Figure 3.

a) The cantilever motion during oscillation in bimodal imaging with only a small portion interacting with the sample. Note that the oscillation of the second eigenmode cannot be seen, as it is assumed infinitely small by the model proposed here. The stiffness profile of the interacting tip is plotted for three tip shapes: b) punch, c) paraboloid, d) cone. The weight functions used for integrating these stiffness profiles are shown for e) the first eigenmode (assuming a large amplitude) and f) the second eigenmode (assuming a small amplitude). The weight function for the second eigenmode diverges where the interaction stiffness is nonzero, reflecting the sensitivity of bimodal AFM to nanomechanical properties. The binomial approximation is illustrated for both weight functions; they are very accurate approximations in the interaction range, and can be used to derive a simple analytical solution for bimodal imaging. Note: the three different x-axes shown in these plots are interchangeable; they reflect the most appropriate parameterization of the tip position in each case.