# Hyperpolarized Xenon-129 MRI, Fluorine-19 PFG NMR, SAXS and electron microscopy studies of the impact on structure and gas phase mass transport of introducing controlled macroporosity into mesoporous, alumina, catalyst support pellets

Stefano Collins, Severine Humbert, Dina Lofficial, Christophe Vallee, Christopher Parmenter, Huw E.L. Williams, Arthur Harrison, Max Filkins, Galina E Pavlovskaya, Thomas Meersmann, et al.

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Stefano Collins, Severine Humbert, Dina Lofficial, Christophe Vallee, Christopher Parmenter, et al. Hyperpolarized Xenon-129 MRI, Fluorine-19 PFG NMR, SAXS and electron microscopy studies of the impact on structure and gas phase mass transport of introducing controlled macroporosity into mesoporous, alumina, catalyst support pellets. Chemical Engineering Science, 2026, 321 (Part B), pp.122843. 10.1016/j.ces.2025.122843. hal-05454428

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# Hyperpolarized Xenon-129 MRI, Fluorine-19 PFG NMR, SAXS and electron microscopy studies of the impact on structure and gas phase mass transport of introducing controlled macroporosity into mesoporous, alumina, catalyst support pellets

Stefano Collins<sup>a,b,c</sup>, Severine Humbert<sup>b</sup>, Dina Lofficial<sup>b</sup>, Christophe Vallee<sup>b</sup>, Christopher Parmenter<sup>d</sup>, Huw E.L. Williams<sup>e</sup>, Arthur Harrison<sup>c</sup>, Max Filkins<sup>c,d</sup>, Galina E. Pavlovskaya<sup>c</sup>, Thomas Meersmann<sup>c,d</sup>, Sean P. Rigby<sup>a,*</sup>

## ARTICLE INFO

### Keywords:

Magnetic resonance imaging

PGSE NMR

Gas sorption

Seeded percolation

Knudsen diffusion

Tortuosity

## ABSTRACT

A novel combination of experimental characterization techniques has been used to assess the impact of the controlled addition of macroporosity to mesoporous catalyst supports for use with diffusion-limited, gas-phase reactions. This is the first study to 2D-map the gas-phase uptake across different internal regions within individual mesoporous pellets (or any other inorganic mesoporous media) using hyperpolarized xenon MRI. A porogen-templating method aimed to control the gas-phase mass transport properties of disordered alumina pellets. The degree of success of the pore structure design was examined using electron microscopy and gas sorption, and, for the resultant mass transport properties, using gas-phase PFG NMR and hyperpolarized xenon MRI. TEM showed that the macropore templating led to the creation of a corona of parallel-aligned alumina platelets around the macropore boundaries. The particular impact of this corona on mass transport was studied using PFG NMR, and found to be significant. The MRI showed that individual pellets had very large length-scale heterogeneities in the macroscopic spatial distribution of gas uptake rates. The data from the pore structural characterization, along with percolation theory, was used to develop a structural model to predict the impact of creating additional macroporosity by the particular fabrication method employed, and was successfully validated using the PFG NMR data for gas self-diffusion and MRI mapping of gas uptake rates across individual pellets.

## 1. Introduction

In the development of supports for heterogeneous catalysts for diffusion-limited reactions, there is a trade-off between providing as much surface area to disperse the catalyst as possible, normally requiring small pores, and keeping the pellet effectiveness factor high, generally needing larger pores, especially when operating in the Knudsen diffusion regime (Prachayawarakorn and Mann, 2007). Large pore structures also tend to be more resistant to coking. There is thus an ongoing search for the optimal pore structure for a given application.

One way to potentially achieve the aforementioned optimal pore structure is to introduce controlled levels of macroporosity to provide more rapid access routes to the catalyst dispersed on the mesoporous surface area. Computer simulations of the reaction process can be used to determine the optimum macropore network for a given catalyst (Wang and Coppens, 2010).

The typical means to introduce controlled macroporosity is to use a pore templating procedure during fabrication. The different types of potential template materials vary in the level of control permitted for pore size and geometry, and concomitant cost. Organic materials, such

<footnote><sup>a</sup> Department of Chemical and Environmental Engineering, Faculty of Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK</footnote>

<footnote><sup>b</sup> IFP Energies Nouvelles, Rond-point de l'échangeur de Solaize, BP 3, 69360 Solaize, France</footnote>

<footnote><sup>c</sup> Sir Peter Mansfield Imaging Centre, University of Nottingham, University Park, Nottingham NG7 2RD, UK</footnote>

<footnote><sup>d</sup> Nanoscale and Microscale Research Centre, University of Nottingham, University Park, Nottingham NG7 2RD, UK</footnote>

<footnote><sup>e</sup> Centre for Biomedical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: sean.rigby@nottingham.ac.uk (S.P. Rigby).</footnote>
as sugar crystals, are cheap but relatively lacking in control (Yang, 2005). Latex spheres can be used to template porous oxide materials with macropores in an ordered array (Barros Filho et al., 2005). Foam-type materials are frequently employed to make highly macroporous ceramic supports for highly diffusion-limited catalytic reactions (Nepryahin et al., 2016). These are often made using polymer-based foams as templates. For example, Yeetsorn et al. (2022) proposed that the cell architectures of polyurethane (PU) and polyvinyl alcohol foams are suitable for catalyst support preparation due to their high porosities of 98–99 %, and typical cell pore sizes in the range 50–200 µm. The mass transport properties of foam-like alumina monoliths have been studied using integrated gas sorption and mercury porosimetry, which demonstrated that the macropores in the alumina foam material improved mass transport rates by a factor of ~ 2 to 3 (Nepryahin et al., 2016). In this work, the contribution to mass transport from macropores, created using varying amounts of porogen, will be assessed using alternative NMR and modelling methods not requiring the use of mercury.

It is suggested that barriers to mass transport often may exist at the boundaries of porous materials, or where the diffusing molecules pass from a region of one modal size of pores to a region of another (Keil, 1999; Tallarek et al., 2019). The surface barriers typically consist of a thin layer of dramatically reduced diffusivity. These barriers can lead to differences in observed mass transport rates between steady-state self-diffusion and transient uptake methods of measurement. For some materials, the penetration of the boundary of porous crystallites by diffusing molecules is inhibited by the blinding of many surface pores (Hibbe et al., 2011). In this work, electron microscopy will be used to detect surface barriers and the impact of these barriers on mass transport will be assessed by pulsed-field gradient (PFG) NMR modelling.

The various types of fabrication and forming processes for catalyst pellets can introduce varying levels of heterogeneities in pellet pore structures. A variety of imaging methods, including electron microscopy, magnetic resonance imaging (MRI) and computerized X-ray tomography (CXT) have shown that different raw materials and forming methods for pellets can lead to different degrees and types of heterogeneity (Hollewand and Gladden, 1993; Timonen et al., 1995; Ruffino et al., 2005; Mousa et al. 2023a, 2023b). It is thus necessary to fully understand the impact of different forming methods on pellet pore structure. In this work the impact of the use of an emulsion of porogen (stabilized by surfactant) to form macropores on the surrounding mesoporous matrix will be studied.

Pulsed-field gradient (PFG) NMR has been used extensively to study the porous structure of catalyst support pellets (Hollewand and Gladden, 1995a; Wood and Gladden, 2003; Forman et al., 2016; Rottreau et al., 2017). PFG NMR can be used to probe the pore sizes for macroporous materials or the tortuosity of the network of mesoporous solids. The length-scales probed correspond to those of the root-mean-square (rms) displacement of the probe molecule. For liquids this typically corresponds to length-scales of the order ~ 10s-100s µm. This means that PFG NMR can also detect macroscopic heterogeneities over length-scales larger than the rms displacement where exchange does not occur between different regions. In this work, PFG NMR will be performed using sulphur hexafluoride ($SF_6$) gas to probe length-scales of the order of ~ 10s microns largely in the Knudsen regime, rather than molecular diffusion regime in liquid studies.

Computerised X-ray tomography (CXT), in combination with integrated gas sorption and mercury porosimetry, has been used to demonstrate the importance of the particular nature and degree of macroscopic (> few microns) scale heterogeneities in the spatial distribution of pore sizes, arising from different particle feeds, in controlling mass transport within pellets via gas phase diffusion (Mousa et al., 2023a, 2023b). Further, CXT can be used to study gas uptake into complete packed beds of absorbents or into shale cores (Pini, 2014; Pini et al., 2021; Zhang et al., 2021). It can demonstrate the presence of heterogeneities across the packed bed. However, the low density of gas means that the actual gas phase cannot be seen directly with CXT but

only inferred from the accumulation of denser liquid-like adsorbate on the pore surfaces.

In contrast, MRI can be used to study all of structure, transport, adsorption and reaction processes within porous solids. MRI, using relaxation time and diffusion weighted methods, has been previously used to show that catalyst support pellets, formed by a variety of techniques, have macroscopic heterogeneities (>10 µm) in the spatial distribution of local average porosity (voidage fraction), pore size and tortuosity (connectivity) (Hollewand and Gladden, 1993, 1995a,b; Rigby and Gladden, 1996). These structural heterogeneities have been shown to control the mass transfer properties of pellets (Hollewand and Gladden, 1995b; Rigby et al., 1996). More recently MRI has been used in conjunction with cryoporometry to show, through the active utilization of the (long-range) advanced melting effect, that layering in the pore structure of alumina extrudates is associated with macroscopic, anisotropic diffusion properties (Himona et al., 2024). However, these studies were conducted with liquids. Gas phase MRI of packed beds has been performed using sulphur hexafluoride gas but the high pressure required to obtain sufficient signal-to-noise to image meant that the diffusional mass transfer was also in the molecular regime in all pores, just like with liquids (Sankey et al., 2009; Ramskill et al., 2017). Thermally polarized ethane gas has also been imaged with MRI but at elevated pressures (4 bar) (Anikeeva et al., 2023).

However, the use of hyperpolarized xenon as the probe gas for MRI substantially boosts the signal-to-noise ratio compared to normal gases. This has permitted the study of gas uptake at low pressure, and, thence, in the Knudsen regime, into solely mesoporous alumina absorbent pellets within a packed bed (Pavlovskaya et al., 2015). It was found that the spread in conductances obtained in the Knudsen regime was wider than that for molecular diffusion, showing that it is necessary to use methods that can detect low pressure gases. In this work hyperpolarized (hp) xenon MRI will be used to study gas uptake in the Knudsen regime in mesopores, and with higher resolution than in previous work, in order to map, and thereby compare, macroscopic heterogeneities in the spatial variation of low pressure gas uptake rates for a range of different pellets with varying pore structural characteristics, unlike the just single type of extrudate studied in previous work.

While the potential impact of structural heterogeneities over length-scales much larger than the size of macropores, but smaller than the pellet size, is known (Rigby, 2020), it is rarely considered in most pore network modelling studies, which assume the network is homogeneous beyond a distance of a few nodes, or only consider very short range (compared to the pellet size) correlations in parameters such as pore size (Yang et al., 2025). In this work the potential presence, and importance to mass transport, of structural heterogeneities over a range of different length-scales will be considered, and incorporated into a model where necessary.

The parameter known as “tortuosity” is often used to characterize the ‘ease’ or ‘efficiency’ of mass transport within a void space. Tortuosity is, typically, a void space descriptor related to its topology, but it is often conceptually vague and ill-defined (Fu et al., 2021; Ghanbarian et al., 2013; Holzer et al., 2023). In some part this may be because previous work (Hollewand and Gladden, 1995a,b; Rigby and Gladden, 1996; Rigby and Daut, 2002) suggested that, for porous catalyst pellets with structural heterogeneity over various length-scales, the value of tortuosity obtained may depend upon the physical process and boundary conditions of the measurement, namely whether the diffusion process is self-diffusion or driven via a concentration gradient, and the length- and time-scales of the measurement. In this work the mass transport will be studied over different length-scales via two experimental methods that involve diffusion processes with different boundary conditions. PFG NMR involves steady-state self-diffusion, while the hp Xe MRI method involves transient diffusion driven via a concentration gradient. A further complication, with gas-phase diffusion in the Knudsen regime, is the choice of definition for the reference diffusivity in the effective diffusion coefficient (see Theory section) (Fu et al., 2021; Ghanbarian
et al., 2013; Holzer et al., 2023). This will be considered in detail in this work.

This work aims to explicitly determine the usefulness of macropores introduced via templating with porogen. Previous transport modelling work for heterogeneous media by Maxwell (cited in Crank, 1975) (described in Theory section) and Prager (1960) has demonstrated that a medium consisting of one phase, composed of multiple but isolated spherical regions (corresponding here to macropores), with an enhanced conductivity (corresponding here to diffusivity), distributed randomly across a continuous matrix with a lower conductivity (here the mesoporous matrix), similar in form to the structure shown in Fig. 1, will lead to an increase in the overall conductivity of the medium. This is the underlying basis for the design of the pellets studied here. This work will test the extent to which this ideal design can be implemented for a real material with feasible fabrication techniques, and its performance in enhancing gas-phase mass transport.

In this work, due to the presence of structural heterogeneity over a range of length-scales in the alumina catalyst supports studied here, multiple complementary experimental techniques (described in Section 2) for structural characterization and mass transport studies will be used to elucidate the structure-transport relationships that exist at these different length-scales. The range of techniques used, the length-scales which they probe, and the particular structural or transport properties probed are listed in Table 1. From Table 1, it can be seen that, between them, the experimental techniques used probed complementary length-scales (e.g. the PFG NMR and MRI probe mass transport below, and above, ~100 µm, respectively), or different aspects of the pellet structure, but together provide a comprehensive and exhaustive characterization of all potentially important features. As will be explained in the Theory Section (3), the information and numerical data obtained from these experimental techniques will be fed into multi-scale modelling of the mass transport in order to extract the intrinsic relative improvement to mass transport provided specifically by the addition of the macroporosity to the mesoporous pellet matrix, independent of changes in pellet size and also study the sensitivity of the overall mass transport to variations in the parameters characterizing the particular structural

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Schematic diagram (not to scale) showing the basic elements of the structural model for the pellets studied in this work. The core region consists of mesoporous matrix that also contains macropores, while the rind consists of mesoporous matrix alone. The corona is a boundary layer in the mesoporous matrix surrounding the individual macropores.</figcaption></figure>

<caption>**Table 1**</caption>

The various experimental techniques used in this study, their characteristic length-scale ranges, and the main properties that they will be used to study.

<table><thead><tr><td>Experimental technique</td><td>Length-scale probed</td><td>Pellet properties studied</td></tr></thead><tbody><tr><td>TEM</td><td>~1–100s nm</td><td>Alumina platelet sizes and structural arrangement</td></tr><tr><td>SEM/<br>FIB SEM</td><td>~10 nm–<br>100s µm</td><td>Sizes and spatial distribution of macropores</td></tr><tr><td>SAXS</td><td>~1–100s nm</td><td>Alumina platelet and aggregate sizes</td></tr><tr><td>Nitrogen sorption</td><td>~0.1–200 nm</td><td>Mesopore size distribution and mesopore network connectivity (to obtain tortuosity via percolation theory)</td></tr><tr><td>Mercury<br>porosimetry</td><td>~3 nm<br>~100s µm</td><td>Mesopore and macropore size distributions, and macroscopic heterogeneities in pore size distribution</td></tr><tr><td>PFG NMR</td><td>&lt;100 µm</td><td>Mass transport properties (tortuosity)</td></tr><tr><td>hp Xe MRI</td><td>&gt;100 µm</td><td>Mass transport properties (maps of gas uptake rates)</td></tr></tbody></table>

features found using the various characterization methods.

## 2. Materials and methods

### 2.1. Materials

The materials studied in this work are cylindrical, gamma alumina pellets of 2 mm diameter and length, prepared especially for this study by IFP Energies Nouvelles. The pellets have been fabricated via a synthesis method in which differing amounts of a porogen emulsion were used to induce variable amounts of macroporosity within a boehmite matrix. The porogen acted as a template to create macropores and, then, was removed by calcination. The first step consisted of forming a dispersion of boehmite powder in water by use of a dispersing additive, such as an inorganic acid, and mixing. The porogen, such as a refined oil, wad then added to this suspension in the form of an emulsion using a surfactant (Deshmukh et al., 2018; Dong et al., 2022). After neutralization, by addition of an inorganic base, and shaping, the pellets were dried in an oven at a temperature lower than 150 °C to extract the water. Finally, the pellets were calcined at a temperature between 400–700 °C to obtain gamma alumina and to generate the macropores via porogen degradation (Euzen et al., 2002; Zarezadeh-Mehrizi et al., 2018).

The resultant pellets were either mesoporous (A and AA), or bidisperse mesoporous and macroporous (the rest). The different batches used in this work are listed in Table 2. The four main batches studied here were A, B, C and D, as these cover a wide range in porogen content. In some particular experiments, these four were supplemented by further batches listed in Table 2 which were made with similar or intermediate porogen contents in the synthesis.

### 2.2. Methods

The basic physical principles and general experimental procedures underlying the various experimental characterization methods used here are described in more detail elsewhere (Rigby, 2020). This section

<caption>**Table 2**</caption>

Details of batches of alumina catalyst support pellet used in this work.

<table><thead><tr><td>Batch Identifier</td><td>Relative porogen content /arb. units</td></tr></thead><tbody><tr><td>A</td><td>0</td></tr><tr><td>B</td><td>21</td></tr><tr><td>C</td><td>56</td></tr><tr><td>D</td><td>84</td></tr><tr><td>AA</td><td>0</td></tr><tr><td>AB</td><td>10</td></tr><tr><td>BC</td><td>38</td></tr><tr><td>B#</td><td>21</td></tr></tbody></table>
will thus only provide details more specific to the particular experiments conducted on the samples studied here.

### 2.2.1. Scanning electron microscopy (SEM)

The samples were prepared by vacuum impregnation with a highly fluid monomer (methyl methacrylate), followed by its in-situ polymerisation. The polymerisation was carried out under controlled temperature and pressure in an autoclave, producing solid blocks in the form of cylinders. The samples were then polished with a rotary polisher (EcoMet 250) using SiC polishing disks of reference P4000, which is equivalent to a granulometry of roughly 5 µm. The images were taken on a Zeiss SUPRA 40. The 2D images were then processed, using the 'plug im!' software, to output a PSD histogram and a porosity (assuming isotropy in 3D) for each batch.

### 2.2.2. Focused ion beam scanning electron microscopy (FIB SEM)

The same polished samples that had been prepared for SEM were also used for FIB SEM. The serial milling images were acquired using a Zeiss Crossbeam 550 FIB-SEM instrument. The samples were mounted onto a sample holder, placed inside the instrument, and then subjected to a vacuum. The sample was rotated by 54°, to be perpendicular to the ion-beam, whilst ensuring that eucentricity was reached. The coincidence point was then reached, ensuring that the SEM and the FIB were looking at the same location. An automated FIB milling algorithm was subsequently set up and executed to mill through a cross-section of material (50 nm slice thickness) to acquire a 3D image stack. This method was performed on samples from each batch. To address the charging effects, several developments to the experimental procedure were pursued. Since the platinum layer is only deposited on a small region (above the region of interest) of the material, a layer of sticky aluminium foil (conductive) was placed onto the sample surface to reduce charging.

### 2.2.3. Transmission electron microscopy

Two types of preparation were performed, namely the 'cross-sectional' and 'direct' preparations. For the 'cross-sectional' preparation, the sample was first embedded as chips (powder form) in epoxy resin. Subsequently, sections of 70 nm thickness were taken using a high precision ultramicrotome. These sections were placed on the TEM support (5 mm diameter grid). For the 'direct' preparation, the sample was first ground in a mortar and then a spatula tip of the powder was ultrasonically dispersed in ethanol. A drop of the mixture was then placed on the TEM support. For both experiments, the sample was observed in bright field TEM mode on a JEOL F200 microscope.

To visualise the arrangement of crystallites at the outer surface of the pellets, the 'cross-sectional' method could not be used, since this method does not guarantee that the area observed in the microscope corresponds to the outer surface. For this experiment, a "double coating" preparation method was developed. The "double coating" preparation method involved the following steps: the sample was embedded with epoxy resin to strengthen it, the sample was, subsequently, 'trimmed' to obtain a smaller region of interest, the sample was then re-embedded in resin to trap this region of interest, and then trimmed again for shaping. Finally, 70 nm thick sections were taken using the ultramicrotome and analysed in a JEOL F200 microscope.

### 2.2.4. Small angle X-ray scattering (SAXS)

Small angle X-Ray scattering (SAXS) was performed on the SWING beamline of synchrotron SOLEIL (Saclay, France) with an 8 keV incident beam. Batches in the form of spherical pellets (between Kapton foil) of the γ-alumina were analysed. To cover a wide range of q values, two sample-detector distances (1003 mm and 6229 mm) were employed. The scattering images were recorded using an AVIEX PCCD170170 detector. More information on the experiment method is described by Duchène et al. (2023).

### 2.2.5. Nitrogen sorption

Nitrogen physisorption was performed at 77 K using a 3Flex instrument (Micromeritics). Before any experiment, the samples were subjected to pre-treatment under a secondary vacuum ($10^{-5}$ mbar) at 350 °C for 3 h to remove any impurities, physisorbed and chemisorbed water. The pore volume and BET surface area were determined from the nitrogen isotherm. The pore size distributions were determined from the nitrogen adsorption isotherm branch based on the Barrett-Joyner-Halenda (BJH) method (Barrett et al., 1951), using the hemispherical meniscus in the Kelvin equation, and the Harkins-Jura t-layer equation (Rigby, 2020). Previous work (as discussed extensively in Rigby (2020)) has shown that the BJH method using the Kelvin equation provides accurate pore sizes for materials with mesopores of the range of sizes as the aluminas studied here.

### 2.2.6. Mercury porosimetry

The pellets were analysed by mercury intrusion porosimetry using an AUTOPORE IV instrument (Micromeritics). The equilibration time at each pressure was set to 30 s. Before analysis, the samples were subjected to a pre-treatment at 250 °C for 2 h in an oven.

### 2.2.7. $SF_6$ pulsed-field gradient (PFG) NMR

In some refining processes, the diffusion regime of hydrocarbons in catalysts is intermediate, i.e. diffusion is limited simultaneously by Knudsen diffusion and molecular diffusion. As a result, it was necessary to use a probe gas whose Knudsen number, under the conditions of the NMR analysis, was close to that of the molecules representative of the feed or effluent of those processes under process conditions. Sulphur hexafluoride ($SF_6$) was chosen as the diffusing molecule because it meets this criterion (see below). In addition, this gas is inert, easy to use, has very low polarity and therefore interacts little with the surface of the solid. Finally, the $^{19}$F nucleus is favourable for NMR analysis (spin ½, natural abundance of 100 % and good gyromagnetic ratio).

A sample of around 20–30 pellets (for a given batch) was placed inside an NMR tube and subjected to a thermal treatment at 250 °C for 30 min to remove physisorbed water. $SF_6$ was then introduced into the vacuumed NMR tube at a pressure of 2 bars and the tube was placed inside the magnet, where the temperature was set to 305.5 K. Some experiments, were also performed at 315.15 K and 325.15 K. The experimental conditions (pressure and temperature) were chosen to ensure the mass transport was in the regime with an approximate Knudsen number value of 1.72 at 305.5 K and 2 bars, which is typical of conditions during a large scale heterogeneous catalytic reaction.

$^{19}$F PFG-NMR self-diffusion measurements of $SF_6$ were performed on a 400 MHz (9.4 T) JEOL Spectrometer equipped with a 5 mm "Royal Probe" and a field gradient amplifier allowing the application of field gradient pulses with amplitudes up to 90 G/cm. The 13-interval stimulated echo-pulsed bipolar sequence (BPP STE DOSY) was used. The pulse duration of the magnetic field gradient, δ was 0.002 s, the diffusion times used were 0.05–0.20 s, the number of gradient points was 12, and the magnetic field gradient strength was 0.5–20 G/cm.

### 2.2.8. Hyperpolarised (hp) xenon MRI uptake experiments

The method used was similar to that employed by Pavlovskaya et al. (2015). For xenon uptake imaging experiments, samples were prepared with a moisture content of approximately 4–5 wt% water. The moisture content remained below the monolayer threshold (approximately 5 wt %), preventing interference with xenon mass transport while ensuring sufficient xenon relaxation time for imaging by shielding any paramagnetic impurities. After preparation, the pellets were placed in glass tubes, sealed by a valve attached to each tube, and allowed to equilibrate for a minimum of one week before experiments.

Initial experiments attempted to image a horizontal row of pellets, but the protocol evolved to stack pellets vertically for improved individual pellet imaging. This vertical arrangement necessitated a specialised foam preparation. An Open Cell Reticulated Filter Foam, 45 PPI,
A5 × 10 mm Thick, Black was saturated with crystal clear epoxy resin in a 1:1 resin-to-hardener ratio. After 2–3 min in the resin, excess resin was squeezed out of the foam. This preparation preserves macropore accessibility while blocking micropores on the surface of the foam that could interfere with xenon chemical shift measurements. A small hole, slightly larger than the pellet diameter, was drilled vertically down the epoxied, hardened foam. The foam was shaped to fit the dimensions of the NMR cell. Sample transfer to the NMR cell (placed vertically in the epoxied foam) occurred in a controlled glove bag environment under dry, inert conditions. The NMR cell once closed, was placed in the magnet.

Isotopically enriched ¹²⁹Xe (enriched to approximately 85 ± 5 % of ¹²⁹Xe) was used to prepare a 1 % Xe/10 % N₂/89 % He gas mixture. The high-pressure cylinder was regulated to 1700 mbar, and the gas mixture flowed continuously through the SEOP cell. A post-SEOP liquid-nitrogen cold trap was employed to freeze and collect the hp ¹²⁹Xe in large doses. The flow rate of the gas mixture through the cold trap was adjusted to 10 mL s⁻¹ to ensure sufficient collection within a 10–12 min window. During the collection process, the frozen xenon needs to be placed in a permanent magnetic field to minimise polarisation losses. Once sufficient hyperpolarised xenon gas was trapped/frozen, it was delivered in batch mode to the (vacuumed) NMR cell by warming up the cold trap inside a glass of lukewarm water, with a temperature estimated to be 20 °C. A ¹²⁹Xe NMR spectrum was obtained to verify moisture content from the pellet phase chemical shift value that was correlated to a moisture dependent chemical shift calibration curve.

The imaging sequence began with the delivery, and pressure equilibration of highly concentrated, enriched hp xenon gas. Upon gas delivery, a reference image of the pellet phase was acquired to normalise the loadings, as signal intensity can vary between different batches of delivered hp xenon. All images in this work were acquired with chemical shift selection at the pellet phase resonance frequency. The key modification in this sequence, compared to the one used by Pavlovskaya et al. (2015), was the use of a single, soft Gaussian-shaped pulse applied at the pellet phase resonance frequency. A single pulse proved more efficient for depleting the hyperpolarized signal in the fast diffusion within the porous materials under study. A second image of the pellet phase was then recorded following a transport time delay. During this transport time, hyperpolarised xenon from the gas phase exchanged with the “dead” gas in the pellets, generating a newly formed pellet phase chemical shift peak. By repeating this sequence for varying transport times, the xenon uptake could be monitored through transport-weighted images, which showed increasing signal intensity over time. The resulting data enabled the calculation of mass transport coefficients (MTCs) (k) for each image pixel using the Linear Driving Force (LDF) model.

In order to make a comparison of average MTCs, from MR images, between batches three different averaging processes were tested:

Method 1: In this method, signal intensity was first averaged within the region of interest (ROI) for each slice, which was the region of the image containing the pellets. The averaged intensity increase was then fitted against time using the adjusted LDF model. This approach should offer greater accuracy compared to individual pixel fitting due to its stronger signal strength. To account for intra-batch variability, the process was repeated across 3–4 runs for each batch.

Method 2: In this approach, the adjusted LDF model was fitted to individual pixels within the ROI, generating k values for each. The spatially resolved k values were then averaged, with equal weighting for each, across all repeats to account for intra-batch variability.

Method 3: This approach was similar to Method 2, but the average was weighted by the individual fitting error of the k values, instead.

# 3. Theory

## 3.1. Structure-transport model construction

### 3.1.1. Pellet structure

The basic elements of the structural model are depicted schematically in Fig. 1. The reasons for including these elements will become evident from the characterization data given in Section 4. The structural model for bimodal pellets consists of isolated macropores distributed pervasively across a continuous mesoporous matrix core region, but not within a solely mesoporous ‘rind’ (or crust) region located around the pellet periphery. Furthermore, each macropore is surrounded by a barrier of narrower mesopores in the matrix immediately adjacent to it. The microscopy evidence for these pellet basic structural features will be presented in the Results section.

A pore size distribution (PSD) can be obtained from a nitrogen adsorption isotherm using a standard algorithm, such as that of Barrett-Joyner-Halenda (BJH) (Barrett et al., 1951). The PSD thereby obtained is for pore sizes weighted by pore volume. However, to conduct percolation analysis, it is necessary to convert to a distribution weighted by pore number by assuming a relationship between pore diameter and pore length. In previous work it was assumed that pore length $L_p$ was generally related to pore diameter d by (Rigby et al., 2004):

$$L_p = k_i d^\alpha \quad (1)$$

where α is a power law exponent and $k_i$ is a constant.

Below the length-scale of the pellet, at the level of the pores themselves, the network of pores can be represented by a Bethe lattice (also known as a Cayley tree) (Mason, 1982; Parlar and Yortsos, 1988). This structure consists of a network where Z pore bonds are joined at a central node. At the other end of each of these pore bonds is a further junction, forming node i, where the pore joins Z-1 other pore bonds. Each of these pore bonds also has a further similar junction at its other end, and so on over ever more additional generations of nodes out to the periphery of the porous material, thereby creating a branching tree-like structure (hence the alternate name). The value of the model pore connectivity parameter Z can be found from a percolation analysis of the gas sorption data for a porous material. The method for obtaining the value of Z for a Bethe lattice model of a real porous material is described in detail elsewhere by Parlar and Yortsos (1988). Parlar and Yortsos (1988) have provided the equations for the accessibility function for seeded bond or site percolation in Bethe lattices with connectivity Z. This enables percolation analysis to be performed for systems where some vapour pockets, with a pore number fraction of $p_i$, have been left open at the top of the nitrogen adsorption isotherm.

### 3.1.2. Mass transport

The gas diffusivity is obtained from the Kinetic Theory of Gases, such that the diffusivity, $D_i$, is given by:

$$D_i = \frac{l}{3} v, \quad (2)$$

where v is the average molecular speed at the relevant temperature, and l is the characteristic length-scale, which is the molecular mean free path λ in the molecular regime, and some critical pore size d in the Knudsen diffusion regime. The Knudsen regime arises when the mean free path is larger than the pore sizes.

The standard Knudsen formula, as described above, was derived for pores of infinite length. However, a common occurrence for real porous materials is that the key aspect ratio of pores (ie the ratio of pore diameter to pore length) is unity. In this case, the actual Knudsen regime diffusivity, $D_K$, is given by (Bukowski et al., 2021):

$$D_K = \frac{3}{7} D_{K,\infty} \quad (3)$$
where $D_{K,\infty}$ is the Knudsen diffusivity from the 'classical' formula (Eq. (2) above with $L_p=\infty$).

For diffusion through a composite porous medium consisting of a series of different phases, each with different diffusivities, arrayed in layers along the direction of the diffusive flux, the resistance to the diffusive flux also depends upon the thickness, $h_i$, of the particular phase i layer in the direction of the net diffusive flux. Hence, the overall diffusive resistance $R_{tot}$ (=1/conductance, $C_{tot}$) of a composite medium is given by (Nonoyama et al., 2011):

$$R_{tot} = \sum R_i = \frac{h_{tot}}{D_{net}} = \sum \frac{h_i}{D_i} \qquad (4)$$

where:

$$h_{tot} = \sum h_i \qquad (5)$$

Where a composite porous medium is, instead, comprised of one phase denoted a, with diffusivity $D_a$, consisting of isolated (disconnected) (spherical) regions dispersed amongst a continuous matrix of another phase denoted b, with diffusivity $D_b$, then the overall diffusivity for the composite medium, $D_w$ is given by the formula due to Maxwell such that (Crank, 1975):

$$\frac{D_w - D_b}{D_w + 2D_b} = v_a \frac{D_a - D_b}{D_a + 2D_b} \qquad (6)$$

where $v_a$ is the volume fraction of phase a.

Where mass transport processes are studied by the gas uptake method (Rigby, 2020), the data-set obtained from observations of mass uptake (M) over time t can be modelled using the Linear Driving Force (LDF) model (Rigby, 2020):

$$M = M_{\infty}(1 - \exp(-kt)) \qquad (7)$$

where $M_{\infty}$ is the equilibrium mass uptake at long times, and k is the mass transfer coefficient (MTC). For diffusive mass uptake into a spherical pellet, the MTC is given by (Rigby, 2020):

$$k = \frac{15D_e}{a^2} \qquad (8)$$

where $D_e$ is the effective diffusivity of the porous medium, and a is the pellet radius.

For a porous medium, the effective diffusivity is generally given by:

$$D_e = \frac{\varepsilon D_i}{\tau} \qquad (9)$$

where $D_i$ is the appropriate reference diffusivity, $\varepsilon$ is the porosity (=voidage fraction), and $\tau$ is the tortuosity of the porous medium. However, where the observed diffusivity for a porous medium is obtained using pulsed-field gradient (PFG) (aka PGSE) NMR, then the NMR diffusivity is given by (Hollewand and Gladden, 1995a):

$$D_{PFG} = \frac{D_i}{\tau} \qquad (10)$$

This difference is because in the NMR method a spin density term cancels with the porosity term (Hollewand and Gladden, 1995a).

For a porous medium that is represented by a Bethe lattice, consisting of a perfect (ie each potential bond is occupied) branching ($Z > 2$) network of conductances $C_1$, then the ratio of the average conductivity C of the medium to the conductance of a single bond is approximately (at low Z) given by (Stinchcombe, 1974):

$$\frac{C}{C_1} = \frac{Z(Z-2)}{Z-1} \qquad (11)$$

### Small angle X-ray scattering

The intensity of X-ray radiation scattered by a fractal surface is

known to be proportional to a negative power of the wave-vector q, such that (Rieker et al., 1999):

$$I(q) \propto q^{-\eta} \qquad (12)$$

where $q = 4\pi\lambda^{-1}\sin(\varphi/2)$, $\lambda$ (here) is the wavelength of the radiation, and $\varphi$ is the scattering angle. From the value of $\eta$, the fractal nature of the system under investigation can be determined (Avnir, 1989). If the exponent is in the range $1 < \eta < 3$, then it describes the mass fractal of dimension ($D_m$)$_{SAXS} = \eta$, but if in the range $3 < \eta < 4$ then it describes surface fractals of dimension ($D_S$)$_{SAXS} = 6 - \eta$. For $\eta = 4$, Equation (12) leads to Porod's law, where ($D_S$)$_{SAXS} = 2$ and the surface is flat.

## 4. Results

### 4.1. Electron microscopy

#### 4.1.1. Scanning electron microscopy (SEM)

Fig. 2 shows typical SEM images of the core region of batches A-D. While batch A did not use porogen in its synthesis, it still contains a very few isolated macropores (black circular regions) and inclusions of less dense alumina phase (darker grey regions). Consistent with the use of increasing volumes of porogen in their synthesis, batches B-D show increasing levels of macroporosity. The SEM images of batches B-D also show a small number of denser inclusion regions. The particular macroporosities for batches B-D were obtained from image analysis of several images like those in Fig. 2, and the relevant values are shown in Table 3. Image analysis was also conducted to obtain the average pore size of the macropores and these are given in Table 3.

Further SEM studies of the periphery of pellets from batches B-D, such as those shown in Fig. 3, have shown that in this region the pellets possess a 'rind'-like structure wherein there are no observable macropores, probably due to the lack of porogen. This rind was common but not evident on every pellet. The thickness of this rind is variable but typically ~30-50 µm. This rind was probably formed during the drying step in fabricating the pellets, as water was being removed from the boundary region of the pellet. Since the porogen was linked to water via surfactant molecules, then it can also be extracted by capillary wicking even when the temperature of the drying step is actually lower than the boiling point of the porogen itself (Handscombe and Kraft, 2010).

#### 4.1.2. Focused ion beam scanning electron microscopy (FIB-SEM)

Fig. 4(a) shows an image of a single FIB-SEM slice of a pellet from Batch C. In this case, the macropores have not been filled with epoxy resin. A platinum layer (white layer on top) was added to reduce charging effects. The image shows isolated macropores within the mesoporous matrix, as well as a sector of a denser grain in the top right region of the image. It is noted that the cross-sections of macropores near the denser grain have an elongated appearance, with the long axis of the pore parallel to the nearest boundary of the dense grain, such that the long axes of successive pores rotate progressively around the grain periphery. To make a closer study of the shape and degree of isolation of the macropores away from dense grains, a series of FIB-SEM images were obtained, and an example sequence (of every other slice) is shown in Fig. 4(b). The experiment was performed near the aluminium foil/ adhesive and large trenches were cut out on the sides of the region of interest to facilitate the dissipation of the negative charge. In the series of images particular pores can be seen to be roughly circular, growing through the sequence and then diminishing relatively uniformly. Given the more regular appearance of macropores away from the grain, it is proposed that the more elongated, 'discus-like' shape of macropores nearer the dense grains is generated when the aqueous matrix is compressed against the more resilient denser grains during the drying step. However, as seen from the standard SEM images in Fig. 2, the denser grains are generally sparse.
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** SEM images of batches (a) A, (b), B, (c), C and (d) D. The magnification is 1000 times, the scale bar is 10 µm, and the accelerating voltage used is 5.00 kV.</figcaption></figure>

<table><caption>**Table 3**
Results of the analysis of SEM images of batches A-D.</caption><thead><tr><th>Batch</th><th>Macro-Porosity from SEM</th><th>Average Macropore Size /µm</th></tr></thead><tbody><tr><td>A</td><td>0.0</td><td>N/A</td></tr><tr><td>B</td><td>0.09 ± 0.01</td><td>0.28</td></tr><tr><td>C</td><td>0.18 ± 0.02</td><td>0.30</td></tr><tr><td>D</td><td>0.20 ± 0.01</td><td>0.28</td></tr></tbody></table>

### 4.1.3. Transmission electron microscopy (TEM)

Transmission electron microscopy (TEM) experiments were performed on batch D using the two different sample preparation methods explained in the Materials and Methods section above. The first method, denoted the ‘direct’ preparation method has been used to measure the average dimensions and shape of the γ-alumina platelets (or crystallites), from ‘shard’-like samples of the material, obtained by grinding in a mortar. The TEM images, shown in Fig. 5, permitted the approximate measurement of the length and thickness of the platelets. The lengths obtained varied from approximately 7 to 20 nm, with a thickness ranging from approximately 2.5 to 4 nm. The crystal planes of these platelets can also be observed. The mesopore network is formed of the gaps between the packing of individual platelets and agglomerates of these.

Using the second sample preparation method, denoted the ‘cross-sectional’ preparation method, 70 nm thick cross-sectional slices of the material were studied to compare the particular spatial arrangement of

the alumina platelets localised around macropores with that deeper within the mesoporous matrix. The TEM images for batch D shown in Fig. 6 illustrate the arrangement of the alumina platelets around the macropores and the TEM image in Fig. 7 shows the arrangement of the platelets deep inside the matrix. These data suggest that the platelets tend to be characterized by a nematic-like arrangement such that there is a local 1-dimensional, regular array of alumina platelets along the boundary of the macropores. In the TEM images in Fig. 6, long dark lines corresponding to long pores can be seen formed between the aligned platelets. In contrast, no discernible regular pattern is seen in the platelets in the matrix. Similar findings were also obtained for batches B and C.

The more regular arrangement of platelets around the macropores could be explained by the presence of surfactant in the synthesis. Surfactant is used to ensure good dispersion and sphericity of both the platelets and macropores in the final product. It is known that both the amount and type of surfactant used can have an impact on the porous structure of the material due to charge-charge interactions between the surfactant and the material surface (Jung et al., 2023; Korneeva et al., 2012). Mo et al. (2024) studied the effect of varying the amount of surfactant on a macro-mesoporous spherical alumina porous structure, similar to the materials under study here. Notably, in their study, significant variations in the macroporous structure were observed when altering the levels of surfactant. By increasing the level of surfactant, both the pellet size and the internal pore sizes were reduced, due to the increased surface tension of the aqueous phase. Since the surfactant has an impact on the macropore-matrix interface, it is therefore possible that
<figure><img src="image_5.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** SEM images of batches (a) C and (b) D, where each show two pellets with a 'rind'-like structure around the periphery which is denser with no observable macroporosity. The magnification is 250 times, the scale bar is 100 µm and the accelerating voltage used is 5.00 kV.</figcaption></figure>

the platelets are repulsed (or attracted) by the surfactant present at the macropore/matrix interface. The surfactant may also preferentially adsorb on a particular facet of the alumina crystallite. These interactions likely have the effect of orienting the platelets in an arrangement which occupies the smallest volume: the first few layers of platelets will be stacked on top of each other in the direction of the macropore/matrix interface.

## 4.2. Small angle X-ray scattering (SAXS)

Fig. 8 shows the SAXS data for Batches A-D in the form of pellets. For all batches, the SAXS data have three main regions, one with steeply rising *I* for *q*⁻¹ below ~ 2–3 nm, one with steeply rising *I* for *q*⁻¹ above ~ 15 nm, and one relatively flatter in-between. It is noted that the upper intensity region of the SAXS curve for the pellet form of Batch A rises much less steeply over the intermediate range of *q* such that it lies below the corresponding curve for the other batches at low *q*-values. This difference results from the general much lower numbers of macropores in pellets of Batch A. A modified version of the Debye model has been developed to interpret the scattered intensities through the use of multiscale Boolean models of spheroids (Duchène et al., 2023). Numerical

computation of the scattered intensities of multiscale Boolean models of spheroids allows one to take into account anisotropic primary grains such as alumina platelets. This model was fitted to the data in Fig. 8 to obtain the characteristic dimensions of the alumina platelets/crystallites, assuming an ellipsoidal shape. These data are given in Table 4. It can be seen that the thickness of the platelets is typically ~ 6 nm for all batches.

## 4.3. Mercury-Helium pycnometry and nitrogen sorption

The pellet bulk volume and skeletal density were obtained by mercury-helium pycnometry and the results are shown in Table 5.

Nitrogen sorption isotherms were obtained for all batches. Examples of raw nitrogen sorption isotherms for batches A-D are shown in Fig. 9 (a). It can be seen that the hysteresis loops close at the top of all the isotherms suggesting a gap in pore sizes between those in the mesoporous matrix and the macroporosity seen in the SEM images. Hence, specific volumes for the mesoporosity could, thus, be defined. From Fig. 9(a), it can be seen that this parameter increases in value with increased porogen content. Together with the mercury-helium pycnometry data, the specific mesopore volume can be used to determine the porosity of the mesoporous matrix and these values are given in Table 5 for batches A-D. The differential PSDs obtained from these isotherms are shown in Fig. 9(b). The median pore size was obtained from the volume-weighted BJH PSD and the values for batches A-D are given in Table 5. The volume weighted coefficients of variation were also determined from the BJH PSDs for batches A-D and these statistics are also given in Table 5.

In Fig. 9(a) there is a notable contrast in the shapes of the desorption knees between that of batch A and those of batches B-D. The knee for batch A is sharper, and more like a right-angle, while those for batches B-D are more rounded. This is because, while, for batch A, desorption of condensate filling the mesopores at the top of the adsorption isotherm is limited to occur only from the exterior surface of the pellets, in contrast, batches B-D possess many macropores that remain empty at the top of a conventional adsorption isotherm and so can provide vapour seeds to promote more rapid (ie at higher pressure) evaporation of condensate, as the pressure drops below the critical value. Hence, the percolation knees are more rounded for batches B-D. Percolation analysis, using the accessibility functions for seeded percolation provided by Parlar and Yortsos (1988), was performed on the nitrogen sorption data for all 8 samples listed in Table 2 in the Materials Section. The free fitting parameters in the percolation modelling were the pore connectivity, pore length exponent, and the seed pore fraction. An example of the fit of the experimental sorption data to the theoretical accessibility function is shown in Appendix A. It was found that good fits could be obtained for all batches and the resultant pore bond connectivities are shown in Table 6. The pore connectivities obtained for all batches were very similar, with a mean of 2.18 ± 0.04 (1 std. err.). This suggests that the topology of the mesoporous matrix pore network is very similar for all batches. The pore connectivity is an overall average pore co-ordination number for the whole (large) lattice. Hence, since this value is above 2 for the samples in this work, then some of the individual lattice nodes will have a co-ordination number above 2. As the mesopore network is modelled as a Bethe lattice, then the relationship between the pore network connectivity and the overall conductance of the network (compared to that of a single pore bond, and, hence, the tortuosity) can be obtained from the relationship derived by Stinchcombe (1974) given as Eq. (11) in the Theory Section.

Despite the relatively close numerical similarity between pore connectivities across the eight batches, there was some correlation between pore connectivity and median pore size. The coefficient of determination for a linear regression of median pore size and connectivity was statistically-significant at 0.5815 (F-statistic = 8.34, significance *p* = 0.028).
<figure><img src="image_7.png" /><figcaption>Fig. 4. (a) FIB SEM slice of batch C. The macropores have not been filled with epoxy resin. The magnification is 10020×, the scale bar is 1 µm and the accelerating voltage used is 2.00 kV. (b) Series of alternate FIB-SEM images of a trench in a pellet from Batch C. The magnification is 10180 × and the accelerating voltage used was 5.00 kV.</figcaption></figure>

4.4. *Mercury porosimetry*

Mercury porosimetry data were obtained for fragmented samples of
pellets from Batches A-D, analysed using the modified (for alumina)
Kloubek correlations (Rigby, 2000), and an example for Batch A is given
in Fig. 10, with data for Batches B-D given in the Appendix, Fig. A2. The
data for fragmented samples exhibited two steps, one at large sizes
reflecting intrusion into the gaps between the powder particles and one
at smaller sizes reflecting intrusion into the pores within the powder
particles. The first step suggests that the powder particles were in the
range ~ 10–100 µm since the gap sizes of random particle packings are
typically of the same order as the particle sizes. The intra-pellet pore
volume from mercury porosimetry is given in Table 5. For Batches A-D a
close (within the experimental error of the calibration) superposition is
obtained for the intrusion and extrusion curves at lower pore sizes. The
slightly sharper knee at the top of the extrusion curve, compared to that
for intrusion, is probably due to a slight delay in retraction arising from
the need to initiate “snap-off”, as seen experimentally previously for
sol–gel silica spheres, and observed in simulations of mercury retraction
in such materials (Rigby et al., 2011). The significant deviation of the
extrusion curve from the intrusion curve, on the start of entrapment, is
abrupt and the retraction curve then runs near-horizontal and parallel to
the intermediate plateau of the intrusion curve. This suggests that
mercury entrapment is occurring primarily in the larger mesopores and
macropores only. It was found that fragmentation did not result in a
significant decrease in entrapment levels compared to those for whole
pellet samples (Collins, 2025). This suggested that macroscopic struc-
tural heterogeneities (over length-scales of the order of the powder
particle size and larger) are not responsible for the entrapment. This
 contrasts with what has been observed previously for some sol–gel sil-
icas where the entrapment disappears entirely, and the intrusion and
extrusion curves overlay each other along their whole lengths, when the
pellets were fragmented to similar powder particle sizes as here (Rigby

and Edler, 2002).

4.5. *Pulsed-Field gradient (PFG) NMR studies of the self-diffusion of SF<sub>6</sub>*

PFG NMR studies of the self-diffusion of SF₆ in 9 batches of alumina were conducted at 2 bar and the three different temperatures of 303.5, 315.15 and 325.15 K. An example of a PFG log-attenuation plot is given in Appendix Fig. A3. For all samples, at all temperatures, the form of the log-attenuation plots was linear, and thus straight-line fits were performed to obtain the diffusivity. To determine the nature of the limiting observed diffusion process the diffusivities for each batch were plotted against the square-root of the temperature at which they were obtained, and these data are shown in Fig. 11 for batches A-D. Also shown in Fig. 11 are fits of straight lines obtained from linear regression of the experimental data. By eye, it can be seen that good fits to straight lines were obtained and the coefficients of determination (r²) for all the fits shown in Fig. 11 were greater than 0.9995. This suggests that the limiting diffusion process is Knudsen diffusion as the diffusivity for this process has a square-root dependency on temperature (Al-Bayaty et al., 1994).

The mean free paths for the SF₆ molecule at the experimental tem-
peratures and pressure were 20.9–22.2 nm, which is larger than the
median pore size and most mesopores in the materials. The root mean
square displacements for the SF₆ molecule, during the PFG NMR
experiment, ranged from 60 to 104 µm, depending on the particular
batch and temperature. From the SEM images in Fig. 2, it is clear that the
volume explored by a typical molecule will be a region of the void space
including many macropores. The PFG diffusivities obtained for eight
batches of alumina were plotted against the median mesopore size from
nitrogen adsorption for each of the three temperatures employed, and
the results are shown in Fig. 12.

These data were then fitted to straight lines using linear regression
and the fitted parameters and statistics for the fits are shown in Table 7.
<figure><img src="image_8.png" /><figcaption>Fig. 4. (continued).</figcaption></figure>

At all three temperatures, good straight-line fits were obtained, with coefficients of determination all exceeding 0.93, and that pass the *F*-test (all *p* < 0.001). Table 7 shows that the error bars for the respective

gradient and intercept overlap for the data for 315 and 325 K, and, in Fig. 12, it can be seen that the lines are beginning to converge.

It is noted that the straight lines in Fig. 12 clearly do not pass through
<figure><img src="image_9.png" /><figcaption>Fig. 5. (a) A typical example of a TEM image of a ‘shard’ of batch D, together with (b) an enlargement of the region indicated by a red rectangle in (a). The scale bar for (a) is 50 nm.</figcaption></figure>

the origin, as might be expected from Eq. (2) if the median mesopore size was the critical pore size for Knudsen diffusion. The non-zero intercept suggests that there is a constant off-set $d_o$ between the measured median pore size $d_m$ and the critical pore size for diffusion, while the constant slope suggests that the tortuosity is the same for all batches, since then the data would be consistent with the functional form:

$$D_{PFG} = \frac{vd_m}{3\tau} - \frac{vd_o}{3\tau} \qquad (13)$$

as has been observed in Fig. 12. The values of the pore size off-set and tortuosity were obtained from the gradients and intercepts of the straight-line fits to the data in Fig. 12 using Eq. (13), and these parameters are also shown in Table 7. It can be seen that, for the three temperatures, the pore size off-sets are the same, within experimental error. At the same time, the tortuosity declines slightly with temperature but

<figure><img src="image_10.png" /><figcaption>Fig. 6. (a) Cross-sectional TEM image of a region around macropores within a pellet from batch D showing the nematic arrangement of platelets around macropores. (b) Close-up image of the region contained within the red rectangle in (a). The red arrow indicates region of aligned platelets.</figcaption></figure>

the form of the data suggests a limiting value of around ~ 4.26, as the temperature rises, since the error bars for the values at 315 and 325 K overlap. This limiting behaviour probably occurs because, as the temperature rises, so does the mean free path of molecules, such that eventually all mesopores, irrespective of diameter, become firmly in the Knudsen regime, with all transitional behaviour in mesopores eradiated. It is noted that the value of tortuosity obtained is similar to the typical value of tortuosity of ~ 4 for catalyst pellets, as reported by Satterfield (1996).

The values of the critical pore size obtained from the fits to the data in Fig. 12 and Eq. (13) are given in Table 5. It can be noted that these critical pore sizes are similar to the size of the alumina platelets observed in the TEM images. Hence, the PFG NMR data suggests that the key,
<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Cross-sectional TEM image of the internal mesoporous matrix region of a pellet of batch D showing the random arrangement of platelets in the matrix. The scale bar is 100 nm.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 8.** SAXS data for Batches A-D in the form of pellets.</figcaption></figure>

<table><caption>**Table 4**
The results of the model fitting of the SAXS data shown in Fig. 8.</caption><thead><tr><th>Batch</th><th>Thickness/<br>nm</th><th>Height/<br>nm</th><th>Length/<br>nm</th><th>Radius of<br>gyration/nm</th><th>Aspect<br>ratio</th></tr></thead><tbody><tr><td>A</td><td>6.1</td><td>6.1</td><td>9.2</td><td>2.8</td><td>1.5</td></tr><tr><td>B</td><td>5.5</td><td>5.5</td><td>16.6</td><td>4.1</td><td>3</td></tr><tr><td>C</td><td>5.5</td><td>5.5</td><td>19.3</td><td>4.7</td><td>3.5</td></tr><tr><td>D</td><td>5.8</td><td>5.8</td><td>46.5</td><td>10.6</td><td>8</td></tr></tbody></table>

observed rate-controlling factor for mass transport over length-scales of less than ~ 100 µm probed by this technique is the critical mesopore size. This is because the mass transfer process, probed via this technique, occurs through the macropores and mesoporous walls in series. However, the effective controlling (denoted ‘critical’) pore size is not the median (as plotted in Fig. 12) but the median minus the pore size offset

(of ~ 7 nm). Once this offset is subtracted from the measured median pore size, to obtain the critical pore size, then the variation in the observed diffusivity is proportional to the corresponding variation in critical pore size, as would be expected from the Knudsen model.

The SF₆ self-diffusion coefficients for the three different temperatures were correlated separately against the pore bond connectivity from nitrogen adsorption for the batches listed in Table 6. The regression analysis for each yielded a significant coefficient of determination. However, when a multiple linear regression (MLR) was performed for the median pore diameter and pore bond connectivity against the self-diffusivity for a given temperature, while the coefficient for the median diameter remained statistically significant (p < 0.05), that for the pore bond connectivity did not (p>>0.05). This suggests that the SF₆ self-diffusion coefficients depend strongly on the median pore diameter, but the individual correlation with pore bond connectivity arises because, as mentioned above, this parameter is itself correlated with median pore diameter too.

### 4.6. Hyperpolarised (hp) xenon MRI

Xenon-129 spectra were obtained for samples of batches A-D following equilibration with hyperpolarized xenon gas, and these are shown in the Supplementary Material. Gas uptake experiments were also conducted. Fig. 13 shows an example of the temporal series of normalized MR images of the region of interest (ROI) for hp xenon uptake into stacks of pellets from batch B. The individual voxels within the ROI were each fitted to the LDF model (Eq. (7)), to obtain spatially-resolved maps of the MTC. Examples of these maps for batches A-D are shown in Fig. 14.

From Fig. 14, it can be seen that individual pellets can be resolved in the stack, and that each pellet is composed of approximately 3–4 by 4–5 pixels (of size 0.37 mm × 0.52 mm), giving some information on the spatial variation of the MTC across an individual pellet. The average error in the fits to the LDF model for individual pixels was estimated for each batch. The raw variance in the fitted LDF values across the pixels of an image was assumed to be the sum of variances resulting from independent distributions in structural heterogeneity and random noise. To obtain the coefficient of variation in the LDF MTC due to structural heterogeneity, the variance for the fitting error was subtracted from the raw variance in fitted LDF MTCs to give the remainder variance. Table 8 shows a comparison of the resultant coefficients of variation (CoV) due to structural heterogeneity in the fitted MTCs in the ROI of the images, obtained by image analysis Method 2 (described in Method section), which can be compared with the coefficient of variation of the volume-weighted PSD obtained from nitrogen adsorption, for batches A-D, given in Table 5. It can be seen that the CoVs for the nitrogen PSDs are ~ 1–3 times the corresponding CoVs obtained from analysis of the xenon MRI-derived maps via Method 2. The similarity of the order of magnitude of the degree of heterogeneity in the nitrogen pore sizes and the spread of MTCs in the MRI maps from Method 2, even despite the relatively large voxel volumes, suggests that heterogeneities in the spatial distribution of local average pore sizes are likely to be responsible for the macroscopic heterogeneities in the spatial distribution of MTC. The different pixel weighting used in Method 3 removes the variability. To compare the relative mass transport performance of batches with increasing macroporosity, Table 8 also presents the ratio of the LDF MTCs for each batch to that of batch A. It can be seen that increasing macroporosity leads to an increase in MTC relative to the solely mesoporous pellet.

From Table 8, it is noted that the coefficients of variation in the image MTC k-values are similar for the four batches A-D. This suggests a similar degree of spatial heterogeneity in all batches. Hence, adding the macroporosity to the pellets has not led to additional macroscopic heterogeneities.
<table><caption>**Table 5**

Pellet characteristic parameters from mercury-helium pycnometry, nitrogen sorption, and PFG NMR, for batches A-D.</caption><thead><tr><th>Batch</th><th>Bulk volume/ (ml g<sup>-1</sup>)</th><th>Skeletal density/ (ml g<sup>-1</sup>)</th><th>Meso-porosity</th><th>Median pore diameter of BJH PSD /nm</th><th>Coefficient of variation of BJH PSD</th><th>Critical pore size /nm</th><th>Hg porosimetry pore volume/ (ml g<sup>-1</sup>)</th><th>Pellet radius /mm</th></tr></thead><tbody><tr><td>A</td><td>0.790</td><td>3.31</td><td>0.600</td><td>10.0</td><td>0.50</td><td>3.0</td><td>0.43</td><td>0.845</td></tr><tr><td>B</td><td>0.915</td><td>3.30</td><td>0.644</td><td>10.5</td><td>1.07</td><td>3.5</td><td>0.55</td><td>0.910</td></tr><tr><td>C</td><td>1.11</td><td>3.27</td><td>0.651</td><td>11.5</td><td>1.07</td><td>4.5</td><td>0.69</td><td>0.955</td></tr><tr><td>D</td><td>1.22</td><td>3.27</td><td>0.669</td><td>13.0</td><td>1.33</td><td>6.0</td><td>0.81</td><td>0.955</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** (a) Conventional nitrogen sorption isotherms obtained at 77 K for batches A(■), B(▲), C(●) and D(◆), and the (b) differential PSD obtained using the BJH algorithm from the isotherms in (a).</figcaption></figure>

### 4.7. Modelling of diffusion

Given that the MLR analysis showed that there was not a statistically-significant contribution from pore bond connectivity above and beyond the correlation of median pore size with the PFG diffusivity, it was assumed that the connectivity of the mesoporous matrix was not significantly different across all batches. If the average pore connectivity across the eight batches, obtained from the seeded percolation analysis developed by Parlar and Yortsos (1988), is used to predict the tortuosity

<table><caption>**Table 6**

Pore bond connectivities obtained from seeded percolation analysis of conventional nitrogen sorption isotherms for 8 batches of alumina pellets.</caption><thead><tr><th>Batch</th><th>Median pore diameter/nm</th><th>Pore connectivity</th></tr></thead><tbody><tr><td>A</td><td>10.0</td><td>2.03</td></tr><tr><td>B</td><td>10.5</td><td>2.26</td></tr><tr><td>C</td><td>11.5</td><td>2.29</td></tr><tr><td>D</td><td>13.0</td><td>2.24</td></tr><tr><td>AA</td><td>9.1</td><td>2.07</td></tr><tr><td>AB</td><td>8.9</td><td>2.04</td></tr><tr><td>BC</td><td>11.2</td><td>2.27</td></tr><tr><td>B#</td><td>10.3</td><td>2.21</td></tr></tbody></table>

<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Mercury porosimetry intrusion (solid line) and extrusion (●) data for powder form of a sample from Batch A. The raw data have been analysed using the modified Kloubek correlations (Rigby 2000, 2020).</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** A plot of the SF<sub>6</sub> self-diffusion coefficient (m<sup>2</sup> s<sup>-1</sup>) against the square root of temperature (K<sup>1/2</sup>) for batches A-D. The data all shows good linearity with the coefficients of determination r<sup>2</sup> ≥ 0.9995 for all batches.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Fig. 12.** A plot of the SF<sub>6</sub> self-diffusion coefficient (m<sup>2</sup> s<sup>-1</sup>) against the median mesopore diameter from nitrogen adsorption for batches A-D. The straight lines shown are the results of linear regressions of the corresponding data-sets for the different temperatures.</figcaption></figure>

with Eq. (11) then the value obtained is 3.1<sup>+2.1</sup><sub>-0.9</sub>, with the error coming from the standard error in the connectivity. It is noted that this value of tortuosity is similar to that of ~ 3.5 obtained via the random walk method (which mimics self-diffusion) with displacements equivalent to the modal pore size for 3D cluster-cluster aggregate (CCA) models with voidage fractions in the same range as the mesoporous network of the alumina pellets (Rigby and Gladden, 1996). The value of tortuosity from

seeded percolation analysis is similar, within experimental error, to the limiting value for (standard) Knudsen diffusion obtained from the SF<sub>6</sub> PFG NMR of 4.3 ± 0.4. Therefore, the latter experimental measurement will be taken as the tortuosity of the alumina matrix. It is considered that the corona and mesoporous matrix may differ in the conductance of individual network elements but not the ratio given in Eq. (11), which determines the tortuosity.

The individual macropores accompanied by their mesoporous barrier corona, as shown in Fig. 1, were ultimately treated as a single unit. It was assumed that, for a typical macropore, the direction of the vector for the overall diffusional flux for gas uptake towards the centre of the pellet would cross a given macropore via a random chord, such that the typical straight-line (net) diffusional path-length across a macropore would be given by the average chord length of 4/3 of the radius (Sjöstrand, 2002). Hence, the overall diffusional resistance for a typical macropore and its accompanying corona (on both sides) will be given by Eq (4). The TEM data, such as that in Fig. 6, suggests that the barrier layer of aligned alumina platelets is ~ 2 layers deep, and the length of the platelets can be up to ~ 10 times their thickness. Therefore, the thickness of the corona was taken as 10 nm. The standard Knudsen diffusivity formula was used to predict mass transport in the corona as the TEM data shows that the platelets are much (~10 × ) longer than the gaps between them when aligned. The critical pore sizes for the corona were those listed in Table 5. The thickness of a macropore will be taken as an average chord length. The diffusivity in a macropore will be the relevant molecular diffusivity with a mean free path of 56 nm, corresponding to that of xenon at the temperature of the MRI experiment (293 K). The apparent

<caption>**Table 7**</caption>

Fitted parameters and statistics for the fit of straight lines to the PFG diffusivity against median mesopore size for eight batches of alumina at three different temperatures. Quoted errors are 1 standard error unless otherwise stated.

<table><thead><tr><th>Temperature /K</th><th>Gradient /10<sup>8</sup> m<sup>2</sup>s<sup>-1</sup> nm<sup>-1</sup></th><th>Intercept /10<sup>8</sup> m<sup>2</sup>s<sup>-1</sup></th><th>Tortuosity (&plusmn;95 % CI)</th><th>Pore size offset /nm</th><th>F-statistic</th><th>Significance, p</th><th>Adj. r<sup>2</sup></th></tr></thead><tbody><tr><td>303.50</td><td>1.15 &plusmn; 0.11</td><td>-7.7 &plusmn; 1.2</td><td>5.97 &plusmn; 0.58</td><td>6.6 &plusmn; 1.2</td><td>105.8</td><td>1.8 &times; 10<sup>-5</sup></td><td>0.930</td></tr><tr><td>315.15</td><td>1.49 &plusmn; 0.15</td><td>-10.5 &plusmn; 1.5</td><td>4.9 &plusmn; 0.9</td><td>7.0 &plusmn; 0.7</td><td>101.7</td><td>5.5 &times; 10<sup>-5</sup></td><td>0.935</td></tr><tr><td>325.15</td><td>1.70 &plusmn; 0.18</td><td>-12.0 &plusmn; 1.9</td><td>4.26 &plusmn; 0.44</td><td>7.0 &plusmn; 1.3</td><td>91.97</td><td>2.1 &times; 10<sup>-4</sup></td><td>0.938</td></tr></tbody></table>

<figure><img src="image_17.png" /><figcaption>**Fig. 13.** An example of the temporal series of normalized MR images of the region of interest (ROI) for hp xenon uptake into stacks of pellets from batch B. Only data within the region of interest (ROI) is shown. The ROI is created from the reference images with the highest signal intensity, using the Image Segmenter tool in MATLAB. The scale is normalized signal intensity/.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Examples of spatially-resolved maps of the LDF MTC for batches (a) A, (b) B, (c) C and (d) D, obtained from hp xenon MRI experiments. The scale is LDF MTC (s⁻¹). The gridlines have a pitch equal to the size of the pellets to aid the identification of individual pellets in the images.</figcaption></figure>

<caption>**Table 8**</caption>

Average LDF MTCs (obtained via image analysis Methods 1, 2 and 3) and adjusted coefficients of variation (CoVs) (obtained from Method 2) for the MRI maps of xenon uptake MTCs, for batches A-D. The quoted errors are 95 % confidence intervals.

<table><thead><tr><th>Batch</th><th>LDF MTC<br>Method 1<br>Average</th><th>CoV</th><th>k<sub>i</sub>/k<sub>A</sub></th><th>Method 2<br>Average</th><th>CoV</th><th>k<sub>i</sub>/k<sub>A</sub></th><th>Method 3<br>Average</th><th>CoV</th><th>k<sub>i</sub>/k<sub>A</sub></th></tr></thead><tbody><tr><td>A</td><td>0.93 ± 0.03</td><td>N/A</td><td>1.0</td><td>1.02 ± 0.04</td><td>0.24</td><td>1.0</td><td>0.95 ± 0.03</td><td>N/A</td><td>1.0</td></tr><tr><td>B</td><td>1.11 ± 0.01</td><td>N/A</td><td>1.19 ± 0.04</td><td>1.19 ± 0.04</td><td>0.28</td><td>1.17 ± 0.06</td><td>1.13 ± 0.03</td><td>N/A</td><td>1.19 ± 0.04</td></tr><tr><td>C</td><td>1.28 ± 0.06</td><td>N/A</td><td>1.38 ± 0.08</td><td>1.36 ± 0.05</td><td>0.24</td><td>1.33 ± 0.07</td><td>1.33 ± 0.03</td><td>N/A</td><td>1.40 ± 0.03</td></tr><tr><td>D</td><td>1.42 ± 0.06</td><td>N/A</td><td>1.53 ± 0.08</td><td>1.55 ± 0.05</td><td>0.23</td><td>1.52 ± 0.08</td><td>1.51 ± 0.04</td><td>N/A</td><td>1.59 ± 0.05</td></tr></tbody></table>

diffusivity for the typical macropore plus corona unit from Eq. (4) is given in Table 9. The macropores with their accompanying coronas will be treated as combined macroporous phase. The volume fractions of pellets of each batch that consist of this combined phase are shown in Table 9.

As shown schematically in Fig. 1, the cores of the pellets in the bimodal batches are considered to comprise a macroporous phase, consisting of isolated macropores and their accompanying coronas, pervasively distributed within a continuous, solely mesoporous matrix phase. The overall diffusivity $D_w$ for this composite core was calculated
<table><caption>**Table 9**

Predictions from modelling of mass transport using pellet structural model given in Fig. 1.</caption><thead><tr><th>Batch</th><th>Overall macropore and corona phase diffusivity /10<sup>-6</sup> m<sup>2</sup>s<sup>-1</sup></th><th>Volume fraction of the combined macropore and corona phase</th><th>Diffusivity of the composite core region /10<sup>-8</sup> m<sup>2</sup>s<sup>-1</sup></th><th>Ratio of overall conductances /(-)</th><th>Ratio of Predicted MTCs</th></tr></thead><tbody><tr><td>A</td><td>N/A</td><td>0.0</td><td>4.38</td><td>1.00</td><td>1.00</td></tr><tr><td>B</td><td>1.72</td><td>0.102</td><td>6.46</td><td>1.35 ± 0.24</td><td>1.16 ± 0.21</td></tr><tr><td>C</td><td>1.99</td><td>0.205</td><td>9.27</td><td>1.81 ± 0.31</td><td>1.42 ± 0.25</td></tr><tr><td>D</td><td>2.31</td><td>0.228</td><td>11.4</td><td>2.21 ± 0.39</td><td>1.73 ± 0.30</td></tr></tbody></table>

using the Maxwell formula given in Eq. (6), and the values obtained are shown in Table 9. The diffusivity for the continuous matrix phase was the effective diffusivity obtained using the mesoporosity value given in Table 5, the tortuosity of ~ 4.3 obtained from the PFG NMR (which is within the 95 % confidence intervals of that obtained above from the percolation analysis of 3.1<sup>+2.1</sup><sub>-0.9</sub>), and the reference Knudsen diffusivity for pores with an aspect ratio of unity, since the matrix platelets are disordered, and a critical pore size equal to the median pore size from nitrogen adsorption, since the mesoporous matrix must be the main contributor to this value.

The cores of pellets from batches B-D are considered to be surrounded by a ring of solely mesoporosity with similar pore structural and mass transport properties to the mesoporous matrix within the core. The overall diffusional resistance for the whole pellet, including rind and core, was then obtained using Eq. (4). The rind thickness was taken as 50 µm, and the overall batch average pellet radii are given in Table 5. The overall pellet resistances were used to calculate the ratio of the overall diffusional conductances for the material of pellets of batches B-D, with macropores, to that for the (virtually) solely mesoporous batch A, and the results are shown in Table 9. It can be seen that, overall, the macroporosity improves the conductance by approximately 35, 81 and 121 % for batches B, C and D, respectively.

The LDF MTC, *k*, value was calculated for batch A using the overall porosity of batch A, a tortuosity value of 4.26, and the Knudsen diffusivity for a pore of aspect ratio of unity and critical pore size equal to the nitrogen sorption median value. The pellets of batch A were also assumed to be spherical and have the batch average diameter. The value of the MTC thereby obtained was 0.92 s<sup>-1</sup>, which is within the range of experimental values obtained from MRI experiment, as given in Table 8. The ratios of the LDF MTCs for each of batches B-D to that for A, as the reference denominator, were obtained with the corresponding analogous assumptions for B-D as made for A mentioned above, and these are given in Table 9. It can be seen that these ratios match those obtained experimentally, via MRI and image analysis Methods 1–3, given in Table 8, within the 95 % confidence intervals. From a comparison of the ratios (relative to batch A) of the conductances and MTCs for batches B-D, it can be seen that the slightly larger typical diameter of the more macroporous pellets slightly diminished the enhancement of the intrinsic mass transport properties of the pellet material itself, provided by the increased macroporosity.

The relative importance of the key model structural parameters in determining the overall mass transport rate was assessed using a sensitivity study. Each of the key parameters listed in Table 10 were varied by ± 10 % one at a time, and their impact on the LDF MTC, or ratios, was noted, and these are recorded in Table 10 as percentages of the original model value. The most significant impact was observed when varying

the mesopore matrix tortuosity, which affected the absolute value of the MTC to the same order of magnitude as the perturbation, and also the ratios of MTCs separately to a lesser extent. For the bimodal materials, the relative impact of varying the other model parameters all increased in the order B to D, as might be expected given that the macroporosity increased in that order and the other four model parameters tested related to the nature and impact of the macroporous phase. The relative sizes of the impact on the MTC ratios of varying tortuosity, critical pore size, corona thickness and rind thickness were all roughly the same order, while the impact of macropore average size was ~ 1–2 orders smaller than these. However, it is also noted that the overall largest relative impact on the predicted MTCs came from the mesopore aspect ratio, which was of the order ~ 2.

## 5. Discussion

It has been seen that the introduction of varying amounts of porogen does not simply result in an increasing incidence of macropores but this, particular, way of introducing the macropores also resulted in additional structural features, like the rind and corona. This work has shown that these structural features do impact the overall mass transport rates, though the level of macroporosity itself is still the primary variable determining diffusivity differences between the batches. Overall, for the highest porogen level, and thus the largest macroporosity, the overall gain in mass transport rates by a factor of ~ 2–3 is similar to that observed by another method for an alumina foam monolith (Nepryahin et al., 2016). This is probably because in both cases the macroporosity consisted of isolated voids surrounded by a mesoporous skeleton. The TEM studies suggest that, away from the macropores, the mesoporosity of the alumina remains disordered. However, the level of porogen has affected the general modal mesopore size, and, thus, may have affected the overall density and/or arrangement of the packing of alumina platelets.

The analysis of the gas sorption data using a seeded percolation model, to account for the lack of filling of the macroporosity, at the top of the conventional gas adsorption isotherm, has enabled the derivation of an apparent connectivity for the interconnections within a Bethe bond lattice model of the mesoporous matrix. The low values of model pore bond connectivity, close to 2, probably reflect a high degree of pore blocking arising from the barrier layers of the corona and the rind. The same network structural model has enabled the prediction of a tortuosity for Knudsen diffusion within the mesoporous phase that is very similar (within experimental error) to that obtained experimentally from the SF<sub>6</sub> PFG NMR. The success of the Bethe lattice model in predicting tortuosity from nitrogen sorption data does not imply that the mesoporous void space actually looks like a pore bond version of a Bethe

<table><caption>**Table 10**

Results of sensitivity study of varying key model parameters by ± 10 % on mass transport rates expressed as a percentage of original value (N/A = not applicable).</caption><thead><tr><th rowspan="2">Mass transport parameter</th><th colspan="2">Tortuosity</th><th colspan="2">SF<sub>6</sub> critical pore size</th><th colspan="2">Corona thickness</th><th colspan="2">Rind thickness</th><th colspan="2">Macropore size</th></tr><tr><th>+10 %</th><th>-10 %</th><th>+10 %</th><th>-10 %</th><th>+10 %</th><th>-10 %</th><th>+10 %</th><th>-10 %</th><th>+10 %</th><th>-10 %</th></tr></thead><tbody><tr><td>k<sub>A</sub></td><td>90.91</td><td>111.1</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td></tr><tr><td>k<sub>B</sub>/k<sub>A</sub></td><td>100.2</td><td>99.8</td><td>100.1</td><td>99.9</td><td>100.2</td><td>99.8</td><td>99.8</td><td>100.2</td><td>99.99</td><td>100.009</td></tr><tr><td>k<sub>C</sub>/k<sub>A</sub></td><td>100.3</td><td>99.6</td><td>100.2</td><td>99.7</td><td>100.4</td><td>99.6</td><td>99.7</td><td>100.3</td><td>99.94</td><td>100.06</td></tr><tr><td>k<sub>D</sub>/k<sub>A</sub></td><td>100.4</td><td>99.5</td><td>100.3</td><td>99.6</td><td>100.5</td><td>99.5</td><td>99.6</td><td>100.4</td><td>99.89</td><td>100.14</td></tr></tbody></table>
lattice, as is apparent from the TEM images. However, the instrumentalist success of the model does mean it must capture/embody the minimal aspects of the void space needed to predict tortuosity.

The observation that the SF₆ PFG NMR diffusivity varies linearly with median pore diameter for the various batches of alumina pellet but that that line does not pass through the origin suggests that there is a more or less (within the small scatter about the straight lines in Fig. 12) a constant offset across the various batches between the median pore size and the critical pore size for the controlling diffusion process in the SF₆ PFG NMR experiment. It was noted that the critical pore sizes obtained for each batch, using the value of the offset obtained from this plot, correspond to the typical values found for the widths of alumina platelets in the TEM, and also the SAXS data. This suggests that the critical pore size may correspond to constrictions formed by the close packing of platelets, since the typical gap sizes will then be similar to the particle sizes in the packing. The diffusion process probed by PFG NMR is the self-diffusion of the SF₆ molecules which may involve passage through constrictions that control the observed mass transfer rate. Simulations of the diffusion of tracers within pore networks, derived from electron tomography of SBA-15 and KIT-6 silica, have suggested that only a small number of bottlenecks is sufficient to cause a dramatic decline in the observed diffusivity (Reich et al., 2018a, 2018b). In contrast, the diffusional flux in the xenon uptake experiments could potentially bypass the barrier regions around the macropores by staying within the mesoporous matrix during its passage towards the centre of the pellet. Hence, the critical pore size for Knudsen diffusion is different for the two regions, and, thence, the two types of diffusional process with different boundary conditions (steady-state self-diffusion versus transient diffusive flux). Indeed, previous modelling work (Rigby and Daut, 2002) has suggested that the relative impact of certain types of void space heterogeneity is affected by the volume-averaging out arising due to the steady-state self-diffusion during PFG NMR, but, in contrast, may more significantly impact diffusion under an imposed concentration, as in the MRI transient uptake experiments.

The MRI studies of the uptake of hp xenon have shown that the pellets possess macroscopic heterogeneities in the spatial distribution of MTC, and these are over a large length-scale compared to the size of the pellet. The similarity in the orders of the coefficients of variation of the MTCs from the imaging data, and that for pore size from nitrogen sorption, suggests that heterogeneities in the spatial distribution of local average pore size give rise to the heterogeneity in mass transport. This is consistent with the median mesopore size being the main controlling pore structural parameter for the overall diffusivity of the mesoporous matrix. However, the large spatial extent of the heterogeneities seen in the MRI means that their contribution to overall tortuosity will be low, since the flux then does not have to pass through many such regions enroute to the centre of the pellet, as suggested by the success of using the median pore size. This is consistent with the observation that fragmentation of the pellets does not lead to a complete loss of mercury entrapment, in contrast to what has been observed previously for sol-gel silica spheres of similar size to the pellets studied here, where this did occur (Rigby and Edler, 2002). In the case of the silicas, the fragmentation of the whole pellets led to the removal of the pore shielding responsible for mercury snap-off and entrapment. It is noted that for the silica materials the tortuosity for transient liquid-liquid exchange experiments, conducted by MRI on whole pellets, was larger than that for liquid phase PFG NMR experiments (Rigby and Gladden, 1996). This suggested that, for these pellets, macroscopic heterogeneities (>~40 µm) made a substantial contribution to the overall tortuosity. This is consistent with these same heterogeneities giving rise to the entire entrapment for these pellets.

In contrast, for the alumina Batches A-D studied here the macroscopic heterogeneities seen in the MRI are not responsible for the mercury entrapment, probably because they are so large compared to the overall size of the pellet that there is not much pore shielding occurring over these large length-scales. Instead, the particular pore shielding that

gives rise to the entrapment for the aluminas arises over length-scales smaller than the particle fragments (<~10–100 µm). Pore shielding is associated with tortuosity due to changes in the pore conductance (Salmas and Androutsopoulos, 2001). This is consistent with the tortuosity of the alumina matrix being taken as that from the PFG NMR which probes length-scales only up to the order of 10–100 µm.

## 6. Conclusions

The effected synthesis of an attempted real simulacrum of an ideal void space design can result in unexpected structural deviations from the theoretical ideal. Transmission electron microscopy studies of the regions around the macropores, in the templated materials, have revealed an alignment of the alumina platelets parallel to the perimeter of the macropores, which presents a diffusional barrier that controls the self-diffusion process probed by PFG NMR. The SEM data has revealed a rind around the periphery of the pellets which lacks macropores. The MRI studies have revealed some slight, large length-scale heterogeneities in the macroscopic variation of the mesopore structure of the pellets, though these are not significant enough to control overall mass transport in this case. However, the modelling studies have suggested that the design aim of creating additional macroporosity does lead to a significant increase in overall mass transfer rates, although, this gain is slightly offset by the slightly larger overall size of the macroporous pellets and, also, additionally limited by the co-creation of the rind and corona regions in the pellets which introduce more diffusional resistance. The more statistically-representative SAXS data confirmed the typical sizes of alumina platelets seen in the TEM images. The gas sorption data provided the PSD and pore connectivity for the mesopore network. The data on mercury entrapment for whole and fragmented pellets was found to be consistent with the relatively low level, but very large length-scale, macroscopic structural heterogeneity seen in the MRI experiments. Hence, the novel combination of characterization methods has enabled a detailed assessment of the effectiveness of the fabrication method, employing porogen to improve mass transport, to be made over several different length-scales.

### CRediT authorship contribution statement

**Stefano Collins:** Writing – original draft, Validation, Methodology, Investigation, Conceptualization. **Severine Humbert:** Writing – original draft, Validation, Methodology, Investigation. **Dina Lofficial:** Writing – review & editing, Writing – original draft, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization. **Christophe Vallee:** Writing – review & editing, Writing – original draft, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization. **Christopher Parmenter:** Supervision, Methodology, Investigation. **Huw E.L. Williams:** Writing – original draft, Supervision, Methodology, Investigation. **Arthur Harrison:** Methodology, Investigation. **Max Filkins:** Methodology, Investigation. **Galina E. Pavlovskaya:** Writing – review & editing, Writing – original draft, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Data curation, Conceptualization. **Thomas Meersmann:** Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization. **Sean P. Rigby:** Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements

SC, SPR, HELW, GEP and TM thank IFPEN for the provision of the PhD studentship for SC. The authors are grateful to Thomas Bizien (SOLEIL synchrotron – SWING Beamline) for assistance and to the SOLEIL staff for smoothly running the facility. SEM and TEM experiments were conducted at IPFEN by Hanane DE FREITAS, Julie POULI-ZAC and Anaïs KAPOUDJIAN. Some of the mercury porosimetry and

nitrogen sorption experiments were performed by Naima TALBI, Larysa RATEL, Denis ROUX, Nicolas ALONSO, Fabien GEORGET and Joel VIVAS DIAS (IFPEN). PFG-NMR experiments of SF₆ were performed by Laurent LEMAITRE, Florent MOREAU and Frederic FILALI (IFPEN). We acknowledge funding for the Zeiss Crossbeam 550 FIB-SEM instrument from the EPSRC (EP/S021434/1). We thank Lorelei Robertson at the nmRC for assistance with some of the data acquisition.

Appendix

<figure><img src="image_19.png" /><figcaption>Fig. A1. Example of a fit of the theoretical accessibility function (solid line) for seeded bond percolation to experimental data (●) derived from nitrogen sorption isotherms for a sample from batch B. The dashed line represents y = x.</figcaption></figure>

The PFG NMR data was fitted to the log-attenuation equation:

$$ \ln(\psi) = \ln\left(\frac{I}{I_0}\right) = -(\gamma\delta G)^2 \left(\Delta - \frac{\delta}{3} - \frac{\tau}{2}\right) D_{\text{PFG}} \quad (A1) $$

where I is the experimental observed intensity, $I_0$ is the intensity with no attenuation, $\gamma$ is the gyromagnetic ratio, $\delta$ is the gradient pulse duration, G is the gradient strength, $\Delta$ is the diffusion time and $D_{\text{PFG}}$ is the fitted diffusivity.
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. A2.** Mercury porosimetry intrusion (solid line) and extrusion (●) data for powder form of samples from Batches (a), B, (b) C and (c) D. The raw data has been analysed using the Kloubek correlations mentioned in the Theory section.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. A3. Example of a PFG-NMR attenuation plot of experimental data (□) fitted to a mono-exponential model (Eq. A1. The regression fit (red solid line) yields D<sub>PFG</sub> = (2.90 ± 0.02) × 10<sup>-7</sup> m<sup>2</sup> s<sup>-1</sup>, where the uncertainty is the standard error in the fit. Sample information: Batch A at 305.5 K, Δ = 20 ms. The data point in orange is not considered.</figcaption></figure>

# Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.ces.2025.122843.

## Data availability

The data that has been used is confidential.

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