Study On Measurement For Spring Constants Of Microcantilevers In Atomic Force Microscopy

Atomic force microscopy (AFM) is widely used as a tool for detection of small forces, and becomes an important measuring instrument in the research field of nanotribology and bio-mechanics. The spring constant is an important bridge in converting AFM signals to forces loaded on the tips, and it is of great importance to develop a reliable and easy method for cantilevers' spring constant measurement. This dissertation is focusing on dynamic method for determination of spring constants of AFM cantilevers.A simplified theoretical modeling as well as a FEA method is developed for researching the frequency response of rectangular and v-shaped cantilevers in vacuum. Simulation on some common used rectangular and v-shaped cantilevers has been down for validation. Modeling for interpreting the relationship between young's modulus and the base frequency of out-of-plane bending vibration is well developed, and validated through a FEA method.It is well known that the frequency response of an elastic beam is strongly dependent on the fluid in which it is immersed. The frequency response of cantilevers immersed in fluid is detailed discussed in the dissertation for the convenience of research on quantitative analysis for dynamic properties of microcantilevers and the preparation for theoretical fundaments of spring constant determination through dynamic tests methods. A simplified one-dimension modeling for frequency response of microcantilevers immersed in fluid is developed to ease the problem as the damping vibration of a spring-mass system at first. Consequently, the formula indicating the ration between the natural frequency of cantilevers in vacuum and fluid is presented. Because the simulation based on the simplified method show a evident difference against experimental experiences, the fluid-structure interaction (FSI) problem of microcantilevers immersed in fluid is discussed lately. Two methods are presented for solving the FSI problem of microcantilevers, which are air-spring modeling method and computational fluid dynamic (CFD) method, comparative simulation have been down on a rectangular cantilever and a v-shaped cantilever, which indicates a good agreement with the experience. A dynamic method for determination of cantilever spring constants is presented based on the CFD modeling presented in this dissertation, which involves three steps. The first step is measuring the resonance frequency of a micro cantilever immersed in air. The second step is calculation of resonance frequencies of the microcantilever with different young's modulus, and the young's modulus can be determined after comparison with the measured values. The third step is taking the determined young's modulus to the FEA model of the microcantilever to determine its spring constant. The spring constants of a NSC14/No Al, a NSC11/No Al-large cantilever and a NSC11/No Al-small cantilever have been measured based in the dynamic method.The effective spring constant of microcantilevers are differing from theoretical ones, and terrible errors could happen with out consideration of it. The effect of cantilever tilt angle, in-plane deformation, contact stiffness and tip position on effective spring constants of cantilevers are detailed discussed. Theoretical formulae have been presented to correct the spring constants. Related experiments on the cantilevers used in this dissertation have been done for validation. The effect of those factors on AFM based nanotribology measurements are discussed, and a correction method in lateral force calibration is presented, which is of great value to the users and designers of AFM.