On the Influence of Ion Excluded Volume (Steric) Effects on the Double

Apr 20, 2010 - Department of Mechanical Engineering, University of Nevada, Las Vegas, ... Advances in Colloid and Interface Science 2017 244, 36-53 ...
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J. Phys. Chem. C 2010, 114, 8389–8397

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On the Influence of Ion Excluded Volume (Steric) Effects on the Double-Layer Polarization of a Nonconducting Spherical Particle in an AC Field Hui Zhao* Department of Mechanical Engineering, UniVersity of NeVada, Las Vegas, NeVada 89154 ReceiVed: December 29, 2009; ReVised Manuscript ReceiVed: March 8, 2010

The dipole moment of a charged, dielectric, spherical particle under the influence of a uniform alternating electric field is computed by solving the modified Poisson-Nernst-Planck (PNP) equations (Bikerman’s mean-field model) (Philos. Mag. 1942, 33, 384-397) accounting for excluded volume effects of the finite ion size as a function of the double-layer thickness, the electric field frequency, the particle’s surface charge, and the volume fraction of the ions in the bulk characterizing the excluded volume repulsion. In the limit of thin electric double layers, we carry out an asymptotic analysis to develop simple models calculating dipole moments which are in favorable agreement with the modified PNP model. Our results reveal that excluded volume effects, imposing a maximum on the counterion concentration, reduce the dipole moment at high frequencies and possibly enhance the dipole moment at low frequencies, assuming that the particle bears the same zeta potential. Excluded volume effects often become significant in highly concentrated salt solutions or near highly charged surfaces. The modified PNP model considering the ion size generally improves the theoretical predictions in comparison to experimental data, and a possible explanation for such improvement is suggested. 1. Introduction With the advance and development of microfabrication and nanotechnology, electrokinetic phenomena attract increasing interest and find various new applications in lab-on-a-chip devices and other emerging technologies including nextgeneration electronics and photovoltaic cells.2-8 In particular, due to its salient features (i.e., low voltage, the precise ability to control magnitude of the forces9), dielectrophoresis, where a particle can migrate toward predetermined locations by judiciously generating nonuniform AC electric fields,10-12 has been widely used to manipulate, separate, position, and assemble nanoparticles13-16 and biomolecules.17,18 When a charged, dielectric particle is suspended in an electrolyte solution, counterions are attracted to form an electric double layer surrounding the charged particle.19,20 An applied electric field redistributes counterions inside the double layer and therefore polarizes the double layer. The double-layer polarization accordingly modifies the charge distribution along the particle which when viewed far away appears as a dipole moment. Understanding of the particle’s polarization and the ability to precisely predict the dipole moment characterizing the polarization is of importance to design effective devices to manipulate nanoparticles and biomolecules. The standard electrokinetic theory consists of the classical Poisson-Nernst-Planck (PNP) equations to describe the doublelayer polarization. The PNP model has been used extensively to calculate the dipole moment of spherical particles,21-32,34-36 long (two-dimensional) cylindrical particles with the electric field transverse to their axis,37 and elongated cylindrical particles with the electric field along their axis.38 The standard PNP model predicted high-frequency and low-frequency dispersions, explained the disappearance of the low-frequency dispersion in the presence of thick double layers,35,39 and favorably agreed * Corresponding author. E-mail: [email protected].

with experiments on the polarization of short rod-shaped doublestranded DNA molecules in a dilute solution.38 Despite successful qualitative agreements with experiments, frequently the PNP model underestimated the low-frequency dispersion amplitude and the characteristic frequency.40-42 Moreover, recent experiments revealed that the dielectrophoretic tension inside actin filaments induced by the polarization under the action of an AC field is orders of magnitude larger than that predicted by the PNP model in a concentrated solution.18 The constant discrepancy between the theory and experiments suggests that certain key underlying physics might be missing in the PNP model. A fundamental assumption of the PNP model is that the electrolyte solution is dilute, and ions are treated as noninteracting, point-size charges. In the absence of an external force, the PNP equations can be integrated to the famous PoissonBoltzmann distribution which describes the ion concentration inside the double layer. However, the Boltzmann distribution predicts a nonphysical counterion concentration at high zeta potentials (ζ*).43,44 No doubt, the finite size of ions sets a maximum on the counterion concentration. With this maximum concentration constraint, a condensed layer of counterions likely appears near the charged surface. It is also recognized that the PNP model cannot predict the existence of such a layer. Indeed, this shortcoming of the PNP model was noticed a long time ago. Attempts to account for excluded volume effects characterizing the repulsion among finite-sized ions were made by proposing modified chemical potentials to develop modified PNP models. For a detailed literature review on various modified PNP models and their history, interested readers are referred to Bazant et al.45 (termed steric effects in their papers). So far, the modified PNP model has been solved near an infinitely long charged plate,43,44,46,47 an infinitely long charged cylinder,48,49 and a charged sphere.50,51 The modified PNP model has been used to successfully explain the flow reversal in AC electroosmosis at high frequency observed in experiments which

10.1021/jp101121z  2010 American Chemical Society Published on Web 04/20/2010

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Zhao In the above, all variables are dimensionless. C( are ion concentrations, and φ is the electric potential. We use R*T*/F* as the electric potential scale, the bulk concentration C*0 as the concentration scale, and R*T* as the chemical potential scale. R* is the ideal gas constant; T* is the temperature; and F* is the Faraday constant. Throughout the manuscript, the superscript * denotes the dimensional form of the various variables. Variables without the superscript * are dimensionless. At equilibrium (∇µ( ) 0), one can easily integrate eq 1 and obtain the classical Boltzmann distribution

Figure 1. Schematic of the geometry and the coordinate system.

C( ) e-φ

(2)

52,53

otherwise cannot be predicted by the standard PNP model. It also predicted that the usual maximum mobility based on the PNP model disappears, and excluded volume effects significantly increase the electrophoretic mobility at large zeta potentials, attributed to the reduction of the surface conduction by excluded volume effects.54,55 Recall that the particle’s polarization was generally underestimated by the standard PNP model and consider that excluded volume effects can increase the electrophoretic mobility. It is worthwhile investigating excluded volume effects on the dipole moment. So far only Aranda et al.56,57 have considered excluded volume effects on a particle’s polarization by solving the modified PNP model for a range of bulk concentrations and surface charges. In contrast, here we perform a detailed, physically based asymptotic analysis on the modified PNP model and derive simple analytical models to calculate dipole moments over a broad range of frequency in the limit of thin double layers. In particular, we extend the standard Dukhin-Shilov theory accounting for the effect of ion diffusion to include the excluded volume effects, and an approximate expression for the dipole moment is derived using the modified PNP model for the low-frequency range and thin double layers where excluded volume effects are expected to have the most prominent impacts. In addition, we also calculate the dipole moments using zeta potentials as the boundary condition. The use of zeta potentials helps us to identify and capture different consequences of excluded volume effects on ion migration, convection, and diffusion, respectively, with respect to a particle’s polarization. The manuscript is organized as follows. The modified PNP model accounting for ion excluded volume effects is introduced in section 2. In section 3, we solve the modified PNP model with a regular perturbation expansion in terms of the applied electric field. In section 4, in the limit of thin double layers, we derive simple, approximate expressions for the dipole moment coefficient. In section 5, we discuss, respectively, the results using zeta potentials and surface charges as boundary conditions. Section 6 concludes. 2. Modified Poisson-Nernst-Planck Model Consider a uniformly negatively charged spherical particle with radius a* and permittivity ε*2 freely suspended in a symmetric 1-1 electrolyte with permittivity ε*1 (Figure 1). In the absence of an external electric field, ions are at equilibrium. In other words, the electrochemical potential µ( of each species is spatially uniform where the subscripts (+) and (-) denote, respectively, the cations and anions. In the dilute solution theory, assuming that ions are independent pointlike charges, the chemical potential of an ion admits the simple form19,20

µ( ) ln C( ( φ

(1)

The Poisson equation is reduced to

∇2φ ) -

(C+ - C-) 2λD2

)

sinh(φ) λD2

(3)

where λD ) (1/a*)[(ε*1 R*T*)/(2F*2C*0 )]1/2 is the dimensionless Debye screening length normalized with the particle’s radius a*. Equation 3 is often referred to as the Poisson-Boltzmann (PB) equation and can be solved by specifying either the zeta potential ζ or the surface charge σ on the particle’s surface. Equation 2 indicates that the Boltzmann distribution gives an unrealistic counterion concentration close to the surface at high zeta potentials. To correct this deficiency, a modified PNP model accounting for excluded volume effects of the finite ion size has been developed by assuming that ions and water molecules occupy a square cell with a uniform grid size 43-46,58 In this modified PNP model, the modified electroa*. i chemical potential is given

µ( ) ln C( ( φ - ln(1 - ν/2(C+ + C-))

(4)

The last term in eq 4 represents excluded volume effects, and ν ) 2C*0 a*3 i is the effective volume fraction of bulk ions. This term imposes a constraint on counterion concentrations inside the double layer which cannot exceed 2/ν to ensure the existence of such chemical potentials. The equilibrium ion concentrations can be derived analytically by assuming ∇µ( ) 0, and they are

C( )

e-φ 1 + 2ν sinh2(φ/2)

(5)

Equation 5 indicates that when φ is large the counterion concentration saturates at 2/ν. Substituting eq 5 into the Poisson eq 3, one ends up with

∇2φ )

λD2(1

sinh(φ) + 2ν sinh2(φ/2))

(6)

The corresponding boundary conditions are

φ ) ζ at r ) 1 or

(7)

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dφ ) σ at r ) 1 dr

(8)

φ ) 0 at r f ∞

(9)

and

When ν ) 0, eq 6 reduces to the standard Poisson-Boltzmann eq 3. Unfortunately, eq 6 cannot be further solved analytically, and one has to resort to numerical techniques. At high surface charges or large zeta potentials, counterions saturate near the particle’s surface and collapse into a condensed layer. Inside this condensed layer, the co-ion concentration is negligible, and the counterion concentration is presumably equal to its maximum concentration (2/ν). Under these conditions, the approximate electric potential inside this condensed layer can be derived analytically with boundary conditions 7 and 8, and the solution is

φ)-

σ 1 r2 1 + - 2 +ζ-σ+ 2 2 r 6λDν 3λDνr 2λDν

(10)

Figure 2a depicts, respectively, the approximate electric potential (dashed lines) computed by eq 10 and that of eq 6 (solid lines) as a function of r for several surface charges when ν ) 0.2 and λD ) 0.1. Equation 10 approximates the electric potential very well inside the condensed layer where counterions saturate. However, the zeta potential and the surface charge are not independent. Only one of them can be specified to solve eq 6, and the other is determined after the electrical potential is calculated by eq 6. Therefore, eq 10 cannot be used to predict the electric potential inside the condensed layer a priori. But importantly, following the approach laid out in Lopez-Garcia et al.,51 one can estimate the thickness of the condensed layer by calculating the location of the local minimum of eq 10 ((dφ)/ (dr) ) 0). Easily, it is predicted to be

lc ) (1 + 3λD2ν|σ|)1/3 - 1

(11)

Figure 2b plots the counterion concentration as a function of r when ν ) 0.2 and λD ) 0.1. The symbols denote the “edge” of the condensed layer estimated by eq 11. Evidentially a condensed layer with a saturated counterion concentration of 2/ν ) 10 is formed. In addition, let λDf0, and eq 11 yields lc ∼ λD2ν|σ|. Next, we assume that the total charge inside the condensed layer balances the surface charge of the particle. According to the Gauss law, σ ∼ (ζ/lc). Hence, lc grows like |ζ|1/2 which is in agreement with Kilic et al.43 and Khair and Squires54 using different arguments. In summary, the modified PNP model successfully prohibits the unbounded growth in counterions and predicts a condensed layer in contact with the surface besides the traditional double layer. 3. Dipole Moment Coefficient Now we consider that a uniform AC electric field -E0ejωteˆz is imposed far from the dielectric, charged particle. The electric field drives the charged particle to oscillate around its center b (t) ) U0ejωt parallel to the electric field. To with a velocity U facilitate the computation, we use the Galilean transformation to fix the origin of the coordinate system at the particle’s center.

Figure 2. (a) Electric potential of the equilibrium double layer as a function of r for several surface charges. (b) The counterion concentration of the equilibrium double layer as a function of r for several surface charges. The solid and dashed lines in (a) correspond, respectively, to the electric potential predicted by the modified PNP model and that approximated by eq 10. The symbols in (b) denote the “edges” of the condensed layers estimated by eq 11.

b is not known a priori and needs to be The particle’s velocity U determined as part of the solution process. We use the spherical (r, θ, φ) coordinate system with its origin fixed at the center of the particle. In the above, θ is the angle between eˆr and eˆz. Figure 1 depicts the geometry and the coordinate system. To calculate the dipole moment of the spherical particle in the presence of the AC electric field, one needs to solve a set of partial different equations: the Stokes equation for the velocities; the Poisson equation for the electric potential; the modified Nernst-Planck equations accounting for excluded volume effects for each ion species, termed the modified PNP model. The modified Nernst-Planck equations are, respectively44

∂C νC∇C - mu b · ∇C ) ∇ · ∇C + F∇φ + ∂t 1 - νC

(

)

(12) and

∂F νF∇C - mu b · ∇F ) ∇ · ∇F + C∇φ + ∂t 1 - νC

(

)

(13)

To derive eq 12 and eq 13, one assumes that ionic diffusivities of cation and anion are the same. In the above, m ) (ε*1 R*2T*2)/ (F*2η*D*) + is the cation’s mobility; η*is the solvent’s dynamic viscosity; b u is the liquid’s velocity vector; and D*+ is the cation’s diffusivity. To facilitate the analytical analysis, we have defined

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the local salt concentration C ) (C+ + C-)/2 and the local charge density F ) (C+ - C-)/2 to replace ion concentrations C(. Here we consider a nonconducting, spherical particle subjected to a weak, external electric field. In other words, the external electric field disturbs only slightly the electric potential and the ion concentrations of the equilibrium electric double layer. Thus, one can use a regular perturbation expansion in terms of the applied electric field about the equilibrium double layer, and the dipole moment can be computed from the firstorder electric potential far from the particle. Witness that the first-order equations are linear in the perturbed quantities. Accounting for the linearity of the firstorder governing equations, one can decompose the first-order problem into two sub problems: (1) the E problem consisting of a sphere held stationary in the presence of the same electric field as the original problem at infinity and (2) the U problem consisting of a sphere at rest in a uniform flow field in the absence of the external electric field.35,36,59 Accordingly, the solutions of the first-order problem can be written as the superposition

X(1) ) (X(1)E + U0X(1)U)eiωt

(14)

Consider that most equations (but not the Nernst-Planck equations) are identical to these in Zhao and Bau35 and Zhao.36 For brevity, we do not present detailed first-order equations and only present the first-order modified Nernst-Planck equations

(

increases. The mesh was refined a few times to guarantee that the computational results are mesh-independent. The particle and its adjacent electric double layer perturb the electric field. Far from the particle, the perturbed field appears like a field induced by a dipole, and the electric potential admits the form E0(-r + f/r2)cos θ, where f is the dipole coefficient. The real part of f (Re(f)) can be deduced from the behavior of the first-order electric potential as a function of r sufficiently far from the particle, which remains a constant over 1 , r , R. When ν ) 0, we calculated the dipole moment f for various values of λD and ζ and obtained excellent agreement with computational results using the standard PNP model reported in the literature34,35 to verify our computational algorithm. Moreover, our computed dipole moments agreed well with those predicted by the analytical models in the limit of thin double layers (Figure 3). 4. Dipole Moment for Thin Double Layers In the limit of thin double layers λD , 1, ion transport inside the double layer plays a critical role in determining the dipole moment coefficient. At high frequencies, ions’ migration and convection are dominant over diffusion. A surface conduction model thereafter can be developed to calculate the dipole moment. 4.1. Surface Conductance (Maxwell-Wagner (MW)) Model. When one is studying the dielectric properties of dilute suspensions and calculating dielectrophoretic forces, often the dipole coefficient of a spherical particle is approximated as10-12

∂C(1) ) ∇ · ∇C(1) + F(0)∇φ(1) + F(1)∇φ(0) + ∂t

f)

)

ν(C(0)∇C(1) + C(1)∇C(0)) - ν2C(0)2∇C(1) + mu b(1)∇C(0) (1 - νC(0))2 (15)

ε¯*2 - ε¯*1 ε¯*2 + 2ε¯*1

where

ε¯*i ) ε*i - j

and

(

∂F(1) ) ∇ · ∇F(1) + C(0)∇φ(1) + C(1)∇φ(0) + ∂t ν(F(0)∇C(1) + F(1)∇C(0)) (1 - νC(0))2 ν2(F(0)C(0)∇C(1) + F(1)C(0)∇C(0) - C(1)F(0)∇C(0)) + (1 - νC(0))2 mu b(1)∇F(0) (16)

)

In the above, C(0), F(0), and φ(0) are, respectively, the local salt concentration, the local charge density, and the electric potential at equilibrium in the absence of the external electric field (section 2). The above first-order equations with boundary conditions can be solved by the finite element software Comsol 3.5.33 The computational domain consisted of a finite domain 0 e r e R. We selected R ) 104 since an increase in R by a factor of 10 resulted in variations smaller than 1%, suggesting that R is sufficiently large to render the computational results reasonably R-independent. To resolve the detailed structure of the electric double layer, nonuniform elements were used with a dense mesh concentrated next to the particle’s surface and the elements’ size gradually increasing as the distance from the particle

(17)

κ*i ω*

(18)

In the above, jε*1 and jε*2 are, respectively, the complex permittivities of the electrolyte and the particle; and κ*1 and κ*2 are, respectively, the conductivities of the electrolyte and the particle. The electrolyte’s conductivity is given by κ*1 ) (2F*2D*C + * 0 )/ (R*T*).19 The particle’s effective conductivity is κ*2 ) κ2(i)* + 2σs(DL)*/a*, where κ2(i)* and σs(DL)*are, respectively, the intrinsic conductivity of the particle and the surface conductivity of the diffuse layer.60 Here, we assume that κ(i)* 2 ) 0. When the electric double layer is thin (λD , 1)54

σs(DL)* ) λD*(

∫1∞ (C+(0) + C-(0) - 2) + (0) (0) m(C+ - C)(φ(0) - ζ)dr)κ*1

(19)

When ωf0, f ) (κ*2 - κ*1 )/(κ*2 + 2κ*1 ) ) (2Du - 1)/(2Du + 2), where Du ) σs(DL)*/(a*κ*1 ) is the Dukhin number.19 4.2. Low-Frequency (Dukhin-Shilov (DS)) Model. Ions’ migration and convection often deplete ions into the bulk at one side of the particle and withdraw ions from the bulk at the other side of the particle, resulting in a concentration gradient. At low frequencies, ions have sufficient time to diffuse. As a result, the process of diffusion changes the strength of the dipole coefficient. However, the surface conduction model (section 4.1)

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Figure 3. Dipole coefficient Re(f) as a function of the frequency (a) for several of the effective volume fractions of the ions νwhen λD ) 0.01 and ζ ) -7; (b) for different zeta potentials where λD ) 0.01 and ν ) 0.01; (c) for different λD when ζ ) -7 and ν ) 0.1. The solid lines, the dashed lines, and the symbols correspond, respectively, to the predictions from the surface conduction model, the DS model, and the modified PNP model.

does not take diffusion in the electrolyte solution into account, is applicable only when ω* . D*/a*2, and fails to predict the low-frequency dispersion.34,37,61,62 To understand the role of the diffusion, Dukhin and Shilov63 (i.e., see Grosse and Shilov;64 Zhao and Bau;37 Zhao36) used an asymptotic analysis to calculate the dipole coefficient in the limit of thin double layers λD , 1. In the Appendix, we extended the Dukhin-Shilov theory to include the excluded volume effects. The dipole moment coefficient is given

f)

A B

(20)

where A ) (1 + Wj)(1 + W)(2R+ + 2R- - 2) + (1 + W + Wj)(1 - ν)(U-(2R+ - 1) + U+(2R- - 1))

(21)

and B ) 2((1 + Wj)(1 + W)(R+ + R- + 2) + (1 + W + Wj)(1 - ν)(U-(R+ + 1) + U+(R- + 1)))

(22)

Detailed expressions of A and B are given in the Appendix. At high frequencies (ω . 1), eq 20 can be rewritten as

f ) f∞ )

2Du - 1 2Du + 2

(23)

where we already used the identity R+ + R- ) 2Du. Not surprisingly, the high-frequency limit of the Dukhin-Shilov theory is equal to the low-frequency limit of the MW model. Figure 3a depicts the dipole coefficient Re(f) as a function of the frequency for several effective volume fractions ν when λD ) 0.01 and ζ ) -7. The solid lines, the dashed lines, and the symbols correspond, respectively, to the predictions from the MW model, the DS model, and the modified PNP model. As expected, the MW model fails to predict the dipole coefficient at low frequencies where the diffusion is important, and the DS model is only applicable at low frequencies. The dipole coefficients predicted from the modified PNP model agree well, respectively, with those of the MW model at high frequencies and the DS model at low frequencies. Figure 3b and Figure 3c plot, respectively, the dipole coefficient Re(f) as a function of the frequency for different zeta potentials ζ when λD ) 0.01 and ν ) 0.01 and for several double-layer lengths λD when ζ ) -7 and ν ) 0.1. Interestingly, the dipole moment predicted from the DS model deviates significantly from that by the modified PNP model at high zeta potentials and with large double-layer lengths. In stark contrast, the MW model still agrees well with the modified PNP model. This difference can be explained: the approximate dipole moment of expression 20 relies on the assumption that the chemical potential is constant across the double layer. For the standard PNP model, this assumption was justified only when λD2e|ζ|/2 , 1by Chew.65 This justification should also hold for the modified PNP model. Thus, it is not surprising that the DS model fails at high zeta potentials and with large double-layer lengths.

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Figure 5. Dukhin number as a function of the volume fraction ν for (a) σ ) -30 and (b) σ ) -80 when λD ) 0.1. The solid line, dashed line, and the dash-dotted line correspond, respectively, to the Du number, the contribution to the Du number from the convection, and the contribution to the Du number from the migration. Figure 4. Dipole coefficient Re(f) as a function of the frequency: (a) ζ ) -10, (b) ζ ) -15 when λD ) 0.1. The solid line, the dashed line, and the dash-dotted line denote, respectively, ν ) 0.01, ν ) 0.1, and ν ) 0.5.

5. Results and Discussion For an arbitrary double-layer thickness or for a larger zeta potential, the simple models in section 4 are not applicable. Instead, the dipole moment has to be numerically computed by solving the modified PNP model. Figure 4 depicts the dipole coefficient Re(f) as a function of the frequency when λD ) 0.1. Figures 4a and 4b correspond, respectively, to ζ ) -10 (ζ* ) -250mV in terms of dimensional zeta potential) and ζ ) -15 (ζ* ) -375mV). The solid line, the dashed line, and the dash-dotted line denote, respectively, ν ) 0.01, ν ) 0.1, and ν ) 0.5. The dipole coefficient remains negative at high frequencies, arises, attains a maximum, decreases, and asymptotically approaches its low-frequency limit as the frequency decreases. This frequency dependence is similar to that predicted by the standard PNP model, and the detailed underlying mechanisms have been extensively explained and discussed in our previous works.35-37 For brevity, we do not repeat conclusions here. Instead, we focus on excluded volume effects. The finite ion size prevents the counterion concentration from unboundedly growing inside the double layer. The reduction of the total number of the counterions by excluded volume effects mitigates the surface conduction. Thus, it is reasonable that the maximum of the dipole coefficient decreases, and the high-frequency dispersion shifts to a lower frequency as ν increases. Remarkably, when ζ ) -15, the low-frequency limit of the dipole moment increases with ν, qualitatively different from that of ζ ) -10. This qualitatively different behavior is readily explained: at high zeta potentials, although surface conduction is reduced by excluded volume effects, the bulk diffusion, which is opposite to the direction of the surface conduction, is also

mitigated, and the effect of the diffusion accordingly diminishes. It is likely that the interplay between excluded volume effects on the diffusion and on the surface conduction results in a higher dipole moment coefficient associated with a larger ν at a higher zeta potential ζ. Recall that in section 2 we introduced the structure of the equilibrium double layer accounting for excluded volume effects. In contrast to the standard PNP model, the zeta potential not only depends on the surface charge but also is a function of ν, the effective volume fraction of ions in the bulk which is a function of the bulk concentration too. In other words, the zeta potential is not a unique particle’s surface property any more. Thus, it is more appropriate to directly use the surface charge as the boundary condition to calculate the electric potential at equilibrium. In the following, the surface charge is used as the boundary condition, and its effect on the dipole moment is investigated. Excluded volume effects have much complicated impacts on the dipole moment. On one hand, excluded volume effects set a maximum on the counterion concentration near the surface which seems to reduce the surface conduction. On the other hand, excluded volume effects increase the thickness of the condensed layer, which leads to a higher zeta potential. The increase of the zeta potential can be simply illustrated by the following argument: in section 2, we knew that the thickness of the condensed layer lc ∼ λD2ν|σ| and σ ∼ (ζ/lc). Hence, one can immediately conclude |ζ| ∼ λD2νσ2. The larger ν, the higher ζ. Frequently, a higher zeta potential increases the electroosmotic flow which can enhance the convection, potentially increasing the surface conduction. To quantitatively characterize excluded volume effects on the surface conduction, Figure 5 depicts the Dukhin number, which defines the relative importance of the surface conduction to the bulk conductivity, as a function of the volume fraction ν for σ ) -30 (a) and σ ) -80 (b) when λD ) 0.1 where

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∫1∞ r2(C+(0) + C-(0) - 2) + (0) (0) mr2(C+ - C)(φ(0) - ζ)dr (24)

The surface conduction consists of two components: migration and convection.54 The first and second terms in eq 24 represent, respectively, the contributions from the ions’ migration and ions’ convection. In Figure 5, the solid line, dashed line, and the dashdotted line correspond, respectively, to the Du number, the contribution to the Du number from the convection, and the contribution to the Du number from the migration. As expected, the ions’ migration decreases as ν increases due to the reduction of the total number of ions inside the double layer. In contrast, the ions’ convection increases as ν increases owing to the enhancement of the electro-osmotic velocity. Because of the interplay of the migration and the convection, one would expect that at low surface charges (i.e., σ ) -30) the reduction of the contribution of the migration to surface conduction by excluded volume effects is dominant, and the dipole moment accordingly decreases as ν increases. While at high surface charges (σ ) -80), the enhancement due to the increase of the zeta potential on convection exceeds the loss of the migration. Consequently, the dipole moment increases as ν increases. Figure 6 plots the dipole coefficient Re(f) as a function of the frequency when λD ) 0.1 for two types of surface charges: (a) σ ) -30 and (b) σ ) -80. The solid line, the dashed line, and the dash-dotted line denote, respectively, ν ) 0.01, ν ) 0.1, and ν ) 0.5. Figure 6 reveals intriguing impacts of excluded volume effects on dipole moments for different surface charges and agrees with our speculations. Aranda-Rascon et al.57 reported that the dipole moment increases over a broad range of frequencies as the excluded volume effects increase for the parameters (i.e., surface charge densities and bulk concentrations) used in their work. In contrast, Figure 6 showed that only when the surface charge density is high the increase of the dipole moment due to the excluded volume effects is observed. 6. Conclusion We theoretically investigated the double-layer polarization of a charged, dielectric, spherical particle in the presence of an AC electric field using the modified Poisson-Nernst-Planck equations accounting for crowding or excluded volume effects due to the finite size of ions which often happens near highly charged surfaces or in highly concentrated salt solutions. Finitesize counterions collapse and form a condensed layer with a saturated concentration near the particle’s surface which reduces the surface conduction and hence decreases the peak value of the dipole moment. However, at low frequencies, the situation is much complicated. Excluded volume effects also reduce the strength of the bulk diffusion opposite to the direction of the surface conduction, likely increasing the dipole moment. At large zeta potentials, the gain of the dipole moment by the reduction of the diffusion exceeds the loss of the dipole moment by the decrease of the surface conduction, and excluded volume effects surprisingly enhance the dipole moment. Moreover, in the limit of thin double layers, we extended the standard Maxwell-Wagner model and the Dukhin-Shilov model to incorporate ion excluded volume effects. The calculated dipole moment from these simple models favorably agreed with that predicted by the modified PNP model under certain conditions. Interestingly, the low-frequency DS model was only applicable when λD2e|ζ|/2 , 1, much worse than the MW model

Figure 6. Dipole coefficient Re(f) as a function of the frequency when λD ) 0.1. (a) σ ) -30 and (b) σ ) -80. The solid line, the dashed line, and the dash-dotted line denote, respectively, ν ) 0.01, ν ) 0.1, and ν ) 0.5.

which agreed well with the modified PNP model as long as λD , 1. In other words, the DS model is restricted only to the polarization of moderately charged particles but is still valid in highly concentrated salt solutions. For a more realistic scenario, we assumed that the particle’s surface charge was kept constant and revealed that ion crowding effects decrease the strength of the migration but increase the amplitude of the zeta potential by thickening the condensed layer, which, in turn, boosts the electro-osmotic flow, leading to a stronger convection. For moderately or highly charged particles, the enhancement of the dipole moment due to the ions’ convection exceeds the loss of the dipole moment owing to the ions’ migration in the presence of excluded volume effects. Such enhancement resulting in a higher dipole coefficient over a broad range of frequency at least qualitatively bridges the discrepancy between the experimental results and theoretical predictions, in particular, at low frequencies where the predicted dipole moment is much smaller than that deduced from the experiments. Our results were based on one type of modified chemical potential accounting for excluded volume effects. Other possible chemical potentials can also be proposed to consider crowding effects. In principle, each new chemical potential possibly leads to a new set of the modified PNP models, and the dipole moment can presumably be computed with new PNP models. Recently, Bazant et al.45 presented a detailed review of the history of various modified PNP models, and we refer interested readers to Bazant et al.45 for a lucid exposition. In their detailed summary, other modified PNP models were also solved and compared. Different PNP models qualitatively agreed with each other very well. Therefore, we are confident that this simple and straightforward model used here qualitatively captured the underlying physics of excluded volume effects on the dipole moment, and our major conclusions should prevail even if other

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Zhao

more complicated chemical potentials characterizing excluded volume effects are used. Finally, when one uses the classic PNP model to calculate the dipole moment, often the particle’s surface is assumed to be uniformly charged with a constant zeta potential and be ideal, precluding any reaction and adsorption. Furthermore, the macroscopic properties of suspending electrolyte such as permittivity and viscosity remain uniform in space. In reality, due to the complicated nature of nanoparticles and electrolyte, all the mentioned assumptions might break down at certain circumstances. Additional improvement or correction of the PNP model incorporating other effects may deserve attention, and future investigations might be able to further help to interpret experimental data.

(1) ∂µ( ∂r

|

r)1

( ( ( )

(1) ∂µ( ∂ (0) )sin θ G( sin θ∂θ ∂θ

jωC(1) ) ∇2

C(1) 1-υ

(25)

To derive eq 25, we substitute C(0) ) 1 and F(0) ) 0 outside the double layer and replace the time derivative with jω in eq 15. Accounting for the neutrality, the electric potential obeys the Laplace equation

∇2φ(1) ) 0

C(1) ) Kc

e

2

r

1 + (1 + j)Wr cos θ 1 + W + jW

∞ (0) (1) (1) (1) where G(0) ( ) ∫1 (C( - 1)dr and µ( ) (C )/(1 - ν) ( φ . (1) To complete the solution process, the tangent velocity uθ still needs to be decided. The velocity consists of both an electro-osmotic component and a diffusio-osmotic component induced by the bulk concentration gradient along the particle. In the limit of thin electric double layers, the surface is assumed to be locally flat, and the electro-osmotic velocity component is well-known as

uθ(1)E ) (ζ - φ(0))∇θφ(1)

φ

(

)

f ) -r + 2 cos θ r

(30)

On the other hand, the bulk concentration gradient creates an osmotic pressure which also can drive a flow. The osmotic pressure can be computed from the normal component of the Stokes equation

-

dφ dp 1 - 2 F(0) )0 dr dr 2λD

(31)

In the above, F(0) ) (sinh φ(0))/(1 + 2ν sinh2 φ(0)/2). Equation 31 can be integrated to

p)

ln(1 - ν + ν cosh φ(0)) (1) C ν

(32)

Inserting the pressure into the tangential component of the Stokes equation and integrating twice, the diffusio-osmotic velocity component becomes

(27)

uθ(1)D ) (r

(28)

where W ) [(1 - ν)ω/2]1/2. Kc and f (the dipole coefficient) are integration constants that still need to be determined. In regard to the boundary conditions on the particle’s surface, we follow the approach of Grosse and Shilov.64 Briefly, we assume that the bulk reaches a local equilibrium with the double layer. In other words, the perturbed chemical potentials are constant across the thin double layer. We write down the ion conservation equations in terms of the actual and far field variables, take the difference, and integrate the resulting equation in terms of the radial coordinate from r ) 1 to infinity to obtain the boundary conditions

∫r∞ g(s)ds + ∫0r g(s)sds)∇θC(1)

(33)

where g(s) ) (ln(1 - ν + ν cosh φ(0)))/(ν). To facilitate the derivation, we define the following new parameters: R( ) G( ( m ∫1∞(ζ - φ(0))(1 - C(0) ( )dr and U( ) G(/(1 - ν) + m ∫1∞(r ∫r∞g(s)ds + ∫0r g(s)sds)(1 - C(0) ( )dr. The boundary conditions eq 29 now become

and

(1)

(29)

(26)

The solutions of eq 25 and eq 26 can be easily integrated -(1+j)W(r-1)

))

∫1∞ (1 - C((0))uθ(1)dr

Appendix Dukhin and Shilov63 (i.e., see Grosse and Shilov;64 Zhao and Bau;37 Zhao36) used an asymptotic analysis to calculate the dipole coefficient in the limit of thin double layers λD , 1. Briefly, in the thin double-layer limit, most of the electrolyte remains neutral except for a thin and locally planar electric double layer adjacent to the particle’s surface. At low frequencies, due to the concentration polarization, the bulk concentration outside the electric double layer does not remain a constant any more, rather than satisfies

+

r)1

(1) ∂µ( ∂r

|

r)1

) (2R(φ(1)(1) + 2U(C(1)(1)

(34)

By substituting the expressions of chemical potentials outside the double layer, we can obtain Kcand f. The expression of f is given in the main text (section 4.2). References and Notes (1) Bikerman, J. J. Philos. Mag. 1942, 33, 384–397. (2) Velev, O. D.; Kaler, E. W. Langmuir 1999, 15, 3693–3698. (3) Stone, H. A.; Stroock, A.; Ajdari, A. Annu. ReV. Fluid Mech. 2004, 36, 381–411. (4) Squires, T. M.; Quake, S. ReV. Mod. Phys. 2005, 77, 977–1026. (5) Yu, B.; Meyyappan, M. Solid-State Electron. 2006, 50, 536–544. (6) Singh, A.; Li, X.; Protasenko, V.; Galantai, G.; Kuno, M.; Xing, H.; Jena, D. Nano Lett. 2007, 7, 2999–3006. (7) Saunder, B. R.; Turner, M. J. AdV. Colloid Interface Sci. 2008, 138, 1–23.

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