Triple-Walled Carbon Nanotube for Strain Sensing

*Minesh Patel, Anand Joshi and Ajay Patel* 

#### **Abstract**

This manuscript deals with the triple-walled carbon nanotube used for strain sensing at the nanolevel. The strain sensing occurs because of the shift of resonant frequency due to external loading on the pre-strained resonator. In the present work, the strain effect on triple-walled carbon nanotubes of different lengths and diameters are investigated under various conditions. The sensitivity of these sensors are improved with the decrease of tube length, while the key recurrence is upgraded with the expansion of tube distance across, in the case of axial strain sensing. The effect on the frequency is not significant with variation in the middle and outer wall length. The triple-walled carbon nanotube of the same length has higher frequency for bridge boundary condition when compared to the cantilevered boundary condition, so bridge boundary condition triple-walled carbon nanotube is more favorable for strain sensing.

**Keywords:** triple-walled carbon nanotubes, finite element method, strain, vibration, frequency

#### **1. Introduction**

Strain sensors have a broad range of applications in engineering, manufacturing and for measuring various quantities, such as stress, pressure, torque and vibration. Regardless of its exceptional characteristics, it has many limitations that consider traditional stress sensors like semiconductor and metallic stress gauge, measurement limits, low sensitivity, difficulties embedded in material structures, reduced fatigue life and sensitivity to environmental conditions. These limits have increased the demands for using innovative smart materials, e.g. silicon [1], nanoparticles [2–4], nanowires [5, 6], graphene [7–9] and carbon nanotubes (CNTs) [10–15].

Carbon nanotubes offer limitless opportunities as high frequency mechanical resonators for a number of applications. Nanotubes are ultralight, which is ideal for ultralow mass detection and ultrasensitive force detection [16, 17]. CNTs are potential contenders for a nanobeam-based strain sensor because of their optimal surface and vast viewpoint proportion. What's more, inferable from their high solidnessto-weight proportion, the ultra-affectability of CNTs additionally prompts an express data about strain by estimating the recurrence movement [18]. Li and Chou [19] have figured the key recurrence movement of CNTs with connected strain and weight in view of a continuum auxiliary mechanics demonstration. Their outcomes demonstrate that the vibrational recurrence diminishes with expanding pivotal strain in a SWCNT, which contradicts the experiments of Sazonova et al.

*Triple-Walled Carbon Nanotube for Strain Sensing DOI: http://dx.doi.org/10.5772/intechopen.81083* 

[20]. Truly, as indicated by the hypothesis of continuum vibration [21], when a strain is connected in a beam or string, one ought to expect a significant increase of vibrational recurrence. Patel and Joshi [22–27] have investigated the influence of resonant frequency of double-walled carbon nanotubes with mass sensing, surface deviation, atomic vacancy defect and pinhole defect. Patel et al. [28] have analyzed variation of mass attached on external walled of TWCNT.

#### **2. Modeling of triple-walled carbon nanotubes**

 In this study, the continuum mechanics method is employed. For a TWCNT, a shell model is assumed such that each of the nested tubes is a specific shell and coupled with adjacent tubes through Van der Waals interaction. Strain analysis of TWCNTs is carried out using spring elements which are modeled as an interlayer interaction of the nanotube to designate the Van der Waals potential. The current model is simulated by taking into account the effective wall thickness of tubes and Van der Waals interaction between various carbon nanotubes to find the results. A cantilever boundary condition is considered for the simulation purpose.

#### **3. Finite element procedure**

Initial strain i.e. strain which is independent of stress, may be due to several causes, shrinkage, crystal growth, or most frequently temperature changes will result in an initial strain vector. All the effects of initial strains or stresses are incorporated in the nodal force vectors or nodal load vectors and it does not change the stiffness matrix.

The total potential energy is sum of energy due to bond expanse interaction, energy due to bending, the energy due to dihedral angle torsion, energy due to out-of-plane torsion and energy due to non-bonded Van der Waals interaction. It is shown in Eq. (1).

$$\text{U}\begin{array}{rcl}\text{U}\_{\text{Total}} &=& \sum \text{U}\_{r} + \sum \text{U}\_{\theta} + \sum \text{U}\_{\mathcal{Q}} + \sum \text{U}\_{\theta} + \sum \text{U}\_{\text{vdw}} \\\end{array} \tag{1}$$

For bond systems, the key contributions to the whole static energy come from the first four expressions of Eq. (1). Under the statement of small deformation, the harmonic approximation is adequate for describing the energy [29]. The terminologies for each form of energy is defined by,

$$\left\| U\_r \right\| = \left\| \frac{1}{2} k\_r \left( r - r\_0 \right)^2 \right\| = \left\| \frac{1}{2} k\_r \left( \Delta r \right) \right\|^2 \tag{2}$$

$$\mathcal{U}\_{\theta} = \frac{1}{2}k\_0 \{\theta - \theta\_0\}^2 = \frac{1}{2}k\_{\theta} \{\Delta\theta\}^2 \tag{3}$$

$$\mathbf{U}\_{\tau} = \mathbf{U}\_{\mathcal{Q}} + \mathbf{U}\_{\mathcal{w}} = \frac{1}{2} k\_{\tau} \left(\boldsymbol{\Delta}\boldsymbol{\mathcal{Q}}\right)^{2} \tag{4}$$

where, kr, kθ, and kτ are the bond enlarging, bond twisting and torsional resistance force constants, respectively, while Δr, Δθ and Δ∅ represent bond enlarging or stretching increment, bond angle variant and angle deviation of bond twisting respectively.

*Proceedings of the 4th International Conference on Innovations in Automation...* 

The VdW potential is described the interlayer interaction. In the present paper, a number of concentric elastic cylindrical layers of SWCNTs are treated as TWCNTs.

$$U(R) \ = \ 4e \left[ \left( \frac{\rho}{R} \right)^{12} - \left( \frac{\rho}{R} \right)^{6} \right] \tag{5}$$

where R = atomic distance and ε = 3.8655 × 10<sup>−</sup>13 N nm and ρ = 0.34 nm, respectively. In order to calculate the vibration explanation of TWCNTs, equations are developed that describe the dynamics equilibrium of the finite element model. The group or element equation created by the global stiffness and mass matrices can be gathered. The equation becomes,

$$\mathcal{L}\_1(d\_2 - d\_1) \ = \mathcal{K}\_1 d\_1 + \mathcal{M}\_1 \check{d}\_1 \tag{6}$$

$$-c\_1(d\_2 - d\_1) + c\_2(d\_3 - d\_2) \ = K\_2 d\_2 + M\_2 \check{d}\_2 \tag{7}$$

$$
\sigma\_3(d\_3 - d\_2) \ = \, \, K\_3 d\_3 \, + \, M\_3 \check{d}\_3 \tag{8}
$$

where Mi are mass matrices and ki are stiffness matrices. d1 = Y1eiwt, d2 = Y2eiwt , d3 = Y3eiwt .

Further, the same can be represented as:

$$
\begin{pmatrix} c\_1 + K\_1 & -c\_1 & 0 \\ -c\_1 & c\_1 + c\_2 + K\_2 & -c\_2 \\ 0 & -c\_2 & c\_2 + K\_3 \end{pmatrix} \begin{pmatrix} Y1 \\ Y2 \\ Y3 \end{pmatrix} - \alpha\_n^2 \begin{pmatrix} M\_1 & 0 & 0 \\ 0 & M\_2 & 0 \\ 0 & 0 & M\_3 \end{pmatrix} = \begin{array}{c} 0 \end{array} \tag{9}
$$

 **Figures 1** and **2** show schematic diagrams of cantilever and bridged TWCNTs, which represent a finite element model of TWCNT with the spring element.

 To model the layer to layer interactions and describe the Van der Waals (VdW) potentials between carbon atoms on different layers using ANSYS, a spring element COMBIN40 is utilized. COMBIN40 is an assemblage of a spring-slider and damper in parallel, joined to a gap in series. The spring stiffness coefficient of Eqs. (2)–(4) are taken to be equal to kr = 6.52 × 10<sup>−</sup><sup>7</sup> N nm<sup>−</sup><sup>1</sup> , kθ = 8.76 × 10<sup>−</sup>10 N nm rad−<sup>2</sup> and ks = 2.78 × 10<sup>−</sup>10 N nm rad−<sup>2</sup> [30].

**Figure 1.**  *Cantilevered TWCNTs.* 

*Triple-Walled Carbon Nanotube for Strain Sensing DOI: http://dx.doi.org/10.5772/intechopen.81083* 

**Figure 2.**  *Bridged TWCNTs.* 

#### **4. Results**

 The models developed for the analysis has fixed free and bridged boundary conditions along with variation in initial strain and tube lengths (L3/L2/L1 = 1, L = 15,10,5). The inner tube radius R1 = 0.65 nm, middle tube radius R2 = 1.0 nm and outer tube radius R3 = 1.35 nm. In **Figure 4**, the fundamental frequencies of TWCNTs as function of connected strain at nano level are shown. The fundamental frequencies as reported in **Table 1** are in gigahertz for the nanotube lengths. The shorter nanotubes often ascend to higher crucial frequencies. The principal frequency increases when the TWCNTs are subjected to a connected strain.


#### **Table 1.**

*Frequency variation for cantilever CNTs with length ratio L3/L2/L1 = 1.* 


#### **Table 2.**

*Fundamental frequency for fixed-free and fixed-fixed TWCNTs with variable strain for dissimilar length ratio (outer, middle and inner tube radius respectively R3 = 1.35, R2 = 1.0, R1 = 0.65).* 

**Figure 3.** 

*Effect of applied initial axial strains.* 

**Figure 4.** 

*Variation of fundamental frequency of TWCNT for various length ratio.* 

**Figure 5.**  *Change in the frequency with different strains.* 

*Triple-Walled Carbon Nanotube for Strain Sensing DOI: http://dx.doi.org/10.5772/intechopen.81083* 

**Figure 6.** 

*Variation of fundamental frequency of Fixed-fixed & Fixed free TWCNTs for same length of tube.* 


#### **Table 3.**

*Frequency for fixed-free and fixed-fixed TWCNTs with dissimilar pre strain for L3/L2/L1 = 1 (Outer, Middle and Inner tube radius respectively R3 = 0.6102, R2 = 0.4746, R1 = 0.3390 nm).* 

 A function of initial strain versus fundamental frequency with the variation in length ratio is shown in **Table 2**. For equal length of all tubes of TWCNTs, the impact of the length of nanotube in terms of change in strain is greater in shorter TWCNTs. The same has been depicted in **Figure 3**. The frequency of variable length TWCNT is higher compared to the equal length TWCNT. However, the variation in length ratio does not show any significant changes in frequency as shown in **Figure 4**.

**Figure 5** shows the fundamental frequency versus strain. **Figure 6** depicts the variations in frequencies for different configurations with the same length of nanotube.

 The model developed was analyzed for fixed free TWCNTs and bridged TWCNTs with the variation in initial strain and lengths L3/L2/L1 = 1, L = 15. The inner radius R1 = 0.3390 nm of inner tube, middle tube radius R2 = 0.4746 nm and outer tube radius R3 = 0.6102 nm where R1, R2 and R3 are the inner, middle and outer tube radius of the center line. In **Table 3** cantilever and bridge TWCNTs frequency variation versus initial strain is shown.

### **5. Conclusion**

1. Strain sensitivity of TWCNT is enhanced with reduction in tube length for strain applied along the length.

