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

Transcription

1 Colloids and Surfaces A: Physicochemical and Engineering Aspects 195 (2001) 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. Física Aplicada I, Facultad de Ciencias, Uni ersidad de Málaga, Málaga, Spain b Dpto. Física Aplicada, Facultad de Ciencias, Uni ersidad de Granada, 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 -potential of the particles. The Onsager relation between the sedimentation potential and the electrophoretic mobility of spherical colloidal particles in concentrated suspensions, derived by Ohshima for low -potentials, is also analyzed as well as its appropriate range of validity. On the other hand, the above-mentioned Ohshima s theory has also been modified to include the presence of a dynamic Stern layer (DSL) on the particles surface. The starting point has been the theory that Mangelsdorf and White (J. Chem. Soc. Faraday Trans. 86 (1990) 2859) developed to calculate the electrophoretic mobility of a colloidal particle, allowing for the lateral motion of ions in the inner region of the double layer (DSL). The role of different Stern layer parameters on the sedimentation velocity and potential are discussed and compared with the case of no Stern layer present. For every volume fraction, the results show that the sedimentation velocity is lower when a Stern layer is present than that of Ohshima s prediction. Likewise, it is worth pointing out that the sedimentation field always decreases when a Stern layer is present, undergoing large changes in magnitude upon varying the different Stern layer parameters. In conclusion, the presence of a DSL causes the sedimentation velocity to increase and the sedimentation potential to decrease, in comparison with the standard case, for every volume fraction. Reasons for these behaviors are given in terms of the decrease in the magnitude of the induced electric dipole moment on the particles, and therefore on the relaxation effect, when a DSL is present. Finally, we have modified Ohshima s model of electrophoresis in concentrated suspensions, to fulfill the requirements of Shilov Zharkhik s cell model. In doing so, the well-known Onsager reciprocal relation between sedimentation and electrophoresis previously obtained for the dilute case is again recovered but now for concentrated suspensions, being valid for every -potential and volume fraction Elsevier Science B.V. All rights reserved. Keywords: Sedimentation velocity; Sedimentation potential; Concentrated suspensions; Onsager reciprocal relation * Corresponding author /01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S (01)

2 158 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) Introduction It is well-known that when a colloidal suspension of charged particles is settling steadily in a gravitational field, the electrical double layer surrounding each particle is distorted because of the fluid motion, giving rise to a microscopic electric field (the relaxation effect). As a consequence, the falling velocity of the particle, i.e. the sedimentation velocity, is lower in comparison with that of an uncharged particle. On the other hand, these electric fields superimpose to yield a macroscopic electric field in the suspension, i.e. the sedimentation field or sedimentation potential gradient (usually called sedimentation potential). A general sedimentation theory for dilute colloidal suspensions, valid for non-conducting spherical particles with arbitrary double layer thickness and -potential, was developed by Ohshima [1] on the basis of previous theoretical approaches [2 9]. In his paper Ohshima removed the shortcomings and deficiencies already reported by Saville [10] concerning Booth s method of calculation of the sedimentation potential. Furthermore, he presented a direct proof of the Onsager reciprocal relation that holds between sedimentation and electrophoresis. On the other hand, a great effort is being addressed to improve the theoretical results predicted by the standard electrokinetic theories dealing with different electrokinetic phenomena in colloidal suspensions. One of the most relevant extensions of these electrokinetic models has been the inclusion of a dynamic Stern layer (DSL) onto the surface of the colloidal particles. Thus, Zukoski and Saville [11] developed a DSL model to reconcile the differences observed between potentials derived from static electrophoretic mobility and conductivity measurements. Mangelsdorf and White [12], using the techniques developed by O Brien and White for the study of the electrophoretic mobility of a colloidal particle [13], presented in 1990 a rigorous mathematical treatment for a general DSL model. They analyzed the effects of different Stern layer adsorption isotherms on the static field electrophoretic mobility and suspension conductivity. More recently, the theory of Stern layer transport has been applied to the study of the low frequency dielectric response of colloidal suspensions by Kijlstra et al. [14], incorporating a surface conductance layer to the thin double layer theory of Fixman [15,16]. Likewise, Rosen et al. [17] generalized the standard theory of the conductivity and dielectric response of a colloidal suspension in AC fields of DeLacey and White [18], assuming the model of Stern layer developed by Zukoski and Saville [11]. Very recently, Mangelsdorf and White presented a rigorous mathematical study for a general DSL model applicable to time dependent electrophoresis and dielectric response [19,20]. In general, the theoretical predictions of the DSL models improve the comparison between theory and experiment [14,17,21,22], although there are still important discrepancies. Returning to the sedimentation phenomena in colloidal suspensions, a DSL extension of Ohshima s theory of the sedimentation velocity and potential in dilute suspensions, has been recently published [23]. The results show that whatever the chosen set of Stern layer parameters or -potential may be, the presence of a DSL causes the sedimentation velocity to increase and the sedimentation potential to decrease, in comparison with the standard prediction (no Stern layer present). On the other hand, the theory of sedimentation in a concentrated suspension of spherical colloidal particles, proposed by Levine et al. [9] on the basis of the Kuwabara cell model [24], has been further developed by Ohshima [25]. In that paper, Ohshima derived a simple expression for the sedimentation potential applicable to the case of low -potential and non-overlapping of the electric double layers. He also presented an Onsager reciprocal relation between sedimentation and electrophoresis, valid for the same latter conditions, using an expression for the electrophoretic mobility of a spherical particle previously derived in his theory of electrophoresis in concentrated suspensions [26]. This theory is also based on the Kuwabara cell model in order to account for the hydrodynamic particle particle interactions, and uses the same boundary condition on the electric potential at the outer surface of the cell, as that of

3 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) Levine et al. s theory of the electrophoresis in concentrated suspensions [27]. Recalling the attention on the DSL correction to the electrokinetic theories, it seemed of interest to explore the effects of extending the standard Ohshima s theory of the sedimentation velocity and potential in a concentrated suspension of charged spherical colloidal particles [25], to include a DSL model. Thus, the chosen starting point has been the method proposed by Mangelsdorf and White in their theory of the electrophoretic mobility of a colloidal particle, to allow for the adsorption and lateral motion of ions in the inner region of the double layer (DSL) [12]. Finally, the aims of this paper can be described as follows. First, we have obtained a numerical solution of the standard Ohshima s theory of sedimentation in concentrated suspensions, for the whole range of -potential and volume fraction, and non-overlapping double layers. Furthermore, we have extended the latter standard theory to include a DSL on the surface of the particles, and analyzed the effects of its inclusion on the sedimentation velocity and potential. And then, we have analyzed the Onsager reciprocal relation that holds between sedimentation and electrophoresis in concentrated suspensions, for both standard and DSL cases. It can be concluded that the presence of a Stern layer provokes a rather slow increase on the magnitude of the sedimentation velocity of a colloidal particle, whatever the values of Stern layer, particle and solution parameters used in the calculations. On the other hand, the presence of a Stern layer causes the sedimentation potential to decrease with respect to the standard prediction. interactions (see Fig. 1). According to this model, each spherical particle of radius a is surrounded by a concentric virtual shell of an electrolyte solution, having an outer radius of b such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction throughout the entire suspension, i.e. = a 3. (1) b In fact, a is the radius of the hydrodynamic unit, i.e. a rigid particle plus a thin layer of solution linked to its surface moving with it as a whole. The surface r=a is usually called slipping plane. This is the plane outside which the continuum equations of hydrodynamics are assumed to hold. As usual, we will make no distinction between the terms particle surface and slipping plane. Before proceeding with the analysis of the modifications arising from the DSL correction to the standard model, it will be useful to briefly review the basic standard equations and boundary conditions. Concerned readers are referred to Ohshima s paper for a more extensive treatment. Consider a charged spherical particle of radius a and mass density p immersed in an electrolyte solution composed of N ionic species of valencies z i, bulk number concentrations n i, and drag coefficients i (i=1,, N). The axes of the coordinate system (r,, ) are fixed at the centre of the particle. The polar axis ( =0) is set parallel to g. 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 [25] on the basis of the Kuwabara cell model to account for the hydrodynamic particle particle Fig. 1. Schematic picture of an ensemble of spherical particles in a concentrated suspension, according to the Kuwabara cell model [24].

4 160 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) The particle is assumed to settle with steady velocity U SED, the sedimentation velocity, in the electrolyte solution of viscosity and mass density o in the presence of a gravitational field g. For the spherical symmetry case, both U SED and g have the same direction. In the absence of g field, the particle has a uniform electric potential, the -potential, at r=a, where r is the radial spherical coordinate, or equivalently, the modulus of position vector. A complete description of the system requires a knowledge of the electric potential (r), the number density or ionic concentration n i (r) and the drift velocity v i (r) of each ionic species (i= 1,, N), the fluid velocity u(r), and the pressure p at every point r in the system. The fundamental equations connecting these quantities are [1,25]: 2 (r)= (r) (2) rs o N (r)= z i en i (r) (3) i=1 2 u(r) p(r) (r)+ o g=0 (4) u(r)=0 (5) v i =u 1 i i (i=1,, N) (6) i (r)= i +z i e (r)+k B T ln n i (r) (i=1,, N) (7) [n i (r)v i (r)]=0 (i=1,, N), (8) where e is the elementary electric charge, K B the Boltzmann s constant and T is the absolute temperature. Eq. (2) is Poisson s equation, where rs is the relative permittivity of the solution, o the permittivity of a vacuum, and (r) is the electric charge density given by Eq. (3). Eqs. (4) and (5) are the Navier Stokes equations appropriate to a steady incompressible fluid flow at low Reynolds number in the presence of electric and gravitational body forces. Eq. (6) expresses that the ionic flow is caused by the liquid flow and the gradient of the electrochemical potential defined in Eq. (7), and it can be related to the balance of the hydrodynamic drag, electrostatic, and thermodynamic forces acting on each ionic species. Eq. (8) is the continuity equation expressing the conservation of the number of each ionic species in the system. The drag coefficient i is related to the limiting conductance io of the ith ionic species by [13] i = N Ae 2 z i o i (i=1,, N), (9) where N A is Avogadro s number. At equilibrium, that is, in the absent of the gravitational field, the distribution of electrolyte ions obeys the Boltzmann distribution n i =n i exp z i e (i=1,, N), (10) K B T and the equilibrium electric potential satisfies the Poisson Boltzmann equation 1 d d r 2 = el (r) (11) r 2 dr dr rs o el N (r)= i=1 z i en i (r), (12) being el the equilibrium electric charge density. The unperturbed or equilibrium electric potential must satisfy these boundary conditions at the slipping plane and at the outer surface of the cell (a)= (13) d (b)=0 (14) dr where is the -potential. As the axes of the coordinate system are chosen fixed at the center of the particle, the boundary conditions for the liquid velocity u and the ionic velocity of each ionic species at the particle surface are expressed by the following equations u=0 at r=a (15) v i rˆ =0 at r=a (i=1,, N) (16) which mean, respectively, that the fluid layer adjacent to the particle surface is at rest, and that there are no ion fluxes through the slipping plane (rˆ is the unit normal outward from the particle surface). According to the Kuwabara

5 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) cell model, the liquid velocity at the outer surface of the unit cell satisfies the conditions: u r = U SED cos at r=b (17) = u=0 at r=b, (18) which express, respectively, that the liquid velocity is parallel to the sedimentation velocity, and the vorticity is equal to zero. Following Ohshima, we will assume that the electrical double layer around the particle is only slightly distorted due to the gravitational field about their equilibrium values. Thus, the following perturbation scheme for the above-mentioned quantities can be used, n i (r)=n i (r)+ n i (r) (i=1,, N) (19) (r)= (r)+ (r) (20) i (r)= i + i (r) (i=1,, N) (21) where the superscript is related to the state of equilibrium. The perturbations in ionic number density and electric potential are related to each other through the perturbation in electrochemical potential by i = z i e +K B T n i (i=1,, N). (22) n i In terms of the perturbation quantities, the condition that the ionic species are not allowed to penetrate the particle surface in Eq. (16), transforms into i rˆ =0 at r=a (i=1,, N), (23) when a DSL is not considered. Besides, for the case of negligible overlapping of double layers on the outer surface of the unit cell, this extra condition holds: i =0( n i =0, =0) (i=1,, N). (24) For the spherical case and following Ohshima [25], symmetry considerations permit us to introduce the radial functions h(r) and i (r), and then write u(r)=(u r, u, u ) = 2 r hgcos, 1 r d dr (rh)g sin, 0 (25) i (r)= z i e i (r)(g rˆ) (i=1,, N), (26) to obtain the following set of ordinary coupled differential equations and boundary conditions at the slipping plane and at the outer surface of the cell: L(Lh)= e N dy r dr n i z i2 exp( z i y) i (r), (27) i=1 with y=e /KT, L( i (r))= dy d i zi dr dr 2 i e h (i=1,, N) r (28) h(a)= dh (a)=0, dr Lh(r)=0 at r=b (29) d i (a)=0 dr (i=1,, N) (30) i (b)=0 (i=1,, N), (31) L being a differential operator defined by L d2 d dr 2+ r dr 2 r 2. (32) In addition to the previous 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 [25]. 3. Extension to include a dynamic Stern layer We now deal with the problem of including the possibility of adsorption and ionic transport in the inner region of the double layer of the particles. We will follow the method developed by Mangelsdorf and White [12] in their theory of the electrophoresis and conductivity in a dilute colloidal suspension. This theory allows for the adsorption and lateral motion of ions in the latter inner region using the well-known Stern model. According to this method, the condition that ions cannot penetrate the slipping plane no longer maintains, and therefore, the evaluation of the fluxes of each ionic species through the slipping plane permits us to obtain the following new slipping plane boundary conditions for the functions i (r),

6 162 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) d i dr (a) 2 i a i(a)=0 (i=1,, N) (33) i = [en i ]/(ae 10 pki )( i / it )exp[(z i e/k B T)( d /C 2 ] N, N A (N A 10 3 c j /10 pkj )exp[( z j e/k B T)( d /C 2 )] j=1 (34) in terms of the so-called surface ionic conductance parameters i of each ionic species, comprising the effect of a mobile surface layer. These parameters depend on, the -potential ; the ratio between the drag coefficient i of each ionic species in the bulk solution and in the Stern layer it ; the density of sites N i available for adsorption in the Stern layer; the pk i of ionic dissociation constant for each ionic species (the adsorption of each ionic species onto an empty Stern layer site is represented as a dissociation reaction in this theory [12]), the capacity C 2 of the outer Stern layer, the radius a of the particles, the electrolyte concentration through c j, i.e. the equilibrium molar concentration of type j ions in solution, and the charge density per unit surface area in the double layer d. It is worth noting that the other boundary conditions expressed by Eqs. (29) and (31) remain unchanged when a DSL is assumed. A numerical method similar to that proposed by DeLacey and White in their theory of the dielectric response and conductivity of a colloidal suspension in time-dependent fields [18], has been applied to solve the above-mentioned set of coupled ordinary differential equations of the sedimentation theory in concentrated colloidal suspensions. Furthermore, both standard and DSL cases have been extensively analyzed. In a recent paper [23], we successfully employed the latter numerical scheme to solve the standard theory of sedimentation in dilute colloidal suspensions. All the details and steps of the numerical procedure can be found in that reference. 4. Calculation of the sedimentation velocity and potential Let us describe now how the sedimentation velocity and potential for a concentrated suspension can be calculated. According to the condition for the fluid velocity at the outer surface of the unit cell, the fluid velocity has to be parallel to the sedimentation velocity (see Eqs. (17) and (25)). Thus, we can obtain the sedimentation velocity U SED, once the value of function h has been determined at the outer surface of the cell, i.e. U SED = 2h(b) g. (35) b For the case of uncharged particles ( =0), the sedimentation velocity is given by the wellknown Stokes formula [25] ST U SED = 2a 2 ( p o ) g. (36) 9 As regards the sedimentation potential E SED,it can be considered as the volume average of the gradient of the electric potential in the suspension volume V, i.e. E SED = V 1 (r)dv. (37) V Following Ohshima [25], the net electric current i in the suspension can be expressed in terms of the sedimentation potential and the first radial derivatives of i functions at the outer surface of the unit cell, i =K ESED + 1 N K z i2 e 2 n i i=1 i d i dr (b)n g, (38) where K is the electric conductivity of the electrolyte solution in the absence of the colloidal particles. If now we impose, following Saville [10] and Ohshima [25], the requirement of zero net electric current in the suspension, we finally obtain E SED = 1 N K z i2 e 2 n i d (b)n i g. (39) i=1 i dr Likewise, we define the scaled sedimentation potential E* SED as in the dilute case by 3 ek * = 2 rs o K B T( p o ) E SED E SED. (40) g

7 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) Onsager reciprocal relation between sedimentation and electrophoresis in concentrated suspensions It is well-known that an Onsager reciprocal relation holds between sedimentation and electrophoresis. A direct proof of this relationship was derived by Ohshima et al. [1] for dilute suspensions, and is given by E SED = ( p o ) g, (41) K where is the electrophoretic mobility of a colloidal particle. Furthermore, this relation is also satisfied when a DSL is incorporated to the theories of sedimentation and electrophoresis in dilute colloidal suspensions [23]. On the other hand, the electrophoretic mobility is usually represented by a scaled quantity * [13] defined by *= 3 e. (42) 2 rs o K B T Eq. (41) can then be rewritten in terms of the scaled quantities to give a simple convenient expression for the Onsager relation, namely, E* SED = *. (43) Very recently Ohshima derived an Onsager relation between sedimentation and electrophoresis in concentrated suspensions, applicable for low potentials and non-overlapping of double layers [25]. In that paper, Ohshima used an expression for the electrophoretic mobility OHS of a spherical colloidal particle, derived according to his theory of the electrophoresis in concentrated suspensions [26]. The Onsager relation he found is given by E SED = (1 )( p o ) (1+ /2)K OHSg, (44) or equivalently, E* SED = (1 ) (1+ /2) * OHS, (45) where Eqs. (40) and (42) have been used. In the limit when volume fraction tends to zero, Eqs. (44) and (45) converges to the well-known Eqs. (41) and (43) which describes the Onsager relation between sedimentation and electrophoresis in dilute suspensions. However, very recently Dukhin et al. [28] have pointed out that the Levine Neale cell model [27], employed by many authors to develop theoretical electrokinetic models in multiparticle systems, in particular those of sedimentation, electrophoresis and conductivity in concentrated suspensions [9,26,29 32], presents some deficiencies. According to Dukhin et al. [28] the Levine Neale cell model is not compatible with certain classical limits concerning, specially, the volume fraction dependence in the exact Smoluchovski s law in concentrated suspensions. Instead of the Levine Neale cell model, Dukhin et al. propose to use the Shilov Zharkikh cell model [33] which not only agrees with the latter Smoluchovski s result but also correlates with the electric conductivity of the Maxwell Wagner theory [34]. It is worth noting that Ohshima s theory of the electrophoretic mobility in concentrated suspensions [26] incorporates the Levine Neale boundary condition on the electric potential at the outer surface of the unit cell. This condition states that the local electric field has to be parallel to the applied electric field E at the outer surface of the cell. Then, it seemed quite interesting to compare the changes in Ohshima s Onsager relation for concentrated suspensions, if any, that could arise from the consideration of a different boundary condition on the electric potential according to the Shilov Zharkikh cell model, which is based on arguments of non-equilibrium thermodynamics. Following Ohshima s theory of electrophoresis in concentrated suspensions [26], the boundary condition for the perturbed electric potential at the outer surface of the unit cell is expressed by rˆ = E rˆ at r=b. (46) However, according to the Shilov Zharkikh cell model, the latter condition changes to = E r at r=b. (47) being E the macroscopic electric field. For low -potentials and non-overlapping of double layers, Eq. (22) becomes [26,32]

8 164 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) i =z i e, (48) and consequently, Eq. (46) transforms into i rˆ = z i eerˆ. (49) Following Ohshima, spherical symmetry considerations permit us to write i (r)= z i e i (r)(e rˆ) (i=1,, N), (50) which is analogous to Eq. (26) for sedimentation. Now, according to Eq. (50), Eq. (49) finally becomes d i (b)=1. dr (51) However, following the Shilov Zharkikh boundary condition given by Eq. (47), a different result can be obtained, i.e. i (b)=b, (52) where Eq. (50) has been used reading E instead of E. If now we change in Ohshima s theory of the electrophoretic mobility in concentrated suspensions, the boundary condition given by Eq. (51) for that in Eq. (52), a quite different numerical result for the electrophoretic mobility is obtained (we will call it SHI ). Furthermore, if we confine ourselves to the analytical approach of low -potentials developed in Ohshima s papers of sedimentation [25] and electrophoresis [26] in concentrated suspensions, an Onsager reciprocal relation different to that by Ohshima (Eqs. (44) and (45)), is found, i.e. E* SED = * SHI. (53) It should be noted that this new Onsager relation has exactly the same form as the well-known Onsager relation connecting sedimentation and electrophoresis in dilute suspensions (see Eqs. (41) and (43)). Likewise, we have numerically confirmed that this Onsager relation also holds for the whole range of -potentials unlike that of Eq. (44). In conclusion, we can state that the Onsager reciprocal relation between sedimentation and electrophoresis, previously derived for the dilute case, also holds for concentrated suspensions if Shilov Zharkikh s boundary condition is considered. In the next section, we will present numeri- Fig. 2. Ratio of the standard sedimentation velocity to the Stokes sedimentation velocity of a spherical colloidal particle in a KCl solution at 25 C, as a function of particle volume fraction and dimensionless -potential. cal computations clearly showing that the latter Onsager relation is also maintained when a DSL is included in the theories of sedimentation and electrophoresis in concentrated suspensions, for whatever conditions on the values of the -potential and Stern layer parameters. 6. Results and discussion 6.1. Sedimentation elocity In Fig. 2 we show some numerical results of the ratio of the standard sedimentation velocity U SED ST to the Stokes velocity U SED, for a spherical colloidal particle in a KCl solution as a function of dimensionless -potential and volume fraction. As we can see, the sedimentation velocity ratio rapidly decreases when the volume fraction increases whatever the value of -potential we choose. This behavior reflects that the higher the volume fraction, the higher the hydrodynamic particle particle interactions. However, at fixed volume fraction the sedimentation velocity ratio seems to be less affected when -potential increases, showing a rather slow decrease due to the increasing importance of the relaxation effect.

9 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) As regards the DSL correction to the standard sedimentation velocity, we represent in Fig. 3 the ratio of the standard sedimentation velocity U SED to the DSL sedimentation velocity (U SED ) DSL as a function of dimensionless -potential and volume fraction. The values of the Stern layer parameters that we have chosen for the numerical computations are indeed rather extreme, but our intention is to show maximum possible effects of the incorporation of a DSL into the standard model. When a DSL is present, the induced electric dipole moment on the particle decreases in comparison with the standard prediction for the same conditions, and so does the relaxation effect [34]. As a consequence, the particle will achieve a larger sedimentation velocity than it would in the absence of a Stern layer (note that the sedimentation velocity ratio is always 1). On the other hand, it should be noted that for a given volume fraction there is a minimum in the ratio, or in other words, a maximum deviation from the standard prediction when that ratio is represented as a function of -potential. In fact, both standard and DSL sedimentation velocities present a maximum deviation from the Stokes prediction (uncharged spheres) when they are represented against -potential for a given volume fraction. This maximum deviation can be related to the concentration polarization effect [34]. In other words, as -potential increases from the region of low -values, the relaxation effect increases as well causing a progressive reduction of the sedimentation velocity. If is further increased, the induced electric dipole moment generated on the falling particle tends to be diminished due to ionic diffusion fluxes in the diffuse double layer. These fluxes arise from the formation of gradients of neutral electrolyte outside the double layer at the front and rear sides of the hydrodynamic unit while falling under gravity, giving rise to a decreasing magnitude of the induced electric dipole moment. In other words, the relaxation effect [34] would be less important. The final result is a decrease in the magnitude of the microscopic electric field generated by the distorted hydrodynamic unit, i.e. particle plus double layer, and then, a smaller reduction of the sedimentation velocity at very high -potentials. When a DSL is considered, a new ionic transport process develops in the perturbed inner region of the double layer, giving rise to an increasing importance of the above-mentioned concentration polarization effect at every -potential. Consequently, the reduction on the sedimentation 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 Sedimentation potential Fig. 3. Ratio of the standard sedimentation velocity to the DSL sedimentation velocity of a spherical colloidal particle in a KCl solution at 25 C, as a function of particle volume fraction and dimensionless -potential. In Fig. 4 the standard sedimentation potential is represented as a function of dimensionless -potential and volume fraction, for the same conditions as those in Fig. 2. The constant C e is defined at the bottom of the picture. It is worth noting the decrease in the magnitude of the sedimentation potential as the volume fraction decreases. Obviously, the lower the volume fraction, the lower the number of particles contributing to the generation of the sedimentation field. We can also see the

10 166 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) 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 -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 -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 suspension of spherical particles in a KCl solution at 25 C, as a function of volume fraction and dimensionless -potential. presence of a maximum when the sedimentation potential is represented against the -potential for a given volume fraction, being a consequence of the above-mentioned concentration polarization effect [34]. As -potential increases, the strength of the dipolar electric moment induced on the distorted particles while settling in the gravitational field increases as well, giving rise to a larger contribution to the sedimentation potential. As -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 -potential and volume fraction. Several remarkable features can be observed in this picture. First, the DSL correction to the sedimentation potential gives always rise to lower values of the sedimentation potential than those predicted by the standard model of sedimentation for the same conditions (note that the sedimenta Onsager reciprocal relation between sedimentation and electrophoresis in concentrated suspensions In Fig. 6 we display, for the case of no DSL present, the scaled sedimentation potential and the scaled electrophoretic mobility multiplied by the factor C defined in the picture, as a function of dimensionless -potential for different volume fractions. Both quantities have been numerically Fig. 5. Ratio of the DSL sedimentation potential to the standard sedimentation potential in a colloidal suspension of spherical particles in a KCl solution at 25 C, as a function of volume fraction and dimensionless -potential.

11 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) Fig. 6. Plot of the scaled standard electrophoretic mobility and sedimentation potential in a colloidal suspension of spherical particles in a KCl solution at 25 C, as a function of dimensionless -potential for different volume fractions. For nonzero volume fractions, E* SED in open symbols; * OHS in solid symbols (Ohshima s model). and independently calculated with Ohshima s models of sedimentation [25] and electrophoresis [26] in concentrated colloidal suspensions. The results clearly indicate that in the limit when volume fraction tends to zero Ohshima s Onsager relation for low -potentials, Eq. (45), converges to the well-known Onsager relation Eq. (43) previously derived for the dilute case, which is valid for the whole range of -values. In other words, the scaled sedimentation potential is numerically coincident with the scaled electrophoretic mobility in that limit (note that in this case the factor C =1). For the remaining volume fractions, the Onsager reciprocal relation proposed by Ohshima for concentrated suspensions would be a good approximation for low and low volume fraction, as observed in Fig. 6. On the other hand, as pointed out in a previous section, we have modified Ohshima s model of electrophoresis in concentrated suspensions to fulfill the requirements of Shilov Zharkikh s cell model. In doing so, we have obtained the same expression for the Onsager reciprocal relation between sedimentation and electrophoresis as that previously derived for the dilute case, but now for concentrated suspensions. In other words, the scaled sedimentation potential is numerically coin- 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 -potential approximation. On the contrary, it remains valid for the whole range of values. In Fig. 7 the scaled sedimentation potential and the scaled electrophoretic mobility are displayed as a function of dimensionless -potential for different volume fractions. Again, both quantities have been independently calculated by numerically solving on the one hand Ohshima s theory of sedimentation in concentrated suspensions, and on the other, Ohshima s theory of electrophoresis in concentrated suspensions including now the Shilov Zharkikh boundary condition (Eq. (52)) instead of that by Levine Neale (Eq. (51)). As we can see, the numerical agreement between each set of results is excellent whatever the values of volume fraction or -potential have been chosen. This is also true when a DSL approach is used, as shown in Fig. 8 for the same conditions as those of Fig. 5. Fig. 7. Plot of the scaled standard electrophoretic mobility and sedimentation potential in a colloidal suspension of spherical particles in a KCl solution at 25 C, as a function of dimensionless -potential for different volume fractions. For nonzero volume fractions, E* SED in open symbols; * SHI in solid symbols (Shilov Zharkikh s model).

12 168 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) Acknowledgements Financial support for this work by MEC, Spain (Project No. MAT ), and INTAS (Project ) is gratefully acknowledged. References Fig. 8. Plot of the scaled DSL electrophoretic mobility and sedimentation potential in a colloidal suspension of spherical particles in a KCl solution at 25 C, as a function of dimensionless -potential for different volume fractions. For nonzero volume fractions, E* SED in open symbols; * SHI in solid symbols (Shilov Zharkikh s model). 7. Conclusions In this work, we have presented numerical calculations concerning the sedimentation velocity and potential in concentrated suspensions for arbitrary -potential and non-overlapping double layers of the particles. Furthermore, we have extended the standard Ohshima s theory of sedimentation in concentrated suspensions, to include a DSL into the model. The results show that regardless of the particle volume fraction and -potential, the mere presence of a DSL causes the sedimentation velocity to increase and the sedimentation potential to decrease in comparison with the standard predictions. On the other hand, we have analyzed the Onsager reciprocal relation between sedimentation and electrophoresis derived by Ohshima for concentrated suspensions, and compared it with the Onsager relation obtained according to the Shilov Zharkikh cell model. We have confirmed that the Shilov Zharkikh cell model fulfills the same Onsager relation in concentrated suspensions as that previously derived for the dilute case, for whatever conditions of -potential and volume fraction, including a DSL as well. [1] H. Ohshima, T.W. Healy, L.R. White, R.W. O Brien, J. Chem. Soc. Faraday Trans. 2 (80) (1984) [2] F. Booth, J. Chem. Phys. 22 (1954) [3] J.T.G. Overbeek, Kolloid Beih. 54 (1943) 287. [4] F. Booth, Proc. R. Soc. London Ser. A 203 (1950) 514. [5] D. Stigter, J. Phys. Chem. 84 (1980) [6] P.H. Wiersema, A.L. Loeb, J.T.G. Overbeek, J. Colloid Interf. Sci. 22 (1966) 78. [7] S.R. De Groot, P. Mazur, J.T.G. Overbeek, J. Chem. Phys. 20 (1952) [8] D.C. Henry, Proc. R. Soc. London Ser. A 133 (1931) 106. [9] S. Levine, G. Neale, N. Epstein, J. Colloid Interf. Sci. 57 (1976) 424. [10] D.A. Saville, Adv. Colloid Interf. Sci. 16 (1982) 267. [11] C.F. Zukoski IV, D.A. Saville, J. Colloid Interf. Sci. 114 (1986) 45. [12] C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 86 (1990) [13] R.W. O Brien, L.R. White, J. Chem. Soc. Faraday Trans. 274 (1978) [14] J. Kijlstra, H.P. van Leeuwen, J. Lyklema, J. Chem. Soc. Faraday Trans. 88 (1992) [15] M. Fixman, J. Chem. Phys. 72 (1980) [16] M. Fixman, J. Chem. Phys. 78 (1983) [17] L.A. Rosen, J.C. Baygents, D.A. Saville, J. Chem. Phys. 98 (1993) [18] E.H.B. DeLacey, L.R. White, J. Chem. Soc. Faraday Trans. 277 (1981) [19] C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 94 (1998) [20] C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 94 (1998) [21] J. Kijlstra, H.P. van Leeuwen, J. Lyklema, Langmuir 9 (1993) [22] F.J. Arroyo, F. Carrique, T. Bellini, A.V. Delgado, J. Colloid Interf. Sci. 210 (1999) 194. [23] F. Carrique, F.J. Arroyo, A.V. Delgado, J. Colloid Interf. Sci. 227 (2000) 212. [24] S. Kuwabara, J. Phys. Soc. Jpn. 14 (1959) 527. [25] H. Ohshima, J. Colloid Interf. Sci. 208 (1998) 295. [26] H. Ohshima, J. Colloid Interf. Sci. 188 (1997) 481. [27] S. Levine, G. Neale, J. Colloid Interf. Sci. 47 (1974) 520. [28] A.S. Dukhin, V.N. Shilov, Y.B. Borkovskaya, Langmuir 15 (1999) [29] M.W. Kozak, E.J. Davies, J. Colloid Interf. Sci. 112 (1986) 403.

13 F. Carrique et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 195 (2001) [30] M.W. Kozak, E.J. Davies, J. Colloid Interf. Sci. 127 (1989) 497. [31] M.W. Kozak, E.J. Davies, J. Colloid Interf. Sci. 127 (1989) 166. [32] H. Ohshima, J. Colloid Interf. Sci. 212 (1999) 443. [33] V.N. Shilov, N.I. Zharkikh, Y.B. Borkovskaya, Colloid J. 43 (1981) 434. [34] S.S. Dukhin, V.N. Shilov, Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes, Wiley, New York, 1974.