Porosity and Structure of Hierarchically Porous Ni/Al2O3 Catalysts for CO2 Methanation

CO2 methanation is often performed on Ni/Al2O3 catalysts, which can suffer from mass transport limitations and, therefore, decreased efficiency. Here we show the application of a hierarchically porous Ni/Al2O3 catalyst for methanation of CO2. The material has a well-defined and connected mesoand macropore structure with a total porosity of 78%. The pore structure was thoroughly studied with conventional methods, i.e., N2 sorption, Hg porosimetry, and He pycnometry, and advanced imaging techniques, i.e., electron tomography and ptychographic X-ray computed tomography. Tomography can quantify the pore system in a manner that is not possible using conventional porosimetry. Macrokinetic simulations were performed based on the measures obtained by porosity analysis. These show the potential benefit of enhanced mass-transfer properties of the hierarchical pore system compared to a pure mesoporous catalyst at industrially relevant conditions. Besides the investigation of the pore system, the catalyst was studied by Rietveld refinement, diffuse reflectance ultraviolet-visible (DRUV/vis) spectroscopy, and H2-temperature programmed reduction (TPR), showing a high reduction temperature required for activation due to structural incorporation of Ni into the transition alumina. The reduced hierarchically porous Ni/Al2O3 catalyst is highly active in CO2 methanation, showing comparable conversion and selectivity for CH4 to an industrial reference catalyst. Catalysts 2020, 10, 1471; doi:10.3390/catal10121471 www.mdpi.com/journal/catalysts Catalysts 2020, 10, 1471 2 of 22


Synthesis and Structural Characterization
The Ni/Al 2 O 3 catalyst with a hierarchical meso-and macropore structure (denoted as Ni/Al 2 O 3 -h) was prepared based on the route by Herwig et al. [15]. According to inductively coupled plasma optical emission spectrometry (ICP-OES) analysis, the Ni mass fractions of the xerogel sample was 18.0 wt.% for Ni/Al 2 O 3 -h.
We analyzed the crystallographic structure of the calcined Ni/Al 2 O 3 -h sample via Rietveld refinement of powder X-ray diffraction (PXRD) data to understand the influence of the co-gelation synthesis procedure on the resulting structure. For the synthesis of Ni/Al 2 O 3 -h, the formation of a NiAl 2 O 4 spinel phase was already known [15], while crystallographic details were not reported yet. However, this is important to properly understand the material as NiAl 2 O 4 is a spinel structure and most transition Al 2 O 3 , e.g., γ or η-Al 2 O 3 , are typically described as defect-spinel structures [32][33][34][35]. As Ni/Al 2 O 3 -h is synthesized by co-gelation, an initially homogeneous distribution of Ni and Al can be expected. This raises the question of whether an Al 2 O 3 transition form, NiO or NiAl 2 O 4 as reported by Herwig et al. [15], is obtained upon calcination. From a qualitative analysis of the PXRD one can already exclude the formation of crystalline NiO. To get more insights into the formation of the Al 2 O 3 transition form or NiAl 2 O 4 we used Rietveld refinement. The resulting patterns for two different refinements of the same PXRD data set are shown in Figure 1a,b. An initial approach for the refinement was to use structure models of γ-Al 2 O 3 by Zhou et al. and NiAl 2 O 4 by O'Neill et al. without refinement of atomic parameters, which is shown in Figure 1a [33,36]. Compared to this, Figure 1b shows another refinement with the same structural models as above, but with refinement of occupancy of the Al positions of the γ-Al 2 O 3 transition form. From the refined patterns, one can Catalysts 2020, 10, 1471 4 of 22 directly see that the occupancy refinement (Figure 1b) led to a better fit compared to fixed occupancies ( Figure 1a). Other potentially matching Al 2 O 3 transition forms such as η-Al 2 O 3 by Zhou et al. were also tested for refinement but led to worse results [33]. The results obtained from the Rietveld refinements are summarized for both refinements in the Supplementary Materials Table S1. The observed better fit was also confirmed by the Rwp and GoF fit criteria of the refinement. For both refinements, the main phase was the Al2O3 transition form with a phase fraction of 95.5 wt.% versus 84.3 wt.% for the refined occupancy vs. the fixed one. In both cases, the crystallite sizes obtained by microstructure analysis were similar and resembled nanocrystalline materials of around 7 nm size for the Al2O3 transition form as well as for NiAl2O4. The nanocrystalline nature is typical for Al2O3 transition forms [34,35]. Our focus in the refinement results was on the occupancy of the Al positions. For a better structural understanding, the Al2O3 structure is illustrated in Figure 1c. The O position of the Al2O3 phase was fixed as it is generally reported that in all defect spinel-type Al2O3 transition forms (γ-, η-, θ-Al2O3) this position is fully occupied and the transition forms can be described by different Al occupancies of the octahedral and tetrahedral voids of the O sublattice [33,35]. In Figure 1c, the octahedral voids of the Al1 position are shown in blue and the tetrahedral voids of the Al2 position are orange. For clarity, the Al3 position is not shown, while it resembled a distorted octahedral position. In comparison with both refinements, the occupancy of the tetrahedral void was equal with 0.84 in the literature-reported structure and 0.839(4) in the refinement. However, the distorted Al3 position in the analysis with refined occupancy was strongly decreased compared to the literature structure and we considered this position as barely occupied. The results obtained from the Rietveld refinements are summarized for both refinements in the Supplementary Materials Table S1. The observed better fit was also confirmed by the R wp and GoF fit criteria of the refinement. For both refinements, the main phase was the Al 2 O 3 transition form with a phase fraction of 95.5 wt.% versus 84.3 wt.% for the refined occupancy vs. the fixed one. In both cases, the crystallite sizes obtained by microstructure analysis were similar and resembled nanocrystalline materials of around 7 nm size for the Al 2 O 3 transition form as well as for NiAl 2 O 4 . The nanocrystalline nature is typical for Al 2 O 3 transition forms [34,35]. Our focus in the refinement results was on the occupancy of the Al positions. For a better structural understanding, the Al 2 O 3 structure is illustrated in Figure 1c. The O position of the Al 2 O 3 phase was fixed as it is generally reported that in all defect spinel-type Al 2 O 3 transition forms (γ-, η-, θ-Al 2 O 3 ) this position is fully occupied and the transition forms can be described by different Al occupancies of the octahedral and tetrahedral voids of the O sublattice [33,35]. In Figure 1c, the octahedral voids of the Al1 position are shown in blue and the Catalysts 2020, 10, 1471 5 of 22 tetrahedral voids of the Al2 position are orange. For clarity, the Al3 position is not shown, while it resembled a distorted octahedral position. In comparison with both refinements, the occupancy of the tetrahedral void was equal with 0.84 in the literature-reported structure and 0.839(4) in the refinement. However, the distorted Al3 position in the analysis with refined occupancy was strongly decreased compared to the literature structure and we considered this position as barely occupied. As the Ni/Al 2 O 3 -h material was prepared via co-gelation, a homogeneous distribution of Ni was expected. However, the refined Al1 occupancy showed an increased occupation with 1.156(7) compared to the literature value of 0.58. This indicates that Ni was not only present in the NiAl 2 O 4 spinel phase, but also present in the octahedral void of the Al 2 O 3 transition form. This higher occupancy indicates an increased electron density in the octahedral void, which could be caused by a mixed occupancy of this position by Al and Ni, as in the ideal NiAl 2 O 4 structure. A refinement of a mixed occupancy is limited by the data quality and requires further studies. This is partially supported by results from DRUV/vis spectroscopy, as shown in Figure 2. In several literature sources, the absorption band at around 380 nm was assigned to octahedrally coordinated Ni 2+ , while the bands at about 550 to 650 nm were assigned to tetrahedrally coordinated Ni 2+ within the NiAl 2 O 4 structure [11,[37][38][39]. Therefore, both coordination environments of Ni 2+ should be present within the Ni/Al 2 O 3 -h sample. Rietveld refinement indicates that Ni 2+ was already integrated into the Al 2 O 3 transition form by occupying the octahedral Al1 site, while we also observed the formation of NiAl 2 O 4 secondary phase. The incorporation of Ni already into the Al 2 O 3 transition form upon calcination resembled a potential problem regarding the reducibility of the material. However, based on the current results, a detailed structural knowledge about the incorporation pathway was missing and needs to be investigated.
Catalysts 2020, 10, x FOR PEER REVIEW 5 of 22 expected. However, the refined Al1 occupancy showed an increased occupation with 1.156 (7) compared to the literature value of 0.58. This indicates that Ni was not only present in the NiAl2O4 spinel phase, but also present in the octahedral void of the Al2O3 transition form. This higher occupancy indicates an increased electron density in the octahedral void, which could be caused by a mixed occupancy of this position by Al and Ni, as in the ideal NiAl2O4 structure. A refinement of a mixed occupancy is limited by the data quality and requires further studies. This is partially supported by results from DRUV/vis spectroscopy, as shown in Figure 2. In several literature sources, the absorption band at around 380 nm was assigned to octahedrally coordinated Ni 2+ , while the bands at about 550 to 650 nm were assigned to tetrahedrally coordinated Ni 2+ within the NiAl2O4 structure [11,[37][38][39]. Therefore, both coordination environments of Ni 2+ should be present within the Ni/Al2O3-h sample. Rietveld refinement indicates that Ni 2+ was already integrated into the Al2O3 transition form by occupying the octahedral Al1 site, while we also observed the formation of NiAl2O4 secondary phase. The incorporation of Ni already into the Al2O3 transition form upon calcination resembled a potential problem regarding the reducibility of the material. However, based on the current results, a detailed structural knowledge about the incorporation pathway was missing and needs to be investigated. The H2-TPR curves for Ni/Al2O3-h are shown in Figure 3 a. The reduction of NiO typically occurs between 473 K and 573 K, depending on the particle size, support material, and confinement within pores [40,41]. Consistently, a physical mixture (PM) of NiO powder and the Ni-free xerogel support showed a reduction peak with maximum at 580 K ( Figure 3a). Compared to this mixture, the maximum reduction temperature (Tred) of Ni/Al2O3-h was 1111 K. This remarkably higher Tred was likely caused by the incorporation of Ni into the Al2O3 or the NiAl2O4 structure ( Figure 1) [40], and has been suggested by Herwig et al. [15] and Titus et al. [42] for Ni/Al2O3 xerogel catalysts. Within the Al2O3 or the NiAl2O4 structure, Ni 2+ is strongly bound within the framework of the crystal, which hampers its reduction to Ni 0 . The H2 consumption during TPR of Ni/Al2O3-h amounted to 1800 µmol g -1 , which represented a reduction of only 59% of the Ni atoms based on ICP-OES results, when NiO was assumed to react with H2 in a 1:1 molar ratio. This shows that even the high temperatures of up to 1173 K were not enough to fully reduce Ni within the sample. Nevertheless, reduction during TPR resulted in color change of the powder from blue to black (Figure 3b), indicative of the transformation of Ni 2+ to Ni 0 , consistent with UV-vis spectroscopy results ( Figure 2). The H 2 -TPR curves for Ni/Al 2 O 3 -h are shown in Figure 3 a. The reduction of NiO typically occurs between 473 K and 573 K, depending on the particle size, support material, and confinement within pores [40,41]. Consistently, a physical mixture (PM) of NiO powder and the Ni-free xerogel support showed a reduction peak with maximum at 580 K ( Figure 3a). Compared to this mixture, the maximum reduction temperature (T red ) of Ni/Al 2 O 3 -h was 1111 K. This remarkably higher T red was likely caused by the incorporation of Ni into the Al 2 O 3 or the NiAl 2 O 4 structure ( Figure 1) [40], and has been suggested by Herwig et al. [15] and Titus et al. [42] for Ni/Al 2 O 3 xerogel catalysts. Within the Al 2 O 3 or the NiAl 2 O 4 structure, Ni 2+ is strongly bound within the framework of the crystal, which hampers its reduction to Ni 0 . The H 2 consumption during TPR of Ni/Al 2 O 3 -h amounted to 1800 µmol g -1 , which represented a reduction of only 59% of the Ni atoms based on ICP-OES results, when NiO was assumed to react with H 2 in a 1:1 molar ratio. This shows that even the high temperatures of up to 1173 K were not enough to fully reduce Ni within the sample. Nevertheless, reduction during TPR resulted in color change of the powder from blue to black (Figure 3b), indicative of the transformation of Ni 2+ to Ni 0 , consistent with UV-vis spectroscopy results ( Figure 2).

Catalytic Measurements
The catalytic activity in the methanation of CO2 with H2 was investigated for the Ni/Al2O3-h catalyst reduced at two temperatures ( Figure 4). The activity of the catalyst was compared to an industrial Ni/Al2O3 reference catalyst (denoted as SPP2080-IMRC) containing 8.2 wt.% Ni and a Brunauer-Emmett-Teller (BET) surface area of 130 m 2 •g −1 . The activity studies were performed to determine whether the necessary harsh activation conditions of the material resulted in a functional catalyst for CO2 methanation. A substantial increase of the CO2 conversion was found if Ni/Al2O3-h was reduced at 1123 K compared to 923 K (Figure 4a), while the CH4 selectivity increased significantly to >90% (Figure 4b). This finding is conclusive with the results obtained from H2-TPR (Figure 4a), in which the temperature of 923 K represented the onset of the catalyst reduction, whereas 1123 K was close to the maximum H2 consumption. Therefore, it is reasonable to assume that increasing the reduction temperature to 1123 K led to a significantly larger fraction of Ni atoms being reduced to the elemental state and, thus, becoming active. This resulted in a significant enhancement of the CO2 conversion and the CH4 selectivity since Ni 0 was mainly responsible for the hydrogenation activity of the catalyst [43,44].
When reduced at 1123 K, the CO2 conversion of 67% obtained for the Ni/Al2O3-h catalyst at a reaction temperature of 623 K and 673 K was higher than ~38% (623 K) and ~49% (673 K) for the 20 wt.% Ni/ZrO2 catalysts studied by Martínez et al. [45] under similar catalytic conditions, although the CH4 selectivity of >99% reported by Martínez et al. is higher than the values obtained in this study. The Ni/Al2O3-h catalyst reduced at 1123 K is also comparable with the SPP2080-IMRC catalyst in terms of CO2 conversion and CH4 selectivity ( Figure 4). However, given both the substantially higher reduction temperature and Ni loading, the Ni/Al2O3-h catalyst did not compete with the SPP2080-IMRC catalyst in terms of activity. Furthermore, it should be noted that comparison of catalysts here based on conversion is shown purely as a qualitative method to determine whether or not the catalyst was active following synthesis of the hierarchical pore system. To quantitatively compare effectiveness or efficiency of the various catalysts in the future, further kinetic studies are required including consideration of metal loading, dispersion, turnover, and presence of mass transport phenomena over a range of conditions.

Catalytic Measurements
The catalytic activity in the methanation of CO 2 with H 2 was investigated for the Ni/Al 2 O 3 -h catalyst reduced at two temperatures ( Figure 4). The activity of the catalyst was compared to an industrial Ni/Al 2 O 3 reference catalyst (denoted as SPP2080-IMRC) containing 8.2 wt.% Ni and a Brunauer-Emmett-Teller (BET) surface area of 130 m 2 ·g −1 . The activity studies were performed to determine whether the necessary harsh activation conditions of the material resulted in a functional catalyst for CO 2 methanation. A substantial increase of the CO 2 conversion was found if Ni/Al 2 O 3 -h was reduced at 1123 K compared to 923 K (Figure 4a), while the CH 4 selectivity increased significantly to >90% (Figure 4b). This finding is conclusive with the results obtained from H 2 -TPR (Figure 4a), in which the temperature of 923 K represented the onset of the catalyst reduction, whereas 1123 K was close to the maximum H 2 consumption. Therefore, it is reasonable to assume that increasing the reduction temperature to 1123 K led to a significantly larger fraction of Ni atoms being reduced to the elemental state and, thus, becoming active. This resulted in a significant enhancement of the CO 2 conversion and the CH 4 selectivity since Ni 0 was mainly responsible for the hydrogenation activity of the catalyst [43,44].
When reduced at 1123 K, the CO 2 conversion of 67% obtained for the Ni/Al 2 O 3 -h catalyst at a reaction temperature of 623 K and 673 K was higher than~38% (623 K) and~49% (673 K) for the 20 wt.% Ni/ZrO 2 catalysts studied by Martínez et al. [45] under similar catalytic conditions, although the CH 4 selectivity of >99% reported by Martínez et al. is higher than the values obtained in this study. The Ni/Al 2 O 3 -h catalyst reduced at 1123 K is also comparable with the SPP2080-IMRC catalyst in terms of CO 2 conversion and CH 4 selectivity ( Figure 4). However, given both the substantially higher reduction temperature and Ni loading, the Ni/Al 2 O 3 -h catalyst did not compete with the SPP2080-IMRC catalyst in terms of activity. Furthermore, it should be noted that comparison of catalysts here based on conversion is shown purely as a qualitative method to determine whether or not the catalyst was active following synthesis of the hierarchical pore system. To quantitatively compare effectiveness or efficiency of the various catalysts in the future, further kinetic studies are required including consideration of metal loading, dispersion, turnover, and presence of mass transport phenomena over a range of conditions.

Tomography Studies
To investigate the pore structure of the hierarchical meso-/macroporous Ni/Al2O3-h catalyst we performed ET and PXCT studies. The volume renderings for the PXCT and both discrete algebraic reconstruction technique (DART)-reconstructed ETs are shown in Figure 5. For the PXCT volumes in Figure 5a, the upper part of the tomogram was cropped to remove Pt traces from the sample preparation. For original tomogram, see Supplementary Materials Figure S4. The reconstructed tomogram had a voxel size of 38.1 nm and an effective spatial resolution of 78.3 nm, estimated by Fourier shell correlation (FSC) as implemented in the reconstruction script of Odstrčil et al. [46]. The selected region and the associated FSC are shown in the Supplementary Materials Figure S5 and Figure S6, respectively. The measured particle had a maximum diameter of about 25 µm. The sample size and the resolution allowed obtaining information about the whole macropore space of the material. The two DART-reconstructed ET volumes of the Ni/Al2O3-h catalyst are visualized in Figure  5b,c, respectively. The measured particles had a diameter of about 450 nm for ET1 and 400 nm for ET2, while the obtained voxel size after simultaneous iterative reconstruction technique (SIRT) and DART reconstruction was 0.92 nm and 0.64 nm for ET1 and ET2, respectively. The sample size and resolution of the ET volumes allowed us to investigate the mesopores of the material, though on a limited region of the sample. The effective resolutions for ET1 and ET2 were estimated as 4.2 nm and 5.6 nm based on the 10-90% rise distance analysis of an edge feature in the SIRT reconstructed tomogram (see Supplementary Materials Figures S1 and S2). For all three tomograms shown in Figure  5, image analysis was performed as described in the experimental Section 3.4. to obtain labels for the material (blue, white), the total pore space (orange), and separated individually labelled pores (8-bit color coding). For all three tomograms, a slice through the volume is shown to illustrate the total pore space together with the separated pores. The combination of PXCT and ET was essential to cover all relevant length scales of the pore system ranging from mesopores (2 nm to 50 nm) up to macropores in the µm range [28,47]. The different length scales between PXCT and ET are illustrated by the black box in Figure 5, as we can considered the ET volume as a zoom into the material label identified by PXCT. In the PXCT volume (Figure 5a), one can see a well-connected macropore network, while we did not resolve the mesopores of the material as the resolution was limited to 78.3 nm. However, by applying ET one can resolve the mesopores present inside the material label of the PXCT, as shown in Figure 5b,c. The mesopores formed a connected network, similar to the macropores. Based on this multiscale analysis, one can justify the application of the term "hierarchical" for the pore system, as the macropores formed an overall connected system as shown by PXCT, with the mesopores

Tomography Studies
To investigate the pore structure of the hierarchical meso-/macroporous Ni/Al 2 O 3 -h catalyst we performed ET and PXCT studies. The volume renderings for the PXCT and both discrete algebraic reconstruction technique (DART)-reconstructed ETs are shown in Figure 5. For the PXCT volumes in Figure Figure 5b,c, respectively. The measured particles had a diameter of about 450 nm for ET1 and 400 nm for ET2, while the obtained voxel size after simultaneous iterative reconstruction technique (SIRT) and DART reconstruction was 0.92 nm and 0.64 nm for ET1 and ET2, respectively. The sample size and resolution of the ET volumes allowed us to investigate the mesopores of the material, though on a limited region of the sample. The effective resolutions for ET1 and ET2 were estimated as 4.2 nm and 5.6 nm based on the 10-90% rise distance analysis of an edge feature in the SIRT reconstructed tomogram (see Supplementary Materials Figures S1 and S2). For all three tomograms shown in Figure 5, image analysis was performed as described in the experimental Section 3.4. to obtain labels for the material (blue, white), the total pore space (orange), and separated individually labelled pores (8-bit color coding). For all three tomograms, a slice through the volume is shown to illustrate the total pore space together with the separated pores. The combination of PXCT and ET was essential to cover all relevant length scales of the pore system ranging from mesopores (2 nm to 50 nm) up to macropores in the µm range [28,47]. The different length scales between PXCT and ET are illustrated by the black box in Figure 5, as we can considered the ET volume as a zoom into the material label identified by PXCT. In the PXCT volume (Figure 5a), one can see a well-connected macropore network, while we did not resolve the mesopores of the material as the resolution was limited to 78.3 nm. However, by applying ET one can resolve the mesopores present inside the material label of the PXCT, as shown in Figure 5b,c. The mesopores formed a connected network, similar to the macropores. Based on this multiscale analysis, one can justify the application of the term "hierarchical" for the pore system, as the macropores formed an overall connected system as shown by PXCT, with the mesopores identified by ET connected to this macropore network. Quantitative analysis of the tomograms is presented in Section 2.4.
Catalysts 2020, 10, x FOR PEER REVIEW 8 of 22 identified by ET connected to this macropore network. Quantitative analysis of the tomograms is presented in Section 2.4.

Porosity Analysis
The separated pores (8-bit color code), total pore (orange), and material (blue) labels for both the ET and PXCT volumes were quantitatively analyzed to retrieve porosity descriptors for the Ni/Al2O3-h catalyst. From the label analysis using Avizo we retrieved information such as volume and surface area for the material and total pore labels, as well as the equivalent pore diameter deq

Porosity Analysis
The separated pores (8-bit color code), total pore (orange), and material (blue) labels for both the ET and PXCT volumes were quantitatively analyzed to retrieve porosity descriptors for the Ni/Al 2 O 3 -h catalyst. From the label analysis using Avizo we retrieved information such as volume and surface area for the material and total pore labels, as well as the equivalent pore diameter d eq (Equation (4)) for each individual pore label, as shown in Figure 5a-c for the PXCT and ET volumes, respectively. The rounded total pore V p,tot and material volumes V s,tot are listed in Table 1 together with the total porosity ε tomo of each tomogram. All three tomograms show a porosity larger than 40%. With respect to the resolution, ε tomo of the PXCT is defined as the macroporosity ε M of the Ni/Al 2 O 3 -h catalyst based on image analysis. To calculate the total porosity ε tot of the material based on the tomography studies, the mesoporosity visible by ET must also be included. Therefore, the porosity of the material label in PXCT was assumed to further contain the average of the porosity values obtained from ET, which was 44.9%. Taking into account this extra porosity for the PXCT results, a ε tot of 70.4% was obtained. Based on this ε tot , the mesoporosity ε m of the Ni/Al 2 O 3 -h catalyst based on the tomography analysis was 24.1%. Table 1. Results obtained from the quantitative pore analysis based on image analysis of the Ni/Al 2 O 3 -h catalyst. Rounded total material volume V s,tot , total pore volume V p,tot , porosity of the measured tomograms ε tomo , weighted mean pore diameter d eq,w , and fitted mean pore diameter d eq,f for the ET1, ET2, and PXCT. In addition to the porosity information, one can extract information about the pore size distribution from the tomograms. The frequency of the occurrence of each pore diameter for the three tomograms is shown in the Supplementary Materials Figure S7. For the PXCT volumes, one can observe a unimodal pore size distribution based on the frequency analysis with a maximum around 2 µm. The frequency-based pore distribution indicates a bimodal pore size distribution for the ET volumes with a maximum around 3 nm and another at around 22 nm to 25 nm. Based on the qualitative analysis of the ET, we can assign the small pores around 2 nm to 6 nm to mainly isolated pores or artefacts from the pore separation that were not removed by the "Opening" module or caused by the limited effective resolution. These were not considered to contribute to the total porosity. The larger mesopores represent the measurable pores of the connected pore system.
To investigate the contribution of different pore sizes toward the pore volume, the porosity of each individual pore was calculated by Equation (6), and the sum of the individual porosities was plotted depending on d eq, as shown in Figure 6 for (a) ET1, (b) ET2, and (c) PXCT. The obtained pore size distributions were analyzed to obtain the weighted numerical mean pore diameter d eq,w as well as the fitted mean pore diameter d eq,f from the Gaussian fit. Both mean pore diameter values for all three tomograms are summarized in Table 1. The difference between d eq,w and d eq,f was used to estimate the error of the mean pore diameter, while, for further discussion, the values of d eq,w were used. For ET1 and ET2 we obtained d eq,w of 37.4 ± 1.2 nm and 29.2 ± 1.3 nm, respectively. These values represent mesopores in the Ni/Al 2 O 3 -h catalyst, while no significant amount of macropores could be detected by ET. The absence of meaningful macropore data may be due to the limited size of the sample, which can be investigated by ET (e.g., 450 nm diameter), which can potentially be the same order of magnitude as a single large macropore. The average of d eq,w of ET1 and ET2 is the mean mesopore diameter d eq,meso of 33.3 ± 4.1 nm of the Ni/Al 2 O 3 -h catalyst determined from tomography analysis. The mean pore diameter for the macroporosity d eq,macro was similarly determined from the pore size distribution of the PXCT data as 2545 ± 23 nm. While no significant porosity contribution of macropores below 1 µm can be observed, the majority of the macropores ranged from 1 µm up to 3.5 µm, resembling a narrow macropore distribution and, thus, a relatively homogenous pore system. Compared to the study of Tariq et al. on a hierarchical Al 2 O 3 material, we observed a much higher macro-and mesoporosity for our Ni/Al 2 O 3 -h catalysts and a narrower pore size distribution, especially for the macropore space [22]. Furthermore, by the combination of PXCT with ET we could cover the relevant length scales for the hierarchical pore system of the Ni/Al 2 O 3 -h catalyst with only two techniques, instead of a combination of ET (limited sample volume), dual-beam FIB imaging (destructive), and µ-CT (low spatial resolution). PXCT is particularly notable for allowing exceptional spatial resolutions for a hard X-ray imaging method, which can therefore be applied to extended sample volumes in the micron scale. Here the combination of ET and PXCT enabled us to detect pores ranging from about 2 nm to 80 nm (ET) up to 4 µm (PXCT).  The textural properties of Ni/Al2O3-h were also studied using well-established characterization techniques, i.e., N2 sorption and Hg porosimetry. For these conventional techniques, a purely mesoporous Ni/Al2O3-m sample, prepared via the same method but without addition of PEG, was used for comparison. The Ni/Al2O3-m sample contained a similar amount of 18.6 wt.% Ni as the Ni/Al2O3-h. According to the IUPAC definition [48], the N2 sorption isotherms displayed in Figure 7a can be classified as type IV isotherms with a H1 hysteresis, indicating mesoporous materials. Indeed, narrow pore width distributions (Figure 7b) with maxima at 8.3 nm and 10.3 nm were obtained, respectively. The different mesopore widths for these materials were most likely caused by the The textural properties of Ni/Al 2 O 3 -h were also studied using well-established characterization techniques, i.e., N 2 sorption and Hg porosimetry. For these conventional techniques, a purely mesoporous Ni/Al 2 O 3 -m sample, prepared via the same method but without addition of PEG, was used for comparison. The Ni/Al 2 O 3 -m sample contained a similar amount of 18.6 wt.% Ni as the Ni/Al 2 O 3 -h. According to the IUPAC definition [48], the N 2 sorption isotherms displayed in Figure 7a can be classified as type IV isotherms with a H1 hysteresis, indicating mesoporous materials. Indeed, narrow pore width distributions (Figure 7b) with maxima at 8.3 nm and 10.3 nm were obtained, respectively. The different mesopore widths for these materials were most likely caused by the influences of the polymerization-induced phase separation on the gelation process, which was induced in Ni/Al 2 O 3 -h but not in Ni/Al 2 O 3 -m. The existence of mesopores within Ni/Al2O3-h and Ni/Al2O3-m was also confirmed with Hg porosimetry (Figure 8). Additionally, Ni/Al2O3-h exhibited macropores with a narrow width distribution around 1000 nm. This illustrates the bimodal nature of the pore system with distinct meso-and macropore regimes. Approximately two-thirds of the specific total pore volume of Ni/Al2O3-h was contributed by macropores, with the remaining third from the mesopores ( Table 2). Both mesopore width and specific mesopore volume of Ni/Al2O3-h and Ni/Al2O3-m are comparable to the values obtained by Herwig et al. [15], though lower values for macropore width and higher values for specific macropore volume were reported. However, the Ni loading (3.4 wt.%, 23 wt.%, and 33 wt.%) and the amount of polyethylene glycol (PEO) used during gelation, both of which were found to have a large impact on the macropore formation [15], are different from this current work. We decided on a Ni mass fraction of 18 wt.% since this is a common Ni loading for CO2 methanation catalysts [45,49]. Therefore, the comparability of the macropore systems of the materials prepared here with those from Herwig et al. [15] is limited.
The pore widths wP of Ni/Al2O3-h and Ni/Al2O3-m determined from Hg porosimetry were 1-2 nm higher and the specific mesopore volume was slightly lower compared to the results from N2 sorption ( Table 2). This may be due to an inaccurate representation of the smaller mesopores (<10 nm) by Hg porosimetry at high pressure, which results in a higher impact of connectivity effects [50]. Thus, within the mesopore range, Hg intrusion could rather represent a more pore throat width distribution compared to a rather pore body width distribution for N2 sorption, as suggested by Rijfkogel et al. [51].
To experimentally determine the porosity of the Ni/Al2O3 xerogels, He pycnometry was performed and combined with the Hg porosimetry results according to Equation (1). The skeletal density of Ni/Al2O3-h and Ni/Al2O3-m was similar (Table 2), with that of Ni/Al2O3-h being slightly lower, possibly due to a higher degree of pores inaccessible to Hg. Notably, a porosity of 59% was obtained for the purely mesoporous Ni/Al2O3-m. The introduction of PEO during gelation, which generates a substantial amount of macropores, then gave rise to an increase in porosity to 78% for Ni/Al2O3-h. The existence of mesopores within Ni/Al 2 O 3 -h and Ni/Al 2 O 3 -m was also confirmed with Hg porosimetry (Figure 8). Additionally, Ni/Al 2 O 3 -h exhibited macropores with a narrow width distribution around 1000 nm. This illustrates the bimodal nature of the pore system with distinct mesoand macropore regimes. Approximately two-thirds of the specific total pore volume of Ni/Al 2 O 3 -h was contributed by macropores, with the remaining third from the mesopores ( Table 2). Both mesopore width and specific mesopore volume of Ni/Al 2 O 3 -h and Ni/Al 2 O 3 -m are comparable to the values obtained by Herwig et al. [15], though lower values for macropore width and higher values for specific macropore volume were reported. However, the Ni loading (3.4 wt.%, 23 wt.%, and 33 wt.%) and the amount of polyethylene glycol (PEO) used during gelation, both of which were found to have a large impact on the macropore formation [15], are different from this current work. We decided on a Ni mass fraction of 18 wt.% since this is a common Ni loading for CO 2 methanation catalysts [45,49]. Therefore, the comparability of the macropore systems of the materials prepared here with those from Herwig et al. [15] is limited.  Table 2. Specific surface area (ABET) and pore volume (VP) from N2 sorption, specific pore mesopore volume (VP,meso), specific total pore volume (VP,tot), mesopore width (wP,meso), macropore width (wP,macro), mesoporosity (εm) and macroporosity (εM), skeletal density (ρ), and total porosity (εtot) from He pycnometry, and mean mesopore diameter (deq,meso), mean macropore diameter (deq,macro), and porosity information from tomography analysis for calcined Ni/Al2O3-h and Ni/Al2O3-m.  Table 2. Specific surface area (A BET ) and pore volume (V P ) from N 2 sorption, specific pore mesopore volume (V P,meso ), specific total pore volume (V P,tot ), mesopore width (w P,meso ), macropore width (w P,macro ), mesoporosity (ε m ) and macroporosity (ε M ), skeletal density (ρ), and total porosity (ε tot ) from He pycnometry, and mean mesopore diameter (d eq,meso ), mean macropore diameter (d eq,macro ), and porosity information from tomography analysis for calcined Ni/Al 2 O 3 -h and Ni/Al 2 O 3 -m.

Ni/Al 2 O 3 -h Ni/Al 2 O 3 -m
The pore widths w P of Ni/Al 2 O 3 -h and Ni/Al 2 O 3 -m determined from Hg porosimetry were 1-2 nm higher and the specific mesopore volume was slightly lower compared to the results from N 2 sorption (Table 2). This may be due to an inaccurate representation of the smaller mesopores (<10 nm) by Hg porosimetry at high pressure, which results in a higher impact of connectivity effects [50]. Thus, within the mesopore range, Hg intrusion could rather represent a more pore throat width distribution compared to a rather pore body width distribution for N 2 sorption, as suggested by Rijfkogel et al. [51].
To experimentally determine the porosity of the Ni/Al 2 O 3 xerogels, He pycnometry was performed and combined with the Hg porosimetry results according to Equation (1). The skeletal density of Ni/Al 2 O 3 -h and Ni/Al 2 O 3 -m was similar (Table 2), with that of Ni/Al 2 O 3 -h being slightly lower, possibly due to a higher degree of pores inaccessible to Hg. Notably, a porosity of 59% was obtained for the purely mesoporous Ni/Al 2 O 3 -m. The introduction of PEO during gelation, which generates a substantial amount of macropores, then gave rise to an increase in porosity to 78% for Ni/Al 2 O 3 -h.
To compare the results of the porosity analysis of Ni/Al 2 O 3 -h for the different techniques we focused on the pore size distributions and porosity values obtained by the standard techniques and imaging methods. The total porosity ε tot from He pycnometry and Hg porosimetry was 78.0%, with a macroporosity of ε M = 55.0% and a mesoporosity of ε m = 23.0%. The combined values obtained from the tomography studies were 70.4% for ε tot , 46.3% for ε M, and 24.1% for ε m . The macroporosity determination via both methods seemed to be in a similar range, with a difference of 7.4%. This behavior with similar macroporosity differences between X-ray tomography and Hg porosimetry characterization was also found by Andersson et al. and Tariq et al. [22,23]. Compared to the macroporosity determination, the difference between the mesoporosity values obtained from conventional techniques and the imaging techniques was even smaller, with only 1.1%. Tariq et al. [22] reported the major discrepancy in porosity determination for the mesoporosity analysis of Hg porosimetry compared to ET. However, in our case, the mesoporosity information matched well. The reason for the differences in the study of Tariq et al. is that, there, pores below 10 nm diameter could not be properly resolved [22]. This shows that the accuracy of quantitative porosity analysis by tomographic methods is directly linked to the spatial resolution, and it is, therefore, crucial to report on the spatial resolution during such investigations.
Aside from the porosity values, the w P,macro by Hg porosimetry with 1.012 µm was smaller than the d eq,macro determined via PXCT with 2.545 µm. Hg porosimetry is known from literature to overestimate small pore sizes in the macropore range on the costs of larger pore sizes [22,[52][53][54]. This can be explained considering that Hg porosimetry probes the pore throat size/pore opening size by using the Washburn equation [55], while with image analysis using the PXCT data we determined the equivalent diameter assuming spherical pores within the material according to Equation (4). In addition to the w P,macro , the w P,meso of the mesopores determined by Hg porosimetry or N 2 -sorption ranged from 10.3 to 8.3 nm, respectively, and was much smaller than the average 33.3 ± 4.1 nm obtained from ET. This difference can be explained, again, by the different pore size definitions as for macropores. On the other hand, the presence of a hierarchical pore network is difficult to elucidate unambiguously by conventional methods, like N 2 sorption, Hg porosimetry, and He pycnometry. From the visual inspection of the pore structure from the segmented tomography data Figure 5, we can conclude that the Ni/Al 2 O 3 -h catalyst exhibits a well-connected macropores' network, while the mesopores are located within the solid material label of the PXCT. This represents a hierarchical pore system as the macropores form a parent-connected network from which the mesopores are originating. This clearly shows the advantage of imaging methods in performing quantitative and meaningful pore structure analysis, as not only pore sizes and porosity can be analyzed, but also more complex pore network models can be retrieved and co-located in 3D space [20].

Macrokinetics
To obtain information about the macrokinetics of the hierarchically porous catalyst, a model-based investigation was conducted on the discussed sample. The results were compared to an exclusively mesoporous sample. The relevant equations are given in the Supplementary Materials. This model-based investigation aimed to evaluate the relevant conditions in CO 2 methanation where a tailored hierarchical pore system might be advantageous in terms of mass-transport properties, compared to a conventional purely mesoporous material.
The calculated diffusion coefficients (D CO2,eff ) based on PXCT data and Hg-Intrusion data are given in Table 3 at industrially occurring CO 2 methanation conditions. According to these results, the introduction of the macroporous network into the porous catalyst material increased the effective diffusion coefficient by more than one order of magnitude. The reason for this is that in the macropores' molecular diffusion was the main diffusion mechanism, while in the mesopores, Knudsen diffusion prevailed. Thus, much higher catalyst effectiveness factors are expected to be present for a catalyst pellet with hierarchical pore system of industrially relevant size at high temperatures. Table 3. Pore network data based on tomography and Hg porosimetry measurements as well as calculated effective diffusion coefficients of CO 2 of a stoichiometric mixture x H 2 /x CO 2 = 4 : 1 at 600 K and 10 bar. To underline this statement, the effective reaction rates as well as effectiveness factors for a spherical catalyst pellet of 2.5 mm diameter based on the pore network properties given in Table 3 were calculated. The results are shown in the Arrhenius plots in Figure 9. On the one hand, it is evident that mass transport limitations in the exclusively mesoporous sample started at about 550 K. On the other hand, mass transport limitations for the hierarchically porous sample started at about 650 K to 675 K, depending on the data set. However, it has to be noted, that this effect was not solely based on the increased diffusion coefficient in the hierarchically porous material but also on the lower intrinsic reaction rate. This is due to the lower mesoporosity of the hierarchically porous sample, in comparison to the exclusively mesoporous sample. Based on the observed potential mass-transfer limitations from the macrokinetic simulations, further kinetic studies on the Ni/Al 2 O 3 -h and Ni/Al 2 O 3 -m samples should be carried out to experimentally validate whether a hierarchical pore system is superior in terms of minimizing mass transport phenomena, compared to a purely mesoporous material.
To underline this statement, the effective reaction rates as well as effectiveness factors for a spherical catalyst pellet of 2.5 mm diameter based on the pore network properties given in Table 3 were calculated. The results are shown in the Arrhenius plots in Figure 9. On the one hand, it is evident that mass transport limitations in the exclusively mesoporous sample started at about 550 K. On the other hand, mass transport limitations for the hierarchically porous sample started at about 650 K to 675 K, depending on the data set. However, it has to be noted, that this effect was not solely based on the increased diffusion coefficient in the hierarchically porous material but also on the lower intrinsic reaction rate. This is due to the lower mesoporosity of the hierarchically porous sample, in comparison to the exclusively mesoporous sample. Based on the observed potential mass-transfer limitations from the macrokinetic simulations, further kinetic studies on the Ni/Al2O3-h and Ni/Al2O3-m samples should be carried out to experimentally validate whether a hierarchical pore system is superior in terms of minimizing mass transport phenomena, compared to a purely mesoporous material.
(a) (b) (c) Figure 9. Arrhenius plots of effective and intrinsic reaction rates as well as catalyst effectiveness factor η based on pore network properties given in Table 3
The synthesis of 18Ni/Al2O3-h was adapted from Herwig et al. [15]: 2.88 g NiCl2·6H2O, 12.53 g AlCl3·6H2O, and 0.200 g PEG-900k were dissolved in an EtOH-H2O-mixture to give a molar ratio of nM/nEtOH/nH2O = 1/5.4/30.1/3.5·10 −6 , with nM = nNi + nAl. The masses of NiCl2·6H2O and AlCl3·6H2O were calculated for a target Ni mass fraction of 20 wt.%. The solution was cooled in an ice bath to 274 to 276 K. Then, 11.52 g of propylene oxide (PO), representing a molar ratio of nM/nPO = 3.1, were added rapidly under vigorous stirring (600 rpm). The solution was stirred for 3 min in the ice bath, followed by 7 min of stirring at room temperature. The container was then sealed and stored at 323 to 328 K for 24 h. Afterwards, the gel was kept in an EtOH bath for 7 days and then dried at room temperature. Finally, the dried gel was calcined in ambient air at 1123 K for 12 h using a heating rate of 2 K min −1 . Figure 9. Arrhenius plots of effective and intrinsic reaction rates as well as catalyst effectiveness factor η based on pore network properties given in Table 3 for a catalyst pellet of 2.5 mm diameter.
The synthesis of 18Ni/Al 2 O 3 -h was adapted from Herwig et al. [15]: 2.88 g NiCl 2 ·6H 2 O, 12.53 g AlCl 3 ·6H 2 O, and 0.200 g PEG-900k were dissolved in an EtOH-H 2 O-mixture to give a molar ratio of n M /n EtOH /n H2O = 1/5.4/30.1/3.5·10 −6 , with n M = n Ni + n Al . The masses of NiCl 2 ·6H 2 O and AlCl 3 ·6H 2 O were calculated for a target Ni mass fraction of 20 wt.%. The solution was cooled in an ice bath to 274 to 276 K. Then, 11.52 g of propylene oxide (PO), representing a molar ratio of n M /n PO = 3.1, were added rapidly under vigorous stirring (600 rpm). The solution was stirred for 3 min in the ice bath, followed by 7 min of stirring at room temperature. The container was then sealed and stored at 323 to 328 K for 24 h. Afterwards, the gel was kept in an EtOH bath for 7 days and then dried at room temperature. Finally, the dried gel was calcined in ambient air at 1123 K for 12 h using a heating rate of 2 K min −1 . For comparison, an equivalent, purely mesoporous sample (denoted as 18Ni/Al 2 O 3 -m) was prepared as stated above, but without PEG-900k added in the gelation solution. Furthermore, a standard industrial Ni/Al 2 O 3 catalyst (denoted as SPP2080-IMRC) was used as a reference material.
Diffuse reflectance ultraviolet-visible (DRUV/vis) spectroscopy was conducted on a Lambda 650S UV/vis spectrometer (Perkin Elmer Waltham, MA, USA) equipped with a 150 mm Ulbricht sphere. Spectra were recorded in the range between 200 to 800 nm with 1-nm step size and 2-nm slit width. N 2 sorption experiments were conducted on a Sorptomatic 1990 instrument (Porotec, Hofheim am Taunus, Germany). Samples were degassed at 523 K for at least 4 h. The adsorption and desorption isotherms were recorded at 77 K and analyzed using the ASiQwin software (Quantachrome Instruments, Boynton Beach, FL, USA). The total pore volume was determined at p/p 0 ≥ 0.95. Specific surface area and pore width distribution were determined by the BET method [59] (adsorption branch) and BJH method [60] (desorption branch), respectively. An experimental error of 5% was expected for specific surface area, specific pore volume, and pore width.
Hg porosimetry analysis was conducted on a Pascal 140 (ThermoScientific, Waltham, MA, USA) for data points from pressures up to 250 kPa and a Pascal 440 (ThermoScientific, Waltham, MA, USA) for data points from pressures between 250 kPa to 400 MPa. The contact angle used was 140 • with a surface tension of 0.48 N m −1 . The pore width was calculated by the Washburn equation [55]. The mesoand macropore widths with the maximum differential pore volume were referred to as modal mesopore width (w P,meso ) and modal macropore width (w P,macro ), respectively. An experimental error of 5% was expected for specific pore volumes and pore widths.
He pycnometry was performed on a Pycnomatic ACT EVO instrument (Porotec, Hofheim am Taunus, Germany) at room temperature. For analysis, at least 10 pulses of He with a gas pressure of 2 bar were used. The final skeletal density was calculated by taking the average value of four consecutive measurements. An experimental error of 0.002 g cm −3 was expected for the skeletal density. The total porosity (ε tot ) of the sample was calculated from Equation (1) using the skeletal density (ρ), determined via He pycnometry, and the total specific pore volume (V P,tot ), determined from Hg porosimetry. The meso-and macroporosity were calculated likewise, using the specific mesopore and specific macropore volume in the numerator, respectively.
Temperature-programmed reduction (TPR) of catalysts was conducted using the ChemStarPx+ equipment (Quantachrome Instruments, Boynton Beach, FL, USA). Then, 100 mg sample were loaded in a quartz U-tube. First, the sample was heated in a He flow at 423 K for 30 min and subsequently cooled to 323 K. The reduction was performed using 10% H 2 in Ar (50 cm 3 (STP)/min). The temperature was raised using a ramp rate of 10 K/min to 1173 K and held at this level for 30 min. To remove moisture formed during reduction, effluents were sent through a silica gel water trap before reaching the detector. H 2 consumption was measured continuously using a thermal conductivity detector (TCD). Cooling was performed in He flow. Finally, a pulse calibration with H 2 was performed at 373 K. The loop volume was 528 µL and Ar was used as carrier flow.

Catalytic Measurements
Catalytic experiments were performed in an automated lab-scale catalyst testing unit equipped with a tubular quartz glass reactor (inner diameter: 8 mm). The reactor was operated at atmospheric pressure with 100 mg catalyst samples diluted with SiC (carborundum powder, coarse, 46 grit, abcr GmbH, Karlsruhe, Germany) in a ratio of 1:9. The nonpelletized samples were reduced in situ prior to the catalytic experiments in a H 2 /He mixture. The SPP2080-IMRC sample was reduced at 673 K for 8 h using a mixture of V He = 4/1/5 with a total gas flow rate of 200 cm 3 min −1 , respectively. Then, 24 online gas chromatography (GC) measurements were taken (time about 2 h) under each set of conditions after each temperature change and equilibration time, while the last ten measurements were used for calculation of CO 2 conversion and CH 4 selectivity. Water was separated from the reactor effluent with a membrane unit and was not analyzed. No other carbon-containing analytes were detected apart from CH 4 , CO 2, and CO, which were, therefore, used for the overall carbon mass balance. The CO 2 conversion and CH 4 selectivity (both in %) were calculated from the last 10 GC measurements, as defined in Equations (2) and (3).

Tomography Studies
Samples for ET were prepared by grinding in an agate mortar and suspending in EtOH. A droplet of the suspension was added on a 100 × 400-mesh carbon-coated Cu-TEM grid (Quantifoil Micro Tools GmbH, Großlöbichau, Germany) with a support film thickness of 10-20 nm and dried under atmospheric conditions for 30 min, before adding Au colloidal particles (6.5 nm diameter). ET measurements of the sample were performed using a Fischione 2020 tomography holder (Fischione Instruments, Export, PA, USA), on a Titan 80-300 (FEI Company, Hillsboro, OR, USA) microscope with an acceleration voltage of 300 kV in STEM mode. Tilt series of high-angle annular dark field STEM images were collected for two particles with the Xplore3D software 3.1 (FEI Company, Hillsboro, OR, USA) over a tilt range of −72 to 76 • for ET1 and −74 to 78 • for ET2, with an increment of 2 • . Alignment of the tilt series was performed in IMOD [61], by tracking 20 Au particles as fiducial markers, manually refining their position, enabling refinement of magnification and image rotation, and grouping tilt angles every three tilts without applying a distortion correction. A mean residual alignment error of 0.34 and 0.48 pixels was obtained for ET1 and ET2, respectively. After applying a simultaneous iterative reconstruction technique (SIRT) [62] with 100 iterations to the aligned projection series, the reconstructions were denoised by employing the ImageJ [63] plugins Filters/Median 3D and Enhance Local Contrast (ELAHE). Final reconstructions were obtained by ten iterations of discrete algebraic reconstruction technique (DART) [64] using TomoJ plugin in Fiji [63,65]. The reconstructed tomograms had a final voxel size of 0.94 × 0.94 × 0.94 nm 3 for ET1 and 0.64 × 0.64 × 0.64 nm 3 for ET2.
The sample for PXCT was first reduced in a micro-capillary setup under the following conditions: He:H 2 (15:5 cm 3 min −1 ), heated at 20 K/min then held for 30 min at 1023 K, followed by reaction conditions of He:H 2 :CO 2 (15:4:1 cm 3 min −1 ) for 30 min at 773 K. For PXCT the resulting catalyst was mounted on an Al-pin of the OMNY design [66] using a Strata 400 S FIB/SEM (FEI Company, Hillsboro, OR, USA). A suitable particle was Pt-glued to a micromanipulator and then transferred and Pt-glued on the Al-pins. The mounted particle is shown in the Supplementary Materials Figure S3.
PXCT experiments were carried out at the coherence branch I13-1 of Diamond Light Source (DLS, UK) [67,68] at a photon energy of 9.7 keV and a beam size of about 5 µm on the sample. Far-field diffraction patterns were recorded using a MerlinX detector with a Quad system of 512 × 512 pixels and a pixel size of 55 µm, positioned at a sample-detector distance of 4.13 m. For collection of tomography data, two-dimensional projections of the sample were recorded over a rotational range of 180 • in 0.18 • steps (1000 projections), with a field of view of 40 × 22 µm 2 and an exposure time of 30 ms per sampling point. The total acquisition of the tomogram took about 19 h. Ptychographic reconstructions were done on a cropped area of 256 × 256 pixels, corresponding to a pixel size of 37 nm, with the PtyREX package [69] to obtain a 2D projection series. This series was aligned and reconstructed into 3D volumes with a reconstructed voxel size of 38.1 nm and effective isotropic resolution of 78.3 nm using the MATLAB script by Odstrcil et al. [46].
Image processing and analysis of the obtained ET and PXCT tomograms was carried out using Avizo 9.7.0 software (FEI SAS, Thermo Fisher Scientific, Merignac, France). For the DART-reconstructed ET data, the material was first segmented using thresholding. Then, the object was masked using the "Lasso" and "Interpolation" tools to separate pores from background. For the masked object, the pores were labeled by thresholding. The material and pore labels were analyzed by "Label Analysis" to determine the label volumes and surfaces. To analyze individual pores, the pores' label was separated into individual pores by combination of the watershed, distance map, and H-Maxima analysis as implemented in Avizo's "Separate Objects" module. An "Opening" module was used afterwards to remove point artifacts of the pore separation. The separated pores were then analyzed by "Label Analysis" to determine characteristics of the individual pores, focusing on the single pore volume and equivalent pore diameter d eq , which is defined as follows: with V 3D as the voxel-based calculated 3D volume of a single pore. For the reconstructed PXCT data, first the Pt residues from sample preparation were cropped, as shown in the Supplementary Materials Figure S4. Then a nonlocal-means filter was applied on the grayscale images. This was followed by masking the material and pores from the background using the "Convex Hull" module. Subsequently the masked imaged was separated into pores and material by thresholding. This simple segmentation was possible due to sufficient contrast in electron density between air in the pores and the Ni/Al 2 O 3 material. The obtained material and pore labels were further analyzed, as described for the ET data.
The porosity of the ET and PXCT data ε tomo was calculated as follows: with the total volume of the material (V s,tot ) and the total pore volume (V p,tot ) as obtained from label analysis of the unseparated pores. To retrieve the porosity distribution depending on d eq , the porosity of an individual pore is defined as ε ind .
with the individual pore volume V 3D of a single pore. The distribution of the individual porosity depending on d eq was analyzed regarding its weighted numerical mean pore diameter and by fitting a Gaussian function. The difference of the mean values from the weighted numerical analysis and the fit was assumed as the standard deviation of the mean pore diameter from the numerical analysis.

Macrokinetics
Detailed information about the calculation of the diffusion properties and macrokinetics are provided in the Supplementary Materials section "Catalyst Particle Model".

Conclusions
A hierarchically porous Ni/Al 2 O 3 catalyst was shown to be active for CO 2 methanation, having previously shown increased performance in dry reforming of CH 4 . The hierarchical catalyst showed a similar performance for CO 2 methanation as a standard industrial Ni/Al 2 O 3 reference catalyst, with CO 2 conversion of 69% and >90% selectivity for CH 4 at 623 K. However, the catalyst needed to be pretreated at a high temperature of 1123 K to reduce Ni 2+ to Ni 0 . This is due to the incorporation of Ni into spinel-type structures of Al 2 O 3 and NiAl 2 O 4 . A similar behavior is to be expected for other Ni/Al 2 O 3 catalysts prepared by co-gelation. Therefore, co-gelation is likely not a suitable preparation method for applications at moderate temperatures (523-773 K) such as CO 2 methanation, and impregnation methods should be preferred. The hierarchical Ni/Al 2 O 3 catalyst exhibited a volume-based total porosity of 78%, mesoporosity of 23.0%, macroporosity of 55.0%, and mean pore width of about 1 µm for the macropores and 8.3 nm to 10.3 nm for the mesopores based on Hg porosimetry and He pycnometry. Using ET and PXCT for a multiscale analysis of the pore system, we retrieved similar porosity values as from Hg porosimetry. However, the mean pore diameters of the meso-and macropores obtained from the imaging methods were more than twice the size compared to Hg porosimetry. We attributed this to different definitions of the pore width and diameter and the technical limitations of Hg porosimetry. Additionally, we proved and can quantify the hierarchical pore structure of the catalyst and the thoroughly connected macropores by tomography, which is not practical using conventional porosimetry methods. A simulation of diffusion properties from experimental porosity data together with macrokinetic properties revealed potential mass transport limitations. The introduction of macropores and the hierarchical pore structure are suggested to strongly improve the mass transport and can, thus, lead to a higher effectiveness of the catalyst in comparison to a purely mesoporous catalyst. This study demonstrates the great benefit of a combined synthetic, analytic, and computational approach for quantifiable characterization and rational design of porous supported methanation catalysts. The chosen system is ideal for future extensive kinetic studies of the influence of mass transport phenomena, to confirm the results obtained from simulations.
Supplementary Materials: The following are available online at http://www.mdpi.com/2073-4344/10/12/1471/s1. Figure S1: Selected slice of the SIRT-reconstructed ET1. Figure S2: Selected slice of the SIRT-reconstructed ET2. Figure S3: SEM images of the Ni/Al 2 O 3 -h sample mounted on the Al-pin for the PXCT experiment. Figure S4: Uncropped PXCT of the Ni/Al 2 O 3 -h sample and cropped tomogram. Figure S5: Selected region of the PXCT tomogram of the Ni/Al 2 O 3 -h sample and resulting input for the FSC analysis. Figure S6: FSC analysis of the NiAl 2 O 3 -h PXCT. Figure S7: Frequency distribution of different pore diameters obtained from the label analysis of the separated pores from the ET and PXCT experiments of the Ni/Al 2 O 3 -h catalyst. Table S1: Structural parameters obtained from Rietveld analysis for the refinements of Ni/Al 2 O 3 -h.