Electrophoresis of a Membrane-Coated Sphere in a Spherical Cavity

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Research Article

Electrophoresis of a Membrane-Coated Sphere in a Spherical Cavity

Eric Lee, and Yu-Po TangJyh-Ping Hsu

Langmuir, 2004, 20 (21), 9415-9421 • DOI: 10.1021/la048329b Downloaded from http://pubs.acs.org on November 21, 2008

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Electrophoresis of a Membrane-Coated Sphere in a

Spherical Cavity

Eric Lee and Yu-Po Tang

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617

Jyh-Ping Hsu*

Department of Chemical and Materials Engineering, National I-Lan University, I-Lan, Taiwan 26041

Received July 5, 2004. In Final Form: August 8, 2004

The boundary effect on the electrophoresis of particles covered by a membrane layer is discussed by considering a spherical particle in a spherical cavity under the conditions where the effect of double-layer polarization can be significant. The influence of the key parameters of the system under consideration on the electrophoretic mobility of a particle is investigated. These include the surface potential; the thickness of the double layer; the relative size of the cavity; and the thickness, the fixed charge density, and the friction coefficient of the membrane layer. The fixed charge in the membrane layer of a particle is found to have a significant influence on its electrophoretic behavior. For instance, depending upon the amount of fixed charge in the membrane layer, the mobility of a particle may exhibit a local minimum as the thickness of the double layer varies.

Introduction

Electrophoresis has been applied widely to estimate the surface properties of charged entities and in various separation processes. Its fundamental and practical significances have drawn the attention of researchers in many fields since the last century, and it has been studied extensively, both theoretically and experimentally. von Smoluchowski,1_{for example, was able to derive an analytic} expression for the electrophoreitc mobility of a rigid particle. Although it is based on drastic assumptions such as low surface potential, weak applied electric field, and a thin double layer, the result of von Smoluchowski is concise and has been used widely to elaborate experi-mental observations. The analysis of von Smoluchowski was extended by many investigators to take account of more general conditions.2-4 _{In addition to the} electro-phoresis of rigid particles, that of porous particles such as polyelectrolytes and particles comprising a rigid core and an ion-penetrable membrane or polymer layer, the so-called soft or fuzzy particles,5-9_{has also been studied.} The former can be viewed as a limiting case of the latter by letting the size of the rigid core vanish. The presence of a membrane layer on the surface of a particle can have an appreciable influence on its electrophoretic behavior. The density of the ionic cloud of counterions near the rigid core of a particle can increase appreciably because of the presence of a membrane layer, even if it is thin. From the

hydrodynamic point of view, because the presence of an uncharged membrane layer yields an extra drag, the movement of a particle is slowed. However, the influence of the membrane layer can be profound if it is charged. The problem becomes even more complicated when the surface potential is high, when the effect of double-layer polarization can be significant.10,11 _{Solving the} electro-phoresis of a soft particle can be nontrivial because the governing equations in the membrane layer and those in the liquid phase must be solved simultaneously. In general, an appropriate numerical scheme is necessary so that the problem can be solved efficiently and accurately. Ohs-hima12 _{solved analytically the electrophoresis of a soft} particle under the conditions that the effect of double-layer polarization is negligible. Hill et al.10_{and Saville}11 pointed out, however, that this effect cannot be ignored when the surface potential is not low. In practice, electrophoresis is often conducted under situations where the presence of a boundary can be important. A typical example includes the electrophoresis performed in a porous medium to avoid possible natural convection. In separation through electrophoresis, microporous gels or membranes are designed to have a strong enough electric field so that separation efficiency can be raised.13 Appar-ently, the boundary effect can play a role in these cases. While theoretical analyses relevant to the boundary effect on electrophoresis are ample in the literature, many of them are limited to the case of infinitely thin double layers.14,15_Zydney16_{analyzed the electrophoresis of a rigid} sphere at the center of a spherical cavity for the case of low electric potential and finite double-layer thickness. Although this geometry is an idealized one, it is capable

* To whom correspondence should be addressed. Tel.: 886-3-9357400 ext.285. Fax: 886-3-9353731. E-mail: jphsu@ntu.edu.tw.

(1) von Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129. (2) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2

1978, 74, 1607.

(3) Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 514. (4) Overbeek, J. Th. G. Adv. Colloid Sci. 1950, 3, 97.

(5) Donath, E.; Pastuschenko, V. Bioelectrochem. Bioenerg. 1979, 6, 543.

(6) Jones, I. S. J. Colloid Interface Sci. 1979, 68, 451.

(7) Levine, S.; Levine, M.; Sharp, K. A.; Brooks, D. E. Biophys. J.

1983, 42, 127.

(8) Sharp, K. A.; Brooks, D. E. Biophys. J. 1985, 47, 563. (9) Ohshima, H.; Kondo, T. Biophys. Chem. 1991, 39, 191.

(10) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci.

2003, 258, 56.

(11) Saville, D. A. J. Colloid Interface Sci. 2000, 222, 137. (12) Ohshima, H. Adv. Colloid Interface Sci. 1995, 62, 189. (13) Jorgenson, J. W. Anal. Chem. 1986, 58, 743A. (14) Keh, H. J.; Chiou, J. Y. AIChE. J. 1996, 42, 1397. (15) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417. (16) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476.

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of providing insight about the boundary effect on elec-trophoresis. The analysis of Zydney16 _{was extended by} Lee et al.17_{to the case of arbitrary surface potential and} when double-layer polarization can be significant. The latter was further extended by Chu et al.18_{to the case of} an arbitrary level of applied electric field.

In this study, the boundary effect on electrophoresis is investigated for the case of soft particles under conditions where the effect of double-layer polarization can be significant. The sphere-in-spherical-cavity geometry of Zydney16 _{is adopted, and the influences of the key} parameters, including the thickness of the double layer, the relative size of the cavity, and the thickness, the fixed charge density, and the friction coefficient of the membrane layer, on the electrophoretic behavior of a particle are examined.

Theory

The problem under consideration is illustrated in Figure 1, where a spherical particle comprises a rigid core of radius

a and a membrane layer of thickness d is placed at the

center of a spherical cavity of radius c. Let b ) a + d. The cavity is filled with an aqueous solution containing z1:z2 electrolyte, where z1and z2are respectively the valence of cations and that of anions. An electric field E in the positive z direction is applied, and U is the electrophoretic velocity of the particle. The spherical coordinates (r, θ, φ) are adopted with its origin located at the center of the particle.

The electric field of the system under consideration φ can be described by the Poisson equation

F )_∑j_n_j_z_j_{e is the spatial variation of mobile ions, F}_fix_{is the} fix charge density in the membrane layer, is the permittivity of the liquid phase, njand zjare respectively

the number concentration and the valence of ionic species

j, and i ) 1 for a < r < b and i ) 0 for b < r < c. For

convenience, φ is decomposed into the electrical potential in the absence of E or the equilibrium potential φ1and a potential which arises from the applied electric field, φ2, that is, φ ) φ1+ φ2. If the effect of double-layer polarization is significant, the double layer surrounding the particle is no longer spherically symmetric. To account for this effect, the spatial distribution of mobile ions is expressed as2

where gjrepresents a perturbed potential, which accounts

for the effect of fluid flow on the concentration of mobile ions, nj0is the bulk concentration of ionic species j, e is the

elementary charge, kBis the Boltzmann constant, and T is the absolute temperature. The equilibrium potential can be described by

where i ) 1 for a < r < b and i ) 0 for b < r <c. Suppose that the surface of the rigid core of a particle (r ) a) remains at a constant potential, both the electrical potential and the electric field are continuous on the ion-penetrable layer-liquid interface (r ) b), and the cavity wall (r ) c) is held to be uncharged. Therefore, the boundary conditions for the equilibrium potential φ1are

The governing equation for φ2can be obtained by eq 3 and

φ ) φ1+ φ2as

The following boundary conditions are assumed for φ2:

(17) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65.

(18) Chu, J. W.; Lin, W. H.; Lee, E.; Hsu, J. P. Langmuir 2001, 17, 6289.

Figure 1. Schematic representation of the problem considered

-zje(φ1+ φ2+ gj) k_BT

)

-exp

(

-zjeφ1 k_BT

)

]

(8) ∂φ₂ ∂r ) 0, r ) a (9) φ2|r)b-) φ₂|_r)b+, r ) b (10)

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The first condition arises from that the rigid core of a particle is nonconductive and is impermeable to ionic species. The last condition implies that φ2is assumed to reach the applied electric field.

At steady state, the conservation of ionic species yields

where Djis the diffusivity of ionic species j. In terms of

gj, we have

We assume that the surface of the rigid core of a particle is impermeable to ions, both the concentration of ionic species and its flux are continuous on the ion-penetrable membrane layer-liquid interface, and the concentration of ionic species reaches the equilibrium value on the cavity wall. Therefore, the boundary conditions for the concen-tration field are

In these expressions, nˆ is the unit normal vector, δris the

unit vector in the r direction, and fjis the flux of ionic

species j defined by

We assume that the flow field can be described by the Navier-Stokes equation in the creeping flow regime. In terms of the stream function ψ, it can be shown that the governing equation for the flow field is

where i ) 1 for a < r < b and i ) 0 for b < r < c and γ is the frictional coefficient of the membrane layer. E4₎

E2_E2_{, with}

In terms of ψ, the r and the θ components of v, vrand vθ,

can be expressed respectively as vr) -(1/r2sin θ)(∂ψ/∂θ)

and vθ) (1/r sin θ)(∂ψ/∂r). We assume that both the surface

of the rigid core of a particle and that of a cavity are nonslip. Also, the velocity of liquid and both its normal and tangential stresses are continuous on the membrane

layer-liquid interface. These assumptions lead to the following boundary conditions:

For a more concise presentation, the governing equa-tions and the associated boundary condiequa-tions are rewritten in terms of dimensionless variables scaled by the following quantities: the radius of the rigid core of a particle, a, the surface potential on the core, ζa, the bulk number

concentration of ionic species 1, n10, and the electrophoretic velocity based on von Smoluchowski’s theory1_{when an} electric field ζa/a is applied, UE) ζa2/ηa. Also, we define

r* ) r/a,n_j/) nj/n10,E_Z

/

) EZ/(ζa/a), v* ) v/UE, U* ) U/UE, φ₁/) φ1/ζa,φ2

/

) φ2/ζa,gj /

) gj/ζa, and ψ* ) ψ/UEa. In terms

of scaled symbols, the governing equations for the equi-librium electric potential become

where Q ) (Ffixa2/ζa), which is the scaled fix charge density

in membrane layer, and φr) zieζa/kBT, the scaled surface potential of the rigid core of a particle, i ) 1 for a < r <

b, and i ) 0 for b < r < c. The associated boundary

conditions are

The two-dimensional dependent variables φ₂/, g₁/, g₂/, and ψ* can be made one-dimensional by applying the method of separation of variables,. The following expres-sions were used for the present problem: φ₂/) Φ2(r) cos

θ,g₁/) G1(r) cos θ,g2 /

) G2(r) cos θ, and ψ* ) Ψ(r) sin2θ. On the basis of these expressions, it can be shown that the resultant governing equation forφ₂/becomes

r*)b+_/a, r* ) b/a (31) φ*₁) 0, r* ) c/a (32)

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where

The associated boundary conditions are

The governing equations for the concentration field become

where Pej) UEa/Djis the electric Peclet number of ionic

species j. The corresponding boundary conditions are

The governing equation for the flow field becomes

where (λa)2_{) (γa}2_{/η), which is a scaled shear parameter} in the membrane layer, i ) 1 for a < r < b, i ) 0 for b <

r < c, and D4_{) D}2_D2_{) (d}2_/dr*2_{- 2/r*}2₎2_{. The associated} boundary conditions are

Solving the above equations subject to the associated boundary conditions yields the flow, the electric, and the concentration fields. Following the approach of O’Brien and White,2_{the problem under consideration is} decom-posed into two sub-problems. In the first problem, a particle moves with constant velocity in the absence of the applied electric field. If we let F1be the total force acting on the particle, then F1is proportional to its terminal velocity, that is, F1) χU*, where χ is independent of U*. In the second problem, an external electric field is applied, but the particle remains fixed. In this case, if we let F2be the total force acting on the particle, then F2is proportional to the applied field, that is, F2 ) βEz

/

, where β is independent ofE_z/. Because χ is independent of U* and β is independent ofE_z/, we choose|U*| ) 1 and |E_z/| ) 1, for

simplicity, and, therefore, χ ) F1and β ) F2. The forces acting on a particle F comprises the electrical force Fe

and the hydrodynamic force Fd. If we let Fzi be the

magnitude of F in problem i in the z direction, and Fezand

Fdzbe the components of Feand Fdin the z direction, then

The electric force acting on a particle can be evaluated by

where σE _{≡ (EE -} 1_/

2E2I) and E ) -∇φ. Fez can be

expressed as

where δzis the unit vector in the z direction. In terms of

scaled symbols, it can be shown that

Fdcan be evaluated by19

where σH_{≡ -pI + η[∇v + (∇v)}T_{]. F}

(σ H_‚nˆ)‚δ zds (60)

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or, in terms of scaled symbols

The sum of F1and F2vanishes at steady state, and the magnitude of the scaled electrophoretic velocity of a particle,µm / , can be expressed asµm / ) U*/E_z/) -β/χ ) -F2/F1.

Results and Discussions

The governing equations and the associated boundary conditions are solved numerically by a pseudo-spectral method based on Chebyshev polynomials. The perfor-mance of this approach for solving electrophoresis prob-lems in a spherical cavity was found to be satisfactory.17,18 The influence of the key parameters of the system under consideration on the mobility of a particle is investi-gated through numerical simulation. These include the thickness of the double layer; the relative size of cavity; and the thickness, the fixed charge density, and the fric-tion coefficient of the membrane layer. The influence of the scaled fix charge density Q on the electrophoretic behavior of a particle is illustrated in Figure 2, where the variation of the scaled mobility of a particleµm

/ as a function of Q at various κa is presented. This figure re-veals that whileµm

/

varies nonlinearly with Q, in general, theµ_m/-Q relation is roughly linear if κa is sufficiently

large, that is, the concentration of electrolyte is sufficiently high or the double layer surrounding a particle is sufficiently thin. Figure 3 illustrates the variations of the scaled hydrodynamic force F1and the scaled electric force

F2as a function of Q at various κa. This figure indicates that F1is much more insensitive to the variation in both

Q and κa than F2is, which implies that the behavior of µ_m/ in Figure 2 is dominated by F2. As will be discussed later, the nonlinear µm

/

-Q relation in Figure 2 arises mainly from the effect of double-layer polarization that occurs for κa in the range [1.0, 5.0]. This effect becomes insignificant when κa is sufficiently large. On the other hand, if κa is small, both the hydrodynamic retardation

arising from the presence of the double layer and the effect of double-layer polarization are significant. The competi-tion between the electric force and the hydrodynamic force

Figure 2. Effect of fixed charge density Q on mobility at various κa for the case when φr) 1.0, λa ) 5.0, c/a ) 2.0, and d/a ) 0.5.

F_dz) 4 3πζa 2

[

r*2 ∂ ∂r*(D 2 Ψ) - 2r*(D2Ψ)

]

r*)b/a + 4 3πζa 2 (κa) 2 (1 + R)φ_r{r* 2 Φ2[exp(-φrφ*1) -exp(Rφrφ*1)]}r*)b/a (61)

Figure 3. Variation of scaled hydrodynamic force F1(a) and

scaled electric force F2(b), as a function of Q at various κa for

the case of Figure 2.

Figure 4. Variation of scaled mobility µm

/

as a function of κa at various Q for the case when φr) 1.0, λa ) 5.0, c/a ) 2.0, and

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can be further illustrated in Figure 4. This figure shows that if κa is small, the hydrodynamic retardation force dominates. In this case, although the electric force acting on a particle increases with κa, the scalared mobility_|µm

/ | remains roughly constant. If κa is sufficiently large, the electric force dominates, and |µm

/

| increases with the increase in κa. It is interesting to note that if Q is sufficiently high, |µm

/

| exhibits a local minimum as κa varies. This phenomenon arises from the polarization of the double layer surrounding a particle, which induces a local electric field having a direction reverse of that of the applied electric field. The effect of double-layer polarization becomes appreciable when the thickness of the double layer is comparable to the linear size of a particle, that is, κa ranges from 0.1 to 10, and is insignificant when κa is either too small or too large. A similar phenomenon was also observed in the case of rigid particles.17

The influence of the friction coefficient of the membrane layer on the electrophoretic mobility of a particle is illustrated in Figures 5 and 6, and that of the relative thickness of the membrane layer and the relative mag-nitude of the cavity on the mobility of a particle is presented

in Figures 7 and 8, respectively. Figures 5 and 6 indicate that the larger the λa, the smaller the µm

/

is, which is expected. Figure 7 suggests that the thicker the membrane layer, the larger the mobility, and Figure 8 indicates that the smaller the cavity, the smaller the mobility. These are expected because the thicker the membrane layer, the more the total amount of fixed charge in the membrane layer, and the smaller the cavity, the more significant the steric retardation on the movement of a particle.

In practice, the membrane layer arises often from the attachment of polymer or macromolecules on the particle surface. Because the greater the amount of polymer molecules attached, the higher the density of membrane layer, it is highly possible that Q and λa are positively correlated. As mentioned previously, Q is related to the magnitude of the electric force acting on a particle; the higher the Q, the greater the electric force and the larger the absolute value of mobility. On the other hand, λa is related to the hydrodynamic drag; the larger the λa, the greater the hydrodynamic drag and the smaller the mobility. These imply that the density of the membrane layer can have a confound influence on the mobility of a particle, as is illustrated in Figure 9. For example, with

Figure 5. Variation of scaled mobility µ_m/ as a function of λa at various κa for the case when φr) 1.0, Q ) 10.0, c/a ) 2.0,

and d/a ) 0.5.

/

as a function of κa at various λa for the case when φr) 1.0, Q ) 10.0, c/a ) 2.0,

and d/a ) 0.5.

/

as a function of κa at various d/a for the case when c/a ) 2.0, φr) 1.0, Q ) 10.0,

and λa ) 5.0.

/

as a function of κa at various c/a for the case when d/a ) 0.5, φr) 1.0, Q ) 10.0,

and λa ) 5.0.

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the same Q/λa ratio, while the mobility of a particle with

Q ) 1 is greater than that with Q ) 0 (free of fixed charge),

the mobility of a particle with Q ) 5 is smaller than that with Q ) 0.

The present study focused on the influences of the nature of a colloidal particle as well as the presence of a boundary on its electrophoretic behavior. The membrane-coated particle adopted here simulates a wide class of nonrigid entities in practice. Polystyrene latex dispersions, for example, comprise particles that have a hairy-structured surface arising from the extension of the internal polymer chains. Surfactants are usually introduced into a colloidal dispersion to avoid coagulation. In this case, the surfactant molecules adsorbed to particle surface form a gel layer, and the resultant composite entity can be simulated by the present model. Biocolloids such as cells and micro-organisms where their peripheral zone is ion-penetrable and contains dissociable functional groups are also typical examples which can be described by the present model. Boundary effect is of practical significance in phoresis measurements. For instance, capillary

electro-phoresis and gel electroelectro-phoresis, where this effect cannot be neglected, are widely adopted in the separation of entities such as DNA. In this study, the boundary effect is simulated by considering the presence of a cavity. Often, estimating surface properties of colloidal particles such as the zeta potential, charge density, and thickness of the surface layer on the basis of experimental observations is necessary. This can be conducted by matching the present model to experimental data, and the estimated values of the adjustable parameters provide sufficient information. Apparently, the present model can be used directly to predict the electrophoretic behavior of particles of known physical properties under specified conditions. This is of particular importance in design of an electrophoretic separation device.

Conclusions

The electrophoresis of a membrane-coated particle in a spherical cavity is investigated theoretically. As a result of the presence of a membrane layer, particles are capable of exhibiting specific behavior that is not observed for the case of rigid particles. The physical properties of the membrane layer are characterized mainly by its fixed charge density and friction coefficient; both of them are positively correlated with the density of the membrane layer. The former is advantageous and the latter is disadvantageous to the electrophoretic movement of a particle. We show that while the mobility of a particle varies nonlinearly with fixed charge density, in general, they are roughly linearly correlated if the double layer surrounding the particle is sufficiently thin. The nonlinear relation arises mainly from the effect of double-layer polarization, becoming insignificant when the double layer is thin. Depending on the level of fixed charge density, the mobility of a particle may exhibit a local minimum as the thickness of the double layer varies, which is also observed for the case of rigid particles. The presence of a cavity has a negative influence on the electrophoretic movement of a particle.

Acknowledgment. This work is supported by the

National Science Council of the Republic of China. LA048329B

Figure 9. Variation of scaled mobility µ_m/ as a function of κa at various combinations of Q and λa for the case when φr) 1.0,

Electrophoresis of a Membrane-Coated Sphere in a Spherical Cavity

Research Article

Electrophoresis of a Membrane-Coated Sphere in a Spherical Cavity

Eric Lee, and Yu-Po TangJyh-Ping Hsu

More About This Article

Additional resources and features associated with this article are available within the HTML version:

•

Supporting Information

•

Access to high resolution figures

•

Links to articles and content related to this article

Electrophoresis of a Membrane-Coated Sphere in a

Spherical Cavity

Eric Lee and Yu-Po Tang

Jyh-Ping Hsu*

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