# Comprehensive Analytical Approximations of the Pull-In Characteristics of an Electrostatically Actuated Nanobeam under the Influences of Intermolecular Forces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Modeling

^{3})/12 and with an initial gap size between the flexible beam and its respective grounded actuating electrode of d, as shown in Figure 1. The governing differential static equation of motion of the nanobeam resulting bending deflection $w\left(x\right)$ under the influence of the electrostatic, as well as the intermolecular force can be written as follows [10]:

^{−12}F/m is the air permittivity, ${V}_{DC}^{}$ is the applied actuating DC voltage across the nanobeam and its stationary substrate, H is the Hamaker constant defined for a van der Waals (vdW) body-body interaction with values in the range of 10

^{−19}J [21], c = 3 × 10

^{8}m/s is the speed of light and ħ = 1.055 × 10

^{−34}J

^{−s}is Planck’s constant.

#### 2.1. First Case: van der Waals Force Only (${F}_{vdW}\ne 0$ and ${F}_{Casimir}=0$)

_{i}are unknown constant coefficients and ${\mathsf{\Psi}}_{i}\left(x\right)$ denote the beam modal functions satisfying the doubly-clamped boundary conditions at x = 0 and x = 1. Substituting Equation (7) into Equation (6), we get the following discretized equation:

_{1}:

#### 2.2. Second Case: Casimir Force Only (${F}_{vdW}=0$ and ${F}_{Casimir}\ne 0$)

_{1}:

_{1}, only one solution in both cases has been found to be real and, hence, as the unique physical solution of the equilibrium deflection of the nanobeam.

## 3. Results and Discussion

#### 3.1. Closed-Form Solutions for the Detachment Length and Its Respective Minimum Gap Size for an Un-Actuated Nanobeam

#### 3.2. Comparison with the Literature Assuming an Un-Actuated Nanobeam

^{−19}nm

^{2}, thickness h = 3.5 nm and width b = 18 nm is assumed.

#### 3.3. Closed-Form Solutions for the Critical Pull-In Voltage for an Actuated Nanobeam

_{1}becomes imaginary and the system becomes unstable, hence undergoing the pull-in structural instability.

#### 3.3.1. First Case: Van der Waals Force Only (${F}_{vdW}\ne 0$ and ${F}_{Casimir}=0$)

**√**Case with the fringing-field effect and without the mid-plane stretching effect:

_{1}, which becomes imaginary. Then, and for each corresponding ${\alpha}_{vdW}^{}$ and ${\alpha}_{ff}^{}$ value, the corresponding threshold pull-in parameter value of $\left({\alpha}_{e}{V}_{pull-in}^{2}\right)$ is prescribed. Accordingly, a 3D table is constructed correlating the values of the pull-in, van der Waals and electric fringing field parameters all together. The table is first suitably curve fitted with a third order polynomial using the MATLAB curve fitting toolbox as shown in Equation (19), with a calculated RMS error of 0.03, and the outcome equation is then platted as shown in Figure 2.

**√**Case with the mid-plane stretching effect and without the fringing-field effect:

#### 3.3.2. Second Case: Casimir Force Only (${F}_{vdW}=0$ and ${F}_{Casimir}\ne 0$)

**√**Case with the fringing-field effect and without the mid-plane stretching effect:

**√**Case with the mid-plane stretching effect and without the fringing-field effect:

#### 3.4. Comparison with the Literature of the Pull-In Parameter in Presence of Van der Waals and Casimir Forces

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Three-dimensional (3D) plot of the variation of the pull-in parameter with the van der Waals and electric fringing-field parameters.

**Figure 3.**3D plot of the variation of the pull-in parameter with the van der Waals and mid-plane stretching parameters.

**Figure 4.**3D plot of the variation of the pull-in parameter with the Casimir and electric fringing-field parameters.

**Figure 5.**3D plot of the variation of the pull-in parameter with the Casimir and mid-plane stretching parameters.

Parameter | Equation (14) | Results of [11] | Results of [12] |
---|---|---|---|

${\alpha}_{vdW}^{cr}$ | 0.3095 | - | 0.250 |

${w}_{\mathrm{max}}\text{\hspace{0.17em}}@\text{\hspace{0.17em}}{\alpha}_{vdW}^{cr}$ (in nm) | 48.625 | 40.4 | 57.857 |

**Table 2.**Comparison of the critical Casimir parameter using Equation (16) and its corresponding maximum nanobeam deflection with the results of [14].

Parameter | Equation (16) | Results of [14] |
---|---|---|

${\alpha}_{cas}^{cr}$ | 0.249 | 0.230 |

${w}_{\mathrm{max}}\text{\hspace{0.17em}}@\text{\hspace{0.17em}}{\alpha}_{cas}^{cr}$ (in nm) | 37.865 | 39.310 |

**Table 3.**Comparison of the threshold nanobeam detachment length ${L}_{cr}$ and its respective minimum gap size ${d}_{cr}$ obtained using Equation (17) and (18) for the case of van der Waals force and Casimir force, respectively, with the results of [15].

Parameter | Van der Waals Force Case | Casimir Force Case | ||
---|---|---|---|---|

Equation (17) | Results of [15] | Equation (18) | Results of [15] | |

${L}_{cr}$ (in nm) | d = 16 nm | d = 25 nm | ||

298.85 | 312 | 324.77 | 340 | |

${d}_{cr}$ (in nm) | L = 200 nm | |||

5.31 | 5.1 | 9.73 | 9.5 |

**Table 4.**Effect of the mid-plane stretching parameter on the critical intermolecular force parameters as compared with the results of [10].

${\mathit{\alpha}}_{\mathit{s}\mathit{t}}$ | ${\mathit{\alpha}}_{\mathit{v}\mathit{d}\mathit{W}}^{\mathit{c}\mathit{r}}$ | ${\mathit{\alpha}}_{\mathit{c}\mathit{a}\mathit{s}}^{\mathit{c}\mathit{r}}$ | ||
---|---|---|---|---|

Equation (14) | Results of [10] | Equation (16) | Results of [10] | |

6 | 51.5 | 53.5 | 39.3 | 40.89 |

12 | 54.5 | 57.7 | 41.1 | 42.29 |

18 | 57.7 | 62.57 | 42.3 | 44.62 |

24 | 61.1 | 67.73 | 43.9 | 47.29 |

**Table 5.**Variation of the pull-in parameter with the Casimir parameter as obtained using Equation (21) and then as compared with the results of [14].

${\mathit{\alpha}}_{\mathit{c}\mathit{a}\mathit{s}}^{}$ | ${\mathit{\alpha}}_{\mathit{f}\mathit{f}}={\mathit{\alpha}}_{\mathit{s}\mathit{t}}=0$ | ${\mathit{\alpha}}_{\mathit{f}\mathit{f}}=1;\text{}{\mathit{\alpha}}_{\mathit{s}\mathit{t}}=0$ | ||
---|---|---|---|---|

Equation (21) | Results of [14] | Equation (21) | Results of [14] | |

10 | 48.57 | 49.43 | 32.63 | 33.81 |

20 | 30.38 | 31.08 | 19.78 | 20.77 |

30 | 13.37 | 14.07 | 8.18 | 8.91 |

**Table 6.**Variation of the pull-in parameter with the van der Waals parameter as obtained using Equation (20) and then as compared with the results of [11].

${\mathit{\alpha}}_{\mathit{v}\mathit{d}\mathit{W}}^{}$ | ${\mathit{\alpha}}_{\mathit{s}\mathit{t}}^{}=6$ | |
---|---|---|

Equation (20) | Results of [11] | |

10 | 62.57 | 63.21 |

20 | 46.18 | 47.9 |

30 | 29.1 | 33.2 |

**Table 7.**Variation of the pull-in parameter with the Casimir parameter as obtained using Equation (22) and then as compared with the results of [11].

${\mathit{\alpha}}_{\mathit{c}\mathit{a}\mathit{s}}^{}$ | ${\mathit{\alpha}}_{\mathit{s}\mathit{t}}^{}=6$ | |
---|---|---|

Equation (22) | Results of [11] | |

10 | 54.24 | 55.8 |

20 | 34.31 | 35.77 |

30 | 14.68 | 18.2 |

**Table 8.**Variation of the pull-in voltage (in Volts) with both the van der Waals and Casimir parameters as obtained using Equations (20) and (22), respectively, and then as compared with the results of [12].

L (nm) | Van der Waals Force Case | Casimir Force Case | ||||
---|---|---|---|---|---|---|

Equation (20) ${\mathit{\alpha}}_{\mathit{s}\mathit{t}}=0$ | Equation (20) ${\mathit{\alpha}}_{\mathit{s}\mathit{t}}^{}\ne 0$ | Results of [12] ${\mathit{\alpha}}_{\mathit{s}\mathit{t}}^{}\ne 0$ | Equation (22) ${\mathit{\alpha}}_{\mathit{s}\mathit{t}}=0$ | Equation (22) ${\mathit{\alpha}}_{\mathit{s}\mathit{t}}^{}\ne 0$ | Results of [12] ${\mathit{\alpha}}_{\mathit{s}\mathit{t}}^{}\ne 0$ | |

130 | 12.91 | 15.76 | 14.41 | 12.64 | 14.45 | 14.37 |

150 | 9.69 | 11.83 | 10.82 | 9.47 | 10.91 | 10.77 |

180 | 6.72 | 8.28 | 7.51 | 6.53 | 7.54 | 7.43 |

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**MDPI and ACS Style**

Ouakad, H.M.; AlQasimi, J.E. Comprehensive Analytical Approximations of the Pull-In Characteristics of an Electrostatically Actuated Nanobeam under the Influences of Intermolecular Forces. *Actuators* **2018**, *7*, 3.
https://doi.org/10.3390/act7010003

**AMA Style**

Ouakad HM, AlQasimi JE. Comprehensive Analytical Approximations of the Pull-In Characteristics of an Electrostatically Actuated Nanobeam under the Influences of Intermolecular Forces. *Actuators*. 2018; 7(1):3.
https://doi.org/10.3390/act7010003

**Chicago/Turabian Style**

Ouakad, Hassen M., and Jihad E. AlQasimi. 2018. "Comprehensive Analytical Approximations of the Pull-In Characteristics of an Electrostatically Actuated Nanobeam under the Influences of Intermolecular Forces" *Actuators* 7, no. 1: 3.
https://doi.org/10.3390/act7010003