# A Thermoacoustic Model for High Aspect Ratio Nanostructures

^{1}

^{2}

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## Abstract

**:**

_{2}Ga) nanoresonators are potential candidates as nanoresonators, nanoactuators, and for scanning probe microscopy applications.

## 1. Introduction

_{2}Ga metallic nanoneedles [7] and ZnO nanowires and nanobelts [26], may enable better materials for nano-actuation and SPM applications. While many experiments have reported such nanoresonators, only a few theoretical reports exist on the modeling and characterization of nanoresonators’ natural frequency and loss mechanisms [3,27,28,29,30].

_{n}) is the ratio of mean free path of air molecules to the lateral size of the resonator and it depends on the pressure of the air. Three different regimes based on the Knudsen number (K

_{n}) were defined [5]. For K

_{n}smaller than 0.01, continuum regime for air and fluid mechanics equations can be used. For K

_{n}larger than 10, one can use the free molecular flow regime. A cross-over regime where 0.01 < K

_{n}< 10 was also defined as transient between the continuum and molecular flow regimes [5]. Variations on the cross-flow regime has also been reported in the literature [31,32]. For K

_{n}< 10, reports have also used simple continuum models even for the cross-flow regime [27,33,34].

_{2}Ga nanowire in the free molecular regime and suggested an equation for Q factor estimation as ${\mathrm{Q}}_{\mathrm{i}}=\frac{{\mathsf{\omega}}_{\mathrm{i}}\mathsf{\rho}\mathrm{A}}{4\mathrm{bP}}\sqrt{\frac{{\mathsf{\pi}\mathrm{R}}_{0}\mathrm{T}}{2{\mathrm{M}}_{\mathrm{m}}}}$ [8,35], where (i) represents different modes for the lateral vibration of the beam, ${\mathsf{\omega}}_{\mathrm{i}}$ is the angular natural frequency of nanoresonator at different modes; $\mathsf{\rho}$ is the density of nanoresonator; A is the cross-sectional area of the nanoresonator; b is the effective area for damping per unit length; P is the pressure; R

_{o}is the universal gas constant; T is the temperature; and M

_{m}is the molecular weight. Unfortunately, the results of this equation are not consistent with their experimental Q factor findings using laser Doppler vibrometry resulting in a large error in estimated Q factors [8,35]. For the first mode, the experimental Q factors were 1.2, whereas the Q factor using the above-estimated equation was only 0.49, creating an almost 58% difference between the estimated and the experimental values. The errors are similar in other modes.

_{n}and damping coefficient (inverse of quality factor). This equation is just for resonators with a rectangular cross-section and it is not applicable on cylindrical geometries. Also, this theory is only tested on ZnO nanobelts and it has not been confirmed for other materials and geometries. The other limitation of this theory is that it is only for the first mode. It should be noted that while Biedermann et al. claim that there is a linear relationship between the resonance frequency and Q factor, Yum et al. report, and our results presented here, show a nonlinear relationship between these two parameters. While these methods are interesting, a new model that accurately describes the theoretical and experimental values of Q factor is in need of further development.

_{n}< 10 and our simulations show excellent agreement with experimental data. Figure 1 shows the Knudsen number, continuum and free molecular flow for two resonators that are simulated in COMSOL.

## 2. Analytical Model

_{i}is the constant coefficient of natural frequency for different modes; L is the length; E is the elastic modulus; and I is the planar second moment of area of the nanoresonator. Figure 2 shows loading, mechanical properties and free body diagram of a nanoresonator when it is actuated by transverse base motion. Working in the steady state, the actuator applies constant vibration amplitude on the clamping end of a nanoresonator. Therefore, a base harmonic mechanical displacement is considered as the actuation input (E = D·Sin($\mathsf{\omega}$t)) of nanoresonator instead of force actuation. D is the input vibration amplitude. This model does not include air damping and it is applicable to high vacuum conditions. Assuming the nanoresonator as a Bernoulli-Euler beam [38], the equation of motion is given by:

_{i}are the constants applying the boundary conditions as a clamp-free beam, C

_{1}to C

_{4}will be:

_{1}and the values of C

_{1}to C

_{4}are infinite.

## 3. Thermoacoustic Finite Element Model (FEM)

_{solid}is the velocity field of solid elements at the interface. For the continuity equation we have:

_{0}is the background (initial) density. The momentum equation for thermoacoustic region would be:

_{B}is the bulk viscosity. The energy conservation would be:

_{p}represents the heat capacity at constant pressure, k the thermal conductivity, α

_{0}the coefficient of thermal expansion. And finally the equation of state connects these parameters together:

_{T}is the isothermal compressibility. Thermoacoustic model only considers the conduction for temperature variation. Temperature variation can change the air properties including density, coefficient of thermal expansion and isothermal compressibility. These values can affect the condition of acoustic interaction and that is why the local temperature variation around the resonator is important. Other studies have estimated the thickness of thermal and viscous layer as a function of frequency [50]. For all of the frequencies, the stationary temperature is assumed room temperature (20 °C). If the temperature variation is large, there might be some convective heat transfer outside of thermoacoustic layer which in long term may cause a change in stationary temperature.

## 4. Results

_{2}Ga nanowire (cylindrical shape with high density and bending modulus) [35,51] and a zinc oxide (ZnO) nanobelt (square cross-section) [26,27].

_{2}Ga nanowire and ZnO nanobelt. The first resonance frequency of ZnO nanobelt is close to the second mode of Ag

_{2}Ga and the stress at the clamp end for these two modes are close. However, the vibration amplitude and strain energy for the ZnO nanobelt is much higher than for the Ag

_{2}Ga nanowire, suggesting ZnO nanowires can be better choices for SPM as a tip. However, there are other deliberations including the fabrication process that should be considered before final selection.

_{2}Ga nanoneedle. As is shown, the radius of the thermal boundary layer decreases as the frequency increases. For example, the diameter of temperature variation for the fourth mode is in the same scale of the nanoresonator length, but for the first mode, the diameter of temperature variation is almost twice the length of the nanoresonator. Based on the acoustic effects of MEMS resonators that led to improved performance [50], we used a similar concept to define the radius of the thermoacoustic sphere. A smaller thermoacoustic sphere can be used for high frequency modes to make the computations faster and less expensive. In these simulations, the radius of the thermoacoustic sphere (R) for the first mode is 50% larger than the resonator’s length (L), but in higher modes, the radius of the thermoacoustic model and the length of the nanoresonator are equal. Figure 6 compares the results of the thermoacoustic model for the first resonance mode with different thermoacoustic sphere size. The results are different when R = L, R = 1.5 L. However, there are no differences between the results of simulations when R = 1.5 L and R = 2 L. Therefore, if the sphere is large enough that the temperature variation at outer regions of sphere is zero, the size of sphere is optimum and making the sphere bigger will not affect the results. Based on these observations for the final results, R = 1.5 L was selected for the first mode but for modes 2, 3, and 4, R = L was selected as the size of sphere.

^{3}as the density of the nanoresonator were kept constant, and the results are plotted for different diameters. While the analytical model (Equation (11)) suggests linear behavior between the diameter and normal stress, the simulation results show a second-order polynomial fit. As the diameter of the nanoresonator decreases, it gets closer to the free molecular regime and the thermoacoustic model fails to simulate the vibration of nanoresonator in this regime accurately. However, Figure 8 shows a fourth-order polynomial fit for strain energy which is consistent with what is suggested in Equation (12).

^{3}as density. Equations (11) and (12) confirm the same effect for the bending modulus.

^{3}density were kept constant, and length was changed from 10 to 60 µm. The analytical model as well as Figure 11 shows a second-order fit for stress versus the inverse of length. Figure 12 and Equation (12) show a fourth-order fit for strain energy versus the inverse of length. Figure 13 and Figure 14 also show the negligible effect of density on stress and strain energy as it was suggested in Equation (11) and (12).

_{i}is the constant coefficient of natural frequency for different modes, d the diameter, L length, E elastic modulus, and ρ the density of nanowire. This equation is for an individual nanoresonator as a fix-free cantilever vibrating in air and it looks like this equation is similar to the one that has been suggested by Yum et al. [27]. However, by adding the thermal damping, the power of parameters has been modified in Equation 18 and is more accurate.

## 5. Discussion and Conclusions

_{2}Ga and ZnO nanowires and nanobelts have also been used. In carbon nanotubes, the shank lengths are only in a region of a few micrometers long and also cause adhesive effects due to van der Wals forces which can be overcome by using resonating probes. To minimize this influence, different techniques have been evaluated such as: torsional [54], longitudinal [55], and lateral vibrations in SPM and AFM [56]. Compared to carbon nanotubes, the Ag

_{2}Ga offers a versatile probe, room temperature probe fabrication and also an ability to grow probes of any length with high rigidity [7]. These could be useful in places like probing inside a via. In general, these methods employ a 1D probe tip that is oscillated below the natural frequency of the probe’s shank. Once the work piece is contacted by the vibrating free end of the probe tip, it causes a shift in the phase and natural frequency of the probe system [57,58]. All of the earlier solutions do not provide the capability of rapidly and accurately measuring even high aspect ratio micro-scale features such as vias, micro-cooling channels, and fiber optic couplers. In particular, for micro scale probes with spherical contact tips, tip diameters in many cases are larger than 100 μm, which is not effective for holes with less than 200 μm diameter. Second, probe tips smaller than 100 μm in diameter are relatively short and still large compared to the targeted hole diameters such as fuel injector nozzles, chip vias, or micro cooling channels. Furthermore, the measurement speed is low and probes often function only in the touch trigger mode. These probes are susceptible to adhesion and thus, after contacting the work piece, they stick to the measured surface. Procedures to overcome adhesion effects often reduce repeatability and increased time needed to perform the measurement. Using nanoresonators with high strain energy to avoid sticking to the target surface is a feature that is desired. The high frequency vibration can apply large stress at its bonding section to the actuator and make the bonding break. High amplification can act as a virtual tip to move the nanoresonator close to the target surface.

_{2}Ga nanowire and ZnO nanobelt were selected as the potential candidates due to their use as a stylus in SPM. This model is the first of its kind and has excellent agreement with the experimental reports in the literature and thus is useful for predicting the mechanical properties of such 1D nanomaterials. Future developments in this area could be development of a package in COMSOL to describe the molecular flow regime that can accurately predict the resonance frequency and quality factors of nanoresonators. New 2D nanomaterials such as transition metal dichalgogenides (TMDs) with their layered structure and ultra-high strength and ability to withstand deformation up to 25% could be simulated for their resonance frequency and quality factors using this model rather than doing expensive experiments in the future.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Nomenclature

${\mathsf{\omega}}_{\mathrm{i}}$ | Angular natural frequency of nanoresonator at different modes |

$\mathsf{\rho}$ | Density of nanoresonator |

$\mathrm{A}$ | Cross-section of nanoresonator |

$\mathrm{P}$ | Pressure |

${\mathrm{M}}_{\mathrm{m}}$ | Molecular weight |

$\mathrm{T}$ | Temperature |

${\mathrm{R}}_{\mathrm{o}}$ | Universal gas constant |

$\mathrm{b}$ | Effective area for damping per unit length |

${\mathrm{k}}_{\mathrm{i}}$ | The constant coefficient of natural frequency for different modes |

$\mathrm{E}$ | Elastic modulus of nanoresonator |

$\mathrm{L}$ | Length of nanoresonator |

$W$ | Width of nanoresonator |

$\mathrm{h}$ | Height of nanoresonator |

$\mathrm{I}$ | Planar second moment of nanoresonator area |

$\mathrm{x}$ | Special coordinates of nanoresonator |

$\mathrm{y}$ | Lateral displacement of vibrating nanoresonator in time domain |

$\mathrm{t}$ | Time |

$\mathrm{B}\left(\mathrm{x}\right)$ | Deflection of nanoresonator in special coordinates |

$\mathrm{e}$ | Base excitement of nanoresonator |

$\mathrm{w}$ | Actuating angular frequency |

$\mathrm{D}$ | Amplitude of base excitation |

${\mathrm{C}}_{1}$, ${\mathrm{C}}_{2}$, ${\mathrm{C}}_{3}$, ${\mathrm{C}}_{4}$, $\mathrm{C}$, $\mathrm{C}\prime $ | Constant |

$\mathsf{\lambda}$ | Eigenvalue |

$\mathrm{RM}$ | Reaction moment at clamp end of nanoresonator |

$\mathsf{\sigma}$ | Stress |

$\mathrm{d}$ | Diameter of nanowire |

$\mathrm{V}$ | Elastic strain energy |

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**Figure 1.**Effect of nanoresonator size and its surrounding pressure in air on the Knudsen number at 300 °·K. The squares show the condition of nanoresonators that are tested in the thermoacoustic model.

**Figure 3.**Thermoacoustic model of nanoresonators. Triangular prism element for the resonator and tetrahedron for surrounding medium.

**Figure 4.**Frequency response function of Ag

_{2}Ga nanowire and ZnO nanobelt resulted from experimental data and thermoacoustic model. The results of experimental data and thermoacoustic model match.

**Figure 6.**The effect of thermoacoustic sphere radius (R) on the temperature variation distribution and Q factor for the first mode of Ag

_{2}Ga nanowire.

**Figure 7.**Effect of diameter on the normal stress at the clamp end of nanowire. Theoretical model suggested a linear fit while the thermoacoustic model shows a second-order polynomial fit for the relationship of diameter and normal stress at the clamp end.

**Figure 8.**Effect of diameter on the stored strain energy of nanowire. The analytical model matches with the results of the thermoacoustic model ($\mathit{V}\propto {\mathit{d}}^{4}$).

**Figure 9.**Effect of bending modulus on the normal stress at the clamp end of a nanowire. The analytical model corresponds to the results of the thermoacoustic model ($\mathit{\sigma}\propto \mathit{E}$).

**Figure 10.**Effect of bending modulus on the elastic strain energy of nanowire. The analytical model matches with the results of a thermoacoustic model ($\mathit{V}\propto \mathit{E}$).

**Figure 11.**Effect of length on the normal stress at the clamp end of nanowire. The analytical model’s findings match the results of the thermoacoustic model (${\mathit{\sigma}}_{\mathit{x}\mathit{x}}\propto \frac{1}{{\mathit{L}}^{2}}$).

**Figure 12.**Effect of length on the elastic strain energy of nanowire. The analytical model matches the results of the thermoacoustic model ($\mathit{V}\propto \frac{1}{{\mathit{L}}^{4}}$).

Material | Length (μm) | Cross-Section | Bending Modulus (GPa) | Density (kg/m^{3}) |
---|---|---|---|---|

ZnO nanobelt [27] | 31.6 | Rect. (228 × 314 nm^{2}) | 84.8 | 5606 |

Ag_{2}Ga nanowire [35] | 60 | Circle (dia. 206 nm) | 83.2 | 8960 |

**Table 2.**Comparing natural frequencies and Q factors of nanoresonators from experiments and simulations.

Nanoresonator | Vibration Mode | Natural Frequency (kHz) From Equation (10) | Damped Resonance f_{exp} (kHz) | Damped Resonance f_{sim} (kHz) | Δf (kHz) (% Error) | Q_{exp} ([35]) | Q_{sim} | Estimated Q ([35]) |
---|---|---|---|---|---|---|---|---|

Ag_{2}Ga nanowire | 1st mode | 24.7 | 22.5 | 23.4 | 1.1 (4.8) | 1.2 | 0.9406 | 0.49 |

Ag_{2}Ga nanowire | 2nd mode | 154.5 | 151 | 149.5 | 2.5 (1.6) | 6.0 | 4.8362 | 2.90 |

Ag_{2}Ga nanowire | 3rd mode | 432.6 | 428 | 428.8 | −0.8 (0.1) | 12.0 | 12.1473 | 8.17 |

Ag_{2}Ga nanowire | 4th mode | 847.7 | 845 | 844.2 | −6.2 (0.7) | 22.0 | 21.4809 | 16.00 |

ZnO nanobelt | 1st mode | 143.44 | 140.3 | 3.14 (2.2) | 3.85 | 4.0963 | − |

**Table 3.**The mechanical response of the Ag

_{2}Ga nanowire and Zno nanobelt from a thermoacoustic model.

Mechanical Response | Ag_{2}Ga 1st Mode (23.4 KHz) | Ag_{2}Ga 2nd Mode (149.5 KHz) | Ag_{2}Ga 3rd Mode (428.8 KHz) | Ag_{2}Ga 4th Mode (844.2 KHz) | ZnO Nanobelt (140.3 KHz) |
---|---|---|---|---|---|

Stress (MPa) | 5.9 | 76.9 | 286.7 | 680.3 | 98.9 |

Strain energy (J) | 2.0474 × 10^{−17} | 4.6904 × 10^{−18} | 4.0924 × 10^{−14} | 2.4916 × 10^{−14} | 5.9527 × 10^{−15} |

**Table 4.**The effect of material and geometrical properties of a vibrating nanoresonator in vacuum on its mechanical response.

Mechanical Response | Effect of Length | Effect of Diameter | Effect of Bending Modulus | Effect of Density |
---|---|---|---|---|

Normal stress at the clamp end (high vacuum) | $\frac{1}{{\mathrm{L}}^{2}}$ | $\mathrm{d}$ | $\mathrm{E}$ | No effect |

Normal stress at the clamp end (ambient condition) | $\frac{1}{{\mathrm{L}}^{2}}$ | ${\mathrm{d}}^{2}$ | $\mathrm{E}$ | No effect |

Elastic stored energy (high vacuum) | $\frac{1}{{\mathrm{L}}^{4}}$ | ${\mathrm{d}}^{4}$ | E | No effect |

Elastic stored energy end (ambient condition) | $\frac{1}{{\mathrm{L}}^{4}}$ | ${\mathrm{d}}^{4}$ | E | No effect |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Loeian, M.S.; Cohn, R.W.; Panchapakesan, B. A Thermoacoustic Model for High Aspect Ratio Nanostructures. *Actuators* **2016**, *5*, 23.
https://doi.org/10.3390/act5040023

**AMA Style**

Loeian MS, Cohn RW, Panchapakesan B. A Thermoacoustic Model for High Aspect Ratio Nanostructures. *Actuators*. 2016; 5(4):23.
https://doi.org/10.3390/act5040023

**Chicago/Turabian Style**

Loeian, Masoud S., Robert W. Cohn, and Balaji Panchapakesan. 2016. "A Thermoacoustic Model for High Aspect Ratio Nanostructures" *Actuators* 5, no. 4: 23.
https://doi.org/10.3390/act5040023