Nanoparticle gel electrophoresis: Bare charged spheres in polyelectrolyte hydrogels

Journal of Colloid and Interface Science 394 (2013) 1–12

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Nanoparticle gel electrophoresis: Bare charged spheres in polyelectrolyte hydrogels Fei Li, Reghan J. Hill ⇑ Department of Chemical Engineering, McGill University, Montreal, Quebec, Canada H3A OC5

a r t i c l e

i n f o

Article history: Received 30 July 2012 Accepted 9 October 2012 Available online 30 October 2012 Keywords: Nanoparticle gel electrophoresis Electrophoretic mobility Electrokinetic theory

a b s t r a c t Nanoparticle gel electrophoresis has recently emerged as an attractive means of separating and characterizing nanoparticles. Consequently, a theory that accounts for electroosmotic flow in the gel, and coupling of the nanoparticle and hydrogel electrostatics and hydrodynamics, is required, particularly for gels in which the mesh size is comparable to or smaller than the particle radii. Here, we present an electrokinetic model for charged, spherical colloidal particles undergoing electrophoresis in charged (polyelectrolyte) hydrogels: the gel-electrophoresis analogue of Henry’s theory for electrophoresis in Newtonian electrolytes. We compare numerically exact solutions of the model with several independent asymptotic approximations, identifying regions in the parameter space where these approximations are accurate or break down. As previously assumed in the literature, Henry’s formula, modified by the addition of a constant electroosmotic flow mobility, is accurate only for nanoparticles that are small compared to the hydrogel mesh size. We derived an exact analytical solution of the full model by judiciously modifying the theory of Allison et al. [15] for uncharged gels, drawing on the superposition methodology of Doane et al. [14] to account for hydrogel charge. This furnishes accurate and economical mobility predictions for the entire parameter space. The present model suggests that nanoparticle size separations (with diameters [40 nm) are optimal at low ionic strength, with a gel mesh size that is selected according to the particle charging mechanism. For weakly charged particles, optimal size separation is achieved when the Brinkman screening length is matched to the mean particle size. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Nanoparticles are widely used in biomedical applications, including optical imaging [1], sensing [2], quantitative tagging [3], and drug delivery [4]. In addition to the effect on mobility, nanoparticle shape and size affect their electronic and optical properties [4–6]. For example, CdSe nanoparticle dispersions change from blue to red when the particle radius increases from 2.3 to 5.5 nm [7]. Standard nanoparticle syntheses do not generally produce dispersions with uniform properties [8,6], motivating nanoparticle electrophoresis for sizing and separations [6,5,9]. Advantages of gel electrophoresis are that (i) in addition to size and shape, nanoparticles can be separated according to their charge, (ii) the gel eliminates large-scale convection [10], and (iii) multiple runs can be performed on a single gel [5]. Moreover, gel electrophoresis has been used to study nanoparticle mobilities [11,12] and, therefore, to access surface physicochemical properties [13]; it has also been proposed as a novel method to non-invasively study hydrogel microstructure [14].

⇑ Corresponding author. E-mail address: [email protected] (R.J. Hill). 0021-9797/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcis.2012.10.022

According to Grimm et al. [11], the nanoparticle mobility in a polymer gel matrix is principally determined by the particle size with respect to the gel mesh size. However, the mobility is also influenced by particle surface charge, surface coatings, electrolyte concentration, and hydrogel charge [5,14,15]. For example, negatively charged soft nanoparticles can be stationary or migrate towards the negative electrode, depending on the hydrogel architecture (charge and permeability) and nanoparticle surface coating [5,14]. When particles are sufficiently small compared to the mesh size, gel electrophoretic mobilities have been interpreted using formulas for classical microelectrophoresis, corrected for background electroosmotic flow. For example, Hanauer et al. [5] used the wellknown Henry model to infer nanoparticle f-potentials from gel electrophoretic mobilities. Unknown, however, is the degree to which the particles hydrodynamically interact with the gel skeleton. Note also that most nanoparticles are functionalized with soft coatings, further complicating the relationship between the measured mobility, electrokinetic f-potential, and electrical charge. Electrokinetic flows in hydrogels have been modelled using the Brinkman model [16] to account for hydrodynamic coupling of electroosmotic flow to the hydrogel skeleton. Hill [17] calculated the electric-field-induced force exerted on a charged sphere

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immobilized in an uncharged electrolyte-saturated Brinkman medium. In addition to numerically exact calculations, he derived an analytical approximation for the strength of the Brinkmanlet for large particles.1 We will compare the mobility from this asymptotic approximation to the theory of Allison et al. [15], which has no restrictions on the parameters. Note that more recent theoretical analyses have addressed the role of hydrogel elasticity on immobilized particle dynamics [19,20]. Following the classical superposition methodology of O’Brien and White [21], Allison et al. [15] considered gel electrophoresis as the superposition of two dynamic cases: one where a particle translates in a porous medium without an electric field (the so-called U-problem), and another where a charged particle is stationary in an uncharged hydrogel subjected to an electric field (E-problem). In addition to numerical solutions of the full electrokinetic model, they derived an exact analytical solution for the gel electrophoretic mobility of weakly charged spheres (jfj 6 2kBT/e) in uncharged hydrogels. However, many hydrogels are charged or can become charged. For example, the charge density of agarose increases with pH, reaching a maximum when pH > 3.5 [22]. Morevoer, even ideally uncharged hydrogels, such as polyacrylamide, become weakly charged due to chemical reactions [20]. Theoretical analysis with charge on the hydrogel skeleton is more difficult, and presently no exact solutions are available. Doane et al. [14] addressed, in an approximate manner, charged soft nanoparticles undergoing electrophoresis in charged hydrogels. They solved the E-problem by superposing a subproblem E(1) in which a charged particle is placed in an uncharged hydrogel and a subproblem E(2) in which an uncharged particle is placed in a charged hydrogel. However, Doane et al. [14] considered weakly charged particles with a Debye length that is large compared to the particle radius and Brinkman screening length (comparable to the mesh size), enabling the electrical force to be set equal to the bare Coulomb force. This furnished a simple formula for nanoparticle gel electrophoretic mobility. Several approximations in the theory of Doane et al. [14] need to be verified. First, the influence of the ja parameter when ja  1 must be assessed. Note that, with charged hydrogels, ja depends on the added salt concentration and the hydrogel charge. For nanoparticles with radii less than 100 nm, j a can be as large as 100, so approximations based on ja  1, while convenient, may not be accurate. Second, instead of solving the E-problem as a whole, Doane et al. [14] adopted a linear superposition of the foregoing E(1) and E(2) subproblems. In this paper, we remove both approximations by solving the Eproblem directly for all values of ja (with jfj [ 2kBT/e). Guided by the theory of Allison et al. [15] for charged particles in uncharged gels, we neglect charge perturbations, also adopting the linearized Poisson–Boltzmann equation. Thus, our calculations furnish the gel-electrophoresis analogue of Henry’s theory for classical electrophoresis in Newtonian electrolytes. Note that the polydispersity of quantum dots tends to be much smaller than for metallic (e.g., Au, Ag, and Pt) nanoparticles [23–25], so the model may be especially useful for metal-nanoparticle electrophoresis.

2. Theory

there exists a diffuse layer of ions with Debye thickness j1. The suspending medium comprises a uniformly charged polymer skeleton, solvent (water with dielectric constant s), and mobile ions (counter ions of the hydrogel skeleton, and ions from the added z–z electrolyte). The hydrogel is modelled as a Brinkman [16] medium with Darcy permeability ‘2 and charge density qf. There are three (j = 1, 2, 3) mobile ion species: one hydrogel counterion and two added-salt ions (z–z electrolyte). The mobile ion fluxes are given by Nernst-Planck relations

jj ¼ Dj $nj  Dj

zj enj $w þ nj u kB T

ð1Þ

and satisfy steady conservation equations

$  jj ¼ 0:

ð2Þ

Here, u is the solvent velocity, Dj = kBT/(6pgaj), nj, zj and aj are ion diffusion coefficients, ion concentrations, ion valances and ion Stokes radii obtained from limiting conductances or diffusivities; kBT is the thermal energy and e is the fundamental charge. The electrostatic potential w satisfies the Poisson equation

0 s $2 w ¼

X

zj enj þ zf enf ;

ð3Þ

j¼1;2;3

where nf is concentration of hydrogel fixed charges, each with valence zf. The terms on the right-hand side are the net mobile-ion P charge density qm ¼ j¼1;2;3 zj enj and the fixed-ion charge density qf = zfenf. Fluid dynamics are governed by momentum and mass conservation equations (incompressible fluid)

0 ¼ $p þ g$2 u  g‘2 u  qm $w and $  u ¼ 0;

where p is the pressure, and g‘ u and qm $w are the Darcy hydrodynamic drag force and electrostatic body force acting on the fluid. As detailed below, we linearize the foregoing equations for weak electric fields and weak electrostatic potentials, also neglecting perturbations to the equilibrium ion concentrations. From the particle equation of motion, and linearity and symmetry considerations, the electrophoretic mobility can be written

l¼

2C E 3a‘2 ½1 þ a=‘ þ ða=‘Þ2 =9

;

1 By analogy with the well-known Stokeslet [18], we use the term Brinkmanlet to describe the flow driven by a point force applied the fluid in a Brinkman medium.

ð5Þ

where the particle electrophoretic velocity U = lE. Note that CE is the strength of the Brinkmanlet for the E-problem (detailed below) in which the particle is held stationary while subjected to an electric field E. The denominator comes from the Brinkmanlet for the U-problem (detailed below) in which the particle translates through the gel with velocity U in the absence of an electric field. 2.1. U-problem In the U-problem, the particle translates with constant velocity U in the absence of an electric field (E = 0). Although the particle and hydrogel are charged, we neglect all electrokinetic influences, because these are typically weak compared to the hydrodynamic forces. Accordingly, the fluid velocity and pressure satisfy Brinkman’s equations

0 ¼ $p þ g$2 u  g‘2 u and $  u ¼ 0 We consider a charged sphere translating with velocity U in an electrolyte-saturated hydrogel while subjected to a uniform electric field E. The particle is non-conducting with radius a, surface charge density r, and dielectric constant p. At the surface (r = a),

ð4Þ

2

ð6Þ

with boundary conditions u = U at r = a and u = 0 as r ? 1. The solution can be obtained from Brinkman [16] by removing the far-field pressure gradient, giving the hydrodynamic force exerted on the particle by the fluid [26,27]

F h;U ¼ 6pgaU½1 þ a=‘ þ ða=‘Þ2 =9:

ð7Þ

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Fig. 1. Scaled mobility versus ja for particles in uncharged gels. Left panel: j‘ = 0.03125, 0.125, 0.5, 2, 8, 32 (upward); Right panel: ‘/a = 0.025, 0.05, 0.25, 1, 4 (upward). Numerics (circles, for five of the six cases in the left panel, and for four of the five cases in the right panel) are compared with analytical-approximate theories (lines) valid in various regions of the parameter space: Doane et al. [14] (thick double layers, j1  a, ‘, blue dashed lines), this work (thin double layers, j1  a, ‘, red dashed lines), Henry [28] (black line); Hill [17] (large particles, a  j1, ‘, green dashed lines). Blue solid lines are the exact solution of Allison et al. [15] for jfj [ kBT/e.

Fig. 2. Scaled mobility (top), scaled f-potential (bottom, left) and surface charge density r (bottom, right) versus scaled reciprocal electrolyte Debye length jsa: a = 10 nm; ‘ = 5 nm, r = 0.5 lC cm2 (left panels), f = kBT/e (right panels), qf = 0, 1000, 2000  400 C m3 (upward). Numerics (circles) are compared with analytical theories: Doane et al. (j1  a, ‘, blue dashed lines), this work (j1  a, ‘, red dashed lines), modified Henry (black lines), modified Allison et al. (blue solid lines).

2.2. E-problem In the E-problem, the particle is held stationary (U = 0) while subjected to an electric field E. This drives electroosmotic flow, also perturbing the equilibrium electrostatic potential and pressure. However, for a weakly charged particle with a surface potential jfj [ 2kBT/e, perturbations to the equilibrium ion densities and charge density can be neglected. Accordingly, we set nj  n0j ; w ¼ w0 þ w0 ; p ¼ p0 þ p0 , and u = u0 , where the superscript 0 denotes 0 equilibrium, and denotes the respective perturbation.

When E = U = 0, the fluid is at rest (u0 = 0) and the equilibrium fields satisfy

n0j

zj en0j

!

0

$w ¼ 0; kB T X zj en0j þ zf enf ;  0 s $2 w0 ¼ 

$  Dj $

 Dj

ð8Þ ð9Þ

j¼1;2;3

 $p0  q0m $w0 ¼ 0; with boundary conditions

ð10Þ

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F. Li, R.J. Hill / Journal of Colloid and Interface Science 394 (2013) 1–12

Dj $n0j  Dj

zj en0j kB T

!

$w0  er ¼ 0 at r ¼ a;

0 s $w0 jout  er  0 p $w0 jin  er ¼ r n0j ! n1 j

at r ¼ a;

as r ! 1 and w0 ! 0 as r ! 1:

ð11Þ ð12Þ ð13Þ

At equilibrium, the mobile ions are Boltzmann distributed, giving zj ew0 =ðkB TÞ n0j ¼ n1 , which, when linearized for jw0j < kBT/e, furnish j e the well-known linearized Poisson–Boltzmann equation

$2 w0 ¼ j2 w0 ;

ð14Þ

where the squared-reciprocal Debye length

j2 ¼

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

ð15Þ

and the ionic strength

I¼

3 X 1 j¼1

2

z2j n1 j

zf q f ¼ Is þ : 2e

ejðarÞ

ra w ¼ ; 0 s ðja þ 1Þr ra2 qf ejðarÞ r2 j2 a4 e2jðarÞ : þ p0 ¼ 0 s ðja þ 1Þr 20 s ðja þ 1Þ2 r2 0

and

ra : 0 s ðja þ 1Þ

ð19Þ

Interestingly, the charge on the hydrogel only influences the equilibrium electrostatic potential by modifying the Debye length j1 according to the hydrogel countercharge. Note that the polyelectrolyte hydrogel fixed charge and its mobile counter charge are assumed to be uniformly smeared out, in similar manner to the surface charge on the sphere. This may not be the case, so—on the continuum scale that our model is applied—the fixed (counter) charge qf must be accepted as an effective, electrokinetic charge that is not necessarily equal to the titratable charge. We expect qf to reflect the hydrogel chemistry and physical architecture, the latter of which often varies considerably with the hydrogel synthesis. Next, when E – 0, the perturbed electrostatic potential satisfies

$2 w0 ¼ 0; ð16Þ

Note that js is the reciprocal Debye length based on the ionic strength of the added salt Is, whereas j1 is based on the total ionic strength I, which includes the hydrogel counterions. Accordingly, 2

f w0 ðr ¼ aÞ ¼

ð17Þ ð18Þ

ð20Þ

and the fluid velocity and pressure satisfy

0 ¼ gr2 u0  $p0  g‘2 u0  q0m $w0

ð21Þ

with boundary conditions

0 s $w0 jout  er  0 p $w0 jin  er ¼ 0

at r ¼ a;

ð22Þ

u0 ¼ 0 at r ¼ a;

ð23Þ

u0 ¼ U 1

ð24Þ

as r ! 1;

w0 ¼ E  r

as r ! 1;

ð25Þ

where U1 = qf‘2E/g is the undisturbed electroosmotic flow velocity. Accordingly,

Fig. 3. Scaled mobility (top), scaled f-potential (bottom, left) and surface charge density r (bottom, right) versus scaled reciprocal electrolyte Debye length jsa: a = 10 nm; ‘ = 50 nm, r = 0.5 lC cm2 (left panels), f = kBT/e (right panels), qf = 0, 1000, 2000  4 C m3 (upward). Numerics (circles) are compared with analytical theories: Doane et al. (j1  a, ‘, blue dashed lines), this work (j1  a, ‘, red dashed lines), modified Henry (black lines), modified Allison et al. (blue solid lines).

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F. Li, R.J. Hill / Journal of Colloid and Interface Science 394 (2013) 1–12

 a3  E  r; w0 ¼  1 þ c r

ð26Þ

Uðja; a=‘Þ ¼

where c = (s  p)/(p + 2s). Moreover, from linearity and symmetry considerations for an incompressible flow,

u0 ¼ ðhr

1

1

þ hr ÞE þ ðhr  hr ÞE  er er þ U 1 ;

ð27Þ

where h(r) is a scalar function of radial distance that decays as

and

h ! C E r2

F n ðxÞ ¼ ex

as r ! 1;

ð28Þ

Z

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

1

1

giving

u0 ! 2C E r3 E  er er þ C E r 3 E  eh eh þ U 1

as r ! 1

ð29Þ

and

  p0 !  g‘2 C E þ cqf a3 r 2 E  er

with c = (s  p)/(p + 2s). Note that the functions Fn are easily evaluated using the exponential integral E1 and recursion relations. Taking the curl of Eq. (21) with Eq. (27) furnishes

ð30Þ ¼

2

Here, the strength of the Brinkmanlet C = limr?1 hr quantifies the far-field decay of the electric-field-induced velocity and pressure disturbances [17]. For example, the mobility of Doane et al. [14] for ja  1 and j‘  1 infers

h i 3 C E ¼ g1 a‘2 0 s f  g1 a3 ‘2 qf ð‘=aÞ2 þ ‘=a þ 1=3 ; 2

ð31Þ

and for uncharged hydrogels we can determine from the theory of Allison et al. [15] that

h i C E ¼ g1 a‘2 0 s f 1 þ ja=ðj‘ þ 1Þ þ cðjaÞ2 Uðja; a=‘Þ ; where

ð34Þ

hrrrr þ 4hrrr r1  ð‘2 þ 4r 2 Þhrr  2‘2 r 1 hr þ 2‘2 r 2 h

as r ! 1: E

exy dy yn

ð33Þ

ð32Þ

ra2 j2 ðjr þ 1ÞejðarÞ 3 3 ðca r þ 1Þ ðja þ 1Þr 2 g

ð35Þ

with boundary conditions 1

ha

¼

q f ‘2 2g

and hr ¼ 

q f ‘2 2g

at r ¼ a

ð36Þ

and

hr r þ 2h ¼ 0 and hrr r þ 3hr ¼ 0 as r ! 1:

ð37Þ

As shown in Appendix B, the electrical force

F e;E ¼ 4pra2 E 

4 3 4 pa qf E  pca3 qf E; 3 3

ð38Þ

and the hydrodynamic force

Fig. 4. Scaled mobility (top) and scaled f-potential (bottom, left) and surface charge density r (bottom, right) versus scaled reciprocal electrolyte Debye length jsa: a = 10 nm; ‘ = 100 nm, r = 0.5 lC cm2 (left panels), f = kBT/e (right panels), qf = 0, 1000, 2000  1 C m3 (upward). Numerics (circles) are compared with analytical theories: Doane et al. (j1  a, ‘, blue dashed lines), this work (j1  a, ‘, red dashed lines), modified Henry (black lines), modified Allison et al. (blue solid lines).

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F. Li, R.J. Hill / Journal of Colloid and Interface Science 394 (2013) 1–12

F h;E ¼ 4pg‘2 C E E  4pra2 E þ

4 3 4 pa qf E þ pca3 qf E; 3 3

ð39Þ

giving a net electric-field-induced force E

2

E

F ¼ 4pg‘ C E;

particle equation of motion a contribution from the electroosmotic flow driven by the hydrogel charge. Accordingly,

l¼ ð40Þ

2ðC Eð1Þ þ C Eð2Þ Þ 3a‘ ½1 þ a=‘ þ ða=‘Þ2 =9 2

ð43Þ

;

where CE is obtained from the numerical solution of Eq. (35) (in dimensionless form, Appendix A) using the Matlab function bvp.

where CE(1) = CE given by Eq. (32) with j evaluated according to Eq. (15), i.e., including hydrogel counterions, and

3. Results

C Eð2Þ ¼

First, Henry’s mobility [28], modified by the addition of the hydrogel gel electroosmotic flow mobility qf‘2/g, as undertaken by Hanauer et al. [5], is

is furnished by the solution of the E-problem for an uncharged particle in a charged gel [14]. Third, an asymptotic analysis for (thin double layers) ja  1 and j‘  1 with arbitrary ‘/a furnishes a mobility

2 3

l ¼ f0 s fH ðjaÞ=g  qf ‘2 =g;

ð41Þ

where

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

ð42Þ

and E1 is the exponential integral. This is an asymptotic approximation for (large Brinkman screening lengths) ‘  a and ‘  j1 with arbitrary ja. Second, we modify the theory of Allison et al. [15] for uncharged gels to account for electroosmotic flow. Rather than adding the electroosmotic flow mobility, we add to the electrical force in the

3a‘4 qf ½1 þ a=‘ þ ða=‘Þ2 =3 2g

a2 q l¼ f 3g

n o 1 þ 3ð1  v1 Þ½‘=a þ ð‘=aÞ2  1 þ a=‘ þ ða=‘Þ2 =9

ð44Þ

;

ð45Þ

where v1 = 0sf/(qf‘2) is the ratio of the Helmholtz–Smoluchowski slip mobility 0s f/g to the undisturbed hydrogel electroosmotic flow mobility qf‘2/g. This formula is obtained by matching an outer solution of Brinkman’s equations (with a uniform electrical body force qfE) with an inner solution of the Stokes equations (with non-uniform body force from the diffuse charge) that yields the Helmholtz–Smoluchowski slip velocity. Fourth, we recall Doane et al.’s asymptotic approximation for (thick double layers) ja  1 and j‘  1 with arbitrary ‘/a,

Fig. 5. Scaled mobility versus particle radius a : j1 ¼ 1 nm; ‘ = 10 nm, r = 0.5 lC cm2 (top panels), f = kBT/e (bottom panels), qf = 0, 1000, 2000  100 C m3 s (upward). Numerics (circles) are compared with analytical theories: Doane et al. (j1  a, ‘, blue dashed lines), this work (j1  a, ‘, red dashed lines), modified Henry (black lines), modified Allison et al. (blue solid lines).

F. Li, R.J. Hill / Journal of Colloid and Interface Science 394 (2013) 1–12

7

Fig. 6. Scaled mobility versus particle radius a : j1 ¼ 1 nm; ‘ = 50 nm, r = 0.5 lC cm2 (top panels), f = kBT/e (bottom panels), qf = 0, 1000, 2000  4 C m3 (upward). s Numerics (circles) are compared with analytical theories: Doane et al. (j1  a, ‘, blue dashed lines), this work (j1  a, ‘, red dashed lines), modified Henry (black lines), modified Allison et al. (blue solid lines).

h i 2 a2 qf 1 þ 3ð‘=aÞ þ 3ð1  v2 Þð‘=aÞ l¼ ; 3g 1 þ a=‘ þ ða=‘Þ2 =9

ð46Þ

€ ckel mobility (2/ where v2 = (2/3)0sf/(qf‘2) is the ratio of the Hu 3)0s f/g to the undisturbed hydrogel electroosmotic flow mobility qf‘2/g. Note that we have corrected a minor but important oversight in Doane et al.’s theory: the Brinkman drag coefficient in their solution of the so-called U-problem contained an erroneous term arising from a mean pressure gradient that does not exist [26,27]. Finally, using boundary-layer analysis for large particles in uncharged gels, we note that Hill [17] derived

CE ¼

3rða‘Þ2 3f0 s jða‘Þ2 ¼ ; 2gðj‘ þ 1Þ 2gðj‘ þ 1Þ

ð47Þ

valid for ja  1, a/‘  1, and jfj [ kBT/e. Combining the accompanying electrical force with the Darcy drag force (with zero far-field pressure gradient), and adding the electroosmotic flow correction as applied to the theory of Allison et al. [15] above, furnishes an electrophoretic mobility for charged gels with ja  1 and a/‘  1,

going asymptotic theories and the exact analytical solution of Allison et al. [15]. These results are summarized in Fig. 1. Here, the dimensionless mobility U3g/(E20sf) is a function of only two independent dimensionless parameters, e.g., ja and j‘ (left panel) or ja and ‘/a (right panel). Clearly, our numerical solution of Eq. (35) (circles) agrees with the theory of Allison et al. [15] as given by Eq. (32). A careful examination reveals that all the approximate theories agree with the exact solution in the respective domains of validity. For example Doane et al.’s thick-double-layer theory (blue dashed lines) is accurate when j‘  1 with ja  1; Hill’s theory for large particles (green dashed lines) is accurate when ‘/a  1 with ja  1; and our thin-double-layer approximation (red dashed lines) is accurate when j‘  1 with ja  1. Note that these asymptotic formulae are easily evaluated in their respective domains of validity where numerical solutions (including Allison et al.’s theory) can be computationally challenging, e.g., ja J 100.

3.2. Charged gels

3.1. Uncharged gels

Turning now to charged gels, numerical and analytical theories are compared in Figs. 2–4, demonstrating how the mobility depends on the electrolyte concentration (top panels). In these and all other figures below, Eq. (43) (blue solid lines2) reproduces exactly the independent numerical computations (circles). While we could plot results in the manner undertaken for uncharged gels in

Before presenting the results for charged gels, we briefly review results for uncharged gels, comparing our numerics with the fore-

2 For interpretation of color in Figs. 1–10, the reader is referred to the web version of this article.

l¼

90 s f

g

j‘2 =a 3‘2 qf ;  j‘ þ 1 g

ð48Þ

where again, j is evaluated according to Eq. (15).

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F. Li, R.J. Hill / Journal of Colloid and Interface Science 394 (2013) 1–12

Fig. 7. Scaled mobility versus particle radius a : j1 ¼ 10 nm; ‘ = 10 nm, r = 0.5 lC cm2 (top panels), f = kBT/e (bottom panels), qf = 0, 1000, 2000  100 C m3 s (upward). Numerics (circles) are compared with analytical theories: Doane et al. (j1  a, ‘, blue dashed lines), this work (j1  a, ‘, red dashed lines), modified Henry (black lines), modified Allison et al. (blue solid lines).

Fig. 1 (now including fixed charge on the gel), we have chosen instead to plot dimensionless mobilities without scaling with the f-potential; while not as mathematically concise, this approach furnishes a more direct and transparent connection to experiments. Note that the permeability of agarose gel membranes was measured by Johnson and Deen [29] using pressure driven flow. They reported Brinkman screening lengths from 4.7 to 25 nm as the agarose concentration decreased from 7.3% to 2%. Moreover, the concentration of negative charge in polyacrylic acid gels (from streaming potential measurements) has been reported up to 0.3 mol l1 [30], corresponding to a sizeable qf   3  107 C m3. On the other hand, for agarose gels, which are widely used for electrophoresis, charge densities up to qf   1  104 C m3 can be inferred from electroosmotic flow [14]. In Figs. 2–4, which have different values of ‘, we have adjusted the hydrogel charge densities to maintain a fixed product ‘2qf. This ensures that the scaled mobilities remain O(1), and corresponds, approximately, to skeletons in which the charge per unit mass is fixed while the permeability and charge vary with the hydrogel concentration. Here, the scaled mobility is plotted versus the reciprocal electrolyte Debye length j1 scaled s with the particle radius a = 10 nm. Note that, with the particle surface charge density set to a constant r = 0.5 lC cm2, the accompanying f-potentials decrease with increasing jsa (bottom panels). Accordingly, the f-potentials become independent of the hydrogel charge density when the electrolyte ionic strength is greater than the hydrogel counterion ionic strength.

According to this electrokinetic model, the undisturbed electroosmotic flow is independent of jsa. Thus, as jsa increases, the particle mobility increasingly reflects the background electroosmotic flow and hydrodynamic interaction with the skeleton (when ‘/a is small enough). In Fig. 2, the Brinkman screening length ‘ = 5 nm gives ‘/a = 0.5. Under these conditions, Doane et al.’s theory furnishes an accurate mobility prediction at low ionic strength. At high ionic strength, all the analytical approximations are reasonable because the f-potential vanishes at fixed surface charge density; clearly, Henry’s approximation is accurate over a wide range of higher ionic strengths. Similar results are shown in Fig. 3, here for a more permeable hydrogel skeleton with ‘ = 50 nm. Finally, in Fig. 4, ‘ = 100 nm; again, even though Doane et al.’s theory is the most accurate at low ionic strengths, Henry’s modified theory furnishes a satisfactory approximation at all ionic strengths. The influence of particle size on the mobility is examined in Figs. 5–10. Here, numerical and analytical theories are compared with fixed values of js (fixed ionic strength). Note that, by fixing the particle surface charge density and changing the particle size with fixed j, the accompanying f-potentials increase in magnitude, from zero when a ? 0 to a finite value as a ? 1, as indicated by the modified Henry mobilities (Eq. (41)). In Fig. 5, j1 ¼ 1 nm with ‘ = 10 nm, which is representative of s an hydrogel saturated with physiological ionic strength electrolyte [31]. Similarly to Figs. 2–4, the particle surface charge density r = 0.5 lC cm2 with three values of the hydrogel fixed charge qf < 0 (again with ‘2qf = const.) that yield O(1) dimensionless mobilities. With such a thin diffuse layer, Doane et al.’s theory

F. Li, R.J. Hill / Journal of Colloid and Interface Science 394 (2013) 1–12

9

Fig. 8. Scaled mobility versus particle radius a : j1 ¼ 10 nm; ‘ = 50 nm, r = 0.5 lC cm2 (top panels), f = kBT/e (bottom panels), qf = 0, 1000, 2000  4 C m3 s (upward). Numerics (circles) are compared with analytical theories: Doane et al. (j1  a, ‘, blue dashed lines), this work (j1  a, ‘, red dashed lines), modified Henry (black lines), modified Allison et al. (blue solid lines).

for ja  1 is expected to be accurate only when a [ 1 nm, as confirmed by the comparison with numerics (left panel). Under these conditions, Henry’s calculation (Eq. (41)) is accurate for the uncharged gel. With increasing particle radius, the conditions ja  1 and j‘  1 required by Eq. (45) are satisfied when a J 10 nm (right panel). Accordingly, the thin-double-layer approximation (Eq. (45)) furnishes a convenient and accurate approximation, and Doane et al.’s theory captures only the qualitative trends. Note that, in contrast to classical electrophoresis (e.g., Henry’s formula) where the mobility is independent of particle size when ja  1, @ l/@a > 0 when a  ‘. Under these conditions, the dominant influences are the electroosmotic flow and hydrodynamic interaction with the skeleton, both of which are independent of particle charge. Accordingly, size selectivity in gel electrophoresis might be achieved at high ionic strengths by tuning the gel charge and permeability to maximize the hydrogel charge while maintaining a Brinkman screening length that is slightly smaller than the smallest particles. Of course, the gel must be compliant enough to avoid trapping particles, possibly making this ideal situation difficult to achieve in practice. In Fig. 6, we have increased the permeability to achieve ‘ = 50 nm, also decreasing the hydrogel charge densities to maintain O(1) dimensionless mobilities. Now, Henry’s modified theory is quantitative when a [ 50 nm, while our thin-double-layer approximation (Eq. (45)) is quantitative only when a J 50 nm. As expected, Doane et al.’s theory is quantitative only when a [ j1 1 nm.

Next, Fig. 7 shows the result of increasing the diffuse layer thickness to achieve j1 ¼ 10 nm with all other parameters the s same as in Fig. 5. Now, none of the conditions required for Eq. (45) are achieved and, accordingly, even for large particles, Eq. (45) breaks down. Moreover, the conditions for Doane et al.’s theory are achieved only for particles with a  10 nm. In practice, quantitative accuracy of Doane et al.’s theory is achieved for particles with a [ 2 nm, capturing the qualitative trends only for larger particles. Clearly, in this region of the parameter space, none of the analytical-approximate theories are quantitative. Another situation where exact solutions of the full model are vital for larger particles is shown in Fig. 8. Here, j1 ¼ 10 nm and s ‘ = 50 nm, with appropriately adjusted hydrogel charge densities to maintain O(1) dimensionless mobilities. The mobilities of small particles are well predicted by Henry’s modified theory, as applied to interpret nanoparticle mobilities in agarose gels by Hanauer et al. [5]. For particles with a J ‘ = 50 nm, however, none of the analytical-approximate theories are valid, and none furnish accurate mobility predictions. Fig. 9 shows mobilities with large j1 s ¼ 100 nm, corresponding to a low added salt concentration. With ‘ = 10 nm, Doane et al.’s theory furnishes a superior approximation than Henry’s modified theory for particles with a [ 10 nm, because it accounts for hydrodynamic interaction with the hydrogel. Note that the prevailing fpotentials (with fixed r = 0.5 lC cm2) are extraordinarily high at such low ionic strengths, so without hydrodynamic interaction, Henry’s modified theory furnishes large, negative mobilities, similarly to the thin-double-layer approximation (Eq. (45)).

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F. Li, R.J. Hill / Journal of Colloid and Interface Science 394 (2013) 1–12

Fig. 9. Scaled mobility versus particle radius a : j1 ¼ 100 nm; ‘ = 10 nm, r = 0.5 lC cm2 (top panels), f = kBT/e (bottom panels), qf = 0,  1000,  2000  100 C m3 s (upward). Numerics (circles) are compared with analytical theories: Doane et al. (j1  a, ‘, blue dashed lines), this work (j1  a, ‘, red dashed lines), modified Henry (black lines), modified Allison et al. (blue solid lines).

4. Discussion For nanoparticles (i.e., diameter [40 nm), varying the ionic strength can shift the ja parameter from values that are less than or comparable to one to values that are much greater than one. Moreover, the results above show that nanoparticle size-separation efficiency depends not only on hydrogel charge and permeability, but also on the particle charging mechanism. As a general rule, minimizing the ionic strength tends to favour the separation of populations with regulated surface charge density, when the mesh size is large (see Fig. 10, top left); whereas minimizing the ionic strength favours separating populations with regulated f-potential, when the mesh size is small (see Fig. 9, bottom). Thus, depending on the charging mechanism, varying the mesh size while maintaining a low ionic strength may expedite optimal size separations. When the f-potential is regulated, larger negatively charged particles can achieve more positive mobilities (in a negatively charged gel), because these particles bear a lower charge, making them more responsive to the background electroosmotic flow (see Fig. 9, bottom left). On the other hand, when the surface charge density is regulated, a mobility minimum with respect to size can cause larger particles to achieve more negative mobilities (see Fig. 10, top left). Here, increasing the size lowers the f-potential and increases the hydrodynamic drag, again making these particles more responsive to the background electroosmotic flow. With moderately charged nanoparticles, the hydrogel charge serves mainly to shift mobilities independently of the size, thereby having a minor influence on size selectivity. Thus, to minimize the ionic strength, it is desirable to minimize the hydrogel charge.

However, for very weakly charged particles, hydrogel charge is essential to achieve a size separation in a finite time. As quantified by Eq. (46) with f = 0, uncharged particles achieve higher positive mobilities in negatively charged gels, because their hydrodynamic coupling to the electroosmotic flow is stronger than the hydrodynamic coupling to the skeleton. Accordingly, with a fixed electroosmotic flow velocity qf‘2/g, the optimal mesh size is achieved when ‘ a, i.e., the Brinkman screening length should be matched to the mean population size. 5. Conclusion We validated independent numerical and analytical-approximate solutions of the electrokinetic model proposed by Doane et al. [14] for colloidal spheres in charged (polyelectrolyte) hydrogels, pioneered by Allison et al. [15] for uncharged gels. While each analytical-approximate solution was demonstrated to be accurate in its respective domain of validity, much of the parameter space for nanoparticle gel electrophoresis demands more elaborate analysis, as furnished by numerical solutions and, indeed, Eq. (43). Two judicious modifications to the theory of Allison et al. [15] for uncharged gels—as set out by Doane et al. [14] to account for hydrogel charge—furnish mobilities that agree exactly with independent numerical solutions. Thus, accurate nanoparticle gel mobilities in charged hydrogels can be readily obtained for all practically relevant parameters by evaluating Eq. (43).3 3 A self-contained Matlab function to evaluate this function is available from the corresponding author.

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F. Li, R.J. Hill / Journal of Colloid and Interface Science 394 (2013) 1–12

Fig. 10. Scaled mobility versus particle radius a : j1 ¼ 100 nm; ‘ = 50 nm, r = 0.5 lC cm2 (top panels), f = kBT/e (bottom panels), qf = 0, 1000, 2000  4 C m3 s (upward). Numerics (circles) are compared with analytical theories: Doane et al. (j1  a, ‘, blue dashed lines), this work (j1  a, ‘, red dashed lines), modified Henry (black lines), modified Allison et al. (blue solid lines).

The present model neglects charge polarization and electroviscous influences within the skeleton, which are expected to produce an electroosmotic flow velocity that varies with the electrolyte ionic strength, depending on the characteristic microscales and their size with respect to j1. The model also neglects steric interactions (e.g., the frictional coupling force of Doane et al. [14], or the thermodynamic multiplicative factor adopted by Allison et al. [15]) between the particles and skeleton, which are expected to be significant when a  ‘. Therefore, in addition to addressing charge polarization and electroviscous influences within the skeleton, future theoretical modelling should address soft nanoparticle coatings [32], which are often required to prevent particle aggregation and adhesion to the hydrogel skeleton, and particle charge polarization and electroviscous influences, which are expected to be important when the particle f-potential is high and ja  1 [15,21,33]. Clearly, there remain many more opportunities to advance our understanding of nanoparticle gel electrophoresis, using theoretical modelling and experimentation. To help elucidate the separate roles of steric and hydrodynamic influences, gel electrophoretic mobilities should be reported with accompanying gel diffusion coefficients, free-solution electrophoretic mobilities, and hydrodynamic radii [34].

Engineering, McGill University, for support through a McGill Engineering Doctoral Award (MEDA). Appendix A. Scaling and non-dimensionalization Here, distance r and parameters ‘ and j1 are scaled with the particle radius a, and other variables are scaled as follows: 2

u ¼

u 0 s kB T 2 j2 ‘2 with uc ¼ ; uc ge2 a

h ¼

h 0 s kB T j‘2 with hc ¼ ; hc ge

w ¼

w kB T ; with wc ¼ wc e

E ¼

E kB T j with Ec ¼ ; Ec e



r 0 s kB T j with rc ¼ ; rc e q   k T j2 qf ¼ f with qc ¼ 0 s B ; qc e r ¼

C E ¼ Acknowledgments R.J.H. gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Research Chairs program. F.L. thanks the Faculty of

l ¼

CE C Ec

with C Ec ¼

0 s kB T j‘2 a2 ge

;

l 20 s kB T with lc ¼ : 3ge lc

Adopting these dimensionless variables, Eq. (35) becomes

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F. Li, R.J. Hill / Journal of Colloid and Interface Science 394 (2013) 1–12



hrrrr þ ¼





4  1 4 2  2   h  2 þ 2 hrr  2 hr þ 2 h r  rrr r ‘ ‘ r ‘ r 2 r j2 ðj r þ 1Þðcr3 þ 1Þej ð1r Þ ðj þ

1Þ‘2 r2

with boundary conditions

1 h ¼  qf j 2 

and



at r ¼ 1

and 







hr r  þ 2h ¼ 0 and hrr r  þ 3hr ¼ 0 as r  ! 1: From the numerical solution, the strength of the Brinkmanlet 

C E ¼ hc a2 lim h r2 : 

Z

$  rh dV Z

1



g

r¼a

¼ 4pg‘2 C E E  4pra2 E þ

4 3 4 pa qf E þ pca3 qf E; 3 3

where rh is the hydrodynamic stress tensor, I is the identity tensor, and S ¼ ð1=2Þ½$u þ ð$uÞT  is the rate of strain tensor. With a constant solvent dielectric, the Maxwell stress tensor

1 2



re ¼ p0 I þ 0 s $w$w  0 s 1 



q @ s s @ q

  ð$w  $wÞI  qf wI T

1 ¼ p0 I þ 0 s $w$w  0 s ð$w  $wÞI  qf wI; 2 so the electric force in the E-problem

F e;E ¼

Z

re  er dA

r¼a

¼

Z

½p0 I þ 0 s $w$w 

r¼a

¼ 4pra2 E 

1 0 s ð$w  $wÞI  qf wI  er dA 2

4 3 4 pa qf E  pca3 qf E: 3 3

Note that the perturbed pressure satisfies the momentum equation

$p0 ¼ gr2 u  g‘2 u  q0m $w0 ; so

þ r‘ Þhr þ ð2r2  ‘2 Þh þ rhrrr E  er Z rðjaÞ2 ½cðjaÞ3 þ 3eja 1 et þ dtE  er 3ðja þ 1Þ t jr

¼ g 3hrr  ð2r



2

crðjaÞ2 ðj2 r2  jr þ 2ÞejðarÞ E  er 3ðja þ 1Þðr=aÞ3

¼ ðg‘2 C E þ cqf a3 Þr2 E  er :

 0 ½ðp0 þ p0 ÞI þ 2gS  er dA  u þ q $ w dV ¼ m 2 r¼a ‘ r!1 Z Z 1 Z g p0 er dA  r 2 u0  er er dA  q0m $wdV ¼ r!1 r!1 ‘ r¼a   Z g E 3 2 C þ c a q ¼ f r E  er er dA 2 r!1 ‘ ! Z qf ‘2 g E 3 r 2 2C r E  er er þ E  er er dA þ g r!1 ‘ Z 1 q0m $wdV  Z

1

1

1

r¼a

r!1



p0 ðr; hÞ ¼ ðgr‘2 hr þ g‘2 h  cqf a3 r2 ÞE  er

The hydrodynamic force in the E-problem

rh  er dA 

r  d  r hrr þ 2r 1 hr  ð2r 2 þ ‘2 Þh E  er dr 1 Z r Z r 0 2 1  g‘ U  er dr  q0m $w  er dr þ p0

¼ g

where the far-field pressure p0 = 0. As r ? 1,

Appendix B. Force calculation

F h;E ¼

i

gr2 u  g‘2 u  q0m $w0  er dr

 cqf a3 r 2 E  er þ p0 ;

r !1

Z

Z h

1

1 ¼  qf j 2

 hr

p0 ðr; hÞ ¼

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