The previous chapters demonstrate that AFM bending experiments are a reliable tool for the nanomechanical characterization of fibers. A prerequisite, however, is the correct identification and interpretation of the boundary conditions. For some measurements (e.g., Chapter 9), we found stiffness profile shapes that could not be described by any of the common beam theory models or by a mixture of those.
Therefore, there was a need to elucidate the possible sources of such a behavior.
While most of the work in the literature has focused on non-ideal conditions at the fiber supports, Chapter 7 investigates the effect of a slack fiber, which represents an irregularity at the midsection. Our goal was to estimate how a certain degree of slack will affect the apparent mechanical properties and present guidelines on how to de-tect such influences and avoid a misinterpretation of the results. Since this effect is difficult to study experimentally (in the sense that a well-defined and systematically varied slack cannot be readily introduced), we used finite element (FE) analysis to simulate stiffness profiles for fibers with various degrees of slacking. We found that the slack produced a misleading shape of the stiffness profile (Figure 2.7). The dom-inating effect for the profile change was the slack-to-radius ratio. For moderate slack depths (comparable to the radius), the shape of the stiffness profile resembled the simply supported beam model, although the fiber was firmly clamped in the sim-ulations. Evaluation using the SSBM however would lead to an overestimation of Young’s modulus by over one order of magnitude.
In addition, we also investigated large-deformation-measurements. Experimentally, those measurements are often realized by applying the load in the substrate plane.
To see how this affects the measurements, we performed simulations with vertical loading (i.e., in the same direction as the slack) and lateral loading (i.e., perpendic-ular to the slack, Figure 2.8). The simulations revealed that lateral measurements were not significantly influenced by the slack, no matter if performed within the small-deformation regime or the large-deformation regime. Therefore, they could be an experimental solution to deal with samples where slack is an issue.
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1.0 µm 0.7 µm 0.5 µm 0.3 µm 0.1 µm No slack
Figure 2.7Stiffness profiles obtained from the FE simulations. For the sake of clarity, not all simulated profiles are shown here. (Chapter 7)
Figure 2.8Volume element model used for studying the effect of loading the fiber in and perpendicular to the slack direction (Chapter 7).
2.6 Beyond Small Deformations
All previous chapters mainly focused on the linear elastic properties of the fibers.
However, for many applications, also properties beyond the linear elastic regime are important. A possible approach to study these with bending experiments is dis-cussed in Chapter 8. By applying a lateral instead of a vertical deformation, the fibers can be deformed until failure. In this setup, the fiber experiences a combina-tion of bending and tension, and an appropriate evaluacombina-tion is considered.
As the first important step for a quantitative evaluation of these measurements, a lat-eral calibration approach of the AFM setup is presented. In contrast to vertical mea-surements, the lateral calibration is more complicated, less precise, and although several approaches have been reported in the literature, there are no standard tech-niques so far. In our approach, the lateral sensitivity is determined in the same man-ner as the vertical sensitivity, using the slope of the measured signal in the constant compliance regime when applying forces to a hard substrate (Figure 2.9). Using the vertical steps in the structured glass substrates that are also used for the bending experiments ensured that the cantilever hits the calibration substrate at almost the same position as the fibers in the measurements, thus keeping the lever arm (i.e., the distance of the contact point to the reflective side of the cantilever) constant. The lateral spring constant was calculated from the geometry of the cantilever.
Cantilever
Figure 2.9(a) Sketch of the lateral sensitivity calibration using the step of the struc-tured glass substrates. (b) Lateral deflection-displacement curve obtained by push-ing the cantilever against the undeformable substrate (Chapter 8).
We compared vertical and lateral measurements on BTA fibers in the small-deformation regime and found a good agreement between both small-deformation modes, although the lateral measurements were much more prone to scatter, most likely due to the unavoidable uncertainty of the lateral calibration. Building up on these results, we performed large deformation measurements until failure (Figure 2.10).
Only the very thin fibers of one trisamide system could reliably be broken, since thicker fibers detached from the substrate before fracture. Nevertheless, we were able to estimate the flexural strength of the investigated system, which was compa-rable to that of Nylon.
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Heidelberg fit
Figure 2.10Example of a lateral force curve until failure. The nonlinear part in the beginning describes a purely elastic behavior of the material, but includes geomet-ric nonlinearities (Chapter 8).
The work in this chapter demonstrates that lateral experiments can be a powerful addition to the vertical bending setup, but there remain some drawbacks that have to be addressed in future work. The most fundamental requirement in order to ap-ply the lateral experiments to a wide variety of fibers is a suitable approach for a fixation on the substrate. In addition, the lateral calibration remains a major error source despite the presented improvements. Once these issues are solved, the pos-sibility to combine vertical and lateral bending on exactly the same position allows characterization of the mechanical properties within and beyond the linear elastic regime with outstanding reliability.