Sedimentation Velocity and Potential in a Concentrated Colloidal Suspension Effect of a Dynamic Stern Layer

Colloids and Surfaces A: Physicochemical and Engineering Aspects 195 (2001) 157–169 www.elsevier.com/locate/colsurfa

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

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

a Dpto. Fı´sica Aplicada I, Facultad de Ciencias, Uni6ersidad de Ma´laga, 29071 Ma´laga, Spain b Dpto. Fı´sica Aplicada, Facultad de Ciencias, Uni6ersidad de Granada, 18071 Granada, Spain

Abstract

The standard theory of the sedimentation velocity and potential of a concentrated suspension of charged spherical colloidal particles, developed by H. Ohshima on the basis of the Kuwabara cell model (J. Colloid Interf. Sci. 208 (1998) 295), has been numerically solved for the case of non-overlapping double layers and different conditions concerning volume fraction, and n-potential of the particles. The Onsager relation between the sedimentation potential and the electrophoretic mobility of spherical colloidal particles in concentrated suspensions, derived by Ohshima for low n-potentials, is also analyzed as well as its appropriate range of validity. On the other hand, the above-mentioned Ohshima’s theory has also been modified to include the presence of a dynamic Stern layer (DSL) on the particles’ surface. The starting point has been the theory that Mangelsdorf and White (J. Chem. Soc. Faraday Trans. 86 (1990) 2859) developed to calculate the electrophoretic mobility of a colloidal particle, allowing for the lateral motion of ions in the inner region of the double layer (DSL). The role of different Stern layer parameters on the sedimentation velocity and potential are discussed and compared with the case of no Stern layer present. For every volume fraction, the results show that the sedimentation velocity is lower when a Stern layer is present than that of Ohshima’s prediction. Likewise, it is worth pointing out that the sedimentation field always decreases when a Stern layer is present, undergoing large changes in magnitude upon varying the different Stern layer parameters. In conclusion, the presence of a DSL causes the sedimentation velocity to increase and the sedimentation potential to decrease, in comparison with the standard case, for every volume fraction. Reasons for these behaviors are given in terms of the decrease in the magnitude of the induced electric dipole moment on the particles, and therefore on the relaxation effect, when a DSL is present. Finally, we have modified Ohshima’s model of electrophoresis in concentrated suspensions, to fulfill the requirements of Shilov–Zharkhik’s cell model. In doing so, the well-known Onsager reciprocal relation between sedimentation and electrophoresis previously obtained for the dilute case is again recovered but now for concentrated suspensions, being valid for every n-potential and volume fraction. © 2001 Elsevier Science B.V. All rights reserved.

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

* Corresponding author.

0927-7757/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S0927-7757(01)00839-1 158 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169

1. Introduction More recently, the theory of Stern layer transport has been applied to the study of the low It is well-known that when a colloidal suspen- frequency dielectric response of colloidal suspension of charged particles is settling steadily in a sions by Kijlstra et al. [14], incorporating a sur- gravitational field, the electrical double layer surface conductance layer to the thin double layer rounding each particle is distorted because of the theory of Fixman [15,16]. Likewise, Rosen et al. fluid motion, giving rise to a microscopic electric [17] generalized the standard theory of the con- field (the relaxation effect). As a consequence, the ductivity and dielectric response of a colloidal falling velocity of the particle, i.e. the sedimenta- suspension in AC fields of DeLacey and White tion velocity, is lower in comparison with that of [18], assuming the model of Stern layer developed an uncharged particle. On the other hand, these by Zukoski and Saville [11]. Very recently, Man- electric fields superimpose to yield a macroscopic gelsdorf and White presented a rigorous mathe- electric field in the suspension, i.e. the sedimenta- matical study for a general DSL model applicable tion field or sedimentation potential gradient to time dependent electrophoresis and dielectric (usually called sedimentation potential). response [19,20]. In general, the theoretical predic- A general sedimentation theory for dilute col- tions of the DSL models improve the comparison loidal suspensions, valid for non-conducting between theory and experiment [14,17,21,22], al- spherical particles with arbitrary double layer though there are still important discrepancies. thickness and n-potential, was developed by Returning to the sedimentation phenomena in Ohshima [1] on the basis of previous theoretical colloidal suspensions, a DSL extension of Ohshi- ma’s theory of the sedimentation velocity and approaches [2–9]. In his paper Ohshima removed potential in dilute suspensions, has been recently the shortcomings and deficiencies already re- published [23]. The results show that whatever the ported by Saville [10] concerning Booth’s method chosen set of Stern layer parameters or n-poten- of calculation of the sedimentation potential. Fur- tial may be, the presence of a DSL causes the thermore, he presented a direct proof of the On- sedimentation velocity to increase and the sedi- sager reciprocal relation that holds between mentation potential to decrease, in comparison sedimentation and electrophoresis. with the standard prediction (no Stern layer On the other hand, a great effort is being present). addressed to improve the theoretical results pre- On the other hand, the theory of sedimentation dicted by the standard electrokinetic theories deal- in a concentrated suspension of spherical colloidal ing with different electrokinetic phenomena in particles, proposed by Levine et al. [9] on the colloidal suspensions. One of the most relevant basis of the Kuwabara cell model [24], has been extensions of these electrokinetic models has been further developed by Ohshima [25]. In that paper, the inclusion of a dynamic Stern layer (DSL) onto Ohshima derived a simple expression for the sedi- the surface of the colloidal particles. Thus, mentation potential applicable to the case of low Zukoski and Saville [11] developed a DSL model n-potential and non-overlapping of the electric to reconcile the differences observed between n- double layers. He also presented an Onsager re- potentials derived from static electrophoretic mo- ciprocal relation between sedimentation and elec- bility and conductivity measurements. trophoresis, valid for the same latter conditions, Mangelsdorf and White [12], using the techniques using an expression for the electrophoretic mobil- developed by O’Brien and White for the study of ity of a spherical particle previously derived in his the electrophoretic mobility of a colloidal particle theory of electrophoresis in concentrated suspen- [13], presented in 1990 a rigorous mathematical sions [26]. This theory is also based on the treatment for a general DSL model. They ana- Kuwabara cell model in order to account for the lyzed the effects of different Stern layer adsorp- hydrodynamic particle–particle interactions, and tion isotherms on the static field electrophoretic uses the same boundary condition on the electric mobility and suspension conductivity. potential at the outer surface of the cell, as that of F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169 159

Levine et al.’s theory of the electrophoresis in interactions (see Fig. 1). According to this model, concentrated suspensions [27]. each spherical particle of radius a is surrounded Recalling the attention on the DSL correction by a concentric virtual shell of an electrolyte to the electrokinetic theories, it seemed of interest solution, having an outer radius of b such that the to explore the effects of extending the standard particle/cell volume ratio in the unit cell is equal Ohshima’s theory of the sedimentation velocity to the particle volume fraction throughout the and potential in a concentrated suspension of entire suspension, i.e. charged spherical colloidal particles [25], to in- a3 clude a DSL model. Thus, the chosen starting = . (1) point has been the method proposed by Mangels- b dorf and White in their theory of the elec- In fact, a is the radius of the ‘hydrodynamic trophoretic mobility of a colloidal particle, to unit’, i.e. a rigid particle plus a thin layer of allow for the adsorption and lateral motion of solution linked to its surface moving with it as a ions in the inner region of the double layer (DSL) whole. The surface r=a is usually called ‘slipping [12]. plane’. This is the plane outside which the contin- Finally, the aims of this paper can be described uum equations of hydrodynamics are assumed to as follows. First, we have obtained a numerical hold. As usual, we will make no distinction be- solution of the standard Ohshima’s theory of tween the terms particle surface and slipping sedimentation in concentrated suspensions, for plane. the whole range of n-potential and volume frac- Before proceeding with the analysis of the mod- tion, and non-overlapping double layers. Further- ifications arising from the DSL correction to the more, we have extended the latter standard theory standard model, it will be useful to briefly review to include a DSL on the surface of the particles, the basic standard equations and boundary condi- and analyzed the effects of its inclusion on the tions. Concerned readers are referred to Ohshi- sedimentation velocity and potential. And then, ma’s paper for a more extensive treatment. we have analyzed the Onsager reciprocal relation Consider a charged spherical particle of radius that holds between sedimentation and elec- a and mass density z immersed in an electrolyte trophoresis in concentrated suspensions, for both p solution composed of N ionic species of valencies standard and DSL cases. It can be concluded that z , bulk number concentrations n , and drag co- the presence of a Stern layer provokes a rather i i efficients u (i=1, …, N). The axes of the coordi- slow increase on the magnitude of the sedimenta- i nate system (r, q, ) are fixed at the centre of the tion velocity of a colloidal particle, whatever the particle. The polar axis (q=0) is set parallel to g. values of Stern layer, particle and solution parameters used in the calculations. On the other hand, the presence of a Stern layer causes the sedimentation potential to decrease with respect to the standard prediction.

2. Standard governing equations and boundary conditions

The starting point for our work has been the standard theory of the sedimentation velocity and potential in a concentrated suspension of spherical colloidal particles, developed by H. Ohshima Fig. 1. Schematic picture of an ensemble of spherical particles [25] on the basis of the Kuwabara cell model to in a concentrated suspension, according to the Kuwabara cell account for the hydrodynamic particle–particle model [24]. 160 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169

The particle is assumed to settle with steady ve- pressing the conservation of the number of each locity USED, the sedimentation velocity, in the ionic species in the system. p u electrolyte solution of viscosity and mass den- The drag coefficient i is related to the limit- z \ o sity o in the presence of a gravitational field g. ing conductance i of the ith ionic species by For the spherical symmetry case, both USED and [13] g have the same direction. In the absence of g N e 2 z field, the particle has a uniform electric poten- u = A i (i=1, …, N), (9) i \ o tial, the n-potential n,atr=a, where r is the i radial spherical coordinate, or equivalently, the where NA is Avogadro’s number. modulus of position vector. At equilibrium, that is, in the absent of the A complete description of the system requires gravitational field, the distribution of electrolyte a knowledge of the electric potential c(r), the ions obeys the Boltzmann distribution number density or ionic concentration n (r) and i (o) (o) zi e the drift velocity vi(r) of each ionic species (i= n i =n i exp − (i=1, …, N), (10) 1, …, N), the fluid velocity u(r), and the pres- KBT sure p at every point r in the system. The and the equilibrium electric potential (o) sa- fundamental equations connecting these quanti- tisfies the Poisson–Boltzmann equation ties are [1,25]: 1 d d (o) z (o)(r) z(r) r 2 =− el (11) 92C r 2 dr dr m m (r)=−m m (2) rs o rs o N N z (o) % (o) z(r) % z en (r) (3) el (r)= zi en i (r), (12) = i i i=1 i=1 p92 9 z9C z being z (o) the equilibrium electric charge density. u(r)− p(r)− (r)+ og=0 (4) el The unperturbed or equilibrium electric poten- 9 u(r)=0 (5) tial must satisfy these boundary conditions at 1 9v the slipping plane and at the outer surface of vi =u −u i (i=1, …, N) (6) i the cell v (r)=v +z eC(r)+K T ln n (r) i i i B i (o)(a)=n (13) (i=1, …, N) (7) d (o) 9 (b)=0 (14) [ni(r)vi(r)]=0(i=1, …, N), (8) dr where e is the elementary electric charge, KB the where n is the n-potential. Boltzmann’s constant and T is the absolute tem- As the axes of the coordinate system are cho- m perature. Eq. (2) is Poisson’s equation, where rs sen fixed at the center of the particle, the m is the relative permittivity of the solution, o the boundary conditions for the liquid velocity u z permittivity of a vacuum, and (r) is the electric and the ionic velocity of each ionic species at charge density given by Eq. (3). Eqs. (4) and (5) the particle surface are expressed by the follow- are the Navier–Stokes equations appropriate to ing equations a steady incompressible fluid flow at low Reynolds number in the presence of electric and u=0atr=a (15) gravitational body forces. Eq. (6) expresses that v rˆ=0atr=a (i=1, …, N) (16) the ionic flow is caused by the liquid flow and i the gradient of the electrochemical potential which mean, respectively, that the fluid layer ad- defined in Eq. (7), and it can be related to the jacent to the particle surface is at rest, and that balance of the hydrodynamic drag, electrostatic, there are no ion fluxes through the slipping and thermodynamic forces acting on each ionic plane (rˆ is the unit normal outward from the species. Eq. (8) is the continuity equation ex- particle surface). According to the Kuwabara F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169 161

lv cell model, the liquid velocity at the outer surface i(r)=−zi e i(r)(g rˆ)(i=1, …, N), (26) of the unit cell satisfies the conditions: to obtain the following set of ordinary coupled q ur =−USED cos at r=b (17) differential equations and boundary conditions at the slipping plane and at the outer surface of the v=9×u=0atr=b, (18) cell: e dy N which express, respectively, that the liquid veloc- % 2 L(Lh)=−p n i z i exp(−zi y) i(r), (27) ity is parallel to the sedimentation velocity, and r dr i=1 the vorticity is equal to zero. with y=e (o)/KT, Following Ohshima, we will assume that the u electrical double layer around the particle is only dy d i 2 i h L( i(r))= zi − (i=1, …, N) slightly distorted due to the gravitational field dr dr e r about their equilibrium values. Thus, the follow- (28) ing perturbation scheme for the above-mentioned dh quantities can be used, h(a)= (a)=0, Lh(r)=0atr=b (29) dr (o) l ni(r)=n i (r)+ ni(r)(i=1, …, N) (19) d i(a)=0(i=1, …, N) (30) c(r)= (o)(r)+lc(r) (20) dr v v (o) lv i(r)= i + i(r)(i=1, …, N) (21) i(b)=0(i=1, …, N), (31) where the superscript (o) is related to the state of L being a differential operator defined by equilibrium. The perturbations in ionic number d2 2 d 2 density and electric potential are related to each L 2 + − 2. (32) other through the perturbation in electrochemical dr r dr r potential by In addition to the previous boundary condi- z elc+K Tln tions, we must impose the constraint that in the lv = i B i (i=1, …, N). (22) i n (o) stationary state the net force acting on the particle i or the unit cell must be zero [25]. In terms of the perturbation quantities, the condition that the ionic species are not allowed to penetrate the particle surface in Eq. (16), trans- 3. Extension to include a dynamic Stern layer forms into 9lv We now deal with the problem of including the i rˆ=0atr=a (i=1, …, N), (23) possibility of adsorption and ionic transport in when a DSL is not considered. the inner region of the double layer of the parti- Besides, for the case of negligible overlapping cles. We will follow the method developed by of double layers on the outer surface of the unit Mangelsdorf and White [12] in their theory of the cell, this extra condition holds: electrophoresis and conductivity in a dilute col- lv =0(ln =0, lc=0) (i=1, …, N). (24) loidal suspension. This theory allows for the ad- i i sorption and lateral motion of ions in the latter For the spherical case and following Ohshima inner region using the well-known Stern model. [25], symmetry considerations permit us to intro- According to this method, the condition that ions duce the radial functions h(r) and i(r), and then cannot penetrate the slipping plane no longer write maintains, and therefore, the evaluation of the fluxes of each ionic species through the slipping u(r)=(u , uq, u) r plane permits us to obtain the following new 2 q 1 d q slipping plane boundary conditions for the func- = − hgcos , (rh)g sin ,0 (25) r r dr tions i(r), 162 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169 d 2l sion can be calculated. According to the condi- i(a)− i (a)=0(i=1, …, N) (33) dr a i tion for the fluid velocity at the outer surface of the unit cell, the fluid velocity has to be parallel −pKi u u t | l [eNi]/(ae 10 )( i/ i)exp[(zi e/KBT)( d/C2] i = N , to the sedimentation velocity (see Eqs. (17) and 3 % 3 −pKj n | NA10 + (NA10 c j /10 )exp[(−zj e/KBT)( − d/C2)] (25)). Thus, we can obtain the sedimentation ve- j=1 locity U , once the value of function h has (34) SED been determined at the outer surface of the cell, in terms of the so-called surface ionic conduc- i.e. tance parameters l of each ionic species, com- i 2h(b) prising the effect of a mobile surface layer. These USED = g. (35) parameters depend on, the n-potential n; the ra- b u tio between the drag coefficient i of each ionic For the case of uncharged particles (n=0), the species in the bulk solution and in the Stern sedimentation velocity is given by the well- u t layer i; the density of sites Ni available for known Stokes formula [25] adsorption in the Stern layer; the pK of ionic i 2a 2(z −z ) dissociation constant for each ionic species (the ST p o USED = p g. (36) adsorption of each ionic species onto an empty 9

Stern layer site is represented as a dissociation As regards the sedimentation potential ESED,it reaction in this theory [12]), the capacity C2 of can be considered as the volume average of the the outer Stern layer, the radius a of the parti- gradient of the electric potential in the suspen- cles, the electrolyte concentration through c j , sion volume V, i.e. i.e. the equilibrium molar concentration of type j & 1 9c ions in solution, and the charge density per unit ESED =− (r)dV. (37) | V V surface area in the double layer d. It is worth noting that the other boundary conditions ex- Following Ohshima [25], the net electric cur- pressed by Eqs. (29) and (31) remain unchanged rent i in the suspension can be expressed in when a DSL is assumed. terms of the sedimentation potential and the first A numerical method similar to that proposed radial derivatives of i functions at the outer by DeLacey and White in their theory of the surface of the unit cell, dielectric response and conductivity of a col- ! 1 N z 2e 2n d n " loidal suspension in time-dependent fields [18], % i i i i =K ESED + u (b) g , has been applied to solve the above-mentioned K i=1 i dr set of coupled ordinary differential equations of (38) the sedimentation theory in concentrated col- where K is the electric conductivity of the elec- loidal suspensions. Furthermore, both standard trolyte solution in the absence of the colloidal and DSL cases have been extensively analyzed. particles. In a recent paper [23], we successfully employed If now we impose, following Saville [10] the latter numerical scheme to solve the standard and Ohshima [25], the requirement of zero net theory of sedimentation in dilute colloidal sus- electric current in the suspension, we finally ob- pensions. All the details and steps of the numeri- tain cal procedure can be found in that reference. N 2 2 n 1 % z i e n i d i ESED =− u (b) g. (39) K i=1 i dr 4. Calculation of the sedimentation velocity and Likewise, we define the scaled sedimentation potential potential E*SED as in the dilute case by p Let us describe now how the sedimentation 3 eK ESED E*SED = m m z z . (40) velocity and potential for a concentrated suspen- 2 rs oKBT( p − o) g F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169 163

5. Onsager reciprocal relation between (41) and (43) which describes the Onsager relation sedimentation and electrophoresis in concentrated between sedimentation and electrophoresis in di- suspensions lute suspensions. However, very recently Dukhin et al. [28] have It is well-known that an Onsager reciprocal pointed out that the Levine–Neale cell model [27], relation holds between sedimentation and elec- employed by many authors to develop theoretical trophoresis. A direct proof of this relationship electrokinetic models in multiparticle systems, in was derived by Ohshima et al. [1] for dilute sus- particular those of sedimentation, electrophoresis pensions, and is given by and conductivity in concentrated suspensions z z [9,26,29–32], presents some deficiencies. Accord- ( p − o) v ESED =− g, (41) ing to Dukhin et al. [28] the Levine–Neale cell K model is not compatible with certain classical where v is the electrophoretic mobility of a col- limits concerning, specially, the volume fraction loidal particle. Furthermore, this relation is also dependence in the exact Smoluchovski’s law in satisfied when a DSL is incorporated to the theo- concentrated suspensions. Instead of the Levine– ries of sedimentation and electrophoresis in dilute Neale cell model, Dukhin et al. propose to use the colloidal suspensions [23]. Shilov–Zharkikh cell model [33] which not only On the other hand, the electrophoretic mobility agrees with the latter Smoluchovski’s result but is usually represented by a scaled quantity v* [13] also correlates with the electric conductivity of the defined by Maxwell–Wagner theory [34]. It is worth noting that Ohshima’s theory of the electrophoretic mo- 3pe v*= v. (42) bility in concentrated suspensions [26] incorpo- 2m m K T rs o B rates the Levine–Neale boundary condition on Eq. (41) can then be rewritten in terms of the the electric potential at the outer surface of the scaled quantities to give a simple convenient ex- unit cell. This condition states that the local elec- pression for the Onsager relation, namely, tric field has to be parallel to the applied electric field E at the outer surface of the cell. E* =v*. (43) SED Then, it seemed quite interesting to compare Very recently Ohshima derived an Onsager rela- the changes in Ohshima’s Onsager relation for tion between sedimentation and electrophoresis in concentrated suspensions, if any, that could arise concentrated suspensions, applicable for low n- from the consideration of a different boundary potentials and non-overlapping of double layers condition on the electric potential according to [25]. In that paper, Ohshima used an expression the Shilov–Zharkikh cell model, which is based v for the electrophoretic mobility OHS of a spheri- on arguments of non-equilibrium thermodynam- cal colloidal particle, derived according to his ics. Following Ohshima’s theory of electrophore- theory of the electrophoresis in concentrated sus- sis in concentrated suspensions [26], the boundary pensions [26]. The Onsager relation he found is condition for the perturbed electric potential at given by the outer surface of the unit cell is expressed by z z (1− )( p − o) 9lc E =− v g, (44) rˆ=−E rˆ at r=b. (46) SED (1+/2)K OHS However, according to the Shilov–Zharkikh or equivalently, cell model, the latter condition changes to (1−) E* = v* , (45) lc SED (1+/2) OHS =− E r at r=b. (47) where Eqs. (40) and (42) have been used. In the being E the macroscopic electric field. For low limit when volume fraction tends to zero, Eqs. n-potentials and non-overlapping of double lay- (44) and (45) converges to the well-known Eqs. ers, Eq. (22) becomes [26,32] 164 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169 lv lc i =zi e , (48) and consequently, Eq. (46) transforms into 9lv i rˆ=−zi eErˆ. (49) Following Ohshima, spherical symmetry considerations permit us to write lv i(r)=−zi e i(r)(E rˆ)(i=1, …, N), (50) which is analogous to Eq. (26) for sedimentation. Now, according to Eq. (50), Eq. (49) finally becomes d i(b)=1. (51) dr However, following the Shilov–Zharkikh Fig. 2. Ratio of the standard sedimentation velocity to the boundary condition given by Eq. (47), a different Stokes sedimentation velocity of a spherical colloidal particle result can be obtained, i.e. in a KCl solution at 25 °C, as a function of particle volume n fraction and dimensionless -potential. i(b)=b, (52) where Eq. (50) has been used reading E instead cal computations clearly showing that the latter of E. If now we change in Ohshima’s theory of Onsager relation is also maintained when a DSL the electrophoretic mobility in concentrated sus- is included in the theories of sedimentation and pensions, the boundary condition given by Eq. electrophoresis in concentrated suspensions, for (51) for that in Eq. (52), a quite different numeri- whatever conditions on the values of the n-poten- cal result for the electrophoretic mobility is ob- tial and Stern layer parameters. v tained (we will call it SHI). Furthermore, if we confine ourselves to the analytical approach of low n-potentials developed in Ohshima’s papers 6. Results and discussion of sedimentation [25] and electrophoresis [26] in concentrated suspensions, an Onsager reciprocal 6.1. Sedimentation 6elocity relation different to that by Ohshima (Eqs. (44) and (45)), is found, i.e. In Fig. 2 we show some numerical results of the v ratio of the standard sedimentation velocity USED E*SED = *SHI. (53) ST to the Stokes velocity USED, for a spherical col- It should be noted that this new Onsager rela- loidal particle in a KCl solution as a function of tion has exactly the same form as the well-known dimensionless n-potential and volume fraction. As Onsager relation connecting sedimentation and we can see, the sedimentation velocity ratio electrophoresis in dilute suspensions (see Eqs. (41) rapidly decreases when the volume fraction in- and (43)). Likewise, we have numerically confi- creases whatever the value of n-potential we rmed that this Onsager relation also holds for the choose. This behavior reflects that the higher the whole range of n-potentials unlike that of Eq. volume fraction, the higher the hydro- (44). In conclusion, we can state that the Onsager dynamic particle–particle interactions. However, reciprocal relation between sedimentation and at fixed volume fraction the sedimentation electrophoresis, previously derived for the dilute velocity ratio seems to be less affected when n-po- case, also holds for concentrated suspensions if tential increases, showing a rather slow decrease Shilov–Zharkikh’s boundary condition is consid- due to the increasing importance of the relaxation ered. In the next section, we will present numeri- effect. F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169 165

As regards the DSL correction to the standard fraction. This maximum deviation can be related sedimentation velocity, we represent in Fig. 3 the to the concentration polarization effect [34]. n ratio of the standard sedimentation velocity USED In other words, as -potential increases from n to the DSL sedimentation velocity (USED)DSL as a the region of low -values, the relaxation effect function of dimensionless n-potential and volume increases as well causing a progressive reduction fraction. The values of the Stern layer parameters of the sedimentation velocity. If n is further in- that we have chosen for the numerical computa- creased, the induced electric dipole moment gener- tions are indeed rather extreme, but our intention ated on the falling particle tends to be diminished is to show maximum possible effects of the incor- due to ionic diffusion fluxes in the diffuse double poration of a DSL into the standard model. layer. These fluxes arise from the formation of When a DSL is present, the induced electric gradients of neutral electrolyte outside the double dipole moment on the particle decreases in com- layer at the front and rear sides of the hydrody- parison with the standard prediction for the same namic unit while falling under gravity, giving rise conditions, and so does the relaxation effect [34]. to a decreasing magnitude of the induced electric As a consequence, the particle will achieve a dipole moment. In other words, the relaxation larger sedimentation velocity than it would in the effect [34] would be less important. The final absence of a Stern layer (note that the sedimenta- result is a decrease in the magnitude of the micro- tion velocity ratio is always B1). scopic electric field generated by the distorted On the other hand, it should be noted that for hydrodynamic unit, i.e. particle plus double layer, a given volume fraction there is a minimum in the and then, a smaller reduction of the sedimentation n ratio, or in other words, a maximum deviation velocity at very high -potentials. from the standard prediction when that ratio is When a DSL is considered, a new ionic transport process develops in the perturbed inner re- represented as a function of n-potential. In fact, gion of the double layer, giving rise to an both standard and DSL sedimentation velocities increasing importance of the above-mentioned present a maximum deviation from the Stokes concentration polarization effect at every n-poten- prediction (uncharged spheres) when they are rep- tial. Consequently, the reduction on the sedimen- resented against n-potential for a given volume tation velocity is always lower when a DSL is present in comparison with that of the standard case. Another important feature in Fig. 3 is that the relative deviation of the DSL sedimentation velocity from the standard prediction seems to be more important the higher the volume fraction or equivalently, the higher the hydrodynamic particle–particle interactions.

6.2. Sedimentation potential

In Fig. 4 the standard sedimentation potential is represented as a function of dimensionless n-potential and volume fraction, for the same condi-

tions as those in Fig. 2. The constant Ce is deﬁned at the bottom of the picture. It is worth noting the decrease in the magnitude of the sedimentation potential as the volume fraction decreases. Obvi- Fig. 3. Ratio of the standard sedimentation velocity to the DSL sedimentation velocity of a spherical colloidal particle in ously, the lower the volume fraction, the lower the a KCl solution at 25 °C, as a function of particle volume number of particles contributing to the generation fraction and dimensionless n-potential. of the sedimentation ﬁeld. We can also see the 166 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169

tion potential ratio is always less than unity). This can be explained according to the above-mentioned additional decrease in the magnitude of the standard induced electric dipole moment when a DSL is present. Secondly, we can observe an important increase in the ratio tending to unity in the limit of high n-potentials for ﬁxed volume fraction. In other words, there would be no signiﬁcant deviation from the standard model in spite of the presence of a DSL. This behavior is easy to explain because at high n-potential the Stern layer reaches saturation while the diffuse layer charge density continues to rise, rapidly overshadowing the effects of a DSL, and thus, approaching to the standard prediction.

Fig. 4. Standard sedimentation potential in a colloidal suspen- 6.3. Onsager reciprocal relation between sion of spherical particles in a KCl solution at 25 °C, as a function of volume fraction and dimensionless n-potential. sedimentation and electrophoresis in concentrated suspensions presence of a maximum when the sedimentation potential is represented against the n-potential for In Fig. 6 we display, for the case of no DSL a given volume fraction, being a consequence of present, the scaled sedimentation potential and the above-mentioned concentration polarization the scaled electrophoretic mobility multiplied by effect [34]. As n-potential increases, the strength the factor C defined in the picture, as a function n of the dipolar electric moment induced on the of dimensionless -potential for different volume fractions. Both quantities have been numerically distorted particles while settling in the gravitational field increases as well, giving rise to a larger contribution to the sedimentation potential. As n-potential is further increased the relaxation effect seems to become less significant owing to the concentration polarization effect, tending in turn to diminish the dipolar electric moment, and then, the sedimentation potential generated in the suspension. Let us consider now the effects of the inclusion of a DSL into the standard theory of the sedimentation potential. Thus, in Fig. 5 we represent the ratio of the DSL sedimentation potential to the standard sedimentation potential as a function of dimensionless n-potential and volume fraction. Several remarkable features can be observed in this picture. First, the DSL correction to the sedimentation potential gives always rise to lower values of the sedimentation potential than those Fig. 5. Ratio of the DSL sedimentation potential to the standard sedimentation potential in a colloidal suspension of predicted by the standard model of sedimentation spherical particles in a KCl solution at 25 °C, as a function of for the same conditions (note that the sedimenta- volume fraction and dimensionless n-potential. F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169 167

cident with the scaled electrophoretic mobility whatever the volume fraction may be, if Shilov– Zharkikh’s boundary condition (Eq. (52)) is assumed. We have confirmed this result by numerical integration of the theories, as it can be seen in Eq. (7). Likewise, it is worth pointing out that this Onsager relation is not a low n-potential approximation. On the contrary, it remains valid for the whole range of n values. In Fig. 7 the scaled sedimentation potential and the scaled electrophoretic mobility are displayed as a function of dimensionless n-potential for different volume fractions. Again, both quantities Fig. 6. Plot of the scaled standard electrophoretic mobility and have been independently calculated by numeri- sedimentation potential in a colloidal suspension of spherical cally solving on the one hand Ohshima’s theory of particles in a KCl solution at 25 °C, as a function of dimen- sedimentation in concentrated suspensions, and sionless n-potential for different volume fractions. For non- v on the other, Ohshima’s theory of electrophoresis zero volume fractions, E*SED in open symbols; *OHS in solid symbols (Ohshima’s model). in concentrated suspensions including now the Shilov–Zharkikh boundary condition (Eq. (52)) and independently calculated with Ohshima’s instead of that by Levine–Neale (Eq. (51)). As we models of sedimentation [25] and electrophoresis can see, the numerical agreement between each set [26] in concentrated colloidal suspensions. The of results is excellent whatever the values of vol- results clearly indicate that in the limit when ume fraction or n-potential have been chosen. volume fraction tends to zero Ohshima’s Onsager This is also true when a DSL approach is used, relation for low n-potentials, Eq. (45), converges as shown in Fig. 8 for the same conditions as to the well-known Onsager relation Eq. (43) pre- those of Fig. 5. viously derived for the dilute case, which is valid for the whole range of n-values. In other words, the scaled sedimentation potential is numerically coincident with the scaled electrophoretic mobility in that limit (note that in this case the factor C =1). For the remaining volume fractions, the Onsager reciprocal relation proposed by Ohshima for concentrated suspensions would be a good approximation for low n and low volume fraction, as observed in Fig. 6. On the other hand, as pointed out in a previous section, we have modified Ohshima’s model of electrophoresis in concentrated suspensions to fulfill the requirements of Shilov–Zharkikh’s cell model. In doing so, we have obtained the same expression for the Onsager reciprocal relation be- Fig. 7. Plot of the scaled standard electrophoretic mobility and tween sedimentation and electrophoresis as that sedimentation potential in a colloidal suspension of spherical particles in a KCl solution at 25 °C, as a function of dimen- previously derived for the dilute case, but now for sionless n-potential for different volume fractions. For non- concentrated suspensions. In other words, the v zero volume fractions, E*SED in open symbols; *SHI in solid scaled sedimentation potential is numerically coin- symbols (Shilov–Zharkikh’s model). 168 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 157–169

Acknowledgements

Financial support for this work by MEC, Spain (Project No. MAT98-0940), and INTAS (Project 99-00510) is gratefully acknowledged.

References

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