Bending experiments

For the AFM bending experiments, the nanofibers were transferred to glass sub-strates structured with lithographically prepared channels to achieve a free-standing configuration. Figure 5.3(a) shows a schematic representation of the deformation set-up. A representative SEM image of suspended fibers can be found in the Sup-porting Information (SI 5.1). As a first step, we ensured that there was no inden-tation or compression of the nanofibers contributing to the bending experiments.

Therefore, we recorded force-deformation curves on the supported segments of the samples. A typical measurement can be seen in Figure 5.3(b). There was no signifi-cant deformation visible for the typically applied loads.

For the actual bending experiments, we performed force-distance measurements on 30 to 40 positions along the free-standing nanofiber segments. The stiffness at the respective position was calculated from the slope of the force-distance curve. To-wards the midpoint of the channel the measured stiffness decreased, and hence we obtained stiffness profiles along the whole free-standing segment Figure 5.3(c).

From the shape of the profiles, we determined the boundary conditions (i.e., the type of fixation) of each segment and the modulus of the nanofiber using models of classical beam theory.⁵⁸A detailed discussion of the importance of the boundary conditions, of the analytical models, and of the experimental procedure can be found in the literature.⁴⁰The two most commonly applied models are the double-clamped beam model (DCBM), where the ends of the sample are firmly fixed, and the simply supported beam model (SSBM), where the ends are assumed to rotate freely.^53,58,59

(a)

(b)

(c)

Figure 5.2(a) SEM image and size distribution of nanofibers of1a. The average di-ameterdis0.21±0.08µm. The SEM overview image and the inset highlight the nar-row size distribution and homogeneous structure of the nanofibers. (b) SEM image and size distribution of nanofibers of1b(d= 0.4±0.2µm). (c) SEM image and size distribution of fibers of2a(d= 2±1µm). The vertical dashed lines in the histograms indicate the minimal and maximal size of each fiber type for which successful AFM bending measurements were performed.

δ(x)

Figure 5.3 (a) Sketch of the experimental setup. (b) Exemplary force deformation curves obtained on the bare substrate, around the middle of the supported segment (A) and at an intermediate position between edge and midpoint (B). (c) Exemplary stiffness profile of a supported segment of1a. The data was obtained by averaging over 6 measurements on the same segment and fitted with the DCBM (solid line) and SSBM (dashed line). The DCBM is clearly valid.

We observed both types of behavior and could clearly distinguish between both cases. However, effects like a slight slackness of the sample or defects at the fixa-tion points can lead to profiles that resemble a SSBM-type behavior although the ends are clamped, and hence can lead to an inaccurate interpretation of the data.

On the contrary, when a profile matches the DCBM, the experimental conditions have to be in perfect agreement with the modeled clamped conditions, and there-fore the interpretation is much more reliable. To avoid uncertainties, we decided to use only segments that clearly corresponded to the DCBM for the calculation of the modulus.

For the DCBM, the stiffnessk(x) at a given positionxis k(x) = 3L³EI

(L−x)³x³ (5.1)

Here, L is the length of the free-standing segment and x is the position where the load is applied, with 0≤x≤L. EI is the flexural rigidity and defined as the product of the modulus E and the area moment of inertia I. The measured profiles were fitted with EI as the only free parameter. To obtain E, we imaged each individual nanofiber with AFM and calculated I from the respective cross sections. This is a

highly important step, since the flexural rigidity scales quartic with the diameter of the samples. By imaging each individual investigated nanofiber and calculating the correct area moment of inertia, our evaluation of the mechanical data is not based on any assumptions concerning size and shape of the cross section. An exemplary cross section can be found in the Supporting Information (SI 5.2).

It can be seen from Equation (5.1) that the spring constants of multiple nanofiber segments can be normalized so that one obtains

knorm(x) = 3E

We averaged the normalized spring constants of all investigated nanofiber segments and fitted the data with Equation (5.2) (Figure 5.4). The resulting modulus for 1a was 2.3±0.3 GPa. In the same way, we determined the Young’s moduli of1b(2.1± 0.1 GPa) and2a(3.3±0.3 GPa). The quality of the fit shows the excellent agreement of the deformation profiles with the DCBM. Therefore, the error bars are a direct measure of the uncertainty of the determined Young’s modulus which is caused by defects and inhomogeneities of the self-assembled nanofibers and by uncertainties of the measurements. Those may be attributed to the error of the AFM cantilever spring constant (<10%), the channel width (<5%) and the determined area moment of inertia (<20%).

Since the molecular structure suggests highly anisotropic mechanical properties for the nanofibers, the effect of anisotropy should also be discussed. When bent, the upper part of a beam is subjected to compression while the lower part is subjected to extension. This leads to additional shear forces within the beam that become im-portant in anisotropic materials when the length-to-radius ratio of the bent segment becomes _R^L ≤4

qE

G.⁶⁰ In this case, shearing can be accounted for by defining an ap-parent bending modulus Eb that is related to the true elastic modulus E and the shear modulusGvia^45,47

1

for samples with a circular cross section. L is the length and R the radius of the free-standing segment,f_sis the form factor of shear, which is related to the sample’s cross section (e.g., 10/9 for a cylinder).⁶¹For more arbitrary cross sections, the area

3000

Figure 5.4 Results of the spatially resolved bending experiments. The shape of all profiles shows excellent agreement with the DCBM. The larger size of the error bars compared to the individual measurements (as seen in Figure 5.3(c)) is a conse-quence and a direct measure of the distribution and uncertainty of the determined moduli amongst individual specimen of one sample.

and second moment of area cannot be expressed by the radius and therefore 1

In any way, if shearing plays a dominant role, G and consequently E can be de-termined from the slope of 1/Eb versus I/(AL²). However, for all investigated sys-tems, no significant influence of shearing could be observed (Supporting Informa-tion SI 5.3). Therefore, the anisotropy is only of minor importance for the bending behavior under our measurement conditions, and the determined values can be re-garded as the true Young’s moduli in the axial direction of the nanofibers.