The Hertz model describes the forced elastic indentation of a homogeneous, smooth sample of infinite thickness. Even if these requirements are not fulfilled, the Hertz model is still a good approximation, as long as (1) the sample is locally homogenous, (2) the sample roughness is small in comparison to the size of the tip, and (3) the indentation is small in comparison to the sample thickness. On top of the LC beams, for example, these requirements were mostly met. Here, only the spherical tip end touched the comparably smooth and homogenous sample surface during nanoindentation, and the recorded force curves could be clearly separated in a flat non-contact region and in a contact region where the force increased monotonously with increasing indentation. Consequently, the Hertz model fits matched the force curve data with adequate accuracy and thus yielded reliable stiffness values. At other positions, the tip-sample contact was less well defined. Directly adjacent to steep structures (e.g., at the rims of the LC beams), the sample surface was sometimes contacted by the cantilever beam instead of or in addition to the tip end. In other cases, the tip slipped off the beam sideways. On inhomogeneous, soft, and thin regions, such as the remains of the optic nerve axons, the tip sometimes penetrated the sample until it sensed the much stiffer substrate. For these non-Hertzian contacts, a clear separation into non-contact and contact regions was impossible. The force curve data did not follow a Hertzian indentation pathway, and thus the poorly matching Hertz model fits gave erroneous stiffness values, which could not be considered for data analysis. We therefore applied a “fit-quality criterion” to address this issue. The fit-quality can be expressed by the sum of the squared differences between the fitted Hertz model and the experimental data:
Here,
δ represents the fitted indentation value for a given point,
δj stands for the measured indentation at that point, and
σj is a weighting value. We considered each data point to have the same accuracy and set
σj = 1. In a computational least square fit,
χ is minimized by iteratively improving the fitting parameters. The average absolute deviation of the experimental data from the model is given by
ε ≡
/
N, where
N is the number of data points per force curve. The smaller the
ε, the better the model fits the experimental data. For the force map shown in
Figure 1,
ε ranged from 0 to 1.5 nm. We excluded all YMEs obtained from force curves with
ε > 0.3 nm to get more reliable results.
Figure 1F shows the stiffness image after additional application of this fit-quality criterion.