Influence of cell-model boundary conditions on the conductivity and electrophoretic mobility of concentrated suspensions

Transcription

1 Advances in Colloid and Interface Science 118 (2005) Influence of cell-model oundary conditions on the conductivity and electrophoretic moility of concentrated suspensions F. Carrique a, J. Cuquejo a, F.J. Arroyo, *, M.L. Jiménez c, A.V. Delgado c a Departamento de Física Aplicada I, Facultad de Ciencias, Universidad de Málaga, Málaga, Spain Departamento de Física, Facultad de Ciencias Experimentales, Universidad de Jaén, Jaén, Spain c Departamento de Física Aplicada, Facultad de Ciencias, Universidad de Granada, Granada, Spain Availale online 20 July 2005 Astract In the last few years, different theoretical models and analytical approximations have een developed addressing the prolem of the electrical conductivity of a concentrated colloidal suspension. Most of them are ased on the cell model concept, and coincide in using Kuwaara s hydrodynamic oundary conditions, ut there are different possile approaches to the electrostatic oundary conditions. We will call them Levine Neale s (LN, they are Neumann type, that is they specify the gradient of the electrical potential at the oundary), and Shilov Zharkikh s (SZ, Dirichlet type). The important point in our paper is that we show y direct numerical calculation that oth approaches lead to identical evaluations of the conductivity of the suspensions if each of them is associated to its corresponding evaluation of the macroscopic electric field. The same agreement etween the two calculations is reached for the case of electrophoretic moility. Interestingly, there is no way to reach such identity if two possile choices are considered for the oundary conditions imposed to the fieldinduced perturations in ionic concentrations on the cell oundary (r =), dn i (r =). It is demonstrated that the conditions dn i ()=0 lead to consistently larger conductivities and moilities. A qualitative explanation is offered to this fact, ased on the plausiility of counter-ion diffusion fluxes favoring oth the electrical conduction and the motion of the particles. D 2005 Elsevier B.V. All rights reserved. Keywords: Electrical conductivity; Concentrated suspensions; Kuwaara cell model; Shilov-Zharkikh s oundary condition; Levine-Neale s oundary condition Contents 1. Introduction Basic equations Boundary conditions DC conductivity of the suspension Results and discussion Choice of electrostatic oundary conditions Boundary condition for ionic concentrations Addendum. Electrophoretic moility Conclusions Acknowledgments References * Corresponding author. address: fjarroyo@ujaen.es (F.J. Arroyo) /$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi: /j.cis

2 44 F. Carrique et al. / Advances in Colloid and Interface Science 118 (2005) Introduction It is a well known fact that the different electrokinetic phenomena, including static and dynamic electrophoresis, dielectric spectroscopy of suspensions, dc and ac conductivity, etc., are extremely sensitive to the properties of the particle/liquid interface [1]. In the case of dilute suspensions, the effects of the concentration of solids are negligile or can e accounted for y means of terms linearly dependent on the volume fraction (concentration expressed in volume of solids y volume of suspension) / of dispersed material, and the prolem can e solved ased on the case of a single particle in an unounded electrolyte [2 4]. In this paper, we will focus on the evaluation of the static (dc) conductivity of a suspension. If this is dilute, Dukhin and Derjaguin [5] first derived a simple conductivity formula in the case of the so-called thin doule layer or, more properly, large ja limit, that is, when the particle radius a is much larger than the electrical doule layer thickness j 1 (ja X1). Dukhin and Derjaguin s analysis was susequently extended to include aritrary values of ja or of the zeta potential, f [6 8] or complex systems such as suspensions of porous or composite particles [9,10]. While the prolem can e considered as rigorously solved for dilute suspensions, there is an increasing interest in studying concentrated systems, not only ecause they are potentially more useful from the technological point of view, ut also ecause this opens the field to new theoretical elaorations. The fundamental prolem of accounting for hydrodynamic particle particle interactions in concentrated suspensions is usually faced y means of cell models [11,12], which have een applied to a large variety of electrokinetic phenomena, including electrophoresis, sedimentation, electroacoustic and others [13 20]. Note that, since the particle/liquid interface is usually charged, electrostatic interactions etween particles must e considered in addition to hydrodynamic ones. Again, different approaches have een proposed, and several authors have developed formulas for the dc conductivity of concentrated suspensions [16,21,22]. All these contriutions were ased on the so-called Levine Neale (LN) oundary conditions [13], that will e descried elow. After Dukhin et al. [23] pointed out some inconsistencies in LN conditions, a series of papers [24 29] have used oundary conditions according to the now called Shilov Zharkikh cell model [30]. In this paper we intend to descrie oth types of oundary conditions, and to discuss the existence of differences etween the two corresponding ways of calculating the dc conductivity and the electrophoretic moility of concentrated suspensions of spheres. The paper is organized as follows: in the Next section, a rief summary will e given of the asic equations governing the prolem of calculating the conductivity K of a suspension of spheres. Then we will discuss the oundary conditions (oth hydrodynamic and electrostatic) allowing to otain the expression of K and of the electrophoretic moility, u e,of the particles. Finally, in the Results and discussion paragraphs we will analyze the implications of the choice of oundary conditions on the numerically evaluated conductivity and moility data. 2. Basic equations Before proceeding with the calculation of the conductivity of a concentrated suspension, it may e useful to riefly review the asic equations of the full theory. A more extensive mathematical treatment can e found in previous papers [16,25,26]. In the following, use will e made of a reference frame fixed to the particle, with origin at its centre, r denoting the position of any point with respect to that frame. The velocity v(r) of the liquid will e governed y Navier Stokes equations for an incompressile, viscous fluid with viscosity g, sujected to an electrical ody force ecause of the doule layer charge: gl 2 vr ðþ lpðþ q r el ðþlw r ðþ¼0; r livðþ¼0; r ð1þ ð2þ where W(r) is the electric potential at position r, the pressure p(r), and q el (r) is the volume charge density, q el ðþ¼ r XN z i en i ðþ: r ð3þ In Eq. (3), we have assumed that the dispersion medium contains N ionic species with charges ez i and numer concentrations n i (r) (i =1,2,...,N). The relationship etween charge density and potential is given y Poisson s equation, l 2 WðÞ¼ r q elðþ r ; ð4þ e m where e m is the permittivity of the liquid medium, which we assume constant. The velocity of ions includes convective, electrical, and diffusive contriutions: v i ðþ¼vr r ðþ 1 ll ðþ r i ði ¼ 1;...; NÞ; ð5þ where is the drag coefficient of type i ions, related to their diffusion coefficient, D i, as follows ¼ k BT D i ði ¼ 1;...; NÞ; ð6þ where k B is the Boltzmann constant, and T the asolute temperature.

3 F. Carrique et al. / Advances in Colloid and Interface Science 118 (2005) In Eq. (5), l i (r) is the electrochemical potential of the ith ion: l i ðþ¼l r V i þ z i ewðþþk r B Tlnn i ðþ r ði ¼ 1;...; NÞ; ð7þ l V i eing its standard chemical potential. The conservation of the numer of each ionic species leads to: li½n i ðþv r i ðþ r Š ¼ 0 ði ¼ 1;...; NÞ: ð8þ 3. Boundary conditions According to Kuwaara s cell model [12], hydrodynamic interactions etween particles can e accounted for y solving the prolem of a single spherical particle (radius a) surrounded y a spherical liquid shell, concentric with it and with radius, such that the particle/cell volume ratio a 3 / 3 is equal to the volume fraction of solids in the whole suspension, /: / ¼ a 3 : ð9þ A crucial aspect of the prolem is the proper choice of oundary conditions, oth in the inner (r = r =a) and outer (r = ) cell surfaces. The former is effectively the solid/ liquid separation, so-called slip plane: oth the ions and the liquid located eneath that surface are considered immoile, strongly attached to the particle. The fact that the liquid cannot slip on the particle will e expressed as: vðr ¼ aþ ¼ 0: ð10þ Additionally, since the particle is non-conducting, the velocity of ions in the normal direction to the surface is zero: v i Iˆrj r¼a ¼ 0 ði ¼ 1;...; NÞ: ð11þ where rˆ is the outward normal to the surface. In the outer surface of the cell (r =), we will follow Kuwaara s oundary conditions. In the radial direction, the velocity of the liquid will e the radial component of its value far from the particle: v r j r¼ ¼ v e cosh ¼ u e ETcosh; ð12þ where v e is the electrophoretic velocity, and E* is the electric field responsile for the particle motion, that will e evaluated later. h is the angle etween the direction of the field and that of the position vector, r. In addition, the flow is free of vorticity on that surface: l vj r¼ ¼ 0: ð13þ If W 0 (r) is the equilirium (i.e., in the asence of any applied field) potential distriution, its value on r =a is the zeta potential, f: W 0 ðr ¼ aþ ¼ f; ð14þ and the discontinuity of the normal electric displacement yields: dw 0 e m dr j ¼ r ð15þ r¼a where r is the surface charge density of the particles. The condition that the cell is electroneutral leads finally to: dw 0 dr j ¼ 0: ð16þ r¼ The key prolem now is the specification of the electrostatic oundary conditions in the presence of the field. First, it is convenient to write the non-equilirium quantities in terms of their equilirium values plus a fielddependent perturation: n i ðþ¼n r 0 i ðþþdn r i ðþ; r l i ðþ¼l r 0 i ðþþdl r i ðþ; r ði ¼ 1;...; NÞ ð17þ ði ¼ 1;...; NÞ ð18þ WðÞ¼W r 0 ðþþdw r ðþ: r ð19þ In the linear theory of electrokinetic phenomena it is admitted that the applied field strength is low enough for the perturations to e linear in the field, and, furthermore, quadratic and higher terms will e dropped. Following Ohshima [16,31], spherical symmetry considerations permit us to introduce the radial functions h(r), / i (r), and N(r) as follows: vr ðþ¼ 2 r hr ðþetcosh; 1 r dl i ðþ¼ z r i e/ i ðþetcosh; r dwðþ¼ N r ðþetcosh; r d dr ½rhr ðþšetsinh; 0 ; ð20þ ði ¼ 1;...; NÞ; ð21þ ð22þ where E* is a field yet to e defined in terms of the macroscopic electric field. The oundary conditions descried so far yield: ha ð Þ ¼ dh dr j ¼ 0; ð23þ r¼a Lhr ½ ðþš r¼ ¼ 0; ð24þ h ð Þ ¼ u e 2 ; ð25þ d/ i dr j r¼a ¼ 0 ði ¼ 1;...; NÞ; ð26þ where L is a differential operator: Lu d2 dr 2 þ 2 r d dr 2 r 2 ; ð27þ

4 46 F. Carrique et al. / Advances in Colloid and Interface Science 118 (2005) and the ordinary differential equations verified y h, / i, and N are detailed in refs. [16,31]. Let us now consider the oundary conditions associated to dw and N, that will also give us clues on how to define the field E* in Eqs. (20) (22). Specifying the values of dw on the cell oundary r = is essential and, as Dukhin et al. [23] pointed out, it displays the connection etween the macroscopic, or experimentally measured field, and the local electrical properties. One possile choice of oundary condition is that of Levine Neale (LN), (a Neumann type condition): LN : ðldwþiˆrj r¼ ¼ ETIˆr ¼ ETcosh; ð28þ or LN : dnðþ r dr j ¼ 1: ð29þ r¼ Shilov et al. [30] propose a Dirichlet-type condition (SZ hereafter): SZ : dwðþj r r¼ ¼ ETIrj r¼ ¼ ETcosh; ð30þ and in terms of N: SZ : NðÞ ¼ : ð31þ Some condition must e imposed on the ionic perturations dn i (r)atr =. Following Lee et al. [32,33], the numer density of each ionic species must e equal to the corresponding equilirium ionic density. Therefore: dn i ðþj r r¼ ¼ 0; dl i ðþj r r¼ ¼ z i edwðþj r r¼ ; / i ðþ ¼ NðÞ; ði ¼ 1;...; NÞ: ð32þ Ding and Keh [29] suggest that an alternative condition on ionic perturations can e Bdn i ðþ r Br j ¼ 0; r¼ d/ i dr j ¼ dn r¼ dr j ; r¼ ði ¼ 1;...; NÞ: ð33þ In addition, we must impose the constraint that in the steady state the net force acting on the cell must vanish. This condition can e expressed as [31]: g d dr r¼ ½rLhðÞ r Š ¼ q 0 elðþnðþ; ð34þ where q 0 el () is the equilirium charge density at r =. Note that neglecting doule layer overlap is equivalent to set q 0 el ()=0 in Eq. (34). Our target now is to find an expression for the conductivity K of the suspension, considering the different choices for oundary conditions. This will e done in the next section. 4. DC conductivity of the suspension The electrical conductivity, K, of the suspension is defined in terms of the volume averages of the current density jà and the electric field, EÀ: jà ¼ KEÀ; -À ¼ 1 Z - ð35þ dv; V V where the integration is extended to the total volume of the cell. The average field, EÀ, will e: EÀ ¼ 1 Z lwðþdv r ¼ 1 Z ldwðþdv; r ð36þ V V V V where use has een made of the fact that the volume average of the gradient of the equilirium electric potential is zero [7]. Using Eqs. (5) and (18): jr ðþ¼q 0 elðþvr r ðþ XN z i en 0 i ðþ r ldl i ðþ: r ð37þ Noting that li(r j)=j, and using the divergence theorem, jà ¼ 1 Z z i e > r n 0 i V v þ ldl i n0 i Iˆr ds; ð38þ S where S is the outer spherical surface of the cell. Using Eq. (21), and calling C i ¼ 2 r d/ i 3 dr / i ; ð39þ r¼ one otains jà ¼ 1 V þ 1 V z 2 i e2 n 0 i ðþ Z z i en 0 i ðþ > S 3C i 2 / i ðþ 4 3 p2 ET ðrvþiˆr ds: ð40þ Now, taking into account Eqs. (20) and (25): Z > ðrvþiˆrds ¼ 4 3 p3 u e ET: ð41þ S Also, Eq. (22) leads to [34]: EÀ ¼ NðÞ ET; ð42þ and, finally, sustituting (Eqs. (9), (41), (42)) in (40) and using (35): K K m ¼ z 2 i e2 n V i / i ðþ NðÞ 3/ a 3 NðÞ C i 2 h ð Þ z i en V i exp z iew 0 ðþ NðÞ k B T ; z 2 i e2 n V i ð43þ

5 F. Carrique et al. / Advances in Colloid and Interface Science 118 (2005) where we have expressed n 0 i () in terms of the potential at using the Boltzmann factor: n 0 i ðþ ¼ n V i exp z iew 0 ðþ ; ð44þ k B T n V i eing the ulk concentration of the i-th ionic species, and the following expression has een used to evaluate the conductivity of the medium, K m : K m ¼ XN z 2 i e2 n V i : ð45þ If, in addition, we use the conditions (Eq. (32)) for the ionic perturations, K K m ¼ z 2 i e2 n V i 1 3/ a 3 C i 2 h ð Þ z i en V i NðÞ NðÞ exp ziew 0 ð Þ k B T ; z 2 i e2 n V i ð46þ whereas if the conditions chosen for dn i (r =) are those given y Eq. (33), the suitale equation for conductivity is (Eq. (43)) with the additional constraints that d/ i / dr r= =dn/dr r=,(i =1,...,N). Note that if the perturations in Eqs. (20) (22) were expressed in terms of EÀ (that is, if E*=EÀ), then Eq. (42) would yield N()=, which is precisely Eq. (31), corresponding to SZ oundary condition. Thus we reach the important conclusion that if Shilov Zharkikh oundary condition (Eq. (31)) is used, then the field E* is necessarily equal to the macroscopic (average) one EÀ, and N()=. If, on the other hand the Levine Neale oundary condition (Eq. (28)) is chosen, the field E* (that Ohshima calls applied external field, see Ref. [34]) is related to the average one as E* =EÀ / N(), and, in addition, dn /dr r= = 1. Mixing these conditions leads to unphysical results, whereas, properly used, Levine Neale and Shilov Zharkikh conditions should lead to identical evaluations of K. 5. Results and discussion 5.1. Choice of electrostatic oundary conditions A numerical technique ased on the method used y DeLacey and White [4] for solving the prolem of ac conductivity was used to numerically evaluate the conductivity of the suspensions. This method has turned out to e very efficient in the integration of the electrokinetic equations in the static case [26,27]. We will first consider how the choice of the cell oundary condition for dw(r) influences the calculation of K. Fig. 1 shows that LN and SZ oundary conditions (with dn i ()=0) lead to identical results whatever the value of zeta potential (Fig. 1a), ja (Fig. 1) and volume Fig. 1. Ratio of the conductivity increment of the suspension (K K m )to the solution (K m ) as a function of: a) dimensionless zeta potential for ja =10 and / =0.5; ) ja for f =100 mv; c) volume fraction / for f =25, 100 mv, and ja =10. The dispersion medium is KCl, and the particle radius is a =100 nm. Solid symols: Levine Neale oundary conditions. Open symols: Shilov Zharkikh. Ionic perturation condition: dn i ()=0. fraction (Fig. 1c). Note that for the volume fraction involved, increasing f (Fig. 1a) can provoke a very important raise in K, a clear consequence of the large surface conductivity contriution of such highly charged doule layers. As to the effect of ja, the results in Fig. 1 indicate that increasing this quantity provokes a fall in (K K m )/K m ecause the influence of the medium conductivity progressively hides that of the surface conductivity. In fact, it must e recalled here that the net effect of the doule layer conductivity is controlled y the Dukhin numer Du [1],

6 48 F. Carrique et al. / Advances in Colloid and Interface Science 118 (2005) relating the surface conductivity K r to that of the medium, and to the particle radius: Du ¼ Kr K m a : ð47þ Note that Du (and, as a consequence, the contriution of the particles to the conductivity K) is more important the larger the f, the lower the K m and the smaller the size. Finally, Fig. 1c is also self-explaining: increasing the volume fraction of particles with low zeta potential leads to a reduction of the conductivity of the suspension as compared to that of the solution, whereas performing the same operation if f is high yields a higher conductivity the larger the volume fraction Boundary condition for ionic concentrations As mentioned, we have in principle two options in estalishing a oundary condition for the ionic perturation, as specified y Eqs. (32) (dn i ()=0) and (33) (fldn i / flr r= =0). Both are compatile with (and independent of) the previously discussed conditions for dw. Fig. 2 shows our estimations of (K K m )/K m as a function of the dimensionless zeta potential for different ja values and a fixed volume fraction, / =0.5. Three features are immediately clear: i) using fldn i /flr r= =0 leads to lower conductivity increments; ii) the differences etween the results otained with this condition and with dn i ()=0 increase with zeta potential; iii) the relative differences etween the two approaches change with ja. In order to stress the latter point, in Fig. 3 we have plotted K /K m as a function of ja at constant zeta and volume fraction. Items i) and ii) aove can e explained y considering that forcing dn i =0 at r =, instead of fldn i /flr r= =0, opens the possiility of the existence of diffusion fluxes of ions (mainly counter-ions, much more aundant at high f ) at the cell oundary. The increased flow of counter-ions will yield a higher surface conductivity and hence a larger K of the suspension, as compared to the case of no concentration gradients. Clearly, as the zeta potential increases, the effects associated to doule layer polarization will e more intense [35,36], and the differences etween the cases dn i ()=0 and fldn i /flr r= =0 are consistently more significant. At low f, the counter-ion excess concentration is equally low and so will e the eventual diffusion fluxes. The effect of ja (Figs. 2a c and 3) appears to confirm our arguments: the two ways of calculation of K differ mainly in the medium ja regime, where doule-layer polarization effects are known to e more significant for electrokinetics [5]. At low ja the doule layer is so extended that the effect of the external field on it is very small (the volume charge density is very low throughout the ionic atmosphere). On the contrary, at high ja, the Dukhin numer is so small that again doule layer polarization effects are unimportant. In the region in etween these two extremes, the effects of changes in the fluxes of counter-ions Fig. 2. Relative conductivity increment, (K K m )/K m, plotted as a function of the dimensionless zeta potential for suspensions of spheres 100 nm in radius with volume fraction / =0.5, and two choices for ionic perturations in the cell oundary 1: dn i ()=0; 2: fldn i /flr r= =0. ja values: 1 (a), 10 (), 100 (c). in the doule layer will e significant, and the arguments aove explained, concerning the differences etween dn i =0 and fldn i /flr=0, effectively apply. Note (Fig. 3) that at very high ja the conductivity ratio tends to its Maxwell value, as it is only controlled y volume fraction and not y particle charge Addendum. Electrophoretic moility According to Eq. (25), the electrophoretic moility, u e,is equal to 2h()/, so that having solved the velocity function h(r) atr =, u e is immediately otained. Fig. 4 shows some

7 F. Carrique et al. / Advances in Colloid and Interface Science 118 (2005) Fig. 3. Relative conductivity K/K m of suspensions of spherical particles, with volume fraction / =0.5, dimensionless f =6, as a function of ja for the choices dn i ()=0 (1) and fldn i /flr r= =0 (2). MW indicates the Maxwell Wagner limit. representative results calculated with LN and SZ oundary conditions for dw, and with dn i =0 and fldn i /flr =0 for the ionic concentration perturations. It is important to point out that, in this case, not only different choices for dn i ut also different electrostatic oundary conditions for dw largely affect the estimation of u e. While the effect of dn i oundary conditions might e expected, the fact that dw() choice influences the moility and not the conductivity is, at first glance, striking. In previous papers from different authors [19,26], calculations ased on LN and SZ oundary conditions can e rought to coincidence at low f values, if the former are multiplied y the Maxwell factor for uncharged spheres (1 /)/(1+/ /2). This is done in Fig. 5 for the same set of data as in Fig. 4: note that, indeed, oth types of calculation tend to coincide in that region of zeta potentials. When f is aove 50 mv at room temperature the divergence etween Fig. 5. Same as Fig. 4, except that LN data have een multiplied y the Maxwell factor (1 /)/(1+/ /2). them is noticeale, and the Maxwell factor is clearly insufficient. Our previous analysis of the conductivity can help us in elucidating the source of such differences: in the case of K, oth approaches use the same field, EÀ, to define the conductivity in terms of jà. But when it is the electrophoretic moility that is calculated, the SZ model would write, for the velocity of electrophoresis: v e ¼ u SZ e EÀ; ð48þ whereas, according to Levine Neale approach, v e ¼ u LN e ET; ð49þ and oth fields are related (Eq. (42)): ET ¼ NðÞ EÀ; ð50þ so that u sz e should e compared with (u LN e /N()). However, it is advisale to define any kinetic coefficients in terms of model-independent fields; in the case of electrophoresis, this can e done y referring to the macroscopic, average electric field, <E>, so that Eq. (48) would e the Fig. 4. Dimensionless electrophoretic moility, u e *, of spherical particles (a =100 nm) as a function of the dimensionless zeta potential, in KCl solution with ja =10 and for constant / =0.35. Curves laeled (1) and (2) correspond, respectively, to setting dn i ()=0 and fldn i /flr r= =0 for the ionic perturations. LN (solid symols) and SZ (open symols) stand for Levine Neale and Shilov Zharkikh oundary conditions for dw. Note: u e *=(3ge /2e m K B T)u e. Fig. 6. Same as Fig. 4, ut with LN data multiplied y /N() (Eqs. (48) (50)).

8 50 F. Carrique et al. / Advances in Colloid and Interface Science 118 (2005) requested model-independent relationship. A detailed discussion on this suject is given y Dukhin et al. [23]. Fig. 6 demonstrates that in this case, a perfect agreement etween SZ and LN evaluations of the moility can again e found: one can use either of them to calculate u e, ut, since they have different definitions of the field, a connection factor is needed to ring them to coincidence. The difference etween the choices dn i ()=0 and fldn i / flr r= =0, that was evident in conductivity calculations, still persists: the electrophoretic moility is consistently larger in the former case, the more so the larger f. Looking at the prolem for the reference frame fixed in the particle, this means that the liquid moves faster past the steady particle. Again, a qualitative explanation can e given to this fact, ased on the plausile existence of non-zero diffusion fluxes of counter-ions if the condition dn i ()=0 is chosen. These fluxes will drag liquid with them, increasing its velocity with respect to the particle. Turning to the laoratory frame, the electrophoretic velocity will e larger, as oserved. 6. Conclusions It is of utmost importance in modern colloid science to improve on our knowledge of the electrokinetics of concentrated suspensions, a challenging fundamental prolem, and a field of many technological applications. We have discussed in this contriution the usefulness of cell models, considering the essential point of choosing oundary conditions for oth the electrostatic potential and the ionic concentrations. In the case of the potential, Dirichlet and Neumann conditions lead to identical results for the conductivity and the electrophoretic moility of concentrated suspensions of spheres if the field responsile for the perturations is suitaly chosen in each case. Concerning the perturations in ionic concentrations, the differences are apparent in oth conductivity and moility. Experimental data will perhaps e the definitive check for the validity of Dirichlet or Neumann s conditions. Acknowledgments References [1] Lyklema J. Fundamentals of interface and colloid science. Solid Liquid Interfaces, vol. II. London Academic Press; [2] O Brien RW, White LR. J Chem Soc Faraday Trans 1978;2(74):1607. [3] Ohshima H, Healy T, White LR. J Chem Soc Faraday Trans 1983; 2(79):1613. [4] DeLacey EHB, White LR. J Chem Soc Faraday Trans 1981;2(77): [5] Dukhin SS, Derjaguin BV. In: Matijeviç E, editor. Surface and Colloid Science. Electrokinetic Phenomena, vol. 7. New York Wiley; 1974 Chapter 3. [6] Saville DA. J Colloid Interface Sci 1979;71:477. [7] O Brien RW. J Colloid Interface Sci 1981;81:234. [8] O Brien RW. J Colloid Interface Sci 1983;92:204. [9] Liu YC, Keh HJ. J Colloid Interface Sci 1997;192:375. [10] Liu YC, Keh HJ. Langmuir 1998;14:1560. [11] Happel J. AIChE J 1958;4:197. [12] Kuwaara S. J Phys Soc Jpn 1959;14:527. [13] Levine S, Neale GH. J Colloid Interface Sci 1974;47:520. [14] Levine S, Neale GH, Epstein N. J Colloid Interface Sci 1976;57:424. [15] Ohshima H. J Colloid Interface Sci 1998;208:295. [16] Ohshima H. J Colloid Interface Sci 1999;212:443. [17] Marlow BJ, Fairhurst D, Pendse HP. Langmuir 1998;4:611. [18] Ohshima H, Dukhin AS. J Colloid Interface Sci 1999;212:449. [19] Dukhin AS, Ohshima H, Shilov VN, Goetz PJ. Langmuir 1999; 15:3445. [20] Ohshima H. J Colloid Interface Sci 1997;195:137. [21] Midmore BR, O Brien RW. J Colloid Interface Sci 1988;123:486. [22] De Backer R, Watillon A. J Colloid Interface Sci 1976;54:69. [23] Dukhin AS, Shilov VN, Borkovskaya YB. Langmuir 1999;15:3452. [24] Lee E, Yen FY, Hsu JP. J Phys Chem B 2001;105:7239. [25] Carrique F, Arroyo FJ, Delgado AV. Colloids Surf A Physicochem Eng Asp 2001;195:157. [26] Carrique F, Arroyo FJ, Delgado AV. J Colloid Interface Sci 2001; 243:351. [27] Carrique F, Arroyo FJ, Delgado AV. J Colloid Interface Sci 2002; 252:126. [28] Hsu JP, Lee E, Yen FY. J Phys Chem B 2002;106:4789. [29] Ding JM, Keh HJ. J Colloid Interface Sci 2001;236:180. [30] Shilov VN, Zharkikh NI, Borkovskaya YB. Colloid J 1981;43:434. [31] Ohshima H. J Colloid Interface Sci 1997;188:481. [32] Lee E, Chu JW, Hsu J. J Colloid Interface Sci 1999;209:240. [33] Lee E, Chu JW, Hsu J. J Chem Phys 1999;110: [34] Ohshima H. Adv Colloid Interface Sci 2000;88:1. [35] Shilov VN, Delgado AV, González-Caallero F, Horno J, López- García JJ, Grosse C. J Colloid Interface Sci 2000;232:141. [36] Carrique F, Arroyo FJ, Jiménez ML, Delgado AV. J Chem Phys 2003; 118:1945. Financial support for this work y MCyT, Spain (Projects MAT ), and FEDER funds is gratefully acknowledged.