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 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

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 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

* =

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 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

*=

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 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

=−Er 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-

culations concerning the sedimentation velocity

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-

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-

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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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