Fick Diffusion Coefficients of the Gaseous CH4âCO2 System from

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Fick Diffusion Coefficients of the Gaseous CH4−CO2 System from Molecular Dynamics Simulations Using TraPPE Force Fields at 101.325, 506.625, 1013.25, 2533.12, and 5066.25 kPa Joseph R. Vella* ExxonMobil Research and Engineering, 22777 Springwoods Village Parkway, Spring, Texas 77389, United States

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S Supporting Information *

ABSTRACT: Gaseous Fick diffusion coefficients are calculated for the CH4−CO2 system using molecular dynamics. CH4 was modeled using the united-atom transferable potentials for phase equilibria (TraPPE) force field developed by Martin and Siepmann [J. Phys. Chem B. 1998, 102, 2569], and CO2 was modeled using the rigid TraPPE force field developed by Potoff and Siepmann [AIChE J. 2001 47, 1676]. They can adequately predict the compressibility factor of the CH4−CO2 system over a substantial temperature, pressure, and composition range. However, deviations from experimental values are observed at conditions for which the ideal gas mixture assumption is not valid. It is shown that reasonable calculations of the Fick diffusion coefficient are obtained when compared to experimental data, and a correlation is found in the literature at conditions for which the ideal gas mixture behavior is observed. Calculations are also performed where the mixture does not behave as an ideal gas and are compared to calculations from the aforementioned correlation. The methods yield different results, however the lack of experimental data at these conditions does not allow one to say which are more accurate.

1. INTRODUCTION Transport phenomena is a fundamental area of chemical engineering that elucidates the dynamics of mass, energy, and momentum. A deep and quantitative understanding of this area is crucial for the successful development and deployment of important technologies across many disciplines. Several texts are dedicated to transport phenomena 1 and its application in the biomedical and biotechnology industry,2,3 porous materials,4 and the oil and gas industry.5 Mass transfer is one of three main areas that comprise transport phenomena. When a chemical engineer contemplates mass transfer, one would tend to immediately consider Adolf Fick’s laws of diffusion. The first of these laws gives a proportional relationship between the diffusive flux of a species and the concentration gradient of the same species. The “constant” of proportionality in this relation is referred to as the Fick diffusion coefficient. Quotations are placed around the word constant because this coefficient is not truly constant. For instance, the Fick diffusion coefficient has a temperature and composition dependence.6 Accurate values of Fick diffusion coefficients are important for the successful © XXXX American Chemical Society

design of many processes. For example, vapor-phase Fick diffusion coefficients of impurities in natural gas are essential pieces of information for the successful modeling, scale-up, and development of natural gas processing operations. While gaseous Fick diffusion coefficients are often documented from experimental studies,7−16 many conditions of interest make measurements cumbersome or difficult. Thus, alternative approaches can be useful. Classical molecular dynamics (MD) is a widely used simulation technique used to calculate physical properties of a wide range of systems. The accuracy of these calculated properties depends on the reliability of the force field used to describe inter- and intramolecular interactions. In this work, the Fick diffusion coefficients of the CH4−CO2 system are calculated at various conditions from molecular dynamics Special Issue: FOMMS Received: December 13, 2018 Accepted: April 9, 2019

A

DOI: 10.1021/acs.jced.8b01198 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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εij = (εiiεjj)1/2

simulations. This system is chosen because of its prominence in the oil and gas industry. It should be noted that thermodynamic and transport properties of the CH4−CO2 system have been studied extensively from molecular dynamics using various force fields.17−19 However, none of these studies calculated the Fick diffusion coefficients. It is well known that kinetic theory can predict transport properties for low density gases.20,21 Moreover, at these conditions, gases will behave as ideal gases, thus the interactions between particles are negligible. One should expect MD simulations at these conditions to yield accurate results regardless of the force field employed. The accuracy of the force field will play an important role for gases at conditions in which intermolecular interactions are significant, such as at high pressures. This paper is organized as follows. Section 2 will give details about the force fields utilized in this work. Section 3 will describe the details of the simulations used to calculate various properties. Section 4 will present and discuss the results of the simulations. Finally, section 5 will provide concluding remarks.

Lennard-Jones parameters and charges for each site considered in this work are given in Table 1. Table 1. Lennard-Jones Parameters and Partial Charges for the TraPPE Force Fields Used in This Worka ε/kB (K)

σ (nm)

q (e)

148 27 79

0.373 0.280 0.305

0 0.70 -0.35

CH4 parameters are taken from Martin and Siepmann.22 CO2 parameters are taken from Potoff and Siepmann.23 kB is the Boltzmann constant.

3. METHODS In this section, details on the methodologies used to calculate various properties of pure CH4, pure CO2, and CH4−CO2 mixtures are given. All simulations described below used a timestep of 1 fs and were run using LAMMPS (14 May 2016 version).28 3.1. Self-Diffusion Coefficients. The self-diffusion coefficients, Dself, of pure CH4 and CO2 were obtained from simulations using the Einstein relation. 1 d Dself = lim ⟨Δr(t )2 ⟩ (4) 6 t →∞ dt 2 In this equation, ⟨Δr(t) ⟩ is the mean-squared displacement of molecules at time t. The factor of 1/6 comes from the fact that the mean-squared displacement is averaged over three Cartesian coordinates. Simulations were performed in the canonical ensemble (constant NVT). The standard Nosé-Hoover thermostat29,30 was used to maintain constant temperature for CH4. Because CO2 molecules are modeled as rigid bodies, the Nosé-Hoover thermostat with chains31,32 was used with the algorithm described by Kamberaj et al.33 The densities of these simulations were taken from isothermal−isobaric ensemble (constant NPT) simulations at 101.325 kPa (using the NoséHoover thermostat and the Nosé-Hoover barostat34,35 for pure methane systems and the Nosé-Hoover thermostat with chains and the Nosé-Hoover barostat with chains for pure carbon dioxide systems). For pure CH4, three to five independent simulations, each with a single time origin, of 8000 molecules were run for each temperature. Each simulation contained a 200 ps equilibration period followed by 500 ns production period where the meansquared displacement of the molecules are calculated. Calculations were repeated for three other system sizes of 200, 1000, and 27 000 molecules in order to investigate the effect of system size on Dself. For pure CO2, three to five independent simulations, each with a single time origin, of 8000 molecules were run for each temperature. Each simulation contained a 500 ps equilibration period followed by a 10 ns production period. Simulations of CO2 were shorter than those of CH4 because the CO2 force field is computationally more expensive. Calculations were also repeated for system sizes of 200, 500, and 1000 molecules in order to probe the effect of system size. In this work, for all calculations of self-diffusion coefficients, the slope of the mean-squared displacement versus time was

Here, εij and σij are the Lennard-Jones parameters for the interaction between sites i and j. rij is the distance between the sites, and ϵ0 is the vacuum permittivity. qi and qj are the electric charges on sites i and j, respectively. CH4 was modeled using the united-atom TraPPE force field.22 That is, the molecule is treated as a single LennardJones site. CO2 was modeled using the rigid model developed by Potoff and Siepmann.23 The carbon and oxygen atoms are explicitly modeled as Lennard-Jones sites with charges. Each carbon atom is connected to two oxygen atoms with rigid bonds. The CO bond length is 0.116 nm and the OC O bond angle is 180°. Carbon atoms (in CO2) carry a partial positive charge and oxygen atoms carry a partial negative charge. The long-range Coulombic interactions were calculated using a particle−particle−particle mesh (PPPM) solver24 with a 1 × 10−4 relative error in forces. This method is typically more efficient25 than the standard Ewald summation method.26 All interactions were cut off at 1.5 nm. Analytical tail corrections were applied to the potential energy and pressure. It should be noted that the TraPPE webpage27 suggests that a cutoff of 1.4 nm and Ewald summations should be used for the aforementioned potentials. However, these differences are not expected to significantly affect results. The standard Lorentz-Berthelot mixing rules were used to calculate Lennard-Jones parameters for cross-interactions: 1 (σii + σjj) 2

site CH4 C (in CO2) O (in CO2) a

2. FORCE FIELDS In this work, the transferable potentials for phase equilibria (TraPPE) force fields were used to model CH4 and CO2. In the TraPPE formalism, the Lennard-Jones potential is used to describe nonbonded dispersion interactions. Electrostatic interactions are modeled with point charges and Coulomb’s law. The potential energy, Uij, between two nonbonded sites labeled i and j can be written as ÄÅ É ÅÅi y12 i y6ÑÑÑ qiqj ÅÅjj σij zz ÑÑ σ j z ij Uij = 4εijÅÅÅÅjjj zzz − jjjj zzzz ÑÑÑÑ + j rij z ÑÑ 4π ϵ0rij ÅÅj rij z k { ÑÑÖ ÅÅÇk { (1)

σij =

(3)

(2)

and B


É ÅÄÅ ∂ ln γ1 ÑÑÑÑ Å ÑÑ Γ = ÅÅÅÅ1 + x1 ÅÅÇ ∂x1 ÑÑÑÖT , P

Journal of Chemical & Engineering Data taken from the diffusive regime by ensuring the slope is equal to unity on a log−log scale. Once the diffusive regime was found, the slope of the mean-squared displacement was fit using the least squares regression technique. The time at which the system reached the diffusive regime slightly varied depending on the details of the simulation. For D self calculations, the slope was calculated after the first nanosecond of the production period until the end. Representative plots for the mean-squared displacement vs time on a log−log scale can be found in the Supporting Material. 3.2. Compressibility Factors. The compressibility factors, Z, for CH4−CO2 mixtures were calculated at several compositions, pressures, and temperatures using the following relation: Z=

PV NkBT

(7)

Here, γ1 is the activity coefficient of species 1. For ideal diffusive binary mixtures we can approximate DMS to be equal to the Darken diffusion coefficient, DDarken.37 DDarken is related to the self-diffusivities of both components in the mixture through the Darken equation:44 D MS ≈ DDarken = x 2D1self (x1 , x 2) + x1D2self (x1 , x 2)

(8)

For ideal thermodynamic binary mixtures, it is noted that Γ = 1. For a mixture that is ideal both in a diffusive sense and thermodynamic sense, eqs 6 and 8 can be combined to give DFick = D MS ≈ DDarken = x 2D1self (x1 , x 2) + x1D2self (x1 , x 2)

(5)

(9)

In this equation, P is the system pressure, V is the system volume, N is the number of molecules, kB is the Boltzmann constant, and T is the system temperature. Simulations were performed in the isothermal-isobaric ensemble. The Nosé-Hoover thermostat with chains and the Nosé-Hoover barostat with chains were used to maintain constant temperature and pressure for CO2 molecules. For CH4 molecules, the standard Nosé-Hoover thermostat was used to maintain constant temperature, while their positions were dilated in accordance to the aforementioned NoséHoover barostat with chains. This follows the instructions of the LAMMPS documentation on running an NPT simulation for a mixture of rigid and nonrigid bodies.36 Simulations were divided into a 1.5 ns equilibration period, followed by a 5 ns production period. The production period was divided into five independent blocks (each 1 ns) which were used to compute the average Z and the corresponding uncertainty. The compressibility factors of mixtures corresponding to x1 = 0.2711, 0.6051, 0.8470 were calculated at several temperatures and pressures. Here, x1 is the mole fraction of methane in the mixture. Simulations contained 500 CO2 molecules and, depending on the value of x1, either 186, 766, or 2767 CH4 molecules. Larger system sizes containing eight times as many molecules were run to test the effect of system size on the calculation of the compressibility factor. No statistically significant differences were observed (see the Supporting Material). 3.3. Fick Diffusion Coefficients. As stated earlier, the Fick diffusion coefficient, DFick, is defined as the coefficient that relates the diffusive flux to the concentration gradient through a linear relation. In general, calculating DFick from molecular dynamics isn’t as straightforward as calculating Dself. For general binary mixtures, the calculation of DFick requires knowledge of the Maxwell-Stefan diffusion coefficient, DMS, and a quantity called the thermodynamic factor, Γ.37−42 DMS is used in the Maxwell-Stefan model for diffusion for multicomponent systems and relates diffusion of a certain species to its chemical potential gradient. Green-Kubo relations exist for the calculation of DFick.43 However, for most conditions examined in this paper, a simplified approach can be taken. For a binary mixture,

DFick = ΓD MS

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The self-diffusion coefficients can be calculated using eq 4. Note that in eqs 8 and 9 the self-diffusion coefficients are composition-dependent,45 and are not the same as the purecomponent self-diffusivities described in subsection 3.1. Thus, the self-diffusivities of CO2 and CH4 need to be measured in the mixture. As discussed in subsection 3.1, calculations of diffusivities were taken in the diffusive regime. For most mixture simulations, the slope of the mean-squared displacement was calculated using a least-squares regression after the first 5 ns of the production period until the end. A more detailed discussion of the relationship between DFick and DMS for general mixtures can be found in the work by Jamali et al.37 In this article, DFick is first calculated for a binary mixture of CO2 and CH4 using eq 9 when Z = 1. Note that when Z = 1, a gas mixture behaves as an ideal gas mixture, that is intermolecular interactions between all species can be neglected. Because an ideal gas mixture is a specific type of ideal thermodynamic mixture (a mixture where interactions between all species are assumed to be identical), and an ideal diffusive mixture (a mixture where the displacement crosscorrelations between different species are negligible compared to self-diffusivities), eq 9 is valid. Simulations were performed in the canonical ensemble for mixtures containing 50 CO2 and 2500 CH4 molecules at various temperatures and densities. This corresponds approximately to a 2% CO2 mixture (i.e., x1 = 0.98). Constant temperature was maintained using the Nosé-Hoover thermostat with chains for CO2 molecules and the standard Nosé-Hoover thermostat was used for CH4 molecules. For each temperature and pressure, 3-5 independent simulations were used to calculate DFick. Each run was divided into a 1.5 ns equilibration period followed by a 15 ns production period. Calculations for DFick were also performed for two conditions for which Z is not equal to 1. For these cases, DFick was calculated using eq 6 because it is not obvious if eq 9 is valid at these conditions. To use eq 6, DMS and Γ need to be calculated. The calculation of DMS from MD simulations typically requires the calculation of the Onsager coefficients. However, for binary systems, due to the relationships between the Onsager coefficients, DMS can be calculated from the following equation:46,47

(6)

where C



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Table 2. Calculated Self-Diffusion Coefficients, Dself, for CH4 and CO2 for Different System Sizesa CH4 N

L (nm)

200 1000 8000 27000

19.98517 34.17417 68.33277 102.5200

CO2 Dself (cm2/s)

N

L (nm)

± ± ± ±

200 500 1000 8000

19.95420 27.08522 34.12524 68.25048

0.196 0.217 0.221 0.219

0.014 0.004 0.003 0.001

Dself (cm2/s) 0.111 0.109 0.107 0.108

± ± ± ±

0.008 0.003 0.004 0.002

a

N refers to the number of molecules and L is the corresponding box length at 293.2 K. The density of each simulation corresponds to a pressure of 101.325 kPa.

yz 1 ijj m1 jjj x1 + x 2zzzz 6Nx1x 2 k m2 { ÄÅ N d ÅÅÅÅ 1 ÅÅ∑ ri(0) − × lim ÅÅ t →∞ dt ÅÇ i = 1 2

D MS =

ÉÑ2 ÑÑ ∑ ri(t )ÑÑÑÑÑ ÑÑÖ i=1

2533.12 kPa and at 410.93 K and 5066.25 kPa. Simulations were run in the canonical ensemble at a density corresponding to the pressure of interest. Again, temperature was kept constant using the Nosé-Hoover thermostat with chains for CO2 molecules and the standard Nosé-Hoover thermostat was used for CH4 molecules. For both conditions, three to five independent simulations were used. Each run was divided into a 1.5 ns equilibration period followed by a 50 ns production period. The production period was longer in these cases in an attempt to improve the statistics of the mean-squared displacement of the sum of positions in eq 10. For each trajectory, 51 time origins were taken during the production period, each evenly spaced by 1 ns.

N1

(10)

In this equation, N is the total number of particles, N1 is the number of particles of species 1, x1 and x2 are the mole fractions of species 1 and 2, m1 and m2 are the masses of species 1 and 2, and ri(t) is the location of particle i of species type 1 at time t. The angled brackets indicate averages over multiple time origins. As a reminder, in this work, species 1 corresponds to CH4. Equation 10 requires the calculation of the mean-squared displacement of the sum of positions of species 1 over multiple time origins. Because only a single sum of positions is tracked during a simulation (as opposed to tracking N1 particles in a simulation for the calculation of Dself), obtaining adequate statistics for this calculation can be difficult. Similar to the calculation of Dself, for all calculations for DMS, the slope of the mean-squared displacement of the sum of particle positions is taken in the diffusive regime (where the slope is equal to unity on a log−log scale). The slope of the meansquared displacement of the sum of particle positions was fit using the least squares regression technique. A representative plot for the mean-squared displacement of the sum of particle positions is given in the Supporting Material. There are several methods for calculating Γ. For example, they can be calculated using an appropriate equation of state,42,48 or explicitly from simulations using the permuted Widom test particle insertion method,49,50 or the KirkwoodBuff integrals.41,43,51,52 In this work, the activity coefficient is calculated using the Peng-Robinson equation of state53 as implemented in Aspen Plus (version 10).54 It should be noted that using the Peng-Robinson equation of state is the least rigorous method to calculate Γ of the three mentioned because it does not rely on the force field employed in the simulations. However, it is adequate to demonstrate whether or not the binary mixture behaves as an ideal thermodynamic mixture at a given set of conditions. Calculation of Γ through the Widom test particle insertion method or through Kirkwood-Buff integrals directly depend on results from MD simulations and thus, depend on the force field. The Widom test particle insertion method relies on several types of test particle insertions in the NPT ensemble to calculate Γ. Kirkwood-Buff integrals relate Γ to radial distribution functions, which are straightforward to calculate from simulations. For calculation of DFick where Z is not equal to 1, simulations were performed for a binary mixture of 186 CH4 molecules and 500 CO2 (x1 = 0.2711) at 310.93 K and

4. RESULTS AND DISCUSSION This section will provide and discuss the results for the various properties described in section 3. In all cases, calculated properties are compared to experimental data or correlations when available. All error bars and uncertainties correspond to the 95% confidence intervals. 4.1. Self-Diffusion Coefficients. Table 2 gives the calculated self-diffusion coefficients for pure CH4 and pure CO2 at 293.2 K and a density corresponding to 101.325 kPa for various system sizes. It is clear that for almost all system sizes, there is no significant system-size effects observed for Dself. It should be noted that the calculated value for Dself of a 200 CH4 molecule system does not fall within the statistical uncertainty of the other system sizes. However, the upper bound of the 95% confidence interval for the system with 200 CH4 molecules deviates from the lower bound of 95% confidence interval for the system with 1000 CH4 molecules by less than 1.5%. Thus, it is concluded that system-size effects are negligible. This is consistent with the well-known system-size dependence of self-diffusion coefficients studied by Yeh and Hummer.55 In their work, Yeh and Hummer derived a size correction for the calculated self-diffusion coefficient given as self self = Dcalc + D∞

kBTξ 6πηL

(11)

Here, Dself ∞ is the self-diffusion coefficient for infinite system size, Dself calc is the self-diffusion coefficient calculated from a simulation in a cubic box with sides of length L and periodic boundary conditions. η is the viscosity calculated from simulation and ξ is a constant that is approximately equal to 2.837297 for a cubic simulation with periodic boundary conditions. Although we expect η to be small for vapor systems when compared to liquid systems, L is much larger for vapors than for liquids. Moreover, self-diffusion coefficients are orders of magnitude larger for gases when compared to liquids. Thus, it follows that the system-size correction in eq D



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11 will only contribute a small percentage relative to the 56 absolute value of Dself ∞ . A study by Moultos et al. showed that there were significant system-size effects on the calculated values of Dself for both CH4 and CO2. However, these calculations were performed at much higher pressures, 20000 kPa (200 bar) for CH4 and 50000 kPa (500 bar) for CO2. Thus, we expect the Yeh and Hummer correction to only be significant for these very high pressure cases. Figure 1 shows the percent deviation between experimental values12 of Dself and values computed from MD at 101.325 kPa. Figure 2. Percent deviation between experimental compressibility factors and calculated compressibility factors from MD simulations as a function of pressure for the CH4−CO2 mixture with x1 = 0.2711 at 310.93, 410.93, and 510.93 K. Experimental data are taken from Reamer et al.59

Figure 1. Percent deviation between experimental self-diffusion coefficients and calculated self-diffusion coefficients from MD simulations for CH4 (blue) and CO2 (red) at 101.325 kPa. Experimental results are taken from Weismann and DuBro.12

Although the calculations from simulations slightly underpredict Dself for both CH4 and CO2 when compared to experimental data, the deviation does not exceed 10% at the conditions examined. The tendency of the CH4 and CO2 TraPPE force fields to underestimate Dself was shown in a previous work.18 It was shown that the all-atom OPLS CH4 force field57 gives an average absolute relative deviation for Dself of 5.01%. The flexible Cygan CO2 force field58 gives an average absolute relative deviation for Dself of 4.85%. The TraPPE force fields yielded an average absolute relative deviation of 6.96% and 12.56% for Dself for CH4 and CO2, respectively. Note that the calculated percent deviations for Dself in the present work are consistent with the average absolute relative deviations calculated by Aimoli et al.18 However, at 1 atm and the temperatures examined, CH4 and CO2 both behave as ideal gases for which intermolecular interactions are not important. Therefore, we should expect agreement between simulations and experiments. It should be noted that uncertainties for the experimental values are not reported by Weissman and DuBro.12 It is possible that results from experiments and simulations are consistent within their combined uncertainties. 4.2. Compressibility Factors. Figures 2, 3, and 4 show the percent deviations of Z from MD simulations compared to experimental data from Reamer et al.59 for various CH4−CO2 mixtures. Note that in Figure 4, error bars are smaller than the symbol size. Figure 2 contains the largest error bars due to the small system size relative to the ones used to produce Figures 3 and 4. Results are shown for mixtures with mole fractions of methane equal to 0.2711, 0.6051, and 0.8470. At each composition, Z is calculated for different pressures at 310.93, 410.93, and 510.93 K. For all three compositions examined, we first note that the MD predictions at 510.93 K are very accurate, each having a

Figure 3. Percent deviation between experimental compressibility factors and calculated compressibility factors from MD simulations as a function of pressure for the CH4−CO2 mixture with x1 = 0.6051 at 310.93, 410.93, and 510.93 K. Experimental data are taken from Reamer et al.59

Figure 4. Percent deviation between experimental compressibility factors and calculated compressibility factors from MD simulations as a function of pressure for the CH4−CO2 mixture with x1 = 0.8470 at 310.93, 410.93, and 510.93 K. Experimental data are taken from Reamer et al.59 Error bars are smaller than symbol size.

percent deviation from experiments of less than 1%. At this temperature, over the pressure range examined, all three binary mixtures behave as ideal gas mixtures. At 410.93 K, the mixture with x1 = 0.2711 gives perfect agreement with experimental data within statistical uncertainty. For x1 = 0.6051 and 0.8470, MD simulations accurately predict Z at low pressures, but there is a slight overestimation of Z at higher pressures although the overestimation does not ever exceed 10%. The same trend is observed for x1 = 0.6051 and E



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an empirical model that was fit to 340 experimental diffusion coefficients found in the literature. It should be noted that the experimental data used to fit the Fuller correlation are stated to be limited in temperature range for organic systems. Thus, caution should be taken when applying this correlation. The form of the Fuller correlation can be found in the Supporting Material. Figure 5 also gives the percent deviations of experimental values12 relative to the Fuller correlation. The values of DFick from the MD calculations are given in Table 3 along with the corresponding calculated Z. Analogous plots showing the absolute value of DFick as a function of temperature for MD simulation results, experiments, and the Fuller correlation are given in the Supporting Material. Figure 5 shows that the MD predictions agree with experiments, and the Fuller correlation within statistical uncertainty for most temperatures examined at 101.325 kPa. Figures 6 and 7 show that at both 506.625 and 1013.25 kPa, there is also agreement between MD predictions and the Fuller correlation within statistical uncertainty for most temperatures examined. It it noted that the uncertainty associated with the calculations can lead to significant percent deviations, however all deviations are below 25%. The agreement between MD calculations, experiments, and the Fuller correlation should be expected because the CH4− CO2 binary mixture behaves as an ideal gas mixture in this temperature and pressure space. This can be inferred from the results shown in the previous subsection. That is, regardless of temperature and composition, the experimental data of Reamer et al.59 and MD predictions show that Z does not deviate far from unity at pressures less than or equal to 1013.25 kPa. This is confirmed for the aforementioned MD calculations explicitly in Table 3. The effect of system size was examined for DFick. No significant system size effects were found, similar to the self-diffusivities given in Table 2 (see the Supporting Material). It has been stated that for the above calculations that the thermodynamic factor, Γ, is equal to 1. Although this is obviously true for ideal gas mixtures, it is explicitly shown using the Peng-Robinson equation of state.53 Figure 8 shows Γ as a function of x1 at 1013.25 kPa for both 293.1 and 500 K. These are two conditions where DFick was calculated from MD simulations. It is clear from the graphs that Γ does not deviate far from one for all values of x1 at these conditions, validating the assumption used for eq 9. Note that at lower pressures (101.325 and 506.625 kPa), Γ will deviate even less from 1. This is because at lower pressures, the CH4−CO2 mixture will behave more like an ideal gas mixture. 4.4. Fick Diffusion Coefficients at Nonideal Gas Conditions. Table 4 gives the calculated values for Z and DFick for a CH4−CO2 mixture with x1 = 0.2711 at 310.93 K and 2533.12 kPa and 410.93 K and 5066.25 kPa. Calculations were performed using eqs 6 and 10. Corresponding calculations for DFick are also shown using the Fuller correlation. Γ is shown for both conditions as a function of x1 in Figure 9. From the values of Z, it can be seen that CH4−CO2 mixture at these conditions behave less like an ideal gas than the mixtures described in Table 3. The values for DFick from MD simulations differ from the calculations from the Fuller correlation. Considering the calculated uncertainties, at 310.93 K and 2533.12 kPa the difference between the two methods used to calculate the Fick diffusion coefficient

0.8470 at 310.93 K. For x1 = 0.2711 at 310.93 K, there is low deviation from experiments at low pressure but significant underestimation of Z at higher pressures. 4.3. Fick Diffusion Coefficients at Ideal Gas Conditions. Percent deviations for the MD calculations of DFick relative to a correlation by Fuller et al.60 are given in Figures 5, 6, and 7 for the CH4−CO2 system. The Fuller correlation is

Figure 5. Percent deviation of Fick diffusion coefficients calculated from MD and from experiments12 with respect to the Fuller correlation.60 The system corresponds to a CH4−CO2 binary mixture with x1 = 0.98 at 101.325 kPa. Percent deviations of experimental values from Weissman and DuBro12 are shown in red. Percent deviations of MD simulation results are shown in green. Note that error bars for experimental results are not given because uncertainties were not reported.

Figure 6. Percent deviation of Fick diffusion coefficients calculated from MD with respect to the Fuller correlation.60 The system corresponds to a CH4−CO2 binary mixture with x1 = 0.98 at 506.625 kPa.

Figure 7. Percent deviation of Fick diffusion coefficients calculated from MD with respect to the Fuller correlation.60 The system corresponds to a CH4−CO2 binary mixture with x1 = 0.98 at 1013.25 kPa.

F



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Table 3. Calculated Compressibility Factors and Fick Diffusion Coefficient for the CH4−CO2 Binary Mixture with x1 = 0.98 from MD Simulations Described in Subsection 3.3a 101.325 kPa T (K)

Z

293.1 317.4 367.9 400.0 450.0 500.0

0.998 0.998 0.999 0.999 1.000 1.000

D

Fick

506.625 kPa 2

Z

(cm /s)

0.173±0.031 0.181±0.035 0.251±0.030 0.281±0.017 0.355±0.050 0.394±0.092

D

0.993 0.994 0.998 0.998 0.999 1.000

Fick

1013.25 kPa 2

(cm /s)

0.031±0.005 0.041±0.003 0.050±0.008 0.067±0.009 0.075±0.010 0.088±0.015

Z

DFick (cm2/s)

0.985 0.989 0.995 0.997 0.999 1.001

0.017±0.001 0.018±0.004 0.026±0.002 0.032±0.005 0.038±0.005 0.045±0.007

a

Fick diffusion coefficients were calculated using eq 9.

Figure 8. Thermodynamic factors for the the CH4−CO2 mixture as a function of CH4 mole fraction (x1). The blue squares are for calculations at 293.1 K and 1013.25 kP. The green squares are for calculations at 500 K and 1013.25 kPa. The black-dotted line corresponds to where the thermodynamic factor is equal to 1. Activity coefficients were calculated using the Peng-Robinson equation of state53 as implemented in Aspen Plus (version 10).54

especially true in process simulations for which correlations found in the literature are convenient. It should be noted that system-size effects were not examined for the MD results given in Table 3. This is because in all aforementioned simulations the system-size correction was found to be negligible. Examining Figure 9 shows that Γ does not deviate far from 1 for the CH4−CO2 mixture at 310.93 K and 2533.13 kPa and 410.93 K and 5066.25 kPa. Although the deviations at these conditions are larger than those shown for the conditions in Figure 8, it was found that using the Darken equation (eq 9) does not yield results that are statistically different than those using eqs 6 and 10. This comparison can be found in the Supporting Material. This suggests that the Γ for the conditions examined in Table 4 is negligible enough that eq 9 is still valid for calculating DFick. It would be interesting to consider a system where Γ is more significant. Consider a mixture of 250 CH4 molecules and 500 CO2 molecules (x1 ≈ 0.333) at 310.93 K and 10132.5 kPa. At this temperature and pressure, this mixture does not behave like an ideal gas mixture. MD simulations estimate that Z = 0.65 ± 0.02. It also does not behave like an ideal thermodynamic mixture. Figure 10 explicitly shows this by giving Γ as a function of x1 at 310.93 K and 10132.5 kPa calculated from the PengRobinson equation of state. This figure shows that at x1 = 0.333, Γ ≈ 0.8. Thus, at this condition eq 9 should not be accurate.

Table 4. Calculated Compressibility Factors and Fick Diffusion Coefficient for the CH4−CO2 Binary Mixture with x1 = 0.2711 from MD Simulations Described in Subsection 3.3a 310.93 K and 2533.12 kPa DFuller 2

410.93 K and 5066.25 kPa

Z

DFick (cm2/s)

(cm /s)

Z

DFick (cm2/s)

DFuller (cm2/s)

0.92

0.014±0.005

0.008

0.95

0.020±0.008

0.006

a

Fick diffusion coefficients were calculated using eqs 6 and 10. Calculations are compared to values predicted from the Fuller correlation.60

deviates anywhere between 12.5% to 137.5%. At 410.93 K and 5066.25 kPa, the deviation is anywhere between 90% to 345%. Clearly there is a significant discrepancy between the two methods. Unfortunately, to the best of the author’s knowledge, experimental Fick diffusion coefficients for the CH4− CO2 system at these high pressure are not available in the literature. Thus, it is difficult to say which method yields more accurate results. Both approaches have their flaws. MD simulations rely on the accuracy of the TraPPE force fields at these conditions, while the reliability of the Fuller correlation depends on if the correlation is able to be extended to high pressure conditions where no experimental data was used in the fitting procedure. However, this analysis emphasizes the fact that correlations should be rigorously vetted before adopting them for practical use. This is G



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Figure 9. Thermodynamic factors for the the CH4−CO2 mixture as a function of CH4 mole fraction (x1). The blue squares are for calculations at 310.93 K and 2533.12 kPa. The green squares are for calculations at 410.93 K and 5066.25 kPa. The black-dotted line corresponds to where the thermodynamic factor is equal to 1. Activity coefficients were calculated using the Peng-Robinson equation of state53 as implemented in Aspen Plus (version 10).54

Figure 10. Thermodynamic factor for the the CH4−CO2 mixture as a function of CH4 mole fraction (x1). Calculations are performed at 310.93 K and 10132.5 kPa. The black-dotted line corresponds to where the thermodynamic factor is equal to one. Activity coefficients were calculated using the Peng-Robinson equation of state53 as implemented in Aspen Plus (version 10).54

NVT simulations of a mixture of 250 CH4 molecules and 500 CO2 molecules were run at 310.93 K and at a density corresponding to 10132.5 kPa. Temperature was kept constant using the same methodology outlined in subsection 3.3. To improve statistics, 15 independent simulations were run. Each contained a 1.5 ns equilibration period followed by a long production period of 300 ns. The production period was very long in order to take 61 evenly-spaced time origins, each 5 ns apart. The 15 mean-squared displacements of the sum of CH4 positions are shown as a function of time on a log−log plot in Figure 11. It can be seen that the diffusive regime has not been reached as the lines are parallel to the red-dotted line corresponding to a slope of 2 on the log−log plot. Thus, the simulations appear to still be in the ballistic regime. This is despite the long simulation time and multiple time origins. Therefore, DMS cannot be calculated from these lines. It is possible the simulations have not been run long enough to reach the diffusive regime. It is also possible that

Figure 11. Mean-squared displacements of the sum of CH4 molecule positions in a CH4−CO2 mixture with x1 = 0.333 at 310.93 K and 10132.5 kPa. The red-dotted line corresponds to a slope of 2 on the log-log scale, and the black-dotted line corresponds to a slope of 1 on the log−log scale.

more statistics need to be collected. More work needs to be done in order to test these two hypotheses. A new plugin for H



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LAMMPS has recently been developed by Jamali et al.61 and can be used to efficiently explore this case in more detail.

Notes

5. CONCLUDING REMARKS The Fick diffusion coefficient was calculated for the gaseous CH4−CO2 mixture at various temperatures and pressures using molecular dynamics. CH4 was modeled using the united-atom TraPPE force field developed by Martin and Siepmann,22 while CO2 was modeled using the rigid TraPPE force field developed by Potoff and Siepmann.23 It was shown that using these two models together with the standard Lorentz−Berthelot mixing rules provide accurate predictions of the compressibility factor over a wide range of compositions, pressures, and temperatures. However, there seems to be deviation from experimental results at low temperatures and high pressures. It is possible that an alternative mixing rule could address this discrepancy, however this was not examined in this work. Nevertheless, predictions of the compressibility factor from MD simulations agreed well with experiments at 1013.25 kPa and lower. The calculated Fick diffusion coefficients were in reasonable agreement with experiments12 and the Fuller correlation.60 This agreement was expected due to the ideal nature of the mixture. Calculation of the Fick diffusion coefficient for a CH4−CO2 mixture at conditions in which the both the ideal gas mixture and ideal thermodynamic mixture assumptions do not hold was attempted in order to the explore the effect of the potentials. For a CH4−CO2 mixture with x1 = 0.2711 at 310.93 K and 2533.12 kPa and at 410.93 K and 5066.25 kPa calculated DFick from MD simulations were compared to those calculated from the Fuller correlation. Different results were obtained, but no experimental data are available to assess the accuracy of each method. However, the differences emphasize the importance of the caution that should be taken when applying correlations found in the literature, especially when they are applied to conditions not considered in the fitting procedure. Finally calculations were also attempted for a CH4−CO2 mixture with x1 = 0.333 at 310.93 K and 10132.5 kPa However, difficulties arose due to poor statistics or insufficient simulation times (despite a production run of 300 ns). Future work will examine these difficulties in more detail.

ACKNOWLEDGMENTS The author would like to thank Edward Maginn for advice in properly and efficiently implementing the TraPPE force fields in LAMMPS. The author would also like to thank Othon Moultos and Seyed Jamali for helpful advice on calculating Fick diffusion coefficients.

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The author declares no competing financial interest.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.8b01198. Calculated and experimental values of self-diffusion coefficients, compressibility factors and Fick diffusion coefficients; the methodology behind calculation of the diffusion coefficients and activity coefficients; and comparison between two different methods for calculation of Fick diffusion coefficients (PDF)

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REFERENCES

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Corresponding Author

*E-mail: [email protected]. ORCID

Joseph R. Vella: 0000-0002-8666-8719 I



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