Electrophoresis of spheres with uniform zeta potential in a gel modeled as an effective medium

Journal of Colloid and Interface Science 313 (2007) 328–337 www.elsevier.com/locate/jcis

Electrophoresis of spheres with uniform zeta potential in a gel modeled as an effective medium Stuart A. Allison ∗ , Yao Xin, Hongxia Pei Department of Chemistry, Georgia State University, Atlanta, GA 30302-4098, USA Received 7 March 2007; accepted 16 April 2007 Available online 20 April 2007

Abstract The effective medium model [H.C. Brinkman, Appl. Sci. Res. A 1 (1947) 27] is used to calculate the electrophoretic mobility of spheres in a gel with uniform zeta potential on their surface. In the absence of a gel support medium or ion relaxation (the distortion of the ion atmosphere from equilibrium due to the presence of an external flow or electric field), our results reduce to those of Henry [D.C. Henry, Proc. R. Soc. London Ser. A 133 (1931) 106]. The relaxation effect can be ignored for weakly charged particles, or for particles with low absolute zeta potential. Using a procedure similar to that employed by O’Brien and White [R.W. O’Brien, L.R. White, J. Chem. Soc. Faraday Trans. 2 74 (1978) 1607], the relaxation effect is accounted for in the present work and results are presented over a wide range of particle sizes, gel concentrations, and zeta potentials in KCl salt solutions. In the limit of no gel, our results reduce to those of earlier investigations. The procedure is then applied to the mobility of Au nanoparticles in agarose gels and model results are compared to recent experiments [D. Zanchet, C.M. Micheel, W.J. Parak, D. Gerion, S.C. Williams, A.P. Alivisatos, J. Phys. Chem. B 106 (2002) 11758; T. Pons, H.T. Uyeda, I.L. Medintz, H. Mattoussi, J. Phys. Chem. B 110 (2006) 20308]. Good agreement with experiment is found for reasonable choices of the model input parameters. © 2007 Elsevier Inc. All rights reserved. Keywords: Gel electrophoresis of spheres; Nanoparticle gel electrophoresis; Transport in gels; Effective medium modeling of electrophoresis

1. Introduction Electrophoresis, both in free solution and in a gel support medium, is a widely used technique in the biological and colloidal sciences and more recently in nanotechnology [1,2]. The primary utility of the method lies in its simplicity on the one hand, and power on the other to separate biomolecules, colloids, and nanoparticles on the basis of size and charge. As an aid in particle characterization, considerable attention has been devoted to the theory of transport of model particles in electric fields. The theory of electrophoresis is best developed for free solution (no gel support present), particularly for spherical particles [3–7]. Free solution electrophoresis of rodlike [8], ellipsoidal [9–11], arbitrarily shaped [12,13], “hairy” [14–18], and particles containing special surface conductance properties [19,20] have also been treated theoretically. * Corresponding author.

E-mail address: [email protected] (S.A. Allison). 0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2007.04.030

The theory of electrophoresis in gels is more difficult than in free solution due to hydrodynamic and direct interaction of the migrating particles with the gel support medium. When the migrating particles are large or extended (as in the case of high molecular weight DNA), reptation theories have been successful in explaining many, but not all of the features of gel electrophoresis [21,22]. For dilute gels or for particles that are small relative to the pore spacings in the gel, Ogston [23] and related models [24–26] have proven effective in dealing with steric (direct) interactions. Computer simulation methods such as the lattice method [27] and Brownian dynamics [28,29] have also been useful in this regard. These studies, however, have ignored the long range hydrodynamic interaction between the migrating particle and the gel support medium. Since hydrodynamic interaction is better understood in connection with sedimentation and diffusion in a gel, we would like to briefly review how other investigations have dealt with this problem in the recent past. The long range nature of hydrodynamic interaction in the sedimentation or diffusion of particles in a concentrated sus-

S.A. Allison et al. / Journal of Colloid and Interface Science 313 (2007) 328–337

pension, or the transport of dilute particles in a gel where the gel “segments” are modeled explicitly, has proven to be difficult to deal with [30]. Computational methods involving periodic boundary conditions show promise in this regard, but are quite complicated to implement [31]. A simpler approach is the effective medium, EM, model originally developed by Brinkman [32] and Debye and Bueche [33]. In this approach, the fluid surrounding the particle, which includes both solvent and the gel support medium, is treated as a hydrodynamic continuum. A special “screening” term is added to the external force/volume on the fluid in the low Reynolds number Navier– Stokes equation that accounts for the presence of a gel. In a reference frame where the fluid is stationary far from the model particle (fluid velocity, v, equals 0 far from the model particle and equals u0 on the particle surface), the fluid obeys the Brinkman [32], and solvent incompressibility equations η∇ 2 v − ∇p = ηλ2 v − se ,

(1)

∇ · v = 0,

(2)

where η is the solvent viscosity, p is the pressure, λ (units of 1/length) is a screening parameter due to the presence of a gel, and se accounts for the external force/volume on the fluid due to factors other than gel segment viscous drag. Starting from a microscopic model where the support medium consists of uniform “segments” with friction coefficient, ζs , and (local) number density, ns , Felderhof and Deutch [34] were able to derive Eq. (1) following an averaging procedure that involved a mean field approximation. Also, ns ζs = ηλ2

(3)

which makes it possible to estimate λ if the gel concentration is known along with its primary chemical structure. This point shall be discussed in more detail in Section 4. The diffusion of a particle in a gel and the role of both hydrodynamic and steric interactions have been extensively studied [35,36]. The ratio of the translational diffusion constant of the particle in the presence, D, and absence, Dng , of a gel support medium can be written as the product of two factors that represent hydrodynamic interaction, F , and steric effects, S, respectively [37], D = F (Rh , λ)S(φex ), Dng

(4)

where Rh = kB T /6πηDng is the hydrodynamic radius of the diffusing particle, kB is the Boltzmann constant, T is the absolute temperature, λ is the gel screening parameter from Eqs. (1) and (3), and φex is the volume fraction of the fluid domain excluded to penetration by the diffusing particle. In recent years, the EM model has shown considerable promise in describing the electrophoretic relaxation of a single, initially stretched DNA molecule in a gel support medium [38,39]. Stimulated by this work as well as by the success of the EM model in describing diffusion in gels [36,37], we apply the EM model in the present work to the problem of electrophoresis of spheres in a gel support medium. In analogy with Eq. (4),

we write   μ μ = S(φex ), μng μng EM

329

(5)

where μ(μng ) is the electrophoretic mobility in the presence (absence) of the gel. The EM subscript on the right-hand side of Eq. (5) denotes the mobility ratio within the framework of the EM model. The principal objective of the present work is the determination of (μ)EM for a spherical particle containing a centrosymmetric charge distribution. The outline of the paper is as follows. In Section 2, (μ)EM is determined for a model sphere where the distortion of the ion atmosphere from equilibrium is ignored. This is a valid approximation for weakly charged spheres, but breaks down when the reduced absolute zeta potential, |Y |, exceeds approximately 2 [4–7]. In the limiting case of no gel, our result reduces to that of Henry [3] as it should. In Section 3, ion relaxation is included following a procedure based on that of O’Brien and White [7]. This extends the range of validity of our results to spheres with an arbitrarily large |Y | and in the limit of no gel, we obtain results entirely consistent with a number of earlier studies [4–7]. To the best of our knowledge, all of the results dealing with a finite gel (λ > 0) are being reported for the first time. The problem of force balance, which is central to determining the electrophoretic mobility of a model particle, is handled differently when a gel is present. The force balance procedure, as well as some of the technical details of our modeling procedure, is placed in Appendix A. In Section 4, the method is applied to the electrophoresis of gold nanoparticles in a gel [40,41]. In this section, the steric term appearing in Eq. (5) is discussed as well and incorporated into modeling. In Section 5, we summarize the principal results of the work and briefly discuss future directions. 2. Mobility of a weakly charged sphere in an effective medium In this as well as the next section, many of the technical details are relegated to Appendix A and only some of the key points shall be addressed in the main body of this work. Extensive reference shall be made to Appendix A. The medium exterior to the model sphere itself is modeled as a continuous effective medium of dielectric constant εw and gel screening parameter λ (in 1/length). This medium consists of solvent, mobile ions, and gel. The sphere (radius = a) is assumed to be non-conducting, and contains a centrosymmetric charge distribution, with internal dielectric constant εI . Electrophoresis shall be viewed as the superposition of two transport cases. In Case 1, the sphere is translated with velocity u0 in the absence of an external electric field (e = 0). As discussed in Appendix A (see Eq. (A.23)), the scalar potentials, Φα (x), are introduced to represent the deviation of the concentration of α-ions from their equilibrium value due to an external flow or electric field. In the absence of relaxation (nα (x) = nα0 (x)) or an external electric field, the potentials, ψ (1) , (Eq. (A.26)), (1) and Φα vanish. (The “(1)” superscript denotes Case 1 transport.) Also, b equals 0 and Eq. (A.34) completely vanishes.

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Eq. (A.33) then gives

around a spherical particle of charge qQ [43],

(1)

zT = f u0 ,

(6)

Cq 2 Q e−κ(x−a) 4πεw kB T x (1 + κa)    qζ a −κ(x−a) = , e kB T x

y0 (x) =

(1) zT

where is the total force exerted by the sphere on the fluid (Case 1) and f is the translational friction coefficient of the sphere in an effective medium and is given by Eq. (A.10). In Case 2, the sphere is held stationary (u0 = 0), but is subjected to a constant external field, e. When the low dielectric sphere is placed in a constant external electric field, e, the perturbing potential at point x in the fluid (see Eq. (A.26)) is given by [42]  3 a (2) ψ (x) = −γ (7) x · e, x εw − εI γ= (8) . εI + 2εw In the absence of relaxation, the local ion concentrations equal their equilibrium values as in Case 1 transport. From (2) Eq. (A.23), we must set Φα (x) = −ψ (2) (x). The corresponding 1 dimensional radial functions from Eqs. (A.30) and (6) above are φα(2) (x) = −

γ a3 x2

(9)

(2)

and dα = −γ a 3 (Eq. (A.31)). From Eq. (A.32), ω = 0 due to charge neutrality. In this case, b = e and Eq. (A.33) reduces to zT(2) = −qQeff e, Qeff = −4π



 ×

(10) ∞ zα cα0

α

a

ah1 1− 2 3 λ x



x 2 dxe−zα y0

  ah2 2γ a 3 − 2 3 1− 3 . 3λ x x

(11)

In the above two equations, q is the protonic charge, Qeff can be viewed as an effective charge [13], the sum over α is over all mobile ion species present of valence zα and ambient concentration cα0 , y0 (x) is the reduced equilibrium electrostatic potential at distance x from the origin of the sphere (Eq. (A.24)), and h1 and h2 are given by Eqs. (A.3) and (A.4). In Eq. (11); y0 , h1 , and h2 are functions of the radial variable, x. Under steady state conditions, the net force exerted by the sphere on the fluid vanishes. Adding Eqs. (9) and (10) and setting the sum to zero, 0 = f u0 − qQeff e.

(12)

Define the effective medium electrophoretic mobility by the relation, u0 = μEM e, then qQeff . μEM = f

(13)

Ignoring relaxation is only valid for weakly charged particles or for particles with small |y0 |. Expand exp(−zα y0 (x)) in Eq. (11) to first order in zα y0 (x). Also, y0 (x) is approximated with the solution of the linear Poisson–Boltzmann equation

(14) (15)

where C = 4π (in CGS units), or 1/ε0 where ε0 is the permittivity of free space (in MKS units), and ζ is the zeta potential (equilibrium electrostatic potential at x = a), 2q 2 CI , εw kB T 1 I= cα0 zα2 . 2 α

κ2 =

(16) (17)

With these substitutions, Eq. (13) can be written ∞ 4πεw aζ κ 2 μEM = x dxe−κ(x−a) Cf a  ah1 ah2 2a 4 γ h2 × 1− 2 3 − 2 3 + . λ x 3λ x 3λ2 x 6

(18)

Let p = λa, and t = κa. After some straightforward but tedious algebra, Eq. (18) can be put in the form of a dimensionless mobility, ξ , 3Cηq μEM 2εw kB T  Y tp 2 = + γ t Φ(t, p) , 1+ t +p (1 + p + p 2 /9)

ξ=

(19)

where Y = qζ /kB T is a reduced (dimensionless) zeta potential, and Φ(t, p) =

 3

 3

F5 (t) − F5 (t + p) + F5 (t) − F4 (t + p) 2 p p

 + F5 (t) − F3 (t + p) . (20)

The Fn (t)s appearing in Eq. (20) are closely related to the exponential integrals [44]. Fn (t) = et En (t), ∞ e−tx En (t) = dx n . x

(21) (22)

1

Accurate numerical algorithms [44] are available to calculate E1 . For n > 1, use is made of the recurrence formula [44] En (t) =

 1 −t e − tEn−1 (t) . n−1

(23)

This makes it possible to compute the higher order En s and Fn s. Plotted in Fig. 1 is ξ/Y versus log10 (t) for a wide range of dimensionless λ/κ values with γ = 12 . In the limit of no gel, λ/κ = 0 and Eqs. (19) and (20) reduce to the mobility expression of Henry [3] and others [45].

S.A. Allison et al. / Journal of Colloid and Interface Science 313 (2007) 328–337

331

(i−j )

(x) is the j th “distant” solution for Case i and ion α, φα (i) and the dj are the linear coefficients uniquely determined that satisfy boundary conditions at x = a. For j = 0 or N + 1 or (i−j ) N + 2, the outer boundary condition on φα (x) is set to 0. For j = 1–N , δα,j (25) , κx  1, x2 where δα,j is the Kroneker delta. The remaining two homogeneous “distant” solutions (j = N + 1 or N + 2) are associated with the distant behavior of a scalar field, g(x), from which the fluid velocity, v, is derived [7]. It is defined by



 v(x) = curl curl g(x)b + u∞ , (26)

φα(i−j ) (x) =

Fig. 1. ξ/Y versus log10 (κa) in the absence of relaxation. It is assumed εw  εI (γ = 12 in Eq. (19)). Different lines represent different λ/κ values. From the uppermost to lowermost lines, λ/κ = 0.0 (solid line), 0.010 (dashed line), 0.027 (dotted line), 0.072 (solid line), 0.193 (dashed line), 0.519 (dotted line), 1.389 (solid line), 3.727 (dashed line), 10.0 (dotted line).

3. Mobility of a highly charged sphere in an effective medium When the charge or absolute zeta potential of the spherical particle is large, it is necessary to account for the relaxation effect (distortion of the local ion distributions from their equilibrium value) [4–7]. We follow the basic strategy of O’Brien and White that was originally applied to the electrophoresis of a sphere in the absence of a gel, or free solution electrophoresis [7]. Subsequently, this approach was applied to the free solution electrophoresis [16,46] and viscosity [46] of a highly charged spherical particle containing a gel layer on its outer surface. In the present work, we study the transport of a sphere in a stationary gel that is accounted for using the EM model summarized by Eqs. (1) and (2). As discussed in detail in Ref. [7], the coupled equations for the fluid velocity and ion transport (which involves the Φα potentials) are first transformed into 1-dimensional differential equations in the radial variable, x. These are then solved numerically for Cases 1 and 2 transport problems, see Section 2 for a discussion of these transport cases. Let N denote the number of mobile ions species present, which is two in the case of a simple salt. For each transport case, N + 2 homogeneous and one inhomogeneous set of differential equations are solved subject to different distant boundary conditions. The overall solution for Case 1 or 2 is then taken to be a particular linear combination of the above mentioned N + 3 “distant” solutions that satisfy appropriate boundary conditions at x = a. The overall solution of φα (x) for Case i (i = 1 or 2), is written φα(i) (x) =

N+2 

(i)

dj φα(i−j ) (x),

(24)

j =0

where the sum over j extends over the inhomogeneous (j = 0) and different homogeneous (j = 1 to N +2) “distant” solutions,

where b equals u0 (Case 1) or e (Case 2) and u∞ equals −u0 (Case 1) or 0 (Case 2). Substituting Eq. (26) into Eq. (1) and making use of Eqs. (A.20), (21), (30), we obtain η

 d 2 2 (i) ∇ ∇ g (x) − λ2 ∇ 2 g (i) (x) dx    dnα0 (x) φα (x) qzα + δi,2 , = dx x α

(27)

where the “(i)” superscript denotes Case 1 (i = 1) or 2 (i = 2), and nα0 (x) is the equilibrium concentration (Eq. (A.25) of αions at distance x from the center of the sphere). In the limit of no gel (λ = 0), Eq. (27) reduces to Eq. (6.6) of Ref. [7]. The overall solution of g (i) is written as a linear combination of the g (i−j ) in complete analogy to Eq. (24). The “distant” solutions of g (i−j ) are set to zero except for j = N + 1 and N + 2. For κx  1,

2 2 −λx (28) e −1 + , λ λ2 x 1 g (i−N+2) (x) = . (29) x Both Eqs. (28) and (29) are solutions of Eq. (27) when κx  1 (and the right-hand side of Eq. (27) can be set to 0). In the limit λx → 0, the right-hand side of Eq. (28) equals x, and the distant behavior of Ref. [7] (Eq. (7.3)) is retrieved. As discussed in Appendix A, force balance is handled differently when a gel is present. From Eqs. (A.32)–(A.34) and the above results, the total force exerted by the sphere on the fluid can be written g (i−N+1) (x) =



4π (1) zα cα0 dj Ωα(1−j ) − δj,α , (30) qu0 3 N N+2

(1)

zT = f u0 +

α=1 j =0

(2)

zT =

4π qe 3

N N+2   α=1 j =0

∞ Ωα(i−j )

=

2

x dxe a

(2)

zα cα0 dj

−zα y0





(i−j ) 

dφα + dx

−

Ωα(2−j ) − δj,α ,

ah1 1− 2 3 λ x



(31) 

(i−j )

2φα x  (i−j ) 

3δi,2 +

 dφα ah2 δi,2 + dx λ2 x 3

.

(32)

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Fig. 2. ξ/Y versus log10 (κa) with relaxation and Y = 1.0. Different lines represent different λ/κ values. From the uppermost to lowermost lines, λ/κ = 0.0 (open squares and solid line), 0.010 (filled squares and dashed line), 0.027 (open diamonds and dotted line), 0.072 (filled diamonds and solid line), 0.193 (open triangles and dashed line), 0.519 (filled triangles and dotted line), 1.389 (×s and solid line), 3.727 (*s and dashed line), 10.0 (+s and dotted line).

Fig. 3. ξ/Y versus log10 (κa) with relaxation and Y = 3.0. Different lines represent different λ/κ values. From the uppermost to lowermost lines, λ/κ = 0.0 (open squares and solid line), 0.010 (filled squares and dashed line), 0.027 (open diamonds and dotted line), 0.072 (filled diamonds and solid line), 0.193 (open triangles and dashed line), 0.519 (filled triangles and dotted line), 1.389 (×s and solid line), 3.727 (*s and dashed line), 10.0 (+s and dotted line).

(i−j )

Once the scalar fields, φα , have been obtained numerically, (i−j ) it is straightforward to compute Ωα . Defining

4π   (i) zα cα0 dj Ωα(i−j ) − δj,α q 3 N N+2

Ξ (i) =

(33)

α=1 j =0

and adding Eqs. (30) and (31) together, setting the sum to zero, and writing u0 = μEM e where μ is the electrophoretic mobility, μEM = −

Ξ (2) . f + Ξ (1)

(34)

From Eq. (19) we can readily compute a reduced (dimensionless) mobility, ξ , for the more general case of a highly charged sphere in a gel. A stand alone Fortran program has been written to compute the mobility of a sphere in a gel modeled as an effective medium using the protocol described above. It shall be made available free of charge upon request to the authors. As an application, a sphere covering a wide range of the dimensionless variables κa, λ/κ, and Y = qζ /kB T , shall be considered. Fixed parameters are T = 298.15 K, εw = 78.54, η = 0.894 centipoise (8.94 × 10−4 kg/(m s)), γ = 12 (Eq. (8)), and KCl salt. The mobilities/friction coefficients of the mobile ions are needed in (i−j ) the calculation of the φα potentials and are obtained from limiting molar conductivities [5,6]. For K+ and Cl− , the corresponding hydrodynamic radii are 0.1314 nm. For different salts, independent calculations must be carried out if the relaxation effect is significant. Figs. 2–5 summarize the behavior of ξ/Y for a wide range of κa and λ/κ for Y = 1 (Fig. 2), 3 (Fig. 3), 5 (Fig. 4), and 7 (Fig. 5). The results for Y = 1 are almost indistinguishable from those of Fig. 1 where the relaxation effect is not included. This demonstrates that when |Y | is small, the relaxation correction is negligible. The succession of Figs. 2–5 demonstrate that ion relaxation can have a substantial effect on

Fig. 4. ξ/Y versus log10 (κa) with relaxation and Y = 5.0. Different lines represent different λ/κ values. From the uppermost to lowermost lines, λ/κ = 0.0 (open squares and solid line), 0.010 (filled squares and dashed line), 0.027 (open diamonds and dotted line), 0.072 (filled diamonds and solid line), 0.193 (open triangles and dashed line), 0.519 (filled triangles and dotted line), 1.389 (×s and solid line), 3.727 (*s and dashed line), 10.0 (+s and dotted line).

mobility when |Y | is large. In the limit of no gel, our results are in excellent agreement with previous studies [6,7]. One question we would like to address before considering specific applications is the effect of ion relaxation on mobility when a gel is present. The effect in the absence of a gel has been extensively studied [6,7]. Shown in Fig. 6 is a plot of X = (ξ/Y )Y =5 /(ξ/Y )Y =1 versus κa for three different values of λ/κ. This particular mobility ratio is chosen since relaxation has a substantial effect on mobility when Y = 5, but little effect when Y = 1. Also, gel effects on mobility unrelated to the relaxation effect cancel out in X. The “no gel” case (λ/κ = 0.0) is represented by diamonds connected by a solid line. The intermediate data set (squares connected by dashed

S.A. Allison et al. / Journal of Colloid and Interface Science 313 (2007) 328–337

333

ple), its weight concentration, M (typically in g of dry gel material/total volume), and possibly the concentration of crosslinker if present [47]. To make contact with the λ parameter of the EM model, use is made of Eq. (3) along with ideas advanced in Ref. [34]. The gel is modeled as a collection of identical interconnected “segments,” represented by beads, of hydrated radius σ , number density ns , and segment friction factor ζs . As discussed in Ref. [34], how the “segments” are interconnected is unimportant provided the virtual bonds that connect the segments exert negligible friction on solvent flowing past them. Accounting for the hydrodynamic interaction between the segments and making use of Eq. (A.10)   1 2 2 ζs = 6πησ 1 + λσ + λ σ . (35) 9 Fig. 5. ξ/Y versus log10 (κa) with relaxation and Y = 7.0. Different lines represent different λ/κ values. From the uppermost to lowermost lines, λ/κ = 0.0 (open squares and solid line), 0.010 (filled squares and dashed line), 0.027 (open diamonds and dotted line), 0.072 (filled diamonds and solid line), 0.193 (open triangles and dashed line), 0.519 (filled triangles and dotted line), 1.389 (×s and solid line), 3.727 (*s and dashed line), 10.0 (+s and dotted line).

Fig. 6. X versus log10 (κa) for three values of λ/κ. X is defined as (ξ/Y )Y =5 /(ξ/Y )Y =1 and λ/κ = 0.0 (diamonds connected by a solid line), λ/κ = 0.193 (squares connected by a dashed line), and λ/κ = 1.389 (triangles connected by a dotted line).

line) corresponds to λ/κ = 0.193. As discussed in the next section, this corresponds to a gel under typical experimental conditions (a 2% agarose gel (approximately) in a 0.01–0.02 molar monovalent salt solution). The third set (triangles connected by a dotted line) corresponds to a substantially denser gel with λ/κ = 1.389. What these results show is that for small spheres (κa  2) the relaxation effect is independent of gel concentration. For the intermediate case (λ/κ = 0.193) there is an effect for larger particles, but the effect is quite small. Only for large particles in a dense gel does one observe a substantial effect of the gel on relaxation. 4. Application to gold nanoparticles In the literature, the gel is typically characterized by its chemical composition (agarose or polyacrylamide, for exam-

Also, if ρg denotes the dry weight density of gel material (which equals 1.64 g/ml for agarose [48]), ωs denotes the ratio of dry gel volume to hydrated gel volume (which equals 0.625 for agarose [49]), and φ denotes the volume fraction of hydrated gel to the total volume, the weight concentration of gel can be written 4 M = ρg ωs φ = πρg ωs ns σ 3 . (36) 3 Substituting Eqs. (35) and (36) into Eq. (3) and solving for λ yields     5 A 4 A λ= (37) 1+ + 1− , 2σ 9 A 9 where A = 9M/(2ρg ωs ). For the hydrated segment radius, σ , a value based on the fundamental chemical makeup of the gel might be chosen or else it could be left as an adjustable parameter. For the agarose gel of interest in this work, a reasonable value for σ might be the average fiber radius of 1.9 nm which comes from low angle X-ray scattering [50]. Once these parameters are chosen, λ is defined. Using the protocols described in Sections 2 and 3, μEM , where the EM subscript denotes a mobility derived using the effective medium model, can be computed. However, we also need to consider the steric factor, S(φex ), discussed in Section 1. With regard to the gel electrophoresis of spherical particles in dilute gels where hydrodynamic interaction is ignored, the steric function can be readily deduced from Ogston and closely related theory. We can write [23,25,28] 1 2 ≈ e−0.667φex = e−0.0.667(1+a/σ ) φ . (38) 1 + 2/3φex Equation (38) above is very similar to a steric function deduced and used in diffusional studies in congested media [35,51] 

1.09 S(φex ) = exp −0.84φex (39) . S(φex ) ≈

Equation (39) was deduced by a curve fitting procedure over a broad range of φex whereas Eq. (38) is limited to small φex , but has the advantage of being specific to gel electrophoresis. In this work, we shall be primarily interested in dilute gels since in concentrated gels, the detailed structure of the gel also becomes important [27,35]. If φex is small, Eqs. (38) and (39) give similar corrections. Consequently, Eq. (38) shall be used in this

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Fig. 7. μ/μng versus M for gold nanoparticles (radius = 5 nm). Squares and diamonds correspond to experimental [40] and model mobility ratios, respectively. M represents the gel concentration in g of “dry” gel material/ml. In modeling, σ = 1.9 nm.

work. Two independent mobility studies of gold nanoparticles shall now be considered [40,41]. This analysis shall serve as a benchmark test of the ability of our modeling protocol to account for the effect of the gel on the electrophoretic mobility of small particles in the nanometer size range. Zanchet et al. [40] report mobility ratios, (μ/μng ), of monodisperse gold nanoparticles with a radius of 5 nm in agarose gel for a range of gel concentrations. The buffer system is “0.5× TBE” which consists of 44.5 mM Tris base plus 44.5 mM boric acid plus 1 mM EDTA. At a pH of 9.0, EDTA is present primarily in the form of a trianion. Using pKa s of 9.14 and 8.08 for boric acid and protonated Tris, it is readily deduced that the ionic strength equals 16.1 mM. Because the particles are fairly small (κa = 2.08) and λ/κ is also small (λ/κ = 0.360 when M = 0.023 g/ml), the gel should have little effect on the relaxation effect in light of Fig. 6. Since we are also considering the mobility ratio, μ/μng , the actual charge on the gold nanoparticle, which is unknown, is unimportant. Plotted in Fig. 7 are experimental (filled squares) and model (unfilled diamonds) mobility ratios versus gel concentration. Experimental points were taken from Fig. 1 of Ref. [40] and model mobilities were computed using Eqs. (5), (19), and (38) using parameters discussed previously. In this example, we set σ equal to 1.9 nm, which comes from X-ray scattering [50]. For M  0.03 g/ml, agreement between modeling and experiment is good. It should be emphasized that all model parameters were set in advance and none were adjusted to obtain the model points displayed in the figure. Above a gel concentration of 0.04 g/ml, the model tends to underestimate the retarding effects of the gel on the particle. Recently, Pons et al. [41] reported (among other things) absolute electrophoretic mobilities of Au nanoparticles in agarose gel as a function of agarose concentration. Furthermore, data is provided for several different sizes which bracket the 5 nm particle considered previously. The buffer consists of 10 mM Tris borate buffer plus 2 mM EDTA at pH 9.0. Under these conditions, the ionic strength is estimated to be 14.28 mM.

Fig. 8. − log10 (μ) versus M for gold nanoparticles of different radii. Squares, diamonds, and triangles represent experimental data [41] with a = 2.4, 4.8, and 7.6 nm, respectively. Solid, dashed, and dotted lines represent the corresponding model fits with σ = 2.4 nm.

In order to reproduce a “free solution” mobility of about −1.07 × 10−8 m2 /V s for a Au nanoparticle with a radius of 2.4 nm, modeling requires Y = −0.75 or ζ = −19.3 mV. The corresponding charge can be estimated from the initial slope, (dΛ0 (x)/dx)x=a , and gives −4.67 in protonic units. For an absolute zeta potential of 0.75, the relaxation effect should be negligible. For this data set, we did not obtain a good fit as a function of gel concentration by setting σ = 1.9 nm. However, the results are very sensitive to the choice of this parameter. Setting σ equal to 2.4 nm gives a good fit for all three particle sizes as shown in Fig. 8. Squares, diamonds, and triangles represent experimental data with a = 2.4, 4.8, and 7.6 nm, respectively. Solid, dashed, and dotted lines represent the corresponding model fits with σ = 2.4 nm. It should be noted that both Refs. [40,41] present mobility versus M data for Au nanoparticles of 5 nm radius (or close to that value) in agarose gels. Although the buffer conditions are not identical, they are very similar. Despite the similarities in the two studies, the dependence of μ on gel concentration is stronger in Ref. [40] than in Ref. [41]. In modeling, this is reflected in the different values of σ used to fit the two data sets. At this time, we are not certain what factor(s) is (are) responsible for these differences. One might question the relevance of using a dielectric sphere model to model the electrophoresis of (conducting) gold nanoparticles. This important point is closely related to the more general question of the effect of electrostatic boundary conditions on the electrophoretic mobility of spherical particles of uniform zeta potential. In Refs. [6,7], the (free solution) electrophoresis of (dielectric) spherical particles of uniform, but arbitrary zeta potential, was obtained by solving the equations that govern ion distribution, fluid velocities, and electrostatic potential, in a rigorous manner. Section 3 of the present work basically follows the approach of these earlier studies. In Ref. [6], it is mentioned that the mobility is independent of the interior dielectric constant of the sphere. Since the electrostatic potential outside of a conducting sphere is identical to that of a dielectric sphere with εI  εw [42], the same conclusion

S.A. Allison et al. / Journal of Colloid and Interface Science 313 (2007) 328–337

would be expected to apply to a conducting particle. O’Brien and White [7] addressed the point in greater detail and concluded that “mobility depends only on the particle size and shape, the properties of the electrolyte solution in which it is suspended, and the charge inside or electrostatic potential on, the hydrodynamic shear plane in the absence of an applied field or any macroscopic motion.” On this basis we feel the dielectric sphere model should be applicable to a conducting sphere as well. 5. Summary The principal objective of this work has been to calculate the electrophoretic mobility of model spheres in a gel with uniform zeta potential in the framework of the effective medium, EM, model [32–34]. As discussed in Sections 1 and 4, it is necessary to also account for steric effects [23,24,35–37] in making a comparison between model and experimental mobilities. In the absence of ion relaxation, our results reduce to those of Henry [3]. For highly charged particles, however, the relaxation effect should not be ignored [4–7]. Using a numerical procedure similar to that employed by O’Brien and White [7], the relaxation effect is accounted for in the present work and results are presented over a wide range of particle sizes, gel concentrations, and zeta potentials in KCl salt solutions. In the limit of no gel, our results reduce to those of earlier investigations [6,7]. Since the relaxation effect depends on the mobility of the mobile salt ions present, separate calculations would have to be carried out for different salts. Our Fortran program is available, free of charge, upon request to the authors. The procedure is then applied to the mobility of Au nanoparticles in agarose gels measured in two independent studies [40,41]. Good agreement with experiment is found for reasonable choices of the model input parameters. However, different gel segment radii must be used in the two studies in order to achieve good fits. The present work should be of use in characterizing nanoparticles such as quantum dots [40,41], QDs, and also QD–DNA [40] or QD–protein [41] complexes. Following our earlier work on modeling the free solution mobility of peptides [52], where the macromolecule is represented as a string of beads, we are currently extending that line of investigation to account for a gel. Appendix A. Force balance on a spherical particle A detailed study of the translation of an uncharged sphere in an effective medium, EM, where the fluid obeys Eq. (1), is given by Stigter [38]. Since this study is particularly relevant to the present work, it shall be useful to summarize some key results from this work. Assume the sphere, radius = a, translates through a stationary EM with velocity u0 . Choosing the origin to coincide with the center of the sphere, the fluid velocity u(x) can be written [38] u(x) = U (a, λ, x) · u0 , where U (a, λ, x) =

 a

h1 (a, λ, x)I + h2 (a, λ, x)nn , 2 3 λ x

(A.1)

(A.2)

1 3 + 3λx + 3λ2 x 2 e−λ(x−a) 2

 − 3 + 3aλ + a 2 λ2 ,

3 h2 (a, λ, x) = − 3 + 3λx + λ2 x 2 e−λ(x−a) 2

 + 3 + 3aλ + a 2 λ2 ,

335

h1 (a, λ, x) =

(A.3)

(A.4)

I is the 3 × 3 identity tensor, and nn = xx/x 2

(A.5)

is the unit position dyadic. In the limit x → a, U(a, λ, x) → I. In the limit of large x (x  1/λ and x  a) a U (a, λ, x) ≈ 2 3 I . (A.6) λ x Far from the translating sphere, the fluid velocity falls off as x −3 . This can be contrasted with the x −1 dependence seen for a translating sphere in a Newtonian fluid in the absence of a gel. The corresponding pressure, p(x), is given by [38]

ηa p(x) = 2 3 + 3aλ + a 2 λ2 u0 · n. (A.7) 2x The integral of the hydrodynamic stress tensor, σ h ,

σ H = −pI + 2η ∇u + ∇uT (A.8) over the surface of the spherical particle, Sp , is related to the total force exerted by the particle on the fluid, zT . This in turn, equals the translational friction constant, f , times the particle velocity  zT = − σ H · n dSx = f u0 . (A.9) Sp

Stigter has shown [38]   1 f = 6πηa 1 + λa + λ2 a 2 . 9

(A.10)

Next we consider the more general problem of a highly charged particle either translating through an EM with velocity u0 (Case 1); or stationary, but subjected to a constant external electric field, e (Case 2) [7]. Because an electrical force is also present, Eq. (A.9) must be generalized to account for the electric stress, σ E .   zT = − σ H · n dSx − σ E · n dSx . (A.11) Sp

Sp

Now in the fluid domain, Eq. (1) can be written

0 = ∇ · σH + σE + σG ,

(A.12)

where ∇ · σ H = η∇ 2 v − ∇p,

(A.13)

∇ · σ E = se,

(A.14)

∇ · σ G = −ηλ2 v,

(A.15)

where v is the fluid velocity, se is the external (electric) force/volume acting on the fluid, and other quantities have been

336

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defined previously. Assume we have a single isolated sphere, and integrate Eq. (A.12) over the entire fluid domain, Ω. Applying the divergence theorem to the ∇ · σ H and ∇ · σ E terms and using Eq. (A.11)   zT = − σ H · n dS + ηλ2 v dV . (A.16) Ω

S∞

In (A.16), S∞ denotes a surface very far from the center of a particle. This result is valid whether a gel is present or not (λ = 0). In the absence of a gel, Eq. (A.16) has been used to compute total force and ultimately electrophoretic mobilities of highly charged spheres [7]. The presence of a gel complicates the problem. On the basis of Eqs. (A.6) and (A.8), we do not expect the fluid velocity to make a contribution to the surface integral on the rhs of Eq. (A.16) due to its x −3 distance dependence. However, the pressure term is expected to fall off as x −2 (Eq. (A.7)) and this does contribute to zT . In order to use Eq. (A.16) to compute zT , both v and p would have to be obtained by numerical solution. It is possible to avoid the necessity of determining p directly using a different strategy to compute zT [53]. Begin with the differential form of the Lorentz reciprocal theorem [54]

s T · v + ∇ · (v · σ H ) = s T · v  + ∇ · v  · σ H , (A.17) where unprimed quantities denote the actual fields, primed quantities denote some arbitrary field, and sT denotes the total external force/volume on the fluid due to electric and gel sources, s T = s e − ηλ2 v.

(A.18)

Both primed and unprimed fields must be solutions of Eq. (1) subject to the appropriate boundary conditions. For the primed fields, choose an uncharged particle of radius a (which is the same as the actual particle) translating with velocity u0 . Thus v is given by Eq. (A.1). Also choose λ = λ. Integrating Eq. (A.17) over all space, applying the divergence theorem and making use of Eqs. (A.9) and (A.14), 

 I − U (a, λ, x) · s e (x) dVx . z T = f u0 + (A.19) Ω

The external (electric) force/volume, se , can be written [12] s e (x) = s  (x) + ∇π(x), where s  (x) = q

 α

π(x) = kB T

zα nα0 (x)(∇Φα (x) + e),



nα (x),

(A.20)

(A.21) (A.22)

α

q is the protonic charge, the sum over α is over all mobile ion species present, nα0 (x) is the equilibrium concentration of α ions of valence zα at x, nα (x) is the corresponding nonequilibrium concentration of α ions, e is a constant external electric field, kB is the Boltzmann constant, and T is the temperature.

The potential, Φα , represents the departure of nα from its equilibrium value, nα0 , which arises due to the presence of an external electric or flow field [7,12]. nα (x) = nα0 (x)e−qzα (Φα (x)+ψ(x))/kB T

 ≈ nα0 (x) 1 − qzα Φα (x) + ψ(x) /kB T .

(A.23)

Let Λ(x) and Λ0 (x) denote the electric potential at x in the presence and absence of a perturbing electric or flow field, respectively. The potential Λ0 (x) arises from the interaction of the charge on the sphere (assumed to be spherically symmetric in the present work) with the mobile ions in the fluid. Defining a dimensionless equilibrium potential y0 (x) = qΛ0 (x)/kB T

(A.24)

we can write nα0 (x) = cα0 e−zα y0 (x) ,

(A.25)

where cα0 is the ambient concentration of α ions. The perturbing potential, ψ , appearing in Eq. (A.23) is related to Λ and Λ0 by Λ(x) = Λ0 (x) + ψ(x) − e · x.

(A.26)

Substituting Eq. (A.20) into (A.19), we obtain 

 I − U (a, λ, x) · s  (x) dVx , z T = f u0 + ω +

(A.27)

 ω=

Ω

 π(x) I − U (a, λ, x) · n(x) dSx

(A.28)

S∞

Far from Sp , Eqs. (A.22) and (A.23) yield 

cα0 kB T − qzα Φα (x) . π(x) ≈

(A.29)

α

Also, [I − U] → I far from Sp . Following O’Brain and White, the three dimensional potential, Φα (x), can be reduced to a one dimensional potential, φα (x) by (b is a constant vector) 1 (A.30) φα (x)b · x. x Choose S∞ in Eq. (A.26) to be a sphere of radius R (R  a or 1/λ). At large distance, φα has the asymptotic form (dα is a constant) Φα (x) =

φα (R) = dα /R 2 . Using Eqs.(A.29)–(A.31) in Eq. (A.28) 4π  cα0 zα dα . ω = − qb 3 α

(A.31)

(A.32)

In the above equation b = u0 (Case 1) or e (Case 2) and dα shall be determined from the boundary conditions (see Section 3). Finally, the volume integral in Eq. (A.27) can be reduced to a single integral over the radial variable, x, by carrying out the angular integrations. Making use of Eqs. (A.2)–(A.4), (21), (25), and (30) in (A.27) zT = f u0 + ω + t,

(A.33)

S.A. Allison et al. / Journal of Colloid and Interface Science 313 (2007) 328–337

  ∞ 4πq  ah1 2 −zα y0 zα cα0 x dxe t= 1− 2 3 3 α λ x a

  dφα ah2 − 2 3 e+ b . dx λ x (A.34) In Eq. (A.34), the variables y0 , h1 , h2 , and φα depend implicitly on x. 



2φα dφα + × 3e + b x dx



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