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The Influence of a Dynamic Stern Layer on the Primary Electroviscous Effect
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
206, 334 –337 (1998)
CS985677
NOTE The Influence of a Dynamic Stern Layer on the Primary Electroviscous Effect The theory developed by Watterson and White (Watterson, I. G., and White, L. R., J. Chem. Soc., Faraday Trans. 2 77, 1115 (1981)) to calculate the primary electroviscous coefficient of a suspension of charged spherical colloidal particles has been extended, by considering the presence of a dynamic Stern layer onto the particle surface, following the method developed by Mangelsdorf and White (Mangelsdorf, C. S., and White, L. R., J. Chem. Soc. Faraday Trans. 80, 2859 (1990)) for electrophoresis. The presence of mobile ions causes the primary electroviscous coefficient to decrease compared to when the Stern layer ions are inmobile. A separate dependence of the primary electroviscous coefficient on k21 (Debye length) and a (particle radius) has been found. © 1998 Academic Press Key Words: electroviscous effect; dynamic Stern layer.
1. INTRODUCTION The viscosity h of a colloidal suspension is greater than that of the suspending fluid h0. At very low particle concentrations, when interactions between them can be neglected, 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 are 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,
F
h 5 h0 1 1
G
5 f , 2
[1]
where f is the volume-per-volume fraction of the particles. It is interesting to note that there is no effect of the 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 Von Smoluchowski (2) and is called the primary electroviscous effect (3). We can write
F
h 5 h0 1 1
G
5 ~1 1 p! , 2
[2]
The edl extends a distance k21 (Debye length) from the surface of the particle,
k2 5
O N
n `i z 2i ,
[3]
i51
where e is the elementary charge, «0 the vacuum permittivity, «r the dielectric constant of the liquid medium, k the Boltzmann constant, T the absolute temperature, zi the valence, and n` i the bulk number density of the ith ionic species (i 5 1, . . . , N). The perturbation in the flow field around an uncharged particle has the characteristic dimension of the particle radius a. When the ratio ka of particle radius 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 ka where the edl thickness is large compared with the 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 to ka . 10. Later, Booth (6) derived an expression for the primary electroviscous coefficient, following his own treatment of electrophoresis (7), valid for all ka values. However, Booth’s theory is restricted to small values of z-potential and small Peclet numbers (8). For small Peclet numbers the edl is only slightly distorted from its equilibrium shape. Russel (9) extended Booth’s analysis to larger values of Peclet numbers (Pe ! ka with ka @ 1), but his theory was still restricted to small z-potentials. The most recent theory on the primary electroviscous effect has been elaborated by Watterson and White (10), and it is valid for all ka and z-values. They solved numerically the equations that govern the phenomenon and concluded that a maximum in p appeared at very high z-potentials, being more marked at low ka values. The authors pointed out the convenience of accomplishing experimental work “in order to observe this maximum, which serve as a severe test of the fundamental equations underlying the theories of dynamical processes in colloidal systems.” However they recognized that “the rather large values of z where the maximum occurs may render the experimental observations of p maximum difficult.” On the other hand, they did not try to explain this maximum.
2. THEORY The Watterson-White theory considered an insolated spherical particle immersed in an infinite electrolyte solution composed of N ionic species of charge zie, bulk number density n` i , and drag coefficient li. In the absence of any movement, the particle is considered having a radial surface charge distribution s0 and a corresponding z-potential at the slipping plane, from which the laws of continuum mechanics hold (10). The dimensionless equations,
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 z-potential) and the properties of the electrolyte. 0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
4pe2 « 0« rkT
334
L 4F˜ 5
22 x2
S DO ˜0 dC dx
N
i51
˜
0 ˜i c˜izie2ziC f
[4]
335
NOTE and
˜ i 5 zi L 2f
S DF ˜0 dC dx
G
d f˜ i 1 l˜ i~ x 2 3F˜ ! , dx
[5]
where the differential operators L4 and L2 are defined in (10), were numerically solved by these authors with the boundary conditions F˜~x! O ¡ 0, x3`
x
dF˜~ x! O ¡ 0, dx x 3 `
f˜ i ~ x! O ¡ 0,
F˜~ x!?x5 k a 5
d f˜ i ~ x! dx
x3`
U
x5ka
U
ka dF˜~ x! , 3 dx
50
x5 k a
5
~i 5 1, . . . , N!
1 3
[6]
[7]
following the O’Brien and White method (11). The primary electroviscous coefficient finally obtained was p5
6 C˜ 2 1, 5~ k a! 3 N11
[8]
f˜ i ~ x! O ¡ 0, x3`
where C˜N11 is an asymptotic coefficient of the function F˜(x), F˜ ~ x! 5
apply to the electrophoresis theory developed by O’Brien and White (11) 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 compared to when they are immobile. Considering that the fundamental equations that are the basis of the theory of electrophoresis are the same used to describe the primary electroviscous effect, we have accomplished a treatment for this phenomenon following Mangelsdorf and White. An appropriate model of the region behind the slipping plane is due to Stern (29). According to the model, this region is divided into the inner Stern layer, where no charge density can exist and extends to a distance b1 from the particle surface, and into the outer Stern layer, where the ions have lost part of their hydration sheath and moved closer to the particle surface as a result of some strong surface interaction, which extends from b1 to a distance b1 1 b2 from the particle surface. The original theory developed by Watterson and White (10) assumed that these Stern layer ions are immobile. Assuming a thin Stern layer, modified boundary conditions in the slipping plane can be obtained,
U
df˜ i ~ x! dx
x5k a
2
U
6di f˜ ~ x! ka i
x5k a
50
~i 5 1, . . . , N!.
[12]
The quantity di is the surface conductivity parameter which encapsulates the effect of allowing i ions to move behind the slipping plane,
˜ N12 ˜ N11 C C . 2 1 x x4
[9]
di 5
S D
s s0i li z ie z exp kT az ien `i l ti~ b 1!
.
[13]
We have introduced the following dimensionless groups, c˜ i 5
c iz i
O
l˜ i 5
N
c jz
« 0« rkT l h 0z ie 2 i
˜ 05 C
eC 0 kT
z˜ 5
ez kT
[10]
2 j
j51
O N
eN A x 5 kr
F˜ 5 k F
f˜ i 5
i51
h0
c iz i
f i,
Details on the process to obtain conditions [12], based upon evaluating the i ions flow through the slipping plane, can be found in Ref. (28). On the other hand, conditions [6] are the same. The value of di depends on the Stern layer adsorption isotherm that we use through the parameter sS0i i.e., the Stern layer charge density due to ith ionic species. In this paper we have studied only one possibility that considers adsorption of ions onto available surface area. The description of this adsorption isotherm can be found in Ref. (28). The expression for sS0i is
[11]
where NA is Avogadro’s number, ci is the molar concentration of i ion, C is the potential distribution around the colloidal particle at equilibrium, F(r) and fi(r) are functions defined in Ref. (10), and r is the distance from the center of the particle. 0
3. RESULTS AND DISCUSSION As has been pointed out, p from Eq. [8] plotted against z shows a maximum for every ka value. In their theory of electrophoresis, O’Brien and White (11) found a similar result plotting electrophoretic mobility against z. Many papers on the electrokinetics of polystyrene latexes (12–19) have proven that the standard electrokinetic model cannot explain the observed behavior. Zukoski and Saville (20) presented extensive experimental results on the electrophoretic mobility and electrical conductivity and found that the z-potentials inferred from the suspension conductivity were larger than those derived from electrophoresis. They developed (21) a dynamic Stern layer model in an attempt to reconcile the differences observed and concluded that Stern layer transport could account for the discrepancies (22). Recently (23) another explanation based upon the influence of a dynamic Stern layer mechanism into the edl has been proposed, with good results (19). Although the notion of a dynamic Stern layer has been a part of colloid science for many years (24), only recently has the real importance of this phenomenon been pointed out (25–27). The existence of edl involves the local presence of excess charge that may move tangentially within the Stern layer. The convincing evidence for lateral ionic mobility into the Stern layer led Mangelsdorf and White (28) to
s s0i 5
F
n `i 2z ie 0 z ieN i exp C ~ b 1! Ki kT
OH F N
11
j51
G
n `j 2z je 0 exp C ~ b 1! Kj kT
GJ
,
[14]
where Ni is the total number of Stern layer sites per unit area and Ki is the dissociation constant in the Stern layer for i ions. The problem we must solve is to obtain the solutions (F˜(x) and f˜ i(x)) of Eqs. [4] and [5] with the boundary conditions [6] and [12], and to calculate the electroviscous coefficient using Eq. [8]. For this purpose, we have used O’Brien and White’s method (11) too. A self-contained FORTRAN subroutine has been written to calculate the primary electroviscous coefficient p with the inclusion of a mobile Stern layer. In the immobile surface-layer problem, graphs of the primary electroviscous coefficient as a function of z-potential reveal p-maximum (10), being the value of the maximum approximately proportional to 1/ka. This is, a priori, an unexpected result, and a tentative explanation of the maximum has been tried. However, due to the rather large values where the maximum occurs, the experimental observation is difficult. For a fixed z-potential the electric body force on the fluid contained into the edl is higher the lower the ka, therefore p takes larger values, as can be seen in Fig. 1. When a mobile Stern layer is considered (see Table 1 of Ref. (10) for the data used) the effect is lowered; this tendency is more important when ka slightly increases. Actually, the effects of Stern layer are negligible at very low ka because only relatively few ions are adsorbed into the Stern layer (see Fig. 2). At higher ka, the number of ions adsorbed into the Stern layer is higher.
336
NOTE
FIG. 1. Primary electroviscous coefficient as a function of ka (ez/kT 5 24.7). (a) No dynamic Stern layer, (b) dynamic Stern layer (a 5 250 nm).
FIG. 3. Primary electroviscous coefficient as a function of reduced z-potential (y 5 ez/kT), for ka 5 30 and different particle radius: (a) no dynamic Stern layer, (b) a 5 5 nm, (c) a 5 20 nm, (d) a 5 100 nm, (e) a 5 250 nm, (f) a 5 500 nm.
Consequently the electric body force on the diffuse layer should decrease and the primary electroviscous effect will be less important. When ka is further increased the Stern layer will reach saturation while the diffuse-layer charge density continues to rise, overshadowing the effects of the mobile Stern-layer ions, and the curve labeled b in Fig. 1 should converge to the corresponding curve labeled a in the same figure. Figure 3 shows the dependence of the primary electroviscous coefficient on the z-potential at a fixed ka value. As can be seen, when no dynamic Stern layer is considered, the curve reaches the known maximum at very high z values. When we introduce a dynamic Stern layer into the problem a new result is obtained: there exists a dependence on the particle radius and on the bulk concentrations of the ions (through k) separately, instead of the product ka only, which is the case for an immobile surface layer. This result is straightforward from the dependence of parameter di on the particle radius a and on n` i (Eq. [13]) and can be derived by dimensional analysis because new dimensionless groups appear, like Ni/(aKi), which measures the number of i-type ions that are adsorbed in their corresponding vacancies. However, the most important fact is that the p-maximum moves to higher z-potential values and tends to disappear when the particle radius increases. This behavior could be explained as follows. As the particle radius increases the available surface area for the adsorption of counterions in-
creases, according to the Stern layer adsorption isotherm considered, and the number of adsorbed counterions will be higher. Therefore, the influence of the Stern layer will be more important. In the case of the absence of Stern layer adsorption, for ka fixed, an increase of z-potential can accommodate more counterions on the edl and the distortion of the flow field is higher, causing p increases. But when z is greater than a certain value, the edl cannot accept more counterions because of the electrostatic repulsion between them. Consequently, the electrostatic body force increases and the total effect is to reduce the distortion of the flow into the edl. As a result, the primary electroviscous coefficient decreases from a certain z-potential value. We could say then that the edl “gets rigid,” and p tends to zero because the system approximates the Einstein Model (1). When the Stern layer transport is included in the model, the counterions can also accommodate into the Stern layer until it is saturated (see Fig. 4),), and consequently the p-maximum presents at higher z values. Elsewhere (30) we suggested that the primary electroviscous effect theory should incorporate the presence of a dynamic Stern layer. In this work this mechanism has been added to the Watterson-White theory and the numerical results have shown some interesting results that must be experimentally tested in order to check the theory. In a future work we will present an experimental research on this subject.
FIG. 2. Charge density as a function of ka (ez/kT 5 24.7). (a) Into the Stern layer (a 5 250 nm), (b) into the diffuse layer.
FIG. 4. Charge density as a function of reduced z-potential (y 5 ez/kT), for ka 5 32.86. (a) Into the Stern layer, (b) into the diffuse layer.
337
NOTE
ACKNOWLEDGMENTS Financial support by DGICYT PB95-0481 (Ministerio de Educacio´n y Cultura, Spain) and DGUI FQM-0231 (Junta de Andalucı´a, Spain) is gratefully acknowledged. E.R.R. expresses his gratitude to the Junta de Andalucı´a 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 “Rheology. Theory and Applications” (F. Eirich, Ed.), Vol. 3, Academic Press, New York, 1960. 4. Krasny-Ergen, W., Kolloid Z. 74, 172 (1936). 5. Finkelstein, B. N., and Chursin, M. P., Acta Physicochim. U.R.S.S. 17, 1 (1942). 6. Booth, F., Proc. Roy. Soc. A 203, 533 (1950). 7. Booth, F., Trans. Faraday. Soc. 44, 955 (1948). 8. Peclet number expresses the ratio between the hydrodynamic forces due to shear and the Brownian forces, which tend to restore the equilibrium configuration of the system, and, therefore, measures the extent to which the movement of the fluid, relative to the particle, disturbs the ionic atmosphere. 9. Russel, W. B., J. Fluid. Mech. 85, 673 (1978). 10. Watterson, I. G., and White, L. R., J. Chem. Soc., Faraday Trans. 2 77, 1115 (1981). 11. O’Brien, R. W., and White, L. R., J. Chem. Soc., Faraday Trans. 2 74, 1607 (1978). 12. Meijer, A. E. J., Van Megen, W. J., and Lyklema, J., J. Colloid Interface Sci. 66, 99 (1978). 13. Baran, A. A., Dudkina, L. M., Soboleva, N. M., and Chechik, O. S., Kolloid Z. 43, 211 (1981). 14. Van der Put, A. G., and Bijsterbosch, B. H., J. Colloid Interface Sci. 92, 499 (1983). 15. Goff, J. R., and Luner, P., J. Colloid Interface Sci. 99, 468 (1984).
16. Midmore, B. R., and Hunter, R. J., J. Colloid Interface Sci. 122, 521 (1988). 17. Chow, R. S., and Takamura, K., J. Colloid Interface Sci. 125, 226 (1988). 18. Elimelech, M., and O’Melia, Ch., Colloids Surf. 44, 165 (1990). 19. Rubio-Herna´ndez, F. J., J. Non-Equilib. Thermodyn. 21, 30 (1996). 20. Zukoski, C. F., and Saville, D. A., J. Colloid Interface Sci. 107, 322 (1985). 21. Zukoski, C. F., and Saville, D. A., J. Colloid Interface Sci. 114, 32 (1986). 22. Zukoski, C. F., and Saville, D. A., J. Colloid Interface Sci. 114, 45 (1986). 23. Van der Linde, A. J., and Bijsterbosch, B. H., Croatica Chim. Acta 63, 455 (1990). 24. Dukhin, S. S., and Semenikhin, N. M., Kolloid Z. 32, 366 (1970). 25. Lyklema, J., Colloids Surf. 92, 41 (1994). 26. Dukhin, S. S., Adv. Colloid Interface Sci. 61, 17 (1995). 27. Matsumura, H., and Dukhin, S. S., Bull. Electrotech. Lab. 60, 31 (1996). 28. Mangelsdorf, C. S., and White, L. R., J. Chem. Soc. Faraday Trans. 86, 2859 (1990). 29. Stern, O., Z. Electrochem. 30, 508 (1924). 30. Rubio-Herna´ndez, F. J., Go´mez-Merino, A. I., Ruiz-Reina, E., and Carnero-Ruiz, C., Colloids Surf. A 140, 293 (1998). Francisco-Jose´ Rubio-Herna´ndez1 Emilio Ruiz-Reina Ana-Isabel Go´mez-Merino Departamento de Fı´sica Aplicada II Universidad de Ma´laga Ma´laga, Spain Received January 21, 1998; accepted May 22, 1998
1 To whom correspondence should be addressed at Departamento de Fı´sica Aplicada II, Universidad de Ma´laga, Campus de El Ejido, 29013-Ma´laga, Spain. E-mail: [email protected].
206, 334 –337 (1998)
CS985677
NOTE The Influence of a Dynamic Stern Layer on the Primary Electroviscous Effect The theory developed by Watterson and White (Watterson, I. G., and White, L. R., J. Chem. Soc., Faraday Trans. 2 77, 1115 (1981)) to calculate the primary electroviscous coefficient of a suspension of charged spherical colloidal particles has been extended, by considering the presence of a dynamic Stern layer onto the particle surface, following the method developed by Mangelsdorf and White (Mangelsdorf, C. S., and White, L. R., J. Chem. Soc. Faraday Trans. 80, 2859 (1990)) for electrophoresis. The presence of mobile ions causes the primary electroviscous coefficient to decrease compared to when the Stern layer ions are inmobile. A separate dependence of the primary electroviscous coefficient on k21 (Debye length) and a (particle radius) has been found. © 1998 Academic Press Key Words: electroviscous effect; dynamic Stern layer.
1. INTRODUCTION The viscosity h of a colloidal suspension is greater than that of the suspending fluid h0. At very low particle concentrations, when interactions between them can be neglected, 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 are 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,
F
h 5 h0 1 1
G
5 f , 2
[1]
where f is the volume-per-volume fraction of the particles. It is interesting to note that there is no effect of the 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 Von Smoluchowski (2) and is called the primary electroviscous effect (3). We can write
F
h 5 h0 1 1
G
5 ~1 1 p! , 2
[2]
The edl extends a distance k21 (Debye length) from the surface of the particle,
k2 5
O N
n `i z 2i ,
[3]
i51
where e is the elementary charge, «0 the vacuum permittivity, «r the dielectric constant of the liquid medium, k the Boltzmann constant, T the absolute temperature, zi the valence, and n` i the bulk number density of the ith ionic species (i 5 1, . . . , N). The perturbation in the flow field around an uncharged particle has the characteristic dimension of the particle radius a. When the ratio ka of particle radius 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 ka where the edl thickness is large compared with the 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 to ka . 10. Later, Booth (6) derived an expression for the primary electroviscous coefficient, following his own treatment of electrophoresis (7), valid for all ka values. However, Booth’s theory is restricted to small values of z-potential and small Peclet numbers (8). For small Peclet numbers the edl is only slightly distorted from its equilibrium shape. Russel (9) extended Booth’s analysis to larger values of Peclet numbers (Pe ! ka with ka @ 1), but his theory was still restricted to small z-potentials. The most recent theory on the primary electroviscous effect has been elaborated by Watterson and White (10), and it is valid for all ka and z-values. They solved numerically the equations that govern the phenomenon and concluded that a maximum in p appeared at very high z-potentials, being more marked at low ka values. The authors pointed out the convenience of accomplishing experimental work “in order to observe this maximum, which serve as a severe test of the fundamental equations underlying the theories of dynamical processes in colloidal systems.” However they recognized that “the rather large values of z where the maximum occurs may render the experimental observations of p maximum difficult.” On the other hand, they did not try to explain this maximum.
2. THEORY The Watterson-White theory considered an insolated spherical particle immersed in an infinite electrolyte solution composed of N ionic species of charge zie, bulk number density n` i , and drag coefficient li. In the absence of any movement, the particle is considered having a radial surface charge distribution s0 and a corresponding z-potential at the slipping plane, from which the laws of continuum mechanics hold (10). The dimensionless equations,
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 z-potential) and the properties of the electrolyte. 0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
4pe2 « 0« rkT
334
L 4F˜ 5
22 x2
S DO ˜0 dC dx
N
i51
˜
0 ˜i c˜izie2ziC f
[4]
335
NOTE and
˜ i 5 zi L 2f
S DF ˜0 dC dx
G
d f˜ i 1 l˜ i~ x 2 3F˜ ! , dx
[5]
where the differential operators L4 and L2 are defined in (10), were numerically solved by these authors with the boundary conditions F˜~x! O ¡ 0, x3`
x
dF˜~ x! O ¡ 0, dx x 3 `
f˜ i ~ x! O ¡ 0,
F˜~ x!?x5 k a 5
d f˜ i ~ x! dx
x3`
U
x5ka
U
ka dF˜~ x! , 3 dx
50
x5 k a
5
~i 5 1, . . . , N!
1 3
[6]
[7]
following the O’Brien and White method (11). The primary electroviscous coefficient finally obtained was p5
6 C˜ 2 1, 5~ k a! 3 N11
[8]
f˜ i ~ x! O ¡ 0, x3`
where C˜N11 is an asymptotic coefficient of the function F˜(x), F˜ ~ x! 5
apply to the electrophoresis theory developed by O’Brien and White (11) 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 compared to when they are immobile. Considering that the fundamental equations that are the basis of the theory of electrophoresis are the same used to describe the primary electroviscous effect, we have accomplished a treatment for this phenomenon following Mangelsdorf and White. An appropriate model of the region behind the slipping plane is due to Stern (29). According to the model, this region is divided into the inner Stern layer, where no charge density can exist and extends to a distance b1 from the particle surface, and into the outer Stern layer, where the ions have lost part of their hydration sheath and moved closer to the particle surface as a result of some strong surface interaction, which extends from b1 to a distance b1 1 b2 from the particle surface. The original theory developed by Watterson and White (10) assumed that these Stern layer ions are immobile. Assuming a thin Stern layer, modified boundary conditions in the slipping plane can be obtained,
U
df˜ i ~ x! dx
x5k a
2
U
6di f˜ ~ x! ka i
x5k a
50
~i 5 1, . . . , N!.
[12]
The quantity di is the surface conductivity parameter which encapsulates the effect of allowing i ions to move behind the slipping plane,
˜ N12 ˜ N11 C C . 2 1 x x4
[9]
di 5
S D
s s0i li z ie z exp kT az ien `i l ti~ b 1!
.
[13]
We have introduced the following dimensionless groups, c˜ i 5
c iz i
O
l˜ i 5
N
c jz
« 0« rkT l h 0z ie 2 i
˜ 05 C
eC 0 kT
z˜ 5
ez kT
[10]
2 j
j51
O N
eN A x 5 kr
F˜ 5 k F
f˜ i 5
i51
h0
c iz i
f i,
Details on the process to obtain conditions [12], based upon evaluating the i ions flow through the slipping plane, can be found in Ref. (28). On the other hand, conditions [6] are the same. The value of di depends on the Stern layer adsorption isotherm that we use through the parameter sS0i i.e., the Stern layer charge density due to ith ionic species. In this paper we have studied only one possibility that considers adsorption of ions onto available surface area. The description of this adsorption isotherm can be found in Ref. (28). The expression for sS0i is
[11]
where NA is Avogadro’s number, ci is the molar concentration of i ion, C is the potential distribution around the colloidal particle at equilibrium, F(r) and fi(r) are functions defined in Ref. (10), and r is the distance from the center of the particle. 0
3. RESULTS AND DISCUSSION As has been pointed out, p from Eq. [8] plotted against z shows a maximum for every ka value. In their theory of electrophoresis, O’Brien and White (11) found a similar result plotting electrophoretic mobility against z. Many papers on the electrokinetics of polystyrene latexes (12–19) have proven that the standard electrokinetic model cannot explain the observed behavior. Zukoski and Saville (20) presented extensive experimental results on the electrophoretic mobility and electrical conductivity and found that the z-potentials inferred from the suspension conductivity were larger than those derived from electrophoresis. They developed (21) a dynamic Stern layer model in an attempt to reconcile the differences observed and concluded that Stern layer transport could account for the discrepancies (22). Recently (23) another explanation based upon the influence of a dynamic Stern layer mechanism into the edl has been proposed, with good results (19). Although the notion of a dynamic Stern layer has been a part of colloid science for many years (24), only recently has the real importance of this phenomenon been pointed out (25–27). The existence of edl involves the local presence of excess charge that may move tangentially within the Stern layer. The convincing evidence for lateral ionic mobility into the Stern layer led Mangelsdorf and White (28) to
s s0i 5
F
n `i 2z ie 0 z ieN i exp C ~ b 1! Ki kT
OH F N
11
j51
G
n `j 2z je 0 exp C ~ b 1! Kj kT
GJ
,
[14]
where Ni is the total number of Stern layer sites per unit area and Ki is the dissociation constant in the Stern layer for i ions. The problem we must solve is to obtain the solutions (F˜(x) and f˜ i(x)) of Eqs. [4] and [5] with the boundary conditions [6] and [12], and to calculate the electroviscous coefficient using Eq. [8]. For this purpose, we have used O’Brien and White’s method (11) too. A self-contained FORTRAN subroutine has been written to calculate the primary electroviscous coefficient p with the inclusion of a mobile Stern layer. In the immobile surface-layer problem, graphs of the primary electroviscous coefficient as a function of z-potential reveal p-maximum (10), being the value of the maximum approximately proportional to 1/ka. This is, a priori, an unexpected result, and a tentative explanation of the maximum has been tried. However, due to the rather large values where the maximum occurs, the experimental observation is difficult. For a fixed z-potential the electric body force on the fluid contained into the edl is higher the lower the ka, therefore p takes larger values, as can be seen in Fig. 1. When a mobile Stern layer is considered (see Table 1 of Ref. (10) for the data used) the effect is lowered; this tendency is more important when ka slightly increases. Actually, the effects of Stern layer are negligible at very low ka because only relatively few ions are adsorbed into the Stern layer (see Fig. 2). At higher ka, the number of ions adsorbed into the Stern layer is higher.
336
NOTE
FIG. 1. Primary electroviscous coefficient as a function of ka (ez/kT 5 24.7). (a) No dynamic Stern layer, (b) dynamic Stern layer (a 5 250 nm).
FIG. 3. Primary electroviscous coefficient as a function of reduced z-potential (y 5 ez/kT), for ka 5 30 and different particle radius: (a) no dynamic Stern layer, (b) a 5 5 nm, (c) a 5 20 nm, (d) a 5 100 nm, (e) a 5 250 nm, (f) a 5 500 nm.
Consequently the electric body force on the diffuse layer should decrease and the primary electroviscous effect will be less important. When ka is further increased the Stern layer will reach saturation while the diffuse-layer charge density continues to rise, overshadowing the effects of the mobile Stern-layer ions, and the curve labeled b in Fig. 1 should converge to the corresponding curve labeled a in the same figure. Figure 3 shows the dependence of the primary electroviscous coefficient on the z-potential at a fixed ka value. As can be seen, when no dynamic Stern layer is considered, the curve reaches the known maximum at very high z values. When we introduce a dynamic Stern layer into the problem a new result is obtained: there exists a dependence on the particle radius and on the bulk concentrations of the ions (through k) separately, instead of the product ka only, which is the case for an immobile surface layer. This result is straightforward from the dependence of parameter di on the particle radius a and on n` i (Eq. [13]) and can be derived by dimensional analysis because new dimensionless groups appear, like Ni/(aKi), which measures the number of i-type ions that are adsorbed in their corresponding vacancies. However, the most important fact is that the p-maximum moves to higher z-potential values and tends to disappear when the particle radius increases. This behavior could be explained as follows. As the particle radius increases the available surface area for the adsorption of counterions in-
creases, according to the Stern layer adsorption isotherm considered, and the number of adsorbed counterions will be higher. Therefore, the influence of the Stern layer will be more important. In the case of the absence of Stern layer adsorption, for ka fixed, an increase of z-potential can accommodate more counterions on the edl and the distortion of the flow field is higher, causing p increases. But when z is greater than a certain value, the edl cannot accept more counterions because of the electrostatic repulsion between them. Consequently, the electrostatic body force increases and the total effect is to reduce the distortion of the flow into the edl. As a result, the primary electroviscous coefficient decreases from a certain z-potential value. We could say then that the edl “gets rigid,” and p tends to zero because the system approximates the Einstein Model (1). When the Stern layer transport is included in the model, the counterions can also accommodate into the Stern layer until it is saturated (see Fig. 4),), and consequently the p-maximum presents at higher z values. Elsewhere (30) we suggested that the primary electroviscous effect theory should incorporate the presence of a dynamic Stern layer. In this work this mechanism has been added to the Watterson-White theory and the numerical results have shown some interesting results that must be experimentally tested in order to check the theory. In a future work we will present an experimental research on this subject.
FIG. 2. Charge density as a function of ka (ez/kT 5 24.7). (a) Into the Stern layer (a 5 250 nm), (b) into the diffuse layer.
FIG. 4. Charge density as a function of reduced z-potential (y 5 ez/kT), for ka 5 32.86. (a) Into the Stern layer, (b) into the diffuse layer.
337
NOTE
ACKNOWLEDGMENTS Financial support by DGICYT PB95-0481 (Ministerio de Educacio´n y Cultura, Spain) and DGUI FQM-0231 (Junta de Andalucı´a, Spain) is gratefully acknowledged. E.R.R. expresses his gratitude to the Junta de Andalucı´a for the conceded FPD-96 grant.
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1 To whom correspondence should be addressed at Departamento de Fı´sica Aplicada II, Universidad de Ma´laga, Campus de El Ejido, 29013-Ma´laga, Spain. E-mail: [email protected].