Langmuir 2007, 23, 12061-12066
12061
Primary Electroviscous Effect in a Moderately Concentrated Suspension of Charged Spherical Colloidal Particles Hiroyuki Ohshima* Faculty of Pharmaceutical Sciences and Institute of Colloid and Interface Science, Tokyo UniVersity of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan ReceiVed June 14, 2007. In Final Form: September 16, 2007 On the basis of the standard theory of the primary electroviscous effect in a moderately concentrated suspension of charged spherical particles in an electrolyte solution presented by Ruiz-Reina et al. (Ruiz-Reina, E.; Carrique, F.; Rubio-Herna´ndez, F. J.; Go´mez-Merino, A. I.; Garcı´a-Sa´nchez, P. J. Phys. Chem. B 2003, 107, 9528), which is applicable for the case where overlapping of the electrical double layers of adjacent particles can be neglected, the general expression for the effective viscosity or the primary electroviscous coefficient p of the suspension is derived. This expression is applicable for a suspension of spherical particles of radius a carrying arbitrary zeta potentials ζ at the particle volume fraction φ e 0.3 for the case of nonoverlapping double layers, that is, at κa > 10 (where κ is the Debye-Hu¨ckel parameter). A simple approximate analytic expression for p applicable for particles with large κa and arbitrary ζ is presented. The obtained viscosity expression is a good approximation for moderately concentrated suspensions of the particle volume fraction φ e 0.3, where the relative error is negligible for κa g100 and even at κa ) 50 the maximum error is ∼20%. It is shown that a maximum of p, which appears when plotted as a function of the particle zeta potential, is due to the relaxation effect as in the case of the electrophoresis problem.
Introduction Einstein1 showed that an effective viscosity ηs of a dilute suspension of uncharged spherical colloidal particles in a liquid is greater than the viscosity η of the original liquid and derived the following expression for ηs:
5 ηs ) η 1 + φ 2
(
)
(1)
where φ is the particle volume fraction. If the liquid contains an electrolyte and the particles are charged, the effective viscosity ηs is further increased due to the presence of the electrical double layer around the particles. This phenomenon, when the double layer interactions between the particles are neglected, is called the primary electroviscous effect,2-10 and the effective viscosity ηs is expressed as
{
5 ηs ) η 1 + (1 + p)φ 2
}
(2)
where p is the primary electroviscous coefficient. The standard theory for this effect and the governing electrokinetic equations for a dilute suspension of charged spherical particles were given by Watterson and White,8 who also provided numerical solutions of the electrokinetic equations. This theory predicts a maximum of p when plotted as a function of the particle zeta potential. Rubio-Herna´ndez et al.11,12,14,15 and Sherwood et al.13 pointed * Telephone and fax: +81-4-7121-3661. E-mail:
[email protected]. tus.ac.jp. (1) Einstein, A. Ann. Phys. 1906, 19, 289. (2) Smoluchowski, M. Kolloid- Z. 1916, 18, 194. (3) Krasny-Ergen, W. Kolloid-Z. 1936, 74, 172. (4) Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 533. (5) Russel, W. B. J. Fluid Mech. 1978, 85, 673. (6) Lever, D. A. J. Fluid Mech. 1979, 92, 421. (7) Sherwood, J. D. J. Fluid Mech. 1980, 101, 609. (8) Watterson, I. G.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 1115. (9) Hinch, E. J.; Sherwood, J. D. J. Fluid Mech. 1983, 132, 337. (10) Duhkin, A. S.; van de Ven, T. G. M. J. Colloid Interface Sci. 1993, 158, 85.
out the importance of surface conductance16-18 in the electroviscous effect. The effective viscosity of a suspension of particles of types other than rigid particles has also been theoretically investigated. Taylor19 proposed a theory of the effective viscosity of a suspension of uncharged liquid drops. This theory has recently been extended to the case of charged liquid drops by Ohshima,20 who also derived approximate analytic expressions for p for the case of rigid particles (see also ref 21). Ohshima’s theory20 was also able to reproduce the maximum of p predicted by the theory of Watterson and White.8 Garcı´a-Salinas and de las Nieves22 presented a theory which predicts a maximum of p when plotted as a function of κa, where κ is the Debye-Hu¨ckel parameter and a is the particle radius. They also studied the influence of counterion type and diffusion on the primary electroviscous effect.23 Natraj and Chen24 developed a theory for charged porous spheres, and Allison et al.25 and Allison and Xin26 discussed the case of polyelectrolyte-coated particles. (11) Rubio-Herna´ndez, F. J.; Ruiz-Reina, E.; Go´mez-Merino, A. I. J. Colloid Interface Sci. 1998, 206, 334. (12) Rubio-Herna´ndez, F. J.; Ruiz-Reina, E.; Go´mez-Merino, A. I. J. Colloid Interface Sci. 2000, 226, 180. (13) Sherwood, J. D.; Rubio-Hernndez, F. J.; Ruiz-Reina, E. J. Colloid Interface Sci. 2000, 228, 7. (14) Rubio-Herna´ndez, F. J.; Ruiz-Reina, E.; Go´mez-Merino, A. I. Colloids Surf., A 2001, 192, 349. (15) Rubio-Herna´ndez, F. J.; Ruiz-Reina, E.; Go´mez-Merino, A. I.; Sherwood, J. D. Rheol. Acta 2001, 40, 230. (16) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7, Chapter 2. (17) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1986, 114, 32. (18) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1986, 114, 45. (19) Taylor, G. I. Proc. R. Soc. London, Ser. A 1932, 138, 41. (20) Ohshima, H. Langmuir 2006, 22, 2863. (21) Ohshima, H. Theory of Colloid and Interfacial Electric Phenomena; Elsevier: Amsterdam, 2006. (22) Garcı´a-Salinas, M. J.; de las Nieves, F. J. Langmuir 2000, 16, 7150. (23) Garcı´a-Salinas, M. J.; de las Nieves, F. J. Colloids Surf., A 2003, 222, 65. (24) Natraj, V.; Chen, S. B. J. Colloid Interface Sci. 2002, 251, 200. (25) Allison, S.; Wall, S.; Rasmusson, M. J. Colloid Interface Sci. 2003, 263, 84. (26) Allison, S.; Xin, Y. J. Colloid Interface Sci. 2006, 299, 977.
10.1021/la701768a CCC: $37.00 © 2007 American Chemical Society Published on Web 10/25/2007
12062 Langmuir, Vol. 23, No. 24, 2007
Ohshima
For concentrated suspensions, hydrodynamic interactions among particles must be considered. The hydrodynamic interactions between spherical particles can be taken into account by means of Happel’s cell model.27 This model assumes that each sphere of radius a is surrounded by a virtual shell of outer radius b and the particle volume fraction φ is given by
φ ) (a/b)3
(3)
There are two different boundary conditions to be satisfied at the outer cell surface: Happel’s27 and Simha’s28 boundary conditions. On the basis of Happel’s cell model (eq 3) with Simha’s boundary conditions (not with Happel’s boundary condition), the following equation for the effective viscosity ηs of a concentrated suspension of uncharged particles of volume fraction φ is derived:29
{
5 ηs ) η 1 + φ S(φ) 2
}
(4)
with
S(φ) ) )
4(1 - φ7/3) 4(1 + φ
) - 25φ(1 + φ ) + 42φ
10/3
4/3
4(1 - φ7/3) 4(1 - φ5/3)2 - 25φ(1 - φ2/3)2
5/3
(5)
Equation 4 was first derived by Simha,28 and S(φ) is called Simha’s function. As φ f 0, S(φ) tends to 1 so that eq 4 reduces back to Einstein’s eq 1. This is not true when Happel’s boundary condition is used. Zholkovskiy et al.30 discussed the difference between these two boundary conditions. Sherwood31 pointed out that if the mean shear rate is correctly evaluated, then Einstein’s result is obtained. The standard theory of the primary electroviscous effect has been extended to cover the case of concentrated suspensions of charged spherical particles with thin electrical double layers under the condition of nonoverlapping electrical double layers of adjacent particles by Ruiz-Reina et al.29 and Rubio-Herna´ndez et al.,32 who write
{
5 ηs ) η 1 + (1 + p)φ S(φ) 2
}
Figure 1. Cell model for a concentrated suspension of spherical particles. Each sphere of radius a is surrounded by a virtual shell of outer radius b. The particle volume fraction φ is given by φ ) (a/b)3.
Hu¨ckel parameter and a is the particle radius), we derive the general expression for the effective viscosity or the electroviscous coefficient p of a concentrated suspension of charged spherical colloidal particles and a simple approximate analytic expression for p applicable for particles with arbitrary zeta potentials at large κa for the case of nonoverlapping double layers.
Basic Equations Consider a suspension of charged spherical particles of radius a in an electrolyte solution (Figure 1). The origin of the spherical polar coordinate system (r, θ, φ) is held fixed at the center of one particle. Let the electrolyte be composed of N ionic mobile species of valence zi, bulk concentration (number density) n∞i , and drag coefficient λi (i ) 1, 2, ..., N). The drag coefficient λi of the ith ionic species is related to the limiting conductance Λ0i of that ionic species by
λi )
NAe2|zi| Λi0
(7)
(6)
where p is the primary electroviscous coefficient for a concentrated suspension. Later, Ruiz-Reina et al.33 considered the secondary electroviscous effect by taking into account the effects of overlapping of the electrical double layers of adjacent particles, and Carrique et al.34 studied the influence of a dynamic Stern layer for concentrated suspensions. In the present article, on the basis of the theory of the primary electroviscous effect of Ruiz-Reina et al.29 for moderately concentrated suspensions of charged spherical particles with thin electrical double layers, that is, large κa (where κ is the Debye(27) Happel, J. J. Appl. Phys. 1957, 28, 1288. (28) Simha, R. J. Appl. Phys. 1952, 23, 1020. (29) Ruiz-Reina, E.; Carrique, F.; Rubio-Herna´ndez, F. J.; Go´mez-Merino, A. I.; Garcı´a-Sa´nchez, P. J. Phys. Chem. B 2003, 107, 9528. (30) Zholkovskiy, E. K.; Adeyinka, O. B.; Masliyah, J. H. J. Phys. Chem. B 2006, 110, 19726. (31) Sherwood, J. D. J. Phys. Chem. B 2007, 111, 3370. (32) Rubio-Herna´ndez, F. J.; Carrique, F.; Ruiz-Reina, E. AdV. Colloid Interface Sci. 2004, 107, 51. (33) Ruiz-Reina, E.; Garcı´a-Sa´nchez, P.; Carrique, F. J. Phys. Chem. B 2005, 109, 5289. (34) Carrique, F.; Garcı´a-Sa´nchez, P.; Ruiz-Reina, E. J. Phys. Chem. B 2005, 109, 24369. (35) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Permagon: London, 1966.
where e is the elementary electric charge and NA is Avogadro’s number. The main assumptions in our analysis are as follows. (i) The Reynolds number of the liquid flow is small enough to ignore inertial terms in the Navier-Stokes equation, and the liquid can be regarded as incompressible. (ii) The applied shear field is weak so that the electrical double layer around the particle is only slightly distorted. (iii) The slipping plane (at which the liquid velocity relative to the particle becomes zero) is located on the particle surface. (iv) No electrolyte ions can penetrate the particle surface. (v) We treat the primary electroviscous effect, ignoring double layer interactions between the particles. This corresponds to the assumption of nonoverlapping electrical double layers. Imagine that a linear symmetric shear field u(0)(r) is applied to the system so that the velocity u(r) at position r outside the particle is given by the sum of u(0)(r) and a perturbation velocity. We can express u(0)(r) as
u(0) ) R‚r
(8)
where R is a symmetric traceless tensor so that u(0)(r) becomes irrotational, namely
Expression of the Primary ElectroViscous Effect
Langmuir, Vol. 23, No. 24, 2007 12063
∇ × u(0) ) 0
(9)
u(r) ) u(0)(r) at r ) b
(10)
Equation 21 follows from assumption (iii) that the slipping plane is located at r ) a. Equation 22 is Simha’s boundary condition,28 which means that the perturbation velocity field is zero at the outer cell surface. The boundary condition for the ionic flow vi(r) at r ) a is given by
Also, we have from assumption (i),
∇‚u(0) ) 0 ∇‚u ) 0
(11)
From symmetry considerations, u(r) takes the
form32
u(r) ) u (r) + ∇ × ∇ × [(R‚∇) f(r)] (0)
vi(r)‚n|r)a ) 0 (12)
where f(r) is a function of r (r ) |r|) and the second term on the right-hand side corresponds to the perturbation velocity. We introduce a function h(r), defined by
h(r) )
d 1 df dr r dr
( )
Linearized Equations For a weak shear field (assumption (ii)), one may write
(13)
h(r) r × u(0)(r) ) u(r) ) u (r) + ∇ × r dh 2h (0) 1 dh h - r(r‚u(0)) (14) 1- u + 2 dr r dr r r
ni(r) ) ni(0)(r) + δni(r)
(24)
ψ(r) ) ψ(0)(r) + δψ(r)
(25)
µi(r) ) µ(0) + δµi(r)
(26)
Fel(r) ) Fel(0)(r) + δFel(r)
(27)
(0)
)
(
)
where h(r) is a function of r only and we have used eq 9 (irrotational flow), eq 10 (incompressible flow), and (r‚∇)u(0) ) u(0) (linear flow). The basic equations for the liquid flow u(r) and the velocity vi(r) of the ith ionic species are given by
η∇ × ∇ × u(r) + ∇p(r) + Fel(r) ∇ ψ(r) ) 0
where the quantities with superscript (0) refer to those at equilibrium (i.e., in the absence of the shear field) and δni(r), δψ(r), δµi(r), and δFel(r) are perturbation quantities. These small quantities are related to each other by
δµi(r) ) zie δψ(r) + kT
(15)
1 vi(r) ) u(r) - ∇µi(r) λi
(16)
∇‚(ni(r) vi(r)) ) 0
(17)
∆δψ(r) ) -
µi(r) )
δFel(r)
1
)-
ro
δni(r) ni(0)(r) N
∑zie δni(r)
(18)
Fel(r) ro
(19)
∆ψ(r) ) N
Fel(r) )
zie ni(r) ∑ i)1
(20)
(29)
ro i)1
∂ δµ | ) 0 at r ) a ∂r i r)a + zie ψ(r) + kT ln ni(r)
(28)
Equation 23 as combined with eqs 16 and 21 yields
with
µ∞i
(23)
which follows from assumption (iv).
and then eq 12 can be rewritten as
(
(22)
(30)
It follows from assumption (v) that δψ(r) ) 0 and δni(r) ) 0 at the outer cell surface at r ) b. We thus have from eq 28
δµi(r) ) 0 at r ) b
(31)
With the help of eqs 24-27, the electrokinetic eqs 15-17 reduce to20 N
where eq 15 is the Navier-Stokes equation at low Reynolds numbers (Stokes equation), p(r) is the pressure outside the particle, Fel(r) is the charge density resulting from the mobile charged ionic species given by eq 20, ψ(r) is the electric potential, µi(r) and ni(r) are, respectively, the electrochemical potential and the concentration (number density) of the ith ionic species, µ∞i is a constant term in µi(r), r is the relative permittivity of the electrolyte solution, o is the permittivity of a vacuum, k is the Boltzmann constant, and T is the absolute temperature. Equation 16 expresses that the flow vi(r) of the ith ionic species is caused by the liquid flow u(r) and the gradient of the electrochemical potential µi(r), given by eq 18. Equation 17 is the continuity equation for the ith ionic species, and eq 19 is Poisson’s equation. The following boundary conditions for u on the particle surface r ) a and at the outer cell surface at r ) b must be satisfied.
u(r) ) 0 at r ) a
(21)
η∇ × ∇ × ∇ × u(r) )
∇δµi(r) × ∇ni(0)(r) ∑ i)1
(
)
1 ∇‚ ni(0)(r) u(r) - ni(0)(r) ∇δµi(r) ) 0 λi
(32)
(33)
We assume that the distribution of electrolyte ions at equilibrium ni(0)(r) obeys the Boltzmann equation and the equilibrium double layer potential ψ(0)(r) outside the particle satisfies the Poisson-Boltzmann equation, namely
[ ] N
1 d r2 dr
( ) r2
dy dr
κ2
)-
zin∞i exp(-zjy) ∑ i)1 N
zi2n∞i ∑ i)1
(34)
12064 Langmuir, Vol. 23, No. 24, 2007
with
κ)
(
N
1
∑ kT i)1
zi2e2n∞i
r o
y)
Ohshima
Equation 41 can formally be integrated to give
)
1/2
(35)
∫br
(0)
eψ kT
(36)
where κ is the Debye-Hu¨ckel parameter and y(r) is the scaled equilibrium potential outside the sphere. The boundary conditions for ψ(0)(r) are
ψ (a) ) ζ (0)
dψ(0) dr
|
)-
r)a+
dψ(0) dr
|
h(r) ) C1r3 + C2r +
r)b-
(37) σ ro
(38)
)0
(39)
where ζ is the zeta potential (assumption (ii)) and σ is the particle surface charge density. Equations 38 and 39 state that the unit cell as a whole is electrically neutral. Further, symmetry considerations permit us to write
φi(r) (0) r‚u (r) r2
δµi(r) ) -zie
(40)
L
() [
φi dy zi dφi λi 3h + 1) r dr r dr e r
(
(41)
)]
G(r) ) -
2e dy 2
ηr
d 1 d 4 d2 4 d 4 L≡ r ) 2+ dr r4 dr r dr r2 dr
(
)(
)∫ ( )} ∫ ( ) {
(
a 3
|
){
)}
(
The Electroviscous Coefficient p Following the theory of Ruiz-Reina et al.,29 it can be shown that the effective viscosity ηs of a concentrated suspension of rigid particles is given by
(
)
3C3
ηs ) η 1 + φ
a3
where C3 is calculated from h(r) via
C3 )
( |
b 4 d2 h 10 dr2
r)b-
+
b d3h 3 dr3
(53)
| )
(54)
r)b-
which coincides with C3 in eq 51. The electroviscous coefficient p, defined by eq 6, is thus given by
p)
(43) p)
[(
3a2 50
∫ab
(44)
The boundary conditions for u(r) (eqs 21 and 22) can be expressed in terms of h(r) and φi(r) as follows.
h(a) )
)
6C3 5a3 S(φ)
-1
(55)
By evaluating C3 from eq 54 and substituting the result into eq 55, we obtain
N
zi2n∞i exp(-ziy)φi ∑ dr i)1
(
+
3 r2 2a5 b φ5/3r5 dy zi dφi + + 1- 5 5/3 5 3 a 3 + 2φ 15r a dr r dr λi 3h 1 r x5 dy zi dφi λi 3h 1dr + a r2 - 3 dx + 1e r 5 dx x dx e x r (52)
φi(r) ) -
(42)
with
C4
+ r r4 x5 rx2 r3 x7 + G(x) dx (51) 70 70r4 30r2 30 2
where the integration constants C1-C4 are determined so as to satisfy eqs 38-41. Similarly, integration of eq 42 subject to eqs 49 and 50 yields
Here, φi(r) is a function of r only. In terms of φi(r) and h(r), eqs 32 and 33 can be rewritten as
L(Lh) ) G(r)
C3
(45)
1-
)
5r2 2r5 + 3a2 3a5 φ5/3 - φ7/3 7r5 5r7 1- 5+ 7 7/3 1-φ 2a 2a
(
)(
)]
G(r) dr (56)
which is the required expression for the electroviscous coefficient p for a moderately concentrated suspension of charged spherical particles.
Approximation for Low ζ and Large ka
dh 1 ) dr r)a+ 3
(46)
h(b) ) 0
(47)
We derive an approximate formula for the effective viscosity ηs applicable for the case where the zeta potential ζ is low. We use the following approximate solution to eq 34 for the equilibrium potential ψ(0)(r):
(48)
ψ(0)(r) ) ζ
|
dh )0 dr r)b-
Similarly, the boundary conditions for δµi(r) (eqs 30 and 31) reduce to
|
dφi dr
r)a+
)0
φi(b) ) 0
(49) (50)
a κb cosh[κ(b - r)] - sinh[κ(b - r)] + O(ζ2) (57) r κb cosh[κ(b - a)] - sinh[κ(b - a)]
and h(r) in eq 51 may be replaced by
h(r) )
[
1 5a3 5r 3r3 S(φ) 2 - 3 + 5 6 r 2b 2b 2 3a 1 - φ5/3 1 7r 5r3 - 5+ 7 7/3 4 5 1-φ r 2b 2b
(
)(
)]
(58)
Expression of the Primary ElectroViscous Effect
Langmuir, Vol. 23, No. 24, 2007 12065
where ζh is the scaled zeta potential. By using the above results, we finally obtain the following approximate expression for p applicable for moderately concentrated suspensions of φ e 0.3 at arbitrary ζ:
p)
[ (
)
72 1 + e- ζh/2 2 S(φ) Q(φ) R(φ) m+ ln + 2 2 (κa) m1 + e+ζh/2 ln 2 1 + 2 Q(φ) F
{
}(
)] 2
(66)
with
m( ) Figure 2. Dependence of S(φ), Q(φ), R(φ), and their product S(φ) Q(φ) R(φ) upon the particle volume fraction φ.
( )
For the case where κa . 1, it follows from eqs 56-58 that N
p)
zi2n∞i λi ∑ i)1
( )∑ 6rokT 5ηe
2
N
L(κa,φ)
zi2n∞i
() eζ
kT
[
R(φ) ) 1 -
(60)
]
2
(62)
In the limit of φ f 0 and κa . 1, eq 59 agrees with Booth’s result for the dilute case4 (see also eq 82 in a previous paper20). The product S(φ) Q(φ) R(φ) appearing in eq 60 takes the value of 1 at φ f 0 and increases with φ, tending to 3 as φ f 1 (Figure 2).
Approximation for Arbitrary ζ and Large ka Consider the case where the electrolyte is the z-z symmetrical type (z1 ) -z2 ) z > 0 and n∞1 ) n∞2 ) n∞) but may have different ionic motilities λ1 ) λ+ and λ2 ) λ-. We employ an approximation method developed in a previous paper.20 We may assume that the particles are positively charged, that is, ζ > 0 without loss of generality. We denote φi(r) for cations and anions by φ+(r) and φ-(r), respectively. For κa . 1, we may regard (r - a)/a as of the order of 1/κa and φ((r) ≈ φ((a) in the integrand of eq 56. Also, under the condition of nonoverlapping double layers at large κa, the following large κa approximate solution to eq 34 can be used:
y(r) ) 2 ln
(
)
1 + γ exp[-κ(r - a)] 1 - γ exp[-κ(r - a)]
(63)
with
γ ) tanh(ζh/4) ζh )
zeζ kT
p)
(59)
(61)
7(φ5/3 - φ7/3) 2(1 - φ7/3)
(68)
9(m+ + m- ) 2(κa)
2
S(φ) Q(φ) R(φ)
(zeζ kT )
2
(69)
Results and Discussion
3(1 - φ5/3) 3 + 2φ5/3
2 (1 + 3m- )(eζh/2 - 1) κa
which agrees with eq 59.
5 S(φ) Q(φ) R(φ) L(κa,φ) ) (κa)2 Q(φ) )
(67)
where F corresponds to Dukhin’s number16 in the electrophoresis problem. For low ζ, eq 66 reduces to
2
i)1
with
F)
2rokT λ( 3ηz2e2
(64) (65)
Equation 56 is the general expression for the primary electroviscous coefficient p applicable for moderately concentrated suspensions of charged spherical particles of radius a (i.e., at φ e 0.3) at arbitrary ζ for the case where overlapping of the electrical double layers of adjacent particles can be neglected (i.e., at κa > 10). This expression is equivalent to the exact numerical results of Ruiz-Reina et al.29 and those of the Watterson-White theory8 (if φ tends to 0). On the basis of eq 56, we have derived eq 66 for p. Equation 66 is an approximate formula applicable for all values of ζ at large κa and φ e 0.3. For the dilute case of φ f 0, eq 66 reduces to
p)
[ (
) (
)(
)]
m72 1 + e-ζh/2 2 1 + e+ζh/2 m+ ln + ln 2 2 1 + 2F 2 (κa)
2
(70)
which agrees with a result obtained in a previous paper (eq 93 in ref 20) (Note that e-ζh/2 in the second term in eq 93 in ref 20 should be e+ζh/2.) For the low zeta potential case at φ f 0, eq 69 becomes
L(κa,0) )
5 (κa)2
(71)
which is also obtained from eq 70 by taking the low zeta potential limit. It is found that eq 66 is in good agreement with the exact numerical results of Ruiz-Reina et al.29 and those of Watterson and White6 for the limiting case of φ f 0. Some examples of the calculation of p plotted as a function of the scaled particle zeta potential zeζ/kT for a suspension of positively charged spherical particles (ζ > 0) of particle volume fraction φ in a KCl aqueous solution at 25 °C are given in Figures 3 and 4 in comparison with the exact numerical results,6,29 in which Λ+° ) 73.5 × 10-4 m2 Ω-1 mol-1 for K+ and Λ-° ) 76.3 × 10-4 m2 Ω-1 mol-1 for Cl- (that is, m+ ) 0.176 and m- ) 0.169). Figure 3 (which corresponds to Figure 2 in a previous paper20) gives the results for the dilute case φ f 0 calculated from eq 66
12066 Langmuir, Vol. 23, No. 24, 2007
Ohshima
is negligible for κa g 100 and the maximum error is ∼20% even at κa ) 50. These figures also show the low zeta results calculated via eq 59, which is a good approximation for zeζ/kT e 2. It is also seen that p exhibits a maximum around zeζ/kT ≈ 6. This is due to the relaxation effects, which become appreciable for high zeta potentials, as in the case of the electrophoresis problem. The relaxation effect is taken into account in eq 66 through a parameter F, which corresponds to Dukhin’s number16 in the electrophoresis problem. On the other hand, the low ζ approximation (eq 59), which ignores the relaxation effect, does not reproduce a maximum. For a further increase in zeζ/kT, eq 66 approaches a nonzero limiting value of p given by
p) Figure 3. Primary electroviscous coefficient p as a function of scaled zeta potential zeζ/kT for a suspension of charged spherical particles for several values of κa for the limiting case of particle volume fraction φ f 0 in an aqueous KCl solution at 25 °C. Solid lines are large κa results calculated using eq 66, and dotted lines are low ζ results calculated using eq 59. Filled circles are exact numerical results of Watterson and White.8
72 S(φ) Q(φ) R(φ) m+(ln 2)2 2 (κa)
(72)
which depends only on the drag coefficient λ+ of co-ions. The electroviscous coefficient p given by eq 66 depends on the particle volume fraction φ. The φ dependence of p comes mainly from the product S(φ) Q(φ) R(φ) of three functions S(φ), Q(φ), and R(φ), which increases from 1 at φ f 0 to 3 at φ f 1 (Figure 2). For small φ, we have
S(φ) Q(φ) R(φ) ) 1 +
25 115 5/3 625 2 φφ + φ + ... (73) 4 6 16
which is approximated well by
S(φ) Q(φ) R(φ) ) 1 + 5.13φ - 3.96φ2
(74)
with the relative errors being less than 1%. Experimentally, the φ dependence of p may be determined from a plot of experimentally obtained values of p (measured at different φ) as a function of φ by a curve-fitting procedure. It thus becomes possible to make a comparison of theory and experiment. Note also that the obtained expression for p is applicable for the case of large κa, where p takes small values and thus a high accuracy in determination of p is needed. Figure 4. Primary electroviscous coefficient p as a function of scaled zeta potential zeζ/kT for a suspension of charged spherical particles for several values of particle volume fraction φ at κa ) 50 in an aqueous KCl solution at 25 °C. Solid lines are large κa results calculated using eq 66, and dotted lines are low ζ results calculated using eq 59. Filled circles are exact numerical results of Ruiz-Reina et al.29
with φ f 0 (i.e., eq 70) for several values of κa in comparison with the corresponding numerical results of Watterson and White.8 Figure 4 shows the results calculated with eq 66 at κa ) 50 for several values of φ in comparison with the exact numerical results of Ruiz-Reina et al.29 It is seen that the relative error of eq 66
Conclusion The principal result in the present paper is an analytic approximate expression (eq 66) for the primary electroviscous coefficient p for a moderately concentrated suspension of charged spherical particles of the particle volume fraction φ e 0.3. Equation 66 is applicable for large κa and arbitrary ζ under the condition of nonoverlapping double layers, where the relative error of eq 66 is negligible for κa g 100 and even at κa ) 50 the maximum error is ∼20%. It is shown that p exhibits a maximum when plotted as a function of the particle zeta potential due to the relaxation effect. LA701768A
