Sedimentation velocity and potential in a concentrated ...hera.ugr.es/doi/15087980.pdf · by a concentric virtual shell of an electrolyte solution, having an outer radius of b such

Colloids and Surfaces

A: Physicochemical and Engineering Aspects 195 (2001) 157–169

Sedimentation velocity and potential in a concentratedcolloidal suspension

Effect of a dynamic Stern layer

F. Carrique a,*, F.J. Arroyo b, A.V. Delgado b

a Dpto. Fısica Aplicada I, Facultad de Ciencias, Uni�ersidad de Malaga, 29071 Malaga, Spainb Dpto. Fısica Aplicada, Facultad de Ciencias, Uni�ersidad de Granada, 18071 Granada, Spain

Abstract

The standard theory of the sedimentation velocity and potential of a concentrated suspension of charged sphericalcolloidal 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 conditionsconcerning volume fraction, and �-potential of the particles. The Onsager relation between the sedimentationpotential and the electrophoretic mobility of spherical colloidal particles in concentrated suspensions, derived byOhshima for low �-potentials, is also analyzed as well as its appropriate range of validity. On the other hand, theabove-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. FaradayTrans. 86 (1990) 2859) developed to calculate the electrophoretic mobility of a colloidal particle, allowing for thelateral motion of ions in the inner region of the double layer (DSL). The role of different Stern layer parameters onthe sedimentation velocity and potential are discussed and compared with the case of no Stern layer present. Forevery volume fraction, the results show that the sedimentation velocity is lower when a Stern layer is present than thatof Ohshima’s prediction. Likewise, it is worth pointing out that the sedimentation field always decreases when a Sternlayer is present, undergoing large changes in magnitude upon varying the different Stern layer parameters. Inconclusion, the presence of a DSL causes the sedimentation velocity to increase and the sedimentation potential todecrease, in comparison with the standard case, for every volume fraction. Reasons for these behaviors are given interms of the decrease in the magnitude of the induced electric dipole moment on the particles, and therefore on therelaxation effect, when a DSL is present. Finally, we have modified Ohshima’s model of electrophoresis inconcentrated suspensions, to fulfill the requirements of Shilov–Zharkhik’s cell model. In doing so, the well-knownOnsager reciprocal relation between sedimentation and electrophoresis previously obtained for the dilute case is againrecovered but now for concentrated suspensions, being valid for every �-potential and volume fraction. © 2001Elsevier Science B.V. All rights reserved.

Keywords: Sedimentation velocity; Sedimentation potential; Concentrated suspensions; Onsager reciprocal relation

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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 agravitational field, the electrical double layer sur-rounding each particle is distorted because of thefluid motion, giving rise to a microscopic electricfield (the relaxation effect). As a consequence, thefalling velocity of the particle, i.e. the sedimenta-tion velocity, is lower in comparison with that ofan uncharged particle. On the other hand, theseelectric fields superimpose to yield a macroscopicelectric 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-conductingspherical particles with arbitrary double layerthickness and �-potential, was developed byOhshima [1] on the basis of previous theoreticalapproaches [2–9]. In his paper Ohshima removedthe shortcomings and deficiencies already re-ported by Saville [10] concerning Booth’s methodof calculation of the sedimentation potential. Fur-thermore, he presented a direct proof of the On-sager reciprocal relation that holds betweensedimentation and electrophoresis.

On the other hand, a great effort is beingaddressed to improve the theoretical results pre-dicted by the standard electrokinetic theories deal-ing with different electrokinetic phenomena incolloidal suspensions. One of the most relevantextensions of these electrokinetic models has beenthe inclusion of a dynamic Stern layer (DSL) ontothe surface of the colloidal particles. Thus,Zukoski and Saville [11] developed a DSL modelto reconcile the differences observed between �-potentials derived from static electrophoretic mo-bility and conductivity measurements.Mangelsdorf and White [12], using the techniquesdeveloped by O’Brien and White for the study ofthe electrophoretic mobility of a colloidal particle[13], presented in 1990 a rigorous mathematicaltreatment for a general DSL model. They ana-lyzed the effects of different Stern layer adsorp-tion isotherms on the static field electrophoreticmobility and suspension conductivity.

More recently, the theory of Stern layer trans-port has been applied to the study of the lowfrequency dielectric response of colloidal suspen-sions by Kijlstra et al. [14], incorporating a sur-face conductance layer to the thin double layertheory of Fixman [15,16]. Likewise, Rosen et al.[17] generalized the standard theory of the con-ductivity and dielectric response of a colloidalsuspension in AC fields of DeLacey and White[18], assuming the model of Stern layer developedby Zukoski and Saville [11]. Very recently, Man-gelsdorf and White presented a rigorous mathe-matical study for a general DSL model applicableto time dependent electrophoresis and dielectricresponse [19,20]. In general, the theoretical predic-tions of the DSL models improve the comparisonbetween theory and experiment [14,17,21,22], al-though there are still important discrepancies.

Returning to the sedimentation phenomena incolloidal suspensions, a DSL extension of Ohshi-ma’s theory of the sedimentation velocity andpotential in dilute suspensions, has been recentlypublished [23]. The results show that whatever thechosen set of Stern layer parameters or �-poten-tial may be, the presence of a DSL causes thesedimentation velocity to increase and the sedi-mentation potential to decrease, in comparisonwith the standard prediction (no Stern layerpresent).

On the other hand, the theory of sedimentationin a concentrated suspension of spherical colloidalparticles, proposed by Levine et al. [9] on thebasis of the Kuwabara cell model [24], has beenfurther 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 electricdouble 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 histheory of electrophoresis in concentrated suspen-sions [26]. This theory is also based on theKuwabara cell model in order to account for thehydrodynamic particle–particle interactions, anduses the same boundary condition on the electricpotential 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 inconcentrated suspensions [27].

Recalling the attention on the DSL correctionto the electrokinetic theories, it seemed of interestto explore the effects of extending the standardOhshima’s theory of the sedimentation velocityand potential in a concentrated suspension ofcharged spherical colloidal particles [25], to in-clude a DSL model. Thus, the chosen startingpoint has been the method proposed by Mangels-dorf and White in their theory of the elec-trophoretic mobility of a colloidal particle, toallow for the adsorption and lateral motion ofions in the inner region of the double layer (DSL)[12].

Finally, the aims of this paper can be describedas follows. First, we have obtained a numericalsolution of the standard Ohshima’s theory ofsedimentation in concentrated suspensions, forthe whole range of �-potential and volume frac-tion, and non-overlapping double layers. Further-more, we have extended the latter standard theoryto include a DSL on the surface of the particles,and analyzed the effects of its inclusion on thesedimentation velocity and potential. And then,we have analyzed the Onsager reciprocal relationthat holds between sedimentation and elec-trophoresis in concentrated suspensions, for bothstandard and DSL cases. It can be concluded thatthe presence of a Stern layer provokes a ratherslow increase on the magnitude of the sedimenta-tion velocity of a colloidal particle, whatever thevalues of Stern layer, particle and solutionparameters used in the calculations. On the otherhand, the presence of a Stern layer causes thesedimentation potential to decrease with respectto the standard prediction.

2. Standard governing equations and boundaryconditions

The starting point for our work has been thestandard theory of the sedimentation velocity andpotential in a concentrated suspension of spheri-cal colloidal particles, developed by H. Ohshima[25] on the basis of the Kuwabara cell model toaccount for the hydrodynamic particle–particle

interactions (see Fig. 1). According to this model,each spherical particle of radius a is surroundedby a concentric virtual shell of an electrolytesolution, having an outer radius of b such that theparticle/cell volume ratio in the unit cell is equalto the particle volume fraction throughout theentire suspension, i.e.

�=�a

b�3

. (1)

In fact, a is the radius of the ‘hydrodynamicunit’, i.e. a rigid particle plus a thin layer ofsolution linked to its surface moving with it as awhole. The surface r=a is usually called ‘slippingplane’. This is the plane outside which the contin-uum equations of hydrodynamics are assumed tohold. As usual, we will make no distinction be-tween the terms particle surface and slippingplane.

Before proceeding with the analysis of the mod-ifications arising from the DSL correction to thestandard model, it will be useful to briefly reviewthe 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 radiusa and mass density �p immersed in an electrolytesolution composed of N ionic species of valencieszi, bulk number concentrations ni

�, and drag co-efficients �i (i=1, …, N). The axes of the coordi-nate system (r, �, �) are fixed at the centre of theparticle. The polar axis (�=0) is set parallel to g.

Fig. 1. Schematic picture of an ensemble of spherical particlesin a concentrated suspension, according to the Kuwabara cellmodel [24].


The particle is assumed to settle with steady ve-locity USED, the sedimentation velocity, in theelectrolyte solution of viscosity � and mass den-sity �o in the presence of a gravitational field g.For the spherical symmetry case, both USED andg have the same direction. In the absence of gfield, the particle has a uniform electric poten-tial, the �-potential �, at r=a, where r is theradial spherical coordinate, or equivalently, themodulus of position vector.

A complete description of the system requiresa knowledge of the electric potential �(r), thenumber density or ionic concentration ni(r) andthe drift velocity vi(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. Thefundamental equations connecting these quanti-ties are [1,25]:

�2�(r)= −�(r)rso

(2)

�(r)= �N

i=1

zi eni(r) (3)

��2u(r)−�p(r)−��(r)+�og=0 (4)

� � u(r)=0 (5)

vi=u −1�i

�i (i=1, …, N) (6)

i(r)= i�+zi e�(r)+KBT ln ni(r)

(i=1, …, N) (7)

�[ni(r)vi(r)]=0 (i=1, …, N), (8)

where e is the elementary electric charge, KB theBoltzmann’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 thepermittivity of a vacuum, and �(r) is the electriccharge density given by Eq. (3). Eqs. (4) and (5)are the Navier–Stokes equations appropriate toa steady incompressible fluid flow at lowReynolds number in the presence of electric andgravitational body forces. Eq. (6) expresses thatthe ionic flow is caused by the liquid flow andthe gradient of the electrochemical potentialdefined in Eq. (7), and it can be related to thebalance of the hydrodynamic drag, electrostatic,and thermodynamic forces acting on each ionicspecies. Eq. (8) is the continuity equation ex-

pressing the conservation of the number of eachionic 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=NAe2�zi �

� io (i=1, …, N), (9)

where NA is Avogadro’s number.At equilibrium, that is, in the absent of the

gravitational field, the distribution of electrolyteions obeys the Boltzmann distribution

ni(o)=ni

� exp�

−zi e� (o)

KBT�

(i=1, …, N), (10)

and the equilibrium electric potential � (o) sa-tisfies the Poisson–Boltzmann equation

1r2

ddr�

r2 d� (o)

dr�

= −� el

(o)(r)rso

(11)

� el(o)(r)= �

N

i=1

zi 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 atthe slipping plane and at the outer surface ofthe 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, theboundary conditions for the liquid velocity uand the ionic velocity of each ionic species atthe particle surface are expressed by the follow-ing equations

u=0 at r=a (15)

vi � r=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 thatthere are no ion fluxes through the slippingplane (r is the unit normal outward from theparticle surface). According to the Kuwabara


cell model, the liquid velocity at the outer surfaceof the unit cell satisfies the conditions:

ur= −USED 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, andthe vorticity is equal to zero.

Following Ohshima, we will assume that theelectrical double layer around the particle is onlyslightly distorted due to the gravitational fieldabout their equilibrium values. Thus, the follow-ing perturbation scheme for the above-mentionedquantities can be used,

ni(r)=ni(o)(r)+ ni(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 ofequilibrium. The perturbations in ionic numberdensity and electric potential are related to eachother through the perturbation in electrochemicalpotential by

i=zi e �+KBT ni

n i(o) (i=1, …, N). (22)

In terms of the perturbation quantities, thecondition that the ionic species are not allowed topenetrate the particle surface in Eq. (16), trans-forms into

� i � r=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 unitcell, this extra condition holds:

i=0( ni=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 thenwrite

u(r)= (ur, u�, u�)

=�

−2r

h g cos �,1r

ddr

(rh)g sin �, 0�

(25)

i(r)= −zi e�i(r)(g � r) (i=1, …, N), (26)

to obtain the following set of ordinary coupleddifferential equations and boundary conditions atthe slipping plane and at the outer surface of thecell:L(Lh)= −

e�r

dydr

�N

i=1

ni�z i

2 exp(−zi y)�i(r), (27)

with y=e� (o)/KT,

L(�i(r))=dydr�

zi

d�i

dr−

2�i

ehr�

(i=1, …, N)

(28)

h(a)=dhdr

(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�d2

dr2+2r

ddr

−2r2. (32)

In addition to the previous boundary condi-tions, we must impose the constraint that in thestationary state the net force acting on the particleor the unit cell must be zero [25].

3. Extension to include a dynamic Stern layer

We now deal with the problem of including thepossibility of adsorption and ionic transport inthe inner region of the double layer of the parti-cles. We will follow the method developed byMangelsdorf and White [12] in their theory of theelectrophoresis and conductivity in a dilute col-loidal suspension. This theory allows for the ad-sorption and lateral motion of ions in the latterinner region using the well-known Stern model.According to this method, the condition that ionscannot penetrate the slipping plane no longermaintains, and therefore, the evaluation of thefluxes of each ionic species through the slippingplane permits us to obtain the following newslipping plane boundary conditions for the func-tions �i(r),


d�i

dr(a)−

2 i

a�i(a)=0 (i=1, …, N) (33)

i=[eNi ]/(ae 10−pKi)(�i/� i

t)exp[(zi e/KBT)(�d/C2]

NA103+ �N

j=1(NA103c j

�/10−pKj)exp[(−zj e/KBT)(�−�d/C2)],

(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. Theseparameters depend on, the �-potential � ; the ra-tio between the drag coefficient �i of each ionicspecies in the bulk solution and in the Sternlayer � i

t; the density of sites Ni available foradsorption in the Stern layer; the pKi of ionicdissociation constant for each ionic species (theadsorption of each ionic species onto an emptyStern layer site is represented as a dissociationreaction in this theory [12]), the capacity C2 ofthe outer Stern layer, the radius a of the parti-cles, the electrolyte concentration through c j

�,i.e. the equilibrium molar concentration of type jions in solution, and the charge density per unitsurface area in the double layer �d. It is worthnoting that the other boundary conditions ex-pressed by Eqs. (29) and (31) remain unchangedwhen a DSL is assumed.

A numerical method similar to that proposedby DeLacey and White in their theory of thedielectric response and conductivity of a col-loidal suspension in time-dependent fields [18],has been applied to solve the above-mentionedset of coupled ordinary differential equations ofthe sedimentation theory in concentrated col-loidal suspensions. Furthermore, both standardand DSL cases have been extensively analyzed.In a recent paper [23], we successfully employedthe latter numerical scheme to solve the standardtheory 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 andpotential

Let us describe now how the sedimentationvelocity and potential for a concentrated suspen-

sion can be calculated. According to the condi-tion for the fluid velocity at the outer surface ofthe unit cell, the fluid velocity has to be parallelto the sedimentation velocity (see Eqs. (17) and(25)). Thus, we can obtain the sedimentation ve-locity USED, once the value of function h hasbeen determined at the outer surface of the cell,i.e.

USED=2h(b)

bg. (35)

For the case of uncharged particles (�=0), thesedimentation velocity is given by the well-known Stokes formula [25]

USEDST =

2a2(�p−�o)9�

g. (36)

As regards the sedimentation potential ESED, itcan be considered as the volume average of thegradient of the electric potential in the suspen-sion volume V, i.e.

ESED= −1V�

V

��(r)dV. (37)

Following Ohshima [25], the net electric cur-rent �i� in the suspension can be expressed interms of the sedimentation potential and the firstradial derivatives of �i functions at the outersurface of the unit cell,

�i�=K��ESED+1

K� �N

i=1

�z i2e2ni

�

�i

d�i

dr(b)n

g�

,

(38)

where K� is the electric conductivity of the elec-trolyte solution in the absence of the colloidalparticles.

If now we impose, following Saville [10]and Ohshima [25], the requirement of zero netelectric current in the suspension, we finally ob-tain

ESED= −1

K� �N

i=1

�z i2e2ni

�

�i

d�i

dr(b)n

g. (39)

Likewise, we define the scaled sedimentationpotential ESED* as in the dilute case by

ESED* =3�eK�

2rsoKBT(�p−�o)��ESED�

�g� . (40)


5. Onsager reciprocal relation betweensedimentation and electrophoresis in concentratedsuspensions

It is well-known that an Onsager reciprocalrelation holds between sedimentation and elec-trophoresis. A direct proof of this relationshipwas derived by Ohshima et al. [1] for dilute sus-pensions, and is given by

ESED= −�(�p−�o)

K� g, (41)

where is the electrophoretic mobility of a col-loidal particle. Furthermore, this relation is alsosatisfied when a DSL is incorporated to the theo-ries of sedimentation and electrophoresis in dilutecolloidal suspensions [23].

On the other hand, the electrophoretic mobilityis usually represented by a scaled quantity * [13]defined by

*=3�e

2rsoKBT. (42)

Eq. (41) can then be rewritten in terms of thescaled quantities to give a simple convenient ex-pression for the Onsager relation, namely,

ESED* =*. (43)

Very recently Ohshima derived an Onsager rela-tion between sedimentation and electrophoresis inconcentrated suspensions, applicable for low �-potentials and non-overlapping of double layers[25]. In that paper, Ohshima used an expressionfor the electrophoretic mobility OHS of a spheri-cal colloidal particle, derived according to histheory of the electrophoresis in concentrated sus-pensions [26]. The Onsager relation he found isgiven by

ESED= −�(1−�)(�p−�o)

(1+�/2)K� OHSg, (44)

or equivalently,

ESED* =(1−�)

(1+�/2)OHS* , (45)

where Eqs. (40) and (42) have been used. In thelimit when volume fraction tends to zero, Eqs.(44) and (45) converges to the well-known Eqs.

(41) and (43) which describes the Onsager relationbetween sedimentation and electrophoresis in di-lute suspensions.

However, very recently Dukhin et al. [28] havepointed out that the Levine–Neale cell model [27],employed by many authors to develop theoreticalelectrokinetic models in multiparticle systems, inparticular those of sedimentation, electrophoresisand conductivity in concentrated suspensions[9,26,29–32], presents some deficiencies. Accord-ing to Dukhin et al. [28] the Levine–Neale cellmodel is not compatible with certain classicallimits concerning, specially, the volume fractiondependence in the exact Smoluchovski’s law inconcentrated suspensions. Instead of the Levine–Neale cell model, Dukhin et al. propose to use theShilov–Zharkikh cell model [33] which not onlyagrees with the latter Smoluchovski’s result butalso correlates with the electric conductivity of theMaxwell–Wagner theory [34]. It is worth notingthat Ohshima’s theory of the electrophoretic mo-bility in concentrated suspensions [26] incorpo-rates the Levine–Neale boundary condition onthe electric potential at the outer surface of theunit cell. This condition states that the local elec-tric field has to be parallel to the applied electricfield E at the outer surface of the cell.

Then, it seemed quite interesting to comparethe changes in Ohshima’s Onsager relation forconcentrated suspensions, if any, that could arisefrom the consideration of a different boundarycondition on the electric potential according tothe Shilov–Zharkikh cell model, which is basedon arguments of non-equilibrium thermodynam-ics. Following Ohshima’s theory of electrophore-sis in concentrated suspensions [26], the boundarycondition for the perturbed electric potential atthe outer surface of the unit cell is expressed by

� � � r= −E � r at r=b. (46)

However, according to the Shilov–Zharkikhcell 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]


i=zi e �, (48)

and consequently, Eq. (46) transforms into

� i � r= −zi eE r. (49)

Following Ohshima, spherical symmetry consider-ations permit us to write

i(r)= −zi e�i(r)(E � r) (i=1, …, N), (50)

which is analogous to Eq. (26) for sedimentation.Now, according to Eq. (50), Eq. (49) finallybecomes

d�i

dr(b)=1. (51)

However, following the Shilov–Zharkikhboundary condition given by Eq. (47), a differentresult can be obtained, i.e.

�i(b)=b, (52)

where Eq. (50) has been used reading �E� insteadof E. If now we change in Ohshima’s theory ofthe 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 weconfine ourselves to the analytical approach oflow �-potentials developed in Ohshima’s papersof sedimentation [25] and electrophoresis [26] inconcentrated suspensions, an Onsager reciprocalrelation different to that by Ohshima (Eqs. (44)and (45)), is found, i.e.

ESED* = SHI* . (53)

It should be noted that this new Onsager rela-tion has exactly the same form as the well-knownOnsager relation connecting sedimentation andelectrophoresis in dilute suspensions (see Eqs. (41)and (43)). Likewise, we have numerically confi-rmed that this Onsager relation also holds for thewhole range of �-potentials unlike that of Eq.(44). In conclusion, we can state that the Onsagerreciprocal relation between sedimentation andelectrophoresis, previously derived for the dilutecase, also holds for concentrated suspensions ifShilov–Zharkikh’s boundary condition is consid-ered. In the next section, we will present numeri-

Fig. 2. Ratio of the standard sedimentation velocity to theStokes sedimentation velocity of a spherical colloidal particlein a KCl solution at 25 °C, as a function of particle volumefraction and dimensionless �-potential.

cal computations clearly showing that the latterOnsager relation is also maintained when a DSLis included in the theories of sedimentation andelectrophoresis in concentrated suspensions, forwhatever 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 theratio of the standard sedimentation velocity USED

to the Stokes velocity USEDST , for a spherical col-

loidal particle in a KCl solution as a function ofdimensionless �-potential and volume fraction. Aswe can see, the sedimentation velocity ratiorapidly decreases when the volume fraction in-creases whatever the value of �-potential wechoose. This behavior reflects that the higher thevolume fraction, the higher the hydro-dynamic particle–particle interactions. However,at fixed volume fraction the sedimentationvelocity ratio seems to be less affected when �-po-tential increases, showing a rather slow decreasedue to the increasing importance of the relaxationeffect.


As regards the DSL correction to the standardsedimentation velocity, we represent in Fig. 3 theratio of the standard sedimentation velocity USED

to the DSL sedimentation velocity (USED)DSL as afunction of dimensionless �-potential and volumefraction. The values of the Stern layer parametersthat we have chosen for the numerical computa-tions are indeed rather extreme, but our intentionis to show maximum possible effects of the incor-poration of a DSL into the standard model.When a DSL is present, the induced electricdipole moment on the particle decreases in com-parison with the standard prediction for the sameconditions, and so does the relaxation effect [34].As a consequence, the particle will achieve alarger sedimentation velocity than it would in theabsence of a Stern layer (note that the sedimenta-tion velocity ratio is always �1).

On the other hand, it should be noted that fora given volume fraction there is a minimum in theratio, or in other words, a maximum deviationfrom the standard prediction when that ratio isrepresented as a function of �-potential. In fact,both standard and DSL sedimentation velocitiespresent a maximum deviation from the Stokesprediction (uncharged spheres) when they are rep-resented against �-potential for a given volume

fraction. This maximum deviation can be relatedto the concentration polarization effect [34].

In other words, as �-potential increases fromthe region of low �-values, the relaxation effectincreases as well causing a progressive reductionof the sedimentation velocity. If � is further in-creased, the induced electric dipole moment gener-ated on the falling particle tends to be diminisheddue to ionic diffusion fluxes in the diffuse doublelayer. These fluxes arise from the formation ofgradients of neutral electrolyte outside the doublelayer at the front and rear sides of the hydrody-namic unit while falling under gravity, giving riseto a decreasing magnitude of the induced electricdipole moment. In other words, the relaxationeffect [34] would be less important. The finalresult is a decrease in the magnitude of the micro-scopic electric field generated by the distortedhydrodynamic unit, i.e. particle plus double layer,and then, a smaller reduction of the sedimentationvelocity 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 anincreasing importance of the above-mentionedconcentration polarization effect at every �-poten-tial. Consequently, the reduction on the sedimen-tation velocity is always lower when a DSL ispresent in comparison with that of the standardcase. Another important feature in Fig. 3 is thatthe relative deviation of the DSL sedimentationvelocity from the standard prediction seems to bemore important the higher the volume fraction orequivalently, the higher the hydrodynamic parti-cle–particle interactions.

6.2. Sedimentation potential

In Fig. 4 the standard sedimentation potentialis represented as a function of dimensionless �-po-tential and volume fraction, for the same condi-tions as those in Fig. 2. The constant Ce is definedat the bottom of the picture. It is worth noting thedecrease in the magnitude of the sedimentationpotential as the volume fraction decreases. Obvi-ously, the lower the volume fraction, the lower thenumber of particles contributing to the generationof the sedimentation field. We can also see the

Fig. 3. Ratio of the standard sedimentation velocity to theDSL sedimentation velocity of a spherical colloidal particle ina KCl solution at 25 °C, as a function of particle volumefraction and dimensionless �-potential.


Fig. 4. Standard sedimentation potential in a colloidal suspen-sion of spherical particles in a KCl solution at 25 °C, as afunction of volume fraction and dimensionless �-potential.

tion potential ratio is always less than unity). Thiscan be explained according to the above-men-tioned additional decrease in the magnitude of thestandard induced electric dipole moment when aDSL is present.

Secondly, we can observe an important increasein the ratio tending to unity in the limit of high�-potentials for fixed volume fraction. In otherwords, there would be no significant deviationfrom the standard model in spite of the presenceof a DSL. This behavior is easy to explain be-cause at high �-potential the Stern layer reachessaturation while the diffuse layer charge densitycontinues to rise, rapidly overshadowing the ef-fects of a DSL, and thus, approaching to thestandard prediction.

6.3. Onsager reciprocal relation betweensedimentation and electrophoresis in concentratedsuspensions

In Fig. 6 we display, for the case of no DSLpresent, the scaled sedimentation potential andthe scaled electrophoretic mobility multiplied bythe factor C� defined in the picture, as a functionof dimensionless �-potential for different volumefractions. Both quantities have been numerically

presence of a maximum when the sedimentationpotential is represented against the �-potential fora given volume fraction, being a consequence ofthe above-mentioned concentration polarizationeffect [34]. As �-potential increases, the strengthof the dipolar electric moment induced on thedistorted particles while settling in the gravita-tional field increases as well, giving rise to a largercontribution to the sedimentation potential. As�-potential is further increased the relaxation ef-fect seems to become less significant owing to theconcentration polarization effect, tending in turnto diminish the dipolar electric moment, and then,the sedimentation potential generated in thesuspension.

Let us consider now the effects of the inclusionof a DSL into the standard theory of the sedimen-tation potential. Thus, in Fig. 5 we represent theratio of the DSL sedimentation potential to thestandard sedimentation potential as a function ofdimensionless �-potential and volume fraction.Several remarkable features can be observed inthis picture. First, the DSL correction to thesedimentation potential gives always rise to lowervalues of the sedimentation potential than thosepredicted by the standard model of sedimentationfor the same conditions (note that the sedimenta-

Fig. 5. Ratio of the DSL sedimentation potential to thestandard sedimentation potential in a colloidal suspension ofspherical particles in a KCl solution at 25 °C, as a function ofvolume fraction and dimensionless �-potential.


Fig. 6. Plot of the scaled standard electrophoretic mobility andsedimentation potential in a colloidal suspension of sphericalparticles in a KCl solution at 25 °C, as a function of dimen-sionless �-potential for different volume fractions. For non-zero volume fractions, ESED* in open symbols; OHS* in solidsymbols (Ohshima’s model).

cident with the scaled electrophoretic mobilitywhatever the volume fraction may be, if Shilov–Zharkikh’s boundary condition (Eq. (52)) is as-sumed. We have confirmed this result bynumerical integration of the theories, as it can beseen in Eq. (7).

Likewise, it is worth pointing out that thisOnsager relation is not a low �-potential approxi-mation. On the contrary, it remains valid for thewhole range of � values.

In Fig. 7 the scaled sedimentation potential andthe scaled electrophoretic mobility are displayedas a function of dimensionless �-potential fordifferent volume fractions. Again, both quantitieshave been independently calculated by numeri-cally solving on the one hand Ohshima’s theory ofsedimentation in concentrated suspensions, andon the other, Ohshima’s theory of electrophoresisin concentrated suspensions including now theShilov–Zharkikh boundary condition (Eq. (52))instead of that by Levine–Neale (Eq. (51)). As wecan see, the numerical agreement between each setof 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 asthose of Fig. 5.

and independently calculated with Ohshima’smodels of sedimentation [25] and electrophoresis[26] in concentrated colloidal suspensions. Theresults clearly indicate that in the limit whenvolume fraction tends to zero Ohshima’s Onsagerrelation for low �-potentials, Eq. (45), convergesto the well-known Onsager relation Eq. (43) pre-viously derived for the dilute case, which is validfor the whole range of �-values. In other words,the scaled sedimentation potential is numericallycoincident with the scaled electrophoretic mobilityin that limit (note that in this case the factorC�=1). For the remaining volume fractions, theOnsager reciprocal relation proposed by Ohshimafor concentrated suspensions would be a goodapproximation for low � and low volume fraction,as observed in Fig. 6.

On the other hand, as pointed out in a previoussection, we have modified Ohshima’s model ofelectrophoresis in concentrated suspensions tofulfill the requirements of Shilov–Zharkikh’s cellmodel. In doing so, we have obtained the sameexpression for the Onsager reciprocal relation be-tween sedimentation and electrophoresis as thatpreviously derived for the dilute case, but now forconcentrated suspensions. In other words, thescaled sedimentation potential is numerically coin-

Fig. 7. Plot of the scaled standard electrophoretic mobility andsedimentation potential in a colloidal suspension of sphericalparticles in a KCl solution at 25 °C, as a function of dimen-sionless �-potential for different volume fractions. For non-zero volume fractions, ESED* in open symbols; SHI* in solidsymbols (Shilov–Zharkikh’s model).


Fig. 8. Plot of the scaled DSL electrophoretic mobility andsedimentation potential in a colloidal suspension of sphericalparticles in a KCl solution at 25 °C, as a function of dimen-sionless �-potential for different volume fractions. For non-zero volume fractions, ESED* in open symbols; SHI* in solidsymbols (Shilov–Zharkikh’s model).

Acknowledgements

Financial support for this work by MEC, Spain(Project No. MAT98-0940), and INTAS (Project99-00510) is gratefully acknowledged.

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7. Conclusions

In this work, we have presented numerical cal-culations concerning the sedimentation velocityand potential in concentrated suspensions for ar-bitrary �-potential and non-overlapping doublelayers of the particles.

Furthermore, we have extended the standardOhshima’s theory of sedimentation in concen-trated suspensions, to include a DSL into themodel. The results show that regardless of theparticle volume fraction and �-potential, the merepresence of a DSL causes the sedimentation veloc-ity to increase and the sedimentation potential todecrease in comparison with the standardpredictions.

On the other hand, we have analyzed the On-sager reciprocal relation between sedimentationand electrophoresis derived by Ohshima for con-centrated suspensions, and compared it with theOnsager relation obtained according to theShilov–Zharkikh cell model. We have confirmedthat the Shilov–Zharkikh cell model fulfills thesame Onsager relation in concentrated suspen-sions as that previously derived for the dilute case,for whatever conditions of �-potential and volumefraction, including a DSL as well.


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[32] H. Ohshima, J. Colloid Interf. Sci. 212 (1999) 443.

[33] V.N. Shilov, N.I. Zharkikh, Y.B. Borkovskaya, Colloid J.43 (1981) 434.

[34] S.S. Dukhin, V.N. Shilov, Dielectric Phenomena and theDouble Layer in Disperse Systems and Polyelectrolytes,Wiley, New York, 1974.

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Sedimentation velocity and potential in a concentrated ...hera.ugr.es/doi/15087980.pdf · by a concentric virtual shell of an electrolyte solution, having an outer radius of b such