## Abstract

The vibration of tightly helical nanosprings affected by van der Waals (vdW) interactions is investigated based on continuum modelling. Explicit solutions are derived to clarify the influence of initial pitch, stiffness and the number of nanosprings on the period, frequency and amplitude of the vibration. Unlike classic linear/nonlinear springs, the waveform of the vibration is always asymmetric for tightly helical nanosprings due to the asymmetry of vdW attraction and repulsion. The at most three equilibrium positions for the nanosprings strongly depend on the deformation due to competition between the vdW interactions and the elastic energy of the nanosprings. This study provides physical insights into the origin of the novel dynamic properties of such nanosprings.

## 1. Introduction

Self-assembled helical structures have been extensively detected in one-dimensional nanowires (e.g. helical carbon nanotubes (CNTs), silicon nanowires, helical graphene ribbons, etc.) [1–3] and various protein molecules [4,5]. Owing to their unique helical structures, such structures exhibit some special electronic [6,7], mechanical [8,9] or magnetic properties [10,11] and possess significant potential to make resonators, nanosprings, electromagnetic wave absorbers, etc. A fundamental understanding of their static and dynamic mechanical properties is of major importance for designing new nanoelectromechanical devices, biomaterials, microelectronics and microsystems. The static force–displacement relationship [12–15] and dynamic response [16,17] of such helical nanowires and helical CNTs (which are defined as nanosprings here) have been studied based on molecular dynamics (MD) simulations, first-principles calculations, continuum modelling and experiments in previous work. Alexandre & Douglas [18] derived Young's modulus and Poisson's ratio of the nanosprings from the Kirchhoff rod model based on some experimental data. Zhang & Zhao [9] reported the mechanical stiffness of a single nanospring with square and round configurations using a finite-element method, in which the significant influence of the configuration and the size of the section on the spring stiffness is discussed in detail.

However, few of the theoretical and numerical models of the nanosprings considered the van der Waals (vdW) interactions, which would dominate their dynamic properties at the nanoscale. For example, Ru [19] studied the infinitesimal buckling of a double-walled CNT under axial compression and found that the critical axial strain is dependent on the vdW interactions between the inner and outer tubes. Bartell [20] found that the bond angles in ethylene derivatives could be affected by the intramolecular vdW forces. Recently, softly and tightly helical structures have been observed in some experiments [21] and have shown a pronounced longitudinal periodicity of 1–1.2 nm, in which the minimum value of the helix pitch *h* between the intertube walls can be up to 3.5 Å [21]. In this case, the vdW interactions between the adjacent helices would significantly affect the mechanical properties of tight nanosprings. Subsequently, the static nonlinear responses of tightly defected helical CNTs under tension have been derived by a molecular mechanics model [12], MD simulations [22] and first-principles calculations [23]. The results showed a strong nonlinear force–displacement relation even at the beginning of the deformation because of the vdW interactions. In view of their asymmetric force–displacement relation and tiny size, tightly helical nanosprings exhibit some unique properties and functions. They hold promise for diverse applications in nanodevices and biological areas as biosensors, biological force probes and functional elements in nanoelectronics and nanoelectromechanical systems [24–26]. However, the influence of vdW interactions on the dynamic properties of tight nanosprings is still not clear and should be further clarified.

In this paper, the vibration of tight nanosprings affected by vdW interactions is studied by continuum modelling. The influence of pitch, stiffness and the number of the nanosprings on the period and amplitude of the vibration is clarified in detail. The established analytical solutions should be of great help for understanding the novel dynamic properties of nanosprings in their potential applications in nanoelectromechanical devices.

## 2. The continuum model of the vibration of nanosprings based on the van der Waals interactions

### (a) The cohesive energy between two parallel nanowires (and two parallel carbon nanotubes)

To analyse the vibration of nanosprings, sketches of single/multiple nanospring systems are shown in figures 1 and 2, respectively. From figure 1*a,b*, the equilibrium equation of the single-nanospring system with vdW interactions is described as follows:
*h*_{0} and *h*_{1} are the pitches (or the equilibrium positions) of the nanospring without/with vdW interactions (figure 1*a*,*b*), respectively; *k* is Hooke's stiffness of the nanospring and *F*_{vdW} (*h*_{1}) is the vdW force between adjacent helices of the nanospring. Note that the vdW force between every other helix is neglected (see §3a) because the distance is greater than 9 Å.

The energy of the vdW interactions can be described by the 6–12 LJ potential [24,25] of
*d* is the distance between the interacting atoms and *σ* is a parameter that is determined by the equilibrium distance.

The cohesive energy per unit length between two parallel nanowires can expressed as [27]
*ρ*_{1} is the nanowire density (which represents the number of atoms per unit length), *d*^{2} = *x*^{2} + *h*^{2} and *h* is the distance between the two parallel nanowires.

Substituting equation (2.2) into equation (2.3) gives
*S* is the half-circle length of the spring.

Therefore, the vdW force between two parallel nanowires is obtained
*, σ *= 3.345 Å, *ρ _{l}* = 0.7037 Å

^{−1},

*S*= 100 Å and

*h*is the distance between adjacent helices.

To study the dynamic properties of the tightly helical CNTs (in this paper, it should be stressed that the difference between the softly and tightly helical CNTs is reported to be *h* < 10 Å and *h *≥ 10 Å, respectively) (figure 3), the cohesive energy per unit length between two paralleled CNTs should be given by [27]
*a*_{0} is the complicated function of the radii (*r*_{1} and *r*_{2}) of the two CNTs and *h, F*_{2} and *F*_{5} are the functions of 2(*a*_{0}*r*_{1})/(*r*_{1} + *a*_{0}).

Defining

Substituting equation (2.11) into equation (2.5) gives
*F*_{CNT-vdW} represents the vdW force between two parallel CNTs.

### (b) The vibration of nanosprings based on the van der Waals interactions

In this paper, the nanosprings of helical nanowires and helical CNTs are both analysed and each structure has 10 pitches. The half-circle length of the helical nanowires is 100 Å, while that of the helical (10,10) CNTs is 94.7027 Å [12]. By hanging a mass *m *= 10^{−10} kg (all masses are used as *m *= 10^{−10} kg) on the end of the nanospring, the pitch will be stretched from *h*_{1} to *h*_{2} in order to maintain an equilibrium (figure 1*c*). In this case, the equilibrium equation can be expressed as
*mg* is hung from the nanospring (here *l *= 50 Å in figure 1*d* in this paper). The acceleration of the mass is determined by the second derivative of the displacement *x* (figure 1*d*) with respect to time *t* and can be denoted by the acceleration of

Substituting equation (2.13) into equation (2.14) gives
*n* is the number of nanosprings in the system.

## 3. The analytical results for the vibration of nanosprings

In this section, the effect of different parameters on the vibration of nanosprings is analysed in detail.

### (a) The effect of different parameters on the vibration of helical nanowires

#### (i) The effect of prestress from van der Waals interactions (*h*_{0})

In order to produce a vibration, the static-equilibrated nanospring with a weight *mg* hanging from it (figure 1*c*) is stretched to reach an extension of distance *l* (*l* represents point A in figure 1*d*) and then is released freely for a given *k *= 1 nN nm^{−1}. The amplitudes and periods of the nanospring for different *h*_{0} are derived from equation (2.15), as shown in figure 4*a*. Figure 4*b* shows that the peak value of the displacement *P* (*P* represents the maximum point B (figure 1*d*) of the vibration in figure 4*a*) slightly increases at first with increasing *h*_{0} (2 Å <* h*_{0} < 4 Å), and then it increases sharply in the range 4 Å <* h*_{0} < 8 Å. Finally, it approaches *l* for *h*_{0} > 9 Å. Unlike the classic linear/nonlinear springs, the waveform of the vibration is always asymmetric due to the asymmetry of vdW attraction and repulsion in figure 4*a*. For *h*_{0} < 7.3 Å, *P* is always less than *l* because the total work of the vdW force is negative in the process from A to B (figure 1*d*). On the contrary, the peak value is slight greater than *l* for *h*_{0} > 7.3 Å because the total work of the vdW force is positive in the process from A to B. When *h*_{0} > 7.3 Å, the value of *P* is equal to *l*, which means the total work of the vdW force is zero in the process from A to B. That is to say, the amplitude from O to A is identical to that from O to B in figure 1*d*, while the waveform of the vibration is still asymmetric. When *h*_{0} > 8 Å, the vibration becomes symmetric because the vdW interactions are too small to affect the vibration of the system.

Figure 4*c* shows the distribution of the vibration period *T* and frequency *f* of figure 4*b* with *h*_{0}. The value of *T*(*f*) increases (decreases) sharply for 2 Å <* h*_{0} < 8 Å and then decreases (increases) smoothly for *h*_{0} > 8 Å. When *h*_{0} > 9 Å, the value of *T*(*f*) converges to a constant of 63 µs (1.59 × 10^{4} Hz). The reason is that the vibration becomes symmetric because the effect of the vdW interactions on the vibration is too small and can be neglected for *h*_{0} > 9 Å, which results in a constant of *T *= 63 µs for given different *h*_{0}. That is to say, the nanospring for *h*_{0} > 9 Å is the same with a classic spring, in which the different *h*_{0} (*h*_{0} > 9 Å) has no effect on the *T*(*f*). This indicates that the equivalent stiffnesses of the nanosprings are variable and nonlinear, where the equivalent stiffnesses contain the effect of vdW interactions on the nanosprings. The relationship between the external force *F* and the total displacement *x* of the nanospring can be written as
*k*_{e} = *F*/*x* (in other words, *k*_{e} is the slope of the *F*–*x* curve under small deformation), in which *k*_{e} should be the function of *x* and *h* when the vdW force cannot be neglected [12].

#### (ii) The effect of stiffness *k*

Similar to classic springs, the stiffness *k* of the nanosprings also plays a key role in their vibration. Figure 5*a* shows the waveform of the vibration for different *k*. For small values of *k* in figure 5*b*, the values of *P* are always less than zero and all curves are symmetric for a given equilibrium position. On the contrary, the values of *P* are larger than zero and all curves are asymmetric for large values of *k* in figure 5*c*. To determine the critical value of *k*, figure 5*b* displays the relationship of *k* and *P*, in which the critical value *k *= 0.27 nN nm^{−1}. For *k* < 0.27 nN nm^{−1}, a weight *m* is hung on the nanospring and then the system can remain static, in which the vdW attractive force and the tensile force from the nanospring both balance the system (in this case, the vdW force is far larger than the tensile force of the nanospring).When the nanospring is stretched to reach an extension of *l *=* *5 nm (figure 1*d*) and then is released freely, the vdW attractive force becomes too small and can be ignored. In this case, the weight *m* cannot be balanced at all by the nanospring at this position (*l *=* *5 nm). Thus, the weight *m* moves further down and then reaches a new equilibrium position where the vdW force becomes too small and can be neglected. Afterwards, the vibration will be produced based on this new equilibrium position, in which the waveform of the vibration becomes symmetric due to no vdW interaction. For *k* > 0.27 nN nm^{−1}, the vdW force is always considerable and cannot be neglected in the process of the vibration, which leads to the asymmetric waveform in figure 5*b*. Figure 5*d* shows that *T*(*f*) nonlinearly decreases (increases) with increasing *k*. This indicates that the equivalent stiffnesses of the nanosprings (see equation (3.1)) are also variable and nonlinear.

#### (iii) The effect of the number (*n*) of nanosprings

For a multi-nanospring system (figure 2), the adjacent helices belong to different nanosprings. To clearly analyse its dynamic property, the vdW force between the non-adjacent helices is ignored here. The vdW force between the adjacent helices can be obtained by equation (2.6). The waveform of the vibration for different *n* for a given *h*_{0} = 9 Å (figure 2*a*) is shown in figure 6*a*. Figure 6*b* shows that the peak value of the displacement *P* decreases sharply as *n *≤ 3 and then increases slightly for *n* > 3. For *n *< 3, the total work of the vdW force is negative and decreases with increasing *n*. The possible reason for this is that the effect of the vdW force on the vibration is higher than that of the stiffness of *nk*, so it means that *P* decreases with increasing *n* (*n *≤ 3). On the contrary, the stiffness *nk* of the whole nanospring increases linearly and dominates the dynamic properties, so it means that *P* increases slightly with increasing *n* (*n *≥ 3)*.*

Figure 6*c* shows the period *T* of the vibration increases and the frequency *f* decreases with increasing *n*, which results from the increase in the equivalent stiffness of nanospring.

### (b) The vibration of helical carbon nanotubes

To study the vibration of helical CNTs, a tightly helical (10,10) CNT is chosen as an example. From our previous work [12], the stiffness of the helical (10,10) CNT approaches *k *= 2.09 nN nm^{−1} and the half-circle length approximates 94.7027 Å, when *h *< 10 Å. The amplitudes and periods of the CNT spring for different *h*_{0} are derived from equation (2.15), as shown in figure 7*a*. Figure 7*b* shows the relation between the peak value of the displacement *P* and *h*_{0} (figure 1*a*). Figure 7*c* shows the distribution of the vibration period *T* and frequency *f* with *h*_{0}. The effect of *h*_{0} on the vibration of the CNT spring (figure 7*a*–*c*) is similar to that of the helical nanowire in figure 4*a–c*. The same reason as that given in §3a(i) can be used to explain the similar phenomenon.

However, it is difficult to determine the correct values of *h*_{1} and *h*_{2} (figure 1*b,c*) for a given *k *= 2.09 nN nm^{−1} and *S *= 94.7027 Å. From equation (2.13), the values of *h*_{1} and *h*_{2} can be derived directly. Three coloured lines are shown in figure 7*d*. The red line is the curve from *F *= –*k*(10*h*–10*h*_{0}) (see equation (2.13)) and represents the relationship between *h* and the force without any weight and vdW interactions. The blue line is the curve from *F *= *mg* – *k*(10*h*–10*h*_{0}) (see equation (2.13)) and represents the relationship between *h* and the force with the weight (vdW interactions are not considered). The black line represents the curve of *F*_{vdW}(*h*) (see equation (2.13)) from vdW interactions. When *h*_{0} < 6.68 Å, the red line and the blue line both have one intersection point with the black line in figure 7*d*, in which point 1 represents *h*_{1} and point 2 represents *h*_{2} (figure 1*b*,*c*). However, it should be noted that the red line has two intersection points and the blue line has three intersection points with the black line when *h*_{0} = 6.68 Å. Therefore, we have to carefully determine the correct *h*_{1} and *h*_{2}. In this case, we choose the values of *h* at point 4 and point 7 as *h*_{1} and *h*_{2}, respectively. Although the CNT spring can be balanced at point 3 and point 4, the value of *h* at point 4 is closer to *h*_{0} = 6.68 Å than that at point 3. To be further balanced by the vdW interactions for *h*_{0} = 6.68 Å, the pitch *h* has to reach the position of point 4 first. Therefore, the value of *h* at point 4 should be chosen as *h*_{1}. When *h*_{1} is determined, the value of *h* at point 7 should be equal to *h*_{2} because *h*_{2} is always larger than *h*_{1} (figure 1*c*). When *h*_{0} > 6.68 Å, we only need to choose the values of *h* at the last two intersection points 8 and 9 as *h*_{1} and *h*_{2}, respectively.

## 4. Discussion

The present analytical model can be used to correctly characterize the dynamic properties of helical nanowires and the different helical CNTs. Furthermore, it can also be applied to describe the vibration of other configurations of nanosprings such as helical multi-walled CNTs, in which the vdW force between two different multi-walled CNTs can be obtained from previous work [27]. It should be stressed that helical CNTs are only considered as an elastic material in this paper. Unlike classic linear/nonlinear springs, the at most three equilibrium positions for nanosprings strongly depend on the deformation due to competition between the vdW interaction and the elastic energy of the nanosprings. It should also be stressed that buckling of the helical nanowires and CNTs is neglected in this study.

## 5. Conclusion

The dynamic properties of nanosprings affected by vdW interactions are investigated based on continuum modelling. The influence of pitch, stiffness and the number of the nanosprings on the period and amplitude of the vibration is clarified in detail. Unlike classic linear/nonlinear springs, the waveform of the vibration is always asymmetric due to the asymmetry of vdW attraction and repulsion. The conclusion can be summarized as follows:

(1) For a given stiffness

*k*= 1 nN nm^{−1}of the nanosprings and a given mass*m*= 10^{−10}kg, the maximum amplitude of the vibration mainly increases with increasing equilibrium distance*h*_{0}(*h*_{0}< 8 Å) and the vibration is always asymmetric, while it converges to a constant for*h*_{0}> 9 Å and the vibration tends to be symmetric. The value of the period*T*(frequency*f*) increases (decreases) sharply for 2 Å <*h*_{0}< 8 Å and then decreases (increases) smoothly for*h*_{0}*h*_{0}> 9 Å, the value of*T*(*f*) converges to a constant.(2) For a given mass

*m*= 10^{−10}kg and a given*h*_{0}= 3.556 Å, the maximum amplitude of the vibration increases with increasing the stiffness (*k*) of the nanosprings (*k*< 0.27 nN nm^{−1}), while it converges to a constant. The period*T*(frequency*f*) nonlinearly decreases (increases) with increasing*k*.(3) For a given

*k*= 1 nN nm^{−1}and a given*h*_{0}= 3.556 Å, the period*T*(frequency*f*) nonlinearly increases (decreases) with increasing number of nanosprings*n*.(4) Unlike classic linear/nonlinear springs, the atmost three equilibrium positions for tightly helical CNTs strongly depend on the deformation due to the competition between their vdW interactions and their elastic energy.

## Data accessibility

The datasets supporting this article have been uploaded as part of the electronic supplementary material.

## Authors' contributions

J.Z. and S.B. performed all the calculations and interpreted the initial results. J.Z. and P.Y. interpreted the final results and edited the final manuscript. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

We gratefully acknowledge support by the National Natural Science Foundation of China (grant nos. 11572140 and 11302084), the Programs of Innovation and Entrepreneurship of Jiangsu Province, the Fundamental Research Funds for the Central Universities (grant nos. JUSRP11529 and JG2015059), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (NUAA) (grant no. MCMS-0416G01) and the Open Fund of Key Laboratory for Intelligent Nano Materials and Devices of the Ministry of Education (NUAA) (grant no. INMD-2015M01).

## Acknowledgements

We gratefully acknowledge support from the ‘Thousand Youth Talents Plan’.

## Footnotes

Electronic supplementary material is available online at http://dx.doi.org/10.6084/m9.figshare.c.3491568.

- Received April 3, 2016.
- Accepted September 8, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.