one-half
the
most
(I=
0.5).
independent
(ref.
27)
very
small
and
practical
fi of
particles
original
value
in
1)
s
(20)
t1/2
= T&i
here
any
only
co11 isions
particle
interactions between
collision
efficiency
E(rT.r2)
=
where
in
between Eq.
an
by
ignored.
particles
defined
in
Eq.
17
(Eq.
is approximate
20
of
radius
is
equivalent
r
are
since
considered)
_
(21)
hydrodynamic, particles
21 reduces
to
The
to
S t1/2 t1/2
t,,2 the
as
are
primary
are
(ref.
van
der
considered_
i
21)
Waals
and
electrostatic
For water
at ambient
interactions
temperature
(20°C)
17
V2
= E(r,
,r2)
N is
the
where
2.10’1 7 number of
The number of 10’
cm-a
have
(22)
corresponding
particle
number densities
efficiency
smaller
Consequently,
by the
and Hull electrolyte of
with
is
actions
occur
equally of
valid
slow
10 and
11
the
Their
polydisperse
particles
(ref.
is
systems
population
of
practical
Honig
the
critical
in a monodisperse of
energy
and
waters
coagulation,
interest.
criterion
strength
24)
all
rapid of
order
A co1 1 ision
and computed
coagulation
ionic
for to
of
Natural
29).
dispersion
van der Waals
a polydisperse
the
rate
van der Waais The
work described
given
tend
to
collision
rate
applications concentration forces
coagulation
particles
both
on the
sediment
layer
of
for
the
the
system
onset
i t can
particles
the
Their
be used
for
of
electrolyte,
attraction.
and when hydrodynamic
thus
to
inter-
evaluate
any combination
can
of
of
coagulating
serve
this
in Table
particles
results
particles
of
of
the
unlike
purpose3.
treatment, onset
determined
solely
for
is
experiments
with
account
into
long-range the
abrupt,
the
particles.
hydrodynamic
interacting
dependence
particles.
purification,
rapid
Brohnian
hydrodynamic,
functional
is
relative
mobility
and
the
of
important.
Once co1 1 isions
forces)
of
For
prediction
coagulation
coagulation. from
the
for
two approaching
water
hydrodynamic
coagulation
take the
only
of
improve theory
and the
to
of
occur, the
Hamaker
so a quantitative
criterion
of
exist.
From well-controlled From the
of
discharges)
the
to account
and rate
sizes
been
classical
between
potential
collision
relative
determine
The onset
stability
the
has
1)
interactions
in ocean
is
paper (ref.
efficiencies
(wastewater
rate
(modified
constant.
layer
van der Waals
affect
practical Double
co1 1 ision
and double
short-range
forces
in this
by Smoluchowski’s
The computed
diffusion.
of
for
of
example,
(ref.
from
surfaces_
of
the
cme3
interactions
onset
for
= E - 55 hrs.
ONS
CONCLUSI
collision
of
a stable
in seconds.
is,
t,,2 10’
9,
is
u and A.
The aim of
the
the
charge
and
of
transition
in Figs.
a function
between
stability
VII,
the
is
t l/2 sludge
lOs-
hydrodynamic at
charge
time order
the
curves
constant
surface
analysis
1,
the
cm3 and sewage
imp1 ies
only
ignored
coagulation
particle
the
in
30)
of
0.001
concentration
particles
rapid
than
bend
(ref,
per
in primary
to a half-life
purposes. given
parti cl es
particles
an
‘effective’
calculations sizes
can
The curves
Interpolation
can
an
initially
monodisperse
Hamaker constant presented
be estimated. in Figs.
be used
here
the
Tables
5 and 6 are for
intermediate
can
suspension
be determined.
collision
efficiencies
1 and 2 and Figs. given
in parametric
values
of
the
5 and 6 form
Hamaker
18 The
constant. sizes.
Dynamic which
of
Equation involve
TABLE
E(rl
to
and
obtain
coagulating
is
.r,)
interest
so
thris it
realistic
obtained
can
be
results
particles,
(ref-
for
all
efficiencies
der
Uaals
in
modeling
for
Brownian
diffusion;
5
3
1
lo-* 10-3
I_0040 I_0042 I_0053 1_0098 1.0248 1.0691 I-1983
;::I: IO0 10 102 der
Slaals
and
‘2”1
retardation
1.0027 I_0028 l-0035 1.0064 1.0157 1.0435 1.1254
10
1.0022 l-0023 l.0025 1.0037 1.0075 1.0189 l-0540
:-x8;: 1 :a030 1.0040 1.0116 l.0251 1.0905
hydrodynamic
20
50
100
1_0021 l.0022 l-0023 l.0029 l.0049 1.0120 1.0300
1.0021 1.0021 1_0021 1.0024 1.0032 1.0049 1.0142
1.0020 1.0020 1 .ooto 1.0022 -1-0026 l-0040 1_0082
forces
5
3
1
10-4
0.2409
lC-’
10-z lOi=
0.2791 O-3286 0.3867
:: 102
5.5546 O-5477 0-7194
TABLE
10
20
50
100
O-4810
0_36lS 0,-k;
O-2971 0.3401 Cl.3931 0.4512
0:5207 0.6841 O-7430 O-8266
0.6562 0.5806 O-7700
0.5150 0.5981 O-7335
0.8763 o-8936
0.6198 0.6620 0.7060
0-8796
0.8659 0.8862 0.9070 O-9341
0.9101 O-9237
2
Collision den-
efficiencies waa1s
‘2~‘l
for
Brarnirn
diffusion;
retardation
parameter
==l
forces 1
3.
5
10
20
50
1_0021 1.0021 1.0024
1.0011 1.0011 1_0012 1_0019
100
A/&T) 10-b 10-x IO-’ 10-1 100 10 10’ der
Uaals
r2/r1 A/
1.0041
1.0027
l-0043
1_oots
1.0061 1.0159 1.0568 l-1866 IS546
1.0041 I-0108 l-0375 1.1198 l-354?
and
hydrodynamic
1
3
l-0024 1.0026 1.0035 l-0083 : xz5 1:2572
1.0022 1.0023 1.0028 l-0056 1.0169 1.0520 l-1537
l-0040 1.0101 1.0296
l-0045 l-0127
l-0866
l-0379
IO
20
50
O-6276 0.6630 O-7112
l_OOll 1.0011 l_OOll I_0015 I.0028 1.0071 1.0201
forces 5
100
CkT> IO-' 10-3 10-z 10-x 10’ :x2
General
parametera=O.l
A/&T)
van
particle
the
31)-
A/CkT)
“a”
of
into of
forces
‘2”l
van
pairs
incorporated
’
Coiiision van
function
(r1,r21,
O-2410 0.2797 O-3331
O-4078
0.5181 O-7025 l.0955
o-2996
0.3647
0.4862
O-3411 o-3992
0.4090 o-4694
0.5298 o-5879
O-4755
O-5450
O-6553
0.5775
0.6384
~- _-
0.7362
0.8217 0.9682
X-zz: o:a441 0.8740 0.9019
X-t;;: 0:slos o.Yz8l 0.9437 0.9603 0.9844
physical
processes
19 TABLE
3
Apprmtimations hydrodynamic E(r,,r2)
=
for collision efficiencies and van der UaaIs forces a +
bx
*
Retardation
parameter
A/(kT)
a
10-2
Retardation
bx
0.25950
*0.88410 2 C r2/rl
IO-’
*valid
for
VIII.
ACKNOWLEDGEMENTS authors
NA82RAD00004 Quality
accounting
for
‘2/r,
10-e
dx
1O-6
2.6953 2.9043 3.0409
1.3845
1 cx
3.0439 3.0405 3.0423 Z-8985 2.4525 1.4595 O-34764
0.31685 0.39643 0.51062 *Ct.69014
Assessment
cx
-4.9962 -5.3031 -5-4760 -5.4055 -5.0576 -4-3310 -2.4753
o=
0.21770
x =
Brorrnian diffusion for lc r2frl< 100)
o-1
2.9593 3.0338 3.0339 2.9251 2-6954
a
The
dx’.
lO-2
parameter
A/(M)
*
Q= bx
0.21811 0.25878 0.3r151 0.37254 0.44285
10-1 1 10 102
10-p 10-s 70-z 10-1 1 10 10’
cx’
in (valid
10-u
dx
1O-6
Z-8666 2.9227 3.0742 3 -0677
-5.2527 -5.3293 -5.5222 -5-4343 -4-6867 -2.7279 -0_5300
c 100
gratefully
Division and
Laboratory
acknowledge
- National
the Mellon at
Ocean
Foundation
the
financial
Service through
- NOAA
support
of
through
Grant
a grant
to the
the Ocean No.
Environmental
Caitech.
REFERENCES 1 H, Smoiuchowski, Physik Z., 17(1316)557. W.U. Keck Laboratory, California Institute 2 J.R. Hunt, Report No. AC-5-80, of Technology, Pasadena, California, 1980. 3 J. Chandrasekhar, Rev. Mod. Phys., 15(1943)1. 4 A.L, Spielman, J. Colloid Interface Sci., 33(1970)5625 G.K. Batchelor, J. Fluid Mech., 74(1376)1. 6 A. Einstein, The Theory of the Brownian Movement, Dover, New York, 1926. 7 M. Stimson and G-B_ Jeffrey, Proc. Roy. Sot. London Ser. A, 111(1926)110. 8 H.C. Hamaker, Physica, 4(1937)1058. 3 J.H. Schenkel and J-A. Kitchener, Trans. Faraday Sot., 56(1960)161. 10 H.B.G. Casimir and D. Polder. Phys. Rev., 73(1948)360. 11 J. Mahanty and B.W. Ninham, Dispersion Forces, Academic Press, New York, 1976. 12 G.R. Zeichner
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and W-R, Schowalter, J. Colloid Interface Sci., 71(1979)237. of Aerosols, Pergaman Press, New York, 1364. N.A. Fuchs, The Mechanics H. Brenner, Advan. Chem. Eng.. 6(1966)328S. Twomey, Atmospheric Aeroscis, Elsevier, Amsterdam/New York, 1377. A. Schmidt-Ott and H, Burtscher, J. Colloid Interface Sci., 89(1982)353J. Lyklema, Advan. Colloid Sci., 2(1968)84. E.M. Lifshitz, Sov. Phys, JETP, i(1356)73. B. Vincent and S.G. Whittington, in E. Hatijevic (Ed,), Surface and Colloid Science, Plenum,Press, New York, 1982.
20 20 F. Gelbard. Y. Tamhour and J.H. Seinfeld, J. Colloid Interface Sci., 76(1980)54121 E.J_W. Verwey and J.T.G. Overbeek, Theory of the Stability of Lyophobic Co1 loids, Elsevier, Amsterdam, 1948. 22 W, Stumm and J-J. Morgan, Aquatic Chemistry, Wiley-Interscience, 1981. 23 J. Lyklema and H.P. Van Leeuwen, Adv. Colloid interface Sci., 16(1982)227. 24 1-A. Valioulis, Ph.D. Thesis, California institute of Technology, Pasadena, California, 1983. S. Levine and L.N. McCartney, J. Colloid Interface Sci., 25 G.M. Bell, 33(1970)335. 26 B.V. Derjaguin, Discuss. Faraday Sot., 18(1994)85. 27 R-H, Ottewill and J-N, Show, Disc, Faraday Sot,, 42(1966)194, 28 E.P. Honig. G-J. Roeberson and D.H. Wiersema, J. Colloid interface Sci., 36(1971)97. 29 C.R. O’Helia, Environ, Sci, Tech., 14, 9(198o)lo52. 30 E-P. Honig and P-H. Mull, J, Colloid Interface Sci., 36(1971)258. 31 1-A. Valiouiis anti E.J. List, Environ. Sci. Tech. (in press).