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Electrophoresis of a Sphere in a Spherical Cavity at Arbitrary Electrical Potentials Jhih-Wei Chu, Wen-Hsun Lin, Eric Lee, and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, ROC Received February 15, 2001. In Final Form: May 25, 2001 The electrophoretic behavior of a sphere in a spherical cavity is examined theoretically. We show that if the applied electric field is low, the mobility exhibits a local minimum as the thickness of double layer varies. This local minimum disappears if the applied electric field is high. If double layer is thin, the mobility becomes independent of the surface potential of the particle and the applied electric field. For low surface potentials, the mobility decreases with the increase in the applied electric field, but the reverse in true if the surface potential is high. If an electric field is applied in the direction from the bottom of the particle to its top, counterions are concentrated on the bottom of the particle. The presence of cavity wall will affect the thickness of double layer and the degree of double-layer polarization. For the case of thick double layer and small (particle radius/cavity radius) ratio, the contours of net space charge density near the cavity wall become elliptical.
Introduction The electrophoretic behavior of a charged colloid particle in an applied electric field depends on the physicochemical properties of both the particle and the liquid phase, the geometry of the system, and the magnitude of the applied electric field. A complete description of the problem under consideration involves a set of coupled partial differential equations, known as electrokinetic equations.1 Due to its nonlinear nature, solving these equations analytically is nontrivial, if not impossible. Available results in the literature are mainly based on idealized conditions. Typical assumptions made in the relevant analyses include the following: (i) The particle with the adjacent liquid within the shear surface is treated as a rigid sphere. (ii) The properties of the liquid phase, such as density, electric conductivity, electric permittivity, and viscosity are constant, and are position independent. (iii) The particle is nonconducting, and the charge on its surface distributes uniformly. (iv) The classic Gouy-Chapman theory is applicable for the description of the space charge density and electric potential within the electric double layer. (v) The boundary effects are negligible; i.e., an isolated particle in an infinite liquid is considered. (vi) The level of the applied electric field is low, and its effect can be simulated by considering a perturbation from the corresponding equilibrium state. In this case, the equation governing the transport of ions can be approximated by a linear expression. (vii) The electrical potential of the system under consideration is low relative to the thermal energy, and the equation governing the spatial variation of electrical potential can be approximated by a linear expression. (viii) The velocity of a particle is slow, and the resultant fluid flow has a negligible effect on the distributions of electrical potential and ion concentration. That is, the effects of double-layer polarization and relaxation can be neglected. In general, this assumption is realistic if κa .1, κ and a being respectively the reciprocal Debye length and the radius of the particle. On the basis of these * To whom correspondence should be addressed. Fax: 886-223623040. E-mail:
[email protected]. (1) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, U.K., 1989; Vols. 1 and 2.
assumptions, Smoluchchowski2,3 was able to derive the relation between the ζ potential and the electrophoretic mobility, U/E, of a particle
ζ U ) E 4πη
(1)
where U is the magnitude of the electrophoretic velocity, E is the strength of the applied electric field, and and η are respectively the dielectric constant and the viscosity of the liquid phase. Due to assumption viii, eq 1 is not applicable for medium values of κa. The effects of doublelayer polarization and relaxation were analyzed by Overbeek4 and Booth,5 and they found that the deviation from eq 1 increases with ζ. However, their quantitative validity was limited to low ζ. For a general ζ, a numerical scheme is necessary. Wiersema6 and O’Brien and White,7 for example, solved numerically the electrokinetic equations and obtained qualitatively similar results. They found that, if ζ is sufficiently high, U/E has a local minimum as κa varies, and the higher the ζ, the lower the local minimum. The result of Wiersema, however, diverges at high ζ, and that of O’Brien and White is restricted to assumption vi. The electrophoretic behavior of a polyion in an infinite fluid was examined recently by Allison and Nambi;8 the governing equations were solved by a combined DIE/finite difference algorithm. Their work was also limited to assumption vi; the same problem was also solved by a boundary element method.9 Allison and Nambi concluded that the accuracy of the spatial variations of electrical potential and ion density is crucial in the estimation of the electrophoretic mobility of a charged entity. (2) Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129. (3) Dukhin, S. S.; Derjaguin, B. V. Surface and Colloid Science; Wiley: New York, 1974; Vol. 7. (4) Overbeek, J. Th. G. Adv. Colloid Sci. 1950, 3, 97. (5) Booth, F. Proc. R.. Soc. London, Ser. A 1950, 203, 514. (6) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1966, 22, 78. (7) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (8) Allison, S. A.; Nambi, P. Macromolecules 1994, 27, 1413. (9) Allison, S. A. Macromolecules 1996, 29, 7391.
10.1021/la010244c CCC: $20.00 © 2001 American Chemical Society Published on Web 09/08/2001
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If the presence of a boundary needs to be considered, the problem becomes even more complicated. Previous results10-17 were also limited to assumptions i-viii. Often, it was assumed that the equation describing the electric field can be decoupled from the hydrodynamic equation. This is applicable only if κa f ∞, i.e., thin double layers. Zydney18 incorporated electric forces into the hydrodynamic equation and, thus, his results are applicable to all κa, but the effects of double-layer polarization and relaxation were neglected. Zydney18 examined the electrophoresis of a sphere at the center of a spherical cavity, and under the condition of low surface potential, the essentially one-dimensional problem was solved analytically. Although the geometry considered is an idealized one, it is capable of providing insights in the electrophoretic behavior of a colloid particle in a porous medium. The simple geometry was also adopted to simulate a concentrated colloidal dispersion.19-23 Lee et al.24-26 extended the analysis of Zydney by taking the effects of doublelayer polarization and relaxation into account. They showed that, similar to the case of an infinite fluid, if the surface potential is sufficiently high, the electrophoretic mobility has a local minimum as κa varies. Although assumptions vii and viii were eliminated in their study, the results of Lee et al.24-26 were limited to assumption vi. If assumption vi is applicable, the equation describing the transport of ions can be approximated by a linear expression. This implies that the magnitude of the applied electrical field will not affect the scaled mobility. In this case, the electrokinetic equations become linear, and the problem under consideration can be divided into two subproblems, which implies that tedious iterations can be avoided in the calculation of the electrophoretic mobility.7,24-26 In practice, however, since the magnitude of the electric field can have a significant effect on the spatial distribution of ionic concentrations, a more general treatment is highly desirable. This is done in the present study. Here, a pseudospectral scheme is adopted to solve the general electrokinetic equations for the case when a boundary is present. In particular, the effect of the strength of the applied field is discussed. The present analysis is based on the geometry of Zydney,18 which significantly reduces the magnitude of the computational effort required to solved the problem. Theory Referring to Figure 1, we consider a rigid, nonconducting spherical particle of radius a at the center of a noncon(10) Jorgenson, J. W. Anal. Chem. 1986, 58, 743A. (11) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417. (12) Keh, H. J.; Chiou, J. Y. Am. Inst. Chem. Eng. J. 1996, 42, 1397. (13) Morrison, F. A.; Stuhel, J. J. J. Colloid Interface Sci. 1970, 33, 88. (14) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377. (15) Keh, H. J.; Lien, L. C. J. Fluid Mech. 1991, 224, 305. (16) Keh, H. J.; Horng, K. D.; Kuo, J. J. Fluid Mech. 1991, 231, 211. (17) Feng, J. J.; Wu, W. I. J. Fluid Mech. 1994, 264, 41. (18) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476. (19) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (20) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1986, 112, 403. (21) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 127, 497. (22) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 129, 166. (23) Ohshima, H. J. Colloid Interface Sci. 1997, 188, 481. (24) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1997, 196, 316. (25) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65. (26) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1999, 209, 240.
Chu et al.
Figure 1. Schematic representation of the system under consideration where a sphere of radius a is placed at the center of a spherical cavity of radius b. A uniform electric field E is applied in the z direction. (r, θ, φ) denotes the spherical coordinates adopted; the origin is located at the center of the cavity.
ducting spherical cavity of radius b. A uniform electric field E with strength E is applied in the z direction. The spherical coordinates (r, θ, φ) with the origin located at the center of the cavity are adopted. The electrokinetic equations include the equation for the conservation of ions, that for the electrical potential, and the hydrodynamic equation. The conservation of ions leads to
{[
] }
∂nj zjenj )∇ B Dj nj + ∇ B φ - njb v ∂t kBT
(2)
where ∇ B is the gradient operator, nj, Dj, and zj are respectively the number concentration, the diffusivity, and the valence of ionic species j, e is the elementary charge, φ is the electric potential, b v is the fluid velocity, and kB and T are respectively the Boltzmann constant and the absolute temperature. We assume that the electrical potential can be described by the Poisson equation
∇2φ ) -
F
M
)-
∑ j)1
zjenj
(3)
where is the permittivity of the liquid phase, F is the space charge density, and M is the total number of ionic species. Suppose that the flow field around the particle can be described by the Navier-Stokes equations in the creeping flow regime with electrical body forces included. We have
∇ Bb v) 0
(4)
∂v b v - ∇p b - ∇F bφ Ff ) η∇2b ∂t
(5)
In these expressions p is the pressure and Ff and η are the fluid density and viscosity, respectively. We assume that the motion of the particle is slow so that the system is at a quasi-steady state, and the terms that involve the time derivative in the eq 5 can be neglected.
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For convenience the electrical potential φ is decomposed into the potential that would exist in the absence of the applied electric field φ1 (i.e., the equilibrium electrical potential) and the potential arising from the applied electric field φ2. Similar to the treatment of O’Brien and White,9 the distortion of double layer is simulated by considering a deviation function for electrical potential gj. We have
∇2φ1 ) -
F1
N
)-
∑ j)1
zjenj0
( ) zjeφ1
exp -
(6)
kBT
(
)
zje(φ1 + φ2 + gj) nj ) nj0 exp kBT
∇2φ2 ) -
F2
)-
(
N
∑ j)1
zjenj0
(
exp N
∑ j)1
(7)
) ( ))
zje(φ1 + φ2 + gj)
-
kBT
zjenj0
exp -
zjeφ1 kBT
(8)
φ ) φ1 + φ2
(9)
F ) F1 + F2
(10)
zje zje 1 1 ∇φ1∇gj ) u∇φ + u∇gj + ∇φ ∇g + ∇2gj kBT Dj Dj kBT 2 j zje ∇g ∇g (11) kBT j j where nj0 is the bulk concentration of ion species j. The pressure term in eq 5 can be eliminated by adopting the stream function representation24
1 B )(F∇ B (φ1 + φ2)) bı φE4ψ ) - (sin θ∇ η
(12)
where bıφ is the unit vector in the φ direction, ψ denotes the stream function, and the operator E4 is defined by E4 ) E2E2, where
sin θ ∂ 1 ∂ ∂2 E ) 2+ 2 ∂r r ∂θ sin θ∂θ 2
(
)
(12a)
The r and the θ components of velocity b v, vr and vθ, can be described respectively by
vr ) -
1 1 ∂ψ ∂ψ vθ ) r sin θ ∂r r2 sin θ ∂θ
(13)
Suppose that both the particle and the cavity are remained at constant potentials characterized by the ζ potentials ζa and ζb, respectively. Therefore, the boundary conditions associated with φ1 are
φ 1 ) ζa r ) a
(14)
φ 1 ) ζb r ) b
(14a)
The boundary conditions associated with the electrical potential induced by the applied electric field φ2 are assumed as
∂φ2 )0 r)a ∂r
(15)
∂φ2 ) -Ez cos θ r ) b ∂r
(15a)
The boundary conditions associated with eq 2 are assumed as
gj ) -φ2 r ) b
(16)
∂gj )0 r)a ∂r
(16a)
Note that the partition of φ into φ1 and φ2 if for convenience. Strictly speaking, φ1 and φ2 have only mathematical, but no physical, meaning. Since we assume that both the surfaces of particle and cavity are remained at constant potential, the boundary conditions assumed for electric field should satisfy φ ) ζa at r ) a and φ ) ζb at r ) b. Under the applied electric field, the particle moves in the z direction with velocity U of magnitude U. Suppose that the cavity is remained fixed. Therefore, the boundary conditions associated with the Navier-Stokes equation are
vr ) U cos θ vθ ) -U sin θ r ) a
(17)
v r ) 0 vθ ) 0 r ) b
(17a)
In terms of stream function ψ, these boundary conditions become
1 ∂ψ ψ ) - Ur2 sin2 θ ) -Ur sin2 θ r ) a (18) 2 ∂r ψ)
∂ψ )0 r)b ∂r
(18a)
The symmetric nature of the problem suggests that
∂φ1 ∂φ2 ∂g1 ∂g2 ∂ψ ) ) ) )ψ) )0 θ)0 θ)π ∂θ ∂θ ∂θ ∂θ ∂θ (19) Suppose that the liquid phase contains a single electrolyte (M ) 2). Let z1 and z2 be respectively the valences of cations and anions, with R ) - z2/z1. Then the electroneutrality in the bulk liquid-phase implies n20 ) (n10/R). If we define the reciprocal Debye length, κ, as 2
κ)[
nj0(ezj)2/kBT]1/2 ∑ j)1
(20)
then it can be shown that
n10z1 )
(κa)2kBT (1 + R)e2a2z1
(21)
Let UE ) (ζa2/ηa) be the magnitude of the velocity of the particle predicted by the Smoluchowski’s theory when an electric field of strength ζa/a is applied. For a simpler mathematical manipulation, all the governing equations are rewritten in the corresponding
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scaled forms. In the discussion below, a symbol with an asterisk denotes a scaled quantity. For example, the scaled number concentration of cations and anions, n1* and n2*, can be rewritten respectively as
function and the associated boundary conditions are
[( (
)
∂g2* (κa)2 ∂g1* ∂φ* n1* + Rn2* ∂r* ∂θ (1 + R) ∂r* ∂g2* ∂g1* ∂φ/ n1* + Rn2* sin θ (27) ∂θ ∂θ ∂r*
E*4ψ* ) -
) ]
n1* ) exp[- φr(φ1* + φ2* + g1*)]
(22)
n2* ) exp[Rφr(φ1* + φ2* + g2*)]
(22a)
1 ψ* ) - U*r*2 sin2 θ r* ) 1 2
(27a)
where φr ) ζaz1e/kBT denotes the scaled surface potential of particle. The governing equation for φ1 becomes
∂ψ* ) - U*r* sin2 θ r* ) 1 ∂r*
(27b)
and
∇/2φ1* ) -
2
(ka) 1 [exp(-φrφ1) - exp(aφrφ1)] (23) (1 + a) φr
If we assume that the particle is positively charged and the cavity is uncharged, then the boundary conditions for φ1* are
φ1* ) 1 r* ) 1
(23a)
φ1* ) ζb/ζa, r* ) 1/λ
(23b)
where r* ) r/a and λ ) a/b. It should be pointed out that the charged conditions assumed here are for illustration, and the present analysis is readily applicable to other possible charged conditions on particle and cavity surfaces. The variation of φ2* is described by
ψ* )
(κa) 1 {(n1* - n2*) (1 + R) φr [exp(-φrφ1*) - exp(Rφrφ1*)]} (24)
FEz )
∂φ2* ) -Ez* cos θ r* ) 1/λ ∂r*
(24b)
where Ez* ) Eza/ζa. The scaled gj, j ) 1 and 2, is described by 2
∇* g1* - φr∇*φ1*∇*g1* ) Pe1u*∇*φ1* + Pe1u*∇*φ2* + Pe1u*∇*g1* + φr∇*φ2*∇*g1* + φr∇*g1*∇*g1* (25)
Rφr∇*φ2/∇*g2* - Rφr∇*g2/∇*g2* (26)
∂φ1* ∂(φ1* + φ2*) cos θ ∂r* r*)1 ∂r* 1 ∂(φ1* + φ2*) sin θ r*2 sin θ dθ r* ∂θ r*)1
) 2πζa2Ez*KE
( ) (
∂g2* ∂g1* )0 ) 0 r* ) 1 ∂r* ∂r*
(26a)
g1* ) - φ2* g2* ) - φ2* r* ) 1/λ
(26b)
In these expressions, Pej ) UEa/Dj is the electric Peclet number for ionic species j, UE ) ζa2/ηa being the characteristic velocity. The scaled equation for stream
(29)
∂φ1* ∂(φ1* + φ2*) cos θ ∂r* r*)1 ∂r* 1 ∂(φ1* + φ2*) sin θ r*2 sin θ dθ (29a) r* ∂θ r*)1
∫0π
)
In spherical coordinates, the hydrodynamic force FDz can be calculated by
FDz ) ηπ
(
2
)
∫0π r4 sin3 θ∂r∂ r2Esinψ2 θ r)a dθ -
(
)
dθ ∫0π r2 sin2 θF∂φ ∂θ r)a
π
(
2
)
∂ E* ψ* dθ ∫0π r*4sin 3θ∂r* r*2 sin2 θ r*)1
) πζa2
(κa)2 - πζa2 (1 + R)φr
(
∫0π(r*2 sin2 θ(n1* - n2*) ∂θ2 )r*)1 dθ
) πζa2 U*KDf -
with
)
where
∇*2g2* + Rφr∇*φ1*∇*g2* ) Pe2u*∇*φ1* + Pe2u*∇*φ2* + Pe2u*∇*g2* -
(28)
( ) (
∫0π
FEz ) 2πζa2
Ez*KE ) (24a)
∫s∫σ(- ∇Bφ) dAB
where σ is the surface charge density. In spherical coordinates we have
The associated boundary conditions are
∂φ2* ) 0 r* ) 1 ∂r*
(27c)
The sum of the external forces acting on the particle in the z direction includes the electrostatic force FEz, and the hydrodynamic force FDz. The former can be calculated by
2
∇*2φ2* ) -
∂ψ* ) 0 r* ) 1/λ ∂r*
∂φ *
)
(κa)2 E *K (1 + R)φr z De
) FDf* + FDe*
(30)
where FDf* ) πζa2U*KDf, FDe* ) πζa2Ez*KDe(κa)2/(1 + R)φr, and
U*KDf )
(
(
/2
))
E ψ* ∂ ∫0π r*4 sin3 θ∂r* r*2 sin2 θ
r*)1
dθ
(30a)
Electrophoresis of a Sphere in a Spherical Cavity
Ez*KDe )
∫0π(r*2 sin2 θ(n1* - n2*) ∂θ2 )r*)1 dθ ∂φ *
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(30b)
FDf* and FDe* are respectively the scaled viscous force and the scaled electric body force exert on the particle. The mobility of the particle can be evaluated by the fact that the net force exerted on it vanishes at steady state; that is,
FDz + FEz ) 0
(31)
It can be shown that the scaled mobility of the particle, Um*, is
Um* )
2 U* (κa) [KDe/(1 + R)φr] - 2KE ηU ) ) ζaE Ez* KDf
(32)
Note that since KE, KDe, and KDf are all functions of the applied electric field and the scaled terminal velocity of the particle, the computation of the scaled mobility involves an iterative procedure. For a specified Ez*, an initial guess for U*, Ui*, is assumed which is used to calculate KE by eq 29, and KDf and KDe are calculated by eq 30. An updated Ui*, Ui+1*, is then evaluated by eq 32. This is continued until a convergent U* is obtained. A pseudospectral method based on Chebyshev polynomials24,25 that is readily applicable to the present problem is adopted for the resolution of the governing equations. The method adopted has a fast rate of convergence, and the convergent properties are independent of the associated boundary conditions. Also, the mini-max property typically associated with the Chebyshev polynomial is maintained. In the present case, the computational domain is two-dimensional, and the pseudospectral method is applied in both the r and the θ directions. For instance, the Nth-order × Mth-order approximation to the function fNM(r, θ) is expressed by N
fNM(r, θ) )
M
∑ ∑ fNM(ri, θj)gi(r)gj(θ)
(33)
i ) 0j ) 0
where fNM(ri, θj) is the value of fNM at the kth collocation point, where k ) [(N - 1)i + j]. The interpolation polynomials gi(r) and gj(θ) depend on the collocation points, and these points are determined by mapping the computational domain onto the square [-1,1] × [-1,1] through
r)
b+a b-a y+ 2 2
π θ ) (x + 1) 2
(34) (34a)
The N + 1 interpolation points in the interval [-1,1] are chosen to be the extreme values of an Nth-order Chebyshev polynomial TN(y)
yj ) cos
(πjN)
j ) 0, 1, ..., N
(35)
The corresponding interpolation polynomial gi(y) is
gj(y) )
Figure 2. Variation of scaled mobility Um* as a function of κa at various levels of scaled applied field Ez* for the case the sphere is positively charged with φr ) 1 and the cavity uncharged. Key: λ ) 0.5; R ) 1; Pe1 ) Pe2 ) 0.01. The linear model24-26 is based on the linearized approximations of eqs 22, 22a, 24-26, and 27.
(-1)j+1(1 - y2)[dTN(y)/dy] cjN2(y - yj)
j ) 0,1, ..., N (36)
Figure 3. Variation of scaled mobility Um* as a function of κa at various levels of scaled applied field Ez* for the case φr ) 3. Key: same as Figure 2.
where cj is defined by
cj )
{
2, j ) 0, N 1, 1 e j e N - 1
(37)
Both the partial derivatives and the integration of fNM(r, θ) are evaluated on the basis of eq 33. The corresponding nonlinear system is then solved with a Newton-Raphson iteration scheme. Double precision is used throughout the computation, and grid independence is checked to ensure that the mesh used is fine enough. For each κa, the calculations consist of 39 × 25 nodal points for (Nr × Mθ). Results and Discussion The multiplication of Dj with the terms in the bracket on the right-hand side of eq 2 is the well-known NernstPlanck equation, which is the special case for ionic flux when the convection (or polarization) effect arising from the fluid velocity b v is neglected. This equation leads to a Boltzmann distribution for each ionic species. This is satisfactory for the case when the applied electric field is
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Figure 4. Variation of scaled mobility Um* as a function of κa at various levels of scaled applied field Ez* for the case φr ) 5. Key: same as Figure 2.
Figure 5. Variation of 2KE as a function of κa at various levels of scaled applied field Ez* for the case φr ) 3. Key: same as Figure 2.
weak, since both φ2 and gj in eq 7 is much smaller than φ1. On the other hand, if the applied electric field is strong, the convection term in eq 2 may not be neglected, and the magnitudes of φ2 and gj can be appreciable. Apparently, the Boltzmann distribution is no longer valid, and the general Poisson equation, eq 3, should be considered. In this case, eqs 6-10 needed to be solved simultaneously for electric field. The problem of weak applied electric field was considered in our previous study,24,25 and the problem of weak applied electric field and low surface potential of particle was analyzed by Zydney.18 The behavior of the system under consideration is examined through numerical simulation. In a study of the drift velocity of particle in water when a strong electric field is applied, Pikhitsa et al.27 found that specific behavior is observed. The strength of electric field in their experiments ranges from 250 to 1000 V/cm. Let us consider a particle of radius 10 µm and surface potential 0.0771 V; that is, φr ) 3. If the strength of applied electric field Ez ) 77.1 V/cm, then Ez* ) 1, and if Ez ) 385 V/cm, then Ez* (27) Pikhitsa, P. V.; Tsargorodskaya, A. B.; Kontush, S. M. J. Colloid Interface Sci. 2000, 230, 334.
Chu et al.
Figure 6. Variation of 2KE as a function of κa at various levels of scaled applied field Ez* for the case φr ) 5. Key: same as Figure 2.
Figure 7. Contours for the net scaled ion concentrations CD ()n1* - n2*) for the case φr ) 3, κa ) 0.01, λ ) 0.5, Ez* ) 0.01, Pe1 ) Pe2 ) 0.01, and R ) 1.
) 5. For the case φr ) 5, if Ez ) 128.5 V/cm, then Ez* ) 1, and if Ez ) 642 V/cm, then Ez* ) 5. Therefore, the upper limit for Ez* in the numerical simulation is chosen to be on the order of 10. It should be pointed out that the socalled strong applied electric field is relative to that induced by a charged particle. The experimental observation of Pikhitsa et al.27 also reveals that previous results based on weak applied electric field (Ez* , 1) are insufficient to correlate their data. Figure 2 shows the variation of the scaled mobility Um* as a function of κa at various levels of scaled applied field Ez* for the case of low surface potential (φr ) 1); the corresponding results for higher surface potentials are presented in Figures 3 and 4. For comparison, the results predicted by the linear model of Chu et al.24-26 which are based on the linearized approximations of eqs 22, 22a, 24-26, and 27 are also presented in Figures 2-4. As suggested by Figure 2, if the surface potential is low, then using the linearized model is appropriate even if the applied electric field is in a medium range. Figures 3 and 4 reveal that, however, as the surface potential becomes high, even the applied electric field is low and the deviation
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Figure 8. Contours for the net scaled ion concentrations CD ()n1* - n2*) for the case of Figure 7 except κa ) 5.6 and Ez* ) 1.
Figure 10. Contours for the net scaled ion concentrations CD ()n1* - n2*) for the case of Figure 9 except Ez* ) 5.
Figure 9. Contours for the net scaled ion concentrations CD ()n1* - n2*) for the case of Figure 7 except Ez* ) 1.
Figure 11. Variation of scaled mobility Um* as a function of κa at various λ ()a/b) for the case φr ) 1, Ez* ) 3, Pe1 ) Pe2 ) 0.01, and R ) 1.
of the linear model from the present nonlinear model is significant. For example, Figure 3 shows that if Ez* is low, Um* has a local minimum as κa varies, which does not appear in Figure 2. This local minimum disappears, however, if Ez* is sufficiently high. Figure 3 also shows that the deviation of the linear model from the corresponding nonlinear model increases with the level of Ez*. However, if κa f 0, both models lead to the same Um* regardless of the levels of φr and Ez*, as can be seen from Figures 2-4. The disappearance of the local minimum of Um* against κa curve in Figure 3 for the case when Ez* is high can be explained by the behavior of KE as κa varies. According to eqs 29 and 30, KE is a measure for the contribution to electrophoretic mobility by the electrostatic force. In eq 29, (∂φ1*/∂r*)r*)1 increases with κa, but the reverse is true for (∂φ2*/∂θ*)r*)1. This is because the larger the κa, the faster the spatial variation of electrical potential in the double layer, i.e., the larger the -∇*φ1*. This has the effect of reducing the influence of the applied electric field. The variation of 2KE as a function of κa at various levels of scaled applied field Ez* for the case φr ) 3 is presented in Figure 5, and that for a higher φr is shown in Figure 6. These figures reveal that if Ez* is low, KE has
a local minimum as κa varies, and the higher the φr, the more pronounced the local minimum. However, if Ez* is sufficiently high, the local minimum disappears. This is because if Ez* is low, KE is dominated by (∂φ1*/*r*)r*)1, and it is dominated by (∂φ2*/∂θ*)r*)1 if Ez* is high. It is interesting to note that if φr is low, Um* decreases with the increase in Ez* (Figure 2), but the reverse in true if φr is high (Figures 3 and 4). This is mainly due to the combined effect of (κa)2KDe/(1 + R)φr, 2KE, and KDf in the definition of Um*, eq 32. It should be pointed out that the numerical scheme adopted in the present study becomes inefficient if κa is greater than about 10, and therefore, the results for κa f ∞ are not shown. This is because that if κa is large, the double layer surrounding a particle becomes thin, and the spatial variation in the electrical potential inside is very steep. If κa is large, solving the governing equations numerically is not recommended, and a semianalytical approach such as perturbation method is suggested. Figures 7-10 illustrate the contours for the net scaled ion concentrations, CD, defined as (n1* - n2*) for various combinations of κa and Ez*. Since the particle is positively charged, CD is negative. Figure 7 shows that if Ez* is low, the contour is almost circular even if the double layer is
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Figure 12. Contours for the net scaled ion concentrations CD ()n1* - n2*) for the case φr ) 1, κa ) 5.6, λ ) 0.667, Ez* ) 3, Pe1 ) Pe2 ) 0.01, and R ) 1.
Chu et al.
Figure 14. Variation of the percent deviation of the linear model from the present nonlinear model as a function of κa at various levels of scaled applied field Ez* for the case of Figure 3.
Figure 13. Contours for the net scaled ion concentrations CD ()n1* - n2*) for the case φr ) 1, κa ) 5.6, λ ) 0.667, Ez* ) 3, Pe1 ) Pe2 ) 0.01, and R ) 1.
thick; that is, double-layer polarization is insignificant. This is expected since the velocity of the particle is slow if Ez* is low. Figure 8 suggests that if the double layer is thin, its polarization is also insignificant even if the applied electric field is appreciable. As stated previously, this is because - (∂φ1*/∂r*)r*)1 increases with κa, which has the effect of reducing the influence of the applied electric field. Double-layer polarization becomes significant, however, if Ez* is sufficiently high or the double layer is sufficiently thick, as suggested by Figures 9 and 10. These figures reveal that the concentration of anions (counterions) near the bottom of the particle is higher than that near the top of the particle; that is, anions are concentrated near the bottom of the particle. This is because the movement of the particle is in the direction pointing toward the top of the particle. Figures 11-13 simulate the effect of the presence of a boundary (cavity wall) on the electrophoretic behavior of a particle. The variation of scaled mobility Um* as a function of κa at various λ ()a/b) is shown in Figure 11. Figures 12 and 13 illustrate the contours for the net scaled ion concentrations CD defined as n1* - n2* for two different
Figure 15. Variation of scaled mobility Um* as a function of scaled strength of applied electric field Ez* at different κa for the case λ ) 0.5, Pe1 ) Pe2 ) 0.01, and R ) 1. Key: (a) φr ) 3; (b) φr ) 5.
Electrophoresis of a Sphere in a Spherical Cavity
combinations of κa and λ. Figure 11 shows that, for fixed surface potential and applied electric field, Um* decreases with the increase in λ. That is, the closer the particle to the cavity wall, the smaller the mobility. Apparently, the flow field is influenced by the presence of the boundary, which has the effect of reducing the terminal velocity of the particle. The presence of the boundary will also affect both the variation in the thickness of double layer and the degree of double-layer polarization. A comparison between Figures 8 and 12, for example, reveals that the rate of decrease in the thickness of double layer is slower if λ is larger. As can be seen in Figure 13, if the double layer is thick and λ is small, the effect of the top and the bottom parts of the cavity wall on CD is more significant than that of the left and the right parts of the cavity wall. This leads to the elliptical-shaped contours near cavity wall. This may be due to the fact that the movement of the particle is in the z direction. Figure 14 illustrates the variation of the percent deviation in the scaled mobility based on the linear model of Chu et al.24-26 from that based on the present general nonlinear model as a function of κa at various levels of the applied electric field Ez* for the case of Figure 3. This figure reveals that the percent deviation has a local maximum for κa in a medium range. In general, the effect in the applied electric field is insignificant for Ez* , 1, but it becomes appreciable if Ez* is on the order of unity. Furthermore, numerical calculation reveals that if φr is
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6, then the maximal percent deviation exceeds 50% at Ez* ) 0.01. That is, the nonlinear effect in the governing equation of electric field can be significant even if the applied electric field is low. The variation of the scaled mobility Um* as a function of the scaled strength of applied electric field Ez* at different κa is presented in Figure 15. In general, Um* increases with Um*, as expected. However, if double layer is thick, the influence of Ez* on Um* is inappreciable for the range of applied electric field examined, but it becomes significant if the double layer is thin. Under the conditions assumed, a dramatic increase in Um* is observed as Ez* is higher than unity. In summary, the electrophoresis of a sphere in a spherical cavity is investigated for the case of an arbitrary level of both the applied electric field and the potential on particle surface. The results obtained provide new insights about the boundary effect on the electrophoretic behavior of a spherical particle. In addition to some new observations, which are not found for the case of low applied electric field, the exact shape of a disturbed ionic cloud under the polarization effect, which characterizes the present nonlinear phenomenon, is determined. Acknowledgment. This work is supported by the National Science Council of the Republic of China. LA010244C
