Verbiest G J, Rost M J
JARA-FIT and II. Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany.
Huygens-Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands.
Ultramicroscopy. 2016 Dec;171:70-76. doi: 10.1016/j.ultramic.2016.07.018. Epub 2016 Aug 16.
To make the forces in an Atomic Force Microscope that operates in a dynamic mode with one or multiple vibrations applied to the cantilever, quantitative, one needs to relate a change in resonance frequency of the cantilever to a specific tip-sample interaction. Due to the time dependence of the force between the tip and sample caused by the vibrations, this task is not only difficult, but in fact only possible to solve for certain limiting cases, if one follows common theoretical approaches with a Taylor expansion around the deflection point. Here, we present an analytical method for calculating the resonance frequencies of the cantilever that is valid for any tip-sample interaction. Instead of linearizing the tip-sample interaction locally, we calculate an averaged, weighted linearization taking into account all positions of the tip while vibrating. Our method bridges, therefore, the difficult gap between a free oscillating cantilever and a cantilever that is pushed infinitely hard into contact with a surface, which describes a clamped-pinned boundary condition. For a correct description of the cantilever dynamics, we take into account both the tip mass and the tip moment of inertia. Applying our model, we show that it is possible to calculate the modal response of a cantilever as a function of the tip-sample interaction strength. Based on these modal vibration characteristics, we show that the higher resonance frequencies of a cantilever are completely insensitive to the strength of the tip-sample interaction.
要使在动态模式下运行的原子力显微镜中的力实现量化，即对施加在悬臂上的一种或多种振动进行量化，就需要将悬臂共振频率的变化与特定的针尖 - 样品相互作用联系起来。由于振动导致针尖与样品之间的力随时间变化，这项任务不仅困难，而且实际上，如果遵循围绕偏转点进行泰勒展开的常见理论方法，仅在某些极限情况下才能解决。在此，我们提出一种计算悬臂共振频率的分析方法，该方法对任何针尖 - 样品相互作用均有效。我们不是在局部对针尖 - 样品相互作用进行线性化，而是在考虑针尖振动时的所有位置的情况下，计算一种平均加权线性化。因此，我们的方法弥合了自由振荡悬臂与被无限用力推向与表面接触的悬臂（这描述了一种夹支 - 铰支边界条件）之间的困难差距。为了正确描述悬臂动力学，我们同时考虑了针尖质量和针尖转动惯量。应用我们的模型，我们表明可以计算悬臂的模态响应作为针尖 - 样品相互作用强度的函数。基于这些模态振动特性，我们表明悬臂的较高共振频率对针尖 - 样品相互作用强度完全不敏感。
