Langmuir 2008, 24, 6453-6461
6453
Primary Electroviscous Effect in a Dilute Suspension of Soft 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 January 4, 2008. ReVised Manuscript ReceiVed February 18, 2008 A theory for the primary electroviscous effect in a dilute suspension of soft particles (i.e., particles coated with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte solution is presented. The general expression for the effective viscosity ηs of the suspension and the primary electroviscous coefficient p, which is further expressed in terms of a function L, is given. On the basis of the general expressions, we derive approximate analytic expressions for ηs and p, which are applicable when the density of the fixed charges distributed within the surface layer is low. Further we obtain a simple approximate analytic expression (without involving numerical integrations) for p applicable for most practical cases. It is found that the function L exhibits a minimum when plotted as a function of κa (κ is the Debye–Hückel parameter and a is the particle core radius), unlike the case of a suspension of hard particles, in which case L decreases as κa increases, exhibiting no minimum. The presence of a minimum for the case of a suspension of soft particles is due to the fact that L is proportional to 1/κ2 at small κa and to κ2 at large κa. Because of the presence of this minimum, the difference in L between soft and hard particles becomes very large for large κa.
Introduction The effective viscosity ηs of a dilute suspension of uncharged colloidal particles in a liquid is greater than the viscosity η of the original liquid. Einstein1 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, then the effective viscosity ηs is further increased. This phenomenon is called the primary electroviscous effect2–17 and the effective viscosity ηs can be 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 as well as their numerical solutions were given by Watterson and * E-mail:
[email protected]. (1) Einstein, A. Ann. Phys. 1906, 19, 289. (2) Smoluchowski, M. Kolloid-Z. 1916, 18, 194. (3) Krasny-Ergen, W. Kolloidzeitschrift 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. (11) Rubio-Hernández, F. J.; Ruiz-Reina, E.; Gómez-Merino, A. I. J. Colloid Interface Sci. 1998, 206, 334. (12) Rubio-Hernández, F. J.; Ruiz-Reina, E.; Gómez-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-Hernández, F. J.; Ruiz-Reina, E.; Gómez-Merino, A. I. Colloids Surf., A 2001, 192, 349. (15) Rubio-Hernández, F. J.; Ruiz-Reina, E.; Gómez-Merino, A. I.; Sherwood, J. D. Rheol. Acta 2001, 40, 230. (16) Ohshima, H Langmuir 2006, 22, 2863. (17) Ohshima, H Theory of Colloid and Interfacial Electric Phenomena; Elsevier: Amsterdam, 2006.
White.8 Ohshima16,17 derived an approximate analytic expression for p in a dilute suspension of spherical particles with arbitrary zeta potentials and large κa (where κ is the Debye–Hückel parameter and a is the particle radius). For concentrated suspensions, interactions among particles must be considered.18–27 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.19 and Rubio-Hernández et al.22 Ohshima27 has recently derived an approximate analytic expression for p in a moderately concentrated suspension of spherical particles with arbitrary zeta potential and large κa. The electroviscous effect in a suspension of particles of types other than rigid particles has also been theoretically investigated. Taylor28 proposed a theory of the electroviscous effect in a suspension of uncharged liquid drops. This theory has been extended to the case of charged liquid drops by Ohshima.16 Natraj and Chen29 developed a theory for charged porous spheres, and Allison et al.30 and Allison and Xin31 discussed the case of polyelectrolyte-coated particles. In the present article, we present a theory of the primary electroviscous effect in a dilute suspension of soft particles, i.e., (18) Simha, R. J. Appl. Phys. 1952, 23, 1020. (19) Ruiz-Reina; Carrique, F.; Rubio-Hernández, F. J.; Gómez-Merino, A. I.; García-Sánchez, P. J. Phys. Chem. B 2003, 107, 9528. (20) Zholkovskiy, E. K.; Adeyinka, O. B.; Masliyah, J.H. J. Phys. Chem. B 2006, 110, 19726. (21) Sherwood, J.D. J. Phys. Chem. B 2007, 111, 3370. (22) Rubio-Hernández; F, J.; Carrique, F.; Ruiz-Reina, E. AdV. Colloid Interface Sci. 2004, 107, 51. (23) Ruiz-Reina, E.; García-Sánchez, P.; Carrique, F. J. Phys. Chem. B 2005, 109, 5289. (24) Carrique, F.; García-Sánchez, P.; Ruiz-Reina, E. J. Phys. Chem. B 2005, 109, 24369. (25) García-Salinas, M. J.; de las Nieves, F. J. Langmuir 2000, 16, 7150. (26) García-Salinas, M. J.; de las Nieves, F. J. Colloids Surf., A 2003, 222, 65. (27) Ohshima, H Langmuir 2007, 23, 12061. (28) Taylor, G. I. Proc. R. Soc. London, Ser. A 1932, 138, 41. (29) Natraj, V.; Chen, S. B. J. Colloid Interface Sci. 2002, 251, 200. (30) Allison, S.; Wall, S.; Rasmusson, M. J. Colloid Interface Sci. 2003, 263, 84. (31) Allison, S.; Xin, Y. J. Colloid Interface Sci. 2006, 299, 977.
10.1021/la800027m CCC: $40.75 2008 American Chemical Society Published on Web 05/20/2008
6454 Langmuir, Vol. 24, No. 13, 2008
Ohshima
ionic species is related to the limiting conductance Λ0i of that ionic species by
λi )
Figure 1. A spherical soft particle. a ) radius of the particle core. d ) thickness of the polyelectrolyte layer covering the particle core. b ) a + d.
particles covered with an ion-penetrable surface layer of charged or uncharged polymers. Electrokinetic phenomena in a suspension of soft particles are quite different from that of hard particles without surface structures.32–52 For soft particles, the zeta potential (i.e., the potential at the particle core surface) becomes less important and instead the following two potentials play an essential role in their electrokinetics, that is, the Donnan potential and the potential at the boundary between the polyelectrolyte layer and the surrounding electrolyte solution (which we call the surface potential of a soft particle). We derive expressions for the effective viscosity and the primary electroviscous coefficient of a dilute suspension of soft particles.
Basic Equations Consider a dilute suspension of spherical soft particles, i.e., spherical hard particles covered with an ion-penetrable layer of polyelectrolytes in an electrolyte solution of volume V. We assume that the uncharged particle core of radius a is coated with an ion-penetrable layer of polyelectrolytes of thickness d. The polymer-coated particle has thus an inner radius a and an outer radius b ) a + d (Figure 1). The origin of the spherical polar coordinate system (r, θ, φ) is held fixed at the center of one particle. We consider the case where dissociated groups of valence Z are distributed with a uniform density N in the polyelectrolyte layer so that the density of the fixed charges Ffix in the surface layer is given by Ffix ) ZeN, where e is the elementary electric charge. Let the electrolyte be composed of M ionic mobile species of valence zi, bulk concentration (number density) ni∞, and drag coefficient λi (i ) 1, 2, ..., M). The drag coefficient λi of the ith (32) Donath, E.; Pastuschenko, V. Bioelectrochem. Bioenerg. 1980, 7, 31. (33) Wunderlich, R. W. J. Colloid Interface Sci. 1982, 88, 385. (34) Levine, S.; Levine, M.; Sharp, K. A.; Brooks, D. E. Biophys. J. 1983, 42, 127. (35) Sharp, K. A.; Brooks, D. E. Biophys. J. 1985, 47, 563. (36) Donath, E.; Voigt, A. J. Colloid Interface Sci. 1986, 109, 122. (37) Ohshima, H. J. Colloid Interface Sci. 1994, 163, 474. (38) Ohshima, H. AdV. Colloid Interface Sci. 1995, 62, 189. (39) Saville, D. A. J. Colloid Interface Sci. 2000, 222, 137. (40) Ohshima, H. J. Colloid Interface Sci. 2001, 233, 142. (41) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 258, 56. (42) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 263, 478. (43) Lopez-Garcia, J. J.; Grosse, C.; Horno, J. J. Colloid Interface Sci. 2003, 265, 327. (44) Lopez-Garcia, J. J.; Grosse, C.; Horno, J. J. Colloid Interface Sci. 2003, 265, 341. (45) Dukhin, S. S.; Zimmermann, R.; Werner, C. J. Colloid Interface Sci. 2004, 274, 309. (46) Hill, R. J. Phys. ReV. E 2004, 70, 051406. (47) Ohshima, H. Colloid Polym. Sci. 2005, 283, 819. (48) Hill, R. J.; Saville, D. A. Colloids Surf., A 2005, 267, 31. (49) Dukhin, S. S.; Zimmermann, R.; Werner, C. AdV. Colloid Interface Sci. 2006, 122, 93. (50) Duval, J. F. L.; Ohshima, H. Langmuir 2006, 22, 3533. (51) Ohshima, H. Electrophoresis 2006, 27, 526. (52) Ohshima, H. Colloid Polym. Sci. 2007, 285, 1411.
NAe2|zi|
(3)
Λi0
where NA is Avogadro’s number. We adopt the model of Debye-Bueche,53 in which the polymer segments are regarded as resistance centers distributed in the polyelectrolyte layer, exerting frictional forces -γu on the liquid flowing in the polymer layer, where u is the liquid velocity relative to the particle and γ is the frictional coefficient. The main assumptions in our analysis are as follows. (i) The Reynolds numbers of the liquid flows outside and inside the polyelectrolyte layer are small enough to ignore inertial terms in the Navier–Stokes equations and the liquid can be regarded as incompressible. (ii) The applied shear field is weak so that 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 core surface. (iv) No electrolyte ions can penetrate the particle core. (v) The polyelectrolyte layer is permeable to mobile charged species. (vi) The relative permittivity r takes the same value both inside and outside the polyelectrolyte layer. 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 core is given by the sum of u(0)(r) and a perturbation velocity. Thus u(r) obeys the boundary condition
u(r) f u(0)(r) where r ) |r|. We can express
rf∞
as u(0)(r)
(4)
as
u(0)(r) ) r · r where r is a symmetric traceless tensor so that irrotational, viz.,
(5) u(0)(r)
becomes
∇ × u(0) ) 0
(6)
Also we have from assumption (i) the following continuity equations
∇ · u(0) ) 0 ∇·u)0
(7) (8)
From symmetry considerations, u(r) takes the form54
u(r) ) u(0)(r) + ∇ × ∇ × [(r · ∇ )f(r)]
(9)
where f(r) is a function of r and the second term on the righthand side corresponds to the perturbation velocity. We introduce a function h(r), defined by
h(r) )
d 1 df dr r dr
( )
(10)
then eq 9 can be rewritten as
[ h(r)r r × u (r)] ) (1 - dhdr - 2hr )u + r1 ( dhdr - hr )r(r · u
u(r) ) u(0)(r) + ∇ ×
(0)
(0)
(0)
2
) (11)
where h(r) is a function of r only and we have used eq 6 (irrotational flow), eq 7 (incompressible flow), and (r · ∇)u(0) ) u(0) (linear flow). (53) Debye, P.; Bueche, A. J. Chem. Phys. 1993, 1, 13. (54) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Permagon: London, 1966.
The ElectroViscous Effect
Langmuir, Vol. 24, No. 13, 2008 6455
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) + γu(r) + ∇ p(r) + Fel(r) ∇ ψ(r) ) 0, a < r < b (12) η ∇ × ∇ × u(r) + ∇ p(r) + Fel(r) ∇ ψ(r) ) 0, r > b (13)
σH ) -pI + η[∇u + (∇u)T]
must be continuous at r ) b, where I is the unit tensor. The pressure p(r) is thus continuous at r ) b. The boundary condition for the ionic flow vi(r) at r ) a is given by
vi(r) · n|r)a ) 0
(14)
∇ · (ni(r)vi(r)) ) 0
(15)
Linearized Equations For a weak shear field (assumption ii), one may write
with
µi(r) ) µi + zieψ(r) + kT ln ni(r) Fel(r) + ZeN , ∆ψ(r) ) εrε0 Fel(r) ∆ψ(r) ) , εrε0
ab
i)1
u · n|r)b+ ) u · n|r)b-
(21)
u × n|r)b+ ) u × n|r)b-
(22)
where n is the unit normal outward from the particle core surface. 3. The normal and tangential components of the stress tensor σ are continuous at r ) b, viz.
(σ · n) · n|r)b+ ) (σ · n) · n|r)b-
(23)
(σ · n) × n|r)b+ ) (σ · n) × n|r)b-
(24)
where σ is the sum of the hydrodynamic stress σH and the Maxwell stress σE. 5. The potential ψ(r) and the electric field -∇ψ(r) are continuous at r ) b. The continuity of ψ(r) results from assumption vi. Thus the normal and tangential components of Maxwell’s stress σE are continuous at r ) b. Therefore eqs 23 and 24 imply that the normal and tangential components of the hydrodynamic stress σH, which is given by
(36) ∇ · (ni(0)(r)u(r) -
1 (0) n (r) ∇ δµi(r)) ) 0 λi i
(37)
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, viz.
(
)
∑
(
)
∑
(
)
(
)
M zieψ 1 d 2 dψ(0) 1 ZeN r ) zieni∞ exp , 2 dr dr ε ε kT εrε0 r r 0 i)1 a < r < b (38) M zieψ 1 d 2 dψ(0) 1 r ) zieni∞ exp , 2 dr dr εrε0 i)1 kT r
which are rewritten as
r>b
(39)
[ [
6456 Langmuir, Vol. 24, No. 13, 2008
{
κ2
1 d 2 dy r )r2 dr dr
( )
( )
with
∑ zini∞ exp(-ziy) + ZN i)1
M
∑ zi ni 2
∞
i)1
M
2
κ
1 d 2 dy r )r2 dr dr
}
M
∑ zini
∞
exp(-ziy)
i)1
M
∑ zi ni
∞
2
i)1
κ)
(
]
∑
,
a b 2
(54) (55)
h(b+) ) h(b-) dh dh ) dr r)b+ dr r)b-
(44)
rf∞
φi(r)
(51)
h(a) )
where eq 44 states that the particle core surface is uncharged and eq 45 is the continuity condition of electric potential (which results from assumption vi). Further, symmetry considerations permit us to write
δµi(r) ) -zie
γη
2e dy zi2ni∞ exp(-ziy)φi 2 dr ηr i)1
(42)
dψ(0) )0 dr r)a+ (0)
(50)
The boundary conditions for u(r) (eqs 20-24) can be expressed in terms of h(r) as follows.
eψ(0)(r) kT
|
)]
with
where κ is the Debye–Hückel parameter and y(r) is the scaled equilibrium potential. The boundary conditions for ψ(0)(r) are
(0)
(
M
1 z 2e2ni∞ εrε0kT i)1 i y(r) )
() [
L
G(r) ) -
)
M
]
Ohshima
|
d2h dr2 r)b+ )
|
| |
|
r)b+ )
{
d2h dr2
|
(56) (57) (58)
r)b-
|
d3h dh 2h(b-) + λ2 1 3 r)bdr r)bb dr h(r) f 0 as r f ∞ dh h f 0 as r f ∞ dr
}
(59) (60) (61)
Equation 59 results from the continuity condition of pressure p(r). Similarly, the boundary conditions for δµi(r) (eqs 33 and 34) can be expressed in terms of φi(r) as
|
dφi )0 dr r)a+ φi(r) f 0 as r f ∞
(48) (49)
(62) (63)
Equations 48 and 49 can formally be integrated to give
[
( [ ( ] [
)
(
)
x5 x x2 r x2 3 3x 3 9x 3x2 9 + 3 + cosh[λ(r x)] + + + + 7 4 sinh[λ(r 2 4 2 4 2 4 3 6 3 6 4 3 2 5 2 5 3 5 4 5λ r 5λ λr λr λr λr λr λr λr λr λr b3 6b2 r 15b 18b2 3b3 45 45b 3b3 45b 15 18b2 x)] G(x) dx + C1 - 2 + 2 2 + 4 2 - 4 3 + 4 4 - 6 3 + 6 4 cosh[λ(r - b)] + 3 2 - 3 3 - 4 3 + 5 2 + 5 4 + λ λr λr λr λr λr λr λr λr λr λr λr 45 1 1 3 3 3 3 1 ) sinh[λ(r - b)] + C2 - 2 4 + 2 2 2 - 4 3 3 + 4 2 4 cosh[λ(r - b)] + 3 3 2 - 3 2 3 + 5 3 4 sinh[λ(r - b)] + λ7r4 λr λbr λbr λbr λbr λbr λbr 1 1 3 3 3 3 C3 + sinh(λr) - 3 3 cosh(λr) + C4 + cosh(λr) - 3 3 sinh(λr) , a < r < b (64) λ2r2 λ4r4 λr λ2r2 λ4r4 λr 5 2 3 7 D3 D4 r r x rx x h(r) ) ∞ + G(x) dx + D1r3 + D2r + 2 + 4 , r > b (65) 4 2 70 70r 30 30r r r
h(r) )
]
∫br
{(
∫
(
( )
)
)
) } {(
(
(
)
)
}
]
where the integration constants C1-C4 and D1-D4 are determined so as to satisfy eqs 54-61. It follows from eqs 60 and 61 that D1 ) D2 ) 0. Similarly, integration of eq 50 subject to eqs 62 and 63 yields
The ElectroViscous Effect
Langmuir, Vol. 24, No. 13, 2008 6457
φi(r) ) -
(
r2 2a5 + 5 15r3
) { ∫a∞ dy dr
)}
zi dφi λi 3h 1 dr + + 1r dr e r 5
(
∫ar
( ) { r2 -
x5 dy zi dφi λi 3h + 1e x r3 dx x dx
(
)} dx
(66)
The Electroviscous Coefficient p Following the standard theory of Watterson and White,8 it can be shown that the effective viscosity ηs of a suspension of soft particles in an electrolyte solution of volume V is given by
(
ηs ) η 1 + φ
3D3 b3
)
(67)
with
φ)
(4π ⁄ 3)b3 V
(68)
where φ is the particle volume fraction of soft particles of outer radius b and D3 is calculated from h(r) via
D3 ) lim [r2h(r)]
(69)
rf∞
The result is
D3 )
5b3L2 b5 + 6L1 20L1
∫b
∞
{
L3 -
( )}
5L2 r 2 2L1 r + 3 b 3 b
()
5
G(r) dr -
a5 2 2 3λ b L1
∫a
b
{
L4 -
}
3b3L5(r) r 2 r 5 15 r 2 - L6 + 2 L7(r) G(r) dr 3 a a 2a λ ab a (70)
()
()
()
where the definitions of L1-L4, L5(r), L6, and L7(r) are given in the Appendix. We first consider the case of a suspension of uncharged soft particles, i.e., ZeN ) 0. In this case D3 ) 5b3L2/L1 and eq 67 becomes
5 ηs ) η 1 + Ωφ 2
(
)
(71)
where
Ω)
L2 L1
(72)
Equation 71 as combined with eq 72 is the effective viscosity ηs of a dilute suspension of uncharged soft particles and the coefficient Ω expresses the effect of the uncharged polymer layer coating the particle core upon the effective viscosity of the suspension. For a suspension of charged soft particles, we write
{
}
(73)
-1
(74)
5 ηs ) η 1 + (1 + p)φΩ 2 The primary electroviscous coefficient p is thus given by
p)
6D3 3
5b Ω
By substituting eq 70 into eq 74, we obtain
3b2 p) 50L2
∫b
∞
{
( )}
5L2 r 2 2L1 r L3 + 3 b 3 b
()
5
-1)
6D3L1 5b3L2
( )∫ {
2 a G(r) dr - 2 5λ L2 b
5
b
a
}
3b3L5(r) r 2 r 5 15 r 2 L4 - L6 + 2 L7(r) G(r) dr (75) 3 a a 2a λ ab a
()
()
()
which is the required general expression for the electroviscous coefficient p for a dilute suspension of charged spherical soft particles.
Approximation for Low Fixed Charge Densities We derive an approximate formula for the effective viscosity ηs applicable for the case where the fixed charge density ZeN is low. We use the following approximate solution to eqs 40 and 41 for the equilibrium potential ψ(0)(r):
ψ(0)(r) )
{
[ (
)
b ψ(0)(r) ) ψ0 exp[-κ(r - b)], r with
}]
1 + κb -κ(b-a) sinh[κ(r - a)] a cosh[κ(r - a)] ZeN 1e , + 1 + κa κr r εrε0κ2 r>b
a
