Localized Electroosmosis (LEO) Induced by Spherical Colloidal

Article pubs.acs.org/Langmuir

Localized Electroosmosis (LEO) Induced by Spherical Colloidal Motors Tso-Yi Chiang and Darrell Velegol* The Pennsylvania State University, Department of Chemical Engineering, University Park, Pennsylvania 16802, United States ABSTRACT: Experimental data show that the speed of colloidal (catalytic) motors decreases as the size of the motor particles increases. However, previous electrokinetic models have shown that the colloidal motor speed for spheres is independent of size, at least for the case of infinitesimally thin double layers and reaction-limited catalysis. Although a size dependence of motor speed has been calculated for diffusion-limited catalysis, most motor experiments are done in the reaction-limited regime. This apparent contradiction led us to examine how motor speed (U) changes with distance (δ) from a wall, starting from the usual electrokinetic equations. A key finding is that interactions between a colloidal motor and a nearby wall produce a localized electroosmotic (LEO) flow field that can significantly alter the motor speed near the wall. Because large motor particles typically settle closer to the wall than small motors, LEO thus provides at least one explanation of the size dependence of motor speed. Furthermore, LEO provides a new method of creating flow fields in capillaries and microchannels.

■

INTRODUCTION Colloidal motors composed of two metals move in a solution of hydrogen peroxide by a mechanism called autoelectrophoresis.1−3 In autoelectrophoresis, an electric field is induced in solution by electrochemical reactions that occur on the particle surface. Experiments have shown that the speed of colloidal motors decreases as the size of the particle increases.4,5 Although a size dependence of motor speed has been calculated for diffusion-limited catalysis,6 most motor experiments are done in the reaction-limited regime,3 for which previous electrokinetic models have shown that the speed of spherical motor particles should be independent of size for infinitesimally thin double layers.6,7 One important factor that has not been considered in detail for colloidal motors is the effect of a nearby wall. It was first suggested by Wheat et al. that the interactions between the motor and the solid substrate below could be important, which can thus affect the observed motor speed.5 Recently, it has also been shown that the observed motor speed changes with substrates having different magnitudes of zeta potentials.8 Here, we employ modeling with the electrokinetic equations to calculate the speed (U) of spherical colloidal motors that are a certain distance (δ) from a wall. We find that interactions between a colloidal motor and a nearby wall produce a localized electroosmotic (LEO) flow field that can significantly alter the motor speed near the wall. Unlike the electroosmotic flow caused by an applied electric field, the electroosmotic flow caused by a colloidal motor arises from the wall directly below the motor, and even then, it exists primarily within one particle diameter along the wall, which is why we refer to it as localized electroosmosis. Because large motor particles typically settle © 2014 American Chemical Society

closer to the wall than small motors, LEO thus provides at least one explanation for the size dependence of the motor speed. The electrophoretic velocity of a charged particle in an applied electric field has been studied for decades. In principle, the electric field and the flow field are coupled in the full electrokinetic equations, and the coupled PDEs need to be solved to obtain the electrophoretic velocity.9,10 The full electrokinetic equations are complicated to solve, so most commonly, simplifying assumptions are made. One common assumption is to assume infinitesimal electrical double layers (EDLs), an approach commonly used by Anderson and coworkers,11 Keh and co-workers,12−18 and others.19,20 For a spherical particle of radius (a) in a solution having a certain Debye length (κ−1), this means that κa → ∞. Under this assumption, the electric field (E) is uncoupled from the velocity field, so the electric potential is calculated outside the EDL using the Laplace equation. This assumption has enabled the calculation of the Smoluchowski velocity not only for spherical particles11 but also for particles of arbitrary shape having uniform zeta potentials21 as well as for nonuniform particle zeta potentials.22−24 Recently, researchers have begun to study the electrophoresis of particles not in an applied electric field but in an electric field that is self-generated by catalytic action on the particle surface. In 2004, Paxton et al. fabricated bimetallic nanorods that can move autonomously in hydrogen peroxide (H2O2).25 The chemical energy in H2O2 is converted by electrochemical reactions to an electric field in the vicinity of the particle, and Received: June 14, 2013 Revised: February 7, 2014 Published: February 12, 2014 2600

dx.doi.org/10.1021/la402262z | Langmuir 2014, 30, 2600−2607

Langmuir

Article

thus the particles have access to a tremendous amount of “fuel” from the solution. For a typical particle size (a ≈ 10−6 m), surface reaction rate (R ≈ 1018 ions/m2 s), diffusion coefficient (DH2O2 ≈ 10−9 m2/s), and bulk reactant concentration (cH2O2,∞ ≈ 5.3 × 1026 molecules/m3 = 3%), one typically finds that the Thiele modulus26 (aR/DH2O2cH2O2,∞)1/2 < 0.01 holds, meaning that the process is reaction-limited, not diffusion-limited. To model colloidal motor transport, the electrokinetic equations have been solved to predict the motor speed (U) based on independently measurable parameters. Kline et al. developed a model to describe the self-generated electric field arising from gold and silver disks, showing the origin of experimentally observed bands of particles and establishing methods for modeling colloidal motors.27 Golestanian et al. developed a general model for motors moving in bulk solution by different autoelectrokinetic mechanisms, with the driving force being a gradient of electric potential, temperature, or solute concentration.7 They found that with infinitesimal EDLs and reaction-limited conditions, motor speed should not depend upon particle size for spherical motors in the bulk solution. Yariv obtained the analytical solution for motor speed with infinitesimal EDLs using linearized equations with the surface reaction kinetics for both diffusion-limited and reactionlimited motors in the bulk.6 Moran et al. developed a model describing the bulk speed of cylindrical motors considering the finite thickness of EDL and solved for the full equations numerically.28 Later, the group developed a more detailed model for cylindrical motors with the surface reaction rate being second order in proton concentration and with a change of zeta potentials due to the surface reaction taken into consideration.29 However, none of these references have found a size dependence on spherical motor speed for infinitesimal EDLs and reaction-limited catalysis. As discussed later, allowing for finite EDL thickness might seem like it would cause larger particles to move faster for the same surface charge density, which is opposite to the trend observed in experiments. One common assumption in the models above is that the colloidal motors are in an unbounded solution. Interactions between colloidal motors and bounding walls have not yet been considered. However, in the microscopy observations reported in the literature, the colloidal motors are settled near a glass slide.30 Particles settle until electrostatic repulsion balances gravitational attraction.30 For particle−surface separations of δ > 200 nm that are predicted for most experiments in the literature, we calculate that the van der Waals forces are small, especially when retardation is considered. The motors attain an average equilibrium distance (δ) from the wall, with just a small amount of Brownian fluctuation around that average. Our essential hypothesis is shown in Figure 1. Because small particles maintain a larger distance from the wall, we hypothesize that the self-generated electric field around the particle drives little flow at the bounding wall. That is, small particles will behave in a manner more consistent with their being in the bulk (Figure 1a). In contrast, larger particles approach more closely, so we hypothesize that motor−wall interactions are more important. We test our hypothesis using a modeling approach, solving the electrokinetic equations for a spherical colloidal motor (Janus motor) near a wall. Not until the mid-1980s did researchers begin to study boundary effects on the electrophoretic motion of colloidal spheres in an applied electric field. Keh and Anderson first studied the effect of a wall on the

Figure 1. Schematic of localized electroosmosis (LEO) induced by a Janus motor. The LEO flows change the speed of colloidal motors of different sizes when they are near a wall. The curves represent electric field lines induced by the transport of protons. (a) For a small particle, electrostatic repulsion keeps the particle far above the wall. The transport of protons is only slightly affected by the wall, so the particle speed is similar to the case in an unbounded solution. (b) A large particle settles closer to the wall as a result of gravity. Because ions cannot penetrate the wall, electric field lines are squeezed, inducing electroosmotic flow on the wall that alters the observed motor speed. Arrows indicate the directions of the induced electric field (E), the electrophoretic velocity (uep) of the negatively charged motor, and the electroosmotic flow velocity (veo) induced on the negatively charged wall.

electrophoretic speed of a spherical particle using the method of reflections.12 A few years later, Keh and Chen solved for the exact solution to the speed of charged colloidal spheres parallel to a dielectric plane in a uniform applied electric field.13 In 2003, Yariv and Brenner studied the near-contact electrophoretic motion of a sphere parallel to a planar wall using matched asymptotic expansions, obtaining a result consistent with Keh and Chen for motion parallel to a wall.31 Thus, whereas results exist for colloidal motors undergoing autoelectrokinetic motion in bulk solution and for particles undergoing applied-field electrokinetics near a wall (as well as in bulk solution), there are no calculated results for autoelectrokinetic motion near a wall. In this article, we introduce the idea of LEO flows and develop a model including wall effects to predict the speed (U) of a spherical colloidal motor that is a certain distance (δ) from a wall. Like other interesting findings from broken symmetry,32,33 in the present work we break the common symmetry assumption of an unbounded solution. The model is based on the usual electrokinetic equations under the infinitesimal EDL approximation. We use the Laplace equation to evaluate the E field arising from the surface reactions around the particle and near the plate. We then calculate the final speed of the motor as the electrophoretic velocity plus the local electroosmotic flow velocity (eq 6). This approximation is based on Keh and Chen’s results.13 The electrophoretic velocity is calculated using a known equation from the literature (eq 7). Results from our model show the same trend as that observed in experiments; that is, motor speed decreases as the size of the motor particle increases, and the change can be 50%. Our calculations reveal that not only does the existence of the wall change speed but it can also change the direction of the motor. Significantly, not only does LEO provide an explanation of differences in motor speed with size but also it provides a path 2601

dx.doi.org/10.1021/la402262z | Langmuir 2014, 30, 2600−2607

Langmuir

Article

for driving flows in confined spaces such as capillaries and tight pores.

■

(VDW) forces because a calculation reveals that VDW energies are typically 200 nm.37 In our model, we assume that the reaction and the flow are at steady state and thus independent of time. Also, we employ the infinitesimal EDL approximation. In actuality, κ−1 is typically about 180 nm considering that equilibrium water has dissolved carbon dioxide, which gives carbonic acid. Thus, the resulting pH value is about 5.5 or 5.6 for deionized water, giving the bulk concentration of protons as n∞ = 1.90 × 1021 ions/m3. For a 1.0 μm radius, κa is about 5. Nevertheless, assuming an infinite κa allows us to obtain an estimate of the physics. We note that if one were to reason the result in a manner similar to Henry’s law (Appendix) with a finite Debye length (κ−1) then one would surmise that the motor speed increases as the size of the motor particle increases, an effect opposite to that observed in experiments or found in our model. Thus, considering a finite κ−1 would not help to explain the size dependence of motor speed following the same trend as what is observed experimentally. Although the motor speed was found to decrease dramatically at higher ionic strength in experiments,3 one explanation is that the presence of salt reduces the magnitude of the zeta potential of the motor.38 Furthermore, high ionic strength has been found to reduce the surface reaction rate, which thus decreases the speed of colloidal motors.38 Solving for the Electric Field. With the thin EDL approximation, the electric potential (ψ) in the outer region is calculated using the Laplace equation. The use of the Laplace equation requires that the solution be electroneutral and that it has a uniform conductivity (or uniform composition). As shown by Prieve’s work for our reaction-limited case, where one of the ion fluxes is zero (here the anion flux is zero),19 electroneutrality holds. Furthermore, even for the case when the reaction rate is higher at 6% H2O2, we find using eq 14 of ref 19 that the change in conductivity is less than 10%. Thus, we expect only small errors in using the Laplace equation. At the edge of the EDL, the boundary condition for the electric field is obtained by matching with a solution calculated for within the EDL as a result of the surface electrochemical reactions. This inner solution for thin EDL and small flux on the surface was found to be27

ELECTROKINETIC MODELING

Consider a Janus motor composed of two metals, labeled α and β, near a wall. The particle has a distance of closest approach

Figure 2. Schematic of a α−β Janus motor at a distance δ from the wall. The oxidation reaction takes place on the α metal, and the reduction reaction occurs on β. A flow of electrons is generated from α to β in the metal, with protons moving in solution in the same direction. The transport of protons induces an electric field, causing the movement of the colloidal motor, which almost always has a nonzero zeta potential. With a wall below the motor, the induced electric field is affected compared to the bulk.

(δ), as shown in Figure 2. The metals catalyze a redox reaction couple for species A and B: A red → A ox + H+ + e− Box + H+ + e− → Bred

(1)

With the oxidation of species A on α and the reduction of species B on β, a flow of electrons is generated from α to β through the metal, along with proton flow migrating in solution in the same direction. The transport of protons induces an electric field (E) in solution. This E field acts on any charged surfaces (and their EDLs) in the region, including the colloidal motor, and causes transport. In a solution of H2O2, both Ared and Box are H2O2, Aox is oxygen gas, and Bred is water. The two half reactions give the overall decomposition reaction of H2O2. α and β can be any two metals having different mixed potentials for the redox reaction.2 Common metals used for a colloidal motor are platinum−gold (Pt−Au, where α = Pt and β = Au),1−5,25 gold−silver (Au−Ag, where α = Au and β = Ag),34 and nickel−gold (Ni−Au, where α = Ni and β = Au).35 Although our calculation is for any particle−wall gap (δ), one way to predict the value of δ is by balancing the gravitational force Fg = 4πa3(ρp − ρf)g/3 and the electrostatic repulsion Fes ≈ 4πεakζpζw exp(−κδ) between the spherical particle and the wall with like charge,36 giving ⎛ ⎞ 3εζpζw ⎟ δ ≈ κ −1 ln⎜⎜ 2 −1 ⎟ ρ − ρ κ a ( ) g ⎝ ⎠ p f

E=

JkT 2eD H+n∞

(3)

Here J = Rn is the normal flux on the particle surface, where R is the surface reaction rate, n is the unit normal vector pointing from the particle to the fluid phase, k is the Boltzmann constant, T is the temperature, e is the proton charge, DH+ is the proton diffusion coefficient (∼9.0 × 10−9 m2/s), and n∞ is the bulk concentration of protons. Equation 3 relates the induced E field to the normal flux on the particle surface and has a similar form to Bard’s eq 4.2.1.39 We nondimensionalize all of the equations. We use ϕ = Zeψ/ kT, r ̅ = r/(a + δ), x̅ = x/(a + δ), z̅ = z/(a + δ), ∇̅ = (a + δ)∇, and j = R(a + δ)/DH+n∞, where Z is the valence of the ion. The origin is set at the center of the sphere. Starting from the Laplace equation, the nondimensionalization gives

(2)

∇̅ 2 ϕ = 0

where ε is the fluid permittivity, ζp is the zeta potential of the particle, ζw is the zeta potential of the wall, ρp is the density of the particle, ρf is the density of the fluid, and g is the gravitational acceleration. Here, we neglect van der Waals

(4)

The boundary conditions are that the electric field normal to the wall beneath the motor is zero, which assumes that the dielectric constant of water is much greater than the wall, and 2602

dx.doi.org/10.1021/la402262z | Langmuir 2014, 30, 2600−2607

Langmuir

Article

Solving for the Flow Field. The flow field (v) is solved using Stokes equation in the absence of the particle but in the presence of the E field resulting from the particle. We use the dimensionless flow field v ̅ = v/U∞, where U∞ = ε ⟨ζpE⟩/η is the electrophoretic speed of the motor in an unbounded solution. This gives

that the electric field into or out of the Janus motor at r = a arises from reactions with a given rate. Mathematically, we can express these as r ̅ → ∞, ϕ → 0 x ̅ = − 1,

∂ϕ =0 ∂x ̅

j ∂ϕ π r ̅ = λ, 0 ≤ θ < , − = 2 ∂r ̅ 2 j ∂ ϕ π r ̅ = λ, < θ ≤ π , − =− ∂r ̅ 2 2

η∇̅ 2 v̅ − ∇̅ p = 0, ∇̅ · v̅ = 0

where p is the pressure. The boundary conditions are x ̅ → ∞ , v̅ → 0 εζ ∂ϕ kT x ̅ = −1, v̅ = w iz ηU∞ Ze(a + δ) ∂z ̅

(5)

where λ = a/(a + δ). The boundary condition at r ̅ = λ is simply the dimensionless form of eq 3 because a + κ−1 ≈ a with the thin EDL approximation. These equations enable us to find the E field outside the particle. Solving for the Janus Motor Transport. We calculate the final particle speed along the z direction as u ≈ uep + v|r = 0

(6)

■

RESULTS AND DISCUSSION Figures 3−5 show the electric fields and field lines around the colloidal motor. Figure 3 shows the electric field lines around a

εζp E η

(9)

The fluid velocity approaches zero when it is far away from the wall. On the substrate surface, there is induced electroosmotic flow that can be described using the E field calculated in the previous subsection. Here, we assume that the pressure gradient is negligible because it has been shown to be small for the system;12 we discuss this more in the next section. We solved eqs 8 and 9 numerically using COMSOL. The result of calculating the fluid flow velocity is that we can evaluate v(r = 0) and use eqs 6 and 7 to calculate the final speed of the motor.

where uep is the electrophoretic speed of the particle, given the electric field around the particle. The value v(r = 0) is the fluid speed in the z direction at the center of the particle, given the same electric field, but in the absence of the particle. With Keh and Chen’s results,13 we assume that it is a good approximation to neglect the nonlinear interactions between flow fields on the particle and on the wall surface for λ ≤ 0.9. (See Table 1 and Figure 2 in their work.) When the Janus motor approaches the wall, the most important change that happens is that the electric field arising from the surface reactions on the motor causes an electroosmotic flow on the wall surface, thus changing v(r = 0). At the same time, the electric field around the particle is altered somewhat such that the average E field around the particle, ⟨E⟩ = ∫ ∫ SE·iz dS/(4πa2) where iz is the unit vector in the z direction, changes by 0.9, where the electric field and the flow field start to couple. The hydrodynamically coupled region (λ > 0.9) is neglected in this article. In contrast, for the case of a self-generated electric field, the electroosmotic flow is zero for infinite separations but monotonically increases with λ. One of the main differences is that the electroosmotic flow caused by an applied electric field is present in the bulk solution (infinite separations between the particle and the wall) whereas that induced by a self-generated electric field is local to being underneath the particle, when the particle is close to the wall (otherwise it is very nearly zero). We have shown that the motor speed is affected by the wall as the motor particle settles. Because of the fact that both the induced electroosmotic flow and the particle−plate separation (δ) are affected by ζw, we examine how ζw affects the motor speed at a fixed dimensionless separation (λ). Figure 8 shows the dimensionless motor speed U/U∞ versus λ with different ratios of ζw/ζp. The motor remains roughly at its bulk speed 2605

dx.doi.org/10.1021/la402262z | Langmuir 2014, 30, 2600−2607

Langmuir

■

1/2

For the applied electric field case, Ez,sub(0) = 1.09(δ/a)(2 −2)/2 = 1.09 (δ/a)−0.2929, whereas for the self-generated case Ez,sub(0) = 1.25(δ/a)−0.6777. It is interesting that the exponents are irrational numbers. Thus, the E field varies more strongly with the gap for the self-generated case than for the applied field case. A possible explanation is that the self-generated case produces a roughly constant current through the gap region and the applied field case produces a roughly constant potential. We have found that the electric field lines are not symmetric at the top and bottom of a Janus motor because of the wall and the flow field is also nonsymmetric. This asymmetry may cause the rotation of the motor, although it has been shown by Anderson that the angular velocity is zero in a nonuniform E field for a uniform zeta potential on a spherical particle in an unbounded solution.22 We have not examined this phenomenon in this article, and we note that assuming no rotation leads to only a small error in translational velocity for λ ≤ 0.9.13 There are other possible causes of torques, including asymmetric colloidal forces (due to Janus spheres) and asymmetric weighting (due to the deposition of a layer of metal on one side of the particle to make a Janus particle). However, we know that motors in a H2O2 solution are usually found in experiments to have straight runs over distances of tens of micrometers. Because of the fact that motors are found to have one face (for a Janus sphere) or end (for a rod) leading, the net torque would seem to be close to zero. Otherwise, a motor would change direction and speed on a time scale of 1/ Dr (where Dr is the rotational diffusion coefficient), whereas experimentally we see that the speed remains constant even though the direction changes in two dimensions. With doublets and cylinders, gravity helps to resist rotation more strongly because, for instance, a cylinder would have to “lift up” to rotate perpendicular to the long axis. A hypothesis for future study is that a motor moves with an equilibrium angle45,46 at which the net torque resulting from potential energy effects such as the asymmetric weight and dynamic effects such as electrokinetic flows equals zero.

Article

APPENDIX: ESTIMATING THE SIZE DEPENDENCE OF THE MOTOR SPEED USING HENRY’S LAW For spherical particles with low zeta potentials, the electrophoretic speed (u) and the surface charge density (ρs) can be shown as u=

2 εζE∞ f (κa) 3 η

(10)

⎛ 1 + κa ⎞ ⎟ ρs = εκζ ⎜ ⎝ κa ⎠

(11)

where E∞ = E∞iz is the applied electric field, and f(κa) is the function showing the double-layer polarization. Consider two cases in which particle 2 is larger than particle 1 and thus κ2a2 > κ1a1. Assuming constant ρs, ε, and E∞, we find ⎛ 1 + κ1a1 ⎞⎛ a 2 ⎞ f (κ2a 2) u2 =⎜ >1 ⎟⎜ ⎟ u1 ⎝ 1 + κ2a 2 ⎠⎝ a1 ⎠ f (κ1a1)

(12)

because κ−1 remains roughly the same with the change in motor size. We find that the electrophoretic speed increases with increasing a, which is opposite to both the experimental results and our model. Furthermore, this effect is not strong enough to change the motor speed by 50% when doubling the size, which is something that has been shown in experiments.

■

AUTHOR INFORMATION

Corresponding Author

*Phone: (814) 865-8739. Fax: (814) 865-7846. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We gratefully acknowledge support from the National Science Foundation through IDR grant CBET 1014673 and MRSEC grant DMR-08-20404.

■

CONCLUSIONS We have developed a model for self-swimming colloidal motors near a wall. The model is derived from the usual electrokinetic equations and reveals that localized electroosmotic (LEO) flow−a self-generated flow at the wall only near the particle−is important. For motors moving near a wall with the same sign of the zeta potential, LEO manifests itself by slowing down the particle, consistent with experimental observations. Thus, LEO must be taken into account in designing flows of colloidal motors in confined channels. Furthermore, the presence of LEO might alter how some experiments are done for colloidal motors in the future, with observations occurring a few diameters or more from a wall. The fact that the electroosmotic flow can be induced very close to the wall could enable colloidal motors to be agents for localized pumping in confined systems. It is interesting that whereas confining surfaces are usually thought to act as resistance to flow, thereby decreasing fluid speeds, for LEO a wall actually acts as a pump. As a result, LEO causes localized mixing between a colloidal motor and a confining surface. The same physics would apply between two colloidal motors. If the colloidal motors are reactive, perhaps for mineral particles47 or for particles with enzymes on their surfaces, this mixing would enhance the reaction rate for diffusion-limited cases.

■

REFERENCES

(1) Paxton, W. F.; Sen, A.; Mallouk, T. E. Motility of Catalytic Nanoparticles through Self-Generated Forces. Chem.Eur. J. 2005, 11, 6462−6470. (2) Wang, Y.; Hernandez, R. M.; Bartlett, D. J., Jr.; Bingham, J. M.; Kline, T. R.; Sen, A.; Mallouk, T. E. Bipolar Electrochemical Mechanism for the Propulsion of Catalytic Nanomotors in Hydrogen Peroxide Solutions. Langmuir 2006, 22, 10451−10456. (3) Paxton, W. F.; Baker, P. T.; Kline, T. R.; Wang, Y.; Mallouk, T. E.; Sen, A. Catalytically Induced Electrokinetics for Motors and Micropumps. J. Am. Chem. Soc. 2006, 128, 14881−14888. (4) Dhar, P.; Fischer, Th. M.; Wang, Y.; Mallouk, T. E.; Paxton, W. F.; Sen, A. Autonomously Moving Nanorods at a Viscous Interface. Nano Lett. 2006, 6, 66−72. (5) Wheat, P. M.; Marine, N. A.; Moran, J. L.; Posner, J. D. Rapid Fabrication of Bimetallic Spherical Motors. Langmuir 2010, 26, 13052−13055. (6) Yariv, E. Electrokinetic Self-Propulsion by Inhomogeneous Surface Kinetics. Proc. R. Soc. A 2011, 467, 1645−1664. (7) Golestanian, R.; Liverpool, T. B.; Ajdari, A. Designing Phoretic Micro- and Nano-Swimmers. New J. Phys. 2007, 9, 126−133. (8) Wang, W.; Chiang, T. Y.; Velegol, D.; Mallouk, T. E. Understanding the Efficiency of Autonomous Nano- and Microscale Motors. J. Am. Chem. Soc. 2013, 135, 10557−10565. 2606

dx.doi.org/10.1021/la402262z | Langmuir 2014, 30, 2600−2607

Langmuir

Article

(36) Israelachvili, J. N. Intermolecular and Surface Forces, 3rd ed.; Academic Press: Burlington, MA, 2011. (37) Velegol, D.; Anderson, J. L. Determining the Forces between Polystyrene Latex Spheres Using Differential Electrophoresis. Langmuir 1996, 12, 4103−4110. (38) Sabass, B.; Seifert, U. Nonlinear, Electrocatalytic Swimming in the Presence of Salt. J. Chem. Phys. 2012, 136, 214507−214520. (39) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; Wiley: New York, 2001. (40) Anderson, J. L.; Prieve, D. C. Diffusiophoresis Caused by Gradients of Strongly Adsorbing Solutes. Langmuir 1991, 7, 403−406. (41) Scales, P. J.; Grieser, F.; Healy, T. W.; White, L. R.; Chan, D. Y. C. Electrokinetics of the Silica-Solution Interface: A Flat Plate Streaming Potential Study. Langmuir 1992, 8, 965−974. (42) Kirby, B. J.; Hasselbrink, E. F., Jr. Zeta Potential of Microfluidic Substrates: 1. Theory, Experimental Techniques, and Effects on Separations. Electrophoresis 2004, 25, 187−202. (43) Bonnefont, A.; Argoul, F.; Bazant, M. Z. J. Electroanal. Chem. 2001, 500, 52−61. (44) Solomentsev, Y.; Velegol, D.; Anderson, J. L. Conduction in the Small Gap between Two Spheres. Phys. Fluids 1997, 9, 1209−1217. (45) Kretzschmar, I.; Song, J. H. Surface-Anisotropic Spherical Colloids in Geometric and Field Confinement. Curr. Opin. Colloid Interface Sci. 2011, 16, 84−95. (46) Ke, H.; Ye, S.; Carroll, R. L.; Showalter, K. Motion Analysis of Self-Propelled Pt-Silica Particles in Hydrogen Peroxide Solutions. J. Phys. Chem. A 2010, 114, 5462−5467. (47) McDermott, J. J.; Kar, A.; Daher, M.; Klara, S.; Wang, G.; Sen, A.; Velegol, D. Self-Generated Diffusioosmotic Flows from Calcium Carbonate Micropumps. Langmuir 2012, 28, 15491−15497.

(9) O’Brien, R. W.; White, L. R. Electrophoretic Mobility of a Spherical Colloidal Particle. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607−1626. (10) Teubner, M. The Motion of Charged Colloidal Particles in Electric Fields. J. Chem. Phys. 1982, 76, 5564−5573. (11) Anderson, J. L. Colloid Transport by Interfacial Forces. Annu. Rev. Fluid Mech. 1989, 21, 61−99. (12) Keh, H. J.; Anderson, J. L. Boundary Effects on Electrophoretic Motion of Colloidal Spheres. J. Fluid Mech. 1985, 153, 417−439. (13) Keh, H. J.; Chen, S. B. Electrophoresis of a Colloidal Sphere Parallel to a Dielectric Plane. J. Fluid Mech. 1988, 194, 377−390. (14) Chen, S. B.; Keh, H. J. Electrophoresis in a Dilute Dispersion of Colloidal Spheres. AIChE J. 1988, 34, 1075−1085. (15) Chen, S. B.; Keh, H. J. Axisymmetric Electrophoresis of Multiple Colloidal Spheres. J. Fluid Mech. 1992, 238, 251−276. (16) Keh, H. J.; Chen, S. B. Particle Interactions in Electrophoresis I. Motion of Two Spheres along Their Line of Centers. J. Colloid Interface Sci. 1989, 130, 542−555. (17) Keh, H. J.; Yang, F. R. Particle Interactions in Electrophoresis III. Axisymmetric Motion of Multiple Spheres. J. Colloid Interface Sci. 1990, 139, 105−116. (18) Keh, H. J.; Yang, F. R. Particle Interactions in Electrophoresis IV. Motion of Arbitrary Three-Dimensional Clusters of Spheres. J. Colloid Interface Sci. 1991, 145, 362−389. (19) Prieve, D. C. Changes in Zeta Potential Caused by a dc electric current for thin double Layers. Colloids Surf., A 2004, 250, 67−77. (20) Squires, T. M.; Bazant, M. Z. Induced-Charge Electro-Osmosis. J. Fluid Mech. 2004, 509, 217−252. (21) Morrison, F. A., Jr. Electrophoresis of a Particle of Arbitrary Shape. J. Colloid Interface Sci. 1970, 34, 210−214. (22) Anderson, J. L. Effect of Nonuniform Zeta Potential on Particle Movement in Electric Fields. J. Colloid Interface Sci. 1985, 105, 45−54. (23) Fair, M. C.; Anderson, J. L. Electrophoresis of Nonuniformly Charged Ellipsoidal Particles. J. Colloid Interface Sci. 1989, 127, 388− 400. (24) Feick, J. D.; Velegol, D. Electrophoresis of Spheroidal Particles Having a Random Distribution of Zeta Potential. Langmuir 2000, 16, 10315−10321. (25) Paxton, W. F.; Kistler, K. C.; Olmeda, C. C.; Sen, A.; St. Angelo, S. K.; Cao, Y.; Mallouk, T. E.; Lammert, P. E.; Crespi, V. H. Catalytic Nanomotors: Autonomous Movement of Striped Nanorods. J. Am. Chem. Soc. 2004, 126, 13424−13431. (26) Masel, R. I. Chemical Kinetics and Catalysis; Wiley-Interscience: New York, 2001. (27) Kline, T. R.; Iwata, J.; Lammert, P. E.; Mallouk, T. E.; Sen, A.; Velegol, D. Catalytically Driven Colloidal Patterning and Transport. J. Phys. Chem. B 2006, 110, 24513−24521. (28) Moran, J. L.; Wheat, P. M.; Posner, J. D. Locomotion of Electrocatalytic Nanomotors Due to Reaction Induced Charge Autoelectrophoresis. Phys. Rev. E 2010, 81, 065302(1)−065302(4). (29) Moran, J. L.; Posner, J. D. Electrokinetic Locomotion Due to Reaction-Induced Charge Auto-Electrophoresis. J. Fluid Mech. 2011, 680, 31−66. (30) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989; Chapters 4 and 12. (31) Yariv, E.; Brenner, H. Near-Contact Electrophoretic Motion of a Sphere Parallel to a Planar Wall. J. Fluid Mech. 2003, 484, 85−111. (32) Squires, T. M.; Bazant, M. Z. Breaking Symmetries in InducedCharge Electro-Osmosis and Electrophoresis. J. Fluid Mech. 2006, 560, 65−101. (33) Khair, A. S.; Squires, T. M. Surprising Consequences of Ion Conservation in Electro-Osmosis over a Surface Charge Discontinuity. J. Fluid Mech. 2008, 615, 323−334. (34) McDermott, J. J.; Velegol, D. Simple Fabrication of Metallic Colloidal Doublets Having Electrical Connectivity. Langmuir 2008, 24, 4335−4339. (35) Fournier-Bidoz, S.; Arsenault, A. C.; Manners, I.; Ozin, G. A. Synthetic Self-Propelled Nanorotors. Chem. Commun. 2005, 441−443. 2607

dx.doi.org/10.1021/la402262z | Langmuir 2014, 30, 2600−2607