Theoretical study on the sensitivity of dynamic acoustic force measurement through monomodal and bimodal excitations of rectangular micro-cantilever

In numerical calculations, we obtain oscillations of a rectangular micro-cantilever excited at its resonance frequencies in measurement of dynamic acoustic forces. We introduce two excitation schemes for micro-cantilever actuation which are single- and bimodal-frequency excitation schemes (figure 1). In single-frequency excitation scheme, external driving force is applied to the micro-cantilever at a single frequency, that is its first or second eigenmode frequency. On the other hand, bimodal-frequency excitation scheme consists of application of driving forces at the first two eigenmode frequencies simultaneously. The EOMs are approximated by using point-mass object in consideration of the dynamic model of forced and damped harmonic oscillator [24]. The EOMs for instantaneous deflection z_i (t) which are the second order differential equations are introduced below for single-frequency (Equation (1)) and bimodal-frequency (equation (2)) excitation schemes.

$$\begin{eqnarray}&&{m}_{e}\ddot{{z}_{i}}(t)=-{k}_{i}{z}_{i}(t)-\displaystyle \frac{({m}_{e}{\omega }_{i})}{{Q}_{i}}\dot{{z}_{i}}(t)+{F}_{{exc},i}+{F}_{{Acoustic}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{m}_{e}\ddot{{z}_{i}}(t)=-{k}_{i}{z}_{i}(t)-\displaystyle \frac{({m}_{e}{\omega }_{i})}{{Q}_{i}}\dot{{z}_{i}}(t)+{F}_{{exc},1}+{F}_{{exc},2}+{F}_{{Acoustic}}\end{eqnarray} \tag{ 2 }$$

where m_e , k_i , Q_i , ω_i are the effective mass, the spring constant, the quality factor and the angular resonance frequency (ω_i = 2 π f_ei ) of the flexural mode i of the micro-cantilever. i is equal to 1 and 2 for the first and second eigenmode respectively. The micro-cantilever operates in air, whose influence on oscillation characteristic is represented by the quality factor [11]. F_exc,i is the excitation force at eigenmode frequency f_ei (equation 3). Since we investigate resonant dynamics of the micro-cantilever under dynamic acoustic force, frequency of external driving force is equal to eigenmode frequency of the micro-cantilever. In consideration of damping effect, maximum amplitude sensitivity to dynamic acoustic force is obtained by driving the micro-cantilever at its resonance frequencies.

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In our case, responses of the micro-cantilever to acoustic forces at different frequencies f_Ac are investigated for both excitation schemes. F_Acoustic is formulated as cosine wave at angular frequency (ω_Ac = 2π f_Ac ) with acoustic force strength F_Ac (equation (4)).

$$\begin{eqnarray}&&{F}_{{exc},i}={F}_{i}\cos ({\omega }_{i}t)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{F}_{{Acoustic}}={F}_{{Ac}}\cos ({w}_{{Ac}}t)\end{eqnarray} \tag{ 4 }$$

Deflections of the micro-cantilever z_s (t) for single-frequency excitation scheme (equation 5) and z_b (t) for bimodal-frequency excitation scheme (equation 6) are approximated as follows [24]:

$$\begin{eqnarray}&&{z}_{s}(t)={z}_{i}(t)\approx {A}_{i}\cos ({\omega }_{i}t-{\phi }_{i})\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{z}_{b}(t)=\sum _{i=1}^{2}{z}_{i}(t)+E(\epsilon )\approx \sum _{i=1}^{2}{A}_{i}\cos ({\omega }_{i}t-{\phi }_{i})\end{eqnarray} \tag{ 6 }$$

where A_i is the oscillation amplitude and ϕ_i is the phase shift of the ith eigenmode. The term E() contains contributions of oscillations at higher modes. More interestingly, since we explore the effect of multi-frequency excitation on measurement sensitivity, observables A_i and ϕ_i obtained using both excitation schemes are compared and evaluated in consideration of mode interaction. Different eigenmodes of the micro-cantilever under excitation forces at eigenmode frequencies are coupled by the virial of the excitation forces in the presence of dynamic acoustic forces. Moreover, observables of the second eigenmode are calculated in order to evaluate measurement sensitivity of acoustic forces at higher frequencies.

Acoustic force acting on the micro-cantilever determines oscillation characteristic of the micro-cantilever under excitation force. Changes in time-varying observables such as amplitudes and phase shift are linked to energy conservation and dissipation processes via analytical expressions [21]. Therefore, dissipated power and stored energy of the micro-cantilever undergoing acoustic forces could be utilized as indicator parameters for measurement sensitivity. For both excitation schemes, energy is supplied externally to the micro-cantilever by excitation force in the steady-state. There exist energy losses due to dissipative interaction with the operating environment and acoustic forces. Phase shift between free oscillation and response in the presence of acoustic force is directly related to the amount of energy dissipation.

The external work done E_ext,i by the excitation force and energy exchange with the operating environment E_env,i per period ($T=\tfrac{2\pi }{{\omega }_{i}}$) for i^th mode are described as follows [25]:

$$\begin{eqnarray}&&{E}_{{ext},i}={\int }_{0}^{T}{F}_{{exc},i}\dot{{z}_{i}}(t)\,{dt}=\pi {A}_{i}{F}_{i}\sin ({\phi }_{i})\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{E}_{{env},i}=\displaystyle \frac{{k}_{i}}{{\omega }_{i}{Q}_{i}}{\int }_{0}^{T}{\left(\dot{{z}_{i}}(t)\right)}^{2}\,{dt}=\displaystyle \frac{\pi {\left({A}_{i}\right)}^{2}{k}_{i}}{{Q}_{i}}\end{eqnarray} \tag{ 8 }$$

The energy dissipation per cycle is determined using energy conservation principle. The energy dissipation E_diss,i per period for i^th mode is calculated by using following equations [25]:

$$\begin{eqnarray}&&{E}_{{diss},i}={E}_{{ext},i}-{E}_{{env},i}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{E}_{{diss},i}=\pi {A}_{i}\left({F}_{i}\sin ({\phi }_{i})-\displaystyle \frac{{k}_{i}{A}_{i}}{{Q}_{i}}\right)\end{eqnarray} \tag{ 10 }$$

Dissipated power through interaction with acoustic force is also utilized as an indicator of energy dissipation. Since deflections of the micro-cantilever could be treated with diverse periods, dissipated power P_i is also defined beside the term of dissipated energy [21]. Alternatively, virial term V_i represents the stored energy of the micro-cantilever experiencing dynamic acoustic force and energy conservation process.

$$\begin{eqnarray}&&{P}_{i}=\displaystyle \frac{{E}_{{diss},i}}{T}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{P}_{i}=\displaystyle \frac{1}{2}{F}_{i}{A}_{i}{\omega }_{i}\sin ({\phi }_{i})-\displaystyle \frac{{k}_{i}{\left({A}_{i}\right)}^{2}{\omega }_{i}}{2{Q}_{i}}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{V}_{i}=-\displaystyle \frac{1}{T}{\int }_{0}^{T}{F}_{{exc},i}{z}_{i}(t)\,{dt}=-\displaystyle \frac{1}{2}{F}_{i}{A}_{i}\cos ({\phi }_{i})\end{eqnarray} \tag{ 13 }$$

Dissipated power and virial per cycle for ith mode in single- and bimodal-frequency excitations are calculated using the analytical expressions ((Equation (12)) and (equation (13))) respectively [8].

In our calculations, the EOMs ((Equation (1)) and (equation (2))) are solved numerically using Fourth order Runge-Kutta method.

Acoustic force frequencies and their spectral regions with respect to the eigenmode frequencies are introduced in table 1. The acoustic force frequency in region A, f_a is selected as quite lower than the first eigenmode frequency. The acoustic force frequency f_i in region E is selected as higher than the second eigenmode frequency. The acoustic force frequency f_e is set to the average value of the eigenmode frequencies f_e1 and f_e2. Region B and D indicate the frequency bands arranged around the first and second eigenmode frequencies respectively. Frequency region B contains three acoustic force frequencies which are f_b , f_c and f_d . Besides, frequency region D possesses three acoustic force frequencies which are f_f , f_g and f_h . In region B and D, f_c and f_g are equal to the first and second eigenmode frequency respectively. In our study, we explore responses of the micro-cantilever to acoustic forces at the frequencies around eigenmode frequencies. Because of that reason, diverse frequency regions with respect to the eigenmode frequencies are constructed. Even if the eigenmode frequencies are well-separate, we investigate the effect of multifrequency excitation on the observables and energy quantities by comparing its results with the ones obtained using a single-frequency excitation scheme. Accordingly, this allocation of the frequency bands improves assessment of the interaction of flexural modes in bimodal-frequency excitation.

Micro-cantilever properties including dimensional parameters, i.e. length(l), width(w) and thickness(t), and coefficients of EOMs for both excitation schemes are presented in table 2 as given in [24].

The effective mass of the micro-cantilever m_e is determined using the expression (equation (14)) as introduced in [26]. The mass density ρ of the micro-cantilever is equal to 2320kg m⁻³ as in [24].

$$\begin{eqnarray}&&{m}_{e}=\displaystyle \frac{33}{140}{lwt}\rho ,\end{eqnarray} \tag{ 14 }$$

Acoustic force strengths are determined by multiplying the one-side surface area of the micro-cantilever (A_surface = lxw) with effective sound pressure levels (Pa) [7]. Sound pressure levels in the range of 34.9-54.9 dB correspond to acoustic force strengths, ranging between 10 pN to 100 pN [27]. The lower and upper limits of the range are determined based on degree of observable sensitivities in numerical simulations. Dynamic acoustic force is assumed to act on the free end of the micro-cantilever in consideration of validity of point-mass model (figure 1). Magnitudes of excitation forces at the first two eigenmode frequencies (F₁ and F₂) are set to 10.6 and 35.2 pN for single- and bimodal-frequency excitations respectively. The time interval of numerical simulation is between 19.0 and 19.20 ms. Initial conditions such as initial deflections and initial velocities are set to zero and applied to solve the second order differential equations ((Equation (1)) and (equation (2))) numerically in Matlab.

Time-dependent values of amplitude and phase shift are determined over a particular period of an oscillation [28]. In the algorithm, for z₁ oscillations, we determine an amplitude by taking average of first two amplitudes of an oscillation in the time interval between 19.14 and 19.18 ms [24]. For z₂ oscillations, we calculate an amplitude by taking average of first two amplitudes in the time interval between 19.15 and 19.16 ms. Besides, phase shift is determined using free oscillation and response of the micro-cantilever under acoustic force.

figures 4(a)–(d) depict the amplitude and phase shift curves as a function of acoustic force strength for the frequencies given in table 1. To visualize mode interactions in multifrequency excitations, results obtained using the excitation schemes are given in the same sub-plots in figure 4 for the critical frequencies. As the acoustic force strength is increased from 10 pN to 100 pN, gradual increase (approximately 2.2 nm) in A₁ is obtained for the frequency f_c for both excitation schemes (figure 4(a)). For the other frequencies in region B (f_b and f_d ), the same trend takes place in amplitude curves. Amplitude sensitivity does not exist for the acoustic force frequencies in the other regions. Moreover, phase shift becomes more evident for the frequency f_d (ϕ₁ of approximately 38.5° for the acoustic force strength of 100 pN in bimodal-frequency excitation) as it is seen from figure 4(b). Herein, amplitude response to acoustic force strength is much more significant than phase shift at the first eigenmode for measurement sensitivity.

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As acoustic force strength is increased gradually in the range of 10-100 pN, A₂ increases steadily for the frequency f_g and f_i (figure 4(c)). There does not exist any notable phase shift sensitivity to acoustic force strength for the frequency f_i , whereas maximum phase shift of around 60° is obtained for the frequency f_g . For the frequency f_i , resonant micro-cantilever under multifrequency operation responds to dynamic acoustic forces at the frequencies quite higher than second eigenmode frequency. Maximum phase shift ϕ₂ of around 178° is obtained for the frequency f_h (figure 4(d)). In addition, a steeper phase shift transition of around 148° in the acoustic force strength range of 25-50 pN is observed for the frequency f_h in bimodal-frequency excitation. Herein, free oscillation has much more interaction conditions with the dynamic acoustic forces at the frequency f_h leading to sharper and larger phase shifts. As it is seen from figure 4(c) and (d), A₂ and ϕ₂ sensitivities are enhanced notably for the acoustic force strength range of 35-65 pN at the frequency f_h using bimodal-frequency excitation scheme. For this case, sensitivity improvement is achieved by driving the micro-cantilever at higher eigenmode frequency. Additionally, it should be noted that fluctuating behavior and lower amplitude sensitivity are observed for the acoustic force frequency f_f as it is illustrated in figure 4(c). Based on these results, excitation parameters such as magnitudes of driving forces are to be optimized to increase observable sensitivities.

Virial and energy conservation theorems are used to explain energy dissipation process during measurement of dynamic acoustic forces. Results of dissipated power and virial with respect to acoustic force strength for the frequencies given in table 1 are plotted in figure 4(e-h). Virial at the first eigenmode V₁ for both excitations decreases steadily for the acoustic force frequencies in region B as it is seen from figure 4(e). Negative values of virial represent energy dissipation from stored kinetic energy of the resonant micro-cantilever, which increases with interaction through larger acoustic force strength. Rates of dissipated powers at the first eigenmode P₁ decrease in the negative range as acoustic force strength increases (figure 4(f)). Negative values of dissipated power indicate transferring energy from supplied energy to the dissipative operating environment in the presence of dynamic acoustic forces based on the energy conservation principle (equation (9)). Dissipated power P₂ increases continuously for the frequency f_g as acoustic force strength increases gradually (figure 4(h)). On the other hand, there does not exist any change in virial V₂ with respect to acoustic force strength for the frequency f_g . Dissipated energy of the micro-cantilever undergoing tip-sample interaction force is related with phase shift in AFM operations [29]. In our case, significant level of power dissipation due to dynamic acoustic forces is evident for the frequency f_g at which considerable amount of amplitude and phase shift are obtained (figure 4(c)(d)(h)). On the other hand, dissipated power curve P₂ for the frequency f_h increases and decreases as acoustic force strength ascents (figure 4(h)). This erratic behavior is due to different dissipation grades of kinetic energy from the mechanically oscillating system. Moreover, except for the frequency f_i , notable fluctuations in energy quantities are not observed for the acoustic force frequencies outside the region D.

Analytical expressions of energy quantities such as supplied energy (Equation (7)), energy transferred to the operating environment (Equation (8)), and dissipated energy owing to dynamic acoustic force (equation (10)) are derived by using virial theorem and energy conservation principle. Using those equations, the energy quantities for the critical acoustic force frequencies in regions B and D are calculated and the results are plotted in figure 5. Phase shift is related to the amount of energy quantities for dynamical system of resonant micro-cantilever under tip-sample interaction force based on analytical expressions [30]. For our case, most of the supplied energy dissipates due to interaction with dynamic acoustic forces at the frequency f_h (figure 5(f)). On the other hand, as it is seen from figure 5(a), supplied energy is transferred to the operating environment for the frequency f_b in consideration of smaller phase shifts at the first eigenmode (figure 4(b)). Moreover, there do not exist considerable deviations among amounts of energy exchanged with the operating environment E_env,i for both excitation schemes (figure 5(a-f)). Unlike z-trajectories, velocities ($\dot{{z}_{i}}$) of the resonant micro-cantilever are not influenced by mode interaction in multi-frequency excitation. For the frequencies in region B, E_ext,i and E_diss,i obtained using single- and bimodal-frequency excitation schemes deviate from each other as acoustic force strength increases (figure 5(a-c)). We would like to emphasize that interaction of flexural modes is evident in bimodal-frequency excitation based on energy levels for those frequencies. Ideally, energy levels at the first and second eigenmodes depend on amplitudes, influenced by acoustic force strengths.

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