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

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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 trans- port has been applied to the study of the low It is well-known that when a colloidal suspen- frequency dielectric response of colloidal suspen- sion of charged particles is settling steadily in a sions by Kijlstra et al. [14], incorporating a sur- gravitational field, the electrical double layer sur- face 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.
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