This thesis addresses development of a new force spectroscopy tool, correlation force spectroscopy (CFS), for the measurement of the mechanical properties of very small volumes of material (molecular to µm³) at kHz-MHz time-scales. CFS is based on atomic force microscopy (AFM) and the principles of CFS resemble those of dual-trap optical tweezers. CFS consists of two closely-spaced micro-cantilevers that undergo thermal fluctuations. Measurement of the correlation in thermal fluctuations of the two cantilevers can be used to determine the mechanical properties of the soft matter, e.g. a polymeric molecule, that connects the gap between the two cantilevers. Modeling of the correlations yields the effective stiffness and damping of the molecule. The resolution in stiffness is limited by the stiffness of the cantilever and the frequency by the natural frequency of the cantilevers, but, importantly, the damping resolution is not limited by the damping of the cantilever, which has enabled high-resolution measurements of the internal friction of a polymer. The concept of CFS was originally presented by Roukes' group in Caltech [Arlett et al., Lecture Notes in Physics, 2007]; I developed the first practical versions of CFS for experimentation, and have used it in two applications (1) microrheology of Newtonian fluids and (2) single molecule force spectroscopy. To understand the correlation in thermal fluctuations of two cantilevers I initially validated the theoretical approach for analyzing correlation in terms of deterministic model using the fluctuation-dissipation theorem [Paul and Cross, PRL, 2004]. I have shown that the main advantages of such correlation measurements are a large improvement in the ability to resolve stiffness and damping. Use of CFS as a rheometer was validated by comparison between experimental data and finite element modeling of the deterministic vibrations of the cantilevers using the known viscosity and density of fluids. Work in this thesis shows that the data can also be accurately fitted using a simple harmonic oscillator model, which can be used for rapid rheometric measurements, after calibration. The mechanical properties of biomolecules such as dextran and single stranded DNA (ssDNA) are also described. CFS measurements of single molecule properties of ssDNA reveal the internal friction of the molecule in solution. / Ph. D.

A number of microcantilever systems of fundamental importance are explored using theoretical and numerical methods to quantify and provide physical insights into the dynamics of experimentally accessible systems that include a variety of configurations and viscous fluids. It is first shown that the correlated dynamics of both a laterally and vertically offset cantilever pair can be accurately predicted by numerical simulations. This is verified by comparing the correlated dynamics yielded by numerical simulations with experimental measurement. It is also demonstrated that in order to obtain these accurate predictions, geometric details of the cantilever must be included in the numerical simulation to directly reflect the experimental cantilever. A microrheology technique that utilizes the fluctuation-dissipation theorem is proposed. It is shown that by including the frequency dependence of the fluid damping, improvements in accuracy of the predictions of the rheological properties of the surrounding fluid are observed over current techniques. The amplitude spectrum of a 2-D cantilever in a power-law fluid is studied. The resulting amplitude spectrum yielded a curve similar to an overdamped system. It is observed that the amplitude and noise spectrum yield the same qualitative response for a 2-D cantilever in a shear thinning, power-law fluid. The correlated dynamics of a tethered vertically offset cantilever pair is investigated. It is shown that for a range of stiffness ratios, which is the ratio of the spring constant of the tethering relative to the cantilever spring constant, the change in the correlated dynamics of a Hookean spring tethered cantilever pair can be seen in the presence of fluid coupling. The dynamics of a spring-mass tethered, vertically offset cantilever pair is qualitatively studied by simplifying the model to an array of springs and masses. The resulting study found that the correlated dynamics of the displacement of mass of the tethered object yielded newly observed features and characteristics. It is shown that the curve shape of the cross-correlation of the displacement of the mass of the tethered object is similar to that of the auto-correlation of the displacement of the mass representing a step forced cantilever. The cross-correlation of the displacement of the mass of the tethered object, however, is found to be significantly more dependent on the stiffness ratio than the auto-correlation of the displacement of the mass representing a cantilever for t > 0. At t = 0, it is observed that the mass of the tethered object yields the same finite value for the cross-correlation for all studied values of the stiffness ratio. This characteristic is a result of the symmetry of the studied spring-mass system. / Ph. D.