Langmuir 2006, 22, 2863-2869
2863
Primary Electroviscous Effect in a Dilute Suspension of Charged Mercury Drops 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 September 21, 2005. In Final Form: January 15, 2006 The standard theory of the primary electroviscous effect in a dilute suspension of charged spherical rigid particles in an electrolyte solution (Watterson, I. G.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 1115) is extended to cover the case of a dilute suspension of charged mercury drops of viscosity ηd. A general expression for the effective viscosity or the electroviscous coefficient p of the suspension is derived. This expression tends to that for the case of rigid particles in the limit of ηd f ∞. We also derive an approximate analytical viscosity expressions applicable to mercury drops carrying low zeta potentials at arbitrary κa (where κ is the Debye-Hu¨ckel parameter and a is the drop radius) and to mercury drops as well as rigid spheres with arbitrary zeta potentials at large κa. It is shown that the large-κa expression of p for rigid particles predicts a maximum when plotted as a function of zeta potential. This result for rigid particles agrees with the exact numerical results of Watterson and White. It is also shown that in the limit of high zeta potential the effective viscosity of a suspension of mercury drops tends to that of uncharged rigid spheres given by Einstein’s formula (Einstein, A. Ann. Phys. 1906, 19, 289), whereas in the opposite limit of low zeta potential the effective viscosity approaches that of a suspension of uncharged liquid drops derived by Taylor (Taylor, G. I. Proc. R. Soc. London, Ser. A 1932, 138, 41).
Introduction A suspension of colloidal particles in a liquid has an effective viscosity ηs that is greater than the viscosity η of the original liquid. Einstein1 showed that the effective viscosity ηs of a dilute suspension of uncharged spherical colloidal particles of particle volume fraction φ is given by
5 ηs ) η 1 + φ 2
(
)
(1)
For the case of a dilute suspension of uncharged liquid drops of viscosity ηd, Taylor2 obtained
{
ηs ) η 1 +
5ηd + 2η
φ
2(ηd + η)
}
(2)
As ηd f ∞, eq 2 reduces back to eq 1. If the liquid contains an electrolyte and the particles are charged, then the effective viscosity ηs is further increased because of the presence of the electrical double layer around the particles. This phenomenon is called the primary electroviscous effect, and
{
5 ηs ) η 1 + (1 + p)φ 2
}
(3)
can be written, where p is the primary electroviscous coefficient. A number of authors3-18 have proposed theoretical treatments * E-mail:
[email protected]. Tel and Fax: +81-4-7121-3661. (1) Einstein, A. Ann. Phys. 1906, 19, 289. (2) Taylor, G. I. Proc. R. Soc. London, Ser. A 1932, 138, 41 (3) Smoluchowski, M. Kolloid-Z. 1916, 18, 194. (4) Krasny-Ergen, W. Kolloidzeitschrift 1936, 74, 172. (5) Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 533. (6) Russel, W. B. J. Fluid Mech. 1978, 85, 673. (7) Lever, D. A. J. Fluid Mech. 1979, 92, 421 (8) Sherwood, J. D. J. Fluid Mech. 1980, 101, 609. (9) Watterson, I. G.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 1115. (10) Hinch, E. J.; Sherwood, J. D. J. Fluid Mech. 1983, 132, 337. (11) Duhkin, A. S.; van de Ven, T. G. M. J. Colloid Interface Sci. 1993, 158, 85.
of the primary electroviscous effect in a suspension of rigid particles. The standard theory for this effect and the governing electrokinetic equations were given by Watterson and White,9 who also provided numerical solutions of the electrokinetic equations. Rubio-Herna´ndez et al.12,13,15,16 and Sherwood et al.14 pointed out the importance of the additional surface conductance19-21 in the primary electroviscous effect. Ruiz-Reina et al.17 and Rubio-Herna´ndez et al.18 considered the effective viscosity of moderately concentrated colloidal suspensions. The electroviscous effect in a suspension of particles of types other than rigid particles has also been theoretically investigated. Natraj and Chen22 developed a theory for charged porous spheres, and Allison et al.23 discussed the case of polyelectrolyte-coated particles. In the present article, we treat mercury drops, whose electrokinetic behaviors are quite different from those of rigid particles in that liquid flow exists inside the particles.24-29 We write the effective viscosity ηs of a dilute suspension of charged mercury drops as (12) Rubio-Herna´ndez, F. J.; Ruiz-Reina, E.; Go´mez-Merino, A. I. J. Colloid Interface Sci. 1998, 206, 334. (13) Rubio-Herna´ndez, F. J.; Ruiz-Reina, E.; Go´mez-Merino, A. I. J. Colloid Interface Sci. 2000, 226, 180. (14) Sherwood, J. D.; Rubio-Herna´ndez, F. J.; Ruiz-Reina, E. J. Colloid Interface Sci. 2000, 228, 7. (15) Rubio-Herna´ndez, F. J.; Ruiz-Reina, E.; Go´mez-Merino, A. I. Colloids Surf., A 2001, 192, 349. (16) Rubio-Herna´ndez, F. J.; Ruiz-Reina, E.; Go´mez-Merino, A. I.; Sherwood, J. D. Rheol. Acta 2001, 40, 230. (17) 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. (18) Rubio-Herna´ndez, F. J.; Carrique, F; Ruiz-Reina, E. AdV. Colloid Interface Sci. 2004, 107, 51. (19) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7, Chapter 2. (20) Zukoski, C. F.; D. A. Saville, D. A. J. Colloid. Interface. Sci. 1986, 114, 32. (21) Zukoski, C. F.; D. A. Saville, D. A. J. Colloid. Interface. Sci. 1986, 114, 45. (22) Natraj, V; Chen, S. B. J. Colloid Interface Sci. 2002, 251, 200. (23) Allison, S; Wall, S.; Rasmusson, M. J. Colloid Interface Sci. 2003, 263, 84.
10.1021/la0525628 CCC: $33.50 © 2006 American Chemical Society Published on Web 02/18/2006
2864 Langmuir, Vol. 22, No. 6, 2006
{
ηs ) η 1 +
5ηd + 2η
Ohshima
(1 + p)φ
2(ηd + η)
}
u(0)(r) ) R ‚ r
(4)
where p is the primary electroviscous coefficient of this suspension. As p f 0, eq 4 becomes eq 2, whereas as ηd f ∞, eq 4 tends to eq 3. The purpose of the present article is to extend the standard theory of Watterson and White9 to cover the case of a dilute suspension of charged mercury drops and to derive analytic expressions for the effective viscosity ηs and the primary electrocviscous coefficient p of the suspension.
(8)
where r ) |r| and r is a symmetric tensor and thus u(0)(r) satisfies
∇ × u(0) ) 0
(9)
Also, assume that u(0)(r), u(r), and uin(r) are incompressible flows (assumption (i)),
∇ ‚ u(0) ) 0
Basic Equations Consider a dilute suspension of charged spherical mercury drops of radius a and viscosity ηd in an electrolyte solution under a linear shear filed. 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). Because electroneutrality holds in the bulk solution phase,
(10)
∇‚u)0
(11)
∇ ‚ uin ) 0
(12)
It follows from eq 10 that r is a traceless tensor. From symmetry considerations, u(r) takes the form30
u(r) ) u(0)(r) + ∇ × ∇ × [(r ‚ ∇)f(r)]
(13)
N
zin∞i ) 0 ∑ i)1
(5)
where f(r) depends only on the radial distance r. We introduce a function h(r) of r only defined by
can be written. The drag coefficient λi of the ith ionic species is related to the limiting conductance Λ0i of that ionic species by
λi )
NAe2|zi| Λ0i
(6)
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 numbers of the liquid flows outside and inside the drop are small enough to ignore inertial terms in the NavierStokes equation, and the liquid can be regarded as incompressible. (ii) The applied shear field is weak, so the electrical double layer around the particle is only slightly distorted. (iii) The drop remains spherical during motion under the shear field. (iv) The slipping plane (at which the liquid velocity relative to the drop becomes zero) is located on the drop surface. (v) No electrolyte ions can penetrate the drop surface, which implies that no electrochemical reactions occur on the drop surface (i.e., neither neutralization of ions nor ionization of atoms of the drop takes place). This assumption is equivalent to postulating that a mercury drop behaves like an ideally polarizable conductor. Consequently, one may assume that neither electrostatic charge nor field nor current exists inside the drop and that the drop surface is always equipotential. Let u(r) and uin(r) be, respectively, the velocity at position r outside the drop and that inside the drop. Imagine that a linear symmetric shear field is applied to the system so that the velocity u(r) at position r outside the particle obeys the boundary condition
u(r) f u(0)(r) as r f ∞
( )
(14)
Then, eq 13 can be rewritten as
[hrr × u ] dh 2h 1 dh h ) (1 - - )u + ( - )r(r ‚ u dr r r dr r
u(r) ) u(0) + ∇ ×
(0)
(0)
(0)
2
) (15)
where we have used eq 9 (symmetric flow), eq 10 (incompressible flow), and (r ‚ ∇)u(0) ) u(0) (linear flow). Similarly, the velocity uin(r) inside the drop takes the form
uin(r) ) u(0) + ∇ ×
(
) 1-
[
hin r × u(0) r
)
]
(
)
dhin 2hin (0) 1 dhin hin u + 2 r(r ‚ u(0)) dr r dr r r (16)
where hin(r) is a function of r only. We assume that u(r) and uin(r) obey the following NavierStokes equations (assumption (ii)):
(7)
with (24) Ohshima, H; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1643. (25) Baygents, J. C.; Saville, D. A. J. Chem. Soc., Faraday Trans. 1991, 87, 1883. (26) Baygents, J. C.; Saville, D. A. J. Colloid Interface Sci. 1991, 146, 9. (27) Ohshima, H. J. Colloid Interface Sci. 1997, 189, 376. (28) Ohshima, H. J. Colloid Interface Sci. 1999, 218, 535. (29) Lee, E.; Hu, J.-K.; Hsu, J.-P. J. Colloid Interface Sci. 2003, 257, 250.
d 1 df dr r dr
h(r) )
η∇ × ∇ × u(r) + ∇p(r) + Fel(r)∇ψ(r) ) 0
(17)
ηd∇ × ∇ × uin(r) + ∇pin(r) ) 0
(18)
1 vi(r) ) u(r) - ∇µi(r) λi
(19)
∇ ‚ (ni(r)vi(r)) ) 0
(20)
with (30) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon: London, 1966.
Primary ElectroViscous Effect in Mercury Drops
Langmuir, Vol. 22, No. 6, 2006 2865
µi(r) ) µ∞i + zieψ(r) + kT ln ni(r)
(21)
Fel(r) ro
(22)
zieni(r) ∑ i)1
(23)
∆ψ(r) ) N
Fel(r) )
where r is the relative permittivity of the electrolyte solution, o is the permittivity of a vacuum, vi is the velocity of the ith ionic species outside the drop, k is the Boltzmann constant, T is the absolute temperature, p(r) and pin(r) are, respectively, the pressure outside the drop and that inside the drop, Fel(r) is the charge density resulting from the mobile charged ionic species given by eq 23, 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, and µ∞i is a constant term in µi(r). Equation 19 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 21. Equation 20 is the continuity equation for the ith ionic species, and eq 22 is Poisson’s equation. The following boundary conditions for u and uin on the drop surface r ) a must be satisfied. u and uin are equal, and their normal components vanish, viz.,
to those at equilibrium (i.e., in the absence of the shear field), µ(0) i is a constant independent of r and δni(r), δψ(r), and δµi(r) are perturbation quantities. One may assume that the distribution of electrolyte ions at equilibrium n(0) i (r) obeys the Boltzmann equation and that the equilibrium double layer potential ψ(0)(r) outside the drop satisfies the Poisson-Boltzmann equation, both being functions of r only, viz.,
(
)
zieψ(0) kT
(34)
F(r) 1 d 2dψ(0) r )2 dr dr ro r
(35)
∞ n(0) i (r) ) ni exp -
(
)
with N
F(0) el (r)
∑ i)1
)
N
zien(0) i (r)
Equation 35 then becomes
∑ i)1
)
zien∞i
(
exp -
)
zieψ(0) kT
[ ]
(36)
N
u|r)a+ ) u|r)a-
(24)
u ‚ n|r)a+ ) uin ‚ n|r)a- ) 0
(25)
1 d r2 dr
κ2
( ) r2
dy
)-
dr
N
(26)
where σ and σin are, respectively, equal to the hydrodynamic stresses of the liquids outside and inside the drop because it follows from assumption (v) that the electric stress is always zero inside the drop and the tangential electric stress component outside the drop is also zero on the drop surface. Thus,
σ ) -p(r)I + η[∇u(r) + ∇u(r)T]
with
κ)
(
1
N
∑
rokT i)1
y(r) )
zi2e2n∞i
σin ) -pin(r)I + ηd[∇uin(r) + ∇uin(r) ]
(28)
where I is the unit tensor. Finally, the boundary condition for the ionic flow vi is given by
(29)
which follows also from assumption (v).
(39)
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(0)(a) ) ζ
(40)
ψ(0)(r) f 0 as r f ∞
(41)
where ζ is the zeta potential (assumption (iv)). The small quantities are related to each other by
Linearized Equations
ni(r) )
(38)
eψ(0)(r) kT
For a weak shear field (assumption (ii)),
δni(r)
+ δni(r)
(30)
ψ(r) ) ψ(0)(r) + δψ(r)
(31)
µi(r) ) µ(0) + δµi(r)
(32)
Fel(r) ) F(0) el (r) + δFel(r)
(33)
n(0) i (r)
)
1/2
(27) T
vi ‚ n|r)a ) 0
(37)
zi2n∞i ∑ i)1
where n is the unit normal outward from the drop surface. (ii) The tangential component of the stress tensor is continuous, viz.,
(σ ‚ n) × n|r)a+ ) (σin ‚ n) × n|r)a-
zin∞i exp(-ziy) ∑ i)1
may be written, where the quantities with superscript (0) refer
δµi(r) ) zieδψ(r) + kT
∆δψ(r) ) -
δFel(r) ro
)-
1
n(0) i (r)
(42)
N
∑zieδni(r)
(43)
ro i)1
By substituting eqs 30-33 into eqs 17, 18, and 20 and neglecting the products of the small quantities, we obtain from
2866 Langmuir, Vol. 22, No. 6, 2006
Ohshima
eqs 17 and 18 after taking the curl to eliminate ∇ p(r) N
η∇ × ∇ × ∇ × u(r) )
∇δµi(r) × ∇n(0) ∑ i (r) i)1
(44)
where eq 52 corresponds to assumption (v) (eq 29) and eq 60 comes from the fact that δni(r) f 0 as r f ∞. Equation 49 subject to eq 60 is solved to give
hin(r) ) A(r3 - a2r)
ηd∇ × ∇ × ∇ × uin(r) ) 0
(45)
where A is a constant. From eq 60,
and from eq 20
(
)
1 (0) ∇ ‚ n(0) i (r) u(r) - ni (r)∇δµi(r) ) 0 λi
φi(r) r ‚ u(0)(r) r2
L(Lh) ) G(r)
(48)
L(Lhin) ) 0
(49)
)]
φi dy zi dφi λi 3h + 1) r dr r dr e r
(
(50)
N
∑ 2 dr i)1
G(r) ) -
zi2n∞i
exp(-ziy)φi
(51)
ηr L≡
d 1 d 4 d2 4 d 4 r ) 2+ dr r4 dr r dr r2 dr
(52)
In deriving eqs 48-50, we have again used ∇‚ u(0) ) 0 (incompressible flow), ∇ × u(0) ) 0 (symmetric flow), (r ‚ ∇)u(0) ) u(0) (linear flow), and ∇ × ∇ × ∇u ) L(Lh)(r/r) × u. The boundary conditions given in eqs 7 and 24-26 can be rewritten in terms of F(r) and Fin(r) as follows:
(
) 6aA
(62)
r)a-
|
(
) |
ηa d2h 1 - ) r)a+ 3 5ηd - 2η dr2
(63)
r)a+
h(r) )
(5ηd + 2η)a3 6(ηd + η)r2
(
-
ηda5 2(ηd + η)r4
3
2
+
5
)
7
r x rx x + G(x) dx + + ∫r∞ - 70 30 30r2 70r4
{
(5ηd + 2η)a3x2 x5 + 20(ηd + η)r2 60(ηd + η)r2 30r2 ηda5
(5ηd - 2η)a7
ηda5x2
140(ηd + η)r
20(ηd + η)r
+ 4
4
-
)
x7 G(x) dx (64) 70r4
Similarly, the integration of eq 50 subject to eqs 59 and 60 yields
φi(r) ) -
(
)∫ { ( ){
)}
dy zi dφi λi 3h + 1dr + dr r dr e r 3h 1 r dy 2 x5 zi dφi λi r - 3 dx (65) + 15 a dx x dx e x r
r2 2a5 + 5 15r3
∫
∞
a
(
(
)}
(53)
General Expression for the Primary Viscous Coefficient p
dh f 0 as r f ∞ dr
(54)
Following the theory of Watterson and White,9 we consider the volume-averaged stress tensor 〈σ〉 defined by
| )|
∫V σ dV
a 3
(55)
hin(a) ) 0
(56)
where 〈σ〉 stands for the average of the stress tensor σ over the suspension volume V. We use the following identity:9,30
(57)
〈σ〉 ) -〈p(r)〉 I + η(〈∇u(r)〉 + 〈∇u(r)T〉) + 1 [σ + p(r)I - η(∇u(r) + ∇u(r)T)] dV V V
h(a) )
|
dh 1 dhin - ) dr r)a+ 3 dr r)a-
d2h 2 dh 2h + dr2 a dr a2
dr2
h(r) f 0 as r f ∞ h
η
|
d2hin
Equation 48 subject to eqs 55 and 63 can formally be integrated to give
∫a∞
with
2e dy
) 2a2A and
r)a-
dh dr
(47)
Here, φi(r) is a function of r only. By using eqs 15, 16, and 47, eqs 44-46 can be rewritten in terms of φi(r), h(r), and hin(r) as
() [
|
may be written. Equations 57, 58, and 62 can be combined to give
Furthermore, symmetry considerations permit us to write
L
dhin dr
(46)
δµi(r) ) -zie
(61)
r)a+
(
) ηd
d2hin dr2
+
r)a-
(58) The boundary conditions for φi(r) are given below
|
(59)
φi(r) f 0 as r f ∞
(60)
dφi dr
r)a+
1 V
(66)
∫
)|
2 dhin 2hin - 2 a dr a
〈σ〉 )
) η(〈∇u(r)〉 + 〈∇u(r)T〉) + 1 [σ - η(∇u(r) + ∇u(r)T)] dV (67) V V
∫
where we have omitted the term in 〈p(r)〉 because the average pressure is necessarily zero. The integral in the second term on the right-hand side of eq 67 may be calculated for a single drop as if the other were absent and then multiplied by the particle number Np in the volume V because the integrand vanishes beyond
Primary ElectroViscous Effect in Mercury Drops
Langmuir, Vol. 22, No. 6, 2006 2867
the double layer around the drop and the suspension is assumed to be dilute. We transform the volume integral into a surface integral over an infinitely distant sphere S containing a single isolated drop at its center. This procedure avoids the integration on the internal stresses within the drop. Then, eq 67 becomes
Np V
∫S 1r {σ ‚ rr - (ur + ru)} dS
〈σ〉 ) η(〈∇u〉 + 〈(∇u)T〉) +
which is the required expression for the primary electroviscous coefficient p for a dilute suspension of mercury drops.
Low-Zeta-Potential Approximation We derive an approximate formula for the effective viscosity ηs applicable to the case where the zeta potential ζ is low. In this case, the Poisson-Boltzmann equation (eq 37) may be linearized to yield
) η(〈∇u〉 + 〈(∇u)T〉) + 1 3φ {σ ‚ rr - (ur + ru)} dS (68) 3 S r 4πa
a ψ(0)(r) ) ζ e-κ(r-a) + O(ζ2) r
∫
where
(77)
and eq 64 reduces to 3
φ)
(4π/3)a Np V
(69)
is the volume fraction of the mercury drops. By using the asymptotic form of F(r) given by eq 64, the second term on the right-hand side of eq 68 becomes
ηCr ∫S 1r {σ ‚ rr - (ur + ru)} dS ) 6φ a3
3φ 4πa3
h(r) )
(70) p) (71)
rf∞
6(ηd + η)r2
{
(72)
(
)
3C 6φ ηCr ) 2ηr 1 + φ 3 3 a a
(
(73)
)
{
}
6(ηd + η) C -1 5ηd + 2η a3
(75)
By evaluating C from eqs 64 and 71, we obtain
10(5ηd
)-
3ηda2e 5η(5ηd
{ ( N
) (
5ηd + 2η r2 + 3ηd a2 2(ηd + η) r5 G(r) dr 3ηd a5
∑zi2ni∞∫a + 2η) i)1
(
3ηd
1 dy
∞
a2
+
)}
N
zi2n∞i
() eζ
2
kT
(79)
3ηd
2 1 1 5ηd + 2η κa 1 η + e E3(κa) + 5 ηd + η 60κa 10 ηd + η
2ηd + η κa ηd(2ηd - η) κa e E6(κa) + e E7(κa) + ηd + η (ηd + η)2 2 ηdη 1 5ηd + 2η κa e E (κa) + (eκaE6(κa))2 7 8 2 2 η + η (η + η) d d
6
)} a5
ηd(5ηd + 2η)
2
(eκaE7(κa))2 +
49 ηd (eκaE8(κa))2 2 ηd + η
(ηd + η) 2 1 5ηd + 2η 2κa 1 5ηd + 2η 2κa e E3(2κa) e E6(2κa) 30 ηd + η 60 ηd + η 2
2 7 ηd(5ηd + 2η) 2κa 27 ηd e E (2κa) + e2κaE10(2κa) 8 20 (η + η)2 10 ηd + η d (80)
and En(κa) is the nth-order exponential integral, defined by
{
En(κa) ) an - 1
exp(-ziyi) 1 -
r2 dr 2(ηd + η) r5
) (
5ηd + 2η r2
ηe2
L(κa)
3 5ηd + 2η κa 2 7ηd + 3η κa e E4(κa) + e E5(κa) 10 ηd + η 5 ηd + η
(74)
The primary electroviscous coefficient p, defined by eq 4, is thus given by
1-
}( ) ∑
6(ηd + η) rokT 5ηd + 2η
zi2n∞i λi ∑ i)1
2
3C a3
ηs ) η 1 + φ
∞ ∫ a + 2η)
(78)
( ) ( ) ( ) ( ) ( ) { } { } ( ) { } ( ) ( ) ( ) { } ( )
L(κa) )
so that the effective viscosity ηs of a dilute suspension of mercury drops is given by
3ηda2
( )
+ O(ζ)
is obtained, where
Equation 68 thus becomes
p)
2(ηd + η)r4
i)1
〈∇u〉 + 〈(∇u)T〉 ) 2r
p)
ηda5
N
Also, to lowest order in the concentration of drops,
〈σ〉 ) 2ηr +
-
Thus
where C is defined by
C ) lim [r2h(r)]
(5ηd + 2η)a3
-κr
∫a∞ ern
dr
(81)
For the special case of rigid spheres (ηd f ∞), eq 80 reduces
φi(r) dr (76) to
2868 Langmuir, Vol. 22, No. 6, 2006
Ohshima
1 1 3 14 + eκaE3(κa) - eκaE4(κa) + eκaE5(κa) 60κa 2 2 5 25 κa κa κa 4e E6(κa) + 2e E7(κa) + (e E6(κa))2 2 49 κa 1 κa 2 2 30(e E7(κa)) + (e E8(κa)) - e2κaE3(2κa) 2 6 5 2κa 7 27 e E6(2κa) - e2κaE8(2κa) + e2κaE10(2κa) (82) 12 4 10
L(κa) )
φ((a) ) (
(
)
η ηd + η
[
2aλ( κen∞
× (1 - e-ζh/2)
1+
]
8n∞ {λ+ (1 - e-ζh/2)2 + λ-(1 - eζh/2)2} 5κ2(ηd + η) (88)
By substituting eq 88 into eq 85, we find that the electroviscous coefficient p is given by
which agrees with Booth’s result.5,31
Large-Ka Approximate Formula For 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-, a large-κa approximate formula for the electroviscous coefficient p on the basis of the approximation method employed in refs 24 and 32 can be derived. We denote φi(r) for cations and anions by φ+(r) and φ-(r), respectively. Equation 65 can be rewritten as
φ((r) ) φ((a) + ∆φ((r)
( [
p)
)
18η2 × 5ηd + 2η m+(1 - e-ζh/2)2 + m- (1 - eζh/2)2
5(ηd + η) + 6η{m+(1 - e-ζh/2)2 + m-(1 - eζh/2)2}
]
(89)
where
m( )
(83)
2rokT λ( 3ηz2e2
is the scaled ionic drag coefficient. We find from eq 4 that the effective viscosity p is given by
with
φ((a) ) -
a2 3
{
)a
∫a
(1 - e
2
)}
∫a∞ dy dr
3h z dφ( λ( 1( r dr e r
∞
-zy
)
{
(
2φ( r3
( )} dr
1 dφ( λ( d h - 2 ze dr r r dr
(84)
It can be shown24,32 that φ((r) ≈ φ( (a) for κa . 1. For large κa, (r - a)/a may be regarded to be on the order of 1/κa, and the quantity in the integrand of eq 76 may be expanded, obtaining
p)-
3ηda2z2en∞
∫
∞
5η(5ηd + 2η) a 5ηd + 6η r - a 2 ηd a
(
)( ) }
{( )(
[
5 ηs ) η 1 + φ × 2
dr
1 2η r - a + a r2 ηd dy -zy {e φ+(a) + e+zyφ-(a)} dr (85) dr
)
{
5ηd + 2η + 6η[m+(1 - e- ζh/2)2 + m-(1 - eζh/2)2]
()
{
ηs ) η 1 +
5ηd + 2η
(1 - eζh/2)φ-(a)] (86)
p) zeζ kT
(91)
)
(92)
which is the effective viscosity of a suspension of uncharged spherical rigid particles derived by Einstein.1 For comparison, we give below the corresponding large-κa approximate expression for p for the case of rigid spheres. One may assume that the particles are positively charged (i.e., ζ > 0). On the basis of the same approximation method given above,
where
ζh )
}
which agrees with the effective viscosity of a suspension of uncharged liquid drops derived by Taylor2 (eq 2). In the opposite limit of high zeta potentials, eq 90 tends to
(
)
φ
2(ηd + η)
5 ηs ) η 1 + φ 2
(
(90)
This is the required expression for the large-κa approximate formula for the effective viscosity ηs of a suspension of charged mercury drops. In the limit of low zeta potential, eq 90 reduces to
We use
d h 1 η 4zen∞ )+ [(1 - e-ζh/2)φ+(a) 2 dr r a ηd + η 5(ηd + η)κa
}]
5ηd + 5η + 6η[m+ (1 - e-ζh/2)2 + m- (1 - eζh/2)2]
[ (
) (
)(
)]
m1 + e-ζh/2 2 1 + e-ζh/2 72 m+ ln + ln 2 2 1 + 2F 2 (κa)
(87)
2
(93)
is finally obtained, with is the scaled zeta potential. Then eq 84 becomes (31) Honig, E. P.; Pu¨nt, W. F. J.; Offermans, P. H. G. J. Colloid Interface Sci. 1990, 134, 169. (32) Ohshima, H.; Healy. T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1613.
F)
2 (1 + 3m-)(eζh/2 - 1) κa
(94)
A similar expression has been obtained by Sherwood8 and Hinch and Sherwood.10
Primary ElectroViscous Effect in Mercury Drops
Langmuir, Vol. 22, No. 6, 2006 2869
Figure 1. Function L(κa) for a suspension of mercury drops (eq 80) in comparison with that for rigid spheres (eq 82).
Results and Discussion We have developed a theory of the primary electroviscous effect in a dilute suspension of charged mercury drops on the basis of the standard theory for a dilute suspension of rigid spheres given by Watterson and White.9 We derived a general expression for the electroviscous coefficient p of the suspension (eq 76). We have derived a low-zeta approximate analytical expression for the primary electroviscous coefficient p (eq 79) and the large-κa approximate expression for p (eq 89) as well as the corresponding expressions for the case of rigid spheres (eq 93). In Figure 1, we plot L(κa), which describes the primary electroviscous effect for low zeta potentials, for mercury drops (eq 80), and that for rigid spheres (eq 82). It is seen that the difference in L(κa) between mercury drops and rigid spheres is small for small κa (κa < 1) but becomes very large for large κa. Indeed, the large-κa asymptotic form of L(κa) for mercury drops is given by
L(κa) )
(
) ()
1 1 η 2 +O 5 ηd + η κa
(95)
whereas for rigid spheres it is given by
L(κa) )
( )
1 5 +O 2 (κa) (κa)3
(96)
That is, in the limit of κa f ∞, L(κa) remains finite for mercury drops but becomes zero for rigid spheres. In Figure 2, we plot the large-κa expression for p as a function of scaled zeta potential zeζ/kT for several values of κa for the case of rigid spheres calculated via eq 93 in comparison with the low-zeta results calculated using eq 82. Figure 2 shows that p has a maximum. This agrees with the exact numerical results of Watterson and White.9 The appearance of a maximum in p at intermediate κa values is due to the relaxation effect, as in the case of electrophoresis. The relaxation effect is characterized by a parameter F defined by eq 94, which corresponds to Dukhin’s number. For further increases in ζ, the function p approaches a nonzero limiting value given by
p)
72 m (ln 2)2 2 + (κa)
(97)
In Figure 3, we plot the large-κa limiting form of p as a function of scaled zeta potential zeζ/kT for the case of mercury drops calculated via eq 89 in comparison with the low-zeta results calculated using eq 80 and see that p shows no maximum. However, this does not necessarily mean that there is no maximum in p for finite κa values. Also note that as κa f ∞, p tends to zero
Figure 2. Primary electroviscous coefficient p as a function of scaled zeta potential zeζ/kT for a suspension of spherical rigid particles. Solid lines are large-κa results calculated using eq 93, and dotted lines are low-zeta results calculated using eq 82.
Figure 3. Primary electroviscous coefficient p as a function of scaled zeta potential zeζ/kT for a suspension of charged mercury drops. The solid line is the large-κa limiting form of p (eq 89), and the dotted lines are low-zeta results calculated using eq 80.
as 1/(κa)2 for rigid spheres (eq 96) but becomes a nonzero limiting value given by eq 90 for mercury drops. In the limit of very high zeta potentials, the effective viscosity of a suspension of mercury drops and that for charged rigid spheres both tend to that of uncharged rigid spheres (eq 1). That is, mercury drops with very high zeta potentials behave like rigid spheres. This phenomenon, called the “solidification effect”, is observed in other electrokinetic phenomena (i.e., electrophoresis and sedimentation potential32). Finally, we must mention that the present theory, which is based on the assumption that the drop surface is always equipotential, can also be applied to a liquid drop in general, provided that its surface is always equipotential.
Conclusions We have extended a theory of the primary electroviscous effect in a dilute suspension of charged spherical rigid particles in an electrolyte solution of Watterson and White9 to cover the case of a dilute suspension of charged mercury drops. We derived a general expression for the electroviscous coefficient p of the suspension (eq 76). We then derived an approximate analytical expression for the primary electroviscous coefficient applicable to mercury drops carrying low zeta potentials at arbitrary κa (eq 79) and expressions applicable to mercury drops as well as rigid spheres with arbitrary zeta potentials at large κa values (eqs 89 and 93). LA0525628
