Gel electrophoresis and size selectivity of charged colloidal particles in a charged hydrogel medium

Author’s Accepted Manuscript Gel electrophoresis and size selectivity of charged colloidal particles in a charged hydrogel medium S. Bhattacharyya, Simanta De

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S0009-2509(15)00729-0 http://dx.doi.org/10.1016/j.ces.2015.11.012 CES12670

To appear in: Chemical Engineering Science Received date: 22 July 2015 Revised date: 17 October 2015 Accepted date: 17 November 2015 Cite this article as: S. Bhattacharyya and Simanta De, Gel electrophoresis and size selectivity of charged colloidal particles in a charged hydrogel medium, Chemical Engineering Science, http://dx.doi.org/10.1016/j.ces.2015.11.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Gel electrophoresis and size selectivity of charged colloidal particles in a charged hydrogel medium S. Bhattacharyya∗, Simanta De Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, West Bengal, India

Abstract The electrophoresis of a charged colloid particle embedded in a charged hydrogel medium is studied based on the numerical computation of Stokes-Nernst-Planck-Poisson equations. In this study, no prior assumption on surface charge density of the particle, Debye length and imposed external electric field is made. We have compared our computed results for a lower range of surface charge density with the existing analytical solution based on the weakly charged particle and found them in good agreement when the fixed charge density of the hydrogel is low. Nonlinearity effects in gel electrophoresis is pronounced for a thick Debye length and higher values of particle ζ-potential. Even in the absence of a gel medium (free-solution), the numerical procedure used in this work yields mobilities different from the previous theoretical analysis based on the Debye-H¨ uckel approximation when the scaled ζ-potential exceeds 2 and the Debye length is in the order of the particle size. The strong background electroosmotic flow (EOF) for a high fixed charge density of the polyelectrolyte hydrogel with a mesh size comparable to the particle radius drags the particle along the direction of the EOF. In this case, the electrophoretic velocity of the particle varies with its size. The particle electrophoretic velocity and forces are determined for a wide range of intrinsic parameters values. Keywords: Charged Hydrogel, Mobility, Electroosmosis, Nernst-Planck equations.

1. Introduction Hydrogels are water-saturated cross-linked polymers having a porous structure with molecular-scale porosity. Electrophoresis in hydrogel medium is widely used to separate biomolecules such as DNA and proteins in electrophoretic applications (G¨org, 1994; Yoshioka et al., 2003). Electrokinetic transport in a permeable membrane with charged inclusions has importance in the context of biological cells, dialysis and fuel cells (Hill, 2006). Gel electrophoresis is also important in remediating a contamination in situ (Jones et al., 2011). The widespread applications of gel electrophoresis and its advantages over the conventional free electrophoresis in a liquid medium has already been discussed in several articles e.g., Li and Hill (2013), Bhattacharyya et al. (2014). Gel electrophoresis is complicated compared to free electrophoresis due to the occurrence of long range hydrodynamic interactions and short range steric effects on account of friction between the gel skeleton and the particles. The long range effects can be modeled through the effective medium approach (Johnson et al., 1996), in which the hydrogel is modeled as a continuum and the hydrodynamics is governed by the Brinkmann equation. If the particle size is comparable to the gel mesh size then the steric interaction becomes important. Brady (1994) proposed that the diffusivity in a gel can be expressed as a product of factors which accounts for the hydrodynamic effect and the steric effect. This model was validated experimentally, theoretically and computationally by several authors namely, Johansson and L¨ofroth (1993), Allison et al. (2007), Tsai et al. ∗ Corresponding

author Email addresses: [email protected] (S. Bhattacharyya), [email protected] (Simanta De)

Preprint submitted to Chemical Engineering Science

November 23, 2015

(2011) and Hsu et al. (2012). Doane et al. (2010) introduced a frictional coupling force to model the steric interactions between the particle and the hydrogel skeleton. Based on the effective medium approach, the long range interactions in gel electrophoresis have been addressed by several authors namely, Hill (2006), Allison et al. (2007), Tsai et al. (2011), Tsai and Lee (2011), Hsu et al. (2012), Hsu et al. (2013) and Bhattacharyya et al. (2014). When the electrostatic force and hydrodynamic drag experienced by the particle is balanced then the particle migrates at a constant velocity; the electrophoretic velocity. Allison et al. (2007) provided an analytical formulae to determine the elecrtophoretic mobility of a particle with low and high surface charge. The double layer polarization (DLP) effects on mobility has been analyzed by Tsai et al. (2011) based on a first-order perturbation analysis. Hsu et al. (2012) found through the first-order perturbation analysis under a weak field assumption that the DLP effect in gel electrophoresis is low compared to the free-solution electrophoresis and it is further diminished with the increase of gel concentration. Recently, Bhattacharyya et al. (2014) determined the mobility and forces and validated the formulae of Allison et al. (2007) by considering the double layer polarization effects. There it was found that the double layer polarization and relaxation effect in uncharged electrophoresis is significant when the particle is highly charged and the gel is highly permeable. Agarose gel has been used for gel electrophoresis because of its favorable permeability. Many hydrogels e.g., agarose gel are charged or acquire charges due to chemical reactions (Li and Hill (2013) and the references there-in). In presence of an electric field, the mobile counterions in gel get transported towards the oppositely charged electrodes and therefore induce an electroosmotic flow (EOF). This EOF affects the electrophoretic transport of a charged particle. Mohammadi and Hill (2010) analyzed the electrokinetic transport of uncharged inclusions in a charged hydrogel due to the impact of electroosmotic flow theoretically. The size selectivity of nanoparticles coated with a charged polymer layer in gel electrophoresis was demonstrated experimentally by Hanauer et al. (2007). Doane et al. (2010) made experimental and theoretical studies to quantify the mobility of weakly charged PEGylated metal nanoparticles in a charged hydogel medium. Subsequently, Li and Hill (2013) extended this study for weakly charged bare nanoparticles in a charged or uncharged hydrogel. There they obtained an expression by modifying the theory of Allison et al. (2007) for determining the mobility of a charged particle in a charged hydrogel medium. The electrophoresis of soft particles in polyelectrolyte hydrogels is studied under the Debye-H¨ uckel approximation by Li et al. (2014). Li et al. (2014) have shown that the dielectric permittivity of the particle has an impact on gel electrophoresis. Their analysis shows that the DLP effect diminishes the magnitude of the mobility. It has been predicted in that paper that the double layer polarization and relaxation have a strong impact for a metallic particle (large dielectric permittivity), but these effects are negligible at a thin Debye length for a perfectly dielectric particle (zero permittivity). It may be noted that for a soft particle, as considered by Li et al. (2014), the nonlinear effects such as DLP and relaxation become less important as compared to a bare particle. A similar problem, as considered by Doane et al. (2010) and Li et al. (2014), is analyzed through a different approach by Allison et al. (2014). In this paper, we have considered the electrophoresis of a charged particle in a charged hydrogel medium. Our analysis is based on the computation of the coupled Stokes-Nernst-Planck-Poisson equations in their full form. The numerical study based on the computation of the Nernst-Planck equations for ion transport on colloid dispersions is rather limited owing to its several complexities. Recently, Shih and Yamamoto (2014) performed direct numerical simulations to study the electrokinetics of charged colloids in an AC electric field. We developed the computer code, which we have validated by comparing with several experimental and theoretical solutions. The mechanism of electrokinetic transport of a nanoparticle in a charged hydrogel is studied by considering the interactions of electroosmosis, double layer polarization (DLP), hydrodynamic drag and electrostatic force. We have highlighted the impact of the double layer polarization and interaction of the background EOF on gel electrophoresis by comparing with the existing analytical solutions. The present analysis does not consider the steric interactions with the gel skeleton. One of the objectives of the present study is to demonstrate the size selectivity of nanoparticles by electrophoresis in a charged hydrogel medium. Our results show that by suitably tuning the fixed charge density of the hydrogel and ionic concentration of added electrolyte, a size-selectivity can be achieved.

2

2. Mathematical Model We consider the electrophoresis of a nonconductive, impermeable charged spherical particle of radius a in a charged hydrogel under an applied electric field E0 . In presence of the applied electric field, the particle moves with a constant velocity UE∗ relative to the surrounding medium. This problem is equivalent to that of a stationary sphere experiencing an incoming flow at a uniform velocity of −UE∗ far from the particle surface in a frame of reference fixed at the center of the particle (Fig.1a). A spherical polar coordinate (r, θ, ψ) is adopted with the origin at the center of the sphere and the initial line (θ = 0) is the z-axis along which the electric field is imposed. We assume the problem to be axially symmetric with z-axis as the axis of symmetry. The hydrogel is modeled as a Brinkman medium with screening length , which characterize the hydrodynamic permeability of the medium. We consider a uniformly charged hydrogel skeleton with fixed charge density ρf which is of the same sign as that of the surface charge density of the particle. The equations governing this electrokinetic phenomena are the Darcy-Brinkman extended Stokes equation with electric body force term for fluid flow, the Nernst-Planck equations for ion transport, and the Poisson equation for the electric field. For simplicity, we have considered a symmetric z-z electrolyte with valance zi = ±Z for i = 1, 2 , respectively. Following Saville (1977), we have scaled the variables as follows. The radius of the sphere a is considered as the length scale, the thermal potential φ0 = kB T /Ze is the potential scale, U0 = (e φ20 /μa) is the velocity scale, εe φ20 /a2 is the pressure scale and the bulk ionic number n0i is the scale for ionic concentration. Here e is the elementary electric charge, kB is the Boltzmann constant, T is the absolute temperature, εe is the permittivity of the medium and μ is the viscosity. The velocity scale U0 corresponds to the Smoluchowski velocity for surface potential φ0 under an electric field φ0 /a. The non-dimensional parameter β = a/ provides a measure of the permeability  of the gel medium. A higher value of β implies a lower permeability of the gel medium. Here κs = 2ZI0 e/εe φ0 is the inverse of the EDL thickness based on the bulk concentration I0 of the added electrolyte. The non-dimensional form of the Stokes-Brinkamn equations for Newtonian fluid to describing the motion of ionized fluid in uniformly charged hydrogel is (κs a)2 ρe ∇φ = 0 (1) ∇p − ∇2 u + β 2 u + 2 along with the equation of continuity for incompressible fluid ∇·u=0

(2)

where u = (u, v) is the velocity vector, with u and v being the cross-radial and radial components respectively, t is the time and p is the pressure and ρe = (n1 − n2 ) is the scaled charge density, scaled by I0 e with ni being the ionic concentration of ith ionic species with valence zi of the added electrolyte. The non-dimensional form of the Nernst-Planck equation governing the transport of the ith ionic species is given by P e (u · ∇ni ) − ∇2 ni ∓ ∇ · (ni ∇φ) = 0

(3)

For simplicity we have considered that the diffusivity of the ions are equal i.e., Di = D for i = 1, 2, with that P e = εe φ20 /μD. The electric potential satisfies the Poisson equation ∇2 φ = −

(κs a)2 ρe − β 2 Q f 2

(4)

the nondimensional quantity Qf measure the fixed charge density (immobile) of the hydrogel scaled by εe φ0 /2 i.e., ρf = Qf εe φ0 /2 . The non-dimensional parameters governing the electrophoresis of a particle in a gel media are the Peclet uckel parameter κs a, parameter related to the gel permeability is Number P e = εe φ20 /μD, the Debye-H¨ β = a/ and the scaled electric field Λ = E0 a/φ0 . The Peclet number P e = εe φ20 /μD, measures the ratio of advective to diffusion transport of ions. We have assumed the diffusion coefficients D = 2 × 10−9 m2 /s, and permittivity of the medium εe = 695.39 × 1012 C/Vm with that P e = 0.23. 3

-

-

E0

-

ex

-

-

-

-

-

eθ

-

-

-

O

er

-

-

θ

- - -

-

-

-

ψ

-

-

ez

-

-

-

(a)

(b)

Figure 1: (a) Schematic description of the geometry and the spherical coordinate system. (b) Grid distribution around the solid sphere.

On the surface of the particle (r = 1), either the surface charge density or the surface potential can be prescribed. ∂φ = −σ, (∇ni ± ni ∇φ) · er = 0 u = 0, φ = ζ or ∂r Far from the particle (r = R) u = −UE ez , φ = −Λr cos θ, ni = 1 where ζ and σ are respectively, the non-dimensional surface potential and surface charge density which are scaled by φ0 and εe φ0 /a and R is the radius of the outer boundary. When the charged particle migrates under the influence of an electric field, the forces acting on the particle are electrostatic force and hydrodynamic force. The axi-symmetric nature of our problem suggests that only the z-component of these forces need to be considered. The electrostatic and hydrodynamic forces along the flow direction can be calculated by integrating on the surface of the particle the Maxwell stress tensor σ E and hydrodynamic stress tensor σ H respectively, and are given by   E  σ · er · ez dS (5) FE∗ = S

∗ FD =



 H  σ · er · ez dS

S

(6)

    where σ E = εe EE − (1/2)E 2 I and σ H = −pI + μ ∇u + (∇u)T . Here E = −∇φ, E 2 = E · E and I is the unit tensor. Here the superscript ∗ denotes dimensional quantities. The unit vectors along radial and axial directions are denoted by er and ez , respectively. These forces can be computed based on the formula given by Bhattacharyya et al. (2014). The forces are scaled by εe φ20 and the scaled forces are denoted by FE and FD . The velocity UE is determined through the balance of forces. The electrophoretic mobility based on the EM model is defined as μ∗E = UE /E0 . The dimensional mobility is scaled by 2εe φ0 /3μ to obtain the non-dimensional mobility μE . Here we have determined the electrophoretic velocity of the particle due to hydrodynamic interactions through the balance of forces. Steric interactions can be determined by the model discussed in Hsu et al. (2012). The governing equations are solved in a coupled manner through a control volume approach over a staggered grid arrangement. Numerical method adopted here is similar to the method described in Bhattacharyya et al. (2014) in the context of electrophoresis in an uncharged gel medium. In contrast to the previous algorithm, here we have used an upwind discretization for the convection and electromigration terms in the ion transport equation. This upwind discretization imparts stability to the numerical scheme when the electromigration and convection terms in the ion transport equations are significant. This 4

makes the solver second-order accurate in space variables. The discretized equations are solved through the pressure-correction based iterative SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm. The procedure is based on a cyclic guess-and-correct operation 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. At every time step, the iteration process is continued till the difference between the values of each variables i.e., u, ni and φ in successive iteration becomes lower than 10−5 . A time-dependent numerical solution is achieved by advancing the variables through a sequence of short time steps. We started the motion from the initial stationary condition and achieved a steady-state after a large time step for which the variables become independent of time. At each time step, the Poisson equation for electric field is solved by the Successive-Over-Relaxation method (SOR). The variables near the surface of the rigid sphere varies more rapidly than elsewhere. In order to account for this fast change, we considered a non-uniform grid distribution along the r-direction, however, a uniform grid is considered along the θ-direction. We considered a finer mesh around the solid sphere. The grid size in radial direction is then increased gradually through an arithmetic progression as we move away from the sphere. We have considered a sufficient number of grid points within the Debye layer, and because of that the number of grid points varies with the variation of the Debye length. A grid independency test by comparing with several existing theoretical and experimental results are conducted to ensure convergence of the numerical solution. For this purpose, different meshes varying from 75 × 500 to 150 × 1200 in the cross-radial and radial directions are considered. The minimum grid size in the r-direction i.e., δr in the dense grid as 0.005 which is subsequently increased up to 0.04 far away from the surface of the sphere along with δθ = 0.021 is found to be the optimal choice. The distribution of grids near the particle is shown in Fig.1(b). The outer boundary of the computational domain was placed at a far radial distance so as to have no further effect on the forces experienced by the particle due to the variation of the outer boundary size. We have estimated the size of the computational domain by evaluating the drag for a hydrodynamic Stokes U = 6πμU a(1 + β + β 2 /9). This estimation for flow in a porous medium for which the drag is known i.e., FD the size of the outer boundary was used to determine correct outer boundary size for the gel electrophoresis problem. The distance of the outer boundary from the particle center is varied between 20 to 25 times the radius of the particle. 6

7

O Brien and White (1978) Shih and Yamamoto (2014) Liu et al. (2014) Present result

5 4

6 5

μE

μE

κa 3

4

2

κa

3

1

2

0 -1 1

O’Brien and White (1978) Liu et al. (2014) Wiersema et al. (1966) Present result

2

3

ζ

4

1 1

5

(a)

2

3

ζ

4

5

(b)

Figure 2: Variation of mobility for free-solution electrophoresis (β = 0, ρf = 0) with particle ζ-potential for (a) κa = 0.1, 0.3, 1, 2; (b) κa = 4, 10, 20, 30, 50, 70, 100. The inset figure show the comparison with analytic solutions for κa = 0.1, 0.5, 1.

The electrophoretic velocity of the particle (UE ) is obtained by solving the balance of drag and electric forces experienced by the particle iteratively. The iteration process starts with an initial assumption for the electrophoretic velocity. The drag and electric forces are obtained by computing the governing equations. Iteration process continues till the balance of forces is established. 5

In order to validate our algorithm, we have made comparisons of our computed mobility for gel electrophoresis as well as free-solution electrophoresis with several existing results. In Fig.2(a) we made a comparison for a lower range of Debye length of our computed mobility for free-solution electrophoresis by setting β = 0 and ρf = 0 in the governing equations with the mobility as obtained by O’Brien and White (1978), Shih and Yamamoto (2014) and Liu et al. (2014). The maximum percentage difference between our results and the results due to O’Brien and White (1978) is found to be 4%, which occurs at ζ = 5 when κa = 0.1. In the inset of Fig.2(a), a comparison of our computed results for free-solution (β = 0, ρf = 0) with the analytic solution for mobility due to Ohshima (2001) and the mobility based on the Henry function (Hunter, 2001) is presented. We find that the analytic solutions for ζ ≥ 2 when κa ∼ O(1) over estimates our computed results. We have also presented in Fig.2(b) a comparison of our computed mobility for free-solution electrophoresis for low to high values of ζ-potential at larger range of κa with several existing solutions i.e., Wiersema et al. (1966), O’Brien and White (1978) and Liu et al. (2014). We found a good matching of our computed mobility with O’Brien and White (1978) at higher range of κa even at larger values of ζ-potential. Comparisons of mobility for free-solution electrophoresis with several other published results is shown in the supplementary material (Fig.S1). We have compared our results for electrophoretic mobility in gel electrophoresis with the analytical solutions presented by Li et al. (2014) for the charged hydrogel case and Allison et al. (2007) and Tsai et al. (2011) for uncharged hydrogel (ρf = 0). We found (Fig.3a-d) a good agreement of our computed solution for smaller values of particle ζ-potential and hydrogel charge density with those studies. The comparison of computed mobility in uncharged hydrogel with the experimental results due to Park and Hamad-Schifferli (2008) is shown in the supplementary material (Fig.S2). 3. Results and Discussion We have developed a computer code based on the numerical method as illustrated above to analyze the electrophoresis in a charged gel medium for a wide range of electrokinetic parameters. The electrokinetics are governed by the non-dimensional parameters β, Λ and ρf , κs a, which represent respectively, the permeability, external electric field, fixed charged density of the hydrogel and ionic concentration of the added electrolyte. In addition, the variation of particle surface potential or charge density is also studied. The scaling of variables suggests that at fixed values of the parameter β and Λ, the electric field increases at the same rate as the rate by which the particle size is lowered and the gel mesh size too reduces at an equal rate. Following Li and Hill (2013), we have considered the gel fixed charge density up to ρf = −5 × 104C/m3 and mesh size  up to 100nm. The electrolyte concentration is varied between 3.7 × 10−4 mM to 93.2mM i.e., κ−1 s to vary between 1 to 500nm. 3.1. Electrophoretic mobility as a function of intrinsic parameters The variation of mobility with the Debye length at different values of the hydrogel fixed charge density and particle ζ-potential is shown in Fig.3(a). We have considered the surface potential of the particle and the fixed charge density of the gel to be of the same sign. We have computed the expression (42) for mobility as provided by Li et al. (2014) based on the Debye-H¨ uckel approximation and is indicated in the figures to elucidate the effects of Debye layer polarization and relaxation. We have also indicated the analytical solution as obtained by Allison et al. (2007) and Tsai et al. (2011) for the uncharged gel (ρf = 0). In order to compare with Li et al. (2014), we have considered the Debye layer thickness κ−1 to be based on the total ionic strength including the hydrogel counterions i.e., κ−1 is based on the ionic strength I = I0 −Zρf /2e. The variation in Debye length at a fixed value of the hydrogel fixed charge density (ρf ) is achieved by varying the ionic concentration of the added electrolyte. The EOF induced by the fixed charge density of the hydrogel is in opposite direction to the electrostatic force due to the negative surface charge of the particle. The intrinsic mobility of the negatively charged particle is opposite to the direction of the imposed electric field. We find that the change in the scaled mobility due to the variation of the Debye layer thickness is significant when the Debye length is comparable to the size of the particle. The magnitude of the electrophoretic mobility of the particle in an uncharged or weakly charged hydrogel increases with the decrease of Debye 6

length and approaches a constant value when the Debye length become thin. An excellent agreement of our computed results with the existing analytic solutions due to Li et al. (2014) and Allison et al. (2007) for |ζ| ≤ 2 is encouraging. As expected, our results for ζ = −3, where Debye-H¨ uckel approximation may not be valid, deviates from the solution due to Li et al. (2014). The results for uncharged gel (ρf = 0) as presented by Allison et al. (2007) with a correction indicated in the corrigendum for large values of ζ is also shown in Fig.3(a). The mobility values of Allison et al. (2007) is extracted from the Fig.3 in their paper. We find a good matching with these results for an uncharged gel particularly at larger κa. The mobility based on the first-order perturbation analysis of Tsai et al. (2011) for an uncharged gel also shows a good agreement (Fig.3a). We find from Fig.3(a) that a large difference between the solution based on Li et al. (2014) analysis occurs at ζ = −3 for an uncharged gel. Due to the convective motion of ionic species, the double layer surrounding the particle may not remain spherical. The double layer polarization is pronounced when the Debye length is in the order of the particle size and it diminishes as the Debye layer becomes thinner. The high charge density of the particle attracts more counterions due to a shielding effect and the impact of ion convection around the particle for thicker Debye layer becomes significant, which makes the ion transport equation depend on the the equation for fluid flow and vice-versa. The impact of advection on ion transport is illustrated later in this section. This nonlinear effects, which grows with the rise of particle charge density, leads to a discrepancy between the present solution and the solution based on the Debye-H¨ uckel approximation. Present result Tsai et al. (2011) Li et al. (2014) Allison et al. (2007)

5 4

Present result Li et al. (2014) Allison et al. (2007)

10 ζ = -3

8

6

2

ρf

4

μE

μE

-μE

6 3

4 |ζ|

2

ζ = -2

0 1

|

0.1

ζ = -1 10

20

κa

30

40

50

Allison et al. (2007) Li et al. (2014) Present result

8

-25000

-2

κsa =50

-4 0

-50000

ρf (C/m3)

(a)

|ζ|

0

κsa =1

-2 0

2

-25000

ρf (C/m )

(b)

3

-50000

(c) 40

Present result Li et al. (2014)

10

Present result Li et al. (2014) 30

8

μE

μE

9

|ζ|

20

10

7

0

6

| 2.6

10

20

κa

30

40

0.5

50

(d)

|ζ| 2

β

4

6

8 10

(e)

Figure 3: Mobility as a function of (a) Debye-H¨ uckel parameter κa when β = 1, ρf = 0, −1000C/m3 for different values of ζ = −1, −2, −3; (b) ρf for κs a = 1; (c) ρf for κs a = 50 when β = 1 for different ζ = −1, −2, −3. (d) Mobility versus κa when β = 1, ρf = −50000C/m3 for different ζ = −1, −2, −3; (e) Mobility versus β when κa = 50, ρf = −50000C/m3 for different ζ = −1, −2, −3. All these computations are done for a = 50nm and Λ = 0.5.

When the background electroosmotic flow becomes strong for large values of ρf , the negatively charged particle migrates along the direction of the external electric field i.e., μE ≥ 0 (Fig.3b,c). Fig.3(b),(c) shows that the electrophoretic mobility of the particle changes from negative to positive as ρf is increased. Mobility 7

reversal in gel electrophoresis has also been observed by Doane et al. (2010). The value of ρf beyond which the particle velocity changes its direction increases with the rise of the surface potential of the particle. An increase in ρf produces a linear increment in the electrophoretic velocity of the particle. We find from Fig.3(b),(c) that an increases in ζ-potential increases the magnitude of the mobility when the particle moves opposite to the direction of the applied electric field. However, mobility decreases with |ζ| for large ρf when the particle migrates along the direction of the applied electric field (Fig.3b,c). Li and Hill (2013) have provided a physical explanation for having a higher positive mobility of an uncharged particle in a negatively charged hydrogel medium. The increase in the electrolyte concentration (i.e., increase of κs a) leads to an enhanced screening effect, which results in an increment in mobility when the particle translates in the negative direction i.e., μE < 0, it reduces the mobility when the particle is convected along the direction of the imposed electric field by the strong background EOF. This justifies the findings as presented in Fig.3(b) and 3(c) for κs a = 1 and 50, respectively. Our computed results for ζ = −1, −2 (Fig.3b,c) are in close agreement with the solution due to Li et al. (2014) for lower values of the hydrogel fixed charge density. A discrepancy from the analytic solution even at ζ = −1 occurs as the fixed charge density of the hydrogel (ρf ) is raised. We find that our computed solutions are closer to the solution due to Li et al. (2014) when the Debye layer becomes thinner (Fig.3c,d). It may be noted that in the previous studies i.e., Doane et al. (2010), Li and Hill (2013) and Li et al. (2014) have neglected the interaction of background EOF on the ionic concentration. Li et al. (2014) have shown that the ion concentration and charge density perturbation for a particle with low dielectric permittivity vanishes as the Debye layer becomes infinitely thin. This justifies the close agreement of our solution with the solution due to Li et al. (2014) at higher κa as seen in Fig.3(c),(d). We have shown later in this section that the ion convection effect is low for the case of thin Debye layer. The difference between our computed solutions and the results due to Li et al. (2014) is almost negligible for a thin Debye layer (κa = 50) when gel is low permeable i.e., β > 1 (Fig.3e). Li et al. (2014) presented an asymptotic expression for mobility by modifying the asymptotic analysis of Li and Hill (2013) when κa  1 and κ  1. We find that the mobility decreases as the gel becomes more dense. The EOF in the hydrogel medium is proportional to −ρf E0 2 /μ, thus the EOF effect reduces as the permeability of the gel medium is reduced i.e., increase of β. The convective transport of ions is important when the charge density is high and the gel is highly permeable. 200 150

150

3 Pe = 0 Pe > 0

FE

FE

100

100

50

1.5

|ζ|

|ζ|

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-50

0 -50 0

κsa

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ζ = -1 ζ = -2 ζ = -3

2.5

-UE /Λ

200

Pe = 0 Pe > 0

0.5 -100 -25000

ρf (C/m3) (a)

-50000

0

-25000

ρf (C/m3) (b)

-50000

0| 0.01

0.2

0.4

Λ

0.6

0.8

1

(c)

Figure 4: Variation of electric force with ρf when Λ = 0.5 (a) κs a = 1; (b)κs a = 50. (c) Variation of UE /Λ with Λ for for κs a = 1, 50 and ρf = −1000C/m3 . Here β = 1, a = 50nm and ζ = −1, −2, −3.

To illustrate the effect of convection on ion transport, we have compared in Fig.4(a), (b) the electric force with the case in which the convective term is absent in the ion transport equation (i.e., P e = 0). It is evident from these results that the impact of advection on ion transport grows as the fixed charge density of the gel is increased. The advective transport of ions is stronger when the Debye layer is thicker. This clearly indicates that the Poisson-Boltzmann equation, which does not take into account the advection of ions, is no longer sufficient to analyze the electrokinetics with finite Debye length, instead the Nernst-Planck equations coupled with the equations for fluid flow and electric field are to be considered. Li et al. (2014) 8

also indicated the need to compute the Nernst-Planck equations for ion conservation and Poisson equation for electric field to include the nonlinear effects. The double layer deformation at higher range of hydrogel charge density is also illustrated later in this section. The dependence of the electrokinetics due to a variation of electric field from a low to moderate range is illustrated in Fig.4(c) by showing the variation of the ratio UE /E0 (the mobility) as a function of the applied electric field for a thick as well as a thin Debye layer. We find that mobility is uninfluenced by the variation of the imposed electric field for both thick and thin EDL cases. The experimental results, as found by Doane et al. (2010), also shows a linear variation of electrophoretic velocity for a lower range of applied electric field. The applied electric field, in general, is considered below 106 V/m, beyond which the Joule heating effect in gel electrophoresis becomes substantial. The effect of a higher range of applied electric field is illustrated later. 3.2. Size-selectivity and trapping of particles 50 40 -1

-2

ζ= -3

κsa

30 20 10

|

0.1 0

-5000

-10000

ρf (C/m3)

-15000

Figure 5: Critical value of hydrogel fixed charge density as a function of electrolyte concentration at different ζ-potential for which the particle velocity becomes zero. Here β = 1, Λ = 0.5 and a = 50nm at different ζ(= −1, −2, −3).

In Fig.5 we present an estimate for the critical value of the hydrogel fixed charge density as a function of the ionic concentration of electrolytes for which the particle velocity becomes zero in a highly permeable gel medium. We present the critical ρf for different values of ζ-potential. Beyond this critical ρf , the negatively charged particle migrates along the direction of the applied electric field i.e., μE > 0. Here the gel screening length is the same as the particle radius i.e., β = 1. The critical value of ρf becomes higher with the decrease of gel permeability. It is evident from this result that the critical ρf rises with the increase of electrolyte concentration and surface potential of the particle. In Fig.6(a), (b) the size dependency of the mobility is considered for a fixed value of the particle surface charge density (σ). Imposed electric field, which is constant for all the cases presented in Fig.6(a)-(d), is taken to be strong enough to induce a potential drop across the particle in the order of the thermal potential. We have shown later in this section that the particle velocity for the moderate range of imposed electric field is linear with respect to the applied field when the hydrogel charge density is high. In this case, the electrophoretic velocity of the particle reduces with the increase of its size. It may be noted that an increment in particle size at a fixed value of the surface charge density the particle surface potential remain constant. Fig.6(a), (b) shows that the variation of particle mobility with its size is significant when a thick Debye length is considered. Thus, the ionic concentration of the added salt has a role in particle sorting through gel electrophoresis for a hydrogel with a high fixed charge density and a large mesh size. However, the variation of mobility with particle size becomes low (Fig.6b) when the permeability of the hydrogel becomes low i.e., at a low mesh size. Our results for mobility dependence on the particle size shows that the size selectivity through gel electrophoresis is possible for a permeable hydrogel with a high fixed charge density when the ionic concentration of the added electrolyte is low. The particle transport is dominated by the background electroosmotic flow when the fixed charge density and permeability of the hydrogel are high i.e., 2 ρf is high. In this case, the electrophoretic velocity of the 9

10

3 2

8

κs

-1

1

μE

μE

6

κ-1s 4

0 -1

= 10 nm κ-1s = 50 nm

-2 2

-3

σ = -0.01 C/m 2 σ = -0.005 C/m 2

0 5

10

15

20

25

30

35

-4 40

45

50

5

10

σ = 0.01 C/m2 2 σ = 0.005 C/m 15 20 25 30 35

a (nm)

40

45

50

a (nm)

(a)

(b) 4

10 3 9 2 |ζ|

7 6

|ζ|

1 0

5

10

15

20

25

-1

= 40 nm

|ζ|

4 5

= 30 nm

= 50 nm

μE

μE

8

30

35

40

45

-2 5

50

= 10 nm

|ζ| 10

15

20

25

30

a (nm)

a (nm)

(c)

(d)

35

40

45

50

Figure 6: Variation of mobility with particle radius a at ρf = −50000C/m3 when (a)  = 50nm for different σ(= 2 2 −0.005C/m2 , −0.01C/m2 ) and κ−1 s (= 5nm, 10nm, 50nm); (b) for different σ(= −0.005C/m , −0.01C/m ), (= 10nm, 30nm) −1 and κ−1 = 50nm for different ζ(= −1, −2, −3) and (= 40nm, 50nm); (d) κ−1 = 50nm for s (= 5nm, 10nm, 50nm); (c) κs s different ζ(= −1, −2, −3) and (= 10nm, 30nm).

particle becomes strongly size-dependent as the hydrodynamic drag as well as the electric force experienced by the particle varies with the size of the particle. Fig.6(c), (d) shows the dependence of particle mobility on its size at different values of the surface potential when the fixed charge density of the hydrogel is high. The mobility is scaled by εe φ0 /μ, thus a slight change in the scaled mobility produces a large change in the electrophoretic velocity. The study of Li and Hill (2013) shows that the separation of populations is favoured in low ionic strength of the added electrolyte. We have considered the electrolyte concentration in the lower range (κ−1 s = 50nm). We present results for different values of the hydrogel screening length. It is evident from these results that the size dependency of the mobility is significant only when the hydrogel screening length is close to the maximum size of the particle. The size dependency of the mobility for low ζ-potential is less significant when lower values of the hydrodynamic penetration length of the gel media is considered (Fig.6d). However, Fig.6(d) shows that at large ζ-potential (ζ = −3), an efficient size selectivity is possible for low values of hydrogel mesh size i.e.,  = 10nm. Li and Hill (2013) have also made a similar observation. 3.3. Force, Streamline and Charge density The dependence of the scaled electric force (or drag), scaled by εe φ20 , experienced by the particle in electrophoresis on different parameters is illustrated in Fig.7(a)-(c). It may be noted that the computed values of the forces acting on the particle are equal in magnitude for the particle in electrophoresis. Fig.7(a) shows that the dependence of electric force on κs a as well as ζ-potential are similar as that of the particle mobility. The increase in ionic concentration of the electrolyte (or surface charge density of the particle) creates a stronger Columbic force due to a stronger shielding effect. At higher values of the hydrogel fixed 10

600

250

ζ = -1 ζ = -2 ζ = -3

|ζ| 500

200

400 150

200 150

300

100

200

FE

FE

50

FE

ρ f = -1000 C/m3 ρ f = -10000 C/m3 ρ f = -50000 C/m3

100

κ-1 s

100

|ζ|

0

0

0

-100 -50 -100 | 0.1

50

-50

-200 10

20

κsa

30

40

-300 0.5

50

2

4

(a)

6

β

8

-100 0

10

-25000

ρf (C/m3)

(b)

-50000

(c)

Figure 7: Variation of electric force when a = 50nm, Λ = 0.5 with (a) κs a when β = 1 for different values of ρf (= −1000C/m3 , −10000C/m3 , −50000C/m3 ); (b) β when ρf = −50000C/m3 for different κ−1 s (= 1nm, 5nm, 50nm); (c) ρf when β = 1 and κ−1 s = 1nm. Results are presented for different values of ζ(= −1, −2, −3).

charge density, the electroosmotic flow exerts a force on the particle whose magnitude is proportional to the mobile charge density. It is evident from Fig.7(b) that the electric force depends on the permeability of the hydrogel medium. For the case of low permeable hydrogel (large β), the effect of the background EOF even for large values of hydrogel fixed charge density becomes so small that the electric force FE remains negative for a negatively charged particle with |ζ| ≤ 3. Fig.7(c) shows that the force increases linearly with the increase of fixed charge density of the hydrogel and reduces with the rise of the particle surface potential. 3.5

1.3 |ζ|

3 2.5

Ω

Ω

1.2

2

|ζ|

1.5

1.1

|

0.1

10

20

κsa

30

40

1 -20000

50

(a)

-30000

-40000

ρf (C/m ) 3

-50000

(b)

Figure 8: Variation of the ratio of the drag forces on a charged and uncharged particle of migrating at the same velocity UE when a = 50nm, β = 1, Λ = 0.5 with (a) κs a at ρf = −50000C/m3 for different values of ζ(= −1, −2, −3); (b) ρf at κ−1 s = 1nm for different values of ζ(= −1, −2, −3).

The hinderance produced by the electric double layer around the charged particle is illustrated in Fig.8(a), (b) by presenting the drag factor Ω. The factor Ω is the ratio between the drag of a charged and uncharged particle moving at the same speed. The Stokes drag for an uncharged particle in a porous medium can U = 6πμU a(1 + β + β 2 /9). We have determined the drag factor Ω when the particle be determined by FD mobility is positive. The hinderance effect grows with the increase of the ζ-potential and ion concentration. The shielding effect grows with the increase of particle charge density and/or increase of electrolyte ionic strength. The non-zero charge density in the medium produces an electrohydrodynamical pressure field. However, the drag factor reduces as the fixed charge density of the hydrogel is increased. Fig.8(b) shows that the drag factor tends to 1 when the fixed charge density becomes sufficiently large. Thus, the Debye layer effect on the particle hydrodynamics becomes negligible when the background EOF becomes strong. The streamline patterns and distribution of mobile net charge density (n1 − n2 ) around the particle are 11

(a)

(b)

(c)

Figure 9: Streamlines and net mobile charge density ρe around the charged particle when ζ = −1, β = 1, Λ = 0.5, κs a = 1 and a = 50nm for different ρf . (a) ρf = 0; (b) ρf = −10000C/m3 ; (c) ρf = −50000C/m3 .

shown in Fig.9(a)-(c) for different values of the gel fixed charge density including an uncharged gel. The gel screening length is considered to be large and the Debye layer is thicker. When ρf = 0, the negatively charged particle moves along the negative z-axis whereas, it moves at a higher velocity along the positive z-axis for large values of ρf . The Debye layer is concentric to the particle for an uncharged or weakly charged gel. However, the Debye layer is deformed for high ρf . The double layer polarization at this high value of hydrogel fixed charge density is evident. The double layer polarization is more important when Debye layer is thick. At thick Debye layer, the fluid motion is strong enough to overcome the ion diffusion and drag the mobile counterions downstream of the particle. 3.4. Nonlinear dependence of velocity on imposed electric field 2

10

ρf = 0 ρf = -250 C/m3

1.5

10

8 |ζ|

8

κ-1 s

1

UE

7

UE

-UE

ζ = -1 ζ = -2 ζ = -3

9

6 5

4

4

0.5

2

3 0 0.5

6

0.75

1

Λ (a)

1.25

1.5

2 0.5

0.75

1

Λ (b)

1.25

1.5

0 0.5

0.75

1

Λ

1.25

1.5

(c)

Figure 10: Variation of electrophoretic velocity with non-dimensional electric field Λ for a = 100nm (a) when  = 100nm and κ−1 = 100nm for different ζ(= −1, −2, −3) and ρf (= 0, −250C/m3 ); (b) when  = 100nm and ρf = −12500C/m3 for s −1 = 100nm, ρf = −12500C/m3 for different different ζ(= −1, −2, −3) and κ−1 s (= 2nm, 10nm, 100nm); (c) when ζ = −1, κs (= 10nm, 30nm, 50nm, 70nm, 100nm).

Most of the experimental and theoretical studies on electrophoresis are considered for that range of the external electric field in which the mobility is independent of the magnitude of the external electric field. Recently, Shih and Yamamoto (2014) showed a nonlinear response on electrophoresis when the electric field exceeds 105 V /m. We now address the situation where the gel electrophoresis may not vary linearly with the applied electric field. Large applied electric fields are commonly encountered in lab-on-a-chip technologies (Figliuzzi et al., 2014). We have considered the applied electric field to be strong enough to create a potential drop across the particle bigger than the thermal potential i.e., 2aE0 ≥ φ0 or Λ = E0 a/φ0 ≥ 0.5. Our results (Fig.10a-c) show that the velocity varies nonlinearly with the imposed electric field for Λ > 0.5 when the 12

hydrogel is uncharged or weakly charged. The nonlinear variation of UE with Λ is prominent as the particle ζ-potential is increased. Nonlinearity is also prominent for a lower range of Debye length. Particle velocity varies linearly with the applied field when the fixed charge density of the hydrogel is large. At higher values of the fixed charge density (ρf ), the electrokinetics of the particle is dominated by the background electroosmotic flow and the EOF varies linearly with the applied electric field. For this, a linear variation of UE with Λ is found when the fixed charge density and permeability of the hydrogel is high. 10

10

8 ζ = -1 ζ = -2 ζ = -3

4

μE , μSE

S

μE , μE

8 6 β

2

6

κ-1s

4

0 2 -2

|

0.1

10

20

κsa

30

40

-10000

50

(a)

-20000

-30000

ρf (C/m ) 3

-40000

-50000

(b)

Figure 11: Variation of mobility with (a) κs a when Λ = 0.5, ρf = −50000 for different β(= 1, 3) and ζ(= −1, −2, −3); (b) ρf when ζ = −1, β = 1, Λ = 0.5, a = 50nm for different κ−1 s (= 1nm, 10nm, 50nm). Here solid lines are computed mobility and dashed-symbols lines (or symbols) are superposed (μS E = μE |ζ=0 − Qf ) mobility values.

We now attempt to estimate the mobility of the particle embedded in a charged hydrogel medium through a linear superposition of particle mobility in an uncharged hydrogel medium (ρf = 0) with the electroosmotic mobility of the hydrogel (−Qf ). In analyzing the gel electrophoresis in a charged hydrogel medium, Doane et al. (2010) determined the electric force on the charged particle under the Debye-Huckel approximation by linear superposition of the electric force on an uncharged particle embedded in charged gel medium and the electric force on a charged particle in an uncharged gel. The limitation of the present linear superposition technique is illustrated in Fig.11(a),(b). As expected, the present superposition technique is proper when the electrophoresis is dominated by the background EOF i.e., the hydrogel charge density (ρf ) and the mesh size of the gel is high. However, Fig.11(a),(b) shows that the electrophoretic mobility based on the linear superposition principle underestimate the computed mobility values for thick Debye layer with small β when large ρf is considered. When the hydrogel charge density is high, the strong background EOF drags the negatively charged particle to migrate along the direction of the applied electric field. The counterions in the double layer surrounding the particle are dragged downstream by the motion of the particle relative to the medium. The formation of counterion plume along the negative z-axis at high value of the hydrogel charge density is evident from Fig.9(c). This creates an induced electric field which acts along the direction of the imposed field. This induced electric field due to the deformation of the double layer is strong when the double layer is thick and the hydrogel is highly permeable. The superposition technique does not take into account this interaction of the background EOF on double layer of the particle. This might be the reason for the electrophoretic mobility based on the linear superposition principle to underestimate the computed mobility for thick Debye layer with small β as seen in Fig.11(a),(b). For a low permeable hydrogel (β > 1), the superposition principle does not produce a correct estimate of the mobility as the effect of electroosmosis on the particle electrokinetics is low. For a dense hydrogel, the diffusion dominated shielding effect becomes strong, which produces hindrance on the electrophoresis of the particle. This phenomena has not been taken into account in this linear superposition technique.

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4. Conclusion We have made a numerical study on electrophoresis of a charged colloid through a charged hydrogel medium by solving the governing equations based on the first principle of electrokinetics. Thus, no restriction on Debye length, ζ-potential, hydrogel charge density or imposed external electric field is invoked. The agreement of our computed solutions for mobility with the existing analytic solution due to Li et al. (2014), based on the Debye-H¨ uckel approximation, for lower range of particle ζ-potential and hydrogel fixed charge density is encouraging. However, the solution for mobility due to Li et al. (2014) over-estimates our computed solution when the convective transport of ions and double layer polarization becomes strong. Present results correctly takes into account the Debye layer polarization by convection and electromigration of ions, relaxation by the molecular diffusion of ions and their interactions with the background EOF. The main findings of our study can be highlighted as follows: 1. Electrophoresis varies linearly with the applied electric field for moderate range of applied electric field. However, a nonlinear variation of the electrophoretic velocity with the imposed electric field is found when the hydrogel is weakly charged under an applied electric field which is strong enough to create a potential drop across the particle larger than the thermal potential. 2. The effect of convection on the ion distribution for a charged hydrogel is not negligible when the gel is highly permeable. The ion convection around the particle suspended in a gel with low friction coefficient grows with the rise of the particle ζ-potential when the Debye length is in the order of the particle size. The shielding effect becomes strong for a dense gel medium, higher electrolyte concentration or high values of ζ-potential. In those cases, the hindrance effect on the electrophoresis of the particle grows. For higher values of the gel fixed charge density and gel mesh size, the electrophoresis is dominated by the background EOF and the flow around the the particle can be described through the ordinary Stokes flow. 3. We find that a size selectivity through gel electrophoresis is possible when the background electroosmotic flow is strong and the ionic concentration of the added electrolyte is low. The mobility dependence on the particle size is more prominent for higher values of the particle charge density. 4. Our numerical solutions show that a linear superposition of the electrophoretic mobility for the uncharged gel medium with the electroosmotic mobility of the charged gel medium produces an accurate estimate of the electrophoretic mobility of the colloid in charged hydrogel when the fixed charge density of the hydrogel is strong enough to induce a background EOF which drags the negatively charged particle along the direction of the EOF. In the present analysis, the steric interactions with the gel skeleton and the effect of dielectric permittivity of the particle are not considered. References Allison, S. A., Li, F., Hill, R. J., 2014. The electrophoretic mobility of a weakly charged soft sphere in a charged hydrogel: Application of the lorentz reciprocal theorem. J. Phys. Chem. B 118 (29), 8827–8838. Allison, S. A., Xin, Y., Pei, H., 2007. Electrophoresis of spheres with uniform zeta potential in a gel modeled as an effective medium. J. Colloid Interface Sci. 313 (1), 328–337 [Corrigendum 2008. J. Colloid Interface Sci. 325 (1), 296.]. Bhattacharyya, S., De, S., Gopmandal, P. P., 2014. Electrophoresis of a colloidal particle embedded in electrolyte saturated porous media. Chem. Eng. Sci. 118, 184–191. Brady, J. F., 1994. The long-time self-diffusivity in concentrated colloidal dispersions. J. Fluid Mech. 272, 109–134. Doane, T. L., Cheng, Y., Babar, A., Hill, R. J., Burda, C., 2010. Electrophoretic mobilities of pegylated gold nps. J. Am. Chem. Soc. 132 (44), 15624–15631. Figliuzzi, B., Chan, W. H. R., Moran, J., Buie, C. R., 2014. Nonlinear electrophoresis of ideally polarizable particles. Physics of Fluids (1994-present) 26 (10), 102002. G¨ org, A., 1994. Gel electrophoresis: Proteins: Edited by M. J. Dunn, Bios Scientific Publishers, Oxford 1993. FEBS Letters 344 (23), 266–270. Hanauer, M., Pierrat, S., Zins, I., Lotz, A., S¨ onnichsen, C., 2007. Separation of nanoparticles by gel electrophoresis according to size and shape. Nano Lett. 7 (9), 2881–2885. Hill, R. J., 2006. Transport in polymer-gel composites: Response to a bulk concentration gradient. J. Chem. Phys 124 (1), 014901.

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Hsu, J.-P., Huang, C.-H., Tseng, S., 2012. Gel electrophoresis: Importance of concentration-dependent permittivity and doublelayer polarization. Chem. Eng. Sci. 84, 574–579. Hsu, J.-P., Huang, C.-H., Tseng, S., 2013. Gel electrophoresis of a charge-regulated, bi-functional particle. Electrophoresis 34 (5), 785–791. Hunter, R. J., 2001. Foundations of colloid science. Oxford University Press, New York. Johansson, L., L¨ ofroth, J.-E., 1993. Diffusion and interaction in gels and solutions. 4. hard sphere brownian dynamics simulations. J. Chem. Phys 98 (9), 7471–7479. Johnson, E. M., Berk, D. A., Jain, R. K., Deen, W. M., 1996. Hindered diffusion in agarose gels: test of effective medium model. Biophys. J. 70 (2), 1017–1023. Jones, E. H., Reynolds, D. A., Wood, A. L., Thomas, D. G., 2011. Use of electrophoresis for transporting nano-iron in porous media. Groundwater 49 (2), 172–183. Li, F., Allison, S. A., Hill, R. J., 2014. Nanoparticle gel electrophoresis: Soft spheres in polyelectrolyte hydrogels under the debye–h¨ uckel approximation. J. Colloid Interface Sci. 423, 129–142. Li, F., Hill, R. J., 2013. Nanoparticle gel electrophoresis: Bare charged spheres in polyelectrolyte hydrogels. J. Colloid Interface Sci. 394, 1–12. Liu, Y.-W., Pennathur, S., Meinhart, C. D., 2014. Electrophoretic mobility of a spherical nanoparticle in a nanochannel. Physics of Fluids (1994-present) 26 (11), 112002. Mohammadi, A., Hill, R. J., 2010. Steady electrical and micro-rheological response functions for uncharged colloidal inclusions in polyelectrolyte hydrogels. Proc. R. Soc. A 466 (2113), 213–235. O’Brien, R. W., White, L. R., 1978. Electrophoretic mobility of a spherical colloidal particle. Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics 74, 1607–1626. Ohshima, H., 2001. Approximate analytic expression for the electrophoretic mobility of a spherical colloidal particle. Journal of colloid and interface science 239 (2), 587–590. Park, S., Hamad-Schifferli, K., 2008. Evaluation of hydrodynamic size and zeta-potential of surface-modified au nanoparticledna conjugates via ferguson analysis. The Journal of Physical Chemistry C 112 (20), 7611–7616. Saville, D., 1977. Electrokinetic effects with small particles. Annu. Rev. Fluid Mech. 9 (1), 321–337. Shih, C., Yamamoto, R., 2014. Dynamic electrophoresis of charged colloids in an oscillating electric field. Phys. Rev. E 89 (6), 062317. Tsai, P., Huang, C.-H., Lee, E., 2011. Electrophoresis of a charged colloidal particle in porous media: boundary effect of a solid plane. Langmuir 27 (22), 13481–13488. Tsai, P., Lee, E., 2011. Gel electrophoresis in suspensions of charged spherical particles. Soft Matter 7 (12), 5789–5798. Wiersema, P., Loeb, A., Overbeek, J. T. G., 1966. Calculation of the electrophoretic mobility of a spherical colloid particle. Journal of Colloid and Interface Science 22 (1), 78–99. Yoshioka, H., Mori, Y., Shimizu, M., 2003. Separation and recovery of dna fragments by electrophoresis through a thermoreversible hydrogel composed of poly (ethylene oxide) and poly (propylene oxide). Anal. Biochem. 323 (2), 218–223.

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