Detecting the mass and position of an adsorbate on a drum resonator

Abstract

The resonant frequency shifts of a circular membrane caused by an adsorbate are the sensing mechanism for a drum resonator. The adsorbate mass and position are the two major (unknown) parameters determining the resonant frequency shifts. There are infinite combinations of mass and position which can cause the same shift of one resonant frequency. Finding the mass and position of an adsorbate from the experimentally measured resonant frequencies forms an inverse problem. This study presents a straightforward method to determine the adsorbate mass and position by using the changes of two resonant frequencies. Because detecting the position of an adsorbate can be extremely difficult, especially when the adsorbate is as small as an atom or a molecule, this new inverse problem-solving method should be of some help to the mass resonator sensor application of detecting a single adsorbate. How to apply this method to the case of multiple adsorbates is also discussed.

1. Introduction

The ultimate goal of any detection method is to achieve the level of resolving a single quantum of a measured entity [1]. To pursue this goal, various mechanisms are used. For example, the presence of a single NO₂ molecule on graphene is detected by measuring the Hall resistivity change during the adsorption and desorption processes [1]; a single biotin–streptavidin binding on a nanowire is detected by measuring the conductance change [2]; a single biotin–streptavidin binding on a nanorod is also detected by measuring the shift of its surface plasmon resonance [3]. As for an inertial mass resonator, the resonance frequency shift owing to the adsorption/binding of a particle is the detecting mechanism. The ultimate limit for the mass resonator sensitivity is imposed by the thermodynamic fluctuation [4,5], which has been theoretically proved to be well below a single dalton (1 dalton ≈1.65×10⁻²⁴ g is the mass of a proton) [5]. Enormous efforts have been made to push the mass resonator sensitivity towards the limit. There are three major approaches to enhance the mass resonator sensitivity. Because the resonant frequency of a beam is proportional to $h/L^2\times \sqrt{E/\rho }$ (h and L are the thickness and length; E and ρ are the Young modulus and mass density) [6–8], the first is to scale down the structure size, which makes the factor of h/L² larger and at the same time the fractional change in mass is also larger in a smaller resonator for the same adsorbate. The second is to use the materials with large E/ρ, such as carbon nanotube (CNT) [9–11], graphene [7] and silicon carbide [12]. Both approaches result in higher resonant frequencies. Therefore, a small fractional change in resonant frequency is still absolutely large enough to be detected [13]. As a two-dimensional material, the monolayer graphene has reached the limit of the scaling-down strategy and has its whole volume exposed to surface adsorbates [1]. At the same time, the very small interstitial space owing to the close packing of carbon atoms in graphene does not allow small atoms such as helium to pass through [14]. This impermeability property, which makes the adsorption of small atoms possible, is a necessity for pushing the sensitivity of a graphene mass resonator towards the limit. Graphene is also the strongest material ever tested [15], which can take the failure strain up to 12%. The large tensile in-plane stress inside a suspended graphene, which can be applied or simply result from the uncontrolled forces applied during the exfoliation process [16], can further increase the resonant frequency. Sometimes, the binding between an adsorbate and graphene/CNT is not strong enough to prevent the adsorbate diffusion [17,18]. The adsorbate movement can be a significant source of noise and error in the measurement of the mass resonator [17]. The presence of large tensile stress can significantly reduce or suppress the adsorbate movement [18]. Unlike the above two approaches, which in essence increase resonant frequency, the third one is to increase the quality factor. The quality factor (Q) determines the ‘sharpness’ of the resonance, which is inversely proportional to damping and is approximately given as [19]

$$Q=\frac{1.09f_o}{\mathrm{\Delta }f},$$ 1.1

where f_o is the resonant frequency and Δf is its shift. Clearly, for a given resonant frequency, larger Q means that smaller Δf can be detected. Although various models and mechanisms are proposed [20,21], a clear picture of the quality factor variation of small-scale structures still remains elusive [21]. The in-plane tension is shown to increase the quality factor significantly by both simulation [18] and experiment [22]. It is always effective to increase the quality factor by putting the mass resonator in a vacuum environment, which reduces or eliminates the damping owing to ambient air [23]. A quality factor as large as 100 000 has been achieved in a graphene resonator [19].

The sensitivity of micro/nanomechanical mass resonators has been steadily improving with roughly an order of magnitude per year [4]. Hanay et al. [24] used silicon-based nanoelectromechanical systems technology to develop a mass resonator capable of detecting a single protein with a mass of around 1 million dalton (MDa); Jensen et al. [9] developed a CNT-based mass resonator with the capability of detecting a single gold atom with a mass of 196.967 Da; Chiu et al. [10] developed a CNT-based mass resonator capable of detecting a single xenon atom (131.293 Da) and a single argon atom (39.948 Da). Sensitivity for detecting a 1 Da mass has recently been achieved by Chaste et al. [11] in their CNT-based mass resonator. Although these achievements are very impressive, there is a fundamental problem remaining in these mass resonators: all the frequency shift measurements actually cannot give the mass of an individual atom, molecule or nanoparticle [24]. As demonstrated by Jensen et al. [9], their CNT-based mass resonator actually does not measure the mass of a gold atom even though the resonator has the capability of detecting a smaller frequency shift induced by the adsorption of a gold atom. The atom/molecule/particle mass and its position are the two (major) convolving factors of determining the resonant frequency shift of a resonator. To know the position, there are three major experimental approaches: (i) measuring the trajectory of a sprayed particle [4]; (ii) occluding some portions of resonator so that the particle must land at a specific location [9]; and (iii) direct measurement of the position of an adsorbate [25]. These measurement approaches not only require extra experimental set-up, but also (sometimes) are extremely difficult if not impossible to be applied. For example, Dohn et al. [25] determined the position of a micrometre-sized particle by using optical imaging. However, in the resonator described by Burg et al. [26] that had an embedded microfluidic channel where the biomolecules are pumped in, the optical method did not work, because of the smaller size of a biomolecule and the small contrast between a biomolecule and solution. The exact position of the biomolecule is thus a major uncertainty in this experiment [26]. The following insightful comment by Knobel [13] fully summarizes the motivation of this study: before a practical mechanical mass spectrometer can be made, the most important problem to be solved is to determine the atom/molecule/particle position.

In this study, we present a method to determine the mass and position of an adsorbate on a circular resonator, which is also often referred to as a drum resonator [27–29]. In a forward problem, the adsorbate mass and position are given to calculate the resonant frequency [30–34]. However, in the real application of a mass resonator, resonant frequencies are the measured quantities; the adsorbate mass and position are the unknowns to be determined [35]. An inverse problem is thus encountered: how can we use the measured resonant frequencies to determine the adsorbate mass and position? This inverse problem is solved based on the following two mechanisms: (i) the adsorbate mass and position have different impacts on the same resonant frequency and (ii) for a circular membrane that has infinite resonant frequencies, the same adsorbate mass and position have different impacts on different resonant frequencies [25]. In essence, we use two resonant frequencies to uniquely determine the adsorbate mass and position. The methods of using multiple resonant frequencies to determine the mass and position of a single adsorbate have been developed [24,36]. However, unlike the method of probability density function [24] or a minimizing procedure which requires at least four resonant frequencies to be measured [36], this study presents a straightforward method to attack this inverse problem and its accuracy is also demonstrated.

2. Model development

Figure 1a is a schematic of a graphene layer deposited on a circular hole with radius r_a. Figure 1b shows the polar coordinate system and an adsorbate is located at (r_o,θ_o). The membrane potential energy, U, is given as follows:

$$U=\frac{F}{2}{\int }_0^{r_a}{\int }_0^{2\pi }[{\left(\frac{\mathrm{\partial }w}{\mathrm{\partial }r}\right)}^2+{\left(\frac{1}{r}\frac{\mathrm{\partial }w}{\mathrm{\partial }\theta }\right)}^2]r\mathrm{d}r\mathrm{d}\theta ,$$ 2.1

where F is the tensile force per unit length at the boundary and w is the membrane displacement. The kinetic energy, T, is given as follows:

$$T=\frac{m}{2}{\int }_0^{r_a}{\int }_0^{2\pi }{\left(\frac{\mathrm{\partial }w}{\mathrm{\partial }t}\right)}^{2}r\mathrm{d}r\mathrm{d}\theta +\frac{\mathrm{\Delta }M}{2}{{\left(\frac{\mathrm{\partial }w}{\mathrm{\partial }t}\right)}^2|}_{r=r_o,\theta ={\theta }_o}.$$ 2.2

Figure 1.

(a) Schematic of a graphene layer deposited on a substrate with a circular hole. (b) The polar coordinate system. The circle radius is r_a and an adsorbate is located at (r_o,θ_o).

Here m is the membrane mass per unit area and m=ρh (ρ, h are the mass density and thickness, respectively). ΔM is the adsorbate mass and ∂w/∂t|_r=ro,θ=θo is the corresponding velocity at (r_o,θ_o). Here, the effect of an adsorbate is modelled as a concentrated mass [32–34]. To write down the above expressions for the potential and kinetic energies, the following five assumptions are used. (i) The continuum modelling applies to graphene, which is corroborated by the atomistic studies of the monolayer graphene [37,38]. (ii) The potential energy of equation (2.1) does not include the bending energy, which is another way of saying that the suspended graphene is modelled as a membrane structure. The atomistic simulation shows that monolayer graphene has a bending stiffness close to that of a lipid bilayer of a cell [37], which may have some potential in biological application. Because of large applied or residual in-plane tension, the potential energy owing to tension is often several orders of magnitude larger than the bending energy [15]. With the presence of large in-plane tension, a plate structure behaves like a membrane one [39]. Therefore, the suspended graphene is often modelled as a membrane structure [15,22,27–29,31,40–42]. (iii) Equation (2.1) does not include the potential energy contributions induced by an adsorbate or heating by assuming that the pre-existing in-plane tension is very large compared with those induced by adsorbate or heating. Because of the adsorption-induced electronic and mechanical distortion of a graphene structure [43,44], an adsorbate can impart a tension to a suspended graphene, which causes the increase in resonant frequency [42]. At the same time, a suspended graphene is often actuated by laser heating [22,27–29], which induces thermal stress and thus changes the in-plane tension. The in-plane tension inside a graphene is quite often remarkably high and higher than the fracture strength of most materials [15,18]. Besides advantages such as reducing adsorbate diffusion, and increasing resonant frequencies and the quality factor as mentioned above [18,22], the presence of large in-plane tension also has other advantages. For example, the suspended graphene is usually non-flat with ripples [42], which causes local elastic distortion and thus complicates our analysis. Also, tension or heating (keeping in mind that graphene has negative thermal expansion coefficient) can reduce or suppress the ripples [41]. Another advantage is associated with the following fourth assumption. (iv) The geometric nonlinearity owing to large deflection is not considered. When the deflection amplitude is (very) large, the nonlinear Duffing equation becomes the governing equation [31,42], which, again, can further complicate our analysis. The presence of in-plane tension can delay the onset of nonlinearity, which is to say that the system stays in a linear range even when the amplitude is relatively large [42]. (v) Because r_a is the radius of a hole on a substrate, the assumption here is that no peeling-off occurs. When a graphene sheet is deposited on a substrate, adhesion is responsible for the binding [40], which is induced by the van der Waals force [45]. With a large deflection amplitude, peeling-off can occur [46], which means that the actual radius of the graphene sheet is varying and larger than r_a. This moving boundary together with the induced degenerate modes as discussed later in detail can make the analysis extremely difficult. Koenig's experiment shows that the adhesion of graphene sheet is ‘ultrastrong’ [40], which together with the fourth assumption of small deflection ensures this no peeling-off assumption.

The following non-dimensionalization scheme is introduced:

$$\xi =\frac{r}{r_a},W=\frac{w}{r_a}\mathrm{and}{\xi }_o=\frac{r_o}{r_a}.$$ 2.3

Now the potential energy and kinetic energy become the following forms:

$$U=\frac{F{r}_a^2}{2}{\int }_0^1{\int }_0^{2\pi }[{\left(\frac{\mathrm{\partial }W}{\mathrm{\partial }\xi }\right)}^2+{\left(\frac{1}{\xi }\frac{\mathrm{\partial }W}{\mathrm{\partial }\theta }\right)}^2]\xi \mathrm{d}\xi \mathrm{d}\theta$$ 2.4

and

$$T=\frac{m{r}_a^4}{2}{\int }_0^1{\int }_0^{2\pi }{\left(\frac{\mathrm{\partial }W}{\mathrm{\partial }t}\right)}^2\xi \mathrm{d}\xi \mathrm{d}\theta +\frac{\mathrm{\Delta }M{r}_a^2}{2}{{\left(\frac{\mathrm{\partial }W}{\mathrm{\partial }t}\right)}^2|}_{\xi ={\xi }_o,\theta ={\theta }_o}.$$ 2.5

Before we start to compute the resonant frequencies of the membrane with an adsorbate, we need to have a brief discussion on the membrane vibration pattern, which is important not only to the computation but also to the mass resonator application. By the method of separation of variables, W can be expressed as follows [47–49]:

$$W(\xi ,\theta ,t)=V(\xi ,\theta )\cos (\omega t)=R(\xi )\mathrm{\Theta }(\theta )\cos (\omega t),$$ 2.6

where ω is the membrane resonant frequency/eigenfrequency; V (ξ,θ) is the spatial part, which is decomposed further into two parts, R and Θ. For a uniform membrane with no adsorbate, the governing equation is the following [47–49]:

$$m\frac{{\mathrm{\partial }}^{2}W}{\mathrm{\partial }{t}^2}-F{▽}^{2}W=0,$$ 2.7

where ▽² is the Laplace operator defined as

$${▽}^2=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}+\frac{1}{r^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\theta }^2}=\frac{{\mathrm{\partial }}^2}{a^2\mathrm{\partial }{\xi }^2}+\frac{1}{a^2\xi }\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }+\frac{1}{a^2{\xi }^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\theta }^2}.$$ 2.8

By substituting equation (2.6) into equation (2.7), the Bessel equation is obtained, which is a classical problem and the detailed solution process can be found elsewhere [47–49]. Here we summarize only some of results, as follows [47,48]:

$$V(\xi ,\theta )=R(\xi )\mathrm{\Theta }(\theta )=\sum _{n=0}^{\mathrm{\infty }}\sum _{i=1}^{\mathrm{\infty }}[a_{ni}{J}_n({\lambda }_{n,i}\xi )\cos (n\theta )+b_{ni}{J}_n({\lambda }_{n,i}\xi )\sin (n\theta )],$$ 2.9

where n and i are two positive integers; a_ni and b_ni are the (unknown) constants associated with the modes. J_n is the Bessel function of the first kind and λ_n,i is the value corresponding to the ith zero of J_n. When n≠0, which is the so-called degenerate case [48], there are two modes at a resonant frequency: $J_n({\lambda }_{n,i}\xi )\cos (n\theta )$ and $J_n({\lambda }_{n,i}\xi )\sin (n\theta )$, which thus means that the linear combination of these two modes, i.e. $a_{ni}{J}_n({\lambda }_{n,i}\xi )\cos (n\theta )+b_{ni}{J}_n({\lambda }_{n,i}\xi )\sin (n\theta )$, is also the mode. When n≠0, the mode is θ-dependent and thus non-axisymmetric. Only when n=0 can the corresponding modes be axisymmetric [47–49]. The three lowest axisymmetric mode shapes of J₀(λ_0,1ξ), J₀(λ_0,2ξ) and J₀(λ_0,3ξ) with λ_0,1=2.4048, λ_0,2=5.5201 and λ_0,3=8.6537 [50] and the three lowest non-axisymmetric modes of $J_1({\lambda }_{1,1}\xi )\cos (\theta )$, $J_2({\lambda }_{2,1}\xi )\cos (2\theta )$ and $J_3({\lambda }_{3,1}\xi )\cos (3\theta )$ with λ_1,1=3.8317, λ_2,1=5.1356 and λ_3,1=6.3802 [50] are plotted in figure 2. When those non-axisymmetric modes are excited, this θ-dependent property will make the detection of mass and its position much more difficult. In addition to the non-axisymmetric modes introducing another variable θ, the mode shape of $a_{ni}{J}_n({\lambda }_{n,i}\xi )\cos (n\theta )+b_{ni}{J}_n({\lambda }_{n,i}\xi )\sin (n\theta )$ also becomes dependent on the initial conditions, which can make the identification of resonance very difficult, because the resonant peak magnitude can change from time to time owing to the change of the initial conditions. As shown in the resonance spectrum of a circular graphene membrane in the experiment described by Barton et al. [29], multiple peaks cluster around their predicted resonant frequencies because of the presence of the degenerate modes. These non-axisymmetric degenerate modes are induced by the asymmetry caused by surface contamination [29]. In the application of the mass resonator sensor, these non-axisymmetric modes should be avoided. On the other hand, when the boundary is well defined [28], which means that no peeling-off occurs, the non-axisymmetric modes will not be excited as long as the membrane initial configuration and excitation force are axisymmetric [49]. Because of the presence of very large residual tensile stress, the graphene layer has a flat initial configuration and a spatially uniformly distributed pressure/load can be easily exerted, which can effectively prohibit the appearance of non-axisymmetric modes.

Figure 2.

Axisymmetric and non-axisymmetric modes. (a–c) The axisymmetric modes of (a) J₀(λ_0,1ξ), λ_0,1=2.4048; (b) J₀(λ_0,2ξ), λ_0,2=5.5201; and (c) J₀(λ_0,3ξ), λ_0,3=8.6537. (e–g) The non-axisymmetric modes of (d) $J_1({\lambda }_{1,1}\xi )\cos (\theta )$, λ_1,1=3.8317; (e) $J_2({\lambda }_{2,1}\xi )\cos (2\theta )$, λ_2,1= 5.1356; and (f) $J_3({\lambda }_{3,1}\xi )\cos (3\theta )$, λ_3,1=6.3802. (Online version in colour.)

If only the axisymmetric modes are excited, equation (2.9) can now be written in a simpler form, as follows [47–49]:

$$V(\xi ,\theta )=V(\xi )=\sum _{i=1}^{\mathrm{\infty }}{a}_i{J}_0({\lambda }_{0,i}\xi ),$$ 2.10

where a_i is the unknown constant associated with each mode. In the computation, only finite mode shapes are used. As shown later, we need to compute only the first two eigenfrequencies and six modes in total are used, which is shown to be accurate enough. Correspondingly, the potential and kinetic energies associated with the axisymmetric deformation become the following:

$$U=\pi r_a^{2}F{\int }_0^1{\left(\frac{\mathrm{\partial }W}{\mathrm{\partial }\xi }\right)}^2\xi \mathrm{d}\xi$$ 2.11

and

$$T=\pi r_a^{4}m{\int }_0^1{\left(\frac{\mathrm{\partial }W}{\mathrm{\partial }t}\right)}^2\xi \mathrm{d}\xi +\frac{\mathrm{\Delta }M{r}_a^2}{2}{{\left(\frac{\mathrm{\partial }W}{\mathrm{\partial }t}\right)}^2|}_{\xi ={\xi }_o}.$$ 2.12

By substituting equations (2.6) and (2.10) into equations (2.11) and (2.12), minimizing the functional T−U, i.e. ∂(T−U)/∂a_i=0 [49,51], leads to the following eigenvalue problem:

$$\mathbf{K}-{\lambda }^2\mathbf{M}=0,$$ 2.13

where $\lambda =\sqrt{m{r}_a^2{\omega }^2/F}$ is the dimensionless eigenfrequency; K and M are the matrices defined as

$${\mathbf{K}}_{ij}=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{a}_i\mathrm{\partial }{a}_j}\left[{\int }_0^1{\left(\frac{\mathrm{d}V}{\mathrm{d}\xi }\right)}^2\xi \mathrm{d}\xi \right]={\int }_0^1\frac{\mathrm{d}{J}_0({\lambda }_{0,i}\xi )}{\mathrm{d}\xi }\times \frac{\mathrm{d}{J}_0({\lambda }_{0,j}\xi )}{\mathrm{d}\xi }\xi \mathrm{d}\xi$$ 2.14

and

$$\begin{array}{rl}& {\mathbf{M}}_{ij}=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{a}_i\mathrm{\partial }{a}_j}\left({{\int }_0^1{V}^2\xi \mathrm{d}\xi +\alpha V^2|}_{\xi ={\xi }_o}\right)\\ & ={\int }_0^1{J}_0({\lambda }_{0,i}\xi )J_0({\lambda }_{0,j}\xi )\xi \mathrm{d}\xi +\alpha J_0({\lambda }_{0,i}{\xi }_o)J_0({\lambda }_{0,j}{\xi }_o),\end{array}$$ 2.15

where the dimensionless parameter α is defined as $\alpha =\mathrm{\Delta }M/(2\pi m{r}_a^2)$, which physically indicates half of the ratio of the adsorbate mass to the total mass of the membrane.

The eigenvalue problem of equation (2.13) is formulated by the Rayleigh–Ritz method, whose accuracy is dependent on the accuracy of the deflection functions used to approximate the membrane mode shape. Instead of the Bessel functions, Timoshenko [51] used the following cosine functions to approximate the mode shape:

$$V(\xi )=a_1\cos \left(\frac{\pi }{2}\xi \right)+a_2\cos \left(\frac{3\pi }{2}\xi \right)+\cdots +a_i\cos \left[\frac{(2i-1)\pi }{2}\xi \right].$$ 2.16

The comparison of the first four J₀(λ_0,iξ) with $\cos [(2i-1)\pi \xi /2]$ is plotted in figure 3. Clearly, as seen in figure 3, when i=1, the difference between J₀(λ_0,1ξ) and $\cos (\pi \xi /2)$ is very small, which results in a 0.5% error in Timoshenko's first eigenfrequency computation compared with the exact one [51]. For i=2,3 and 4, the shape difference between J₀(λ_0,iξ) and $\cos [(2i-1)\pi \xi /2]$ enlarges significantly, which results in a much larger difference in the eigenfrequency computation. The same problem also appears when polynomials are used to approximate the mode shapes of the membrane [49]. Furthermore, even this 0.5% computation error of the first eigenfrequency is unacceptable in the micro/nano mass resonator application. One of the great advantages of the micro/nano mass resonator sensor is its ability to detect tiny fractional changes. For a nanoresonator with an eigenfrequency of gigahertz (10⁹ Hz, GHz) or higher [11,12], this 0.5% difference means a megahertz (MHz) difference. As the exact mode shape of the Bessel function is used in our computations, equation (2.13) gives the exact values. For a nanoscaled mass resonator with a circular configuration, the Bessel functions must be used as the mode shapes.

Figure 3.

(a) Mode shapes of J₀(λ_0,1ξ) and J₀(λ_0,2ξ) in comparison with $\cos (\pi \xi /2)$ and $\cos (3\pi \xi )/2$. A solid triangle indicates the peak of J₀(λ_0,2ξ). (b) Mode shapes of J₀(λ_0,3ξ) and J₀(λ_0,4ξ) in comparison with $\cos (5\pi \xi /2)$ and $\cos (7\pi \xi )/2$. The circles indicate the nodes.

3. Results and discussion

For axisymmetric excitation, the adsorbate mass and its location are the only two factors impacting the system eigenfrequencies. Here, the essential idea of using the two eigenfrequency changes to uniquely determine the adsorbate mass and location is based on the following fact: the response of different eigenfrequencies to the adsorbate mass and its location is different. Figure 4 plots the variations of the first (λ₁) and the second (λ₂) eigenfrequencies as the adsorbate moves from the centre towards the edge. In figure 4, α is fixed as α=0.01. λ₁ monotonically increases from 2.3112 at ξ_o=0 to λ₁=λ_0,1=2.4048 at ξ_o=1. In 0≤ξ_o≤0.44, λ₂ monotonically increases from 4.9872 to λ₂=λ_0,2=5.5201, then in 0.44≤ξ_o≤0.7, λ₂ monotonically decreases from 5.5201 to 5.4447, and finally in 0.7≤ξ_o≤1, λ₂ monotonically increases from 5.4447 to 5.5201. These variations of λ₁ and λ₂ correspond exactly to the mode shape variation as given in figure 3. The magnitude of the first mode shape of J₀(λ_0,1ξ) monotonically decreases from 1 to 0 as ξ changes from 0 to 1. The effect of adsorbate mass is thus largest at the centre and becomes zero at the edge. Therefore, λ₁ is minimum when the adsorbate is at the centre. When the adsorbate is at the edge, λ₁ recovers λ_0,1, which is the eigenfrequency of a membrane with no adsorbate. For the higher mode of J₀(λ_0,iξ) (i≥2), there are i−1 nodes whose displacements are zero. When an adsorbate is at the nodes, the effective mass becomes zero and the eigenfrequency is thus unchanged. For the second mode of J₀(λ_0,2ξ), there is one node at ξ=λ_0,1/λ_0,2=2.4048/5.5201≈0.44, which is marked as a circle in both figures 3 and 4. As seen in figure 3, J₀(λ_0,2ξ) also has a peak at ξ=0.7, which is marked as a solid triangle. The presence of a node and a peak is responsible for the λ₂ variation pattern as seen in figure 4.

Figure 4.

(a) Variation of the first resonance frequency (λ₁) as the adsorbate moves from the membrane centre towards the edge. (b) The corresponding variation of the second resonance frequency (λ₂). The circle corresponds to the node at which λ₂ has no changes. Here, the dimensionless concentrated mass is fixed as α=0.01.

Now let us present how to determine the adsorbate mass and its location by using the different responses of two eigenfrequencies. For α=0.01 and ξ_o=0.2, the first two eigenfrequencies are uniquely determined by equation (2.13) as λ₁=2.3273 and λ₂=5.2992. In the real-world application of mass resonator sensors, the adsorbate mass (α) and its location (ξ_o) are the unknowns to be determined; the eigenfrequencies of λ₁ and λ₂ are the measured quantities. As a forward problem, once α and ξ_o are given, the eigenfrequencies are uniquely determined. However, as an inverse problem, there are infinite combinations of α and ξ_o that yield the same eigenfrequency for any single resonance mode. In figures 5 and 6, α is varied from 0 to 0.02 with a step size of dα=4×10⁻⁴ and ξ_o is varied from 0 to 1 with a step size of dξ_o=2×10⁻². Figure 5 plots the first eigenfrequency of λ₁ as a function of α and ξ_o. The level plane is the one with the fixed value of λ₁=2.3273. The intersection of the two planes is a curve, which physically indicates that there are infinite combinations of α and ξ_o resulting in the eigenfrequency of λ₁=2.3273. Figure 6 plots the second eigenfrequency of λ₂ as a function of α and ξ_o. The level plane is the one with the fixed value of λ₂=5.2992. Again, the intersection of the two planes is a curve, which is the infinite combinations of α and ξ_o resulting in the eigenfrequency of λ₂=5.2992. When the two intersecting curves obtained in figures 5 and 6 are projected into the α−ξ_o plane as shown in figure 7, there is an intersection marked as a circle, which happens to be exactly (α,ξ_o)=(0.01,0.2). Again, the physical reason for the two curves to intersect is that the two eigenfrequencies have different responses to the adsorbate mass and its location. It is noteworthy that, even when an adsorbate lands on the nodal ring of the membrane, the above method can still work. Because only the axisymmetric modes are used, the position detected by the method is the radial position.

Figure 5.

The λ₁ variation as a function of α and ξ_o. The level plane is the one with the constant of λ₁=2.3273. The intersection of the two planes is marked with a solid curve.

Figure 6.

The λ₂ variation as a function of α and ξ_o. The level plane is the one with the constant of λ₂=5.2992. The intersection of the two planes is marked with a dashed curve.

Figure 7.

The projections of the two intersection curves obtained in figures 5 and 6 into the α−ξ_o plane. The intersection of the two curves is marked with a circle, which corresponds to (α,ξ_o)=(0.01,0.2) exactly.

Equation (2.13) is for an undamped system, which has also been used to model many CNT and graphene resonators [14,16,30,31,42]. In reality, because the membrane energy dissipates, damping exists. With the presence of damping, the eigenvalue problem becomes the following:

$$\mathbf{K}\mathbf{G}+\mathbf{D}\frac{\mathrm{d}\mathbf{G}}{\mathrm{d}t}+f\mathbf{M}\frac{{\mathrm{d}}^2\mathbf{G}}{\mathrm{d}{t}^2}=0,$$ 3.1

where the matrices K and M remain the same as those defined in equations (2.14) and (2.15); f is a constant defined as $f=m{r}_a^2/F$; D is the damping matrix defined as D_ij=(ca²/F)M_ij (c is the coefficient of damping [48]); G(t) is the vector associated with time-varying variables and defined as G^T(t)=(g₁(t),g₂(t),…,g_n(t)). Equation (3.1) is the so-called non-gyroscopic damped system [52], and the detailed computation procedure of finding its eigenvalues is presented in [6]. The quality factor, Q, has the relationship of Q∝c⁻¹. There are many studies for the damping mechanisms of small-scaled CNT and graphene resonators [18,19,22,27–29]. c is found by the so-called half-power method from the frequency response curve obtained from experiment [48], which is treated as a known parameter in equation (3.1). c is usually different in different mode and size-dependent [27–29]. The above method of using two resonant frequencies is also applicable to the non-gyroscopic damped system.

Because the model and method are developed for the one adsorbate case, we address the important issue of how to apply the method in a real experiment scenario. In the experiment, the previously adsorbed atoms/molecules/particles can be cleaned off by passing a large electric current through a resonator, which generates Joule heating and boils off the adsorbates [11]. However, it is impossible in a micro/nanomechanical mass resonator experiment to realize the scenario of just one adsorbate landing on a resonator during the spraying process. There are multiple adsorbates on a resonator surface [4,9–11,24]. When the number of adsorbates (N) is large and unknown, which usually is the case in a real experiment, our inverse problem-solving method can still apply in theory with much more complexity and much less efficiency. Furthermore, because the method requires 2N resonant frequencies to be measured to solve the inverse problem, it becomes much more difficult or even impossible in an experiment when N is large. Fortunately, in many micro/nanomechanical mass resonator applications, we do not have to solve the problem of multiple adsorbates. The underlying rationale is in the following three conditions. First, the state-of-art micro/nanomechanical resonators are so sensitive that they can detect the shifts in resonant frequencies induced by a single adsorption event. The step-wise decrease in resonant frequency recorded in the experiments indicates the discrete nature of adsorbates arriving at the micro/nanomechanical resonator surface one by one, which is also the hallmark of sensing the individual adsorption events of one protein [4,24], one atom [9,10] and one molecule [11]. By building a histogram of count versus frequency shift for the ensemble of sequential single gold atom adsorption, Jensen et al. [9] could identify with a certain confidence level that the gold atomic mass ranges between 0.1 and 1 zg compared with the true value of 0.327 zg (1 zg=10⁻²¹ g). Similarly, by building histograms of event probability versus frequency shift for the ensembles of sequential single protein adsorption, Naik et al. [4] and Hanay et al. [24] could tell the masses of oligomers consisting of a ‘nominally pure’ bovine serum albumin protein and the different isoform masses of a human IgM antibody. Second, we need the assumption that the previously adsorbed particles have little or no impact on the step-wise decrease in resonant frequency observed in the above experiments [4,9–11,24], the mass and position of an incoming particle being (solely) responsible for the decrease in resonant frequency. For the case of multiple (unknown) adsorbates, α in equation (2.15) becomes the following:

$$\alpha =\frac{\mathrm{\Delta }M}{2\pi m{r}_a^2+\sum _i^N{M}_i}\approx \frac{\mathrm{\Delta }M}{2\pi m{r}_a^2},$$ 3.2

where N is the (unknown) number of previously adsorbed particles and M_i is the corresponding (unknown) mass; ΔM is the mass of an incoming particle. When the mass of adsorbates is much smaller than that of the resonator, i.e. $\sum _i^N{M}_i\ll \pi m{r}_a^2$, the approximation in equation (3.2) holds. As discussed above, one great advantage of micro/nanomechanical resonator sensors is that they can detect the tiny fractional shifts in resonant frequencies and very tiny masses can thus be sensed. In Chiu's experiment with noble gases as adsorbates [10], the mass of the CNT resonator is around 1000 zg, the atomic masses of xenon and argon are 0.218 zg and 0.066 zg, which corresponds to α=1.09×10⁻⁴ and α=3.3×10⁻⁵, respectively. In Hanay et al.'s experiment [24], α is around 10⁻⁴. In these micro/nanomechanical mass resonator applications, the mass of the previously adsorbed particles has (almost) no impact on the resonant frequency, which is also an implicit assumption used in the above statistics methods [4,9,24]. The third condition is that there is enough time to measure the resonant frequencies between two individual adsorption events, which is why the step-wise decrease in resonant frequency can be monitored. As mentioned above, the resonant frequency of a micro/nanomechanical resonator can be as high as GHz or higher [11,12], the time required to measure the resonant frequency is thus very small. On the other hand, the adsorption rate is relatively slow, for example the argon adsorption rate is 0.09 atoms per second [10]. In conjunction with these three conditions, the inverse problem-solving method is viable in the real application of micro/nanomechanical mass resonators. The inverse problem-solving method also has the advantage of conducting mass sensing with just one datum of one single adsorption event. In the above statistics methods, tens or hundreds of data are needed to ‘decouple’ the two convolving parameters (mass and position of an adsorbate) by assuming a certain distribution rule such as Gaussian [4,9,24]. In contrast, our method can determine the mass and position of an adsorbate with one adsorption event.

4. Conclusion

The method of using two resonant frequencies to detect both the mass and position of an adsorbate on a circular membrane is presented. There are infinite combinations of the mass and its position which can result in the same resonant frequency shift. In this particular application, we show that using two resonant frequencies can uniquely and accurately determine the mass and its position. The two resonant frequencies used here are associated with the lowest two of the axisymmetric mode. As higher axisymmetric modes have higher resonant frequencies and thus better sensitivity, this method can be easily extended to this application scenario by simply calculating the other two different resonant frequencies. The non-axisymmetric modes, which make the theoretical and experimental analyses much more difficult, should be avoided in the drum resonator application. Only the mass effect of an adsorbate is considered by assuming very large in-plane tension. If the adsorbate-induced tension is considered, more resonant frequencies are needed to solve the inverse problem. The possibility of applying this inverse problem-solving method to the general/real scenario of multiple adsorbates in a micro/nanomechanical resonator experiment is also discussed.

Funding statement

The research was supported by the National Natural Science Foundation of China (NSFC nos 11023001 and 11372321).