Article pubs.acs.org/Langmuir
Nonlinear Thermokinetic Phenomena Due to the Seebeck Effect Hideyuki Sugioka* Frontier Research Center, Canon Inc., 30-2, Shimomaruko 3-chome, Ohta-ku, Tokyo Japan ABSTRACT: We propose a novel mechanism to produce nonlinear thermokinetic vortex flows around a circular cylinder with ideally high thermal conductivity in an electrolyte. That is, the nonlinear thermokinetic slip velocity, which is proportional to the square of the temperature gradient [(∇T)20], is derived based on the electrolyte Seebeck effect, heat conduction equation, and Helmholtz−Smoluchowski formula. Different from conventional linear thermokinetic theory, our theory predicts that the inversion of the temperature gradient does not change the direction of the thermokinetic flows and thus a Janus particle using this phenomenon can move to the both hotter and colder regions in a temperature gradient field by changing the direction of its dielectric end. Our findings bridge the gap between the electro- and thermo-kinetic phenomena and provide an integrated physical viewpoint for the interface science.
I. INTRODUCTION In the presence of temperature gradient, ∇T, a particle in a solution moves in the direction of the temperature gradient with a velocity u = −DST∇T, where D is the ordinary diffusion coefficient, ST = DT/D is the Soret coefficient, and DT is the thermodiffusion coefficient. This thermokinetic phenomenon is important since it can be used for various biomedical applications such as thermally driven segregation of macromolecular solutions,1 light tweezers,2 and pattern formations.3 In particular, although it is not recognized well, nonlinear thermokinetic phenomena, which velocity is proportional to the square of the temperature gradient, are attractive since the nonlinear phenomena may increase the performance of the applications largely and open the new era to control particles and the flow around them. However, to the best of my knowledge, there is no theory that matches with the requirements, although various proposed theories have certainly succeeded to explain linear thermophoretic effects to some extent. For example, the various classical thermophoretic theories are reviewed in ref 4; the theory considering the dependence of the surface tension (Marangoni force) on temperature explains that thermophoresis in polymer solution is independent of molecular weight and concentration,5 although the dependence on particle size is still argued;6 and the recent theories considering the dependence of the dielectric phoretic force on temperature with the assumption of an electric double layer due to the existence of a surface charge succeed in explaining various boundary problems.3,7−9 Further, Jiang et al.10 observed that the Janus particle (the particle containing two different surfaces on its two sides) moves in the direction of its dielectric end by the laser irradiation, and this phenomenon is successfully explained by the self-thermophoresis termed by the authors; i.e., the directed motion of the Janus particle is considered to be induced by the temperature gradient generated by itself. Moreover, just recently, Majee and Würger11 have interestingly © 2014 American Chemical Society
pointed out that particles can be charge by the adsorption of the irradiated laser light due to the electrolyte Seebeck effect and can be applied to the pattern formation and selective transport by the application of external electric fields. However, all these theories are intrinsically linear theory, and the sign of the Soret coefficient is intrinsically invariant. Further, around a metal cylinder, linear effects are present yet weakened because of the isothermal surface. Thus, in this study, we focus on providing a description of a novel nonlinear thermokinetic theory based on the electrolyte Seebeck effect. In particular, we propose a novel mechanism to produce nonlinear thermokinetic flow around a circular cylinder with ideally high thermal conductivity in an electrolyte under a temperature gradient field along with the thin-doublelayer approximation,12−15 as a first step.
II. THEORY II-1. Nonlinear Thermokinetic Theory. Figure 1 shows the schematic view of the basic concept of the nonlinear thermokinetic phenomena. In particular, Figure 1a shows a geometry of our considering system in a homogeneous temperature gradient (∇T)0 > 0 [typically, (∇T)0 = 1 K/ μm]; i.e., we consider the two-dimensional (2D) system placing a circular cylinder of radius a (typically, 1 μm) at a center position [(x,y) = (0,0), r = 0, and T = 0] in an unbounded electrolyte solution of the concentration C0. Further, Figure 1b,c,d shows the initial heat and electric fluxes, final electric field, and thermo-kinetic flow field, respectively; the color map shows the temperature field [calculated from eq 3], and the red and blue colors correspond to the hottest and coldest temperature (typically, T = +3 and −3 K at x = +3 and −3 μm). Namely, by the application of a homogeneous temperReceived: February 13, 2014 Revised: June 10, 2014 Published: July 8, 2014 8621
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effect. Here, the expression of E might be complex if we consider the inside region of the electric double layer; e.g., for one-dimensional problem, Chikina et al.16 consider such system (in their eqs 18−33); however, since we use the thin-doublelayer approximation,12−15 we can assume that ion concentrations are homogeneous in the outside region of the electric double layer at the final state and in the whole region at the initial state; and thus, for our current problem, we can simply describe it as (2)
E = S ∇T
where S is the Seebeck coefficient. Thus, we initially observe an induced electric field normal to the metal surface, as shown in Figure1a. Note that although we may need to explain eq 2 more, we will discuss later for readability. [Please see eqs 23−26.] Thus, by solving eqs 1 and 2 with the appropriate boundary condition that T = 0 at r = a, we obtain ⎛ a2 ⎞ T = ⎜1 − 2 ⎟(∇T )0 r cos θ ⎝ r ⎠
Eri = S Figure 1. (a) Basic concept of a nonlinear thermokinetic phenomenon based on the Seebeck effects. By the application of (∇T)0, a local temperature gradient ∇T becomes normal to the metal surface, and the electric field E = S∇T is initially generated due to the Seebeck effect, as shown in panel b; because of this induced electric field, ions move along the electric field lines, an electric double layer forms, and finally the electric field becomes parallel to the surface, as shown in panel c; finally, ions slip along the tangential electric field and it generates quadorapolar macroscopic vortex flows, as shown in panel d. Here, the color map shows the temperature field (calculated from eq 3) and the red and blue colors correspond to the hottest and coldest temperature (typically, +3 K and −3 K); a = 1 μm and (∇T)0 = 1 K/ μm.
⎛ S ∂T a2 ⎞ = −S⎜1 − 2 ⎟(∇T )0 sin θ r ∂θ ⎝ r ⎠
Eθi =
Eir
(4)
(5)
Eiθ
where and are the initial normal and tangential electric fields. It should be noted that the detail derivation of eqs 3−5 is very similar to that of eqs 12−14, which is described later. Because of the induced electric field in Figure 2b, ions move along the electric field lines, and the left and right sides are covered with the positive and negative ions, respectively; i.e., an inhomogeneous electric double layer forms on the metal surface, as shown in Figure 1c. Consequently, we finally observe an electric field parallel to the metal surface. The tangential electric field Efθ (≡ −(1/r)(∂Φf/∂θ)) is calculated from eqs 1 and 2 with the appropriate boundary conditions under the assumption of the thin-electric-double layer approximation; i.e., by using cylindrical coordinate (r, θ), we can describe our problem as follows:
ature gradient (∇T)0 (> 0) at r = ∞, field lines of local temperature gradient around the metal are promptly deformed and become normal to the metal surface according to the steady heat diffusion equation that ∇2 T = 0
⎛ a2 ⎞ ∂T = S⎜1 + 2 ⎟(∇T )0 cos θ ∂r ⎝ r ⎠
(3)
(1)
∂ 2Φf
since generally we can assume that the thermal conductivity of the metal cylinder is considered to be infinity (λm = ∞) compared to that of the electrolyte (λw), as shown in Figure 1a. The local temperature gradient field ∇T also causes a macroscopic electric field E due to the electrolyte Seebeck
∂r 2
∂Φf ∂r
2 1 ∂Φf 1 ∂ Φf + 2 =0 + r ∂r r ∂θ 2
=0
(at r = a)
(6)
(7)
Figure 2. Dependence of UThermo on (∇T)0 and a. 0 8622
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Φf → −S(∇T )0 r cos θ
(at r → ∞)
(8)
U0Thermo =
where Φf is an electric potential for the final (steady) state. Obviously, the final electric field Ef is described by superposition of the initial electric field obtained by Ei and the electric field Echarge generated by the charge of the counterions that form the electric double layer in Figure 1c. From eq 8, we can assume Φf = Φ̃f (r ) cos θ
∂r 2
1 ∂Φ̃f 1 − 2 Φ̃f = 0 r ∂r r
+
(9)
(10)
This equation is known as the Euler equation, and by transforming it with r = ez or z = ln r, we obtain the general solution as Φ̃f = C1′r + C2′r −1
(11)
where C′1 and C′2 are the parameters that are determined from the boundary conditions eqs 7 and 8; i.e., C′1 = −S(∇T)0 and C′2 = −a2S(∇T)0. Thus, we obtain ⎛ a2 ⎞ Φf = −S⎜1 + 2 ⎟(∇T )0 r cos θ ⎝ r ⎠ Erf = − Eθf = −
⎛ a2 ⎞ = S⎜1 − 2 ⎟(∇T )0 cos θ ∂r ⎝ r ⎠
urcylinder(r , θ ) = 2 (12)
(13)
(14)
outside ζ = Φiinside , r = a − Φ f , r = a = + 2S(∇T )0 a cos θ
(15)
Eθf, r = a = −2S(∇T )0 sin θ
(16)
It should be noted that eqs 18−22 show the novel nonlinear thermokinetic phenomena although they are predicted based on the predictions for ICEK phenomena; i.e., an electrolyte Seebeck effect bridges between the ICEK and thermokinetic phenomena and our theory provides a new integrated viewpoint for the interface science between solid and liquid. II-3. Differential Form on the Seebeck Effect for a Homogeneous System. Equation 2 is considered to be a differential form for a homogeneous ion system; thus, it might be obvious to some extent for some researchers at least in the framework of the thin double layer approximation. However, eq 2 is important from the viewpoint of fundamental physics thus here we will explain the derivation in a little more detail based on Wü rger’s formulations18 for thermoelectro diffusion systems; i.e., eq 2 is obtained based on the well-known ion currents
Φoutside f,r=a
where [= −2S(∇T)0a cos θ] is a final electric potential inside at the outside-edge and Φinside = 0) is an initial f,r=a (= −ST electric potential at the inside-edge. Please note that we assume that thermal conductivity of metal is infinite since the thermal conductivity of metal is approximately 100 to 1000 times higher than that of electrolyte (typically, water); thus, we set that T = 0 at the inside-edge of the double layer (i.e., on the real metal surface). Finally, as shown in Figure 1d, ions slip along the tangential electric field and generate quadorapolar macroscopic vortex flows. As known well, this slip velocity is described by the Helmholtz-Smoluchowski formula, εζ f Eθ , a η
⎡ ⎤ S±̂ C± ez ∇T ⎥ J± = ∓ezD±⎢∇C± ± C±∇Φ + ⎢⎣ ⎥⎦ kTa kTa
for a thermoelectric diffusion system every small local region
(17)
J = J+ + J− = 0
where ε is the permittivity and η is the viscosity of the liquid (typically, ε = 80ε0 and η = 1 mPa s for water; ε0 is the vacuum permittivity). Thus, from eqs 15−17, we obtain uθslip = 2U0Thermo sin 2θ
(20)
2a3 Thermo U0 sin 2θ (21) r3 Further, from ref 17, we predict that the Janus particle in a temperature gradient will move in the direction of its dielectric end at a velocity 9 Thermo thermo U Janus = U0 (22) 64
Therefore, the zeta potential (that is defined by the potential difference between the outside- and inside-edges of the electric double layer) and tangential electric field at the outside-edge are obtained as
uθslip = −
a(a 2 − r 2) Thermo U0 cos 2θ r3
uθcylinder (r , θ ) =
∂Φf
⎛ 1 ∂Φ a2 ⎞ = −S⎜1 + 2 ⎟(∇T )0 sin θ r ∂θ ⎝ r ⎠
(19)
is a characteristic velocity of the nonlinear thermokinetic phenomena. That is, the nonlinear thermokinetic slip velocity, which is proportional to the square of the temperature gradient [(∇T)20], is derived based on the electrolyte Seebeck effect, heat conduction equation, and Helmholtz−Smoluchowski formula. II-2. Relation between Induced Charge Electrokinetic Phenomena and the Nonlinear Thermokinetic Phenomena. Interestingly, eqs 18 and 19 are formally identical with the equations of induced charge electro-kinetic (ICEK) phenomena13,14 by the replacement of S(∇T)0 with an applied electric field E0, and the derivation process clearly shows that we can derive the other thermokinetic relations for the different problems by replacing E0 with S(∇T)0 in the corresponding equations for ICEK problems. In other words, by replacing the characteristic velocity of ICEK phenomena [UICEK (= εaE20/η)] 0 Thermo with U0 , we obtain the corresponding relations. Namely, from ref 14 we predict that the radial and azimuthal flow velocities for the conductive cylinder in a temperature gradient are
By substituting eq 9 into eq 6, we obtain ∂ 2Φ̃f
ε[S(∇T )0 ]2 a η
18,19
(23)
by assuming that at (24)
where the suffix “+” (“−”) shows the positive (negative) ion, C± is the concentration, ±ze is the charge of the ions, k is the Boltzmann constant, D± (= D) is the diffusion coefficient, Ta (≡ Tm + T) is an absolute temperature, Tm (e.g., ≃ 300 K) is an absolute temperature of the metal, and S±̂ is the “Eastman
(18)
where 8623
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Further, if the Joule loss PJ = σE2 and the viscous loss Pv = ηU20/λ2D for an unit volume are negligible compare to the heat flux Fh = λ∇T, the Seebeck electric field E = S∇T is justified since temperature distribution is stable, where σ is electric conductivity, λ is a thermal conductivity, and λD is the Debye screening length for the electric double layer; e.g., for aqueous dilute solution, if we assume that E ∼ 103 V/m for ∇T ∼ 1 K/ μm, S ∼ 1 mV/K, λD ∼ 0.1 μm, U0 ∼ 1 μm/s, σwater = 5.5 μ S/ m, λwater = 0.582 W/mK, and ηwater = 1 mPas, we obtain Pwater = j 5.5 W/m3, Pv = 0.1 W/m3, and Fh = 0.58 MW/m 2; thus, fortunately, although the conductivity increases as the ion concentration increases, the Joule and viscous losses are negligible compared to the value of the heat flux at least in the dilute limit. Further, the Joule loss generates only during the transient charging process of the electric double layer, while the viscous loss continues to generate in the steady state.
entropy of transfer”;20 i.e., by substituting eq 23 into eq 24, we obtain ⎡ ez (C+ + C −)∇Φ ezD⎢ −∇(C+ − C −) − kTa ⎣ −
⎤ 1 ̂ (S+C+ − S−̂ C −)∇T ⎥ kTa ⎦ (25)
= 0.
Thus, we obtain eq 2 for a homogeneous region (i.e., C+ = C− = C0) on the thin-double-layer problem with S=
C+S+̂ − C −S−̂ S ̂ − S−̂ ( ≡ Se̅ ) = + ez(C+ + C −) 2ez
(26)
It should be noted that the “S” and eq 2 are formally identical with the ordinary Seebeck coefficient that is determined for the bulk region and the simple relation (E = S∇T) that is used for the experimental measurement, respectively. However, the reason for ∇C± = 0 is different from each other; thus, the meaning is different from each other. Specifically, the relation (E = S∇T) and the Seebeck coefficient “S” in ref 19 are derived by using the charge neutrality at the far from boundaries; and usually the relation is used in an one-dimensional problem. Further, we describe the Seebeck effect on a finite double layer problem in Appendix A; of course, the advanced nonlinear thermokinetic theory supports the results of the simple nonlinear thermokinetic theory using the thin double layer approximation. However, we need to say that the explanation of this part is a little bit conceptual and it might be unclear for the expert readers in this field; thus, we strongly recommend the expert readers to read Appendix A; but concisely, by substituting Φ = Φother + Φeq into eq 25, we can separate eq 25 into two equations:
III. RESULTS Figure 2 shows the dependence of UThermo on the temperature 0 gradient[(∇T)0] and the radius (a) at S = 0.21, 0.5, and 1 mV/ K. As shown in Figure 2a, the sign of UThermo does not change 0 for the inversion of (∇T)0; this is because U0Thermo is proportional to (∇T)20. Further, UThermo is proportional to a 0 as shown in Figure 2b. Here, S = 0.21 and 1 mV/K are the Seebeck coefficients for HCl and tetrabutylammonium nitrate (TBAN), respectively, in water.11 Further, since the temperature gradient in the range of ∇T ≤ 5 K/μm is common in the thermokinetic experiments,10,21 the predictions in Figure 2a are realistic and they are observable in an appropriate experiment; specifically, we predict that UThermo = 0.12, 0.70, and 2.83 μm/s 0 (UThermo = 0.62, 3.54, and 14.17 μm/s) at S = 0.21, 0,5, and 1 0 mV/K, respectively, under the conditions that a = 1 μm (a = 5 μm) and (∇T)0 = 2 K/μm. It should be noted that although the order of UThermo might be comparable with the character0 istic thermodiffusion movement of metal due to a surface charge,18,22 we believe that we can distinguish the nonlinear thermokinetic phenomena from the linear thermokinetic phenomena by the characteristics U0 ∝ (S∇T)2; e.g., a thermophoretic slip flow around a particle on a wall is observed experimentally by Weinert and Braun;3 thus, in similar experiments, if the direction of a slip flow is invariable for the change of the direction of a thermal gradient, we can recognize the nonlinear thermokinetic phenomena. Figure 3 shows calculated results of flow fields around a fixed two face cylinder that has the metal and dielectric faces in a temperature gradient in a microfluidic channel by the boundary element method.15,23 In Figure 3, the velocities at the metal surface are provided by eq 18 and those at the dielectric surface and the upper and lower wall surfaces are assumed to be zero. Surprisingly, as shown in Figures 3a, b, the nonlinear thermokinetic phenomenon with the broken symmetries such as two face cylinders and Janus particles can produce a pumping function in the both normal and inverse directions; e.g., under the conditions that the channel width w = 10 μm, the channel length L/w = 2.25, and a/w = 0.15, the average flow velocity Up that is measured at the inlet is estimated as Up/Uthermo = ±0.16. 0 Although it might be difficult to observe the pumping effect experimentally because of a small pressure problem due to a small activity area, we believe that the characteristic asymmetrical vortex flows are observable at least. This is a resulting phenomenon of the nonlinear thermokinetic phenomena and interestingly by using a microcircular channel, we may produce the macroscopic circular flow in a temperature gradient.
⎡ ⎤ ez ezD±⎢ −∇(C+ − C −) − (C+ + C −)∇Φeq ⎥ = 0 kTa ⎣ ⎦ ⎡ ez ⎤ 1 ̂ ezD±⎢ − (C+ + C −)∇Φother − (S+C+ − S−̂ C −)∇T ⎥ kTa ⎣ kTa ⎦ =0
where Φeq is a equilibrium potential that describes ion concentrations by the Boltzmann equation and Φother is the potential of the other part. Here, the first equation is satisfied automatically from the Boltzmann equation and thus we obtain −∇Φother =
S+̂ C+ − S−̂ C − ∇T ez(C+ + C −)
This is the real meaning of eq 2 and the readers should be care for the real usage; further, please note that the equilibrium potential is localized in the inside region of the double layer region and becomes zero in the outside region. Furthermore, in Appendix B, we discuss a classical thermophoretic approach based on the diffusiophoretic velocity to clarify our problem in detail. In addition, in Appendix C, we derive a linear thermokinetic velocity due to a net charge and the Seebeck electric field. Additionally, we discuss the pioneering work and the one-dimensional problem in Appendix D to clarify our problem. 8624
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distinguish the nonlinear thermophoresis from the ordinary thermophoresis by the lateral motion, even if there is a linear thermophoretic velocity due to the net charge of the particle. Please note that the ordinary linear thermophoretic velocity is represented by eq 74 in Appendix B, and it shows that the relevant small parameter is the ratio of Debye length and particle size. Further, by using a resistive heater or circular hot air with a relatively weak light source for the observations, we can distinguish the nonlinear thermophoresis from the selfthermophoresis.10 However, for a thermophoretic experiment using a small Janus particle (≤ 1 μm), we need to care about the Brownian motion; specifically, similar to the experiment of ref 10, if we use 1 μm Au-silica Janus particles, we need to observe the Brownian trajectories and show that the diffusion coefficient of the Janus particle increases as the temperature gradient increases, without using a strong light source. Thus, to examine our prediction more directly, we may need to use a relatively large particle (e.g., 3 μm) with a large temperature gradient (e.g., 10 K/μm); this is because the corresponding Seebeck electric field [E = S(∇T)0] becomes 10 kV/m for S = 1 mV/K of the TBAN−water electrolyte and the corresponding large characteristic velocity (UThermo = 0.85 mm/s) makes it possible to perform ordinary observations of the electrokinetic phenomena.17 Specifically, we may consider the experimental setup similar to that used by Bonetti et al.,19 although we need to miniturize it and it might be a challenging study; e.g., by using an upper sapphire substrate cooled by circular cold water, a lower sapphire substrate heated by a thin film resistance, and a 5 μm spacer, we can prepare an experimental cell having the TBAN−water electrolyte and 3 μm Au−silica Janus particles in a 5 μm gap. Thus, by setting a cold and hot surface at 25 and 75 °C, respectively, we can obtain the homogeneous temperature gradient (∇T)0 = 10 K/μm with the heat flux Ff = λwater(∇T)0 = 5.8 MW/m2. Therefore, we believe that through the sapphire window we can observe a lateral motion of Janus particle and vortex flows around the particles and thus we can examine our theory, although the directions of the dielectric end of the particles are random. Further, we suggest that we should use a dilute electrolyte (10−6 to 10−3 M) in experiment since the decrease of velocity in a dense electrolyte (> 100 mM) is wellknown in ICEK phenomena.24 Furthermore, we believe that a experiment for the distilled water without additional ions is also interesting to examine the relation between the selfthermophoresis10 and the nonlinear thermophoresis due to the Seebeck effect.
Figure 3. Control of the flow direction by the direction of the dielectric surface.
Further, a Janus particle using this nonlinear thermokinetic phenomenon can move to the both hotter and colder regions at the velocity of eq 22 in a temperature gradient field by changing the direction of the dielectric face end. In other words, the sign of Soret coefficient can be controlled by the direction of the dielectric face. This phenomenon is attractive as a model of a living thing and a novel nanomachines; i.e., if it can change the surface condition freely it can move freely in the circumstance of a temperature gradient.
IV. DISCUSSION In this time, we did not consider the self-themophoretic system that uses temperature differences in itself and moves by absorbing laser power.10 However, we believe that we can expand our theory to the self-themophoretic system and other general thermokinetic systems in the same manner; i.e., by calculating the initial temperature (T) and the final potential (Φf) by the heat conduction equation [eq 1], the Laplace equation [eq 6], and appropriate boundary conditions (the Dirichlet and Neumann boundary conditions, respectively), we can derive the inhomogeneous zeta potentials ζ and tangential electric fields Efθ around the particles on the basis of the electrolyte Seebeck effect [eq 2]; thus, we generally obtain the themophoretic slip velocities around the concerned particles. It should be noted that different from the theory of the selfthermophoresis that uses the temperature differences between two faces and considers the empirical Soret coefficient,10 our theory is based on more fundamental physics [i.e., eq 23]. In the future, we will specifically expand our theory to other systems and show innovative applications. In Figure 3, we predict that irrespective of the direction of the temperature gradient, a fixed Janus particle can generate a flow in the direction of its dielectric end; thus an unfixed Janus particle can move in the direction of its metal end. Please note that according to our theory, even if we apply the temperature gradient in the ±y direction in Figure 3, we can also predict the same motion of the Janus particle; i.e., the Janus particle moves in the (±x) direction that is perpendicular to the applied temperature gradient. This prediction is useful to examine our theory experimentally as a fist step, since the thermophoretic motion in the lateral direction is not known; i.e., we can
V. CONCLUSION In conclusion, on the basis of the electrolyte Seebeck effect, we have proposed a theory of nonlinear thermokinetic phenomena in which the characteristic velocities are proportional to the square of the applied temperature gradient. We believe that our findings provide the physical concept to integrate the electroand thermokinetic phenomena.
■
APPENDIX A: THE SEEBECK EFFECT ON A FINITE DOUBLE LAYER PROBLEM The thin double layer approximation is an established method, and the corresponding treatment of the finite double layer is also well-known in the field of colloid science; however, some researchers might be worried about the treatment of the diffusion term in eq 23 when they consider the problem of a 8625
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finite electric double layer; thus, here we introduce how we can treat the diffusion term in a self-consistent manner. Please note that in many points the treatment might be similar to that of the ordinary induced-charge electro-kinetic phenomena; however, the basic physics is different; thus, here we clarify the problem. Namely, to consider the finite electric double layer problem, we assume that
Φ = Φf + Φeq
∇2 Φeq = −
ρ ε
C± = C0e∓ez / kTaΦeq
C+ + C − − C0 2 we obtain the relation δC =
C+S+̂ − C −S −̂ = C0(S+̂ − S −̂ ) + δC(S+̂ − S −̂ ) + (C+ − C −)
Thus, if we can assume that
(27)
δC ≪ C0 ,
∂r
(29)
C+ + C − ≃ 2C0
(40)
S+̂ − S−̂ = Se̅ 2ez
(41)
In this case, by considering the divergence of eq 34 with eq 1 we obtain −∇2 Φf ≃ Se̅ ∇2 T = 0
(42)
Therefore, the condition of eq 34 is satisfied automatically, and thus eq 25 is self-consistently satisfied in the same approximation as that of ref 11. Further, by substituting eqs 39 and 40 into eq 25, we obtain ⎡ ⎤ C ez ezD⎢ −∇(C+ − C −) − 2C0 ∇Φ − 0 (S+̂ − S−̂ )∇T ⎥ kTa kTa ⎣ ⎦
(32)
(43)
≃0
Please note that eq 43 is intrinsically the same equation as eq 5 of the Supporting Information of ref 11, and the obtained solution (Φ = Φf + Φeq) satisfies eq 43 completely. On the other hand, from eqs 28 and 29, we obtain the Poisson−Boltzmann equation
⎡ ez ⎤ S±̂ C± ∓ezD±⎢ ± ∇T ⎥ C±∇Φf + ⎢⎣ kTa ⎥⎦ kTa
∇2 Φeq = κ 2 Φeq
(44)
where κ ≡ (1/λD) = ((2C0(ez) )/(εkTa)) and λD is the Debye screening length. Thus, by considering Φeq = Φ̃eq cos θ, we obtain 2
Namely, the diffusion and electro-phoretic terms concerning Φeq in eq 23 automatically become zero. Therefore, by assuming J = J+ + J− = 0, we generally obtain
̃ ∂ 2Φeq
⎡ ez ⎤ 1 ̂ ezD±⎢ − (C+ + C −)∇Φf − (S+C+ − S−̂ C −)∇T ⎥ = 0 kTa ⎣ kTa ⎦
∂r 2
(33)
+
̃ ⎛ 1 ∂Φeq 1⎞ ̃ − ⎜κ 2 + 2 ⎟Φeq =0 ⎝ r ∂r r ⎠
1/2
(45)
Thus, by using the boundary condition of eq 30, we obtain
Namely, even for the inhomogeneous problem that includes the diffusion term, we obtain
Φeq =
(34)
2Sa (∇T )0 K1(κr ) cos θ K1(κa)
(46)
Please note that the solution of eq 45 is known as a modified Bessel equation [K1(κr)]. Consequently, based on eq 43, we have obtained the complete potential Φ (= Φf + Φeq) for the finite double layer problem concerning the nonlinear thermokinetic phenomena. Obviously, the new solution shows that the solution in the outside region of the electric double layer is the same as the previous predictions [i.e., eqs 12 to 16]. Further, in this case, it is well-known that the slip velocity of eqs 17 to 19 is justified when λD ≪ a. In other words, the detail of the inside structure of the electric double layer is not important for the determination of the slip velocity, which determines a macroscopic flow field.
with C+S+̂ − C −S−̂ ≡ S′ ez(C+ + C −)
(39)
(30)
⎡ ⎤ S±̂ C± ez ∇T ⎥ = J± = ∓ezD±⎢∇C± ± C±∇(Φeq + Φf ) + ⎢⎣ ⎥⎦ kTa kTa
S=
C+S+̂ − C −S−̂ ≃ C0(S+̂ − S−̂ )
S≃
Thus, by substituting eqs 27 and 31 into eq 23, we obtain
Ef = −∇Φf = S∇T
(38)
Thus, by substituting eqs 39 and 40 into eq 35, we obtain
where Φeq is the equilibrium potential, C0 is a bulk concentration of the ions, and ρ [= ez(C+ − C−)] is a charge. Please note that in eq 27 we just use the principal of superposition; i.e., eqs 6 and 28 are separated equations from ∇2(Φeq + Φf) = −ρ/ε; further, eq 29 is the Boltzmann equation. From eq 26, we obtain ez ∇C± ± C±∇Φeq = 0 kTa (31) ez ez ρ = −2ezC0 sinh Φeq ≃ −2ezC0 Φeq kTa kTa
(C+ − C −) ≪ C0
we obtain
(28)
Φeq + Φf = 0 (at r = a), Φeq = 0, = 0 (at r = ∞)
S+̂ − S −̂ 2
(37)
with the boundary conditions that
∂Φeq
(36)
(35)
where S is generally a valuable parameter depending on the ion concentrations (C±); thus, we can use eq 2 in the whole region with the general definition of S in eq 35. Here, we consider the same approximation as that of ref 11 and its Supporting Information as a first step; i.e., by considering 8626
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Of course, one can solve the finite double layer problem more specifically by using an ordinary procedure; namely, by considering the steady Stokes equation that η∇2u = ∇p − ρE and ∇·u = 0 with the principal of the superposition that u = ueq + uf and p = peq + pf, we can separate it into two equations; i.e., 2
η∇ uf = ∇pf ,
∇·uf = 0
η∇2 ueq = ∇peq − ρE ,
On the other hand, the solution of eq 47 is known as ufθ = 2C2r−3 sin 2θ,14,25 where C2 is an unknown parameter, which is f determined by the boundary condition that ueq θ (a) + uθ(a) = 0; 2 3 i.e.. C2 = ((ε[S(∇T)0] a)/η)a ; thus, we obtain uθslip = 2
(47)
∇·ueq = 0
■
APPENDIX B: CLASSICAL THERMOPHORETIC APPROACH BASED ON THE DIFFUSIOPHORETIC VELOCITY Thermokinetic phenomena of metal are usually treated in linear approximation;22 however, the linear approach may cease to be valid in the vicinity of heat-conducting metal. Thus, here we reconsider the classical thermophoretic approach based on the diffusiophoretic velocity. Further, although there are several kind of theories for classical (linear) thermophoresis, here we consider the diffusiophoretic theory;22,26 i.e., by considering the treatment of refs 22 and 26, we start from ∂Φf ⎞ ∂ 2u eq 1 ∂peq 1 ⎛ ∂Φeq + +ρ ⎜ η 2θ = ⎟ (θ component) ∂θ ⎠ r ∂θ r ⎝ ∂θ ∂r
eq ∂ ⎡1 ∂ 1 ∂ur ⎤ 1 ∂peq − ρEθ η ⎢ (ruθeq) − ⎥= ∂r ⎣ r ∂r r ∂θ ⎦ r ∂θ
−η
(49)
eq ∂peq 1 ∂ ⎡1 ∂ 1 ∂ur ⎤ (ruθeq) − − ρEr ⎢ ⎥= ∂r r ∂θ ⎣ r ∂r r ∂θ ⎦
(r component)
(50)
where Φ = Φf + Φeq ⎛ 2Sa ⎞ ⎛ a2 ⎞ = −S⎜1 + 2 ⎟(∇T )0 r cos θ + ⎜ ⎟(∇T )0 K1(κr ) ⎝ r ⎠ ⎝ K1(κa) ⎠
(58)
0=
(51)
cos θ
(52)
∂Φ ∂r ⎛ 2Sa ⎞ ⎛ ∂K (κr ) a2 ⎞ = +S⎜1 − 2 ⎟(∇T )0 cos θ − ⎜ ⎟(∇T )0 1 ∂r ⎝ r ⎠ ⎝ K1(κa) ⎠
Er = −
∂r
0=
= −ρEθ
∂peq ∂r
(θ component)
(r component)
εΦeq (r )Eθ η
= −2
(r component)
1 ∂peq 1 ∂Φeq ≡0 +ρ r ∂θ r ∂θ
(59)
(61)
(62)
Thus, by substituting eq 62 into eq 58, we obtain (54)
∂ 2uθeq ∂r
2
= +ρ
1 ∂Φf r ∂θ
(63)
Namely, at least for our problem, the term ((1/r)(∂peq/∂θ) + ρ(1/r)(∂Φeq/∂θ)) in eq 58 does not produce a thermophoretic velocity. Consequently, at least for our problem, the classical themophoretic velocity based on the diffusiophoresis is negligible. Nevertheless, if we dare to consider the diffusiophoresis,22,26 we need to return back to eq 54; i.e., we consider
(55)
where ρ ≃ −ε(∂2Φeq/∂r2), Eθ ≃ Efθ(a + λD) ≃ −2(1 − (λD/a)) S(∇T)0 sin θ ≃ −2S(∇T)0, and Er(r) ≃ Eeq r (r) = −(∂Φeq/∂r). Thus, by using the boundary condition that ueq θ (∞) = 0 and (∂ueq θ /∂r)(∞) = 0, we obtain uθeq(r ) =
∂r
That is,
η − ρEr
∂Φeq
⎛ ⎞ 1 ∂Φeq 1 ∂peq ez 1 ∂Φeq ≡ −ρ = ⎜2ezC0 sinh Φeq ⎟ r ∂θ kTa ⎠ r ∂θ r ∂θ ⎝
Thus, by assuming that (∂/∂θ) = 0 and neglecting the curvature of the cylinder in eqs 49 and 50, we obtain ∂ 2u eq η 2θ
∂r
+ρ
where p0 is a constant. Thus, we obtain
(53)
cos θ
∂peq
Please note that eqs 58 and 59 correspond to eqs 49 and 50, although we usually neglect the first term on the left-side as a proper approximation as shown in eqs 54 and 55. From eq 59 with the relation that ρ = ez(C+ − C−) = −2ezC0 sinh(ez/ kT)Φeq, we obtain ez peq = 2kTaC0 cosh Φeq + p0 kTa (60)
1 ∂Φ r ∂θ ⎛ 2Sa ⎞ ⎛ K (κr ) a2 ⎞ = −S⎜1 + 2 ⎟(∇T )0 sin θ + ⎜ ⎟(∇T )0 1 r ⎝ r ⎠ ⎝ K1(κa) ⎠
Eθ = −
sin θ
(57)
Please note that because the proposed nonlinear thermophoresis has a strong similarity with the ordinary ICEK phenomena, one can find a similar argument in the ordinary ICEK study.
(48)
where E = −∇Φ = −∇(Φeq + Φf) and the boundary condition is that ueq + uf = 0 at r = a and ∇ueq = 0 and ueq = 0 at r = ∞. Further, the each component of eq 48 can be described as
(θ component)
ε[S(∇T )0 ]2 a ⎡ a3 K (κr ) ⎤ ⎥ sin 2θ ⎢ 3 − 1 η K1(κa) ⎦ ⎣r
ε[S(∇T )0 ]2 a ⎛ K1(κr ) ⎞ ⎟ sin 2θ ⎜ η ⎝ K1(κa) ⎠
η
(56) 8627
∂ 2uθeq ∂r
2
= +ρ
1 ∂Φeq r ∂θ
(64)
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interaction between the fixed charge and the external electric field due to the Seebeck effect; i.e., from eq 17, we obtain
where ρ = ez(C+ − C −) = −2ezC0 sinh
ez Φeq ≃ −εk 2 Φeq kTa
uθslip,fix = −
(65) 2
2 Φeq )
1 ∂Φeq ε ∂(κ =− ∂θ r ∂θ 2r ∂r 2 2 ε ∂T ∂(κ Φeq ) =− ∂T 2r ∂θ ≃ −εκ 2 Φeq
(66)
Here, if we admit the concept of the diffusiophoresis,22,26 we need to consider that Φeq of the last term in eq 66 is not dependent on temperature; i.e., ∂ 2uθeq ∂r 2
≃−
2 εΦeq ∂Ta ∂κ 2 2ηr ∂θ ∂Ta
UT = (68)
2C (ez)2 ∂κ 2 κ2 =− 0 2 =− ∂Ta Ta εkTa
(69)
Φeq ≃ α e−κr
(70)
∂r
2
≃ κ 2 Φeq
UT′ =
∂r 2
≃B
(71)
e−2κr B 1 ∂ ⎛⎜ 2 ∂U ⎞⎟ B ∂ 2U r = ≃ r (2κ )2 r 2 ∂r ⎝ ∂r ⎠ (2κ )2 ∂r 2 (72)
where B=+
εζ 2κ 2e 2κa λD(∇T )0 cos θ ηTa
QS(∇T )0 2 εζ = S(∇T )0 6πμa 3 μ
(77)
εζ S(∇T )0 μ
(78)
■
APPENDIX D: DISCUSSION ON PIONEERING WORK AND THE ONE-DIMENSIONAL THERMOKINETIC PROBLEM Chikina et al. performed pioneering work using a perturbation method for one-dimensional thermoelectro diffusion systems considering Seebeck effects in electrolyte,16 and thus some researchers may consider that we should also use their method. In fact we tried but soon understand that we should not use their method at least for our problem. The main reason is that our method using an equilibrium potential is much more accurate than the Chikina et al.’s perturbative approach. Specifically, we consider the 1D thermokinetic problem in the direction of x (−W0 ≤ x ≤ +W0); i.e., we consider a closed cell of length 2W0, which includes an electrolyte solution in a homogeneous temperature gradient [(∇T)0 > 0] but does not include a circular cylinder as shown in Figure 1a. Of course, the calculation procedure is almost same as the procedure of Appendix A, except the
(73)
and we use the well-known relation of Yukawa potential (U = e−2κr/r). Thus, by using the boundary condition that ueq θ = 0 at r = a and (∂ueq θ /∂r) = 0 at r = ∞, we obtain uθeq(∞) ≃ −
(76)
Therefore, there will be a competition between linear and nonlinear effects; i.e., from eq 15 an induced zeta potential ζinduced due to the Seebeck effect is approximately 2S(∇T)0a; e.g., in the case of the TBAN−water electrolyte (S = 1 mV/K) with a particle of radius a (= 10 μm), ζinduced = 1, 10, and 100 mV at (∇T)0 = 0.1, 1, and 10 K/μ, respectively. Namely, the nonlinear thermophoresis becomes dominant only under the existence of a large temperature gradient (e.g., > 1 K/μm), when there is a substantial net charge on the particle. Further, please note that, in our understanding, it is difficult to assume a net charge on the noble metal such as Au and Ag except the effect of a surfactant for stabilization, although a significant linear thermophoretic effect is reported for the Au and Ag particles using a surfactant by the thermal field-flow fractionation.27
and αe−κa = ζ is a zeta potential by a homogeneous fixed charge. Thus, from eqs 67−70, we obtain ∂ 2uθeq
(75)
Please note that it is well-known that eq 77 is correct in the limit of a/λD → 0, while in the limit of a/λD → ∞ (i.e., thindouble-layer approximation) the expression becomes
Please note that eq 70 is obtained from the Poisson− Boltzmann equation
∂ 2Φeq
2Sεζ (∇T )0 sin θ η
where Q (= 4πεaζ) is a total net surface charge of a particle of radius a; thus, we obtain
(67)
≃ +2λD(∇T )0 cos θ r = a + λD
≃
6πμaUT = QS(∇T )0
where ∂Ta ∂T ≃ ∂θ ∂θ
r = a + λD
where ζ is the zeta potential due to a fixed surface charge. Subsequently, we obtain the same results from eq 63 under the appropriate boundary conditions. It should be noted that according to our theory the linear thermophoresis (due to a net charge and the Seebeck electric field) is observable not only for the thermally conductive particles but also for the thermally insulative particles. Namely, on the level of Debye-Hückel approximation, we derive the thermophoretic velocity UT for the three-dimensional problem as follows:
Thus, we obtain ∂ 2u eq η 2θ
εζ f Eθ η
B e−2κa εζ 2 ⎛ λD ⎞ ⎜ ⎟(∇T )0 cos θ = − 4ηTa ⎝ a ⎠ (2κ )2 a (74)
Consequently, the diffusiophoretic velocity approaches zero at (λD/a) → 0, and thus we can neglect the linear effect in the thin double layer approximation.
■
APPENDIX C: LINEAR THERMOPHORETIC VELOCITY DUE TO A FIXED NET CHARGE AND THE SEEBECK ELECTRIC FIELD Equation 63 shows that if there is a nonzero surface net charge, a linear thermophoretic velocity exists because of the 8628
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outside regions simultaneously. In particular, our 2D problem is much more complicated; thus, we need to use a more sophisticated method. Please note that although in our 2D problem we can describe an initial electric field simply as Ei = S∇T because of the homogeneity of the initial concentration, the solution of the final electric field in the ion diffusion region (eqs 52 and 53) is too complex to describe by one apparent Seebeck effect; instead, we need to define it as Ef ≡ Eθeθ + Erer ≡ Sa,θ(∇T)θeθ + Sa,r(∇T)rer, where eθ and er [(∇T)θ and (∇T)r] are a unit vector (a component of ∇T) in the direction of θ and r, respectively; however, in our opinion, it is useless since it is too complex to use; rather, we should understand a simple treatment using an equilibrium potential with the principal of superposition, as described in Appendix A. Moreover, if we need the more precise analytical solution at large ψD, we can use the well-known Gouy-Chapman’s solution as the solution of the nonlinear Poisson−Boltzmann equation for the large gap problem (W0 = ∞) or the solution of ion-conserving PoissonBoltzmann theory28 for the finite gap problem. Consequently, we could not use Chikina et al.’s method directly, but we believe that we have succeeded in elaborating their intended idea and contributing to the better understanding of the thermokinetic field; but at the same time, we need to admit that there is a lot of work to do in this field in the future. In particular, we may need to reconsider the boundary condition of the edge positions (eq 82) with the deep considerations on experimental facts since here we just use Chikina et al.’s condition.
boundary condition, but for the convenience of the reader we rewrite the main equations: ⎡ ez (C+ + C −)∇x Φ ezD⎢ −∇x (C+ − C −) − kTa ⎣ −
⎤ 1 ̂ (S+C+ − S−̂ C −)∇x T ⎥ kTa ⎦ (79)
=0 Φ = Φf + Φeq ,
ρ ∇2x Φeq = − , ε
C± = C0e∓ez / kTa Φeq (80)
with the boundary conditions that Φeq = 0, ∇x Φeq = 0
−∇x Φ = 0
(at x = 0)
(at x = ±W0)
(81) (82)
where ∇x = (∂/∂x). Please note that we just use Chikina et al.’s boundary condition16 in eq 82 as a first step. Similar to the 2D problem, by using the relation that ez −∇x (C+ − C −) = (C+ + C −)∇x Φeq kTa (83) and the Poisson-Boltzmann equation ∇2xϕeq = κ2Φeq, we easily obtain Φf = −xS(∇T )0 ,
S≡
C+S+̂ − C −S−̂ ez(C+ + C −)
■
(84)
right Φeq = ψDright e−κ(W0− x) (at x ≥ 0), left Φeq
=
ψDleft e−κ(W0+ x)
(at x ≤ 0)
Corresponding Author
*E-mail: [email protected].
(85)
Notes
left where ψright D and ψD are determined by the boundary condition. Thus, by using the boundary condition (eq 82), we easily obtain that ψright = λDS(∇T)0 > 0 and ψleft D D = −λDS(∇T)0 < 0. Therefore, we obtain
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS I am grateful to Prof. M. Sano for helpful discussions on the thermophoretic flow of a Janus particle.
light left Φ = Φf + Φeq + Φeq
= S(∇T )0 [λD(e−κ(W0− x) − e−κ(W0+ x)) − x]
∂Φ E=− = S(∇T )0 [1 − 2e−κW0 cosh κx] ∂x
■
(86)
C+S+̂ − C −S−̂ [1 − 2e−κW0 cosh κx] ez(C+ + C −)
REFERENCES
(1) Duhr, S.; Braun, D. Optothermal Molecule Trapping by Opposing Fluid Flow with Thermophoretic Drift. Phys. Rev. Lett. 2006, 97, 038103. (2) Braun, D.; Libchaber, A. Trapping of DNA by Thermophoretic Depletion and Convection. Phys. Rev. Lett. 2002, 89, 188103. (3) Weinert, F.; Braun, D. Observation of Slip Flow in Thermophoresis. Phys. Rev. Lett. 2008, 101, 168301. (4) Goldhirsch, I.; Ronis, D. Theory of Thermophoresis I. General Considerations and Mode-Coupling Analysis. Phys. Rev. A 1983, 27, 1616. (5) Würger, A. Thermophoresis in Colloidal Suspensions Driven by Marangoni Forces. Phys. Rev. Lett. 2007, 98, 138301. (6) Vigolo, D.; Brambilla, G.; Piazza, R. Thermophoresis of Microemulsion Droplets: Size Dependence of the Soret Effect. Phys. Rev. E 2007, 75, 040401(R). (7) Rasuli, S.; Golestainian, R. Soret Motion of a Charged Spherical Colloid. Phys. Rev. Lett. 2008, 101, 108301. (8) Fayolle, S.; Bickel, T.; Würger, A. Thermophoresis of Charged Colloidal Particles. Phys. Rev. Lett. 2008, 77, 041404. (9) Morthomas, J.; Würger, A. Hydrodynamic Attraction of Immobile Particles Due to Interfacial Forces. Phys. Rev. E 2010, 81, 051405.
(87)
Please note that if we define an apparent Seebeck coefficient Sa by E ≡ Sa∇xT, we obtain Sa =
AUTHOR INFORMATION
(88)
for the 1D thermokinetic problem. We believe that eqs 86−88 are precise, and they are analytical solutions that Chikina et al. tried to obtain. Actually, these solutions are exact for the problems corresponding to small zeta potential problems (i.e., (ez/kTa)Φeq < 1). Further, please note that Sa ≃ ((Ŝ+−Ŝ−)/2ez) = Se̅ in the outside region of the ion diffusion region (−W0 + λD < x < W0 − λD) since C+ (= C−) ≃ C0 and 2e−kW0 cosh kx ≃ 0 in the outside region of the ion diffusion region (i.e., in the bulk region). However, Chikina et al.’s apparent Seebeck coefficient [Stot in their paper16] is described as a constant. Thus, their perturbative method is not suitable to describe a thermokinetic situation of both inside and 8629
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(10) Jiang, H.; Yoshinaga, N.; Sano, M. Active Motion of a Janus Particle by Self-Thermophoresis in a Deforcused Laser Beam. Phys. Rev. Lett. 2010, 105, 268302. (11) Majee, A.; Würger, A. Charging of Heated Particles Using the Electrolyte Seebeck Effect. Phys. Rev. Lett. 2012, 108, 118301. (12) Fair, M.; Anderson, J. Electrophoresis of Non-uniformly Charged Ellipsoidal Particles. J. Colloid Interface Sci. 1989, 127, 388. (13) Bazant, M.; Squires, T. Induced-Charge Electrokinetic Phenomena: Theory and Microfluidic Applications. Phys. Rev. Lett. 2004, 92, 066101. (14) Squires, T.; Bazant, M. Induced-Charge Electro-osmosis. J. Fluid Mech. 2004, 509, 217. (15) Sugioka, H. Suppression of Reverse Flows in Pumping by Induced-Charge Electro-osmosis Using Asymmetrically Stacked Elliptical Metal Posts. Phys. Rev. E 2008, 78, 057301. (16) Chikina, I.; Shikin, V.; Varlamov, A. A. Seebeck Effect in Electrolytes. Phys. Rev. E 2012, 86, 011505. (17) Gangwal, S.; Cayre, O.; Bazant, M.; Velev, O. Induced-Charge Electrophoresis of Metallodielectric Particles. Phys. Rev. Lett. 2008, 100, 058302. (18) Würger, A. Transport in Charged Colloids Driven by Thermoelectricity. Phys. Rev. Lett. 2008, 101, 108302. (19) Bonetti, M.; Nakamae, S.; Roger, M.; Guenoun, P. Huge Seebeck Coefficients in Nanoaqueous Electrolytes. J. Chem. Phys. 2011, 134, 114513. (20) Eastman, E. Theory of the Soret Effect. J. Am. Chem. Soc. 1928, 50, 283. (21) Jiang, H.; Wada, H.; Yoshinaga, N.; Sano, M. Manipulation of Colloids by a Nonequilibrium Depletion Force in a Temperature. Phys. Rev. Lett. 2009, 102, 208301. (22) Giddings, J.; Shinude, P.; Semenov, S. Thermophoresis of Metal Particles in a Liquid. J. Colloid Interface Sci. 1995, 176, 454. (23) Sugioka, H. Basic Analysis of Induced-Charge Electrophoresis Using the Boundary Element Method. Colloids Surf., A 2011, 376, 102. (24) Bazant, M. Z.; Kilic, M. S.; Storey, B. D.; Ajdari, A. Towards an Understanding of Induced-Charge Electrokinetics at Large Applied Voltages in Concentrated Solutions. Adv. Colloid Interface Sci. 2009, 152, 48. (25) Gamayunov, N. I.; Murtsovkin, V. A.; Dukhin, A. S. Pair Interaction of Particles in Electric-Field. 1. Features of Hydrodynamic Interaction of Polarized Particles. Colloid J. USSR 1986, 48, 197−203. (26) Ruckenstein, E. Can Phoretic Motions Be Treated As Interfacial Tension Gradient Driven Phenomena? J. Colloid Interface Sci. 1981, 83, 77. (27) Shiundu, P.; Munguti, S.; Williams, S. Retention Behavior of Metal Particle Dispersions in Aqueous and Nonaqueous Carriers in Thermal Filed-Flow Fractionation. J. Chromatogr. A 2003, 983, 163. (28) Sugioka, H. Ion-Conserving Poisson−Boltzman Theory. Phys. Rev. E 2012, 86, 016318.
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Nonlinear Thermokinetic Phenomena Due to the Seebeck Effect Hideyuki Sugioka* Frontier Research Center, Canon Inc., 30-2, Shimomaruko 3-chome, Ohta-ku, Tokyo Japan ABSTRACT: We propose a novel mechanism to produce nonlinear thermokinetic vortex flows around a circular cylinder with ideally high thermal conductivity in an electrolyte. That is, the nonlinear thermokinetic slip velocity, which is proportional to the square of the temperature gradient [(∇T)20], is derived based on the electrolyte Seebeck effect, heat conduction equation, and Helmholtz−Smoluchowski formula. Different from conventional linear thermokinetic theory, our theory predicts that the inversion of the temperature gradient does not change the direction of the thermokinetic flows and thus a Janus particle using this phenomenon can move to the both hotter and colder regions in a temperature gradient field by changing the direction of its dielectric end. Our findings bridge the gap between the electro- and thermo-kinetic phenomena and provide an integrated physical viewpoint for the interface science.
I. INTRODUCTION In the presence of temperature gradient, ∇T, a particle in a solution moves in the direction of the temperature gradient with a velocity u = −DST∇T, where D is the ordinary diffusion coefficient, ST = DT/D is the Soret coefficient, and DT is the thermodiffusion coefficient. This thermokinetic phenomenon is important since it can be used for various biomedical applications such as thermally driven segregation of macromolecular solutions,1 light tweezers,2 and pattern formations.3 In particular, although it is not recognized well, nonlinear thermokinetic phenomena, which velocity is proportional to the square of the temperature gradient, are attractive since the nonlinear phenomena may increase the performance of the applications largely and open the new era to control particles and the flow around them. However, to the best of my knowledge, there is no theory that matches with the requirements, although various proposed theories have certainly succeeded to explain linear thermophoretic effects to some extent. For example, the various classical thermophoretic theories are reviewed in ref 4; the theory considering the dependence of the surface tension (Marangoni force) on temperature explains that thermophoresis in polymer solution is independent of molecular weight and concentration,5 although the dependence on particle size is still argued;6 and the recent theories considering the dependence of the dielectric phoretic force on temperature with the assumption of an electric double layer due to the existence of a surface charge succeed in explaining various boundary problems.3,7−9 Further, Jiang et al.10 observed that the Janus particle (the particle containing two different surfaces on its two sides) moves in the direction of its dielectric end by the laser irradiation, and this phenomenon is successfully explained by the self-thermophoresis termed by the authors; i.e., the directed motion of the Janus particle is considered to be induced by the temperature gradient generated by itself. Moreover, just recently, Majee and Würger11 have interestingly © 2014 American Chemical Society
pointed out that particles can be charge by the adsorption of the irradiated laser light due to the electrolyte Seebeck effect and can be applied to the pattern formation and selective transport by the application of external electric fields. However, all these theories are intrinsically linear theory, and the sign of the Soret coefficient is intrinsically invariant. Further, around a metal cylinder, linear effects are present yet weakened because of the isothermal surface. Thus, in this study, we focus on providing a description of a novel nonlinear thermokinetic theory based on the electrolyte Seebeck effect. In particular, we propose a novel mechanism to produce nonlinear thermokinetic flow around a circular cylinder with ideally high thermal conductivity in an electrolyte under a temperature gradient field along with the thin-doublelayer approximation,12−15 as a first step.
II. THEORY II-1. Nonlinear Thermokinetic Theory. Figure 1 shows the schematic view of the basic concept of the nonlinear thermokinetic phenomena. In particular, Figure 1a shows a geometry of our considering system in a homogeneous temperature gradient (∇T)0 > 0 [typically, (∇T)0 = 1 K/ μm]; i.e., we consider the two-dimensional (2D) system placing a circular cylinder of radius a (typically, 1 μm) at a center position [(x,y) = (0,0), r = 0, and T = 0] in an unbounded electrolyte solution of the concentration C0. Further, Figure 1b,c,d shows the initial heat and electric fluxes, final electric field, and thermo-kinetic flow field, respectively; the color map shows the temperature field [calculated from eq 3], and the red and blue colors correspond to the hottest and coldest temperature (typically, T = +3 and −3 K at x = +3 and −3 μm). Namely, by the application of a homogeneous temperReceived: February 13, 2014 Revised: June 10, 2014 Published: July 8, 2014 8621
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effect. Here, the expression of E might be complex if we consider the inside region of the electric double layer; e.g., for one-dimensional problem, Chikina et al.16 consider such system (in their eqs 18−33); however, since we use the thin-doublelayer approximation,12−15 we can assume that ion concentrations are homogeneous in the outside region of the electric double layer at the final state and in the whole region at the initial state; and thus, for our current problem, we can simply describe it as (2)
E = S ∇T
where S is the Seebeck coefficient. Thus, we initially observe an induced electric field normal to the metal surface, as shown in Figure1a. Note that although we may need to explain eq 2 more, we will discuss later for readability. [Please see eqs 23−26.] Thus, by solving eqs 1 and 2 with the appropriate boundary condition that T = 0 at r = a, we obtain ⎛ a2 ⎞ T = ⎜1 − 2 ⎟(∇T )0 r cos θ ⎝ r ⎠
Eri = S Figure 1. (a) Basic concept of a nonlinear thermokinetic phenomenon based on the Seebeck effects. By the application of (∇T)0, a local temperature gradient ∇T becomes normal to the metal surface, and the electric field E = S∇T is initially generated due to the Seebeck effect, as shown in panel b; because of this induced electric field, ions move along the electric field lines, an electric double layer forms, and finally the electric field becomes parallel to the surface, as shown in panel c; finally, ions slip along the tangential electric field and it generates quadorapolar macroscopic vortex flows, as shown in panel d. Here, the color map shows the temperature field (calculated from eq 3) and the red and blue colors correspond to the hottest and coldest temperature (typically, +3 K and −3 K); a = 1 μm and (∇T)0 = 1 K/ μm.
⎛ S ∂T a2 ⎞ = −S⎜1 − 2 ⎟(∇T )0 sin θ r ∂θ ⎝ r ⎠
Eθi =
Eir
(4)
(5)
Eiθ
where and are the initial normal and tangential electric fields. It should be noted that the detail derivation of eqs 3−5 is very similar to that of eqs 12−14, which is described later. Because of the induced electric field in Figure 2b, ions move along the electric field lines, and the left and right sides are covered with the positive and negative ions, respectively; i.e., an inhomogeneous electric double layer forms on the metal surface, as shown in Figure 1c. Consequently, we finally observe an electric field parallel to the metal surface. The tangential electric field Efθ (≡ −(1/r)(∂Φf/∂θ)) is calculated from eqs 1 and 2 with the appropriate boundary conditions under the assumption of the thin-electric-double layer approximation; i.e., by using cylindrical coordinate (r, θ), we can describe our problem as follows:
ature gradient (∇T)0 (> 0) at r = ∞, field lines of local temperature gradient around the metal are promptly deformed and become normal to the metal surface according to the steady heat diffusion equation that ∇2 T = 0
⎛ a2 ⎞ ∂T = S⎜1 + 2 ⎟(∇T )0 cos θ ∂r ⎝ r ⎠
(3)
(1)
∂ 2Φf
since generally we can assume that the thermal conductivity of the metal cylinder is considered to be infinity (λm = ∞) compared to that of the electrolyte (λw), as shown in Figure 1a. The local temperature gradient field ∇T also causes a macroscopic electric field E due to the electrolyte Seebeck
∂r 2
∂Φf ∂r
2 1 ∂Φf 1 ∂ Φf + 2 =0 + r ∂r r ∂θ 2
=0
(at r = a)
(6)
(7)
Figure 2. Dependence of UThermo on (∇T)0 and a. 0 8622
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Φf → −S(∇T )0 r cos θ
(at r → ∞)
(8)
U0Thermo =
where Φf is an electric potential for the final (steady) state. Obviously, the final electric field Ef is described by superposition of the initial electric field obtained by Ei and the electric field Echarge generated by the charge of the counterions that form the electric double layer in Figure 1c. From eq 8, we can assume Φf = Φ̃f (r ) cos θ
∂r 2
1 ∂Φ̃f 1 − 2 Φ̃f = 0 r ∂r r
+
(9)
(10)
This equation is known as the Euler equation, and by transforming it with r = ez or z = ln r, we obtain the general solution as Φ̃f = C1′r + C2′r −1
(11)
where C′1 and C′2 are the parameters that are determined from the boundary conditions eqs 7 and 8; i.e., C′1 = −S(∇T)0 and C′2 = −a2S(∇T)0. Thus, we obtain ⎛ a2 ⎞ Φf = −S⎜1 + 2 ⎟(∇T )0 r cos θ ⎝ r ⎠ Erf = − Eθf = −
⎛ a2 ⎞ = S⎜1 − 2 ⎟(∇T )0 cos θ ∂r ⎝ r ⎠
urcylinder(r , θ ) = 2 (12)
(13)
(14)
outside ζ = Φiinside , r = a − Φ f , r = a = + 2S(∇T )0 a cos θ
(15)
Eθf, r = a = −2S(∇T )0 sin θ
(16)
It should be noted that eqs 18−22 show the novel nonlinear thermokinetic phenomena although they are predicted based on the predictions for ICEK phenomena; i.e., an electrolyte Seebeck effect bridges between the ICEK and thermokinetic phenomena and our theory provides a new integrated viewpoint for the interface science between solid and liquid. II-3. Differential Form on the Seebeck Effect for a Homogeneous System. Equation 2 is considered to be a differential form for a homogeneous ion system; thus, it might be obvious to some extent for some researchers at least in the framework of the thin double layer approximation. However, eq 2 is important from the viewpoint of fundamental physics thus here we will explain the derivation in a little more detail based on Wü rger’s formulations18 for thermoelectro diffusion systems; i.e., eq 2 is obtained based on the well-known ion currents
Φoutside f,r=a
where [= −2S(∇T)0a cos θ] is a final electric potential inside at the outside-edge and Φinside = 0) is an initial f,r=a (= −ST electric potential at the inside-edge. Please note that we assume that thermal conductivity of metal is infinite since the thermal conductivity of metal is approximately 100 to 1000 times higher than that of electrolyte (typically, water); thus, we set that T = 0 at the inside-edge of the double layer (i.e., on the real metal surface). Finally, as shown in Figure 1d, ions slip along the tangential electric field and generate quadorapolar macroscopic vortex flows. As known well, this slip velocity is described by the Helmholtz-Smoluchowski formula, εζ f Eθ , a η
⎡ ⎤ S±̂ C± ez ∇T ⎥ J± = ∓ezD±⎢∇C± ± C±∇Φ + ⎢⎣ ⎥⎦ kTa kTa
for a thermoelectric diffusion system every small local region
(17)
J = J+ + J− = 0
where ε is the permittivity and η is the viscosity of the liquid (typically, ε = 80ε0 and η = 1 mPa s for water; ε0 is the vacuum permittivity). Thus, from eqs 15−17, we obtain uθslip = 2U0Thermo sin 2θ
(20)
2a3 Thermo U0 sin 2θ (21) r3 Further, from ref 17, we predict that the Janus particle in a temperature gradient will move in the direction of its dielectric end at a velocity 9 Thermo thermo U Janus = U0 (22) 64
Therefore, the zeta potential (that is defined by the potential difference between the outside- and inside-edges of the electric double layer) and tangential electric field at the outside-edge are obtained as
uθslip = −
a(a 2 − r 2) Thermo U0 cos 2θ r3
uθcylinder (r , θ ) =
∂Φf
⎛ 1 ∂Φ a2 ⎞ = −S⎜1 + 2 ⎟(∇T )0 sin θ r ∂θ ⎝ r ⎠
(19)
is a characteristic velocity of the nonlinear thermokinetic phenomena. That is, the nonlinear thermokinetic slip velocity, which is proportional to the square of the temperature gradient [(∇T)20], is derived based on the electrolyte Seebeck effect, heat conduction equation, and Helmholtz−Smoluchowski formula. II-2. Relation between Induced Charge Electrokinetic Phenomena and the Nonlinear Thermokinetic Phenomena. Interestingly, eqs 18 and 19 are formally identical with the equations of induced charge electro-kinetic (ICEK) phenomena13,14 by the replacement of S(∇T)0 with an applied electric field E0, and the derivation process clearly shows that we can derive the other thermokinetic relations for the different problems by replacing E0 with S(∇T)0 in the corresponding equations for ICEK problems. In other words, by replacing the characteristic velocity of ICEK phenomena [UICEK (= εaE20/η)] 0 Thermo with U0 , we obtain the corresponding relations. Namely, from ref 14 we predict that the radial and azimuthal flow velocities for the conductive cylinder in a temperature gradient are
By substituting eq 9 into eq 6, we obtain ∂ 2Φ̃f
ε[S(∇T )0 ]2 a η
18,19
(23)
by assuming that at (24)
where the suffix “+” (“−”) shows the positive (negative) ion, C± is the concentration, ±ze is the charge of the ions, k is the Boltzmann constant, D± (= D) is the diffusion coefficient, Ta (≡ Tm + T) is an absolute temperature, Tm (e.g., ≃ 300 K) is an absolute temperature of the metal, and S±̂ is the “Eastman
(18)
where 8623
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Further, if the Joule loss PJ = σE2 and the viscous loss Pv = ηU20/λ2D for an unit volume are negligible compare to the heat flux Fh = λ∇T, the Seebeck electric field E = S∇T is justified since temperature distribution is stable, where σ is electric conductivity, λ is a thermal conductivity, and λD is the Debye screening length for the electric double layer; e.g., for aqueous dilute solution, if we assume that E ∼ 103 V/m for ∇T ∼ 1 K/ μm, S ∼ 1 mV/K, λD ∼ 0.1 μm, U0 ∼ 1 μm/s, σwater = 5.5 μ S/ m, λwater = 0.582 W/mK, and ηwater = 1 mPas, we obtain Pwater = j 5.5 W/m3, Pv = 0.1 W/m3, and Fh = 0.58 MW/m 2; thus, fortunately, although the conductivity increases as the ion concentration increases, the Joule and viscous losses are negligible compared to the value of the heat flux at least in the dilute limit. Further, the Joule loss generates only during the transient charging process of the electric double layer, while the viscous loss continues to generate in the steady state.
entropy of transfer”;20 i.e., by substituting eq 23 into eq 24, we obtain ⎡ ez (C+ + C −)∇Φ ezD⎢ −∇(C+ − C −) − kTa ⎣ −
⎤ 1 ̂ (S+C+ − S−̂ C −)∇T ⎥ kTa ⎦ (25)
= 0.
Thus, we obtain eq 2 for a homogeneous region (i.e., C+ = C− = C0) on the thin-double-layer problem with S=
C+S+̂ − C −S−̂ S ̂ − S−̂ ( ≡ Se̅ ) = + ez(C+ + C −) 2ez
(26)
It should be noted that the “S” and eq 2 are formally identical with the ordinary Seebeck coefficient that is determined for the bulk region and the simple relation (E = S∇T) that is used for the experimental measurement, respectively. However, the reason for ∇C± = 0 is different from each other; thus, the meaning is different from each other. Specifically, the relation (E = S∇T) and the Seebeck coefficient “S” in ref 19 are derived by using the charge neutrality at the far from boundaries; and usually the relation is used in an one-dimensional problem. Further, we describe the Seebeck effect on a finite double layer problem in Appendix A; of course, the advanced nonlinear thermokinetic theory supports the results of the simple nonlinear thermokinetic theory using the thin double layer approximation. However, we need to say that the explanation of this part is a little bit conceptual and it might be unclear for the expert readers in this field; thus, we strongly recommend the expert readers to read Appendix A; but concisely, by substituting Φ = Φother + Φeq into eq 25, we can separate eq 25 into two equations:
III. RESULTS Figure 2 shows the dependence of UThermo on the temperature 0 gradient[(∇T)0] and the radius (a) at S = 0.21, 0.5, and 1 mV/ K. As shown in Figure 2a, the sign of UThermo does not change 0 for the inversion of (∇T)0; this is because U0Thermo is proportional to (∇T)20. Further, UThermo is proportional to a 0 as shown in Figure 2b. Here, S = 0.21 and 1 mV/K are the Seebeck coefficients for HCl and tetrabutylammonium nitrate (TBAN), respectively, in water.11 Further, since the temperature gradient in the range of ∇T ≤ 5 K/μm is common in the thermokinetic experiments,10,21 the predictions in Figure 2a are realistic and they are observable in an appropriate experiment; specifically, we predict that UThermo = 0.12, 0.70, and 2.83 μm/s 0 (UThermo = 0.62, 3.54, and 14.17 μm/s) at S = 0.21, 0,5, and 1 0 mV/K, respectively, under the conditions that a = 1 μm (a = 5 μm) and (∇T)0 = 2 K/μm. It should be noted that although the order of UThermo might be comparable with the character0 istic thermodiffusion movement of metal due to a surface charge,18,22 we believe that we can distinguish the nonlinear thermokinetic phenomena from the linear thermokinetic phenomena by the characteristics U0 ∝ (S∇T)2; e.g., a thermophoretic slip flow around a particle on a wall is observed experimentally by Weinert and Braun;3 thus, in similar experiments, if the direction of a slip flow is invariable for the change of the direction of a thermal gradient, we can recognize the nonlinear thermokinetic phenomena. Figure 3 shows calculated results of flow fields around a fixed two face cylinder that has the metal and dielectric faces in a temperature gradient in a microfluidic channel by the boundary element method.15,23 In Figure 3, the velocities at the metal surface are provided by eq 18 and those at the dielectric surface and the upper and lower wall surfaces are assumed to be zero. Surprisingly, as shown in Figures 3a, b, the nonlinear thermokinetic phenomenon with the broken symmetries such as two face cylinders and Janus particles can produce a pumping function in the both normal and inverse directions; e.g., under the conditions that the channel width w = 10 μm, the channel length L/w = 2.25, and a/w = 0.15, the average flow velocity Up that is measured at the inlet is estimated as Up/Uthermo = ±0.16. 0 Although it might be difficult to observe the pumping effect experimentally because of a small pressure problem due to a small activity area, we believe that the characteristic asymmetrical vortex flows are observable at least. This is a resulting phenomenon of the nonlinear thermokinetic phenomena and interestingly by using a microcircular channel, we may produce the macroscopic circular flow in a temperature gradient.
⎡ ⎤ ez ezD±⎢ −∇(C+ − C −) − (C+ + C −)∇Φeq ⎥ = 0 kTa ⎣ ⎦ ⎡ ez ⎤ 1 ̂ ezD±⎢ − (C+ + C −)∇Φother − (S+C+ − S−̂ C −)∇T ⎥ kTa ⎣ kTa ⎦ =0
where Φeq is a equilibrium potential that describes ion concentrations by the Boltzmann equation and Φother is the potential of the other part. Here, the first equation is satisfied automatically from the Boltzmann equation and thus we obtain −∇Φother =
S+̂ C+ − S−̂ C − ∇T ez(C+ + C −)
This is the real meaning of eq 2 and the readers should be care for the real usage; further, please note that the equilibrium potential is localized in the inside region of the double layer region and becomes zero in the outside region. Furthermore, in Appendix B, we discuss a classical thermophoretic approach based on the diffusiophoretic velocity to clarify our problem in detail. In addition, in Appendix C, we derive a linear thermokinetic velocity due to a net charge and the Seebeck electric field. Additionally, we discuss the pioneering work and the one-dimensional problem in Appendix D to clarify our problem. 8624
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distinguish the nonlinear thermophoresis from the ordinary thermophoresis by the lateral motion, even if there is a linear thermophoretic velocity due to the net charge of the particle. Please note that the ordinary linear thermophoretic velocity is represented by eq 74 in Appendix B, and it shows that the relevant small parameter is the ratio of Debye length and particle size. Further, by using a resistive heater or circular hot air with a relatively weak light source for the observations, we can distinguish the nonlinear thermophoresis from the selfthermophoresis.10 However, for a thermophoretic experiment using a small Janus particle (≤ 1 μm), we need to care about the Brownian motion; specifically, similar to the experiment of ref 10, if we use 1 μm Au-silica Janus particles, we need to observe the Brownian trajectories and show that the diffusion coefficient of the Janus particle increases as the temperature gradient increases, without using a strong light source. Thus, to examine our prediction more directly, we may need to use a relatively large particle (e.g., 3 μm) with a large temperature gradient (e.g., 10 K/μm); this is because the corresponding Seebeck electric field [E = S(∇T)0] becomes 10 kV/m for S = 1 mV/K of the TBAN−water electrolyte and the corresponding large characteristic velocity (UThermo = 0.85 mm/s) makes it possible to perform ordinary observations of the electrokinetic phenomena.17 Specifically, we may consider the experimental setup similar to that used by Bonetti et al.,19 although we need to miniturize it and it might be a challenging study; e.g., by using an upper sapphire substrate cooled by circular cold water, a lower sapphire substrate heated by a thin film resistance, and a 5 μm spacer, we can prepare an experimental cell having the TBAN−water electrolyte and 3 μm Au−silica Janus particles in a 5 μm gap. Thus, by setting a cold and hot surface at 25 and 75 °C, respectively, we can obtain the homogeneous temperature gradient (∇T)0 = 10 K/μm with the heat flux Ff = λwater(∇T)0 = 5.8 MW/m2. Therefore, we believe that through the sapphire window we can observe a lateral motion of Janus particle and vortex flows around the particles and thus we can examine our theory, although the directions of the dielectric end of the particles are random. Further, we suggest that we should use a dilute electrolyte (10−6 to 10−3 M) in experiment since the decrease of velocity in a dense electrolyte (> 100 mM) is wellknown in ICEK phenomena.24 Furthermore, we believe that a experiment for the distilled water without additional ions is also interesting to examine the relation between the selfthermophoresis10 and the nonlinear thermophoresis due to the Seebeck effect.
Figure 3. Control of the flow direction by the direction of the dielectric surface.
Further, a Janus particle using this nonlinear thermokinetic phenomenon can move to the both hotter and colder regions at the velocity of eq 22 in a temperature gradient field by changing the direction of the dielectric face end. In other words, the sign of Soret coefficient can be controlled by the direction of the dielectric face. This phenomenon is attractive as a model of a living thing and a novel nanomachines; i.e., if it can change the surface condition freely it can move freely in the circumstance of a temperature gradient.
IV. DISCUSSION In this time, we did not consider the self-themophoretic system that uses temperature differences in itself and moves by absorbing laser power.10 However, we believe that we can expand our theory to the self-themophoretic system and other general thermokinetic systems in the same manner; i.e., by calculating the initial temperature (T) and the final potential (Φf) by the heat conduction equation [eq 1], the Laplace equation [eq 6], and appropriate boundary conditions (the Dirichlet and Neumann boundary conditions, respectively), we can derive the inhomogeneous zeta potentials ζ and tangential electric fields Efθ around the particles on the basis of the electrolyte Seebeck effect [eq 2]; thus, we generally obtain the themophoretic slip velocities around the concerned particles. It should be noted that different from the theory of the selfthermophoresis that uses the temperature differences between two faces and considers the empirical Soret coefficient,10 our theory is based on more fundamental physics [i.e., eq 23]. In the future, we will specifically expand our theory to other systems and show innovative applications. In Figure 3, we predict that irrespective of the direction of the temperature gradient, a fixed Janus particle can generate a flow in the direction of its dielectric end; thus an unfixed Janus particle can move in the direction of its metal end. Please note that according to our theory, even if we apply the temperature gradient in the ±y direction in Figure 3, we can also predict the same motion of the Janus particle; i.e., the Janus particle moves in the (±x) direction that is perpendicular to the applied temperature gradient. This prediction is useful to examine our theory experimentally as a fist step, since the thermophoretic motion in the lateral direction is not known; i.e., we can
V. CONCLUSION In conclusion, on the basis of the electrolyte Seebeck effect, we have proposed a theory of nonlinear thermokinetic phenomena in which the characteristic velocities are proportional to the square of the applied temperature gradient. We believe that our findings provide the physical concept to integrate the electroand thermokinetic phenomena.
■
APPENDIX A: THE SEEBECK EFFECT ON A FINITE DOUBLE LAYER PROBLEM The thin double layer approximation is an established method, and the corresponding treatment of the finite double layer is also well-known in the field of colloid science; however, some researchers might be worried about the treatment of the diffusion term in eq 23 when they consider the problem of a 8625
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finite electric double layer; thus, here we introduce how we can treat the diffusion term in a self-consistent manner. Please note that in many points the treatment might be similar to that of the ordinary induced-charge electro-kinetic phenomena; however, the basic physics is different; thus, here we clarify the problem. Namely, to consider the finite electric double layer problem, we assume that
Φ = Φf + Φeq
∇2 Φeq = −
ρ ε
C± = C0e∓ez / kTaΦeq
C+ + C − − C0 2 we obtain the relation δC =
C+S+̂ − C −S −̂ = C0(S+̂ − S −̂ ) + δC(S+̂ − S −̂ ) + (C+ − C −)
Thus, if we can assume that
(27)
δC ≪ C0 ,
∂r
(29)
C+ + C − ≃ 2C0
(40)
S+̂ − S−̂ = Se̅ 2ez
(41)
In this case, by considering the divergence of eq 34 with eq 1 we obtain −∇2 Φf ≃ Se̅ ∇2 T = 0
(42)
Therefore, the condition of eq 34 is satisfied automatically, and thus eq 25 is self-consistently satisfied in the same approximation as that of ref 11. Further, by substituting eqs 39 and 40 into eq 25, we obtain ⎡ ⎤ C ez ezD⎢ −∇(C+ − C −) − 2C0 ∇Φ − 0 (S+̂ − S−̂ )∇T ⎥ kTa kTa ⎣ ⎦
(32)
(43)
≃0
Please note that eq 43 is intrinsically the same equation as eq 5 of the Supporting Information of ref 11, and the obtained solution (Φ = Φf + Φeq) satisfies eq 43 completely. On the other hand, from eqs 28 and 29, we obtain the Poisson−Boltzmann equation
⎡ ez ⎤ S±̂ C± ∓ezD±⎢ ± ∇T ⎥ C±∇Φf + ⎢⎣ kTa ⎥⎦ kTa
∇2 Φeq = κ 2 Φeq
(44)
where κ ≡ (1/λD) = ((2C0(ez) )/(εkTa)) and λD is the Debye screening length. Thus, by considering Φeq = Φ̃eq cos θ, we obtain 2
Namely, the diffusion and electro-phoretic terms concerning Φeq in eq 23 automatically become zero. Therefore, by assuming J = J+ + J− = 0, we generally obtain
̃ ∂ 2Φeq
⎡ ez ⎤ 1 ̂ ezD±⎢ − (C+ + C −)∇Φf − (S+C+ − S−̂ C −)∇T ⎥ = 0 kTa ⎣ kTa ⎦
∂r 2
(33)
+
̃ ⎛ 1 ∂Φeq 1⎞ ̃ − ⎜κ 2 + 2 ⎟Φeq =0 ⎝ r ∂r r ⎠
1/2
(45)
Thus, by using the boundary condition of eq 30, we obtain
Namely, even for the inhomogeneous problem that includes the diffusion term, we obtain
Φeq =
(34)
2Sa (∇T )0 K1(κr ) cos θ K1(κa)
(46)
Please note that the solution of eq 45 is known as a modified Bessel equation [K1(κr)]. Consequently, based on eq 43, we have obtained the complete potential Φ (= Φf + Φeq) for the finite double layer problem concerning the nonlinear thermokinetic phenomena. Obviously, the new solution shows that the solution in the outside region of the electric double layer is the same as the previous predictions [i.e., eqs 12 to 16]. Further, in this case, it is well-known that the slip velocity of eqs 17 to 19 is justified when λD ≪ a. In other words, the detail of the inside structure of the electric double layer is not important for the determination of the slip velocity, which determines a macroscopic flow field.
with C+S+̂ − C −S−̂ ≡ S′ ez(C+ + C −)
(39)
(30)
⎡ ⎤ S±̂ C± ez ∇T ⎥ = J± = ∓ezD±⎢∇C± ± C±∇(Φeq + Φf ) + ⎢⎣ ⎥⎦ kTa kTa
S=
C+S+̂ − C −S−̂ ≃ C0(S+̂ − S−̂ )
S≃
Thus, by substituting eqs 27 and 31 into eq 23, we obtain
Ef = −∇Φf = S∇T
(38)
Thus, by substituting eqs 39 and 40 into eq 35, we obtain
where Φeq is the equilibrium potential, C0 is a bulk concentration of the ions, and ρ [= ez(C+ − C−)] is a charge. Please note that in eq 27 we just use the principal of superposition; i.e., eqs 6 and 28 are separated equations from ∇2(Φeq + Φf) = −ρ/ε; further, eq 29 is the Boltzmann equation. From eq 26, we obtain ez ∇C± ± C±∇Φeq = 0 kTa (31) ez ez ρ = −2ezC0 sinh Φeq ≃ −2ezC0 Φeq kTa kTa
(C+ − C −) ≪ C0
we obtain
(28)
Φeq + Φf = 0 (at r = a), Φeq = 0, = 0 (at r = ∞)
S+̂ − S −̂ 2
(37)
with the boundary conditions that
∂Φeq
(36)
(35)
where S is generally a valuable parameter depending on the ion concentrations (C±); thus, we can use eq 2 in the whole region with the general definition of S in eq 35. Here, we consider the same approximation as that of ref 11 and its Supporting Information as a first step; i.e., by considering 8626
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Of course, one can solve the finite double layer problem more specifically by using an ordinary procedure; namely, by considering the steady Stokes equation that η∇2u = ∇p − ρE and ∇·u = 0 with the principal of the superposition that u = ueq + uf and p = peq + pf, we can separate it into two equations; i.e., 2
η∇ uf = ∇pf ,
∇·uf = 0
η∇2 ueq = ∇peq − ρE ,
On the other hand, the solution of eq 47 is known as ufθ = 2C2r−3 sin 2θ,14,25 where C2 is an unknown parameter, which is f determined by the boundary condition that ueq θ (a) + uθ(a) = 0; 2 3 i.e.. C2 = ((ε[S(∇T)0] a)/η)a ; thus, we obtain uθslip = 2
(47)
∇·ueq = 0
■
APPENDIX B: CLASSICAL THERMOPHORETIC APPROACH BASED ON THE DIFFUSIOPHORETIC VELOCITY Thermokinetic phenomena of metal are usually treated in linear approximation;22 however, the linear approach may cease to be valid in the vicinity of heat-conducting metal. Thus, here we reconsider the classical thermophoretic approach based on the diffusiophoretic velocity. Further, although there are several kind of theories for classical (linear) thermophoresis, here we consider the diffusiophoretic theory;22,26 i.e., by considering the treatment of refs 22 and 26, we start from ∂Φf ⎞ ∂ 2u eq 1 ∂peq 1 ⎛ ∂Φeq + +ρ ⎜ η 2θ = ⎟ (θ component) ∂θ ⎠ r ∂θ r ⎝ ∂θ ∂r
eq ∂ ⎡1 ∂ 1 ∂ur ⎤ 1 ∂peq − ρEθ η ⎢ (ruθeq) − ⎥= ∂r ⎣ r ∂r r ∂θ ⎦ r ∂θ
−η
(49)
eq ∂peq 1 ∂ ⎡1 ∂ 1 ∂ur ⎤ (ruθeq) − − ρEr ⎢ ⎥= ∂r r ∂θ ⎣ r ∂r r ∂θ ⎦
(r component)
(50)
where Φ = Φf + Φeq ⎛ 2Sa ⎞ ⎛ a2 ⎞ = −S⎜1 + 2 ⎟(∇T )0 r cos θ + ⎜ ⎟(∇T )0 K1(κr ) ⎝ r ⎠ ⎝ K1(κa) ⎠
(58)
0=
(51)
cos θ
(52)
∂Φ ∂r ⎛ 2Sa ⎞ ⎛ ∂K (κr ) a2 ⎞ = +S⎜1 − 2 ⎟(∇T )0 cos θ − ⎜ ⎟(∇T )0 1 ∂r ⎝ r ⎠ ⎝ K1(κa) ⎠
Er = −
∂r
0=
= −ρEθ
∂peq ∂r
(θ component)
(r component)
εΦeq (r )Eθ η
= −2
(r component)
1 ∂peq 1 ∂Φeq ≡0 +ρ r ∂θ r ∂θ
(59)
(61)
(62)
Thus, by substituting eq 62 into eq 58, we obtain (54)
∂ 2uθeq ∂r
2
= +ρ
1 ∂Φf r ∂θ
(63)
Namely, at least for our problem, the term ((1/r)(∂peq/∂θ) + ρ(1/r)(∂Φeq/∂θ)) in eq 58 does not produce a thermophoretic velocity. Consequently, at least for our problem, the classical themophoretic velocity based on the diffusiophoresis is negligible. Nevertheless, if we dare to consider the diffusiophoresis,22,26 we need to return back to eq 54; i.e., we consider
(55)
where ρ ≃ −ε(∂2Φeq/∂r2), Eθ ≃ Efθ(a + λD) ≃ −2(1 − (λD/a)) S(∇T)0 sin θ ≃ −2S(∇T)0, and Er(r) ≃ Eeq r (r) = −(∂Φeq/∂r). Thus, by using the boundary condition that ueq θ (∞) = 0 and (∂ueq θ /∂r)(∞) = 0, we obtain uθeq(r ) =
∂r
That is,
η − ρEr
∂Φeq
⎛ ⎞ 1 ∂Φeq 1 ∂peq ez 1 ∂Φeq ≡ −ρ = ⎜2ezC0 sinh Φeq ⎟ r ∂θ kTa ⎠ r ∂θ r ∂θ ⎝
Thus, by assuming that (∂/∂θ) = 0 and neglecting the curvature of the cylinder in eqs 49 and 50, we obtain ∂ 2u eq η 2θ
∂r
+ρ
where p0 is a constant. Thus, we obtain
(53)
cos θ
∂peq
Please note that eqs 58 and 59 correspond to eqs 49 and 50, although we usually neglect the first term on the left-side as a proper approximation as shown in eqs 54 and 55. From eq 59 with the relation that ρ = ez(C+ − C−) = −2ezC0 sinh(ez/ kT)Φeq, we obtain ez peq = 2kTaC0 cosh Φeq + p0 kTa (60)
1 ∂Φ r ∂θ ⎛ 2Sa ⎞ ⎛ K (κr ) a2 ⎞ = −S⎜1 + 2 ⎟(∇T )0 sin θ + ⎜ ⎟(∇T )0 1 r ⎝ r ⎠ ⎝ K1(κa) ⎠
Eθ = −
sin θ
(57)
Please note that because the proposed nonlinear thermophoresis has a strong similarity with the ordinary ICEK phenomena, one can find a similar argument in the ordinary ICEK study.
(48)
where E = −∇Φ = −∇(Φeq + Φf) and the boundary condition is that ueq + uf = 0 at r = a and ∇ueq = 0 and ueq = 0 at r = ∞. Further, the each component of eq 48 can be described as
(θ component)
ε[S(∇T )0 ]2 a ⎡ a3 K (κr ) ⎤ ⎥ sin 2θ ⎢ 3 − 1 η K1(κa) ⎦ ⎣r
ε[S(∇T )0 ]2 a ⎛ K1(κr ) ⎞ ⎟ sin 2θ ⎜ η ⎝ K1(κa) ⎠
η
(56) 8627
∂ 2uθeq ∂r
2
= +ρ
1 ∂Φeq r ∂θ
(64)
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interaction between the fixed charge and the external electric field due to the Seebeck effect; i.e., from eq 17, we obtain
where ρ = ez(C+ − C −) = −2ezC0 sinh
ez Φeq ≃ −εk 2 Φeq kTa
uθslip,fix = −
(65) 2
2 Φeq )
1 ∂Φeq ε ∂(κ =− ∂θ r ∂θ 2r ∂r 2 2 ε ∂T ∂(κ Φeq ) =− ∂T 2r ∂θ ≃ −εκ 2 Φeq
(66)
Here, if we admit the concept of the diffusiophoresis,22,26 we need to consider that Φeq of the last term in eq 66 is not dependent on temperature; i.e., ∂ 2uθeq ∂r 2
≃−
2 εΦeq ∂Ta ∂κ 2 2ηr ∂θ ∂Ta
UT = (68)
2C (ez)2 ∂κ 2 κ2 =− 0 2 =− ∂Ta Ta εkTa
(69)
Φeq ≃ α e−κr
(70)
∂r
2
≃ κ 2 Φeq
UT′ =
∂r 2
≃B
(71)
e−2κr B 1 ∂ ⎛⎜ 2 ∂U ⎞⎟ B ∂ 2U r = ≃ r (2κ )2 r 2 ∂r ⎝ ∂r ⎠ (2κ )2 ∂r 2 (72)
where B=+
εζ 2κ 2e 2κa λD(∇T )0 cos θ ηTa
QS(∇T )0 2 εζ = S(∇T )0 6πμa 3 μ
(77)
εζ S(∇T )0 μ
(78)
■
APPENDIX D: DISCUSSION ON PIONEERING WORK AND THE ONE-DIMENSIONAL THERMOKINETIC PROBLEM Chikina et al. performed pioneering work using a perturbation method for one-dimensional thermoelectro diffusion systems considering Seebeck effects in electrolyte,16 and thus some researchers may consider that we should also use their method. In fact we tried but soon understand that we should not use their method at least for our problem. The main reason is that our method using an equilibrium potential is much more accurate than the Chikina et al.’s perturbative approach. Specifically, we consider the 1D thermokinetic problem in the direction of x (−W0 ≤ x ≤ +W0); i.e., we consider a closed cell of length 2W0, which includes an electrolyte solution in a homogeneous temperature gradient [(∇T)0 > 0] but does not include a circular cylinder as shown in Figure 1a. Of course, the calculation procedure is almost same as the procedure of Appendix A, except the
(73)
and we use the well-known relation of Yukawa potential (U = e−2κr/r). Thus, by using the boundary condition that ueq θ = 0 at r = a and (∂ueq θ /∂r) = 0 at r = ∞, we obtain uθeq(∞) ≃ −
(76)
Therefore, there will be a competition between linear and nonlinear effects; i.e., from eq 15 an induced zeta potential ζinduced due to the Seebeck effect is approximately 2S(∇T)0a; e.g., in the case of the TBAN−water electrolyte (S = 1 mV/K) with a particle of radius a (= 10 μm), ζinduced = 1, 10, and 100 mV at (∇T)0 = 0.1, 1, and 10 K/μ, respectively. Namely, the nonlinear thermophoresis becomes dominant only under the existence of a large temperature gradient (e.g., > 1 K/μm), when there is a substantial net charge on the particle. Further, please note that, in our understanding, it is difficult to assume a net charge on the noble metal such as Au and Ag except the effect of a surfactant for stabilization, although a significant linear thermophoretic effect is reported for the Au and Ag particles using a surfactant by the thermal field-flow fractionation.27
and αe−κa = ζ is a zeta potential by a homogeneous fixed charge. Thus, from eqs 67−70, we obtain ∂ 2uθeq
(75)
Please note that it is well-known that eq 77 is correct in the limit of a/λD → 0, while in the limit of a/λD → ∞ (i.e., thindouble-layer approximation) the expression becomes
Please note that eq 70 is obtained from the Poisson− Boltzmann equation
∂ 2Φeq
2Sεζ (∇T )0 sin θ η
where Q (= 4πεaζ) is a total net surface charge of a particle of radius a; thus, we obtain
(67)
≃ +2λD(∇T )0 cos θ r = a + λD
≃
6πμaUT = QS(∇T )0
where ∂Ta ∂T ≃ ∂θ ∂θ
r = a + λD
where ζ is the zeta potential due to a fixed surface charge. Subsequently, we obtain the same results from eq 63 under the appropriate boundary conditions. It should be noted that according to our theory the linear thermophoresis (due to a net charge and the Seebeck electric field) is observable not only for the thermally conductive particles but also for the thermally insulative particles. Namely, on the level of Debye-Hückel approximation, we derive the thermophoretic velocity UT for the three-dimensional problem as follows:
Thus, we obtain ∂ 2u eq η 2θ
εζ f Eθ η
B e−2κa εζ 2 ⎛ λD ⎞ ⎜ ⎟(∇T )0 cos θ = − 4ηTa ⎝ a ⎠ (2κ )2 a (74)
Consequently, the diffusiophoretic velocity approaches zero at (λD/a) → 0, and thus we can neglect the linear effect in the thin double layer approximation.
■
APPENDIX C: LINEAR THERMOPHORETIC VELOCITY DUE TO A FIXED NET CHARGE AND THE SEEBECK ELECTRIC FIELD Equation 63 shows that if there is a nonzero surface net charge, a linear thermophoretic velocity exists because of the 8628
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outside regions simultaneously. In particular, our 2D problem is much more complicated; thus, we need to use a more sophisticated method. Please note that although in our 2D problem we can describe an initial electric field simply as Ei = S∇T because of the homogeneity of the initial concentration, the solution of the final electric field in the ion diffusion region (eqs 52 and 53) is too complex to describe by one apparent Seebeck effect; instead, we need to define it as Ef ≡ Eθeθ + Erer ≡ Sa,θ(∇T)θeθ + Sa,r(∇T)rer, where eθ and er [(∇T)θ and (∇T)r] are a unit vector (a component of ∇T) in the direction of θ and r, respectively; however, in our opinion, it is useless since it is too complex to use; rather, we should understand a simple treatment using an equilibrium potential with the principal of superposition, as described in Appendix A. Moreover, if we need the more precise analytical solution at large ψD, we can use the well-known Gouy-Chapman’s solution as the solution of the nonlinear Poisson−Boltzmann equation for the large gap problem (W0 = ∞) or the solution of ion-conserving PoissonBoltzmann theory28 for the finite gap problem. Consequently, we could not use Chikina et al.’s method directly, but we believe that we have succeeded in elaborating their intended idea and contributing to the better understanding of the thermokinetic field; but at the same time, we need to admit that there is a lot of work to do in this field in the future. In particular, we may need to reconsider the boundary condition of the edge positions (eq 82) with the deep considerations on experimental facts since here we just use Chikina et al.’s condition.
boundary condition, but for the convenience of the reader we rewrite the main equations: ⎡ ez (C+ + C −)∇x Φ ezD⎢ −∇x (C+ − C −) − kTa ⎣ −
⎤ 1 ̂ (S+C+ − S−̂ C −)∇x T ⎥ kTa ⎦ (79)
=0 Φ = Φf + Φeq ,
ρ ∇2x Φeq = − , ε
C± = C0e∓ez / kTa Φeq (80)
with the boundary conditions that Φeq = 0, ∇x Φeq = 0
−∇x Φ = 0
(at x = 0)
(at x = ±W0)
(81) (82)
where ∇x = (∂/∂x). Please note that we just use Chikina et al.’s boundary condition16 in eq 82 as a first step. Similar to the 2D problem, by using the relation that ez −∇x (C+ − C −) = (C+ + C −)∇x Φeq kTa (83) and the Poisson-Boltzmann equation ∇2xϕeq = κ2Φeq, we easily obtain Φf = −xS(∇T )0 ,
S≡
C+S+̂ − C −S−̂ ez(C+ + C −)
■
(84)
right Φeq = ψDright e−κ(W0− x) (at x ≥ 0), left Φeq
=
ψDleft e−κ(W0+ x)
(at x ≤ 0)
Corresponding Author
*E-mail: [email protected].
(85)
Notes
left where ψright D and ψD are determined by the boundary condition. Thus, by using the boundary condition (eq 82), we easily obtain that ψright = λDS(∇T)0 > 0 and ψleft D D = −λDS(∇T)0 < 0. Therefore, we obtain
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS I am grateful to Prof. M. Sano for helpful discussions on the thermophoretic flow of a Janus particle.
light left Φ = Φf + Φeq + Φeq
= S(∇T )0 [λD(e−κ(W0− x) − e−κ(W0+ x)) − x]
∂Φ E=− = S(∇T )0 [1 − 2e−κW0 cosh κx] ∂x
■
(86)
C+S+̂ − C −S−̂ [1 − 2e−κW0 cosh κx] ez(C+ + C −)
REFERENCES
(1) Duhr, S.; Braun, D. Optothermal Molecule Trapping by Opposing Fluid Flow with Thermophoretic Drift. Phys. Rev. Lett. 2006, 97, 038103. (2) Braun, D.; Libchaber, A. Trapping of DNA by Thermophoretic Depletion and Convection. Phys. Rev. Lett. 2002, 89, 188103. (3) Weinert, F.; Braun, D. Observation of Slip Flow in Thermophoresis. Phys. Rev. Lett. 2008, 101, 168301. (4) Goldhirsch, I.; Ronis, D. Theory of Thermophoresis I. General Considerations and Mode-Coupling Analysis. Phys. Rev. A 1983, 27, 1616. (5) Würger, A. Thermophoresis in Colloidal Suspensions Driven by Marangoni Forces. Phys. Rev. Lett. 2007, 98, 138301. (6) Vigolo, D.; Brambilla, G.; Piazza, R. Thermophoresis of Microemulsion Droplets: Size Dependence of the Soret Effect. Phys. Rev. E 2007, 75, 040401(R). (7) Rasuli, S.; Golestainian, R. Soret Motion of a Charged Spherical Colloid. Phys. Rev. Lett. 2008, 101, 108301. (8) Fayolle, S.; Bickel, T.; Würger, A. Thermophoresis of Charged Colloidal Particles. Phys. Rev. Lett. 2008, 77, 041404. (9) Morthomas, J.; Würger, A. Hydrodynamic Attraction of Immobile Particles Due to Interfacial Forces. Phys. Rev. E 2010, 81, 051405.
(87)
Please note that if we define an apparent Seebeck coefficient Sa by E ≡ Sa∇xT, we obtain Sa =
AUTHOR INFORMATION
(88)
for the 1D thermokinetic problem. We believe that eqs 86−88 are precise, and they are analytical solutions that Chikina et al. tried to obtain. Actually, these solutions are exact for the problems corresponding to small zeta potential problems (i.e., (ez/kTa)Φeq < 1). Further, please note that Sa ≃ ((Ŝ+−Ŝ−)/2ez) = Se̅ in the outside region of the ion diffusion region (−W0 + λD < x < W0 − λD) since C+ (= C−) ≃ C0 and 2e−kW0 cosh kx ≃ 0 in the outside region of the ion diffusion region (i.e., in the bulk region). However, Chikina et al.’s apparent Seebeck coefficient [Stot in their paper16] is described as a constant. Thus, their perturbative method is not suitable to describe a thermokinetic situation of both inside and 8629
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(10) Jiang, H.; Yoshinaga, N.; Sano, M. Active Motion of a Janus Particle by Self-Thermophoresis in a Deforcused Laser Beam. Phys. Rev. Lett. 2010, 105, 268302. (11) Majee, A.; Würger, A. Charging of Heated Particles Using the Electrolyte Seebeck Effect. Phys. Rev. Lett. 2012, 108, 118301. (12) Fair, M.; Anderson, J. Electrophoresis of Non-uniformly Charged Ellipsoidal Particles. J. Colloid Interface Sci. 1989, 127, 388. (13) Bazant, M.; Squires, T. Induced-Charge Electrokinetic Phenomena: Theory and Microfluidic Applications. Phys. Rev. Lett. 2004, 92, 066101. (14) Squires, T.; Bazant, M. Induced-Charge Electro-osmosis. J. Fluid Mech. 2004, 509, 217. (15) Sugioka, H. Suppression of Reverse Flows in Pumping by Induced-Charge Electro-osmosis Using Asymmetrically Stacked Elliptical Metal Posts. Phys. Rev. E 2008, 78, 057301. (16) Chikina, I.; Shikin, V.; Varlamov, A. A. Seebeck Effect in Electrolytes. Phys. Rev. E 2012, 86, 011505. (17) Gangwal, S.; Cayre, O.; Bazant, M.; Velev, O. Induced-Charge Electrophoresis of Metallodielectric Particles. Phys. Rev. Lett. 2008, 100, 058302. (18) Würger, A. Transport in Charged Colloids Driven by Thermoelectricity. Phys. Rev. Lett. 2008, 101, 108302. (19) Bonetti, M.; Nakamae, S.; Roger, M.; Guenoun, P. Huge Seebeck Coefficients in Nanoaqueous Electrolytes. J. Chem. Phys. 2011, 134, 114513. (20) Eastman, E. Theory of the Soret Effect. J. Am. Chem. Soc. 1928, 50, 283. (21) Jiang, H.; Wada, H.; Yoshinaga, N.; Sano, M. Manipulation of Colloids by a Nonequilibrium Depletion Force in a Temperature. Phys. Rev. Lett. 2009, 102, 208301. (22) Giddings, J.; Shinude, P.; Semenov, S. Thermophoresis of Metal Particles in a Liquid. J. Colloid Interface Sci. 1995, 176, 454. (23) Sugioka, H. Basic Analysis of Induced-Charge Electrophoresis Using the Boundary Element Method. Colloids Surf., A 2011, 376, 102. (24) Bazant, M. Z.; Kilic, M. S.; Storey, B. D.; Ajdari, A. Towards an Understanding of Induced-Charge Electrokinetics at Large Applied Voltages in Concentrated Solutions. Adv. Colloid Interface Sci. 2009, 152, 48. (25) Gamayunov, N. I.; Murtsovkin, V. A.; Dukhin, A. S. Pair Interaction of Particles in Electric-Field. 1. Features of Hydrodynamic Interaction of Polarized Particles. Colloid J. USSR 1986, 48, 197−203. (26) Ruckenstein, E. Can Phoretic Motions Be Treated As Interfacial Tension Gradient Driven Phenomena? J. Colloid Interface Sci. 1981, 83, 77. (27) Shiundu, P.; Munguti, S.; Williams, S. Retention Behavior of Metal Particle Dispersions in Aqueous and Nonaqueous Carriers in Thermal Filed-Flow Fractionation. J. Chromatogr. A 2003, 983, 163. (28) Sugioka, H. Ion-Conserving Poisson−Boltzman Theory. Phys. Rev. E 2012, 86, 016318.
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