Vibrational behavior of atomic force microscope beam via different polymers and immersion environments

Abstract

This paper investigates the dynamic behavior of a V-shaped atomic force microscope (AFM) beam using different polymers and immersion environments. Polystyrene (PS), polypropylene (PP), polycarbonate (PC), acrylonitrile butadiene styrene (ABS), and high-impact polystyrene (HIPS) polymers have been used as the polymeric samples. Based on the importance and applications of these polymers in micro- and nanotechnology, it is necessary to find the mechanical properties in nanoscale. Considering these samples for AFM beam as new research can be interesting. Nanoindentation for extension and retraction regimes has been done by NT-MDT SOLVER P47 scanning probe microscope. For calibration of computing in nanoscale, polyethylene and EPDM rubber have been applied. For the mathematical modeling of the dynamic behavior of the V-shaped AFM beam, Timoshenko beam theory has been applied, and for modeling the contact between beam and samples, DMT (Derjaguin–Muller–Toporov) contact theory has been used to consider the adhesion due to a soft sample. Air, water, methanol, and acetone have been considered as beam environments. Finite element modeling (FEM) has been used to obtain the frequency response function (FRF) and resonant frequency of the beam. Mathematica software (version 8) has been used for programming equations in FEM. The results show that increasing the elasticity modules of the samples increases the resonant frequency, but the amplitude of FRF of vertical movement of the beam declines. By increasing the liquid viscosity as the immersion environments, the resonant frequency amplitude of FRF of vertical movement of the beam decreases. The results of theoretical modeling have been compared with the experimental method (by JPK Instruments-Nano Wizard 2 Atomic Force Microscope). The results show agreement.

Appendices

Appendix A. Timoshenko beam model

Here mass, stiffness, and damping matrices (Fig. 27) are introduced using Timoshenko beam principle [24]:

$$ \begin{aligned} & [k]_{e} = \\ & \frac{{2\,\overline{EI} }}{{(1 + \phi )L^{3} }}\left[ {\begin{array}{*{20}lll} {L^{2} (2 + A_{1} + \phi /2)} \hfill &\quad {L^{2} (1 - \phi /2)} \hfill &\quad { - L(3 + A_{1} )} \hfill &\quad { - L(3 + A_{1} )} \hfill \\ {} \hfill &\quad {L^{2} (2 - A_{1} + \phi /2)} \hfill &\quad { - L(3 - A_{1} )} \hfill &\quad { - L(3 - A_{1} )} \hfill \\ {} \hfill & {} \hfill &\qquad 6 \hfill &\qquad 6 \hfill \\ {Sym} \hfill & {} \hfill & {} \hfill &\qquad 6 \hfill \\ \end{array} } \right] \\ \end{aligned} $$

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$$ [m]_{e} = [m_{t} ]_{e} + [m_{r} ]_{e} $$

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where $[m_{t} ]_{e}$ and $[m_{r} ]_{e}$ represent the mass matrix for shear inertia and rotatory inertia effects, respectively.

$$ [m_{t} ]_{e} = \left[ {\begin{array}{*{20}l} {t_{11} } \hfill & {t_{12} } \hfill & {t_{13} } \hfill & {t_{14} } \hfill \\ {} \hfill & {t_{22} } \hfill & {t_{23} } \hfill & {t_{24} } \hfill \\ {} \hfill & {} \hfill & {t_{33} } \hfill & {t_{34} } \hfill \\ {Sym} \hfill & {} \hfill & {} \hfill & {t_{44} } \hfill \\ \end{array} } \right] $$

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$$ \begin{aligned} t_{11} & = \frac{{\rho A_{i} L^{3} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{240} + \frac{\phi }{105} + \frac{1}{168}} \right) + \frac{{\rho A_{j} L^{3} }}{{(1 + \varphi )^{2} }}\left( {\frac{{\phi^{2} }}{240} + \frac{\phi }{140} + \frac{1}{280}} \right), \\ t_{12} & = \frac{{ - \rho A_{i} L^{3} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{240} + \frac{\phi }{120} + \frac{1}{280}} \right) - \frac{{\rho A_{j} L^{3} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{240} + \frac{\phi }{120} + \frac{1}{280}} \right), \\ t_{13} & = \frac{{ - \rho A_{i} L^{2} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{40} + \frac{5\phi }{{84}} + \frac{1}{28}} \right) - \frac{{\rho A_{j} L^{2} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{60} + \frac{9\phi }{{280}} + \frac{1}{60}} \right), \\ t_{14} & = \frac{{\rho A_{i} L^{2} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{60} + \frac{9\phi }{{280}} + \frac{1}{70}} \right) + \frac{{\rho A_{j} L^{2} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{40} + \frac{3\phi }{{70}} + \frac{1}{60}} \right), \\ t_{22} & = \frac{{\rho A_{i} L^{3} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{240} + \frac{\phi }{140} + \frac{1}{280}} \right) + \frac{{\rho A_{j} L^{3} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{240} + \frac{\phi }{105} + \frac{1}{168}} \right), \\ t_{23} & = \frac{{\rho A_{i} L^{3} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{40} + \frac{3\phi }{{70}} + \frac{1}{60}} \right) + \frac{{\rho A_{j} L^{3} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{60} + \frac{9\phi }{{280}} + \frac{1}{70}} \right), \\ t_{24} & = \frac{{ - \rho A_{i} L^{2} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{60} + \frac{9\phi }{{280}} + \frac{1}{60}} \right) - \frac{{\rho A_{j} L^{2} }}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{40} + \frac{5\phi }{{84}} + \frac{1}{28}} \right), \\ t_{33} & = \frac{{\rho A_{i} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{4} + \frac{8\phi }{{15}} + \frac{1}{7}} \right) + \frac{{\rho A_{j} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{12} + \frac{\phi }{6} + \frac{3}{35}} \right), \\ t_{34} & = \frac{{ - \rho A_{i} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{12} + \frac{3\phi }{{20}} + \frac{9}{140}} \right) - \frac{{\rho A_{j} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{12} + \frac{3\phi }{{20}} + \frac{9}{140}} \right), \\ t_{44} & = \frac{{\rho A_{i} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{12} + \frac{\phi }{6} + \frac{3}{35}} \right) + \frac{{\rho A_{j} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{4} + \frac{8\phi }{{15}} + \frac{2}{7}} \right), \\ \end{aligned} $$

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$$ [m_{r} ]_{e} = \left[ {\begin{array}{*{20}l} {r_{11} } \hfill & {r_{12} } \hfill & {r_{13} } \hfill & {r_{14} } \hfill \\ {} \hfill & {r_{22} } \hfill & {r_{23} } \hfill & {r_{24} } \hfill \\ {} \hfill & {} \hfill & {r_{33} } \hfill & {r_{34} } \hfill \\ {Sym} \hfill & {} \hfill & {} \hfill & {r_{44} } \hfill \\ \end{array} } \right] $$

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$$ \begin{aligned} r_{11} & = \frac{{\rho I_{i} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{4} + \frac{\phi }{5} + \frac{1}{10}} \right) + \frac{{\rho I_{j} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{12} - \frac{\phi }{30} + \frac{1}{30}} \right), \\ r_{12} & = \frac{{\rho I_{i} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{12} - \frac{\phi }{12} - \frac{1}{60}} \right) + \frac{{\rho I_{j} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{12} - \frac{\phi }{12} - \frac{1}{60}} \right), \\ r_{22} & = \frac{{\rho I_{i} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{12} - \frac{\phi }{30} + \frac{1}{30}} \right) + \frac{{\rho I_{j} L}}{{(1 + \phi )^{2} }}\left( {\frac{{\phi^{2} }}{4} + \frac{\phi }{5} + \frac{1}{10}} \right), \\ r_{13} & = r_{14} = \frac{{\rho I_{i} }}{{(1 + \phi )^{2} }}\left( {\frac{3\phi }{{10}}} \right) + \frac{{\rho I_{j} }}{{(1 + \phi )^{2} }}\left( {\frac{\phi }{5} - \frac{1}{10}} \right), \\ r_{23} & = r_{24} = \frac{{\rho I_{i} }}{{(1 + \phi )^{2} }}\left( {\frac{\phi }{5} - \frac{1}{10}} \right) + \frac{{\rho I_{j} }}{{(1 + \phi )^{2} }}\left( {\frac{3\phi }{{10}}} \right), \\ r_{33} & = r_{34} = r_{44} = \frac{{\rho I_{i} }}{{L(1 + \phi )^{2} }}\left( \frac{3}{5} \right) + \frac{{\rho I_{j} }}{{L(1 + \phi )^{2} }}\left( \frac{3}{5} \right), \\ \end{aligned} $$

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where $\phi = \frac{12\,EI}{{kGA\,L^{2} }},A_{i} = \frac{{EI_{i} - EI_{j} }}{{EI_{i} + EI_{j} }},\quad \overline{EI} = (EI_{i} + EI_{j} )/2$. We employ the proportional damping matrix as [13]:

$$ C = \varphi^{ - T} C_{{{\text{diagonl}}}} \varphi^{ - 1} ,\;C_{{{\text{diagonl}}}} = {\text{diag}}\left( {\frac{{\omega_{1} }}{{Q_{1} }},\frac{{\omega_{2} }}{{Q_{2} }},\ldots ,\frac{{\omega_{n} }}{{Q_{n} }}} \right),\;\varphi = [\varphi_{1} ,\varphi_{2} ,\ldots ,\varphi_{n} ] $$

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Fig. 27

Tapered element

Appendix B. Hamaker constant

Based on the EDLVO principle, $A_{132}$ as the Hamaker constant between material 1 and 2 in the medium 3 can be expressed as [15]:

$$ A_{132} = \left( {\sqrt {A_{11} } - \sqrt {A_{33} } } \right)\left( {\sqrt {A_{22} } - \sqrt {A_{33} } } \right) $$

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Here, silicon tip has been supposed as the material 2 and polymeric samples are considered as the material 1. Air, water, methanol, and acetone have been used as the medium. Based on [15, 16], Table 8 cab be expressed as:

Table 8 Hamaker constants of materials [25, 26]