Colloids and Surfaces
A:Physicochemical and Engineering Aspects 195 (2001) 157–169
Sedimentation velocity and potential in a concentrated
colloidal suspension
Effect of a dynamic Stern layer
F.Carrique
a,
*,F.J.Arroyo
b
,A.V.Delgado
b
a
Dpto.Fı´sica Aplicada I,Facultad de Ciencias,Uniersidad de Ma´laga,29071Ma´laga,Spain
b
Dpto.Fı´sica Aplicada,Facultad de Ciencias,Uniersidad de Granada,18071Granada,Spain
Abstract
The standard theory of the sedimentation velocity and potential of a concentrated suspension of charged spherical
colloidal particles,developed by H.Ohshima on the basis of the Kuwabara cell model (J.Colloid Interf.Sci.208
(1998) 295),has been numerically solved for the case of nonoverlapping double layers and different conditions
concerning volume fraction,and potential of the particles.The Onsager relation between the sedimentation
potential and the electrophoretic mobility of spherical colloidal particles in concentrated suspensions,derived by
Ohshima for low potentials,is also analyzed as well as its appropriate range of validity.On the other hand,the
abovementioned Ohshima’s theory has also been modiﬁed to include the presence of a dynamic Stern layer (DSL)
on the particles’ surface.The starting point has been the theory that Mangelsdorf and White (J.Chem.Soc.Faraday
Trans.86 (1990) 2859) developed to calculate the electrophoretic mobility of a colloidal particle,allowing for the
lateral motion of ions in the inner region of the double layer (DSL).The role of different Stern layer parameters on
the sedimentation velocity and potential are discussed and compared with the case of no Stern layer present.For
every volume fraction,the results show that the sedimentation velocity is lower when a Stern layer is present than that
of Ohshima’s prediction.Likewise,it is worth pointing out that the sedimentation ﬁeld always decreases when a Stern
layer is present,undergoing large changes in magnitude upon varying the different Stern layer parameters.In
conclusion,the presence of a DSL causes the sedimentation velocity to increase and the sedimentation potential to
decrease,in comparison with the standard case,for every volume fraction.Reasons for these behaviors are given in
terms of the decrease in the magnitude of the induced electric dipole moment on the particles,and therefore on the
relaxation effect,when a DSL is present.Finally,we have modiﬁed Ohshima’s model of electrophoresis in
concentrated suspensions,to fulﬁll the requirements of Shilov–Zharkhik’s cell model.In doing so,the wellknown
Onsager reciprocal relation between sedimentation and electrophoresis previously obtained for the dilute case is again
recovered but now for concentrated suspensions,being valid for every potential and volume fraction.© 2001
Elsevier Science B.V.All rights reserved.
Keywords:Sedimentation velocity;Sedimentation potential;Concentrated suspensions;Onsager reciprocal relation
www.elsevier.com/locate/colsurfa
* Corresponding author.
09277757/01/$  see front matter © 2001 Elsevier Science B.V.All rights reserved.
PII:S0927 7757( 01) 00839 1
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169158
1.Introduction
It is wellknown that when a colloidal suspen
sion of charged particles is settling steadily in a
gravitational ﬁeld,the electrical double layer sur
rounding each particle is distorted because of the
ﬂuid motion,giving rise to a microscopic electric
ﬁeld (the relaxation effect).As a consequence,the
falling velocity of the particle,i.e.the sedimenta
tion velocity,is lower in comparison with that of
an uncharged particle.On the other hand,these
electric ﬁelds superimpose to yield a macroscopic
electric ﬁeld in the suspension,i.e.the sedimenta
tion ﬁeld or sedimentation potential gradient
(usually called sedimentation potential).
A general sedimentation theory for dilute col
loidal suspensions,valid for nonconducting
spherical particles with arbitrary double layer
thickness and potential,was developed by
Ohshima [1] on the basis of previous theoretical
approaches [2–9].In his paper Ohshima removed
the shortcomings and deﬁciencies already re
ported by Saville [10] concerning Booth’s method
of calculation of the sedimentation potential.Fur
thermore,he presented a direct proof of the On
sager reciprocal relation that holds between
sedimentation and electrophoresis.
On the other hand,a great effort is being
addressed to improve the theoretical results pre
dicted by the standard electrokinetic theories deal
ing with different electrokinetic phenomena in
colloidal suspensions.One of the most relevant
extensions of these electrokinetic models has been
the inclusion of a dynamic Stern layer (DSL) onto
the surface of the colloidal particles.Thus,
Zukoski and Saville [11] developed a DSL model
to reconcile the differences observed between 
potentials derived from static electrophoretic mo
bility and conductivity measurements.
Mangelsdorf and White [12],using the techniques
developed by O’Brien and White for the study of
the electrophoretic mobility of a colloidal particle
[13],presented in 1990 a rigorous mathematical
treatment for a general DSL model.They ana
lyzed the effects of different Stern layer adsorp
tion isotherms on the static ﬁeld electrophoretic
mobility and suspension conductivity.
More recently,the theory of Stern layer trans
port has been applied to the study of the low
frequency dielectric response of colloidal suspen
sions by Kijlstra et al.[14],incorporating a sur
face conductance layer to the thin double layer
theory of Fixman [15,16].Likewise,Rosen et al.
[17] generalized the standard theory of the con
ductivity and dielectric response of a colloidal
suspension in AC ﬁelds of DeLacey and White
[18],assuming the model of Stern layer developed
by Zukoski and Saville [11].Very recently,Man
gelsdorf and White presented a rigorous mathe
matical study for a general DSL model applicable
to time dependent electrophoresis and dielectric
response [19,20].In general,the theoretical predic
tions of the DSL models improve the comparison
between theory and experiment [14,17,21,22],al
though there are still important discrepancies.
Returning to the sedimentation phenomena in
colloidal suspensions,a DSL extension of Ohshi
ma’s theory of the sedimentation velocity and
potential in dilute suspensions,has been recently
published [23].The results show that whatever the
chosen set of Stern layer parameters or poten
tial may be,the presence of a DSL causes the
sedimentation velocity to increase and the sedi
mentation potential to decrease,in comparison
with the standard prediction (no Stern layer
present).
On the other hand,the theory of sedimentation
in a concentrated suspension of spherical colloidal
particles,proposed by Levine et al.[9] on the
basis of the Kuwabara cell model [24],has been
further developed by Ohshima [25].In that paper,
Ohshima derived a simple expression for the sedi
mentation potential applicable to the case of low
potential and nonoverlapping of the electric
double layers.He also presented an Onsager re
ciprocal relation between sedimentation and elec
trophoresis,valid for the same latter conditions,
using an expression for the electrophoretic mobil
ity of a spherical particle previously derived in his
theory of electrophoresis in concentrated suspen
sions [26].This theory is also based on the
Kuwabara cell model in order to account for the
hydrodynamic particle–particle interactions,and
uses the same boundary condition on the electric
potential at the outer surface of the cell,as that of
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169 159
Levine et al.’s theory of the electrophoresis in
concentrated suspensions [27].
Recalling the attention on the DSL correction
to the electrokinetic theories,it seemed of interest
to explore the effects of extending the standard
Ohshima’s theory of the sedimentation velocity
and potential in a concentrated suspension of
charged spherical colloidal particles [25],to in
clude a DSL model.Thus,the chosen starting
point has been the method proposed by Mangels
dorf and White in their theory of the elec
trophoretic mobility of a colloidal particle,to
allow for the adsorption and lateral motion of
ions in the inner region of the double layer (DSL)
[12].
Finally,the aims of this paper can be described
as follows.First,we have obtained a numerical
solution of the standard Ohshima’s theory of
sedimentation in concentrated suspensions,for
the whole range of potential and volume frac
tion,and nonoverlapping double layers.Further
more,we have extended the latter standard theory
to include a DSL on the surface of the particles,
and analyzed the effects of its inclusion on the
sedimentation velocity and potential.And then,
we have analyzed the Onsager reciprocal relation
that holds between sedimentation and elec
trophoresis in concentrated suspensions,for both
standard and DSL cases.It can be concluded that
the presence of a Stern layer provokes a rather
slow increase on the magnitude of the sedimenta
tion velocity of a colloidal particle,whatever the
values of Stern layer,particle and solution
parameters used in the calculations.On the other
hand,the presence of a Stern layer causes the
sedimentation potential to decrease with respect
to the standard prediction.
2.Standard governing equations and boundary
conditions
The starting point for our work has been the
standard theory of the sedimentation velocity and
potential in a concentrated suspension of spheri
cal colloidal particles,developed by H.Ohshima
[25] on the basis of the Kuwabara cell model to
account for the hydrodynamic particle–particle
interactions (see Fig.1).According to this model,
each spherical particle of radius a is surrounded
by a concentric virtual shell of an electrolyte
solution,having an outer radius of b such that the
particle/cell volume ratio in the unit cell is equal
to the particle volume fraction throughout the
entire suspension,i.e.
=
a
b
3
.(1)
In fact,a is the radius of the ‘hydrodynamic
unit’,i.e.a rigid particle plus a thin layer of
solution linked to its surface moving with it as a
whole.The surface r=a is usually called ‘slipping
plane’.This is the plane outside which the contin
uum equations of hydrodynamics are assumed to
hold.As usual,we will make no distinction be
tween the terms particle surface and slipping
plane.
Before proceeding with the analysis of the mod
iﬁcations arising from the DSL correction to the
standard model,it will be useful to brieﬂy review
the basic standard equations and boundary condi
tions.Concerned readers are referred to Ohshi
ma’s paper for a more extensive treatment.
Consider a charged spherical particle of radius
a and mass density
p
immersed in an electrolyte
solution composed of N ionic species of valencies
z
i
,bulk number concentrations n
i
,and drag co
efﬁcients
i
(i =1,…,N).The axes of the coordi
nate system (r,,) are ﬁxed at the centre of the
particle.The polar axis (=0) is set parallel to g.
Fig.1.Schematic picture of an ensemble of spherical particles
in a concentrated suspension,according to the Kuwabara cell
model [24].
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169160
The particle is assumed to settle with steady ve
locity U
SED
,the sedimentation velocity,in the
electrolyte solution of viscosity and mass den
sity
o
in the presence of a gravitational ﬁeld g.
For the spherical symmetry case,both U
SED
and
g have the same direction.In the absence of g
ﬁeld,the particle has a uniform electric poten
tial,the potential ,at r=a,where r is the
radial spherical coordinate,or equivalently,the
modulus of position vector.
A complete description of the system requires
a knowledge of the electric potential (r),the
number density or ionic concentration n
i
(r) and
the drift velocity v
i
(r) of each ionic species (i =
1,…,N),the ﬂuid velocity u(r),and the pres
sure p at every point r in the system.The
fundamental equations connecting these quanti
ties are [1,25]:
2
(r)=−
(r)
rs
o
(2)
(r)=
N
i =1
z
i
en
i
(r) (3)
2
u(r)−p(r)−(r)+
o
g=0 (4)
u(r)=0 (5)
v
i
=u −
1
i
i
(i =1,…,N) (6)
i
(r)=
i
+z
i
e(r)+K
B
T ln n
i
(r)
(i =1,…,N) (7)
[n
i
(r)v
i
(r)] =0 (i =1,…,N),(8)
where e is the elementary electric charge,K
B
the
Boltzmann’s constant and T is the absolute tem
perature.Eq.(2) is Poisson’s equation,where
rs
is the relative permittivity of the solution,
o
the
permittivity of a vacuum,and (r) is the electric
charge density given by Eq.(3).Eqs.(4) and (5)
are the Navier–Stokes equations appropriate to
a steady incompressible ﬂuid ﬂow at low
Reynolds number in the presence of electric and
gravitational body forces.Eq.(6) expresses that
the ionic ﬂow is caused by the liquid ﬂow and
the gradient of the electrochemical potential
deﬁned in Eq.(7),and it can be related to the
balance of the hydrodynamic drag,electrostatic,
and thermodynamic forces acting on each ionic
species.Eq.(8) is the continuity equation ex
pressing the conservation of the number of each
ionic species in the system.
The drag coefﬁcient
i
is related to the limit
ing conductance
i
o
of the ith ionic species by
[13]
i
=
N
A
e
2
z
i
i
o
(i =1,…,N),(9)
where N
A
is Avogadro’s number.
At equilibrium,that is,in the absent of the
gravitational ﬁeld,the distribution of electrolyte
ions obeys the Boltzmann distribution
n
i
(o)
=n
i
exp
−
z
i
e
(o)
K
B
T
(i =1,…,N),(10)
and the equilibrium electric potential
(o)
sa
tisﬁes the Poisson–Boltzmann equation
1
r
2
d
dr
r
2
d
(o)
dr
=−
el
(o)
(r)
rs
o
(11)
el
(o)
(r)=
N
i =1
z
i
en
i
(o)
(r),(12)
being
el
(o)
the equilibrium electric charge density.
The unperturbed or equilibrium electric poten
tial must satisfy these boundary conditions at
the slipping plane and at the outer surface of
the cell
(o)
(a)= (13)
d
(o)
dr
(b)=0 (14)
where is the potential.
As the axes of the coordinate system are cho
sen ﬁxed at the center of the particle,the
boundary conditions for the liquid velocity u
and the ionic velocity of each ionic species at
the particle surface are expressed by the follow
ing equations
u=0 at r=a (15)
v
i
rˆ =0 at r=a (i =1,…,N) (16)
which mean,respectively,that the ﬂuid layer ad
jacent to the particle surface is at rest,and that
there are no ion ﬂuxes through the slipping
plane (rˆ is the unit normal outward from the
particle surface).According to the Kuwabara
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169 161
cell model,the liquid velocity at the outer surface
of the unit cell satisﬁes the conditions:
u
r
=−U
SED
cos at r=b (17)
=×u=0 at r=b,(18)
which express,respectively,that the liquid veloc
ity is parallel to the sedimentation velocity,and
the vorticity is equal to zero.
Following Ohshima,we will assume that the
electrical double layer around the particle is only
slightly distorted due to the gravitational ﬁeld
about their equilibrium values.Thus,the follow
ing perturbation scheme for the abovementioned
quantities can be used,
n
i
(r)=n
i
(o)
(r)+n
i
(r) (i =1,…,N) (19)
(r)=
(o)
(r)+(r) (20)
i
(r)=
i
(o)
+
i
(r) (i =1,…,N) (21)
where the superscript (o) is related to the state of
equilibrium.The perturbations in ionic number
density and electric potential are related to each
other through the perturbation in electrochemical
potential by
i
=
z
i
e+K
B
Tn
i
n
i
(o)
(i =1,…,N).(22)
In terms of the perturbation quantities,the
condition that the ionic species are not allowed to
penetrate the particle surface in Eq.(16),trans
forms into
i
rˆ =0 at r=a (i =1,…,N),(23)
when a DSL is not considered.
Besides,for the case of negligible overlapping
of double layers on the outer surface of the unit
cell,this extra condition holds:
i
=0(n
i
=0,=0) (i =1,…,N).(24)
For the spherical case and following Ohshima
[25],symmetry considerations permit us to intro
duce the radial functions h(r) and
i
(r),and then
write
u(r)=(u
r
,u
,u
)
=
−
2
r
h g cos ,
1
r
d
dr
(rh)g sin ,0
(25)
i
(r)=−z
i
e
i
(r)(g rˆ) (i =1,…,N),(26)
to obtain the following set of ordinary coupled
differential equations and boundary conditions at
the slipping plane and at the outer surface of the
cell:
L(Lh)=−
e
r
dy
dr
N
i =1
n
i
z
i
2
exp(−z
i
y)
i
(r),(27)
with y=e
(o)
/KT,
L(
i
(r))=
dy
dr
z
i
d
i
dr
−
2
i
e
h
r
(i =1,…,N)
(28)
h(a)=
dh
dr
(a)=0,Lh(r)=0 at r=b (29)
d
i
dr
(a)=0 (i =1,…,N) (30)
i
(b)=0 (i =1,…,N),(31)
L being a differential operator deﬁned by
L
d
2
dr
2
+
2
r
d
dr
−
2
r
2
.(32)
In addition to the previous boundary condi
tions,we must impose the constraint that in the
stationary state the net force acting on the particle
or the unit cell must be zero [25].
3.Extension to include a dynamic Stern layer
We now deal with the problem of including the
possibility of adsorption and ionic transport in
the inner region of the double layer of the parti
cles.We will follow the method developed by
Mangelsdorf and White [12] in their theory of the
electrophoresis and conductivity in a dilute col
loidal suspension.This theory allows for the ad
sorption and lateral motion of ions in the latter
inner region using the wellknown Stern model.
According to this method,the condition that ions
cannot penetrate the slipping plane no longer
maintains,and therefore,the evaluation of the
ﬂuxes of each ionic species through the slipping
plane permits us to obtain the following new
slipping plane boundary conditions for the func
tions
i
(r),
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169162
d
i
dr
(a)−
2
i
a
i
(a)=0 (i =1,…,N) (33)
i
=
[eN
i
]/(ae 10
−pK
i
)(
i
/
i
t
)exp[(z
i
e/K
B
T)(
d
/C
2
]
N
A
10
3
+
N
j =1
(N
A
10
3
c
j
/10
−pK
j
)exp[(−z
j
e/K
B
T)(−
d
/C
2
)]
,
(34)
in terms of the socalled surface ionic conduc
tance parameters
i
of each ionic species,com
prising the effect of a mobile surface layer.These
parameters depend on,the potential ;the ra
tio between the drag coefﬁcient
i
of each ionic
species in the bulk solution and in the Stern
layer
i
t
;the density of sites N
i
available for
adsorption in the Stern layer;the pK
i
of ionic
dissociation constant for each ionic species (the
adsorption of each ionic species onto an empty
Stern layer site is represented as a dissociation
reaction in this theory [12]),the capacity C
2
of
the outer Stern layer,the radius a of the parti
cles,the electrolyte concentration through c
j
,
i.e.the equilibrium molar concentration of type j
ions in solution,and the charge density per unit
surface area in the double layer
d
.It is worth
noting that the other boundary conditions ex
pressed by Eqs.(29) and (31) remain unchanged
when a DSL is assumed.
A numerical method similar to that proposed
by DeLacey and White in their theory of the
dielectric response and conductivity of a col
loidal suspension in timedependent ﬁelds [18],
has been applied to solve the abovementioned
set of coupled ordinary differential equations of
the sedimentation theory in concentrated col
loidal suspensions.Furthermore,both standard
and DSL cases have been extensively analyzed.
In a recent paper [23],we successfully employed
the latter numerical scheme to solve the standard
theory of sedimentation in dilute colloidal sus
pensions.All the details and steps of the numeri
cal procedure can be found in that reference.
4.Calculation of the sedimentation velocity and
potential
Let us describe now how the sedimentation
velocity and potential for a concentrated suspen
sion can be calculated.According to the condi
tion for the ﬂuid velocity at the outer surface of
the unit cell,the ﬂuid velocity has to be parallel
to the sedimentation velocity (see Eqs.(17) and
(25)).Thus,we can obtain the sedimentation ve
locity U
SED
,once the value of function h has
been determined at the outer surface of the cell,
i.e.
U
SED
=
2h(b)
b
g.(35)
For the case of uncharged particles (=0),the
sedimentation velocity is given by the well
known Stokes formula [25]
U
SED
ST
=
2a
2
(
p
−
o
)
9
g.(36)
As regards the sedimentation potential E
SED
,it
can be considered as the volume average of the
gradient of the electric potential in the suspen
sion volume V,i.e.
E
SED
=−
1
V
V
(r)dV.(37)
Following Ohshima [25],the net electric cur
rent i in the suspension can be expressed in
terms of the sedimentation potential and the ﬁrst
radial derivatives of
i
functions at the outer
surface of the unit cell,
i=K
E
SED
+
1
K
N
i =1
z
i
2
e
2
n
i
i
d
i
dr
(b)
n
g
,
(38)
where K
is the electric conductivity of the elec
trolyte solution in the absence of the colloidal
particles.
If now we impose,following Saville [10]
and Ohshima [25],the requirement of zero net
electric current in the suspension,we ﬁnally ob
tain
E
SED
=−
1
K
N
i =1
z
i
2
e
2
n
i
i
d
i
dr
(b)
n
g.(39)
Likewise,we deﬁne the scaled sedimentation
potential E
SED
* as in the dilute case by
E
SED
* =
3eK
2
rs
o
K
B
T(
p
−
o
)
E
SED
g
.(40)
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169 163
5.Onsager reciprocal relation between
sedimentation and electrophoresis in concentrated
suspensions
It is wellknown that an Onsager reciprocal
relation holds between sedimentation and elec
trophoresis.A direct proof of this relationship
was derived by Ohshima et al.[1] for dilute sus
pensions,and is given by
E
SED
=−
(
p
−
o
)
K
g,(41)
where is the electrophoretic mobility of a col
loidal particle.Furthermore,this relation is also
satisﬁed when a DSL is incorporated to the theo
ries of sedimentation and electrophoresis in dilute
colloidal suspensions [23].
On the other hand,the electrophoretic mobility
is usually represented by a scaled quantity * [13]
deﬁned by
*=
3e
2
rs
o
K
B
T
.(42)
Eq.(41) can then be rewritten in terms of the
scaled quantities to give a simple convenient ex
pression for the Onsager relation,namely,
E
SED
* =*.(43)
Very recently Ohshima derived an Onsager rela
tion between sedimentation and electrophoresis in
concentrated suspensions,applicable for low 
potentials and nonoverlapping of double layers
[25].In that paper,Ohshima used an expression
for the electrophoretic mobility
OHS
of a spheri
cal colloidal particle,derived according to his
theory of the electrophoresis in concentrated sus
pensions [26].The Onsager relation he found is
given by
E
SED
=−
(1−)(
p
−
o
)
(1+/2)K
OHS
g,(44)
or equivalently,
E
SED
* =
(1−)
(1+/2)
OHS
*,(45)
where Eqs.(40) and (42) have been used.In the
limit when volume fraction tends to zero,Eqs.
(44) and (45) converges to the wellknown Eqs.
(41) and (43) which describes the Onsager relation
between sedimentation and electrophoresis in di
lute suspensions.
However,very recently Dukhin et al.[28] have
pointed out that the Levine–Neale cell model [27],
employed by many authors to develop theoretical
electrokinetic models in multiparticle systems,in
particular those of sedimentation,electrophoresis
and conductivity in concentrated suspensions
[9,26,29–32],presents some deﬁciencies.Accord
ing to Dukhin et al.[28] the Levine–Neale cell
model is not compatible with certain classical
limits concerning,specially,the volume fraction
dependence in the exact Smoluchovski’s law in
concentrated suspensions.Instead of the Levine–
Neale cell model,Dukhin et al.propose to use the
Shilov–Zharkikh cell model [33] which not only
agrees with the latter Smoluchovski’s result but
also correlates with the electric conductivity of the
Maxwell –Wagner theory [34].It is worth noting
that Ohshima’s theory of the electrophoretic mo
bility in concentrated suspensions [26] incorpo
rates the Levine–Neale boundary condition on
the electric potential at the outer surface of the
unit cell.This condition states that the local elec
tric ﬁeld has to be parallel to the applied electric
ﬁeld E at the outer surface of the cell.
Then,it seemed quite interesting to compare
the changes in Ohshima’s Onsager relation for
concentrated suspensions,if any,that could arise
from the consideration of a different boundary
condition on the electric potential according to
the Shilov–Zharkikh cell model,which is based
on arguments of nonequilibrium thermodynam
ics.Following Ohshima’s theory of electrophore
sis in concentrated suspensions [26],the boundary
condition for the perturbed electric potential at
the outer surface of the unit cell is expressed by
rˆ =−E rˆ at r=b.(46)
However,according to the Shilov–Zharkikh
cell model,the latter condition changes to
=−Er at r=b.(47)
being E the macroscopic electric ﬁeld.For low
potentials and nonoverlapping of double lay
ers,Eq.(22) becomes [26,32]
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169164
i
=z
i
e,(48)
and consequently,Eq.(46) transforms into
i
rˆ =−z
i
eErˆ.(49)
Following Ohshima,spherical symmetry consider
ations permit us to write
i
(r)=−z
i
e
i
(r)(E rˆ) (i =1,…,N),(50)
which is analogous to Eq.(26) for sedimentation.
Now,according to Eq.(50),Eq.(49) ﬁnally
becomes
d
i
dr
(b)=1.(51)
However,following the Shilov–Zharkikh
boundary condition given by Eq.(47),a different
result can be obtained,i.e.
i
(b)=b,(52)
where Eq.(50) has been used reading E instead
of E.If now we change in Ohshima’s theory of
the electrophoretic mobility in concentrated sus
pensions,the boundary condition given by Eq.
(51) for that in Eq.(52),a quite different numeri
cal result for the electrophoretic mobility is ob
tained (we will call it
SHI
).Furthermore,if we
conﬁne ourselves to the analytical approach of
low potentials developed in Ohshima’s papers
of sedimentation [25] and electrophoresis [26] in
concentrated suspensions,an Onsager reciprocal
relation different to that by Ohshima (Eqs.(44)
and (45)),is found,i.e.
E
SED
* =
SHI
*.(53)
It should be noted that this new Onsager rela
tion has exactly the same form as the wellknown
Onsager relation connecting sedimentation and
electrophoresis in dilute suspensions (see Eqs.(41)
and (43)).Likewise,we have numerically conﬁ
rmed that this Onsager relation also holds for the
whole range of potentials unlike that of Eq.
(44).In conclusion,we can state that the Onsager
reciprocal relation between sedimentation and
electrophoresis,previously derived for the dilute
case,also holds for concentrated suspensions if
Shilov–Zharkikh’s boundary condition is consid
ered.In the next section,we will present numeri
Fig.2.Ratio of the standard sedimentation velocity to the
Stokes sedimentation velocity of a spherical colloidal particle
in a KCl solution at 25 °C,as a function of particle volume
fraction and dimensionless potential.
cal computations clearly showing that the latter
Onsager relation is also maintained when a DSL
is included in the theories of sedimentation and
electrophoresis in concentrated suspensions,for
whatever conditions on the values of the poten
tial and Stern layer parameters.
6.Results and discussion
6.1.Sedimentation elocity
In Fig.2 we show some numerical results of the
ratio of the standard sedimentation velocity U
SED
to the Stokes velocity U
SED
ST
,for a spherical col
loidal particle in a KCl solution as a function of
dimensionless potential and volume fraction.As
we can see,the sedimentation velocity ratio
rapidly decreases when the volume fraction in
creases whatever the value of potential we
choose.This behavior reﬂects that the higher the
volume fraction,the higher the hydro
dynamic particle–particle interactions.However,
at ﬁxed volume fraction the sedimentation
velocity ratio seems to be less affected when po
tential increases,showing a rather slow decrease
due to the increasing importance of the relaxation
effect.
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169 165
As regards the DSL correction to the standard
sedimentation velocity,we represent in Fig.3 the
ratio of the standard sedimentation velocity U
SED
to the DSL sedimentation velocity (U
SED
)
DSL
as a
function of dimensionless potential and volume
fraction.The values of the Stern layer parameters
that we have chosen for the numerical computa
tions are indeed rather extreme,but our intention
is to show maximum possible effects of the incor
poration of a DSL into the standard model.
When a DSL is present,the induced electric
dipole moment on the particle decreases in com
parison with the standard prediction for the same
conditions,and so does the relaxation effect [34].
As a consequence,the particle will achieve a
larger sedimentation velocity than it would in the
absence of a Stern layer (note that the sedimenta
tion velocity ratio is always 1).
On the other hand,it should be noted that for
a given volume fraction there is a minimum in the
ratio,or in other words,a maximum deviation
from the standard prediction when that ratio is
represented as a function of potential.In fact,
both standard and DSL sedimentation velocities
present a maximum deviation from the Stokes
prediction (uncharged spheres) when they are rep
resented against potential for a given volume
fraction.This maximum deviation can be related
to the concentration polarization effect [34].
In other words,as potential increases from
the region of low values,the relaxation effect
increases as well causing a progressive reduction
of the sedimentation velocity.If is further in
creased,the induced electric dipole moment gener
ated on the falling particle tends to be diminished
due to ionic diffusion ﬂuxes in the diffuse double
layer.These ﬂuxes arise from the formation of
gradients of neutral electrolyte outside the double
layer at the front and rear sides of the hydrody
namic unit while falling under gravity,giving rise
to a decreasing magnitude of the induced electric
dipole moment.In other words,the relaxation
effect [34] would be less important.The ﬁnal
result is a decrease in the magnitude of the micro
scopic electric ﬁeld generated by the distorted
hydrodynamic unit,i.e.particle plus double layer,
and then,a smaller reduction of the sedimentation
velocity at very high potentials.
When a DSL is considered,a new ionic trans
port process develops in the perturbed inner re
gion of the double layer,giving rise to an
increasing importance of the abovementioned
concentration polarization effect at every poten
tial.Consequently,the reduction on the sedimen
tation velocity is always lower when a DSL is
present in comparison with that of the standard
case.Another important feature in Fig.3 is that
the relative deviation of the DSL sedimentation
velocity from the standard prediction seems to be
more important the higher the volume fraction or
equivalently,the higher the hydrodynamic parti
cle–particle interactions.
6.2.Sedimentation potential
In Fig.4 the standard sedimentation potential
is represented as a function of dimensionless po
tential and volume fraction,for the same condi
tions as those in Fig.2.The constant C
e
is deﬁned
at the bottom of the picture.It is worth noting the
decrease in the magnitude of the sedimentation
potential as the volume fraction decreases.Obvi
ously,the lower the volume fraction,the lower the
number of particles contributing to the generation
of the sedimentation ﬁeld.We can also see the
Fig.3.Ratio of the standard sedimentation velocity to the
DSL sedimentation velocity of a spherical colloidal particle in
a KCl solution at 25 °C,as a function of particle volume
fraction and dimensionless potential.
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169166
Fig.4.Standard sedimentation potential in a colloidal suspen
sion of spherical particles in a KCl solution at 25 °C,as a
function of volume fraction and dimensionless potential.
tion potential ratio is always less than unity).This
can be explained according to the abovemen
tioned additional decrease in the magnitude of the
standard induced electric dipole moment when a
DSL is present.
Secondly,we can observe an important increase
in the ratio tending to unity in the limit of high
potentials for ﬁxed volume fraction.In other
words,there would be no signiﬁcant deviation
from the standard model in spite of the presence
of a DSL.This behavior is easy to explain be
cause at high potential the Stern layer reaches
saturation while the diffuse layer charge density
continues to rise,rapidly overshadowing the ef
fects of a DSL,and thus,approaching to the
standard prediction.
6.3.Onsager reciprocal relation between
sedimentation and electrophoresis in concentrated
suspensions
In Fig.6 we display,for the case of no DSL
present,the scaled sedimentation potential and
the scaled electrophoretic mobility multiplied by
the factor C
deﬁned in the picture,as a function
of dimensionless potential for different volume
fractions.Both quantities have been numerically
presence of a maximum when the sedimentation
potential is represented against the potential for
a given volume fraction,being a consequence of
the abovementioned concentration polarization
effect [34].As potential increases,the strength
of the dipolar electric moment induced on the
distorted particles while settling in the gravita
tional ﬁeld increases as well,giving rise to a larger
contribution to the sedimentation potential.As
potential is further increased the relaxation ef
fect seems to become less signiﬁcant owing to the
concentration polarization effect,tending in turn
to diminish the dipolar electric moment,and then,
the sedimentation potential generated in the
suspension.
Let us consider now the effects of the inclusion
of a DSL into the standard theory of the sedimen
tation potential.Thus,in Fig.5 we represent the
ratio of the DSL sedimentation potential to the
standard sedimentation potential as a function of
dimensionless potential and volume fraction.
Several remarkable features can be observed in
this picture.First,the DSL correction to the
sedimentation potential gives always rise to lower
values of the sedimentation potential than those
predicted by the standard model of sedimentation
for the same conditions (note that the sedimenta
Fig.5.Ratio of the DSL sedimentation potential to the
standard sedimentation potential in a colloidal suspension of
spherical particles in a KCl solution at 25 °C,as a function of
volume fraction and dimensionless potential.
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169 167
Fig.6.Plot of the scaled standard electrophoretic mobility and
sedimentation potential in a colloidal suspension of spherical
particles in a KCl solution at 25 °C,as a function of dimen
sionless potential for different volume fractions.For non
zero volume fractions,E
SED
* in open symbols;
OHS
* in solid
symbols (Ohshima’s model).
cident with the scaled electrophoretic mobility
whatever the volume fraction may be,if Shilov–
Zharkikh’s boundary condition (Eq.(52)) is as
sumed.We have conﬁrmed this result by
numerical integration of the theories,as it can be
seen in Eq.(7).
Likewise,it is worth pointing out that this
Onsager relation is not a low potential approxi
mation.On the contrary,it remains valid for the
whole range of values.
In Fig.7 the scaled sedimentation potential and
the scaled electrophoretic mobility are displayed
as a function of dimensionless potential for
different volume fractions.Again,both quantities
have been independently calculated by numeri
cally solving on the one hand Ohshima’s theory of
sedimentation in concentrated suspensions,and
on the other,Ohshima’s theory of electrophoresis
in concentrated suspensions including now the
Shilov–Zharkikh boundary condition (Eq.(52))
instead of that by Levine–Neale (Eq.(51)).As we
can see,the numerical agreement between each set
of results is excellent whatever the values of vol
ume fraction or potential have been chosen.
This is also true when a DSL approach is used,
as shown in Fig.8 for the same conditions as
those of Fig.5.
and independently calculated with Ohshima’s
models of sedimentation [25] and electrophoresis
[26] in concentrated colloidal suspensions.The
results clearly indicate that in the limit when
volume fraction tends to zero Ohshima’s Onsager
relation for low potentials,Eq.(45),converges
to the wellknown Onsager relation Eq.(43) pre
viously derived for the dilute case,which is valid
for the whole range of values.In other words,
the scaled sedimentation potential is numerically
coincident with the scaled electrophoretic mobility
in that limit (note that in this case the factor
C
=1).For the remaining volume fractions,the
Onsager reciprocal relation proposed by Ohshima
for concentrated suspensions would be a good
approximation for low and low volume fraction,
as observed in Fig.6.
On the other hand,as pointed out in a previous
section,we have modiﬁed Ohshima’s model of
electrophoresis in concentrated suspensions to
fulﬁll the requirements of Shilov–Zharkikh’s cell
model.In doing so,we have obtained the same
expression for the Onsager reciprocal relation be
tween sedimentation and electrophoresis as that
previously derived for the dilute case,but now for
concentrated suspensions.In other words,the
scaled sedimentation potential is numerically coin
Fig.7.Plot of the scaled standard electrophoretic mobility and
sedimentation potential in a colloidal suspension of spherical
particles in a KCl solution at 25 °C,as a function of dimen
sionless potential for different volume fractions.For non
zero volume fractions,E
SED
* in open symbols;
SHI
* in solid
symbols (Shilov–Zharkikh’s model).
F.Carrique et al./Colloids and Surfaces A:Physicochem.Eng.Aspects 195(2001)157–169168
Fig.8.Plot of the scaled DSL electrophoretic mobility and
sedimentation potential in a colloidal suspension of spherical
particles in a KCl solution at 25 °C,as a function of dimen
sionless potential for different volume fractions.For non
zero volume fractions,E
SED
* in open symbols;
SHI
* in solid
symbols (Shilov–Zharkikh’s model).
Acknowledgements
Financial support for this work by MEC,Spain
(Project No.MAT980940),and INTAS (Project
9900510) is gratefully acknowledged.
References
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7.Conclusions
In this work,we have presented numerical cal
culations concerning the sedimentation velocity
and potential in concentrated suspensions for ar
bitrary potential and nonoverlapping double
layers of the particles.
Furthermore,we have extended the standard
Ohshima’s theory of sedimentation in concen
trated suspensions,to include a DSL into the
model.The results show that regardless of the
particle volume fraction and potential,the mere
presence of a DSL causes the sedimentation veloc
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decrease in comparison with the standard
predictions.
On the other hand,we have analyzed the On
sager reciprocal relation between sedimentation
and electrophoresis derived by Ohshima for con
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Shilov–Zharkikh cell model.We have conﬁrmed
that the Shilov–Zharkikh cell model fulﬁlls the
same Onsager relation in concentrated suspen
sions as that previously derived for the dilute case,
for whatever conditions of potential and volume
fraction,including a DSL as well.
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Double Layer in Disperse Systems and Polyelectrolytes,
Wiley,New York,1974.
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