Nanoparticle gel electrophoresis: Soft spheres in polyelectrolyte hydrogels under the Debye–Hückel approximation

Journal of Colloid and Interface Science 423 (2014) 129–142

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Nanoparticle gel electrophoresis: Soft spheres in polyelectrolyte hydrogels under the Debye–Hückel approximation Fei Li a, Stuart A. Allison b, Reghan J. Hill a,⇑ a b

Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 0C5, Canada Department of Chemistry, Georgia State University, Atlanta, GA 30302-3965, United States

a r t i c l e

i n f o

Article history: Received 15 December 2013 Accepted 9 February 2014 Available online 4 March 2014 Keywords: Soft nanoparticles Gel electrophoresis Gel electrophoretic mobility Electrokinetic theory

a b s t r a c t A mathematical model for electrophoresis of polyelectrolyte coated nanoparticles (soft spheres) in polyelectrolyte hydrogels is proposed, and evaluated by comparison to literature models for bare-sphere gel electrophoresis and free-solution electrophoresis. The utilities of approximations based on the bare-particle electrophoretic mobility, free-solution mobility, and electroosmotic flow in hydrogels are explored. Noteworthy are the influences of the particle–core dielectric constant and overlap of the polyelectrolyte shell. The present theory, which neglects ion-concentration and charge-density perturbations, indicates that the gel electrophoretic mobilities of metallic-core nanoparticles in polyelectrolyte gels can be qualitatively different than for their non-metallic counterparts. These insights will be beneficial for interpreting nanoparticle gel-electrophoresis data, optimizing electrophoretic separations, and engineering nanoparticles for technological applications. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Nanoparticles are ideal elements for constructing nanostructured materials and devices with adjustable physical and chemical properties [1,2]. In practice, their surfaces are not smooth and impenetrable [3,4], but often have ion-penetrable porous shells, such as polyelectrolytes, that modify particle shape and size, prevent agglomeration, and impart chemical functionality [5–9]. These porous shells are widely modelled as uniformly distributed resistance centers [10,11] or Brinkman porous media [12,13] with uniformly distributed charge. Due to the frictional force exerted by polyelectrolyte segments on the fluid, the electrophoretic behavior of these ‘soft’ colloidal nanoparticles is different from their bare counterparts. Gel electrophoresis is widely used to separate biomolecules [14–16] and nanoparticles [17–19] based on shape, size, and surface charge. It is also used to study nanoparticle mobility, which reflects particle physicochemical properties [18,20,21]. As suggested by Morris [22], the separation mechanism is the sieving effect of gel pores [19,23] and particle charge. By interpreting the electrophoresis of proteins in starch and polyacrylamide gels, Ferguson [24] and Morris [22] proposed an empirical relationship describing the linear dependence of the logarithm of mobility on gel

⇑ Corresponding author. Fax: +1 514 398 6678. E-mail address: [email protected] (R.J. Hill). http://dx.doi.org/10.1016/j.jcis.2014.02.010 0021-9797/Ó 2014 Elsevier Inc. All rights reserved.

concentration. Using the theory of Ogston [25] for random fiber networks, Rodbard and Chrambach [26] applied the Ferguson relationship to estimate protein size, molecular weight, and free solution mobility. Though generally in agreement with experiments, this relationship considers little of particle intrinsic properties, such as the charging mechanism and surface ligands. Studies of quantum dots [27–29] and nanoparticles [30–33] indicate that these properties impact the gel electrophoretic mobility. When the gel concentration is very low (e.g., 0.2%), freesolution electrophoresis theories have been adapted for gels [18]. Allison et al. [34] solved the problem of a rigid particle with arbitrary uniform surface charge translating in uncharged hydrogels, regarding the gel as an effective medium, and Tsai and Lee [35] applied Kuwabara’s unit cell model for concentrated particle dispersions in uncharged porous media, also addressing polarization and overlap of the diffuse double layers. However, many hydrogels, e.g., agarose, are charged, and even ideally uncharged hydrogels, such as polyacrylamide, become weakly charged due to chemical reactions [36–38]. Electrolyte- and buffer-ion adsorption is another source of fixed charge. Note that agarose has been adopted for nanoparticle gel electrophoresis because of its favorable permeability [39]. In some studies, the skeleton behaves as if it bears zero charge, whereas in others it presents ostensible electroosmotic flow, indicating a finite charge. Doane et al. [40] addressed soft-nanoparticle electrophoresis in charged hydrogels with the assumption that the electrostatic potential is weak and that the Debye length is large compared to

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the particle radius and gel mesh size. In their theory, a soft sphere is approximated as a bare sphere with a radius equal to the hydrodynamic radius of the soft sphere, and an effective surface potential ascertained from the core surface potential [41]. Considering the electro-osmotic flow induced by the hydrogel charge, the closed-form solution of Doane et al. [40] furnished a remarkably satisfactory quantitative interpretation of experimental data for PEGylated gold nanoparticles (Au NPs). More recently, Li and Hill [42] obtained numerically exact solutions of this model for bare spheres in charged gels, identifying regions where various independent asymptotic approximations are valid. No theory has yet addressed the approximations of Doane et al. [40] applied to the electrophoresis of soft spheres in charged gels. The interactions between ‘soft’ colloidal nanoparticles and polyelectrolyte hydrogels are complex. Firstly, hydrogels and polyelectrolyte shells are not homogeneous. For example, from electron micrographs of agarose gels, fibers can lay randomly, and the pores are heterogeneous [43,44]. Hill and Saville [45] examined the influence of polyelectrolyte layer uniformity on particle mobility, and found that the mobilities of spheres with uniform step-like and non-uniform Gaussian-like layers are similar when the layers have the same hydrodynamic thickness. Recently, similar results were reported by Ohshima [46] when applying a soft step function to model an inhomogeneous polyelectrolyte segment distribution; and Chou et al. [47] examined the influence of polyelectrolyte layers with radially varying fixed-charge and hydrodynamic resistance. Next, when the particle size is comparable to the gel mesh size, steric forces are expected to become as important as the primary hydrodynamic and electrical forces. Drawing an analogy to hindered diffusion in agarose gels [48–50], Allison et al. [34] introduced a gel steric factor Sð/Þ to capture the influences of steric interactions on gel electrophoresis. This approach was recently adopted by Hsu et al. [51] to apply free-solution electrophoresis theory to gel electrophoresis. In a different approach, Doane et al. [40] introduced a frictional coupling force to model the steric interaction between the particle and the hydrogel skeleton, successfully quantifying the influences of gel concentration and particle size on measured gel-electrophoretic mobilities. Note that particle shell and hydrogel charge densities may vary with ionic strength and pH; for example, Fatin-Rouge et al. [52] computed the charge density of agarose gels as a function of pH, finding that it reaches a maximum when pH > 3:5. Gel charge density has been estimated from the Donnan potential using glass-micropipette electrodes [53–55]. Finally, we note that buffer depletion and temperatures changes may be important for gel electrophoresis undertaken without buffer recirculation and heat exchange; in these cases, parameters that are often taken to be constants, such as permittivity and viscosity, become gel-concentration dependent [51]. In this paper, we develop a soft-sphere gel electrophoresis theory that combines the conventional free-solution electrophoresis and bare-sphere gel electrophoresis. This theory predicts electrophoretic mobilities of polymer-coated (uncharged or charged) particles in charged and uncharged hydrogels. In addition to providing fundamental insights into how nanoparticle and gel characteristics impact gel electrophoretic mobility, the soft-sphere theory also provides strategies for nanoparticle size and charge selection, and predicting nanoparticle and gel properties.

with radius a, a surface charge density r, and a dielectric constant p . The core is coated with a layer of ion-penetrable polyelectrolyte with thickness L. Within and around the charged core is a diffuse layer of mobile counterions. The suspending medium comprises a uniformly charged polymer skeleton, solvent (water with dielectric constant s ), and mobile ions (counterions of the hydrogel skeleton, and ions from the added z–z electrolyte). Both the shell (region 1) and hydrogel (region 2) are modelled as uniform porous/Brinkman media with characteristic Brinkman screening lengths ‘1 and ‘2 [12], and fixed-charge densities qf ;1 and qf ;2 . Numerous approximations are adopted in the following theoretical development. Among these are that particle core, coating, and gel are uniformly charged, furnishing an electrostatic potential jwj K kB T=e (kB T is the thermal energy and e is the fundamental charge), which motivates the well-known Debye–Hückel approximation and a neglect of ion-concentration and charge-density perturbations. Following O’Brien and White [56] and many others, we solve the problem by superposing two sub-problems: a U-problem where the soft particle translates in the porous medium in the absence of an electric field, and an E-problem where the particle is stationary while subjected to the electric field. In the usual manner, these are superposed to satisfy the particle equation of motion. 2.2. U-problem Hill and Li [57] studied the translation of a soft particle in porous media based on the Brinkman model [12]. Neglecting electrical body forces, fluid dynamics in the ith (i ¼ 1; 2) region are furnished by solving Brinkman’s equations 2 0 ¼ gr2 ui  $pi  g‘2 i ðui  V i Þ  g‘2 ui xi dol and $  ui ¼ 0:

ð1Þ

Here, g is the fluid shear viscosity, pi are the pressures, ui are the solvent velocities, V i are constant velocities that can be set for a stationary shell (V 1 ¼ V 2 ¼ 0) or a shell that translates with the particle core as a rigid body (V 1 ¼ U and V 2 ¼ 0), x1 ¼ 1 and x2 ¼ 0, and dol ¼ 1 for a completely overlapping shell and hydrogel, or dol ¼ 0 for a non-overlapping shell and hydrogel. Hill and Li [57] showed that

ui ¼ Ai ðrÞU þ Bi ðrÞU  er er ;

ð2Þ

pi ¼ C i ðrÞU  er ;

ð3Þ

where h i h i ð1Þ ð1Þ ð2Þ ð2Þ Ai ¼ ai h1 ðiki rÞr 1 þ h1;r ðiki rÞ þ bi h1 ðiki rÞr 1 þ h1;r ðiki rÞ  ki r 3 þ ci ;

ð4Þ

h i h i ð1Þ ð1Þ ð2Þ ð2Þ Bi ¼ ai h1 ðiki rÞr 1  h1;r ðiki rÞ þ bi h1 ðiki rÞr 1  h1;r ðiki rÞ þ 3ki r 3 ;

ð5Þ

2

2

C i ¼ gki ki r 2  gki ci r þ gj2i ðV i =UÞr;

ð6Þ

and

h1 ðiki rÞ ¼ ie

ki r

ð2Þ h1 ðiki rÞ

ki r

ð1Þ

Here,

h

1

ðki rÞ

2

þ ðki rÞ

i

ð7Þ

;

h i 1 2 ¼ ie ðki rÞ  ðki rÞ :

ji ¼ ‘1 i ; ki ¼

ð8Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2i þ j22 xi dol ; h1ð1Þ ðiki rÞ and hð2Þ 1 ðiki rÞ are spherið1Þ

cal Hankel functions of the first and second kind, and h1;r ðiki rÞ and 2. Materials and methods 2.1. Theory We consider a spherical core–shell particle translating under an electric field E with a constant velocity U in an electrolyte-saturated hydrogel. The soft particle has a non-conducting rigid core

ð2Þ h1;r ðiki rÞ

are their radial derivatives. Note that the constants V i in the pressure have been introduced to permit the Brinkman medium in the shell with permeability ‘21 to translate with the core (V 1 ¼ U and V 2 ¼ 0). The tractions are

    t i ¼ T i  er ¼ g Bi r1 þ Ai;r U þ g 2Bi;r  Bi r 1 þ Ai;r  g1 C i U  er er ;

ð9Þ

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h

where the Newtonian stress tensors T i ¼ pi I þ g $ui þ ð$ui ÞT

i

with I the identity tensor. Constants k1 ; k2 ; a1 ; a2 ; b1 and c1 are calculated from the boundary conditions (see Appendix A) using a standard matrix inversion function in Matlab. The hydrodynamic force on the soft particle is

F Uh ¼

Z

t 1 dA þ

Z

b

r¼a

r¼a

gj21 ðu1  V 1 ÞdV ¼ F Uh;n ð1  dol Þ þ F Uh;o dol ;

ð10Þ

r2 w0i ¼ j2 w0i 

qf ;i þ qf ;2 ðxi dol  1Þ

0 s

ð17Þ

;

where the squared-reciprocal Debye length

which becomes

F Uh;n

0

zj ewi =ðkB TÞ The mobile ions are Boltzmann distributed, n0j;i ¼ n1 , j e 1 where nj are the bulk ion concentrations. In this paper, the body forces on the fluid are calculated according to the equilibrium body forces furnished by linearized Poisson–Boltzmann equations (jw0 j K kB T=e)



2 1 2 ð1Þ ¼ 4pgU a2 j22 b h1 ðij2 bÞ  k2 j22 3 3

 ð11Þ

j2 ¼

zf ;2 qf ;2 e 2Ie2 ¼ j2s þ 0 s kB T 0 s kB T

ð18Þ

with ionic strength with a non-overlapping shell (dol ¼ 0), and

F Uh;o

  1 ¼ 4pgU k2 j22 þ j22 a3 3

ð12Þ

with a completely overlapping shell (dol ¼ 1). Because the electrical forces are neglected, the total force on the soft particle is the sum of the hydrodynamic force F Uh and a frictional coupling force F Uf due to steric interaction with the hydrogel skeleton [40]:

F U ¼ F Uh þ F Uf ¼ F Uh;n ð1  dol Þ þ F Uh;o dol  cf U;

ð13Þ

2

where cf ¼ 2pb ng kB Tu1 is a steric friction coefficient with ng the f hydrogel segment number density and uf a frictional-coupling velocity parameter [40]. We verified the analytical solution of Hill and Li [57] by solving the U-problem numerically. 2.3. E-problem The E-problem requires the solutions of an equilibrium state (identified by superscripts 0) and a perturbed state (identified by superscripts 0 ) from the application of a uniform electric field. For a weakly charged particle with a surface potential jfj K 2kB T=e, perturbations to the equilibrium mobile-ion concentrations and fixed-charge densities can be neglected, so nj 

n0j ;

0

0

q  q ;w ¼ w

þ w0 ; p ¼ p0 þ p0 , and u ¼ u0 . The fluxes of the jth (j ¼ 1; 2; 3) mobile-ion in the ith region are given by the Nernst-Planck relationships with steady conservation equations 0

jj;i ¼ Dj $n0j;i  Dj

zj en0j;i kB T

$w0i þ n0j;i u0i and $  j0j;i ¼ 0:

ð14Þ

Here, Dj ¼ kB T=ð6pgaj Þ; n0j;i ; zj and aj are the jth mobile-ion diffusion coefficients, equilibrium concentrations (in the ith region), valences, and Stokes radii obtained from limiting conductances or diffusivities. The electrostatic potentials w0i satisfy Poisson equations

0 s r2 w0i ¼ q0m;i þ qf ;i þ qf ;2 xi dol ;

and $  u0i ¼ 0:

3 zf ;2 qf ;2 1X z2 n1 ¼ Is þ : 2 j¼1 j j 2e

ð19Þ

Note that js is the reciprocal Debye length based on the ionic strength of the added salt Is , whereas j is based on the total ionic strength I, which includes the hydrogel counterions. Accordingly, the equilibrium electrostatic potentials are

ejr ejr qf ;1 þ qf ;2 ðdol  1Þ þ D2 þ ; jr jr 0 s j2 ejr w02 ¼ D3 : jr

w01 ¼ D1

ð20Þ ð21Þ

The constants D1 ; D2 and D3 are computed to satisfy the boundary conditions using a standard matrix inversion function in Matlab. Applying an electric field, the perturbed electrostatic potentials satisfy Laplace equations (recall, n0j ¼ q0 ¼ 0)

r2 w0i ¼ 0;

ð22Þ

and the fluid velocities and pressures satisfy momentum and mass conservation equations 0 2 0 0 0 0 ¼ gr2 u0i  $p0i  g‘2 i ui  g‘2 ui xi dol  qm;i $wi and $ ui ¼ 0

ð23Þ

with boundary conditions (no-slip and continuity conditions) that are detailed in Appendix A. The solution of Eq. (22) is

 a 3  E  r; w0i ¼  1 þ c r where

the

dielectric-contrast

ð24Þ (dipole

strength)

parameter

c ¼ ðs  p Þ=ðp þ 2s Þ spans the range from 1 for high-dielectric constant (metallic) particles to 0:5 for low-dielectric constant particles. From linearity and symmetry considerations for incompressible flow [58],



  u0i ¼  hi;r þ hi r 1 þ qf ;2 ‘22 =g E þ hi;r  hi r 1 E  er er ;

ð25Þ

ð15Þ where

where 0 is the permittivity of a vacuum, and nf ;i are the polyelectrolyte shell and hydrogel fixed-charge concentrations with valences zf ;i . The terms on the right-hand side are the net mobileP ion charge densities q0m;i ¼ j¼1;2;3 zj en0j;i and the fixed-charge charge densities qf ;i þ qf ;2 xi dol ¼ zf ;i enf ;i þ zf ;2 enf ;2 xi dol (xi and dol have the same definitions as in the U-problem). Fluid dynamics are governed by momentum and mass conservation equations (incompressible fluid)

0 ¼ $p0i  q0m;i $w0i

I¼

ð16Þ

The boundary conditions (flux and continuity conditions, detailed in Appendix A) allow for either a constant surface charge density or a constant surface potential.

h2 ðrÞ ! C E r 2 as r ! 1;

ð26Þ

giving

u02 ! 2C E r 3 E  er er þ C E r3 E  eh eh þ U 1 as r ! 1:

ð27Þ

Note that hi;r ðrÞ; hi;rr ðrÞ; hi;rrr ðrÞ denote radial derivatives, and C E ¼ limr!1 h2 ðrÞr 2 is termed the strength of the Brinkmanlet. Eq. (23) is solved by taking the curl with Eq. (25), furnishing



2 2 2 h1;rrrr þ 4h1;rrr r 1  k1 þ 4r 2 h1;rr  2k1 r 1 h1;r þ 2k1 r2 h1   þ g1 j0 s r 2 ½D1 ðjr þ 1Þejr þ D2 ðjr  1Þejr  ca3 r 3 þ 1 ¼ 0 when a < r < b;

ð28Þ

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F. Li et al. / Journal of Colloid and Interface Science 423 (2014) 129–142

and



2 2 2 h2;rrrr þ 4h2;rrr r 1  k2 þ 4r 2 h2;rr  2k2 r1 h2;r þ 2k2 r 2 h2   þ g1 j0 s D3 r2 ðjr þ 1Þejr ca3 r 3 þ 1 ¼ 0 when r > b

ð29Þ

with boundary conditions that are detailed in Appendix A. Eqs. (28) and (29) are solved for hi ðrÞ (in dimensionless form) using the Matlab function bvp with the relative error tolerance set to 106 . The total force on the stationary particle is the sum of the hydrodynamic force F h;E and the electrical force F e;E on the core surface (r ¼ a), and the hydrodynamic and electrical forces acting on the shell:

F E ¼ F h;E þ F e;E þ

Z

b r¼a



qf ;1 $w1 þ g‘2 1 u1 dV

¼ F En ð1  dol Þ þ F Eo dol ;

ð30Þ

where it can be shown that

8 4 2 3 pg‘2 2 b h2 ðr ¼ bÞE þ pcqf ;2 a E; 3 3 4 E 3 F Eo ¼ 4pg‘2 2 C E þ pð1 þ cÞqf ;2 a E 3

E F En ¼ 4pg‘2 2 C E

ð31Þ

Fig. 1. The contribution of ion-concentration and charge-density perturbations to the electrical force (see Eq. (35)) according to the theory of Mohammadi and Hill [38] for uncharged spheres in incompressible, uniformly charged hydrogels: ja ¼ 1; 10; 100, and 1000 (increasing upward). This vanishes for low and highdielectric constant particles, and is otherwise small for the nanoparticles of interest in gel electrophoresis with ja K 1.

ð32Þ

are for non-overlapping (dol ¼ 0) and completely overlapping (dol ¼ 1) shells, respectively; and C E and h2 ðr ¼ bÞ are from the numerical solutions of Eqs. (28) and (29). Note that the second term in Eq. (32) corrects an omission in the force calculated by Li and Hill [42] for bare particle gel electrophoresis; however, this correction vanishes for (i) particles in uncharged gels and (ii) spheres with a metallic core (c ! 1). We have found that the role of the particle dielectric constant is rather subtle, and can become a source of considerable confusion, particularly when interpreting the mobilities of metallic nanoparticles, for which it is tempting to set c ¼ 1. We will therefore proceed to identify these sources of error, and to propose a judicious choice of the dielectric constant that avoids the force being corrupted by the absence of ion-concentration and charge-density perturbations; this is particularly important for particles with high-dielectric-constant cores, e.g., metallic nanoparticles. In uncharged media, including free solutions and uncharged gels, the only term contributing to the force according to Eq. (32) is from the Brinkmanlet (far-field fluid velocity disturbance). Moreover, according to a model that neglects ion-concentration and chargedensity perturbations, the Brinkmanlet depends on the dielectric contrast parameter c ! 0:5 as p =s ! 0 and c ! 1 as p =s ! 1. However, O’Brien and White [56] proved mathematically that the mobility is independent of the dielectric properties. This remarkable observation (for free-solution electrophoresis) is also evident from numerical solutions of the full-electrokinetic models for soft-sphere free-solution electrophoresis [13] and for bare-particle gel-electrophoresis [34]. A physical explanation is that the perturbed electric field—and therefore the electromigration ion fluxes—around a low-dielectric constant particle are tangent to the solid-solvent interface. With an high dielectric constant, however, this electric-field drives ion fluxes into and out of the surface, thereby necessitating ion diffusion fluxes to satisfy a no-flux interfacial boundary condition. To avoid cumbersome numerical calculations, these ion fluxes are absent in the electrokinetic models that neglect ion-concentration and charge-density perturbations. Thus, if such theories are applied to high-dielectric constant particles, the dielectric contrast parameter influencing the far-field velocity disturbance should be set to c ¼ 0:5, irrespective of the actual dielectric contrast. For charged gels, matters are more complicated. Mohammadi and Hill [38] showed that the electrical body force arising from

the dielectric polarization does not affect the far-field fluidvelocity disturbance when the electrical body force on the fluid is the gradient of a scalar, q0m;2 $w2 as r ! 1. Nevertheless, this body force modifies the pressure and, therefore, influences the force. Mohammadi and Hill [38] computed the disturbances around an uncharged, bare sphere embedded in a charged gel, furnishing an analytical solution for the hydrodynamic force that accounts for ion-concentration and charge-density perturbations. For an uncharged sphere embedded in an incompressible, but charged, skeleton, Mohammadi and Hill’s theory furnishes1

h i F EMH ¼ 6pa‘2 qf 1 þ a=‘ þ ða=‘Þ2 =3 E þ

2pa3 qf ½ðja þ 1Þ2 þ 1E :   ðja þ 1Þ2 þ 1 þ p =s ðja þ 1Þ

ð33Þ

Thus, comparing this force to a calculation that neglects ionconcentration and charge-density perturbations [42]

h i 4 F ELH ¼ 6pa‘2 qf 1 þ a=‘ þ ða=‘Þ2 =3 E þ pa3 ð1 þ cÞqf E; 3

ð34Þ

the contribution of ion-concentration and charge-density perturbations is

DF E ¼ F EMH  F ELH "

# ðja þ 1Þ2 þ 1 2 ¼  ð1 þ cÞ 2pa3 qf E:   ðja þ 1Þ2 þ 1 þ p =s ðja þ 1Þ 3 ð35Þ

This is independent of the gel permeability and, as shown in Fig. 1, is bounded in magnitude by ð1  2cÞð2=3Þpa3 qf E, which occurs when ja ! 1, and vanishes when either ja ! 0; p =s ! 1, or p =s ! 0. Note also that the positive sign of DF E indicates, as expected, that the charge polarization diminishes the magnitude of F E . Based on the foregoing observations for bare particles, we have chosen to evaluate the Brinkmanlet contribution to the force with c ¼ 0:5 for all values of p =s , but compute the non-Brinkmanlet contribution according to whether c  0:5 or 1. According to this procedure, the calculations of Li and Hill [42] for bare spheres in 1 Mohammadi and Hill [38] provide a formula for compressible gels, where the force depends on the skeleton shear modulus and Poisson ratio.

F. Li et al. / Journal of Colloid and Interface Science 423 (2014) 129–142

133

charged gels are valid for metallic particles, i.e., when c  1, even though their results are reported with c  0:5. Moreover, the approximate theory of Doane et al. [40] remains valid for the Au NPs to which they applied their theory.

and PEGylated Au NPs with core diameter 2a  5:74 nm [40], the overlapping model might be used when the gel concentration J 4%.

2.4. Mobility

3.1. Free-solution soft-sphere electrophoresis

Following O’Brien and White [56], superposing the foregoing Uand E-problems satisfies the particle equation of motion

First, we test the gel-electrophoresis model by applying it to situations where the gel is a pure electrolyte. This is achieved by setting the fixed-charge to zero and the Brinkman permeability to a value that is large compared to all the other length scales. The soft-sphere electrophoresis problem was solved exactly by Hill et al. [13], and several approximations exist in the literature for

F U þ F E ¼ 0;

ð36Þ

which (from Eqs. (13) and (30)) furnishes U ¼ lE, where the gel electrophoretic mobility

3. Results and discussion

2

l¼

E 2 4 3 8 4 3 4pg‘2 2 C þ 3 pca qf ;2  3 pg‘2 b h2 ðr ¼ bÞð1  dol Þ þ 3 pa qf ;2 dol i : 1 2  2 2 1 2 2 ð1Þ 2 2 4pg 3 a2 j2 b h1 ðij2 bÞ  3 k2 j2 ð1  dol Þ þ 4pg 3 j2 a3  k2 j2 dol  2pb ng kB Tu1 f

h

This simplifies to 2

ln ¼ 

4pg‘2 C E þ 43 pcqf ;2 a3  83 pg‘2 b h2 ðr ¼ bÞ h 2 i2 ; 2 2 1 2 2 ð1Þ 2 4pg 3 a2 j2 b h1 ðij2 bÞ  3 k2 j2  2pb ng kB Tu1 f

ð38Þ

with a non-overlapping shell, and

lo ¼ 

E 4 3 4pg‘2 2 C þ 3 pð1 þ cÞqf ;2 a ; 1 2  2 2 4pg 3 j2 a3  k2 j2  2pb ng kB Tu1 f

ð39Þ

with a completely overlapping shell. Recall, the E-problem requires knowledge of the fluid velocity and pressure, and depends on the solution of the multi-point boundary-value problem (Eqs. (28) and (29)). Alternatively, one can apply a Lorentz reciprocal theorem, which requires only the electrical body force from the E-problem, and the velocity and pressure fields from the U-problem. The resulting mobility expression is cumbersome, but a self-contained Fortran function has been written to evaluate this mobility. In this paper, the results are calculated with p ¼ 2 for non-metallic cores (c  0:5) and p ¼ 100s for their metallic counterparts (c  1:0),2

s ¼ 78:5; g ¼ 8:9  104

Pa s, and T ¼ 298 K. Modelling polymer segments as spheres with radius as , e.g., poly(ethylene glycol) (PEG) segment radius aPEG  0:0175 nm [40], the Brinkman screening lengths ‘i are approximated by

 2 ‘i 2 ¼ ; 9/s F s as

ð40Þ

where /s ¼ ð4=3Þpns a3s is the volume fraction with ns the segment number density, and F s  1 þ 3ð/s =2Þ1=2 when /s  1 is a dimensionless drag coefficient [13]. For hydrogels, an empirical dependence of permeability ‘22 on volume fraction /g is [60]

 2  ‘2 3  ¼ ln /g þ 0:931 ; 20/g ag

ð41Þ

where ag is the fiber radius. When the hydrogel mesh size ‘2 is less than the particle core diameter 2a, it is helpful to interpret the soft-sphere mobility with models for a completely overlapping shell. For example, for agarose hydrogels with ag  1:9 nm [61] 2

Values of [59].

p

as large as 3:0  106 for Au NPs have been reported in the literature

ð37Þ

various special cases. Recall, our model is simplified by (i) the Debye–Hückel approximation, (ii) a neglect of ion-concentration and charge-density perturbations, and (iii) an adoption of a steplike shell profile. The dimensionless mobility3 is plotted as a function of the dimensionless shell thickness L=a for several values of ja P 1 in Fig. 2. Here, the core is uncharged (r ¼ 0) with the mobility magnitude increasing with the shell thickness because of the accompanying increase in the polyelectrolyte-coating charge. Note that our calculations (circles) agree with the full electrokinetic model of Hill et al. [13] (lines) because the electrostatic potential is low enough to justify the Debye–Hückel approximation and neglect of ion-concentration and charge-density perturbations. The influences of core and coating charge are shown in Fig. 3. In the left panel, the coating is uncharged, and the core bears a surface charge that maintains a fixed potential f ¼ kB T=e. As expected, increasing the coating thickness decreases the mobility magnitude, because the hydrodynamic drag of the layer hinders electroosmotic flow driven by the diffuse double layer. In the right panel, the coating bears a charge density that is proportional to the electrolyte ionic strength, with an opposite sign to the core charge. These mobilities are approximately equal to the linear superposition of the mobilities in Fig. 2 (charged coating with uncharged core) and those in the left panel (charged core with uncharged coating). Again, the present soft-sphere theory (circles) is validated by the full electrokinetic model of Hill et al. [13] (lines). The breakdown of the Debye–Hückel approximation and neglect of ion-concentration and charge-density perturbations is demonstrated in Fig. 4, where the dimensionless mobility is plotted as a function of the bulk ionic strength for spheres with a large radius and several polyelectrolyte-coating charge densities. Here, the electrostatic potential is large enough at low ionic strengths for the Debye–Hückel approximation and our neglect of ionconcentration and charge-density perturbations to break down. Nevertheless, the numerical fidelity of our calculations (circles) is demonstrated by comparison to the analytical approximation of Levine et al. [62] (dashed lines) for ja ! 1 with jL 1. Note that Ohshima [41] derived an analytical expression for electrophoresis of charged spheres with uncharged shells and low f-potential. As expected, this agrees exactly with the present

3 The mobility can be positive (negative), indicating that the particle moves towards the negative (positive) electrode; we refer to the mobility magnitude to indicate the migration speed.

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Fig. 2. Free-solution electrophoresis of polyelectrolyte-coated spheres with uncharged cores. Dimensionless electrophoretic mobility versus dimensionless r ¼ 0; nf ;1 ¼ I=10; zf ;1 ¼ 1, shell thickness L=a: a ¼ 3 nm, ‘1 ¼ 1 nm, ja ¼ 1; 4; 8; 12; 16 (increasing upward), and p ¼ 2. The present soft-sphere theory (circles) is compared with the full electrokinetic theory of Hill et al. [13] (lines).

model when the coating charge is zero. Mobilities scaled with the Smoluchowski mobility 0 s f=g (for the bare core) are plotted as a function of the dimensionless shell thickness for various values of ja and f in Fig. 5. When the surface potential is low (f ¼ 0:1kB T=e), the theories agree; however, when the surface potential is very high (f ¼ 6kB T=e), our calculations and the theory of Ohshima [41] overestimate the mobility when ja 1 and L=a  1. Of course, this is because both theories neglect charge-density perturbations. Nevertheless, even with f  6kB T=e, mobilities are still reliably predicted when ja  1 or L=a 1. 3.2. Bare-sphere gel electrophoresis Recall, applying the present soft-sphere theory to bare spheres

Fig. 4. Free-solution electrophoresis of polyelectrolyte-coated spheres with uncharged cores. Dimensionless electrophoretic mobility versus bulk ionic strength: a ¼ 500 nm, L ¼ 7:9 nm, ‘1 ¼ 2:6; 1:8; 1:3; 0:89 nm (decreasing downward), r ¼ 0; qf ;1 ¼ ½0:45; 0:89; 1:7; 3:6  107 C m3 (increasing downward), and p ¼ 2. The present soft-sphere theory (circles) is compared with the full electrokinetic model of Hill et al. [13] (solid lines), and the thin-layer approximation of Levine et al. [62] (ja ! 1; jL 1 and arbitrary L=‘1 , dashed lines).

2a2 ð1 þ cÞqf 2C E h i þ 3a‘2 ½1 þ a=‘ þ ða=‘Þ 9g 1 þ a=‘ þ ða=‘Þ2 =9   2 0  s f ja i 1þ þ cðjaÞ2 Uðja; a=‘Þ ¼ h j‘ þ 1 3g 1 þ a=‘ þ ða=‘Þ2 =9 h i qf ‘2 1 þ a=‘ þ ða=‘Þ2 =3 2a2 ð1 þ cÞqf i þ h i;  h g 1 þ a=‘ þ ða=‘Þ2 =9 9g 1 þ a=‘ þ ða=‘Þ2 =9

l¼

ð42Þ

where [34]

Uðja; a=‘Þ ¼

in charged gels revealed an omission in the electrical force F E calculated by Li and Hill [42]. Accordingly, their Eqs. (38) and (40) (for the electrical force) must be corrected by adding ð4=3Þpa3 ð1 þ cÞqf E; and their Eqs. (5) and (43) (for the mobility) must be corrected by n h io adding 2a2 ð1 þ cÞqf = 9g 1 þ a=‘ þ ða=‘Þ2 =9 . Thus, their exact

and

formula for the gel electrophoretic mobility for bare spheres becomes

F n ðxÞ ¼ ex

Z 1

3‘2 h a i F 5 ðjaÞ  F 5 ja þ 2 ‘ a 3‘ h a i þ F 5 ðjaÞ  F 4 ja þ a ‘ h a i þ F 5 ðjaÞ  F 3 ja þ ; ‘

1

exy dy: yn

ð43Þ

ð44Þ

Fig. 3. Free-solution electrophoresis of soft spheres. Dimensionless electrophoretic mobility versus dimensionless shell thickness L=a: a ¼ 3 nm, ‘1 ¼ 1 nm, f ¼ kB T=e; nf ;1 ¼ 0 (left), I=10 (right), zf ;1 ¼ 1; ja ¼ 1; 4; 8; 12; 16 (increasing upward), and p ¼ 2. The present soft-sphere theory (circles) is compared with the full electrokinetic theory of Hill et al. [13] (lines).

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Fig. 5. Free-solution electrophoresis of spheres with uncharged coatings and charged cores. Electrophoretic mobility scaled with the Smoluchowski mobility 0 s f=g versus dimensionless shell thickness L=a: a ¼ 3 nm, ‘1 ¼ 1 nm, f ¼ 0:1kB T=e (left), 6kB T=e (right), ja ¼ 0:01; 100 (increasing downward), and p ¼ 2. The present soft-sphere theory (circles) is compared with the full electrokinetic theory of Hill et al. [13] (solid lines), and the approximation of Ohshima [41] (dashed lines).

Fig. 6. Scaled mobility versus particle radius a for bare particles in uncharged and charged hydrogels: ‘2 ¼ 10 nm, r ¼ 0:5 lC cm2, qf ;2 ¼ ½0; 100; 200 kC m3 (upward), j1 ¼ 1 nm, and p ¼ 2 (top) and 100s (bottom). The soft-sphere theory (circles) is compared with the corrected bare-particle theory of Li and Hill [42] (Eq. (42), blue lines), s the thick-double-layer theory of Doane et al. [40] (j1 a; ‘, Eq. (47), blue dashed lines), the thin-double-layer theory (j1  a; ‘, Eq. (48), red dashed lines), and the modified theory of Henry [63] (Eq. (45), black lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

with E1 the exponential integral. The other asymptotic approximations of Li and Hill [42] are modified as

Note that the modified Henry mobility [42] remains

2 3

l ¼ 0 s ffH ðjaÞ=g  qf ‘2 =g;

ð45Þ

where

1 5 1 1 fH ðjaÞ ¼ 1 þ ðjaÞ2  ðjaÞ3  ðjaÞ4 þ ðjaÞ5 16 48 96 96   1 1 þ ðjaÞ4  ðjaÞ6 eja E1 ðjaÞ 8 96

ð46Þ

h i 20 s f  3qf ‘2 1 þ a=‘ þ ða=‘Þ2 =3 i l¼ h 3g 1 þ a=‘ þ ða=‘Þ2 =9 þ ang kB T l1 f þ

h

2a2 ð1 þ cÞqf

9g 1 þ a=‘ þ ða=‘Þ2 =9

i

ð47Þ

136

for

F. Li et al. / Journal of Colloid and Interface Science 423 (2014) 129–142

ja  1 and j‘  1, and

3.3. Soft spheres with uncharged shells

n

h io a2 qf þ 3 qf  0 s f=‘2 ‘=a þ ð‘=aÞ2 h i l¼ 3g 1 þ a=‘ þ ða=‘Þ2 =9 þ

h

2a2 ð1 þ cÞqf

9g 1 þ a=‘ þ ða=‘Þ2 =9

i

ð48Þ

ja 1 and j‘ 1. As shown in Fig. 6, which compares the mobilities of bare spheres with fixed surface charge density and a low-dielectric constant (top panels) with the mobilities of high-dielectric constant spheres (bottom panels), the core dielectric constant can have a significant influence on the mobility of large particles in highly charged gels. For spheres with p s , the mobility magnitude tends to increase with particle size, as observed experimentally for silver nanoparticles [18]. As discussed above, this is because the far-field pressure, which contributes to the force, is coupled to the electro-static dipole. Figure 7 compares bare-sphere mobilities from the soft-sphere theory developed in this paper with the earlier theories for bare spheres (Eqs. (42)–(48)). Here, the surface charge density (left panels) and surface potential (right panels) are regulated for low-dielectric constant particles and metallic particles. Again, the particle dielectric constant impacts the mobility, particularly when maintaining a fixed surface potential. for

Nanoparticles are often functionalized with polymeric shells to control colloidal stability, chemical functionality, and dispersion rheology [64,65]. With a porous shell, the electrophoretic mobility is affected by hydrodynamic and electrical forces, which may increase or decrease the mobility according to the shell charge. The mobility of particles bearing uncharged coatings when dispersed in charged gels is shown in Fig. 8 as a function of the reciprocal electrolyte Debye length scaled with the particle radius (a ¼ 3 nm) for several values of the polymer-coating thickness L. Here, the soft-sphere theory is compared with the foregoing bare-sphere theories, modified to mimic a soft particle by having the same hydrodynamic radius

ah ¼ acða=‘1 ; a=‘2 ; L=aÞ;

ð49Þ

where cða=‘1 ; a=‘2 ; L=aÞ is a dimensionless friction coefficient [57], and an effective surface potential [40,41]

fe ¼

f 1  ðj‘1 Þ2



1 ejL  cos ðL=‘1 Þ j‘1



1 þ tanh ðL=‘1 Þ j‘1

 :

ð50Þ

In the top panels, the core surface charge density is fixed, whereas in the bottom panels the core surface potential is fixed. The hydrogel bears a fixed charge that has the same sign as the core surface charge, so the electroosmotic flow is in the opposite direction of the intrinsic particle mobility. The mobility of the negatively charged particle increases with increasing ionic strength,

Fig. 7. Scaled mobility versus scaled reciprocal electrolyte Debye length js a for bare particles in uncharged and charged hydrogels: a ¼ 10 nm, ‘2 ¼ 5 nm, r ¼ 0:5 lC cm2 (left), f ¼ kB T=e (right), qf ;2 ¼ ½0; 400; 800 kC m3 (increasing upward), and p ¼ 2 (top) and 100s (bottom). The soft-sphere theory (circles) is compared with the corrected bare-particle theory of Li and Hill [42] (Eq. (42), blue lines), the thick-double-layer theory of Doane et al. [40] (j1 a; ‘, Eq. (47), blue dashed lines), the thindouble-layer theory (j1  a; ‘, Eq. (48), red dashed lines), and the modified theory of Henry [63] (Eq. (45), black lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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137

Fig. 8. Scaled electrophoretic mobility of spheres with uncharged shells in charged hydrogels versus the reciprocal electrolyte Debye length scaled with the core radius js a: a ¼ 3 nm, L=a ¼ 0; 0:5; 1; 2; 4 (increasing upward), as ¼ 0:0175 nm, ns ¼ 10 (left), 0:1 M (right), ‘2 =a ¼ 6; r ¼ 0:5 lC cm2 (top), f ¼ kB T=e (bottom), qf ;2 ¼ 8 kC m3, dol ¼ 0, and p ¼ 100s . The soft-sphere theory (blue circles and lines) is compared with several approximations using the soft-sphere hydrodynamic radius (Eq. (49)) and effective f-potential (Eq. (50)): corrected bare-particle theory of Li and Hill [42] (Eq. (42), black solid lines); modified thick-double-layer theory of Doane et al. [40] (Eq. (47), blue dashed lines); and modified thin-double-layer theory of Li and Hill [42] (Eq. (48), red dashed lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Electrophoretic mobility of PEGylated Au NPs versus PEG-ligand molecular weight for several agarose concentrations (see the legends) with 42 mM (left) and 10:5 mM (right) ionic strength TAE buffer [40]. Measurements (symbols) are compared with theory (lines) evaluated with the parameters listed in Table 2.

and eventually changes sign when the electroosmotic drag is large enough to overwhelm the other forces, as observed experimentally for PEGylated Au NPs by Doane et al. [40]. According to Hill et al. [13], the free-solution electrophoretic mobility of highly charged spheres with uncharged shells can exhibit a local mobility magnitude maximum at intermediate values of js a. However, by neglecting polarization and relaxation, and maintaining a

relatively thick shell (L=a > 0:0625) and low surface charge density (rja=½0 s kB T=ðeaÞ K 84), these influences are absent for gel electrophoresis, as seen in Fig. 8. Next, we interpret the gel electrophoresis experiments of Doane et al. [40] using the full soft-sphere theory. Figure 9 shows the mobility of PEGylated Au NPs as a function of the PEG (uncharged shell) molecular weight with different agarose

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F. Li et al. / Journal of Colloid and Interface Science 423 (2014) 129–142

Table 1 Summary of the PEG layer characteristics used in the soft-sphere theory to interpret the mobility data in Fig. 9 [40]. M PEG (kDa)

L (nm)

‘1 (nm)

L=‘1

L=a

1 2 5 10

8.2 11.5 18.7 27.6

2.2 2.3 2.7 3.2

3.7 5.0 6.9 8.6

2.9 4.0 6.5 9.6

Table 2 Summary of the best-fit parameters to interpret the mobility data in Fig. 9 using the full soft-sphere gel electrophoresis model. Panel

uf (lm s1)

j1 (nm)

f (mV)

Left Right

998 926

3.60 8.31

91.0 97.2

hydrogel concentrations and two electrolyte ionic strengths. Because the PEG chains are uncharged, the problem is reduced to the mobility of a soft particle with an uncharged shell. Following Doane et al. [40], we adopt a constant Au NP core radius a  2:87 nm, a PEG-chain grafting number N PEG  246, a PEGsegment hydrodynamic radius as  0:0175 nm with molecular weight M s  71 Da, and an agarose segment molecular weight Mg ¼ 1000 kDa. The shell properties are listed in Table 1. Note that the hydrogel charge densities are obtained from the mobility of vitamin B12, which (due to its negligible charge and small size) is assumed to be a passive tracer of the gel electroosmotic flow. For agarose concentrations [2%, we adopt the mobility from Eq. (38) for a non-overlapping shell. The best-fit frictional-coupling velocity lf , Debye length j1 , and surface potential f are listed in Table 2. Compared to the best-fitting parameters of Doane et al. [40], which were obtained using Eq. (47), the frictional-coupling velocity uf and Debye length j1 are somewhat larger. Note that the best-fit Debye length still doubles with a quartering of the ionic strength from 42 mM to 10.5 mM. The core surface potentials, however, are significantly larger, now falling in the range 91 to 97 mV, whereas Doane et al. [40] found f  60 mV. Note that such an high electrostatic potential is strictly beyond the acceptable range of validity for the Debye–Hückel approximation and our neglect of polarization and relaxation effects (ion-concentration and charge-density perturbations). Accordingly, Fig. 10

examines the importance of polarization and relaxation by comparing the theory of Ohshima [41] with the full electrokinetic theory of Hill et al. [13] for free-solution electrophoresis when ja K 1 and qf ;1 ¼ 0. Here, the polarization and relaxation influences are weakened by the shell, suggesting that the soft-sphere gel electrophoresis theory may also be reasonably accurate even when f  6kB T=e with L=a J 3. Clearly, a definitive conclusion must come from a much more intricate (numerical) solution of the gel-electrophoresis model, solving the ion conservation equations and Poisson equation [13,45]. Nevertheless, to the level of accuracy afforded by the Debye–Hückel approximation and our neglect of polarization and relaxation effects, our calculations attest to the general robustness of the approximate analytical model proposed by Doane et al. [40] for small metallic cores with soft, uncharged coronas. 3.4. Soft spheres with charged shells Polyelectrolyte shells are charged by the dissociation of ionizable groups, the type and concentration of which may greatly affect the apparent charge. For uncharged polymers, such as PEG, modifying the end groups can also endow the layer with charge, e.g., –COOH terminal groups provide a negatively charged shell, and –NH2 terminal groups provide a positively charged shell. Many terminally grafted fluorophores can also impart charge to the soft corona, giving rise to electroosmotic flow [66]. Note, however, that the present model approximates the ionizable groups as being uniformly distributed; a similar approach was adopted by Zhang and Hill [66] to model electroosmosic flow at PEGylated lipid bilayer membranes. The influence of the shell charge density on the mobility of a soft particle with a metallic core, such as silver or gold, in a low ionic strength electrolyte is examined in Fig. 11. Here, Eq. (41) is used to estimate the hydrogel permeability with a weight fraction wg ¼ 3% (/g ¼ wg =1:025) and fiber radius ag ¼ 1:9 nm. For a negatively charged shell, increasing the fixed-charge concentrations increases the shell fixed-charge density and reverses the migration direction. This is consistent with the experiments of Hanauer et al. [18], where the gel electrophoretic mobility of silver particles was examined with an increasing ratio of SH-PEG-COOH to SHPEG-OCH3. For a positively charged shell, the mobility increases with the shell charge. Compared to free-solution electrophoresis, e.g., as furnished by Hill et al. [13], a charged hydrogel impacts particle migration through hydrodynamic drag and electro-osmotic flow. Therefore,

Fig. 10. Free-solution electrophoresis of spheres with uncharged coatings. Dimensionless electrophoretic mobility versus dimensionless shell thickness L=a: a ¼ 3 nm, ‘1 ¼ 2:5 nm, f ¼ 4kB T=e (left), 6kB T=e (right), ja ¼ 0:25; 0:5; 1 (increasing upward), and p ¼ 2. The theory of Ohshima [41] (dashed lines) is compared with the full electrokinetic theory of Hill et al. [13] (solid lines).

F. Li et al. / Journal of Colloid and Interface Science 423 (2014) 129–142

139

Fig. 11. Dimensionless electrophoretic mobility versus the shell fixed-charge concentration in polyelectrolyte hydrogels: a ¼ 20 nm, L=a ¼ 0:5; as ¼ 0:0175 nm, ns ¼ 0:1 M, ‘2  30:1 nm, r ¼ 0:05 lC cm2, zf ;1 ¼ 1 (left), þ1 (right), qf ;2  3:09 kC m3, Is ¼ 0:1; 1; 100 mM (left: increasing upward; right: increasing downward), dol ¼ 1, and p ¼ 100s . Symbols are numerical calculations, and lines are the analytical solution based on the Lorentz reciprocal theorem.

Fig. 12. Dimensionless electrophoretic mobility of polyelectrolyte-coated spheres versus the dimensionless shell thickness L=a in polyelectrolyte hydrogels: a ¼ 3 nm, ‘1 =a ¼ 1=3; ‘2 =a ¼ 6 (left), 2 (right), f ¼ kB T=e; nf ;1 ¼ 0:1 mM, zf ;1 ¼ 1; qf ;2 ¼ 8 (top left), 72 (top right), 16 (bottom left), 144 kC m3 (bottom right), js a ¼ 0:1; 1; 10 (increasing upward), dol ¼ 0 (left), 1 (right), and p ¼ 100s . The present soft-sphere theory (blue circles and lines) is compared with the bare-sphere theory of Li and Hill [42] (Eq. (42), blue dashed lines), modified theory of Hill et al. [13] (Eq. (51), black solid lines), and porous-sphere theory (Eq. (52), red dashed lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

if we add to the electrical force in the particle equation of motion for free-solution electrophoresis a contribution from the polyelectrolyte hydrogel, then an approximation of the soft-particle gelelectrophoretic mobility is

h i 9‘22 qf ;2 1 þ ah =‘2 þ ðah =‘2 Þ2 =3  2a2h ð1 þ cÞqf ;2 h i ; l ¼ lE  9g 1 þ ah =‘2 þ ðah =‘2 Þ2 =9

ð51Þ

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Fig. 13. Dimensionless gel electrophoretic mobility of spheres with uncharged shells versus core radius: L ¼ 10 nm, as ¼ 0:0175 nm, ns ¼ 0:1 M, ‘2 ¼ 10 (top), 50 nm (bottom), r ¼ 0:5 lC cm2, qf ;1 ¼ 0; qf ;2 ¼ ½0; 100; 200 (top, increasing upward), ½0; 4; 8 kC m3 (bottom, increasing upward), j1 ¼ 1 (left), 10 nm (right), dol ¼ 0, s and p ¼ 100s . The present soft-sphere theory (blue circles and lines) is compared with the bare-sphere theory of Li and Hill [42] (Eq. (42), black solid lines), and poroussphere theory (Eq. (52), red dashed lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

where lE denotes the free-solution electrophoretic mobility. According to this crude approximation, the gel electrophoretic mobility for porous spheres (L=a 1 and ja 1) is

l¼

qf ;1 ‘21 qf ;2 ‘22  ; g g

ð52Þ

where qf ;1 is the porous sphere fixed-charge density. The dimensionless mobility of polyelectrolyte-coated particles in polyelectrolyte hydrogels is shown in Fig. 12 as a function of the dimensionless shell thickness at various electrolyte ionic strengths. Here, we have maintained a fixed shell permeability (‘1 =a ¼ 1=3), a constant core surface potential, and a constant shell fixed-charge density, while varying the hydrogel properties. For a positively charged shell, increasing the shell thickness increases the hydrodynamic drag force and the electrical force, eventually reducing the mobility magnitude and reversing the direction of motion. Similarly to uncharged shells, the particle mobility increases with the electrolyte ionic strength. For an highly permeable hydrogel (‘2 =a 1, left panels), Eqs. (51) and (52) are in better agreement with the soft-sphere theory for a non-overlapping gel and coating (Eq. (38)) when js a J 1 and L=‘2 K 1. A less permeable hydrogel is examined in the right panels (with the same electro-osmotic mobility), but now compared to the soft-sphere theory with an overlapping gel and coating (Eq. (39)). With a smaller mesh size and higher charge density, Eq. (51) deviates significantly from the soft-sphere theory (Fig. 12, bottom right). Comparing the top and bottom panels indicates that

maintaining the same hydrogel permeability and doubling the hydrogel charge density simply shifts the mobilities. Results for low-dielectric constant particles (not shown here) show that the mobility development is similar to the high-dielectric constant particles above. Note, however, that better agreement between the modified theory of Hill et al. [13] and the present soft-sphere theory is achieved when ‘2 =a 1. The modified soft-particle theory also furnishes a better approximation of the soft-sphere theory for low-dielectric constant particles, particularly when js a J 1 and ‘2 =a 1. Gel electrophoresis is used to separate biomolecules and nanoparticles, with the separation efficiency quantified by mobility differences. For example, a mobility difference of 0:1 lm s1 cm V1 leads to a band distance of 0:18 cm in 30 min at 150 V (15 cm electrode spacing). Figures 13 and 14 show the mobility of spheres with an uncharged or charged shell as a function of the particle core radius with a constant surface charge; these are compared to the values for bare spheres without a shell. In Fig. 13, coating the particle with an uncharged shell attenuates size selectivity, particularly when j1 is small. Similarly to bare spheres, the s hydrogel charge has a weak influence on size selectivity, since it merely shifts the mobilities. Maintaining the same hydrogel electro-osmotic mobility, separation efficiency is improved with a more permeable hydrogel. In Fig. 14, we maintain all the other parameters the same while endowing the shell with a negative charge. The mobility exhibits similar trends, but showing no obvious improvement. Thus, size separation might be improved with an highly permeable gel and a low-ionic-strength electrolyte.

F. Li et al. / Journal of Colloid and Interface Science 423 (2014) 129–142

141

Fig. 14. Dimensionless gel electrophoretic mobility of spheres with charged shells versus core radius: L ¼ 10 nm, as ¼ 0:0175 nm, ns ¼ 0:1 M, ‘2 ¼ 10 (top), 50 nm (bottom), r ¼ 0:5 lC cm2, nf ;1 ¼ 0:5 mM, zf ;1 ¼ 1; qf ;2 ¼ ½0; 100; 200 (top, increasing upward), ½0; 4; 8 kC m3 (bottom, increasing upward), j1 ¼ 1 (left), 10 nm (right), s dol ¼ 0, and p ¼ 100s . The present soft-sphere theory (blue circles and lines) is compared with the bare-sphere theory of Li and Hill [42] (Eq. (42), black lines), and poroussphere theory (Eq. (52), red dashed lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

4. Conclusion We developed and validated numerical and analytical solutions of an electrokinetic model for soft-sphere polyelectrolyte-gel electrophoresis. Such a theory is necessary to interpret the gel electrophoretic mobilities of functionalized nanoparticles. Bare-sphere gel electrophoresis theories [34,40,42] and free-solution electrophoresis theories [13,41,62] were modified and demonstrated to be useful in their respective parameter spaces. Compared to freesolution electrophoresis [13], the soft-sphere theory accounts for the electroosmotic flow of a polyelectrolyte hydrogel. Our model demonstrates that the particle dielectric constant can significantly influence the mobility, particularly when the core surface potential is regulated. In this paper, polarization and relaxation processes were neglected, as were elastic perturbations to the gel skeleton. We showed that these approximations are most severe for metallic-core nanoparticles in charged gels, even when the Debye–Hückel approximation is valid. This is because ion-concentration perturbations are necessary to satisfy an ion-impenetrable boundary condition.

for support through a McGill Engineering Doctoral Award (MEDA). Appendix A. Boundary conditions Boundary conditions for the U-problem are:

u1 ¼ U at r ¼ a; u1 ¼ u2 at r ¼ b; t 1 ¼ t 2 at r ¼ b; u2 ! 0 as r ! 1: These furnish b2 ¼ c2 ¼ 0, and

A1 ¼ 1 at r ¼ a; B1 ¼ 0 at r ¼ a; A1 ¼ A2 at r ¼ b;

Acknowledgments

B1 ¼ B2 at r ¼ b; R.J.H. gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Centre for Self Assembled Chemical Structures (CSACS). F.L. also thanks the Faculty of Engineering, McGill University,

A1;r ¼ A2;r at r ¼ b; B1;r  C 1 =ð2gÞ ¼ B2;r  C 2 =ð2gÞ at r ¼ b:

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F. Li et al. / Journal of Colloid and Interface Science 423 (2014) 129–142

Boundary conditions for the E-problem are:

Dj $n0j;1  Dj

0 s $w01 jr¼a

þ

zj en0j;1 kB T

!

$w01  er ¼ 0 at r ¼ a;

 er  0 p $w01 jr¼a  er ¼ r or w01 ¼ f at r ¼ a;

n0j;1 ¼ n0j;2 at r ¼ b;

[10] [11] [12] [13] [14] [15] [16] [17]

w01 ¼ w02 at r ¼ b;

[18]

$w02  er  $w01  er ¼ 0 at r ¼ b;

[19]

n0j;2 ! n1 j as r ! 1;

[20] [21]

w02 ! 0 as r ! 1:

[22]

Boundary conditions for the perturbations (primed variables) in the E-problem are:

0 s $w01 jr¼a

þ

 er  0 p $w01 jr¼a  er ¼ 0 at r ¼ a;

w01 jr¼aþ ¼ w01 jr¼a at r ¼ a;

[23] [24] [25] [26] [27] [28] [29] [30]

u01 ¼ 0 at r ¼ a; w01 ¼ w02 at r ¼ b;

$w02  er  $w01  er ¼ 0 at r ¼ b;

[31]

u01 ¼ u02 at r ¼ b;

[32]

rh1;rh ¼ rh2;rh at r ¼ b;

[33] [34] [35] [36] [37] [38] [39] [40]

p01 ¼ p02 at r ¼ b; w02 ! E  r as r ! 1; u02 ! U 1 as r ! 1: Here, rhi;rh are the tangential components of the hydrodynamic stress acting on a surface with unit normal in the r direction, and p0i are dynamic pressures, and U 1 ¼ qf ;2 ‘22 E=g is the undisturbed electroosmotic flow velocity. Boundary conditions for Eqs. (28) and (29) are: 1

h1 a

qf ;2 ‘22 ¼ 2g

h1 ¼ h2

qf ;2 ‘22 and h1;r ¼  at r ¼ a; 2g

and h1;r ¼ h2;r at r ¼ b;

h1;rr ¼ h2;rr at r ¼ b;

2 1  h1;rrr þ k1 h1 b þ h1;r þ qf ;2 ‘22 =g ¼ h2;rrr

2 1 þ k2 h2 b þ h2;r þ qf ;2 ‘22 =g at r ¼ b; h2;r r þ 2h2 ¼ 0 and h2;rr r þ 3h2;r ¼ 0 as r ! 1: References [1] [2] [3] [4] [5] [6]

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