Electrokinetics of Concentrated Suspensions of Spherical
Colloidal Particles: Effect of a Dynamic Stern Layer
on Electrophoresis and DC Conductivity
F. Carrique,∗,1F. J. Arroyo,†and A. V. Delgado‡
∗Departamento de F´ısica Aplicada I, Facultad de Ciencias, Universidad de M´alaga, 29071 M´alaga, Spain;†Departamento de F´ısica, Facultad de Ciencias Experimentales, Universidad de Ja´en, 23071 Ja´en, Spain; and‡Departamento de F´ısica Aplicada,
Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Received March 26, 2001; accepted August 4, 2001; published online October 5, 2001
In this paper the theory of the electrophoretic mobility and electri-cal conductivity of concentrated suspensions of spherielectri-cal colloidal particles, developed by H. Ohshima (J. Colloid Interface Sci.188,
481 (1997);J. Colloid Interface Sci.212,443 (1999)), has been re-vised and extended to include the effect of a dynamic Stern layer on the surface of the particles. The starting point has been the the-ory developed by C. S. Mangelsdorf and L. R. White (J. Chem. Soc., Faraday Trans.86,2859 (1990)) dealing with the calculation of the electrophoretic mobility of a colloidal particle, when lateral motion of ions in the inner region of the double layer is possible (dynamic Stern layer (DSL)). The effects of Stern layer parameters on the electrophoretic mobility are first discussed and compared with the case when a Stern layer is absent. The numerical results show that regardless of the values of the Stern layer and solution pa-rameters chosen, the presence of a DSL causes the electrophoretic mobility to decrease in comparison with the standard case (no Stern layer present) for every volume fraction. Furthermore, the stronger the hydrodynamic particle–particle interactions as volume fraction increases, the lower the mobility for a given zeta potential, both mechanisms tending to increase the retarding forces that brake the electrophoretic motion. Concerning direct current conductiv-ity calculations, results show that the presence of a DSL causes the electrical conductivity to increase in comparison with the standard case (no Stern layer present) for every volume fraction and zeta po-tential. Obviously, the additional conductivity contribution of every particle in the system is related to the presence of an extra mobile layer, the DSL. The treatment is based on the use of a cell model to account for hydrodynamic and electrical interactions between particles. We also discuss the use of either Levine–Neale or Shilov– Zharkikh boundary conditions, leading to different results for the mobility and direct current conductivity in conditions of both low (where analytical expressions can be reached) and arbitrary zeta po-tentials. The analogies and discrepancies between both approaches are discusesd. °C2001 Academic Press
Key Words:electrophoretic mobility; electrical conductivity; con-centrated suspensions; electrokinetic equations; electric double layer; dynamic Stern layer.
1To whom correspondence should be addressed. E-mail: [email protected].
1. INTRODUCTION
A great deal of effort has recently been devoted to improv-ing the results of the standard electrokinetic theories dealimprov-ing with different electrokinetic phenomena in dilute colloidal sus-pensions. One of the most remarkable extensions of these elec-trokinetic models has been the inclusion of a dynamic Stern layer (DSL) onto the surface of the colloidal particles. Thus, Zukoski and Saville (1) developed a DSL model to reconcile the differences observed between zeta potentials derived from electrophoretic mobility and static conductivity measurements. Shortly after, Mangelsdorf and White (2), using the technique developed by O’Brien and White for the study of the elec-trophoretic mobility of a colloidal particle (3), included a general DSL model in the study of electrophoresis. They analyzed the role of different Stern-layer adsorption isotherms on both elec-trophoretic mobility and suspension conductivity. More recently, Kijlstra et al. (4) applied the theory of Stern-layer transport to the study of the low-frequency dielectric response of colloidal suspensions, extending the thin-double-layer theory of Fixman (5, 6). Likewise, Rosen et al. (7) generalized the standard theory of the conductivity and dielectric response of a colloidal sus-pension in alternating electric fields of DeLacey and White (8), assuming the model of the Stern layer developed by Zukoski and Saville. Very recently, Mangelsdorf and White (9, 10) de-veloped a general DSL model to be applied to electrophoresis and dielectric response in oscillating electric fields. In general, the DSL models seem to improve the agreement between theory and experiments (4, 7, 11, 12) as compared with the standard predictions in dilute suspensions, although there are still impor-tant discrepancies (in particular, the DSL theory of the primary electroviscous effect seems to increase the separation between calculated and measured data; see Refs. (13–16).
On the other hand, relatively few theoretical studies have dealt with the more practical situation of concentrated suspensions. Focusing on the problem of electrophoresis, Levine and Neale (17) developed a mobility expression for spherical particles with low zeta potentials in concentrated suspensions on the basis of the Kuwabara cell model (18), in order to account for the
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hydrodynamic particle-particle interactions. Kozak and Davies (19, 20) also studied the electrokinetics of concentrated suspen-sions and derived a mobility expression valid for arbitrary zeta potential and nonoverlapping double layers. Likewise, Ohshima (21) derived a general mobility expression for spherical parti-cles in concentrated suspensions tending toward that of Levine and Neale for low zeta potentials, and to that of Kozak and Davies for all zeta potentials and nonoverlapping double lay-ers. Ohshima’s result is also based on the Kuwabara cell model as that of Levine and Neale. However, very recently Dukhin
et al. (22) have pointed out that the Levine–Neale cell model,
employed by many authors to develop theoretical electrokinetic models in multiparticle systems, including sedimentation, elec-trophoresis, and conductivity in concentrated suspensions (19– 21, 23–25), has some deficiencies. According to Dukhin et al. (22) the Levine–Neale cell model is not compatible with the volume fraction dependence of the exact Smoluchowski law in concentrated suspensions. Instead of the Levine–Neale cell model, Dukhin et al. suggest using the Shilov–Zharkikh cell model (26), which is based on arguments of nonequilibrium thermodynamics, and not only agrees with Smoluchowski’s re-sult but also correlates with the electrical conductivity of the Maxwell–Wagner theory (27). Thus, it appeared quite interest-ing to explore in more detail the consequences arisinterest-ing from the inclusion of the Shilov–Zharkikh cell model into Ohshima’s theory of the electrophoretic mobility and direct current (DC) conductivity of concentrated suspensions.
In the present paper, we first solve the electrokinetic equations (with Levine–Neale and Shilov–Zharkikh boundary conditions) to obtain numerical data of electrophoretic mobility and DC conductivity for arbitrary zeta potential and volume fraction, when nonoverlapping double layers are assumed.
The DSL correction to the electrokinetic theories is then dealt with. A DSL extension of Ohshima’s theory of the sedimentation velocity and potential in dilute (28) and concentrated (29) sus-pensions has been recently carried out. In this work, we extend the standard Ohshima’s theory of the electrophoretic mobility (21) and DC conductivity of spherical particles in a concentrated suspension (25) to include a DSL model. As in previous papers (28, 29), we will use the method that Mangelsdorf and White developed to allow for the adsorption and lateral motion of ions in the inner region of the double layer (2).
In summary, the aim of this investigation can be described as follows. First, to derive a new mobility formula for low zeta potentials according to the Shilov–Zharkikh cell model for the description of concentrated suspensions. Second, to obtain a numerical solution of the standard Ohshima’s theory of elec-trophoresis in concentrated suspensions for the whole range of zeta potential, volume fraction, and nonoverlapping double lay-ers, and also, when this theory is modified, to allow for the considerations of the Shilov–Zharkikh cell model. A similar analysis will be carried out using the similar problem of the electrical conductivity of suspensions. And finally, we extend the standard theory of electrophoresis and DC conductivity in
concentrated suspensions (with Shilov–Zharkikh’s conditions) to include a DSL on the surface of the particles.
2. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
Before proceeding, it will be useful to briefly review Ohshima’s standard theory of electrophoresis in concentrated suspensions (21), and to show the notation used in this pa-per. Concerned readers are referred to Ohshima’s paper for a complete treatment. The standard theory of the electrophore-sis in a concentrated suspension of spherical colloidal particles was developed by Ohshima on the basis of the Kuwabara cell model (18) to account for the hydrodynamic particle–particle interactions (see Fig. 1). According to this model, each spher-ical particle of radius a is surrounded by a concentric shell of an electrolyte solution, having an outer radius b such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction throughout the entire suspension:
φ=(a/b)3. [1]
The surface r =a is usually called the “slipping plane.” This
is the plane outside which the continuum equations of hydro-dynamics are assumed to hold. Let us consider now a charged spherical particle of radius a immersed in an electrolyte solu-tion composed of N ionic species of valencies zi, bulk number
concentrations n∞i , and drag coefficientsλi(i =1, . . . ,N ). The
axes of the spherical coordinate system (r, θ, ϕ) are fixed at the center of the particle. The latter is assumed to move in an elec-tric field E with a velocity ve, the electrophoretic velocity, in the
electrolyte solution of viscosityη. The electrophoretic mobility
ueis defined by ve=ueE. The polar axis (θ=0) is set parallel
to E. In the absence of the field the particle has a uniform electric potential, the zeta potentialζ, at r=a. A complete solution to
the problem would require knowledge of the electric potential 9(r), the number density of each type of ion ni(r), and the drift
velocity vi(r) of each ionic species (i=1, . . . ,N ), the fluid
ve-locity v(r), and the pressure p at every point r in the system. The fundamental equations governing the problem are (2, 3, 8)
∇29(r)= −ρ(r)
εrsε0
, [2]
ρ(r)=
N X
i=1
zieni(r), [3]
η∇2v(r)− ∇p(r)−ρ∇9(r)=0, [4]
∇ ·v(r)=0, [5]
vi=v−
1 λi∇µi
(i=1, . . . ,N ), [6]
µi(r)=µ∞i +zi e9(r)+KBT ln ni(r) (i =1, . . . ,N ), [7] ∇ ·[ni(r)vi(r)]=0 (i =1, . . . ,N ), [8]
FIG. 1. Schematic picture of an ensemble of spherical particles in a concentrated suspension according to the Kuwabara cell model (Ref. (18)).
where e is the elementary electric charge, KB is Boltzmann’s
constant, and T is the absolute temperature. Equation [2] is Poisson’s equation, whereεrsis the relative permittivity of the
solution,ε0is the permittivity of a vacuum, andρ(r) is the
elec-tric charge density given by Eq. [3]. Equations [4] and [5] are the Navier–Stokes equations appropriate to a steady incompressible fluid flow at low Reynolds number in the presence of an elec-trical body force. Equation [6] expresses that the ionic flow is caused by the liquid flow and the gradient of the electrochem-ical potential defined in Eq. [7], and it can be related to the balance of the hydrodynamic drag and electrostatic and thermo-dynamic forces acting on each ionic species. Equation [8] is the continuity equation expressing the conservation of the number of each ionic species in the system. The drag coefficientλi is
related to the limiting conductance30
i of the i th ionic species
by (3)
λi=
NAe2|zi| 30
i
(i =1, . . . ,N ), [9]
where NAis Avogadro’s number. At equilibrium, that is, in the
absence of the electric field, the distribution of electrolyte ions
obeys the Boltzmann distribution
n(0)i =n∞i exp
µ
−zie9(0) KBT
¶
(i =1, . . . ,N ), [10]
and the equilibrium electric potential9(0)satisfies the Poisson– Boltzmann equation
1
r2 d dr
µ r2d9
(0)
dr ¶
= −ρ (0) el (r )
εrsε0
, [11]
ρ(0) el (r )=
N X
i=1
zie n(0)i (r ), [12]
ρ(0)
el being the equilibrium electric charge density.
The unperturbed or equilibrium electrical potential must obey the following boundary conditions at the slipping plane and at the outer surface of the cell,
9(0)(a)=ζ, [13]
d9(0)
As the coordinate system is set fixed at the center of the par-ticle, the boundary conditions for the liquid velocity v and the ionic velocity of each ionic species at the particle surface are expressed by the equations
v=0 at r=a, [15]
vi·ˆr=0 at r =a (i =1, . . . ,N ). [16]
Equation [15] expresses that the fluid layer adjacent to the par-ticle surface is at rest, and Eq. [16] that there are no ion fluxes through the slipping plane (ˆr is the unit normal outward from the particle surface). According to the Kuwabara cell model, the liquid velocity at the outer surface of the unit cell must satisfy the conditions
vr = −vecosθ= −ueE cosθ at r=b, [17]
ω= ∇ ×v=0 at r=b, [18]
meaning, respectively, that at that surface the liquid velocity is parallel to the electrophoretic 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 owing to the electric field (we assume that the external field is low enough for this condition to be valid; this condition is most often fulfilled in practical situations), so that a linear perturbation scheme for the above-mentioned quantities can be used,
ni(r)=n (0)
i (r )+δni(r) (i=1, . . . ,N ), [19] 9(r)=9(0)(r )+δ9(r), [20] µi(r)=µ
(0)
i +δµi(r) (i=1, . . . ,N ), [21]
(as usual the superscript(0)refers to equilibrium). The
perturba-tions in ionic number density and electric potential are related to each other through the perturbation in electrochemical potential by
δµi =zieδ9+KBTδni ±
n(0)i (i=1, . . . ,N ). [22]
In terms of the perturbation quantities, the condition that the ionic species cannot penetrate the particle surface in Eq. [16] (DSL not yet considered) transforms into
∇δµi·ˆr=0 at r =a (i =1, . . . ,N ). [23]
According to Ohshima (21), the boundary condition for the perturbed electric potential at the outer surface of the unit cell is expressed by
∇δ9·ˆr= −E·ˆr at r=b. [24]
However, according to the Shilov–Zharkikh cell model (26), the
latter condition changes to
δ9 = −hEi ·r at r =b [25]
and provides, as Dukhin et al. pointed out (22), the connec-tion between the macroscopic, experimentally measured electric fieldhEi, and local electric properties. The differences between both choices of boundary conditions stem from the way in which the macroscopic (experimentally observable) electric field is de-fined in connection with local properties. Thus, for Levine and Neale (17) the local electric field at r =b is parallel to the
ex-ternally applied electric field. In contrast, Shilov and Zharkikh define a macroscopic field as an average of −∇δ9 performed in such a way that Onsager reciprocity relationships hold, no matter the particle concentration (22, 26). For nonoverlapping double layers, Eq. [22] becomes at the outer region of the cell (21, 25)
δµi =zieδ9, [26]
and, correspondingly, Eq. [24] transforms into
∇δµi·ˆr= −zieE ·ˆr at r =b [27]
and Eq. [25] into
δµi = −ziehEi ·r at r =b. [28]
All Ohshima’s equations in the rest of the paper will also be valid for the Shilov–Zharkikh cell model if the applied electric field E is substituted by the macroscopic electric fieldhEi.
Spherical symmetry considerations led Ohshima to introduce the radial functions h(r ), φi(r ), and Y (r ) then write
v(r)=(vr, vθ, vϕ)= µ
−2
rh E cosθ,
1
r d
dr(r h) E sinθ,0 ¶
,
[29] δµi(r)= −zieφi(r )(E·ˆr) (i =1, . . . ,N ), [30]
δ9= −Y (r )(E·ˆr) [31]
to obtain the following set of coupled ordinary differential equa-tions and boundary condiequa-tions at the slipping plane and at the outer surface of the cell (21),
L(Lh)= − e
ηr d y dr
N X
i=1
n∞i z2i exp(−ziy)φi(r ), with y= e9(0)
KBT,
[32]
Lφi(r )= d y dr
µ zidφi
dr −
2λi e
h(r ) r
¶
(i =1, . . . ,N ), [33]
LY (r )= 1
εrsεoKBT N X
i=1
with L being a differential operator defined by
L ≡ d 2
dr2 +
2
r d dr −
2
r2, [35]
h(a)=dh
dr(a)=0, Lh(r )=0 at r =b, [36]
h(b)= ueb
2 , [37]
dφi
dr (a)=0 (i =1, . . . ,N ), [38] dφi
dr (b)=1. [39]
However, if we consider the Shilov–Zharkikh boundary condi-tion given by Eq. [25], a different result is found,
φi(b)=b, [40]
where Eq. [30] has been used. In addition to the latter boundary conditions, we must impose the constraint that in the stationary state the net force acting on the particle or the unit cell must be zero (21).
A numerical method similar to that proposed by DeLacey and White in their theory of the dielectric response and conductivity of a colloidal suspension in time-dependent fields (8) has been applied to solve the above-mentioned set of ordinary differential equations of the theory of the electrophoresis in concentrated colloidal suspensions. The numerical computations are shown and discussed in Section 5.
3. CALCULATION OF THE CONDUCTIVITY OF THE SUSPENSION
The electrical conductivity, K∗, of the suspension is defined by
hii = 1 V
Z
V
i(r) d V =K∗hEi, [41]
withhiibeing the electric current density in the suspension, and
hEithe macroscopic electric field (i.e., minus the average of the gradient of the electrical potential9(r) in each position of the system), with total volume V:
hEi = −1 V
Z
V
∇9(r) d V. [42]
According to O’Brien (30) and Ohshima (25), and recall-ing the assumption of nonoverlapprecall-ing double layers, the current density can be written in terms of the perturbation quantities
(Eqs. [19]–[21]) as
hii = − N X
i=1 zie λiV
Z
V ¡
KBT∇δni+n∞i zie∇δ9 ¢
d V
−NP V
N X
i=1 zie n∞i
λi Z
S
{r· ∇δµi(r)−δµi(r)}ˆr d S, [43]
where S is the outer spherical surface of the cell, and NP the
number of particles in the (total) volume V . Note that because double layers are not allowed to overlap, it will be assumed that
n(0)i ∼=n∞i at the cell surface. Using this approximate equality, and theφifunctions defined in Eq. [30], the following expression
for the average current density can be reached:
hii = N X
i=1 zie2n∞i
λi
hEi − NP V
N X
i=1 zie n∞i
λi
× ½
−zie µ
rdφi dr −φi
¶
r=b ¾ Z
V
(hEi ·ˆr)ˆr d S. [44]
Now, using the value of the conductivity of the supporting solution K∞,
K∞= N X
i=1 zie2n∞i
λi
, [45]
and the result
Z
V
(hEi ·ˆr)ˆr d S=(4/3)πb2hEi, [46]
Eq. [44] becomes
hii = K∞ 1−
4πNP V
PN i=1
z2
in∞i Ci λi
PN i=1
z2
in∞i λi
hEi, [47]
where the coefficients Ci were defined by Ohshima as
Ci= − b2
3
µ rdφi
dr −φi ¶
r=b
. [48]
From Eq. [47], after introducing the volume fraction of solids φ=4πa3NP/3V , the ratio between the conductivities of the suspension and the dispersion medium can be written as follows:
K∗ K∞ =
1−
3φ
a3 PN
i=1 z2
in∞i Ci λi
PN i=1
z2
in∞i λi
. [49]
differs from that deduced by Ohshima (Eq. [58] in Ref. (25)):
K∗ K∞ =
1+
3φ
a3 PN
i=1 z2
in∞i Ci λi
PN i=1
z2
in∞i λi −1 . [50]
We confirm in Section 5 that the correct limiting cases full-filled by Ohshima’s conductivity formula are also exhibited by our Eq. [49], in particular the low-zeta-potential case. However, it is worth pointing out that for moderate or high zeta potentials the predictions of both conductivity expressions are very differ-ent, as can be deduced by numerical integration of the theory.
4. EXTENSION TO INCLUDE A DYNAMIC STERN LAYER
Let us now consider the possibility of adsorption and ionic transport in the inner region of the double layer of the particles. As previously mentioned, we will follow the method developed by Mangelsdorf and White in their theory of the electrophoresis and conductivity in a dilute colloidal suspension (2). This theory allows for adsorption and lateral motion of ions in the inner region using the well-known Stern model. Therefore, we will assume a Stern layer that is thin compared to either a or the double-layer thicknessκ−1, whereκis defined by (31)
κ= " N
X
i=1 n∞i z2ie2
,
εrsε0KBT #1/2
. [51]
Now the condition that ions cannot penetrate the slipping plane is no longer valid, and thus, the evaluation of the fluxes of each ionic species through the slipping plane gives rise to new slip-ping plane boundary conditions for the functionsφi(r ), replacing
Eq. [38],
dφi dr (a)−
2δi
a φi(a)=0 (i =1, . . . ,N ), [52]
δi=
[eNi]
ae10−pKi ³ λi λt i ´ exp h zie
KBT
σd
C2∗ i
NA103+ PN
j=1 NA103c∞j
10−pK j exp
h −zje
KBT
³ ζ−σd
C2∗ ´i
(i=1, . . . ,N ) ,[53]
in terms of the so-called surface ionic conductance parameters δi of each ionic species, comprising the effect of a mobile
sur-face layer. In fact it is the small thickness of the Stern layer in comparison with the other length scales that permits slipping plane boundary conditions to be used, including the effects of a mobile surface layer (2). These parameters depend on the zeta potential ζ, the ratio between the drag coefficient λi of each
ionic species in the bulk solution and in the Stern layerλt i, the
density Ni of sites available for adsorption in the Stern layer,
the pKiof 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), the ca-pacity C2∗of the outer Stern layer, the radius a of the particles, the electrolyte concentration c∞j (the equilibrium molar concen-tration of type j ions in solution), and the charge density per unit surface area in the double layerσd. It is worth noting that
the other boundary conditions remain unchanged when a DSL is assumed.
5. RESULTS AND DISCUSSION
Low-Zeta-Potential Approximations
Following the method described in Ref. (21) for low zeta po-tentials but taking into account the Shilov–Zharkikh cell model instead of the Levine–Neale one, we can solve Eqs. [32] and [33] subject to the boundary conditions expressed by Eqs. [38] and [40] to obtain
φi(r )=Y (r )=
1 1+φ/2
µ r+ a
3
2r2 ¶
(i =1, . . . ,N ). [54]
Substituting Eq. [54] in Eq. [32], the latter reduces to
L(Lh)= − εrsε0κ 2
η(1+φ/2)
µ
1+ a
3
2r2 ¶
d9(0)
dr , [55]
where 9(0) is now the solution of the linearized Poisson–
Boltzmann equation, and is given by (21)
9(0)(r )=ζa r
κb cosh [κ(b−r )]−sinh [κ(b−r )]
κb cosh [κ(b−a)]−sinh [κ(b−a)]. [56]
By solving Eq. [55] subject to the boundary conditions given by Eq. [36], and considering also the condition of zero net force act-ing on the particle in the stationary state (21), the electrophoretic mobility can be derived with the help of Eq. [37] to give
ue=
2εrsε0ζ
3η
Z b
a H (r )
µ
1+ a
3
2r2 ¶
dr+F, [57]
with
H (r )= − (κa) 2
6(1+φ/2)
·
1−3r
2
a2 +
2r3
a3 − a3 b3 × µ 2 5− r3 a3 +
3r5 5a5 ¶¸ 1 ζ µ d9(0)
dr ¶
[58]
and
F = 2εrsε0(κa) 2
9η 9
(0)(b) ·
1+b
3
a3 −
9b2 5a2 −
a3
5b3 ¸
. [59]
We show in Fig. 2 the ratio of the electrophoretic mobility to the Smoluchowski mobility for zero volume fraction, as a
FIG. 2. Ratio of the electrophoretic mobility to the Smoluchowski mobility for zero volume fraction, as a function ofκa and volume fraction of the
sus-pensions. Solid lines show Ohshima’s mobility formula, Eqs. [33] and [34] in Ref. (17). Dotted lines show mobility expressed by Eqs. [57]–[59] in this paper.
function ofκa and the volume fraction of the suspensions. In the
figure, solid lines display the mobility according to Ohshima’s mobility formula (Eqs. [33]–[34] in Ref. (21), Levine–Neale’s cell model), and dotted lines show the mobility calculated from our Eqs. [57]–[59] (Shilov–Zharkikh’s cell model). It is worth pointing out that the volume fraction dependence of the mobility-κa curves is quite different when both approximations are
com-pared, especially in the region of higher values ofκa and volume
fraction. The nature of such discrepancies is based on the ex-istence of two different definitions of electrophoretic mobility, being identical when dilute suspensions are concerned but con-siderably different in the case of concentrated suspensions. The reader is referred to Ref. (22), where the details of this contro-versy are given. Briefly, the difference refers again to the choice of the electric field to be used to obtain the mobility from the elec-trophoretic velocity. Thus, Ohshima (21) assumes that this field is precisely the externally applied one, E; on the other hand, the Shilov–Zharkikh cell model uses an averagehEiof the electric field, that for concentrated suspensions differs from the exter-nally applied field E that, following Ohshima, would exist in the solution at large distance from the particle. The Shilov–Zharkikh choice is again compatible with nonequilibrium thermodynam-ics (22).
In the case of DC conductivity, it can be seen from Eqs. [48]– [50] that the calculation of the conductivity ratio requires the previous knowledge of the functionsφiand their first derivatives
on the cell surface r =b. For low zeta potentials, Ohshima (25)
proved that rather simple expressions can be found for Ci, so
that Eqs. [49] and [50] become, respectively,
K∗ K∞=
1−φ 1+φ/2
1−3φ
µ eζ KBT
¶
L(κa, φ)
PN i=1
z3
in∞i λi
PN i=1
z2
in∞i λi
[60]
and
K∗ K∞ =
1−φ 1+φ/2
1+3φ
µ eζ KBT
¶
L(κa, φ)
PN i=1
z3
in∞i λi
PN i=1
z2
in∞i λi
−1
,
[61]
with L(κa, φ) defined by (25)
L(κa, φ)= − 1
3a3ζ(1−φ)(1+φ/2)
× Z b
a µ
a3
2 +r
3 ¶µ
1−a
3
r3 ¶
d9(0)
dr dr, [62]
where9(0)is the potential distribution in the equilibrium double
layer, Eq. [56]. Note that our Eq. [60] is in agreement with O’Brien’s conductivity formula for low zeta potential and dilute solutions (30). Furthermore, when particles are uncharged (ζ = 0), Eqs. [60] and [61] lead to the well-known Maxwell formula for uncharged spheres (32):
K∗ K∞ =
1−φ
1+φ/2. [63]
Let us also mention that in the limit of infinitesimally thin double layers (κa→ ∞), the predictions of Eqs. [60] and [61] are iden-tical and independent of zeta potential, because L(κa, φ)→0, just as when dilute suspensions are considered (30). However, when the zeta potential is no longer low, the results of the two conductivity expressions, Eqs. [60] and [61], are quite differ-ent, as shown in the next section where numerical data are presented.
Numerical Calculations
In Fig. 3, the electrophoretic mobility is represented as a func-tion of zeta potential for different volume fracfunc-tions. In this figure, numerical calculations of the electrophoretic mobility following Ohshima’s and Shilov–Zharkikh’s models are shown for com-parison. We can observe some remarkable features in these plots. First, for a given zeta potential in the region of not very high zeta values, Ohshima’s mobility is higher than the Shilov–Zharkikh prediction at every volume fraction. Likewise, Ohshima’s mo-bility maximum shifts to lower zeta potentials as volume fraction increases, whereas the Shilov–Zharkikh one not only shifts to the opposite zeta region, it also broadens and almost disappears. Both mobilities also diminish when volume fraction increases at a fixed zeta potential, owing to the increasing importance of the hydrodynamic particle–particle interactions. On the other hand, the numerical data in the region of low zeta potentials of Fig. 3 confirm, according to Dukhin et al. (33), the zero-frequency limit of the analytical relationship between the dynamic elec-trophoretic mobilities in concentrated suspensions, derived by
FIG. 3. Scaled electrophoretic mobility of a spherical particle as a function of zeta potential for different volume fractions of the suspensions in a KCl solu-tion at 25◦C. Particle radius, 100 nm;κa=100. For nonzero volume fractions, solid lines show results calculated by numerical integration of Ohshima’s theory of electrophoresis in concentrated suspensions; dotted lines show the same but with Shilov–Zharkikh’s boundary conditions.
Ohshima, ue−OHS(34), and Shilov, ue−SHI(27):
ue−SHI=ue−OHS
1−φ
1+φ/2. [64]
Concerning DC conductivities, Fig. 4 shows numerical re-sults of the ratio between the conductivities of the suspension and the electrolyte solution, as a function of dimensionless zeta potential for different volume fractions of the suspensions. In
FIG. 4. Ratio of the suspension conductivity K∗to the conductivity of the supporting electrolyte solution K∞, as a function of the dimensionless zeta potential for different volume fractions of the suspensions in a KCl solution at 25◦C. Particle radius, 100 nm;κa=100. Solid lines show results obtained by numerical solution of Ohshima’s conductivity theory; dashed lines show the same but with Shilov–Zharkikh’s boundary conditions.
the figure, solid lines correspond to Ohshima’s conductivity for-mula (Levine–Neale cell model), Eq. [50], and dashed lines to our Eq. [49] (Shilov–Zharkikh cell model). As above, we use, respectively, the subscripts OHS and SHI to denote these con-ductivity ratios. It is worth pointing out that for low zeta po-tential the conductivities obtained with both formulae are in very good numerical agreement, as expected (see also the ap-proximate analytical expressions, Ohshima’s Eq. [61] and our Eq. [60], both valid for low ζ), but they differ considerably for moderate-to-high zeta values. Furthermore, the higher the volume fraction, the higher the deviation between the ratios at every zeta potential. Likewise, Ohshima’s conductivity ratio is a monotonically decreasing function of the volume fraction for a fixed zeta potential. On the other hand, our prediction changes from a monotonically decreasing behavior with volume frac-tion at low zeta values to a monotonically increasing funcfrac-tion of volume fraction for the highest zeta values studied. Similarly, Fig. 5 displays the usual “conductivity increment” (2, 30) as a function of the dimensionless zeta potential for different volume fractions, with the same considerations as those given in Fig. 4. As observed, the conductivity increment tends to the limiting value of (−3/2) when volume fraction andζ tend to zero, thus verifying the standard result for the conductivity increment in dilute suspensions (30). On the other hand, the predictions of the conductivity increment according to both models show op-posite behavior in the region of high zeta potentials as volume fraction increases. While Ohshima’s conductivity increment is always lower than the corresponding prediction for the dilute case in that zeta region, our conductivity ratio is always higher than that of the dilute case. This different behavior reveals how sensitive the theoretical predictions have turned out to be to ap-parently small changes in boundary conditions, in particular the one expressing the behavior of the perturbed electric potential at the outer surface of the unit cell (see Eqs. [24] and [25]).
FIG. 5. Same as Fig. 4, but for the conductivity increment, (K∗−K∞)/
FIG. 6. Ratio of the DSL electrophoretic mobility to the standard elec-trophoretic mobility of a spherical particle, as a function of zeta potential for two volume fractions and different values of the DSL parameter N−(expressed as eN−, inµC/cm2). Dispersion medium, KCl solution at 25◦C. Particle radius,
100 nm;κa=100. Other DSL parameters are shown in the figure.
Effect of a Dynamic Stern Layer
Regarding the DSL corrections, let us first consider their effect on the electrophoretic mobility in concentrated suspensions. We present in Fig. 6 the ratio of the DSL corrected to the standard electrophoretic mobility, both calculated according the Shilov– Zharkikh cell model, as a function of the dimensionless zeta potential, for two extreme volume fractions, and different val-ues of the Stern layer parameter N−, the density of counterion adsorption sites. Recall that the higher this parameter, the more significant the role of Stern-layer conductance in the electroki-netics of the system. Figure 6 demonstrates that the presence of a DSL reduces the mobility, the effect being more pronounced when the role of the DSL is increased on increasing the values of the parameter N−.
A qualitative explanation for these facts can be given, taking into account the rather complex and interrelated mechanisms re-sponsible for the electrophoretic mobility dependence onζ. As the latter rises, the electrokinetic charge increases as well and so does the electrophoretic velocity. On the other hand, the strength of the dipole moment induced on the particles by the electric field also increases with zeta (35), tending to decrease the mobility. The presence of a conducting Stern layer will favor the forma-tion of the dipole by ionic migraforma-tion. As a consequence, the mobility will be further reduced as compared to the standard sit-uation. As the zeta potential is increased, the charge in the Stern layer also rises, and the ratio (u∗e)DSL/u∗edecreases, as observed
in Fig. 6. However, at sufficiently high zeta potentials, the Stern-layer charge must tend to saturate, and hence, the diffuse Stern-layer should play the essential role. As a consequence, the trend of the mobility ratio changes, and the latter increases until eventually reaching a value close to unity. Another important feature of Fig. 6 is that the relative deviation of the DSL electrophoretic
FIG. 7. Same as Fig. 6, but for differentκa values, and the DSL parameters
indicated.
mobility from the standard prediction appears to be less signifi-cant the higher the volume fraction, or equivalently, the more in-tense the hydrodynamic particle–particle interactions. This can be explained by considering that such interactions are so impor-tant that the presence or not of a DSL ceases to be an essential factor in interpreting electrokinetic behavior. This is confirmed by the weaker variations of the mobility ratio withζ when the suspensions are concentrated.
When the double-layer thickness is decreased (κa increased),
while keeping all other parameters unaltered (Fig. 7), the rela-tive mobility reduction brought about by the Stern-layer finite
FIG. 8. Ratio of the standard suspension conductivity (no DSL) to the DSL suspension conductivity (both calculated with Shilov–Zharkikh’s boundary con-ditions) as a function of dimensionless zeta potential, for two volume fractions and different values of the DSL parameter N−(expressed as eN−, inµC/cm2).
Dispersion medium, KCl solution at 25◦C. Particle radius, 100 nm;κa=100. Other DSL parameters are shown in the figure.
FIG. 9. Same as Fig. 8, but for differentκa values, and for the DSL
param-eters indicated.
conductance also decreases. Since in Fig. 7 we assume that a is constant for all cases, increasingκ is equivalent to raising the ionic concentration in the medium. It is hence to be expected that the change in the diffuse layer (for a given zeta potential) will also rise: As a consequence, the effects of DSL will be increas-ingly hidden by those of the diffuse atmosphere. This brings the mobility ratio (u∗e)DSL/u∗e closer to one, as numerically shown
in Fig. 7.
Regarding the DSL correction to the conductivity of concen-trated suspensions, we represent in Fig. 8 the ratio of the stan-dard suspension conductivity (no DSL) to the DSL value, both calculated according to the Shilov–Zharkikh cell model (sim-ilar effects are predicted for Ohshima’s theory), as a function of the dimensionless zeta potential, for low and high volume fractions and the same values of Stern layer parameters as in Fig. 6. As observed, in the presence of a conducting Stern layer, the conductivity of the suspensions is higher than that deduced from the standard predictions (note that the conductivity ratio is always less than unity). Also, the higher the volume fraction, the lower the conductivity ratio, clearly indicating the important role of the mobile Stern layer in the explanation of the conduc-tivity of the suspension. In fact, note how reducing eN−leads to closer proximity between standard and DSL calculations of the conductivity. This behavior is easy to explain because in the latter case a new ionic transport process develops in the per-turbed inner region of the double layer, giving rise to a higher conductivity at every zeta potential and volume fraction.
On the other hand, the relative deviation of the DSL suspen-sion conductivity from the standard prediction seems to be more important the lower the zeta potential, for every volume fraction (Fig. 8). Furthermore, the conductivity ratio tends to unity in the limit of high zeta potentials for every volume fraction: The same arguments used before concerning the Stern-layer charge saturation apply in the case of electrical conductivity.
Finally, Fig. 9 allows us to analyze the effect of the parameter κa on the conductivity differences. As the ionic concentration is
reduced (κdecreases), the conductivity in the diffuse layer must also decrease, so the relative contribution of the Stern layer to the overall conductance is likely to be more significant: hence the larger diferences between K∗and (K∗)DSLobserved in the
figure.
6. CONCLUSIONS
In this work we have first derived a general expression for the conductivity of a concentrated suspension valid for arbitrary zeta potential and volume fraction when nonoverlapping of double layers is assumed. Likewise, approximate analytical expressions for the electrophoretic mobility and electrical conductivity in concentrated suspensions, valid for low zeta potential and arbi-traryκa and volume fraction, have also been obtained. These
ex-pressions have been calculated following the method developed by Ohshima in his theory of the electrophoresis and conduc-tivity of concentrated suspensions, substituting the conditions imposed by the Levine–Neale cell model by those according to the Shilov–Zharkikh cell model. Furthermore, numerical calcu-lations are presented for both quantities in arbitrary conditions (with the only restriction being nonoverlapping double layers) of volume fraction and zeta potential, for both Ohshima and Shilov–Zharkikh’s models. In addition, we have extended the theory corresponding to Shilov–Zharkikh’s boundary conditions to include a DSL into the model. The results show that regardless of the particle volume fraction and zeta potential, the presence of a DSL gives rise to a decrease in the electrophoretic mobil-ity, and an increase in the DC conductivmobil-ity, in comparison with the standard predictions. These DSL effects on the conductiv-ity are relatively more important the higher the volume fraction and the lower the electrolyte concentration. On the other hand, the higher the volume fraction and the higher the electrolyte concentration, the lower the relative influence of DSL effects on the electrophoretic mobility. Furthermore these Stern-layer effects tend to decrease on increasing the zeta potential in the high-zeta-potential region.
ACKNOWLEDGMENTS
Financial support for this work by MEC, Spain (Projects MAT98-0940 and BFM 2000-1099), and INTAS (Project 99-00510) is gratefully acknowledged.
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