# Do Pores Exist?--Foundational Issues in Pore Structural Characterisation
## 4 Virtual Pores
For the SD feed pellets, the macroscopic heterogeneities led to the pattern of mercury intrusion and entrapment having a'shrinking core' form, but where some obvious round SD feed particles were left completely unintruded (dark) behind the (bright white) advancing mercury intrusion front. However, despite the high level of complexity of the methanol synthesis pellet structure, it was found that a simple random pore bond network could predict, from serial  
Figure 8: Comparison, for spray-dried (SD) feed methanol synthesis catalyst pellet, of the observed fractional decline in the nitrogen gas uptake mass transfer coefficient (\(k\mathcal{A}_{0}\)) following blockage of different fractions (%) of the void space with mercury entrapment (\(\blacksquare\)), or water adsorption (\(\blacktriangle\)), and that expected for a random arrangement of entrapped pore liquid from the Prager model (\(\bullet\)), as described in [85]. The pressures and pore sizes correspond to the ultimate values achieved in the mercury intrusion scanning curves that led to the entrapment of mercury (white pixels), as seen in the CXT images of the pellet (grey pixels) obtained thereafter. Reprinted with permission from Ref. [85]. 2023, Elsevier. nitrogen overcondensation and (sorption) data, the changes in the mass transport coefficient from the linear driving force model for kinetic nitrogen gas uptake following mercury entrapment, as shown schematically in Figure 9[85]. The mass transfer coefficient was proportional to the ratio of the porosity to the tortuosity of the pellet. A percolation theory-based analysis of the nitrogen sorption hysteresis loops was used to obtain the pore connectivity and lattice size (\(L\)) of the random PNM [86]. Given that tortuosity is a measure of the extension of the path length of the diffusional flux beyond that of the straight-line distance across the porous medium, then the relative change in tortuosity can be related to the relative change in effective network size, as measured by apparent lattice size. Hence, the change in the measured tortuosity, \(\tau\), for mass uptake into the modified pellet void space, relative to that for the original empty pellet (denoted by the subscript 0), can be related to the change in the apparent size of the percolation model lattice size (\(L-L_{0}\)) following mercury entrapment via the following relation [85]:  
\[\frac{\tau_{0}}{\tau}=1-\gamma(L-L_{0})w\bigg{(}\frac{S}{V}\bigg{)} \tag{1}\]  
where \(w\) is the pellet feed particle size and \(S/V\) is the external surface area-to-volume ratio of the finished pellet; the ratio of the geometry factors for the feed particles and whole pellet is denoted as \(\gamma\), which would be equal to unity if the feed particles and pellet had the same geometrical shape. As can be seen from Figure 9, the experimental findings for SD methanol synthesis pellets, and also for similar pellets made with a differently sized roll-compacted (RC) feed, were found to be consistent with Equation (1). This work was also able to show that a relatively simple, abstract model could capture the key structural aspect of a complex real material determining mass transport processes within the void space [85]. The data shown in Figure 9 suggested that the apparent lattice size of the PNM appeared to increase following mercury entrapment. This is because the lattice size of a random PNM is related to a particular aspect of the originating gas sorption data. The roundedness of the knee in the gas desorption isotherm is considered to be the physical manifestation of the lattice-size effect on the percolation accessibility function for model random bond networks. A more rounded desorption knee implies a higher prevalence of surface clusters of larger pores, relative to interior, bulk pores, as is present in smaller random pore bond network models. The CXT images in Figure 8 show the presence of surface clusters of larger pores in the real pellet, as these become filled with entrapped mercury at lower intrusion pressures.