Molecular Dynamic Simulation of Self- and Transport Diffusion for CO2

The comparisons between the simulated and experimental adsorption isotherms are as shown in Figure 4a, b; both the simulated and experimental data fit...
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Molecular Dynamic Simulation of Self- and Transport Diffusion for CO2/CH4/N2 in Low-rank Coal Vitrinite Yu Song, Bo Jiang, and Meijun Qu Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b03676 • Publication Date (Web): 22 Jan 2018 Downloaded from http://pubs.acs.org on January 22, 2018

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Molecular Dynamic Simulation of Self- and Transport Diffusion for CO2/CH4/N2 in Low-rank Coal Vitrinite Song Yu a,b, Jiang Bo a,b*, Qu Meijun a,b a Key Laboratory of Coal bed Methane Resource & Reservoir Formation Process, Ministry of Education China University of Mining and Technology, Xuzhou 221008, China; b School of resources and earth science, China University of Mining and Technology, Xuzhou 221116, China

Abstract The self-, Maxwell-Stefan-, and transport diffusions of CO2, CH4, and N2 in coal vitrinite macromolecule were simulated through molecular dynamic. Results indicated that these diffusion coefficients increase slowly when T340 K independent of the adsorbate numbers and types. The self- ([CO2]> [N2]> [CH4] in order) and transport diffusion coefficients ([N2]> [CO2]> [CH4] in order) decrease with the increasing adsorbate number. The diffusion activation energy (∆E) of vitrinite-n CO2 (5.07, 5.73, and 15.96 kcal/mol for vitrinite-5 CO2, vitrinite-10 CO2, and vitrinite-22 CO2 respectively) is lower than vitrinite -n CH4 (8.15, 8.97, and 17.09 kcal/mol for vitrinite-5 CH4, vitrinite-10 CH4, and vitrinite-17 CH4 respectively). At the saturation adsorption state, the ∆E of vitrinite-7 N2 (12.03 kcal/mol) is the lowest compared with vitrinite-22 CO2 and vitrinite-17 CH4, indicating that the diffusion process for N2 is the easiest to inspire among these three gases. The swelling ratio ([CO2]> [CH4]> [N2] in order) increases with the increasing temperature, indicating that high temperature is conducive for the swelling equilibrium. While the ∆E of pressure dependence first decreases with the increasing pressure until the peak pressure (0.5~1.0, 1.5~2.0, and 2.5~3.5 MPa for CO2, CH4, and N2 respectively) then increases significantly, indicating that the diffusion energy barrier decreases with the increasing pressure. Keywords Molecular dynamic; CO2/CH4/N2; Selctive diffusion; Low-rank coal vitrinite

1 Introduction Knowledge of gas transport properties of coal is important for coalbed methane (CBM) production and recovery by CO2/N2 injection into coal seams.1 CBM, mainly consists of CH4 and also contains a small amount of heavy hydrocarbons, CO2, H2O, and other gases,2 is a natural gas mainly existing as the adsorption state.3-5 In enhanced coalbed methane (E-CBM) engineering, CH4 first desorbed from the micropores surface, then diffuses into fractures of coal bed, and finally is extracted through permeation flow.6 Harpalani and Chen7 and Zhao et al.8 revealed that CBM diffusion through the coal matrix/coal macromolecue is assumed to be driven through the concentration gradient (also the gradient of the chemical potential) and is usually characterized using Fick’s First and Second Law of Diffusion. Moreover, Shi and Durucan,5 Pan et al.9 and Hu et al.10 proposed that CBM diffusion within the coal matrix is dominated by four microscopic diffusion mechanisms in coals, including Self-diffusion (randomized Brownian motions from one site to another and characterized by self-diffusion coefficients),

Fickian

diffusion

(dominated

by

molecule-molecule

collisions

and

characterized

by

Fickian/Transport diffusion coefficients), Knudsen diffusion (dominated by molecule-wall collisions and characterized by Knudsen diffusion coefficients), and Surface diffusion (transport through physically adsorbed layer characterized by Surface/Maxwell-Stefan diffusion coefficient). The investigations on the diffusion characteristics of various scales for CBM’s main components (CH4) and invading gases (CO2/N2) have dual advantages of not only enhancing the safety and yields of the co-mining project and coalbed methane (CBM) but also effectively reducing greenhouse gas emissions by trapping CO2. International scholars have conducted in-depth efforts on the diffusions of CH4 and CO2 in the coal and have made significant progress both through the experiments and the molecular simulations since last decades. Saghafi et al.11 measured the diffusion coefficients of CO2 and CH4 in sub-bituminous to bituminous rank, ranging from 0.66 to 1.45% in mean maximum vitrinite reflectance from Sydney Basin coals using the and found that CO2 diffusivity (diffusion coefficient) in the Sydney Basin coals varies from 1.2×10−6 to 10.2×10−6 cm2/s, which is twice as quickly as CH4. By the bulk technique, Jian et al.12 obtained the diffusion coefficients of CO2 (2×10-4~8 ×10-3 mm2/s) in the coals (Ro,max=0.93~2.54%) and proposed that this diffusion coefficients decreases with the decreasing mass fraction of CO2 until to the background level. Using the freezing core sampling technique (FCST), Wang et al.13 investigated the low temperature influence on methane adsorption capacity and kinetic properties in crushed coal and proposed that the low temperature (under 273.15 K) greatly inhibits methane 1

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transport in coal while enhancing methane adsorption capacity in coal. Tang et al.14 treated the adsorption–diffusion process as a unified process and then investigated the temperature influence effect on this process. The results indicate that for the diffusion process with increasing temperature, the equivalent diffusion coefficient (Deq) increases with increasing temperature. Yue et al.15 studied the dynamic thermal process for methane sorption in coal. Blind experiments show that the temperature increase of the tested coal sample induced by external energy brought via the filled methane and the possible Joule-Thomson effect is within 0.5 °C in this test and thus will not be considered during methane adsorption in coal.13-15 Cui et al.,1 determined the apparent diffusivities of CO2, CH4, and N2 through numerically simulation using a bidisperse model (0.5–1 nm in size) in the macropore and micropore based on the experimental gas adsorption data and discovered that the apparent micropore diffusivity of CO2 (3×10-5~2×10-2 s-1 at 0.15~4.27 MPa) is generally one or two order of magnitude higher than those of CH4 (2.5×10-5~3.5×10-4 s-1 at 0.12~4.39 MPa) and N2 (2×10-5~9×10-4 s-1 at 0.45~7.20 MPa) because their kinetic diameters have the relation: CO2 (0.33)< N2 (0.36)< CH4 (0.38 nm). Additionally, it is also found that the apparent diffusivity strongly decreases with an increasing gas pressure.16 Increasing the gas pressure will cause the increase in the methane adsorption amount. As adsorption swelling may narrow some micropore entrances 1 and enhance the diffusion energy barrier of adsorbate in micropores, consequently reducing the diffusivities.17,18 Thus, the decreasing diffusivity with the increasing pressure may be attributed to coal matrix swelling caused by gas adsorption.1 Based on the pore classification proposed by International Union of Pure and Applied Chemistry (IUPAC), coal pores are composed of microporpus (50 nm in size).19 Our previous investigations have proven that the specific surface area of low-medium rank coals (Ro,max=0.65~1.34%) is mainly provided by micro-pores (accounting for 96.64~99.56%), and pore volume mainly by macro-pores (99.68~ 99.91%).20,21 Thus, great attentions have been paid to the CH4 diffusion in microporous, as well as the corresponding unipore,3,22 bidisperse,9 and modified bidisperse (also known as Fickian diffusion-relaxation model)1,23 models. These models can be used to calculate the diffusion coefficients through the solutions of the Fick’s second law but various in the unit size. The unipore models (d ~0.3 nm) are consisted of spherical particles with a uniform size (particle size φ~0.5 mm), 3,22 while the bidisperse models (d: 0.5–1 nm) are characterized by spheres of two distinct sizes, i.e. macro spherical particles (φ: 0.35–0.7 mm) and microspheres containing an assemblage of uniform size (φ: 0.04–0.06 mm).9,24 The apparent diffusivities obtained through these physical porous models are 10-14-10-11 m2/s for CH4 and 10-12-10-10 m2/s for CO2 respectively. Although, Staib et al.23 proposed that the Fickian diffusion-relaxation model, as established by Berens and Hopfenberg,25 that visualises the secondary stage as a coal relaxation is more consistent with current understanding of coal sorption than the previous unipore and bidisperse models. However on the one hand, bidisperse model is inappropriate for examining the pressure dependence of diffusion coefficients at high pressures and the unipore model with a single diffusion coefficient cannot describe the sorption kinetics over a range of pressures.23 On the other hand, these porous models just consider the porous structures of coal and has the limitations due to the absences of the aliphatic side chains and functional groups in coal surface. Assuming that the motions of atoms follow Newton’s first law and that interactions among atoms are described by empirical potential functions, molecular simulation can provide a general method to explore various natural processes at the molecular level based on the coal macromolecular models.26 This method takes the aliphatic side chains and functional groups into consideration and has long been an efficient approach to explore the underlying mechanism of macroscopic processes between coal and gas, such as adsorption,20,21,27-29, solvent swelling,8,30-32 and diffusion.8,10,33,34 As the phenomenon decrease of diffusion coefficient with time increasing was found,16 scholars have proposed a time-dependent gas diffusion model recently and suggested that the diffusion of methane in heterogeneous coal matrix may obey the anomalous time and space subdiffusion, rather than Fickian 2

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second law.17,18 The true diffusion coefficient obtained by the counterdiffusion experiment decreases first and then

131 132 133 134 135

2 Vitrinite Model

increases with increasing methane pressure.35 Generally, there are three distinct diffusion coefficients originating from the researchers’ efforts to compare measurements made with different methods.36 They are self-, corrected (or Maxwell-Stefan), and transport diffusion coefficients. Hu et al.10 calculated the self-diffusion coefficiens of CO2 (10-9 m2/s, P=1 atm, T=298 K) and CH4 (1.2×10-9 m2/s, P=1 atm, T=298 K) in the Wiser bituminous model and found that the calculated sorption heat of CO2 was larger than that of CH4, which could be the reason why CO2 can replace adsorbed CH4 gas in the coal seam. Xiang et al.34 investigate the self-diffusion coefficients of CO2 (3.35×10-12), CH4 (1.42×10-11), and H2O (1.86×10-11 m2/s) at 298.15 K in the Yanzhou coal model through the molecular dynamic simulations and proposed that interactions between coal molecules and adsorbates were gradually weakened with increasing temperatures and that high temperatures were non-conducive to adsorption. Based on the molecular dynamic method and Wiser model, Zhao et al.8 conducted the calculations of self- and transport diffusion for CO2 and CH4 and the results revealed that the self-, corrected, and transport diffusion coefficients of CO2 are all larger than CH4, while the calculated diffusion activation energy of CO2 is smaller than CH4. Hu et al.33 also used the dynamic simulation method to explore the self- and multicomponent diffusions of CO2 and CH4 in Wiser model and proposed that the self-diffusion coefficients of CO2 and CH4 decrease with gas concentration but increase with the increasing temperature. As mentioned above, the molecular dynamic efforts for diffusions are mainly based on the whole coal model (such as the Wiser models), scholars around the world have only minimally reported on the diffusion of gas in the maceral vitrinite and the microscopic diffusion mechanism between vitrinite macromolecules and small gas molecules. Coal maceral compositions however will significantly impact the microscopic behaviors.23,37,38 Through hand sampling and separation technologies,39,40 the diffusion behaviors of small molecules in the vitrinite have been explored but to a lesser degree than whole coal analysis. The maceral structural differences impart very significant behavioral differences. Staib et al.23 and Karacan37 proposed that maceral composition effects kinetics in coal. CH4 diffusion studies contrasting these two macerals also report faster kinetics in inertinites than in vitrinites.38 It may possible to identify specific diffusion behaviour in coals of differing maceral composition fromthe coal's temporal swelling response to CO2.23 Song et al.41 calculated the diffusion coefficients of CO2 (1.5×10-10 m2s-1 at 298.15 K), CH4 (1.37×10-11 m2s-1 at 298.15 K), and H2O (1.6×10-10 m2s-1 at 298.15 K) on the coal vitrinite macromolecule and proposed that the coal vitrinite model obtained through 13C NMR and FT IR can characterize the surface diffusion and solid solution diffusion effect of small gas molecules more accurately in adsorption pores than the the whole coal models and perfect graphite models. In addition, to authors’ knowledge, the self- and transport diffusions of N2 in coal have not been reported. Thus, to better understand the diffusions of CBM’s main components (CH4) and invading gases (CO2/N2) in coal vitrinite, the molecular mechanics (MM), molecular dynamic (MD), and grand canonical monte carlo (GCMC) were conducted to investigate the self-, corrected and transport diffusion coefficients of CO2, CH4, and N2 under the periodic boundary conditions (PBCs) based on the vitrinite macromolecule built in our previous work. The effects of gas concentration, pressure, and temperature on these three diffusion coefficients are also discussed. This work, is expected to be of broad interest to CBM exploration, E-CBM engineering, and CO2 capture, utilization, and sequestration (CCUS). Coal, by nature, is a heterogeneous porous solid material with plenty of aliphatic side chains and functional groups.42 In this paper, the vitrinite macromolecule (C214H180O24N2, the maximum vitrinite reflectivity is 0.58% in accordance with ASTM Standard D2798-11) was constructed through solid-state cross-polarization magic angle spinning (CP/MAS)

13

C NMR, fourier-transform infrared (FTIR), and high resolution transmission electron 3

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microscopy (HRTEM). The aromatics cluster size can be determined through the HRTEM and 13C NMR while the aliphatic carbons and heteroatoms were determined by FTIR. The details can be cited in our previous work.41,43 The H atoms were added to the plane macrmolecule until a saturated state in the Materials Studio 8.0 software (Fig. 1a). The selection of force field, which describes interactions among atoms, is very crucial for molecular simulation. An appropriate force field should be able to reproduce experimental results quantitatively. We have used different force fields, such as polymer-consistent force field (PCFF), Dreiding, condensed phase optimized molecular potential for atomistic simulation studies (COMPASS), and universal force field to calculate the density and porosity of the vitrinite model. The results indicated that the density and porosity computed through Dreiding force field was the closest to the experimental values. Therefore, we selected the Dreiding force field to formulate the atomic interactions in all the simulations. The potential function of the Dreding force field takes the following form: 10,33,34,41,43-46,63 (1) (2)

EVal =EBo+EAn+ETor+EIn

(3)

where Etotal is the total energy, EVal and ENon are the valence energy and non-bond energy, respectively. EBo, EAn, ETor, and EIn are the energy of band, angel, torsion, and inversion, respectively. EVan, Eele, and Ehy are the energy of van der Waals, electrostatic, and hydrogen bond, respectively. Firstly, the plane model was optimized using the Geometry Optimization Task and Energy Task in the Forcite module for five times respectively.8 The final configuration Geometry Optimization Task and Energy Task is shown in Fig. 1b. The Etotal (850.09 kcal/mol) is dominated by ENon (496.27 kcal/mol) especially the EVan (496.96 kcal/mol), indicating that the van der Waals force plays a significant role in the stability of the vitrinite macromolecule. Compared with the plane model, the proportion of the ETo and EAn increases significantly, indicating that the optimization processes increases the crooked degree of the vitrinite structure. X

159 160 161 162 163 164 165 166 167 168 169 170 171 172 173

Etotal= EVal +ENon ENon=EVan+Eele+Ehy

X

28. 84Å

136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158

29. 61Å

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Fig. 1 Vitrinite macromolecular plane model (a) and the configuration after geometry and energy optimization (b)

3 Calculation Methods 3.1 Annealing Kinetics To overcome the energy barrier of the molecular structure after the configuration optimization in the molecular field, the Annealing Kinetics simulation was needed to obtain the optimal energy configuration.8,34 The initial temperature, highest temperature, and heating rates were set at 300K, 600 K, and 60 K/per time respectively. Then the MD simulation in the NVT ensemble was conducted at each temperature segement for 1000 ps. The Nose method was utilized as the temperature control program.47 The number of cycles in the annealing simulation is set to 10. At the end of each cycle, the output configuration is calculated by MD to ensure that it is in the lowest energy state.48 Several annealing simulations were progressive conducted until the total energy tends to be stable, indicating that the structure is gradually approaching the true vitrinite macromolecule. The total energy and configuration after annealing simulations are shown in Fig. 2 (a) and Fig .2 (b). The total energy was finally stable at 744.59 kcal/mol, which is significantly lower than the configuration of Geometry and Energy Optimization. The Eele 4

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decrease contributes significantly to the ETo reduction, indicating that the annealing process increases the uniform distribution of charge. Furthermore, the EAn and ETo increase compared with the configuration of Geometry and Energy Optimization, indicating that the structure was furtherly complicated. The PBCs were added in the Amorphous Cell module and the initial density was set at 0.70 g/cm3. The density step was set at 0.01 g/cm3. Then the energy terms under each density was output until the final density 1.60 g/cm3. The total energy at each density step was shown in Fig. 2c, the total energy firstly decreases then increases with the increasing density. The lowest energy was stable at 1.20 g/cm3 then the the curve starts to shake when density > 1.2 g/cm3. In the true coal reservoirs, the coal vitrinite macromolecule actually was’t at the lowest energy state due to the relaxation caused by tectonic stress and solvent swelling action,48,49 thus the 1.24 g/cm3, the energy minimum point after the first shake was considered as the true density, which is in good line with the measured helium density (1.25 g/cm3). The cell parameters now are a=b=c=17.15Å and α=β=γ=90°. Furthermore, the EAn and E To furtherly increases and Eele furtherly decreases compared with the configuration after annealing simulation. The process of geometry optimization~energy optimization~annealing simulation~density simulation is an advanced process approach to obtain the optimal configuration most closely to the true coal vitrinite macromolecule. (a) X

765 760 755 750 745 0

188

1

2

3 4 5 6 Annealing Calculation

7

8

9

26 . 54Å

Total Energy/ kcal/mol

770

(c)

1300 1200

Total Energy (ETo)/ kcal/mol

1100

X

1000 900 800 700 600 0.6

189 190 191 192 193 194 195 196 197 198 199 200 201

17. 15Å

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0.8

1.0 1.2 1.4 1.6 Density/ cm3/g Fig. 2 Total energy change after annealing simulation (a), the final configuration after annealing simulation (b), total energy change after density simulation (c), and the final configuration after density simulation (d)

3.2 GCMC The adsorption state of CO2/CH4/N2 was obtained through the GCMC method in the sorption module. The parameter settings are as follows: The adsorption configuration and adsorption isotherm were calculated through the Task of Locate and Adsorption isotherm in the Dreding force field respectively. The maximum number of iterations and equilibration step were set at 50000 and 10000 respectively, and the convergence criterion was 5×10-4 kcal/mol. The Ewald sum method was used for electrostatic action with a precision of 4.186×10-3 kJ/mol. To ensure the balance of the system, 2×107 GCMC steps were adopted. The former 107 steps were used to reach the balance state and the latter 107 steps to calculate the statistical average.41 Three calculations were conducted in triplicate to obtain the final results with a variation coefficient of < 0.01. Concerning the geothermal gradient of the coal reservoirs, the temperature was set from 298 to 368K with an interval of 10K. To ensure the GCMC 5

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results in line with the experimental results, the maximum pressure for these three gases was set below their critical values (4.5992 MPa for CH4, 7.3773 MPa for CO2 and N2).50,51 The Metropolis rules were adopted to accept or reject the generation, disappearance, translation, and rotation of the small gas molecules to ensure the lowest energy state of the system, where the accept probabilities of the Exchange, Conformer, Roating, and Translating were 40%, 20%, 20%, and 20% respectively. The EVal and its sub items (EBo, EAn, ETo, and EIn) are 0 for CO2, CH4, and N2 during the adsorption process, indicating that the adsorption of vitrinite for these three adsorbates are physical adsorptions. The energy change and the adsorption configurations for different amounts of CH4, CO2, and N2 are as shown in Fig. 3a~l. The ENon firstly decreases with the increasing number of the adsorbates, indicating that the vitrinite attracts the adsorbates at this process. Then it decreases, indicating that the vitrinite no longer attracts the adsorbates and begins to exclude the adsorbates. The adsorption number at the lowest ENon indicates the maximum adsorption amount. The adsorption of vitrinite reaches the saturation state after absorbing 17 CH4, 22 CO2 and 7 N2 per vitrinite macromolecule respectively, indicating that the adsorption ability order manifests as [CO2]> [CH4]> [N2]. Furthermore, the ENons were -200.32, -281.45, and -77.48 kcal/mol for vitrinite-CH4, vitrinite-CO2, and vitriniteN2 respectively, indicating the adsorption affinity manifests as [CO2]> [CH4]> [N2]. The van der waals force plays a dominated role in the physical adsorption process. The optimumadsorption sites for CH4 and CO2 are basic coincident at any adsorption state (5 CO2 and 5 CH4, 10 CO2 and 10 CH4, and their saturation states), indicating that the CO2 can replace CH4 by occupying CH4’s adsorption sites. The adsorption systems when absorbing different amounts of adsorbates are idtntified as vitrinite+ n adsorbatre. The adsorption configurations vitrinite+5 CH4s (b), vitrinite+10 CH4s (c), vitrinite+17 CH4s (d), vitrinite+5 CO2s (f), vitrinite+10 CO2s (g), vitrinite+22 CO2s (h), vitrinite+N2 (j), vitrinite+3 N2s (k), and vitrinite+7 N2s (l) were used as the initial configuration for the MD simulation to investigate the dominance of concentration on the self-diffusion coefficients. 0

(a)

-25 Energy/ kcal/mol

-50 -75 -100 -125 -150 -175

ENon

-200

Eele

-225

0

224

2

EVan

X

4 6 8 10 12 14 16 18 20 22 24 Nabs/ molecules/unit.celle (CH4)

0

(e)

-50 Energy/ kcal/mol

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

-100 -150 -200 -250 -300

225

ENon

EVan

Eele 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Nabs/ molecules/unit.celle (CO2)

X

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(i)

-10 -20 Energy/ kcal/mol

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-30 -40 -50 -60 -70

ENon

-80

Eele 0

226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255

1

EVan

2 3 4 5 6 7 8 9 10 11 12 13 Nabs/ molecules/unit.celle (N2)

Fig. 3 Total energy change during the adsorption process (a, e, i), the adsorption configuration when absorbing 5 CH4 (b), 10 CH4 (c), 17 CH4 (d), 5 CO2 (f), 10 CO2 (g), 22 CO2 (h), 1 N2 (j), 3 N2 (k), and 7 N2 (l)

3.3 MD Simulation The adsorption system was first minimized using the smart and steepest descent methods.10 Then the MD simulation was conducted sequentially for 1000 ps in the NVE (T = 298 K) and NVT ensemble to approach the dynamic equilibrium state. The mean square displacement (MSD) and radial distribution function (RDF) was finally obtained through the MD calculation for 1000ps in the NPT ensemble to ensure the stable linear relationship between MSD and time,10 which is also adequate long to relax to the equilibrium state. The initial velocities of the adsorbate molecules were determined based on the Maxwell– Boltzmann distribution law.8 Time step was 1fs for all the MD simulations. The equations of motion were integrated by the velocity Verlet algorithm. The temperature and pressure were controlled through the methods of Berendsen and Andersen respectively. The parameters are as follows: the maximum iterations: 50000, convergence limit: 5.0×10-4, energy difference: 0.0001 kcal/mol, force difference: 0.005 kcal/mol. Both the electrostic energy and van der Waals energy were obtained through the Atom based method with the cut off distance of 12.5Å. The charge distributions of the force field were calculated through the QEq method.

3.4 Self-, Corrected, and Transport Diffusion The transport diffusion coefficients (Fick diffusion coefficients) were macroscopic defined according to the concentration gradient drived theory.9 The driving force of the diffusion process is actually chemical potential gradient. Thus, the diffusion flux is negatively correlated to the chemical potential gradient.52-54 The Maxwell diffusion theory revealed that the Maxwell-Stefan coefficient of one-component can be obtained through the solution of the velocity autocorrelation function55: ∞

Dc =

1 vi (t ) • vi (0) dt 3 • V • c ∫0

(4)

where c is the concentration, mol/m3. The transport diffusion coefficients (Dt) is the synthetic results of thermodynamics and dynamics of host-guest system. The thermodynamics factors reflect the thermodynamics attributions.54 To establish the relationship between Maxwell-Stefan diffusion coefficients (Dc) and transport diffusion coefficients, the thermodynamics factors ( Γ ) should be defined to transform the concentration gradient to chemical potential gradient, which can further simplified to56,57: Γ =

∂ ln f ∂ ln c

（5）

where f is the fugacity, MPa; the relationship between pressure and fugacity of single component is:

256 257 258 259 260 261 262

X

ln

 Z + (1 + 2) • b  f a = Z − 1 − ln( Z − b) − • ln   p 2 2 •b  Z + (1 − 2) • b 

(6)

where p is the pressure, MPa, a, b, and Z are the attraction parameter, van der Waals covolume, and compressibility factor respectively. The parameters a, b, and Z can be obtained through the Peng-Robinson equation.58 At a certain temperature: a (T ) = a (Tc ) • α (Tr • ω ) , b(T ) = b(Tc ) (7) where α is the scaling factor and ω is the acentric factor. k is the is a characteristic constant of each substance. Tc

7

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is the critical temperature, 190.58, 126.1, and 304.19 K for CH4, N2, and CO2 respectively. The functional form of α (Tr, ω) was determined by using the literature vapor pressure values59,60 and Newton's method to search for the values of α to be used in equation (7), so that the following function: f L=f V (8) could be satisfied along the vapor pressure curve with the following convergence criterion of 10-4 kPa to obtain a stable α value at each temperature. For all substances examined the relationship between α and Tr, can be linearized by the following equation: α = 1 + k • (1 − Tr ) (9) According to Peng and Robinson,58 the characteristic constant could be correlated against the acentric factors: k = 0.37464 + 1.54226 • w − 0.26992 • w2 （0Ds (CH4), which is in the opposite order to their kinetic diameters σ nm)>σ

CO2

CH4

(0.38 nm)>σ

N2

(0.36

(0.33 nm). As the kinetic diameter is a sensitive measure of ability to move in highly restrictive

environments.65 Thus, the pores of [N2]>[CH4] while the transport diffusion cofficient as [N2]>[CO2]>[CH4] at a given temperature. This discrepancy is attributed to the fact that the selfdiffusion coefficients are determined by the kinetic diameter, the polarity, and elongated shape while transport diffusion coefficients by the gas-coal behaviours and adsorption enthalpy (i,e, adsorption affinity). The increasing trendency is consistent with the values calculated through unipore model by Charrière et al.,3 however, the magnitude orders here are significantly higher than the values of Charrière et al.3 (10-12 m2/s in magnitude order) (Fig. 6d, e). This discrepancy is due to the fact that the unipore model (d~0.3 nm) are consisted of spherical particles with a uniform size (particle size φ~0.5 mm) however, the vitrinite model here is composed of micropores (< 1.7 nm here actually) with the circumstances of aliphatic chains and functional groups. On the other hand, the coal matrix micropores of unipore models are approximated by graphite slit pore model, which is composed of aromatic carbons. In our previous investigations, the adsorption capacities of aromatic rings for CO2 and CH4 are higher than those of the functional groups and aliphatic chains.43 Thus the adsorption affinity of unipore model for these gases is higher than this vitrinite macromolecule, also resulting in the lower transport diffusion coefficients. Additionally, the model adopt here takes the non-bond interactions (van der Waals force and electrostatic force) into consideration, thus, it is more appropriete to depict the heterogeneity in porous and chemical structures of coal vitrinite. 4.3.2 Activation Energy and Vitrinite Swelling The transport diffusion of adsorbates in adsorbents is a process of activation. The diffusion reactivity is in accordance with Arrhenius law.10 As is shown in Fig 7a~c, ln (Dt) has a good linear relationship with 1000/T 11

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(R2=0.81~0.95), which is also consistent with equation (14). The activation energy (∆E) independent of the temperatures can be calculated through the slopes of the ln (Dt)~1000/T curves at various temperatures. The temperature dependence of duffusion coefficients can also explained by the activation energy, increasing temperature will increase the kinetic energy of the diffusion particles, thus, more and more particles can conquer this constant energy barrier during the temperature rising process, leading to the high diffusion coefficients. The ∆Es for vitrinite-5CO2, vitrinite-10CO2, and vitrinite-22CO2 are 5.07, 5.73, and 15.96 kcal/mol respectively, which is lower than the values of CH4-vitrinite (8.15, 8.97, and 17.09 kcal/mol for vitrinite-5CH4, vitrinite-10CH4, and vitrinite-17CH4 respectively), indicating that the CO2 diffusion is easier to occur due to its relatively lower diffusion energy barrier at any adsorption state. For the saturation state, the diffusion ∆E of vitrinite-7N2 (12.03 kcal/mol) is the lowest compared with those of vitrinite+22CO2 and vitrinite-17CH4 and more N2 molecules can conquer this barrier, indicating that the diffusion process for N2 is the easier to inspire than CO2 and CH4. The lower diffusion energy barrier of N2 is related to the weaker Van der Waals force as mentioned in above Section 1.3.2, which is also in line with the higher diffusion coefficients than CO2 and CH4. For each gas, the ∆E increases with the increasing adsorbate number, indicating that the more the diffusion particles, the more difficult the diffusion to occur.

4.2 4.0 3.8 3.6 3.4 3.2

425

2.7

2.8

2.9

3.0

3.1

3.2

3.3

t

3.8 3.4 3.2 3.0 2.8

3.4

2.7

2.8

7

Vitrinite+10CO2

6

Vitrinite+22CO2

5 4 3 1 0

2.9

3.0

3.1

3.2

3.3

300

310

320

330

t

/T)+6 2 .78, RVitrinite+7N 2 =0.94 )=-0 .79ln (1/T ln( )+6.9 D) 6, R 2 t =1.5 =0.9 1 0ln (1/ T)+ 8.6 6, 2 R= 0.9 5

ln(D

4.4 4.2 4.0 3.8 3.6

3.4

2.7

2.8

340

350

360

370

Temperature/K

3.0

3.1

3.2

3.3

3.4

5

Vitrinite+10CH4

3.0

Vitrinite+17CH4

2.5

3 2 1

(f)

3.5

Vitrinite+5CH4

290

2.9

1000/T

6

4

Vitrinite+1N2 (c) Vitrinite+3N2

)=-0.7 1ln(1

t

Vitrinite+1N2 Vitrinite+3N2 Vitrinite+7N2

2.0 1.5 1.0 0.5

0

290

ln(D

4.6

(e)

(d) Vitrinite+5CO2

4.8

1000/T

9 8

5.0

Vitrinite+10CH4

ln(D )=

ln( D

3.6

2.6

)=-0.9

Vitrinite+5CH4 (b)

8ln(1 Vitrinite+17CH4 /T)+6 .83, R 2 =0.91 -1.0 t t ) =8ln( 2.1 1 /T 3ln )+7.0 (1/ 5, R 2 T)+ =0.8 1 9.7 5, 2 R= 0.9 5

4.0

1000/T

2

426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442

ln(D

ln(Dt)

ln(Dt)

4.4

4.2

Swelling ratio/%

4.6

4.4

ln(Dt)

(a) ln(D )= -0.61ln t (1/T)+6 .38, R 2 =0.92 ln(D )= -0.69ln t (1/T)+ 6 .53, R 2 ln( =0.92 D) t =1.9 Vitrinite+5CO2 2ln (1/ Vitrinite+10CO2 T)+ 9.6 Vitrinite+22CO2 3, 2 R= 0.9 4

4.8

Swelling ratio/%

410 411 412 413 414 415 416 417 418 419 420 421 422 423 424

Swelling ratio/%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 17

300

310

320 330 340 350 Temperature/K

360

370

0.0 290

300

310

320 330 340 350 Temperature/K

360

370

Fig. 7 Relationship between ln (Dt) and 1000/T for CO2 (a), CH4 (b), and N2 (c) diffusion in vitrinite macromolecule and the swelling ratio for CO2 (d), CH4 (e), and N2 (f)

The higher transport diffusion coefficients of CO2 and N2 than CH4 indicates that the injection of these two gases into the coalbed with high speed to can effectively replace the CH4 during ECBM engineering. However, the GCMC results above suggest that the adsorption ability of vitrinite for these three gases manifests as [CO2]> [CH4]> [N2] and the dominant adsorption sites of CO2 and CH4 are basically consistent. Thus, the replacement mechanism of CO2-ECBM differs from N2-ECBM. The CO2 can replace CH4 through forming competitive adsorption and selective diffusion with CH4. The N2 could achieve it by reducing the partial pressures of CH4, which can furtherly promote the CH4 desorption from the adsorption sites. The swelling ratio caused by the diffusion process for different amounts of CO2, CH4, and N2 diffusing in vitrinite macromolecule are as shown in Fig. 7 d~f. The swelling ratio order of these small gas molecules manifests as [CO2]>[CH4]>[N2] at any adsorption state, which is consistent with the compatibility ability of these gases with the vitrinite macromolecule. Previous invesitations have proposed that for different adsorption gases, carbon dioxide exhibits its preferential adsorption property and causes coal to engender the maximum adsorption-swelling increment.

51,68,69

At 298 K, the swelling ratio for CO2 and CH4 are 1.56% and 0.97%

respectively, which is similar to the values calculated by Zhao et al., (2.01% and 1.04% for CO2 and CH4 12

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443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482

Energy & Fuels

respectively) through Wiser model and within the scopes of the experimental values proposed by Karacan., (2003) (0.75% to 4.18% with the pressure up to 15 atm) of high-volatile bituminous coal from the Pittsburgh seam during the CO2 sequestration

37

. The swelling ratio of 1.56% for CO2 here is also in line with the volume increase of

1.31% in a CO2 atmosphere proposed by Reucroft et al., (1986) through dilatometric experiments 70. Even at the 368 K, the maximum swelling ratio of 9% for CO2 is consistent with the manximum values porposed by Reucroft et al., (1986) (the increases of the order of 9% necessary to account for the surface areas usually found by CO2 adsorption on coal). Furthermore, the swelling ratio increases with the increasing diffusion particle number, indicating that only enough solvent molecules can completely replace the small molecules in coal and completely extend the network of vitrinite macromolecules to achieve equilibrium swelling state

70

. Brochard et al., (2011)

discovered that the differential swelling was almost proportional to the CO2 mole fraction during the competitive adsorption process through the molecular simulation, which on the other way demonstrates the concentration dependence of the swelling

71

. These comparisons indicate the reasonability and validity of the swelling ratio

results. The swelling ratio increases with the increasing temperature independent of the adsorbate number, indicating that increasing temperature is conducive for the swelling equilibrium. The temperature dependence for the swelling ratio is determined by the chemical thermodynamics that the dissolution process requires negative gibbs free energy reduction (∆G0) during the these diffusion process, thus, the higher temperture is favorable to ∆G 348 K, which is independent of the adsorbate type and adsorption state, indicating that this slow increase rate when T>348 K may be caused by the structural changes under the high temperature. As the swelling process is to overcome the non-bond effect (mainly composed of Eele and EVan), the highest swelling ratio can also in line with the lowest Eele and EVan of vitrinite-CO2 systems.

4.4 Pressure Dependence 4.4.1 Pressure Effect There is evident disagreement on whether the pressure promotes or delays the diffusion process of the CO2, CH4, and N2. Clarkson and Bustin22 proposed that different models display the different pressure dependence of diffusion coefficients, which is determined by the used model, even using the same data. By unipore model, these scholars such as Ciembroniewicz and Marecka,72 Clarkson and Bustin22, Charrière et al.,3 and Jian et al.,12 found that the diffusion coefficients increases with the increasing pressure. Furtherly, the results through the modified unipores of Shi and Durucan5 and Cui et al.1 indicated the decrease trend with the increasing pressure. However, Busch et al.61 also used the two exponentials unipore models and the results show that the diffusion coefficients of CO2 and CH4 decrease for “slow” term at elevated pressures (≤6.38 MPa), as well as the resuts of Pone et al.73 (≤3.10 MPa). Here, the diffusion coefficients of pressure dependence are calculated through the MD simulation at 298 K (Fig. 8a~f). The results indicate that the both the self- and teansfort diffusion coefficients first increase and then decrease with the increasing pressure independent of the adsorbate type and adsorption state. The peak pressure lies in 0.5~1.0, 1.5~2.0, and 2.5~3.5 MPa for CO2, CH4, and N2 respectively. The comparisons bwtween our transport diffusion coefficients and the results through bidispere and unipore model in Fig. 8d~e also indicate the similar trend in spite of deviations in the magnitude order attributed to different pore sizes between the bidispere/unipore model and this vitrinite macromolecule model.

13

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Vitrinite+22CO2

7 6 5 4 3 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Pressure/MPa

10

-11

(d) 2.5 -10 10

80

2.0

70 60 22

Calarkson et al. (unipore model) Cui et al.1(bidisperse model) Vitrinite+5CO2

50 40

Vitrinite+10CO2

30

Vitrinite+22CO2

1.5 1.0 0.5

20 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Vitrinite+5CH4 (b)

4.5

Vitrinite+10CH4

4.0

Vitrinite+17CH4

3.5 3.0 2.5 2.0 1.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

(c)

9 8 7 6 5

Vitrinite+1N2 Vitrinite+3N2

4

Vitrinite+7N2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Pressure/MPa 45 10-11

Pressure/MPa (e) 1.0

40

10-10 0.8

35 30 25

Vitrinite+5CH4 0.6

Vitrinite+10CH4 Vitrinite+17CH4

20 15

0.4

Calarkson et al.23 (unipore model)

0.2 Cui et al.1 (bidisperse model) 10 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

95 90 85 80 75 70 65 60 55 50 45 40

(f)

Vitrinite+1N2 Vitrinite+3N2 Vitrinite+7N2

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Pressure/MPa

Pressure/MPa

Pressure/MPa

4.4.2 Activation Energy and Vitrinite Swelling Opposite to the diffusion coefficients, the activation energy first decreases with the increasing pressure until the peak pressure (0.5~1.0, 1.5~2.0, and 2.5~3.5 MPa for CO2, CH4, and N2 respectively) (Fig. 9a~c), indicating that the diffusion energy barrier decreases and more and more diffusion particles could conquer this energy barrier, thus the self- and transport diffusion coefficients increase at low pressure regions. The diffusion type is dominated by the surface diffusion of the absorbed gases in the micropores by jumping among adjacent adsorption sites in intermediate micropores.8 Then when the pressue is higher than the peak pressures, the activation energy increases with the increasing pressure and difficult to conquer for increasing numbers of CO2, CH4, and N2 molecules, which is also consistent with the decreasing self- and transport diffusion coefficients. Then the configurational diffusion of the gaseous molecules dominated the diffusion process at this stage. The diffusion particle diffuses by moving from one fovea to another fovea in the ultra-micropores. Still, the activation energy of these three adsorbates manifests as [CO2]> [CH4]> [N2] at any adsorption state, which is also consistent with the temperature

16 14 Vitrinite+5CO2

10

Vitrinite+10CO2 Vitrinite+22CO2

8 6 4

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

12 10

Vitrinite+5CH4

8

Vitrinite+10CH4 Vitrinite+17CH4

6 4 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Pressure/MPa 3.5

2.5

8

2.5

(d) Vitrinite+5CO2

2.0

Vitrinite+10CO2 Vitrinite+22CO2

2.0 1.5 1.0

1.5

(c)

Vitrinite+1N2 Vitrinite+3N2

6

Vitrinite+7N2

4

2

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Pressure/MPa

Pressure/MPa

Swelling ratio/%

3.0

(b)

2.5

(e) Vitrinite+5CH4 Vitrinite+10CH4 Vitrinite+17CH4

1.0 0.5

2.0

Swelling ratio/%

12

14

Activation energy (∆ ∆ E)/kcal/mol

(a)

18

Activation energy (∆ ∆ E)/kcal/mol

dependence of diffusion coefficients.

499

1.5

(f) Vitrinite+1N2 Vitrinite+3N2 Vitrinite+7N2

1.0 0.5 0.0

0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

500 501 502 503 504 505

Self-dissusion cofficients (Ds)/(m2/s)

8

5.0

2 Transfort dissusion cofficients/(m /s)

2

(a)

Vitrinite+10CO2

2 Transfort dissusion cofficients/(m /s)

Vitrinite+5CO2

9

Page 14 of 17

Fig. 8 Pressure dependence of self- and transport diffusion coefficients at 298~368 K obtained through the MD method

Activation energy (∆ ∆ E)/kcal/mol

484 485 486 487 488 489 490 491 492 493 494 495 496 497 498

2 Transfort dissusion cofficients/(m /s)

483

Self-dissusion cofficients (Ds)/(m2/s)

10-11

Swelling ratio/%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Self-dissusion cofficients (Ds)/(m /s)

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Pressure/MPa

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Pressure/MPa

0

1

2

3

4

5

Pressure/MPa

Fig. 9 Activation energy of CO2 (a), CH4 (b), and N2 (c) diffusing in vitrinite macromolecule and swelling ratio of CO2 (d), CH4 (e), and N2 (f) diffusing in vitrinite macromolecule

The swelling ratio caused by the pressure independence of diffusion coefficients manifest as slow increase at the low pressure regions and significant increase at the high pressure regions for CO2, CH4, and N2 in spite of the adsorption states (Fig. 9d~f), indicating that the coal swelling is only significant at the higher pressures. Busch et 14

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506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549

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al. (2004) also reported that the effect of swelling on the sorption rate is only significant at high pressures or high surface coverage 61. Zhao et al., (2016) discovered that at relatively low pressure, the coal matrix swells seem to be ignorable and the dominant transport mechanism is the surface diffusion. While with pressure rising, the coal swells quickly and the diffusion activation energies enhance, so the configurational diffusion gradually becomes dominant 8. However, either for the CO2 or CH4, the increase rates of the swell ratio for vitrinite in this paper is more significantly than for the Wiser model. This difference is due to the fact that the vitrinite used here swelled and deswelled on the time scale of the experiment has rapid coal structure changes than the liptite, clarite microlithotypes, and clay + inertite region 74. The coal swelling may cause the micropores shrinkage and narrow the channels of the diffusion particles.1 Thus, the higher pressure will cause the diffusion channels narrow and delay the gas diffusion in micropores, which is also compatible with the low diffusion coefficients at high pressure regions. By comparision in the activation energy and swelling ration for these three gases, the vitrinite-CO2 system has the highest diffusion barrier and narrowest diffusion channels, thus the pressure required to reduce the diffusion coefficients is lower (peak pressure) than CH4 and N2. However, the swelling ratios and the energy barrier of vitrinite-N2 are lower at a given pressure than CO2 and CH4, thus, the pressure required to decrease the diffusion coefficients are higher than CO2 and CH4.

5 Conclusions The self- and transport diffusions for CO2, CH4, and N2 in low-rank coal vitrinite were conducted through molecular dynamic. The main conclusions are as follows: 1) The GCMC results show that the adsorption ability order manifests as [CO2]> [CH4]> [N2]. The adsorption amount decreases with the increasing temperature at a given pressure for these three adsorbates. The CO2 is easier to diffuse in micro-pores, followed by N2, indicating that differences exist for various gas types in pore tortuosity factors due to the different diameters. 2) These diffusion coefficients increase slowly when T340 K, indicating that the movement of CO2, CH4, and N4 enhanced with the increasing temperature. The self- and transport diffusion coefficients for these three gases all decrease with the increasing adsorbate number. The ∆Es for vitrinite-n CO2 are lower than the values of vitrinite-n CH4, indicating that the CO2 diffusion is easier to occur than CH4 due to its relatively lower diffusion energy barrier. At the saturation state, the diffusion ∆E of vitrinite-7N2 (12.03 kcal/mol) is the lowest compared with those of vitrinite-22CO2 and vitrinite-17CH4, indicating that the diffusion process for N2 is the easiest to inspire than CO2 and CH4. 3) The swelling ratio order of these small gas molecules manifests as [CO2]> [CH4]> [N2] at any adsorption state, which increases with the increasing temperature, indicating that higher temperature is conducive for the swelling equilibrium. While the ∆E first decreases with the increasing pressure until the peak pressure, indicating that the diffusion energy barrier decreases with the increasing pressure, thus the self- and transport diffusion coefficients increase at low pressure regions. The swelling ratio of pressure independence manifest as slow increase at the low pressure regions and significant increase at the high pressure regions for CO2, CH4, and N2, indicating that the coal swelling is only significant at the higher pressures.

Acknowledges This work is supported by the National Natural Science Foundation of China (No. 41430317, 41402136), Major Projects of National Science and Technology (2016ZX05044001-02), and Basic research project of Jiangsu province (Natural Science Foundation) (BK20140183).

Author Information Corresponding Author: Jiang Bo *E-mail: [email protected]

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