Coupling Effect of Van Der Waals, Centrifugal, and Frictional Forces On

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Coupling eﬀect of van der Waals, centrifugal, and frictional forces on a GHz rotation–translation Cite this: Phys. Chem. Chem. Phys., 2019, 21,359 nano-convertor†

Bo Song,a Kun Cai, *ab Jiao Shi, ad Yi Min Xieb and Qinghua Qin c

A nano rotation–translation convertor with a deformable rotor is presented, and the dynamic responses of the system are investigated considering the coupling among the van der Waals (vdW), centrifugal and frictional forces. When an input rotational frequency (o) is applied at one end of the rotor, the other end exhibits a translational motion, which is an output of the system and depends on both the geometry of the system and the forces applied on the deformable part (DP) of the rotor. When centrifugal force is Received 25th September 2018, stronger than vdW force, the DP deforms by accompanying the translation of the rotor. It is found that Accepted 26th November 2018 the translational displacement is stable and controllable on the condition that o is in an interval. If o DOI: 10.1039/c8cp06013d exceeds an allowable value, the rotor exhibits unstable eccentric rotation. The system may collapse with the rotor escaping from the stators due to the strong centrifugal force in eccentric rotation. In a practical rsc.li/pccp design, the interval of o can be found for a system with controllable output translation.

1 Introduction components.18–22 Hertal et al.23 investigated the significant deformation of carbon nanotubes (CNTs) by surface vdW forces With the rapid development in nanotechnology, miniaturization that were generated between the nanotube and the substrate. of devices from the microscale to nanoscale becomes feasible.1 Huang et al.24 found that a strain gradient in graphene can The concepts of typical nanodevices, such as the resonator,2–5 induce a non-zero net vdW force, which is sufficient to actuate oscillator,6–11 and rotary motor,12–16 have been proposed for over the directional movement of molecular mass on the graphene

Published on 27 November 2018. Downloaded 1/16/2019 8:29:19 PM. a decade, and some of them have been realized in nanofabrica- surface. Besides, self-assembly of low dimensional materials is tion. For example, a nano-resonator has been used as a balance often controlled by vdW force.25–29 to measure the mass of a small molecule.4,5 The oscillator has a When relative sliding occurs between two components in a potential application in memory as an on–off. A few models for device, frictional force will aﬀect the sliding state. In particular, the rotary nanomotor have been proposed, which may be applied the value of frictional force on the nanoscale increases with the to drive the motion of a nano vehicle. relative sliding speed.8,16,30–33 Friction even becomes a major On the nanoscale, the motion of a component in a system factor in the dynamic response of such devices as nano- depends on the external forces exerted. One of the forces is oscillators or nanomotors. Centrifugal force only appears on a van der Waals (vdW) force.17 Although not as strong as ionic or rotating component. For example, in a rotary nanomotor, the covalent bonds, vdW force always plays an important role in the atoms on the rotor are subjected to centrifugal force, which is mechanical properties, electrical transport properties, and the proportional to the square of the rotational speed. At gigahertz dynamic response of the nanodevices at interfaces between rotation, the rotor may be damaged due to heavy centrifugal force on the atoms.34–38 In this study, a model for a rotation–translation nano- a College of Water Resources and Architectural Engineering, Northwest A&F convertor is proposed potentially as a nano on–off or a sensor University, Yangling 712100, China. E-mail: [email protected] b Centre for Innovative Structures and Materials, School of Engineering, for measuring the rotational speed of the rotor. Considering the 2 RMIT University, Melbourne 3083, Australia excellent mechanical strength of sp carbon materials, e.g., 39 40 c Research School of Engineering, The Australian National University, ACT 2601, carbon nanotubes and graphene, and the extremely low Australia friction between neighbouring layers,32,33,39–43 we choose CNTs d State Key Laboratory of Structural Analysis for Industrial Equipment, as the rotor and stators in the system. Like origami,44 four Dalian University of Technology, Dalian 116024, China † Electronic supplementary information (ESI) available: Movie 1 – medium DP at graphene ribbons are used to form a deformable part (DP) for 500 K-input 50 GHz-[5, 5.18]ns.avi. Movie 2 – medium DP at 500 K-input 70 GHz- connecting both the input part and the output part of the rotors 45,46 [3.02, 3.09]ns.avi. See DOI: 10.1039/c8cp06013d (Fig. 1). Different from a traditional transmission system,

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Fig. 1 Schematic of a rotation–translation convertor with a deformable carbon rotor. At the left end of the L-rotor (the input part), an input rotational frequency, i.e., o, is exerted. ‘‘d’’ is the distance between the right edges of the R-rotor and the R-stator2, and it is the output of the system. Dimensions of

the deformable part (DP) are labeled as Lx, Ly,andLz, respectively, with the unit of Å. Three DPs with different sizes are considered.

as a rotational speed is applied on the input part of the rotor, it 2.2 Methods will also drive the rotation of the remaining parts. During The molecular dynamics simulation approach was adopted to rotation, the centrifugal force on the DP causes deformation find the deformation of configuration of the rotor during rotating. of the DP and synchronously produces translation of the output Simulations were accomplished using the open code LAMMPS.47 part. The effects of the vdW, centrifugal and frictional forces on The interaction among atoms in the carbon/hydrogen system is the dynamic behavior of the system are investigated by mole- described by the AIREBO potential.48 In each simulation, the cular dynamics simulations. initial configuration of the system is modified by minimizing its potential energy. Further, some atoms are fixed in their degrees of Published on 27 November 2018. Downloaded 1/16/2019 8:29:19 PM. 2 Model and methodology freedom, e.g., three rings of atoms on the left end of the rotor, four rings of atoms on each stator (Fig. 1). The next step is to relax the 2.1 Model of the rotary device system using a canonical (NVT) ensemble for 100 ps with a In this work, the model shown in Fig. 1 is adopted to illustrate timestep of 0.5 fs. After relaxation using the Nose–Hoover the vdW eﬀect and the centrifugal eﬀect on the nanoscale. In the thermostat,49,50 a specified rotational frequency is exerted on model, a deformable part (DP) made from graphene ribbons is the L-rotor at the atoms that were previously fixed in relaxation. used to connect the L-rotor and the R-rotor made from CNTs. During rotating, the timestep is set as 1 fs, and some essential Three CNT-based stators are fixed to constrain the motion of physical quantities, e.g., variation of potential energy (VPE), the rotors. At the left end of the L-rotor, a constant rotational centrifugal force of the DP, and displacement of the right end of frequency, o, is input. The distance between the right ends of the R-rotor, are recorded for analysis. The value of VPE can be the R-stator2 and the R-rotor is output to show the axial motion calculated by subtracting the initial value of potential energy of of the R-rotor. The detailed parameters are listed in Table 1. the component from its current value.

Table 1 Initial parameters of the rotary system. Dimension unit: Å

Component Chirality Radius Length Ring/layer Number of atoms L-rotor (6,6) 4.068 41.811 35 420C R-rotor (6,6) 4.068 95.921 80 948C + 12H L-stator (11,11) 7.458 12.298 11 242C R-stator1 (11,11) 7.458 12.298 11 242C R-stator2 (11,11) 7.458 12.298 11 242C Small DP 15.623(x) Â 16.664(y) Â 16.700(z) 4 104C Â 2(up + down) + 80C Â 2(left + right) Medium DP 24.145(x) Â 26.620(y) Â 26.520(z) 4 252C Â 2(up + down) + 228C Â 2(left + right) Large DP 32.666(x) Â 31.540(y) Â 31.540(z) 4 400C Â 2(up + down) + 376C Â 2(left + right)

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Temperature effects were considered by considering the that at 83 ps. In particular, the DP becomes a narrow ring, in system at low temperature (e.g., 8 K), room temperature (e.g., which the two sides of the ring attach. For the narrow ring, 300 K), and high temperature (e.g., 500 K). The input rotational it cannot deform further. This means that the rotor rotating at frequency of the L-rotor, i.e., o, was varied from 10 GHz to 60 GHz or higher frequency should have the same configuration, 100 GHz or higher if necessary. i.e., a narrow ring. To show the vdW effect on the deformation and output of At 8 K, the fluctuation of VPE tends to be less than 0.2 eV, the system, the VPE of the DP is defined and calculated using which is far less than that at 300 K or 500 K. This is because the following equation, thermal vibration of the atoms in the DP at higher temperature contributes more to VPE. Meanwhile, it should be mentioned VPE(t) = PE(t) À PE(t ), (1) 0 that the fluctuations of VPE at 40 GHz and 50 GHz are more drastic than those of the remaining cases. The reason will be where PE(t) and PE(t0) are the potential energy of the DP at given later. times t and t0, respectively. In the AIREBO potential function, three items are contained, i.e., the REBO item, which is used for By comparing the stable values of VPE at a specific tempera- describing the bonding interaction among neighboring atoms, ture, e.g., at 8 K, we find that the values are not always higher the torsion item, for evaluating the dihedral angle effect on the when the input rotational frequency is higher. For example, the potential, and the final item is the 12-6 type Lennard-Jones (L-J) stable values of VPE at o = 60 GHz and 70 GHz are much larger potential,17 which is used for evaluating the vdW effect, i.e., than those at 80 GHz and 90 GHz. Similarly, at 300 K, the value of VPE at 60 GHz is the highest among the ten cases. The PE(t)=PREBO(t)+PTorsion(t)+PL-J(t). (2) reason is that the DP becomes a narrow ring, the two sides of the ring attach together, which leads to a decreasing potential The interaction cut-off distance for the L-J potential is set energy of the DP at 80 GHz or 90 GHz rotation (Fig. 3b). Once as 1.02 nm. the DP becomes a narrow ring, the two sides are difficult to separate, and the DP cannot deform with o. Hence, 70 GHz can 3 Results and discussion be considered as the upper limit of the system with a deformable rotor at this temperature. 3.1 Variation of configuration of the medium DP Another phenomenon is that VPE becomes negative in some As deformation of the DP can be evaluated by the value of VPE, cases, e.g., o = 70 GHz at 300 K and 500 K, and 60 GHz at the histories of the VPE of the medium DP under different 500 K. We know that the edge of the DP comprises unsaturated conditions are first drawn in Fig. 2. From the convergent values carbon atoms, which are easily bonded together.34 For a pair of of the VPE curves, a conclusion can be made, in general, that unsaturated carbon atoms being bonded together, the decreas- the value of VPE increases when the input rotational frequency ing potential energy is B4.75 eV (see the first item in the right (o) increases. The fluctuation of VPE depends on both the part of eqn (2)). According to the snapshot of Fig. 3c, the input rotation and temperature. The fluctuation decreases and medium DP with o = 70 GHz at 500 K is bonded together by

Published on 27 November 2018. Downloaded 1/16/2019 8:29:19 PM. tends to be stable after B5 ns in most cases. The fluctuation of nine new C–C bonds. In particular, the sharp drops of VPE in VPE demonstrates the variation of the DP’s configuration. Fig. 2c illustrate the bonding process, i.e., each drop means one Higher fluctuation, e.g., in the first 1 ns, means the DP under- or more new bonds being generated. In this case, there are nine goes large deformation. In particular, the DP is under axial new bonds, but ten new bonds are generated in the same DP torsion due to friction from the stators. For example, for the DP with o = 60 GHz at the same temperature. Self-bonding of the with o = 60 GHz at 300 K, its configuration at 15 ps in Fig. 3a is DP depends on two major factors, i.e., a stator-friction-induced

obviously diﬀerent from the initial one (0 ps), but it is similar to higher torque moment (Mst in Fig. 3d) that leads to the approach

Fig. 2 Histories of VPE of the medium DP with diﬀerent rotational frequency at diﬀerent temperatures: (a) 8 K, (b) 300 K, and (c) 500 K.

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Fig. 3 Variation of configurations of the medium DP in the system under diﬀerent conditions. (a) Axial translation of the right rotor when rotating with o = 60 GHz at 300 K. (b) Stable configurations of the DP under diﬀerent conditions. (c) Formation of nine bonds in the DP when rotating with o = 70 GHz at 500 K. (d) Attraction of the R-stator1 exerted on the DP with o = 10 GHz at 300 K. (e) Collapse process of the system with o = 100 GHz at 500 K.

of the unsaturated carbon atoms in the DP and a relatively low narrow ring. Hence, self-bonding occurs at o = 60 or 70 GHz, centrifugal force that cannot pull the deformed DP into a but not at o = 80 or 90 GHz.

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The final characteristics of the VPE curves are that the values in Fig. 4b, the rotary inertia of the DP jumps from 18.39 kg nm2 jump up quickly within the first 1 ns when o = 100 GHz at 8 K at 4610 ps up to 22.63 kg nm2 at 4640 ps and down to and 500 K. Before giving the reason, we recall that the value of 18.46 kg nm2 at 4820 ps. By observing the snapshots in Fig. 3d, VPE only depends on the deformation of the DP. From the we know that the DP deforms during the period, and most atoms snapshot at 83 ps in Fig. 3a, the DP has become a narrow ring, in the DP are further away from the z-axis at 4640 ps than at and the distance between the middle parts of its two vertical 4610 ps. According to eqn (3), the value of rotary inertia at sides is nearly 0.34 nm, which means that the ring has no more 4640 ps is higher than that at 4610 ps. space for further deformation in the present situation. Hence, Finally, the fluctuation of rotary inertia at 20 GHz r o r 50 GHz the drastic jumping up of VPE at 100 GHz at both temperatures is much stronger than the remaining cases. In particular, at 8 K, must be caused by the other deformation of the rotor. Fig. 3e the fluctuation of rotary inertia of the DP can be neglected when shows the variation process of the system configuration when o = 80 or 90 GHz. The reason for this is that the configuration o = 100 GHz at 500 K. It indicates that the system collapses within of the DP with the input rotational frequency between 20 GHz 260 ps. The jump up of VPE is caused by further deformation of and 50 GHz varies periodically. But, the configuration with the DP together with the L-/R-rotor after escaping from the right o = 90 GHz is very stable during rotating. This demonstrated that stators (snapshot at 245 ps in Fig. 3e). the input rotational frequency in the interval, i.e., 20–50 GHz, can produce a breathing state of the DP’s configuration 3.2 Rotary inertia of the medium DP during rotating (Movie 1, ESI†). To quantitatively illustrate the deformation of the medium DP during rotating, the value of rotary inertia of the DP about the 3.3 Centrifugal force on the rotary medium DP z-axis is calculated by the following equation, The centrifugal force on the DP can be calculated using the P 2 following equation, i.e., Jz = mi Â (ri + di) , (3) P P 2 2 Fc = Fci = mioi Â ri = mDP Â o Â rc, (4) where mi is the mass of the ith atom in the DP, ri is the initial

distance between the ith atom and the z-axis, and di is the axial where mDP is the mass of the DP, oi is the rotational frequency displacement of the atom. Hence, we find that the value of of the ith atom about the z-axis, oIn is the input rotational

Jz will increase when an atom moves away from the z-axis. For frequency, and rc is the eccentricity of the DP’s centroid. As both

example, the value of rotary inertia of the DP in the initial state mDP and o are constants, the magnitude of centrifugal force on

in Fig. 3a must be less than that of the same DP at 15 ps, the DP is proportional to its eccentricity, and the variation of Fc

because most atoms move away from the axis of the rotor. Fig. 4 must be produced by the variation of rc. suggests that, in general, the value of rotary inertia of the DP is Deformation of the DP leads to variations of the VPE and the higher when the input rotational frequency is higher. But three rotary inertia. But, it does not mean that the centrifugal force special characteristics should be demonstrated. on the DP varies simultaneously. If, for example, the rotor Firstly, in some cases, e.g., o = 60 or 70 GHz at 8 K, the value together with the DP has an ideal rotation, i.e., the rotating part Published on 27 November 2018. Downloaded 1/16/2019 8:29:19 PM. of rotary inertia is the highest among the first nine cases, i.e., has symmetric deformation about the z-axis without eccentri- o o 100 GHz. A similar phenomenon also appears at 500 K. city, the value of centrifugal force on the DP should be zero. The ring becomes very narrow (Fig. 3b), most atoms move away If the DP has a stable configuration and a constant eccentricity,

from the z-axis, i.e., di 4 0 (in eqn (3)), and this results in the the value of centrifugal force should be a positive constant. highest value of rotary inertia. However, in an atomic system, random thermal vibration of the Secondly, the value of rotary inertia exhibits sudden jumps atom always exists. During rotating of the DP, thermal vibration when o = 10 GHz at 300 K (Fig. 4b) or 500 K (Fig. 4c). For example, of atoms always exists. Hence, even if the rotating part looks

Fig. 4 Rotary inertia of the medium DP under diﬀerent conditions at diﬀerent temperatures: (a) 8 K, (b) 300 K, and (c) 500 K.

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Fig. 5 Spectrum of centrifugal force on the medium DP with diﬀerent rotational frequencies at diﬀerent temperatures: (a) 8 K, (b) 300 K, and (c) 500 K.

symmetric about the z-axis, thermal vibration still leads to this is that the DP is undergoing eccentric rotating, i.e., the two eccentricity of the DP. Under high speed rotating, non-zero slim tubes connected by the DP bend during rotating (Movie 2, centrifugal force appears. The spectrum of centrifugal force on ESI†). This means that the eccentricity of the DP, i.e.,themagni-

the DP under diﬀerent conditions is drawn in Fig. 5. tude of rc in eqn (4), increases obviously. This does not occur in At lower speeds of rotation, the DP is subjected to a very any other cases when the system does not collapse. Hence, 70 GHz

Published on 27 November 2018. Downloaded 1/16/2019 8:29:19 PM. small centrifugal force. For example, at 8 K, the centrifugal can be considered as an input rotational frequency that is very force on the medium DP is not higher than 45 eV ÅÀ1 when o is close to the eigen frequency with respect to the current system. The not higher than 20 GHz. However, when o increases to 30 GHz, DP will have higher eccentricity when rotating with the eigen the peak value of centrifugal force is over 200 eV ÅÀ1. But, frequency. before o approaches 50 GHz, the maximum of centrifugal force Now, the reason for the collapse of the rotating part with does not exceed 1000 eV ÅÀ1. When o is between 60 GHz and o = 100 GHz at 8 K or 500 K can be explained as follows: the 90 GHz, the peak value of the centrifugal force is over strong centrifugal force pulls the DP away from the rotating 2000 eV ÅÀ1. At 100 GHz, the system collapses, and the axis. Due to the left end of the L-rotor being fixed for input maximum of the centrifugal force is more than ten times that rotation, the right part, i.e., the R-rotor, is under tension toward at 90 GHz. The magnitude of centrifugal force increases slightly the left (snapshot at 151 ps in Fig. 3e). When the axial stretching with the temperature. forcecanovercomethepotentialbarriers at the right edges of the The maximum value of centrifugal force, which appears in R-stator2 and the R-rotor (snapshot at 235 ps in Fig. 3e), the right the first 1 ns, indicates the maximum eccentricity of the DP. end of the R-rotor will continue to move left (snapshot at 255 ps After reaching the peak value, the centrifugal force tends to in Fig. 3e) until complete collapse occurs (snapshot at 260 ps in fluctuate in a narrow range. In this way, the state of the DP can Fig. 3e). Here, 100 GHz can be considered as the upper limit of be illustrated by the mean value and the standard deviation of input rotational frequency for the system at this temperature. the centrifugal force at this stage. For example, the mean value indicates the stable eccentric rotation of the DP. But the 3.4 Axial translation of the R-rotor standard deviation reflects two facts. One is the periodic axial In the model shown in Fig. 1, the R-rotor connects with the DP. waving of the two edges of the narrow DP. The other is the When the DP deforms, the left end of the R-rotor will move with periodic radial vibration of the centroid of the DP. it. During rotating at high speed, the DP becomes a narrow ring It can be found that the maximum of the mean value at a (Fig. 3b), and the R-rotor moves left (the snapshots at 15/75/83 ps specific temperature appears when o = 70 GHz. The reason for in Fig. 3a). Large axial displacement of the right end of the

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Fig. 6 Axial motion of the R-rotor under diﬀerent conditions at diﬀerent temperatures: (a) 8 K, (b) 300 K, and (c) 500 K.

R-rotor occurs simultaneously. A nanodevice, e.g., an on–off, can be made according to the present model with a variable value of d. For example, when d 4 4 nm, the device is in the ‘‘on’’ state, and it is in the ‘‘off’’ state when d o 2.5 nm. Controlling the value of d is significant for such a device. Fig. 6 illustrates the variation of d, i.e., the distance between the right ends of the R-stator2 and the R-rotor, during rotating with the DP under diﬀerent conditions. In general, the value of d decreases with increasing o. At 8 K, the values of d with o = 60 GHz and 70 GHz are the lowest and remain unchanged soon after no more than 1 ns. After that, the DP has no chance to return to its original configuration. Hence, the value of d fluctuates slightly. Similarly, at 500 K, the value of d shows a sudden jump up when o = 60 or 70 GHz due to the generation of the new bonds in the rotating DP (e.g., Fig. 3c). Before self- bonding, the DP becomes a very narrow ring, which leads to a

lower value of d. Once the unsaturated atoms on one side of the Fig. 7 The curves of d versus o at diﬀerent temperatures. Initial value of Published on 27 November 2018. Downloaded 1/16/2019 8:29:19 PM. narrow ring bond together, those internal atoms neighboring the d is 3.909 nm. new bonded atoms are subjected to strong repulsion. Hence, the other side of the ring becomes a wide oval. This is the reason why the value of d jumps up suddenly. but no new bond is being generated in the DP. Fig. 7 indicates The minimum of d appears when o = 80 GHz at 500 K, when that the value of d decreases linearly with increasing o between the DP becomes a very narrow ring (Fig. 3b). Even when the 20 GHz and 50 GHz. When a specific value of d between 4 nm rotor has a higher value of o, the DP cannot deform further. and 4.4 nm is required, the input rotational frequency can be If o is between 20 GHz and 50 GHz, the value of d fluctuates designed accurately. with an amplitude of B0.5 nm. At a lower temperature, e.g., 300 K, the amplitude of fluctuation is lower. The sudden jumps 3.5 Size eﬀect of DP on the output of the system occurring at 300 K or 500 K when o = 10 GHz are caused by the As aforementioned, only the medium DP was involved. Fig. 6 sharp variation of the DP configuration (Fig. 3d). Under cen- and 7 indicate that the minimum of d (40) depends on the size

trifugal force (Fc in Fig. 3d), the DP tends to become a narrow of the DP. To show the effect of the size of the DP on the critical ring with a very small distance between the two rotors. But the value of o and the stable value of d, three models of the DP vdW effect, which generates the bending stiffness of the ribbon shown in Fig. 1 were considered at 300 K. that forms the DP, and the attraction from the R-stator1 (Fig. 3d) After a series of tests, we found that the small DP collapses result in a larger distance between the two rotors, i.e., a larger when o 4 190 GHz (Fig. 8a), which is higher than 110 GHz of

vdW force (FvdW) leads to a higher value of d. If centrifugal force the medium DP. But, the large DP remains stable only when is higher, d reduces. In this case, the DP is in a critical equili- o is no higher than 60 GHz. Otherwise, e.g.,ato = 70 GHz, the brium state under the centrifugal and the vdW forces. rotor system collapses (Fig. 8b). These states can be read from It is necessary to demonstrate that the final stable value of the curves in Fig. 8. This means that a larger DP can maintain d depends obviously on the input rotational frequency but stable rotation only at a lower input speed. Without considering slightly on the temperature if the system is in stable rotation the collapse of the DP, e.g., by constraining the maximum axial

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Fig. 8 Axial translation of the R-rotor when connected to three diﬀerent DPs: (a) small DP, (b) large DP, and (c) curves of d vs. o for the small, medium and large DPs.

of the R-rotor, the maximum deformation of the DP will also (4) High speed rotation (e.g., 80 GHz r o r 90 GHz of the determine the maximum axial displacement of the R-rotor. medium DP): The deformation of the DP is strongly dependent In each curve in Fig. 8c, there is a sharp decrease of d. For on the centrifugal force, and slightly on the vdW force or the example, it occurs between 160 GHz and 170 GHz for the system friction from stators. The DP becomes a narrow ring, and rotation is with the small DP, or between 50 GHz and 60 GHz for the system stable. Axial translational motion of the rotor remains unchanged with the medium DP, or between 30 GHz and 40 GHz for the with respect to o. system with the large DP. The two values form a gap that divides (5) Collapse (e.g., o Z 110 GHz of the medium DP): the states of the DP into stable rotation and unstable rotation. In a Eccentric rotation appears easily, and strong centrifugal force stable rotation, eccentricity of the DP is small and majorly caused on the deformed DP pulls the rotor away from the right stators by irregular deformation of the DP. In an unstable rotation, the DP until the system collapses. is in one of three diﬀerent states. The first state is that the DP may Briefly, if the system is not broken, the rotation of the DP be in an obvious eccentric rotation owing to bending deformation can be divided into stable rotation and unstable rotation states. of the two rotors. Second, self-bonding of the DP may occur. The In a stable rotation, eccentricity of the DP is small and majorly final state is that the DP becomes a narrow ring. caused by irregular deformation of the DP. The output translation is stable and controllable. In an unstable rotation, the DP 4 Conclusions might be in one of three diﬀerent states, i.e., obvious eccentric

Published on 27 November 2018. Downloaded 1/16/2019 8:29:19 PM. rotation owing to bending deformation of the two rotors, self- The eﬀects of vdW, frictional and centrifugal forces exerted bonding of the DP, and the DP becoming a narrow ring. on the deformable rotor of a rotation–translation convertor are evaluated using molecular dynamics simulations. For the deformable part (DP) in the rotor, its configuration is mainly Conflicts of interest determined by the input rotational frequency o.Theaxial The authors declare that there is no conflict of interest. translation of the right rotor, as an output of the system, depends on the state of the DP’s configuration. In general, the value of o determines the state of the DP. Conclusions can be drawn based Acknowledgements on the DP states, i.e., Financial support from the National Key Research and Develop- (1) Low speed rotation (e.g., o o 20 GHz of the medium DP): In this state, the centrifugal force is weaker than the vdW force, ment Plan, China (Grant No.: 2017YFC0405102), and Open and the DP undergoes small deformation during rotating. The project of State Key Laboratory of Structural Analysis for Indus- axial translation of the rotor does not occur. trial Equipment, China (Grant No.: GZ111) are acknowledged. (2) Medium speed rotation (e.g., 20 GHz r o r 50 GHz of the medium DP): The rotor has a stable rotation, and the axial References motion of the rotor increases linearly with o. The deformation of DP is controllable. 1 S. Lin, E. K. Lee, N. Nguyen and M. Khine, Thermally- (3) Mid-high speed rotation (e.g., 60 GHz r o r 70 GHz of induced miniaturization for micro- and nanofabrication: the medium DP): The deformation of the DP in the rotor is progress and updates, Lab Chip, 2014, 14, 3475–3488. unstable due to strong friction from the stators and relatively 2 P. Poncharal, Z. L. Wang, D. Ugarte and W. A. Heer, low centrifugal force, and the DP can be either a wide oval, or a Electrostatic deflections and electromechanical resonances narrow ring, or having self-bonding. of carbon nanotubes, Science, 1999, 283, 1513–1516.

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