Argon, nitrogen, and carbon dioxide physisorption isotherms

Figures 1 to 4 show the complete collection of Ar, N₂, and CO₂ physisorption isotherms of the sample set. The corresponding textural parameters are summarized in the section ‘Characteristic parameter collection’ in the appendix (Tables 6–8).

Figure 1. Ar, N₂, and CO₂ physisorption isotherms (parts a to i) measured on the mudrock sample set at the operational temperature of the adsorptive. The molar uptake is recalculated to liquid basis, assuming bulk liquid density of the respective fluid (see the section ‘Adsorptive Properties’ in the appendix, Table 3) for better comparison. The greater uptake of N₂ relative to Ar at p/p ⁰ << 0.1 shifts N₂ isotherms above those of Ar in the micropore range. CO₂ physisorption isotherms are either lower than or equal to Ar physisorption isotherms on a liquid basis.

Figure 2. Ar, N₂, and CO₂ physisorption isotherms (parts a to i) measured on mudrocks and clays at the operational temperature of the adsorptive. Further explanation can be found in Fig. 1.

Figure 3. Ar, N₂, and CO₂ physisorption isotherms (parts a to i) measured on engineered materials, silicas, and Fe oxides at the operational temperature of the adsorptive. Further explanation can be in found Fig. 1.

Figure 4. Ar and N₂ physisorption isotherms (parts a and b) measured on the Fe oxides at the operational temperature of the adsorptive. Further explanation can be found in Fig. 1.

Scrutiny of the isotherm set revealed subtle differences in isotherm shape at p/p ⁰ < 0.1. Ar isotherms display smoother curvature than N₂ isotherms which indicates slightly greater uptake in the lowest pressure range (Figs 1–4). Starting at the lowest partial pressure, the N₂ uptake increases faster than that for Ar, but levels to approximately the same slope displayed by the Ar isotherm. The initially higher volumetric uptake of N₂ shifts the whole isotherm above that of Ar. This difference reverses when plotted on a molar basis due to the difference in molar volume of both adsorptives at their respective experimental conditions (not shown). Isotherms converge toward greater partial pressure at which a steep upswing occurs. A complete convergence is not observed for the sample set as the upswings can occur at slightly lower p/p ⁰ for N₂ compared to Ar (cf. BCM, YPC-Cf-CuTrien). At the limiting partial pressures, N₂ isotherms clearly exceed the Ar isotherms, except for the SiAl pellets and the vycor cpg-50 nm.

Ar (87 K) and N₂ (77 K) physisorption isotherms of natural porous materials are of type IV (Thommes et al., Reference Thommes, Kaneko, Neimark, Olivier, Rodriguez-Reinoso, Rouquerol and Sing2015), lacking a plateau at high partial pressures as observed for the controlled porous glasses (Figs 1–3). Generally, Ar and N₂ isotherms display qualitatively similar shapes including pronounced hysteresis of either type H3 or type H4 (Thommes et al., Reference Thommes, Kaneko, Neimark, Olivier, Rodriguez-Reinoso, Rouquerol and Sing2015). Micropore-dominated samples (Cu-Trien exchanged materials) are characterized by a type H4 hysteresis loop whereas meso- and macropore-dominated samples possess a type H3 hysteresis loop. Hysteresis loop closure points of mudrock and clay isotherms are identical for the respective absorptive with closing points at p/p⁰ = 0.45 for nitrogen and p/p⁰ = 0.4 for argon. Below the hysteresis critical pressure, adsorption is fully reversible for both adsorptives except for the HM, Bossier(b) (BM(b)) and both Longmaxi samples, which display slight low-pressure hysteresis.

Isotherms of the engineered materials and the kaolinite resemble a type IVa isotherm with individual hysteresis loops of either type H1 (vycor cpg-50nm, SiAl-pellets, kaolinite) or type H2(a) (vycor cpg-20nm, vycor cpg-5nm). Accordingly, hysteresis loops close at pore-size specific partial pressures. The hysteresis loop of the kaolinite isotherm, for example, closes at p/p⁰ ~ 0.8 and is narrow compared to those of the engineered materials (cf. Fig. 2h, Fig. 3a–d). The upswing in adsorption occurs at a slightly lower partial pressure for N₂ than for Ar in the pore network of engineered materials. The desorption branches and hysteresis closure points are approximately equal for both adsorptives for all engineered materials as well as the kaolinite. All non-porous materials (muscovite, silicas and hematites) are characterized by a type II isotherm displaying little or no hysteresis. The small amount of hysteresis is attributed to inter-particle condensation and partially to instrumental reasons rather than to mesopores which are unlikely for these materials.

CO₂ physisorption isotherms were measured only for (micro-) porous samples (mudrocks, clays, and engineered materials) and are shown as inset in Fig. 1, Fig. 2a–g and Fig. 3a–d. Independent of the solid, the CO₂ uptake increased gradually lacking any peculiar feature. The uptake was mostly lower than those of Ar and N₂ at the same partial pressure. However, for some samples (BM(a), LM(a), vycor cpg(20 nm), SiAl pellets) the CO₂ uptake was similar to that of Ar on a liquid basis.

Equilibrium tests

Bertier et al. (Reference Bertier, Schweinar, Stanjek, Ghanizadeh, Clarkson, Busch, Kampman, Prinz, Amann-Hildebrand, Krooss and Pipich2016) demonstrated the effect of equilibration time and diffusion control on N₂ physisorption in the pore network of mudrocks with special emphasis on low-pressure hysteresis. In line with Schlumberger and Thommes (Reference Schlumberger and Thommes2021), those authors highlighted a substantial underestimation of textural parameters for equilibration times which were too short.

Equilibration is particularly crucial in the micropore range when micropores control the access to larger pores (Schlumberger and Thommes, Reference Schlumberger and Thommes2021) as it is often observed in mudrocks and clays through pore network effects. In contrast to N₂, Ar suffers less from diffusional constraints in tight pore networks because of the higher measurement temperature and the lack of specific interaction with surface groups preventing a shift in filling pressure to lower p/p ⁰, thus avoiding slower diffusion for Ar (Cychosz et al., Reference Cychosz, Guillet-Nicolas, García-Martínez and Thommes2017; Thommes, Reference Thommes, Čejka, van Bekkum, Corma and Schüth2007).

The effect of equilibration time on Ar physisorption measurements was studied on the OCM and LM(a) mudrocks with equilibration times between 1 and 13 min and 1 min to 10 min per pressure point, respectively (Fig. 5). These were the maximum times that could be measured with a single dewar of liquid nitrogen.

Figure 5. (a) Ar physisorption isotherm (87 K) of the Opalinus Clay (OCM) and the Longmaxi(a) (LMa) mudrock measured as a function of the equilibration time (16 Pa min⁻¹ for 1–13 min and 2–10 min, respectively); (b) plots of characteristic parameters (A _BET = BET specific surface area, V ₀ = limiting micropore volume, and V _Gv = Gurvich pore volume) of OCM and LM(a) mudrocks as function of equilibration time. Error bars show two standard deviations obtained from repetitive measurements at 5 min equilibration time (OCM) or interpolation (Longmaxi(a)).

Equilibration time tests on the OCM and LM(a) mudrock revealed sample-specific and non-monotonic effects in the increase of textural parameters (Fig. 5). The OCM reached a plateau after 10 min of equilibration time whereas the LM(a) plateaued from ~5 min equilibration time on. Raising the equilibration time from 1 to 13 min increased the limiting pore volume and the specific surface area by a factor of~1.04, while the Gurvich pore volume increased by a factor of ~1.09 for the OCM. This slight increase was achieved at the expense of an additional 1251 min of measurement time. In fact, an equilibration time of 5 min or 10 min produced a maximum relative difference of 1.2% for the LM(a) and 1.4% for the OCM in terms of textural parameters. Accounting for the estimated uncertainty of the Ar physisorption measurements showed that 95% confidence intervals overlap beyond 5 min of equilibration time for each characteristic parameter of both mudrocks (Fig. 5b). This indicated that beyond equilibration times of 5 min, textural parameters were not statistically different, suggesting a negligible control of the equilibration criterion on Ar physisorption for these particular mudrocks, if equilibration times were at least 5 min.

The YPC-K mudrock has been analysed with a focus on the micropore region when testing very long equilibration times (5 vs. 60 min) (Fig. 6). A slight shift of uptake toward greater partial pressure was recognized. This detectable difference quantified to ~10% in terms of micropore volume and BET area, which is consistent with the general isotherm offset. It remained unclear whether this difference was statistically insignificant because confidence intervals overlap with respect to BET area and no conclusion can be drawn (Fig. 6b). Note, however, that the 95% confidence intervals overlapped only in the extreme tails by at most 4%. The difference in micropore volume remained significant as 95% confidence intervals did not overlap each other.

Figure 6. (a) Ar physisorption isotherms measured at 87 K for 5 min and for 60 min equilibration time on the YPC-Kortemark mudrock. The inset image shows an enlarged section of the low-pressure region (p/p ⁰ <10⁻¹); (b) Textural parameters plotted as functions of equilibration time (time_eq) including two standard deviations obtained from uncertainty interpolation. The Gurvich volume decreases of the 60 min sample are attributed to a saturation pressure offset (see the section ‘Factors of uncertainty and the importance of saturation pressure’).

Though the shift of the isotherms (see inset of Fig. 6a) was not substantial, it could still be indicative of an equilibration issue that causes underestimation of textural parameters. Therefore, ambiguity persisted whether a real effect at extreme conditions was detected or simply measurement uncertainty combined with sample heterogeneity. The latter cannot be ruled out as the test was not performed on the identical material (sample heterogeneity) and other unknown factors (p ⁰ offset; see the section ‘Factors of uncertainty and the importance of saturation pressure’) may have contributed to the observed difference. The above demonstrates that an effect of equilibration time is detectable in the isotherm data though it should not be overvalued if no low-pressure hysteresis is observed. Systematic testing underlined the notion that equilibration times are sample-specific and a general recommendation would be questionable, in particular for geological materials with complex pore structures. It is recommended instead that individual testing of equilibration criteria is performed in order to interpret data (and trends) reliably on an absolute and relative basis. However, from a practical point of view and conservatively accounting for uncertainty, an equilibration criterion of 16 Pa min⁻¹ was acceptable in the present study as no significant low-pressure hysteresis was observed and other factors probably have a more substantial impact on isotherm measurements.

Reproducibility and precision of Ar physisorption measurements

The repeatability of Ar physisorption measurements was tested on the mesoporous SiAl pellets (n = 10) and three mudrocks covering a representative range of uptake (HM, BCM, OCM) (n = 5–10) (Fig. 7). Coefficients of variation are defined as relative standard deviations and they are summarized together with the absolute standard deviations for the selection of representative samples in Table 4.

Figure 7. Ar physisorption isotherms (87 K) of repetitive measurements on: (a) the SiAl pellets (n = 10); and (b) the Haynesville mudrock (n = 5); the precision is represented by two standard deviations (2σ) shown for individual data points.

Considering the entire isotherm range, it is noted that the CVs varied with relative pressure. The CV for the SiAl pellets was <1% at pressures below the level of capillary condensation and evaporation, respectively. Maximum coefficients of variation were obtained at the p/p ⁰ of sudden capillary condensation (2.1% at p/p⁰ = 0.825) and evaporation (2.4% at p/p⁰ = 0.756). For the Haynesville mudrock, the average CV was ~ 3.4% for the adsorption branch and 2.6% for the desorption branch. The highest CV values of 6.2% were observed in the ultra-low-pressure range.

Statistical analysis of the main textural parameters (A _BET, V ₀, V _Gv) yielded CVs of between 0.3 and 1.5% for the SiAl pellets and 0.3 and 3.5% for the Haynesville mudrock. The largest CV was observed for the micropore volume of the OCM amounting to 5.9%. For the other mudrocks, smaller CVs were identified for all textural parameters. Generally, repeatability and precision were better for the SiAl pellets compared to the less sorbing mudrocks.

Explicit uncertainty data for Ar physisorption measurements in geoscience have not been found in the literature. For N₂ physisorption isotherms, Kuila and Prasad (Reference Kuila and Prasad2013) reported coefficients of variation of 2–6% for specific surface areas of mudrocks. Lahn et al. (Reference Lahn, Bertier, Seemann and Stanjek2020) determined maximum coefficients of variation of 2.1% for Gurvich total pore volumes, 5.4% for BET areas and 5.3% for micropore volumes of moistened samples. For dry mudrocks, coefficients of variation ranged from 0.46% to at most 2.9% (calculated from physisorption data of Lahn et al., Reference Lahn, Bertier, Seemann and Stanjek2020). This is in good agreement with the data presented. Furthermore, when considering a coefficient of variation of 5.9% (V ₀ of OCM) on absolute scale, it yielded an absolute variation in micropore volume of 0.7 μL g⁻¹ for the OCM. This is regarded as negligible when it comes to any application purposes involving micropore volumes as in pore network modeling. Thus, repeatability and precision were rated as good to excellent at constant experimental conditions. Therefore, the CV values presented here were regarded as small enough to allow interpretation of small differences between samples and the statistical analysis provided fair confidence in the Ar data presented.

Uncertainty factors and the importance of saturation pressure

Isotherm analysis revealed four samples deviating outside of the confidence limits in textural parameters and isotherm shape (see the section ‘Primary Experimental Results’). Ar physisorption isotherms of BCM, YPC-A, YPC-E, and YPC-Cf-CuTrien displayed isotherm-crossing at high p/p ⁰ (molar basis) resulting in a less steep approximation of the saturation pressure (Fig. 8b). This crossing was not observed for the remaining sample set for which N₂ isotherms crossed Ar isotherms only in the low-pressure region (molar basis). This peculiarity was reflected in the Gurvich (total) pore volumes and BJH pore-size distributions. The difference between Ar- and N₂-based Gurvich (total) pore volumes ranged from 24 to 33% which is beyond the average deviation threshold reported in the literature (McKee, Reference McKee1959). In support, divergence in cumulative PSDs (adsorption and desorption branches) increased as a function of pore size (Fig. 8c). In fact, there are several parameters that can cause uncertainty during an isotherm measurement. Such factors included : (1) sample type (e.g. mineral fractionation, particle size etc.); (2) sample treatment (drying (P _vac, T _dry), weighing etc.); (3) gas purity; (4) instrumental technicalities (volume calibration, pressure transducer etc.); (5) equilibration criterion; (6) the operator’s mood and care; to (7) environmental conditions such as room temperature and atmospheric pressure fluctuations (not exhaustive). Schlumberger and Thommes (Reference Schlumberger and Thommes2021) additionally suggest (8) helium entrapment in micropores (systematic dead-volume error) as well as (9) thermal transpiration (corrected here) as factors of uncertainty. The systematic differences in this study were probably the result of saturation-pressure fluctuations induced by changes in the atmospheric pressure over the course of an isotherm measurement (i.e. ~1 day).

Figure 8. (a) Saturation pressure (P _sat) fluctuations recorded during N₂ physisorption measurements performed on five (n = 5) porous and non-porous samples using a Gemini VII 2340 instrument (Micromeritics); (b) Ar (p ⁰ offset) and N₂ physisorption isotherms of the Boom Clay sample plotted on a molar basis. The inset shows an enlarged section of the high partial pressure region; (c) cumulative BJH-PSDs (adsorption branch) computed from Ar and N₂ physisorption isotherms measured on the Boom Clay sample.

Direct p ⁰ measurements provided evidence of the variation in saturation pressure during a physisorption measurement of up to 3 kPa (FB-105FD) (Fig. 8a). These p ⁰ fluctuations are a direct result of atmospheric pressure changes that effect temperature oscillation in the liquid nitrogen bath; i.e. the boiling point of liquid nitrogen changes. Because the saturation pressure was not measured during the Ar physisorption measurements, such fluctuations were not immediately accounted for when the isotherm was produced.

Atmospheric pressure fluctuations have substantial effects on the amount adsorbed as a temperature difference of 0.1 K varies the saturation pressure by ~1.07 kPa. Hence, an incorrect saturation pressure assumption yields either over- or underestimation of the amount adsorbed at a particular partial pressure because the thermodynamic state of the pore fluid (e.g. liquid density or cross-sectional area) is conjectured incorrectly and the p/p ⁰ is shifted depending on the offset direction. In detail, the p ⁰ offset provokes a shift in filling pressure of a particular pore size. In each case, the adsorbed amount (for Ar in this case) is assigned to an incorrect pore size as the pore size is related directly to p/p ⁰ via the Kelvin equation (Seemann et al., Reference Seemann, Bertier, Krooss and Stanjek2017). The effect on liquid density is, however, negligible (<1%). Because the p ⁰ influence scales with pressure, the high-pressure region and associated textural parameters (V _Gv, PSD) are influenced more heavily than the low partial pressure region (V ₀, A _BET) (Fig. 8b,c). The assumption that beyond multilayer-formation, fluid–fluid interaction governs the sorption mechanism rather than solid–fluid interaction, suggests that N₂ and Ar adsorb in the same mesopore space (Kruk and Jaroniec, Reference Kruk and Jaroniec2000; Rouquerol et al., Reference Rouquerol, Rouquerol, Sing, Llewellyn and Maurin2014; Thommes et al., Reference Thommes, Mitchell and Pérez-Ramírez2012). Therefore, similar PSDs should be obtained. Assuming that N₂ isotherms are accurate (p ⁰ measured) allowed to estimate the saturation pressure offset based on N₂ PSDs. In fact, the largest p ⁰ offset was estimated for the Boom Clay yielding a 13.3 hPa offset in saturation pressure (1026.03 hPa) though p ⁰ offsets were typically <5.3 hPa for the YPC-E, YPC-A and YPC-Cf-E. Considering the Boom Clay, the saturation pressure offset caused a negligible difference in V ₀ and A _BET of 1% and 0.37%, respectively, which was within the estimated 95% confidence interval for the BCM. On the most conservative assumption, a maximum relative difference of 46% has been obtained for the Gurvich pore volume whereas BET area and micropore volume changed by ~5% for the same sample (BCM). The maximum estimated differences in V _Gv and A _BET were based on all the Ar physisorption isotherms ever measured in the study laboratory. Even though a rough and conservative approximation, the theoretical consideration conveys the view that the saturation pressure constituted the greatest uncertainty factor, especially for high partial pressures, for the Ar physisorption data presented, as supported by the view of Schlumberger and Thommes (Reference Schlumberger and Thommes2021).

Direct p ⁰ measurements provided evidence for the above and emphasized the importance of continuous p ⁰ measurements instead of relying on a constant literature value for partial pressure transformation (Fig. 8a). The p ⁰ offset was visible directly in the apparent pore-size distribution and a cross-check with an independent physisorption measurement is highly recommended. Consequently, this observation underlined that pore-size distributions and Gurvich total pore volumes have to be reviewed critically and cautiously in order to avoid misinterpretation caused by environmentally induced errors; yet they are good indicators for such a technical issue. Based on the above, the BCM, YPC-A, YPC-E, and YPC-Cf-CuTrien qualify as significantly affected by saturation pressure offsets and their Gurvich (total) pore volumes and BJH PSDs were excluded from the following comparison.

Comparison and characterization of N₂, Ar, and CO₂ isotherms

Figures 1–4 display Ar, N₂, and CO₂ physisorption isotherms of mudrocks, clays, engineered materials and non-porous oxides. Ar and N₂ isotherms are qualitatively similar in shape (e.g. Fig. 1), yet some generally discriminating features were detected: (1) a differently shaped low-pressure region; (2) deviating hysteresis loop closure points; and (3) different upswing at high partial pressures.

(1) Differences in the low-pressure region

N₂ physisorption isotherms exhibited a substantially steeper increase in uptake in the low-pressure region (p/p ⁰ < 0.1) compared to the well rounded and smoother Ar isotherms. This feature was universal for the presented sample set and is commonly found in literature independent of the materials studied (Neimark et al., Reference Neimark, Ravikovitch, Grün, Schüth and Unger1998; Payne et al., Reference Payne, Sing and Turk1973; Schlumberger and Thommes, Reference Schlumberger and Thommes2021; Sotomayor et al., Reference Sotomayor, Cychosz and Thommes2018; Thommes, Reference Thommes2004; Thommes et al., Reference Thommes, Köhn and Fröba2002a, Reference Thommes, Köhn and Fröba2002b, Reference Thommes, Morell, Cychosz and Fröba2013). The difference in low-pressure uptake was attributed to distinct and molecule-specific sorptive–sorbent interactions which are particularly prominent in the low partial pressure region, i.e. where volume filling of micropores and multilayering prevail (Dubinin, Reference Dubinin, Cadenhead, Danielli and Rosenberg1975; Rouquerol et al., Reference Rouquerol, Rouquerol, Sing, Llewellyn and Maurin2014; Sotomayor et al., Reference Sotomayor, Cychosz and Thommes2018). In contrast, the mesopore region showed only minor differences. This implies that the influence of surface forces on the sorption behavior becomes less dominant beyond micropore filling and multi-layer formation (Rouquerol et al., Reference Rouquerol, Rouquerol, Sing, Llewellyn and Maurin2014). However, surface forces probably transcend beyond the first few adsorbate layers and continue to influence adsorption but in a less significant manner compared to the low-pressure region. Thus, fluid–fluid interaction should be governing in the mesopore region, but latest at higher adsorbate loading.

Enhanced sorptive–sorbent interactions are linked to overlapping force fields and specific surface sites in combination with the adsorptive–specific polarizability and polar momentum. N₂ possesses a weak permanent quadrupole moment strengthening adsorption through specific interactions in addition to the action of van der Waals forces (Cychosz et al., Reference Cychosz, Guillet-Nicolas, García-Martínez and Thommes2017; Drain, Reference Drain1953). In contrast, Ar lacks any permanent polar momentum reducing its interaction with a solid surface to pure van der Waals forces.

The existence of a quadrupole moment as for N₂ and CO₂ in combination with e.g. OH-groups or crystal defects can cause specific interaction forces that come on top of the ever-present van der Waals forces. Polarization results in enhanced sorptive–sorbent interaction and evokes a reduction in the filling pressure of micropores for N₂ (Thommes et al., Reference Thommes, Kaneko, Neimark, Olivier, Rodriguez-Reinoso, Rouquerol and Sing2015). The listed features are present in mudrocks and clays, which often have a complex and heterogeneous surface chemistry (Bai et al., Reference Bai, Kang, Chen, Chen, You, Li and Chen2020; Zhang et al., Reference Zhang, Xiong, Li, Wei, Jiang, Lei and Wu2017). In consequence, a substantial difference in uptake (isotherm shape) was expected and confirmed by the presented data in the region where these properties control fluid adsorption.

Enhanced sorptive–sorbent interactions of N₂ relative to Ar were substantiated by the characteristic energy parameter (E ₀) calculated from DA theory. E ₀ (N₂) data were systematically greater providing qualitative evidence of the different energy levels involved in the adsorption process of both gases in micropores (Fig. 9). This was further supported by the characteristic energy of CO₂ adsorption. E ₀ (CO₂) was even greater than that of N₂ due to its stronger permanent polar momentum (Table 3 in section ‘Adsorptive Properties’ in the appendix). Clearly, the adsorption energy trend was substrate independent as illustrated by the heterogeneous selection of samples (Fig. 9). However, ratios of characteristic energies were inconsistent. Thus, they are not exclusively governed by the adsorptive but by a combination of adsorptive and adsorbent.

Figure 9. Characteristic energies of adsorption (E ₀) calculated using DA theory from N₂ (77 K), Ar (87 K), and CO₂ (273 K) physisorption isotherms of representative mudrocks (Bossier(a) = BM(a), Opalinus Clay = OCM), clays (SWy2-CuTrien, YP-Cf), and engineered materials (vycor cpg-20nm; SiAl pellets).

The non-systematic variation could be attributed to the complexity of the pore network architecture, its constituents and eventually the share of an individual pore size to the total microporosity. Hence, a multi-factor control is expected to contribute to the adsorption distortion of N₂ and CO₂ while Ar is exclusively governed by pore geometry (size and shape).

Note that the characteristic energies are ‘fingerprints’ of the true physical energies and should be understood qualitatively. For a quantitative assessment of adsorption enthalpies, microcalorimetry is the analytical tool of choice. Fernandez-Colinas et al. (Reference Fernandez-Colinas, Denoyel, Grillet, Rouquerol and Rouquerol1989a) employed microcalorimetry on activated carbons and could corroborate quantitatively that the greater polarization of N₂ yields appreciably larger differential enthalpies of adsorption compared to Ar (Rouquerol et al., Reference Rouquerol, Rouquerol, Sing, Llewellyn and Maurin2014). This again supports the claim that N₂ sorption is enhanced relative to Ar in micropores.

Micropore volume

Micropore volumes of porous materials were obtained by means of DA theory and the α_s method (Table 5). Micropores in mudrocks are commonly attributed to organic matter and clay minerals, both exerting control over sorption properties and storage capacity of fluids such as CH₄ or CO₂ (Busch et al., Reference Busch, Bertier, Gensterblum, Rother, Spiers, Zhang and Wentinck2016; Chalmers and Bustin, Reference Chalmers and Bustin2008; Ziemiański et al., Reference Ziemiański, Derkowski, Szczurowski and Kozieł2020). To assess the microporosity of mudrocks several researchers used DA theory (Chen and Xiao, Reference Chen and Xiao2014; Chen et al., Reference Chen, Gai and Xiao2021; Clarkson and Haghshenas, Reference Clarkson and Haghshenas2016; Clarkson et al., Reference Clarkson, Freeman, He, Agamalian, Melnichenko, Mastalerz, Bustin, Radliński and Blach2012, Reference Clarkson, Solano, Bustin, Bustin, Chalmers, He, Melnichenko, Radliński and Blach2013; Furmann et al., Reference Furmann, Mastalerz, Bish, Schimmelmann and Pedersen2016; Lahn et al., Reference Lahn, Bertier, Seemann and Stanjek2020; Mastalerz et al., Reference Mastalerz, He, Melnichenko and Rupp2012; Shabani et al., Reference Shabani, Krooss, Hallenberger, Amann-Hildenbrand, Fink and Littke2020). It is considered suitable for geological materials because of its development for geometrically and chemically heterogeneous surfaces (Dobruskin, Reference Dobruskin1998; Kruk et al., Reference Kruk, Jaroniec and Choma1997). Despite IUPAC’s recommendation to use Ar or CO₂ for the assessment of microporosity, N₂ is regularly employed to quantify microporosity and it has been shown to correlate weakly with CO₂-based micropore volumes for a limited number of mudrocks and clays (Fink et al., Reference Fink, Krooss, Gensterblum and Amann-Hildenbrand2017; Hu et al., Reference Hu, Gaus, Seemann, Zhang, Littke and Fink2021; Lahn et al., Reference Lahn, Bertier, Seemann and Stanjek2020; Ziemiański et al., Reference Ziemiański, Derkowski, Szczurowski and Kozieł2020).

Ar-based DA micropore volumes are shown as a function of N₂-based DA micropore volumes (Fig. 10a). Note that the fitting ranges of the DA equation differed significantly for the two adsorptives. While N₂ isotherms could be fitted to p/p ⁰ < 0.05, the smoother and more rounded Ar isotherms allowed data fitting up to p/p ⁰ ~ 0.2. Exemplary fits of the DA equation are shown in the section ‘Micropore size distributions’ (Fig. 13b). It has been noticed that Ar physisorption isotherms measured down to ultra-low pressures exhibit two linear regions (range 1: ~10^–7< p/p⁰ <0.001; range 2: 0.001 < p/p⁰ < 0.1) separated by a step that was not considered as a textural feature (see the section ‘Primary Experimental Results’). The artificial step was not considered in the numerical fit, but fits converged with experimental data at lower partial pressure. However, the step affected, in particular, the exponent of the DA theory biasing fits when included in the fitting routine.

Figure 10. (a) Cross-plot of V ₀ obtained from N₂ and Ar physisorption isotherms measured at normal boiling point temperatures. Error bars represent two standard deviations and were obtained from interpolation of repeated measurements of representative mudrocks and the SiAl pellets; (b) cross-plot of V ₀ obtained from CO₂ and Ar physisorption isotherms measured at normal boiling point temperature (Ar) and 273 K (CO₂), respectively. No error bars are assigned because data for CO₂ isotherms are missing. The dashed line indicates unity.

DA micropore volumes varied between 3.9 μL g⁻¹ and 94.7 μL g⁻¹ for argon and from 2.5 μL g⁻¹ to 96.3 μL g⁻¹ for nitrogen (Appendix Tables 6 and 7). Ar-based values were, on average, 4% larger than those of N₂ except for the SiAl pellets, for which a 2% larger N₂-based micropore volume was calculated (intercept statistically equivalent to zero). While a linear trend showed on absolute scale, ratios (μVol_Ar/μVol_N2) scattered non-systematically. Deviation from unity was sample-specific but independent of the kind of sorbent. The largest differences were observed for the Bossier mudrocks (BM(a), BM(b)), the Callovo Oxfordian mudrock (Cox), and the Haynesville mudrock (HM); BM(a) showing the largest deviation with 36%.

DA micropore volumes of Ar were also compared to CO₂-based DA micropore volumes (Fig. 10b). CO₂-based DA micropore volumes ranged between 3.9 μL g⁻¹ and 69.1 μL g⁻¹ (Table 8). The maximum fitting range for CO₂ isotherms was up to p/p⁰ ≤ 0.036. V ₀ values were mostly smaller than those obtained from Ar isotherms, yet exceptions were identified for Bossier(a) (BM(a)), Longmaxi(a) (LM(a)), Opalinus Clay (OCM), Tournemire (TM), YPC-Aalbeke (YPC-A), and the clay fraction of the YPC-Elverdinge (YPC-Cf-E). The micropore volumes of LM(a) and the YPC-A deviated by >35%. Notably, the micropore volumes of the Cu-Trien exchanged clays were considerably larger for argon than for carbon dioxides. The intercalation of Cu-Trien leads to a widening of the interlayer distance to 1.3–1.35 nm for smectites, probably making the near-edge interlayer space accessible to nitrogen according to Kaufhold et al. (Reference Kaufhold, Dohrmann, Klinkenberg, Siegesmund and Ufer2010, Reference Kaufhold, Dohrmann, Ufer, Kleeberg and Stanjek2011) and therefore to argon; i.e. edge pores created by the stacking disorder of crystallites (irregular stacking and overlap of crystallites) increase accordingly. Assuming that the intercalation of Cu-Trien resulted in a major contribution of these edge pores to total microporosity may explain the large imbalance between V₀(Ar) and V₀(CO₂), as CO₂ access is restricted to a maximum pore size of 1 nm in a conventional low-pressure instrument (Thommes et al., Reference Thommes, Kaneko, Neimark, Olivier, Rodriguez-Reinoso, Rouquerol and Sing2015). The third ‘outlier’ (YPC-A) showed a significantly larger V ₀ of CO₂. The YPC-A possesses the largest CEC of the mudrock set. According to Michot (Reference Michot2018), high-energy sites will disturb micropore analysis as they appear as apparent micropores, masking the “true” microporosity. Considering exchangeable cations as high-energy sites (relative) and the strong quadrupole moment of CO₂ may qualitatively explain this phenomenon. Generally, the data scattered more strongly than those of Ar and N₂ and the average difference amounted to 30%. However, deviation from unity revealed no systematic trend and was again sample-specific and independent of the sorbent class.

In fact, if surface chemistry was a key factor in micropore volume control of the presented samples, one would expect a clustering of the sorbents according to material type. Clustering was not observed suggesting that material-related surface chemical effects are either weak and undetectable or overprinted. The observed underestimation of N₂ and CO₂-based micropore volumes could further imply that N₂ and CO₂ either (1) do not penetrate similar pore space as Ar, or (2) the access is kinetically constrained at operational temperature.

With regard to point 1, it is noted that the difference in accessibility of N₂, CO₂, and Ar may be associated with solid composition (see the section ‘Compositional control on differences in Ar, N₂, and CO₂-based micropore volumes’) or difference in sample preparation. While ultra-high vacuum (p < 13 mPa) was used in the case of Ar, samples for N₂ and CO₂ measurements were outgassed at p < 5 Pa. According to Cychosz et al. (Reference Cychosz, Guillet-Nicolas, García-Martínez and Thommes2017) a pressure level of ~5 Pa is not sufficient to fully evacuate the entire micropore network. In particular, micropores have a high water-retention capacity and may only be outgassed at elevated temperatures, above 378 K, and at very low pressures (Ziemiański et al., Reference Ziemiański, Derkowski, Szczurowski and Kozieł2020). In turn, compartments of the pore network may be blocked by the remaining water and left undetectable by N₂ and CO₂ (Seemann et al., Reference Seemann, Bertier, Krooss and Stanjek2017). As a consequence, micropore volumes would be underestimated by N₂ and CO₂ relative to Ar.

Concerning point 2, note that N₂ (at 77 K) suffers stronger diffusional constraints than Ar in pores smaller than 0.7 nm (Cazorla-Amorós et al., Reference Cazorla-Amorós, Alcañiz-Monge, de la Casa-Lillo and Linares-Solano1998). This constraint relates to the required thermal energy, necessary to diffuse in narrow micropores which is mounted by Ar but not N₂ at their operational temperature. The phenomenon was demonstrated by Reichenbach and Klank (Reference Reichenbach and Klank2014) who investigated N₂ uptake of a zeolite (pore width (PW) = 0.4 nm) as function of temperature revealing pseudo-equilibria at 77 K and significant uptake when the diffusion barrier was broken. Supposing that a non-negligible amount of pore volume is either built or connected via such small constrictions may explain the observed difference for nitrogen.

CO₂ is not kinetically restrained but limited by the accessible pore size of 1 nm. However, considering the milder outgassing conditions for N₂ and CO₂ makes it more likely that the materials were not sufficiently outgassed and water resides in small micropores. In addition, the possibility of helium entrapment during the free space measurement can be an issue for samples containing significant amounts of pores inaccessible to nitrogen or argon (Thommes, Reference Thommes, Čejka, van Bekkum, Corma and Schüth2007; Thommes et al., Reference Thommes, Kaneko, Neimark, Olivier, Rodriguez-Reinoso, Rouquerol and Sing2015). Helium-entrapment influences the isotherm shape in the ultra-low pressure region, and thereby disturbs an unambiguous quantification of the microporosity (Silvestre-Albero et al., Reference Silvestre-Albero, Silvestre-Albero, Llewellyn and Rodríguez-Reinoso2013). Besides, systematic errors in dead-volume cause cumulative dosing errors, becoming important at high p/p ⁰ (Thommes et al., Reference Thommes, Kaneko, Neimark, Olivier, Rodriguez-Reinoso, Rouquerol and Sing2015).

Specific surface area

The specific surface areas of all samples were calculated from Ar and N₂ physisorption isotherms using BET theory. Fitting ranges were constrained by Rouquerol plots (upper p/p ⁰ limit) and goodness of fit (lower p/p ⁰ limit). Monolayer capacities were checked for consistency and molecular cross-sectional areas of nitrogen (0.162 nm²) and argon (0.142 nm²) were used according to ISO9277 (Deutsche Institut für Normung e.V., 2012) (Table 3). BET areas and monolayer capacities computed for micro/mesoporous materials are conceived in the framework of Rouquerol et al. (Reference Rouquerol, Llewellyn and Rouquerol2007), thus representing apparent specific surface areas and strong retention capacities. As remarked by Rouquerol et al. (Reference Rouquerol, Llewellyn and Rouquerol2007), the evaluation of the specific surface area is not straightforward and needs to be performed cautiously, if (1) micropores are present, or (2) capillary condensation occurs in the BET-fitting range (Schlumberger and Thommes, Reference Schlumberger and Thommes2021). The latter was not the case for either N₂ or Ar in the studied sample set because of delayed capillary condensation (Fig. 1). Micropores were present in all studied porous samples contributing to different degrees to total uptake (Rouquerol et al., Reference Rouquerol, Llewellyn and Rouquerol2007).

Figure 11a shows a cross-plot of N₂ vs Ar BET areas including the unity line and error bars (2σ), respectively. BET areas ranged from 0.50 m² g⁻¹ to 219 m² g⁻¹ for nitrogen and 0.50 m² g⁻¹ to 189 m² g⁻¹ for argon (appendix Tables 6 and 7). N₂-based BET areas were systematically larger than or equal to Ar-based BET areas, yet (relative) differences were non-systematic and sample-dependent. The average difference amounted to ~8% with an intercept statistically equal to zero. The largest relative difference of 24% was observed for the non-porous silica (FB105FD).

Figure 11. (a) Cross-plots of A _BET calculated from N₂ and Ar physisorption isotherms measured at normal boiling point temperatures, respectively. Error bars represent two standard deviations and were obtained from interpolation of repetition measurements of representative mudrocks and the SiAl pellets; (b) BET areas of this data set combined with literature data (n _tot = 70)(Klank, Reference Klank2021; Payne et al., Reference Payne, Sing and Turk1973; Sotomayor et al., Reference Sotomayor, Cychosz and Thommes2018; Thommes et al., Reference Thommes, Köhn and Fröba2002a, Reference Thommes, Köhn and Fröba2002b). Dashed lines indicate unity.

The data set presented was augmented by literature values (sorbents: carbons, MCM-41, Al₂O₃, kaolinite, silicas, metal-organic frameworks, controlled pore glasses etc.) (Fig. 11b). Both data sets are in good agreement and while the divergence from unity increased on an absolute scale, relative deviations from unity remained non-systematic, though average differences rose to ~12% (intercept statistically equivalent to zero). Note that only BET areas were compared and no further information from the literature data were scrutinized. Cluster analysis results were ambiguous and yielded neither a class- nor cross-class dependence for the calculated BET areas and their variance. Thus, divergences seem not to be substrate-specific as would be implied by a dominantly surface chemistry-controlled process.

According to Sotomayor et al. (Reference Sotomayor, Cychosz and Thommes2018) orientation effects promoted by specific surface sites cause the largest uncertainty factor (~20%; Thommes et al., Reference Thommes, Kaneko, Neimark, Olivier, Rodriguez-Reinoso, Rouquerol and Sing2015) on the cross-sectional area of N₂. Thus, an unambiguous stipulation of the cross-sectional area of nitrogen is not straightforward. It is traditionally calculated from liquid configuration and amounts to 0.162 nm². Employing the traditional cross-sectional area, Sotomayor et al. (Reference Sotomayor, Cychosz and Thommes2018) and Schlumberger and Thommes (Reference Schlumberger and Thommes2021) documented deviations between Ar- and N₂-based A _BET of up to 25%. Similarly, Kruk and Jaroniec (Reference Kruk and Jaroniec2000) issued specific surface area differences of 29% for MCM-41. Klank (Reference Klank2021) published even larger deviations of up to 32% for a macroporous Al₂O₃ and 57% for a cellular concrete. The range is in line with the data set presented. Maximum differences of 24% (non-porous) and 17% (porous) were observed utilizing the traditional cross-sectional area of N₂.

Several researchers reported N₂ cross-sectional areas different from that of liquid configuration. Jelinek and Kovats (Reference Jelinek and Kovats1994) obtained an effective cross-sectional area of N₂ of 0.135 nm² (for fully hydroxylated, spherical silica). Using 0.135 nm² as the cross-sectional area, Sotomayor et al. (Reference Sotomayor, Cychosz and Thommes2018) computed a deviation for specific surface areas of oxides of <4% and for fully hydroxylated materials of <1%. Adapting the molecular cross-sectional area of Jelinek and Kovats (Reference Jelinek and Kovats1994) for this data set yielded nominally stronger deviations averaging 10% instead of the previous 8%. The approach of Sotomayor et al. (Reference Sotomayor, Cychosz and Thommes2018) may hold for oxides but it is unlikely to be suitable for mudrocks of locally varying and heterogeneous surface chemistry. It is more likely that orientation effects are not detectable in mudrocks and clays because of the narrow local distribution of specific sorption points such as surface groups or crystal defects. If these are spaced too close together, orientation effects will no longer be detectable as an effective change in molecular orientation is no longer possible in large numbers. Furthermore, the contribution of ‘disoriented’ molecules to the BET area could be either: (1) fairly low; or (2) simply masked by the overestimation of the monolayer capacity because of microporosity.

Eventually, the presented data show that A _BET values for Ar and N₂ can be regarded as equivalent at least for microporous mudrocks and clays and when used rigorously as an operational measure for texture, especially, because absolute differences in A _BET can be fairly small (Fig. 11). Still, it is acknowledged that for specific classes of sorbents (non-porous, clearly defined surface chemistry, and distribution of surface sites) a cross-sectional area different from 0.162 nm² is probable and orientation effects are observable (Jelinek and Kovats, Reference Jelinek and Kovats1994). Hence, the importance of individual sample- and class-dependent comparison remains and is further emphasized by the observed ‘random’ differences of this study.

Gurvich pore volume and porosity

Gurvich (total) pore volumes were computed for porous samples at a predefined BJH pore size of 250 nm (N₂: p/p⁰ = 0.9923; Ar: p/p⁰ = 0.9916) (Appendix Tables 6 and 7). Gurvich pore volumes were obtained assuming that capillary condensation has taken place and the adsorbate approaches its liquid bulk density at operational temperature. As the isotherms of the geological materials did not reach a plateau at maximum partial pressure, no recalculation to porosity was performed.

The maximum uptake of non-porous materials was intentionally not interpreted as Gurvich pore volume, because this uptake reflected more the kind of particle packing and less a textural property of the material itself due to condensation in the interparticle space. The resulting packing density is not sufficiently reproducible, and hence, quantitative estimates of the Gurvich pore volumes have little relevance.

Gurvich pore volumes of Ar are displayed as a function of N₂-based Gurvich pore volumes supplemented by a unity line (Fig. 12). The Gurvich pore volumes ranged between 16.8 μL g⁻¹ and 633.4 μL g⁻¹ for argon and 17.7 μL g⁻¹ to 608.0 μL g⁻¹ for nitrogen. For the majority of samples N₂-based Gurvich pore volumes were slightly larger than those obtained from argon physisorption isotherms. The average difference of 5% (porous materials only) was significantly exceeded by individual samples; i.e. deviations by up to 20% and 33% were detected for the YPC-E and BCM samples, respectively. The Gurvich pore volumes of the non-porous materials were included in the plot but not in the calculation of the average difference between Ar and N₂ Gurvich pore volumes. No systematic trend was evident when ratios were compared (V _GV(N₂)/V _GV(Ar)).

Figure 12. Cross-plot of V _Gv obtained from a pore size of 250 nm from N₂ and Ar physisorption isotherms measured at normal boiling point temperature, respectively. Error bars represent two standard deviations and were obtained from interpolation of repeated measurements of representative mudrocks and the SiAl pellets. The dashed line indicates unity. Gurvich pore volumes of the non-porous (np) silicas and Fe oxides are only shown for completeness but not considered in the analysis for reasons outlined in the section ‘Gurvich pore volume and porosity’.

According to Rouquerol et al. (Reference Rouquerol, Rouquerol, Sing, Llewellyn and Maurin2014) the validity of the Gurvich rule was confirmed for a wide range of adsorptives on mesoporous sorbents. The level of agreement is ~5% deviation from the mean value as calculated from the data reported by McKee (Reference McKee1959). Variations between pore volumes of N₂ and Ar have been documented by multiple researchers, albeit the discrepancies frequently exceed the level of 5% (Hartmann and Vinu, Reference Hartmann and Vinu2002; Kruk and Jaroniec, Reference Kruk and Jaroniec2000; Šolcová et al., Reference Śolcová, Matĕjová, Topka, Musilová and Schneider2011).

The origin of these discrepancies can be manifold ranging from technical inaccuracies to physical effects. Theoretically, molecular sieving and packing effects or a false (bulk) density assumption are possible in microporous materials. A correlation of Gurvich pore volume ratios and microporosity ratios could not be identified in this study suggesting that the origin of the difference in total pore volume does not originate from processes associated with the micropore space, e.g. molecular packing.

A violation of the rigidity assumption of the sorbents, as alleged by Rouquerol et al. (Reference Rouquerol, Rouquerol, Sing, Llewellyn and Maurin2014) for some materials, seems unlikely given the nature of the investigated sorptive–sorbent system. It is more likely that the assumption of a single bulk liquid density is too simplistic as it has been demonstrated to vary for supercritical fluids as a function of pore size (Schoen and Thommes, Reference Schoen and Thommes1995; Schoen et al., Reference Schoen, Thommes and Findenegg1997; Thommes and Findenegg, Reference Thommes and Findenegg1994). Moreover, when the uncertainty of constant p ⁰, hence stable temperature, is taken into account, it appears likely that in some instances the assumption of bulk liquid density of argon at its normal boiling point is questionable. Nonetheless, the agreement between Gurvich pore volumes of both adsorptives was good, making a discussion on the origin of the offset dispensable and speculative; except for the aforementioned mudrocks and clays of strongest deviation. The strongest discrepancies in Gurvich pore volumes are accompanied by similar differences in BJH PSD and are attributed to the saturation pressure offset (see the section ‘Factors of uncertainty and the importance of saturation pressure’).

Micropore size distributions

Micropore size distributions (μ-PSD) were calculated for Ar physisorption isotherms using DA theory in combination with Medek’s approach for computing differential pore-size distributions (Dubinin and Astakhov, Reference Dubinin and Astakhov1971; Janssen and Van Oorschot, Reference Janssen and Van Oorschot1989; Medek, Reference Medek1977) (Fig. 13a). Medek (Reference Medek1977) derived the differential micropore volume distribution from the theory of volume filling according to Equation 1:

(1) $$ \frac{d\unicode{x03B8}}{d{r}_{\mathrm{e}}}=\frac{dV}{d{V}_0d{\mathrm{r}}_{\mathrm{e}}}=3n{\left(\frac{\mathrm{k}}{n}\right)}^n{r}_{\mathrm{e}}^{-\left(3n+1\right)}\exp \left(-{\left(\frac{\mathrm{k}}{E}\right)}^n{r}_{\mathrm{e}}^{-3n}\right) $$

where θ is the fractional micropore volume filling obtained as a ratio of V and V ₀ being the micropore volume at a specific p/p ⁰ and the DA limiting pore volume. n and E are the DA exponent and the characteristic energy of adsorption obtained from least-squares fitting of the DA equation, respectively, and k is the interaction constant of the adsorptive (k _Ar = 2.34 kJ nm³ mol^–1). r _e is the equivalent pore radius obtained according to Equation 2:

(2) $$ {r}_{\mathrm{e}}=\frac{{\left(\frac{\mathrm{k}}{E}\right)}^{\frac{1}{3}}}{\varGamma \left(\frac{3n+1}{3n}\right)} $$

Figure 13. (a) Differential micropore size distributions (μ-PSD) of mudrocks computed using DA theory from Ar physisorption isotherms. G(I) to G(III) represent the groupings of mudrocks according to their primary pore size; (b) Ar DA plots of selected mudrocks. Data points represented by ‘x’ are not used for numerical fitting due to an instrumental step as explained earlier; (c) Ar α_s plots of selected mudrocks using the reduced isotherm published by Kruk and Jaroniec (Reference Kruk and Jaroniec2000). α_s plots display two linear ranges indicated by black and green regression lines.

Here, $ \Gamma $ is the gamma function that can be approximated by a polynomial approximation. Janssen and Van Oorschot (Reference Janssen and Van Oorschot1989) proved that Ar physisorption in combination with DA theory yields physically realistic micropore size distributions for (mono modal) zeolites. Unfortunately, micropore size distributions of mudrocks are seldom reported in the literature because of difficulties in their experimental assessment (e.g. Wang and Ju, Reference Wang and Ju2015). Experimental issues are associated with technical difficulties in performing accurate, ultra-low pressure measurements that are essential for micropore analysis but are limited by the device (Schlumberger and Thommes, Reference Schlumberger and Thommes2021). Moreover, reliable models are not available for all materials of interest that accurately describe fluid–solid interaction under significant confinement.

A comparison of Ar, N₂, and CO₂-based μ-PSDs was omitted because of the polar momentum of N₂ and CO₂ and their incomplete low-pressure isotherms (no data p/p⁰ < 0.001 for N₂ and no data p/p⁰ < 0.00003 for CO₂). In contrast to modern DFT models, DA theory does not account for specific fluid–wall interactions, though it was developed for chemically heterogeneous surfaces (Kruk et al., Reference Kruk, Jaroniec and Choma1997). Therefore, the pore-size distribution is negatively affected in the form of a shift in filling pressure for polar molecules impeding a direct correlation of filling pressure and pore size for N₂ and CO₂ (Schlumberger and Thommes, Reference Schlumberger and Thommes2021; Thommes et al., Reference Thommes, Kaneko, Neimark, Olivier, Rodriguez-Reinoso, Rouquerol and Sing2015).

The DA equation described the micropore filling of Ar in mudrocks and clays well up to the limiting micropore filling pressure of p/p⁰ ≈ 0.2. Yet, it failed to assess particular pore-size populations. The reason is that instead of representing local adsorption, it describes average adsorption, i.e. the isotherm is generated by a broad micropore size population producing a continuous isotherm shape. Hence, the model extracts the average micropore size and the corresponding adsorption energy represented by the monomodal distribution (Dobruskin, Reference Dobruskin1998). Physically, the DA-based micropore size distribution is considered as an energy distribution by Dobruskin (Reference Dobruskin1998). Further, the theory underestimates the true micropore size because of the assumption of bulk liquid density. In fact, a liquid configuration is not realized in micropores because of molecular packing restrictions (Cychosz et al., Reference Cychosz, Guillet-Nicolas, García-Martínez and Thommes2017). In turn, the μ-PSDs presented should only be considered for a relative comparison in order to discern textural trends instead of the true physical texture.

Based on the μ-PSDs, three groups of mudrocks were identified of which group I was characterized by broad micropore size distributions and slightly lower primary pore sizes (1.62–1.65 nm). In contrast, group II revealed a narrower distribution with nominally larger primary pore sizes of 1.73–1.74 nm. Group III comprised only two mudrocks of major pore sizes between 1.86–1.9 nm. They were described by the largest primary pore size with a narrow distribution.

The width of the distributions is controlled by the exponent (n) in the DA equation and it increases inversely to n. Following Dobruskin (Reference Dobruskin1998), n is a measure of surface heterogeneity: the smaller n the greater the surface heterogeneity. Thus, a larger degree of heterogeneity of micropore hosting components may be attributed to the mudrocks of group I. This could be interpreted as a result of a mixed contribution of mineralogical and organic constituents to the micropore size distribution. Indeed, mudrocks of group I are defined as gas shales implying that they are thermally mature (Hu et al., Reference Hu, Gaus, Seemann, Zhang, Littke and Fink2021; Merkel et al., Reference Merkel, Fink and Littke2015; Seemann et al., Reference Seemann, Bertier, Krooss and Stanjek2017). It was emphasized by Mastalerz et al. (Reference Mastalerz, Schimmelmann, Drobniak and Chen2013), Chen and Xiao (Reference Chen and Xiao2014) and Furmann et al. (Reference Furmann, Mastalerz, Bish, Schimmelmann and Pedersen2016) that thermal maturation of organic matter can cause overprinting of initial organic matter porosity and the creation of secondary porosity in the micropore range. Accordingly, a mixed contribution of clay- and organic matter-hosted microporosity to the pore-size distribution is probable as clay minerals (especially 2:1 clay minerals) are the main contributors to microporosity in mudrocks.

Mesopore-size distribution of porous materials

Meso and (partial) macropore size distributions (PSD) of porous materials were assessed by means of BJH theory. The statistical thickness of the adsorbate layer (t-curve) prior to capillary condensation was modeled by Harkins and Jura’s general isotherm equation ( $ t=\sqrt{\Big(}13.99/\left(0.034-\mathit{\log}\left(p/{p}^0\right)\right)\Big) $ obtained on a non-porous Al₂O₃) for nitrogen (Harkins and Jura, Reference Harkins and Jura1944). For the PSD inversion of argon physisorption isotherms the t-curve was obtained from the isotherm of a macroporous LiChrosphere-Si1000 silica provided in form of numerical values between 1.29×10⁻⁵ < p/p⁰ < 0.993 by Kruk and Jaroniec (Reference Kruk and Jaroniec2000).

PSDs were calculated from the adsorption and desorption branches of the Ar and N₂ physisorption isotherms. A lower inversion limit of p/p⁰ = 0.3 p/p ⁰ was applied for the adsorption branch. The hysteresis loop closure was excluded from PSD inversion of the desorption branch; i.e. PSDs were derived for p/p ⁰≥ 0.45 and 0.40 for N₂ and Ar, respectively. It is generally accepted that pore-size distribution models represent simplifications with respect to geometry and that they relate an ‘equivalent’ pore width to sorptive uptake. In turn, PSDs are model-dependent; hence they are apparent rather than physically true (Sonwane and Bhatia, Reference Sonwane and Bhatia2000). Kruk and Jaroniec (Reference Kruk and Jaroniec2000) as well as Thommes (Reference Thommes2004) showed that uncalibrated BJH theory underestimates pore size by up to 30% for pores which are <7.5–10 nm in size. Furthermore, BJH does not take into account the correct thermodynamic state of the pore fluid, its dependence on pore size, the occurrence of hysteresis and the delay in capillary condensation as a result of metastable adsorption films. Despite its apparent character, its shortcomings, and the lack of suitable DFT model, the PSDs presented were computed by BJH theory and used on a comparative basis (see the section ‘Quantitative Evaluation of Isotherms and Discussion’).

Cumulative and differential PSDs are presented for representative mudrocks (Bossier(a), Opalinus Clay), clay minerals (kaolinite, montmorillonite) and engineered materials (vycor cpg-20nm, SiAl pellets) (Figs 14–16). These exemplary samples were chosen based on their prominent features in PSD curves and the representative character of the data set.

Figure 14. Meso- and macro-pore-size distributions calculated from the adsorption (a, c) and desorption (b, d) branches of Ar and N₂ isotherms using BJH-theory in combination with a thickness curve from Kruk and Jaroniec, Reference Kruk and Jaroniec2000: (a) PSD obtained for the Bossier (a) mudrock (BM(a)) from the adsorption branch; (b) PSD obtained for the Bossier (a) mudrock from the desorption branch; (c) PSD obtained for the Opalinus Clay (OCM) from the adsorption branch; (d) PSD obtained for the Opalinus Clay from the desorption branch. The differential pore-size distributions are shown as insets in the respective parts.

Figure 15. As in Fig. 14, but for: (a) PSD obtained for the kaolinite from the adsorption branch; (b) PSD obtained for the kaolinite from the desorption branch; (c) PSD obtained for the montmorillonite from the adsorption branch; (d) PSD obtained for the montmorillonite from the desorption branch.

Figure 16. As in Fig. 14, but for: (a) PSD obtained for the vycor cpg-20nm from the adsorption branch; (b) PSD obtained for vycor cpg-20nm from the desorption branch; (c) PSD obtained for SiAl pellets from the adsorption branch; (d) PSD obtained for the SiAl pellets from the desorption branch.

Cumulative and differential PSDs of mudrocks showed fair agreement with local deviations detectable for adsorption and desorption branch-based PSDs (Fig. 14). Cumulative PSD curves for Bossier(a) and Opalinus Clay complied qualitatively and local quantitative differences in uptake were at most 17% (BM(a)) and 10% (OCM); i.e. cumulative PSD curves of Ar and N₂ isotherms converged toward larger pore sizes. Differential PSDs were isotherm branch-specific for both mudrocks. Adsorption branch-based PSDs were broad with minor detail for N₂-based PSDs; i.e. quantitative differences were present and peak positions of Ar-based PSDs were shifted toward smaller pore sizes. Differential PSDs calculated from the desorption branch agreed well with each other for BM(a) and OCM. BM(a) displayed no major peak, though differential PSD curves suggest that Ar and N₂-based PSDs converged toward smaller mesopores (~ 5 nm). The major pore size of the OCM revealed a shift from 12.7 nm (Ar) to 13.1 nm (N₂) corresponding to ~3% of Ar-based PSDs toward smaller values (Fig. 14d). Larger deviations were observed for the Cox mudrock showing a deviation of 7.6 nm (=19.5%) from 39 nm (Ar) to 31.4 nm (N₂) for its primary pore size in adsorption mode (data not shown). Furthermore, cumulative PSDs (adsorption and desorption branches) of the BCM, YPC-A, YPC-E and the Cu-Trien exchanged clay fraction of the YPC-Elverdinge (YPC-Cf-CuTrien) displayed continuous and sample-specific divergence for larger pore sizes. The above was demonstrated for the Boom Clay (Fig. 8c). PSDs of the remaining samples coincided well for both isotherm branches and no systematic deviations are detected.

Figure 15 displays the PSDs of kaolinite and montmorillonite (SWy2-Na). Cumulative and differential PSDs were qualitatively similar, yet they differed quantitatively and in resolution. The Ar physisorption isotherm of the kaolinite has been recorded with much higher point spacing (~2 fold). Ar-based cumulative PSDs of kaolinite were lower than N₂-based PSDs above 14 nm. Differential PSDs computed for the adsorption branch were broad and peaked at ~37 nm but showed a fair match up to 50 nm though Ar-based data scattered between 30 nm and 50 nm. Differential PSDs of the desorption branch aligned excellently yielding a difference of 1 nm (~3%) in major pore size (PW(Ar) = 30.3 nm; PW(N₂) = 31.3 nm). Moreover, the differential PSD revealed significantly more detail (shoulder and sub-peaks) for argon compared to the ‘smeared’ N₂-based PSD (Fig. 15b). Cumulative PSDs of montmorillonite overlapped in the mesopore range, but diverged continuously toward larger pore sizes (PW > 50 nm) for the desorption branch (Fig. 15d). The difference in cumulative uptake maximized to 12 μL g⁻¹ (~18% of Ar PSD) at the measured pore-size threshold of ~230 nm. In contrast, adsorption branch-based cumulative PSDs converged again above 100 nm. Corresponding differential PSDs were unspecific and defined by a broad peak in the macropore range. Quantitative differences were obvious for pores above 15 nm. The differential PSD of the desorption branch resembled the broad peak of the adsorption branch.

Engineered materials indicated sample-specific differences in cumulative and differential PSDs (Fig. 16). While the PSDs were qualitatively similar, quantitative disparities were noticed for both types of samples and PSDs. Ar-based cumulative PSDs of the vycor cpg-20nm were smaller than the corresponding N₂-based PSDs for adsorption (PW $ \ge $ 10 nm) and desorption (PW $ \ge $ 14 nm). The maximal divergence in uptake was 30% and corresponded to a pore size of ~22 nm for both isotherm branches. Differential PSDs exhibited a larger deviation for adsorption- than for desorption-based PSDs (Fig. 16a,b). The first population of pores (PW(Ar) = 15.8 nm; PW(N₂) = 12.8 nm) shifted by 3 nm (~24% of Ar PSD) toward larger pore sizes for argon. For the second population, a repositioning by 8.1 nm (~26% of Ar PSD) was detected for the adsorption branch (PW(Ar) = 31.1 nm; PW(N₂) = 23.0 nm). In comparison, the deviations for the desorption-based differential PSDs were ~0.2 nm to 0.7 nm which corresponded to a consistent relative shift of <3% (population I : PW(Ar) = 9.5 nm vs. PW(N₂) = 9.7 nm; population II : PW(Ar) = 23.5 nm vs. PW(N₂) = 22.8 nm). Similarly, PSDs of SiAl pellets demonstrated better agreement for the desorption branch than for the adsorption branch. Generally, Ar-based cumulative PSDs exceeded N₂-based PSDs. This difference in uptake amounted to constant 5%. Differential PSDs obtained from the adsorption branch deviated by 4 nm (~32%) in peak position (PW(Ar) = 8.5 nm; PW(N₂) = 12.5 nm). In contrast, an inverse difference of 1.2 nm (~13% of Ar PSD) relocation from 9.5 nm(Ar) to 8.3 nm(N₂) was identified for desorption branch-based PSDs.

Few data are available for mudrocks with respect to PSDs computed from Ar physisorption isotherms. The PSDs of Zhang et al. (Reference Zhang, Xiong, Li, Wei, Jiang, Lei and Wu2017), Chen et al. (Reference Chen, Liu, Ding, Zheng and Lu2022) and Delle Piane et al. (Reference Delle Piane, Ansari, Li, Mata, Rickard, Pini, Dewhurst and Sherwood2022) (no N₂ data presented) were obtained from non-local density functional theory for Ar isotherms only and revealed no prominent pore size. Still, the corresponding isotherm data follow the same trend as described in the section ‘Argon, nitrogen, and carbon dioxide physisorption isotherms’. As pointed out, fair agreement existed for the majority of samples, but (1) shifts in BJH diameter (maxima of differential PSD curves) and (2) differences in cumulative uptake were discovered.

The shift in the BJH diameter has been reported in the literature for engineered materials (Choma et al., Reference Choma, Górka and Jaroniec2008; Neimark et al., Reference Neimark, Ravikovitch, Grün, Schüth and Unger1998; Rathouský and Thommes, Reference Rathouský and Thommes2007; Šolcová et al., Reference Śolcová, Matĕjová, Topka, Musilová and Schneider2011; Thommes et al., Reference Thommes, Smarsly, Groenewolt, Ravikovitch and Neimark2006, Reference Thommes, Morell, Cychosz and Fröba2013). Similar to the data presented, the magnitude of shift and direction expose no systematics. However, the shifts found in the literature are generally small and, at most, 0.7 nm (Thommes et al., Reference Thommes, Smarsly, Groenewolt, Ravikovitch and Neimark2006). The shifts presented (adsorption branch) are up to 7.6 nm (19.5%) for mudrocks and 4 nm (32%) for engineered materials and increase with pore size (Fig. 14, Fig. 16). Such systematics are not found in the reported literature but the lower shift obtained from published work is attributed to the fact that the physisorption measurements were probably performed in a liquid argon bath with separate p ⁰ measurements making the measurements considerably more reliable (see the section ‘Factors of uncertainty and the importance of saturation pressure’). The shifts in (primary) peak position are significantly lower for the desorption branch-based PSD than for the adsorption branch-based PSD. This is highlighted by samples characterized by pore blocking an supported by the most frequent pore width of the OCM (Figs. 14d, 15, 16). In fact, the agreement in peak position is at the level of 3% for Ar and N₂-based PSD for the desorption branch. The difference in peak position is equivalent to a maximum of 1 nm but often lower and close to the limit of detection (LOD) when the molecular size of the adsorptive is considered as the LOD. This provides good confidence in Ar and N₂ PSDs obtained from the desorption branch. The larger variation of adsorption branch PSDs could be a result of the inaccurate representation of the adsorption process by BJH theory and therefore related to the neglect of the fluid-specific delay in capillary condensation. Because these processes affect the inverted pore sizes (+ associated volumes), inaccurate accounting of them may cause such variations in PSDs. But, it is emphasized that large shifts, especially prominent in larger pore size, can be induced by technical issues associated with p ⁰ fluctuations (see the section ‘Factors of uncertainty and the importance of saturation pressure’). Minor local differences are probably associated with incorrect assumptions of the thermodynamic state of the pore fluid (bulk parameters) and its neglect of pore-size dependence (e.g. curvature effects and pore-size dependence of surface tension, e.g. Feng et al., Reference Feng, Wu, Bakhshian, Hosseini, Li and Li2020). Furthermore, the selection of the statistical thickness curve and uncontrolled experimental variations can cause such small variations. In the present study, the statistical thickness curve of Kruk and Jaroniec (Reference Kruk and Jaroniec2000) was chosen for PSD inversion of Ar physisorption isotherms.

This yielded good results for the present study, but systematic testing of thickness models (see appendix Fig. 18) revealed that the model choice makes a difference at the lower percent level (8.54 ± 0.08 nm; CV ≈ 1%) when obtaining a pore-size distribution. This is supported by testing different t-curves on a large mudrock data set, supposing that BJH theory is used in its valid partial pressure range (Bertier et al., Reference Bertier, Schweinar, Stanjek, Ghanizadeh, Clarkson, Busch, Kampman, Prinz, Amann-Hildebrand, Krooss and Pipich2016). However, it is more important that the thickness model resembles as closely as possible the solid-sorptive system of interest in terms of solid–fluid interaction. For the materials studied, oxides (Al₂O₃ and SiO₂) were regarded as the best available proxy at the moment of study.

In addition, Kuila and Prasad (Reference Kuila and Prasad2013) demonstrated that BJH partial pore volumes can vary between 10 and 20% which the authors associated with uncontrolled experimental conditions (not better defined). A systematic reproducibility study (25 laboratories) of Klobes et al. (Reference Klobes, Meyer and Munro2006) corroborates the above and the authors quantified a variation of 6.4% for pore volumes obtained from BJH theory.

Thus, the level of agreement complies with the reported results of this study implying that Ar and N₂ PSDs (mesopore range) can be regarded as equivalent under (ideal) experimental conditions. Putting the local differences in Ar and N₂ PSDs into perspective of experimental/technical uncertainties and the purpose of PSD analysis conveys that such differences should be recognized and evaluated critically. This is because of the PSDs’ sensitivity toward external factors (technical issues) unrelated to the sorbent texture, which have a more significant impact on PSD inversion compared to theory-based issues (see the section ‘Factors of uncertainty and the importance of saturation pressure’). However, it is again emphasized that the agreement for desorption data is substantially better than for adsorption data.

Compositional control on differences in Ar, N₂, and CO₂-based micropore volumes

A systematic trend of micropore volume toward compositional parameters was not identified (Fig. 17). Bivariate plots of V ₀ of Ar, N₂, and CO₂ versus TOC, total- and specific clay content (taken from Table 1) indicated no obvious relationship. The cation exchange capacity (CEC) displayed a weak linear trend (Fig. 17b). The CEC trend was equivalent for all three adsorptives indicating no preferential attraction of the exchangeable cations, despite the polarity of N₂ and CO₂. However, Środoń and McCarty (Reference Środoń and McCarty2008) and Kuligiewicz and Derkowski (Reference Kuligiewicz and Derkowski2017) indicated that, at the given drying temperature, exchangeable cations – depending on their enthalpy of hydration – still carry a hydration shell, questioning a preferential interaction of any of the adsorptives with these cations. In addition, the CEC trend is probably a pseudo-trend originating from the smectites of the mudrocks. Smectite is a major donor of exchangeable cations, besides organic matter, but it is also a main contributor to microporosity (Środoń and McCarty, Reference Środoń and McCarty2008; Woodruff and Revil, Reference Woodruff and Revil2011). Consequently, the CEC vs. V ₀ trend may reflect the influence of smectite on microporosity and not CEC.

Figure 17. DA micropore volumes calculated for Ar, N₂, and CO₂ physisorption isotherms and percentage difference of V ₀ (Ar and N₂) plotted as function of (a) total organic carbon content (TOC), (b) cation exchange capacity (CEC), (c) total clay content, and (d) mixed-layer clay minerals including illite, smectite, and muscovite (here: I-S).

The percentage difference of the contribution of the micropore volume to the Gurvich total pore volume ((V ₀(Ar)-V ₀(N₂))/V _GV) did not disclose any systematics or correlation either (Fig. 17). This may imply that none of Ar, N₂, or CO₂ has a considerably better accessibility toward micropores hosted by any of these components. Otherwise, one would expect systematic differences in the micropore volumes and their contribution to the Gurvich total pore volume for a specific adsorptive for either TOC-rich mudrocks like the Longmaxi mudrocks (LM(a), LM(b)) or mudrocks rich in (expandable) clays like the Ypresian Clay samples (YPC-A, YPC-E, YPC-K) and especially pure swelling clays (SWy2). Differences in V ₀ were, however, inconclusive because they varied between 4 and 6% for the TOC-rich Longmaxi while the clay-rich Ypresian Clay samples differ by 7–16% in microporosity (Ar, N₂).

The above presumes that one of these components mainly hosts adsorptive accessible micropores and that micropores are not filled with tightly bound water. Furmann et al. (Reference Furmann, Mastalerz, Bish, Schimmelmann and Pedersen2016) described a dominant control of TOC and thermal maturity on micropore volume (CO₂). Thereby, the authors noted inconsistent trends (positive and negative) that indicate strong sample specificity. This is corroborated by a systematic study of Kuila et al. (Reference Kuila, McCarty, Derkowski, Fischer, Topór and Prasad2014) in which the correlation of TOC vs. V ₀ depends on the individual sample. Similarly, Ross and Bustin (Reference Ross and Bustin2009) observed no systematic variation of micropore volume (CO₂) as function of TOC for Jurassic shales, though it was observed for the Devonian-Mississipian shales (Muskwa and Besa River). Ross and Bustin (Reference Ross and Bustin2009) indicated that the main contributors to microporosity in mudrocks are organic matter as well as clay minerals. Neither total nor specific clay content (illite-smectite) disclosed any obvious control over micropore volumes determined from Ar, N₂, or CO₂ isotherms. This suggests that Ar does not have significantly easier access to clay-specific microporosity as it does to the interlayer space of dry, swelling clays which could provide additional micropore space unavailable to N₂ (kinetic restrictions) (Everett and Powl, Reference Everett and Powl1976; Loganathan et al., Reference Loganathan, Bowers, Yazaydin, Schaef, Loring, Kalinichev and Kirkpatrick2018; Ziemiański et al., Reference Ziemiański, Derkowski, Szczurowski and Kozieł2020). In turn, a dominant control by clay minerals on Ar physisorption is excluded (Fig. 17c–d).

Finally, the individual contribution of organic matter and clay minerals to microporosity varies with sample (e.g. formation) because of evolution- and component-dependent effects of, for example, type of organic matter, thermal maturity, particle- and crystallite size as well as internal- and assemblage architecture. Furthermore, as addressed above, the possibility of insufficient removal of water from the micropores prior to N₂ and CO₂ measurements remains and potentially disturbs an unbiased comparison.

Sorption mechanism

The complex pore network of mudrocks accommodates various interconnected pore classes indicated by a broad PSD and often no distinct step in adsorption. The desorption mechanism in the pore network of mudrocks has been a matter of debate as it has implications for the interpretation of pore-size distributions (Bertier et al., Reference Bertier, Schweinar, Stanjek, Ghanizadeh, Clarkson, Busch, Kampman, Prinz, Amann-Hildebrand, Krooss and Pipich2016). Classical differentiation in pore body (adsorption) and pore throat (desorption) distribution is only possible when equilibrium adsorption and evaporation is encountered. Still, this is seldom the case in geological materials.

Thommes et al. (Reference Thommes, Smarsly, Groenewolt, Ravikovitch and Neimark2006) demonstrated that the complementary use of (1) different adsorptives or (2) isotherm measurements at different temperatures proves effective in sorption mechanism identification. Adapting Thommes et al. (Reference Thommes, Smarsly, Groenewolt, Ravikovitch and Neimark2006)’s approach showed that all mudrocks and expandable clays studied (original + Cu-Trien exchanged) were characterized by a combination of pore network percolation and free evaporation at high partial pressure and cavitation. Cavitation was represented by a fixed closure point of the hysteresis loop that differed for N₂ and Ar (Fig. 1). The forced closure would translate into deviating PSDs which are not shown because of their artificial character (Bertier et al., Reference Bertier, Schweinar, Stanjek, Ghanizadeh, Clarkson, Busch, Kampman, Prinz, Amann-Hildebrand, Krooss and Pipich2016; Groen et al., Reference Groen, Peffer and Pérez-Ramírez2003). In the case of kaolinite, the vycor cpgs, and the SiAl pellets, no cavitation was detected. Instead identical hysteresis closure points of both adsorptives and approximately similar PSDs were diagnostic of pore blocking.

At ~4 nm, the cavitation pore size of Ar is slightly smaller than for nitrogen (PW(N₂) ≈ 5–7 nm) (Nguyen et al., Reference Nguyen, Do and Nicholson2011; Schlumberger and Thommes, Reference Schlumberger and Thommes2021). Note that the pore-size threshold for cavitation of N₂ varies slightly in the literature, but specification of a definite pore size would be speculative for mudrocks and clays in any case. Still, the data proved that desorption in mudrocks and expandable clays was unambiguously controlled by a combination of network percolation and free evaporation at high partial pressure and cavitation at the hysteresis closure point.

The level of control of an individual mechanism was sample dependent. Thus, the pore volume associated with the hysteresis loop closure varied with sample indicating the variable and individual level of control of pore necks in the pore network of mudrocks and clays. Such analysis conveys that a sample-specific but non-negligible part of the pore network of mudrocks is only accessible via small meso- or micropores of non-specifiable size. This can have a tremendous effect on, for example, matrix transport or storage capacity as such constrictions represent the bottle neck for any fluid passing. Furthermore, the above supports the notion of the apparent character of BJH PSDs because the theory does not account for cavitation. This implies that certain pore populations and their pore volumes are not quantitatively accounted for in BJH PSDs. Hence, pore volumes will be underestimated when obtained from the desorption branch of physisorption isotherms measured on mudrocks and clays (Figs 14, 15).