We developed resonance compensating chirp mode (RCCM), an atomic force microscopy (AFM) technique to measure the frequency dependence of the complex shear modulus of live cells over a large bandwidth (quasi-) continuously. RCCM works by applying a continuous frequency sweep (chirp) to the z-scanner and recording the resulting cantilever deflection at high speed. From this data, the frequency-resolved complex shear modulus is extracted. To reach a high maximum frequency, we iteratively shaped the chirp signal to compensate for scanner resonances. This allowed us to measure at frequencies five times higher than the resonant frequency of the scanner. Using a high-speed AFM with small cantilevers, we measured the complex shear modulus of live fibroblast cells in a continuous range between 5 Hz and 30 kHz. We found that the modulus and the loss tangent exhibit a power-law behavior throughout this frequency range. A short chirp duration of 200 ms allowed us to map live cells and generate spatially resolved images of the power-law parameters within minutes. These maps represent a unique combination of high spatial and frequency resolution, low measurement duration, and high maximum frequency.

Immersion microscopy optics may include liquid droplets (e.g., water) to control the light pathway and the numerical aperture of an optical system. Changing the distances between the optical system and an object slide for image focusing also changes the shape (especially the diameter) of the droplet and the surface energy, thus leading to forces acting on both optics and object slides. We examine these effects analytically and derive a numerical model using numerical integration of a recursive integral to predict the force resulting from a liquid droplet changing its shape in the system. Our solutions show that an alteration of the distance leads to a time-dependency of the droplet surface, which is reflected in the corresponding surface and meniscus energies. With this, we can calculate the hydrostatic force that pulls both optical surfaces closer to each other and simulate the time-dependent equilibration of the system.