## 2 Results and Discussion
The images of the pure samples of AC and CCF (Figures S1a and S2a, respectively) show similar features to those of the EG sample indicating also in these cases a highly porous structure consisting of low-density areas (black areas). The addition of OD to the porous matrices results in the images (Figures S1b and S2b) showing an increase in purple and green-yellow areas, indicating higher-density regions. These areas correspond to the uniform distribution of OD on both AC and CCF. For the AC/CNTs/OD composite, the CT microtomography image (Figure S1c) predominantly displays higher-density green-yellow areas, which is attributed to the presence of non-porous CNTs on the external surface of AC. In contrast, this effect is not observed in the CCF/CNTs/OD composite, suggesting that the CNTs are not distributed on the external surface of CCF. This is likely due to the macroporous structure of CCF, which may cause the CNTs to incorporate into the large pores instead. Figure 3: Microtomography CT images of: (**a**) EG; (**b**) EG/OD; and (**c**) EG/CNTs/OD. Green line is coronal (COR), blue line is sagittal (SAG) and red line is transverse (TRA) cross-sections of the samples. Figure 2: Photographs from filter paper after leaching tests at 80 \({}^{\circ}\)C for: activated carbon, expanded, graphite and ceramic carbon foam samples. In order to study the incorporation and distribution of CNTs on the carbon matrices, SEM images were collected and are presented in Figure 4. In the case of the AC/CNTs/OD sample (Figure 4b), the smooth and porous surface of AC (Figure 4a) is fully covered by CNTs which are uniformly dispersed on it and this finding is consistent with the microtomography CT images (Figure S1). Also, in the case of the EG/CNTs/OD sample (Figure 4e,f), CNTs are shown to be uniformly dispersed on the EG inert macroporous surface between its graphitic sheets. Finally, the SEM images of CCF (Figure S3a,b) reveal its macroporous structure, which probably accommodates the CNTs/OD mixture in its depth and it is not possible to distinguish carbon nanotubes with SEM in this particular structure, which is also consistent with the CT microtomography image (Figure S2) where the image of the CCF/CNTs/OD sample mainly consists of air voids. Overall, these findings from the SEM and CT microtomography analysis reveal distinct differences in the incorporation and distribution of CNTs across the various carbon matrices, suggesting that the interactions between CNTs and each carbon matrix are influenced by the distinct surface properties and porosity of the materials. Differential Scanning Calorimetry (DSC) was used in order to investigate the ability of carbon/OD and carbon/CNTs/OD composites to store thermal energy and Figure 5 shows the DSC curves for all systems with the different carbon matrices. Table 1 summarizes the thermal properties derived from the DSC curves and Figure 6 shows comparative graphs for the \(\Delta\)Hm% and the \(\Delta\)Hs% of the composites according to each wt% OD loading. The expected theoretical values for \(\Delta\)Hm and \(\Delta\)Hs, according to OD loading, is also marked in this figure (green lines in Figure 6). Figure 4: Scanning Electron Microscopy (SEM) images of: (**a**) AC matrix; (**b**) AC/CNTs/OD composite; (**c**,**d**) EG matrix; and (**e**,**f**) CCF/CNTs/OD composite. Figure 5: DSC curves for: **(a)** activated carbon samples; **(b)** ceramic carbon foam; and **(c)** expanded graphite samples.