Article pubs.acs.org/Langmuir
Electrophoresis of a Particle at an Arbitrary Surface Potential and Double Layer Thickness: Importance of Nonuniformly Charged Conditions Jyh-Ping Hsu,* Hsiao-Ting Huang, and Li-Hsien Yeh Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617
Shiojenn Tseng Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137 ABSTRACT: Recent advances in material science and technology yield not only various kinds of nano- and sub-micro-scaled particles but also particles of various charged conditions such as Janus particles. The characterization of these particles can be challenging because conventional electrophoresis theory is usually based on drastic assumptions that are unable to realistically describe the actual situation. In this study, the influence of the nonuniform charged conditions on the surface of a particle at an arbitrary level of surface potential and double layer thickness on its electrophoretic behavior is investigated for the first time in the literature taking account of the effect of double-layer polarization. Several important results are observed. For instance, for the same averaged surface potential, the mobility of a nonuniformly charged particle is generally smaller than that of a uniformly charged particle, and the difference between the two depends upon the thickness of double layer. This implies that using the conventional electrophoresis theory may result in appreciable deviation, which can be on the order of ca. 20%. In addition, the nonuniform surface charge can yield double vortex in the vicinity of a particle by breaking the symmetric of the flow field, which has potential applications in mixing and/or regulating the medium confined in a submicrometer-sized space, where conventional mixing devices are inapplicable.
1. INTRODUCTION Recent advances in material science and technology yield not only various kinds of nano- and submicro-scaled particles but also particles of various charged conditions. The characterization of these particles can be challenging because conventional analytical tools such as electrophoresis are usually based on drastic assumptions. For instance, particles are almost always assumed to be uniformly charged in electrophoresis measurements,1,2 and the measured mobility is converted to the corresponding surface potential based on some theoretical models. This becomes unrealistic for entities such as Janus-type particles.3−5 Here, using more sophisticated model, which is capable of simulating the charged conditions on the particle surface, is necessary. Assuming nonuniform surface potential and thin double layer, Anderson6 concluded that the electrophoretic mobility of a rigid, nonconducting colloidal particle is sensitive to the distribution of its surface potential. This analysis was extended to the case of nonspherical particles, such as ellipsoidal particles, by Fair and Anderson,7 and by Keh and Hsieh8 to the case of arbitrary double-layer thickness, but at a low surface potential. Qian et al.9 investigated the electrophoresis of a particle having a positiondependent surface charge density in a nanotube. In addition to particles such as Janus-type particles, nonuniformly charged particles can also be prepared by applying an electric field to conductive particles immersed in an electrolyte solution. Once they are fully polarized, the particles behave like © 2012 American Chemical Society
an insulator due to the presence of the surrounding electric double layer.10 Adopting Helmholtz’s model, Levich11 studied the electrokinetic behavior of polarizable metallic colloidal particles. On the basis of this analysis, Simonov and Shilov12 evaluated the mobility of an ideally polarizable metallic particle taking account of the effect of double-layer polarization (DLP), but the flow field surrounding the particle was not discussed. Under the conditions of thin double layer and low surface potential, Bazant and Squires13 found that the electroosmotic flow field near a conducting cylinder can be asymmetric. They defined the term ICEO (induced charge electro osmosis flow) to describe the effect of an applied electrical field on the induced double layer near a conducting surface. ICEO was also observed experimentally by Gamayunov et al.14 The ICEO for the case of arbitrary double layer thickness was partially addressed by Murtsovkin;15 however, the flow field was solved only in the thin double layer limit. The nonuniform charged conditions on the surface of a particle render it to have various applications. The anisotropic chemical composition of properly designed Janus particles, for example, enables them to migrate in a concentration field, thereby having great potential in many fields of application.16,17 Another example is the mixing of liquid in microfluidics and Received: November 14, 2011 Published: January 3, 2012 2997
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nanofluidics, where the Reynolds number is very small, and how to improve the degree of mixing becomes challenging. One possible solution is ICEO, which can enhance the mixing of liquid, and is widely utilized in lab-on-a-chip devices.18 Nonuniformly charged particles have also been used widely in areas such as biochemical sensing, separation, and particle production.19,20 Considering the emerging applications of nonuniformly charged micro- and nanoparticles, detailed understanding of the electric, the concentration, and the flow fields near such particles is necessary. To this end, classic electrophoresis analysis needs be extended to take account of the nonuniform charged conditions of the surface of a particle. Assuming low surface potential and infinitely thin double layer, Keh and Anderson21 investigated the electrophoretic behavior of a nonuniformly charged sphere. Their analysis was extended later to the case of arbitrary double layer thickness.22 However, up to now, the electrophoresis of nonuniformly charged particles at arbitrary levels of surface potential and double layer thickness has not been addressed. Because this problem becomes inevitable in cases such as conductive particles and Janus-type particles, extending available results to take it into consideration is highly desirable. Furthermore, previous analyses for inducedcharge electrokinetics are too simplified, and therefore are unable to accurately describe the corresponding flow field, and, as a result, their applications in microfluidics and nanofluidics are limited. In particular, because the thickness of double layer is not considered in previous analyses, the effect of DLP, one of the most interesting and important phenomena of electrophoresis, is overlooked. In this study, the importance of the nonuniform charged conditions on the surface of a particle on its electrophoretic behavior is discussed. The influence of those conditions on the effect of DLP and the behavior of the flow field near the particle surface are also examined.
Figure 1. Electrophoresis of a rigid, nonuniformly charged sphere of radius a as a response to an applied uniform electric field E; Ωw is the boundary of a spherical computation domain of radius b (b ≫ a); r, θ, and z are the cylindrical coordinates adopted with the origin at the center of the particle, Θ is the solid angle; E is in the z direction; Ωp is the surface of the particle.
∇ and ∇2 are the gradient operator and the Laplace operator, respectively; ε and η are the permittivity and the viscosity of the liquid phase, respectively; ρ∇ϕ denotes the electric body force acting on the fluid, with ρ being the space charge density; zj, Jj, and Dj are the valence, the flux, and the diffusivity of ionic species j, respectively, j = 1 and 2 denoting cations and anions, respectively; e, kB, and T are the elementary charge, Boltzmann constant, and the absolute temperature, respectively; p is the hydrodynamic pressure; U is the particle velocity. Suppose that E is relatively weak compared to the electrical field established by the particle, as is usually the case in practice. Then, a perturbation method is applicable,6 where each dependent variable is decomposed into an equilibrium component arising from the particle in the absence of E, and a perturbed component arising from the application of E. If we denote the equilibrium component with a subscript “e” and the perturbed component with a prefix δ, then ρ = ρe + δρ, u = ue + δu, p = pe + δp, ϕ = ϕe + δϕ, nj = nje + δnj. Due to the convective motion of the ionic species, the double layer surrounding the particle may no longer remain spherical, known as DLP. This effect can be taken into account by expressing nj as24
2. THEORY Referring to Figure 1, we consider the electrophoresis of a rigid, nonuniformly charged spherical particle of radius a subject to an applied uniform electric field E. For convenience, a spherical computation domain of radius b (b ≫ a) and boundary Ωw is defined. r, θ, and z are the cylindrical coordinates adopted with the origin at the center of the particle, and the solid angle Θ is defined. E is in the z direction, and Ωp is the surface of the particle. The liquid phase is an aqueous, incompressible Newtonian fluid containing binary z1:z2 electrolyte, with z1 and z2 being the valence of cations and that of anions. Let α = −z2/z1, and Ez be the strength of E. Assuming that the system considered is at a pseudo steady state, the electrical potential, ϕ, the number concentration of ionic species j, nj, and the fluid velocity u can be described by23 ∇2 ϕ = −
2 z en ρ j j =−∑ ε ε j=1
(2)
∇· Ji = 0
(3)
−∇p + η∇ u − ρ∇ϕ = 0
(4)
∇·u = 0
(5)
(6)
Here, gj is a hypothetical potential, which needs be solved simultaneously with the governing equations of the electrical and flow fields. For a simpler mathematical treatment, eqs 1−6 are rewritten in scaled forms as
(1)
⎞ ⎛ zje nj∇ϕ⎟ + nj(u − U) Jj = − Dj⎜∇nj + kBT ⎠ ⎝
2
⎡ zje(ϕe + δϕ + gj) ⎤ ⎥ nj = njo exp⎢ − ⎢⎣ ⎥⎦ kBT
∇*2 ϕe* = −
(κa)2 [exp( − ϕe*) (1 + α)
− exp(αϕe*)] 2998
(7)
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Table 1. Percentage Deviation in the Scaled Mobility of a Nonuniformly Charged Particle from the Corresponding Uniformly Charged Particle, Δμ* = [(μ* − μ*β=0)/μ*β=0] × 100%, for Various Combinations of κa and β at ϕ*p = 2
Figure 2. Variation of the scaled electrophoretic mobility μ* as a function of κa for the case of a rigid sphere of scaled constant potential ϕ*e = 1 at the center of an uncharged spherical cavity. Curve: present result with λ = 0.1; discrete symbols: analytical result of Zydney.30
κa
β=2
β=3
β=4
0.1 0.3 0.5 0.8 1 2 3 5 7 10
−0.02 −0.88 −1.46 −2.59 −3.10 −4.23 −5.16 −4.78 −3.86 −4.01
−0.39 −2.09 −3.24 −5.84 −7.29 −9.81 −10.53 −9.98 −9.84 −8.87
−0.71 −3.93 −6.99 −10.41 −12.12 −14.86 −18.30 −18.19 −16.74 −16.66
∇*2 g1* − ∇*ϕe*·∇*g1* = Pe1(u* − U*) ·∇*ϕe*
(9)
∇*2 g2* − α∇*ϕe*·∇*g2* = Pe 2(u* − U*)·∇*ϕe*
(10)
∇*2 u* − ∇*δp* + ∇*ϕe*∇*δϕ* + ∇*2 δϕ*∇*ϕe* = 0
(11)
∇*· u* = 0
(12)
n*j = exp( − ϕe*)[1 − (δϕ* + g *j )]
(13)
Here, ∇* = a∇ and ∇* = a ∇ are the scaled gradient operator and the scaled Laplace operator, respectively; κ = 2 [∑j=1 njo(ezj)2/εkBT]1/2 is the reciprocal Debye screening length; n*j = nj/nj0, ϕ*e = ϕe/ϕref, δϕ* = δϕ/ϕref, and g*j = gj/ϕref with ϕref = kBT/ez1 being the thermal potential; Pej = ε(kBT/z1e)2/ ηDj is the electric Peclet number of ionic species j, j = 1, 2; u* = u/U0 and U* = U/U0 with U0 = ε (kBT/z1e)2/aη being a 2 reference velocity; δp* = δp/pref with pref = εϕref /a2 being a reference pressure. We assume the following boundary conditions for the flow field: 2
2
2
u* = (U /U 0)ez on Ω p
(14)
u* = 0 on Ω w (15) where ez is the unit vector in the z direction and U is the speed of the particle. These conditions imply that the particle surface is nonslip, and the flow field at a point far away from the particle is uninfluenced by its presence. To simulate the induced charge on a conductive particle, the following boundary conditions are assumed for the electric and concentration fields: Figure 3. Variation of the scaled electrophoretic mobility μ* as a function of κa at various values of β with two levels of the averaged scaled surface potential ϕ*p. (a) ϕ*p = 1, (b) ϕ*p = 2.
ϕe* = ϕp* + β cos(Θ) on Ω p
(16)
ϕe* = 0 on Ω w
(17)
n ·∇*δϕ* = 0 on Ω p
(18)
n ·∇*δj* = 0 on Ω p
(19)
n ·∇*δϕ* = − Ez* on Ω w
(20)
g *j = − δϕ* on Ω w
(21)
2
( κa ) [exp(−ϕe*) − exp(αϕe*)]δϕ* ∇*2 δϕ* = (1 + α) +
(κa)2 [exp( −ϕe*)g1* (1 + α)
− exp(αϕe*)g2*]
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Figure 5. Contours of the perturbed anions distribution δn2 = n2 − n2e (no./m3) at κa = 3 and ϕ*p = 2 on the plane θ = π/2 for two levels of β, where R = r/a. (a) β = 0; (b) β = 4.
Figure 4. Contours of the perturbed cations distribution δn1 = n1 − n1e (no./m3) at κa = 3 and ϕ*p = 2 on the plane θ = π/2 for two levels of β, where R = r/a. (a) β = 0; (b) β = 4.
In these expressions, ϕ*p = ϕp/ϕref is the scaled averaged surface potential of the particle with ϕp = ∫ Ωpϕ dΩp being the corresponding averaged surface potential; β is a parameter measuring the nonuniformity of the surface potential; n is the unit outer normal vector; E*z = Ez/(ϕref/a) is the scaled strength of the applied electrical field. Equation 16 states that the surface potential of the particle is position dependent, and eq 17 implies that the potential at a point far away from the particle is uninfluenced by its presence. Equations 18 and 19 imply that the particle is nonconductive and ion-impenetrable, respectively. These are valid, for instance, for Janus-type and fully polarized particles. Equations 20 and 21 state that, at a point far away from the particle, the electric field comes solely from the applied electric and the ionic concentration reaches the bulk value, respectively. Let Fe and Fd be the electric and the hydrodynamic forces acting on the particle, respectively, Fe and Fd be the z 2 components of these forces, respectively, and F*e = Fe/εϕref 2 and F*d = Fd/εϕref be the corresponding scaled quantities. Then,25
Fe* =
⎛ ∂ϕ* ∂δϕ*
∫Ω* ⎜⎜ ∂ne p
⎝
∂Z
⎛ ∂ϕ*e ∂δϕ* ⎞ ⎞ ⎟nz⎟⎟ d Ω*p −⎜ ⎝ ∂t ∂t ⎠ ⎠
and
Fd* =
∫Ω* (σH *·n)·ezdΩ*p p
(23)
Here, Ω*p is the dimensionless surface area of the particle scaled by a2; Z = z/a; n and t are the magnitude of n and that of the unit tangential vector t, respectively; nz is the z component of n; 2 σH* = σH/(εϕref /a2) is the scaled shear stress tensor with σH being the corresponding shear stress tensor. The present problem is solved through the following trialand-error method: (i) Assume an arbitrary particle velocity Ug and solve eqs 7−12. (ii) Evaluate the forces acting on the particle by eqs 22 and 23. (iii) Check whether F*e + F*d = 0; if this is true, then Ug is the desired particle velocity. If this is not the case, then another Ug is assumed and go back to step (ii).
3. RESULTS AND DISCUSSION The present problem is solved numerically by FlexPDE,26 which is found to be sufficiently efficient and accurate for the resolution of similar problems.27 To verify its applicability, the electrophoresis of a rigid particle of constant surface potential
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Figure 6. Contours of the perturbed anions distribution δn1 = n1 − n1e (no./m3) and δn2 = n2 − n2e (no./m3) in the direction of the applied electric field on the plane θ = π/2 at κa = 3 and ϕ*p = 2, where dotted lines denote the boundary of double layer. (a) β = 0; (b) β = 4.
Figure 7. Contours of the net perturbed ionic distribution δn = δn1 − δn2 (no./m3) at κa = 3 and ϕ*p = 2 on the plane θ = π/2. (a) β = 0; (b) β = 4.
at the center of a spherical cavity, which was solved analytically by Zydney,28 is solved numerically by the software adopted. Figure 2 summarizes the results obtained by the two approaches. As can be seen, the present numerical approach successfully reproduces the analytical result; the slight deviation seen in the range where κa takes a medium large value arises from the fact that the effect of DLP was not considered in Zydney.28 To examine the electrophoretic behavior of a particle under various conditions, a series of numerical simulation is conducted. In particular, the influences of the averaged surface potential, the nonuniformity of the surface potential, and the thickness of double layer on the mobility of a particle are discussed. For illustration, we assume that λ = 0.1, which is small enough to neglect the presence of the computation domain,29 and the liquid phase is an aqueous KCl solution, yielding Pe1 = Pe2 = 0.235 and α = 1. 3.1. Variation of Mobility. Figure 3 shows the variation of the scaled mobility of a particle μ* as a function of the thickness of double layer, measured by κa, at various values of the parameter β, which measures the degree of nonuniformity of the surface potential, for two levels of the averaged surface potential ϕ*p. For convenience, the radius of the particle a is fixed in the simulation and the reciprocal Debye length κ varies, that is, the concentration of electrolytes varies. Figure 3a reveals that for the same level of ϕ*p, μ* depends highly on β. Note
that if β = 0, then the particle surface is uniformly charged. In this case, μ* is seen to increase monotonically with increasing κa for the level of ϕ*p considered. This is because the larger the κa, the thinner the double layer, yielding a higher absolute value of the potential gradient on the particle surface, and therefore, a greater electric driving force acting on the particle. The behavior of μ* for a nonuniformly charged particle (β ≠ 0) is quite different from that for a uniformly charged particle. In the former, the larger the β the smaller the μ*, in general. This behavior has not been reported by previous models in the literature. The influence of β on μ* is seen to depend highly on the thickness of double layer (or concentration of electrolyte). Table 1 summarizes the percentage deviation in the μ* of a nonuniformly charged particle from the corresponding uniformly charged particle, Δμ* = [(μ* − μ*β=0)/μ*β=0] × 100%, under various conditions. It is interesting to see in this table that Δμ* has a local maximum at κa ≅ 3. As will be explained latter, the presence of this local maximum in Δμ* arises from the effect of DLP. For the case where β ≠ 0, μ* shows a local minimum as κa varies, and the larger the β, the more apparent the presence of this local minimum. Again, this can be explained by the presence of the effect of DLP. 3001
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Figure 8. Scaled flow field at ϕ*p = 2 on the plane θ = π/2 for various values of κa. (a) κa = 0.5, β = 4; (b) κa = 1, β = 4; (c) κa = 3, β = 4; (d) κa = 10, β = 4; (e) κa = 10, β = 0.
The qualitative behavior of μ* shown in Figure 3b, where ϕ*p takes a higher value, is similar to that seen in Figure 3a. Quantitatively, the μ* in the former is larger than that in the latter, which is expected because the higher the averaged surface potential, the greater the electrical driving force acting on a
particle. As in the case of Figure 3a, a local maximum might also present as κa varies, although it is unapparent. 3.2. Double-Layer Polarization. Figure 4 illustrates the contours of the perturbed concentration of cations, δn1, on the plane θ = π/2 at two values of β; the corresponding contours of 3002
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anions, δn2, are shown in Figure 5. After E is applied, the cations (anions) inside the double layer surrounding the particle move in the direction of E (−E). Because the velocity of these ions is slower than that of the particle, they tend to accumulate near its bottom region. As seen in Figures 4a and 5a, for the case of a uniformly charged particle, this yields positive (negative) values of δn1 and δn2 near the bottom (top) region of the particle. However, as seen in Figures 4b and 5b, the result becomes quite different when the particle is nonuniformly charged. In this case, the cations inside the lower part of the double layer of the particle tend to migrate toward the z direction as E is applied. However, as seen in Figure 4b, because their movement is hindered by the positively charged upper half of the particle, they accumulate near Z = −0.8. Similarly, anions accumulate near Z = 0.2, as shown in Figure 5b. The variations of the perturbed ionic concentrations, δn1 and δn2, along the Z axis on the plane θ = π/2 are presented in Figure 6. This figure reveals that the perturbed concentration of the anions near the upper half of the particle for the case of nonuniformly charged particle is much higher than that for the case of uniformly charged particle. That is, the applied electric field drives the anions inside the double layer near the upper half of the nonuniformly charged particle toward the −E (or −z) direction. This has the effect of reducing the electrical potential on the upper half the particle, thereby enhancing the induced electric field. Under the conditions assumed, the particle moves upward, and the perturbed concentration of anions near its bottom is higher than that of cations. Figure 7 shows the contours of the net ionic distribution δn = δn1 − δn2 (no./m3) at two levels of β on the plane θ = π/2. As seen in Figure 7a, where the particle is uniformly charged (β = 0), δn > 0 (δn < 0) near the top (bottom) region of the particle and δn = 0 at Z ≅ 0. This implies that an internal electrical field is established, the direction of which is opposite to that of the applied electric field, thereby reducing the mobility of the particle. This phenomenon is known as DLP. Note that the |δn(β = 4)| in Figure 7a is larger than that in Figure 7b, implying that the effect of DLP for the nonuniformly charged case is more significant than that for the uniformly charged case. The presence of a local maximum in Δμ* near κa = 3 seen in Table 1 implies that the effect of DLP is most significant at a medium thick double layer, which is consistent with previous result.30 This is because if κa is small, then the concentration of ionic species inside the double layer is low, and the internal electrical field arising from its deformation is insignificant compared to the applied electrical field. On the other hand, if κa is large, then because the double layer is thin, its deformation becomes unimportant. 3.3. Flow Field. When an external electric field is applied, the movement of the cations and anions inside the double layer surrounding a particle in opposite directions generates vortex flow near the particle surface. The behavior of this flow depends upon the level of κa. As seen in Figure 8a, if κa is small (low ionic concentration), only one vortex is present on each side of the particle. In this case, the flow field is dominated by the movement of the particle. On the other hand, if κa is sufficiently large, then both the movement of the particle and that of the ionic species inside the double layer contribute appreciably to the flow field. In this case, two vortices are observed on each side of the particle, as seen in Figure 8d. This implies that the degree of mixing can be improved by adjusting the bulk electrolyte concentration (or κ) and the linear size of the particle, a.
4. CONCLUSIONS The importance of the nonuniform charged conditions of a particle on its electrophoretic behavior is analyzed by considering the electrophoresis of a rigid spherical particle with a surface potential simulating the charged conditions of a Janus type particle or a conductive particle. We show that the distribution of the surface potential of a particle can influence both qualitatively and quantitatively its electrophoretic behavior, implying that assuming an averaged surface potential is unsatisfactory. The distribution of the surface potential is capable of influencing the ionic distribution near a particle, thereby enhancing the effect of DLP, one of the most interesting and significant phenomena in electrophoresis. In general, the nonuniform distribution in the surface potential of a particle has the effect of reducing its mobility. Due to the competition of the movement of a particle and the migration of the ionic species inside the double layer surrounding it, the flow field near its surface depends highly upon its linear size and the bulk electrolyte concentration. As a result, the degree of mixing of the liquid near the particle can be raised by adjusting the bulk electrolyte and/or the particle size so that double vortices are formed on each side of the particle, which has potential applications in microfluidics and nanofluidics.
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AUTHOR INFORMATION
Corresponding Author
*Tel: 886-2-23637448. Fax: 886-2-23623040. E-mail: jphsu@ ntu.edu.tw.
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ACKNOWLEDGMENTS This work is supported by the National Science Council of the Republic of China. REFERENCES
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