Sedimentation velocity and potential in a concentrated colloidal suspension Effect of a dynamic Stern layer

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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,Uni￿ersidad de Ma´laga,29071Ma´laga,Spain
b
Dpto.Fı´sica Aplicada,Facultad de Ciencias,Uni￿ersidad 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 non-overlapping 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
above-mentioned 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 well-known
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.
0927-7757/01/$ - 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 well-known 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 non-conducting
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 non-overlapping 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 non-overlapping 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 above-mentioned
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
T￿n
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 well-known 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 so-called 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 time-dependent fields [18],
has been applied to solve the above-mentioned
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
* =
3￿eK
￿
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 well-known 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
￿*=
3￿e
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 non-overlapping 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 well-known 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 non-equilibrium 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
￿￿=−￿E￿r at r=b.(47)
being ￿E￿ the macroscopic electric field.For low
￿-potentials and non-overlapping 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 well-known
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 above-mentioned
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 above-men-
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 above-mentioned 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 well-known 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.MAT98-0940),and INTAS (Project
99-00510) is gratefully acknowledged.
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7.Conclusions
In this work,we have presented numerical cal-
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and potential in concentrated suspensions for ar-
bitrary ￿-potential and non-overlapping double
layers of the particles.
Furthermore,we have extended the standard
Ohshima’s theory of sedimentation in concen-
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model.The results show that regardless of the
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