Article pubs.acs.org/JPCC
Thermophoretic Microgear on Grounds of Physicochemical Hydrodynamics Semen Semenov*,† and Martin Schimpf‡ †
Institute of Biochemical Physics RAS,119334 Moscow, Kosygin St. 4, Russia Boise State University, Boise, Idaho 83725, United States
‡
ABSTRACT: We use methods of physicochemical hydrodynamics to study the working principles of a thermophoretic microgear in which an axially symmetric temperature gradient causes rotational movement of a heated rotor in the shape of a pinion confined in a cylindrical cavity filled with liquid. Calculations indicate that the rotation of the microgear arises from differences in the physicochemical properties along the edges of the teeth. The angular velocity of the gear under load decreases linearly with the load torque. The resulting characteristics of the thermophoretic gear are compared with that of a thermoosmotic engine having comparable parameters. Optimal geometrical parameters of both systems are discussed. It is shown that performance of the two devices is similar.
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BACKGROUND The ability of an external temperature gradient to drive the movement of a solid body within a liquid environment, referred to as thermophoresis, relies on the transformation of heat flux into motion of the liquid near the liquid−solid interface.1 The liquid motion originates with a slip flow along the solid surface in response to a temperature-induced gradient of the surface pressure, which is established near the liquid−solid interface to cancel the volume force due to the liquid−solid interaction. In molecular liquids, the pressure gradient is due to the thermal expansion of liquid and the related asymmetry in liquid−solid interactions. This slip flow in turn causes a “macroscopic” flow in adjacent layers of liquid. The movement of liquid adjacent to a fixed solid surface in response to a temperature gradient is referred to as thermoosmosis. The basic physical mechanism of thermoosmosis in confined channels is well-described in the seminal papers of Derjaguin et al.2 and Anderson.3 Numerous simulations confirm the basic principles formulated in.2,3 Wold and Hafskjold4 used computer simulations to examine the thermoosmotic flow profile established in a closed slit pore. Liu et al.5 describe a thermoosmotic pump consisting of a slit pore connecting two reservoirs, while Liu and Li6 discuss the same effect within a slit consisting of two different materials connected in series between two reservoirs. Hannaoui et al.7 also outlined the closed-loop flow profile raised in a confined space. A comprehensive review of molecular rotors can also be found in the literature.8 Tu and Ou-Yang9 describe a theoretical thermoosmotically driven 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 between atoms in the two tubes. In a temperature © 2016 American Chemical Society
gradient the directional rotation forms a molecular motor when the motion of the outer tube along the nanotube axis is inhibited. 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. Rotational motion of the nanorotor is described as being caused by 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-Holek12 and co-workers model a system 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. In that work a simulation model that combines the actions of frictional, van der Waals, and thermal forces is used to reproduce the 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 a 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. Several recent articles discuss micro- or nanoscale engines and motors based on thermoosmosis and thermophoresis.14−16 Yang and co-workers14,15 describe a microengine in which circular motion is produced by anisotropic thermophoresis of its parts. The solid structure contains a sequence of saw teeth in Received: June 1, 2016 Revised: September 2, 2016 Published: September 8, 2016 22597
DOI: 10.1021/acs.jpcc.6b05525 J. Phys. Chem. C 2016, 120, 22597−22604
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The Journal of Physical Chemistry C a closed circular shape. This 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 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 the active particles at the surface. A difference in opposing forces along the long and short edges of the teeth results in a torque that rotates the gear. However, this model is inconsistent with the theory of Anderson,3 in which the viscous stresses resulting from slip flow associated with the temperature-induced pressure gradient cancel the surface forces. Consequently, the motion of the solid does not arise directly from a thermophoretic force, but rather from the thermoosmotic motion of liquid along the solid surface. Maggi et al.16 show that microfabricated gears floating on a liquid−air interface can efficiently convert absorbed light into rotational motion. They assumed thermocapillarity to be the underlying mechanism. To increase light absorption, the gears are coated on one side with a layer of amorphous carbon. The authors attribute differences in the rotational speed of different microgears to differences in the carbon coatings. Also, a small linear velocity of individual microgears is superimposed on the rotation. The authors assert that thermocapillary propulsion is one of the strongest mechanisms for light actuation at the micro- and nanoscale, but they conclude that thermophoresis plays no role in their system because a much larger power density would be required to achieve the observed rotations, with obvious heating problems. In this regard we note that thermocapillary phenomena and thermoosmosis have the same physical basis. These effects are described by the same combined equations [see eqs 42−44 in ref 3]. Thus, droplet thermophoresis in liquids is attributed to the gradient in surface tension, while thermophoretic movement of solid particles is attributed to the temperature-induced pressure gradient in the surface layer. The thermocapillary motion of droplets is simply more intensive than solid particle thermophoresis based on hydrodynamic differences.3 In our recent work on thermoosmotically driven micro- or nanoscale engines17 we modeled 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 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 the temperature gradient, as well as differences in physicochemical properties of the two stator materials. The theory we developed in that work relies on the application of physicochemical hydrodynamics to phoretic effects in liquids.3,17,18 We now apply this theory to the microgears described by Yang and Ripoll15 and Maggi et al.16 The various devices discussed below possess some degree of similarity. However, our thermoosmotic engine17 necessitates nonuniformity in the stator surface, which leads to nonuniform thermophoretic properties. The system as whole is placed in a constant temperature gradient. By contrast the model proposed
for the working microgears15,16 do not rely on nonuniform surface properties, although such nonuniformities are assumed to exist. Below we apply our theory17 to elucidate the potential role of surface nonuniformities in the action of thermophoretic microgears. While there is no difference in the principle of action for the microgear devices under consideration,15,16 we mention the microgear of Yang and Rippol15 confined to a cylindrical cavity in order to apply directly the mathematics developed for our model. As we show below, the confined system may also be preferred if one considers the work under load.
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THEORY OF THERMOOSMOSIS AND THERMOPHORESIS IN MOLECULAR LIQUIDS In models based on physicochemical hydrodynamics3 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 and the resulting pressure gradient. In the molecular liquids, 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 wall or particle. 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 fixed in the laboratory frame, the macroscopic or mesoscopic movement of liquid is referred to as thermoosmotic flow.3,4 The most important characteristic of phoretic and osmotic transport is that the external field applies no direct force to the particle or channel wall that includes the fluid in the interfacial region; to the outer fluid the moving particle or the fixed channel wall appears to be force free and torque free. In the work below we follow the method of Anderson,3 which utilizes the Navier−Stokes equation for the flow profile near a solid surface, as described previously.17,18 According to Anderson the length scale of the particle and channel is ordersof-magnitude larger than the thickness of the interfacial region. This difference in length scale allows for the introduction of slip velocity as the principal parameter characterizing the process. Of course, in reality the boundary conditions for the Navier− Stokes equation are the zero tangent velocities at the solid surfaces. However, it is mathematically convenient to mention separately the thin surface layer where the surface forces act and the flow profile changes sharply, achieving a plateau within several molecular radii of the solid surface. This plateau value is called the slip velocity and is used as the boundary condition for the using the Navier−Stokes equation to solve for flow in the bulk liquid. In this method we define a one-dimensional liquid flow profile uy(x) within the interfacial region, where x is the coordinate normal to the solid surface, y = Rφ is the respective tangential coordinate (φ is the azimuth angle in the cylindrical coordinates), and R is the constant radius of the circular cylinder, as illustrated in Figure 1. The use of a single dimension of flow is based on the assumption of a very large radius of curvature for the solid surface compared to sizes of the surrounding liquid molecules and thickness of the liquid surface layer.3 This approximation has wide application in the field because liquid surface layers are usually very thin compared to particle and channel sizes. 22598
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dependence of the interface potential on distance from the surface. The “partial” flow profile in the surface layer is described by the Navier−Stokes equation η
∂ 2uy ∂x 2
( ) Φ(x)
∂ =−
1 v0l
∂y
(4)
eq 4 expresses the balance of forces within the surface layer; there is no other force outside the surface layer except for hydrodynamic friction. The parameter 1 corresponds to the v0l
numeric volume density of the solvent molecules. The temperature dependence of the partial molecular volume is expressed as v0l(T) = v0l(T0)[1 + αT(T − T0)] where T0 is the 1 ∂v reference temperature and αT = v ∂T0l is the cubic thermal
Figure 1. Designations of the coordinate axes and principal parameters.
0l
expansion coefficient. The temperature-related changes are very weak, as the change of temperature is tens of degrees at most and the temperature expansion coefficient αT ≈ 10−3. Consequently, the temperature dependence of the partial molecular volume can be ignored except for in eq 4. In eq 4, the temperature dependence of the molecular volume is responsible for the pressure gradient that induces thermophoretic effects. The boundary conditions for eq 4 are3
A certain dependence of the solvent−wall interaction potential on the azimuth coordinate is the important element of our speculations. Since our aim is to construct a theory describing the physical mechanism of rotation, we simplify the calculations where it is possible. For this reason, we first approximate a smooth profile for the channel between the external wall and the gear teeth, which allows some partial derivatives in the hydrodynamic and heat transport equations to be ignored, except at discrete points. This approximation means that the azimuth coordinate, whether angular or Cartesian, is included in the respective equations as an inconsequential parameter only. In doing so, we first obtain the solution of a one-dimensional problem containing that parameter and then substitute this solution into the “exact” two-dimensional equations to determine the accuracy of our approximation. In our designations, the inconsequential variable is omitted for brevity wherever possible. With this approach the Navier−Stokes equation takes the form η
∂ 2uy(x) ∂x 2
∂P = ∂y
uy(x = 0) = 0;
(1)
=0
(5)
∂x
2
=
αT Φ(x) ∂T ∂y v0l
(6) ∂T ∂y
is the temperature gradient
tangential to the solid−liquid interface. Of course, the temperature gradient also has a component normal to the interface. However, this component causes only a very small thermal expansion of the liquid and does not cause any flow in a one-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
(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 very small, the solution of eq 2 takes the following form:17,18 Φ(x) v0l
∂ 2uy
Here, T is temperature and
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 of liquid near the solid−liquid boundary is obtained using the condition of mechanical (hydrostatic) equilibrium in the direction normal to the surface:
P(x , y) = P0(y) −
∂x
In principle, the interaction potential may also depend on the tangential coordinate if physical parameters such as temperature vary along the surface, causing a respective tangential force. However, any dependence of the interaction potential on the longitudinal coordinate is canceled by the respective surface pressure gradient, assuming the principle of local equilibrium applies.18 In this case, the “surface” Navier−Stokes equation [eq 4] that describes the transverse flow profile in the nonisothermal surface layer takes the following form:
η
∂P 1 ∂Φ =− ∂x v0l ∂x
∂uy(x = ∞)
us = uy(x = ∞) = −
αT ∂T ηv0l ∂y
∫0
∞
x dx Φ(x)
(7)
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 flow profile within a thin circular channel will form the basis for the physicochemical hydrodynamics of the thermophoretic gear developed below. Molecule-wall interaction potentials in liquids can be defined by summing the intermolecular London−van der Waals potential
(3)
Here 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, as discussed below. The two right-hand terms in eq 3 correspond to the macroscopic (bulk) flow profile with pressure P0(y) and the thin layer near the surface where pressure is governed by the 22599
DOI: 10.1021/acs.jpcc.6b05525 J. Phys. Chem. C 2016, 120, 22597−22604
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The Journal of Physical Chemistry C Φwl = −εwl(σwl /r )6
Substituting eq 9 into eq 7 and using the definition19
(8)
σ
over all the molecules of the solid body, where εwl is the energetic parameter of the molecule−wall interaction, σwl is the minimal molecular approach distance between molecules of the liquid and wall, and r is the distance between molecular centers. This potential is a form of the Lennard-Jones potential used widely in numerical simulations. The solid surfaces with the same numerical density of the “active” molecules possessing the same potentials defined by eq 8 have identical properties in a pure liquid. The surfaces of neighboring teeth in the microgear described by Yang and Rippol15 are not uniform because the density of the “active beads” is different at different edges. According to Figure 1, the number of the active beads at the short edge is 6. Using the teeth parameters indicated in ref 15, the internal gear radius R1 = 19a (a is the separation between the active beads responsible for the wall−solvent interactions) corresponding to the radial position of the tooth root, the external radius corresponding to the summit R2 = 25a, the angle between the adjacent clefts θ = 45° for the eight-tooth gear, and the cosine theorem, we obtain the length of the long tooth edge equal to 17.727a. Thus, the long edge can contain at most 17 active beads. In Figure 1 from ref 15, the number of the respective beads is 20. Thus, the figure is not the real image of the mentioned gear. Further, as the length of the long edge is not a whole number, a gap between beads should exist somewhere along the long edge, which means that either the long edge is nonhomogeneous or the separation between beads is not equal to a, as was assumed initially. In the latter case, the edges also should be considered to be made from different materials. The Hamaker potential19 of interaction between molecules of liquid and the solid wall is obtained when the integration of the molecular potentials given by eq 8 is made over the body of the solid wall: Φ(ξ) = −
εwl σwl 3 ⎛ 1 1 ξ ⎞ + ln ⎟ ⎜ + 6 v0w ⎝ ξ 2+ξ 2 + ξ⎠
3
A wl = εwl vwl for the Hamaker constant, we obtain the following 0w
expression for the slip velocity at the wall surface: us =
2 2 − ln 3 αT σwl A wl ∂T 12 ηv0l ∂y
(10)
Note that while eqs 9, 10) 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 10 was obtained under 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.20 The assumption of constant local density is also used in theories of particle thermophoresis.21−24 For example, the assumption is used in a kinetic approach to thermodiffusion of colloidal particles by Bringuier and Bourdon.21,22 We used this assumption in both hydrodynamic and thermodynamic approaches to thermodiffusion in polymer and molecular solutions.18,23,24 We note that eq 10 for the slip velocity differs from the analogous equation for the thermodiffusion coefficient in polymer solutions18 only in the numeric coefficient, which is related to a difference in the shape of the potential. In further calculations, we approximate the region between the gear teeth and cylindrical cavity wall as a narrow channel with slowly changing transversal width and curvature (see Figure 2). That channel consists of a periodic structure of
Figure 2. Channel profile with h(y) mimicking the teeth profile. The curvature is ignored for sake of clarity. (9)
Here v0w is the partial molecular volume of the wall material x and ξ = σ is the reduced distance from the wall surface to the
summits and clefts with repeat elements defined by the distance between clefts. The height of the tooth is smaller than the mean channel width and the tooth width is comparable to the mean channel width. The use of deep teeth is neither preferable in the theory presented in ref 15 nor the approach used here. In the former case the azimuthal component of the thermophoretic force causing the gear motion decreases with increasing radial component of thermophoretic force. In our hydrodynamic approach, the circulation of liquid inevitably established within deep recesses between the teeth will dissipate energy necessary for gear movement.
wl
surface of the nearest solvent molecule. The potential given by eq 9 assumes a planar wall surface. For the surface having recesses or projections, the Hamaker potential will have a distinct shape as the domain of integration is changing. This is the case in ref 15, where the solvent−wall interaction is due to presence of several tens of “active beads” at the tooth surface. While the solvent molecule near the flat gear surface interacts roughly speaking with two wall molecules at the edge, it interacts with one wall molecule at the summit and three molecules at the cleft. Thus, the Hamaker potential of the gear surface will be distinct at these three locations, which means that the gear surface is always nonhomogeneous in the hydrodynamic approach. Certainly, this mechanism can make a visible contribution in systems containing a relatively small number of active beads at the gear surface, which is just the case for ref 15. However, one cannot say a priori how significant this effect will be in the larger systems studied in ref 16. Thus, this mechanism of inhomogeneity cannot be excluded from consideration in the theory of gear rotation observed in ref 16.
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TEMPERATURE DISTRIBUTION The stationary temperature distribution in a system containing a “smooth” thermophoretic gear with constant radius R is described by the equation of thermal conductivity. We begin with a system having cylindrical symmetry, take into account the teeth profile, then estimate the error caused by the teeth as a small perturbation. Thus, we initially define the gear as a long cylinder with radius R, ignoring edge effects. We will then insert the predetermined dependence of R on azimuth angle φ. For our system, the ratio of convective/conductive heat transport is 22600
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The Journal of Physical Chemistry C about 10−6. Consequently, the influence of convective heat on the temperature and flow profile is ignored. In cylindrical coordinates, the cylindrically symmetric equation of thermal conductivity takes the form Κg
1 ∂ ∂T (r ) r = −Q r ∂r ∂r
with a repeat element defined by the distance between clefts. It allows for simplifying the mathematics but introduces no changes in the mentioned physical mechanism. In working the hydrodynamic problem, we define the liquid flow through this element and extend the result to the entire channel. In a shallow channel with slow changes in width, the flow profile is described by the following form of the Navier−Stokes equation:17
(11)
where Kg is the thermal conductivity of the gear material, r is the radial coordinate expressing the distance from the gear axis, T is temperature, and Q is the heat supplied to a unit volume of gear material per unit time. The temperature distribution within the gear material is expressed as Q r2 Tg(r ) = A − Κg 4
η
∂T (r ) ∂T (r ) at r = R = Κl ∂r ∂r
(12)
(13)
(14)
where B and C are constants of integration. Constant B is calculated using the condition Tl(r ) = T0 at r = R1
(15)
where R1 is internal radius of the cylindrical cavity within which the cylindrical gear is placed. Using eqs 12−15, we obtain QR r ln + T0 2Κl R1
(16)
Q R(R1 − R ) + T0 2Κl
(17)
r=R
QR ∂h ≈− 1 2Κl ∂y
uy[x = h(y)] = u + us
(22)
∂P0 h2 x ⎛ x ⎞⎟ x ⎜1 − + (u + us) ⎝ ⎠ ∂y 2η h h h
(23)
h
u y (x ) d x =
∂P0 h3 h + (u + us) = J ∂y 3η 2
(24)
we obtain the differential equation that can be used to calculate the longitudinal pressure distribution: ∂P0 3η 3η = 3 J − 2 (u + us) ∂y h 2h
(25)
(18)
where h = R1 − R ≪ R1 is the width of the channel between the gear surface and the inner surface of the cylindrical cavity. eq 18 yields the following relative error: Κg nh0 2 Κg ∂ 2R2 ≈ 4Κl ∂y 2 16π 2 Κl R2
(21)
∫0
Introducing the azimuth coordinate y = Rφ (R is the constant radius of the circular cylinder [see Figure 1]) and taking into account the presence of the teeth we can write the relevant surface temperature gradient as ∂T ∂y
(20)
The flow rate of liquid J should be the same at any channel cross-section. Integrating eq 23
The temperature at the gear-liquid boundary is Tl(r = R ) =
∂P0 ∂y
uy(x = 0) = 0
u y (x ) =
2
Tl(r ) = −
=−
Here x is the transverse coordinate in the channel, h(y) is the “instant” channel width (see Figure 2), η is the kinematic viscosity, and P0 is the hydrodynamic pressure in the bulk of the channel [see eq 3]. The position x = 0 is placed at the inner wall of the cylindrical cavity and the x axis is directed to the center of the gear. Parameter u is the linear velocity of the gear surface and us is the thermophoretic slip velocity at the gear surface defined by eq 10. The slip velocity will change slowly along the y-axis, while the linear velocity at the solid surface of the gear should be constant. The respective slip velocity at the opposite wall is zero, since the wall is thermostated.15 In the following hydrodynamic derivations we follow the approach developed by Rayleigh and Reynolds.25 The flow profile in the channel is defined as
where Kl is the thermal conductivity of the surrounding liquid. The temperature distribution in the liquid is expressed as Tl(r ) = B + C ln r
∂x 2
with boundary conditions
where A is the constant of integration. At the boundary between gear surface and surrounding liquid the flux of heat should be continuous, i.e. Κg
∂ 2uy
P0(y) − P0(0) = 3Jη −
∫0
3η 2
y
dy′ 3
h (y′)
∫0
y
−
3ηu 2
∫0
y
dy′ 2
h (y′)
dy′ us(y′) h2(y′)
(26)
Here y = 0 is placed at the cleft of the gear (see Figure 2). In the annular channel between the gear and cylindrical wall, the pressure at all clefts should be identical due to the symmetry of the system, i.e., P0(l) − P0(0) = 0, where l is the width of tooth face calculated as the distance between neighboring clefts (see Figure 3) . Integration of eq 26 yields the following expression for the flow rate:
(19)
where n is the number of gear teeth and h0 is the height of the teeth. For a moderate number of shallow teeth h (n = 8, R0 = 0.2 in ref 15) and comparable thermal conductivities (Kg and Kl) the relative error is less than 1%.
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HYDRODYNAMIC PROBLEM The channel between the gear surface and internal wall of the cylindrical cavity is a periodic structure of summits and clefts
J=u 22601
h−2 2 h−3
+
ush−2 2 h−3
(27) DOI: 10.1021/acs.jpcc.6b05525 J. Phys. Chem. C 2016, 120, 22597−22604
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where h−n = ∫ h−n(y) dy ; n = 1, 2, 3. 0
us(y) = DT (y)
∂T ∂y
(34)
Combining eqs 10 and 34, we obtain 2 2 − ln 3 αT σwl A wl (y) 12 ηv0l
DT (y) =
(35)
Using eqs 18 and 34 the integrals in the numerator of eq 33 can be
written
QR
h−1us = − 2Κ1 ∫
as
0
l DT (y) ∂h h ∂y
Figure 3. Geometrical parameters of the gear tooth: h0 is the height and l is the face width of the tooth equal to the distance between adjacent clefts. The curvature is ignored for sake of clarity.
h−2us = − 2Κ1 ∫
Substituting eq 27 into eq 25 we obtain the closed expression for the longitudinal pressure gradient:
integrals can be written as −
−2 ⎞ ⎞ ∂P0 3η ⎛ h−2 3η ⎛ u h = u 2 ⎜⎜ −3 − 1⎟⎟ + 2 ⎜⎜ s −3 − us⎟⎟ ∂y 2h ⎝ h h 2h ⎝ h h ⎠ ⎠
l
QR
l
QR D and− 2Κ1 T l
⎡ ⎞⎤ x ⎛ ⎞ ⎛ u h−2 x⎞ 3 ⎢ ⎛ h−2 u⎜⎜ −3 − 1⎟⎟ + ⎜⎜ s −3 − us⎟⎟⎥ ⎜1 − ⎟ ⎝ h⎠ 4 ⎢⎣ ⎝ h h ⎠ ⎝h h ⎠⎥⎦ h x + (u + us) (29) h The viscous stress at the moving gear surface is expressed as u y (x ) =
∂uy(x = h) (30)
u0 = −
The balance of moments for the gear under load moment M is
R
∫S σy dS = M
∂x
(31)
u0 = −
−2 −2 ⎡ 7 h−1 7h−1us 3 ( h−2)2 ⎤ 3 h ush ⎥+ = u⎢ − − ⎢⎣ 4 4 h−3 ⎥⎦ 4 4 h−3 M = ηLn (32)
u=
h−2h−2us h−3
− 7h−1us +
7 h−1 − 3
u0 =
∫
R − dh h −
∫R
=0
= 0, where R− is the distance from the
us hM − 0 l 8πRLη
(36)
1 l
∫0
l
DT (y)
QR1 ∂h dy 2Κl ∂y
(37)
Qnh0 − (DT − DT+) 4π Κl
(38)
Note that in this approximation the velocity of the gear surface does not depend on parameter ls because the temperature gradient is inversely proportional to the width of the tooth profile and both the descending and ascending parts contribute equally. With higher teeth, however, a dependence on ls will appear because the reverse proportionality disappears.
4M ηLn
■
( h−2)2 h−3
R − dh R − h2
QR1DT 2Κ l
Following the approach used in the literature,15,16 with a triangular tooth profile having a summit placed at y = ls, the thermodiffusion coefficient has different constant values D±T at the ascending and descending sections of the tooth profile:
Solving for u we obtain 3
dy . When the side surface of the
where the slip velocity is expressed though the thermodiffusion coefficient and temperature gradient using eqs 34 and 35. To better understand the underlying mechanism, consider the idle rotation of the thermophoretic gear. We can combine eqs 18 and 34 to obtain
where S in the side surface area of the gear. In eqs 30 and 31 only the azimuth stresses are responsible for gear rotation, while radial stresses are canceled by small deformations in the gear material. The respective surface of integration is normal to the radius vector in the cylindrical coordinates. Consequently, the integration is made over the side surface of the circular cylinder (see Figure 1). The surface area of the circular cylinder is S = 2πRL; its differential is dS = L dy. eq 31 allows for the derivation of load characteristics of the gear as an engine. Combining eqs 29-31) we obtain ∂uy(x)
and
cleft to the gear axis. In these designations R+ is the respective R +R distance to the tooth summit, R = +2 − is the mean radius of the gear, and h0 = R+ − R− is the tooth height (see Figures 1 and 3). These derivations show that the self-supporting rotation of the gear, as described by eq 33, is possible only if the surface properties of the gear surface are nonuniform, as for example, when the adjacent side surfaces of the teeth are made of different materials. As discussed above, the use of deep teeth is not preferable. When the teeth height is much smaller than the mean channel width, eq 33 yields the following simple expression for the linear velocity of the gear surface:
(28)
l
∂x
dy
gear is made of uniform material, DT is a constant and these
where ush−2 = ∫ h−2(y)us(y) dy . By substituting eq 28 into eq 0 23, we obtain the flow profile:
σy = η
l DT (y) ∂h 0 h2 ∂y
DISCUSSION AND CONCLUSIONS Equation 38 indicates that the idle velocity of the side gear surface is governed by the temperature gradient and the difference in thermodiffusion coefficients at adjacent edges. Qnh Comparing eqs 34 and 38 indicates that parameter 0 defines 4πΚ
(33)
l
where h−1us = ∫ h−1(y)us(y) dy . The slip velocity is expressed 0 though the thermodiffusion coefficient DT and the temperature gradient:
l
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The Journal of Physical Chemistry C
The load characteristic of the gear can be obtained directly from eqs 33−35):
the effective temperature gradient created by heating the gear. Consider the interrelation between parameters n and h0, as they are not independent. As discussed above, an effective device necessitates that the teeth profile be smooth and shallow. Thus, while the temperature gradient increases with the number of teeth, it has an upper limit. At the point where the width of the πR tooth face is double the tooth height, l = 2h0 and n = h . In
ω(M ) =
(40)
where Aw l are the Hamaker constants defining the respective liquid-wall interaction energies for the tooth edges. This expression is similar in structure to the corresponding equation obtained in.17 A low slope in the load characteristic is preferable when the frequency of rotation ω (M) changes slowly with load moment. eq 40 indicates that such a characteristic is possible when the gap between the gear teeth and the internal wall of the cylindrical cavity h is sufficiently small and the cylinder length L sufficiently large. For liquids, αT ≈ 10−3K−1, |Aw−l − Aw+l| ≈ 10−13 erg, η ≈ 10−2 g/cm·s, σwl ≈ 2 × 10−7 cm, and v0l ≈ 10−21 cm3 (see ref 16). Using these values and a typical temperature QR gradient∇T = 4Κ ≈ 104 K/cm , eq 40 yields ±
0
that case, eq 38 takes the form u0 =
QR − (DT − DT+) 4Κl
(39)
The value of the optimal effective temperature gradient be estimated using data from ref 16. On the basis of
QR can 4Κ l
incident power 15 μW on a single gear, a typical absorption coefficient for the coating of 2%, a coating thickness of ≈0.1 μm, and R ≈ 6 μm, the heat supplied to the gear is Q ≈ 30 kW/ cm3. We note that the thickness of the coating was used rather than that of the polymeric substrate because the low absorbance and thermal conductivity of the latter leads to it playing a minimal role in the observed effect. Thus, the substrate provides only a small nonuniformity in the gear surface under a more imperfect carbon coating. Using a typical solvent thermal conductivity Kl ≈ 0.12−0.14 × 10−2 W/cm·K, the calculated effective temperature gradient QR is 4Κ ≈ 2 × 104 K/cm , which is comparable to the external
l
ω(M = 0) ≈
10−4 cm/s R
(41)
On the basis of eq 41 a rotor radius R ≈ 10−5 cm is the minimum size that can be expected to yield reliable predictability. For such a rotor, the idle-induced angular velocity of rotation defined by eq 41 is ω(M = 0) ≈ 10 s−1.
■
l
temperature gradients that can be obtained in thermodiffusion experiments.26 For typical DT values of 10−7 cm2/s·K in liquids, we obtain a characteristic linear velocity for the side gear surface of u0 ≈ 2 × 10−3 cm/s, which yields an angular velocity u
2 2 − ln 3 αT σwl (A w−l − A w+l ) QR hM − 12 ηv0lR 4Κl 8πR2Lη
AUTHOR INFORMATION
Corresponding Author
*(S.S.) Telephone: +7(495)939-74-39. E-mail:
[email protected].
2 × 10−3
for the gear of R0 ≈ 6 × 10−4 ≈ 3 s−1. This value is in excellent agreement with empirical data,16 considering the approximations and simplifications used in these calculations. The estimations above assume that all the teeth have identical inhomogeneities. Of course, in ref 16, inhomogeneities will not necessarily be symmetrically distributed over the teeth edges. Rotation will be affected by nonuniform inhomogeneities even when the properties of a single edge are distinct from all other edges. In such cases, the velocity of rotation will be reduced. However, in ref 16, the parameters of the polymeric gear and carbon coating are sharply different, and the difference D−T − D+T could also be several times larger than we have estimated. According to ref 27, the Hamaker constants are ≈(5−7) × 10−13 erg for polystyrene and ≈(50−100) × 10−13 erg for carbon. Thus, the system used in ref 16 could produce reasonable velocities of rotation, despite nonuniform inhomogeneities among the teeth. Of course, our consideration means not that the rotation of the gears in ref 16 is necessarily caused by the inhomogeneity of the gear teeth. We show only that this rotation may be caused by this mechanism. In contrast, our consideration shows that the rotation of the gear in ref 15 cannot be caused by the thermophoretic force and is related to inhomogeneity of the teeth surface. These estimations show that the system studied by Maggi et al.16 provides idle angular velocities comparable to those predicted for our thermoosmotic engine17 when thin wellabsorbing coatings are used. Still, the geometry used by Maggi et al.15 is not optimal for driving a load. Confined systems that consist of either a long cylinder or pinion may be more effective than a freely floating thin gear.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by funding from Boise State University and the Russian Academy of Sciences. REFERENCES
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