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J. Phys. Chem. B 2010, 114, 10114–10125
Electrophoresis of a Soft Particle within a Cylindrical Pore: Polarization Effect with the Nonlinear Poisson-Boltzmann Equation Cheng-Hsuan Huang, Wen-Li Cheng, Yan-Ying He, and Eric Lee* Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan ReceiVed: January 20, 2010; ReVised Manuscript ReceiVed: May 31, 2010
Electrophoresis of a soft particle along the centerline of a cylindrical pore is investigated theoretically in this study. The soft particle consists of an inner hard sphere covered by a concentric porous layer with fixed charge uniformly distributed in it. The polarization effect, the deformation of ion clouds surrounding the particle due to convection flow, is taken into account properly by adopting the full nonlinear Poisson-Boltzmann equation. The study reveals that recent investigation in the literature without consideration of the polarization effect could severely overestimate the particle mobility up to nearly two times if the fixed charge in the porous layer is high. The boundary effect in terms of the reduction of particle mobility is very significant when the double layer is thick and diminishes as it gets very thin. The effect of the highly charged cylindrical wall is analyzed, in particular, among other factors of electrokinetic interest. The presence of the cylindrical wall retards the particle motion in general, as compared with an isolated particle. With the generation of an electroosmotic flow, however, the charged wall can either enhance the particle motion or deter it, depending on the surface potential on the wall and the double-layer thickness. The thinner the double layer, the more significant the influence of the osmotic flow on the particle motion in general. The direction of particle motion may even change twice as the reciprocal of the double-layer thickness increases when both the wall and the particle are highly charged. This is due to the competition between the electric driving force of the charged particle and the hydrodynamic retarding force from the background electroosmotic flow. This has direct impact in practical applications of nanofluidics when a weak electric field is applied. Conducting operations near these critical double-layer thicknesses should be avoided in practice. 1. Introduction Fundamental understanding of the electrophoresis in a cylindrical pore is essential to its smooth operations in various practical applications, ranging from separation techniques such as the capillary electrophoresis (CE) to the recent fast development in the field of microfluidics and nanofluidics such as biosensors. Electrokinetic transport in fluidic channels can facilitate the control and separation of ionic species, such as DNA and proteins, among other biomolecules of interest.1,2 In nanometerscale electrokinetic systems, the thickness of the electric double layer is comparable to the characteristic channel dimensions, which yields nonuniform velocity profiles and strong electric fields transverse to the flow. In such channels, both the streamwise and transverse electromigration fluxes can contribute to the separation and dispersion of biomolecules. As for the involvement of a nanopore, Kasianowicz et al.3 demonstrated that a DNA molecule can be detected as a transient decrease in the ionic current when it passes through a nanopore, and the measurement of the passage duration allows for the determination of the polymer length. Interesting effects of biological and synthetic nanopores were reported recently in experiments motivated by a new electrical DNA sequencing technique with embedded electrodes in the nanopores.4 Research interest is rejuvenated as a result over the fundamental electrophoretic behavior of a particle in a cylindrical pore, especially * To whom correspondence should be addressed. Tel: 886-2-23622530, Fax: 886-2-23622530, E-mail:
[email protected].
for the colloidal particles encountered frequently in biochemistry or bio-organisms in biology.2,4,5 In the field of corresponding theoretical analysis of a charged rigid particle in a cylindrical channel, Keh and Anderson6 studied the electrophoresis of a charged rigid particle within an uncharged wall under the assumption of a very thin double layer. Later, Keh and Chiou7 extended it to consider the effect of a charged cylindrical wall, still assuming a very thin double layer. Anderson and Ennis8 studied the effect of a finite double-layer thickness, but the zeta potential had to be low to make possible the theoretical treatment there. Shugai and Carnie9 later investigated the problem again with the numerical method proposed by Teubner.10 No restriction of double-layer thickness was imposed. Moreover, the double layer of the particle was allowed to touch the cylindrical boundary. The zeta potential still had to be low though. In their conclusion, they revealed that particle mobility can fall by more than 50% for a thick double layer and a sphere half of the radius of the pore. Hsu and Chen11 further took out the restriction of low zeta potential and considered the electrophoresis of a rigid sphere with an arbitrary zeta potential and double-layer thickness in a cylindrical pore. In general, the presence of a charged wall was found to generate an electroosmotic flow of the suspending fluid, which affects the particle motion greatly, whereas an uncharged wall retards the particle motion due to viscous drag. A lot of colloid particles in practice are not rigid, however. Instead, they form a composite structure, which consists of a hard inner sphere covered by a concentric porous layer of finite thickness.12-15 They are frequently referred to as “soft” particles. The soft particle is encountered very often in the field of
10.1021/jp100550p 2010 American Chemical Society Published on Web 07/22/2010
Electrophoresis of a Soft Particle biochemistry.12,14 A living bio-organism or a bionanoparticle falls into this category, for example.16 Ohshima17,18 presented a series of theoretical studies on the electrokinetic behaviors of soft particles, using the Brinkman model19 to describe the porous structure. Approximate analytical formulas were usually obtained, which are valid for dilute suspensions, assuming that the polarization effect is negligible. Hill et al.,16 in particular, considered the electrophoresis of a single soft particle immersed in an unbounded electrolyte solution with the polarization effect, using a numerical technique similar to the one introduced by O’Brien and White.20 Characteristic behaviors of electrokinetic interest were extensively explored there. They found, among other things, that the polarization effect can be very significant when the charge within the soft particle is high. Cheng et al.21 investigated further the boundary effect by analyzing the electrophoresis of a soft particle normal to a conducting solid plane. The study of soft particles has been very active due to its potential in modeling various systems of practical interest. Very recently, Hsu et al.22 considered the electrophoresis of a soft particle along the centerline of a cylindrical pore. They claimed that under the assumption of low surface potential at the inner hard core, it was justified to use the simplified linearized Poisson-Boltzmann equation; hence, the polarization effect, which slows down the particle motion usually, was negligible. Their deduction was based on the experience of treating rigid hard spheres, where it was found that the polarization effect, the deformation of the double layer when the particle is in motion, is significant when the particle surface potential is high, whereas it is negligible when it is low. However, there are two possible sources of space charges in a soft particle system, the one dissociated from the surface of the inner hard core and an additional one from the fixed charge spots within the outer porous layer. Not only the surface potential of the inner hard core can lead to a polarization effect; the counterions dissociated from the porous layer can also contribute to it. As a result, when the fixed charge density is high, the assumption of the linearized Poisson-Boltzmann equation does not hold anymore. Severe overestimation of the particle mobility may take place due to the polarization effect, which normally retards the particle motion. As the polarization effect cannot be taken into account by the linearized version, the adoption of the full nonlinear Poisson-Boltzmann equation is essential to analyze a soft particle system appropriately when the porous layer is highly charged. This is true when a highly charged wall is involved as well. Therefore, we present here a study on the electrophoretic behavior of a composite soft particle moving along the centerline in a cylindrical pore. The composite soft particle consists of an inner hard core that can bear arbitrary levels of surface potential, covered by a concentric outer porous layer of finite thickness and uniformly charged at arbitrary levels as well. A pseudospectral method23 based on Chebyshev polynomials is used to solve the resulting general electrokinetic equations numerically. This method has proven to be very powerful in solving various electrokinetic systems of interest.21,24,25 The influence of the polarization effect at high electric potentials is examined, a major improvement compared with the corresponding low zeta potential approach by Hsu et al. recently.22 We focus on the polarization effect from the porous layer in particular by assuming low surface potential at the inner hard core for most situations considered. This choice also makes easy the comparison with literature reports for limiting cases. In addition to the polarization effect, the impact of other key parameters is examined as well, such as the double-layer
J. Phys. Chem. B, Vol. 114, No. 31, 2010 10115
Figure 1. Schematic representation of the problem considered where a soft sphere of radius b comprising a rigid core of radius a and a porous layer of thickness δ is placed on the axis of a long cylindrical pore of radius c. A uniform electric field E parallel to the axis of the pore is applied in the Z-direction.
thickness of the colloid, the properties of the porous layer, and the ratio of pore-to-particle radii. Both the charged and the uncharged walls are considered. In particular, the nonlinear effect of a charged wall at high surface potential is investigated in detail, focusing on the involvement of an electroosmotic flow. As mentioned earlier, the results of this study have potential applications in microfludic and nanofluidic operations, such as biosensors or lab-on-a-chip devices,26,27 where electrophoresis is often adopted as a driving force to move the colloidal particle through the micro- or nanochannels. Further consideration of other forces on the nanoscale pertinent to the specific nanofluidic system of interest may be necessary though. The fundamental framework presented here is applicable nonetheless in general. It also provides more information about the general electrokinetic behavior of composite soft particles, which has great potential in treating systems of biochemical or biological interests, for instance. The conventional technique of capillary electrophoresis (CE) is also closely related to the system studied here and thus may benefit from the general observations found here. 2. Theory 2.1. Description of the Model. The system under study here is illustrated in Figure 1, where a soft spherical particle of radius b with a concentric rigid core of radius a and an ion-penetrable porous layer of thickness δ (δ ) b - a) moves with velocity U along the axis of a cylindrical pore of radius c in response to an applied uniform electric field E in the Z-direction. The cylindrical pore is filled with an aqueous solution containing z1:z2 electrolytes, where z1 and z2 are the valences of cations and anions. The electroneutrality in the bulk liquid phase requires that n20 ) n10/R, where n10 and n20 are, respectively, the bulk concentrations of cations and anions and R ) -z2/z1.
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Huang et al.
Spherical coordinates (r,θ,φ) are adopted to describe the liquid phase and the membrane layer, and the cylindrical coordinates (R,Θ,Z) are applied to a region external to the soft sphere as well as the cylindrical wall. The origin of the spherical coordinate is located at the center of the soft particle, and the symmetric nature of the problem suggests that only half of the (r,θ) domain needs to be considered. The main assumptions in our analysis are as follows. (i) The Reynolds numbers of the liquid flow are small enough to ignore inertial terms in the Navier-Stokes equations, and the liquid can be regarded as incompressible. (ii) 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 with the fields that occur in the double layer, with |E| , ζaκ, the characteristic electric field measured by the zeta potential divided by the double-layer thickness. (iii) No electrolyte ions can penetrate the particle core. (iv) The polymer layer is permeable to mobile charged species. (v) The permittivity ε takes the same value both inside and outside of the polymer layer. The governing equations for the problem here are, respectively, the Poisson equation, the conservation of ionic species, the Stokes equation outside of the porous layer (r > b), and the Brinkman equation inside (a < r < b)
∇2φ ) -
2 F + Ffix zjenj Ffix )ε ε ε j)1
∑
a
