Negative Thermophoresis in Concentric Carbon Nanotube

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Negative thermophoresis in concentric carbon nanotube nanodevices Jiantao Leng, Zhengrong Guo, Hongwei Zhang, Tienchong Chang, Xingming Guo, and Huajian Gao Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b02815 • Publication Date (Web): 14 Sep 2016 Downloaded from http://pubs.acs.org on September 16, 2016

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Negative thermophoresis in concentric carbon nanotube nanodevices Jiantao Leng1, Zhengrong Guo1, Hongwei Zhang1, Tienchong Chang1,* , Xingming Guo1, Huajian Gao2 1

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, People’s Republic of China 2

School of Engineering, Brown University, Providence, RI 02912, USA

ABSTRACT Positive and negative thermophoresis in fluids has found widespread applications from mass transport to molecule manipulation. In solids, although positive thermophoresis has been recently discovered in both theoretical and experimental studies, negative thermophoresis has never been reported. Here we reveal via molecular dynamics simulations that negative thermophoresis does exist in solids. We consider the motion of a single walled carbon nanotube nested inside of two separate outer tubes held at different temperatures. It is found that a sufficiently long inner tube will undergo negative thermophoresis, whereas positive thermophoresis is favorable for a relatively short inner tube. Mechanisms for the observed positive thermophoresis and negative thermophoresis are shown to be totally different. In positive thermophoresis, the driving force comes mainly from the thermally induced edge force and the interlayer attraction force, while the driving force for negative thermophoresis is mainly due to the thermal gradient force. These findings have enriched our knowledge of the fundamental driving mechanisms for thermophoresis in solids and may stimulate its further applications in nanotechnology. *

Email address: [email protected].

ACS Paragon Plus Environment

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Keywords: Negative thermophoresis, thermal gradient force, edge force, interlayer attraction force, molecular dynamics simulations


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The discovery1 and understanding2 of thermophoresis have provided ample opportunities for applications including thermal precipitation,3, energy conversion,8,

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particle separation,5,

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drug delivery,7

biomolecule manipulation,10-14 and interaction detection,15 etc.

Thermophoresis was first discovered in liquids in 1856 by Ludwig1 who observed salt diffusing in water against a temperature gradient. The phenomenon was confirmed by Soret16 who believed that the temperature gradient leads to a homogeneous salt concentration gradient in liquid. Thermophoresis in aerosols was first reported in 1870 by Tyndall17 who found that dust particles can be pushed away from a heated surface in a dust-filled room. However, in spite of the wide applications3-7, 10-14, 18 of thermophoresis in fluids, it was not until more than a century later that thermophoresis at solid-solid interfaces was first predicted. In 2006, Schoen et al.19, 20 demonstrated via molecular dynamics (MD) simulations that an axial temperature gradient along a single-walled carbon nanotube (SWCNT) would drive a confined nanoparticle to move toward the direction of decreasing temperature. The phenomenon was immediately confirmed in an elegant experimental study by Barreiro et al.9 Such a temperature gradient can also induce the motion of many kinds of nanoscale particles, including fullerene clusters,21 capsule-like nanotubes,22 graphene sheet23 and water droplets,24-26 indicating that thermophoresis in solids has the potential of opening up new opportunities in nanoscale mass transport and energy conversion technologies.9 In contrast, there also exists negative thermophoresis, i.e., induced motion towards a direction of increasing temperature. Negative thermophoresis in fluids was first noticed in 1967 by Dwyer27 in a theoretical solution, and the name was coined by Sone.28 The phenomenon was soon confirmed by a series of numerical calculations.29-31 Very recently,


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negative thermophoresis in fluid was observed in experiments.11, 32, 33 However, to this date, the existence of negative thermophoresis in solids still has not been verified in either theoretical or experimental studies, despite its potential applications in nanotechnology. Here, we demonstrate via MD simulations that negative thermophoresis can occur at solid-solid interfaces. By investigating the motion of a SWCNT nested inside of two separate outer tubes held at different temperatures, we find that the direction of motion of the inner tube is dependent on the geometry of the device. A sufficiently long inner tube with both ends extending out of the outer tubes will undergo negative thermophoresis, while positive thermophoresis is favorable for a relatively short inner tube. Mechanisms for the observed geometry dependent thermophoresis will be discussed in detail.

Simulation models and methods Two models (i.e., Model I and Model II, as shown in Figure 1) are considered in our simulations. In both models, a narrow inner SWCNT is commensurately nested in two open-ended coaxial CNTs. The only difference between the structures of the two models is that the inner tube in Model II is substantially longer and extends out of the outer tubes, while that in Mode I is relatively short with both ends confined inside the outer tubes. Unless specifically stated, the inner tube is a (5, 5) tube, while the two outer tubes are (10, 10) tubes held at 300 and 500 K. All simulations are carried out using the LAMMPS package.34 A time step of 0.5 fs is used in all simulations. A reactive empirical bond order (REBO) potential35 is used to describe the in-plane C-C bond interactions and the interlayer vdW interactions are described by a


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Lennard-Jones (L-J) 6-12 potential (with a well-depth of ɛ = 2.968 meV and an equilibrium distance of σ = 0.3407 nm). We note that the qualitative behavior of the inner tube in the present system should be insensitive to the choice of atomic potentials because the relative sliding behavior of concentric CNTs is governed by the structure as well as the relative strengths of the intertube and intratube interactions, as Tangney et al.36, 37 showed in studying the interlayer friction between concentric CNTs.

Figure 1. Models for molecular dynamics simulations. An inner SWCNT (whose length is different in the two models) is nested inside two concentric outer SWCNTs (whose configurations are exactly the same in the two models). In Model I, the inner nanotube is 36.83 nm long, while that in Model II is 99.28 nm long. The two outer nanotubes, each being 19.35 nm long, are symmetrically placed relative to the inner nanotube, with their neighboring ends spaced at 15 nm apart. In the simulation, the temperature of the left outer tube is lower than that of the right tube.

The shrink-wrapped boundary condition34 is imposed in the longitudinal direction of the inner tube to allow its free motion in the outer tubes, while fixed boundary conditions are used in the other two perpendicular directions. Under these non-periodical boundary conditions, atoms do not interact across the boundary or move from one side of the box to the other. The two outer tubes are constrained against rotation and motion of their centers of mass


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in any direction, and maintained at different temperatures via a Berendsen thermostat to form a stable temperature difference.38, 39 We first fix the longitudinal motion of the inner tube by removing the velocity of its center of mass in all directions, and relax the system for 1.7 ns in a NVT (canonical) ensemble (in which the system temperature is set to the mean temperature of the two outer tubes) to allow a stable temperature gradient to be developed along the inner tube. We then start the simulation in an NVE (microcanonical) ensemble by releasing the artificial constraint on the longitudinal motion of the inner tube.

Results Geometry dependent thermophoretic motion. When the simulations start, both of the inner tubes in the two models start to move spontaneously along the outer tubes at an accelerated speed (see Figures 2a and 2b for snapshots of the motion of (5,5)(10,10) devices). However, the initial directions of motion of the inner tubes in the two models are opposite to each other, one toward the cooler side whereas the other toward the hotter side. In Model I, the inner tube drifts toward the low temperature CNT (LTCNT), corresponding to positive thermophoresis, while in Model II, the inner tube drifts toward the high temperature CNT (HTCNT), i.e. negative thermophoresis. For convenience, we shall regard the initial direction of motion of the inner tube as its intended direction of motion. In both cases, the initially resting inner tube accelerates until its rear end hits the end of one of the outer CNTs and then bounces back and decelerates while moving in the opposite direction, much like a ball bouncing on the ground under gravity (Figures 2c and 2d). As soon as the motion of the inner tube in the opposite direction decelerates to zero speed, it starts to accelerate toward the


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intended direction again. It is also seen from Figures 2c and 2d that the speed of the inner tube in both models depends on the diameter and the temperature difference. However, the initial moving direction is only determined by the length of the inner tube. In this letter, we will only focus our attention on the driving mechanism of the positive and negative thermophoretic motion, and will leave more detailed analyses of various system parameters (including length, diameter and temperature difference) to future work.

Figure 2. Snapshots of thermophoretic motion in Model I (a) and Model II (b) of (5,5)(10,10) devices. The displacements and velocities of the inner tubes in Model I and Model II for devices with different configurations under different temperature differences are shown in (c) and (d), respectively. The elargement of the circled area of the curve for the (5,5)(10,10) device under 300-500 K temperature difference in Figure 2d will be presented in Figure 6d to clearly show the trap behavior of the inner tube.


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The behavior of the inner tube clearly shows that there exists a persistent driving force for the inner tube to move toward the cooler region in Model I but toward the hotter region in Model II. This means that the direction of the driving force in the present device can be designed by tuning the length of the inner tube (or the distance between the outer tubes, as will be shown shortly). In particular, our results show that the directional motion toward the high temperature region (i.e., negative thermophoresis) can be achieved via the design of the geometry of a nanodevice, which is extremely important for the nanoscale actuation and energy conversion.

Mechanics of geometry dependent thermophoresis. Thermophoretic motion of a shorter carbon nanotube nested inside or outside a longer concentric tube with a temperature gradient has been extensively studied,9, 21, 22, 40, 41 and it is found that the motion is always toward cooler regions. For the present devices, different temperatures are imposed on the two outer tubes which provide driving forces for the inner tube, and the motion is dependent on the device geometry. This immediately raises the question about the driving mechanism for the observed geometry dependence of the thermophoretic motion. To understand where the driving force comes from, we show in Figures 3a-3e the longitudinal distribution of the lateral interlayer van der Waals force (the inerlayer van der Waals force projected onto the tube axis) exerted on the inner tube by the outer tubes when the inner tube is accelarated toward its intended direction of motion. In order to eliminate the effect of the potential corrugation, the value of the force is calculated by averaging the force on each ring over the time period in which the inner tube slides a lattice length (~0.246 nm) in


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the outer tubes. Figure 3f defines the ring number on the inner tube. It is clearly seen that the inner tubes in the two devices experience different force distributions. To gain a deeper insight into the driving mechanism, we split the force distribution into six regions: from Region I to Region VI (Figures 3a and 3b). Despite the similar force distribution profiles of the two models in Regions II-V, the force distributions of the two models are quite different in Regions I and VI where the interlayer contacts in the two models are different. In Model I, the ends of the inner tube are in contact (near the equilibrium distance of van der Waals interactions) with the walls of the outer tubes, while in Model II, it is the ends of the outer tubes that are in contact with the wall of the inner tube. In what follows, we will discuss in detail the forces exerted on the inner tube by the outer tubes in these regions.

Figure 3. The distributions of lateral force on the inner tube in two models. (a) and (b) The overview of force distribution for Model I and Model II, respectively. The colored regions indicate the positions of the outer tubes. (c) and (d) Typical force distritubions in Regions II and V for Model I and Model II. (e) Force distribution near the ends of the inner tube in


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Model I. (f) Definition of the carbon ring number on the inner tube.

The difference in temperature between the two outer CNTs can generate a temperature gradient on the inner tube (typical temperature distributions along the inner tubes of the two models are shown in Figure 4). Such a temperature gradient may induce thermal gradient forces

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on the inner tube atoms (i.e., atoms in Regions II and V in Figures 3a and 3b)

whenever they interact with the outer tubes. In addition, these atoms in contact with the outer tubes would experience frictional forces when the inner tube is sliding against the outer tube.36, 40 The total force exerted on Region II or Region V can thus be expressed as Fg + f, with Fg and f being the gradient and frictional forces, respectively. In fact, there are also random forces exerted on the inner tube atoms from their collision with the outer tubes. However, since the random forces cannot contribute a net driving force for a persistent directional motion, here we exclude the effect of the random force on the driving force.

Figure 4. Temperature gradients developed on the inner tubes in Model I (a) and Model II (b). Results for the inner tubes on three positions where they are accelerated to the intended


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direcition are presented.

Previous studies have shown that the thermal gradient force on a carbon nanotube nested inside or outside another one with a thermal gradient is toward the direction of decreasing temperature,9,

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implying that the latter feels a reaction force toward the direction of

increasing temperature. This can also be considered as a result of the net current of phononic excitations 9, 44 induced by the tempearture gradient on the inner tube that hits the cooler outer tube and pulls the hotter outer tube. That is to say, the thermal gradient force in Model I provides a resistance force, while in Model II, it is a driving force. This is confirmed by our calculations. In Model I, the line densities of FII and FV are about -0.10 pN/nm, while in Model II, they are about 0.063 pN/nm. In other words, FII + FV provides a resistance force to the inner tube of about 1.81 pN in Model I, but a driving force of about 2.25 pN in Model II. Since the thermal gradient force is opposite to the sliding direction of the inner tube in Model I, there should be some other forces that drive the tube to move. Now we turn our attention to the interlayer attraction forces exerted on Regions III and IV in Model I (see Figure 3a) and those on Regions I, III, IV, VI in Model II (see Figure 3b), where the ends of the outer tubes contact with the wall of the inner tube. When an end of the inner tube extends out of the outer tube (see Figure 5), an interlayer attraction force (Fa) is induced between the end of the outer tube and the protruding part of the inner tube.36, 45-48 This means that the forces on Regions III and IV in Model I and those on Regions I, III, IV, VI in Model II, i.e., FIII and FIV in Figure 3a and FI, FIII, FIV, FVI in Figure 3b can be expressed as Fa + f.36, 40 Here we take eight rings of the inner tube atoms in contact


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with each outer tube edge to account for the interlayer attraction forces exerted on the inner tube, as the edge attraction occurs generally within a range of three to four rings of atoms. In Model I, FIII + FIV = (-866.822 + 860.843) pN = -5.98 pN, which pushes the inner tube toward the cool region and is thus a driving force.

Outer tube

Fe

Inner tube

Fa

Figure 5. Illustration of the origin of the interlayer attraction force Fa on an inner tube when it extends out of an outer tube, and the origin of the edge force Fe when the end of the inner tube contacts with the wall of the outer tube.

Unlike in Model I, in Model II the interlayer attraction forces Fa from an outer tube appear in pair, approximately equal in magnitube but opposite in direction, and thus almost cancel each other and cannot generate an significant driving force. In the present calculation, FI + FIII + FIV + FVI = (868.09 – 867.67 + 862.01 – 861.55) pN = 0.88 pN (where the non-zero summation of the interlayer attraction forces may have resulted in part from the interlayer friction,36, 40 and in part from the slight difference in temperatures between the regions of the inner tube contacting with the outer tube ends49). Besides the forces mentioned above, there is another force called the thermally induced edge force (Fe)40, 41, 50 acting on each end of the inner tube in Model I (see Regions I and VI in Figures 3a and 3e) where the inner tube end is in contact with the outer tube wall. This edge force is induced by the differential atomic vibrations of the outer tube atoms in contact with and those not in contact with the inner tube (see Figure 5), and is linearly dependent on the


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temperature of the outer tube.40, 50, 51 Here we take five rings of atoms on the inner tube to calculate the edge force. It is found that FI is smaller than FVI, and the unbalanced edge force,

FI + FVI, is (7.191-11.976) = -4.79 pN, providing a driving force comparable to FIII + FIV (= -5.98 pN). It is thus clear that the driving mechanism in the two models are totally different. In Model I, the net driving force is a result of competitions among the thermal gradient force, the frictional force, the interlayer attraction force and the thermally induced edge force. The interlayer attraction force and the thermally induced edge force provide the driving force, whereas the thermal gradient force acts as a resisting force similar to the frictional force. This is why the inner tube in Model I tends to travel toward the cooler region (i.e., experiences positive thermophoresis). In contrast, in Model II, the net driving force results from the thermal gradient force, the frictional force, and the interlayer attraction force. It is mainly the thermal gradient force that drives the inner tube to move towards the hotter region (i.e., undergoes negative thermophoresis), whereas the interlayer attraction forces almost cancel each other out.

Discussion Thermal gradient. Both numerical40 and analytical41 studies have clearly shown that the thermal gradient force increases linearly with the temperature gradient, with a coeficient of 5.33 µeV/atom/K. However, in the present system, the gradient force is relatively small. The reason is that the real temperature gradient in the inner tube is very small (about 0.3 K/nm, see Figure 4) compared with the apparent temperature gradient between the two outer tubes


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((500-300)/35 ≈ 8.6 K/nm). This is due to the fact that the intralayer thermal conductivity of a CNT is orders higher than the interlayer thermal conductivety,52-56 which leads to a very small temperature gradient along the inner tube but a quite high temperature difference between the outer and inner tubes. Temperature dependent interlayer attraction force and edge force. Owing to an entropy difference,49 the interlayer attraction force on the inner tube decreases with the temperature. In spite of the existence of the frictional force, we can see that the interlayer attraction force from the cooler outer tube is higher than that from the hotter one in both models (see Figure 6a, open squares). In trying to eliminate the effect of friction on the calculation of the interlayer attraction force, we imposed a constant small velocity of 2.5 nm/ns at the center of the inner tube in the z direction (yet maintain the temperatures of two outer tubes at 300 and 500 K, respectively) to obtain an approximately quasi-static mean value of the interlayer attraction force. The difference between the results (Figure 6a, open squares) obtained for the thermophoretic motion and the corresponding ones (Figure 6a , solid squares) for the imposed motion should be the friction force (which is estimated about 0.5 pN from each end of the outer tubes, on the same order of the previously calculated result40). The calculated interlayer attraction force in four regions exhibits a clear decreasing trend with temperature. To more clearly show how the temperatures of both the inner tube and outer tube influence the attraction force, by imposing a constant velocity of 2.5 nm/ns on the inner tube of the Model I system, and separately varying the temperatures of the inner tube and two outer tubes, we obtain an approximately linear temperature dependence of the attraction force on the


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temperatures of both the inner and outer tubes (see Figure 6b).

Figure 6. (a) Amplitude of the interlayer attraction forces (open squares) in Regions I, III, IV and VI in Figure 4(f) and the corresponding forces (solid squares) under uniform motion of the inner tube. (b) Dependence of the interlayer attraction force on the temperatures of the inner and outer tubes. (c) Dependence of the edge force on the temperature. (d) The random force can help the inner tube to escape out of the trap (the overall motion behavior of the inner tube can be seen in Figure 2d).

Similar procedure to the calculation of the interlayer attraction force can be used to obtain the edge force acting on the inner tube in Model I. The results show that the edge force is approximately linearly dependent on the outer tube temperature and less sensitive to the inner tube temperature (shown in Figure 6c). This means that the linear temperature dependence of the edge force on a caped CNT is the same as that of the edge force on an open-ended CNT40


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or on a graphene layer50 from an adjacent layer. Role of random force. As has been mentioned, the long term average of the random forces vanishes and thus it has no net contribution to the driving force. However, at the initial motion of the inner tube at rest, the random force may help it get out the possible trap of potential energy corrugation,57-59 if the driving force is small compared with the potential barrier. As shown in Figure 6d, when its velocity decelerates to zero, the inner tube in Model II may fall into a potential trap (possibly due to its larger length which may lead to a larger structural distortion (see Figure 2b) and thus a larger interlayer resistance), and oscillate in a unit lattice with a vibration amplitude of about half of the lattice length la.60 With the aid of the random force, the inner tube can escape from the trap and start to move forward at some point. Geometry dependence of driving force. To give a clearer picture of the geometry dependent thermophoresis, we investigate the effect of a change in distance between the two outer tubes on the driving force of the device under consideration. The inner and outer tubes used in the simulations are the same as those in Model II, with distance between the outer tubes varying from 10 to 80 nm, and the temperatures of the two outer tubes still set to 300 and 500 K, respectively. The two ends of the inner tube is exposed outside of the outer tubes when the distance is smaller than 60 nm, i.e., the configuration of the device is similar to Model II, while when the distance is larger than 60 nm, the two ends of the inner tube is covered inside of the outer tubes, i.e., the configuration of the device is similar to Model I. It is seen from Figure 7 that the driving force first decreases with the distance between the outer tubes, toward the hotter side, in Model II like configurations. The decrease of the driving force is due to the decrease of the thermal gradient in the inner tube caused by the


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increase of the distance between the outer tubes. However, when the two ends of the inner tube are covered inside of the outer tubes to form a Model I like configuration, the driving force reverses its direction, i.e., toward the cooler side, and maintains almost a constant magnitude. This is because the driving force mainly comes from the interlayer attraction force and the edge force, both of which are strongely dependent on the outer tube temperatures (which are independent of the distance between outer tubes) but less dependent on the inner tube temperature (which may change with the distance between outer tubes). The results in Figure 7 clearly show that the direction of the driving force on the inner tube can be reversed by a change in the distance betweent the outer tubes, which means that such a device can generate repeated mechanical operations61 through the geometry dependent thermophoresis of the inner tube by varying the relative axial position of the outer tubes.

Figure 7. Dependence of the driving force on the distance between the outer tubes. Exellent agreement between theoretical prediction and MD calculations confirms the reported driving mechanisms.

Theoretical model for the driving force. From the above discussion, we know that the


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driving force in Model I device can be expressed as F I = ∑ Fe + ∑ Fa + ∑ Fg ,

(1)

while in Model II, F II = ∑ Fa + ∑ Fg .

(2)

The gradient force on the inner tube, i.e., the reaction force on the outer tube caused by the gradient on the inner tube, can be calculated as Fg = ηin ∇TπdmLcont/A0 (see Supporting Information), where ηin is a potential coefficient, Lcont the contact length between tubes, dm the mean diameter of the inner and outer tubes, ∇T the temperature gradient along the tube, and

A0 the area per atom. The edge force on an inner tube end is calculated as Fe = (ηedge−ηin) πdmTouter /A0 (see Supporting Information), where Touter is the outer tube temperature. The attraction force is linearly dependent on both of the inner and outer tube temperatures, i.e., Fa = F0 − αinTinner – αoutTouter, where F0 is the attraction force at zero temperature, and αin and αout are numerically determined coefficients for the device composed of (5, 5) and (10, 10) tubes. By establishing an equivalent thermal circuit of the device under consideration, we can determine the temperature gradient along the inner tube and thus determine all above force components (details are given in Supporting Information), and the initial driving force for the device considered in Figure 7 can be obtained as F II =

71.14L2 pN, for δ < l−2L = 60 nm, (189nm + L)( L + δ )

F I = −15.53 −

(l − δ ) ( 29.36δ − 41.78(l − δ ) ) pN, for δ > l−2L = 60 nm. (378nm + l − δ )(l + δ )

(3)

(4)

It is seen from Figure 7 that the theoretical predictions from Equations. (3) and (4) agree well with the MD simulations, further confirming the above revealed driving mechanism.


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Conclusion By investigating the thermophoretic motion of a single walled carbon nanotube nested inside of two separate outer tubes held at different temperatures, we have shown that the direction of motion of the inner tube is dependent on the system geometry. When both ends of the inner tube extended out of the outer tubes, the inner tube undergoes negative thermophoresis (i.e., moves towards higher temperature), whereas positive thermophoresis is favorable when the ends of the inner tube stay inside of the outer tubes. To the best of our knowledge, this is the first time that negative thermophoresis is observed at solid-solid interfaces. The mechanisms for the positive and negative thermophoretic motions in the devices under consideration are quite different. In positive thermophoresis, the interlayer attraction force and a thermally induced edge force serve as the driving force, whereas the gradient force produces a resisting force. By contrast, in negative thermophoresis, the driving force comes mainly from the thermal gradient force. A proposed theoretical model further comfirms the simulation revealed mechanisms. Since its discovery in liquids more than a century ago, thermophoresis has long been studied for a variety of applications ranging from mass transport to molecule manipulation. The newly discovered example of negative thermophoresis in solids fills an important knowledge gap and is expected to stimulate further interests on this intriguing phenomenon along with potential applications in nanoscale actuation and energy conversion systems.


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Associated Content Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: [A theoretical model for predicting the positive and negative thermophoretic forces.]

Author Information Corresponding author *E-mail: [email protected]

Author Contributions T.C. supervised the project and initiated the study. J.L. performed simulations. Z.G. and H.Z. provided assistance in the simulations. J.L., T.C. and X.G analyzed the data. J.L, T.C., and H.G. wrote the manuscript. All authors contributed to the discussion of the results.

Notes The authors declare no competing financial interest.

Acknowledgements The authors acknowledge the financial supports from the NSF of China (Grant No. 11425209) and Shanghai Pujiang Program (Grant No. 13PJD016). The MD simulations were carried out on the computing platform of the International Center for Applied Mechanics in Energy


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Engineering (ICAMEE), Shanghai University.

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TOC Graphic 252x85mm (120 x 120 DPI)


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Negative Thermophoresis in Concentric Carbon Nanotube

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