Primary Electroviscous Effect with a Dynamic Stern Layer: Low κa Results

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Journal of Colloid and Interface Science 226, 180 184 (2000) doi:10.1006/jcis.2000.6804, available online at http://www.idealibrary.

1 Journal of Colloid and Interface Science 226, (2000) doi: /jcis , available online at on Primary Electroviscous Effect with a Dynamic Stern Layer: Low κa Results F. J. Rubio-Hernández, 1 E. Ruiz-Reina, and A. I. Gómez-Merino E.T.S. Ingenieros Industriales, Departamento de Física Aplicada II, Grupo de Física de Coloides, Universidad de Málaga, Campus de El Ejido, E Málaga, Spain Received October 22, 1999; accepted February 28, 2000 The classical treatments of the primary electroviscous effect show important discrepancies with respect to the experimental data. A possible better agreement may be found if the contribution of the ions adsorbed on the Stern layer, which can move tangentially near the particle surface, is taken into account. This contribution has been incorporated into the Watterson White theory. A study of the influence of the Stern-layer parameters on the primary electroviscous coefficient has been made. C 2000 Academic Press Key Words: primary electroviscous effect; dynamic Stern layer. 1. INTRODUCTION The viscosity η of a colloidal suspension is greater than that of the suspending fluid η 0. At low particle concentrations, this behavior is the consequence of an increase of energy dissipation during laminar shear flow due to the perturbation of the streamlines by the colloidal particles. Assuming that the particles were spherical, rigid, uncharged, and small when compared to the dimensions of the measuring apparatus and large when compared to the size of the solvent molecules, Einstein (1) calculated the dependence of the viscosity of a suspension on the volume fraction at low particle concentrations, [ η = η ], 2 φ [1] where φ is the volume-per-volume fraction. It is interesting to note that there is no effect of particle size. This is because the theory is formulated for dilute suspensions and neglects the effects of interactions between colloidal particles. When the particles are charged and the fluid is an electrolyte, an arrangement of charges in the interface appears that is referred to as the electrical double layer (edl). The flow fields in the vicinity of the particles are further modified due to the electrostatic body force exerted by the particle on the fluid within the edl. This distortion of the edl leads to increased dissipation of energy and a further increase in the viscosity. This effect was first considered by Smoluchowski (2) and is called the primary 1 To whom correspondence should be addressed. Fax: fjrubio@uma.es. electroviscous effect (3). We can write η = η 0 [ (1 + p)φ ], [2] where p, the primary electroviscous coefficient, is a function of the charge on the particle (or, more conventionally, the potential in the slipping plane or ζ potential) and properties of the electrolyte. The edl extends a distance of κ 1 (Debye length) from the surface of the particle, κ 2 = 4πe2 ε 0 ε r kt N ni zi 2, [3] where e is the elementary charge, ε 0 the vacuum permitivity, ε r the dielectric constant of the liquid medium, k the Boltzmann constant, T the absolute temperature, z i the valence, and n i the bulk number density of the ith ionic species (i = 1,...,N). The perturbation in the flow field around an uncharged particle has the characteristic dimension of the particle radius a. When the ratio κa of particle size to edl thickness is large, the region of extrahydrodynamic perturbation due to surface charge on the particle is confined to a thin layer near the surface of the particle. In this limit the electroviscous coefficient p will tend to zero as the electrostatic body force can cause little extramodification of the flow field. In the other limit of small κa, where the edl thickness is large compared with particle size, substantial alteration of the flow field is caused by the electrostatic body force and p can become very large. Theoretical treatments of the primary electroviscous effect have been proposed by several authors. First theories (2, 4, 5) were limited for κa > 10. Later Booth (6) derived an expression for the primary electroviscous coefficient, following his own treatment of electrophoresis (7), valid for all κa values. However, his theory is restricted to small values of the ζ potential and small Peclet numbers. The Peclet number measures the extent to which the movement of the fluid, relative to the particle, disturbs the ionic atmosphere. For small Peclet numbers the edl is only slightly distorted from its equilibrium shape. Russell (8) extended Booth s analysis to larger values of Peclet numbers (Pe κa with κa 1), but his theory was still restricted to /00 $35.00 Copyright C 2000 by Academic Press All rights of reproduction in any form reserved. 180

2 PRIMARY ELECTROVISCOUS EFFECT 181 small ζ values. The most recent theory on the primary electroviscous effect has been elaborated by Watterson and White (9) and is valid for all κa and ζ values by solving numerically the equations that governs the phenomenon. On the other hand, many papers on the electrokinetics of polystyrene model colloids (10 17) have evidenced that the standard electrokinetic model cannot explain the observed experimental behavior. Zukoski and Saville (18) presented extensive experimental results on the electrophoretic mobility and lowfrequency electrical conductivity and found that the ζ -potentials inferred from the suspension conductivity were larger than those derived from electrophoresis. They developed (19) a dynamic Stern layer (DSL) model in an attempt to reconcile the observed differences and concluded that Stern-layer transport could account for the discrepancies (20). Recently (21), another explanation based upon the influence of a DSL mechanism into the edl has been proposed and given good results (17). Althought the notion of DSL has been a part of colloid science for many years (22), only recently has the real importance of this phenomenon been pointed out (23 25). The existence of DSL involves the presence of local excess charge that may move tangentially whitin the Stern layer. An appropiate model of the region behind the slipping plane is due to Stern (26). According to the model, this region is divided in the inner Stern layer, where no charge density can exist and extends to a distance β 1 from the particle surface, and the outer Stern layer, where the ions have lost part of its hydration sheath and moved closer to the particle surface as a result of some strong surface interaction, which extends from β 1 to a distance β 1 + β 2 from the particle surface. The convincing evidence for lateral ionic mobility in the Stern layer led Mangelsdorf and White (27) to apply to electrophoresis theory developed by O Brien and White (28) a treatment based upon a general dynamic Stern-layer model. They concluded that the presence of mobile Stern-layer ions causes the electrophoretic mobility to decrease in comparison with the case when surface conduction is absent. In a recent work (29), we have incorporated a DSL in the theory of the primary electroviscous effect developed by Watterson and White (9), following the treatment of Mangelsdorf and White (27) for electrophoresis. The DSL appears in the model as a modified boundary condition that contains the Stern-layer parameter δ i which encapsulates the effect of allowing type i ions to move behind the slipping plane σ 0 S i δ i = az i eni λ i λ t i (β 1) exp ( zi eζ kt ), [4] where a is particle radius and λ i /λ t i (β 1) is the ratio of ionic drag coefficients in the bulk and in the Stern layer for i type ions. The value of δ i depends of the Stern-layer adsorption isotherm that we use through the parameter σ 0 S i, i.e., the Stern-layer charge density due to ith ionic species. We have studied one possibility, which considers adsorption of ions onto available surface area. The description of this adsorption isotherm can be found in Ref. (27). The expresion for σs 0 i is σ 0 S i = z i en i n i 1 + N j=1 K { i n j exp [ z i e kt 0 (β 1 ) ] K j exp [ z i e kt 0 (β 1 ) ]}, [5] where 0 (β 1 ) is the potential in the plane defined by the distance β 1, N i the total number of Stern-layer sites per unit area and K i the dissociation constant in the Stern layer for type i ions. In this paper, we present new numerical results on the primary electroviscous coefficient in the range of low κa, i.e., when thick edl condition is accomplished. As was pointed out, in this case the effect is considerably high. A detailed study on the influence of the different Stern-layer parameters has been made. RESULTS AND DISCUSSION The equations that govern the phenomena are the Navier Stokes equations for an incompresible fluid, the Poisson Boltzmann equation for the potential distribution, the continuity equation and force balance for each ionic species. Under the asumptions that there are no ion fluxes through the slipping plane and that the shear field disturbs only slightly the ionic atmosphere around the particles, Watterson and White (9) linearized the equations and applied the symmetry of the problem to find a set of coupled ordinary linear differential equations which we show in adimensional form, L 4 F = d { [ 1 d x 4 d ( )]} 1 d x 4 x 4 (x 4 F) = 2 ( d 0 ) N c x 2 i z i e z i 0 φ i [6] L 2 φ i = d2 φ i 2 d = z i ( d x with boundary conditions φ i (x) F(x) + 2 d φ i x 6 x φ 2 i )[ ] d φ i + λ i (x 3 F) d 0 [7] N = c i e z i 0, [8] x d F(x) F(x) x=κa = κa 3, d F(x) = 1 x=κa 3 [9] d φ i (x) = 0 (i = 1,...,N) [10] x=κa

3 182 RUBIO-HERNÁNDEZ, RUIZ-REINA, AND GÓMEZ-MERINO 0 (x) 0 (κa) = ζ, [11] where x is the adimensional distance from the particle surface κa, 0 (=e 0 /kt) is the adimensional potential distribution at equilibrium, ζ (=eζ/kt) is the adimensional zeta potential, F(x) is a function related with the velocity field of the fluid, and φ i is related with the perturbed potentials [9] of i ions. The quantities c i and λ i are defined by c i = c i z i N j=1 c j z 2 j λ i = ε 0ε r kt η 0 z i e 2 λ i. [12] In a previous work (29), we have shown that the correct boundary condition for the functions φ i (x) when a DSL is taken into account become φ i (x) d φ i (x) 6δ i x=κa κa φ i (x) x=κa = 0 (i = 1,...,N), [13] where δ i is the surface conductance parameter, the explicit form of which was given above. The primary electroviscous coefficient is calculated from the expresion (9) p = 6 5(κa) 3 C N+1 1, [14] where C N+1 is an asymptotic coefficient of the function F(x) F(x) C N+1 x 2 + C N+2 x 4, when (x κa) 1. [15] The problem we must solve is to obtain numerically the solutions ( F(x) and φ i (x)) of Eqs. [6] [8] with the boundary conditions [9, 11, 13] and to calculate the electroviscous coefficient using Eqs. [14] and [15]. For this purpose, we have used the O Brien and White method (28). The data used to generate the plots are the same as in (27) and (29) (see Table 1), except for the molar concentrations of ions, which is 10 5 M, given a κa value of We observe on the figures of the primay electroviscous coefficient that the magnitude of the effect is considerably greater in the low κa region, as would be expected. However, the relative influence of the DSL is lower, because the contribution of the diffuse part of the edl is very high. The p maximum presents at very high ζ value, and the primary electroviscous coefficient can be considered as a monotically increasing function on ζ potential under experimental conditions. Figure 1 shows the dependence of the primary electroviscous coefficient on the reduced ζ potential at a fixed κa value. As can be seen, when we introduce a DSL into the problem, there exists TABLE 1 Data Used for the Generation of the Plots Unless Explicitly Indicated in the Figures Particle radius a = 100 nm Temperature T = K Bulk viscosity η = cp Bulk relative permitivity ε r = Outer Stern-layer capacity C 2 = 130 µf/cm 2 Electrolyte molar concentration c = 10 5 mol/l Limiting conductance of counterion K + 0 K + = cm 2 /(ohm gr.equiv.) Limiting conductance of coion Cl 0 Cl = cm 2 /(ohm gr.equiv.) pk of dissociation constant for pk + = 2 counterion pk of dissociation constant for coion pk = 1 λ Counterion drag coefficient ratio + λ t + (β 1) = 1 λ Coion drag coefficient ratio λ t (β 1) = 1 Maximum counterion charge in the en + =80 µc/cm 2 Stern layer Maximum coion charge in the en =80 µc/cm 2 Stern layer a separate dependence on the particle radius and on the bulk concentrations of ions (through the Debye length κ 1 ), instead of the product κa only, which is the case for an inmobile surface layer. This result is straightforward from the dependence of δ i parameter (Eqs. [4] and [5]) on the particle radius a and on n i. A study on the influence of the different Stern-layer parameters has been made. Considering Eq. [13], we observe that when δ i tends to zero, the mobile surface-layer problem reduces to the immobile surface-layer one. This occurs under the following conditions: (i) when few Stern-layer sites are available for adsorption (i.e., en i 0). Figure 2 shows the effect on p, as a function of ζ, of the counterion Stern-site density. The effect of increasing counterion site density is to increase the number of Stern-layer ions and consequently to increase the number of FIG. 1. Primary electroviscous coefficient as a function of reduced ζ potential for κa = 1.04 and different values of particle radius: (a) no surf. cond., (b) 5, (c) 2 (d) 10 (e) 250 nm.

4 PRIMARY ELECTROVISCOUS EFFECT 183 FIG. 2. Primary electroviscous coefficient as a function of reduced ζ potential for different values of counterion site density. Values of en + : (a) no surf. cond., (b) (c) 5, (d) 15, (e) 80 µc/cm 2. FIG. 4. Primary electroviscous coefficient as a function of reduced ζ potential for different values of counterion dissociation constant. Values of pk + : (a) no dynamic Stern layer, (b) 1, (c) (d) 1, (e) 2. mobile conducting ions. Thus, we observe that p decreases as N + increases and that the maximum in p moves to higher values, broadens, and eventually dissappears. (ii) when the Sternlayer drag coefficients are large compared to their bulk solution value (i.e., λ i /λ t i (β 1) 0). Effectively, Fig. 3 illustrates the effect on the primary electroviscous coefficient, as the ratio of counterion drag coefficient in the diffuse layer to that in the Stern layer is varied. As the ratio decreases, the Stern-layer counterion drag coefficient becomes large and the corresponding Stern-layer ionic mobility reduced, causing the Stern-layer ions contribution to decrease. Thus, primary electroviscous coefficient increases. (iii) when there is weak Stern-layer ion binding (i.e., K i is large). Figure 4 illustrates the variation of the primary electroviscous coefficient with ζ -potential, for a range of counterion dissociation constants. On decreasing the counterion dissociation constant, the p maximum shifts to lower ζ potentials. Increasing the binding strength of the counterion to the surface site will increase the number of ions in the Stern layer, becoming saturated at low ζ potentials. (iv) when large ζ values lead to a saturated Stern layer and diffuse-layer excess. In this case the curves for mobile Stern-layer problem should converge to those of the immobile Stern-layer problem. This behavior presents at very large ζ values as 18, as can be seen in the figures. Finally, when the Stern-layer coion parameters are varied the influence on the curves corresponding to the mobile surfacelayer model are negligible, as can be seen in Fig. 5, possibly because very few coions are adsorbed onto the Stern layer. FIG. 3. Primary electroviscous coefficient as a function of reduced ζ potential for different values of counterion drag ratio: (a) no dynamic Stern layer, (b) (c) 0.5, (d) 1. FIG. 5. Primary electroviscous coefficient as a function of reduced ζ potential for different values of coion site density. Values of en + : (a) no dynamic Stern layer, (b) (c) 5, (d) 15, (e) 80 µc/cm 2.

5 184 RUBIO-HERNÁNDEZ, RUIZ-REINA, AND GÓMEZ-MERINO ACKNOWLEDGMENTS Financial support by CICYT PB (Ministerio de Educación y Cultura, Spain) and DGUI FQM-0231 (Junta de Andalucía, Spain) is gratefully acknowledged. E.R.R. expresses his gratitude to the Junta de Andalucía for the conceded FPD-96 grant. REFERENCES 1. Einstein, A., Ann. Physik 19, 289 (1906); 34, 591 (1911). 2. Von Smoluchowski, M., Kolloid Z. 18, 194 (1916). 3. Conway, B. E., and Dobry-Duclaux, A., in Rhelogy, Theory and Applications (F. Eirich, Ed.), Vol. 3. Academic Press, New York, Krasny-Ergen, W., Kolloid Z. 74, 172 (1936). 5. Finkelstein, B. N., and Chursin, M. P., Acta Physicochim. URSS 17, 1 (1942). 6. Booth, F., Proc. R. Soc. A 203, 533 (1950). 7. Booth, F., Trans. Faraday Soc. 44, 955 (1948). 8. Russel, W. B., J. Fluid Mech. 85, 673 (1978). 9. Watterson, I. G., and White, L. R., J. Chem. Soc., Faraday Trans. 2 77, 1115 (1981). 10. Meijer, A. E. J., van Megen, W. J., and Lyklema, J., J. Colloid Interface Sci. 66, 99 (1978). 11. Baran, A. A., Dudkina, L. M., Soboleva, N. M., and Chechik, O. S., Kolloid Zh. 43, 211 (1981). 12. Van der Put, A. G., and Bijsterbosch, B. H., J. Colloid Interface Sci. 92, 499 (1983). 13. Goff, J. R., and Luner, P., J. Colloid Interface Sci. 99, 468 (1984). 14. Midmore, B. R., and Hunter, R. J., J. Colloid Interface Sci. 122, 521 (1988). 15. Chow, R. S., and Takamura, K., J. Colloid Interface Sci. 125, 226 (1988). 16. Elimelech, M., and O Melia, Ch., Colloids Surf. 44, 165 (1990). 17. Rubio-Hernández, F. J., J. Non-Equilib. Thermodyn. 21, 30 (1996). 18. Zukoski, C. F., and Saville, D. A., J. Colloid Interface Sci. 107, 322 (1985). 19. Zukoski, C. F., and Saville, D. A., J. Colloid Interface Sci. 114, 32 (1986). 20. Zukoski, C. F., and Saville, D. A., J. Colloid Interface Sci. 114, 45 (1986). 21. Van der Linde, A. J., and Bijsterbosch, B. H., Croatia Chim. Acta 63, 455 (1990). 22. Dukhin, S. S., and Semenikhin, N. M., Kolloid Zh 32, 366 (1970). 23. Lyklema, J., Colloids Surfaces 92, 41 (1994). 24. Dukhin, S. S., Adv. Colloid Interface Sci. 61, 17 (1995). 25. Matsumura, H., and Dukhin, S. S., Bull. Electrotech. Lab (1996). 26. Stern, O., Z. Elektrochem (1924). 27. Mangelsdorf, C. S., and White, L. R., J. Chem. Soc. Faraday Trans. 86, 2859 (1990). 28. O Brien, R. W., and White, L. R., J. Chem. Soc. Faraday Trans. 2 74, 1607 (1978). 29. Rubio-Hernández, F. J., Ruiz-Reina, E., and Gómez-Merino, A. I., J. Colloid Interface Sci. 106, 334 (1998).

An experimental study on the influence of a dynamic Stern-layer on the primary electroviscous effect

Colloids and Surfaces A: Physicochemical and Engineering Aspects 159 (1999) 373 379 www.elsevier.nl/locate/colsurfa An experimental study on the influence of a dynamic Stern-layer on the primary electroviscous