Thermoosmosis as Driving Mechanism for Micro- or Nanoscale

Oct 23, 2015 - Thermoosmosis as Driving Mechanism for Micro- or Nanoscale Engine Driven by External Temperature Gradient. Semen Semenov† and Martin ...
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Thermoosmosis as Driving Mechanism for Micro- or Nanoscale Engine Driven by External Temperature Gradient Semen Semenov*,† and Martin Schimpf*,‡ †

Institute of Biochemical Physics RAS, Kosygin St. 4, 119334 Moscow, Russia Boise State University, Boise, Idaho 83725, United States

‡

ABSTRACT: We study thermoosmosis as the physical basis for creating a micro- or nanoscale engine. As a model system, we consider concentric cylinders in which an annular cavity has been created. The outer cylinder, which acts as a stator, comprises two halves composed of different materials. The inner cylinder forms a rotor that is driven by thermoosmotic flow within the annular cavity etched into the stator and filled with a nonionic liquid. The circular thermoosmotic flow is established within the enclosed cavity in response to an external temperature gradient directed transverse to the cylindrical axis. The circular flow induces rotation of the rotor through hydrodynamic friction at the rotor surface. The velocity and direction of the rotation depends on the magnitude of both the temperature gradient and difference in physicochemical properties of the two stator materials. The thermoosmotic engine can be used to drive a mechanical load in microand nanoscale devices. The drive load characteristic of the engine is expressed as the dependence of the angular velocity on the load torque.

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INTRODUCTION The ability of an external temperature gradient to drive liquid flow relies on the transformation of heat flux into motion of the liquid.1 The flow of liquid within a ribbon-shaped channel having a high aspect ratio when exposed to a temperature gradient applied across the thin dimension is widely discussed in literature.2−5 Wold and Hafskjold2 describe the closed-loop flow profile established in a closed slit pore. Liu et al.3 describe a thermoosmotic pump consisting of a slit pore connecting two reservoirs, while Liu and Li discuss the same effect for a slit consisting of two different materials connected in series between two reservoirs. In discussing the influence of confinement in thermophoretic parameters, Hannaoui et al.5 outline the closed-loop flow profile raised in such a confined space. Each of these works, which utilize computer simulations, is based on the establishment of a nonisothermal hydrodynamic flow created by the interaction of liquid molecules with the wall material. The physical mechanism of liquid flow through confined channels under the action of a temperature gradient is well studied.6,7 Such flow, termed thermoosmosis, is caused by a temperature-induced pressure gradient established at the solid− liquid boundary. The pressure gradient causes so-called slip flow in a very thin layer at the interface. This slip flow in turn causes a “macroscopic” flow in adjacent layers of liquid. A comprehensive review of molecular rotors can be found elsewhere.8 There are a number of studies of molecular motors based on temperature gradients. Tu and Ou-Yang9 describe a theoretical molecular motor constructed from concentric single-walled carbon nanotubes, where the inner tube is longer than the outer tube. The interaction between inner and outer tubes is derived by summing the Lennard-Jones potentials © XXXX American Chemical Society

between atoms in the two tubes. It is shown that directional rotation forms a molecular motor in a temperature gradient that varies in time when the motion of the outer tube along the nanotube axis is prohibited. Jones et al.10 examine the rotational dynamics of light driven nanorotors made of asymmetrically shaped nanotube bundles and gold nanorod aggregates trapped in optical tweezers. The authors identify the rotational motions caused by either unbalanced radiation pressure or polarization torque. Becton and Wang11 investigate the thermophoretic motion of square nanoflakes on a graphene surface by computer simulations for the purposes of nanomanipulations. Santamaria-Holek et al.12 model a nanosystem consisting of two coaxial carbon nanotubes of disparate lengths, where a longitudinal temperature gradient induces motion of the shorter nanotube along the track of the longer nanotube. A simulation model combining the actions of frictional, van der Waals, and thermal forces is used to reproduce the combination of linear and rotational motions observed in experiments. Jiang et al.13 describe a thermophoretically driven microrotor propelled by a localized temperature gradient. The rotor consists of spherical Janus particles made of silica with half of the particle surface coated by gold. One particle is tethered at the surface, while another rotates around the tethered particle in responses to a light beam heating the gold surface. Finally, Yang and co-workers14,15 describe a microengine in which circular thermophoretic motion is produced by anisotropic thermophoresis of its parts. The solid structure contains a sequence of saw teeth in a closed circular shape. This Received: September 5, 2015 Revised: October 13, 2015

A

DOI: 10.1021/acs.jpcc.5b08670 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C microgear is placed with a fixed axis at the center of a cylindrical cavity filled with liquid. The cavity wall is thermostated, while the gear is heated or cooled. The temperature at each tooth summit differs from that at the clefts, resulting in a temperature gradient along the long and the short edges of each gear tooth. According to their model the temperature gradient tangential to the surface of the gear induces a thermophoretic force parallel to the surface and proportional to the number of particles at the surface. Because the number of active particles along the long and short edges of the teeth is different, so are the forces, producing a torque that rotates the gear. Forces normal to the long and short gear edges, which are not associated with the temperature gradient, produce torques in opposite directions that cancel each other. The model of Yang and co-workers is inconsistent with that of Anderson,7 who showed that when thermophoretic or thermoosmotic motion in liquids is driven by a temperature-induced pressure gradient established in the thin surface layer of liquid near a solid−liquid boundary, hydrodynamic friction in the resulting surface “slip” flow cancels the respective surface forces associated with the pressure gradient. The engine we propose is driven purely by thermoosmosis, opening new possibilities for the design of microengines and the potential to gain new insights into the mechanism of thermoosmosis in molecular liquids.

the assumption of the solid surface having a very large radius of curvature compared to that of the surrounding molecules, allowing the solid surface to be considered a quasi-planar interface. This approximation has wide application in the field because interacting layers of molecular liquid at the surface of colloidal particles are typically very thin compared to the particle size. In this approach the Navier−Stokes equation takes the following form: η

∂ 2uy(x) ∂x 2

∂P ∂y

=

(1)

where η is the dynamic viscosity of the liquid and P(x,y) is the pressure distribution. The transverse pressure distribution P(x) in the surface layer near the solid−liquid boundary is obtained using the condition of mechanical (hydrostatic) equilibrium in the direction normal to the surface: ∂P 1 ∂Φ =− ∂x v0l ∂x

(2)

Here, v0 is the partial molecular volume of the liquid and Φ(x) is the molecule−wall interaction potential. In molecular liquids, where changes in v0 with temperature and pressure are negligible, the solution of eq 2 takes the following form:16

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P(x , y) = P0(y) −

THEORY Thermoosmosis in Molecular Liquids. In hydrodynamic theory,7 the only difference between thermoosmosis and thermophoresis in liquids is the frame of reference. Both phenomena are based on the flow of liquid established in the layer near a particle or wall surface as a result of interaction forces at the interface. The characteristic length of action for this force is on the order of several molecular radii. The liquid flow is caused by a temperature-induced longitudinal pressure gradient established along the surface of the particle or wall. When the solid surface belongs to a movable solid particle, the temperature-induced flow causes particle movement referred to as thermophoresis. When the surface is the inner wall of a channel in a fixed laboratory frame, the macroscopic or mesoscopic movement of liquid is referred to as thermoosmotic flow.6,7 In the work below we follow the method of Anderson7 based on the Navier−Stokes equation for the flow profile near the solid surface and in the bulk volume. In this method a onedimensional liquid flow profile uy(x) is defined, where x is the coordinate normal to the solid surface and y is the respective tangential coordinate (see Figure 1). The approach is based on

Φ(x) v0l

(3)

where P0(y) is a constant of integration defined by the pressure far from the surface layer. This pressure distribution is determined by the specific shape of the system and will be discussed below. Because we consider the Navier−Stokes equation to be linear we can solve the two right-hand terms in eq 3 separately. These two solutions are the volume (i.e., macroscopic) region of the flow profile corresponding to P0(y) and the surface region corresponding to Φ(x)/v0l. The latter region is defined by the very thin layer where the interface potential Φ(x) acts. The “partial” flow profile within the surface layer is described by the Navier−Stokes equation η

∂ 2uy ∂x 2

( ) Φ(x)

∂ =−

1 v0l

∂y

(4)

Equation 4 expresses the balance of forces within that layer; there is no other force outside the surface layer except for hydrodynamic friction. The boundary conditions for eq 4 are7 uy(x = 0) = 0;

∂uy(x = ∞) ∂x

=0

(5)

In principle, the interaction potential may also depend on physical forces along the tangential coordinate. For example, a varying temperature along the particle surface creates a tangential force. However, any dependence of the interaction potential on tangential parameters is canceled by the creation of a surface pressure gradient, provided the condition of local equilibrium is fulfilled.16 Consequently, the “surface” Navier− Stokes equation (eq 4) that describes the transverse flow profile in a nonisothermal surface layer takes the following form:

η

Figure 1. System of coordinates used in the hydrodynamic problem and the flow profile for the idle rotation of the engine. B

∂ 2uy ∂x

2

=

αT Φ(x) ∂T ∂y v0l

(6) DOI: 10.1021/acs.jpcc.5b08670 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C where αT = (1/v0l)(∂v0l/∂T) is the cubic thermal expansion coefficient, T is temperature, and ∂T/∂y is the temperature gradient tangential to the solid−liquid interface. Of course, the application of an external and unidirectional temperature gradient also contains a component normal to the interface, but this component causes only a very small thermal expansion of the liquid; it cannot cause any flow and/or mass transfer (i.e., thermal diffusion) in a single-component liquid. Using the boundary conditions defined by eq 5 the solution of eq 6 at distances much greater than the molecular size of the liquid is α ∂T us = uy(x = ∞) = − T ηv0l ∂y

∫0

Thermoosmotic Flow Profile in a Ribbon-Shaped Channel. Consider a ribbon-shaped channel such as that used in sedimentation field−flow fractionation,18 where the channel is joined at the long ends to from a continuous annular channel. In such a channel, the flow profile is described by the following form of the Navier−Stokes equation: η

x dx Φ(x)

εwl σwl 3 ⎛ 1 1 ξ ⎞ + ln ⎟ ⎜ + 6 v0w ⎝ ξ 2+ξ 2 + ξ⎠

=

∂P0 ∂y

(10)

uy(x = 0, h) = us±

(7)

(11)

Here, x = 0, h are the radial coordinates of the fixed channel walls and us± are the respective slip velocities at the different walls of the channel, which may differ if the two walls are made of different materials. For a closed ribbon-shaped channel whose ends are not connected to form an endless ring, eq 10 is valid along the entire length of the channel except at the ends where transverse flows are created by confining walls that define the channel. Such end effects are negligible in the bulk of the channel, provided the channel length is significantly greater than the channel width. The flow profile in such a channel is defined as

(8)

Here, εwl is the intermolecular energy of the molecule−wall interaction, v0w is the partial molecular volume of the wall material, ξ = x/σwl is the reduced distance from the wall surface to the surface of the nearest solvent molecule, and σwl is the minimal molecular approach distance between molecules of the liquid and wall. The Hamaker potential is obtained by integrating the intermolecular London−van der Waals potentials, defined as

u y (x ) =

∂P0 h2 x ⎛ x ⎞ x ⎜ − 1⎟ + (us+ − us−) + us− ⎠ ∂y 2η h ⎝ h h

(12)

If one of the channel walls is moveable, then the boundary condition takes the form of zero viscous stress at the moveable wall surface:

η

Φwl = −εwl(σwl /r )6

∂uy ∂x

(x = 0 or h) = 0

(13)

The first term on the right-hand side of eq 12 is related to the “macroscopic” longitudinal pressure gradient in the channel, while the other terms on the right-hand side define the flows driven by the slip velocities that result from the temperatureinduced pressure gradient at the channel surface. The pressure gradient is calculated using the condition of zero mass transport along a closed channel:

over all molecules of the solid surface, where r is the distance between molecular centers. The London−van der Waals potential is a type of Lennard-Jones potential used widely in numerical simulations. Substituting eq 8 into eq 7 and using the definition17 Awl = εwl(σwl3/v0w) for the Hamaker constant, we obtain the following expression for the slip velocity at the wall surface: 2 2 − ln 3 αT σwl A wl ∂T us = 12 ηv0l ∂y

∂x

2

with boundary conditions

∞

Parameter us is termed the slip velocity and defines the boundary condition in the “macroscopic” Navier−Stokes equation at the solid−liquid interface. This equation for the volumetric flow profile will be used below to characterize the thermoosmotic engine developed in a thin circular channel. Molecule−wall interaction potentials in liquids can be defined by the Hamaker potential:17 Φ(ξ) = −

∂ 2uy

∫0

(9)

h

u y (x ) d x = 0

(14)

Using eqs 12 and 14, we obtain

Note that, while eqs 8 and 9 contain the numeric density of the wall molecules, the surface area and dimensions of the wall are not a factor in the slip velocity. Equation 9 was obtained with the assumption that the spatial distribution of liquid molecules near the wall is undisturbed by interactions with the wall, an assumption widely used in the study of regular solutions.19 The assumption of constant local density is also used in theories of particle thermophoresis.20−23 For example, the assumption is used in a kinetic approach to the thermodiffusion of colloidal particles by Bringuier and Bourdon.20,21 We used the assumption in both hydrodynamic and thermodynamic approaches to thermodiffusion in polymer solutions.16,22,23 We note that eq 9 for the slip velocity differs from the analogous equation for the thermodiffusion coefficient in polymer solutions16 only in the numeric coefficient, which is related to the difference in the potential shape, which is consistent with the similarity between thermophoresis and thermoosmosis.

∂P0 3η = 2 (us+ + us−) ∂y 4h

(15)

Equation 15 indicates that the pressure drop will be different for two channels having the same dimensions but made of different wall materials, due to a difference in parameter u+s + u−s . If the two channels are attached at their longitudinal ends to form a continuous annular channel, then the pressure change at the connection points will cause spillage of liquid from the region with higher pressure at the connection point to the region with lower pressure. In this way a closed-loop flow can be established. If one of the walls of the annular channel is moveable, the flow profile is defined by u y (x ) = C

⎞ ∂P0 h2 ⎛ x 2 + ⎜ 2 − 1⎟ + us ∂y 2η ⎝ h ⎠

(16) DOI: 10.1021/acs.jpcc.5b08670 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C with boundary condition defined by eqs 11 and 13. The similar flow profile is shown in Figure 1. In order to fulfill the condition of mass conservation (eq 14) we obtain ∂P0 h2 3 = us+ ∂y 2η 2

(17)

and the velocity of the movable wall is us+ + us− (18) 2 Note that in this system u−s is the velocity of thermophoresis at the movable wall, while in a channel with fixed walls eq 13 defines the slip velocity induced by thermoosmosis. The flow defined by eq 18 may be used as a thermoosmotic conveyor that carries a load at the surface of the liquid. The same principle can be used to drive a thermoosmotic engine. We note that results from the approach used in this section are consistent with those obtained elsewhere,2−5 where numeric simulations are used rather than the simple analytical solutions used in this work. Thermoosmotic Engine. Consider a system consisting of a cylindrical rotor placed coaxially in a cylindrical cavity made by aligning two semicylindrical recesses, as illustrated in Figure 2, u = uy(x = 0) + us− =

Figure 3. Assembled thermoosmotic engine.

uy(x = 0) = us + u

(19)

uy(x = h) = us−

(20)

For the right semicylindrical recess the boundary conditions are uy(x = 0) = us − u

(21)

uy(x = h) = us+

(22)

For both channels the point x = 0 is placed at the surface of the cylindrical rotor, while the stator wall is placed at x = h. Parameter u is the linear velocity of the rotor wall with a positive value indicating counterclockwise rotation. The velocity profiles in the left and right semiannular channels are uyl(x) =

∂P0 h2 x ⎛ x ⎞ x ⎜ − 1⎟ + (us− − us − u) + us + u ⎠ ∂y 2η h ⎝ h h (23)

uyr(x) =

Figure 2. Semicylindrical recess that comprises half of the stator assembly.

∂P0 h2 x ⎛ x ⎞ x ⎜ − 1⎟ + (us+ − us + u) + us − u ⎝ ⎠ ∂y 2η h h h (24)

The linear velocity u can be derived using the torque balance for a rotating rotor with applied load torque M, which is assumed to be positive:

where the two halves are made of different materials. The assembled system is illustrated in Figure 3. An annular channel is formed between the inner cylinder, which is freely rotatable around a common axis, and the outer wall of the cylinder. An alternate version is a cylindrical “Janus cavity” made of a single material with two halves that are coated with different materials. When the annular channel contains a liquid, rotation of the inner cylinder can be driven by circular thermoosmotic flow along the annular channel in the presence of a temperature gradient established normal to the cylindrical axis and parallel to the bond plane. The hydrodynamics of this process is described above. The slip velocity at the surfaces of the left and right semicylindrical recesses are defined as u−s and u+s , respectively, while us is the slip velocity at the wall of the rotor. The boundary conditions for the left semicylindrical recess are

⎡ M = ηlR ⎢ ⎢⎣

∫0

πR

∂uyl(x = 0) ∂x

dy −

∫0

πR

∂uyr(x = 0) ∂x

⎤ dy ⎥ ⎥⎦ (25)

Here, l and R are the height and radius of the cylindrical rotor, respectively. The right-hand terms under the integration signs define the viscous stresses applied to the surface of the rotating rotor by the liquid.7 Substituting eqs 23 and 24 into eq 25, we obtain u= D

us− − us+ hM − 2 2ηlR

(26) DOI: 10.1021/acs.jpcc.5b08670 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C ± where us± = (1/πR)∫ πR 0 us (y) dy is the average slip velocity along the considered half-pipe associated with thermoosmosis. The pressure distribution along the semiannular channels is derived using the condition of mass conservation analogous to that defined by eq 14:

∫0

h

uyl(x) dx +

∫0

h

uyr(x) dx = 0

different. Thus, the model can serve to guide a range of potential thermoosmotic engine and actuator designs. The idle frequency of rotation is defined as ω(M = 0) =

−3

(28)

ω(M = 0) ≈

Equation 28 yields the following load characteristic of the thermoosmotic engine: ω(M ) =

us−

− 2R

us+

−

hM 2ηlR2

10−4 cm/s R

(33)

A rotor radius R ≈ 10 cm is the minimum size in which the proposed model can be expected to yield reliable predictability without frictional heating. For such a rotor the induced angular velocity of rotation defined by eq 33 is ω(M = 0) ≈ 10 s−1.

(29)

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AUTHOR INFORMATION

Corresponding Authors

*Phone: +7(495)939-74-39. E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Haase, R. Thermodynamics of Irreversible Processes; AddisonWesley: New York, 1968. (2) Wold, I.; Hafskjold, B. Nonequilibrium Molecular Dynamics Simulations of Coupled Heat and Mass Transport in Binary Fluid Mixtures in Pores. Int. J. Thermophys. 1999, 20, 847. (3) Liu, C.; Lv, Y.; Li, Z. Fluid transport in nanochannels induced by temperature gradients. J. Chem. Phys. 2012, 136, 114506. (4) Liu, C.; Li, Z. Molecular Dynamics Simulation of Composite Nanochannels as Nanopumps Driven by Symmetric Temperature Gradients. Phys. Rev. Lett. 2010, 105, 174501. (5) Hannaoui, R.; Galliero, G.; Hoang, H.; Boned, Ch. Influence of confinement on thermodiffusion. J. Chem. Phys. 2013, 139, 114704. (6) Derjaguin, B. V.; Dukhin, S. S.; Koptelova, M. M. Capillary osmosis through porous partitions and properties of boundary layers of solutions. J. Colloid Interface Sci. 1972, 38, 584−95. (7) Anderson, J. L. Colloid Transport by Interfacial Forces. Annu. Rev. Fluid Mech. 1989, 21, 61. (8) Kottas, G. S.; Clarke, L. I.; Horinek, D.; Michl, J. Artificial Molecular Rotors. Chem. Rev. 2005, 105, 1281−1376. (9) Tu, Z. C.; Ou-Yang, Z. C. A molecular motor constructed from a double-walled carbon nanotube driven by temperature variation. J. Phys.: Condens. Matter 2004, 16, 1287−1292. (10) Jones, P. H.; Palmisano, F.; Bonaccorso, F.; Gucciardi, P. G.; Calogero, G.; Ferrari, A. C.; Marago, O. M. Rotation Detection in Light-Driven Nanorotors. ACS Nano 2009, 3 (10), 3077−3084. (11) Becton, M.; Wang, X. Thermal Gradients on Graphene to Drive Nanoflake Motion. J. Chem. Theory Comput. 2014, 10, 722−730. (12) Santamaría-Holek, I.; Reguera, D.; Rubi, J. M. CarbonNanotube-Based Motor Driven by a Thermal Gradient. J. Phys. Chem. C 2013, 117, 3109−3113. (13) Jiang, H. R.; Yoshinaga, N.; Sano, M. Active Motion of a Janus Particle by Self-Thermophoresis in a Defocused Laser Beam. Phys. Rev. Lett. 2010, 105, 268302. (14) Yang, M.; Liu, R.; Ripoll, M.; Chen, K. Microscale thermophoretic turbine driven by external diffusive heat flux. Nanoscale 2014, 6 (22), 13550−4. (15) Yang, M.; Ripoll, M. A self-propelled thermophoretic microgear. Soft Matter 2014, 10, 1006.

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where ∇T is the macroscopic temperature gradient in the system and eq 25 takes the following form ω(M ) =

−13

−5

where ω = u/R is the angular velocity of the thermoosmotic engine under load. To obtain a more useful expression for the load characteristic we must calculate the average slip velocities us±. In order to simplify the following calculations we ignore the difference in thermal conductivities of the two materials so that the temperature gradient in the system can be assumed to lie in the plane separating the semicylindrical recesses where different slip velocities occur and directed perpendicular to the cylindrical axis. In this case, the longitudinal temperature gradient ∂T/∂y in eq 6 can be written as y ∂T = ∇T sin ∂y R

−1

(32)

For liquids, αT ≈ 10 K , |Aw−l − Aw+l| ≈ 10 erg, η ≈ 10−2 g/cm·s, σwl ≈ 2 × 10−7 cm, and v0l ≈ 10−21 cm3.16 Using these values and a typical temperature gradient ∇T ≈ 104 K/cm, eq 31 yields

(27)

Substituting eqs 23 and 24 into eq 27, we obtain ∂P0 h2 3 = (2us + us− + us+) ∂y η 4

2 2 − ln 3 αT σwl (A w−l − A w+l ) ∇T 12 ηv0lR

2 2 − ln 3 αT σwl (A w−l − A w+l ) M ∇T − 12 ηv0lR 2ηlR2

(31)

Note that Aw±l are the two Hamaker constants defining the respective liquid−wall interaction energies within the cylindrical cavity.

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DISCUSSION AND CONCLUSIONS The expressions for linear velocity and angular frequency of the rotor contain neither the slip velocity of the rotor material us nor the longitudinal pressure gradient ∂P0/∂y in the system. Thus, the angular velocity under a given load torque is independent of the material used in the rotor. Consequently, when the thermal conductivities of the system components are similar, rotor rotation is determined only by difference in the slip velocities associated with the two materials that compose the outer wall. We note that a functioning thermoosmotic engine does not require the annular channel to be built from the semicylindrical components used in this work to illustrate the proposed mechanism. Rotation can also be obtained by placing the rotor between any two opposing walls having different slip velocities. The channels do not even need to contain the same liquid nor do the walls need to be parallel. The only requirement is that different viscous stresses are produced at the opposing sides of the cylindrical rotor so that the respective slip velocities are E

DOI: 10.1021/acs.jpcc.5b08670 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C (16) Schimpf, M. E.; Semenov, S. N. Mechanism of Polymer Thermophoresis in Nonaqueous Solvents. J. Phys. Chem. B 2000, 104, 9935. (17) Israelashvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1992. (18) Schimpf, M. E.; Caldwell, K.; Giddings, J. C. Field-Flow Fractionation Handbook; Wiley Interscience: New York, 2000. (19) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774. (20) Bringuier, E. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 67, 011404. (21) Bringuier, E.; Bourdon, A. J. Non-Equilib. Thermodyn. 2007, 32, 221. (22) Semenov, S.; Schimpf, M. Theory of Soret Coefficients in Binary Organic Solvents. J. Phys. Chem. B 2014, 118 (11), 3115−3121. (23) Semenov, S.; Schimpf, M. Statistical Thermodynamics of Material Transport in Nonisothermal Suspensions. J. Phys. Chem. B 2015, 119 (8), 3510−3516.

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DOI: 10.1021/acs.jpcc.5b08670 J. Phys. Chem. C XXXX, XXX, XXX−XXX