Introduction

The study of gas flow in nanoconfined spaces is fundamental to a variety of applications including natural gas extraction, microfluidic devices, energy storage, etc. Previous investigations have shown that the nanoconfined gas flow is different from the bulk gas flow due to the gas-wall interactions. In addition to the viscous flow (the type of gas transport in macropores), there are other types of gas transport in nanopores including slip flow, Knudsen diffusion, and surface diffusion, which are related to (or dominated by) the gas-wall interaction. All these types of gas transport contribute to the mass flux in the nanopores. Two major concerns related to the nanoconfined gas transport are: (1) the determination of the type of free gas transport and (2) the quantification of the contribution of each type of gas transport to the total gas flux.

A common way to classify the gas transport is by the Knudsen number (Kn), which is defined as the ratio of the mean free path (MFP, denoted by λ) of gas molecules between collisions and the pore diameter or width1,2. The classification of the gas transport widely reported in literatures are summarized in Table 13,4,5,6. However, there remains two important issues needed to be clarified for accurately determining the type of nanoconfined gas transport using Table 1. The first one relates to the determination of the real value of Kn. According to the definition of Kn, both the MFP of gas molecules and the pore size affect the value of Kn. The pore size can be obtained from experimental results7, while the MFPs of gas molecules in nanopores are much more difficult to be determined directly from experiments. For the bulk phase of real gas molecules with the molecular diameter denoted by D, the MFPs (λ) can be obtained based on kinetic theory, and is given as follows8,9:

$$\lambda =\frac{\mu }{\rho }\sqrt {\frac{{\pi M}}{{2ZRT}}}$$
(1)

where µ viscosity of gas, Pa·s, ρ is the gas density, kg/m3, M is the molecular mass of gas, kg/mol, Z is the gas compressibility factor, R is the gas constant (8.314 J/mol/K), and T is the temperature, K. Equation (1) has been widely used to calculate the MFP of gas (methane (CH4)) and corresponding values of Kn in nanopores5,10,11. However, as noted by Xie et al.12, the results derived from Eq. (1) does not represent the true value of MFP within pores. This is because Eq. (1) was formulated considering only the interactions among gas molecules, without taking into account the interactions between gas molecules and the pore surfaces. Hence, it is necessary to find a way to obtain the real MFP of gas molecules in the nanopores pores.

Table 1 Classification of free gas flow regime and type of free gas transport based on the Knudsen number.
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The second issue related to Talele 1 is the specific value of Kn (first column in Table 1) classifying the flow regime and the type of gas transport. As pointed by Chambre et al.4 and Beskok et al.13, the classification presented in Table 1 is based on empirical information, and its universality has not been proved, meaning the limits of Kn between different flow regimes may vary with pore size or flow conditions13.

Another import issue related to the nanoconfined gas transport refers to the contributions of each type of gas transport to the total mass flux, which directly determines the permeability of porous media. Many analytical models describing the nanoconfined gas flow have been proposed to predict the permeability, especially in terms of the prediction of permeability of shale gas formations14,15,16,17,18. In all these models, there are always some unknown parameters (or empirical coefficients) needed to be determined based on specific assumptions, or be determined by matching the calculated permeability values with the experimental results of flow tests in shale cores19,20. The matching results may not be unique and lack of universality. Hence, the model may fail to predict the accurate mass flux of each type of gas transport.

The prerequisite for quantifying the contribution of gas transport to the total gas flux is obtaining the mass flux of each type. The total mass flux (or velocity profile) in nanopores can be obtained by performing simulations including Lattice Boltzmann Method21, Monte Carlo simulation22, and molecular dynamics (MD) simulations23,24,25,26,27. The mass flux of each type is more difficult to be determined. Yin et al.28. have proposed a model, where the total flux is divided into four parts: the surface diffusion flux, the induced free flow flux caused by surface diffusion, viscous flow flux, and slip flow flux. This model implies that the velocity slip of the free gas results from the surface diffusion (induced free flow flux) and pressure gradient (slip flow flux), while the Knudsen diffusion is not considered. Yu et al.29 have performed MD simulations, and analyzed the velocity of methane in nanopores. They calculated the mass flux of viscous flow, Knudsen diffusion, and surface diffusion based on the velocity profile. They found that the mass flux of Knudsen diffusion and surface diffusion can account for significant proportions in the total mass flux rate at low pressures. This finding is based on the model of mass flux they proposed, where the velocity slip of free gas was assumed to originate from the Knudsen diffusion. The rationality of this assumption needs further discussion, because the velocity slip can not only result from Knudsen diffusion, but can also result from high pressure gradient (that is, viscous flow)24,30. Shan et al.31 divided the total mass flow into free gas flow and adsorbed gas flow. The velocity slip was originated from the surface diffusion and pressure gradient, and Knudsen diffusion was not considered. There are also some other flow models proposed to analyze the contributions of each type of gas transport32,33,34, and the methods are basically the same as the models mentioned above. The results of these models indicate the contributions of each type of gas transport are highly dependent on the method of dividing the total mass flux into several types, and thus may vary with different methods.

The main objective of this study is to precisely determine the type of gas transport in nanopores and accurately quantify the contributions of each type of gas transport to the total gas flux from molecular perspectives. Based on the analysis above, one can found that MD simulation is a powerful method to obtain the MFPs (or Knudsen number) and the gas flux in nanopores. We performed MD simulations to get the trajectories of gas molecules (CH4) in nanopores under different flow conditions. The flow regimes of CH4 were determined by analyzing the gas-gas collisions and gas-wall collisions. The real MFPs of CH4 in nanopores were calculated by analyzing the trajectories of each CH4 molecule, and the corresponding real values of Kn were thus obtained. By matching the flow regime and its corresponding Kn, the accurate limit of Kn for the flow regime can be obtained. Further, the velocities of viscous and slip flow, Knudsen and surface diffusion were calculated so that the contributions of each type of gas transport to the total flux were obtained. The reminder of this paper is organized as follows. In Section “Methodology”, the molecular models of CH4 in nanopores are constructed, and the simulation methods are described. Results and relevant discussion are presented in Section “Results and discussion”. Finally, in Section “Conclusions”, conclusions are drawn based on the analysis of results.

Methodology

The molecular models in this study were built in the commercial software Material Studio35, in which the model Visualizer Tools was used to construct nanopores and CH4 models, and the model Amorphous Cell was used to pack CH4 molecules of specific density in the nanopore. Further, the open-source software LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) was used to perform MD simulations. Details of the model construction, methods of MD simulations and MFP calculations are presented in the following part of this section.

Molecular models

In this study, graphene slits were used to build nanopores with different width, and the gas molecules were CH4. Figure 1 shows the molecular models used to perform MD simulations. Figure 1a presents the model of bulk phase of CH4 molecules, depicted by orange balls. The model of one graphene slit is shown in Fig. 1b, where the gray balls represent carbon atoms, and the sticks are bonds between atoms. The length (Lx) and width (Ly) of the slit are equal to 60.1 Å. Figure 1c shows the nanopore with a specific number of CH4 molecules, and the distance between the two graphene slits is denoted by Lz. The z-axis coordinates of the bottom and upper graphene walls are equal to zero and Lz, respectively. Denoting the diameter of carbon of graphene as dg, the pore width denoted by w is equal to Lz − dg. The number of CH4 molecules varies with the pore width and gas pressure. To guarantee the convergence of simulation results, at least 100 CH4 molecules (in the model with Lz = 2 nm at 1.7 MPa) were used in simulations. More CH4 molecules were used in all other simulations.

Fig. 1
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Molecular models: (a) bulk phase of CH4, (b) graphene slit, and (c) nanopore filled with CH4 molecules.

Simulation methods

For all MD simulations, the TraPPE-UA (Transferable Potentials for Phase Equilibria-United Atom) forcefield was used to describe molecular interactions of CH4 in MD simulations, and the AMBER96 forcefield36,37 was applied to describe the interactions of carbon atoms of the graphene slit. The Lennard-Jones potential was applied for non-bonded interactions between atoms, and the Lorentz-Berthelot mixing rule38 was used to describe the non-bonded interactions between different atoms as shown by Eq. (2)

$$\begin{gathered} {U_{{\text{NB}}}}=4{\epsilon _{ij}}\left[ {{{\left( {\frac{{{\sigma _{ij}}}}{{{r_{ij}}}}} \right)}^{12}} - {{\left( {\frac{{{\sigma _{ij}}}}{{{r_{ij}}}}} \right)}^6}} \right] \hfill \\ {\sigma _{ij}}=\frac{1}{2}\left( {{\sigma _{ii}}+{\sigma _{jj}}} \right),{\epsilon _{ij}}=\sqrt {{\epsilon _{ii}}{\epsilon _{jj}}} \hfill \\ \end{gathered}$$
(2)

where UNB denotes the non-bond interaction energy between atom (or molecule) i and j, kcal/mol; \(\epsilon\)ij is the energy parameter, kcal; σii (σjj) is the diameter of gas molecule or wall atom, Å; σij is the minimum distance between atoms (molecules) when collision occurs, Å; rij denotes the distance between two atoms, Å. The parameters for CH4 molecule and the carbon atom of graphene are listed in Table 2, showing that the diameters of CH4 molecule and carbon atom of graphene (dg) are 3.73 and 3.4 Å, respectively. All non-bonded interactions were truncated at 14 Å. The timestep of MD simulations was set to 1 fs.

Table 2 Non-bond interaction parameters of CH4 and carbon of graphene.
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The nanopore with CH4 molecules inside was used to perform MD simulations and MFP calculations. The positions of all atoms of graphene slits were fixed during simulations. A fixed number of CH4 molecules were inserted into the space between the two graphene slits at a specific temperature. Since the number of CH4 molecules, the volume of nanopore, and temperature were specified, the pressure of the confined gas was also determined and can be calculated during NVT simulations. Then, the model was optimized by minimizing the system energy by a Minimization task in LAMMPS. After that, a 1 ns constant-temperature, constant-volume (NVT) simulation was performed to get the equilibrium structure (Fig. 1c). A Nose-Hoover thermostat was applied to CH4 molecules to control the temperature. After that, another 100 ps NVT simulation was performed to get the trajectory of CH4 molecules. To detect the collisions between molecules, the dumping frequency of the trajectory was set to 100 fs. This means that total 1000 frames of the trajectory of each simulation. In each trajectory, the positions of all CH4 molecules are stored for further analysis. The number of CH4 molecules (at least 100), the timespan of equilibrium simulation (1 ns) and the number of frames of each trajectory (1,000 frames) of production simulation can guarantee the convergence of simulation results.

Methods of the collision detection and MFP calculation

The schematic of the calculation of free path between collisions is depicted in Fig. 2. The blue and yellow circles represent two gas molecules, and the orange planes denote the pore walls. The dashed arrow lines denote the trajectories of the gas molecules. When the distance between two molecules (or the distance between gas molecule and the wall) is equal to or less than the collision diameter, i.e., rij ≤ σij (or rjw ≤ σgw), the two gas molecules (or gas molecule and the wall molecule) collide. As depicted in Fig. 2, Molecules i and j collide with each other at positions Ai and Aj, respectively. Then, the two molecules depart from each other, and Molecule i (j) travels from Ai (Aj) to Bi (Dj), where another collision occurs. Based on the definition of the free path (the distance traveled by a molecule between two successive collisions), the free path of Molecule i (j) between the collisions at Ai (Aj) and Bi (Dj) is the distance of Molecule i (j) traveled from Ai (Aj) to Bi (Dj). Further, for the travel of Molecule i (j)from Bi (Bj) to Ci (Cj), the distance between the two molecules rij is less than the collision diameter σij, and the travel distance between the Bi (Bj) and Ci (Cj) is not included in the calculation of MFP. At Ci (Cj), where rij > σij, another free travel starts and point Ci (Cj) is viewed as the start point of the next free path. Hence, for Molecule i, the total free path from point Ai to point Ci, the total free path is equal to lAiBi, where lAiBi denotes the length of the trajectory path from point Ai to Bi. Similarly, the total free path of Molecule j from point Aj to point Cj is equal to (lAjDj + lEjBj), where lAjDj (lEjBj) denotes the length of trajectory path from point Aj (Ej) to Dj (Bj).

Fig. 2
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Schematic of the calculation of the free paths between collisions in MD simulations.

Based on the description above, the MFP of all gas molecules is as follows:

$$\lambda =\frac{{\sum\nolimits_{{i=1}}^{N} {{l_i}} }}{N}$$
(3)

where li is the mean free path of molecule i (i = 1, 2, 3, …, N) obtained by dividing the total free path and the total number of collisions of molecule i in the time span of the simulation, and N is the total number of gas molecules. It is worth noting that, the MFP obtained by Eq. (3) considers the effect of gas-wall collisions. As pointed by Xie12, when the gas molecule is bounded by walls, the MFP calculation should consider the gas-wall collision, and thus we also considered the gas-wall collision in calculating MFP by MD simulations.

Determination of the collision types and contributions to the total flux

As mentioned in Introduction section, the types of gas transport in nanopores include viscous flow, slip flow, Knudsen diffusion and surface diffusion as depicted in Fig. 3. The first three types describe the free gas transport and the surface diffusion describe the adsorbed gas transport. The surface diffusion (denoted by the blue dashed arrows in Fig. 3) originates from the hopping of gas molecule on the pore surface from one adsorption site to another vacant adsorption site. The three types of free gas transport, as pointed out by Cunningham et al.39, can be identified by different types of molecular collisions, and the detailed descriptions are as follows.

  1. a.

    When the gas density is high (Fig. 3a), the distance between gas molecules is short. The gas-gas collision (black dashed arrows in Fig. 3a) takes the major proportion in the total gas collisions (sum of gas-gas and gas-wall collisions), and thus is the main factor affecting the gas flow. The effects of pore wall on the gas flow originate from the gas-wall collision (green dashed arrows in Fig. 3a). The rebounded molecules (from the wall after gas-wall collisions) are diffusely reflected and thus loss the component of the moment along the flow direction produced by pressure gradient39. When all the rebounded molecules from the wall collide with the surrounding molecules “near” (not “on”) the surface (black arrows starting from the wall in Fig. 3a), the momentum of the rebounded molecules is progressively and smoothly transferred to the free gas, leading to the decrease of gas velocity. This type of flow is defined as viscous flow.

  2. b.

    When the density is medium (Fig. 3b), after the gas molecules collide with the pore wall, the rebounded molecules may not collide with the adjacent surrounding fluid molecules near the wall, but can flow back to the free phase (black dashed arrows originated from the end of the green dashed arrows on the pore wall in Fig. 3c). This type of gas transport is called slip flow.

When the gas density further decreases to a low value (Fig. 3c), the main type of collisions are the consecutive gas-wall collisions as depicted by the green dashed arrows between the two walls in Fig. 3c. This type of gas transport is called Knudsen diffusion.

Fig. 3
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Schematics of flow regimes of different gas densities in a pore: (a) high density; (b) medium density; (c) low density.

Based on the description above, one can conclude that: (1) the viscous flow means that the rebounded molecules from the wall collide with the surrounding molecules near the surface; (2) the slip flow denotes that parts of the rebounded molecules from the wall fly to the free gas area; (3) Knudsen and surface diffusions are driven by the molecule-wall diffusion. In the following part of this section, we introduced the method of analyzing the collisions gas molecules to determine the type of free gas transport.

Since the gas-gas (wall) collisions of viscous flow, slip flow, Knudsen and surface diffusion occur in different areas in a pore, it is necessary to define the specific position of each area. Figure 4a shows the schematic of the areas in a pore: the dark gray and blue balls denote the wall atoms and gas molecules, respectively; the areas colored by grey, green, and light green represent adsorption layer, near surface area, and free gas area, respectively. σgg denotes the diameter of gas molecules and σgw is the minimum distance between the gas molecule and the wall atom when the gas-wall collision occurs (collision diameter). The collisions occur in the pore can be divided into gas-gas and gas-wall collisions, which can be further divided into viscous, slip, Knudsen collisions for free gas molecules and surface collisions for adsorbed gas. The table in Fig. 4b summarizes all occasions of the gas-gas (wall) collisions. The rebound (free) gas in the table refers to the gas molecules rebounded (not rebounded) from the wall. The relevant detailed descriptions are as follows:

  1. (1)

    Viscous collision: includes gas-gas collisions between free gas molecules in all areas (A in Fig. 4a), gas-gas collisions between rebounded and free gas molecules in the near surface area (B in Fig. 4a), and the gas-wall collisions between the free gas (whose last collision is the gas-gas collision in the near surface area) and the wall (F in Fig. 4a).

  2. (2)

    Slip collision: includes the gas-gas collisions between rebounded and free gas molecules in the free gas area (C in Fig. 4a), and the gas-wall collisions between the free gas (whose last collision is the gas-gas collision in the free gas area) and the wall (G in Fig. 4a).

  3. (3)

    Knudsen collision: refers to the rebounded gas-wall collisions between the two walls (D in Fig. 4a).

  4. (4)

    Surface collision: refers to the rebounded gas-wall collisions on the same wall (E in Fig. 4a).

Fig. 4
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Classification of gas-gas and gas-wall collisions: (a) schematic of different areas and various types of collision in a pore and (b) summary of all types of collisions.

According to the previous works6,14,20, the velocity contribution of each type of gas transport can be characterized by collision proportions. For example, the ratio between the velocity of viscous flow and the total velocity is equal to the proportion of the viscous collision among the total collisions. Based on this principle, we analyzed the proportions of gas collisions shown in Fig. 4. For a specific time span, the numbers of viscous, slip, Knudsen and surface collisions are denoted by Nvis, Nslip, NKnu, and Nsurf, respectively, and the corresponding proportions (denoted by pvis, pslip, pKnu, psurf, respectively) can be calculated as follows:

$${p_{{\text{vis}}}}=\frac{{{N_{{\text{vis}}}}}}{{{N_{{\text{total}}}}}},{p_{{\text{slip}}}}=\frac{{{N_{{\text{slip}}}}}}{{{N_{{\text{total}}}}}},{p_{{\text{Knu}}}}=\frac{{{N_{{\text{Knu}}}}}}{{{N_{{\text{total}}}}}},{p_{{\text{surf}}}}=\frac{{{N_{{\text{surf}}}}}}{{{N_{{\text{total}}}}}}$$
(4)

where Ntotal denotes the total number of collisions, and Ntotal = Nvis + Nslip + NKnu + Nsurf. Thereby, the values of pvis, pslip, pKnu, and psurf are in the range of zero to one, and the sum of them is equal to one.

Further, we also calculated the velocities of gas molecules after collisions. Take Molecule i at point Ai in Fig. 2 as an example, and assume that the collision at Ai is viscous collision. Molecule i travels the distance lAiBi within a time span denoted by Δt. Hence, the velocity of Molecule i (vAiBi) is:

$${v_{{A_i}{B_i}}}=\frac{{{l_{{A_i}{B_i}}}}}{{\Delta t}}$$
(5)

Since the velocity vAiBi originates from the viscous collision at point Ai, vAiBi is attributed to the velocity of viscous collision, that is, viscous flow. Denoting the number of viscous collision of Molecule i in the time span as Nvis,i, and the corresponding velocity of viscous flow after each viscous collision as vk vis,i, (k = 1, 2, …, Nvis,i) the average velocity of the viscous flow of Molecule i (denoted by vvis,i) can be obtained as:

$${v_{{\text{vis}},i}}=\frac{{\sum\nolimits_{{k=1}}^{{{N_{{\text{vis}},i}}}} {v_{{{\text{vis}},i}}^{k}} }}{{{N_{{\text{vis}},i}}}}$$
(6)

Further, denoting the total molecular number as N, the average velocity of the viscous flow (denoted by vvis) is:

$${v_{{\text{vis}}}}=\frac{{\sum\nolimits_{{i=1}}^{N} {{v_{{\text{vis}},i}}} }}{N}$$
(7)

Combining Eqs. (6) and (7), one obtains the following equation:

$${v_{{\text{vis}}}}=\frac{{\sum\nolimits_{{i=1}}^{N} {\sum\nolimits_{{k=1}}^{{{N_{{\text{vis}},i}}}} {v_{{{\text{vis}},i}}^{k}} } }}{{N{N_{{\text{vis}},i}}}}$$
(8)

Similarly, the average velocities of slip flow (vslip), Knudsen (vKnud) and surface diffusion (vsurf) can also be obtained using Eq. (8) with the velocity of viscous flow replaced by other types of collisions. The total average velocity (denoted by vave) is obtained as follows:

$${v_{{\text{ave}}}}={v_{{\text{vis}}}}{p_{{\text{vis}}}}+{v_{{\text{slip}}}}{p_{{\text{slip}}}}+{v_{{\text{Knud}}}}{p_{{\text{Knud}}}}+{v_{{\text{surf}}}}{p_{{\text{surf}}}}$$
(9)

Hence, the velocity contribution of viscous flow (Cvis), slip flow (Cslip), Knudsen diffusion (CKnud) and surface diffusion (Csurf) gas transport to the total flux can be obtained as:

$${C_{{\text{vis}}}}=\frac{{{v_{{\text{vis}}}}}}{{{v_{{\text{ave}}}}}},{C_{{\text{slip}}}}=\frac{{{v_{{\text{slip}}}}}}{{{v_{{\text{ave}}}}}},{C_{{\text{Knud}}}}=\frac{{{v_{{\text{Knud}}}}}}{{{v_{{\text{ave}}}}}},{C_{{\text{surf}}}}=\frac{{{v_{{\text{surf}}}}}}{{{v_{{\text{ave}}}}}}$$
(10)

Results and discussion

Validation of the method

To validate the simulation method and parameters used in this study, we calculated the density and MFPs of bulk phase of CH4 (using the model depicted in Fig. 1a) of different pressures at a temperature of 353.15 K, and the results are shown in Fig. 5. The densities obtained by MD simulations (red triangles in Fig. 5a) are in good agreement with the data obtained from the NIST (National Institute of Standards and Technology) Chemistry Webbook (black line in Fig. 5a)40. Figure 5b compares the MFPs obtained by MD simulations (black circles) and by Eq. (1) (red line). The values of density and viscosity used in Eq. (1) were obtained from the NIST database40. The values of Z in Eq. (1) were calculated based on the method described in Reference41. The results obtained from MD simulations generally match the results obtained by Eq. (1) when the pressure is smaller than 30 MPa. The relatively larger deviation between the two methods with the pressure larger than 30 MPa results from the decreasing intermolecular distances as depicted by the snapshots in Fig. 5a. These results validate the accuracy of the methods, which can be used to study the MFP of CH4 in nanopores.

Fig. 5
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Results of density and MFPs of bulk phase CH4: (a) mass density obtained by EOS and MD simulations; (b) MFPs obtained by MD simulations and Eq. (1). The subplots in (a) are the snapshots of CH4 molecules at 1, 20 and 49 MPa, respectively.

Free path distribution of CH4 in nanopores

The free path of CH4 at a specific position in a pore is obtained as follows: denoting the positions of two successive collisions of one specific CH4 molecule as Pi and Pj, respectively, and the length of the free path between Pi and Pj (assuming the CH4 molecule flying from Pi to Pj) as Lij, the free path at Pj (not Pi) is equal to Lij. Figure 6 shows the free path distributions of CH4 in the 5 nm pore (circle-symbolled lines) and the values of MFP obtained by Eq. (1) (dot lines) of different pressures. The position z = 0 nm denotes the center of the pore, and z = ± 2.5 nm denotes the positions of the two wall surfaces (depicted by dash lines in Fig. 6). The values of free path are smaller (larger) at the position near the wall surface (pore center). Further, with the pore pressure increasing from 7 to 25 MPa, the value of free path decreases correspondingly at the same position. It is worth noting that, the gas pressure in nanopores differs from the bulk gas pressure (resulting from the gas-gas interactions) due to the gas-wall interactions, and thus the “gas pressure” should be more specific as the “confined-gas pressure”. For the sake of convenience of expression, the word “confined-gas pressure” was simplified as “pressure” in this study.

Fig. 6
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Free path distributions of CH4 in 5 nm pore of different pressures.

The value of free path is directly related to the frequency of gas collision (number of collisions in 1 ps). Figure 7a shows the collision frequency distribution in the 5 nm pore. Most collisions occur near the wall surface, since the collision frequency at z = ± 2.4 nm is much larger than that in the area from z = − 2.0 to z = 2.0 nm. The higher collision frequency near the wall surface can be explained by two reasons. First, the mass density of adsorbed CH4 near the wall is much larger than that of the free gas as demonstrated by the mass density distributions in Fig. 7b. Higher mass density means that gas molecules have more chances to collide with each other. Second, in addition to the gas-gas collision, the gas-wall collision also occurs in the adsorption layer. Hence, the collision frequency in the adsorption layer is larger than that in the free phase area. In addition, with the pressure increasing from 7 to 25 MPa, the collision frequency increases correspondingly, resulting from the increase of mass density as demonstrated by the curves in Fig. 7b.

Fig. 7
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Distributions of (a) collision frequency and (b) mass density of CH4 of different pressures in 5 nm pore.

Figure 8 compares the free path and mass density distributions of CH4 in the pores with different pore sizes (w = 1.66, 2.66, 3.66, 4.66 nm corresponding to Lz = 2, 3, 4, 5 nm). The pressures of the four cases are the same and equal to 12 MPa, and the temperature is 353.15 K. For the cases of Lz = 4 and 5 nm (red and black circle lines in Fig. 8a), the values of free path in the area from z = − 0.4 to 0.4 nm are basically the same. This is because the mass densities of CH4 in this area (z = − 0.4 to 0.4 nm) are also approximately the same as demonstrated by the black and red lines in Fig. 8b, and thus the free path between two successive gas-gas collisions of the two cases are equal. In the area of z > 1 nm (or z < − 1 nm), the free paths of CH4 in the 4 nm pore are smaller than that in the 5 nm pore, resulting from the higher mass density in the 4 nm pore. Further, when Lz = 2 nm (pink circle line in Fig. 8a), the values of free path is much smaller compared with other three cases, resulting from the relatively higher gas density in all areas (pink line in Fig. 8b).

Fig. 8
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Distributions of (a) free path and (b) mass density of CH4 in the pores of different sizes. The pressure is 12 MPa, and the temperature is 353.15 K.

Mean free path of CH4 in nanopores

The MFP in the nanopores can be obtained by Eq. (3) with known values of free path shown in Fig. 6, and the results are presented in Fig. 9a. Similar to the variation of free path with pressure, the MFP of gas in the nanopore with a specific width decreases with the pressure increasing and with the width decreasing. We also calculated the MFPs using Eq. (1) as depicted by the orange dashed line in Fig. 9a, and the values are larger than those in the nanopores at the same pressure, meaning using the kinetic model (Eq. 1) without considering the effects of wall to calculate the MFP in nanopores can cause great error.

By dividing the obtained MFP by the pore widths (w), the corresponding Knudsen numbers Kn were obtained as shown in Fig. 9b. Due to the larger MFP obtained by Eq. (1), the corresponding values of Kn in the nanopore with the same width obtained by Eq. (1) (dashed lines in Fig. 9b) are larger than those obtained by MD simulations (solid lines in Fig. 9b).

Fig. 9
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Variations of (a) mean free path and (b) Knudsen number with pressure in pores with different widths.

Determination of flow types of free gas in nanopores

Based on the method provided in Section “Methods of the collision detection and MFP calculation”, we calculated the proportions of the viscous, slip, Knudsen, and surface collision of different pressures in the nanopores as shown in Fig. 10. With the pressure in the range of 1 to 45 MPa, viscous and surface collisions are the two main types of collisions, and the sum of their proportions are larger than 95%; Slip and Knudsen collisions take much smaller proportions, and their maximum values are 1.9% and 0.6%, respectively. This means that, within the studied pressure range, the main types of CH4 transport in nanopores are viscous flow and surface diffusion. Further, with the pressure increasing, the proportion of viscous (surface, slip, and Knudsen) increases (decrease) correspondingly. This is because higher pressure accounts for higher gas density, and thus leads to higher probability of gas collisions between free gas molecules (collision type A in Fig. 3a), as well as higher probability of collisions between rebounded gas and free gas in near surface area (collision type B). Concurrently, the probabilities of the rebounded gas flying through near surface area and colliding in the free gas area (collision type C) or colliding with the other wall (collision type D) decrease due to the denser gas in the near surface area. In addition, higher pressure also results in higher gas density in adsorption layer, and decreases the number of unoccupied adsorption sites on wall. This diminishes the probability of gas hopping on the wall (surface diffusion characterized by collision type E), and the proportions of surface collision decreases consequently.

Fig. 10
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Proportions of viscous, slip, Knudsen, and surface collision in pores at different pressures.

Since the accurate Knudsen number and the type of gas transport are determined as described above, the physical-based (namely, collision-based) classification of gas transport in nanopores can be proposed. Figure 11 shows the variation of collision proportions with Kn ranging from 0.06 to 0.14 (corresponding to the pressure decreasing from 45 to 1 MPa). As mentioned in Introduction, surface diffusion exists across all ranges of Kn, and thus Fig. 11 does not include surface diffusion. Based on the variation of collision proportions with Kn, Fig. 11a can be divided into three regions (denoted by A, B, and C, respectively):

  1. (1)

    When Kn ≤ 0.06 (region A), the main type of gas collision is viscous collision, whose proportion is larger than 99%. The slip collision takes a small proportion (less than 1%), and the proportion of Knudsen collision is zero. Hence, both slip flow and Knudsen diffusion can be ignored.

  2. (2)

    When 0.06 < Kn ≤ 0.10 (region B), the proportion of viscous collision decreases, but is still larger than 90%. The maximum proportions of slip collision increase to about 8%. The Knudsen collision remains a small proportion (most less than 0.3%) in this region, and thus can be ignored.

  3. (3)

    When Kn > 0.10 (region C), the proportion of Knudsen collision is approximately the same as that of slip collision in 2 nm pore, and the maximum value reaches to 3%. Therefore, all the viscous flow, slip flow, and Knudsen diffusion exist in this region.

Fig. 11
Fig. 11The alternative text for this image may have been generated using AI.
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Variations of proportions of viscous, slip and Knudsen collision with Knudsen number in nanopores.

Based on the results above, we proposed the collision-based classification of gas transport in nanopores as shown by the solid lines in Fig. 11b. Comparing the collision-based and the empirical (dot lines in Fig. 11b drawn according to Table 1) classifications, one can find that the collision-based classification extends the upper Kn limit of viscous flow from 0.1 to 0.16, and raises the lower Kn limit of slip flow from 0.001 to 0.06. The limits of Knudsen diffusion of the two classifications are the same.

Contributions of each type of gas transport to the total gas flux

To simulate the gas flow in the nanopores, a constant external force in the flow direction (x direction in Fig. 1c) was exerted on each gas molecule to mimic the pressure gradient supporting the flow. Figure 12 shows the collision proportions of gas molecules (calculated by Eq. 4) at the pressure of 1.6 MPa in the 3 nm pore with the external force ranging from 10−5 to 10−3 kcal/mol/Å. The proportions of viscous, slip, Knudsen and surface collision are approximately 0.19, 0.01, 0.01, and 0.79, respectively. This means that the external force (equal or smaller than 10−3 kcal/mol/Å) in the flow direction does not affect the collision proportions.

Fig. 12
Fig. 12The alternative text for this image may have been generated using AI.
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Variations of collision proportion with external force exerted on the gas molecules.

Based on the method described in Section “Methodology”, the velocity of gas transport in the pores with different widths were calculated, and the results are shown in Fig. 13. The external force exerted on the gas molecules were 10−3 kcal/mol/Å. The velocities of viscous flow, slip flow, and surface diffusion are approximately the same, while the velocity of Knudsen diffusion is zero at high pressures, resulting from the disappearance of Knudsen diffusion. The gas pressure does not greatly affect the gas velocity in the pore with Lz = 2 nm (Fig. 13a), while the gas velocity generally increases with the pressure increasing in the pores with Lz = 3, 4, and 5 nm. This means that, in the 2 nm pore, the main factor that affects the flow velocity is the gas-wall interactions instead of pressure gradient. Further, the average velocities obtained by Eq. (8) (green cross lines in Fig. 13) are basically the same as the velocities of the viscous, slip flow and surface diffusion.

Fig. 13
Fig. 13The alternative text for this image may have been generated using AI.
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Variations of the velocity of gas transport with pressure in the pores with width of (a) 2 nm, (b) 3 nm, (c) 4 nm, and (d) 5 nm.

The velocity contributions of each type of gas transport to the total mass flux (average velocity) obtained by Eqs. (9) and (10) are shown in Fig. 14. The viscous flow and the surface diffusion take the major part of the total mass flux, since the sum of their contributions are larger than 98%. The proportions of the slip flow and Knudsen diffusion are much smaller (less than 2%). This means that the nanoconfined gas flux mainly arises from the viscosity flow and surface diffusion. Further, with the pressure increasing, the velocity contributions of viscous flow (slip flow, Knudsen and surface diffusion) increases (decrease) as a result from the increase (decrease) of the corresponding collision proportions shown in Fig. 10.

Fig. 14
Fig. 14The alternative text for this image may have been generated using AI.
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Variations of the velocity contribution of gas transport with pressure in the pores with different pressures.

As mentioned in Section “Determination of the collision types and contributions to the total flux”, some literatures used the collision proportions to represent the contributions of different types of gas transport. This kind of representation needs validation, because the equivalence between the collision proportions and velocity contributions was not justified. Figure 15 compares the velocity contributions and the collision proportions obtained by MD simulations, showing that the values of these two parameters are same to each other since the values in Fig. 15 lie on the diagonal line. This means that the velocity contribution of each type of gas transport is equivalent to the corresponding collision proportion. This can be explained as follows: The velocity contributions of each type of flow are determined by both the velocity and the corresponding collision proportions. As shown by Fig. 13, the velocities (obtained by Eq. 8) of different types flow are basically the same for a specific pressure and temperature. This means that the velocity contributions are mainly determined by collision proportions (shown by Fig. 10). One can find that the variations of velocity contribution and collision proportion are basically the same by comparing Figs. 10 and 14, and thus the values of velocity contributions are equal to those of the corresponding collision proportions.

Fig. 15
Fig. 15The alternative text for this image may have been generated using AI.
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Comparison of the collision proportion and the velocity contribution of gas transport in nanopores. The diagonal line (dashed line) is a guided eye line for convenient comparison.

Conclusions

In this study, the type of gas transport in nanopores and the contributions of each type of gas transport to the total gas flux were analyzed by conducting MD simulations. The gas (CH4) transport in nanopores constructed by graphene slits were simulated. We obtained real values of MFP and Knudsen number of methane gas in nanopores, and proposed a novel collision-based classification of gas transport. In addition, the velocity contributions of each type of gas transport were analyzed. The following conclusions are drawn from this study:

  1. (1)

    The MFP of gas in nanopores is smaller than that of bulk gas phase at the same pressure and temperature due to the gas-wall interactions, which accounts for the much smaller free path of adsorbed gas molecules than that of the free gas. The MFP values predicted by the kinetic model are larger than the real values of gas in nanopores. For the CH4 molecules in nanopores with the width equal or less than 5 nm, the value of Kn is in the range of 0.04 to 0.16 with the pressure within 45 to 1 MPa.

  2. (2)

    The main type of the gas transport in nanopores is viscous flow and surface diffusion with the pressure within 1 to 45 MPa, and the sum of their proportions are larger than 95%. The proportions of slip flow, Knudsen diffusion are much lower with the pressure in the studied range (1 to 45 MPa). The proposed collision-based classification extends the upper Kn limit for viscous flow from 0.1 to 0.16 and raises the lower Kn limit for slip flow from 0.001 to 0.06 compared to empirical classifications.

  3. (3)

    The major contributions of velocity to the gas flux in nanopores arise from the viscous flow and surface diffusion, accounting for approximately 98% of the total mass flux. The contribution of slip flow is smaller than that of Knudsen diffusion, and the contributions of both two types are less than 2%. This addresses the importance of these two transport mechanisms in nanoconfined space.

  4. (4)

    Since the velocities of viscous flow, slip flow, and surface diffusion are approximately the same, and are equal to the average velocity, the velocity contribution is equivalent to the collision proportion.

The results presented in this study provide new insights on the types of gas transport in nanopores of different flow regimes characterized by Knudsen number, and accurately quantify the contributions of each type of gas transport from molecular perspectives. To simplify the analysis, we focused on the gas flow in a simple nanopore model constructed by graphene slits. The real structure and surface characteristics of nanopores are more complex than the graphene slits. Hence, the gas flow in the real nanopores is much complexed than the flow presented in this study leading to different results (gas transport type and velocity contribution). The methods provided in this work can be applied in further study.