Importance of Ionic Polarization Effect on the Electrophoretic

ARTICLE pubs.acs.org/JPCC

Importance of Ionic Polarization Effect on the Electrophoretic Behavior of Polyelectrolyte Nanoparticles in Aqueous Electrolyte Solutions Li-Hsien Yeh,* Kuan-Liang Liu, and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617

bS Supporting Information ABSTRACT:

The electrophoresis of a polyelectrolyte, an entirely porous, charged nanoparticle, in various types of aqueous electrolyte solution is modeled taking account of the presence of multiple ionic species, and its applicability is verified by the experimental data of succinoglycan in the literature. We show that, in addition to the electroosmotic flow around a polyelectrolyte, two types of competing polarization effect are also significant: counterion polarization and co-ion polarization, both of them depend largely on the thickness of the double layer. The presence of these two polarization effects yields profound and interesting electrophoretic behaviors that are distinct to polyelectrolytes. The results gathered provide necessary theoretical background for the interpretation of various types of electrophoresis data in practice. Typical examples include that of a nanopore-based sensing device used, for instance, in DNA sequencing.

’ INTRODUCTION Electrophoresis has been adopted widely to characterize the physicochemical properties of colloidal particles. Among these, the polyelectrolyte,1
4 an entirely porous entity such as DNA, biomacromolecules, and synthetic polymers, belongs to a distinct category. Note that although a polyelectrolyte seems to be a limiting case of a soft particle, which comprises a rigid core and a porous layer, the behaviors of the two can be different remarkably5 due to, for example, the effect of counterion condensation in the latter being less significant than in the former. Recent advances in nanofabrication technique extend from conventional applications of electrophoresis to emerging ones such as nanopore sensing and DNA sequencing.6
8 To model more realistically and gain insights about the mechanisms involved in these applications, extending the analyses based on other types of particles to polyelectrolytes taking effects into account that might be significant in practice is highly desirable. These include, for example, the polarization of the double layer,5,9
11 the presence of boundaries,10
13 the electroosmotic flow surrounding the polyelectrolyte,5 and the condensation of counterions.5,14,15 r 2011 American Chemical Society

In addition, because the liquid phase usually contains various kinds of ionic species in practice16,17 taking account of these effects, which are almost always overlooked in relevant theoretical analyses, is necessary. Recently, several theories based on a continuum model18
22 and molecular dynamics23 were developed to describe the electrophoretic behavior of polyelectrolytes, and under constraints such as a low surface potential and without ionic polarization, the results obtained were in qualitative agreement with some of the experimental observations. Note that in the nanofluidic regime24 because the Debye screening length is often comparable to the linear size of a particle, the effect of double-layer polarization (DLP),25 one of the most interesting and important effects of electrophoresis, can be significant. However, it was not considered in the above-mentioned studies. Because an internal electric field is induced and its direction is opposite to the applied electric field, the effect of DLP is capable Received: September 27, 2011 Revised: December 12, 2011 Published: December 29, 2011 367

dx.doi.org/10.1021/jp2093498 | J. Phys. Chem. C 2012, 116, 367–373

The Journal of Physical Chemistry C

ARTICLE

of influencing both quantitatively and qualitatively the electrophoretic behavior of a particle, especially when its surface potential is high.5,9,25 Here, a continuum model comprising a Poisson equation for the electric potential, multi-ion Nernst
Planck equations for the ionic concentrations,26,27 and modified Navier
Stokes equations for the flow field, combined with a skillful superposition solution procedure5 is developed for the first time to investigate the electrophoresis of a polyelectrolyte with various charged properties and electrokinetic softness in various kinds of aqueous electrolyte solutions. This study aims to elaborate the electrophoretic behavior of a polyelectrolyte usually observed in experiments, especially the dependence of its electrophoretic mobility on the electrolyte concentration. We propose, for the first time, that the electrophoresis of a polyelectrolyte involves an important mechanism, co-ion and counterion polarization, which explains the dependence of its electrophoretic behavior on the types of electrolyte.

weak E, the equations governing the present problem can be summarized, in scaled forms, as follows:5,26 ðkaÞ2



∇ 2 ϕe ¼


N

N

∑ α21

α11 expð
α10 ϕe Þ
hQfix ∑ j¼1

ð2Þ

j¼1



ðkaÞ2



∇ 2 δϕ ¼

N

N

∑ α21

 α21 ðδϕ þ g j Þ expð
α10 ϕe Þ ∑ j¼1

ð3Þ

j¼1

  ∇ 2 gj ¼ α10 ∇ϕe 3 ∇gj þ Pej ðv 3 ∇ ϕe Þ, j

¼ 1, 2, :::, N 





ð4Þ










∇ δp þ ∇ 2 v þ ½ð∇ 2 ϕe þ hQfix Þ∇ δϕ

’ THEORY To simulate the problem under consideration, we consider a charged, permeable spherical polyelectrolyte of radius a moving with the velocity U as a response to a DC electric field E of strength E in an electrolyte solution, which is an incompressible Newtonian fluid containing N kinds of ionic species. The polyelectrolyte is homogeneously structured and bears dissociable function groups, which yield a uniform fix charge density Ffix. If we let zp and Np be the valence and the number concentration of the dissociated functional groups, respectively, then Ffix = zpeNp with e being the elementary charge. The model of Debye
Bueche28 is adopted, where the polyelectrolyte comprises polymer-like segments, and they are regarded as resistance centers distributed uniformly inside the polyelectrolyte, exerting frictional force on the liquid flowing through it. We assume the following: (i) The flow field is in the creeping flow regime. (ii) The applied electric field is relatively weak29 so that the magnitudes of all of the dependent variables of the second and higher order in E can be neglected.18 (iii) The permittivity of the liquid phase, ε, inside the polyelectrolyte is the same as that outside it.30 (iv) The diffusivity of ionic species j, Dj, inside the polyelectrolyte is the same as that outside it. For convenience, we let the liquid velocity at the center of the polyelectrolyte vanish, and that at a point far away from it is
U. To take account of the possible presence of the effect of double-layer polarization (DLP), the spatial number concentration of the ionic species j, nj, is assumed as5 " # zj eðϕ þ gj Þ nj ¼ nj0 exp
, j ¼ 1, 2, :::, N ð1Þ kB T









þ ∇ 2 δϕ ∇ ϕe 
hðλaÞ2 ðv Þ ¼ 0 

ð5Þ



∇ 3v ¼ 0

ð6Þ 

nj ¼ expð
α10 ϕe Þ½1
α10 ðδϕ þ gj Þ, j ¼ 1, 2, :::, N /

/2

ð7Þ ϕ/e

δϕ/ϕr, g/j

= gj/ϕr, Here r = ar, r = a r , = ϕe/ϕr, δϕ* = δp* = δp/[ε(ϕr)2/a2], v* = v/Ur, nj* = nj/nj0, and λ = (γ/η)1/2 with ϕr = kBT/ez1 and Ur = ε(ϕr)2/ηa. h is a step region function: h = 0 for the region outside the polyelectrolyte, and h = 1 for the region inside it. r2 is Laplace operator, v the fluid velocity, δp the external pressure, γ the hydrodynamic frictional coefficient of the 2 1/2 polyelectrolyte, and η the viscosity. k = [∑N j=1 nj0(ezj) /εkBT] 2 is the reciprocal Debye screening length, Qfix = Ffixa /εϕr the scaled fixed charge density of the polyelectrolyte, Pej = ε(ϕr)2/ ηDj the electric Peclet number of ionic species j, and αβγ = zβj nγj0/zβ1 nγ10, where subscript 1 denotes a reference ionic species, one of the ionic species in the system. λ
1 is the softness parameter of the polyelectrolyte material4 or a shielding length characterizing the extent of fluid flow penetrating into that material.31 For biocolloids, microorganisms, and polymer gels, λ
1 ranges typically from 0.1 to 10 nm.4,30 To specify the boundary conditions associated with eqs 2
7, we assume the following. (a) The electrical potential, the ionic concentration of species j, and the fluid velocity are all finite inside the polyelectrolyte, implying that ϕ/e , δϕ*, g/j , and v* are all finite in that region. (b) The electrical potential, the electric field, and both the concentrations and the fluxes of all ionic species are continuous on the polyelectrolyte-liquid interface.18 (c) Both the normal and the tangential components of the fluid velocity and the stress tensor are continuous on the polyelectrolyte-liquid interface.5,18 (d) The electric, the concentration, and the flow fields far away from the polyelectrolyte are uninfluenced by its presence, implying that n 3 r*ϕ/e = 0, n 3 r*δϕ* =
E* cos θ, g/j =
δϕ*, and v* =
U*ez at a point far away from the polyelectrolyte, where E* = E/Er, Er = ϕr/a, and U* = U/Ur with n, θ, and ez being the unit normal vector directed into the fluid phase, the solid angle in polar coordinates, and the unit vector in the direction of E, respectively. For convenience, the present problem is decomposed into two subproblems:5 (I) the polyelectrolyte moves with a constant

where ϕ, gj, nj0, zj, kB, and T are the space electric potential, a hypothetical perturbed potential, the bulk number concentration and the valence of ionic species j, the Boltzmann constant, and the absolute temperature, respectively. For an easier mathematical treatment and physical interpretation of the results obtained, each dependent variable is partitioned into an equilibrium term, symbol with subscript e, and a perturbed term, symbol with prefix δ, representing the values of that variable in the absence of E (or equilibrium value) and arising from the application of E, respectively.5 For example, ϕ = ϕe + δϕ, where ϕe and δϕ are the equilibrium potential and the perturbed potential, respectively. Under the conditions of 368

2

2

dx.doi.org/10.1021/jp2093498 |J. Phys. Chem. C 2012, 116, 367–373

The Journal of Physical Chemistry C

ARTICLE

Table 1. Ionic Diffusivities and Corresponding Electric Peclet Numbers of Five Common Electrolyte Solutionsa Na+

K+

H+

NO3


Cl


OH


Diffusivity (Dj  10
9m2/s) 1.33

1.96

9.31

1.90

2.03

5.30

Species

Peclet number (Pej)

0.351 0.238 0.050 0.246 0.230 0.088

a Key: z1 = 1, ε = 7.08  10
10 CV
1m
1, kB = 1.38  10
23 J/K, T = 298 K, η = 10
3 Kgm
1s
1, and e = 1.6  10
19 C.

species was neglected in their study. Note that if the effect of DLP is not taken into account (dashed curve), then the simulated mobility increases monotonically with decreasing ionic strength, and its trend deviates appreciably from the experiment data of Duval et al.,20 especially when the ionic strength is small. This is because the closer the ka to unity the more significant the effect of DLP,5 and at ka = 1, the percentage difference in the mobility can be on the order of 50% if the presence of multiple ionic species is neglected. As seen in Figure 1, due to the presence of the effect of DLP the slope of μ* does not vary monotonically at ka = 4. This effect, which comes from the convective motion of ionic species, is capable of affecting considerably the electrophoretic behavior of a particle, especially in the nanofluidic applications such as nanopore-based technologies.6
8 Therefore, investigating in detail its dependence on the solution conditions and unraveling relevant mechanisms are necessary. To this end, numerical simulations are performed under various conditions to examine the electrophoretic behavior of a polyelectrolyte. The key factors include the physical properties of the polyelectrolyte, such as its charged conditions and electrokinetic softness, and the types of electrolyte species in an aqueous solution often used in experiments, such as HCl, KCl, NaCl, NaNO3, and NaOH. For illustration, we assume that a = 10 nm and E* = 0.01. The diffusivities and the corresponding electric Peclet numbers of all of the ionic species considered are summarized in Table 1. Figure 2 summarizes the influences of the types of electrolyte solution, the bulk ionic concentrations, the thickness of the double layer (measured by ka), and the charge density and the softness parameter of the polyelectrolytes (measured by Qfix and λa, respectively) on its scaled mobility μ*. Because a is fixed, the variations of ka and λa come from those of the bulk ionic concentration and the hydrodynamic permeability of the polyelectrolyte, respectively. As seen in Figure 2, the qualitative behavior of μ* depends highly on the level of λa. If λa is small, corresponding to the case of a loose-structured polyelectrolyte, |μ*| has a local minimum as ka varies. This behavior was observed experimentally in a study of the capillary electrophoresis of carboxylated nanolatexes (Figure 3a of Oukacine et al.32) and that of the salt dependence of DNA translocation through a solid-state nanopore (Figure 4b of Smeets et al.33), where the translocation time verses KCl concentration was shown, and can be explained by the presence of DLP effect.5 On the other hand, if λa is sufficiently large, corresponding to a more compact polyelectrolyte structure, then |μ*| decreases monotonically with increasing ka, which is often observed in the case where the characteristic size of a polyelectrolyte (i.e., a) is large or its hydrodynamic structure is compact.4 This behavior can be explained by the fact that the effect of DLP is offset by the electroosmotic flow coming from the convective flow of counterions inside the double layer. Note that as ka gets large, the mobility of the polyelectrolyte approaches the analytical solution of Hermans and Fujita,34 a nonzero constant value, which is

Figure 1. Variation of the scaled mobility μ* as a function of NaCl ionic strength (and the corresponding ka) for the case of a negatively charged spherical polyelectrolyte of radius a = 10.8 nm at pH 10.3. Solid (dashed) curve: present result with (without) the effect of multiple ionic species double layer polarization at Ffix/F =
236.9 mol/m3 and λ
1 = 0.554 nm; discrete symbols: experimental data of Duval et al.30

scaled velocity U* in the absence of E and (II) E is applied but the polyelectrolyte is held fixed. This treatment has the advantage that a tedious trial-and-error procedure can be avoided in the evaluation of the scaled electrophoretic mobility of the polyelectrolyte μ* (= U*/E*).5 A more detailed solution procedure is given in the Supporting Information.

’ RESULTS AND DISCUSSION FlexPDE (version 4.24, PDE Solutions, Spokane Valley WA), which was found to be sufficiently efficient and accurate for solving similar problems,5,26,29 is adopted to solve the present problem numerically. The applicability of the present theoretical model is first verified by the experimental data of Duval et al.,30 where the electrophoretic behavior of succinoglycan of radius 10.8 nm, viewed as a spherical polyelectrolyte, at pH 10.3 in a capillary electrophoresis apparatus, was conducted. Because the functional groups of the polyelectrolyte at this high level of pH dissociated almost completely it can be assumed to maintain at a fixed charge density. It should be pointed out that previous theoretical studies almost always assumed that the liquid phase contains cations and anions, one kind each only. While this may be adequate for a medium level of pH and if the presence of the other background ionic species is negligible, it becomes unrealistic, otherwise, as in the case of Duval et al.,30 where the medium pH deviates appreciably from 7. To take this effect into account, we consider four major kinds of ionic species (i.e., N = 4), Na+, Cl
, H+, and OH
. The dependence of the scaled electrophoretic mobility of the polyelectrolyte μ* on NaCl ionic strength is shown in Figure 1. For comparison, the corresponding result for the case where only two kinds of ionic species is considered is also presented (dashed curve). As seen in this figure, the present model (solid curve) describes successfully the general trend of the mobility of the particle. The values of the adjustable parameters are Ffix/F =
236.9 mol/m3 and λ
1 = 0.554 nm, where F is Faraday constant. The estimated value of λ
1 is smaller than that of Duval et al.,30 λ
1 = 0.7 nm. This is expected because the effect of DLP arising from the convective motion of multiple ionic 369

dx.doi.org/10.1021/jp2093498 |J. Phys. Chem. C 2012, 116, 367–373

The Journal of Physical Chemistry C

ARTICLE

Figure 3. Variations of the net perturbed ionic number concentration difference, (δn1
δn2), at R = r/a = 0 as a function of Z (=z/a), the scaled distance in the direction of E, at various values of λa in an aqueous NaCl solution for the case of Figure 2 at Qfix =
20 and ka = 1.

Figure 4. Variations of the scaled forces in subproblem two, F/e2, F/h2, and F/2 (= F/e2 + F/h2), as a function of ka at various values of λa for the case of an aqueous NaCl solution in Figure 2a,c.

(δn1
δn2)= [(n1
n1e)
(n2
n2e)] with δnj and nje being the perturbed and the equilibrium number concentrations of ionic species j, j = 1,2, respectively, along the axis of r = 0 near the polyelectrolyte. (δn1
δn2) is a measure for the degree of double-layer polarization (DLP).5 As seen in this figure, (δn1
δn2) is positive (negative) near the north (south) pole of the polyelectrolyte, implying that there is an excess of perturbed cations (anions) near the north (south) pole, yielding an internal electric field, the direction of which is opposite to that of E. This phenomenon is known as DLP, which has the effect of reducing the electrical driving force acting on a polyelectrolyte and making its mobility smaller. Figure 3 also reveals that the larger the value of λa (i.e., more compact polyelectrolyte structure) the less the degree of DLP. This is because the more compact the structure of a polyelectrolyte the less significant is the convective flow of ionic species inside. Figure 4 illustrates the variations of the scaled electrical force F/e2, the scaled hydrodynamic force F/h2, and the scaled net driving force F/2 (= F/e2 + F/h2) acting on a polyelectrolyte in the second subproblem at varies values of ka for various values of λa.

Figure 2. Variations of the scaled mobility μ* as a function of ka at various values of λa for various types of aqueous electrolyte solution. Solid curves with symbols: negatively charged polyelectrolytes with Qfix =
20; dashed curves with symbols: positively charged polyelectrolytes with Qfix = 20; dotted curves: analytical results of Hermans and Fujita,34 where ka f ∞. (a) λa = 1, (b) 3, and (c) 30.

distinct to a spherical polyelectrolyte. This also verifies that the asymptotic behavior of the present model is correct. The specific behavior of μ* at various levels of λa seen in Figure 2 can be further explained by Figures 3
5. Figure 3 illustrates the typical net perturbed ionic concentration difference, 370

dx.doi.org/10.1021/jp2093498 |J. Phys. Chem. C 2012, 116, 367–373

The Journal of Physical Chemistry C

ARTICLE

Figure 5. (a
c) Contours of the scaled fluid velocity v* near a negatively charged polyelectrolyte in subproblem two at various combinations of ka and λa for the case of an aqueous NaCl solution in Figure 2 at Qfix =
20. (d) Velocity gradient near a loose-structured polyelectrolyte (small λa). (e) Velocity gradient near a compact-structured polyelectrolyte (large λa). (a) ka = 1 and λa = 1, (b) ka = 10 and λa = 1, and (c) ka = 1 and λa = 30.

As seen, |F/e2| decreases with increasing ka, and its slope does not vary monotonically in the region 10 and it has a local maximum as ka varies. These can be explained by the competition between the pressure and the viscous components of F/h2, F/h2,(p), and F/h2,(v), respectively.5 As illustrated in Figure 5, F/h2,(v) is dominated by the electroosmotic flow of counterions inside the double layer. The negative value of F/h2,(v) arises from that, driven by the equilibrium electric field, fluid can flow easily through the polyelectrolyte, and the rate of strain across its surface is negative, as can be seen in Figure 5a,b, and is schematically represented in Figure 5d. Figure 5, panels a and b, also suggests that the larger the ka (thinner double layer) the greater the amount of counterions (co-ions) condensed inside (outside) the polyelectrolyte, thereby enhancing the electroosmotic flow around it, and, therefore, the greater the |F/h2,(v)| and |F/h2|. On the other hand, because it is relatively uneasy for fluid to penetrate into a compactly structured polyelectrolyte, the rate of strain across its surface becomes positive, as seen in Figure 5c, and is schematically represented in Figure 5e. It is interesting to see in Figure 2 that, for the same background electrolyte concentration, the |μ*| of a positively charged polyelectrolyte is different from that of a negatively charged polyelectrolyte with the same charge density, which is unexpected and has never been reported in the literature. This arises from the effect of counterion condensation,14 where most of counterions are attracted into the polyelectrolyte and most of co-ions are expelled from it, making the concentration of the mobile

counterions inside the polyelectrolyte always higher than that of the co-ions. Therefore, |μ*| is dominated mainly by the convective motion of the mobile counterions inside the polyelectrolyte. We define this as the effect of counterion polarization. As illustrated in Figure 6a, due to the application of E the concentration of the perturbed counterions (cations) inside the double layer near the top region of the polyelectrolyte is positive, and is negative near its bottom region. This yields a local electric field Ecounterion, the direction of which is opposite to that of E. Because the smaller the diffusivity of counterions the more significant the effect of counterion polarization, and, therefore, the smaller the |μ*|. This, however, is unable to explain another interesting phenomenon in Figure 2, that is, the values of μ* with the same counterions can be different appreciably if the corresponding co-ions are different. In these cases, another mechanism, the effect of co-ion polarization, which comes from the convective motion of the mobile co-ions inside the double layer, plays the key role. As shown in Figure 6b, due to the electrostatic interaction between counterions and the co-ions, the flow of the mobile counterions inside the double layer making the co-ions flowing simultaneously. Therefore, the perturbed concentration of the co-ions (anions) near the top region of the polyelectrolyte is also positive and that near its bottom region negative. This yields an induced electric field Eco‑ion, which has the same direction as E, implying that the effect of co-ion polarization is capable of raising |μ*|. Because the direction of the convective motion of the unbalanced co-ions inside the double layer is opposite to that of the unbalanced counterions an extra drag force is present for that movement of counterions. Consequently, the larger the ionic diffusivity of co-ions the less significant that drag force, and, therefore, the more significant the effect of coion polarization and the larger the mobility of the polyelectrolyte is. Based on the results summarized in Figure 2, we conclude that the mobility of a polyelectrolyte (μ*) depends largely upon its charged nature (Qfix), the bulk ionic concentration (ka), and the kind of ionic species in the liquid phase. If we let μ/k be the scald electrophoretic mobility of a polyelectrolyte in an aqueous solution containing electrolytes k, then for a 371

dx.doi.org/10.1021/jp2093498 |J. Phys. Chem. C 2012, 116, 367–373

The Journal of Physical Chemistry C

ARTICLE

Figure 6. Variations of the perturbed number concentrations of cations δn1 and anions δn2 at R(= r/a) = 0 as a function of Z (=z/a), the scaled distance in the direction of E, in subproblem two for a negatively charged polyelectrolyte at Qfix =
20, λa = 1, and ka = 1 in various types of aqueous electrolyte solution, (a and b), and that for a positively charged polyelectrolyte at Qfix = 20 and λa = 1 at two levels of ka, (c and d). Insert: typical contours of δn1 and δn2 near the polyelectrolyte in an aqueous NaCl solution for the illustrated case of Qfix =
20 and ka = 1, (a and b), and that of Qfix = 20 and ka = 10, (c and d).

negatively charged polyelectrolyte at small ka, |μ/k | ranked as |μ/HCl| > |μ/KCl| > |μ/NaOH| > |μ/NaCl| > |μ/NaNO3|, and as |μ/NaOH| > |μ/HCl| > |μ/KCl| > |μ/NaCl| > |μ/NaNO3| for a positively polyelectrolyte. As verified by Figure 6, this is because if ka is small, then the effect of counterion condensation, which suppresses the double layer, is significant, and therefore, the rank of |μk*| is dominated mainly by the effect of counterion polarization, and for identical counterion, it becomes dominated by the effect of co-ion polarization. On the other hand, if ka is sufficiently large, then |μ/k | ranked as |μ/HCl| > |μ/NaOH| > |μ/KCl| > |μ/NaCl| > |μ/NaNO3|, regardless of the charged nature of the polyelectrolyte. As proposed by Yeh and Hsu,3 this arises from the fact that as ka gets large (double layer gets thin) the effect of counterion condensation suppressing the double layer becomes relatively insignificant. In this case, because most of the unbalanced counterions (co-ions) gather near the inner (outer) surface of the polyelectrolyte, making the effect of co-ion polarization comparable to that of counterion polarization, and affecting appreciably the behavior of |μ/k |, as can be seen in Figure 6, panels c and d.

most of the effects that might be significant in practice, in particular, the double-layer polarization (DLP), the presence of multiple ionic species, and the types of ionic species. Because these effects were usually neglected and/or overlooked for simplicity in previous theoretical analyses, they are unable to interpret many interesting experimentally observed behaviors. For instance, if the presence of multiple ionic species is neglected, the particle mobility decreases monotonically with increasing ionic strength, which might not be true for the case when that effect is taken into account, especially if the thickness of double layer is comparable to the particle radius. We also show that two types of ionic polarization, which are distinct to polyelectrolytes, are present: counterion polarization and co-ion polarization, both are capable of influencing appreciably the electrophoretic behavior of a polyelectrolyte. The former (latter) has the effect of reducing (raising) its mobility. The presence of those two types of ionic polarization is capable of explaining the influence of ionic species on the electrophoretic behavior of a polyelectrolyte. Taking an asymmetric aqueous NaCl solution (the electric Peclet numbers of Na+ and Cl
are 0.351 and 0.23, respectively) as an example, the difference between the mobility of a positively charged polyelectrolyte and that of a negatively one can be on the order of 20%.

’ CONCLUSIONS In summary, a general model for the electrophoresis of a polyelectrolyte nanoparticle is proposed taking account of 372

dx.doi.org/10.1021/jp2093498 |J. Phys. Chem. C 2012, 116, 367–373

The Journal of Physical Chemistry C

ARTICLE

’ ASSOCIATED CONTENT

bS

Supporting Information. Details of the solution procedure for the superposition method. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail: jphsu@ ntu.edu.tw.

’ ACKNOWLEDGMENT This work is supported by the National Science Council of the Republic of China. ’ REFERENCES (1) Noda, I.; Nagasawa, M.; Ota, M. J. Am. Chem. Soc. 1964, 86, 5075. (2) Ogawa, K.; Nakayama, A.; Kokufuta, E. J. Phys. Chem. B 2003, 107, 8223. (3) Viovy, J. L. Rev. Mod. Phys. 2000, 72, 813. (4) Duval, J. F. L.; Gaboriaud, F. Curr. Opin. Colloid Interface Sci. 2010, 15, 184. (5) Yeh, L. H.; Hsu, J. P. Soft Matter 2011, 7, 396. (6) Howorka, S.; Siwy, Z. Chem. Soc. Rev. 2009, 38, 2360. (7) Dekker, C. Nat. Nanotechnol. 2007, 2, 209. (8) Kowalczyk, S. W.; Hall, A. R.; Dekker, C. Nano Lett. 2010, 10, 324. (9) Tsai, P.; Lee, E. Soft Matter 2011, 7, 5789. (10) Zhang, M. K.; Ai, Y.; Kim, D. S.; Jeong, J. H.; Joo, S. W.; Qian, S. Z. Colloid Surf. B-Biointerfaces 2011, 88, 165. (11) Yeh, L. H.; Fang, K. Y.; Hsu, J. P.; Tseng, S. Colloid Surf. B-Biointerfaces 2011, 88, 559. (12) Qian, S. Z.; Joo, S. W.; Hou, W. S.; Zhao, X. X. Langmuir 2008, 24, 5332. (13) Yalcin, S. E.; Lee, S. Y.; Joo, S. W.; Baysal, O.; Qian, S. J. Phys. Chem. B 2010, 114, 4082. (14) Manning, G. S. J. Chem. Phys. 1969, 51, 924. (15) Manning, G. S. J. Phys. Chem. B 2007, 111, 8554. (16) Onsager, L. Phys. Z. 1926, 27, 388. (17) Onsager, L. Phys. Z. 1927, 28, 277. (18) Ohshima, H. Adv. Colloid Interface Sci. 1995, 62, 189. (19) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 258, 56. (20) Duval, J. F. L.; Ohshima, H. Langmuir 2006, 22, 3533. (21) Ghosal, S. Phys. Rev. Lett. 2007, 98. (22) Dukhin, S. S.; Zimmermann, R.; Duval, J. F. L.; Werner, C. J. Colloid Interface Sci. 2010, 350, 1. (23) Grass, K.; Bohme, U.; Scheler, U.; Cottet, H.; Holm, C. Phys. Rev. Lett. 2008, 100. (24) Schoch, R. B.; Han, J. Y.; Renaud, P. Rev. Mod. Phys. 2008, 80, 839. (25) Obrien, R. W.; White, L. R. J. Chem. Soc. Faraday Trans. 2 1978, 74, 1607. (26) Hsu, J. P.; Tai, Y. H. Langmuir 2010, 26, 16857. (27) Hsu, J. P.; Tai, Y. H.; Yeh, L. H.; Tseng, S. J. Phys. Chem. B 2011, 115, 3972. (28) Debye, P.; Bueche, A. M. J. Chem. Phys. 1948, 16, 573. (29) Yeh, L. H.; Hsu, J. P.; Tseng, S. J. Phys. Chem. C 2010, 114, 16576. (30) Duval, J. F. L.; Slaveykova, V. I.; Hosse, M.; Buffle, J.; Wilkinson, K. J. Biomacromolecules 2006, 7, 2818. (31) Zembala, M. Adv. Colloid Interface Sci. 2004, 112, 59. (32) Oukacine, F.; Morel, A.; Cottet, H. Langmuir 2011, 27, 4040. (33) Smeets, R. M. M.; Keyser, U. F.; Krapf, D.; Wu, M. Y.; Dekker, N. H.; Dekker, C. Nano Lett. 2006, 6, 89. (34) Hermans, J. J.; Fujita, H. K. Nederl. Akad. Wet. Proc. Ser. B 1955, 58, 182. 373

dx.doi.org/10.1021/jp2093498 |J. Phys. Chem. C 2012, 116, 367–373