Adsorbents for Gas Storage: Gas Energy, Sorbent Energy and Their Relationship to Capacity ()
1. Introduction
This article focuses on the generalized properties (GPs) of adsorbents for gas storage (StA); GPs were expected to correlate well with sorbent deliverable capacity. It has recently been shown [1] that the GPs (variables, parameters) of porous materials (PMs) behave like similarity numbers in completely different areas of nature (hydraulics, heat and mass transfer, geometry, etc.). The advantage of generalized variables in many-factorial experiments and series is the simultaneous consideration of changes in all single variables included in the generalized one. The GP proposal [2] was based on the identified earlier fact of the selective influence of technological variables (temperature, pressure, reagents, time) on various indicators of the mechanical strength of molded catalysts [3]. Later, it was extended to the textural properties of various PMs [2]. Combining two or more parameters into one GP allows revealing the nuances of the PM texture and dividing the samples into their series with similar properties. PMs include adsorbents and catalysts, both disordered and ordered [1] [2].
An interesting object for the use of GPs is currently relevant physical adsorbents for gas storage. As a rule, both the processing parameters (pressure P and temperature T) and textural parameters of the sorbent are involved in StA studies. When any process variable changes, it produces work, which is the product of force (P) and displacement in size or shape [2]. The textural properties (displacements) of the sorbent include total surface area (SBET), total pore volume (Vp), and pore size distribution (PSD).
StAs include a wide range of materials: disordered and ordered activated carbons (ACs), metal-organic frameworks (MOFs), porous organic polymers (POPs), etc. These materials have narrow (micro-, ultra-micro-) pores in common. In AC, opposite walls enhance the attraction of adsorbate molecules (the overlapping effect [4]). This effect avoids both the very high pressure and very low (cryogenic) temperature needed to store gases such as H2 and CH4. Sethia and Sayari [5] report this effect at a pore size 0.6 ÷ 0.7 nm for H2-AC pair; for the CH4-AC pair, we found that the effect of the carbon pore size is manifested itself in the region of less than 1.3 nm ( [1], Fig. 8). In MOF, the interpenetration of two or more sublattices in the same framework divides large pores into small ones, which increases its efficiency in storing CO2 [6]. Unfortunately, in works devoted to gas storage adsorbents, the pore diameter Dp is rarely given. The variety of sorbents makes it difficult to choose some universal pore size model. It seems that the recently proposed [1] τ = Vp/SBET, the average thickness of the probing gas layers, can be tested for the role of the universal pore size for any porous material.
Generally, total capacity (absorption) at storage pressure and temperature and gas delivery (excess) capacity G at supply pressure and temperature are given. Capacities are measured in g or cm3 of stored gas per unit (per gram, gravimetric G, g/g, per cm3 or L, volumetric G, cm3/cm3) of either sorbent (option 1) or installation (option 2). The 1-st option is important for the scientific basis for the preparation of sorbents and catalysts, the 2-nd option is used for economic evaluations [7]. In this work, the 1st option will be used.
The dimension of gravimetrical uptake (G, g/g) contains a gram of sorbent. Recently [1], using the micropore volume Vmi (cm3/g) of PM as an example, we have shown that the fluctuations of Vmi in a series of samples depend both on the fluctuations of the surface itself and the density of the adsorbent. The latter was found to be a redundant variable and eliminated by dividing Vmi by SBET, i.e., measuring Vmi in cm3/m2, which gives a noticeable reduction in the standard deviation (STD, %) of Vmi in a series of experiments. In addition, the STD itself becomes a serial GP. The Vmi/SBET ratio is also a GP, a characteristic of the surface topography and a kind of roughness [1]. Note that study of the dependences G, g/g, both on SBET, m2/g, and on Vp, cm3/g, is correct since both independent and dependent variables refer to the same gram of material. The idea of using other dimensions of G instead g/g will also be used in the current paper.
Fiero et al. [8] separated the effects of attraction of gas molecules and gas pressure ( [8], Fig. 6: G, g/g, vs. SBET): the amount of hydrogen stored by AC increases significantly due to adsorption only at moderate pressure: 3 MPa/298K and 0.15 MPa/77K, but at high pressure, the proportion of adsorbed gas is much less than that of just compressed gas ( [8], Figs. 7, 31). Moreover,an increase in SBET after a certain value led to a decrease in gravimetric G; this phenomenon was observed both at low (3 MPa) and high (20 MPa) pressures ( [8], Fig. 6). The same phenomenon was previously noted by Goldsmith et al. [9]. The objectives of this work are: 1) to find the cause of the phenomenon, 2) not to separate, but to combine the parameters of gas pressure and adsorbent texture into generalized variables and evaluate the strength of their correlation with gas adsorption.
A small pore size often indicates a small dimensionless pore shape factor Fd.= Dp*SBET/Vp [1]. Fd values in range < 2 were found [https://doi.org/10.3762/bxiv.2020.89.v] among MCM-41 samples when the work of Putz et al. [10] was discussed. These samples, according to the conclusion of the authors of [10], have a disordered structure. Recall that the factor F = 2000, nm*m2/cm3, or the dimensionless Fd = 2 [1] refers to the split pore model in the expression for the average hydraulic pore diameter Dh = Fd*Vp/SBET. Adsorbents with small split pores and parallel flat walls obviously create good conditions for the physical adsorption of gases. However, Gun’ko et al. [11], Azevedo et al. [12] note that the model of a flat slotted pore does not always allow one to describe the experimental data. For ACs, a model of mixed geometry was proposed [12], which represents the pore space as a set of an indefinite ratio of slit-like and triangular pores. Cardenas et al. [13] proposed “infinitely long pores with a polygonal (triangle, square, pentagon, hexagon, octagon, decagon, and circle) cross sections”, including pores less than 1 nm in diameter, but they did not include a split shape. It is also very hard to predict the shape of MOF pores due to their large-scale flexibility, the presence of defects, and long-range disorder [14]. We expect that by applying the thickness of probing gas layers τ = Vp/SBET we will be able to better appreciate the shape of the pores. This is the next goal of this work.
2. Generalized Energy Parameters of Storage Adsorbents
2.1. Two Forms of Energy in Porous Materials
When studying gas StA, a certain volume of either a storage container or sorbent pores is given, as well as pressure and temperature of the gas. The product of the gas volume V and the pressure P in a closed vessel at a constant temperature T
(1)
is the equation of state for an ideal gas.
In a broad sense, PV is energy (work), where P is force and V is displacement [2]. In the case of a real gas, PV is a natural generalized variable whose deviation from constancy indicates the presence of additional variables (proximity to critical points, molecular dissociation, etc.). In the case of gas storage, an additional variable may be the forces of gas attraction (physical sorption) in narrow pores (overlapping effect [4]). This effect can be fixed by the inversely proportional dependence of the capacitance on the pore size. For a porous material, any textural property (pore volume Vp, surface area SBET, and pore size) can serve as the displacement V in Equation (1). The force in the gas phase is pressure P. To go to the energy of the solid part of the sorbent, imagine a closed container, for example, a children’s rubber balloon, on the walls of which a strain gauge is glued. As the gas pressure increases, the elongation and wall stress will increase until the balloon bursts. Let us change in Equation (1) the pressure of the gas P that destroys the cylinder by a certain tensile strength of its material Pm (the subscript m-mechanical). So, the equation of state will be
(2)
and PmV can be called the energy of the sorbent. This is an example of a process (Pm) - structural (V) generalized variable. Previously [SI-1] for a granule of a porous material, the mechanical strength of the pore walls was accepted as a result (as replica) of gas (steam) pressure that occurs inside the pores of the material during its preparation or operation; the gas pressure leading to the destruction of the wall corresponds to the crushing strength of the wall Pm. In practice, such destruction of the walls of the pores occurs when the high pressure in the catalyst bed is abruptly released or when the wet granules are dried too intensively.
PmV was expected to be constant over a range of process variables in the production of molded materials. Pm was a type of mechanical strength, depending on the loading scheme (device). The choice of the appropriate pore volume (total Vp, meso-Vme, micro-Vmi, etc.) was based on the statistical nature of the crushing of the catalyst tablets under uniaxial compression, each tablet is crushed differently. One of the statistical theories of strength, the theory of the weak link-says that the destruction begins from the weakest point in the product and then develops catastrophically. This point can be in the wall of a pore of any size, so every kind of pore volume V must be tested. Indeed, for many materials and loading schemes (tables in SI-1) existеnсе of PmV ≈ const in the dependences of PmV onx (x is a technological variable) has been established. In the current paper, we have added SBET surface area and pore size, either represented by Dp or by generalized Vp/SBET, as displacements.
2.2. Pore Shape Factor as Number of Gas Layers
Taking into account that Vp/SBET is the average thickness τ (nm) of the probing gas layer (usually N2) and having the average pore diameter Dp, one more meaning of the dimensionless pore shape factor Fd can be found. From the “equation in separated generalized variables” for the average pore diameter [1] we obtain the pore shape factor
(3)
Thus, the shape factor Fd has one more physical meaning: it is the average number of probing gas layers in the pore cross-section. In other words, the average number of probing gas layers is determined by the average pore shape.
As mentioned above, if the Dp value is not given in the work under study, then Vp/SBET can be used instead and (use kinetic diameters σ of the nitrogen probing gas-0.364 nm and stored gases: hydrogen-0.289 nm, methane-0.380 nm, and carbon dioxide-0.330 nm) the average number of layers of these gases in pore cross-section, which (number) is simultaneously a dimensionless pore shape factor, can be estimate
(4)
For 1 < Fd< 2, one of the pore shape models can be represented by two layers of molecules with some distance between neighboring molecules in the bottom row (in contrast to the BET assumption); the molecules of the upper layer are partially immersed between molecules of the lower layer. If they are completely immersed, we get one layer (linear packing) and Fd = 1; if they do not sink at all, we have two parallel layers and Fd = 2 (square packing). When they are partially submerged, we have a triangle packing of molecules inside a smooth-walled split pore (note the packing of molecules, not the shape of the pore [12]). The case Fd< 1 means either one layer of molecules with some distance between them, or gas molecules trapped between layers of a layered adsorbent.
Another imaginary model of pores can be a “stalactite cave”, pores with rough walls, in which, at some small distance between the roughness peaks of opposite walls, overlapping forces arise to attract gas molecules. Such a model can fit any Fd. Another molecular packing model that seems to be acceptable for MOF and COF is the Chinese cherry with pinpoint adsorption active places.
2.3. Elimination of Fluctuations in the Density of the Sorbent in a Series of Samples
Excluding the gram in dimension of adsorbent capacity G, g/g, gives: 1) G/SBET, g/m2, surface activity, which indicates whether we need to activate the surface, for example by depositing some amount of metal to provide an H2 spillover [15], 2) G/Vp, g/cm3, a kind of volumetric capacity, 3) conditional G/Vmi, g/cm3, imaginary filling of micropores, provided that the total amount of gas fills the pores of a certain volume (Vmi, Vme, etc.). Comparing these capacities, we can detect the location of micropore, in form of either cracks or pits on the surface of larger micropores or mesopores.
3. Discussion
3.1. Forces of Attraction and Repulsion of the Pore Wall
1) K. Xia et al. [16] activated by CO2 mesoporous carbon SMK-3 at a temperature of 1223 K for 2, 4, 6, and 8 h to store hydrogen. Activation time was the only variable. With its increase, the adsorption of H2 G, wt.% (1 bar/77 K), the total surface area SBET, the total volume Vp, the volumes of micropores V<2 nm, V1-2 nm, V<1 nm and their diameter DHK increase.
The surface activity G/SBET decreases, so the increase in G is associated with all micropore volumes (directly proportional). The best correlation (R2 = 0.9995) shows G vs. V<1 nm. G versus diameter DHK gives a direct proportionality, which contradicts the idea of attraction. So, we tested the generalized size τ = Vp/SBET (Figure 1).
It can be seen from Figure 1 that a decrease in the generalized pore size from 1.1 to 0.95 nm leads to a strong increase in H2 adsorption (according to the attraction effect [4]), but then (from 0.95 to 0.76 nm) adsorption slows down. It is obvious that repulsive forces arise, and Vp/SBET = 0.95 nm is some critical value for this material and gas.
Imagine the density of a gas under the condition that the amount of gas G fits completely into the pores of a given size (Table 1). Pores > 2 nm were called
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Figure 1. Effect of generalized carbon pore size Vp/SBET on H2 adsorption (1 Bar/77 K). Based on the data K. Xia et al. [16].
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Table 1. Imaginary density G/V, g/cm3, of a gas that completely fills pores of a given size. Based on the data K. Xia et al. [16].
G: capacity, V: volume of pores, subscript: pore size.
mesoporous, and their volume Vme was obtained as Vme = Vp − V< 2 nm. Note that the serial STD of all discussed volumes V<1 nm, V1÷2 nm, etc. as well G, are in the range of 21% ÷ 38%.
First of all, let’s compare STD of gravimetric G (30.92%) and volumetric G/Vp(7.83%) capacities. Excluding gram of carbon from the dimension of the uptake G allows to reduce STD by 4 times; this means that the time of carbon activation significantly affects the density of carbon. The average G/V data illustrate the decrease in gas density with increasing pore size. Impressive is STD (only 0.6%) G/V<1 nm, which can be explained by the attraction of gas by the walls; the highest STD shows G/Vme where there is no gas attraction.
2) Texier-Mandoki et al. [17] presented uptakes H2 by 7 carbon species at 77 K and pressures 1, 3. 7 and 10 bars; given are SBET and Vp obtained by low temperature adsorption of N2; Dp is missing. We calculate the average layer thicknesses N2 τ = Vp/SBET, nm, take them as estimates of pore sizes and then calculate the pore shapes (H2 molecules packings) Fd = τ/σH (σH = 0.289 nm is the kinetic diameter of H2 molecules). Figure 2 shows dependencies H2 uptakes G, wt.%, on the pore size (left) and on the H2 molecules packing τ/σH(right).
It can be seen from Figure 2 (left) that in the narrowest pores (0.46 nm) the walls have a repulsive effect on gas molecules and uptake is low; an increase in pressure suppresses repulsive forces and increases absorption. It can also be seen that, starting from a pore size of 0.53 nm, G, wt.%, is inversely proportional to the size.
Figure 2 (right) shows the effect of the number of gas layers (packing of gas molecules in layers) on G. Noteworthy are two packings of H2 molecules: 1.63 and 1.85, which ensure high adsorption. Packing between them, Fdh= 1.7, is a special case, it gives markedly lower uptake (the sample is Norit R0.8 carbon). This result can be explained by the presence of two PSD peaks (see [18], Fig. 2) at pore diameters d1 < 0.49 nm and d2 > 0.49 nm, which gives an average value of d = 0.49 nm; both real diameters impair adsorption.
Returning to the work of Fiero [8], mentioned in the introductory part, we
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Figure 2. Adsorption of H2 by 7 carbon species at 77 K and 4 pressures. Dependence of uptake G on the generalized pore size Vp/SBET (left) and on the average number of H2 layers τ/σH (right). Based on the data of Taxier-Mandoki et al. [17].
believe that the decrease in the absorption of H2 at a very large (excessive) surface area and volume of micropores is explained by the formation of micropores of such small sizes that repulsive forces appear.
3) Sun et al. [19] noted that ultra-micropore sorbent (<0.7 nm) should be avoided for natural gas storage. Obviously, for methane the forces of attraction act in the range of pore diameters range of 0.7 [19] ÷ 1.3 [1] nm.
3.2. The Sorbent Energy PmV
1) Examples of PmV constancy for various volumes in conventional catalysts and sorbents are given in author’s report [SI-1]. The report is given in SI form because the publisher did not refer to it, and the only evidence in English for its existence is the review by Shchukin et al. [20].
2) Valekar et al. [21] obtained by wet granulation the spheres from MOFs: MIL-100(Fe), MIL-101(Cr), UiO-66(Zr), and UiO-66(Zr)_NH2 with ρ-Al2O3 as a binder. Total surface area SBET (m2/g), total pore volume Vp (cm3/g) from low-temperature N2 adsorption, and sphere crushing strength Pm(N) are given. We tested correlations of Pm with SBET, Vp and Vp/SBET; no correlations were found. In the same time the sorbent energy PmVp showed close relationships with SBET (logarithmic function, R2 = 0.819) and especially with size Vp/SBET (Figure 3).
Figure 3 (left) reflects the effect of pore size reduction on increased attraction of gas molecules. Figure 3 (right) shows that the energy of the material is inversely proportional to and closely related to the packing of N2 molecules (R2 = 0.975). It is noteworthy that the maximum energy (point 1.19/6.56) corresponds to τ/σ ≈ 1, i.e., to the pore shape Fd which provides a single-layer packing of N2 molecules.
3) Dhainaut et al. [22] tested tablets hardness Pm (N) of MOFs: UiO-66, UiO-66-NH2, UiO-67, and HKUST-1. As displacement V for PmV, the given SBET, Vmi, SBET/Vmi, crystal density and bulk density of tablets can be tested. We also note two facts related to Vmi: 1) Vmi of powders and tablets, made from
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Figure 3. MOF. Correlations between sorbent energy PmVp and a size Vp/SBET (left); PmVp and packing of N2 molecules (number layers τ/σ, right). Based on the data from Valekar et al. [21].
them, are about equal (see also Table 3), 2) generalized micropore size (thicknesses of probing gas) Vmi/SBET are almost equal for all powder and tablet samples. To find samples with close sorbent energy PmV graphs like Figure 4 were plotted.
On Figure 4 the highest PmSBET has UiO-67. Similar plot with highest (but little lower) energy for the same sorbent was obtained for displacement Vmi (not shown). According to authors ( [22], Fig. S6) this MOF possess the highest volumetric capacity.
3.3. Gas Energy
Mahmoud [23] in a review, on methane storage, gave total and deliverable uptakes, pressure and temperature for 15 MOFs samples, 12 porous organic polymers (POP) and 20 AC samples, their SBET and Vp. Dp is missing, so we used Vp/SBET. For MOF volumetric G, cm3/cm3, for POPs and AC, gravimetric G, g/g, capacities are presented.
a) MOF. G versus Vp, Vp(P/T), SBET, SBET(P/T) and Vp/SBET.
Among the 15 experiments with MOF, the closest dependence shows deliverable G, cm3/cm3, on the total pore volume Vp (Figure 5, left).
On Figure 5 left, the only separated point (0.94/306) belongs to the sample MOF-519 with the highest pressure in the series of 250 Bar. After excluding this sample, the remaining samples form a straight regression line with R2 = 0.8918. The gas energy Vp(P/T), as an independent variable (Figure 4, right), returns MOF-519 in the MOF family, but separates samples ST-2 (200 Bar/303 K, 200 Bar/289 K). After excluding ST-2, the rest of the samples form a straight line with R2 = 0.8849. The absence of points between the MOF-519 and ST-2 samples on this graph suggests that G = 306 cm3/cm3 is the maximum possible capacity
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Figure 4. Tableted MOF sorbents energy Pm*SBET; Pm is hardness (N) and SBET is total surface area m2/g. Based on the data from Dhainaut et al. [22].
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Figure 5. Dependence of the deliverable capacity G of the MOF on the pore volume Vp (left) and on the gas energy Vp(P/T) (right). Based on the data from Mahmoud review [23].
for the studied MOFs; at very high pressure (dense gas), repulsive forces prevent the capacitance from growing.
When SBET was taken as the displacement, the variable SBET(P/T) combined all the samples into one line G-s (not shown), but with a noticeably larger scattering, R2 = 0.7591. With the exception of MOF-519, which is furthest from the line, R2 = 0.8324 was obtained, which is less than for G vs. Vp(P/T).
G, cm3/cm3, depending on another displacement, the size Vp/SBET, is shown in Figure 6.
Figure 6 shows the absence of any relationship between the capacity of this series of MOF and the pore size. The separated point (0.15/175) is Cu-tbo-MOF 5 material (see below).
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Figure 6. MOF.G vs. Vp/SBET. Based on the data from Mahmoud review [23].
The packing of CH4 molecules (number of layers) τ/σ.
In this MOF series, the τ/σ range is 0.39 ÷ 1.47. The smallest value τ/σ = 0.39, the only value < 1, belongs to the mentioned Cu-tbo-MOF 5 sample. This MOF is characterized by “the presence of very strong CH4 adsorption sites near the organic linker with similar adsorption energetics as the open metal sites” [24]. This packing of knots and molecules resembles a Chinese cherry or a stalactite cave with repeatedly changing distances between floor and ceiling; both can serve as pore models. From the remaining data τ/σ (1.00 - 1.47) it can be concluded that these samples have a flat split shape, allowing 1 ÷ 1.5 layers of methane molecules.
b) POP.
The dependences of G, g/g, on SBET and Vp are directly proportional with R2 = 0.9465 and 0.79 correspondingly. The gravimetric capacity G, g/g, makes it possible to estimate the surface activity G/SBET, g/m2; the most active surface has COF-5. Graphs of G vs. SBET(P/T) and vs. Vp(P/T) look like in Figure 5 (left); “pushed out” sample from the series, is PPN-4. Excluding this sample, we get straight lines with R2 = 0.9787 and R2 = 0.9033 for G vs. SBET(P/T) andG vs. Vp(P/T) respectively, i.e. values higher than for G depending on SBET and Vp.
There was no evidence of an effect of Vp/SBET pore size on G, g/g. Assuming that this is due to fluctuations in the density of POPs, the gravimetric uptake of methane G, g/g, was related to cm3 of the total pore volume Vp, cm3/g. G/Vp, g/cm3, kind of volumetric capacity. The plot of G/Vp versus Vp/SBET was approximated by a power function (Figure 7).
Figure 7 shows a very sharp increase in volumetric capacity as a result of a decrease in pore size. Although the relationship between the variables does not seem very strong (R2 = 0.7338), the exclusion of the not very important (due to high Vp/SBET) COF-5 sample (point 0.77/0.144) raises R2 to 0.8483.
c) Activated carbons.
The processing conditions were: constant P = 35 Bar and T = 296 ÷ 300 K (practically constant), so the gas force in Equation (1) was constant. The
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Figure 7. POPs. Effect of the generalized pore size Vp/SBET on the volumetric capacity G/Vp. Based on the data from Mahmoud review [23].
strengths of the connections (R2) between G and displacements SBET, Vp and Vp/SBET were tested (Table 2).
Among the AC samples, MAXSORB – 3 has the highest SBET = 3140 m2/g and lowest Vp= 0.179 cm3/g, which gives the smallest generalized pore size Vp/SBET = 0.057 nm (0.15 methane layers). The latter may explain the highest G = 0.408, g/g, of this sample and suggest a stalactite pore model. Use the volumetric delivery capacity G/Vp, i.e., elimination of fluctuations in sorbent density, provides a very strong relationship (R2 = 0.9658) between this capacity and pore size Vp/SBET.
Fomkin et al. [25] studied the adsorption of H2 G, wt.%, by 5 different carbon materials at 3 combinations (a, b, c) of pressure and temperature (P/T):(P/T)a= 1.01 Bar/20.38 K = 0.050, (P/T)b = 100 Bar/303 K = 0.330, and (P/T)c = 200 Bar/393 K = 0.509. So, P/T was processing generalized variable. The SBET and Vmi of the samples were measured by the adsorption of N2 and benzene, respectively.
The SBET and Vmi of the samples fluctuated markedly: the STDs were 37% and 68%, respectively. Therefore, as an independent textural characteristic of the AC samples, we took the surface topography (roughness) [1] Vmi/SBET, cm3/m2, the STD of which was lower (29%). The dependent variable was the energy in the form G(P/T) and for each (P/T) the behavior of the function G(P/T) =f(Vmi/SBET) was studied; shapes of curves were similar. For compactness, in order to fit all the data on one graph, the average G(P/T) values for each P/T were calculated, individual G(P/T) were divided by average (normalized, subscript n) and normalized G(P/T)na, G(P/T)nb and G(P/T)nc were plotted as functions of Vmi/SBET (Figure 8).
It can be seen that the shapes of curves are similar. The closest are curves for P/T: 0.0500 and 0.509. This is one more example of similarity in PM [1].
Bambalaza et al. [26] compared the absorption H2 by MOF UiO-66 both by powder and by the same material pressed into granules. Two series of adsorption
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Table 2. Strength of correlations between AC textural properties and CH4 capacities. Based on the data from Mahmoud review [24].
G, capacity, Vp, SBET, total pore volume and surface area, R-squared values represent the scatter around the regression line.
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Figure 8. Dependences of the normalized energies G(P/T)n on the surface roughness Vmi/SBET for 3 thermobaric conditions and 5 adsorbents. Based on the data Fomkin et al. [25].
carried out at temperatures: 77 K (dense gas) and 298 K (≈ STP gas); pressures varied from 0 to 100 Bar. On Fig. 3 (left) in [25] shows that a dense gas at high pressure experiences repulsive forces (absorption decreases), but we want also to know the effect of micropore sizes.
Textural parameters: SBET, Vp, Smi, Vmi and PSD are given, but the average values of Dp and Dmi are missing. Thus, we have used the generalized average pore size Vp/SBET and the micropore size Vmi/Smi, which are the thicknesses of the N2 gas layers. Dividing the sizes by the kinetic diameters of N2 and H2 we obtain the average shape factors of pores and micropores (numbers of layers) Fdn, Fdh, i.e., packings of N2 and H2 molecules (subscripts n and h correspondingly) (Table 3).
It follows from the table that tableting the powder did not lead to a change in the micropore size Vmi/Smi. This closeness is confirmed by the PSD peaks (Fig. 2 in [25]); in the region of micropores, the pore sizes are very close for powder and tablets (3 pairs: d1 = 0.6 and 0.6 nm, d2 = 0.70 and 0.76 nm, and d3 = 1.35 and 1.35 nm, respectively). Assuming a constant Vmi/Smi and using the indicated sizes, we can estimate the individual micropore shape factors fd1n, fd2n and fd3n (Table 3).
The value of the total packings Fdn, Fdh in pellets are higher than in powder, while the packings in micropores fdn, fdh have not changed.
The correlations between gas energy Vp(P/T) and gas uptake by powder and pellets were checked using the ratios the pellet parameters to powder parameters (Pellet/Powder) (Table 4).
The pellet gas energy is 0.84 of the powder energy at both temperatures. This value is close (0.82) to the reduction in both total and excess uptake at 298 K. The lack of correlation at 77 K for both total uptake and excess led us to the idea of introducing a residual amount of gas, which is the difference between the total uptake and excess. As can be seen, the decrease in the amounts of residual gas correlates with the energy of the gas at both temperatures.
Gomez-Gualdrón et al. [7] tested H2 cryo-adsorption using the MOF series: NU-1101, NU-1102, and NU-1103, cycling between 100 Bar/77 K and one of two delivery conditions: 5 Bar/77 K and 5 Bar/160 K. Most of the data were obtained both experimentally and by simulations. We took for analysis, as a rule, the experimental data; from those obtained by simulation, we took the smallest d and largest D pore sizes as well as heat of adsorption Qa. Effect of gas energy Vp(P/T) on the total gravimetric capacity G is shown on Figure 9. Exactly the same dependences (not shown) were got for excesses at 5 Bar/77 K and at 5 Bar/160 K.
From the 9 graphs it follows that the largest both total and deliverable capacities correspond to the highest energy of the gas.
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Table 3. Textural properties of MOF UiO-66 powder and pellets. Based on the data of Bambalaza et al. [26].
Vp, SBET, total pore volume and surface area, Vmi, Smi, micropore volume and surface area, Fd, fd, molecules packing: F, total,f, inmicropores pore,n, nitrogen,h, hydrogen.
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Table 4. Gas energy Vp(P/T) and gravimetric amount H2 stored by powder and pellets under pressure P = 100 Bar, temperatures 77 K and 298 K at different stages of storage. Based on the data of Bambalaza et al. [26].
Vp, total pore volume, P, pressure, T, temperature.
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Figure 9. Effect of gas energy Vp(P/T) on the total capacity of storage adsorbents (from left to right) Nu-1102, Nu-1101 and Nu-1103 at 100 bar/77 K. According to Gomez-Gualdron et al. [7].
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Table 5. MOF pore shape factors and packings of H2 molecules. Based on simulated (d, D) and experimental (SBET, Vp) data from Gomez-Gualdrón et al. [7].
Fd, dimensionless pore shape factor, Vp, SBET, total both pore volume and surface area, Packing is number H2 molecules across pore size.
The reported total surface area SBET and volume Vp, as well as the diameters of the smallest d and largest D pores, make it possible to estimate the dimensionless shape factors (the number of layers N2) Fd of these pores and the packing of hydrogen molecules in them (kinetic diameter σ = 0.289 nm).
Comparing d and D with Vp/SBET in the Table 5, it can be concluded that the proportions of both the smallest and largest pores are very small. Fd-s says that Nu-1101, Nu-1102 have split (Fd ≈ 2) smallest and circular (Fd ≈ 4) largest pores, while Nu-1103 has the smallest pores, having ≈ 50:50% split/circular motifs, and the largest pores with ≈ spherical shape. The average thicknesses of the N2 layer (0.40 ÷ 0.44 nm) and the numbers of H2 layers (packings) are almost the same (1.37 ÷ 1.53).
The dependences of the heat of adsorption Qa on both diameters d and D (all received by simulation) have the form of straight lines with equidistant points and equal gradients, inversely proportional and having strong correlations: R2 = 0.954 for d and R2 = 0.982 for D. It is possible that small pores are located inside the largest ones. The dependence of Qa on the experimental Vp/SBET looks different, namely, as a graph in Figure 9 rotated counterclockwise to 180˚, i.e., confirms that the smaller the pore size, the higher the heat of adsorption.
4. Conclusions
The generalized variables proposed for gas storage adsorbents reveal a number of new facts. On the dependence of the storage capacity H2 on the generalized pore size, the critical size is found; at this size, the action of attractive forces on gas molecules decreases and the action of repulsive forces begins. At very high pressure, the repulsive forces prevent the capacitance from growing.
Series of MOF show independence of volumetric capacity on generalized pore size but for POPs and ACs, this dependence exists. For granular MOFs, the sorbent energy (crush strength times volume) is inversely proportional to and closely related to pore size. The maximum energy corresponds to the shape of the pores, which provides a single-layer packing of molecules. For MOF tablets, the highest capacity corresponds to the highest product of hardness and surface area.
In POPs series, no relation between the gravimetric capacity and the pore size was found, but when gravimetric capacity is converted to volumetric one, a rather strong power-law dependence arises between them. The same thing happened with the carbon series.
It has been found that the energy of the gas can correlate with residual uptake better than with the total and delivery capacity.
Models of pores in the form of a Chinese cherry tree and a stalactite cave are proposed.
Article Highlights
Energy as a generalized product of force and displacement is applied to gas storage physical adsorbents: gas pressure forms gas energy; the mechanical strength of the molded adsorbent forms the energy of the sorbent.
Any textural property of the adsorbent: surface area, volume and size of pores (micropores) can be a displacement that provides the best correlation between energy and storage capacity.
The universal pore size for all porous materials is Vp/SBET, the average layer of the probing gas.
An important result of converting a gravimetric capacity into volumetric capacity is the elimination of fluctuations in the density of the sorbent and, thereby, an improvement in the relationship between energy and capacity.
Declarations
No funds, grants, or other support was received.
Ethics Approval
No figures, tables or text passages have been used that require permission.
Nomenclature
AC: activated carbon
Dh: hydraulic circular pore diameter (m) = 4Vp(m3/g)/SBET (m2/g)
Dp: pore size (nm)
F: generalized pore shape factor (nm*m2/cm3) = Dp (nm)*SBET (m2/g)/Vp (cm3/g)
Fd: generalized pore shape factor (dimensionless) = Dp (m)*SBET (m2/g)/Vp (m3/g)
G: capacity: gravimetric g/g, volumetric cm3/cm3
GP: generalized parameter
MOF: metal-organic framework
P: gas pressure
PM: porous material
Pm: mechanical strength of granule
POP: porous organic polymer
PSD: pore size distribution
SBET: total specific surface area (m2/g)
Smi: micropore surface area (m2/g)
StA: gas storage adsorbent
T: temperature
V: displacement (in Equation (1)), any pore surface area, volume, size.
Vmi: micropore volume (cm3/g)
Vp: pore total specific volume (cm3/g)
x: independent process variable
τ = Vp/SBET: the average thickness of the probing gas layers, nm
Supporting Information
SI, examples of PmV = const for various materials, mechanical test methods and pore volumes, from the author’s report.