Rotation-induced axial oscillation of a composite nanoconvertor at low temperature

Abstract

We propose a model of a nanostructure which can transform an input rotation into an output oscillation. In the model, the rotor has two identical internally hydrogenated deformable parts. The mechanism is that the rotation-induced centrifugal force and van der Waals force drive the recoverable deformation of the hydrogenated deformable parts, which gives rise to the axial translation of the free end of the rotor. Once the two hydrogenated deformable parts deform periodically, the free end of the rotor oscillates periodically in the axial direction. Molecular dynamics simulations are conducted to reveal the dynamic response of the system at low temperature. Four main types of deformation and the first three orders of vibration responses of the hydrogenated deformable parts are analyzed. Synchronous breathing vibration of the two hydrogenated deformable parts produces ideal oscillation with large amplitude. Asynchronous axial vibration of the hydrogenated deformable parts reduces the oscillation amplitude or produces beat vibration. The way to control the amplitude of the axial oscillation/vibration is given.

Keywords

Nanodevice rotation-oscillation convertor molecular dynamics carbon nanotube fast Fourier transform

1. Introduction

Carbon nanotubes (CNTs) have excellent mechanical (Qin et al., 2008; Wong et al., 1997), electrical, and thermal properties, which indicate popular applications in nano–electro–mechanical systems (Servantie and Gaspard, 2006). Typically, CNTs can be adopted in two classified fields, that is rotary nanodevices such as nanomotors and translation nanocomponents such as nano-oscillators. Fennimore et al. (2003) first built a rotary nanomotor by using a multiwalled carbon nanotube (MWCNT) as a key component under external electric field. Tu and Hu (2005) designed a rotary motor driven by axially varying voltage theoretically. Wang et al. (2008) built a rotary motor with one CNT shaft and three or six blades and revealed its dynamic response with semiclassical molecular dynamics (MD) simulations. Cai et al. (2014a) discovered the thermally driven rotary nanomotor from double-walled CNTs. Hosseini-Hashemi and Ilkhani (2016) revealed the exact solution for free vibration of rotating nanotubes based on the nonlocal first-order shear deformation shell theory. Shi et al. (2018) built a model of a rotary nanoring at finite temperature. Recently, Shi et al. (2020) proposed a new way to drive circular nanoflakes to rotate by diamond needles.

In 2000, the ultralow friction between the adjacent walls of the MWCNT was verified by experiments (Cumings 2000; Yu et al., 2000), in which the inner tube partly pulled out of the outer tubes can be drawn back into the outer tubes by van der Waals (vdW) force (Bifano et al., 2012). Inspired by this phenomenon, Zheng et al. (2002) initially presented a gigahertz oscillator model from the MWCNT. In their model, the inner tubes oscillated in the open-ended outer tubes when the inner tubes were first partly pulled out of the fixed outer tubes and then released. Upon the model, Legoas et al. (2003) first adopted the MD method to investigate feasibility of the telescopic oscillation from different combinations of CNTs. Even though the intertube frictional force is extremely low and much less than the asymmetric vdW force from both ends of the outer tube, it is essential to the oscillatory behavior of the nanosystem. Guo et al. (2003) estimated the interlayer sliding frictional force on the inner tube during oscillation. To obtain a stable gigahertz oscillator, Neild et al. (2009) introduced periodic forces on the inner tube to balance the frictional force from the outer tube.

In the abovementioned simulations, the inner tube just has translational motion during oscillation. Because the inner tube can be easily driven to rotate (Bourlon et al., 2004; Cai et al., 2015a; Fennimore et al., 2003; Král and Sadeghpour, 2002; Zhang et al., 2004), Cai et al. (2014b) found that the inner tube with initial gigahertz rotational frequency can be self-excited to oscillate in the outer tube. In the CNT-based motion transmission system, the rotation together with the oscillation of the inner tube was also observed (Cai et al., 2015b; Shi et al., 2016). Song et al. (2019a) proposed the definition of a nano rotation-translation convertor with deformable part (DP) on its rotor for the first time and revealed the mechanism of the rotation-translation transformation via MD simulations. However, they found that the DP may become a self-bonded narrow ring and cannot bounce back to its original shape even after removing the input rotation. Once the DP becomes a narrow ring, the system will have a constant axial shrinkage, and the rotor has no oscillation/vibration again. Hence, Song et al. (2019b) modified the DP to be hydrogenated deformable parts (HDPs) by hydrogenation. According to the simulation results at room temperature, HDP performs differently from DP, and could recover to the original shape when the input rotation is removed. In other words, HDP is reusable.

In this work, we propose a new model for the rotation transmission nanosystem with two identical HDPs on the rotor and found that the system could output ideal periodical oscillation at low temperature. The deformation and vibration types of the two HDPs are revealed, and how their coupling affects the system outputting an ideal oscillation is analyzed. The way to control the amplitude of the axial oscillation/vibration is given for potential application in nano on–off as the memory or chemical sensor if the free end of the rotor is functionalized.

2. Model and methodology

2.1. Model

Figure 1 shows a model of the composite nanosystem with two identical HDPs. The system contains three stators and three rotors connected by two HDPs. The sizes of the system except for the HDPs keep unchanged. The details of the parameters are listed in Table 1.

Table 1.

Initial parameters of the system shown in Figure 1. Dimension unit: Å.

Component	Chirality	Radius	Length	Number of rings (include H)	Number of atoms
L-rotor	(6, 6)	4.07	41.82	35	420C
M-rotor	(6, 6)	4.07	61.49	51	612C
R-rotor	(6, 6)	4.07	110.68	92	1092C + 12H
L-stator	(11, 11)	7.46	19.68	19	374C + 44H
M-stator	(11, 11)	7.46	19.68	19	374C + 44H
R-stator	(11, 11)	7.46	24.60	23	462C + 44H
HDP1/HDP2	x × y × z = 24.60 × 24.14 × 35.51			4	336C × 2 (up + down) + 228C × 2 (left + right) + 112H (boundary) + Extra (N_H)

HDP: hydrogenated deformable part.

In Table 1, Extra(N_H = 1) = 16H, Extra(N_H = 2) = 34H, and Extra(N_H = 4) = 68H. For example as shown in Figure 1, “Extra(N_H = 4) = 68H” means that there are 68 hydrogen atoms bonded inside the up and down layers of each HDP when N_H = 4.

For a HDP, the internal surfaces of the upper and lower sides are partly hydrogenated (Autreto et al., 2014; Pei et al., 2010; Reddy and Zhang, 2014; Zhu and Li, 2013; Zhu and Li, 2014). The C-H bonds lead to inward curvature of the HDP because of the asymmetric layout of the local vdW force (Zhu and Li, 2013).

To control the deformation of the two HDPs, the centrifugal force (Cai et al., 2016a; Cai et al., 2016b) is introduced by the input nonzero rotational frequency (ω) on the L-rotor. If the centrifugal force is strong enough, it will pull the atoms on the HDP away from the rotational axis. A higher value of ω means larger centrifugal force, and further away the atoms from the axis. The M-rotor and the R-rotor also have axial translations when the HDPs are deforming. Therefore, the system can transform the input rotation (ω) into an output translation (d) via the deformation of HDPs. Once the HDPs deform periodically, the R-rotor will oscillate (axially vibrate) periodically, that is the system becomes an oscillator.

In Figure 1, HDP1 is between the L-stator and M-stator with an axial distance of Lz1 + (10 + 15) Å, but HDP2 is between the M-stator and R-stator with an axial distance of Lz2 + (26.81 + 20) Å. A longer axial distance between two neighboring stators possibly provides larger radial deformation in the xy-plane for HDPs, and therefore, the eccentric rotation of HDP2 happens easier. Hence, the two identical HDPs may have different dynamic behaviors in rotating, and their coupling effects may influence the curve of d.

2.2. Methodology

2.2.1. Physical properties of HDPs

To describe the radial displacement of the atoms on the HDP, the moment of inertia (MoI) of the HDP is defined, that is

MoI (t) = \sum_{i = 1}^{M} m_{i} {(x_{i}^{2} + y_{i}^{2})}_{t}

(1)

where x_i, y_i, and m_i are the coordinates and mass of the i-th atom in each HDP with a total of M atoms. For a deformed HDP, the value of MoI depends on two factors, that is eccentricity and deformation of the component. If the component has small eccentricity, the variation of MoI will reflect the magnitude of deformation of the HDP.

The eccentricity (e) of the deformed HDP can be defined as

e (t) = \sqrt{{(\frac{\sum_{i = 1}^{M} m_{i} x_{i}}{\sum_{i = 1}^{M} m_{i}})}_{t}^{2} + {(\frac{\sum_{i = 1}^{M} m_{i} y_{i}}{\sum_{i = 1}^{M} m_{i}})}_{t}^{2}}

(2)

2.2.2. MDs simulation

MD simulations are conducted on the software large-scale atomic/molecular massively parallel simulator with the verlet algorithm (Plimpton, 1995), and the interaction among carbon and/or hydrogen atoms is estimated by the adaptive intermolecular reactive empirical bond order potential (Stuart et al., 2000). There are five major steps in each simulation, that is

Step 1

Build the geometric model with two identical HDPs as shown in Figure 1.

Step 2

Reshape the initial system by minimizing the potential energy.

Step 3

Fix the atoms on the three left rings of L-rotor and four rings of each stator, then relax the system for 100 ps in the NVT (constant number of atoms, constant volume and temperature of system) ensemble with the Nose–Hoover thermostat. The time step is 0.0005 ps.

Step 4

Release the left edge of L-rotor, and exert a specified ω on it. The time step is 0.001 ps.

Step 5

Run at least 15,000 ps and record the necessary physical quantities.

For an atomic-level system that is placed in the NVT ensemble, temperature plays a critical role on the dynamic response. In the present work, temperature is controlled at 8 K.

2.2.3. Fast Fourier transform

The output curves of Lz1, Lz2, and d, which depend on the deformation types of HDPs, may have obvious fluctuation during rotating. To accurately describe the fluctuation, the eigen frequencies of these curves are analyzed by fast Fourier transform (FFT) (Luo and Bao, 2019; Mohlenkamp, 1999) on the following equation

K (t) = \sum_{i = 1}^{I} A_{i} \cos (2 π \cdot Ω_{i} \cdot t + ϕ_{i})

(3)

where K can be Lz1, Lz2, or d, A_i is the amplitude of the i-th mode, Ω_i is the i-th eigen frequency, and ϕ_i is the corresponding phase angle. In this study, FFT operation focuses on the final 3000 ps in each curve. The first three orders of vibration modes and frequencies will be obtained.

3. Results and discussion

3.1. Dynamic behavior for the system with N_H = 1 at 8 K

The evolutions of MoI and e of HDPs with N_H = 1 are shown in Figure 2. At 20 GHz, both MoI and e have low mean values (MVs) and weak fluctuation. The two HDPs deform slightly (Figure 3(a)). When ω becomes higher, both the MV and standard deviation (SD) of either MoI or e become higher. However, the system collapses within 0.2 ns at 70 GHz. If the system is in stable state, a higher MV of MoI means more atoms on the HDP moving away from the z-axis because of stronger centrifugal force, whereas the SD indicates the variation of the shape of HDP as shown in Figure 3(b)–(e).

The fluctuating frequency of MoI means that the HDP shape varies periodically with the same frequency. This kind of variation is called breathing vibration, for example HDP1 in Figure 3(c) with the label of “0 → O → 0”. Obviously, breathing vibration of HDP leads to periodic variation of Lz1 or Lz2.

In Figure 3(c), we discovered that the two tubes connected with HDP2 were not concentric but the shape of the HDP becomes asymmetric. It indicates the shear vibration (along the radial direction) of the HDP, which is also obvious at 71 ps in Figure 3(e).

The third type of deformation of the HDP is illustrated in Figure 3(d), that is axial bending deformation. The HDP shape is not symmetric either. But, the two tubes connected with the HDP keep concentric. When two HDPs have the same bending direction, d varies (either increases or decreases) steeply. Otherwise d varies slightly.

As shown in 63 ps and 95 ps in Figure 3(e), the final type is axial twist deformation and can also lead to asymmetric configurations of HDPs. Generally, at higher ω, the HDPs are subjected to stronger torque moment that is caused by greater friction from stators (Zhu et al., 2008). When the difference between the frictional forces from the neighbor stators is great, the HDP deforms in this style.

The increments of d, Lz1, and Lz2 satisfy the following inequation

Δ d - (Δ L z 1 + Δ L z 2) \geq 0

(4)

where, ΔLz1 and ΔLz2 depend on the deformations of HDP1 and HDP2, respectively. Variation of d is mainly determined by the coupling deformation of the two HDPs. The difference of the left side of inequation (4) is caused by the transverse deformation of the M-rotor and R-rotor when the axial deformation of the M-rotor and the R-rotor are ignored.

Among the four deformations, only shear deformation directly improves eccentricity of the system. Hence, one can find that the eccentricity of HDP2 at 50 GHz is much larger than that of HDP1 (Figure 2(b)). Structurally, the two HDPs are constrained in two pairs of stators with different axial distances, and therefore, their vibrations may be asynchronous or having different fluctuation amplitudes.

To demonstrate the output axial translations of the system, the evolutions of d, Lz1, and Lz2 are recorded in Figure 4. For the two identical HDPs, their axial vibrations are different and influenced by ω.

At 20 GHz, the amplitudes of Lz1, Lz2, and d are no higher than ∼0.1 nm (Figure 4(a)). It means that the R-rotor has no axial vibration. When ω = 30 or 40 GHz, the fluctuation amplitude of d becomes ∼0.5 nm and ∼0.75 nm, respectively. Meanwhile, d fluctuates periodically, which indicates that the axial vibration of the R-rotor is periodic and the amplitude approximately keeps constant. The right end of the R-rotor can work as a switch or oscillator. At 50 GHz, the fluctuation amplitude of d is higher than that at 40 GHz. But this periodic vibration cannot keep stable after 12 ns because of serious eccentric vibration of HDP2 (Figures 2(b) and 3(c)).

At 60 GHz, the amplitude of d becomes lower and ‘‘beat’’ phenomenon happens. In Figure 4(e), the variations of Lz1 and Lz2 between 12 and 12.1 ns are complementary because of the opposite axial bending directions of HDPs. When the value of d is near a peak, the two HDPs have synchronous breathing and axial bending vibration with slight difference in their phase angles. Oppositely, when near a trough, the vibrations of the two HDPs are synchronous but have obvious difference in their phase angles.

3.2. Effect of N_H on axial translations at 8 K

Figure 5 gives the evolutions of d, Lz1, and Lz2 of the systems with different N_H at 8 K. When N_H = 2, the amplitude of the d curve is higher than 0.5 nm and keeps stable when 30 GHz ≤ ω ≤ 40 GHz. At either 20 or 50 GHz, the amplitude of d becomes lower. In particular, at 60 GHz, the amplitude becomes less than 0.1 nm, that is the right end of the R-rotor has no obvious axial oscillation. The system collapses at 70 GHz. When N_H = 4, the amplitude of d is near zero at ω ≤ 30 GHz. At 40 GHz, the two HDPs are opened, and the R-rotor has more than 1 nm of axial displacement during rotating. The amplitude of d increases with ω when 40 GHz ≤ ω ≤ 60 GHz. For example d varies between 5.06 nm and 6.80 nm at 60 GHz, which assures the system working as a switch or oscillator.

From Table 2, one can conclude that N_H schemes have a significant influence on the system. The MV of d decreases with the increase of ω. When ω keeps constant, the system with the higher value of N_H always has the higher value of d because of stronger bending force at the hydrogenated areas of HDPs, and therefore, the axial shrinkage of HDPs becomes more difficult. As N_H is specified, there must exist an ω at which d has maximal amplitude.

Table 2.

Statistics of d with different N_H at 8 K (MV ± SD between 12 ns and 15 ns in the curves in Figures 4 and 5).

Hydrogenation	ω = 20GHz	ω = 30GHz	ω = 40GHz	ω = 50GHz	ω = 60GHz
N_H = 1	7.38(MV) ± 0.05(SD)	6.90 ± 0.3	6.18 ± 0.4	5.36 ± 0.6	4.68 ± 0.3
N_H = 2	7.50 ± 0.04	7.13 ± 0.3	6.56 ± 0.4	5.88 ± 0.2	5.26 ± 0.06
N_H = 4	8.21 ± 0.005	8.21 ± 0.003	7.10 ± 0.1	6.53 ± 0.4	5.94 ± 0.6

MV: mean value; SD: standard deviation.

3.3. Vibration analysis of the system

To quantify the axial vibration of the R-rotor, FFT operation on Lz1, Lz2, and d curves is conducted, and the first three orders of vibration are given in Figure 6 and Table 3.

Table 3.

First three orders of fluctuation frequency (Ω_i) and modified amplitude (A_i) of Lz1, Lz2, and d of the system with different N_H schemes at 8 K.

Ω₁/Ω₂/Ω₃ (GHz)
HDP type	Part	ω = 30 GHz	ω = 40 GHz	ω = 50 GHz	ω = 60 GHz
N_H = 1	Lz1	14.73/29.46/42.48	13.14/26.27/39.41	12.67/26.70/39.37	14.22/28.09/43.00
	Lz2	14.73/29.46/42.48	13.18/26.37/39.55	12.60/26.90/39.16	14.21/28.08/42.98
	d	14.72/29.43/42.78	13.16/26.32/39.49	12.62/26.94/39.22	14.31/28.27/42.93
N_H = 2	Lz1	16.35/32.70/46.71	14.58/29.15/45.08	×	×
	Lz2	16.29/30.49/46.78	14.61/29.21/44.84	×	×
	d	16.34/32.68/46.63	14.58/29.15/43.73	×	×
N_H = 4	Lz1	×	×	15.62/31.22/48.86	16.35/26.36/50.72
	Lz2	×	×	15.66/31.31/49.01	16.28/26.32/50.92
	d	×	×	15.63/31.26/48.93	16.32/26.39/50.69
A₁/A₂/A₃ with unitized A₂
HDP type	Part	ω = 30 GHz	ω = 40 GHz	ω = 50 GHz	ω = 60 GHz
N_H = 1	Lz1	12.35/1.00/0.69	3.42/1.00/0.55	6.86/1.00/0.32	1.90/1.00/0.71
	Lz2	37.67/1.00/5.30	2.72/1.00/1.12	5.56/1.00/0.57	1.40/1.00/1.30
	d	27.20/1.00/0.76	113.62/1.00/10.08	242.24/1.00/5.27	53.33/1.00/9.97
N_H = 2	Lz1	17.53/1.00/2.58	10.53/1.00/0.27	×	×
	Lz2	18.54/1.00/7.11	15.64/1.00/1.11	×	×
	d	31.23/1.00/1.84	32.24/1.00/0.42	×	×
N_H = 4	Lz1	×	×	14.53/1.00/0.63	7.67/1.00/0.49
	Lz2	×	×	33.66/1.00/2.84	14.44/1.00/2.67
	d	×	×	38.77/1.00/0.77	22.26/1.00/0.63

Note: ‘‘×’’ means giving no FFT result because of slight fluctuation.

HDP: hydrogenated deformable part.

For instance, the FFT results for the system with N_H = 1 at 30 GHz are shown in Figure 6(a), which illustrates the first three orders of frequency: 14.7, 29.5 (≈2 × 14.7), and 42.5 (≈3 × 14.7) GHz. According to the amplitudes of the three modes, vibration of the R-rotor is mainly in the first mode. It indicates that the first mode of vibration should be corresponding to the breathing deformation. For the same system rotating at 50 GHz (Figure 6(b)), it also vibrates mainly in the first-order mode, which indicates the breathing deformation of HDPs. But, the first three orders of frequency are lower than those at 30 GHz, respectively.

At 60 GHz, the two HDPs vibrate in the first three modes according to their similar amplitudes (Figure 6(c)). However, the second order mode disappears from Figure 6(c3). The amplitude with respect to the first-order mode is ∼0.1605 nm (=Δd), which is higher than that of Lz1 (ΔLz1 = 0.0975 nm) or Lz2 (ΔLz2 = 0.0699 nm), but relatively lesser than their summation. Accordingly, the first-order phase angles of Lz1, Lz2, and d are 372.4° (=360° + 12.4°), 306.8° (=360° − 53.2°), and 732.2° (=720° + 12.2°), respectively. Hence, Lz1 and Lz2 vary asynchronously because of the difference of their first-order phase angles (=12.4° + 53.2° = 65.6°). Considering the differences of phase angles, inequation (4) is still satisfied, that is Δd = 0.1605 nm > ΔLz1 × cos(12.4° − 12.2°) + ΔLz2 × cos(–53.2° − 12.2°) ≈ 0.1266 nm. The difference is high, which implies that higher order vibration of the two HDPs influences variation of d during rotating. For example the second order frequencies of Lz1 and Lz2 are actually generated by axial bending vibration of the vertical sides of HDPs (Figure 3(c)) because of the inertia effect. The third order vibrations of Lz1 and Lz2 are induced by eccentric rotating of the two HDPs. Coupling of the two HDPs’ vibration leads to slight amplitude of the second order vibration of d, that is the second order vibration mode of the R-rotor disappears when the second order vibration modes of Lz1 and Lz2 offset each other.

In Table 3, the first three orders of frequencies of Lz1, Lz2, and d satisfy the follow equation

Ω_{1} = Ω_{2} / 2 = Ω_{3} / 3

(5)

except that N_H = 4 and ω = 60 GHz.

Correspondingly, in the system with N_H = 1, the amplitudes of Lz1 or Lz2 are different slightly when ω ≥ 40 GHz. But the second (and third) amplitude of d is very low, which implies that the R-rotor vibrates mainly in the first mode. For the system with N_H = 2, the system has large amplitude of axial vibration when ω is between 30 and 40 GHz. Both Lz1 and Lz2 contain three vibration modes with similar amplitudes. But the amplitude of d is mainly determined by the first-order mode, that is the R-rotor vibrates mainly in the axial direction. If the system contains the two HDPs with N_H = 4, the amplitude of d is large, for example >0.5 nm, when ω ≥ 50 GHz. Therefore, to obtain large amplitude axial vibration of the R-rotor at a given ω, the HDPs should vibrate mainly in the first-order mode, and the other modes of vibration should be reduced as much as possible.

4. Conclusions

To realize rotation–oscillation transformation, we propose a new nanoconvertor in this study. The rotor in the convertor has five parts, that is the L-rotor for inputting rotation (ω), the R-rotor for outputting translation (d), and the M-rotor for connecting two identical HDPs. Recoverable deformation of the two HDPs depends on the input rotation and the hydrogenation scheme. Their coupling periodic deformation determines the axial output of the R-rotor at low temperature. Base on numerical results obtained by the MD method, the following conclusions are drawn.

First, the two identical HDPs deform mainly in four types, that is breathing, axial bending, radial shear, and axial twist. The first two types of deformation determine the axial translation of the R-rotor. The latter two may happen with high ω and lead to eccentricity and instability. In a stable system, the latter two asymmetric deformation should be controlled.

Second, when ω is low, the configurations of the two HDPs keep unchanged because of weak centrifugal forces. If ω is too high, the strong centrifugal forces bring in large deformation and obvious eccentricity of the HDPs, which may lead to collapse of the system. Hence, the axial translation of the R-rotor varies only when ω is between [30, 60] GHz in this study.

Third, the two identical HDPs may deform synchronously or not. Once they deform synchronously, the R-rotor can generate ideal oscillation with both stable amplitude and frequency. Otherwise, oscillation may disappear or the beat phenomenon appears. The hydrogenation scheme on HDPs affects the dynamics signals of the system.

In conclusion, to obtain an ideal oscillation with large amplitude from the system rotating at a given frequency, the HDPs in a potentially applicable nanodevice should keep breathing vibration, whereas other forms of vibration should be confined.

Supplemental Material

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was support by National Key Research and Development Plan, China (Grant No.: 2017YFC0405102), National Natural Science Foundation, China (Grant No.: 11372281), and State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China (Grant No.: GZ18111).

ORCID iDs

Bo Song

Kun Cai

Supplemental material

Supplemental material for this article is available online.

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Supplementary Material

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Rotation-induced axial oscillation of a composite nanoconvertor at low temperature

Abstract

Keywords

1. Introduction

2. Model and methodology

2.1. Model

2.2. Methodology

2.2.1. Physical properties of HDPs

2.2.2. MDs simulation

2.2.3. Fast Fourier transform

3. Results and discussion

3.1. Dynamic behavior for the system with NH = 1 at 8 K

3.2. Effect of NH on axial translations at 8 K

3.3. Vibration analysis of the system

4. Conclusions

Supplemental Material

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Supplemental material

References

Supplementary Material

3.1. Dynamic behavior for the system with N_H = 1 at 8 K

3.2. Effect of N_H on axial translations at 8 K