Electrophoretic and Electroosmotic Motion of a Charged Spherical

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Electrophoretic and Electroosmotic Motion of a Charged Spherical Particle within a Cylindrical Pore Filled with Debye-Bueche-Brinkman Polymeric Solution Yu-Fan Lee, Yu-Fen Huang, Shan-Chi Tsai, Hsiao-Yun Lai, and Eric Lee Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b02795 • Publication Date (Web): 21 Nov 2016 Downloaded from http://pubs.acs.org on November 30, 2016

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Electrophoretic and Electroosmotic Motion of a Charged Spherical Particle within a Cylindrical Pore Filled with Debye-Bueche-Brinkman Polymeric Solution Yu-Fan Lee1,†, Yu-Fen Huang1,†, Shan-Chi Tsai†, Hsiao-Yun Lai† and Eric Lee*,†

†Department

of Chemical Engineering

National Taiwan University Taipei 10617, Taiwan Tel: +886-2-23622530 Fax: +886-2-23623040

Keywords:

Capillary

electrophoresis,

Electroosmosis,

Polymer

solution,

Debye-Bueche-Brinkman equation, Double layer polarization 1

*

: These authors contributed equally to this work.

:Corresponding author: Professor Eric Lee

E-mail: [email protected]

ACS Paragon Plus Environment

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Abstract

Electrophoretic and electroosmotic motion of a charged spherical particle within a cylindrical pore filled with a Debye-Bueche-Brinkman (DBB) polymeric solution is investigated theoretically, which is of high relevance in capillary electrophoresis as well as micro- and nanofluidic applications involving polymeric solutions in a microor nanopore. The DBB model describes rheological response of a polymeric solution with linear polymer dissolved in a homogeneous solvent. It is a well-known non-Newtonian model in liquid physics based on rigorous theoretical derivations. By Debye and Bueche, corresponding governing fundamental electrokinetic equations are solved numerically with a patched pseudo-spectral method based on Chebyshev polynomials.

We found that the double layer polarization effect reduces the particle mobility severely when the Debye parameter, κa, is around unity, especially in narrow pores. This is attributed to the extra confinement effect from the nearby wall which tends to sweep the predominant counterions within the double layer to the wake of the moving particle, resulting in a motion-deterring induced electric field. The electrophoretic mobility in a polymer solution is smaller than that in an aqueous electrolyte solution in general due to the much stronger viscous drag effect in a polymer solution. 1


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Moreover, electroosmotic flow (EOF) due to a charged pore wall is found to exhibit a highly non-Newtonian behavior. Unlike the corresponding plug-like flow for a Newtonian solution, an axisymmetric flow with a large local maximum in the velocity profile in the region near the pore wall is observed. This radial-varying velocity profile offers a potential extra separation mechanism which favors the elution of smaller particles in general.

The results obtained here provide fundamental understandings and insights of the electrophoresis and electroosmosis phenomena in a cylindrical pore filled with polymeric solution.

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1. Introduction

Electrophoresis, the motion of a particle in response to an applied electric field, is a very powerful and efficient technique for the separation and characterization of colloidal particles in various fields. The operation is normally conducted within a cylindrical capillary, the so-called capillary electrophoresis (CE), as the heat dissipation problem originated from the Joule heating effect can be minimized, which leads to a much better separation performance without the convective dispersion.1 Much smaller volume of sample consumption and easy integration with the sample-filling and detection regions to form an automatic system are other merits of this configuration. Indeed CE has been the major horse power in the field of biochemical and biomedical analyses with electrophoresis.

In conventional capillary electrophoresis, the capillary diameter ranges normally from 25 to 100 μm, so the boundary effect of the pore wall upon the particle motion is essentially negligible when taking the particle size to be one micron or smaller. However, internal diameters ranging from 200 nanometers to one micron are commercially available nowadays, and nearly cylindrical pores with size as small as 20 nanometers are available as well such as nanoporous alumina disks (Anodisk).2 3


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Moreover, with the rapid development of micro- and nanotechnologies, devices involving channels or pores with dimensions in the nanoscale are very common nowadays in various lab-on-a-chip (LOC) type operations.3, 4 With the channel size comparable to the particle size within, the presence of the channel wall has profound impact on the particle motion and hence cannot be ignored anymore. Therefore, a careful investigation of this boundary effect, both hydrodynamically and electrostatically, is necessary.

In practice, CE is often filled with polymeric solutions1, 4, 5 to provide an extra size-dependent separation mechanism needed in the separation of proteins and DNAs, among other important colloidal particles of interest.4,

6, 7, 8, 9, 10

Numerous

experimental studies have been reported in various aspects of its performance.11, 12, 13, 14

Corresponding theoretical studies about electrokinetic behavior of analyte in a

cylindrical pore filled with (Debye-Bueche-Brinkman) polymeric solution, however, have never been reported in the literature, to the best of the authors’ knowledge. Obviously, a comprehensive theoretical investigation will provide some crucial insight into this complicated system as direct and controlled observation of the particle motion is next to impossible in such tiny systems.

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Literature review does show, however, that theoretical analyses for a cylindrical pore filled with aqueous electrolyte solution have been conducted by several research groups.15, 16, 17, 18, 19, 20, 21, 22, 23, 24 Among them, Ennis and Anderson16 considered the rigid particle case with a thin double layer. They found that the presence of the cylindrical wall hindered the particle motion both hydrodynamically and electrostatically. Liquid droplets, as well as soft and porous particles, have been studied by Lee’s group in recent years.20, 21, 22 The presence of the pore wall is found to further slowdown the motion of the highly charged colloidal entities with arbitrary double layer thickness. Recently Keh and his coworkers23 reported a study valid for the thin double layer situation. The information provided by these reports can be helpful in that they serve as the limiting Newtonian cases of a corresponding non-Newtonian polymeric system.

The system of interest is complicated mainly due to its characteristic non-Newtonian rheological properties of the polymeric fluid, which affects the particle motion via a hydrodynamic mechanism.25 The hydrodynamic (rheological) response of a polymeric solution consists of the impact of the frictional force from the coiled polymer chains as well as the viscous force from the solvent medium, and Debye and Bueche26 conducted a classic theoretical analysis. Based on Einstein’s

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theory, they were able to derive an equation similar in form to the well-known Brinkman equation,27 which is widely accepted as the model for the fluid flow within a porous medium. Note that while the Brinkman model is of semi-empirical nature and valid for heterogeneous systems only, such as the fluid flow in porous media like soil or gel, the Debye-Bueche model is a precise conclusion based on rigorous theoretical derivations valid for homogeneous polymeric solutions. As a matter of fact, in the field of liquid physics, they are often addressed together as the Debye-Bueche-Brinkman (DBB) equation (model).28, 29 This extra drag force due to the presence of polymer chains is proportional to the velocity of the particle migrating through them in general. Note that all the general non-Newtonian response of a polymeric solution is incorporated implicitly, as this extra drag force will affect corresponding shear stress and pressure simultaneously in the modified Stokes equation.

In addition to the obvious relevance to CE, this configuration is also involved and integrated into various LOC formats for biochemical and biomedical applications.4 In particular, unique phenomena pertinent to nanofluidic operations have been reported recently such as the double layer overlapping (DLO) and corresponding non-plug flow,30 which might provide a separation mechanism in radial

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direction.31 We thus decide to conduct a comprehensive and rigorous theoretical study on it by solving the governing fundamental electrokinetic equations for this system with a numerical approach employing the patched pseudo-spectral method based on Chebyshev polynomials,32 a powerful and efficient method for nonlinear electrokinetic problems in general. This is a very challenging task both numerically and mathematically as two different geometries, spherical and cylindrical, are involved in the same configuration.

In summary, we present here a theoretical study for a highly charged rigid spherical particle with arbitrary double layer thickness conducting electrophoretic motion in a cylindrical pore filled with polymeric solution. The impact of the degree of polymer contribution, represented by the Debye-Bueche-Brinkman (DBB) screening length, λ-1, is investigated in detail as it is the key parameter affecting the particle motion here in comparison with the corresponding aqueous electrolyte solution. The boundary effect due to the presence of the wall is analyzed in particular to understand its hydrodynamic and electrostatic impact on the particle motion. Other parameters of electrokinetic interest are explored as well, such as surface charge of the particle, which can be either highly or lowly charged, the ionic strength of the solution and so on. The effect of the electroosmosis flow (EOF) due to a charged wall

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is investigated closely as well, especially the impact of the double layer overlapping (DLO) and corresponding formation of a non-plug (non-Newtonian) flow. The results presented here are also useful for microfluidics and nanofluidics involving polymeric solution in a cylindrical pore.

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2. Theory

2.1. Description of the Model

The system under consideration is shown in Fig. 1, where a spherical colloidal particle of radius a moves along the axis of a cylindrical pore of radius Rb with velocity U subject to an applied electric field Ez in the Z-direction. The cylindrical pore is filled with polymeric solution containing z1 : z2 electrolytes, z1 and z2 being the valences of cations and anions respectively. The electroneutrality in the bulk solution requires that n20 = n10/α, where n10 and n20 are respectively the bulk concentrations of cations and anions and α = -z2/z1. α is set to unity for a binary 1:1 system considered here, such as a KCl electrolyte solution. Spherical coordinates (r,,) describe the rigid sphere, and cylindrical coordinates (R,,Z) depict the region between the particle and the cylindrical wall. The origin of the spherical coordinates is set at the center of the particle. Note that only half of the (r,) domain needs to be considered due to the symmetry nature of the problem here.

The major assumptions in this analysis are summarized briefly as follows. (i) The inertial effect is negligible to justify the creeping flow or the quasi-steady state 9


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situation. (ii) The polymeric electrolyte solution is incompressible. (iii) The applied electric field Ez is weak enough to justify a linear perturbation analysis reported by the classic work of O’Brien and White,33 which is normally true in typical electrophoretic operations. (iv) No penetration of ions across the particle surface is allowed.

The governing equations for the problem here are the fundamental general electrokinetic equations, which consist of the Poisson equation, the conservation of ionic species and the modified Debye-Bueche-Brinkman (DBB) equation with an extra account of the electric force   :

2 z en        j j   , j 1 2

(1)

  z j en j     f j      D j  n j     n j v   0 , k BT    

j = 1, 2,

(2)

 v  0 ,

(3)

2 v  p     v  0 ,

(4)

where ρ, e, kB, and T are respectively the net space charge density, the elementary charge, the Boltzmann constant, the absolute temperature of the system. ε, p, γ and η

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are respectively the permittivity, the pressure, the frictional coefficient and the viscosity of the polymeric solution. nj and zj are respectively the number concentration and the valence of ionic species j, with j = 1 representing cations and j = 2 anions. Dj is the diffusion coefficient for ionic species j. Note that γ is often replaced in practice by the so-called Debye-Bueche screening length, λ-1, defined as

 1  ( /  )1/2 . Equation (3) is the continuity equation indicating that the homogeneous polymeric solution phase is incompressible. Equation (4) is the modified DBB equation taking the electric force into account. Introducing a stream function  and associated standard mathematical manipulations in classic fluid mechanics, the above set of governing equations, Eq. (1)-Eq. (4), can be further simplified. Equation (4) reduces to the following:

E 4  ( a )2 E 2  

sin        ,    r   r 

(5)

where E 4  E 2 E 2 in spherical coordinates. E 2 is defined as follows:

E2 

 2 sin    1    2  . r 2 r   sin   

(6)

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Moreover, by adopting scaled symbols, the governing equations can be rewritten in dimensionless form. The following characteristic variables are chosen for the formation of corresponding dimensionless scaled variables: the particle radius a, the bulk concentration of cations, n10, the thermal electric potential per unit valence of cations, 0 (defined as kBT/z1e), and the characteristic particle velocity, UE = ε02/ηa, which is the prediction by Smoluchowski’s theory34 when an external electric field of

0/a is applied.

The resulting dimensionless variables are listed as follows: r* = r/a, R* = R/a, Z* = Z/a, Ez* = Ez/(0/a), ζa* = ζa/0, ζw* = ζw/0, e* = e/0, δ* = δ/0, gj* = gj/0, nj* = 2

nj/n10, * = /UEa2, v* = v/UE, Pej = UEa/Dj and  1   kBT /  n j 0 (ez j )2 . κ is the j 1

Debye parameter, with κ-1 serving as a measurement of double layer thickness. Pej stands for the Peclet number of ion species j, which measures the convection contribution to the total flux of ionic species j.

2.2. Equilibrium State (No external electric field is applied yet)

At equilibrium state, no external electric field is applied and the system is motionless. Equation (1) can be reduced as follows, in dimensionless form: 12


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 a  exp  *  exp  *     e  e  1    

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2



*2 * e

(7)

where a is the inverse of the dimensionless Debye length. The spherical particle is assumed to have a constant zeta potential, a*. The cylindrical wall (Rb* = Rb/a) can be either chargeless or charged with a constant zeta potential w*. The axial symmetry condition demands that e* /   0 at   0,  .

The presence of the charged colloid does not affect the fluid within the pore infinitely far away from it, so the electric field e* there depends only on R*, the radius direction of the cylindrical pore. That is to say, we can denote e* = e,∞*(R*), where

e,∞*(R*) is the electric potential distribution in absence of the charged particle. The details can be found in the analysis by Huang et al.19, 20, 21, 22 The boundary conditions needed for the corresponding two-dimensional electrophoresis problem with the presence of a spherical particle are thus complete as follows:

e*  1 ,

r*  1,

e*   w* ,

R* 

(8)

Rb , a

(9)

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e*  e,*  R*  , e*  0, 

Z*  

L , a

(10)

  0 , .

(11)

As one cannot assume infinity in the numerical procedure, a finite value of L/a need to be selected with one-dimensional electric potential profile applied at both inlet (Z* = 0) and outlet (Z* = L/a). The solution is then considered to be asymptotically convergent once the mobility calculated does not vary with increasing value of L/a anymore. It turns out that L/a = 6 is long enough and chosen throughout our analysis here.

2.3. Linearized Perturbed State (After an External Electric Field is Applied)

The perturbed state (particle in motion) only slightly deviates from the original equilibrium (motionless) state when a weak electric field is applied. In other words, * = e* + *, nj* = nje* + nj*, etc., and the symbol  represents a perturbed amount. Classic linear perturbation analysis assumes that the perturbed state satisfies the same set of original governing electrokinetic equations, just like the equilibrium state. The effect of double layer polarization due to convection flow around the particle is

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introduced at this stage by defining a position-dependent perturbation function gj* as follows:

 z  n*j  exp   j e*   *  g *j   .  z1 

(12)

where gj* refers to the deformation of the original double layer due to the particle motion. It can be shown that

 n*j  

 zj  exp   e*   *  g *j  . z1  z1 

zj

(13)

Following the classic linear perturbation analysis as proposed by O’Brien and White 33

, the resulting linearized equations governing the perturbed variables are obtained as

follows:

( a) 2 exp  e*    exp e*    *     1     *2

*

( a) 2 exp  e*  g1*   exp e*  g 2*  ,   1    

 e* g1* 1 e* g1*  Pe1  e*  * e*  *   g   * *  *2    , r    r *2 sin    r * r *    r r *2

* 1

15


(14)

(15)

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  * g * 1  * g *  Pe   *  * e*  *  *2 g 2*    e* *2  *2 e 2   *2 2  e  , r    r sin    r * r *    r r

(16)

E *4 *  ( a )2 E *2 * *   g1*  g 2* * *  e exp     exp       e e  2  * * r  a   r        sin  . * 1      g1* g 2* * *  e   exp  e    exp e  *       r  

(17)

To obtain the associated boundary conditions needed, we assume that the particle is non-conductive and impenetrable, and so is the wall of the cylindrical pore. For convenience, we assume that the particle is fixed whereas the wall of the pore (Rb* = Rb/a) moves with velocity -Uiz, where iz is the unit vector along the axis of the pore. The concentrations of ionic species should reach the bulk values at both the inlet and the outlet of the pore. Moreover, a fully developed electroosmosis flow is generated far away from the particle, which leads to Eqs. (24) and (25). Based on these physical deductions, the boundary conditions associated with the present problem are as follows:

 * 0, r *

r* = 1,

 * 0, R*

R* 

 *   EZ* , * Z

(18)

Rb , a

Z*  

(19) L , a

(20)

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g *j r g *j *

R

*

 0,

 0,

g *j   * ,

r* = 1, R* 

(21)

Rb , a

Z*  

(22)

L , a

(23)

 *  U * R*2  * 

 Rb    a 

and

 *  U * R* , R*

 *  U * R*2  *  R* 

and

 *  0, Z *

1 2

1 2

* 0,  *  0, r *

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R*  Z*  

Rb , a

L . a

r* = 1,

(24) (25) (26)

r* = 1.

(27)

In addition, all variables must satisfy the axial symmetry condition at θ = 0, π as well:

*  * g j  *    *  0 ,   

θ = 0, π,

(28)

where  *  0 is a reference value for stream function.

The above coupled differential equations, together with associated boundary conditions, are then solved by a pseudospectral method based on the Chebyshev polynomials.32 The particle mobility can be calculated via force balance, once the flow and electric fields are obtained.33 Details of this approach can be found elsewhere.20, 21, 22

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3. Results and Discussion

The convergence of the numerical scheme in terms of grid-independence is ensured with a 50x50 mesh, which is adopted for all the calculations here. We then compare our calculation results with those available in the literature, for instance, those provided by Ennis and Anderson16 who studied the electrophoretic motion of a poorly charged rigid particle in a cylindrical pore filled with aqueous electrolyte solution, subject to the constraint that the double layer is not allowed to touch the channel wall. An analytical formula valid for the thin double layer is obtained there. As shown in Table 1, the agreement is excellent with the deviation less than 1% when κa is larger than 3, indicating the accuracy of the program used here

3.1. Effect of particle surface charges, zeta potential,  a :

As a model configuration, the diameter of the channel is set to be twice that of the colloidal particle to focus on the impact of a nearby cylindrical wall upon the particle electrophoretic motion. Fig. 2 presents the profiles of the scaled (dimensionless) particle mobility as a function of κa at various (dimensionless) zeta

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potentials in a chargeless pore filled with polymeric electrolyte solution characterized by  a  1 . Note that κ-1 is a measure of the double layer thickness. The larger the κa value is, the thinner the double layer is.

In general, when the double layer gets thinner (larger κa), the particle moves faster. This is expected as the electric driving force is increased with a larger potential gradient across the double layer. This speeding-up effect, however, is somewhat suppressed when κa is around unity. This is due to the convection-induced motion-deterring polarization effect associated with highly charged particles.20, 21, 22 The predominant counterions within the double layer are driven to the wake of the moving particle by the fluid flow around it. An induced electric field is thus generated which slows down the particle motion and is often referred to as the polarization (deformation) effect (of the double layer). The larger the particle zeta potential is, the more severe this polarization effect is, as shown in Fig. 2. The double layer polarization effect results in local minima around κa = 1 when the zeta potential is higher than 3. As a result, we choose  a  3 as the model particle zeta potential for further investigation. Corresponding contour plots for the equilibrium (liquid) space charge density under different double layer thickness are shown Fig. 3 to provide visual insight of the system. Note that the space charge density in the vicinity of the

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particle in Fig. 3(A) for a thick double layer (κa = 0.01) is much smaller than that of Fig. 3(B) for a thin double layer (κa = 5), which is part of the reason why the electric driving force would be smaller hence the particle moves slower. Moreover, the polarization of double layer is clearly shown in Fig. 4(A), where the disturbance of the space charge density is presented. Corresponding streamline plot is shown as well in Fig. 4(B) for completeness.

3.2. Effect of the reciprocal of the Debye-Bueche screening length,  a , the degree of polymer contribution:

Fig. 5 shows the scaled particle mobility, m* , as a function of  a for various values of λa, the reciprocal of the dimensionless Debye-Bueche screening length. The influence of λa is so significant that roughly 40% reduction of mobility is observed for λa = 3 whereas nearly 90% reduction for λa = 10! This is because the extra hydrodynamic friction force due to the presence of polymer chains is proportional to the square of λa by definition. Thus λa = 1, 3, 10 correspond to (λa)2=1, 9, 100 or roughly across three orders of magnitude. This extra drag force is so high with the increase of λa that it results in a sharp reduction in particle mobility.

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Note that as the polarization of double layer is induced mainly by convection, a much slower particle motion hence leads to the seemingly disappearance of polarization effect in terms of local minimum of the mobility profile for λa =10 in Fig. 5.

3.3. Effect of the radius of the cylindrical pore, Rb* :

Fig. 6 shows the impact of the scaled pore radius, Rb* , upon the particle mobility. The narrower the pore is, the slower the particle motion is. Corresponding results for a poorly charged particle in an infinite medium of polymeric solution is shown as well for comparison purpose.35 The presence of the wall generates a drag force which retards the particle motion hydrodynamically. Moreover, for very narrow pore like Rb*  1.2 , the double layer is severely squeezed due to the confinement of the cylindrical wall, which drives many counterions in the lateral region to the wake of it and slows it down by an induced electric field. Together with the hydrodynamic effect, this electrostatic effect ends up with a nearly 90% reduction of mobility for  a  1 in comparison with Rb*  3 , where the double layer has not touched the wall.

This confinement effect is shown in Fig. 7 and Fig. 8 respectively for both

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equilibrium electric potential, e* , and the disturbed charge density distribution,  * . as well. The severe deformation of the double layer is clearly demonstrated in these figures. The uneven distribution (polarization) of counterions within the double layer is clearly shown in Fig. 8. Corresponding equilibrium charge distribution is shown as well in Fig. 9 for completeness. Note that as the disturbance amount is normally of the order of one percent of the corresponding equilibrium value, this plot is essentially the same as the net charge distribution when the particle is in motion.

3.4. Effect of electroosmotic flow (EOF):

We now consider the effect of a charged wall which generates an electroosmosis flow due to the migration of counterions within the electric double layer adjacent to it. For a negatively charged surface, the counterions within the double layer carry positive charges and results in an EOF that enhances the motion of a positively charged particle. On the other hand, a positively charged wall would retard its motion. Note that constant zeta potential of the charged wall is assumed here. Otherwise, it is simply impossible to conduct a meaningful calculation if someone argues about the possible charge-regulation nature of the wall. It is too complicated and meaningless as the focus should be on the particle motion itself.

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Fig. 10 shows the EOF velocity profiles far away from the particle in a thin double layer situation (κa = 10 here) for various values of λa. Note that a is chosen to be half the pore radius here, i.e. Rb*  2 . Plug flows are observed for both λa = 0.1 and aqueous solution (λa = 0) beyond the double layer. The dashed line is the analytical prediction for an aqueous electrolyte solution.36 Our results agree with it excellently as shown. We also compared our results with those by Scales,37 who considered the EOF within a poorly charged capillary wall filled with solid porous material. The agreement is excellent again but not shown here for space-saving. It is interesting to note here in Fig. 10 that EOF fails to penetrate into the mainstream as λa increases, essentially motionless beyond the actual double layer thickness for λa = 10. The extra drag force was so strong that the molecular momentum, or the shear stress, cannot be transferred across it anymore. This indicates that care must be taken in polymeric solution CE in terms of EOF generation. It may not be a plug flow if the DBB screening length under consideration is very small. A nearly 45% reduction is observed for λa = 1 in comparison with an aqueous solution (λa = 0). For λa>3, on the other hand, nearly motionless flow in the mainstream is observed. Corresponding profiles for κa = 5 and κa = 1 are shown respectively in Fig. 11 and 12 for reference.

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Note also that local maxima of velocity profile are observed in the region near the pore wall in Fig. 10 and Fig. 11. This radially-varying velocity profile may provide a possible extra separation mechanism as smaller particles tend to cluster near the wall with larger ones in the mainstream.3, 31

3.5. Joint effect of electrophoresis and electroosmosis flow (EOF):

For a highly charged particle, the effects of electrophoresis and electroosmosis are not arithmetically addable. A consideration of both effects has to be made simultaneously. The calculation results are shown in Fig. 13, where the mobility profiles of a positively charged particle (ζa* = 3) are shown as a function of κa under various charge conditions of the wall. The dashed lines indicate the corresponding profiles for an aqueous solution. As discussed earlier, the negatively charged wall will enhance the particle motion while the positively charged one will retard it. The higher the wall charge is, the more significant the impact upon the particle motion is. Note that when the ζw* is greater than 1, the particle may even move backward, indicated by the negative sign of the mobility, due to the retarding effect of the EOF. That’s why EOF often needs to be suppressed in practical electrophoretic operations.

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The case for λa = 3 is shown in Fig. 14. In general, the particle motion gets slower and slower with increasing λa due to stronger hydrodynamic retarding force. It is interesting to note in Fig. 14 that all mobility profiles nearly coincide with the chargeless situation (w* = 0) when the double layer is very thin, about  a  5 here. This is because as the polymeric solution becomes so viscous (λa =3), the EOF is confined near the wall and no momentum transport to the mainstream is possible. Hence the influence of EOF on the particle motion disappears, regardless how strong the charge condition of the wall (w*) is. This is consistent with the observation in Fig. 11 where the EOF dies down in the mainstream for λa = 3 and κa = 5. The EOF behavior in a polymeric solution is quite different from that in an aqueous solution.

Plots of the equilibrium charge distribution, which are essentially the same as the net charge distribution when the particle is in motion, are shown in Fig. 15 for cases when the wall is either oppositely or likely charged. As shown in Fig. 15(A), the counterions from the likely charged wall (w* = 3) bears the opposite sign of the charged particle thus retards the particle motion. The pouring in of these extra amount counterions reshape the original double layer as shown. Basically these two double layers overlap with each other. On the other hand, when the wall is oppositely charged, the EOF enhances the particle motion as the counterions near the wall has likely

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charges as the particle. Moreover, electrostatically speaking, they and the original couterions surrounding the particle may cancel each other to a significant degree, leading to shrinkage of the original double layer surrounding the particle as shown in Fig. 15(B) (w* = -3).

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4. Conclusions

Electrophoretic and electroosmotic motion of a charged spherical particle within a cylindrical pore filled with polymeric solution is investigated theoretically in this study. Various electrokinetic parameters such as the pore diameter, the particle surface charge, the double layer thickness and the degree of polymer contribution (DBB screening length, λ-1) are examined for their respective impact and joint effect upon the particle motion. Moreover, the electroosmosis flow owing to a charged pore wall is investigated in detail as it is a crucial driving force as well in typical micro-/nanofluidic operations. The results are summarized as follows.

(1) For a highly charged particle, the double layer polarization effect always retards the particle motion and manifests itself when the double layer thickness is comparable to the particle diameter. A local minimum is generally observed in the mobility profile against the double layer thickness. The presence of the cylindrical wall enhances this impact due to further confinement of the double layer.

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(2) A higher degree of polymer contribution reduces the particle mobility due to the extra hydrodynamic drag force induced by the polymer chains. Up to 90% of reduction is observed in some situations.

(3) The electroosmosis flow (EOF) is completely suppressed in the mainstream of the channel if the degree of polymer contribution and the ionic strength are both high enough. This suppression is favorable if the existence of EOF is undesirable. Moreover, local maxima of velocity profiles near the wall (instead of the mainstream) are observed due to the non-Newtonian behavior of the polymeric solution. This radially-varying EOF may provide an extra separation mechanism as smaller particles tend to cluster near the wall while larger ones in the mainstream.

The results presented here are also useful in microfluidic and nanofluidic applications involving cylindrical pores filled with polymeric solution.

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Acknowledgement

This work is supported financially for all the authors by the Ministry of Science and Technology of the Republic of China under the grant number: MOST 104-2221-E-002-177-MY2.

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References

1.

Weinberger, R. Practical capillary electrophoresis; Academic Press2000.

2.

Woudenberg Jr, R. C. Anhydrous proton conducting materials for use in high

temperature polymer electrolyte membrane fuel cells; ProQuest2007. 3.

Kovarik, M. L.; Jacobson, S. C. Nanofluidics in lab-on-a-chip devices.

Analytical chemistry 2009, 81 (17), 7133-7140. 4.

Chung, M.; Kim, D.; Herr, A. E. Polymer sieving matrices in microanalytical

electrophoresis. The Analyst 2014, 139 (22), 5635-54. 5.

Khaledi, M. G. High-performance capillary electrophoresis: theory, techniques,

and applications; Wiley-Interscience1998; Vol. 204. 6.

Ren, J.; Fang, N.; Wu, D. Inverse-flow derivatization for capillary

electrophoresis of DNA fragments with laser-induced fluorescence detection. Anal Chim Acta. 2002, 470 (2), 129-135. 7.

Heller, C. Capillary electrophoresis of proteins and nucleic acids in gels and

entangled polymer solutions. Journal of Chromatography A 1995, 698 (1), 19-31. 8.

Kao, Y.-Y.; Liu, K.-T.; Huang, M.-F.; Chiu, T.-C.; Chang, H.-T. Analysis of

amino acids and biogenic amines in breast cancer cells by capillary electrophoresis using

polymer

solutions

containing

sodium

dodecyl

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sulfate.

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of

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Chromatography A 2010, 1217 (4), 582-587. 9.

Li, Z.; Yang, R.; Wang, Q.; Zhang, D.; Zhuang, S.; Yamaguchi, Y.

Electrophoresis of periodontal pathogens in poly (ethyleneoxide) solutions with uncoated capillary. Analytical biochemistry 2015, 471, 70-72. 10. Song, X.; Li, L.; Qian, H.; Fang, N.; Ren, J. Highly efficient size separation of CdTe quantum dots by capillary gel electrophoresis using polymer solution as sieving medium. ELECTROPHORESIS 2006, 27 (7), 1341-1346. 11. Carrilho, E.; Ruiz-Martinez, M. C.; Berka, J.; Smirnov, I.; Goetzinger, W.; Miller, A. W.; Brady, D.; Karger, B. L. Rapid DNA sequencing of more than 1000 bases per run by capillary electrophoresis using replaceable linear polyacrylamide solutions. Anal Chem. 1996, 68 (19), 3305-3313. 12. Kim, Y.; Yeung, E. S. Separation of DNA sequencing fragments up to 1000 bases by using poly (ethylene oxide)-filled capillary electrophoresis. J Chromatogr A. 1997, 781 (1), 315-325. 13. López-Lorente, A.; Simonet, B.; Valcárcel, M. Electrophoretic methods for the analysis of nanoparticles. TrAC Trends in Analytical Chemistry 2011, 30 (1), 58-71. 14. Radko, S. P.; Chrambach, A. Capillary zone electrophoresis of rigid submicron-sized particles in polyacrylamide solution: Selectivity, peak spreading and resolution. Journal of Chromatography A 1999, 848 (1), 443-455.

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15. Hsu, J.-P.; Chen, Z.-S. Electrophoresis of a sphere along the axis of a cylindrical pore: effects of double-layer polarization and electroosmotic flow. Langmuir 2007, 23 (11), 6198-6204. 16. Ennis, J.; Anderson, J. Boundary effects on electrophoretic motion of spherical particles for thick double layers and low zeta potential. J Colloid Interface Sci. 1997, 185 (2), 497-514. 17. Keh, H.; Anderson, J. Boundary effects on electrophoretic motion of colloidal spheres. J Fluid Mech. 1985, 153, 417-439. 18. Keh, H. J.; Chiou, J. Y. Electrophoresis of a colloidal sphere in a circular cylindrical pore. AIChE J. 1996, 42 (5), 1397-1406. 19. Shugai, A. A.; Carnie, S. L. Electrophoretic motion of a spherical particle with a thick double layer in bounded flows. J Colloid Interface Sci. 1999, 213 (2), 298-315. 20. Huang, C. H.; Cheng, W. L.; He, Y. Y.; Lee, E. Electrophoresis of a Soft Particle within a Cylindrical Pore: Polarization Effect with the Nonlinear Poisson-Boltzmann Equation. J. Phys. Chem. B 2010, 114 (31), 10114-10125. 21. Huang, C. H.; Hsu, H. P.; Lee, E. Electrophoretic motion of a charged porous sphere within micro- and nanochannels. Phys Chem Chem Phys. 2012, 14 (2), 657-667. 22. Huang, C. H.; Lee, E. Electrophoretic Motion of a Liquid Droplet in a

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Cylindrical Pore. J Phys Chem C. 2012, 116 (28), 15058-15067. 23. Chiu, H. C.; Keh, H. J. Electrophoresis of a colloidal sphere with double-layer polarization in a microtube. Microfluidics and Nanofluidics 2016, 20 (4). 24. Tseng, S.; Lin, C.-Y.; Hsu, J.-P.; Yeh, L.-H. Electrophoresis of deformable polyelectrolytes in a nanofluidic channel. Langmuir 2013, 29 (7), 2446-2454. 25. Zhao, C.; Yang, C. Electrokinetics of non-Newtonian fluids: A review. Advances in Colloid and Interface Science 2013, 201–202, 94-108. 26. Debye, P.; Bueche, A. M. Intrinsic viscosity, diffusion, and sedimentation rate of polymers in solution. J Chem Phys. 1948, 16 (6), 573-579. 27. Brinkman, H. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Applied Scientific Research 1949, 1 (1), 27-34. 28. Tchesskaya, T. A generalized Zimm model: Hydrodynamic screening in polymer solutions. J Mol Liq. 2009, 150 (1), 77-80. 29. Ahlrichs, P.; Everaers, R.; Dünweg, B. Screening of hydrodynamic interactions in semidilute polymer solutions: A computer simulation study. Phys Rev E Stat Nonlin Soft Matter Phys. 2001, 64 (4), 040501. 30. Haywood, D. G.; Harms, Z. D.; Jacobson, S. C. Electroosmotic Flow in Nanofluidic Channels. Analytical Chemistry 2014, 86 (22), 11174-11180. 31. Kirby, B. J. Micro-and nanoscale fluid mechanics: transport in microfluidic

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devices; Cambridge University Press2010. 32. Canuto, C.; Hussaini, M. Y.; Quarteroni, A. M.; Thomas Jr, A. Spectral methods in fluid dynamics; Springer Science & Business Media2012. 33. O'Brien, R. W.; White, L. R. Electrophoretic mobility of a spherical colloidal particle. Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics 1978, 74, 1607-1626. 34. Smoluchowski, M. v. Grundriß der Koagulationskinetik kolloider Lösungen. Colloid Polym Sci. 1917, 21 (3), 98-104. 35. Allison, S. A.; Xin, Y.; Pei, H. Electrophoresis of spheres with uniform zeta potential in a gel modeled as an effective medium. J Colloid Interface Sci. 2007, 313 (1), 328-337. 36. Rice, C.; Whitehead, R. Electrokinetic flow in a narrow cylindrical capillary. J Phys Chem. 1965, 69 (11), 4017-4024. 37. Scales, N.; Tait, R. N. Modeling electroosmotic and pressure-driven flows in porous microfluidic devices: zeta potential and porosity changes near the channel walls. J Chem Phys. 2006, 125 (9), 094714.

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Table of Contents Graphic

4

2

z*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0

-2

-4 -2

-1

0

*

1

2

Electroosmosis flow (EOF) of a DBB polymeric fluid has drastic and unique impact upon particle motion.

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Tables

TABLE 1: Comparison of scaled mobility values for κa = 5 at various (dimensionless) channel radius, Rb*.

Ennis and Calculation of Rb*

Relative Anderson’s

this study

deviation (%) prediction 16

1.667

0.430215

0.433291

-0.70979

2

0.558228

0.561924

-0.65774

2.5

0.655523

0.657749

-0.3384

3.333

0.720287

0.721995

-0.23655

5

0.756317

0.757903

-0.2092

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Figures

Fig. 1.

Schematic representation of the system under consideration. The cylindrical wall can be either positively or negatively charged.

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0.8 0.7

Rb= 2, a= 1, w= 0 *

*

0.6 0.5

*m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4

*a= 1,2,3,4

0.3 0.2 0.1 0 -2 10

Fig. 2.

10

-1

a

10

0

10

1

Scaled mobility as a function of a at different ζa* with Rb* = 2 and λa = 1 for a chargeless wall, ζw* = 0.

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(A)

(B)

4

4

e -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -110 -120 -130 -140 -150 -160 -170 -180 -190 -200 -210 -220 -230 -240

e -5E-05 -0.0001 -0.00015 -0.0002 -0.00025 -0.0003 -0.00035 -0.0004 -0.00045 -0.0005 -0.00055 -0.0006 -0.00065 -0.0007 -0.00075 -0.0008 -0.00085 -0.0009 -0.00095

0

-2

-4

Fig. 3.

0

1

*

2

z*

2

z*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0

-2

-4

2

0

1

*

2

Contour plots of the scaled equilibrium space charge density (ρe), with λa = 1, ζa* = 3, ζw* = 0 and Rb* = 2 for (A) κa = 0.01 and (B) κa = 5 respectively.

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(A)

(B)

4

4

* 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

* 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9

0

-2

-4

Fig. 4.

0

1

*

2

z*

2

z*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0

-2

-4

2

0

1

*

2

Contour plots of (A) the scaled space charge density disturbance (δρ*) and (B) the scaled stream function (ψ*), with λa = 1, κa = 1, ζa* = 3, ζw* = 0 and Rb* = 2.

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0.5

*a= 3, *w= 0 R*b= 2 0.4

a= 0,1,3,10 0.3

*m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.2

0.1

0 -2 10

Fig. 5.

10

-1

a

10

0

10

1

Scaled mobility as a function of a at different λa with ζa* = 3 and Rb* = 2 for a chargeless wall, ζw* = 0.

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0.8

a= 3 a= 1 *

0.6

m*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4

R*b= 3 2

0.2

1.5 1.2 0 -2 10

Fig. 6.

10

-1

a

10

0

10

1

Scaled mobility as a function of a at different Rb* with ζa* = 3 and λa = 1. Dashed line: results of Allison et al. 35 with λa = 1.

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2 1.5

e* 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

1 0.5

z*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0

-0.5 -1 -1.5 -2 0

Fig. 7.

0.5

*

1

Contour plot of the scaled equilibrium electric potential (e*), with λa = 1, κa = 1, ζa* = 3, ζw* = 0 and Rb* = 1.2.

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*

2

10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

1.5 1 0.5

z*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

0

-0.5 -1 -1.5 -2 0

Fig. 8.

0.5

*

1

Contour plot of the scaled space charge density disturbance (δρ *), with λa = 1, κa = 1, ζa* = 3, ζw* = 0 and Rb* = 1.2.

9


Langmuir

2

e -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -5.5 -6 -6.5 -7 -7.5 -8 -8.5 -9 -9.5

1.5 1 0.5

z*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 52

0

-0.5 -1 -1.5 -2 0

Fig. 9.

0.5

*

1

Contour plot of the scaled equilibrium space charge density (ρe), with λa = 1, κa = 1, ζa* = 3, ζw* = 0 and Rb* = 1.2.

10


Page 47 of 52

1

a= 0.1 a= 0.3

0.8

0.6

a= 1

V *z

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

0.4

a= 3 0.2

a= 10 0

0

0.5

1

R*b

1.5

2

Fig. 10. Eletroosmosis veloctiy profile far away from the particle as a function of radius at different λa with κa = 10. Dashed line: results of corresponding aqueous solution.

11


Langmuir

1

a= 0.1 a= 0.3

0.8

0.6

a= 1

V *z

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 48 of 52

0.4

a= 3 0.2

a= 10 0

0

0.5

1

R*b

1.5

2


12


Page 49 of 52

1

0.8

a= 0.1 0.6

a= 0.3

V *z

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

0.4

a= 1 0.2

a= 3 a= 10 0

0

0.5

1

R*b

1.5

2


13


Langmuir

1.4

*a= 3 a= 1

1.2 1

w= -3,-2,-1,0,1,2,3 *

0.8 0.6 0.4

*m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 50 of 52

0.2 0 -0.2 -0.4 -0.6 -0.8 -2 10

10

-1

a

10

0

10

1

Fig. 13. Scaled mobility as a function of κa at different ζw* with λa = 1, ζa* = 3 and Rb* = 2. Dashed line: results of corresponding aqueous solution.

14


Page 51 of 52

1.4

*a= 3 a= 3

1.2 1 0.8

w= -3,-2,-1,0,1,2,3 *

0.6 0.4

*m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

0.2 0 -0.2 -0.4 -0.6 -0.8 -2 10

10

-1

a

10

0

10

1

Fig. 14. Scaled mobility as a function of κa at different ζw* with λa = 3, ζa* = 3 and Rb* = 2. Dashed line: results of corresponding aqueous solution.

15


Langmuir

(A)

(B)

4

4 e

-2

0

1

*

9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9

2

z*

0

-4

e

9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9

2

z*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 52 of 52

0

-2

-4

2

0

1

*

2

Fig. 15. Contour plots of the scaled equilibrium space charge density (ρe), with λa = 1, κa = 1, ζa* = 3 and Rb* = 2 for (A) ζw* = 3 and (B) ζw* =-3, respectively.

16


Electrophoretic and Electroosmotic Motion of a Charged Spherical

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