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Electrophoresis of a Sphere at an Arbitrary Position in a Spherical Cavity Jyh-Ping Hsu,* Shih-Hsing Hung, and Chen-Yuan Kao Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. Received May 28, 2002. In Final Form: August 22, 2002 The boundary effect on electrophoresis is of both fundamental and practical significance. Here, we consider the case when a sphere is placed at an arbitrary position in a spherical cavity. We show that the qualitative behavior of the particle depends largely on the charged conditions on both particle surface and cavity surface, and the presence of the latter may have the effect of increasing the mobility of the particle. For example, for the case when both particle and cavity are maintained at constant surface potential, if the particle is close to the cavity surface, charge reversal may occur on the latter. In this case, the closer the particle to cavity surface, the greater its mobility. It is found that the thicker the double layer surrounding the particle and/or the greater the difference between the surface potential of the particle and that of the cavity, the easier it is for the charge reversal to occur.
1. Introduction Electrophoresis is one of the most important electrokinetic phenomena, and it has various applications in practice. It is widely adopted for analysis of the surface properties of entities of micron and submicron sizes such as microorganisms, biological cells, and inorganic colloidal particles. Apparently, a detailed understanding of the phenomenon under general conditions is highly desirable for both fundamental theory and applications considerations. This, however, is not an easy task even under an idealized condition. The difficulty arises from the complicated interactions between hydrodynamic and electric effects, which lead to a set of coupled, nonlinear differential governing equations, the so-called electrokinetic equations. For a rigid, isolated, nonconducting particle in a liquid phase of constant physical properties under the conditions of low electric potential and thin double layer, Smoluchowski1 was able to derive a concise expression for the variation of the electrophoretic velocity of a particle, U, as a function of its zeta potential, ζ, and the applied electric field, E, as
U)
ζE η
(1)
where and η are the permittivity and the viscosity of the liquid phase, respectively. The ratio U/E is known as the electrophoretic mobility of the particle. Morrison2 concluded that if the local radius of curvature of a particle is much larger than the thickness of the double layer surrounding it, eq 1 is shape independent. In many applications of electrophoresis, charged entities move under the influence of adjacent particles or the presence of a boundary. A typical example for the former includes, for example, the electrophoresis of a concentrated dispersion where the interaction between neighboring particles is significant. That for the latter includes, for instance, the electrophoretic separation of proteins where * To whom correspondence should be addressed. Fax: 886-223623040. E-mail:
[email protected]. (1) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1989; Vol. I. (2) Morrison, F. A. J. Colloid Interface Sci. 1970, 34, 210.
charged entities move through a pore in a membrane. In these cases, modification of eq 1 is necessary from a practical point of view. Analyses about the boundary effect on electrophoresis are ample in the literature.3-16 Morrision and Stukel,3 for example, considered the electrophoresis of a sphere normal to a plane for the case of a thin double layer and low surface potential. Keh and Anderson4 studied the electrophoresis of a charged, nonconducting sphere moving parallel to a flat plate, normal to a plane, and along the axis of a cylindrical pore. Shugai and Carnie5 investigated the electrophoresis of a particle normal to a plane for the case of low electrical potential and arbitrary double layer thickness. Zydeny6 and Lee et al.7,8 examined the electropheretic behavior of a spherical particle at the center of a spherical cavity. This geometry is of course an idealized one, but it can be used to simulate the electrophoresis in a porous medium or in a cylindrical geometry. Another merit of that approach is substantial simplification in mathematical treatment can be made. The sphere-spherical cavity model used by Zydney6 and Lee et al.7,8 has the disadvantage that the behavior of a particle which is close to the cavity surface cannot be simulated. This can be crucial, unfortunately, for applications such as electrodeposition or electrophoresis occur in a narrow duct. In the present study, the problem considered by Zydeny and Lee et al. is extended to the case where a particle can assume an arbitrary position in a cavity. (3) Morrison, F. A.; Stukel, J. J. J. Colloid Interface Sci. 1969, 33, 88. (4) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417. (5) Shugai, A. A.; Carnie, S. L. J. Colloid Interface Sci. 1999, 213, 298. (6) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476. (7) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1997, 196, 316. (8) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65. (9) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377. (10) Keh, H. J.; Lien, L. C. J. Chin. Inst. Chem. Eng. 1989, 20, 283. (11) Keh, H. J.; Lien, L. C. J. Fluid Mech. 1991, 224, 305. (12) Keh, H. J.; Chiou, J. Y. AIChE J. 1996, 42, 1397. (13) Keh, H. J.; Jan, J. S. J. Colloid Interface Sci. 1996, 183, 458. (14) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (15) Chu, J. W.; Lin, W. H.; Lee, E.; Hsu, J. P. Langmuir 2001, 17, 6289. (16) Keh, H. J.; Horng, K. D.; Kuo, J. J. Fluid Mech. 1991, 231, 211.
10.1021/la0204996 CCC: $22.00 © 2002 American Chemical Society Published on Web 10/05/2002
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Hsu et al.
∇2Ψ1 ) κ2Ψ1
(3)
( )
where κ is the inverse Debye length defined by
∑j zj2nj°
e2
κ)
1/2
kBT
(3a)
Similarly, the electrical potential associated with the applied electric field, Ψ2, is described by
∇2Ψ2 ) 0
(4)
We consider two types of surface conditions: constant surface potential and constant surface charge density. For the former, the boundary conditions associated with eqs 3 and 4 are
Figure 1. Schematic representation of the problem considered. A charged sphere is placed inside a spherical cavity. An electric field E parallel to the axis of the pore is applied. a and b are respectively the radii of the sphere and of the spherical cavity. The center of the particle is placed at z ) m, and that of the cavity at z ) 0.
2. Theory The system under consideration is illustrated in Figure 1, where a nonconducting, spherical particle of radius a is located at an arbitrary position in a spherical cavity of radius b. The center of the particle is at z ) m, and that of the cavity is at z ) 0. A uniform electric field E in the z-direction is applied. The medium between the particle and the cavity is an aqueous solution containing z1/z2 electrolyte. Suppose that the movement of the particle is slow so that the system is at a quasi-steady state, and the distribution of ions follows the Boltzmann distribution. Suppose that the electrical potential Ψ can be described by the Poisson-Boltzmann equation
∇2Ψ ) -
F
N
∑ j)1
)-
zjenj° exp(-zjeΨ/kBT)
(2)
where ∇2 is the Laplace operator, is the permittivity of the liquid phase, F is the space charge density, N is the number of kinds of ionic species, nj° and zj are the bulk number concentration and the valence of ionic species j, respectively, e is the elementary charge, kB is the Boltzmann constant, and T is the absolute temperature. We assume that the applied electric field is weak relative to the electric fields induced by the particle. In this case, the total electrostatic potential Ψ can be expressed as a linear superposition of the electrical potential in the absence of the applied electric field (i.e., equilibrium potential), Ψ1, and the electrical potential outside the particle that arises from the applied field, Ψ2.17 Suppose that the distortion of the ionic cloud surrounding the particle is negligible. If the electrical potential is low, the Debye-Hu¨ckel approximation is applicable, and Ψ1 can be described approximately by (17) Henry, D. C. Proc. R. Soc. A 1931, 133, 106.
Ψ1 ) ζa on the particle surface
(5)
Ψ1 ) ζb on the cavity surface
(6)
n‚∇Ψ2 ) 0 on the particle surface
(7)
n‚∇Ψ2 ) -Ez cos θ on the cavity surface
(8)
In these expressions, ζa and ζb are respectively the surface potentials of the particle and the cavity, and n is the unit normal directed into the liquid phase. For the case of constant surface charge density, the boundary conditions associated with eqs 3 and 4 become
-σa on the particle surface
(9)
-σb on the cavity surface
(10)
n‚∇Ψ2 ) 0 on the particle surface
(11)
n‚∇Ψ2 ) -Ez cos θ on the cavity surface
(12)
n‚∇Ψ1 )
n‚∇Ψ1 )
In these expressions, σa and σb represent respectively the surface charge densities on particle and cavity surfaces. We assume that the flow field can be described by the Navier-Stokes equation in the creeping flow regime
η∇2u - ∇p ) F∇Ψ
(13)
∇‚u ) 0
(14)
where u, η, and p are respectively the velocity, the viscosity, and the pressure of the liquid phase. The term on the right-hand side of eq 13 represents the electrical body force. For simplicity, we consider an incompressible fluid with constant physical properties. Let U be the magnitude of the particle velocity in the z-direction. The boundary conditions associated with eqs 13 and 14 are assumed as
u ) Uiz on the particle surface
(15)
u ) 0 on the cavity surface
(16)
where iz is the unit vector in the z-direction. Equations 15 and 16 arise from the no-slip conditions on particle and cavity surfaces. The total force acting on the particle comprises the hydrodynamic force and the electrostatic force. The
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axisymmetric nature of the problem under consideration implies that only the forces in the z-direction needed to be evaluated. The electrostatic force acting on the particle in the z-direction, FzE, can be calculated by
FzE )
dS ∫∫σEz dS ) ∫∫σ(- ∂Ψ ∂z ) S
(17)
S
where S denotes the particle surface. The hydrodynamic force acting on the particle in the z-direction, FzD, can be decomposed into two terms: one that arises from the viscous force and one that arises from the hydrodynamic pressure.18 We have
FzD )
∂(u‚t) t dS + ∂n z
∫∫η S
∫∫-pnz dS
(18)
S
where t and n are respectively the unit tangential and unit normal vectors on the particle surface, with n being the magnitude of n and tz, and nz are respectively the z-components of t and n. The mobility of the particle can be determined on the basis of the fact that the total force acting on it vanishes at steady state, that is,
FzD + FzE ) 0
(19)
3. Results and Discussion The behavior of the system under consideration is examined through numerical simulation. The governing equations and the associated boundary conditions are solved by FlexPDE,19 a partial differential equations solver based on a finite element method. The accuracy of the numerical method adopted is controlled by setting the error limit in the software. According to our experience, convergent results can be obtained by setting the error limit to 10-6 for the electric field and 10-4 for the flow field. The performance of the numerical method used is satisfactory for the ranges of the parameters examined. The estimation of the mobility of a particle comprises a trial-and-error procedure, where the velocity of a particle is assumed first, followed by the evaluation of the electric force and the viscous force acting on the particle. Equation 19 is then used to check if the assumed velocity is appropriate. The checking step is based on the criterion |(FzE + FDz)/FEz| e 0.1%. The applicability of the present numerical scheme is also justified by examining the problem discussed by Zydney,6 where the electrophoresis of a sphere at the center of a spherical cavity is analyzed. Figure 2 shows the results predicted by the present method and those based on Zydney’s approach. As can be seen from this figure, the performance of the numerical scheme adopted is satisfactory. For convenience, we define parameter P as P ) 100m/(b - a)%, where m is the z-coordinate of the center of the sphere; P ) 0% if the particle is at the center of the cavity, and P ) 100% if it touches the cavity. Also, a scaled electrophoretic mobility ω is defined as below:
ω)
ηU ζREFEz
(20)
where ζREF is a reference potential; for the case of constant (18) Backstrom, G. Fluid Dynamics by Finite Element Analysis; Studentlitteratur: Sweden, 1999. (19) FlexPDE version 2.22, PDE Solutions Inc., USA. (20) Hsu, J. P.; Kuo, C. Y. Langmuir 2002, 18, 2743.
Figure 2. Variation of scaled electrophoretic mobility ω as a function of λ ()a/b) (a) and κa (b) for the case of a sphere of constant scaled surface potential ζ/a ()ζa/(kBT/e)) placed in an uncharged spherical cavity. κa ) 1 in part a, and λ is 0.4 in part b. Solid line, present result; discrete symbols, result of Zydney.6
surface potential ζREF is the particle surface potential ζa, and for the case of constant surface charge density ζREF is kBT/e. The electrophoretic behaviors of a particle under various types of surface conditions are examined. These include constant surface potential and constant surface charge density. 3.1. Constant Surface Potential. The electrophoretic behaviors of a particle for the case when both particle and cavity surface remain at constant potential are shown in Figures 3-5. Figure 3 shows the variation of the scaled electrophoretic mobility of a particle ω as a function of P for various λ ()radius of particle/radius of cavity) at two levels of scaled surface potential of the particle, ζ/a ()ζa/(kBT/e)). This figure reveals that if λ is small, the larger the P, the smaller the ω; if λ is large, ω is essentially the same as P varies; and the larger the λ, the smaller the ω. These results are expected, because if λ is large and/or P is large, the effect of viscous retardation due to the presence of a cavity wall is significant. Figure 3 also suggests that varying particle surface potential from ζ/a ) 0.5 to ζ/a ) 1 has an inappreciable effect on its mobility. Figure 4 illustrates the variation of the scaled electrophoretic mobility of a particle ω as a function of P for various κa. This figure indicates that if κa is large, the larger the P, the smaller the ω, and if κa is small, the larger the P, the larger the ω. The former is because that,
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Figure 3. Variation of scaled electrophoretic mobility ω as a function of P at various λ ()a/b) for the case of a sphere placed in an uncharged spherical cavity at κa ) 1. Solid lines, ζ/a ) 1; dashed lines, ζ/a ) 0.5.
Figure 4. Variation of scaled electrophoretic mobility ω as a function of P at various κa. Parameters used are the same as those of Figure 3 except that λ ) 0.4 and ζ/a ) 0.5.
Figure 5. Variation of scaled electrophoretic mobility ω as a function of P for various κa at two different ζ/a values. Solid lines, ζ/a ) 1; dashed lines, ζ/a ) 0.5. Parameters used are the same as those of Figure 3.
for fixed ionic strength, a large κa implies a large particle, and therefore, a great viscous retardation arises from the presence of a cavity wall, which yields a small electro-
Hsu et al.
Figure 6. Variation of scaled electrophoretic mobility ω as a function of P at various λ ()a/b) for the case σ/a ) 0.5, σ/b ) σb/(κkBT/e) ) 0, and κa ) 1.
Figure 7. Variation of scaled electrophoretic mobility ω as a function of P at various κa for the case λ ) 0.4. Parameters used are the same as those of Figure 6.
phoretic mobility. It is interesting to note that, however, if a particle is close to a cavity surface, then the closer it is to a cavity surface, the greater its mobility. This can be explained by resorting to the result of Hsu and Kao,20 where the electrical interactions between a spheroidal particle and a spherical cavity both are maintained at constant, positive surface potential. For the case when the electrical potential on the cavity surface is lower than that of the particle surface and the particle is sufficiently close to the cavity, a charge reversal occurs on the former. This implies that the electrical interaction force between the particle and the cavity becomes attractive. Figure 4 suggests that the thicker the double layer, the earlier the charge reversal occurs. The variation of the scaled electrophoretic mobility of a particle ω as a function of P for various κa at two levels of scaled surface potential of particle ζ/a is presented in Figure 5. As can be seen from this figure, if κa is large, the effect of ζ/a on ω is unimportant. This is because if κa is large, the double layer surrounding a particle is thin, and its shielding effect becomes important. Figure 5 also indicates that, if κa is small, the effect of ζ/a on ω is inappreciable if the particle is located near the cavity center (small P). This effect becomes significant, however, as the particle deviates from the cavity center, and the higher the level of ζ/a, the larger the ω. Since the surface
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therefore, a small ω. Figure 8 shows the variation of the scaled electrophoretic mobility of a particle ω as a function of P for various scaled surface charge densities of the particle, σ/a ()σa/(κkBT/e)). This figure reveals that the larger the σ/a, the larger the ω, and the larger the P, the smaller the ω, as expected. A comparison between Figures 3-5 and Figures 6-8 reveals that the qualitative behavior of the electrophoretic mobility of a particle for the case when both the particle and the cavity are maintained at constant surface charge density is different from that when they are maintained at constant surface potential. Again, this is mainly due to the possibility of charge reversal on the cavity surface in the latter when a particle is close to the cavity surface.
Figure 8. Variation of scaled electrophoretic mobility ω as a function of P at various σ/a for the case λ ) 0.4. Curve 1, σ/a ) 1; curve 2, σ/a ) 2; curve 3, σ/a ) 3.
potential of the cavity is fixed, this implies that the greater the difference between the surface potential of the particle and that of the cavity, the easier it is to observe a charge reversal on the cavity surface. 3.2. Constant Surface Charge Density. The electrophoretic behaviors of a particle for the case when both particle and cavity surface are maintained at constant surface charge density are illustrated in Figures 6-8. Figure 6 shows the variation of the scaled electrophoretic mobility of a particle ω as a function of P for various λ. This figure reveals that the larger the λ, the smaller the ω, and the larger the P, the smaller the ω, as expected. The variation of the scaled electrophoretic mobility of a particle ω as a function of P for various κa is illustrated in Figure 7, which suggests that the larger the κa, the larger the ω, and the larger the P, the smaller the ω. These are expected because both a large P and a thick double layer yield a greater viscous retardation and,
4. Conclusion In summary, the presence of a boundary on the electrophoretic behavior of a charged entity is investigated theoretically. In particular, the problem of a sphere in a spherical cavity is discussed. The present analysis extends previous study in that the former can be placed at an arbitrary position in the latter. Two types of boundary conditions are considered, namely, constant surface potential and constant surface charge density. In general, the presence of a cavity wall has the effect of decreasing the mobility of a particle. However, for the case when both the surface of the particle and that of the cavity remain at constant but different potentials, if the former is sufficiently close to the latter, the presence of the latter has the effect of increasing the mobility of the former. This can be explained by the phenomenon of charge reversal occuring on the surface which has a lower potential. Charge reversal does not occur for the base when both the surface of the particle and that of the cavity remain at constant charge density. Acknowledgment. This work is supported by the National Science Council of the Republic of China. LA0204996
