Measurement and interpretation of unary supercritical gas adsorption isotherms in micro-mesoporous solids

Consider a batch adsorber filled with solid adsorbent, which is exposed to an adsorptive gas at elevated pressures (\(T/T_{\rm c}>1\)). The total number of moles of the adsorbate, \(N_{\rm t}\), is given by the sum of the number of moles in the bulk fluid phase, \(N_{\rm b}\), and in the adsorbed phase, \(N_{\rm a}\):where,In Eq. 2, \(\rho_{\rm b}\) is the density of the bulk fluid phase, V and \(\phi\) refer to the adsorber volume and its total voidage, respectively; \(V_{\rm a}=N_{\rm a}/\rho_{\rm a}\) is the volume occupied by the adsorbed phase with average density, \(\rho_{\rm a}\). By reducing the free space available to the bulk phase, \(V_{\rm a}\) also reduces its mass by an amount proportional to \(\rho_{\rm b}\). However, the effect of this “correction” is difficult to estimate a priori, because for a supercritical gas \(\rho_{\rm a}\) is not well defined (see Fig. 1) and \(V_{\rm a}\) doesn’t necessarily correspond to the specific pore volume of the solid. Accordingly, adsorption calculations for supercritical gases are carried out by using the Gibbs adsorption excess [35], which can be obtained by recasting Eqs. 1 and 2:As shown by the left-hand side of Eq. 3, when \(\rho_{\rm b}\ll \rho_{\rm a}\approx \rho_{\rm L}\), \(N_{\rm a}\approx {N_{\rm ex}}\), as it is the case for adsorption at sub-critical conditions. However, for \(\rho_{\rm b}\le \rho_{\rm a}\), the two adsorption quantities may differ significantly and an arbitrary choice of \(\rho_{\rm a}\) may no longer have a negligible impact on the material balance calculation. The right-hand side of Eq. 3 shows that \(N_{\rm ex}\) depends solely on quantities that can (in principle) be accessed either experimentally or numerically, providing a means to describe the extent of adsorption of the given adsorbate without knowledge of \(\rho_{\rm a}\). As anticipated in Section 1, a major point of contention is how to determine the adsorbent pore volume (and hence \(\phi\)) accurately.

One can also ask the following question: given that the space which can be filled with gas is reduced by the presence of the solid, wouldn’t it be more effective to store gas in the vessel devoid of adsorbent? The answer to this question is found by considering another measurable quantity–the so-called net amount adsorbed [14], i.e.where \(V_{\rm s}=V(1-\phi )\) is the skeletal (impenetrable) volume of the adsorbent. Accordingly, when \(N_{\rm net}=0\), \(N_{\rm t} = \rho_{\rm b}V\), meaning that beyond this point the presence of the adsorbent diminishes the gas storage capacity of the container. When \(N_{\rm net}>0\), adsorption is beneficial to gas storage and the maximum in the net adsorption isotherm thus identifies the pressure of maximum adsorptive storage capacity compared to compression alone [23]. The left-hand side of Eq. 4 also indicates that net adsorption is defined without prior knowledge of the adsorbent pore volume (and hence \(V_{\rm s}\)). Therefore, this formulation provides a direct means to evaluate adsorptive storage capacity at high pressures without introducing potential errors associated with pore volume estimates. By the same token, the right-hand side of Eq. 4 indicates that the conversion from net to absolute adsorption only requires knowledge of the combined volume \(V_{\rm a} + V_{\rm s}\). For a microporous solid, the latter is the solid volume that includes the micropores [23] and can be estimated with excellent accuracy without probing the micropore space [3, 29]. It will be apparent from this study that for adsorbents containing mesopores, the conversion from net (or excess) to absolute adsorption is more problematic.

Textural characterisation of the three adsorbents was carried out by physisorption analysis in a Quantachrome Autosorb iQ using Ar at 87K in the pressure range 1\(\times 10^{-7}-0.1\) MPa to obtain estimates of surface area, pore volume and pore volume distribution (Table 1, details provided in Section S1). The measured Ar adsorption isotherms are plotted in Fig. 3. ZIF shows a larger total amount adsorbed (580 cm\(^3\)(STP)/g) as compared to MZ and MC (360–380 cm\(^3\)(STP)/g), indicating a larger total pore volume (\(\nu_{tot}\approx 0.7\) cm\(^3\)/g for ZIF vs. \(\nu_{tot}\approx 0.5\) cm\(^3\)/g for both MZ and MC). Based on the IUPAC classification of physisorption isotherms [42], ZIF shows Type I(a) behaviour–typical of microporous solids, albeit with a small hysteresis loop of Type H2(a), indicating the presence of pores wider than 4 nm. For MZ the isotherm is a composite of Type I(a) (\(P/P_{0}<0.6\)) and Type IV(a) with hysteresis of Type H1 (\(P/P_{0}>0.6\)), which is characteristic of dual-porosity adsorbents with both micro- and meso-pores. MC shows Type IV(a) behaviour with hysteresis of Type H1, which is characteristic of mesoporous adsorbents with a narrow range of uniform mesopores. This characteristic features are reflected in the obtained cumulative pore size distribution curves, which are shown as inset in each panel of Fig. 3. Micropores (\(d<2\) nm) contribute to about 70% and 38% of the total pore volume of ZIF and MZ, respectively, while they are absent in MC. For the latter, approximately 90% of the total pore volume is found in pores of widths \(4-20\) nm. Therefore, the relative amount of mesoporosity increases in the order \(\text{ZIF}<\text{MZ}<\text{MC}\). As given in Table 1, ZIF and MZ possess similar specific surface area (\(1400-1500\) m\(^2\)/g), which is much larger than the value obtained for MC (approx. 200 m\(^2\)/g).

A Rubotherm Magnetic Suspension Balance (MSB) was used to measure the high-pressure adsorption isotherms on the three adsorbents at \(T=308\) K in the pressure range \(0.02-21\) MPa. The detailed experimental procedure, including the analysis of experimental uncertainties, is described in Section S2, while key parameter values are reported in Table 2. Briefly, this instrument provides high-resolution (10 \(\mu\)g) weight measurements at two measuring positions, namely \(\text{MP}_1\) (the basket with the adsorbent) and \(\text{MP}_2\) (the combined measured weight of \(\text{MP}_1\) and a calibrated titanium sinker). These two readings are used to compute the amount of gas adsorbed and the gas bulk density at given pressure and temperature conditions [11]. To this end, three sets of experiments are carried out, which are described next: (i) CO\(_2\) gravimetry without adsorbent; (ii) Helium gravimetry with adsorbent; and (iii) CO\(_2\) gravimetry with adsorbent. For (i) and (ii), the probing gas is charged into the MSB initially evacuated to obtain weight measurements (\(\text{MP}_{1}\) and \(\text{MP}_{2}\)) at several pressure points. Under the assumption of negligible gas adsorption, the following two equations apply: where \(\mathrm{MP}_{1,0}\) and \(\text{MP}_{2,0}\) are the weight measurements under vacuum; \(V_{\rm sk}\) is the volume of the calibrated sinker and \(V_0\) is the volume of the lifted parts in measuring position 1. For experiment (i) \(V_0=V_{\rm met}\) and for experiment (ii) \(V_0=V_{\rm met} + V_{\rm s}\), where \(V_{\rm met}\) and \(V_{\rm s}\) are the volume of the lifted metal parts and the skeletal volume of the adsorbent, respectively. In both cases, the volume \(V_0\) is estimated from the weighted linear regression of \(\text{MP}_{1}\) plotted as a function of \(\rho_{\rm b}\). Measurements obtained from experiment (i) are presented in Fig. S1 and the value of \(V_{\rm met}\) is reported in Table 2 together with its uncertainty, \(u(V_{\rm met})\). Estimates of \(V_{\rm s}\) and \(u(V_{\rm s})\) obtained in the experiments with adsorbent and Helium as probing gas are also presented in Table 2 and will be discussed in Section 5.

The adsorption isotherms were measured by charging the adsorbate (CO\(_2\)) into the MSB initially evacuated to obtain weight measurements at several pressure points. The net mass adsorbed is computed as: while the following equation is used to compute the excess mass adsorbed: where the density of the bulk adsorptive \(\rho_{\rm b}\) at each pressure point is obtained from Eq. 5a, while \(\text{MP}_{1,0}\), \(V_{\rm met}\) and \(V_{\rm s}\) are known from experiments (i) and (ii). The experimental results of the adsorption experiments are reported in terms of the molar net (or excess) amount adsorbed per unit mass of adsorbent:where the subscript i refers to net or ex, \(M_{\rm m}\) is the molar mass of the adsorbate (for CO\(_2\), \(M_{\rm m}=44.01\) g/mol) and \(m_{\rm s}\) is the mass of adsorbent.

The experimental CO\(_2\) net adsorption data are plotted in Fig. 4 as a function of the molar bulk density, \(\rho\), for measurements carried out at \(T=308\) K on (a) ZIF, (b) MZ and (c) MC. The three isotherms show a similar behaviour: after the initial increase in \(n_{\rm net}\), the experimental data outline a maximum and fall off with a further increase in the bulk density. Yet, the maximum net amount adsorbed, the density at which \(n_{\rm net}=0\) and the shape of the descending part of the isotherm differ among the three adsorbents, reflecting the characteristic features of their pore volume distribution. In particular, the isotherms measured on both ZIF and MZ are characterised by a steep initial increase and a sharp maximum, which can be attributed to the presence of micropores. Their contribution to adsorption is also reflected in the location of the crossing point with the abscissa, \(\rho (n_{\rm net}=0)\), which decreases in the order ZIF (9 mol/L) > MZ (8 mol/L) > MC (6 mol/L). This trend indicates that an adsorbent with a large amount of microporosity features a wider pressure range, where gas adsorption is beneficial to gas storage. A similar decreasing trend is observed for the measured isotherm’s maximum, namely ZIF (6.7 mmol/g) > MZ (2.5 mmol/g) > MC (0.8 mmol/g), which directly relates to the material’s storage capacity. As anticipated previously, the location where \(n_{\rm net}=0\) is unaffected by the choice of the specific quantity used to report adsorption (moles per unit mass or per unit volume) and the observed trend has therefore general validity. However, the positive branch of the isotherm, including the maximum net amount adsorbed per unit volume, does depend on the bulk density of the material, which may differ among the three adsorbents considered here. Accordingly, any further calculation of the gas storage capacity, including the comparison of different adsorbents, should be carried out on a volume basis.

A dashed line is plotted in each panel of Fig. 4 that has been obtained upon fitting the adsorption data measured at large bulk density values (\(\rho >15\) mol/L). As a means for comparison, solid lines are also shown in each plot with a slope \(-V_{\rm s}/m_{\rm s}\), corresponding to the adsorbent’s specific skeletal volume. The larger value of the slope of the line fitted to the experimental data reflects the additional contribution of the adsorbed phase to buoyancy during the gravimetric adsorption experiment. In fact, from the definition of the net amount adsorbed, Eq. 6, the slope of the isotherm should approach the value \(-(V_{\rm s}+V_{\rm a})/m_{\rm s}\) once adsorption saturation is reached. For the microporous ZIF, the descending part of the isotherm follows by and large the trend outlined by the dashed line, indicating that this condition may have indeed been attained. On the contrary, the isotherms measured on MZ and MC approach the trend-line only at large density values. The observed divergence of the experimental data from linearity is an indication of the distinct pore-filling behaviour of the adsorbed phase in micro- and meso-pores at supercritical conditions. The evident change in the slope observed for MZ at approximately \(\rho =7.5\) mol/L may be traced back to the characteristic dual-porosity structure of this material. These observations highlight the practical difficulty of applying this graphical approach to the interpretation of supercritical adsorption data measured on materials with a significant amount of mesoporosity. We will further expand on this point when discussing the excess adsorption data, which reveal these effects more strongly.

The results of the Helium gravimetry experiments on the three adsorbents are presented in Fig. 5, where the measured weight is plotted as a function of the gas bulk density. The three materials outline three distinct straight lines with slope \(b=-(V_{\rm met}+V_{\rm s})/m_{\rm s}\), thereby reflecting a difference in their specific skeletal volume. The constant b was obtained by the method of weighted least squares, where the uncertainty in the measured weight, \(u(\text{MP}_1)\), was accounted for; the uncertainty in b was then obtained by error propagation in terms of \(u(\text{MP}_1)\) values (see e.g. [16]). For each adsorbent, estimates of \(V_{\rm s}\) have been computed from the constant b and are reported in Table 2, together with their uncertainty, \(u(V_{\rm s})\). It can be seen that \(u(V_{\rm s})/V_{\rm s}\) is generally less than 0.5%, demonstrating that the material skeletal density, \(\rho_{\rm s}\), can be determined precisely. For the three adsorbents, the following values were obtained: \(\rho_{\rm s}=1.517\pm 0.003\) g/cm\(^3\) (ZIF), \(\rho_{\rm s}=2.532\pm 0.006\) g/cm\(^3\) (MZ) and \(\rho_{\rm s}=1.681\pm 0.008\) g/cm\(^3\) (MC). These values compare favourably with those observed in the literature on either equivalent or similar adsorbent materials, e.g. \(\rho_{\rm s}=1.482\pm 0.002\) g/cm\(^3\) for ZIF-8 [26], \(\rho_{\rm s}=2.349\pm 0.004\) g/cm\(^3\) for ZSM-5 [26] and \(\rho_{\rm s}=1.686\) g/cm\(^3\) for the mesoporous carbon [34]. The observed differences are attributable among other aspects to the degree of sample activation. We note in fact that while the individual Helium gravimetry experiment yields a precise estimation of \(\rho_{\rm s}\), relative differences of approximately \(2{-}3\)% were observed in this study between repeated experiments on the same sample, irrespective of the adsorbent used. These measurements are also shown in Fig. 5 (filled vs. empty symbols) and were carried out after the sample had been newly activated (e.g. after the completion of the adsorption experiment). While in Fig. 5 the differences between these repeated runs may appear to be negligibly small, they would yield small, yet noticeable differences in the computed excess amount adsorbed at large bulk densities. Without loss of generality, we can safely assume that the uncertainty in the computed excess amount adsorbed \(u(n_{\rm ex})\approx \rho_{\rm b}u(V_{\rm s})/(M_{\rm m}m_{\rm s})\), where \(u(V_{\rm s})\) now refers to the difference in \(V_{\rm s}\) between repeated runs. It follows that \(u(n_{\rm ex})=0.1{-}0.2\) mmol/g at \(\rho_{\rm b}=10\) mol/L and \(u(n_{\rm ex})=0.2{-}0.4\) mmol/g at \(\rho_{\rm b}=20\) mol/L. These results also highlight the importance of regenerating the sample in-situ and of carrying out the Helium measurement just before the adsorption experiment. Yet, the accurate estimation of the sample specific skeletal volume in microporous solids is still a matter of debate [3, 10, 14] and our observations extend this discussion to include adsorbents with mesopores.

The excess adsorption isotherms measured on the three adsorbents at \(T=308\) K are plotted in Fig. 6 as a function of the molar bulk density, \(\rho\). These have been obtained by conversion of the net adsorption data using the adsorbent’s skeletal volume obtained by Helium gravimetry. The three isotherms are positive over the pressure range investigated in this study, indicating that the (average) density of the adsorbed phase is larger than the density of the bulk fluid. The excess isotherms reveal more strongly the pore-size dependent adsorption behaviour already indicated by the analysis of the net adsorption isotherms. With increasing fraction of mesoporosity, the location of the isotherm’s maximum moves towards larger bulk density values, namely 2.6 mol/L (ZIF)< 4.6 mol/L (MZ) < 6.7 mol/L (MC). The ZIF and MC excess isotherms are characterised by a narrow maximum, thereby reflecting their rather uniform pore-size distribution, centred at approximately \(1{-}2\) nm and 10 nm, respectively. The broader maximum outlined by the isotherm measured on MZ is thus the direct manifestation of its dual-porosity structure that encompasses both micropores and (small) mesopores. ZIF features again the largest amount adsorbed (\(n_{\rm ex}\approx 8\) mmol/g), while lower, yet similar, values are observed for MZ and MC (\(n_{\rm ex}\approx 3.5\) mmol/g). In light of the textural parameter values reported in Table 1, this observation indicates that surface area is subordinate to pore volume in contributing to the extent of excess adsorption.

Another notable feature of the isotherms shown in Fig. 6 is the characteristic shape beyond their maximum. To facilitate the comparison between the three adsorbents, a dashed line is plotted in each panel of the figure that has been fitted to the data points at large bulk densities (\(\rho >15\) mol/L, corresponding to the last \(4{-}5\) points). These lines are thus equivalent to those shown in Fig. 4, but their slope differs by an amount equivalent to \(-V_{\rm s}/m_{\rm s}\) (Table 2). By analogy with the interpretation carried out on the net adsorption isotherms, the slope of the excess isotherm at adsorption saturation can thus be used to estimate the specific volume of the adsorbed phase [20]. The obtained values are \(V_{\rm a}/m_{\rm s}=0.396\) cm\(^3\)/g (ZIF), \(V_{\rm a}/m_{\rm s}=0.192\) cm\(^3\)/g (MZ) and \(V_{\rm a}/m_{\rm s}=0.138\) cm\(^3\)/g (MC). We note that when the deviations in the measured value of \(\rho_{\rm s}\) between repeated runs are taken into account (\(2{-}3\)%, see Section 5.2), these estimates of \(V_{\rm a}/m_{\rm s}\) may vary of up to \(15{-}30\)%, with the largest uncertainty being observed with the mesoporous materials.

The actual or absolute amount adsorbed in the pores of the material can be obtained by correcting the excess amount adsorbed with an independent estimation of the volume of the adsorbed phase, i.e. \(n_{\rm a}=n_{\rm ex} + \rho {v_{\rm a}}\). In its simplest implementation, this correction uses an estimate of the pore volume, which can be obtained by textural analysis (see Section 4.1). If the solid is microporous, pore filling is attained at relatively low density values, thereby providing a means to compute the absolute amount adsorbed [23]:where \(\nu_{\rm tot}\phi_2\) is the specific micropore volume of the adsorbent. The primary limitation of Eq. 9 is its extension to systems with mesopores, which contribute to the adsorption process, but are unlikely to become fully saturated at supercritical conditions. Early findings by Specovius and Findenegg [36] have demonstrated that thick adsorbed layers can be formed in small mesopores (approx. 10 nm in width) over a temperature range of the order of 10 K above \(T_{\rm c}\) and upon approaching the critical density of the fluid, \(\rho_{\rm c}\). Yet, the same authors have estimated that these layers may only reach up to 5 molecular diameters of adsorbate molecules oriented parallel to the pore surface. Therefore, the estimation of the absolute amount adsorbed in systems with mesopores depends on knowing not only how the adsorbed phase may spread over a range of pore sizes, but also its dependency on the bulk density. Here, we conceptualise a pore-filling process whereby micropores (width \(<2\) nm) and very small mesopores (width \(<3\) nm) are readily filled at low bulk densities (\(\rho <2\) mol/L), while small mesopores (width \(<11{-}12\) nm) experience near-critical condensation starting at \(\rho \approx 5\) mol/L. The mathematical expression adopted to represent this process and its application to the three adsorbents considered in this study are described in Section S3. The absolute adsorption isotherms obtained upon application of this approach are shown in Fig. 7.

It is instructive to compare these results with those described in Sect. 5.3, where we adopted a graphical approach to interpret the descending part of the excess adsorption isotherm. In this model, if adsorption saturation were reached at high pressures, a plot of \(n_{\rm ex}\) vs. \(\rho\) should give a straight line with slope \(-V_{\rm a}/m_{\rm s}\). For ZIF, MZ and MC the obtained estimates of \(V_{\rm a}/m_{\rm s}\) correspond to approximately 55%, 40% and 30% of the total pore volume of the adsorbent, \(\nu_{tot}\), respectively (Table 1). The linearity of the ZIF excess isotherm beyond its maximum would suggests that the pores which drive the adsorption process in this material are filled entirely with the adsorbed phase at bulk density values \(\rho >5\) mol/L. The absolute isotherm shown in Fig. 7 was constructed by assuming that this pore volume includes all pores with width smaller than 3 nm (see also the inset plot in Fig. 3a). On the contrary, the estimate of \(V_{\rm a}/m_{\rm s}\) obtained for ZIF corresponds to the volume found in pores with width below 1.5 nm, yielding approximately 80% of the micropore volume (\(=\nu_{\rm tot}\phi_2\)) and only 55% of the total pore volume. Based on the considerations above, one would expect the adsorbed phase to occupy the entire pore volume available, as it has been previously observed for CO\(_2\) adsorption on 13X zeolite (\(>95\)% micropore filling [29]). The micropore structure of zeolites is determined by the crystal lattice and it features micropores of width 1.2 nm with no distribution of pore sizes. On the contrary, ZIF possesses a distribution of pores over a broader range, including micropores and narrow mesopores (\(<2.5\) nm)–possibly limiting the degree of micropore filling. These observations highlight the difficulty in applying the graphical approach to estimate absolute adsorption with microporous materials characterised by a distribution of pore sizes.

The absence of micropores and/or the presence of an even larger amount of (larger) mesopores complicate the interpretation of the results obtained for MZ and MC and challenge the application of the graphical approach. While the estimate of \(V_{\rm a}/m_{\rm s}\) obtained for MZ matches the volume of micropores (\(=\nu_{\rm tot}\phi_2\)), the characteristic shape of the excess isotherm clearly shows that mesopores do also contribute to the adsorption process. MZ doesn’t have a distribution of micropores (see the inset plot in Fig. 3b). The absolute adsorption isotherm shown in Fig. 7 was obtained upon assuming that micropores are indeed readily filled at low bulk density values (\(\rho <2\) mol/L), but that small mesopores (width less than 12 nm) experience condensation for \(\rho >5\) mol/L, reaching saturation at \(\rho \ge \rho_{\rm c}\). The resulting absolute isotherm has the expected “Type I” shape, which would be impossible to obtain upon direct application of Eq. 9. A similar argument holds for MC, where in the absence of microporosity, the observed adsorption process at elevated pressures is interpreted as the result of near-critical condensation in mesopores with width less than 12 nm. The cumulative pore volume found in these mesopores corresponds to approximately 66% of the total pore volume and is substantially larger than the volume of the adsorbed phase estimated from the graphical method (\(=0.3\nu_{\rm tot}\)). While the graphical approach is obviously inadequate for mesoporous materials, care should also be taken in adopting our proposed method to estimate the absolute adsorption isotherms. A conceptualisation where adsorbed molecules are considered to form as if they were a condensed phase of density \(\rho_{\rm a}\) and volume \(V_{\rm a}\) may be too simplistic [20], because it tacitly assumes that these properties are averaged over the entire mesopore size range and also neglects density profiles within pores themselves.

The classification of adsorption isotherms according to their characteristic shape is a tool of practical value, because it provides a link between the macroscopic behaviour observed experimentally and the textural properties of the adsorbent. However, the IUPAC classification has been traditionally based on the notion of absolute adsorption and does not explicitly consider excess (or net) adsorption. Supercritical adsorption experiments reported in the literature on both crystalline and ordered porous materials provide direct evidence of adsorption behaviour, which can be ascribed to a characteristic distribution of micro- and meso-pores. Some of the traditional adsorbent morphologies that have been investigated include: zeolite crystals, where adsorption is mainly within the micropores [29]; mesoporous silicas, such as Controlled Pore Glass (CPG) [33, 41], SBA-15 [9, 40] and MCM-41 [31]; and mesoporous carbons [1, 36]. These ordered porous solids can be considered as the building blocks for more complex porous structures, including hierarchical adsorbents [37] or disordered materials (e.g. activated carbons), which feature pores across the entire spectrum of sizes. As such, they also represent important reference materials for the textural characterisation of industrial adsorbents by physisorption [8] and for the development of rigorous models with predictive ability at high pressures [12].

With analogy to the classic IUPAC classification [42], we propose to group supercritical adsorption isotherms into three types (Fig. 8). In this endeavour we have solely considered the excess adsorbed amount, because it remains by and large the most widely adopted formulation to report supercritical adsorption isotherms. In this context, we agree with the recommendation that the adsorbent skeletal volume (and its uncertainty) should always be reported alongside the excess adsorption data, so as to enable their conversion to net adsorption [3, 23].

The Type E1 isotherm is given by microporous solids and is the direct counterpart of the Type I isotherm in the original classification of sub-critical isotherms. The adsorption process is governed by the filling of the accessible micropore volume. The enhanced adsorbent-adsorptive interactions in narrow micropores results in the steep initial uptake and micropore filling at \(\rho_{\rm b}/\rho_{\rm c}\ll 1\). The isotherm is thus characterised by an early maximum (M) followed by a linear descending part (starting at point L) with a slope approximately equal to \({\rho_{\rm c}\nu_{\rm tot}\phi_2}\). Materials featuring a distribution of wider micropores and narrow mesopores (\(<\,\sim 2.5\) nm), e.g. activated carbons, also belong to this category, albeit they may show slight deviations from linearity (see Fig. 2).

The Type E2 isotherm is given by mesoporous adsorbents and is the direct counterpart of the Type IV isotherm in the original classification of sub-critical isotherms, albeit without the appearance of a hysteresis loop. Compared to Type E1, the isotherm’s maximum (M) is shifted towards higher bulk density values and its location depends on the average pore size, approaching \(\rho_{\rm b}/\rho_{\rm c}=1\) for large mesopores (\(>\sim 20\) nm) [33]. For measurements at near-critical conditions (\(T-T_{\rm c}=10\) K), the isotherm shows the distinctive features of multilayer adsorption on the mesopore walls, namely an inflection point (B) and a marked dependence of the maximum excess adsorption on temperature [36]. The descending part of the isotherm beyond the maximum is curved; as such, the determination of the adsorbed phase properties is not as straightforward as for Type E1 and a rigorous quantitative discussion of the excess adsorption isotherm would require knowledge of the density profile in the mesopores of the material.

The Type E3 isotherm is given by the adsorption of supercritical gases on micro-mesoporous materials with fairly large mesopores (\(>\sim 20\) nm) and has been observed in this study for the dual-porosity MZ. The isotherm shows features of both Type E1 and E2: it is characterised a steep initial uptake at \(\rho_{\rm b}/\rho_{\rm c}\ll 1\) followed by a broad maximum (located somewhere between Type E1 and Type E2) and a curved descending part. At sufficiently large bulk density values, the isotherm falls linearly with density with a slope of value \(\ge \rho_{\rm c}\nu_{\rm tot}\phi_2\).