Both the surface area of the micropores S_mi and their volume V_mi are measured for 1 gram of material, i.e. in m²/g and cm³/g, respectively. These dimensions reflect the fact that in a series of experiments, technological variables change not only the geometry of the pores, but also the density of the solid material.

Referring to some published studies, we found that often the ratios of micropore volume V_mi to total surface area S_BET are very close within a series of samples [https://doi.org/10.3762/bxiv.2020.89.v1]. This is manifested in low values of STD, %. Going deeper into this issue, we came to the conclusion that V_mi/S_BET can be a useful characteristic of surface topography, which is a kind of roughness [12].

Micropores are located on the surface of the meso/macropores, and their surface is part of the total surface area (let’s take it as S_BET); therefore, it seems quite correct to take the ratio (V_mi/S_BET), cm³/m², in order to exclude fluctuation in the density of materials, but to take into account fluctuations in the total surface. The difference between the values of the standard deviation (STD, %) V_mi (cm³/g) and V_mi/S_BET (cm³/m²) may indicate the presence of through micropores, closed intra-wall space V_iw [3] and surface cracks.

In a broad sense, STD can be applied to a series of any parameters in any process, therefore it is a general purpose serial generalized parameter.

Let us represent the expression for the hydraulic pore size as

D h = F ∗ ( V t / S t ) (1)

If V_t, S_t and D_h have dimensions cm³/g, m²/g and nm respectively, F will be 2000, 4000, and 6000, nm * m²/cm³, for slit, circular and spherical pore models. If the volume, surface area and size are mesured in the same linear units, for example in m³/g, m²/g and m, respectively (we denote such a factor as F_d—dimensionless), then for the above forms, the F_d will be Equations (2), (3) and (4). Consequently F_d = 0.001F.

D_h is often used as the average pore diameter D_p [13] [14] [15]. Substituting D_p instead of D_h in Equation (1), we obtain the equation “in separated variables”

D p = F d ∗ ( V t / S t ) (2)

thus, the average pore diameter is the product of two generalized factors: the dimensionless pore shape factor F_d and the size factor V_t/S_t (volume per unit surface or average adsorbate thickness) of the pore of indefinite shape. The last factor has the dimension of length. That is, it is generalized pore size. This size represents the size of a family of pores of any shapes, but with an equal V_t/S_t ratio.

When the average pore size D_p is measured (calculated) without assuming some pore shape (BJH, DFT models; TEM images, etc.), we can calculate F (or F_d) using Equation (3).

F d = D p ∗ S t / V t (3)

In [https://doi.org/10.3762/bxiv.2020.89.v1], examples are given with F_d from 1 to 14, that is, from fragmented slit shape (F_d < 2) to highly corrugated one ( F d ≫ 6 ). If F_d is in the range from 2 to 4, we can calculate the percentage of slotted and circular pore shape motifs (slotted motifs – 100 * (4 − F_d)/2, %, circular motifs = (100 * (F_d − 2)/2, %); similarly, for F_d = 4 ÷ 6 percent of circular and spherical motifs can be estimated by the formulas: (100 * (6 − F_d)/2, %. and (100 * (F_d − 4)/2), %, respectively.

Еquation similar to (3) can be proposed for micropores pore range (D_mi < 2 nm)

f m i = D m i ∗ ( S m i / V m i ) (4)

The factor V_mi/S_mi (with the dimension of length), which we called the generalized size of micropors does not contain mass, and its STD can be compared with STD of both S_mi (m²/g) and V_mi (cm³/g) in a series of samples.

Earlier, formulas for the length of circular pores were proposed: the length of the volume of the adsorbate L_v [2] and the length of the surface of the adsorbent L_s [3]. The L_s/L_v ratio has been adopted as a kind of rouphness factor; L_s/L_v = 1 indicates a smooth-walled pore, the surface of which can serve as an intrinsic model of a smooth surface.

The practice of using L_v and L_s had shown the need to have some parameters representing the length for pores of any shape. We propose to bring the formulas for the lengths of pores of the circular shape L_s = S_BET/πD_p and L v = V p / ( π / 4 ) D p 2 to a form suitable for any shape, simply discading π and π/4. Let us call these parameters the indices of length: L_si = S_BET/D_p is the pore surface length index, and L v i = V p / D p 2 is the pore volume length index.

1) Węgrzyniak et al. [25] prepared 8 samples of CMK-3 using SBA-15 as a template and two carbon precursors (1st variable): sucrose (CMK-3s series) and poly-furfuryl alcohol (CMK-3f series). The resulting materials were carbonized at 550˚C, 650˚C, 750˚C and 850˚C (2nd variable), and the silica was removed using a hydrofluoric acid solution.

All CMK-3s samples showed better propane to propene conversion compared to CMK-3f. S_BET, V_p, V_mi and PSD were given; it can be seen from the PSD (Figure 2 in [25] ) that the pore diameters of both series are very close. We have calculated V_mi/S_BET, V_t/S_BET for all samples as well as average values and standard deviations of all parameters for both series (Table 4).

First of all, note the closeness of the values of average generalized pore size V_t/S_BET in two series (0.77, 0.76 nm), which coincides with the PSD pattern of

D_mi—mucropore diameter, V_mi = micropore total volume, S_BET—pore total surface area.

S_BET—total surface area, V_mi—micropore volume, V_p—total pore volume.

real pore diameters. Comparing these data with V_t/S_BET = 1.22 nm of SBA-15 one can see that the replica diameters are smaller then the template diameter.

The average surface rouphness V_mi/S_BET of CMK-3s, created by sucrose, (5.3E−05 cm³/m²) repeated that of SBA-15 (5.8E−05 cm³/m²), while poly-furfuryl alcohol creates carbon CMK-3f with a much smoother surface (V_mi/S_BET = 1.1 E−05 cm³/m²).

From the very low STD of S_BET (2.2, 3.1%) it can be concluded that there is very little effect of carbonization temperature on the surface area of both CMKs.

2) Ponomarenko et al. [26] prepared samples of SBA-15 silica, in which the pore shape factor F_d ranged from 4.539 to 8.721, that is, from almost circular to very corrugated [https://doi.org/10.3762/bxiv.2020.89.v1]. Then the samples were used as templates for the fabrication of carbon OMMs. The question arises, to what extent do the latter inherit the shape of silicate nanotubes? The answer was that the pore shapes of the replicas (F_d = 3.949 ÷ 5.706) strongly differ from the template ones (Figure 5).

The pores of the silica matrices were spherical and corrugated (F_{d tem} = 6.5 ÷ 8.5), but the pores of carbon, as a rule, became close to circular (F_{d rep} = 4 ÷ 5). After the dissolution of silicate, the carbon tubes elongate and expand, the pore diameter narrows; as a result, their shape becomes close to circular. Two samples at F_{d tem} = 4.5, stand out strongly from the mass. It was found that these carbon mateials were heated at the highest temperatures and do not have micropores (samples ##1 and 2 in ( [26], Tableon the page 81)).

1) Lee et al. [27] prepared 7 samples of ordered mesoporous carbon using MSU-H silica as a template, sucrose as a carbon source and boric acid (BA) as a pore expander. BA, %, was the only variable. The STD-s of the terms of Equation (2): D_p(BJH), V_p/S_BET and F were 36.8%, 12.5% and 26.1% respectively, i.e. this is the case when the STD for D_p (36.8%) is close to the sum (38.6 %) of the STDs of the generalized shape factor F and the generalized pore size V_p/S_BET.

The diameters of the carbon pores were noticeably smaller, the walls were thicker and much longer than these parameters of the silica template. F-s (3175 ÷ 7123) are directly proportional to BA, %, and inversely proportional to t/D_p. STD = 1% was found for V_mi/S_BET (at STD = 18% for V_mi).

2) Arshad et al. [28] prepared AC samples by activating an empty bunch of fruit with hot CO₂ (BC sample, with circular pores) and then treating BC with 0.5, 1.0 and 2.0 M KOH solutions (samples AC1, AC2 and AC3 respectively). When using 0.5 M KOH solution and converting BC to AC1, the S_BET and V_p parameters increased by a factor of 10 - 15, but D_p (BJH) decreased from 2.4 to 1.9 nm. The question arises how did S_BET and V_p increase? To explain this phenamenon, we checked the pore shape factor F_d and length L_s.

It was found that materials BC and AC (AC1, AC2 and AC3) have the same circular pore shape (F_d ≈ 4.0), therefore, the pore shape cannot be the cause of these changes. As for the length L_s of the pore volume, it increased 17 times, which explains the increase in S_BET and V_p. These parameters and V_mi are directly proportional to L_s. Stretching the walls leads to an increase in V_mi (cm³/g) by a factor of 2.5, but the V_mi/S_BET (cm³/m²) values of samples AC1, AC2 and AC3 differ very little (STD = 4.9%), i.e. all KOH solutions provide the same surface topography.

Comparing STDs of V_p with V_p/S_BET (41.85% and 4.03%), S_mi with S_mi/S_BET (39.49% and 1.95%) and V_mi with V_mi/S_BET (43.41% and 4.9%) one can conclude that KOH effects strongly on parameters connected with mass but very little on parameters connected with total surfaces.

3) Alam and Mokaya [29] prepared 6 samples of microporous zeolite-like carbons using zeolite Y as a hard template, liquid impregnation of it with furfuryl alcohol, followed by chemical vapour deposition (CVD) of acetonitrile at temperatures of 700˚C ÷ 950˚C. V_mi/V_p were 75% - 83%, STD of V_mi = 25.2% and STD of V_mi/S_BET = 2.4%, that is, all temperatures resulted in approximately the same surface topography.

Pagketanang et al. [30] obtained activated carbon (AC) from rubber seed shells by activating KOH, washing with water, and calcining in a stream of N₂ at temperatures T_c = 700˚C, 800˚C and 900˚C (samples AC700, AC800 and AC900). S_BET, V_p, D_p and V_mi data were presented. We tested dependencies V_mi, F_d, L_si and V_mi/S_BET on T_c. The pore shape factor F_d turned out to be inversely proportional to T_c, while L_si and V_mi were directly proportional to it (not shown), therefore the listed parameters did not reveal anything unusual. Such a case was found usig the dependence of T_c on V_mi/S_BET (Figure 6).

Figure 6 shows that, in contrast to the directly proportional dependence of

V_mi on T_c, annealing from 700˚C to 800˚C did not change V_mi/S_BET. T_c = 800˚C distinguishes between two different mechanisms of micropore formation: up to 800˚C, the surface is formed with a constant number of micropores per m² of surface, but then micropores are formed faster than the total carbon surface.

Bastos-Neto et al. ( [31], Tables 1, 2) studied the adsorption (storage) of methane by 10 commercial and non-commercial activated carbons (AC). Among the samples were the sample #8 with the lowest methane uptake (57 mg/g AC) and sample #10 with the highest absorbance (168 mg/g AC). The authors of [31] estimated V_mi: 1) by the Dubinin–Radushkevich method (V_mi DR) and 2) by the Monte Carlo molecular simulation method (V_mi MC).

There schould be a direct proportionality between methane uptake (mg/g) and micropore volume V_mi (cm³/g). In [31], Figure 3 (CH₄ vs. V_mi DR) and Figure 4 (CH₄ vs. V_mi MC) indicate the presence of at least 2 series of samples, but the diffrence between them is not discussed. In search of an explanation, the original texture properties V_mi, S_BET, V_p, and D_p (HK) were supplemented by F_d, V_p/S_BET, L_vi, L_si, and V_mi/S_BET.

The graph of the dependence of the adsorption of CH₄ on V_p/S_BET (not shown) made it possible to isolate only sample #10 (with the highest absorption), its absorption corresponds to the highest V_p/S_BET (0.0011 cm³/m²). The absorption of all other samples occurs at a constant V_p/S_BET = 0.0006 cm³/m².

Figure 7 shows the absorption of CH₄ as a function of the surface topography V_mi/S_BET.

The 8 out of 10 samples form a cloud of close V_mi/S_BET values in a wide range of CH₄ uptakes (80 - 140 mg/g AC). Samples with minimum and maximum adsorbance are clearly separated. Thus, the lowest and highest adbsorption levels are associated with the surface topography.

Then STDs were calculated for all texture parameters. To separate a series, a parameter with a minimal deviation is needed. The minimum STD (8.9%) was obtained for D_p. Figure 8 shows the dependence of the CH₄ uptake on the average pore diameter D_p.

Two series of samples are clearly visible: series 1 with CH₄ uptake inversely proportional to D_p (low D_p range) and series 2 with CH₄ uptake independent on D_p (D_p > 1.31 nm). The series 1 illustrates well the potential theory of adsorption [21]; it can be said that the influence of opposite pore walls starts from D_p = 1.3 nm and increases with its decrease.

1) Al inside SBA-15

Yue et al. [32] prepared thick-walled AlSBA-15 from a mixture of tetraethylorthosilicate and aluminium tri-tert-butoxide (direct inclusion of Al in SBA-15, Si/Al = 10) and compared it with conventional SBA-15. Both series of materials: SBA-15 and AlSBA-15, were processed by 5 different methods (Table 5): calcination (method C), heating to a high temperature with O₂ – H₂O mixtures (V),

C—calcination, V—hot O₂—H₂O, A, B, N—acidic, basic, neutral solutions; V_iw—intra-wall pore volume, F_d—dimensionless pore shape factor, t—wal thickness, D_p—pore diameter.

treatment with acid (A), basic (B), and neutral (N) solutions (12 samples in total) ( [23] p. 123, table 1). Row of Table 5 contains the SBA and AlSBA samples, processed in the same way.

The direct inclusion of Al in SBA-15 (sample #7 compared to #1) leads to an increase in the volume V_iw of intra-wall pores (4.57 against 3.77 cm³/g). This excess remains for the entire series. Apparently, the Al film covers the surface, preventing the escape of vapors and gases from the internal cavity of the walls.

The inclusion of Al in SBA-15 (run #7 vs. #1) has virtually no effect on the pore shape of SBA-15, which is circular/spherical. In the SBA-15 series, calcination (run #2) and basic solution (run #5) make the pore shape of SBA-15 close to slotted (F_d ≈ 2.800), but the presence of Al prevents strong pore flattening and they remain almost circular (runs ##8 and 11).

Treatment with both acidic (A) and neutral solutions (runs ##4 and 6) does not affect SBA-15, but in AlSBA-15 (runs ##10 and 12) they transform a circular shape into a spherical one. In general, any treated AlSBA has higher F_d value then the corresponding SBA-15. Figures F vs. V_iw for the SBA-15 and AlSBA-15 series demonstrate their direct proportionality (not shown). Table 6 shows the calculated STDs of some texture parameters for both series.

An important result of this work is the strong stabilizing effect of the introduced Al on the pore diameter D_p and the parameter t/D_p, which characterizes the mechanical strength of the structure.

By comparing the STD-s of the diameters D_p and composing them F and V_p/S_BET (Equation (3)), we can conclude that processing variables affect F and V_p/S_BET in opposite way; the dependence of F on V_p/S_BET is inversely proportional (not shown). This once again testifies to the selective influence of processing variables on textural parameters [3] [8].

2) Al inside MCM-41 Walls

La-Salvia et al. [33] introduced Al into thin-walled MCM-41. MCM-41 and two samples of mesoporous materials Al-MCM-41 with different Al content (Table 7) were obtained from STAB, Al sulphate hydrate and TEOS by non-hydrothermal synthesis in an alcaline-free media at room temperature and short reaction times. The resulting material was filtered, dried and calcined. The introduction of Al led to increase in the unit cell parameter a_o and wall thickness t but a decrease in

S_BET, V_p, D_p and t—pores total surface area, total volume, average pore diameter correspondingly; F—generalized pore shape factor, V_p/S_BET—generalized pore size, V_iw—intra-wall pore volume.

a_o—unit cell parameter; S_BET, V_p and D_p—pore surface area, volume and diameter, t—wall thickness, V_w—apparent wall volume, V_iw—intrawall pore volume, F_d—pore shape factor.

S_BET and V_p; the pore diameter did not change (Table 7). The question is where will Al fits: outside or inside the walls. We added parameters F_d, the apparent volume of the wall V_w = S_BET * t and the intra-wall pore volume V_iw [3] as well the length index of the adsorbate volume L_vi.

The F_d values show that the pore shape did not chaged and is close to circular, the surface is slightly smoother. The most informative is the behavior of V_w and V_iw. The addition of 1.96% Al increases these volumes slightly, but the addition of 6.25% Al decreases them markedly. This means that Al occupied part of the space inside the walls, L_vi decreased noticeably, which also explains the reduction in S_BET and V_p.

3) Ti makes the pores of MCM-41 slotted

MCM-41 with the pore wall thickness 2.5 nm was grafted by Palumbo et al. [34] with a solution of titanocene dichloride in chloroform. The addition of only 0.65 wt% Ti greatly changed the pore shape of MCM-41: its shape factor changes from a circular-slit (F_d = 3.134) to fragmental slit (F_d =1.494). The change in F_d occured as a result of a doubling of the pore volume length index L_vi, while the surface length index L_si remained practically unchanged.

4) Optimal Mo content

Chanchuey et al. [35] used spherical silica particles (SSP) to prepare an Al-SSP composite (60 wt% Al). The composite was impregnated with Mo precursors to give 1%Mo/Al-SSP and 5%Mo/Al-SSP catalysts. The catalysts were tested in the dehydration of ethylene to ethanol. The 1%Mo/Al-SSP catalyst had better catalytic performence ( [35], Figure 7]. To find a possible relationship between catalytic activity and the textural properties of catalysts, we compiled Table 8.

It follows from the table that the largest pore diameter of 7.2 nm and a close to circular (81.7%) pore shape contribute to 100% conversion of ethylene to ethanol. It can be said that catalysts also play the role of molecular sieve in both size and shape (note: sieving lipase by pore shape is shown in [https://doi.org/10.3762/bxiv.2020.89.v1]).

5) Xu et al. ( [36], p. 1293, Table 1) studied the effect of the Al content on the textural properties and catalytic activity of hierarchical porous aluminosilicate materials. The best Si/Al ratio was found to be 25; the material was crystallized

S_BET, V_p and D_p—total surface area, total pore volume and average pore size; L_vi—pore volume length index; F_d—pore shape factor, Con.—degree of conversion, *) 81.7% = 100 * (6.0 − F_d)/2 [https://doi.org/10.3762/bxiv.2020.89.v1].

within 16, 22, 36 and 48 hours. Data on the surface area, volume and size of micro and mesopore, as well as on the catalytic activity in phenol alkylation with tert-butanol are presented. Our F_d calculations showed that sample C16 (25), which provides the maximum (93.2%) phenol conversion, has a pore shape closest to circular (F = 4863) and both L_si and L_vi have the highest values.

The rare data presented in the article is the size of micropores. The micropore shape factor was in the range f_mi = 1170 - 1593, which means a fragmentary slit shape. Together with S_mi, V_mi, it allows comparing STD-values of terms of Equations (3) and (4). The dependences of F_me on V_me/S_me and f_mi on V_mi/S_mi were tested. The direct correlation between F_me and V_me/S_me (not shown) means that the fluctuations of both parameters are summed up in the STD fluctuations of their product D_me; f_mi versus V_mi gives an inverse correlation (not shown), i.e. they oscillate in opposite directions, and the standard deviation of their product D_mi can be obtained by subtracting the smaller value from the larger one (Table 9).

Proximity is visible: a) for mesopores—the STD of size value (19.8%) with the sum (20.7%) of STD-s of shape factor (9.9%) and the Volume/Surface ratio (10.8%); b) for micropores—the STD of size (8.1%) with difference (8%) between STD-s of shape factor (13.9%) and the Volume/Surface ratio (5.9%). Once again, we note the selective response of meso- and micro-textural parameters to the processing variables.

6) Ding et al. [36] used six γ-Al₂O₃ supports (samples A1 ÷ A6) from various suppliers to prepare a FeK/Al₂O₃ catalyst (samples C1 ÷ C6) for hydrogenation of CO₂ to hydrocarbons. The textural properties and the results of catalytic tests are presented. We plotted the dependence of the pore shape factor of the catalyst F_dc on a similar F_ds value of the support (Figure 9).

The highest CO₂ conversion (54.4%) and selectivity of C5+ hydrocarbons (31.1%) were achieved for the sample with F_ds = 3.988 and F_dc = 3.950, i.e. when the pores of the catalyst inherited the circular pore shape of the support. The worst result (≈40% conversion ( [36], Figure 7)) was obtained by a sample clearly separated in Figure 9, prepared on a support with F_ds = 5.473 (74% of spherical motive) and having F_dc = 4.622 (31% of spherical surface).

7) Zubizarreta et al. [24] investigated the adsorption of H₂ in the two menshioned above (Section 3.2.2) series of Ni-doped carbon xerogels with pore diameters of

16 nm (series CX16) and 45 nm (series CX45). The uptake of H₂ depends on the H₂ spillover, i.e. H₂ transfer from the metal catalyst to the carbon support. The sample with the highest adsorption capacity was CX16AOXI (D_p = 16 nm, designations: A—chemically activated carbon xerogel, OX—oxidized HNO₃, I—Ni doped by impregnation). To identify the causes, we compared this sample with a similarly treated CX45AOXI sample (D_p = 45 nm) using L_si, L_vi, and F_d (Table 10).

The pore length of the sample CX45AOXI (and all CX45 samples with wide pores) is much shorter that of the pores of 16 nm. This creates a very strong surface corrugation of wide pores, D_p = 45 nm, compared to narrow pores, D_p = 16 nm (F_d = 28 against 17.1).

Recall that smooth cylindrical pores have F_d = 4; comparing this number with the obtained values, we can say that for a high adsorption capacity, the surface should be highly corrugated (F_d = 17.1), but F_d = 28 indicates exessive corrugation, which worsens the spillover of H₂.

Perkas et al. [38] improved the CuO-ZnO/TiO₂ catalyst for the WGS reaction by

L_si. L_vi—indices of pore both surface and volume lengths, F_d—generalized shape factor.

sonochemical method. The best results were achieved with catalysts supported on commercial TiO₂ Degussa P25 support. Among the carriers tested, the P25 has the smallest pore diameter (1 nm) and F_d = 0.5, (i.e. slit pores with a very fragmented surface), but the longest pore volume (L_vi = 9E+10 m/g).

Sonochemical treatment led to an increase in D_p from 1 to 8.6 nm and F_d from 0.5 to 1.944, that is, to pores with a slot pores with a smooth surface; L_vi decreased from 9E+10 to 0.31E+10 m/g, i.e. 30 times. Apparently, sonochemical treatment smoothes surface irregularities of the pore, stretching it along the perimeter and tightening it along the axis.