## 2 Materials and Methods  
### Graphene Nanocomite Preparation and Characterization  
The polymer matrix of the studied nanocomposites consisted of multi-block thermoplastic elastomer consisting of hard segments of poly(ethylene terephthalate) (PET) and soft segments of amorphous fatty acid ester sequences based on di-linoleic acid (DLA). The synthesis of both neat copolymers and nanocomposites was carried out in two stages, based on our previous work [26; 27; 28], in a kilo-scale (3.5-dm\({}^{3}\)) Fourne Maschinenbau polycondensation reactor. Briefly, the first stage consisted of transesterification of dimethyl terephthalate (DMT) and ethylene glycol (EG), followed by esterification with DLA, and, finally, polycondensation. After the completion of the reaction, the polymer was discharged from the reactor in the form of an extruded filament, cooled with water, and cut into granules suitable for further processing, such as injection or compression molding. Two different neat co-polymers were prepared with different hard to soft segment ratios, resulting in a more elastic material with 40 wt% of hard segments (PET-DLA 4060) and a stiffer co-polyester with 60 wt% of hard segments (PET-DLA 6040). Additionally, nanocomposites with the same hard to soft segment ratios but with 1 wt% of a commercial graphite nanoplatelet nanofiller Grade-A0-3 by the Graphene Supermarket were prepared via in-situ polymerization. (The producer indicates that the Grade-A0-3 graphene has an average thickness of \(\sim\)12 nm, representing \(>\)30 monolayers; therefore, according to the latest classification [29], the Grade-A0-3 material should be termed as a \"graphite nanoplatelet\"). The synthesis was carried out in stages, in the same fashion as that of the neat polymer, but the nanofiller was first dispersed in the liquid EG and DLA monomers by a combination of bath and probe sonication and high-speed mixing. We have already published detailed analyses of these materials [26; 27], including Fourier transform infrared spectroscopy (FTIR), proton nuclear magnetic resonance (\({}^{1}\)H NMR), differential scanning calorimetry (DSC), water contact angle measurement, tensile testing, moisture vapor transmission tests, bacterial adhesion testing, and a cytocompatibility assessment. For THz-TDS measurements, test samples were prepared using compression molding (Remi-Plast PH10T hot press). For the more elastic PET-DLA 4060 copolymer and nanocomposite specimen, nominally 0.3-mm-thick samples were prepared in the same fashion as the membranes for the prototype heart assist device (Figure 1b): The temperature was 175 \({}^{\circ}\)C and the pressure was 6 bar. For the stiffer PET-DLA 6040 copolymer and nanocomposites, the temperature was increased to 230 \({}^{\circ}\)C, while pressure remained unchanged, and thicker samples (nominally 0.9 mm thick) were prepared. After fabrication, our nanocomposite samples were characterized using both a high-resolution TEM (HRTEM) and Raman methods in order to image graphene nanoflakes and estimate their size, dispersion, and a number of graphene layers in a flake. Figure 2 presents representative TEM images obtained using a JEM-2100 JEOL microscope at different magnification levels of the thin, 1-wt% graphene-PET-DLA 4060 nanocomposite sample. Figure 2a is a low-magnification image covering a 10 \(\times\) 10-\(\upmu\)m\({}^{2}\) area showing the density and dispersion of the flakes. In this randomly selected image we can identify 12 nanofiller flakes that are well separated from each other. Besides the one \"giant\" flake (#1), the other flakes have sizes below 1 \(\upmu\)m, so, in rough approximation, they can be treated as independent and non-interacting particles dispersed in a host copolymer matrix. Figure 2b shows a HRTEM image of one of the smallest (7-nm \(\times\) 130-nm) nanoflakes observed and atomic-scale measurements of interplane distances from two areas that could be well oriented for TEM studies. We can clearly image graphene planes and, in this case, see approximately ten layers.