# Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basis

#### Maxwell’s Relations

## 3. Results and Discussions

^{8}$\mathrm{A}/\mathrm{m}$. At this point, the effect of ${H}_{x}$ dominates the total stiffness of both cases (with and without mass), and further increases in ${H}_{x}$ result in a frequency shift of zero. This observation applies to both the simply supported and the bridged cases. The corresponding diagram converges for all three cases when the magnetic intensity parameter reaches a sufficiently high value.

_{x}= 0), the results in this figure align with the findings of Xu et al. [58]. According to their work, in a DWCNT with a greater length, a smaller frequency is observed. This correlation is based on Equation (24), where the frequency shift and frequency are directly related. Furthermore, from a physical perspective, this implies that a larger outer tube diameter in a system with a constant inner tube diameter is more effective in sensing nano-mass, as it exhibits a more significant frequency shift. Similarly, the same effect can be observed in a system with a fixed outer tube diameter where the inner tube has a smaller diameter.

#### This Study’s Limitations and Potential Future Research Areas

^{−24}. In comparison to the sensitivity values of some of the other models in the literature, this value is bigger than some and smaller than others. For example, Cho et al. [62] introduced a sensitivity on the order of 10

^{−18}for an SWCNT model with a diameter of 2.7 nm and a length of 55 nm. In other works, carried out by Lee et al. [63] for SWCNT models with a diameter of 1.1 nm and lengths of 4.1, 5.6, and 8 nm, the sensitivity was on the order of 10

^{−21}. Chaste et al. [64], using a 10 nm × 10 nm single-layer graphene sheet [65], increased the sensitivity to 10

^{−27}. These differences were due to the use of different control parameters, such as geometry, environmental control parameters, temperature, and so on, as mentioned in the introduction. Some of these control parameters are among the limitations of the current model and should be considered when the results of the current model are compared to those in the literature. For example, in the current model, we did not consider the ambient temperature’s effect on the nanoresonator’s performance, which effectively adds external noise to the system. Meanwhile, the mentioned concept of adding the magnetic field effect to a nano-mass resonator to increase its sensitivity can be extended to other models, such as SWCNTs and nano-beams, due to the similarity of their modeling formulations to those of the current model.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The effect of the magnetic intensity on the frequency shift for various values of the elastic foundation modulus for (

**a**) cantilevered, (

**b**) simply supported, and (

**c**) bridged nanotubes (μ = 1 nm).

**Figure 3.**The effect of the magnetic intensity on the frequency shift for various values of the aspect ratio for (

**a**) cantilevered, (

**b**) simply supported, and (

**c**) bridged nanotubes (μ = 1 nm).

**Figure 4.**The effect of the magnetic intensity on the frequency shift for various values of length for (

**a**) cantilevered, (

**b**) simply supported, and (

**c**) bridged nanotubes (μ = 1 nm, k = 5 × 10

^{8}, ${d}_{2}$ = 0.7 nm, $\frac{{d}_{2}}{{d}_{1}}=2$).

**Figure 5.**The influence of the longitudinal magnetic field on the sensitivity of DWCNT resonators with (

**a**) cantilevered, (

**b**) simply supported, and (

**c**) bridged nanotubes (μ = 1 nm, k = 5 × 10

^{8}).

**Figure 6.**The influence of the longitudinal magnetic field on the sensitivity of DWCNT resonators for the cantilevered, simply supported, and bridged nanotubes (k = 5 × 10

^{8}, ${H}_{x}$ = 1 × 10

^{9}A/m).

**Table 1.**Natural frequencies (×${10}^{12}\mathrm{H}\mathrm{z}$) of DWCNTs with different boundary conditions (${d}_{2}=1.4\mathrm{n}\mathrm{m},{d}_{1}=0.7\mathrm{n}\mathrm{m},\rho =2300\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3})$.

Elishakoff et al. [55] | Xu et al. [54] | Present Study | |
---|---|---|---|

simply supported | 0.46830 | 0.46 | 0.47 |

cantilevered | 0.202 | 0.17 | 0.17 |

bridged | 1.0515 | 1.06 | 1.085 |

**Table 2.**First three natural frequencies ($\times {10}^{12}\mathrm{H}\mathrm{z}$) of embedded hinged–hinged DWCNTs with different stiffness coefficients ($k$) $(\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{k}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}:h=34\mathrm{n}\mathrm{m},{d}_{1}=0.684\mathrm{n}\mathrm{m},\rho =2300\frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^{3}})$.

Mode | $\mathit{k}={10}^{8}\mathbf{N}/{\mathbf{m}}^{2}$ | $\mathit{k}={10}^{9}\mathbf{N}/{\mathbf{m}}^{2}$ | ||
---|---|---|---|---|

Khosrozadeh et al. [56] | Present Study | Khosrozadeh et al. [56] | Present Study | |

1 | 0.5113 | 0.4997 | 0.6665 | 0.6539 |

2 | 1.9328 | 1.9127 | - | 1.9571 |

3 | 4.1670 | 4.1817 | - | 4.2040 |

**Table 3.**Natural frequencies (×${10}^{11}\mathrm{H}\mathrm{z}$) of simply supported DWCNTs with different non-local parameters when outer tube is fixed ${(w}_{2}=0)({d}_{2}=1.4\mathrm{n}\mathrm{m},{d}_{1}=0.7\mathrm{n}\mathrm{m},\rho =2300\frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^{3}})$.

c ($\mathbf{T}\mathbf{P}\mathbf{a}$) | ${\mathit{e}}_{0}\mathit{a}$ (nm) | Marmu et al. [57] | Present Study |
---|---|---|---|

0 | 0 | 1.6625 | 1.6625 |

0.5 | 1.6592 | 1.6592 | |

1 | 1.6496 | 1.6496 | |

0.0694 | 0 | 4.4304 | 4.4304 |

0.5 | 4.4305 | 4.4305 | |

1 | 4.4846 | 4.4846 |

Symbol | Description | Numerical Value |
---|---|---|

$l$ | Length (nm) | 5.55 |

$\rho $ | Density (kg/m^{3}) | 2300 |

$E$ | Young’s modulus (N/m^{2}) | 1 × 10^{12} |

${d}_{2}$ | Outside diameter (nm) | 1.4 |

${d}_{1}$ | Inside diameter (nm) | 0.7 |

$h$ | Thickness (nm) | 0.34 |

${A}_{2}$ | Outside area (nm^{2}) | 1.4954 |

${A}_{1}$ | Inside area (nm^{2}) | 0.7477 |

${I}_{1},{I}_{2}$ | Moments of inertia (nm^{4}) | 0.0566, 0.3879 |

${H}_{x}$ | Magnetic intensity (A/m) | 1 × 10^{8} |

$k$ | Elastic foundation modulus (N/m^{2}) | 5 × 10^{8} |

$\zeta $ | Position of attached mass (nm) | - |

$M$ | Attached mass (g) | - |

${d}_{2}/{d}_{1}$ | Aspect ratio | - |

$\mu $ | Non-local parameter (nm) | 0–1 |

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**MDPI and ACS Style**

Ahmadi, H.R.; Rahimi, Z.; Sumelka, W.
Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field. *Appl. Sci.* **2024**, *14*, 3010.
https://doi.org/10.3390/app14073010

**AMA Style**

Ahmadi HR, Rahimi Z, Sumelka W.
Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field. *Applied Sciences*. 2024; 14(7):3010.
https://doi.org/10.3390/app14073010

**Chicago/Turabian Style**

Ahmadi, Hamid Reza, Zaher Rahimi, and Wojciech Sumelka.
2024. "Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field" *Applied Sciences* 14, no. 7: 3010.
https://doi.org/10.3390/app14073010