12100
Langmuir 2005, 21, 12100-12108
Colloid Vibration Potential and Ion Vibration Potential in a Dilute Suspension of 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 July 9, 2005. In Final Form: October 5, 2005 A general electroacoustic theory is presented for the macroscopic electric field in a dilute suspension of spherical colloidal particles in an electrolyte solution, which consists of the colloid vibration potential (CVP) and the ion vibration potential (IVP), induced by an oscillating pressure gradient field due to an applied sound wave. This is a unified theory that unites previous theories for CVP and those for IVP. Approximate analytic expressions are derived for CVP and IVP. The obtained IVP expression agrees with Debye’s formula that is corrected by taking into account the force acting on the electrolyte ions as a result of the pressure gradient in the sound wave. The obtained CVP expression is correct to the first order of the particle zeta potential and applicable for arbitrary κa, where κ is the Debye-Hu¨ckel parameter and a is the particle radius. It is found that an Onsager relation holds between CVP and dynamic electrophoretic mobility. It is also shown that the CVP from particles with very small κa approaches IVP; that is, in the limit of very small κa a particle behaves like an ion.
Introduction Debye1 predicted an electroacoustic phenomenon that when a sound wave is propagated in an electrolyte solution, the motion of cations and that of anions may differ from each other because of their different masses so that periodic excesses of either cations or anions should be produced at a given point in the solution, generating vibration potentials. This potential is called the ion vibration potential (IVP). Debye1 derived an approximate expression for IVP. Oka2 extended Debye’s theory by accounting for the electrophoretic and relaxation effects as well as the effect of diffusion of ions. Hermans,3 however, showed that the contributions of these effects are small and pointed out the importance of the force acting on the electrolyte ions due to the pressure gradient in the sound wave. A more general expression for IVP was derived by Bugosh et al.4 An excellent review on IVP was given by Zana and Eager.5 The readers are also referred to a book by Dukhin and Goets.6 A similar electroacoustic phenomenon is expected to occur in a suspension of colloidal particles. Since colloidal particles are much larger and carry a much greater charge than electrolyte ions, the potential difference in the suspension is caused by the asymmetry of the electrical double layer around each particle rather than the relative motion of cations and anions. Herman,3 Enderby,7 and Enderby and Booth8 have developed a theory for a potential difference in the colloidal suspension, which is called colloid vibration potential (CVP). O’Brien9 developed a theory of electroacoustics dealing with CVP and its inverse phenomenon termed electrosonic amplitude (ESA). Oh* To whom correspondence should be addressed. Telephone and Fax: +81-4-7121-3661. E-mail:
[email protected]. (1) Debye, P. J. Chem. Phys. 1933, 1, 13. (2) Oka, S. Proc. Phys.-Math. Soc. Jpn. 1933, 13, 413. (3) Hermans, J. J. Philos. Mag. 1938, 25, 426. (4) Bugosh, J.; Yeager, E.; Hovorka, F. J. Chem. Phys. 1947, 15, 592. (5) Zana, R.; Yeager, E. Mod. Aspects Electrochem. 1982, 14, 3. (6) Dukhin, A. S.; Goetz, P. J. Ultrasound for Characterizing Colloids, Particle Sizing, Zeta Potential, Rheology; Elsevier: Amsterdam, 2002. (7) Enderby, J. A. Proc. Phys. Soc. 1951, 207A, 329. (8) Booth, F.; Enderby, J. A. Proc. Phys. Soc. 1952, 208A, 321. (9) O’Brien, R. W. J. Fluid Mech. 1988, 190, 71.
shima and Dukhin10 derived a simple expression for CVP of concentrated suspension of spherical colloidal particles with the help of Onsager’s reciprocal relation. Dukhin et al.11-13 and Shilov et al.14 further developed a general theory of CVP for concentrated suspensions. The readers are also referred to Zana and Eager,5 Marlow and Rowell,15 Hunter,16 and Dukhin and Goets.6 In the above-mentioned electroacoustic thoeries,1-16 IVP and CVP have been treated separately and thus it has not been made clear how theories for IVP and those for CVP are related to each other. In a real colloidal suspension in an electrolyte solution, however, IVP and CVP are both generated simultaneously so that a general unified theory covering both of IVP and CVP is needed. In the present paper, we derive a general electroacoustic theory for the electric field or the potential difference, which will be found to be the sum of CVP and IVP, caused by a sound wave in a dilute suspension of spherical colloidal particles in an electrolyte solution by taking explicitly into account the presence of both of the particles and the electrolyte ions. We obtain a simple analytic approximate expression for CVP for low zeta potentials at arbitrary κa values, where κ is the Debye-Hu¨ckel parameter and a is the particle radius. The mathematical analysis given in the present work is similar to that in static or dynamic electrophoresis problems.16-26 (10) Ohshima, H.; Dukhin, A. S. J. Colloid Interface Sci. 1999, 212, 449. (11) Dukhin, A. S.; Ohshima, H.; Shilov, V. N.; Goetz, P. J. Langmuir 1999, 15, 3445. (12) Dukhin, A. S.; Shilov, V. N.; Ohshima, H.; Goetz, P. J. Langmuir 1999, 15, 6692. (13) Dukhin, A. S.; Shilov, V. N.; Ohshima, H.; Goetz, P. J. Langmuir 2000, 16, 2615. (14) Shilov, V. N.; Borkovskaja, Y. B.; Dukhin, A. S. J. Colloid Interface Sci. 2004, 277, 347. (15) Marlow, B. J.; Rowell, R. L. J. Energy Fuels 1988, 2, 125. (16) Hunter, R. J. Colloids Surf. A: Physicochem. Eng. Aspects 1998, 141, 37. (17) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (18) Ohshima, H.; Healy. T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1613. (19) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1643.
10.1021/la0518593 CCC: $30.25 © 2005 American Chemical Society Published on Web 11/15/2005
Colloid and Ion Vibration Potentials
Langmuir, Vol. 21, No. 26, 2005 12101
The sample volume is small compared with the sound wavelength λ and thus ∇p∞ can be assumed to be practically uniform in the suspension. (v) The slipping plane (at which the liquid velocity relative to the particle becomes zero) is located on the particle surface. (vi) No electrolyte ions can penetrate the particle surface. (vii) The relative permittivity p of the particle is very small compared with that of the liquid (p , r) so that p is practically equal to zero The flow velocity of the liquid u(r, t) at position r and time t and that of the jth mobile ionic species vj(r, t) may be written as Figure 1. Spherical colloidal particle of radius a with velocity U exp(-iωt) in an electrolyte solution under an oscillating pressure gradient field ∇p∞ exp(-iωt) of angular frequency ω due to a sound wave.
Equations for Liquid and Ionic Flows Consider a dilute suspension of spherical particles of radius a moving with a velocity U exp(-iωt) in a liquid of viscosity η and relative permittivity r containing a general electrolyte in an applied oscillating pressure gradient field ∇p∞ exp(-iωt) due to a sound wave propagating in the suspension, where ω is the angular frequency (2π times frequency) and t is time. We treat the case in which ω is low such that the dispersion of r can be neglected. We imagine that ∇p∞ and U are both parallel to the z axis. The origin of the spherical polar coordinate system (r, θ, φ) is held fixed at the instantaneous center of one particle and the polar axis (θ ) 0) is set parallel to the z axis (Figure 1). Let the electrolyte be composed of N ionic mobile species of valence zj, mass mj, volume Vj, bulk concentration (number density) n∞j , and drag coefficient λj (j ) 1, 2, ‚‚‚, N). Since electroneutrality holds in the bulk solution phase N
zjn∞j ) 0 ∑ j)1
(1)
The drag coefficient λj of the jth ionic species is related to the limiting conductance Λ0j of that ionic species by 2
λj )
NAe |zj| Λ0j
(2)
where e is the elementary electric charge and NA is Avogadro’s number. The main assumptions in our analysis are as follows. (i) The particle radius a is much smaller than the sound wavelength λ so that the sound wave is not disturbed by the presence of the particles. (ii) The liquid can be regarded as incompressible. To be exact, this assumption is not consistent with electroacaustic phenomena, but holds good as long as IVP and CVP problems as well as dynamic electrophoresis are concerned. (iii) The applied pressure gradient field ∇p∞ is weak so that U is proportional to ∇p∞ and terms of higher order in ∇p∞ may be neglected. (iv) (20) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299. (21) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1992, 88, 3567. (22) Mangelsdorf, C. S.; White, L. R. J. Colloid Interface Sci. 1993, 160, 275. (23) Sawatzky, R. P.; Babchin. A. J. J. Fluid Mech. 1993, 246, 321. (24) Ohshima, H. J. Colloid Interface Sci. 1996, 179, 431. (25) Ohshima, H. J. Colloid Interface Sci. 1997, 195, 137. (26) Ohshima, H. Langmuir 2005, 21, 9818.
u(r, t) ) u(r) exp(-iωt)
(3)
vj(r, t) ) vj(r) exp(-iωt), (j ) 1, 2, ... N)
(4)
The fundamental electrokinetic equations are
∂ Fo [u(r,t) + Ue-iωt] ) -η∇ × ∇ × u(r,t) - ∇p(r,t) + ∂t N
λjnj(r,t){vj(r,t) - u(r,t)} ∑ j)1 ∇‚u(r,t) ) 0
(5) (6)
∂ mj [vj(r,t) + Ue-iωt] ) ∂t -λj{vj(r,t) - u(r,t)} - ∇µj(r,t) (7) ∂nj(r,t) + ∇‚(nj(r,t)vj(r,t)) ) 0 ∂t
(8)
with
µj(r,t) ) µj∞ + zjeψ(r,t) + kT ln nj(r,t) + p(r,t)Vj (9) Fel(r,t) ro
(10)
zjenj(r,t) ∑ j)1
(11)
∆ψ(r,t) ) N
Fel(r,t) )
where Fo is the mass density of the liquid, p(r, t) is the pressure, ψ(r, t) is the electric potential outside the sphere, µj(r, t) and nj(r, t) are respectively the electrochemical potential and the concentration (the number density) of the jth ionic species, µ∞j is a constant term in µj(r, t), k is Boltzmann’s constant, T is the absolute temperature, and o is the permittivity of a vacuum. Equations 5 and 6 are the Navier-Stokes equations and the equation of continuity for an incompressible liquid flow (assumption (ii)), where the term Fo(u‚∇)u has been omitted (assumption (iii)). Equations 7 and 8 are the equations of motion and of continuity for the flow of the jth ionic species, where d/dt has been approximately replaced by ∂/∂t. The term involving the particle velocity U exp(-iωt) in eqs 5 and 7 arises from the fact that the particle has been chosen as the frame of reference for the coordinate system. Equation 10 is the Poisson equation connecting the charge density Fel(r, t) (eq 11) and the electric potential ψ(r, t). When the ionic mass may be neglected (mj ≈ 0), eq 7 reduces to
-λj{vj(r,t) - u(r,t)} - ∇µj(r,t) ) 0
(12)
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Langmuir, Vol. 21, No. 26, 2005
Ohshima
so that the last term of the right-hand side of eq 5 (after substituting eq 12), which is the body force per unit volume acting on a volume element of the liquid, becomes N
which gives the following boundary condition for u(r):
u(r) f
N
nj(r,t)∇µj(r,t) ) Fel(r,t)∇ψ(r,t) + kT∑nj(r,t) ∑ j)1 j)1
(13)
The second term on the right-hand side of eq 13 is the osmotic pressure of electrolyte ions. Thus, eq 5 becomes
∂ Fo [u(r,t) + Ue-iωt] ) -η∇ × ∇ × u(r,t) - ∇p(r,t) ∂t Fel(r,t)∇ψ(r,t) (14) where the osmotic pressure of electrolyte ions has been included in the pressure term. In the present paper, we assume that the electrolyte ions are essentially point charges with negligible masses and use the approximate Navie-Stokes eq 14 instead of the original Navie-Stokes eq 5. As will be shown later, however, the ion vibration potential involves the small but finite volume and mass of the electrolyte ions so that we must retain ionic mass and volume terms that lead to the ion vibration potential and thus we use eq 7 (instead of eq 12) for the equation of motion of electrolyte ions.
∫0 ∫a dt[(ur cos θ - uθ sin θ +
Fo
π
rd
4 d [Ue-iωt] ) Fh U)e-iωt]2πr2 sin θ dr dθ + Fp πa3 3 dt (15)
(
)
where Fp is the mass density of the particle and U ) |U|. The magnitude of the hydrodynamic force Fh is given by
Fh )
∫0π(σrr cosθ - σrθ sin θ)2πr2 sin θ dθ
(16)
u ) 0 at r ) a
vj‚n|r)a ) 0
Linearized Equations For a weak field ∇p∞ (assumption (iii)), the electrical double layer around the particle is only slightly distorted. Then we may write
nj(r,t) ) n(0) j (r) + δnj(r) exp(-iωt)
(23)
ψ(r,t) ) ψ(0)(r) + δψ(r) exp(-iωt)
(24)
p(r,t) ) p(0)(r) + δp(r) exp(-iωt)
(25)
µj(r,t) ) µ(0) + δµj(r) exp(-iωt)
(26)
where the quantities with superscript (0) refer to those at equilibrium, i.e., in the absence of ∇p∞, and µ(0) j is a constant independent of r, and δnj(r), δψ(r), δµj(r), and δF(r) are perturbation quantities. Note that u(r), U, and vj(r) vanish when ∇p∞ ) 0 so that these can also be considered to be small quantities. We assume that the distribution of electrolyte ions (treated as point charges) at equilibrium n(0) j (r) obeys the Boltzmann equation and the equilibrium double layer potential ψ(0)(r) outside the particle satisfies the PoissonBoltzmann equation, both being functions of r only, viz.
(
∞ n(0) j (r) ) nj exp -
(
-iωFo[u∞(z) + U] ) -
dp∞ dz
u∞(z) f
|∇p | -U as r f ∞ iωFo
with N
F(0) el (r)
)
∑ j)1
N
zjen(0) j (r)
r2 dr (19)
∑ j)1
)
zjen∞j
(
exp -
(28)
)
zjeψ(0) kT
(29)
Here the volume of the electrolyte ions have been ignored. Equation 28 then becomes
1 d
∞
(27)
)
(18)
where U ) |U|. Thus, eqs 17 and 18 yield
)
zieψ(0) kT
F(0)(r) 1 d 2 dψ(0) r ) dr ro r2 dr
The pressure gradient ∇p(r) at position r far from the particle is given by ∇p∞ and thus
where r ) |r|. We now consider u(r) and p(r) far from the particle (i.e., r f ∞), which are in the z direction and depends only on z () rcos θ), so that they may be expressed as u∞(z) and p∞(z), respectively. In this case, from eqs 6 and 14, we obtain
(22)
where n is the outward normal to the particle surface.
Boundary Conditions
(17)
(21)
and the ionic species cannot penetrate the particle surface
where σrr and σrθ are the normal and tangential components of the hydrodynamic stress.
∇p(r) f ∇p∞ as r f ∞
(20)
The other boundary conditions are given from assumptions (v) and (vi). That is, the slipping plane, at which the liquid velocity relative to the particle is zero, is assumed to be located at the particle surface
Equation of Motion of a Particle Consider the forces acting on an arbitrary sphere S of radius r enclosing the particle at its center. The radius r of S is taken to be sufficiently large so that the net electric charge within S is zero. There is then no net electric force acting on S and one needs consider only the hydrodynamic force. The equation of motion for S is thus given by
|∇p∞| - U as r f ∞ iωFo
( ) r2
dy dr
[
N
)
]
zjn∞j exp(-zjy) ∑ j)1
κ2
N
∑ j)1
zj2n∞j
(30)
Colloid and Ion Vibration Potentials
with
κ)
(
1
N
∑ kTj)1
zj2e2n∞j
r o
y(r) )
Langmuir, Vol. 21, No. 26, 2005 12103
with
)
1/2
G(r) ) -
(31)
eψ(0)(r) kT
γ)
(32)
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
(33)
ψ (r) f 0 as r f ∞
(34)
(0)
where ζ is the zeta potential (assumption (v)). The small quantities are related to each other by
∆δψ(r) ) -
δFel(r)
)-
r0
1
∑
x
iωFo ) (1 + i) η
L≡
(36)
(
(39)
Further, symmetry considerations permit us to write
u(r) ≡ (ur,uθ,uφ) ) 2 1 d (rh(r)) ∇p∞ sin θ,0 (40) - h(r) ∇p∞ cos θ, r r dr
| |
(
| |
)
δµj(r) ) - zjeφj(r) E cos θ
(41)
δψ(r) ) -Y(r)E cos θ
(42)
Here h(r), φj(r), and Y(r) are functions of r only and eq 40 automatically satisfies eq 6. In terms of φj(r), Y(r), and h(r), we obtain from eqs 37-39
L(Lh + γ2h) ) G(r) Lφj - βj2(φj - Y) )
(
dφj 2λj h dy z dr j dr e r
(43)
)
(44)
and from eqs 35 and 36
LY ) -
1
dφj ) 0 at r ) a dr
(52)
φj(r) f 0 as r f ∞
(53)
Y(r) f 0 as r f ∞
(54)
dY ) 0 at r ) a dr
(55)
where eq 51 corresponds to the assumption (v) that u ) 0 at r ) a (eq 21); eq 52 follows from eqs 15 and 20; eq 53 states that electrolyte ions cannot penetrate the particle surface (assumption (vi)) (eq 22); eqs 54 and 55 state that δnj(r) f 0 and δψ(r) f 0 as r f∞; and eq 55 corresponds to the assumption (vii) that p , r so that p is practically equal to zero. Equation 43 subject to eqs 50 and 51 can formally be integrated to give
∫
i r r {H*(r′) - H(r′)} G(r′) dr′ + 1+ 2 a 2γa 3γ ∞ i 1 {H*(r) - H(r)} r r′3G(r′) dr′ + 1+ 2 2 2γa 3γ r r i [H(r)H*(r′) - H*(r)H(r′)]G(r′) dr′ + 2γ5r2 a
h(r) )
[
]
]∫
[
∫
[
]∫
∞ i a H(r) {H*(r) - H(r)} a H(r′)G(r′) dr′ + + γ4r2 H(a) 2γa H(a) - H(r) ∞ aΓ {H(a) - H(r)}G(r) dr + 4 2 H(a){H(a) - Γ} a γr 3aΓ H(a) - H(r) (56) 2ηγ4r2 H(a) - Γ
[
]∫
{
}
where
N
zj2e2n∞j exp(-zjy)(φj - Y) ∑ kTj)1 r o
(50)
)
and from eqs 7 and 8
-iωδnj(r) + ∇‚(n(0) j (r)vj(r)) ) 0
dh ) 0 at r ) a dr
a3(Fp - Fo) U 1 1 U as r f ∞ r + 2 |∇p∞| ηγ2 3For2 |∇p∞| (51)
(37)
(38)
x
(49)
N
λju(r) - ∇δµj(r) + iωmjU λj - iωmj
(47)
2 d 1 d 2 d2 2 d r ) + dr r2 dr dr2 r dr r2
η∇ × ∇ × ∇ × u(r) - iωFo∇ × u(r) )
vj(r) )
ωFo 1 ) (1 + i) 2η δ
(48)
By substituting eqs 3, 4, and 23-26 into eq 14, we obtain after taking the curl to eliminate ∇δp(r)
∇δµj(r) × ∇n(0) ∑ j (r) j)1
x
where δ ) (2η/ωFo)1/2 is the penetration depth in the liquid. The boundary conditions are given as follows:
h(r) f zjeδnj(r)
(46)
iωλj ωλj , βj2 ) 2kT kT
βj ) (1 - i)
(35)
N
r0j)1
zj2nj∞ exp(-zjy)φj ∑ ηr dr j)1
h)
δnj(r) δµj(r) ) zjeδψ(r) + kT (0) + Vjδp(r) nj (r)
N
e dy
(45)
H(r) ) (1 - iγr) exp[iγ(r - a)] -
γ2r3 3a
(57)
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Langmuir, Vol. 21, No. 26, 2005
Ohshima
H*(r) ) (1 + iγr) exp[- iγ(r - a)] -
γ2r3 3a
(58)
-
(γa)2 H(a) ) 1 - iγa 3
(59)
2(γa) (Fp - Fo) 9Fo
(60)
( ) ( ) ( ∫ ( )∑
N
1
∑ 3K*j)1 iω
2
Γ)
Co )
zj2e2n∞j λj
∞
3K* kT
r3 +
a
∫a∞
r3 +
a3
N
2
j)1
a3 dy
zj
2 dr
dφj
-
2λj h
dr
e r
)
dr -
zj2e2n∞j {exp(-zjy) - 1}{φj(r) Y(r)} dr (67)
with Similarly, integration of eq 44 (with eqs 52 and 53) and of eq 45 (with eqs 54 and 55) yields
( ) ( )∫ ∫( ) ( ( )∫ (
βj2 φj(r) ) 3
r′3 r - 2 {φj(r′) - Y(r′)} dr′ r
∫a
r
βj2 a3 r+ 2 3 2r 1 3
r
a
∞
a
K∞ )
{φj(r′) - Y(r′)} dr′ +
)
dφj 2λj h r′3 dy zj dr′ 2 dr′ dr′ e r′ r dφj 2λj h ∞ dy a3 1 r+ 2 a zj dr′ (61) 3 dr′ dr′ e r′ 2r r-
Y(r) ) -
K* ) K∞ - iωro
1
∫ 3 kT a r o
r
( )∑ ( )∫ ∑ r-
Y(r′)} dr′ +
r′3
N
2
r 1
)
r+
a
∞
a
zj2e2n∞j ×
3rokT j)1 2r exp(- zjy){φj(r′) - Y(r′)} dr′ (62) 2
Also it immediately follows from eqs 44 and 45 that
(69)
λj
Colloid Vibration Velocity The particle velocity U, which we call colloid vibration velocity, can be obtained from the boundary condition eq 52, viz.
[
U ) 2lim
j)1
N
∑ j)1
zj2e2n∞j
Here K∞ and K* are, respectively, the usual conductivity and the complex conductivity of the electrolyte solution in the absence of the particles. As will be seen later, CVP is expressed in terms of Co.
zj2e2n∞j exp(- zjy){φj(r′) 3
N
(68)
rf∞
]
h(r) 1 + 2 ∇p∞ r ηγ
From eqs 56 and 70, we obtain
U)
[
∫a∞ {H(a) - H(r)}G(r) dr +
2 3γ {H(a) - Γ} 2
H(a) ηγ {H(a) - Γ} 2
φj(r) f Y(r) f
Co r2 Co r2
as r f ∞
(63)
as r f ∞
(64)
where Co is a constant to be determined and -Co/a3 corresponds to the dipole coefficient. That is, outside the double layer, all of φj(r) and Y(r) take the same asymptotic form. With the help of eqs 62-65, we can obtain the following two different expressions for Co:
Co ) lim r2φj(r) ) rf∞
Y(r)} dr -
3
j z ∫a∞ (r3 + a2 ) dy dr ( j dr
dφ
3
1 3
Co ) lim r2Y(r) ) rf∞
∫a∞ (r3 + a2 ){φj(r) -
βj2 3
1
∞ ∫ a 3 kT r
( )∑ a3
)
2λj h dr (65) e r
N
zj2e2n∞j × 2 j)1 o exp(-zjy){φj(r) - Y(r)} dr (66) r3 +
By multiplying z2je2n∞j /λj on both sides of eq 65, summing over j, multiplying -iωro on both sides of eq 66, and adding these two equations, we obtain
(70)
]
∇p∞ (71)
Note that for uncharged particles (ζ ) 0) on the righthand side of eq 71 vanishes so that eq 71 becomes
Uuncharged )
H(a) ηγ2{H(a) - Γ}
∇p∞
(72)
which agrees with an expression for the velocity of an uncharged particle in an oscillating pressure gradient field.27 Macroscopic Electric Field: IVP and CVP Consider a suspension of Np identical spherical particles of radius a in a general electrolyte solution of volume V. The macroscopic electric field E exp(-iωt) in the suspension may be regarded as the average of -∇ψ(r, t) in the suspension over the sample volume V, viz.
Ee-iωt ) -
1 V
∫V ∇ψ(r,t) dV ) - V1 ∫V ∇δψ(r) dV e-iωt (73)
where we have used eq 24 and the fact that the volume average of ∇ ψ(0) is zero. In order to calculate E, we use the requirement that the volume-average current density in the suspension (27) Lamb, H. Hydrodynamics, 6th ed.; Cambridge University Press: Cambridge, U.K., 1975; Chapter XI, Art 363, p 660.
Colloid and Ion Vibration Potentials
〈i(t)〉 )
Langmuir, Vol. 21, No. 26, 2005 12105
∫V i(r,t) dV
1 V
must be zero. The current density i(r, t) at position r and time t can be written as N
N
(75)
i(r,t) ) i(r)e
-iωt
(76)
By substituting eqs 35 and 38 into eq 73 and neglecting the products of perturbation quantities, we obtain
i(r) ) N
∑ j)1
{
zjen(0) j (r)
}{
∇δnj(r)
λju(r) - zje∇δψ(r) - kT
λj - iωmj
i(r) ≈ -
∂t
where the second term comes from the displacement current. We may write
}
-
n(0) j (r)
∫ V S
Far from the particle (r f ∞), eq 77 becomes
(
N
∑ j)1 λ
i(r) ≈
j
- iωmj
∇δnj(r)
kT
){
zjen∞j
}
-
[( N
∫ ∑ V V j)1 λ
zjen∞j
∫ V V
){
λj∇p
j - iωmj
}
[ (
zjen∞j
N
∑ j)1 λ
i-
∇δnj(r)
kT
n∞j
Np
(
∑ j)1 λ
zjen∞j
- iωmj
j
n∞j
){
- iωmj
j
+ Vj∇p
}
∞
∞
)-
[( ) zjen∞j
N
∫S ∑ j)1
λj∇p
iωFo
{∑( ) N
zj2e2n∞j
j)1
λj
[
N
〈i(r)〉 ) K* E +
]
(84)
∑ j)1
( )(
(
)
zjen∞j mj - FoVj λj
Fo
)
mj - FoVj
λj
Fo
∇p∞ + K*E
∇p∞ -
]
3φCo K* ∇p∞ (85) 3 a
where
φ)
(80)
which holds for most practical cases. We note that in a statistically homogeneous suspension the average of ∇δnj is zero. Then by using eq 1, the first term on the righthand side of eq 79 becomes
∑ j)1
}
- iωro∇δψ(r) dV (79)
zjen∞j
]
- iωro ∇p∞
3φ K* Co∇p∞ a3
Further we assume that
N
]
where we have replaced i(r) by its asymptotic form (eq 77) and used eqs 41, 64, and 65. Thus, we find
- zje∇δψ(r) -
ωmj , λj
- zjeδψ(r) -
{r∇δµj(r)‚n - δµj(r)n} -
λj
4πNpCo V
)-
]
}
iωFo
iωro{∇δψ(r)‚n - δψ(r)n} dS
- zje∇δψ(r) -
iωF
){
λjδp(r)
+ Vjδp(r) n + iωroδψ(r)n dV )
∞
∇δnj(r) - Vj∇p∞ + iωro∇δψ(r) dV + kT ∞ nj 1
(83)
- Vj∇p∞ + iωro∇δψ(r) as r f ∞ (78)
n∞j
1
∇δµj(r) + iωro∇δψ(r) as r f ∞
λj
δnj(r)
V
where we have used eqs 1 and 20. By adding and subtracting eq 78 in the integrand of eq 74, we obtain
〈i〉 )
[
ri‚n -
kT
- zje∇δψ(r) -
iωFo
( )
N
∞
λj∇p
∑ j)1
zjen∞j
which will be used later. The integral in the second term on the right-hand side of eq 79 may be calculated for a single particle as if the others were absent and then multiplied by the particle number Np, since the integrand vanishes beyond the particle double layer. We transform the volume integral into a surface integral over an infinitely distant sphere S containing a single isolated particle at its center. Then, the second term on the righthand side of eq 79 becomes
N
Vj∇δp(r) + iωmjU + iωro∇δψ(r) (77)
(82)
and thus eq 78 can also be written as
∂∇ψ(r,t)
zjenj(r,t)vj(r,t) - ro ∑ j)1
i(r,t) )
∇δnj(r) ∇δµj ≈ zje∇δψ(r) + kT as r f ∞ n∞j
(74)
(81)
Note that since the asymptotic form of ∇δnj (obtained from eq 35), when the ionic volumes are neglected, given by
(4/3)πa3Np V
(86)
is the particle volume fraction. By remembering that must be zero, we find that
E ) EIV + ECV with
EIV ) -
1
N
∑ K*j)1
( )(
(87)
)
zjen∞j mj - FoVj λj
Fo
∇p∞
(88)
12106
Langmuir, Vol. 21, No. 26, 2005
ECV )
Ohshima
3φCo ∞ ∇p a3
(89)
where EIV is the ion vibration field and ECV is the colloid vibration field. Equation 87 is the required expression for the macroscopic electric field in a suspension of spherical colloidal particles of radius a and volume fraction φ in an electrolyte solution generated by an oscillating pressure gradient, where Co and K* are given by eqs 67 and 68, respectively.
we find that an Onsager relation holds between ECV and µ(ω), viz.
2roζ{1 - iγa(1 + 1/κa)} 3η{H(a) - Γ} (1 - iγ/κ)
[
Now we derive an approximate mobility formula applicable for the case where the zeta potential ζ is low. In this case, the solution to the Poisson-Boltzmann equation 30 yields
1+
(90) µ(ω) )
and we obtain from eqs 57 and 67
{
Co )
}
N
1+
K* ηγ {H(a) - Γ} 4
{H(a) - H(r)}dr + O(ζ2) )
∫a∞
( ) 1+
a3 dψ(0) 3
2r
dr
a3 dy
×
2r3 dr aΓroκ2
K*ηγ {H(a) - Γ} 4
[ {
)]
( )(
with
β ) κ - iγ
(94)
where En(z) is the exponential integral of order n. Substituting eq 93 into eq 90 yields
ECV ) -
[ { }
9φΓroκ2ζ
eβa E5(βa) 2K* a2ηγ4{H(a) - Γ} (γa)2 E3(βa) - H(a)eκa + iγaE4(βa) 3 2γ2 iγκa 1∇p∞ (95) 3κ β
( )(
)]
By comparing eq 95 with the following expression for the dynamic mobility µ(ω) of a spherical particle in an oscillating electric field22-24
µ(ω) )
[ {
roζ κ2 eβa E5(βa) - iγaE4(βa) 2 η γ [H(a) - Γ]
}
)] (96)
(γa)2 2γ2 iγκa E3(βa) - H(a)eκa + 13 3κ β
( )(
(98)
(99)
roζ(1 - iγa)
(100)
η{ H(a) - Γ}
Results and Discussion
×
eβa E5(βa) - iγaE4(βa) 2K ηγ4{H(a) - Γ} (γa)2 2γ2 iγκa E3(βa) - H(a)eκa + 1(93) 3 3κ β
}
}] 3
by a factor 1/(1 - iγ/κ). This is because eq 100 has been derived by assuming that γ , κ.
{H(a) - H(r)} dr + O(ζ2) (92)
3aΓroκ2ζ
roζ(1 - iγa)
µ(ω) )
By substituting eqs 91 into eq 92, we find that
Co ) -
2.5 κa(1 + 2e-κa)
η{H(a) - Γ} (1 - iγ/κ)
(91)
( )
zj2en∞j ∫a ∑ j)1
∞
1
which differs from O’Brien’s formula,9 viz.
3aΓ H(a) - H(r) + O(ζ) 2ηγ4r2 H(a) - Γ
aΓ
{
2 1+
×
For large κa, eq 98 further tends to
a ψ(0)(r) ) ζ eκ(r-a) + O(ζ2) r
h(r) )
(97)
Note that in a previous paper,24 we have derived an approximate expression for eq 96 without involving exponential integrals. The result is
µ(ω) )
Low Zeta Potential Approximation
φ(Fp - Fo) µ(ω)∇p∞ FoK*
ECV ) -
We have derived general expressions for the macroscopic electric field E, which is the sum of the ion vibration field EIV (eq 88) and the colloid vibration field ECV (eq 97) in a suspension of spherical colloidal particles of radius a and volume fraction φ in an electrolyte solution generated an oscillating pressure gradient. We have shown that an Onsager relation, eq 97 holds between ECV and dynamic electrophoretic mobility µ(ω) for the low ζ case. Note, however, that Onsager’s relation 97 always holds irrespective of the magnitude of the particle zeta potential ζ.9-12 An accurate approximate expression for µ(ω) applicable for high zeta potentials has recently been derived.26 Experimentally, the total potential difference TVP between two points in the suspension is observed. If the pressure difference between these two points is ∆P, then TVP is expressed as the sum of the ion vibration potential (IVP) and the CVP, viz.
TVP ) IVP + CVP with
IVP )
1
N
∑
K*j)1
CVP )
( )(
(101)
)
zjen∞j mj - FoVj λj
Fo
φ(Fp - Fo) µ(ω)∆P FoK*
∆P
(102)
(103)
which are obtained by replacing ∇p∞ with -∆P in the expressions for EIV and ECV (eqs 88 and 97). Note that EIV and ECV themselves are also often referred to as IVP and CVP, respectively. The total electric current TVI ) K*TVP generated in the suspension can also be observed,6 which is expressed as the sum of the ion vibration current (IVI) and the colloid vibration current (CVI), viz.
Colloid and Ion Vibration Potentials
Langmuir, Vol. 21, No. 26, 2005 12107
TVI ) IVI + CVI with N
IVI )
∑ j)1
( )(
CVI )
(104)
)
zjen∞j mj - FoVj Fo
λj
∆P
φ(Fp - Fo) µ(ω)∆P Fo
(105)
(106)
Debye’s equation1,3,4 for IVP is that for the amplitude Ψ of IVP at a point in the suspension, which is expressed in terms of the amplitude ao of the liquid velocity and the velocity g of the applied sound wave. Let us compare our result for IVP with Debye’s equation. In the absence of the particles, the liquid velocity u∞(z) and the electric potential ψ(z, t) are given by
u∞(z,t) ) ao exp(-iω(t - z/g))
(107)
ψ(z,t) ) Ψ exp(-iω(t - z/g))
(108)
Comparison of eq 19 with (U ) 0) and eq 107 yields
ao )
|∇p∞| iωz/g |∇p∞| e ≈ iωFo iωFo
(109)
where we have used the assumption that the sample volume is small compared with the sound wavelength λ ) 2πg/ω (assumption (iv)) and thus ao is practically uniform in the sample volume. Also from eq 108, we find that the electric field E(z, t) is related to ψ(z, t) by
E(z,t) ) -
∂ψ iω ) ψ(z,t) ∂z g
Figure 2. Magnitudes of CVI, IVI, and TVI divided by ∆P as a function of electrolyte concentration n for a suspension of spherical colloidal particles of radius a carrying zeta potential ζ in a KCl solution. Calculated with eqs 104-106 (as combined with eq 98) at ω/2π ) 1 MHz, φ ) 0.01, T ) 298 K, η ) 0.89 mPa s, r ) 78.5, ζ ) 20 mV, a ) 0.1 µm, Fo ) 1 × 103 kg/m3, Fp ) 3 × 103 kg/m3, Λ°+ ) 73.5 × 10-4 m2 Ω1- mol-1, V+ ) 3.7 × 10-6 m3/mol, m+ ) 39.1 × 10-3 kg/mol, Λ°- ) 76.3 × 10-4 m2 Ω1- mol-1, V- ) 22.8 × 10-6 m3/mol, and m- ) 35.5 × 10-3 kg/mol.
(110)
By combining eqs 109 and 110, we obtain
()
1 Φ IVP ) ∆P Fog ao
(111)
It thus follows from eqs 88 and 111 that
Φ)
gao
N
∑
K* j)1
( ) zjen∞j λj
(mj - FoVj)
(112)
which agrees with Debys’s IVP expression corrected by taking into account the force acting on the electrolyte ions as a result of the pressure gradient in the sound wave.1,3,4 In Figures 2 and 3, we give some of the results calculated with eqs 104-106 (as combined with eq 98) for IVI, CVI, and TVI for the typical case where ω/2π ) 1 MHz, φ ) 0.01, ζ ) 20 mV, a ) 0.1 µm, Fo ) 1 × 103 kg/m3, r ) 78.5 (water at 25 °C) , and Fp ) 3 × 103 kg/m3 (Figure 2) or Fp ) 1.1 × 103 kg/m3 (Figure 3) in a KCl solution at 25 °C (Λ°+ ) 73.5 × 10-4 m2 Ω-1 mol-1 and m+ ) 39.1 × 10-3 kg/mol for K+, and Λ°- ) 76.3 × 10-4 m2 Ω-1 mol-1 and m) 35.5 × 10-3 kg/mol for Cl-). For the ionic volumes for K+ and Cl- ions, we have used the values of their partial molar volumes reported by Zana and Yeager,5 that is, V+ ) 3.7 × 10-6 m3/mol (for K+) and V- ) 22.8 × 10-6 m3/mol (for Cl-). It is seen that IVI and CVI may become of the same order of magnitude especially when the particle mass density Fp does not differ much from the liquid mass density Fo (Figure 3) and that IVI increases rapidly with increasing electrolyte concentration n and may exceed CVI at very high electrolyte concentrations.
Figure 3. Same as Figure 2 but for Fp ) 1.1 × 103 kg m-3.
In the present work, we have developed an electroacoustic theory for a dilute suspension of colloidal particles. For concentrated suspensions, the dependence of CVP becomes different from that for dilute suspensions. It has been shown that the φ dependence of CVP is determined mainly by a factor (1 - φ)/(1 + φ/2).10-13 Thus, for concentrated suspensions, an approximate expression for CVI is given by
CVI )
(
)
φ(1 - φ) Fp - Fo µ(ω)∆P Fo (1 + φ/2)
(113)
Also, Dukhin et al. found that the liquid density Fo in eq
12108
Langmuir, Vol. 21, No. 26, 2005
Ohshima
113 must be replaced by the mass density Fs of the suspension.12 Finally, it is of interest to consider the limiting case of a very small particle with κa f 0 and γa f 0, in which case the particle should behave like an ion. In this limit, the particle dynamic mobility µ(ω) tends to Hu¨ckel’s static mobility expression
2roζ ζ µH ) 3η
(114)
where the zeta potential ζ is related to the particle charge Q by
ζ)
Q 4πroa
(115)
which is the Coulomb potential. If the particle of radius a and volume fraction φ carrying charge Q may be regarded as an ion of valence zp, bulk concentration (number density) np, mass density Fp, mass mp, volume Vp, and drag coefficient λp, and then we may put
4 λp ) 6πηa, Q ) zpe, Vp ) πa3, mp ) VpFp, 3 φ ) Vpnp (116) Equation 103 thus becomes
CVP f
zpenp(mp - FoVp) 2a2npQ(Fp - Fo) ∆P ) ∆P 9ηFoK* K*λpFo (117)
which is just the contribution from an ion to IVP, as is seen from a comparison with eq 102. It has thus been demonstrated that a small particle with κa , 1 and γa ,
1 indeed behaves like the corresponding ion. It is therefore suggested that the total vibration current TVI in a suspension containing small particles of radius a, charge Q, mass density Fp, and concentration (number density) np, is given by
2a2npQ(Fp - Fo) TVI ) ∆P + 9ηFo
N
∑ j)1
( )(
)
zjen∞j mj - FoVj λj
Fo
∆P
(118)
The first and second terms on the right-hand side of eq 118 are CVI (from small particles) and IVI, respectively. Note that the above treatment has assumed that the particle volume fraction φ is very low so that the thickness 1/κ of the double layer is much larger than the particle radius a but is much smaller than the interparticle distance d, i.e., a , 1/κ , d. Shilov et al. have recently proposed a new electroacoustic theory applicable for the opposite case (κd , 1).14 Conclusion We have derived a unified electroacoustic theory that unites previous theories for colloid vibration potential (CVP) and those for ion vibration potential (IVP). Analytic expressions for IVP, CVP, IVI, and CVI are given by eqs 102, 103, 105, and 106. An Onsager relation holds between CVP and dynamic electrophoretic mobility µ(ω). An approximate expression for µ(ω) applicable for low ζ and arbitrary κa is given by eq 98 and that applicable for arbitrary ζ and large κa (κa g ca. 30) has recently been derived in ref 26. The interrelation between CVP and IVP, which have been treated separately, has been elucidated. That is, it is shown that the CVP from very small particles reduces to IVP. LA0518593
