ARTICLE pubs.acs.org/Langmuir
Electrophoresis of a Charged Colloidal Particle in Porous Media: Boundary Effect of a Solid Plane Peter Tsai, Cheng-Hsuan Huang, and Eric Lee* Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan ABSTRACT: Electrokinetic treatments such as the electrophoretic technique have been applied successfully to various soil remediation and contaminant removal situations. To understand further the fundamental features involved, the electrophoretic motion of a charged particle in porous media is investigated theoretically in this study, focusing on the boundary effect of a nearby solid plane toward which the particle moves perpendicularly. The porous medium is modeled as a Brinkman fluid with a characteristic screening length (λ1) that can be obtained directly from the experimental data. General electrokinetic equations are used to describe the system and are solved with a pseudospectral method based on Chebyshev polynomials. We found that the particle motion is deterred by the boundary effect in general. The closer the particle is to the boundary, the more severe this effect is. Up to a 90% reduction in particle mobility is observed in some situations. This indicates that a drastic overestimation (10-fold!) of the overall transport rate of particles may occur for large-scale in situ operations in porous media, such as soil remediation utilizing large planar electrodes, should a portable analytical formula valid for bulk systems only be used. Correction factors for various situations in porous media are presented as convenient charts with which to aid engineers and researchers in the field of environmental engineering, for instance, as a realistic estimation of the actual transport rate obtainable. In addition, the results of present study can be applied to biomedical engineering and drug delivery as well because polymer gels and skin barriers both have a porous essence.
1. INTRODUCTION Electrophoresis, the transport of charged colloidal particles in response to an externally applied electric field, has long been utilized in colloid science to explore the surface electric property of particles by measuring the particle velocity experimentally.1 The electric potential of the particle surface can be calculated via a force balance between the electric driving force and the hydrodynamic retarding force on the basis of various theoretical deductions that are valid in different situations.24 It is also a powerful separation technique in many practical applications, such as gel electrophoresis where the separation of large biomacromolecules such as proteins and DNAs can be easily done by migrating samples through polymer gels under an electric field.5,6 It also revealed many novel applications in nanotechnology, such as lab-on-a-chip devices.7,8 In addition to the above-mentioned classic applications, electrophoresis has great potential in many other scientific and engineering areas. Among them, electrophoresis in a porous media is of particular interest in the area of chemical engineering and its related interdiscipline fields such as environmental engineering810 and biomedical engineering.11,12 In environmental engineering, for instance, electrophoresis can be utilized in the remediation treatment of contaminated soil in situ by applying a dc electric field across the wet soil to remove colloidal pollutants directly such as petroleum hydrocarbons13,14 or it can transport remediating agents such as nanoiron through fine-grained sand to remove chlorinated organic compounds in groundwater.8 In addition, the idea of r 2011 American Chemical Society
transporting compounds across the skin using a dc electric field (referred to as iontophoresis in biomedical engineering) has been a very promising novel technique under development in biotechnology and pharmacology for transdermal drug delivery.12 In the above applications, the porous medium through which electrokinetic phenomena take place is often modeled as an effective Brinkman medium. To model the fluid flow in a swarm of particles, Brinkman15 first introduced an extra term representing the corresponding hydraulic drag force induced by the presence of the solid part within a porous medium into the NavierStokes equation, the so-called the Brinkman model. The hydrodynamic permeability of this medium can be characterized by the Darcy permeability Kp (equal to the square of the reciprocal hydrodynamic Brinkman screening length, λ2). This idea was also reported independently by Debye and Bueche16 at nearly the same time when they studied the fluid flow in a dilute polymer solution. Although the Brinkman model is semiempirical in nature, it fits well with the experimental data in general17 and is regarded as the standard approach describing the fluid flow within a porous medium. Recently, it was further extended to cover the corresponding electrokinetic transport phenomena inside the porous medium, with an extra consideration of the electrical driving force.1825 Among them, Ohshima18 and Hill et al.22 theoretically investigated the electrophoresis of an Received: August 18, 2011 Revised: September 30, 2011 Published: October 03, 2011 13481
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Langmuir isolated spherical polymer-coated colloidal particle, where the hydrodynamic permeability of the homogeneous polymer coating on the particle surface is characterized by the Brinkman screening length, λ1,18 and lB.22 Later, Hill20 utilized the same approach to investigate the electrokinetic transport phenomena in polymer gel composites. Duval and co-workers21,23,25 developed a model for the electrophoresis of diffuse soft particles where the nonhomogeneous distribution of soft material within the particle shell is taken into account. Recently, Tsai and Lee24 investigated the gel electrophoresis in suspensions of charged spherical particles. Both the short-range steric effect (due to the direct contact friction between the gel and the migrating particles) and the long-range hydrodynamic effect (due to the hydrodynamic force of the gel exerted upon the liquid suspension) are considered with the particle mobility calculated as the product of the predictions from these two approaches separately. In their study, the Brinkman model is utilized to describe the fluid flow in polymer gels. Excellent agreement is obtained between the results reported there and the experimental data available in the literature for gold nanoparticles.26 It should be noted that the Brinkman model is generally applicable to describing the fluid flow in all kinds of porous media, be it the fine-grained sand in soil remediation or the polymer gel in DNA sequencing techniques.27 In the applications for electrokinetic transport in a porous medium, a nearby confining planar boundary is often encountered, for instance, the plate substrates in soil remediation in situ or the skin barrier in transdermal drug delivery. When the particle is near the planar boundary, the presence of the planar boundary will affect the particle motion significantly. However, corresponding studies for particle transport near the solid boundary in a porous medium are very limited. Motivated by the investigation of the transport phenomena for a nanogold or a latex particle across thin surface layers of a matrix that coats the membrane of living cells,28 Feng et al.29 explored the motion of a sphere near planar confining boundaries in a fiber matrix modeled as a Brinkman medium. Note that the particle was assumed to be chargeless in their work. In reality, however, nano or colloidal particles are essentially all charged when they are immersed in a polar medium such as an electrolyte solution.1 Numerous theoretical studies on electrophoresis in electrolyte solutions have been reported for particles that are either rigid4 or nonrigid,30 a double layer that is either very thin31 or of finite thickness,32 and particle charges that are either low32 or high.2,3 Moreover, if the particle is near a solid boundary, then the presence of a nearby boundary will affect the particle motion electrostatically because of the deformation of the double layer as well as hydrodynamically because of the drag force. Corresponding studies have been conducted by many researchers for particles moving normal to a solid planar boundary in electrolyte solutions.3338 Corresponding studies for the electrophoretic motion of a colloidal particle in a porous medium filled with an electrolyte solution, however, have not been reported in the literature. Considering the vital importance of the general electrokinetic knowledge of this system in related practical applications in environmental protection, biomedical engineering, biotechnology and nanotechnology, and chemical engineering in general, we present here the results of a theoretical study focusing on the boundary effect and its implications in various practices mentioned above. A pseudospectral method based on Chebyshev polynomials is used to solve the resulting general electrokinetic equations numerically. Key parameters are examined for their respective effects on the particle mobility, such as
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Figure 1. Geometric configuration of the system in this study.
the double-layer thickness, the zeta potential of the particle, the permeability of the porous medium, the separation distance between the particle and the solid plane, and so on. The impact on practical applications such as soil remediation is discussed in detail in particular. The analysis and results are presented in subsequent sections.
2. THEORY As shown in Figure 1, we consider a charged particle of radius a moving with a velocity U normal to a conducting solid plane in an applied electric field E. The particle is suspended in a porous medium filled with the electrolyte solution that contains a z1/z2 electrolyte, with z1 and z2 being the charge numbers of the cations and anions, respectively. α is defined as z2/z1. The electroneutrality in the bulk liquid phase requires that n20 = n10/ α, with n10 and n20 being respectively the bulk concentrations of cations and anions. The distance between the center of the particle and the plane is denoted as h. Due to the axisymmetry nature of the problem, the cylindrical coordinates (ϖ, θ, z) can also be used to describe the system. However, the bispherical coordinates (ξ, η, j) are even better here in terms of the ease of applying the boundary conditions, for instance, with the planar surface treated as a sphere with an infinitely long radius. The bispherical coordinates and the cylindrical coordinates (ϖ, θ, z) are related by38 z¼c
sinh η cosh η cos ξ
ϖ ¼c
ð1Þ
sin ξ cosh η cos ξ
ð2Þ
where c is the focal length. For convenience, we define η0 = cosh1(h/a). η = 0 and η = η0 represent, respectively, the planar surface and the particle surface. The governing equations for the problem here are, respectively, the Poisson equation, the modified Brinkman equation with an extra electric body force term and the conservation of ionic species:24 ∇2 ϕ ¼
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F ¼ ε
2
∑ j¼1
zj enj ε
ð3Þ
∇p þ μ∇2 v F∇ϕ γv ¼ 0
ð4Þ
∇3v ¼ 0
ð5Þ
∇ 3 fj ¼ 0
ð6aÞ dx.doi.org/10.1021/la203240b |Langmuir 2011, 27, 13481–13488
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with
zj enj ∇ϕ þ nj v f j ¼ Dj ∇nj þ kB T
ð6bÞ
Note that eq 5 is the continuity equation indicating that the liquid phase is incompressible, and eq 6b is the modified NernstPlanck equation with an extra term, njv, standing for the ion flux from the convection flow when the particle is in motion. The friction coefficient γ in eq 4 stands for the retardation force of the porous medium, which is proportional to the fluid velocity in the Brinkman model adopted here. Major assumptions in our analysis are as follows. First, the solid obstacles that comprised the porous medium are uniformly distributed in the system. The hydrodynamic permeability of the porous medium can be characterized by the Darcy permeability Kp so that the friction coefficient γ can be expressed as μ/Kp. The porous medium is set to be uncharged for a simplification approach. The Reynolds number of the fluid is assumed to be small enough to justify the use of the Stokes equation, which is essentially always satisfied because the colloidal particle is so small. Moreover, the suspending liquid solution is assumed to be incompressible as well. The applied field E is weak so that the particle velocity U is proportional to E and terms of higher order in E may be neglected. In practice, this means that E is small compared to the field that occurs in the double layer, with |E| , ζak, which is the characteristic electric field measured by the zeta potential of the particle divided by the double-layer thickness. The particle surface is assumed to be nonconducting and ionimpenetrable. Finally, the system is at quasi-steady state when the particle is in motion (i.e., in the regime of creeping flow). Because the applied electric field is weak, all variables in the above equations are only slightly distributed in electrophoresis and can be expressed as the equilibrium state value plus a very small disturbance, for example, ϕ = ϕe + δϕ, nj = nje + δnj, and so forth, where subscript e refers to the equilibrium state and δ refers to a very small disturbance. In particular, the ion distribution in the electrolyte solution is assumed to take the following form, which is similar to the Boltzmann distribution zj e ðϕ þ δϕ þ gj Þ , j ¼ 1, 2, ð7Þ nj ¼ nj0 exp kB T e
function defined as vξ = ((cosh η cos ξ)2)/(c2 sin ξ)(∂ψ)/ (∂η), vη = ((cosh η cos ξ)2)/(c2 sin ξ)(∂ψ)/(∂ξ), the governing equations with associated boundary conditions are the following: 2.1. Equilibrium State. ∇ 2 ϕ e ¼
ðkaÞ2 ½expð ϕr ϕe Þ expðαϕr ϕe Þ ð1 þ αÞϕr ð8Þ
ϕe ¼ 1, η ¼ η0
ð9Þ
ϕe ¼ 0, η ¼ 0
ð10Þ
∂ϕe ¼ 0, ξ ¼ 0 or π ∂ξ
ð11Þ
where r*2 = (x2)/(c*2)[(∂2)/(∂η2) + (∂2)/(∂ξ2) (sinh η)/ (x) (∂)/(∂η) + (cos ξ cosh η 1)/(x sin ξ)(∂)/(∂ξ)], x = cosh η cos ξ. Equations 9 and 10 indicate that both the particle and the solid plane assume a constant surface potential, and eq 11 indicates the axisymmetric nature of the problem. 2.2. Perturbed State. ∇2 δϕ ¼
ðkaÞ2 ½expð ϕr ϕe Þ expðαϕr ϕe Þδϕ ð1 þ αÞjr
ðkaÞ2 ½expð ϕr ϕe Þg1 þ α expðαϕr ϕe Þg2 ð1 þ αÞϕr
ð12Þ ∇2 g1 ϕr ∇ϕe 3 ∇g1 ϕr 2 Pe1 v 3 ∇ϕe ¼ 0
ð13Þ
∇2 g2 þ αϕr ∇ϕe 3 ∇g2 ϕr 2 Pe2 v 3 ∇ϕe ¼ 0
ð14Þ
ðkaÞ2 ðcosh η cos ξÞsin ξ E4 ψ ðλaÞ2 E2 ψ ¼ ð1 þ αÞ c 0 1 ∂ϕe ∂g1 ∂g2 expð ϕr ϕe Þ þ α expðαϕr ϕe Þ B ∂η C ∂ξ ∂ξ B C B C ∂ϕ ∂g1 ∂g @ 2 e A expð ϕr ϕe Þ þ α expðαϕr ϕe Þ ∂ξ ∂η ∂η
where an additional perturbed potential gj is used to take into account the convection contribution to the ion flux.24 Standard linear perturbation analysis is then adopted under the assumption that all the variables in electrophoresis still satisfy the same set of equations, eqs 36, and terms that involve products of small quantities such as δϕ and gj can be neglected.2 Corresponding governing equations for electrophoresis can thus be obtained. To simplify the treatment, scaled symbols are introduced and the governing equations are rewritten in dimensionless form. The following symbols are chosen for the characteristic variables: the radius of a particle, a; the zeta potential of a particle, ζa; the bulk number concentration of the cations, n10; and the velocity based on Smoluchowski’s theory31 when an electric field ζa/a is applied, UE, defined as εζa2/μa. Corresponding dimensionless variables are listed as follows: r* = r/a, c* = c/a, h* = h/a, v* = v/UE, nj* = nj/n10, E* = E/(ζa/a), ϕe* = ϕe/ζa, δϕ* = δϕ/ζa, gj* = gj/ζa, (λa)2 = γa2/ μ, ka = (∑2j = 1nj0(ezj)2/εkBT)1/2a, ϕr = ζa/(z1e/kBT), and Pej = UEa/Dj. With the introduction of the dimensionless stream function defined as ψ* = ψ/UEa2, where ψ is the dimensional stream
ð15Þ
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∂δϕ ¼ 0, η ¼ η0 ∂η
ð16Þ
∂δϕ ¼ E, η ¼ 0 ∂η
ð17Þ
∂gj ¼ 0, η ¼ η0 ∂η
ð18Þ
gj ¼ δϕ, η ¼ 0
ð19Þ
1 csin ξ 2 U , η ¼ η0 ψ ¼ 2 x
ð20Þ
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Figure 2. Mesh adopted in this study.
∂ψ c 2 ¼ 3 sin ξ sinh ηU , η ¼ η0 ∂η x
ð21Þ
∂ψ ¼ 0, η ¼ 0 ψ ¼ ∂η
ð22Þ
ψ f 0, η ¼ 0, and ξ ¼ 0 x
ð23Þ
where E*2 = (x2)/(c*2)[(∂2)/(∂η2) + (∂2)/(∂ξ2) + (sinh η)/(x) (∂)/ (∂η) + (1 cos ξ cosh η)/(x sin ξ)(∂)/(∂ξ)], x = cosh η cos ξ, and E*4 = E*2E*2. Equations 16 and 17 indicate that the particle is nonconducting across the surface and the electric field is applied in the z direction. Equations 18 show that the particle surface is ion-impenetrable. Equations 20 and 21 indicate that the particle moves with a scaled velocity U*, and eqs 22 and 23 state that the fluid is stationary both on the plane surface and at infinity. In addition, all variables must satisfy the axisymmetry condition at ξ = 0 and π, which yields ∂gj ∂δϕ ∂ψ ¼ ¼ ¼ ψ ¼ 0, ξ ¼ 0 or π ∂ξ ∂ξ ∂ξ
ð24Þ
It should be noted that the application of a dc electric field in a water-saturated porous medium may induce electrochemical reactions at this solid plane such as at the metal electrodes,13 bringing about the polarization of the electrodes, which leads an the unfavorable reduction of the applied electric field. In reality, the depolarization of electrode surface is generally realized by the addition of an approximate redox couple.13 In this study, we focus on the sole effect of a planar boundary on the electrophoresis of a colloidal particle following the same approaches made by Ennis and Anderson,35 Chih et al.,33 and so on. The possible chemical reaction on the plane is beyond the scope of this article, and the electric field strength is assumed to be constant. A pseudospectral method based on Chebyshev polynomials is then applied to solve the above system of equations. The details can be found elsewhere.24,33 Once the electric field and the corresponding flow field are obtained, the particle mobility can be calculated on the basis of the force balance between the electric driving force and the hydrodynamic retarding force, as proposed by O’Brien and White.2 The mathematical details can be found elsewhere.24
Figure 3. Scaled mobility as a function of ka at different ϕr values with λa = 1 and h* = 200.
3. RESULTS AND DISCUSSION We first conducted a standard mesh refinement test of the convergence of the numerical scheme in terms of grid independence. The outcome shows that a mesh with 40 40 nodal points is accurate enough in that the results do not vary with a finer mesh. Figure 2 shows the mesh adopted throughout the calculations in this study. The accuracy and reliability of the present scheme are further verified by a direct comparison with the numerical results available in the literature for the limiting case reported by Feng et al.,29 where a purely hydrodynamic problem was analyzed for an uncharged rigid sphere moving toward a planar boundary in a Brinkman medium. The agreement is excellent in terms of the hydrodynamic drag force upon the planar boundary. We thus conclude that our calculation results are both accurate and reliable. The reciprocal of λ is adopted to characterize the hydrodynamic response of the porous medium.15 In practice, it is often presented in the dimensionless form, λa. It should be noted that dimensionless hydrodynamic friction parameter λa is a function of particle radius a as well, and the actual range of λa can be over several orders of magnitude, depending on the specific particle radius of interest. The porosity of a porous medium (js), defined as the fraction of the total volume of the medium occupied by the void space, is directly related to λ. Grained sand8,9,13 or the thin surface layers of a matrix that coat the membranes of living cells28 are two important examples of porous media of engineering and scientific importance. In 1947, Brinkman15 theoretically investigated the viscous force exerted by a flowing fluid on a dense swarm of particles of average radius rs. According to his derivation, the relationship between λrs and js can be expressed by the following equation: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 þ 3 3 1 js ð25Þ , js > 0:4 λrs ¼ 4 6 ð1 js Þ Thus, from eq 25 we can obtain the hydrodynamic screening length λ1 and hence λa once the porosity js and rs of a porous medium are known. To facilitate the subsequent analysis, values of λa of 1, 3, and 10 are picked in this study to represent the highly, the moderately, and the poorly permeable porous media, 13484
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Figure 4. Scaled mobility as a function of h* at different ka values with λa = 1 and ϕr = 3.
Figure 5. Scaled mobility as a function of h* at different ka values with λa = 3 and ϕr = 3.
respectively. Note that the corresponding hydraulic force within the porous medium is proportional to the square of λ by definition. When the particle moves in response to an applied electric field, the double-layer structure would be deformed, or distorted, because of the convective motion of the fluid. An induced electric field opposite to the direction of particle motion is generated and hence deters particle motion in general, a phenomenon called the double-layer polarization effect,2,3 which is significant when the particle is highly charged and is maximized when the double-layer thickness is around unity. The measurement of this convection contribution to ion flux is set to be 0.01 (i.e., Pe1 = Pe2 = 0.01), as typically assumed in electrokinetic studies considering the double-layer polarization effect. The scaled mobility of a charged particle (defined as μm* = μm/(εζa/μ)) is shown in Figure 3 as a function of ka at various values of surface potential ϕr with h* = 200 in a highly permeable porous medium (λa = 1). At such a distance far away from the
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Figure 6. Scaled mobility as a function of h* at different λa values with ka = 0.1 and ϕr = 3.
planar boundary, the presence of the surface is hardly “felt” by the particle. The system reduces to a single particle suspended in an infinitely porous medium. Corresponding theoretical predictions in homogeneous electrolyte solutions, λa = 0, are shown in Figure 3 for verification purpose. When ka is very small (very thick double layer), the Henry's function approaches H€uckel’s prediction of particle mobility μm = 2/3 εζa/μ, and when ka gets very large (very thin double layer), the Henry's function approaches the famous Smoluchowski equation μm = εζa/μ. In Figure 3, the scaled mobility tends to increase with increasing ka at a fixed surface potential, whereas it tends to decrease with increasing ϕr at a fixed ka in the porous medium. Both trends are similar to those for the electrolyte solution.2,3 It is expected that all mobility profiles asymptotically converge to a constant value (below the Smoluchowski mobility) when ka gets very large. The difference between this constant and the Smoluchowski mobility is due to the extra hydrodynamic drag force exerted by the porous medium on the particle. When the particle is close to the solid plane, the double layer may be further compressed, or deformed, by the presence of the boundary, and hence the overall impact on the particle mobility is the outcome of the combination of these factors. The mobility of the particles toward the solid plane is thus a function of the particlewall separation distance. With this general understanding in mind, we go on to examine the specific impact of the planar boundary for various situations of interest in Figures 47. 3.1. Influence of the Separation Distance between the Particle and the Solid Plane. To illustrate the influence of the separation distance between the particle and the plane on the mobility of the particle, Figure 4 shows how the scaled mobility depends on particlewall separation distance h* at various values of ka with ϕr = 3 and λa = 1. Note that h* = 1 stands for the attachment of the particle to the plane, hence the mobility is zero. As shown in Figure 4, the mobility tends to decrease with the decrease in h*. In other words, the closer the particle is to the plane, the more profound the boundary retarding effect is. The reduction of the mobility reading with or without the presence of the plane is a good measurement of the boundary effect. For example, when the particlewall separation distance is sufficiently close (h* = 1.1), up to a 90% mobility reduction is observed for ka = 1. This indicates that if we adopt the mobility 13485
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Table 2. Correction Factors as a Function of h* at Various Values of ka for a Moderately Permeable Porous Medium (λa = 3) ka = 0.01
Figure 7. Scaled mobility as a function of ka at different ϕr values with λa = 1, 10 and h* = 3. Dashed line: results of λa = 0.
h*
ka = 0.1 Γ
h*
h*
Γ
1.1
0.088
1.1
0.088
1.1
0.105
1.3
0.190
1.3
0.190
1.3
0.230
1.5 3
0.252 0.348
1.5 3
0.252 0.351
1.5 3
0.309 0.430
5
0.345
5
0.352
5
0.438
10
0.332
10
0.343
10
0.438
30
0.322
30
0.342
30
0.438
of the particle based on an analytical formula such as the Smoluchowski equation, there would be a 10-fold overestimation of the actual mobility! In practical applications such as soil remediation in situ, large planar electrodes are placed in the ground with direct current being transmitted between them. The charged colloidal particles of interest migrate from the bulk space toward the planar electrode carrying the opposite charges. As shown in Figure 4, the migration speed slows down from a constant value in the bulk space gradually as they approach the planar electrode. The closer the particles get to the electrode, the slower the speed is. The actual rate is controlled by that corresponding to roughly h* = 1.1, or 10 times slower than that predicted by the analytical formula valid for the bulk space only. To provide a closer connection of mobility data to experiments, we define a correction factor Γ (= μm/(εζa/μ)) to correct the Smoluchowski prediction. The correction factor as a function of separation distance h* at various values of ka is given in Table 1, which greatly aids researchers or engineers seeking a preliminary estimation of the actual transport rate before a large-scale in situ operation is launched. For completeness, the corresponding results for a moderately (λa = 3) and a poorly permeable medium (λa = 10) are provided in Tables 2 and 3, respectively. These figures can facilitate the experimental maneuver by providing a theoretical prediction for design purposes, for instance. Figure 4 reveals that for a fixed value of ka the electrophoretic velocity of a particle may not always increase
h*
Γ
h*
Γ
h*
Γ
1.1
0.075
1.1
0.075
1.1
0.163
1.3
0.126
1.3
0.127
1.3
0.050
1.5
0.146
1.5
0.147
1.5
0.193
3
0.155
3
0.157
3
0.218
5 10
0.148 0.140
5 10
0.151 0.146
5 10
0.218 0.218
30
0.136
30
0.145
30
0.218
ka = 0.01
ka = 1 Γ
ka = 1
Table 3. Correction Factors as a Function of h* at Various Values of ka for a Poorly Permeable Porous Medium (λa = 10)
Table 1. Correction Factors as a Function of h* at Various Values of ka for a Highly Permeable Porous Medium (λa = 1) ka = 0.01
ka = 0.1
ka = 0.1
ka = 1
h*
Γ
h*
Γ
h*
Γ
1.1
0.029
1.1
0.029
1.1
0.038
1.3
0.036
1.3
0.036
1.3
0.050
1.5
0.038
1.5
0.039
1.5
0.053
3
0.035
3
0.036
3
0.055
5
0.033
5
0.034
5
0.055
10 30
0.032 0.031
10 30
0.033 0.033
10 30
0.055 0.055
monotonously with its distance from the plane, contrary to the situation in the homogeneous electrolyte solution.33,36 In the porous medium, as the particle moves toward the plane, it tends to speed up to a maximum velocity and then slow down if the particle is sufficiently close to the plane when the double layer is thick (small ka). This seemingly complicated behavior is related to the complicated competition between the hydrodynamic resistance and the deformation of the electric double layer there. The particle moves slower as it approaches the plane, as shown in Figure 4, because of the hydrodynamic hindrance effect of the boundary. At the same time, the distribution of counterions inside the double layer is suppressed by the plane as well. This suppression yields a steeper gradient of electric potential in front of the particle, hence increasing its electric driving force. Moreover, even without the presence of a nearby boundary, the double-layer structure may still be deformed or “polarized” by the convection of the fluid (the so-called double-layer polarization effect as mentioned earlier), which deters the particle motion in general. Taking into account these competing factors simultaneously, we find that the mobility of the particle thus exhibits complicated behavior as shown in Figure 5, with local maxima in some situations when the permeability is low. Apart from the qualitative discussion of the respective effect of various factors, the ultimate particle behavior can be determined for sure only by solving the governing electrokinetic equations rigorously, as we did here. 3.2. Effect of the Permeability of the Porous Medium. To shed more light on the influence of the medium permeability on 13486
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Langmuir the particle mobility, we notice that λa stands for an extra local friction force generated within the porous medium, FD = γv, in Brinkman’s model, where γ = μλ2. Note that this extra drag force is proportional to λ2. Figure 6 illustrates the scaled mobility of a charged particle as a function of particlewall separation distance h* with various values of λa at ϕr = 3 and ka = 0.1. As shown in Figure 6, the mobility tends to decrease with decreasing permeability, as indicated by higher values of λa. The increasing extra drag force slows down the particle motion as a result. Moreover, local maxima are observed with various values of h* when λa g 1. This is due to the combination of both the boundary effect and the double-layer polarization, as mentioned earlier. The deterring hydrodynamic boundary effect seems to be hindered in a sense by the presence of the less-permeable medium at some separation distance. No such extreme values are observed at specific h* when λa is lower than unity. The electrophoretic behavior of the particle is essentially like that in the electrolyte solution (λa = 0).2,3 3.3. Effect of the Surface Potential of the Particle in the Presence of a Solid Plane. The scaled mobility of the particle as a function of ka at various values of ϕr with h* = 3 is illustrated in Figure 7.The higher the particle permeability, the slower the particle motion, as expected. Moreover, a thinner double layer (higher ka) tends to enhance the particle mobility in general because a steeper gradient of electrical potential exists near the particle surface, which in turn generates a higher electric driving force. Moreover, more space charges reside near the particle in a thinner double layer, which contributes to the enhancement of particle mobility as well. When the surface potential is very high, say ϕr g 3 in practice, the electric driving force is so strong that the particle moves fast enough to induce an extra convection mechanism for the ion flux. Counterions in the double layer are swept from the front end and tend to cluster in the rear region. The double layer is deformed or polarized as a result, referred to as the double layer polarization here, which slows down the particle motion in general as an induced electric field opposite to the applied one is generated. As a result, the dimensionless particle mobility decreases somewhat with increasing surface potential ϕr, as shown in Figure 7, regardless of the particle permeability. It should be noted, however, that the particle with a higher surface potential still moves faster than that with a lower one, as the dimensional particle mobility, particle velocity per unit applied electric field μm = μm* εζa/μ, is multiplied by the surface potential. It is just not as fast as might be expected without the deterring doublelayer polarization effect. The presence of the planar boundary further slows down the particle motion in general when the particle is close to the solid plane, hence the particle mobility is smaller than that obtained for the corresponding single-particle situation: either Henry’s function is valid for low zeta potentials (2/3 for an infinitely thick double layer) or Smoluchowski’s predication is valid for a thin double layer (unity for an infinitely thin double layer). As discussed above, local maxima may be present, but eventually the deterring hydrodynamic force will dominate when the particle is sufficiently close to the plane. The results presented here extensively explore the effects of various parameters on the electrophoretic behavior of a particle normal to a solid plane in a Brinkman medium filled with an electrolyte solution. In addition to the application to soil remediation as mentioned above, the study results can also be utilized directly in the field of biotechnology in
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gel-electrophoresis analysis or transdermal drug delivery because the cross-linked polymer gels and the skin barriers are both porous in essence.
4. CONCLUSIONS The electrophoretic motion of a charged spherical particle in a porous medium is investigated theoretically by focusing on the boundary effect of a solid plane toward which the particle moves perpendicularly. The effects of key parameters of electrokinetic interest are examined, such as the double-layer thickness, the surface potential of the particle, the permeability of the porous medium, and so on. In particular, the effect of the presence of the planar boundary is examined in detail, including the physically visible deformation of the double layer due to the presence of the boundary and its electrokinetic implications. The resulting general electrokinetic equations are solved with a pseudospectral method based on Chebyshev polynomials. Comparisons with the limiting cases available in the literature show excellent agreement, indicating the accuracy and reliability of the results presented in this study. The presence of a solid planar wall is found to reduce the mobility in general. The closer the particle is to the plane, the more profound the boundary retarding effect is, which is measured as the reduction in particle mobility. Up to a 90% mobility reduction is observed when the thickness of the double layer surrounding the particle is comparable to the particle radius (ka = 1), when the particle is sufficiently close to the solid plane (h* = 1.1). This suggests that an overestimation of the actual mobility by a factor of 10 may occur when the analytical formula valid in the bulk space only is used, such as the famous Smoluchowski equation in a nonporous infinite medium. A correction factor Γ expressed as a function of separation distance h* is introduced to correct the Smoluchowski prediction, which greatly aids engineers seeking a preliminary estimation of the actual transport rate before a large-scale in situ operation is launched, such as in soil remediation. Moreover, we find that the boundary effect is significantly hindered by the porous medium surrounding the particle. The poorer the permeability of the medium, the more profound this hindrance effect. If the particlewall separation distance is sufficiently small in a poorly permeable medium, then the hydrodynamic effect dominates and the influences of the surface potential and the doublelayer polarization become insignificant. ’ AUTHOR INFORMATION Corresponding Author
*Tel: 886-2-23622530. Fax: 886-2-23622530. E-mail: ericlee@ ntu.edu.tw
’ ACKNOWLEDGMENT This work was finically supported by the National Science Council of the Republic of China. ’ NOMENCLATURE a radius of the hard core (m) h distance between the center of the particle and the plane (m) diffusion coefficient of the jth ionic species (m2/s) Dj e elementary electric charge (1.6 1019 C) E applied electric field (V/m) 13487
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Langmuir fj gj Kp kB lB nj nj0 p Pej rs T U v z zj
number flux density of the jth ionic species (no./m2 s) potential function for the concentration of electrolytes (V) Darcy permeability (m2) Boltzmann constant (1.38 1023 J/K) Brinkman screening length (m) number concentration of the jth ionic species (no./m3) bulk concentration of the jth ionic species (no./m3) pressure (N/m2) Peclet number of the jth ionic species dense swarm of particles of average radius (m) absolute temperature (K) electrophoretic velocity of the particle (m/s) fluid velocity (m/s) z coordinate in cylindrical coordinates valence of the jth ionic species
’ GREEK LETTERS α valence ratio between negatively and positive charged electrolytes γ frictional coefficient of the porous medium ε dielectric constant (C/V 3 m) zeta potential of the particle (V) ζa μ viscosity of the fluid (Pa 3 s) dimensionless electrophoretic mobility μ*m θ θ coordinate in cylindrical coordinates k reciprocal of the Debye length or the electric double layer thickness (1/m) ϖ ϖ coordinate in cylindrical coordinates λa dimensionless group characterizing the extra drag in the porous medium F space charge density (C/m3) ϕ electric potential (V) ξ ξ coordinate in bispherical coordinates η η coordinate in bispherical coordinates j j coordinate in bispherical coordinates porosity of a porous medium js ’ SUPERSCRIPTS * dimensionless form ’ SUBSCRIPTS e equilibrium state 0 macroscopic state 1 cation 2 anion ’ MATHEMATICAL OPERATORS 3 gradient operator Laplacian operator 32 Stokes stream function operator E2 Stokes stream function operator, E4 t E2E2 E4
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