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Article

Fractal Strategy for Improving Characterization of N2 Adsorption–Desorption in Mesopores

by
Kunpeng Feng
1,
Gaofeng Liu
1,2,*,
Zhen Zhang
1,*,
Huan Liu
1,
Runsheng Lv
1,2,
Xiaoming Wang
3,
Ping Chang
4,
Jia Lin
4 and
George Barakos
4
1
School of Resources and Environment, Henan Polytechnic University, Jiaozuo 454003, China
2
Collaborative Innovation Center of Coal Work Safety and Clean High Efficiency Utilization, Henan Polytechnic University, Jiaozuo 454003, China
3
Key Laboratory of Tectonics and Petroleum Resources, China University of Geosciences, Wuhan 430074, China
4
WA School of Mines: Minerals, Energy and Chemical Engineering, Curtin University, Kalgoorlie, WA 6430, Australia
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2024, 8(11), 617; https://doi.org/10.3390/fractalfract8110617
Submission received: 23 September 2024 / Revised: 14 October 2024 / Accepted: 19 October 2024 / Published: 23 October 2024
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)
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Versions Notes

Abstract

:
The current studies primarily analyze the heterogeneity and complexity of mesopore structures based on low-temperature nitrogen (N2) adsorption curves and the Frenkel–Halsey–Hill (FHH) fractal model. However, these studies ignore the fact that the low-temperature N2 desorption curve can also reflect the desorption performance of the mesopore structure. In this research, novel fractal indicators for characterizing the adsorption–desorption performance of mesopores based on the fractal dimension from the N2 adsorption curves and N2 desorption curves are proposed. The novel fractal indicators I1 and I2 are applied to evaluate the adsorption–desorption performance of mesopores with pore size 2–5 nm and pore size 5–50 nm, respectively. The fractal indicator I1 shows an increasing trend with coalification, reflecting that the gas adsorption performance of 2–5 nm mesopores is enhanced with coalification. The fractal indicator I2 exhibits a trend of first increasing and then decreasing with coalification, indicating the gas desorption performance of mesopores with pore size 5–50 nm decreases first and then increases. The proposed indicators provide novel analytical parameters for further understanding the gas adsorption–desorption mechanism of porous coal-based or carbon-based materials.

1. Introduction

As a classical porous material, coal contains pores with various shapes and sizes. The pore sizes in coal cover micropores (<2 nm), mesopores (2–50 nm), and macropores (>50 nm) according to the International Union of Pure and Applied Chemistry (IUPAC) [1,2,3]. Mesopores in coal play a significant role in fluid transport and mass transfer, the preparation of coal-based mesopore materials, and energy development [4,5]. Therefore, accurately characterizing the mesopore structure is the key to a deeper understanding of the porous properties of coal and coal-based materials [6,7].
A series of measurement methods are widely used for the characterization of pore structure, including high-pressure mercury intrusion, low-temperature N2 adsorption (LTNA), low-temperature CO2 adsorption (LTCA), small-angle X-ray scattering, etc. Among them, LTNA is a mainstream method for analyzing the mesopores in porous materials [8,9,10,11]. Srisuda et al. applied LTNA to investigate the mesopore structure of amine-functionalized mesoporous silica materials [12]. Zelenka et al. analyzed the transport kinetics of carbon-based solids during adsorption and desorption in micropores and mesopores by using LTNA [13]. Horikawa et al. prepared nitrogen-doped mesoporous titania with a high specific surface area, and LTNA was used to depict the pore volume distributions of the new material [14]. In addition, LTNA is also widely used to characterize the mesopore structure of coals with different coalification [15,16].
According to the current research status of LTNA, the existing studies have mainly used parameters such as the pore size distribution (PSD), pore volume (PV), and pore-specific surface area (SSA) to quantify micropores [17,18]. These parameters (PSD, PV, and SSA) are determined by the BJH and BET methods based on the cylindrical pore assumption. In contrast, the N2 adsorption–desorption curve is the most intuitive reflection of the adsorption–desorption performance of the mesopore structure. In addition, the hysteresis loop of N2 adsorption–desorption curves is widely used for qualitatively analyzing the specific shapes of mesopores [19,20]. The mesopore structure displays obvious complexity and heterogeneity in porous media [21,22]. The complexity and heterogeneity of the mesopore structure directly affect the function of mesopore materials [23,24,25]. Furthermore, existing studies have indicated that fractal theory can effectively characterize the complexity and heterogeneity of pore structure [26,27], and the Frenkel–Halsey–Hill (FHH) fractal model has also been widely used to determine the fractal dimension of mesopore structure [28,29]. However, the fractal dimension is calculated only based on the N2 adsorption curve, ignoring the fact that the nitrogen desorption curve can reflect the desorption performance of the mesopore structure. Therefore, a novel quantitative characterization method targeting the complexity and heterogeneity of the mesopore structure is worth exploring by coupling N2 adsorption–desorption curves and fractal theory.
In this study, a typical porous medium, coal, is taken as the research object, and low-temperature N2 adsorption–desorption experiments are carried out to analyze the mesopore structure. The FHH fractal model is introduced to determine the fractal dimension based on the low-temperature N2 adsorption–desorption curves, and a novel fractal characterization method for evaluating the adsorption–desorption performance of the mesopore structure is established based on the low-temperature N2 adsorption–desorption curves. This paper proposes a fractal strategy for improving the accuracy of the characterization of the complexity and heterogeneity of mesopore structure, which provides a scientific basis for the fractal design strategy of mesoporous materials.

2. Materials and Experimental

2.1. Coal Samples

Six groups of coal samples with different coalification were used to carry out low-temperature N2 adsorption–desorption experiments. The coal samples were obtained from the following locations: Caiyuan Coal Mine (CY), Shandong Province, China; Pingdingshan No. 8 Coal Mine (PD), Henan Province, China; Xiqu Coal Mine (XQ), Yuwu Coal Mine (YW), Wuyang Coal Mine (WY), and Yuxi Coal Mine (YX), Shanxi Province, China. The coal samples were sealed and wrapped before being transported to the laboratory for further analysis. Basic industrial analysis was performed and the maximum vitrinite reflectance (Ro,max) of these six groups of coal samples was measured, as detailed in Table 1. Compared to Vdaf and FCad, the maximum vitrinite reflectance (Ro,max) is the most commonly used indicator for assessing coalification [30,31,32]. The maximum vitrinite reflectance (Ro,max) varies from 0.86% to 2.99%, covering low-, medium-, and high-rank coals.

2.2. Measurement of Low-Temperature N2 Adsorption–Desorption

The coal samples were crushed and screened to a particle size of 0.17–0.25 mm (60–80 mesh) for low-temperature N2 adsorption–desorption measurement, by the standard [33] ISO 15901-2:2022 The LTNA experiments were conducted using a MICROMERITICS ASAP 2460 (Norcross, GA, USA) with a pore size range of 0.35 to 500 nm. Adsorption–desorption curves were obtained at 77 K for relative pressures (P/P0) ranging from 0.01 to 0.99. Pore size distribution (PSD) and pore volume (PV) were determined using the Barrett–Joyner–Halenda (BJH) method, and the Brunauer–Emmett–Teller (BET) method was used to calculate the pore specific surface area (SSA) [2,34].

2.3. Calculation of Fractal Dimension for LTNA

The FHH fractal model was employed to calculate the fractal dimension based on the low-temperature N2 adsorption–desorption data, with the formulas provided in Equations (1) and (2) [35,36,37].
ln V V 0 = C + A ln ln P 0 P
D = 3 + A
where V is the volume of the adsorbed gas at equilibrium pressure P; V0 is the volume of gas in the monolayer coverage; P0 is the saturation pressure of the gas; A is a constant related to the fractal dimension D; and C is the constant.

3. Results and Discussion

3.1. Mesopore Morphology Analysis

Figure 1 illustrates the adsorption–desorption curves for six groups of coal samples, and the region enclosed by the adsorption curve and the desorption curve is the hysteresis loop. By comparing the shapes of the hysteresis loops in the adsorption–desorption curves, the six groups of coal samples can be categorized into Group A and Group B. Group A includes the coal samples CY, PD, and XQ, which have narrow hysteresis loops. Group B comprises the coal samples YW, WY, and YX, which exhibit wide and distinct hysteresis loops.
For the Group A coal samples, the coal matrix adsorbs gas molecules through van der Waals forces in the pores at relative pressures less than 0.50 [38,39]. Moreover, the hysteresis loops overlap within this relative pressure range, corresponding to semi-pores (e.g., cylindrical pores with an open end and wedge-shaped pores) [40,41,42]. When the relative pressure exceeds 0.50, the coal matrix adsorbs gas molecules through the capillary condensation effect [38,39]. A notable characteristic of the coal samples in Group A is the presence of a narrow hysteresis loop within this relative pressure range. The hysteresis loop within this range indicates the presence of open pores (e.g., cylindrical pores with two open ends and parallel plate pores with four open sides) [23,41,43]. Additionally, a slight inflection point is observed in the desorption curve of Group A coal samples at a relative pressure of 0.45–0.50, attributed to a small number of ink-bottle pores in the coal [40,44,45].
For the Group B coal samples, the coal matrix adsorbs gas molecules through van der Waals forces and the capillary condensation effect at relative pressures less than 0.50 and greater than 0.50, respectively. However, there is a conspicuous hysteresis loop throughout the entire range of the desorption isotherm for Group B coal samples, indicating the presence of open pores (e.g., cylindrical pores with two open ends and parallel plate pores with four open sides) in the coal [41,46]. Although the coal samples YW and YX exhibit prominent and wide hysteresis loops, there is a distinct inflection point in the desorption curve of coal samples YW and YX at a relative pressure of 0.45–0.50, corresponding to ink-bottle pores in these samples [40,41].
The above analysis indicates that previous studies mainly employed the hysteresis loop for N2 adsorption–desorption curves to characterize the different shapes of mesopores, qualitatively reflecting their connectivity. However, the mesopore structure displays obvious complexity and heterogeneity due to the irregular development of different sizes, shapes, and numbers of mesopores, which is critical in influencing pore connectivity. Therefore, the quantitative characterization of the complexity and heterogeneity of mesopores can enhance the accuracy of the mesopore structure.

3.2. Mesopore Structure Parameter Analysis

The difference between the isotherm at a relative pressure range of P/P0 < 0.50 and that at P/P0 > 0.50 is distinct, reflecting the differences in pore structure. According to the Kelvin equation [23,40], the pore size corresponding to a relative pressure of P/P0 = 0.50 is 5.0 nm, and the pore structure parameters for samples with a boundary 5.0 nm are presented in Table 2.
According to Table 2, the pore structure parameters (PV and SSA) for the ranges of 2–5 nm and 5–50 nm of different coal samples are shown in Figure 2, indicating that the variation in PV and SSA is closely related to the coalification. For mesopores with a pore size of less than 5.0 nm (Figure 2a), the PV and SSA initially decrease and then increase with the coalification. The PV and SSA of sample PD are the lowest, and the inflection point corresponding to this variation trend is at Ro,max = 1.30%, which reflects the effect of the second coalification jumps on the 2–5 nm mesopore structure [30,47,48]. For mesopores with a pore size between 5.0 and 50 nm (Figure 2b), the PV and SSA also show a trend of first decreasing and then increasing with the coalification. The PV and SSA of YW are the smallest, while the inflection point corresponding to this variation trend is at Ro,max = 2.20%, reflecting the influence of the third coalification jump on the 5–50 nm mesopore structure [49,50,51].

3.3. Estimation of Pore Structure Fractal Characteristics

Figure 3 indicates the fitting process of fractal dimensions D1, D2, D3, and D4 by the FHH fractal model. The fractal dimensions D1 and D2 correspond to the N2 adsorption data with relative pressures less than 0.5 and greater than 0.5, respectively, and the fractal dimensions D3 and D4 correspond to the N2 desorption data for relative pressures less than 0.5 and greater than 0.5, respectively. A larger fractal dimension indicates higher pore structure complexity and higher pore distribution heterogeneity, which would affect the mass transfer and migration of gas. The slope of each fitted line is greater than −1 and less than 0, and the correlation coefficient (R2) exceeds 0.9, indicating that the mesopores of the coal sample exhibit obvious fractal characteristics [52,53].
According to Figure 3, the fractal dimensions D1, D2, D3, and D4 are depicted in Figure 4. The fractal dimension D1 for samples CY, PD, XQ, YW, WY, and YX is 2.6943, 2.3376, 2.2474, 2.4260, 2.7344, and 2.5657. The fractal dimension D2 of samples CY, PD, XQ, YW, WY, and YX is 2.5887, 2.6219, 2.6199, 2.5469, 2.8226, and 2.7624. Furthermore, the variation trend of fractal dimension D3 from the N2 desorption data (P/P0 < 0.5) is consistent with fractal dimension D1 from the N2 adsorption data (P/P0 < 0.5). The variation trend of fractal dimension D4 from the N2 desorption data (P/P0 > 0.5) is consistent with fractal dimension D2 from the N2 adsorption data (P/P0 > 0.5). The above analysis indicates that the values of fractal dimensions estimated by the adsorption data show similar variation trends to those estimated by the desorption data, but the values exhibit differences for the same sample, as shown in Figure 4. Clearly, it is necessary to explore a quantization method for the differences in the values of fractal dimensions between adsorption and desorption data, which is conducive to comprehensively and reasonably characterizing the complexity and heterogeneity of mesopores.

3.4. Fractal Indicators for Analyzing Adsorption–Desorption Performance of Mesopore

According to the analysis in Section 3.3, the fractal dimension calculated from the N2 desorption data is generally larger than the fractal dimension calculated from the N2 adsorption. The differences in fractal dimensions reflect the complexity and heterogeneity of the mesopore structure, which affects the adsorption and desorption capacities of the mesopores for gas. Therefore, we propose using fractal indicators to analyze the adsorption–desorption performance of mesopores based on the fractal dimensions D1, D2, D3, and D4.
The fractal indicator I1 is defined to reflect the complexity of the mesopore structure with pore size 2–5 nm by the ratio of fractal dimension D3 to fractal dimension D1, as shown in Equation (3). A larger value of I1 indicates a more complex mesopore structure, which is less conducive to gas desorption. Another fractal indicator I2 is defined as the ratio of fractal dimension D4 to fractal dimension D2, as shown in Equation (4), which reflects the complexity of the mesopore structure in the range of 5–50 nm. A larger value of I2 indicates a more complex pore structure with a pore size of 5–50 nm, leading to greater transport resistance and a longer residence time of the gas in the pores after gas desorption [54], which weakens gas desorption [55].
I 1 = D 3 / D 1
I 2 = D 4 / D 2
The fractal indicators I1 and I2 of different coal samples are shown in Figure 5. Compared to I1 and I2, fractal indicator I1 is significantly greater than fractal indicator I2, reflecting that the gas adsorption performance of 2–5 nm mesopores is greater than that of 5–50 nm mesopores, which is consistent with the SSA shown in Table 2, and the SSA of 2–5 nm mesopores is generally greater than that of 5–50 nm mesopores.
Figure 5a indicates that fractal indicator I1 shows an increasing trend with coalification, which reflects that the gas adsorption performance of 2–5 nm mesopores is enhanced with coalification, which is not conducive to gas desorption. The index I1 of coal sample WY is smaller than that of YW and YX. The reason is that there are no obvious ink-bottle pores in coal sample WY, as indicated by the adsorption–desorption hysteresis loops, and the presence of ink-bottle pores is not conducive to gas desorption. The detailed reasons for the increase in the fractal indicator I1 with coalification are discussed as follows. Before the second coalification jump (Ro,max < 1.30%), the aliphatic side chains and oxygen-containing functional groups of coal’s macromolecular structure degrade and fall off with coalification [56,57,58], leading to the disappearance of a large number of pores in the aliphatic side chains [32,59]. In addition, the removed side chains and oxygen-containing functional groups form asphaltene and hydrocarbon substances that block the pores [60], which leads to poor pore connectivity, which is not conducive to gas desorption and induces an increase in fractal indicator I1. After the second coalification jump (Ro,max > 1.30%), the macromolecular structure of coal primarily undergoes pyrolysis [61,62], and pyrolysis produces a large number of pores, which causes the PV and SSA with a pore size less than 5 nm to increase. The gas adsorption capacity of the pores increases, which is not conducive to gas desorption, and the fractal indicator I1 continues to increase.
The fractal indicator I2 exhibits a trend of first increasing and then decreasing with coalification (Figure 5b), which shows an opposite trend to that in Figure 2b. When Ro,max < 2.20%, the PV and SSA of 5–50 nm mesopores decrease with coalification, indicating poor pore connectivity, which weakens gas desorption by hindering gas diffusion [63]. When Ro,max > 2.20%, the PV and SSA of 5–50 nm mesopores increase with coalification, and the pore connectivity is enhanced, which is conducive to gas desorption. The fractal indicator I2 also decreases. Therefore, the fractal indicator I2 is a key parameter closely related to the connectivity of 5–50 nm mesopores, and the pore connectivity influences gas desorption by affecting gas diffusion migration.
In summary, this research has proposed novel fractal indicators to analyze the adsorption–desorption performance of mesopore structures. The adsorption–desorption performance of the mesopore structures of different coal samples is evaluated. The proposed indicators provide novel analytical parameters for further understanding the gas adsorption–desorption mechanism of porous coal-based or carbon-based materials.

3.5. Implication

The current investigations have demonstrated the theoretical basis for the fractal design of nanoporous materials, showing that the fractal structure can dominate the fractal function of these materials [64,65]. Accordingly, an abundance of novel nanoporous materials with fractal functions have been created based on the fractal design strategy, including 3D Multifunctional Integumentary Membranes with Capabilities in Cardiac Electrotherapy [66], amorphous materials [67], Waste Biomass-Derived Solar–Thermal Materials [68], etc. In this research, we proposed a fractal strategy for improving the characterization of N2 adsorption–desorption in mesopores, which achieves a more comprehensive and reasonable characterization of the complexity and heterogeneity of mesopore structure. This study provides a scientific basis for fractal design for the development of coal-based materials, including graphene synthesis in electrocatalysis, electrode materials based on modified porous carbon composites [69,70], carbon storage, hydrogen storage, electrochemical energy storage, and electrochemical sensing, etc. [71,72]. In addition, the application of fractal indicators in research related to adsorbents/catalysts other than coal is also worthy of further analysis and exploration.

4. Conclusions

In this study, six different coal samples are utilized to investigate the mesopore structure characteristics by low-temperature N2 adsorption and the FHH fractal model. The fractal evolution characteristics of mesopores were analyzed, and a fractal strategy for improving the characterization of N2 adsorption–desorption in mesopores was proposed. The main conclusions are as follows:
(1)
The PV and SSA of 2–5 nm mesopore structure initially decrease and then increase with coalification and the inflection point corresponding to the second coalification jump. The PV and SSA of mesopores with pore sizes between 5.0 and 50 nm show a trend of first decreasing and then increasing with coalification and the inflection point corresponding to the third coalification jump.
(2)
Two fractal indicators, I1 and I2, are proposed based on the fractal dimension D1, D2, D3, and D4, which are applied to evaluate the adsorption–desorption performance of mesopores with pore sizes of 2–5 nm and 5–50 nm, respectively.
(3)
The fractal indicator I1 shows an increasing trend with coalification, indicating that the gas adsorption performance of 2–5 nm mesopores is enhanced with coalification, which is not conducive to gas desorption. The fractal indicator I2 exhibits a trend of first increasing and then decreasing with coalification, indicating that the gas desorption performance of mesopores with pore sizes of 5–50 nm decreases initially and then increases.
(4)
The proposed indicators provide novel analytical parameters for further understanding the gas adsorption–desorption mechanisms of porous coal-based or carbon-based materials.

Author Contributions

Conceptualization, K.F., Z.Z. and G.L.; data curation, K.F., Z.Z., H.L., R.L., X.W., P.C., J.L. and G.B.; formal analysis, K.F., G.L., Z.Z., H.L., R.L., X.W., P.C., J.L. and G.B.; funding acquisition, G.L.; methodology, K.F., Z.Z., H.L., R.L., X.W., P.C., J.L. and G.B.; supervision, G.L. and Z.Z.; visualization, K.F. and G.L.; writing—original draft, K.F. and G.L.; writing—review and editing, K.F., Z.Z. and G.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the National Natural Science Foundation of China (No. 42230814 and No. 42372204), the China Scholarship Council (No. 202308410549), the Henan Province International Science and Technology Cooperation Project (No. 242102520034), the Henan Province Science and Technology Research Project (No. 242102320365), and the Key Research Projects of Higher Education Institutions in Henan Province (No. 24B170005).

Data Availability Statement

The data presented in this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Fractalfract 08 00617 g001aFractalfract 08 00617 g001b
Figure 1. Isotherms of N2 adsorption–desorption for the coal samples.
Figure 1. Isotherms of N2 adsorption–desorption for the coal samples.
Fractalfract 08 00617 g001aFractalfract 08 00617 g001b
Fractalfract 08 00617 g002
Figure 2. Structure parameters of mesopores for the coal samples: (a) PV and SSA with pore diameter less than 5 nm, (b) PV and SSA with pore diameter 5–50 nm.
Figure 2. Structure parameters of mesopores for the coal samples: (a) PV and SSA with pore diameter less than 5 nm, (b) PV and SSA with pore diameter 5–50 nm.
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Fractalfract 08 00617 g003
Figure 3. Iso plots of ln(V/V0) vs. ln(ln(P0/P)) generated from N2 adsorption–desorption data.
Figure 3. Iso plots of ln(V/V0) vs. ln(ln(P0/P)) generated from N2 adsorption–desorption data.
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Fractalfract 08 00617 g004
Figure 4. Variation trend of fractal dimensions: (a) The trend of changes in D1 and D3, (b) The trend of changes in D2 and D4.
Figure 4. Variation trend of fractal dimensions: (a) The trend of changes in D1 and D3, (b) The trend of changes in D2 and D4.
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Fractalfract 08 00617 g005
Figure 5. Effect of coalification on adsorption–desorption parameter: (a) The effect of coalification on parameter I1, (b) The effect of coalification on parameter I2.
Figure 5. Effect of coalification on adsorption–desorption parameter: (a) The effect of coalification on parameter I1, (b) The effect of coalification on parameter I2.
Fractalfract 08 00617 g005
Table 1. Proximate analysis of the coal samples.
Table 1. Proximate analysis of the coal samples.
CoalProximate Analysis (%)Ro,max (%)
MadAdVdafFCad
CY2.358.1534.5754.930.86
PD0.9710.1025.1166.671.22
XQ0.6111.1418.5969.661.60
YW1.3010.219.3478.902.18
WY1.012.1210.7087.532.39
YX2.818.176.6683.292.99
Table 2. Mesopore structure parameters for various pore size sections.
Table 2. Mesopore structure parameters for various pore size sections.
CoalPV<5.0 nm (×10−2cm3/g)SSA<5.0 nm (m2/g)PV5.0–50 nm (×10−2cm3/g)SSA5.0–50 nm (m2/g)Ro,max/%
CY0.03620.54000.26090.51500.86
PD0.01590.21390.18260.26681.22
XQ0.02320.30840.15960.28261.60
YW0.03970.55640.07010.11262.18
WY0.04290.65100.07860.17102.39
YX0.06170.76120.23210.39592.99
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Feng, K.; Liu, G.; Zhang, Z.; Liu, H.; Lv, R.; Wang, X.; Chang, P.; Lin, J.; Barakos, G. Fractal Strategy for Improving Characterization of N2 Adsorption–Desorption in Mesopores. Fractal Fract. 2024, 8, 617. https://doi.org/10.3390/fractalfract8110617

AMA Style

Feng K, Liu G, Zhang Z, Liu H, Lv R, Wang X, Chang P, Lin J, Barakos G. Fractal Strategy for Improving Characterization of N2 Adsorption–Desorption in Mesopores. Fractal and Fractional. 2024; 8(11):617. https://doi.org/10.3390/fractalfract8110617

Chicago/Turabian Style

Feng, Kunpeng, Gaofeng Liu, Zhen Zhang, Huan Liu, Runsheng Lv, Xiaoming Wang, Ping Chang, Jia Lin, and George Barakos. 2024. "Fractal Strategy for Improving Characterization of N2 Adsorption–Desorption in Mesopores" Fractal and Fractional 8, no. 11: 617. https://doi.org/10.3390/fractalfract8110617

APA Style

Feng, K., Liu, G., Zhang, Z., Liu, H., Lv, R., Wang, X., Chang, P., Lin, J., & Barakos, G. (2024). Fractal Strategy for Improving Characterization of N2 Adsorption–Desorption in Mesopores. Fractal and Fractional, 8(11), 617. https://doi.org/10.3390/fractalfract8110617

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