Porous materials: microporosity induced by the shape and length of the pores

There are insufficient parameters to explain the appearance of microporosity in porous materials. One of the parameters associated with micropores is the generalized pore shape factor F, which includes, as special cases, the known slit, circular and spherical model pore shapes. F covers the shapes between the slit (F = 2000) and spherical (F = 6000), as well as beyond. For the intermediate shape, one can estimate proportions of model motifs. The transition of the shape from one model to another is accompanied by the appearance of micropores (or vice versa); corresponding dependencies are given. Nanotechnologies, such as self-assembly of ordered mesoporous materials (OMMs), the production of carbon OMMs as replicas of silica matrixes, supercapacitors made of carbon nanofibres (CNF), a hybrid CNF/MWCNT for use in lithium-ion batteries, carbon xerogels with Ni additive for storage of H2, catalysis and others are discussed; they cover materials with F in the range 1,100 ÷ 32,600. A pore surface length index Lsi is proposed for any pore shape; it supplements the generalized parameters causing stresses, deformations and micropores. Using F and Lsi, it was


Introduction
A certain size effect should be taken into account when evaluating the strength of a solid product, including nanoporous material. The statistical theory of strength explains this effect: the larger the body, the greater the likehood of defects arising under the action of the applied force.
In porous materials the expected defects were: first, part of the SBA-15 micropores [1], and then part of the intra-wall pores of OMMs [2]. The parameter representing the size was the ratio of the wall thickness t to the average pore diameter Dp.
As a parameter affecting the strength, t/D is known from the works on cellular solids [3], thin silica films [4] and even cakes [5]. The t/Dp represents dimensionless size in the radial direction of the pore; one of the goals of this work is to supplement it with a parameter acting in the axial direction.
The effect of a technological variable can be mentally replaced by a scheme of mechanical forces [6,7]. Data on simultanious changes in both radial and axial dimensions under an action (influence) of some technological variable, being mentally replaced by mechanical force, will help to better understanding the properties of the material.
Miсropores of materials play an important role in adsorption, separation and catalysis.
The microporous part of the total porosity depends on many processing parameters, in particular on the heating temperature and the preparation time of the material, as shown, for example, in Zukerman et al. [8].
It plays a significant role in many processes, for instance, it reduces the crystallization temperature of the active component in supported catalysts, makes crystals smaller and the catalyst more efficient [8].
One application of the microporous materials is the storage of hydrogen. It is known that an adsorbent for hydrogen storage should have a large surface area (SBET) and a small pore size (Dp) [9].
For a circular pore, these two parameters were combined in the adsorbent surface length Ls = SBET/πDp [2], but for non-circular pores this parameter should be adjusted.
Ls was combined with previously proposed adsorbate volume length Lv [6] into the parameter Ls/Lv, which, as was shown in [2], is equal to the ratio of the average pore diameter Dp to the pore hydraulic diameter Dh.
The Ls/Lv = 1 means that the surface is smooth and can serve as an intrinsic reference point of smooth surface [2].
There is a need for knowledge of intermediate (between slit and circular, circular and spherical) pore shapes, as well as shapes beyond the factors of 2000 and 6000. It can be expected that the transition from one shape to another can be accompanied by stress and deformation that cause micropores: pits, cracks and crevices Another detail of the shape that affects the efficiency of materials is the corrugation of pore walls [12], which should also be taken into account. Corrugation pipes to improve heat transfer are widely used in industry [13]. Thus, there is a need for a generalized factor that can cover all forms of pores.
The main goal of this article is to supplement the previously proposed [2] generalized texture parameters of porous materials with new ones and to detect some unknown properties or behavior of porous materials using the proposed parameters.

Result: new generalized parameters.
The generalized pore shape factor F.
Let Dh4 denote the hydraulic diameter of the circular pore Dh4= 4000Vp/SBET, that is, the diameter obtained from expression with a factor 4000. Accordingly, similar expressions with factors of 2000 and 6000 for slit and spherical pores will give Dh2 and Dh6.
We will expand the formula for the hydraulic pore diameter for pores of any shape by introducing the factor F of any (i.e. general) pore shape as a factor for Vp/SBET (in particular, F2 = 2000, F4 = 4000 and F6 = 6000 -these are discrete factors of the slit, circular and spherical geometric shapes, respectively). A similar expression was proposed in the field of cements [14], but without any connection with rouphness and microporosity. F gives a continious series of pore shapes. In particular, formula (2) allows us to estimate how close the real average pore shape is to the known simplest regular geometric shapes (with smooth surfaces [2]): slit, circular and spherical.
The rouphness and shape of the pores are closely related: Dp/Dh2 = F/F2, Dp/Dh4 = F/F4 and Dp/Dh6 = F/F6. The closer the ratio to unity, the closer the pores shape to one of the mentioned regular geometries, and the surface to smooth; the farther the ratios from unity, the more the pore shape becomes transitional from one regular shape to the neighboring one and the more the surface becomes rough.
One can imagine the transition of the slip pores into cylindrical as a transition of a smooth roofing panel to corrugate. That is, the transition of shapes is accompanied by a change in roughness (and vice versa).
The difference between F of the slit and circular, circular and spherical shapes is 2000, so Pore surface length index Lsi = SBET/Dp.
As mentioned above, SBET and Dp were combined in the surface length Ls = SBET/πDp [2] for the circular pore. For the non-circular pores this parameter should be adjusted. Excluding π, the SBET/Dp ratio can be ranked as one of the generalized texture parameters (GTP) of any porous material. Its change induces stress and strain in the porous material in the axial direction. We call it the pore surface length index and denote Lsi.

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Changes in both the axial dimension of Lsi = SBET/Dp and the radial dimension of Dp itself provide additional information about the material. Simultanios axial and radial tension of the material under the action of a force (or technological parameter) may indicate its behavior similar to auxetic material [6]. You can also expect that a change in pore length will affect the shape of the pores and the proportion of micropores.
The t/Dp ratio as a generalized parameter.
The t/Dp ratio as a parameter related to the strength of materials is known in various fields [1,3,4,5]. The appearance of F allows us to transfer t/Dp to the rank of a dimensionless generalized texture parameter (GTP). Dividing Vw/Vp (GTP-1, [6]) into Dp/Dh (GTP-2, [2]) and F, where Vw = SBET*t and Dh = FVp/SBET, we obtain (t/Dp). In other words, dividing one GTP into two others gives another generalized texture parameter, which we will denote as GTP-4. Thus, t/Dp can be considered not only as a parameter that causes stresses, but also as an ordinary texture parameter indicating the nuances of the formation of OMM, regardless of the stress state of the material.

Range of pore shapes.
To track the effect of processing variables on pore shape, let's start with the data of Putz et al. [15]. They synthesized 18 samples of MCM-41 by hydrolysis of TEOS in water and a mixture of water with 2-methoxyethanol. Technological variables were: catalyst (either NH3 or NaOH), template (CTAB, DTAB or a mixture thereof), post-synthesis heating temperatures ( o C) and time (hours) (60°C-9 h, 500°C-6 h, 700°C-6 h). The name of the sample is made up of preparation conditions, for example CTAB-NH3-60. A combination of NaOH with 700°C has been reported to result in a loss of ordered structure.
The texture parameters SBET, Vp and Dp were presented. We added shape factors F and length indices Lsi. The F range was 1280 ÷ 6126, that is, the F series ranged from less than 2000 (the shape of the slit model) to more than 6000 (the shape of the spherical model).
Sorting data by F as a key you can select processing variables that provide the desired pore shape. Sequential data sorting by template, catalyst, and temperature as keys shows that temperature has the most pronounced effect on pore shape: serial samples heated at 60, 500 and Three structural types of CNF were studied: herringbone HCNF1-from Ni/Al2O3 + methane, HCNF2 -from the same catalyst + propane and platelets (PCNF), as well tubular CNF; in tubular CNFs graphene layers are parallel to the filament.
Two samples of CNTs were made: dried (CNT) and calcined at 450°С (CNT450). "Among the studied CNFs, the highest capacitance value was obtained for the platelet-type CNFs; tubular CNFs exhibited the lowest capacitance value" [16]. The average values of pore diameter Dav (Table 1) were calculated by us as arithmetic mean of the diameters of the DpTEM interval measured by the uthors using the TEM method [16]. The F and Lsi were added to the original data (Table 1). DpTEMpore diameter, TEM method; Davmean value of DpTEM, SBET and Vptotal pore surface area and volume; Fpore shape factor, Lsi = SBET/Dp -pore surface length index.
From the data of tubular CNTs it is seen that the spherical shape model (F = 6000) is the most suitable for them, but according to synthesis procedure they are tubular. We suggest that they are corrugated long circular pores.
Annealing CNT at 450 o C makes it slightly more corrugated. F-s of CNF, created by graphene layers, are 2-3 times higher as CNT; they are wider (Dav) and shorter then CNT-s. The shape factor F = 15494 of platelet CNF is the highest. Thus, in this case the highest capacity corresponds to the highest shape factor.
The following example demonstrates shape transitions. Jeong et al. [17] reported the production of mesoporous (high Vme) CNFs from corn starch reinforced by MWCNT for use this hybrid in lithium-ion batteries. In addition to regular pore parameters, they presented both micropore and mesopore volumes Vmi, Vme. We hypothesized that in such a hybrid, the Vmi/Vme ratio may be related to the shape of the pores.
In the work, firstly, two CNF samples were prepared by carbonization of starch at 700 and 1400 o C (Supporting Information File 1, in short SI-1, table S1-1). The pore shape factor in both samples was F ≈ 4000, which indicates a circular pore with a smooth surface.
Indeed, the authors report a smooth surface, but indicate the shape of the pores as a "more stable ladder configuration". An increase in temperature practically did not change the pore shape and 60 min. Sample "Before" (before activation) was mainly microporous (Vmi/Vme =21). Carbon etching leads to an increase in the pore diameter Dp in direct proportion to the etching time over its entire range (SI-1, Fig. S1-2).
After 15 min of etching, all parameters increased, especially SBET. We can see the simultanious expansion of the pore walls in both radial and axial directions, which is typical for a material with a positive Poisson number (auxetic material) [6]. The authors' idea [17] about the ladder construction of the material seems realistic : the steps rotate, mesopores appear (Vmi/Vme = 3), F becomes 6345, that is, with spherical surface areas. (Another model that comes to mind is the mesh: the filaments sometimes converge, leaving micropores, and then diverge, forming mesopores).
Steps From (SI-1, Table S1-2) it can be seen that materials having the same F ≈ 8000: CNT-30m and CNT-60m are very different in SBET, Vp and Dp, however Vmi/Vme (Fig. 1 below) makes their similar. Please note that the most effecient battery sample CNT-30m (F = 7527) has equal volumes of microand mesopores (Vmi/Vme = 1) (perhaps that makes it the best).
It folows from Table 2  Thus, we examined cases with F from 1300 to 32650.
Sieving based on the shape of molecules (as opposed to sieving based on the size of molecules).
We assumed that in the case of complex probe molecules, the shape of the pores of the adsorbent can play the role of a sieve for the separation of molecules in accordance with their shape.
Serra et al. [19 ]  Me-FDU-12). In both series, a sample of SA-amorphous mesoporous silica with largest pore diameter (27.9 nm) and disordered structure was included as a reference point; it provides a highest load (45 mg/g) and, according to our calculations, had close to circular pore shape (F = 3152).
From both series, we select samples with a close average pore diameter ( to exclude the effect of size) and a similar surface chemistry (unmethylated, i.e without "Me") namely namely (in ascending uptake) SBA-16, FDU-12, KIT-6 and SBA-15 (Fig. 2).

Fig. 2:
Lipase loading,%, depending on the pore shape factor F. Materials, placed on an ascending loading: SBA-16, FDU-12, KIT-6 and SBA-15. Based on data from Serra et al. [19 ]. Figure 2 shows that the more complex the shape of the sorbent (the larger the F value), the less lipase can enter the pores. For lipase, as a probe medium, these sorbents are molecular shape sieves.

The effectiveness of chemical reactions and the shape of the pores.
Landau et al. [20] showed the effect of the content of active sulfated tetragonal ZrO2 in the SBA-15 host on: i) the yield of MTBE in the condensation reaction of t-BuOH and MeOH and ii) the conversion of isopropanol in iso-PrOH (dehydration reaction) (  It can be seen that the increase in the catalyst efficiency can be explained not only by an increase in the mass of ZrO2%, but also by an increase in F from 3058 to 4032. As a result, the ZrO2 islands coalesce into a continuous circular surface, and ZrO2 is better dispersed. The ratio F2/F1 = 1.32 is even closer to the ratio of conversion degrees (in %), i.e. 1.45 (1.43), than the ratio of ZrO2 contents (1.24), i.e. ZrO2 influences not only by its mass but also by changing the pore shape. In this case, the best pore shape is circular.  Tcalcination temperature, ACactivated carbon, SBETtotal surface area, Vptotal pore volume, Lsi = SBET/Dpsurface lenghth index, Vmimicropore volume, Dpaverage pore diameter, F = DpSBET/Vppore shape factor. Table 4 shows that an increase in the calcination temperature leads to a decrease in the diameter Dp, but the pore length index Lsi increases; induced stresses cause an increase in the proportion of micropores in the total pore volume and the transition of the pore shape from circular to slit. The AC yield is direct proportional to the pore shape factor (not shown); micropore volume (%) is inversely proportional to F (see graphical abstract).
From the graph (not shown) of the effect of the calcination temperature T on F (direct proportionality), it can be seen that T ≈ 740°C provides a circular pore shape (F = 4000), which, in turn, corresponds to Vmi = 84% and pore size 1.8 nm.
Silica SBA-15 is often used as a matrix in AC synthesis. Zukerman et al. that is, after 1 day heating, the pore shape was spherical, but an additional treatment for 2 days led to the transformation of ≈ 50% of the surface into a cylindrical one, which was due to a decrease in micropore volume.
Ponomarenko et al. [22] prepared SBA-15 silica samples in which pore shape factors ranged from 4539 to 9263, that is, from a slightly tortuous circular shape to a very corrugated.
Then, the samples were used as templates for the preparation of carbon OMMs. The question arises -to what extent do the replicas Frep inherit the Ftem shape of silica nanotubes ?
We found that the pores of the carbon replica are clustered around Frep = 4500 (75% of circular), while the pores of their silica templates are grouped around Ftem = 8000. Based on the original and our calculated data, we can conclude that carbon, deposited inside silica, behaves like a compressed spring. After the silicate is dissolved, the carbon channels elongate (up to 2.5 ÷ 6.5 times), and their wall thickness becomes 2.0 ÷ 9.8 times thicker than that of silica; as a result, their shape becomes close to circular.
The same carbon behavior was observed in Lee al. [23] study. The matrix was 2D hexahonal mesoporous silica MSU-H, the carbon precursor was sucrose mixed with boric acid as an antifoam [6] and only variable; 7 samples were synthesized. In addition to SBET, Vp and Dp, they mesured micropore volume V<2 nm. Our calculations showed that each m 2 of the surface area contains the same micropore volume, i.e., the V<2 nm/SBET ratio is 0.000425 cm 3 /m 2 for all samples, STD = 1.03 %. Range F was 3175 ÷ 7123; F correlates with V<2 nm only in low F range (3175 ÷ 4800), but there are clear inverse-proportional relationships in the entire range between F and t/Dp (Fig.3) as well as between F and Lsi (not shown), that is growth F with thinning and elongation the pore walls. It was found again that regardless on the matrix and carbon precuror the value V<2 nm per m 2 of surface area SBET remainds constant, namely 0.0003 cm 3 /m 2 with STD = 8.2 %. Common to these samples is material of the templatessilica. A similar ratio V≤1 nm/SBET fluctuates greatly, its STD = 43.8%. In the same time V≤1 nm correlates with F in the range 4400 ÷5500 (Fig. 4); the transition of the pore shape from circular to spherical leads to the appearance the smallest pores (or vise versa).
Bastos-Neto et al. [26] looked for commercial and non-commercial (random sampling) activated carbon (AC) as a methane storage; 10 samples were tested. The Vmi micropore volume was estimated from the N2 adsorption -desorption isotherms by two methods: Dubinin-Radushkevitch (VmiDR) and Monte-Carlo molecular simulation (VmiMC).
The two methods gave very different Vmi values, so we checked the dependencies CH4 uptake (mg/g) vs. both Vmi (cm 3 /g). The DR method gave a cloud of points (no correlation), while the % Ni were highly dispersed on the carbon surface. CO2 conversion rates were from 20% to 65%.
Using the presented SBET, Vp and Dp, we found that all 8 carriers and catalysts have F-s = 3995 ÷ 4005, that is, "pure" circular pore shapes, therefore Ni deposition on carbon did not change their circular shape and smooth surface; so the difference in the activity of the samples cannot be explaned by F. At the same time, the conversion shows an increase with increasing particle size of Ni, which is unusual. The activity of the catalysts was explained by "the overflow effect of hydrogen for CNTs supports" [27]. So we started learn the CNTs.
The Ni/CNT parameters were recalculated per 1 g of CNT [27], that is, they were divided by 0.88 (minus 12 % of Ni), therefore in this case the "cat" index means carbon inside the catalyst. Then, the relationship between the catalyst/support ratios for SBET, Vp, Dp and Ls of samples with % CO2 conversion were checked. The most visible was the graph of % CO2 conversion as a function of the ratio (Ls cat/Ls supp) (that is, depending on the stresses arising from the decrease in the length of the surface of the support after the introduction of Ni) is shown below (Fig. 6).   . 7), we found that the best sample has the lowest F (3873), which corresponds to a circular pore shape. Arshad et al. [29] used an empty fruit bunch as a raw material for the preparation of AC as a H2 storage. The procedure includes a calcination of bunch in a stream of CO2 followed It can now be shown (for the sample 3A) that an increase in the surface area of micropores leads to a decrease in spherical motifs and an increase in cylindrical (up to 25%) motifs of pore shape (Fig. 8). Fig. 8: The relationship between the surface area of micropores Smi and the pore shape factor F as a result of washing freshly prepared SBA. Based on data from Thielemann et al. [30].
We assume that this constancy indicates that both parameters are estimated from the same N2 adsorption isotherms. A useful number is Standard Deviation (%) of V<2 nm/SBET, which shows the homogeneity of the samples in the series. The smallest STD = 1.03 % was obtained for the released "compressed spring" of the active carbon replica in the Lee study [23].

Conclusion
Three new generalized texture parameters (GTP) of porous materials have been proposed.
The pore shape factor F is dimensionless contineous parameter covering the total surface area, total pore volume and pore size in their entire range. Special cases are F of slit, circular and spherical pores. Change F manifests itself in the creation of new surface motifs which is accompanied by the appearance of stresses, deformations, new microporores or healing existing ones.
The t/Dp ratio, which was known as the parameter causing wall stresses in the radial direction, was transfered to the status of the generalized parameter, and now it is a regular texture parameter.
The Lsi = SBET/Dp ratio, named the "surface length index", was proposed as parameter that causes stresses and wall deformations in the axial direction for pores of any shape.
All proposed GTPs provide a better understanding of the formation and behavior of porous materials. Carbon samples behavior similar to auxetic materials were discovered. For molecules of complex shapes, molecular sieving according to their shapes has been proposed and demonstrated.