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Mass and Force Sensing of an Adsorbate on a Beam Resonator Sensor.

Zhang Y, Zhao YP - Sensors (Basel) (2015)

Bottom Line: Extra instruments are also required.The accuracy of the inverse problem solving method is demonstrated, and how the method can be used in the real application of a nanomechanical resonator is also discussed.Solving the inverse problem is helpful to the development and application of a mechanical resonator sensor for two reasons: reducing extra experimental equipment and achieving better mass sensing by considering more factors.

View Article: PubMed Central - PubMed

Affiliation: State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China. zhangyin@lnm.imech.ac.cn.

ABSTRACT
The mass sensing superiority of a micro-/nano-mechanical resonator sensor over conventional mass spectrometry has been, or at least is being firmly established. Because the sensing mechanism of a mechanical resonator sensor is the shifts of resonant frequencies, how to link the shifts of resonant frequencies with the material properties of an analyte formulates an inverse problem. Besides the analyte/adsorbate mass, many other factors, such as position and axial force, can also cause the shifts of resonant frequencies. The in situ measurement of the adsorbate position and axial force is extremely difficult if not impossible, especially when an adsorbate is as small as a molecule or an atom. Extra instruments are also required. In this study, an inverse problem of using three resonant frequencies to determine the mass, position and axial force is formulated and solved. The accuracy of the inverse problem solving method is demonstrated, and how the method can be used in the real application of a nanomechanical resonator is also discussed. Solving the inverse problem is helpful to the development and application of a mechanical resonator sensor for two reasons: reducing extra experimental equipment and achieving better mass sensing by considering more factors.

No MeSH data available.


The variations of the first three resonant frequencies as an adsorbate moves from one clamped end to the other. Here, the mass and axial load are fixed as and., and are the amplitudes of the three resonant frequencies, which indicate the difference between the maximum and minimum of those frequencies.
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sensors-15-14871-f003: The variations of the first three resonant frequencies as an adsorbate moves from one clamped end to the other. Here, the mass and axial load are fixed as and., and are the amplitudes of the three resonant frequencies, which indicate the difference between the maximum and minimum of those frequencies.

Mentions: Figure 3 examines the variations of the first three resonant frequencies as the adsorbate moves from one end to the other. In Figure 3, = 0.1 and = 10 are fixed; varies from zero to one. Again, the three resonant frequencies respond differently as the adsorbate moves from one end to the other. As seen in Equation (5), the adsorbate actual mass() and its location() are the two intricate factors determining the effective mass for the system. The variation patterns of the three resonant frequencies are actually based on the mode shapes, as presented in Figure 1b. At the boundaries of = 0, 1 and node(s) (i.e., = 0), the effective mass is zero, and the resonant frequencies are thus the maximum. For the first mode, there is no node, and its modal displacement reaches the maximum at = 0.5, which corresponds to the maximum effective mass and, thus, minimum resonant frequency. For the second mode, which has one node at = 0.5 and is marked as a solid circle in both Figure 1b and Figure 3, the resonant frequency reaches its maximum because = 0, and the effective mass is zero. At the same time, the modal displacement of the second mode reaches the maximum at = 0.27 and = 0.73, which are marked as two solid triangles in Figure 1b and Figure 3; the effective mass becomes maximum, and the second resonant frequency thus reaches its minimum. The node at = 0.5 and the two maximum modal displacements at = 0.27 and = 0.73 are responsible for the variation of the second resonant frequency, as presented in Figure 3. A similar analysis can be applied to explain the variation of the third resonant frequency. Here, is defined as the difference of the maximum and minimum of the i-th resonant frequency, and = 2.6437, = 5.8863 and = 9.8944. The fact that > > indicates that a higher mode has higher mass sensitivity, which has been used as a mechanism to detect the mass and location of an accreted particle on a micromechanical resonator [51]. In summary, Figure 2 and Figure 3 demonstrate two things: (1) that the axial load and mass have different impacts on different resonant frequencies; (2) that for given axial load and mass (including its position), different resonant frequencies respond differently. These two things are the very physical mechanism to solve the inverse problem.


Mass and Force Sensing of an Adsorbate on a Beam Resonator Sensor.

Zhang Y, Zhao YP - Sensors (Basel) (2015)

The variations of the first three resonant frequencies as an adsorbate moves from one clamped end to the other. Here, the mass and axial load are fixed as and., and are the amplitudes of the three resonant frequencies, which indicate the difference between the maximum and minimum of those frequencies.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4541812&req=5

sensors-15-14871-f003: The variations of the first three resonant frequencies as an adsorbate moves from one clamped end to the other. Here, the mass and axial load are fixed as and., and are the amplitudes of the three resonant frequencies, which indicate the difference between the maximum and minimum of those frequencies.
Mentions: Figure 3 examines the variations of the first three resonant frequencies as the adsorbate moves from one end to the other. In Figure 3, = 0.1 and = 10 are fixed; varies from zero to one. Again, the three resonant frequencies respond differently as the adsorbate moves from one end to the other. As seen in Equation (5), the adsorbate actual mass() and its location() are the two intricate factors determining the effective mass for the system. The variation patterns of the three resonant frequencies are actually based on the mode shapes, as presented in Figure 1b. At the boundaries of = 0, 1 and node(s) (i.e., = 0), the effective mass is zero, and the resonant frequencies are thus the maximum. For the first mode, there is no node, and its modal displacement reaches the maximum at = 0.5, which corresponds to the maximum effective mass and, thus, minimum resonant frequency. For the second mode, which has one node at = 0.5 and is marked as a solid circle in both Figure 1b and Figure 3, the resonant frequency reaches its maximum because = 0, and the effective mass is zero. At the same time, the modal displacement of the second mode reaches the maximum at = 0.27 and = 0.73, which are marked as two solid triangles in Figure 1b and Figure 3; the effective mass becomes maximum, and the second resonant frequency thus reaches its minimum. The node at = 0.5 and the two maximum modal displacements at = 0.27 and = 0.73 are responsible for the variation of the second resonant frequency, as presented in Figure 3. A similar analysis can be applied to explain the variation of the third resonant frequency. Here, is defined as the difference of the maximum and minimum of the i-th resonant frequency, and = 2.6437, = 5.8863 and = 9.8944. The fact that > > indicates that a higher mode has higher mass sensitivity, which has been used as a mechanism to detect the mass and location of an accreted particle on a micromechanical resonator [51]. In summary, Figure 2 and Figure 3 demonstrate two things: (1) that the axial load and mass have different impacts on different resonant frequencies; (2) that for given axial load and mass (including its position), different resonant frequencies respond differently. These two things are the very physical mechanism to solve the inverse problem.

Bottom Line: Extra instruments are also required.The accuracy of the inverse problem solving method is demonstrated, and how the method can be used in the real application of a nanomechanical resonator is also discussed.Solving the inverse problem is helpful to the development and application of a mechanical resonator sensor for two reasons: reducing extra experimental equipment and achieving better mass sensing by considering more factors.

View Article: PubMed Central - PubMed

Affiliation: State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China. zhangyin@lnm.imech.ac.cn.

ABSTRACT
The mass sensing superiority of a micro-/nano-mechanical resonator sensor over conventional mass spectrometry has been, or at least is being firmly established. Because the sensing mechanism of a mechanical resonator sensor is the shifts of resonant frequencies, how to link the shifts of resonant frequencies with the material properties of an analyte formulates an inverse problem. Besides the analyte/adsorbate mass, many other factors, such as position and axial force, can also cause the shifts of resonant frequencies. The in situ measurement of the adsorbate position and axial force is extremely difficult if not impossible, especially when an adsorbate is as small as a molecule or an atom. Extra instruments are also required. In this study, an inverse problem of using three resonant frequencies to determine the mass, position and axial force is formulated and solved. The accuracy of the inverse problem solving method is demonstrated, and how the method can be used in the real application of a nanomechanical resonator is also discussed. Solving the inverse problem is helpful to the development and application of a mechanical resonator sensor for two reasons: reducing extra experimental equipment and achieving better mass sensing by considering more factors.

No MeSH data available.