Eur. Phys. J. E (2014) 37: 104 DOI 10.1140/epje/i2014-14104-4
THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Effect of induced electric field on migration of a charged porous particle Partha P. Gopmandal1 , S. Bhattacharyya2,a , and Bhanuman Barman2 1 2
School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164-2920, USA Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, West Bengal, India Received 27 March 2014 and Received in final form 2 September 2014 c EDP Sciences / Societ` Published online: 6 November 2014 – a Italiana di Fisica / Springer-Verlag 2014 Abstract. The effect of ambient fluid flow on a charged porous spherical particle suspended in an aqueous medium is analyzed. The porous particle is ion permeable and fluid penetrable. The induced electric field due to the polarization of the particle’s electric double layer and counterion condensation leads to a hindrance effect on particle migration by producing an electric force. The influence of this retardation force on the hydrodynamics of the particle is studied through the Nernst-Planck equations, which are coupled with the Stokes-Brinkman equation. The interactions of the double-layer polarization, shielding effect, electroosmosis of unbalanced ions and fluid convection are analyzed. The settling velocity and fluid collection efficiency of the charged aggregate is determined. We have studied the electrokinetics for a wide range of fixed charge density and permeability of the particle with no assumption made on the thickness of the double layer relative to the dimension of the particle.
1 Introduction The separation of bioparticles is often accelerated by either considering a non-uniform cross-section of the microtube [1,2] or imposing an external pressure field in capillary electrophoresis [3]. A controlled migration of polymer by an imposed pressure-driven flow and an external force is investigated by Butler et al. [4]. Centrifugation is a powerful technique for separating and analyzing cells, organelles, charged proteins and colloids. Rasa and Philipse [5] experiment on ultracentrifugation of charged colloidal silica spheres demonstrates the existence of a macroscopic electric field and its drastic effects on sedimentation profiles. Recently, numerical studies on advective flow of homogeneous as well as non-homogenous permeable spheres under the action of an electric field for finite values of Reynolds number were made by Yang et al. [6] and Zhang et al. [7]. When a charged particle is immersed in an aqueous media, a layer of diffuse charge clouds (Electric Double Layer, EDL) of surplus counterions forms around the particle. A relative motion between the particle and the surrounding medium creates a deformation of the double layer resulting in the formation of an induced electric field. This phenomenon is referred to as the double-layer polarization (DLP) [8]. This induced electric field creates an electrostatic force on the unbalanced charges and deters the convection of the particle. Theoretical studies and a
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mathematical models on the migration of charged colloids or porous particles in an aqueous media under the action of gravitational force, centrifugation or pressure-driven flow has drawn the interest of several authors. The steady sedimentation of a single charged rigid spherical colloid under the action of gravity was studied by Booth [9], Stigter [10] and Saville [11]. Ohshima et al. [8] derived the general expression for the sedimentation velocity as a function of particle ζ-potential and Debye layer thickness through a linear perturbation analysis. The retardation effect due to the deformation of double layer through a linear perturbation analysis is found to be substantial when the Debye layer thickness is comparable with the radius of the particle. Keller et al. [12] investigated the settling process of a colloidal particle under the influence of a gravitational or centrifugal field through a model based on the Stokes-Poisson-Nernst-Planck equations. There they have documented most of the important studies on settling of a rigid charged particle. In several biological contexts one can encounter not only the dispersion of discrete charged particles but also of the aggregate of charged colloids or monomers [13,14]. The study of sedimentation of aggregates formed due to the electroflocculation or centrifugation of a colloidal suspension is important in the context of separation and purification [15]. It may be noted that the charged porous sphere i.e., permeable to fluids and ions, has proven to be a successful model to describe the electrokinetic behavior of a polyelectrolyte [16]. The penetration of counterions within a permeable aggregate and the counterion
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condensation [17] phenomena creates a characteristic difference between its electrokinetics compared to a rigid particle. The condensed counterions lower the charge density of the polyelectrolyte [18] and thus have an impact on the dynamics of the particle. Based on the linearized electrokinetic equations, the steady-state sedimentation or migration of polyelectrolyte molecules or charged flocs, modeled as a porous sphere, in an electrolyte solution is analyzed by Keh and Chen [19]. Subsequently, Chiu and Keh [20] analyzed the steady sedimentation of a charged soft sphere in the extent of an electrolyte solution. The steady electrokinetic flow of an electrolyte solution due to the electric field and/ or pressure gradient in the fibrous medium, constructed as an array of charged circular cylinders, is studied through a small perturbation analysis by Wu and Keh [21]. The electrokinetics of a charged porous particle is governed by the mutual interactions of double-layer polarization, counterion condensation, electro-osmosis of unbalanced ions and fluid flow [22]. The convection-induced double-layer polarization on the motion of a charged sphere moving near the air-water interface is studied by Hsu and Lee [23] under the assumption of a slight deformation of the double layer. In all those studies, a slight deformation of the double layer is assumed and the distribution of ion is obtained through a small perturbation from its equilibrium Boltzmann distribution. The small perturbation analysis may become questionable for a high charge density of the particle where the strong shielding effect influences the DLP and electro-osmosis of unbalanced ions. The mathematical model to describe the sedimentation of a charged porous particle is similar to those of its electrophoresis [24]. The effect of double-layer polarization (DLP) on the electrophoresis of a soft particle or an entirely porous particle was made by several authors in recent years, namely, e.g., He and Lee [25], Hsu and Tai [26], Huang et al. [27], Yeh and Hsu [28], Uppapalli and Zhao [29], Huang et al. [30], Yeh et al. [31], Hsu et al. [32], Hsu and Lee [33] and Bhattacharyya and Gopmandal [34]. The above studies on permeable particles suggest that the DLP effect diminishes for either a very low or a high value of κa, where 1/κ is the characteristic thickness of the double layer and a is the radius of the particle. All the previous studies dealing with DLP effects on sedimentation of charged particles are based on a small perturbation from the equilibrium Boltzmann distribution. For finite Debye length, the non-zero charge density outside the double layer can lead to a concentration gradient between the double layer and the bulk fluid. This concentration gradient becomes stronger when the charge density of the particle becomes higher. The presence of a non-zero charge density outside the Debye layer makes the ion transport equations and the equations for fluid flow coupled. The convective transport of ions can be well accounted for by the Nernst-Planck model. In this paper, we have analyzed the dynamics of a charged porous particle sedimenting in an aqueous medium under an external force such as gravitational,
Eur. Phys. J. E (2014) 37: 104
centrifugal field and/or pressure-driven flow. The mutual interactions of the double-layer polarization, counterion condensation and fluid inertia on the dynamics of the charged particle are studied. We have also compared the hydrodynamics of the charged particle with the corresponding uncharged case and analyzed the role of the induced field on the electrokinetics of the particle. Based on the nonlinear PNP model (Poisson-Nernst-Planck), we have measured the effects of electric force on the hydrodynamics of the particle. The impact of the governing parameters such as Debye layer thickness, permeability, fixed charge density of the particle, fluid convection on the DLP and counterion condensation is investigated through the nonlinear model. A computational method based on the finite-volume algorithm is adopted in the present analysis. We found that the fluid convection has a strong effect on the ion distribution. The modeling of configurations involving a fluidporous interface can be made through two different formulations, the Navier-Stokes equations for the clear fluid domain and the Brinkman equation for the porous domain. They are usually solved simultaneously with an interface condition between the two media. The Brinkman equation has the advantage over the well-known Darcy’s law in that it contains the stress tensor, therefore the boundary condition is the continuity assumption of momentum and viscous stress along the interface boundary. The implementation of the interface condition [35] is complicated as it contains material-dependent parameters whose value must be determined empirically. Owing to this difficulty, we have adopted a single-domain approach [36] in which the matching of variable values at the interface is inherent in the formulation itself. In this formulation, two sets of equations for the fluid and the porous regions are combined into one set by using a binary parameter and the Dupuit-Forchheimer relationship [35] is used to relate the fluid velocity between the two zones.
2 Mathematical model and numerical methods We considered a charged porous spherical particle of radius a suspended in an aqueous medium. The surrounding fluid flows at a speed U0 from infinity towards the sphere in a frame of reference which is fixed at the center of the sphere (fig. 1). A spherical polar coordinate system (r, θ, ψ) is adopted and the direction of the far-field velocity is along the negative z-axis. The fluid flow within and around the sphere is governed by the equations as described below. The permeable sphere is assumed to be composed of charged monomers and has a uniform porosity and permeability kp . As the present configuration involves a fluid-porous interface, the Brinkman model is used to account for the jump in shear stress. The fluid velocity q is related to the intrinsic (average over the fluid volume only) velocity V by q = V. We adopt a unified single-domain approach. In this approach, the two sets of equations for the fluid and the porous regions are
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force. The dimensionless parameter B1 = en0 φ0 a/μU0 , where n0 is the bulk concentration of ions. The binary parameter B assumes the value 1 within the particle and 0 outside of it. The non-dimensional parameter governing the fluid flow around and through the charged permeable particle is Darcy number Da = κp /a2 , where μ is the fluid viscosity. In the fluid region, the Darcy number Da is assumed to be infinity and the porosity is 1 outside the porous zone and 0 < < 1 within the porous zone. Based on Gauss law, the electric potential φ is determined by the following Poisson equation: ∇2 φ = −
Fig. 1. Sketch of the flow configuration for a charged porous microparticle. A spherical polar coordinate (r, θ, ψ) is used with origin at the center of the sphere.
combined into one set using appropriate switching terms. The single-domain formulation requires no conditions to impose at the fluid-porous interface. The relative permittivity of the fluid inside and outside the porous aggregate is considered to be the same. The equations governing this electrokinetic phenomenon are the Darcy-Brinkman extended Stokes equations for fluid flow, the Nernst-Planck equations for ion transport and Poisson equation for electric field. We assume an axi-symmetric flow with z-axis as the axis of symmetry. The bulk ionic concentration n0 is the scale for ionic concentration. The radius of the sphere a is assumed to be the length scale, the velocity field is scaled by U0 and φ0 (= RT /F ), the thermal potential is the potential scale. The far-field fluid velocity U0 is considered to be of the order of the corresponding electrophoretic velocity of a porous sphere under an electric field φ0 /a. The pressure is non-dimensionalized by μU0 /a, where μ is the fluid viscosity. The structure of the porous particle is assumed to be homogeneous with a constant fixed charge density ρfix . The non-dimensional form of the equations describing the motion of the ionized fluid in and around the porous aggregate is ∇ · q = 0, q − B1 ρ¯e ∇φ = 0. −∇p + ∇2 q − B Da
(1) (2)
For an axi-symmetric flow q = (v, u) is the velocity vector where v is the radial and u is the cross-radial velocity component. The last two terms in the left side of eq. (2) represent respectively, the body force due to the frictional force within the porous medium and the electric driving
(κa)2 ρ¯e − BQfix , 2
(3)
the net charge density is denoted by ρ¯e = zi ni , where ni is the number concentration and zi is the valence of the i-th ionic species. The concentration of each ionic species is scaled by the bulk number concentration n0 . Here we have considered binary symmetric ionic species with i = 1, inverse of the EDL thickness 2 and z1 = 1, z2 =−1. The 2 2 is given by κ = i e (zi ) n0 /e KB T . Here e is the permittivity of the medium, e is the elementary electric charge, KB is the Boltzmann constant, R is the gas constant, T is the absolute temperature and F is the Faraday constant. We assume that the permittivity e is the same for both the regions i.e., within the porous zone and in the clear liquid zone. The non-dimensional parameter Qfix = ρfix a2 /e φ0 determines the fixed charge density of the porous particle. The non-dimensional form of the Nernst-Planck equation governing the transport of the i-th ionic species is given by Pe (q · ∇)ni − ∇2 ni − zi ∇ · (ni ∇φ) = 0.
(4)
In this case, we have considered that the mass diffusion coefficients are approximately equal, D = D1 ∼ = D2 . It may be noted that the present model based on NernstPlanck equations can handle multivalent ions. The nondimensional parameter Peclet Number P e = U0 a/D and Schmidt number can be defined as Sc = ν/D. It may be noted that P e = Re · Sc, where Re = U0 a/ν is the Reynolds number. Far away from the particle, the electric and concentration fields are uninfluenced by the presence of the particle thus, ∂φ/∂r = 0 and ∂ni /∂r = 0 at r = R a. A symmetry condition is imposed along the axis of symmetry θ = 0 and π. It may be noted that in the present formulation no boundary conditions is required at the fluid-porous interface (r = 1). The electrostatic force (FE ) and the hydrodynamic force (FD ) along the flow direction for this axi-symmetric problem can be obtained by integrating the Maxwell stress tensor and hydrodynamic stress tensor on the surface S of the particle, respectively, and are evaluated by the expression as given by Bhattacharyya and Gopmandal [34]. The forces FE and FD are scaled by e φ20 and μU0 a, respectively. The governing equations for the ion transport and fluid, which are cast into conservation law form are in-
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Eur. Phys. J. E (2014) 37: 104 1 Ω
0.9 η
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0.0208 x 0.005 0.0208 x 0.025 0.0104 x 0.0125 Analytical Results
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Fig. 2. Comparison of drag factor Ω and fluid collection coefficient η for the uncharged case at different values of β. The symbols represent the results due to Neale and Epstein [37] for Ω and Adler [38] for η.
tegrated over each control volume in a staggered grid arrangement. Different control volumes are used to integrate different equations. At each interface of the control volume, the Quadratic Upstream Interpolation for Convective Kinematics scheme (QUICK [39]) is used to discretize the convective terms of the unsteady equations, while a linear interpolation between two neighboring cell values is used to discretize the diffusion terms. A pressure correction based iterative method SIMPLE [40] is used to compute the resulting algebraic equations. The procedure is based on a cyclic series of guess and correct operations to solve the governing equations. The pressure link between the continuity and momentum equations are accomplished by transforming the discretized continuity equation into a Poisson equation for pressure correction. This Poisson equation implements a pressure correction for a divergent velocity field. The iteration procedure starts with an assumption for potential at each cell. At every iteration, the electric field is determined by solving the Poisson equation for electric potential i.e., eq. (3). Equation (3) is solved by a line-by-line iterative method along with the successiveover-relaxation (SOR) technique. The stability and accuracy of the present algorithm is tested through a grid independency test and by comparing with the previously published results (fig. 2).
3 Results and discussions Through the dimensionless parameter√β = λs a, the Darcy = number can be expressed as β = 1/ Da, where λ−1 s √ κp is the softness parameter. The softness parameter can range from 10 nm to 0.4 mm [28]. For nano-particle aggregates of porosity close to 0.94, the permeability, based on the Happel’s law, can lead to Da ≥ 0.01 [41]. It has been found by several authors (e.g., Huang et al. [27]) that the electrophoresis becomes independent of the softness parameter if β = λs a lies outside of 1 and 10. For biological cells the fixed charge density i.e., ρfix ≤ 106 C/m3 . We
have considered the scaled fixed charge density Qfix ≤ 100 for a particle of radius a = 100 nm. The parameter κa varies from 1 to 30, which implies that the bulk concentration varies between 0.01 mM and 10 mM for a particle with a = 100 nm. Previous studies have established that the DLP effect is most substantial for O(1) κa. The scaling of the variables suggests that for a fixed set of values of the non-dimensional parameters i.e., κa and Qfix , if we choose smaller (or higher) aggregate size then the DebyeHuckel parameter (κ) will increase (decrease) at the same rate and ρfix should increase (decrease) at the square of the rate by which a is lowered (increased). Here we have taken φ0 = 0.02586 V, e = 695.39 × 10−12 C/Vm, μ = 10−3 Pa s and ρ = 103 kg/m3 . The diffusivity of ions are considered to be equal and is D = 2.0 × 10−9 m2 /s, thus Sc = 500. Based on the mobility expression due to Hermans and Fujita [42] for a charged porous sphere with Qfix = 100 and Da = 1, the electrophoretic velocity under an imposed field φ0 /a is 0.0464 m/s, which corresponds to Reynolds number Re = 0.00464 and Peclet number P e = 2.32 for a particle of radius 100 nm. We have assumed a creeping flow condition with Re < 1. However, the fluid convection effect on ion distribution is included in the present analysis. In order to test the accuracy of our method, we have compared our computed results for drag factor (Ω) and the fluid collection efficiency (η) for an uncharged permeable sphere (Qfix = 0) at various values of the permeability (β) with the formula obtained by Neale and Epstein [37] and Adler [38] i.e., eq. (5) and eq. (7), respectively. Figure 2 shows that our computed solutions are in excellent agreement with those analytical results. We have also made a grid independency study in fig. 2. The drag factor (Ω) is defined as the ratio of drag between a permeable particle and the corresponding Stokes drag of a solid particle. Within the creeping flow limit the drag factor is inversely proportional to the settling velocity of a permeable particle. For an uncharged particle Neale and Epstein [37] obtained an analytic form of the drag factor under the creeping flow condition as Ω=
2β 2 [1 − 2β 2 + 3[1
tanh β β ] tanh β − β ]
.
(5)
For a permeable sphere with intermediate values of permeability, 0 < Ω < 1. Note that as β −→ ∞ (i.e., Da −→ 0), Ω −→ 1. The fluid collection efficiency (η) of a permeable sphere is defined as the ratio of the interior flow through the aggregate to the flow approaching it. Thus π/2 2 0 v sin θdθ . (6) η= U0 The analytical formula of fluid collection efficiency (η) due to Adler [38] for homogenous porous aggregate in linear Stokes flow is given by −1 −1 tanh β 9 3 + 1− . (7) η= 2β 2 2β 2 β
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(b) Fig. 3. Variation of (a) electric force FE and (b) drag factor Ωf (ratio of drag experienced by the charged polyelectrolyte to the Stokes drag of a rigid particle) with Debye layer thickness (κa) for different values of the charge density when Da = 0.1. Here Qfix = 0 refers to the case of an uncharged permeable sphere. Symbols indicate the results based on the Navier-Stokes-Brinkman equation for Re = 0.005.
ration in capillary electrophoresis. The symbols in fig. 3a indicate that the solution is based on the nonlinear NavierStokes-Brinkman equations. Results show that the convection terms in momentum equations have practically no effect. However, the Peclet number, which measures the relative importance of convection to diffusion of ions, is O(1). Because of this, the transport of ions due to convection is not neglected in the present computations. We find that the DLP effect is significant when 1 ≤ κa ≤ 10, which is in accord with several previous studies e.g. Yeh and Hsu [28] and Bhattacharyya and Gopmandal [34]. It may be noted that the variation in the Debye-Huckel parameter (κa) at a fixed value of a implies a variation in the ionic concentration of the electrolyte. We find that FE increases rapidly with the increase of ionic concentration up to a certain critical value of κa. Beyond this critical value, the electric force starts decreasing slowly with the rise of κa. This critical value of κa depends on the fixed charge density (Qfix ). The penetration of counterions and neutralization of particle charge density due to the shielding effect is a diffusion-dominated process, which is insignificant for low ionic concentration (i.e., κa is small). As the ionic concentration of the electrolyte is increased (κa is increased), the diffusion-dominated mechanism becomes stronger. The occurrence of a local maxima in FE is due to the interaction of the double-layer polarization (DLP) effect and counterion condensation. The analytical solution for sedimentation of a charged rigid particle by Oshima et al. [8] and a soft particle with charged rigid core as considered by Chiu and Keh [20] shows that the retardation effect due to the induced electric field vanishes as κa approaches either zero or infinity. It is evident from eq. (3) that the charge density ρe is zero everywhere when κa → ∞, thus the effect of the induced field is absent. When κa → 0, the shielding effect due to the fixed charge density of the particle becomes zero as the counterion concentration in the liquid becomes negligible. It is expected that the lower or upper limits of κa for which the induced electric field effect is negligible depends on the fixed charge density of the particle.
3.1 Effect of κa and Qfix Figure 3a shows the variation of the electric force experienced by the permeable particle (FE ) as a function of Debye layer thickness for different values of the particle charge density Qfix . We present results for Reynolds number Re = 0.005, the corresponding Peclet number is 2.5. It may be noted that the Reynolds number based on the electrophoretic velocity under an applied electric field φ0 /a with Qfix = 100 and a = 100 nm may lead to Re = 0.00464. Several authors (e.g., Xiao et al. [14]) found that the settling velocity of a diatom aggregate in water ranged from 1.88 mm/s to 5.84 mm/s. This settling velocity for a = 100 nm corresponds to the range of Reynolds number between 1.88 × 10−4 and 5.84 × 10−4 . Thus, the range of Reynolds number based on the sedimentation velocity of the aggregate as considered here can be obtained through centrifugation with moderate values of g-factor. Several authors, e.g. Zheng and Yeung [3,43] and Keker et al. [44] considered O(1) Reynolds number for DNA sepa-
We have determined the drag factor (Ωf ), which is the ratio of the drag of the charged particle (FD ) to the Stokes drag due to a rigid uncharged sphere i.e., 6πμU0 a, as a function of the concentration of the ambient electrolyte. The drag factor is inversely proportional to the settling velocity of the aggregate in Stokes flow limit. We find from fig. 3b that the drag factor follows the same pattern as that of the electric force (fig. 3a) i.e., it increases rapidly with the increase of ionic concentration up to a certain critical value of κa. Beyond this critical value the drag factor starts decreasing slowly with the rise of κa. In any case, the drag factor is bigger than the drag factor due to a porous uncharged particle (i.e., Qfix = 0) for the range of κa considered. Our result shows that the drag factor for an uncharged particle (Qfix = 0) with β = 3.16 (i.e., Da = 0.1) is 0.621, which is the same as can be obtained by (5) and the value presented in fig. 2. We find that the drag factor is increased compared to the uncharged particle by about 40% when Qfix = 50. Thus, the retardation
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produced by the presence of a double layer is evident. This retardation effect can be considered to be due to the development of the electrohydrodynamic (EHD) pressure field [45] created by the Coulumbic electrostatic force arising because of unbalanced charges in and around the particle. We find that the hydrodynamic drag increases with the increase of charge density Qfix . This is due to the fact that, as Qfix is increased, a higher number of counterions accumulate near the particle due to the stronger shielding effect which raises the EHD pressure field. The effective charge density of the permeable particle can be determined by 1 π (κa)2 ρ¯e r2 sin θ drdθ 2π 2 r=0 θ=0 + Qfix . (8) Qeff = 4π
Pe=2.5 Pe=0
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We define the charge neutralization factor, Γ as Γ = 1 − Qeff /Qfix .
Qfix=0
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(9)
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The penetration of counterions within the permeable particle is large if the charge neutralization factor Γ is close to one. In fig. 4a we present the variation of Γ as a function of κa for different values of the particle charge density (Qfix ). It is evident from these results that a monotonic increase in Γ occurs with the rise of κa. The decrease in effective charge density of the particle due to the counterion condensation at higher values of κa creates a monotonic reduction in electrostatic force for those values of κa (fig. 3a). Screening of the fixed charge density is more efficient as the particle fixed charge is increased. The shielding effect grows stronger with the increase of Qfix , which leads to a higher charge neutralization factor at larger values of the particle fixed charge density. In order to illustrate the convection effect on ion distribution, we have included the results corresponding to the case where convection effect is neglected i.e., P e = 0. Our results show that the effective charge density of the particle is greatly influenced by the convective transport of ions. The effect of convection is strong when the EDL is thicker i.e., for lower values of κa. The ion convection lowers the rate of the charge neutralization of the aggregate. These results suggest that the hydrodynamic interaction on counterion condensation is not negligible. The fluid penetration inside the aggregate can be measured through the fluid collection efficiency (7). Figure 4b describes the collection efficiency as a function of κa at different values of Qfix . Results show that the effective permeability of the particle becomes low as the EDL thickness decreases. The fluid collection efficiency also decreases with the increase of Qfix . The fluid collection efficiency is lower than the corresponding value due to an uncharged particle (Qfix = 0). The maximum percentage difference is found to be about 60%. The momentum loss due to the EHD pressure field creates a lower rate of fluid flow through the permeable particle. By comparing fig. 4a and b we find that the parameter values for which Γ increases create a reduction in η and vice versa. A higher rate of fluid flow through the particle drags counterions towards
0.2
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Fig. 4. Effect of Debye layer thickness (κa) on the neutralization factor Γ and fluid collection coefficient η at different values of the charge density when Da = 0.1 (β = 3.16) and P e = 2.5. (a) Γ as a function of κa for Qfix = 10, 30, 50 where the arrow points to the increasing direction of Qfix . Result without the ionic convection (P e = 0) is also included. Solid lines: P e = 2.5; dashed lines: P e = 0. (b) η as a function of κa for Qfix = 10, 30, 50, Da = 0.1 and P e = 2.5. The line with Qfix = 0 represents the uncharged case.
the downstream of the particle which in turn lowers the charge neutralization of the particle. In this way, the fluid convection affects the counterion condensation. Figure 5a shows the induced potential along the surface of the particle for κa = 1, 5, 10 when Qfix = 50. The distribution of φs suggests that the induced electric field is along the negative z-axis, i.e. it acts opposite to the direction of the migration of the particle. This electric potential drop is higher for higher values of the DebyeHuckel parameter. Figure 5b shows the form of φs for different values of Qfix . The increase in the fixed charge density produces an enhancement in the induced electric field due to the increased shielding effect. The effect of fixed charge density (Qfix ) on the electric force, drag factor and particle charge neutralization factor is illustrated in fig. 6a-c. It is clear from these results that an increase in Qfix increases the drag factor and hence
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(b) Fig. 5. Variation of the induced potential φs on the surface of the polyelectrolyte for (a) Qfix = 50 at different values of κa = 1, 5, 10. (b) κa = 10 at different values of Qfix = 10, 30, 50. Here P e = 2.5 and Da = 0.1.
20
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reduces its settling velocity. However, the dependence of FE (or FD ) on Qfix has changed its pattern when higher values of Qfix is considered. This type of nonlinear pattern has been reported in the context of electrophoresis by Yeh and Hsu [28] and Hsu and Lee [46], which was explained through the counterion condensation. We find from fig. 6c that as Qfix becomes large, the effective charge density of the particle drastically drops to a low value and approaches a constant. The distribution of the net charge density in and around the particle is shown in fig. 7a-c for different values of the EDL thickness i.e., κa = 1, 5, 10 when the Peclet number is P e = 2.5. The distribution of the net charge density shows that the induced field acts opposite to the direction of the migration of the particle. The deformation of the double layer is prominent for κa > 1. The net charge density within the particle becomes less positive and a higher number of counterions accumulate downstream of the particle as the EDL becomes thinner. The distribution of the net charge density compliments our previous findings, i.e. the value of the neutralization
0.7
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(c) Fig. 6. Variation of (a) scaled electric force (FE ); (b)drag factor Ωf ; and (c) neutralization factor Γ as a function of particle fixed charge density when P e = 2.5, Da = 0.1 and κa = 10.
factor becomes higher (fig. 4a) with the increase of κa. We found in fig. 3a that FE increases at a faster rate with the increase of κa for the lower range of κa. In this range of κa, the DLP effect grows but the neutralization of the particle charge density is not significant. For a higher range of κa, the effective charge density of the particle reduces with the increase of κa. For this, we find in fig. 3a that FE reduces with the increase of κa when κa is large.
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(b) Fig. 8. Variation of the (a) drag factor Ωf and (b) scaled electric force FE as a function of β for different values of charge density at κa = 10 and P e = 2.5.
Fig. 7. Distribution of the net charge density ((n+ − n− )(κa)2 /2 + BQfix ) when Da = 0.1 (β = 3.16), P e = 2.5 and Qfix = 50. (a) κa = 1. (b) κa = 5. (c) κa = 10.
3.2 Effect of permeability The effect of permeability on the drag factor Ωf and electric force (FE ) is illustrated in fig. 8a, b for thin EDL case i.e., κa = 10 when P e = 2.5 for different values of Qfix = 10, 30, 50 along with the uncharged case, i.e. Qfix = 0. Results show that a change in permeability (β) produces a strong effect on Ωf and FE when permeability is in a higher range (β < 5). In that range of the permeability (i.e., β < 5), the electric force decreases at a faster rate with the increase of β. The drag factor for a charged
particle is always higher than its value for an uncharged particle. When the particle is highly permeable, fluid can flow easily through the particle (fig. 8a). The relatively stronger flow through the particle at higher permeability pushes the counterions towards downstream to create a stronger induced field. Simultaneously, the stronger fluid flow through the particle also drags counterions downstream and hence lowers the particle charge neutralization. The combination of these two effects leads to a large FE at low β (high permeability). Increase in β (lowering the permeability) enhances the diffusion-dominated shielding effect and hence reduces the effective charge density of the particle which results in a lower electric force. We introduce a parameter which measures the ratio between the permeability and the Debye length i.e., χ = κa/β and presents the drag factor as a function of χ. In fig. 9 we vary χ by increasing κa at fixed β. The drag factor is found to increase with the increase of β(i.e., decrease of permeability). We find that the fluid transport within the particle is not similar to the combined parameter χ. At a fixed value of χ, the drag factor may vary due to variation of either β or κa. The drag factor shows the occurrence of local maxima at any fixed value of β. The
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Fig. 9. Variation of drag factor as a function of χ = κa/β for fixed values of the charge density Qfix = 50. We have varied χ by varying κa for fixed values of β.
value of χ at which Ωf attains a local maximum changes with the change of the particle permeability. The maximum in drag factor implies the highest hindrance on the particle transport. The effect of permeability on the net charge density in and around the particle is described in fig. 10a-c when κa = 10. We find that the DLP effect is low for a low permeable particle. Besides, the net charge density within the particle is less when permeability is low. These together explain the reduction of FE with the increase of β, i.e. lowering the permeability, as found in fig. 8a. A long plume of counterions downstream of the particle is observed for a high permeable case. The presence of the unbalanced ions creates an EHD pressure field, which results in an enhanced drag compared to an uncharged particle as seen in fig. 8a. It is evident from fig. 10a-c that the convective transport of ions is not negligible even at this small Peclet number, i.e. P e = 2.5. Figure 11a, b shows that the fluid collection efficiency decreases and the neutralization factor increases as the particle permeability decreases. As the particle becomes more permeable, the charge neutralization factor (Γ ) becomes low. At a fixed value of β, the increase in Qfix reduces the rate of fluid flowing through the particle and increases the neutralization factor. The stronger shielding effect at higher Qfix lowers the effective permeability and charge density of the particle.
Fig. 10. Distribution of the net charge density ((n+ − n− )(κa)2 /2 + BQfix ) when κa = 10, P e = 2.5 and Qf ix = 50. (a) Da = 1. (b) Da = 0.1. (c) Da = 0.01.
3.3 Effect of U0 The effect of the migration speed of the particle (U0 ) on the electrokinetics is illustrated in figs. 12a-c when the Debye-Huckel parameter κa is 10. We have varied the Reynolds number between Re = 0.0005 and Re = 0.005 by varying U0 . The corresponding Peclet number varies in the range 0.25 ≤ P e ≤ 2.5. We have presented results in terms of Peclet number as the Reynolds number does not appear explicitly in the present set of equations. The Peclet number measures the ratio of advection to diffusion transport
of ions. It has already been pointed out (fig. 3a) that in this range of Reynolds number, the fluid flow can be well approximated by the Stokes-Brinkman equation. Figure 12a shows that the electric force increases monotonically with the increase of P e. We have shown in fig. 12c that an increment in fluid convection speed lowers the rate of neutralization of the particle charge density, thus increases the effective charge density of the particle. These in combination produce a higher electric force with the rise of P e. The stronger induced electric field also creates a higher EHD
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Fig. 11. Variation of (a) the fluid collection efficiency (η) and (b) Neutralization factor (Γ ) as a function of β for different values of the charge density when κa = 10.
pressure field, which leads to the drag factor (Ωf ) of a charged particle to grow with the increase of P e (fig. 12b). It is evident from fig. 12c that the shielding effect is strongly influenced by the convective transport of ions. The dependence of the induced field and the effective charge density of the particle on the migration speed is illustrated in fig. 13a,b by presenting the scaled net charge density at two different particle velocities with corresponding Reynolds numbers Re = 0.0005, 0.005. Results show that a marginal variation in particle velocity creates a large change in the ion distribution in and around the particle. At low U0 (= 0.005 m/s), the distribution of net charge density shows a ring-like structure. This suggests that the diffusion mechanism is the dominating effect in ion transport. The rate of neutralization of a particle charge is high and hence the particle possesses a less positive charge for a lower Peclet number. As the migration speed is increased, the counterions are pushed downstream to form a long tail of negative charge density downstream of the particle, which results in a stronger induced field as well as a lower neutralization factor. The distribution of charge density justifies our previous results, i.e. increase in FE with the increase of P e (fig. 12a).
Pe
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Fig. 12. Effect on (a) the scaled electric force FE ; (b) scaled drag force FD and (c) neutralization factor Γ for different values of the charge density due to the variation of the migration speed of the particle when Da = 0.1 and κa = 10.
4 Summary The hindrance effect on the migration of a charged particle through an aqueous medium is analyzed in the Stokes flow regime. This retardation depends on the mutual interactions of the double-layer polarization, shielding effect, electro-osmosis of unbalanced ions and fluid flow. Based on the Stokes-Nernst-Planck model, we have determined the impact of the electrokinetic parameters and the intrinsic
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We found that an increase in the fixed charge density lowers the settling velocity or increases the drag factor and reduces the effective permeability. A higher fixed charge density also produces an increment in the charge neutralization factor. However, the dependence of the drag factor and effective charge density on the fixed charge density is not uniform. The effective charge density of the particle drastically reduces when higher values of the particle fixed charge density are considered. One of the authors (S.B.) wishes to thank the Department of Science & Technology, Govt. India, for providing the financial assistance through a project grant.
References
Fig. 13. Distribution of the net charge density ((n+ − n− )(κa)2 /2 + BQfix ) when Da = 0.1, κa = 10 and Qfix = 50. (a) Re = 0.0005, P e = 0.25. (b) Re = 0.005, P e = 2.5.
parameters on the dynamics of the particle and analyzed the hindrance effects. The drag and electric force experienced by the particle strongly depend on the Debye layer thickness. For a lower range of κa, the drag factor, which is the inverse of the particle settling velocity, increases with the increase of κa till the Debye layer thickness remains close to the particle radius. For a higher range of κa, the electric force and drag factor reduces with the increase of κa, however, this reduction occurs at a comparatively lower rate. The fluid penetration inside the particle slows down as the particle charge density grows and/or the Debye layer becomes thinner (increase of κa). The reduced rate of fluid penetration inside the particle enhances the diffusion-dominated counterion condensation effect, which leads to a reduction in electric force. We have also seen that the decrease in permeability of the particle produces a reduction in electric force. These imply that the fluid convection has a strong effect on both DLP and shielding effect even when the flow is in Stokes regime. Our results show that the stronger shielding effect due to a higher fixed charge density creates a stronger induced field. This mutual interactions of DLP, shielding effect and fluid convection can be correctly accounted for by the present Nernst-Planck model.
1. S. Tseng et al., Phys. Chem. Chem. Phys. 15, 11758 (2013). 2. X. Xuan et al., Anal. Chem. 77, 4323 (2005). 3. J.J Zheng, E.S. Yeung, Anal. Chem. 74, 4536 (2002). 4. J.E. Butler et al., Phys. Fluids 19, 113101 (2007). 5. M. Rasa, A.P Phillipse, Nature 429, 857 (2004). 6. Z. Yang et al., J. Colloid Interface Sci. 344, 214 (2010). 7. X.P. Zhang et al., Colloids Surf. A: Physicochem. Engin. Aspects 402, 168 (2012). 8. H. Ohshima et al., J. Chem. Soc., Faraday Trans. 80, 1299 (1984). 9. F. Booth, J. Chem. Phys. 22, 1956 (1954). 10. D. Stigter, J. Phys. Chem. 84, 2758 (1980). 11. D. Saville, Adv. Colloid Interface Sci. 16, 267 (1982). 12. K. Keller et al., J. Colloid Interface Sci. 344, 228 (2010). 13. T. Saito et al., Colloids Surf. A: Physicochem. Engin. Aspects 265, 104 (2005). 14. F. Xiao et al., J. Environ. Sci. 24, 1157 (2012). 15. T. Harif, A. Adin, Water Res. 41, 2951 (2007). 16. J.G. Overbeek, Pure Appl. Chem. 46, 91 (1976). 17. G.S. Manning, J. Phys. Chem. B 111, 8554 (2007). 18. A. Chepelianskii et al., J. Phys. Chem. B 113, 3743 (2009). 19. H.J. Keh, W.C. Chen, J. Colloid Interface Sci. 296, 710 (2006). 20. Y.S Chiu, H.J. Keh, Colloids Surf. A: Physicochem. Engin. Aspects 440, 185 (2014). 21. Y.Y Wu, H.J. Keh, J. Phys. Chem. B 116, 3578 (2012). 22. L.H. Yeh et al., Nanoscale 4, 7575 (2012). 23. J.P Hsu, E. Lee, Phys. Chem. Chem. Phys. 14, 15729 (2012). 24. H. Ohshima, Curr. Opin. Colloid Interface Sci. 18, 73 (2013). 25. Y.Y He, E. Lee, Chem. Engin. Sci. 63, 5719 (2008). 26. J.P. Hsu, Y.H. Tai, Langmuir 26, 16857 (2010). 27. C.H. Huang et al., Phys. Chem. Chem. Phys. 14, 657 (2012). 28. L.H. Yeh, J.P. Hsu, Soft Matter 7, 396 (2011). 29. S. Uppapalli, H. Zhao, Langmuir 28, 11164 (2012). 30. C.H. Huang et al., J. Phys. Chem. B 114, 10114 (2010). 31. L.H. Yeh et al., J. Phys. Chem. C 116, 367 (2012). 32. J.P. Hsu et al., J. Phys. Chem. B 116, 12626 (2012). 33. H.P. Hsu, E. Lee, J. Colloid Interface Sci. 390, 85 (2013). 34. S. Bhattacharyya, P.P. Gopmandal, Soft Matter 9, 1871 (2013).
Page 12 of 12 35. A.D Nield, A. Bejan, Convection in Porous Media (Springer Verlag, New York, 2005). 36. S. Bhattacharyya et al., Chem. Engin. Sci. 61, 4451 (2006). 37. G. Neale, N. Epstein, Chem. Engin. Sci. 28, 1865 (1977). 38. P.M. Adler, J. Colloid Interface Sci. 81, 531 (1981). 39. B.P. Leonard, Comput. Methods Appl. Mech. Engin. 19, 59 (1979). 40. S.V Patankar, Numerical Heat Transfer and Fluid Flow (Hemisphere Publishers, New York, 1980).
Eur. Phys. J. E (2014) 37: 104 41. P. Vainshtein, M. Shapiro, J. Colloid Interface Sci. 284, 501 (2005). 42. J.J Hermans, H. Fujita, K. Nederl, Akad. Wet. Proc. Ser. B 58, 182 (1955). 43. J.J Zheng, E.S. Yeung, Aust. J. Chem. 56, 149 (2003). 44. R. Kekre et al., Phys. Rev. E 82, 050803 (2010). 45. K.H. Kang, D. Li, J. Colloid Interface Sci. 286, 792 (2005). 46. J.P Hsu, E. Lee, Electrochem. Commun. 15, 59 (2013).
THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Effect of induced electric field on migration of a charged porous particle Partha P. Gopmandal1 , S. Bhattacharyya2,a , and Bhanuman Barman2 1 2
School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164-2920, USA Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, West Bengal, India Received 27 March 2014 and Received in final form 2 September 2014 c EDP Sciences / Societ` Published online: 6 November 2014 – a Italiana di Fisica / Springer-Verlag 2014 Abstract. The effect of ambient fluid flow on a charged porous spherical particle suspended in an aqueous medium is analyzed. The porous particle is ion permeable and fluid penetrable. The induced electric field due to the polarization of the particle’s electric double layer and counterion condensation leads to a hindrance effect on particle migration by producing an electric force. The influence of this retardation force on the hydrodynamics of the particle is studied through the Nernst-Planck equations, which are coupled with the Stokes-Brinkman equation. The interactions of the double-layer polarization, shielding effect, electroosmosis of unbalanced ions and fluid convection are analyzed. The settling velocity and fluid collection efficiency of the charged aggregate is determined. We have studied the electrokinetics for a wide range of fixed charge density and permeability of the particle with no assumption made on the thickness of the double layer relative to the dimension of the particle.
1 Introduction The separation of bioparticles is often accelerated by either considering a non-uniform cross-section of the microtube [1,2] or imposing an external pressure field in capillary electrophoresis [3]. A controlled migration of polymer by an imposed pressure-driven flow and an external force is investigated by Butler et al. [4]. Centrifugation is a powerful technique for separating and analyzing cells, organelles, charged proteins and colloids. Rasa and Philipse [5] experiment on ultracentrifugation of charged colloidal silica spheres demonstrates the existence of a macroscopic electric field and its drastic effects on sedimentation profiles. Recently, numerical studies on advective flow of homogeneous as well as non-homogenous permeable spheres under the action of an electric field for finite values of Reynolds number were made by Yang et al. [6] and Zhang et al. [7]. When a charged particle is immersed in an aqueous media, a layer of diffuse charge clouds (Electric Double Layer, EDL) of surplus counterions forms around the particle. A relative motion between the particle and the surrounding medium creates a deformation of the double layer resulting in the formation of an induced electric field. This phenomenon is referred to as the double-layer polarization (DLP) [8]. This induced electric field creates an electrostatic force on the unbalanced charges and deters the convection of the particle. Theoretical studies and a
e-mail: [email protected]
mathematical models on the migration of charged colloids or porous particles in an aqueous media under the action of gravitational force, centrifugation or pressure-driven flow has drawn the interest of several authors. The steady sedimentation of a single charged rigid spherical colloid under the action of gravity was studied by Booth [9], Stigter [10] and Saville [11]. Ohshima et al. [8] derived the general expression for the sedimentation velocity as a function of particle ζ-potential and Debye layer thickness through a linear perturbation analysis. The retardation effect due to the deformation of double layer through a linear perturbation analysis is found to be substantial when the Debye layer thickness is comparable with the radius of the particle. Keller et al. [12] investigated the settling process of a colloidal particle under the influence of a gravitational or centrifugal field through a model based on the Stokes-Poisson-Nernst-Planck equations. There they have documented most of the important studies on settling of a rigid charged particle. In several biological contexts one can encounter not only the dispersion of discrete charged particles but also of the aggregate of charged colloids or monomers [13,14]. The study of sedimentation of aggregates formed due to the electroflocculation or centrifugation of a colloidal suspension is important in the context of separation and purification [15]. It may be noted that the charged porous sphere i.e., permeable to fluids and ions, has proven to be a successful model to describe the electrokinetic behavior of a polyelectrolyte [16]. The penetration of counterions within a permeable aggregate and the counterion
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condensation [17] phenomena creates a characteristic difference between its electrokinetics compared to a rigid particle. The condensed counterions lower the charge density of the polyelectrolyte [18] and thus have an impact on the dynamics of the particle. Based on the linearized electrokinetic equations, the steady-state sedimentation or migration of polyelectrolyte molecules or charged flocs, modeled as a porous sphere, in an electrolyte solution is analyzed by Keh and Chen [19]. Subsequently, Chiu and Keh [20] analyzed the steady sedimentation of a charged soft sphere in the extent of an electrolyte solution. The steady electrokinetic flow of an electrolyte solution due to the electric field and/ or pressure gradient in the fibrous medium, constructed as an array of charged circular cylinders, is studied through a small perturbation analysis by Wu and Keh [21]. The electrokinetics of a charged porous particle is governed by the mutual interactions of double-layer polarization, counterion condensation, electro-osmosis of unbalanced ions and fluid flow [22]. The convection-induced double-layer polarization on the motion of a charged sphere moving near the air-water interface is studied by Hsu and Lee [23] under the assumption of a slight deformation of the double layer. In all those studies, a slight deformation of the double layer is assumed and the distribution of ion is obtained through a small perturbation from its equilibrium Boltzmann distribution. The small perturbation analysis may become questionable for a high charge density of the particle where the strong shielding effect influences the DLP and electro-osmosis of unbalanced ions. The mathematical model to describe the sedimentation of a charged porous particle is similar to those of its electrophoresis [24]. The effect of double-layer polarization (DLP) on the electrophoresis of a soft particle or an entirely porous particle was made by several authors in recent years, namely, e.g., He and Lee [25], Hsu and Tai [26], Huang et al. [27], Yeh and Hsu [28], Uppapalli and Zhao [29], Huang et al. [30], Yeh et al. [31], Hsu et al. [32], Hsu and Lee [33] and Bhattacharyya and Gopmandal [34]. The above studies on permeable particles suggest that the DLP effect diminishes for either a very low or a high value of κa, where 1/κ is the characteristic thickness of the double layer and a is the radius of the particle. All the previous studies dealing with DLP effects on sedimentation of charged particles are based on a small perturbation from the equilibrium Boltzmann distribution. For finite Debye length, the non-zero charge density outside the double layer can lead to a concentration gradient between the double layer and the bulk fluid. This concentration gradient becomes stronger when the charge density of the particle becomes higher. The presence of a non-zero charge density outside the Debye layer makes the ion transport equations and the equations for fluid flow coupled. The convective transport of ions can be well accounted for by the Nernst-Planck model. In this paper, we have analyzed the dynamics of a charged porous particle sedimenting in an aqueous medium under an external force such as gravitational,
Eur. Phys. J. E (2014) 37: 104
centrifugal field and/or pressure-driven flow. The mutual interactions of the double-layer polarization, counterion condensation and fluid inertia on the dynamics of the charged particle are studied. We have also compared the hydrodynamics of the charged particle with the corresponding uncharged case and analyzed the role of the induced field on the electrokinetics of the particle. Based on the nonlinear PNP model (Poisson-Nernst-Planck), we have measured the effects of electric force on the hydrodynamics of the particle. The impact of the governing parameters such as Debye layer thickness, permeability, fixed charge density of the particle, fluid convection on the DLP and counterion condensation is investigated through the nonlinear model. A computational method based on the finite-volume algorithm is adopted in the present analysis. We found that the fluid convection has a strong effect on the ion distribution. The modeling of configurations involving a fluidporous interface can be made through two different formulations, the Navier-Stokes equations for the clear fluid domain and the Brinkman equation for the porous domain. They are usually solved simultaneously with an interface condition between the two media. The Brinkman equation has the advantage over the well-known Darcy’s law in that it contains the stress tensor, therefore the boundary condition is the continuity assumption of momentum and viscous stress along the interface boundary. The implementation of the interface condition [35] is complicated as it contains material-dependent parameters whose value must be determined empirically. Owing to this difficulty, we have adopted a single-domain approach [36] in which the matching of variable values at the interface is inherent in the formulation itself. In this formulation, two sets of equations for the fluid and the porous regions are combined into one set by using a binary parameter and the Dupuit-Forchheimer relationship [35] is used to relate the fluid velocity between the two zones.
2 Mathematical model and numerical methods We considered a charged porous spherical particle of radius a suspended in an aqueous medium. The surrounding fluid flows at a speed U0 from infinity towards the sphere in a frame of reference which is fixed at the center of the sphere (fig. 1). A spherical polar coordinate system (r, θ, ψ) is adopted and the direction of the far-field velocity is along the negative z-axis. The fluid flow within and around the sphere is governed by the equations as described below. The permeable sphere is assumed to be composed of charged monomers and has a uniform porosity and permeability kp . As the present configuration involves a fluid-porous interface, the Brinkman model is used to account for the jump in shear stress. The fluid velocity q is related to the intrinsic (average over the fluid volume only) velocity V by q = V. We adopt a unified single-domain approach. In this approach, the two sets of equations for the fluid and the porous regions are
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force. The dimensionless parameter B1 = en0 φ0 a/μU0 , where n0 is the bulk concentration of ions. The binary parameter B assumes the value 1 within the particle and 0 outside of it. The non-dimensional parameter governing the fluid flow around and through the charged permeable particle is Darcy number Da = κp /a2 , where μ is the fluid viscosity. In the fluid region, the Darcy number Da is assumed to be infinity and the porosity is 1 outside the porous zone and 0 < < 1 within the porous zone. Based on Gauss law, the electric potential φ is determined by the following Poisson equation: ∇2 φ = −
Fig. 1. Sketch of the flow configuration for a charged porous microparticle. A spherical polar coordinate (r, θ, ψ) is used with origin at the center of the sphere.
combined into one set using appropriate switching terms. The single-domain formulation requires no conditions to impose at the fluid-porous interface. The relative permittivity of the fluid inside and outside the porous aggregate is considered to be the same. The equations governing this electrokinetic phenomenon are the Darcy-Brinkman extended Stokes equations for fluid flow, the Nernst-Planck equations for ion transport and Poisson equation for electric field. We assume an axi-symmetric flow with z-axis as the axis of symmetry. The bulk ionic concentration n0 is the scale for ionic concentration. The radius of the sphere a is assumed to be the length scale, the velocity field is scaled by U0 and φ0 (= RT /F ), the thermal potential is the potential scale. The far-field fluid velocity U0 is considered to be of the order of the corresponding electrophoretic velocity of a porous sphere under an electric field φ0 /a. The pressure is non-dimensionalized by μU0 /a, where μ is the fluid viscosity. The structure of the porous particle is assumed to be homogeneous with a constant fixed charge density ρfix . The non-dimensional form of the equations describing the motion of the ionized fluid in and around the porous aggregate is ∇ · q = 0, q − B1 ρ¯e ∇φ = 0. −∇p + ∇2 q − B Da
(1) (2)
For an axi-symmetric flow q = (v, u) is the velocity vector where v is the radial and u is the cross-radial velocity component. The last two terms in the left side of eq. (2) represent respectively, the body force due to the frictional force within the porous medium and the electric driving
(κa)2 ρ¯e − BQfix , 2
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the net charge density is denoted by ρ¯e = zi ni , where ni is the number concentration and zi is the valence of the i-th ionic species. The concentration of each ionic species is scaled by the bulk number concentration n0 . Here we have considered binary symmetric ionic species with i = 1, inverse of the EDL thickness 2 and z1 = 1, z2 =−1. The 2 2 is given by κ = i e (zi ) n0 /e KB T . Here e is the permittivity of the medium, e is the elementary electric charge, KB is the Boltzmann constant, R is the gas constant, T is the absolute temperature and F is the Faraday constant. We assume that the permittivity e is the same for both the regions i.e., within the porous zone and in the clear liquid zone. The non-dimensional parameter Qfix = ρfix a2 /e φ0 determines the fixed charge density of the porous particle. The non-dimensional form of the Nernst-Planck equation governing the transport of the i-th ionic species is given by Pe (q · ∇)ni − ∇2 ni − zi ∇ · (ni ∇φ) = 0.
(4)
In this case, we have considered that the mass diffusion coefficients are approximately equal, D = D1 ∼ = D2 . It may be noted that the present model based on NernstPlanck equations can handle multivalent ions. The nondimensional parameter Peclet Number P e = U0 a/D and Schmidt number can be defined as Sc = ν/D. It may be noted that P e = Re · Sc, where Re = U0 a/ν is the Reynolds number. Far away from the particle, the electric and concentration fields are uninfluenced by the presence of the particle thus, ∂φ/∂r = 0 and ∂ni /∂r = 0 at r = R a. A symmetry condition is imposed along the axis of symmetry θ = 0 and π. It may be noted that in the present formulation no boundary conditions is required at the fluid-porous interface (r = 1). The electrostatic force (FE ) and the hydrodynamic force (FD ) along the flow direction for this axi-symmetric problem can be obtained by integrating the Maxwell stress tensor and hydrodynamic stress tensor on the surface S of the particle, respectively, and are evaluated by the expression as given by Bhattacharyya and Gopmandal [34]. The forces FE and FD are scaled by e φ20 and μU0 a, respectively. The governing equations for the ion transport and fluid, which are cast into conservation law form are in-
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Eur. Phys. J. E (2014) 37: 104 1 Ω
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Fig. 2. Comparison of drag factor Ω and fluid collection coefficient η for the uncharged case at different values of β. The symbols represent the results due to Neale and Epstein [37] for Ω and Adler [38] for η.
tegrated over each control volume in a staggered grid arrangement. Different control volumes are used to integrate different equations. At each interface of the control volume, the Quadratic Upstream Interpolation for Convective Kinematics scheme (QUICK [39]) is used to discretize the convective terms of the unsteady equations, while a linear interpolation between two neighboring cell values is used to discretize the diffusion terms. A pressure correction based iterative method SIMPLE [40] is used to compute the resulting algebraic equations. The procedure is based on a cyclic series of guess and correct operations to solve the governing equations. The pressure link between the continuity and momentum equations are accomplished by transforming the discretized continuity equation into a Poisson equation for pressure correction. This Poisson equation implements a pressure correction for a divergent velocity field. The iteration procedure starts with an assumption for potential at each cell. At every iteration, the electric field is determined by solving the Poisson equation for electric potential i.e., eq. (3). Equation (3) is solved by a line-by-line iterative method along with the successiveover-relaxation (SOR) technique. The stability and accuracy of the present algorithm is tested through a grid independency test and by comparing with the previously published results (fig. 2).
3 Results and discussions Through the dimensionless parameter√β = λs a, the Darcy = number can be expressed as β = 1/ Da, where λ−1 s √ κp is the softness parameter. The softness parameter can range from 10 nm to 0.4 mm [28]. For nano-particle aggregates of porosity close to 0.94, the permeability, based on the Happel’s law, can lead to Da ≥ 0.01 [41]. It has been found by several authors (e.g., Huang et al. [27]) that the electrophoresis becomes independent of the softness parameter if β = λs a lies outside of 1 and 10. For biological cells the fixed charge density i.e., ρfix ≤ 106 C/m3 . We
have considered the scaled fixed charge density Qfix ≤ 100 for a particle of radius a = 100 nm. The parameter κa varies from 1 to 30, which implies that the bulk concentration varies between 0.01 mM and 10 mM for a particle with a = 100 nm. Previous studies have established that the DLP effect is most substantial for O(1) κa. The scaling of the variables suggests that for a fixed set of values of the non-dimensional parameters i.e., κa and Qfix , if we choose smaller (or higher) aggregate size then the DebyeHuckel parameter (κ) will increase (decrease) at the same rate and ρfix should increase (decrease) at the square of the rate by which a is lowered (increased). Here we have taken φ0 = 0.02586 V, e = 695.39 × 10−12 C/Vm, μ = 10−3 Pa s and ρ = 103 kg/m3 . The diffusivity of ions are considered to be equal and is D = 2.0 × 10−9 m2 /s, thus Sc = 500. Based on the mobility expression due to Hermans and Fujita [42] for a charged porous sphere with Qfix = 100 and Da = 1, the electrophoretic velocity under an imposed field φ0 /a is 0.0464 m/s, which corresponds to Reynolds number Re = 0.00464 and Peclet number P e = 2.32 for a particle of radius 100 nm. We have assumed a creeping flow condition with Re < 1. However, the fluid convection effect on ion distribution is included in the present analysis. In order to test the accuracy of our method, we have compared our computed results for drag factor (Ω) and the fluid collection efficiency (η) for an uncharged permeable sphere (Qfix = 0) at various values of the permeability (β) with the formula obtained by Neale and Epstein [37] and Adler [38] i.e., eq. (5) and eq. (7), respectively. Figure 2 shows that our computed solutions are in excellent agreement with those analytical results. We have also made a grid independency study in fig. 2. The drag factor (Ω) is defined as the ratio of drag between a permeable particle and the corresponding Stokes drag of a solid particle. Within the creeping flow limit the drag factor is inversely proportional to the settling velocity of a permeable particle. For an uncharged particle Neale and Epstein [37] obtained an analytic form of the drag factor under the creeping flow condition as Ω=
2β 2 [1 − 2β 2 + 3[1
tanh β β ] tanh β − β ]
.
(5)
For a permeable sphere with intermediate values of permeability, 0 < Ω < 1. Note that as β −→ ∞ (i.e., Da −→ 0), Ω −→ 1. The fluid collection efficiency (η) of a permeable sphere is defined as the ratio of the interior flow through the aggregate to the flow approaching it. Thus π/2 2 0 v sin θdθ . (6) η= U0 The analytical formula of fluid collection efficiency (η) due to Adler [38] for homogenous porous aggregate in linear Stokes flow is given by −1 −1 tanh β 9 3 + 1− . (7) η= 2β 2 2β 2 β
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110 100 Qfix=10 Qfix=30 Qfix=50
90 80
FE
70 60 50 40 30 20 10 0
5
10
15
κa
20
25
30
(a) 0.9
0.85
Qfix=10 Qfix=30 Qfix=50
Ωf
0.8
0.75
0.7
0.65
0.6
Qfix=0 5
10
15
κa
20
25
30
(b) Fig. 3. Variation of (a) electric force FE and (b) drag factor Ωf (ratio of drag experienced by the charged polyelectrolyte to the Stokes drag of a rigid particle) with Debye layer thickness (κa) for different values of the charge density when Da = 0.1. Here Qfix = 0 refers to the case of an uncharged permeable sphere. Symbols indicate the results based on the Navier-Stokes-Brinkman equation for Re = 0.005.
ration in capillary electrophoresis. The symbols in fig. 3a indicate that the solution is based on the nonlinear NavierStokes-Brinkman equations. Results show that the convection terms in momentum equations have practically no effect. However, the Peclet number, which measures the relative importance of convection to diffusion of ions, is O(1). Because of this, the transport of ions due to convection is not neglected in the present computations. We find that the DLP effect is significant when 1 ≤ κa ≤ 10, which is in accord with several previous studies e.g. Yeh and Hsu [28] and Bhattacharyya and Gopmandal [34]. It may be noted that the variation in the Debye-Huckel parameter (κa) at a fixed value of a implies a variation in the ionic concentration of the electrolyte. We find that FE increases rapidly with the increase of ionic concentration up to a certain critical value of κa. Beyond this critical value, the electric force starts decreasing slowly with the rise of κa. This critical value of κa depends on the fixed charge density (Qfix ). The penetration of counterions and neutralization of particle charge density due to the shielding effect is a diffusion-dominated process, which is insignificant for low ionic concentration (i.e., κa is small). As the ionic concentration of the electrolyte is increased (κa is increased), the diffusion-dominated mechanism becomes stronger. The occurrence of a local maxima in FE is due to the interaction of the double-layer polarization (DLP) effect and counterion condensation. The analytical solution for sedimentation of a charged rigid particle by Oshima et al. [8] and a soft particle with charged rigid core as considered by Chiu and Keh [20] shows that the retardation effect due to the induced electric field vanishes as κa approaches either zero or infinity. It is evident from eq. (3) that the charge density ρe is zero everywhere when κa → ∞, thus the effect of the induced field is absent. When κa → 0, the shielding effect due to the fixed charge density of the particle becomes zero as the counterion concentration in the liquid becomes negligible. It is expected that the lower or upper limits of κa for which the induced electric field effect is negligible depends on the fixed charge density of the particle.
3.1 Effect of κa and Qfix Figure 3a shows the variation of the electric force experienced by the permeable particle (FE ) as a function of Debye layer thickness for different values of the particle charge density Qfix . We present results for Reynolds number Re = 0.005, the corresponding Peclet number is 2.5. It may be noted that the Reynolds number based on the electrophoretic velocity under an applied electric field φ0 /a with Qfix = 100 and a = 100 nm may lead to Re = 0.00464. Several authors (e.g., Xiao et al. [14]) found that the settling velocity of a diatom aggregate in water ranged from 1.88 mm/s to 5.84 mm/s. This settling velocity for a = 100 nm corresponds to the range of Reynolds number between 1.88 × 10−4 and 5.84 × 10−4 . Thus, the range of Reynolds number based on the sedimentation velocity of the aggregate as considered here can be obtained through centrifugation with moderate values of g-factor. Several authors, e.g. Zheng and Yeung [3,43] and Keker et al. [44] considered O(1) Reynolds number for DNA sepa-
We have determined the drag factor (Ωf ), which is the ratio of the drag of the charged particle (FD ) to the Stokes drag due to a rigid uncharged sphere i.e., 6πμU0 a, as a function of the concentration of the ambient electrolyte. The drag factor is inversely proportional to the settling velocity of the aggregate in Stokes flow limit. We find from fig. 3b that the drag factor follows the same pattern as that of the electric force (fig. 3a) i.e., it increases rapidly with the increase of ionic concentration up to a certain critical value of κa. Beyond this critical value the drag factor starts decreasing slowly with the rise of κa. In any case, the drag factor is bigger than the drag factor due to a porous uncharged particle (i.e., Qfix = 0) for the range of κa considered. Our result shows that the drag factor for an uncharged particle (Qfix = 0) with β = 3.16 (i.e., Da = 0.1) is 0.621, which is the same as can be obtained by (5) and the value presented in fig. 2. We find that the drag factor is increased compared to the uncharged particle by about 40% when Qfix = 50. Thus, the retardation
Page 6 of 12
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Qfix
0.6
Γ
produced by the presence of a double layer is evident. This retardation effect can be considered to be due to the development of the electrohydrodynamic (EHD) pressure field [45] created by the Coulumbic electrostatic force arising because of unbalanced charges in and around the particle. We find that the hydrodynamic drag increases with the increase of charge density Qfix . This is due to the fact that, as Qfix is increased, a higher number of counterions accumulate near the particle due to the stronger shielding effect which raises the EHD pressure field. The effective charge density of the permeable particle can be determined by 1 π (κa)2 ρ¯e r2 sin θ drdθ 2π 2 r=0 θ=0 + Qfix . (8) Qeff = 4π
Pe=2.5 Pe=0
0.4 0.2 0
1
2
3
4
5
6
7
8
9 10
6
7
8
9
κa
(a)
3 0.3
We define the charge neutralization factor, Γ as Γ = 1 − Qeff /Qfix .
Qfix=0
0.28
(9)
0.26 0.24 0.22
η
The penetration of counterions within the permeable particle is large if the charge neutralization factor Γ is close to one. In fig. 4a we present the variation of Γ as a function of κa for different values of the particle charge density (Qfix ). It is evident from these results that a monotonic increase in Γ occurs with the rise of κa. The decrease in effective charge density of the particle due to the counterion condensation at higher values of κa creates a monotonic reduction in electrostatic force for those values of κa (fig. 3a). Screening of the fixed charge density is more efficient as the particle fixed charge is increased. The shielding effect grows stronger with the increase of Qfix , which leads to a higher charge neutralization factor at larger values of the particle fixed charge density. In order to illustrate the convection effect on ion distribution, we have included the results corresponding to the case where convection effect is neglected i.e., P e = 0. Our results show that the effective charge density of the particle is greatly influenced by the convective transport of ions. The effect of convection is strong when the EDL is thicker i.e., for lower values of κa. The ion convection lowers the rate of the charge neutralization of the aggregate. These results suggest that the hydrodynamic interaction on counterion condensation is not negligible. The fluid penetration inside the aggregate can be measured through the fluid collection efficiency (7). Figure 4b describes the collection efficiency as a function of κa at different values of Qfix . Results show that the effective permeability of the particle becomes low as the EDL thickness decreases. The fluid collection efficiency also decreases with the increase of Qfix . The fluid collection efficiency is lower than the corresponding value due to an uncharged particle (Qfix = 0). The maximum percentage difference is found to be about 60%. The momentum loss due to the EHD pressure field creates a lower rate of fluid flow through the permeable particle. By comparing fig. 4a and b we find that the parameter values for which Γ increases create a reduction in η and vice versa. A higher rate of fluid flow through the particle drags counterions towards
0.2
0.18 Qfix=10 Qfix=30 Qfix=50
0.16 0.14 0.12 0.1
1
2
3
4
5
(b)
κa
10
Fig. 4. Effect of Debye layer thickness (κa) on the neutralization factor Γ and fluid collection coefficient η at different values of the charge density when Da = 0.1 (β = 3.16) and P e = 2.5. (a) Γ as a function of κa for Qfix = 10, 30, 50 where the arrow points to the increasing direction of Qfix . Result without the ionic convection (P e = 0) is also included. Solid lines: P e = 2.5; dashed lines: P e = 0. (b) η as a function of κa for Qfix = 10, 30, 50, Da = 0.1 and P e = 2.5. The line with Qfix = 0 represents the uncharged case.
the downstream of the particle which in turn lowers the charge neutralization of the particle. In this way, the fluid convection affects the counterion condensation. Figure 5a shows the induced potential along the surface of the particle for κa = 1, 5, 10 when Qfix = 50. The distribution of φs suggests that the induced electric field is along the negative z-axis, i.e. it acts opposite to the direction of the migration of the particle. This electric potential drop is higher for higher values of the DebyeHuckel parameter. Figure 5b shows the form of φs for different values of Qfix . The increase in the fixed charge density produces an enhancement in the induced electric field due to the increased shielding effect. The effect of fixed charge density (Qfix ) on the electric force, drag factor and particle charge neutralization factor is illustrated in fig. 6a-c. It is clear from these results that an increase in Qfix increases the drag factor and hence
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14
150
12
125 100 κa=1 κa=5 κa=10
8
FE
φs
10
6
75 50
4
25
2 0 0
0 10 20 30 40 50 60 70 80 90 100
30
60
90
θ
120
150
Qfix
180
(a)
(a) 0.85
5
4
Ωf
0.8
φs
3
0.75
Qfix=10 Qfix=30 Qfix=50
2
1
0.7 10
0 0
30
60
90
θ
120
150
180
30
40
50
60
Qfix
70
80
90 100
(b)
(b) Fig. 5. Variation of the induced potential φs on the surface of the polyelectrolyte for (a) Qfix = 50 at different values of κa = 1, 5, 10. (b) κa = 10 at different values of Qfix = 10, 30, 50. Here P e = 2.5 and Da = 0.1.
20
1
0.9
Γ
0.8
reduces its settling velocity. However, the dependence of FE (or FD ) on Qfix has changed its pattern when higher values of Qfix is considered. This type of nonlinear pattern has been reported in the context of electrophoresis by Yeh and Hsu [28] and Hsu and Lee [46], which was explained through the counterion condensation. We find from fig. 6c that as Qfix becomes large, the effective charge density of the particle drastically drops to a low value and approaches a constant. The distribution of the net charge density in and around the particle is shown in fig. 7a-c for different values of the EDL thickness i.e., κa = 1, 5, 10 when the Peclet number is P e = 2.5. The distribution of the net charge density shows that the induced field acts opposite to the direction of the migration of the particle. The deformation of the double layer is prominent for κa > 1. The net charge density within the particle becomes less positive and a higher number of counterions accumulate downstream of the particle as the EDL becomes thinner. The distribution of the net charge density compliments our previous findings, i.e. the value of the neutralization
0.7
0.6
0.5 10 20 30 40 50 60 70 80 90 100
Qfix
(c) Fig. 6. Variation of (a) scaled electric force (FE ); (b)drag factor Ωf ; and (c) neutralization factor Γ as a function of particle fixed charge density when P e = 2.5, Da = 0.1 and κa = 10.
factor becomes higher (fig. 4a) with the increase of κa. We found in fig. 3a that FE increases at a faster rate with the increase of κa for the lower range of κa. In this range of κa, the DLP effect grows but the neutralization of the particle charge density is not significant. For a higher range of κa, the effective charge density of the particle reduces with the increase of κa. For this, we find in fig. 3a that FE reduces with the increase of κa when κa is large.
Page 8 of 12
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Ωf
0.7 0.6 0.5
Qfix=10 Qfix=30 Qfix=50 Qfix=0
0.4 0.3 0.2
1
2
3
4
5
β
6
7
8
9
10
(a) 140 120
Qfix=10 Qfix=30 Qfix=50
100
FE
80 60 40 20 0 1
2
3
4
5
β
6
7
8
9
10
(b) Fig. 8. Variation of the (a) drag factor Ωf and (b) scaled electric force FE as a function of β for different values of charge density at κa = 10 and P e = 2.5.
Fig. 7. Distribution of the net charge density ((n+ − n− )(κa)2 /2 + BQfix ) when Da = 0.1 (β = 3.16), P e = 2.5 and Qfix = 50. (a) κa = 1. (b) κa = 5. (c) κa = 10.
3.2 Effect of permeability The effect of permeability on the drag factor Ωf and electric force (FE ) is illustrated in fig. 8a, b for thin EDL case i.e., κa = 10 when P e = 2.5 for different values of Qfix = 10, 30, 50 along with the uncharged case, i.e. Qfix = 0. Results show that a change in permeability (β) produces a strong effect on Ωf and FE when permeability is in a higher range (β < 5). In that range of the permeability (i.e., β < 5), the electric force decreases at a faster rate with the increase of β. The drag factor for a charged
particle is always higher than its value for an uncharged particle. When the particle is highly permeable, fluid can flow easily through the particle (fig. 8a). The relatively stronger flow through the particle at higher permeability pushes the counterions towards downstream to create a stronger induced field. Simultaneously, the stronger fluid flow through the particle also drags counterions downstream and hence lowers the particle charge neutralization. The combination of these two effects leads to a large FE at low β (high permeability). Increase in β (lowering the permeability) enhances the diffusion-dominated shielding effect and hence reduces the effective charge density of the particle which results in a lower electric force. We introduce a parameter which measures the ratio between the permeability and the Debye length i.e., χ = κa/β and presents the drag factor as a function of χ. In fig. 9 we vary χ by increasing κa at fixed β. The drag factor is found to increase with the increase of β(i.e., decrease of permeability). We find that the fluid transport within the particle is not similar to the combined parameter χ. At a fixed value of χ, the drag factor may vary due to variation of either β or κa. The drag factor shows the occurrence of local maxima at any fixed value of β. The
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1 0.9 0.8
Ωf
0.7 0.6
β=1 β = 3.16 β=5
0.5 0.4 0.3 0.2 1
2
3
4
5
χ
6
7
8
9
10
Fig. 9. Variation of drag factor as a function of χ = κa/β for fixed values of the charge density Qfix = 50. We have varied χ by varying κa for fixed values of β.
value of χ at which Ωf attains a local maximum changes with the change of the particle permeability. The maximum in drag factor implies the highest hindrance on the particle transport. The effect of permeability on the net charge density in and around the particle is described in fig. 10a-c when κa = 10. We find that the DLP effect is low for a low permeable particle. Besides, the net charge density within the particle is less when permeability is low. These together explain the reduction of FE with the increase of β, i.e. lowering the permeability, as found in fig. 8a. A long plume of counterions downstream of the particle is observed for a high permeable case. The presence of the unbalanced ions creates an EHD pressure field, which results in an enhanced drag compared to an uncharged particle as seen in fig. 8a. It is evident from fig. 10a-c that the convective transport of ions is not negligible even at this small Peclet number, i.e. P e = 2.5. Figure 11a, b shows that the fluid collection efficiency decreases and the neutralization factor increases as the particle permeability decreases. As the particle becomes more permeable, the charge neutralization factor (Γ ) becomes low. At a fixed value of β, the increase in Qfix reduces the rate of fluid flowing through the particle and increases the neutralization factor. The stronger shielding effect at higher Qfix lowers the effective permeability and charge density of the particle.
Fig. 10. Distribution of the net charge density ((n+ − n− )(κa)2 /2 + BQfix ) when κa = 10, P e = 2.5 and Qf ix = 50. (a) Da = 1. (b) Da = 0.1. (c) Da = 0.01.
3.3 Effect of U0 The effect of the migration speed of the particle (U0 ) on the electrokinetics is illustrated in figs. 12a-c when the Debye-Huckel parameter κa is 10. We have varied the Reynolds number between Re = 0.0005 and Re = 0.005 by varying U0 . The corresponding Peclet number varies in the range 0.25 ≤ P e ≤ 2.5. We have presented results in terms of Peclet number as the Reynolds number does not appear explicitly in the present set of equations. The Peclet number measures the ratio of advection to diffusion transport
of ions. It has already been pointed out (fig. 3a) that in this range of Reynolds number, the fluid flow can be well approximated by the Stokes-Brinkman equation. Figure 12a shows that the electric force increases monotonically with the increase of P e. We have shown in fig. 12c that an increment in fluid convection speed lowers the rate of neutralization of the particle charge density, thus increases the effective charge density of the particle. These in combination produce a higher electric force with the rise of P e. The stronger induced electric field also creates a higher EHD
Page 10 of 12
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1
80
0.9
70
0.8
60
0.7
FE
50
0.6
η
Qfix=10 Qfix=30 Qfix=50
Qfix=10 Qfix=30 Qfix=50 Qfix=0
0.5 0.4
40 30 20
0.3
10
0.2
0
0.1 0
1
2
3
4
5
β
6
7
8
9
0.5
1
10
1.5
2
2.5
1.5
2
2.5
1.5
2
2.5
Pe (a)
(a) 1
Qfix=10 Qfix=30 Qfix=50
0.8
0.9 0.8
0.75
Ωf
0.7
0.7
Γ
0.6 Qfix=10 Qfix=30 Qfix=50
0.5 0.4
0.65 Qfix=0
0.3
0.6
0.2
0.5
1
0.1 0
1
2
3
4
5
β
6
7
8
9
(b)
10
(b)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Qfix=10 Qfix=30 Qfix=50
Γ
Fig. 11. Variation of (a) the fluid collection efficiency (η) and (b) Neutralization factor (Γ ) as a function of β for different values of the charge density when κa = 10.
pressure field, which leads to the drag factor (Ωf ) of a charged particle to grow with the increase of P e (fig. 12b). It is evident from fig. 12c that the shielding effect is strongly influenced by the convective transport of ions. The dependence of the induced field and the effective charge density of the particle on the migration speed is illustrated in fig. 13a,b by presenting the scaled net charge density at two different particle velocities with corresponding Reynolds numbers Re = 0.0005, 0.005. Results show that a marginal variation in particle velocity creates a large change in the ion distribution in and around the particle. At low U0 (= 0.005 m/s), the distribution of net charge density shows a ring-like structure. This suggests that the diffusion mechanism is the dominating effect in ion transport. The rate of neutralization of a particle charge is high and hence the particle possesses a less positive charge for a lower Peclet number. As the migration speed is increased, the counterions are pushed downstream to form a long tail of negative charge density downstream of the particle, which results in a stronger induced field as well as a lower neutralization factor. The distribution of charge density justifies our previous results, i.e. increase in FE with the increase of P e (fig. 12a).
Pe
0.5
1
Pe (c)
Fig. 12. Effect on (a) the scaled electric force FE ; (b) scaled drag force FD and (c) neutralization factor Γ for different values of the charge density due to the variation of the migration speed of the particle when Da = 0.1 and κa = 10.
4 Summary The hindrance effect on the migration of a charged particle through an aqueous medium is analyzed in the Stokes flow regime. This retardation depends on the mutual interactions of the double-layer polarization, shielding effect, electro-osmosis of unbalanced ions and fluid flow. Based on the Stokes-Nernst-Planck model, we have determined the impact of the electrokinetic parameters and the intrinsic
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We found that an increase in the fixed charge density lowers the settling velocity or increases the drag factor and reduces the effective permeability. A higher fixed charge density also produces an increment in the charge neutralization factor. However, the dependence of the drag factor and effective charge density on the fixed charge density is not uniform. The effective charge density of the particle drastically reduces when higher values of the particle fixed charge density are considered. One of the authors (S.B.) wishes to thank the Department of Science & Technology, Govt. India, for providing the financial assistance through a project grant.
References
Fig. 13. Distribution of the net charge density ((n+ − n− )(κa)2 /2 + BQfix ) when Da = 0.1, κa = 10 and Qfix = 50. (a) Re = 0.0005, P e = 0.25. (b) Re = 0.005, P e = 2.5.
parameters on the dynamics of the particle and analyzed the hindrance effects. The drag and electric force experienced by the particle strongly depend on the Debye layer thickness. For a lower range of κa, the drag factor, which is the inverse of the particle settling velocity, increases with the increase of κa till the Debye layer thickness remains close to the particle radius. For a higher range of κa, the electric force and drag factor reduces with the increase of κa, however, this reduction occurs at a comparatively lower rate. The fluid penetration inside the particle slows down as the particle charge density grows and/or the Debye layer becomes thinner (increase of κa). The reduced rate of fluid penetration inside the particle enhances the diffusion-dominated counterion condensation effect, which leads to a reduction in electric force. We have also seen that the decrease in permeability of the particle produces a reduction in electric force. These imply that the fluid convection has a strong effect on both DLP and shielding effect even when the flow is in Stokes regime. Our results show that the stronger shielding effect due to a higher fixed charge density creates a stronger induced field. This mutual interactions of DLP, shielding effect and fluid convection can be correctly accounted for by the present Nernst-Planck model.
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