Analytical study of the frequency shifts of micro and nano clamped–clamped beam resonators due to an added mass

Abstract

We present analytical formulations to calculate the induced resonance frequency shifts of electrically actuated clamped–clamped micro and nano (Carbon nanotube) beams due to an added mass. Based on the Euler–Bernoulli beam theory, we investigate the linear dynamic responses of the beams added masses, which are modeled as discrete point masses. Analytical expressions based on perturbation techniques and a one-mode Galerkin approximation are developed to calculate accurately the frequency shifts under a DC voltage as a function of the added mass and position. The analytical results are compared to numerical solution of the eigenvalue problem. Results are shown for the fundamental as well as the higher-order modes of the beams. The results indicate a significant increase in the frequency shift, and hence the sensitivity of detection, when scaling down to nano scale and using higher-order modes.

Appendices

Appendix 1: Parallel plate theory

The expressions of the electrostatic force in clamped–clamped microbeam and carbon nanotube are derived here. The expressions are derived based on the parallel plates assumption for the microbeam and based on a cylinder interacting with a plate for the CNT.

We assume that the electric field lines between the two parallel plates are perpendicular, which means neglecting the fringing field effect close to the edges.

A voltage source V of load is assumed. The electric charge Q and potential energy E inside the capacitor can be written in the following expressions, respectively [39, 48].

$$Q = C({\text{z}}){\text{V}}$$

(35)

$$E = \frac{{V^{2} }}{2}C({\text{z}})$$

(36)

The electrostatic force $F_{e}$ between the two electrodes in Fig. 9 for both cases can be obtained as [39, 48]

$$F_{e} = - \frac{{\partial E({\text{z}})}}{\partial z} = \frac{{V^{2} }}{2}\frac{{\partial C({\text{z}})}}{\partial z}$$

(37)

Fig. 9

Schematic of a parallel-plate capacitor, and b a cylinder-plate capacitor

The expression of the capacitance $C({\text{z}})$ depends on the geometry of the two electrodes. First, the case of two parallel electrodes is considered shown in Fig. 9a. The capacitance is expressed as [39, 48]

$$C({\text{z}}) = \frac{{\varepsilon_{0} Lb}}{z}$$

(38)

where $\varepsilon_{0} = 8.85 \times 10^{ - 12} ({\text{C}}^{2} /{\text{Nm}}^{2} )$ is the air permittivity. We substitute Eq. (38) into Eq. (37) and get the electrostatic force for the microbeam configuration

$$F_{e} = - \frac{{\varepsilon_{0} bLV^{2} }}{{2z^{2} }}$$

(39)

The case of a cylinder and a rectangular plate capacitor, such as in CNTs, as shown in Fig. 9b can be expressed as [48]

$$C({\text{z}}) = \frac{{2\pi \varepsilon_{0} L}}{{\cosh^{ - 1} \left( {1 + \frac{z}{R}} \right)}}$$

(40)

Finally, we substitute Eq. (40) into Eq. (37) to get the attractive force

$$F_{e} = - \frac{{\pi \varepsilon_{0} LV^{2} }}{{\sqrt {z\left( {z + 2R} \right)} \left( {\cosh^{ - 1} \left( {1 + \frac{z}{R}} \right)} \right)^{2} }}$$

(41)

Appendix 2: Noise analysis

Based on [49, 50], we derive the noise analysis procedure used to compare with the results of Tables 2 and 4 and to check that the induced frequencies due to added mass are above the minimum detectable frequency due to thermal fluctuations around the sensor.

The equation of motion of a mass-spring-damper system, of mass m, stiffness K, and damping coefficient c, driven by a noise force $F_{noise}$ is written as

$$m{\ddot{x}} + c\dot{x} + Kx = F_{noise}$$

(42)

The kinetic energy E stored in the resonator is written as

$$E = \frac{1}{2}m\bar{v}_{n}^{2}$$

(43)

where $\bar{v}_{n}^{2}$ is the mean square velocity. Then we write the equation of motion in terms of velocity, we apply the Laplace transformation

$$\bar{v}_{n}^{{}} = \frac{dx}{dt} = sx$$

(44)

$$s \times m\bar{v}_{n}^{{}} + c\bar{v}_{n}^{{}} + \frac{K}{s}\omega = \bar{F}_{noise}$$

(45)

We set $s = j\omega$.Thus, the mean square velocity due to noise generator $\bar{F}_{noise}$ is written as

$$\bar{v}_{n}^{2} = \frac{{\bar{F}_{noise}^{2} }}{{c^{2} + \left( {\omega_{{}} m - \frac{K}{\omega }} \right)^{2} }}$$

(46)

The previous equation can be written as

$$\bar{v}_{n}^{2} = \frac{1}{{c^{2} }}\frac{{\bar{F}_{noise}^{2} }}{{1 + Q^{2} \left( {\frac{\omega }{{\omega_{0} }} - \frac{{\omega_{0} }}{\omega }} \right)^{2} }}$$

(47)

where $\omega_{0} = \sqrt {K/m}$ is the resonance frequency and $Q = \omega_{0} \frac{m}{c}$ is the quality factor.Then, the kinetic energy stored in the resonator is expressed as

$$E = \frac{1}{2}m\bar{v}_{n}^{2} = \frac{1}{4\pi c}\int_{0}^{\infty } {\frac{{\bar{F}_{noise}^{2} Qd\left( {\frac{f}{{f_{0} }}} \right)}}{{1 + Q^{2} \left( {\frac{f}{{f_{0} }} - \frac{{f_{0} }}{f}} \right)^{2} }}}$$

(48)

The integral in (48) is calculated as

$$\int_{0}^{\infty } {\frac{{^{{}} Q_{{}} d\left( {\frac{f}{{f_{0} }}} \right)}}{{1 + Q^{2} \left( {\frac{f}{{f_{0} }} - \frac{{f_{0} }}{f}} \right)^{2} }}} = \frac{\pi }{2}$$

(49)

Based on the Equipartition theorem; E can be expressed as

$$E = \frac{1}{2}k_{B} T$$

(50)

where $k_{B} = 1.3806488 \times 10^{ - 23} \,{\text{m}}^{2} \,{\text{kg}}\,{\text{s}}^{ - 2} \,{\text{K}}^{ - 1}$ is the Boltzmann constant and T the temperature.

Finally, the force noise generator is expressed as

$$\bar{F}_{noise}^{2} = 4k_{B} Tc$$

(51)

Using Eqs. (42) and (51) the noise spectrum equation can be written as

$$\left\langle {x_{n} } \right\rangle = \sqrt {\bar{x}_{n}^{2} } = \sqrt {\frac{{4k_{B} T_{{}} c}}{{c^{2} \omega^{2} + \left( {K - m\omega^{2} } \right)^{2} }}}$$

(52)

The maximum displacement noise spectral density is at the resonance frequency $\left( {\omega = \omega_{0} } \right)$

$$\left\langle {x_{n} } \right\rangle_{\hbox{max} } = \sqrt {\bar{x}_{n}^{2} } = \sqrt {\frac{{4k_{B} T_{{}} Q}}{{2\pi f_{0} K}}}$$

(53)

2.1 Density of frequency noise

The force gradient acting on the resonator of the level $F_{noise}^{{\prime }}$ changes the spring constant and shifts the resonance frequency

$$\omega_{0} = \left( {\frac{K}{m}} \right)^{{\frac{1}{2}}}$$

(54)

$$\omega_{0}^{{\prime }} = \omega_{0} \times \left( {1 + \frac{{F_{noise}^{{\prime }} }}{K}} \right)^{{\frac{1}{2}}}$$

(55)

where $F_{noise}^{{\prime }} = \sqrt {\frac{{\bar{F}_{noise}^{2} }}{{\overline{x}_{noise}^{2} }}} = K^{\prime}$, which is the spring constant due to the noise force generator ($\left[ {F_{noise}^{{\prime }} } \right] = \frac{N}{m}$).Using Taylor series expansion of Eq. (55) and retaining the first term yield

$$\omega_{0}^{{\prime }} \approx \omega_{0} \times \left( {1 + \frac{{F_{noise}^{{\prime }} }}{2K}} \right)$$

(56)

The frequency shift can be expressed as

$$\delta \omega = \omega_{0} - \omega_{0}^{{\prime }}$$

(57)

Then,

$$\left( {\frac{\delta \omega }{{\omega_{0} }}} \right)^{2} = \left( { - \frac{{F_{noise}^{{\prime }} }}{2K}} \right)^{2} = \left( {\frac{{F_{noise}^{{\prime }} }}{2K}} \right)^{2}$$

(58)

$$\left( {\frac{\delta \omega }{{\omega_{0} }}} \right)^{2} = \frac{{\bar{F}_{noise}^{2} }}{{4K\bar{x}_{noise}^{2} }}$$

(59)

Using (51), the force due to noise can be expressed as

$$\bar{F}_{noise}^{2} = \frac{{4k_{B}^{{}} TK}}{{Q\omega_{0} }}$$

(60)

Finally, the smallest detectable frequency shift, in Hz $\delta f_{noise} = \delta \omega /2\pi$, due to noise can be obtained from (59) and (60) as

$$\delta f_{noise} = \frac{1}{{\bar{x}_{noise}^{{}} }}\sqrt {\frac{{f_{0} k_{B} T}}{2\pi QK}}$$

(61)

2.2 Minimum detectable mass change

In order to calculate the minimum detectable mass, one can write the frequency $f_{0}$ and the frequency after mass adsorption $f_{1}$.

$$f_{0} = \frac{1}{2\pi }\sqrt {\frac{K}{m}} ;\quad f_{1} = f_{0} - \delta f_{noise} = \frac{1}{2\pi }\sqrt {\frac{K}{{m + M_{\hbox{min} } }}}$$

(62)

We manipulate (62) in order to find the expression of the minimum detectable mass $M_{\hbox{min} }$

$$M_{\hbox{min} } = \frac{K}{{4\pi^{2} }}\left( {\frac{1}{{f_{1}^{2} }} - \frac{1}{{f_{0}^{2} }}} \right)$$

(63)

Using (53), (61), and (63) for the case of a clamped–clamped beam with the geometrical properties shown in Table 1, we obtain the results in Table 5.

Table 5 Natural frequency, quality factor, maximum noise displacement, minimum detectable frequency, and minimum detectable mass for different modes of vibration

The objective of the noise analysis is to determine the minimum detectable frequency and minimum detectable mass that the sensor is able to detect. In Table 5, we calculate in the fifth column the minimum detectable frequencies based on the developed theory in “Appendix 2”. The results using the perturbation techniques (Num. Pert.) are considered as the most accurate results. For the first mode of vibration of a clamped–clamped microbeam, from Table 5 $\delta f_{{_{noise} }}^{1st} = 60.24_{{}} [{\text{Hz}}]$. In order to find the minimum detectable number of E. coli, which corresponds to the minimum detectable frequency we use the linear relationship between the added mass and the induced frequency shift in Eq. (31). In Table 2 we recall that the frequency shift induced from 50 E. coli is $\delta f_{50E.coli} = 29.77_{{}} [{\text{Hz}}]$.Hence, we need at least 102 E. coli to induce a frequency shift $\delta f_{{_{102E.coli} }}^{1st} = 60.73_{{}} [Hz]$, which is above the noise level $\delta f_{{_{noise} }}^{1st} = 60.24_{{}} [{\text{Hz}}]$. The same method has been used for the different modes of vibration. We find that we need at least 4 E. coli to induce a frequency shift $\delta f_{{_{4E.coli} }}^{3rd} = 11_{{}} [{\text{Hz}}]$, which is above the noise level of the third mode of vibrations $\delta f_{{_{noise} }}^{3rd} = 9.3_{{}} [{\text{Hz}}]$. Finally, for the fifth mode of vibration, the microbeam-based mass sensor is able to detect a single E. coli inducing a frequency shift $\delta f_{{_{1E.coli} }}^{5rd} = 6.67_{{}} [{\text{Hz}}]$, which is above the calculated noise $\delta f_{{_{noise} }}^{5th} = 2.296_{{}} [{\text{Hz}}]$.