Mixed Alkanes in Nanoslit

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Adsorption and phase behavior of pure/mixed alkanes in nano slit graphite pores: an iSAFT application Jinlu Liu, Le Wang, Shun Xi, Dilipkumar Asthagiri, and Walter G Chapman Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02055 • Publication Date (Web): 01 Sep 2017 Downloaded from http://pubs.acs.org on September 6, 2017

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Adsorption and phase behavior of pure/mixed alkanes in nano slit graphite pores: an iSAFT application Jinlu Liu, Le Wang, Shun Xi, Dilip Asthagiri, and Walter G. Chapman∗ Department of chemical and biomolecular engineering, Rice University, Houston,TX E-mail: [email protected]

Abstract Prediction of fluid phase behavior in nano-scale pores is critical for shale gas/oil development. In this work, we use a molecular density functional theory (DFT) to study the effect of moelcular size and shape on partitioning to graphite nano-pores as a model of shale. Here interfacial Statistical Associating Fluid Theory (iSAFT) is applied to model alkane (C1 − C8 ) adsorption/desorption/phase behavior in graphite slit pores for both pure fluids and mixtures. The pure component parameters were fit to bulk saturated liquid density and vapor pressure data in selected temperature ranges. The potential of interaction between the fluid and graphite is modeled with a Steele 10-4-3 potential that is fit to the potential of mean force from single molecule simulations. Good agreement is found between theory and molecular simulation for the density distributions of pure components in slit pores. The critical properties of methane, ethane and their mixtures as well as the shift in bubble point and dew point densities were studied, showing good agreement with simulation. The competitive adsorption of mixtures of normal and branched alkanes in graphite pores was also studied. Heavier

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components more strongly adsorb up to the point that the entropic penalty due to confinement reduces adsorption.

Introduction The exploration and production of shale gas/oil requires reliable prediction of fluid phase behavior at reservoir conditions. The pores in shale are on the nanometer scale; unlike that in conventional sandstones where the pore sizes are in microns. The nano-confinement of hydrocarbons can affect the phase behavior and partitioning of components between the matrix and production fractures due to the strong interaction between the fluid molecules and pore surface. Successful reservoir simulation can only be achieved with an accurate model of hydrocarbon phase behavior, and a number of studies have been performed to better understand fluids in confined geometry. The physics of nano-confinement also affects other applications involving porous material, such as gas storage using metal organic framework, 1 catalysis in zeolites, 2 and gas mixture separation. 3 One of the primary applications in the shale gas/oil industry is to determine the density and phase behavior of the in-place hydrocarbon mixture as a function of temperature and pressure (ρ(T, P )). Standard adsorption and desorption experiments are often used to characterize porous materials through capillary condensation and hysteresis in nanoscale pores. In shale oil production, the bubble point is one of the key parameters to ensure continuous flow of liquid production. Luo et al. 4 used differential scanning calorimetry to measure the bubble point temperature of octane and decane in controlled-pore glasses (CPGs) with pore sizes of 4.3 and 38.1 nm. Cho et al. 5 measured the bubble point pressures of octane-methane and decane-methane mixtures in mesoporous silica (SBA-15 and SBA-16). Both of these experimental studies reveal the impact of confinement on lowering the bubble point for pores less than 5 nm. In addition to experimental work, molecular simulations have been used extensively to

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understand the effect of confinement on fluid properties. Severson and Snurr 6 studied single component adsorption of alkanes in carbon slit pores. Besides the adsorption isotherms, the orientation profiles were also reported showing that short alkanes lay parallel to the walls while longer alkanes have less distinct layering. To study how confinement affects criticality, Akkutlu and Didar 7 used NVT-Gibbs Monte Carlo method, and Pitakbunkate et al. 8 used grand canonical Monte Carlo method to obtain the critical parameters of methane and ethane in graphite slit pores. Their results, though different from each other, showed a large deviation of critical properties from the bulk. The suppressed critical point was shown to cause a large compositional change in gas production. In a series of molecular dynamics studies by Hu et al., 9–11 an activated kerogen model, namely graphene fragments with oxidized functional groups, was proposed in comparison with the simple graphite model. They have studied the behavior of water and hydrocarbon in model shale nano-pores. The activated kerogen model shows mixed wetting properties and water tends to be trapped inside the pores. They have also shown that a rough surface model adsorbs alkanes more evenly while a smooth surface adsorbs alkanes in layers. A more rigorous approach to model kerogen molecules, which is based on analysis of chemical structure, elemental composition and physical properties, was taken by Ungerer et al. 12 and the adsorption isotherms of methane and ethane on different sized kerogen pores were generated. 13 The adsorption isotherms were modeled by an Extended Langmuir model and Ideal Adsorbed Solution Theory, which can capture the adsorption behavior at low pressure but only work well for large sized pores at high pressure. Molecular simulation by atomistic models also helps improve understanding of kerogen morphology and surface properties. Recent work by Vasileiadis et al. 14 has demonstrated the characterization of pore size and pore surface elemental analysis using Ungerer’s model. Computer simulation is a powerful tool to provide detailed and accurate description of real kerogen systems. The high computing times of molecular simulation are rewarded by accurate prediction of system behavior. However, for some purposes such as understanding

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the phase behavior of a multicomponent system, density functional theory is computationally more efficient. Particularly, if one of the components in the mixture is of extremely low concentration, it becomes very challenging to obtain good statistics in molecular simulation, resulting in increased computational cost. On the other hand, a theoretical method such as density functional theory can be validated versus molecular simulation and efficiently extend results to multicomponent mixtures and trace components. The molecular density functional theory applied here combines several approaches to describe short range repulsion, long range attraction, and molecular shape. Alkane molecules are modeled as chains of spherical segments. The basic building block of the theory is an inhomogeneous hard sphere fluid. The fundamental measure theory 15 proposed by Rosenfeld and a few variants thereafter realized a quantitative description of inhomogeneous hard sphere mixtures. Bonding of spheres to form polyatomic molecules is accomplished through extensions of Wertheim’s perturbation theory 16–19 for associating and polymerizing spheres 20 21 . 22 Long range attractions are described using a mean field approximation (MFA) described later. The mean field approximation (MFA), weighted density approximation (WDA) and Wertheim perturbation theory 16–19 allow for modeling molecular size and shape. Spherical molecules such as Lennard-Jones spheres and associating spheres in confined geometry have been studied with DFT by numerous groups such as Peterson et al., 23 Sokolowski and Fischer, 24 Segura et al., 21 Yu and Wu, 25 Tripathi and Chapman, 26 Haghmoradi et al.. 27 The adsorption isotherm kernel generated from DFT has been used to characterize nanoporous carbon and silica 28–30 and has become one standard analysis tool in commercial adsorption instruments. Although small molecules (e.g. nitrogen, argon and krypton) that are nearly spherical are mostly used in nanoporous material characterization, polyatomic molecules in confined media can be more fascinating in that practical systems often involve mixtures such as in polymer synthesis, lubrication and hydrocarbon storage in shale. The development of inhomogeneous statistical associating fluid theory (iSAFT), 22 modified iSAFT 31 and polymer-DFT 25 have largely enhanced our ability to model inhomogeneous systems of

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complex fluids. The implementations of iSAFT or polymer-DFT, to name a few, include interfacial tension of polymers, 32 lipid assembly and confined block copolymers, 33 tethered polymers within or without solvent, 34 surfactant in oil-water interface, 35 formation of micelle in oil/water/surfactant systems, 36 tethered polymer modified pores, 37,38 and polymer adsorption. 39 For applications in shale gas/oil systems, other DFT approaches have also been proposed based on some common bulk equations of state. For example, the adsorption and desorption behavior of hydrocarbon mixtures was studied by Li et al. 40 using a Peng-Robinson equation of state (EoS) free energy functional. A SAFT 41 or PC-SAFT 42 based free energy functional, however, is more promising since molecular shape is included in the free energy functional. To enable iSAFT to match the dispersion contribution of the bulk PC-SAFT EoS, several modifications of the dispersion functional have been proposed. Shen et al. 43,44 used a WDA in the dispersion term to model methane and carbon dioxide adsorption on porous carbon. Klink and Gross 45 included the hard chain pair correlation function in the dispersion functional and incorporated a local correction term to absorb the difference from the PC-SAFT EoS chemical potential. Xu et al. 46 decomposed the dispersion term to a local and non-local contribution where the surface averaged density was used in a weighted density approximated local term and a mean field manner was used to account for the density variation. These developments can lead to an improved prediction of bulk pressure, density and chemical potential, which makes it easier to compare with experiment and simulation. The accuracy of these dispersion terms that reduce iSAFT to PC-SAFT EoS in the bulk are not fully proven for confined fluids. A WDA alone does not account for the long-range attraction sufficiently and “local plus correction”approaches 47 have only been applied to fluid-fluid interfacial systems. On the other hand, an addition of hard sphere repulsion and mean field attractive interactions is still the “simplest and very successful approach” 48 to model dispersion interactions. In our current work, we use a MFA in describing the dispersion contribution in iSAFT. Model parameters are fit to pure component vapor pressure and saturated liquid densities.

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For a fluid with fixed chemical potential (µ), temperature(T ), and volume(V ), the grand potential or grand free energy (Ω) is a minimum in the equilibrium state, with Ω determined by the chemical potential of each species (µi ), intrinsic Helmholtz free energy of the fluid (A[(ρ(r)]) and external potential from the graphite wall(V ext (r)),

Ω[(ρ(r)] = A[ρ(r)] −

n Z X i=1

ρi (r)(µi − Viext (r))dr,

(1)

where n is the total number of species,Viext (r) is the external potential acting on each species, ρi (r) is the molecular density of the ith species. With Ω being a free energy functional, the equilibrium density distribution is obtained by minimizing the grand potential, δ Ω[(ρ(r)] =0 δ ρi (r) equilibrium

for i = 1, 2, ... n.

(2)

The average density is then determined from the equilibrium density profile of each component.

Free energy functional In density functional theory, the main challenge is to approximate the Helmholtz free energy functional and this functional can be formed through a perturbation approach. In the iSAFT method proposed by Tripathi and Chapman, 22 molecules are constructed by bonding the spherical beads or segments to form chains that resemble the structure of molecules. In this work, our interest is only in alkanes, so the molecules are represented by chains of homonuclear spheres. For a mixture of n molecular species, we have n types of segments with mi segments in each chain molecule. The Helmholtz energy is a summation of different contributions, namely ideal gas (Aid ), hard sphere (Ahs ), chain formation (Ach ) and dispersion (Adisp ) contributions,

A[ρ(r)] = Aid [ρ(r)] + ∆Ahs [ρ(r)] + ∆Ach [ρ(r)] + ∆Adisp [ρ(r)]. 7


(3)

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Ideal gas contribution The exact ideal gas free energy for an atomic mixture of n species is

id

A [ρ(r)] = kT

n Z X i=1

dr

mi X α=1

ρ(i) (r) ln(ρ(i) (r)Λ3i ) − 1 α α

(4)

where k is the Boltzmann constant,mi is the segment number in species i, Λi is the thermal de Broglie wavelength and ρα(i) (r) denotes the density of segment α in the ith molecule. A correction of the atomic ideal gas free energy is included in the chain term. For a homonuclear chain, all the segments have the same density, i.e. ρ(i) (r) = ρ(i) (r) = ...... = ρ(i) (r) = ρi (r). α mi β As Λi is temperature dependent and the temperature is a constant here, the ln(Λ3i ) will be canceled by the corresponding term in bulk chemical potential in Eqn. (1). Hard sphere contribution The hard sphere part accounts for the excluded volume effect for an inhomogeneous fluid. We use the original fundamental measure theory 15 , in which the hard sphere free energy functional is hs

∆A [ρ(r)] = kT

Z

Φex,hs [nj (r)] dr,

(5)

and Φex,hs [nj (r)] is the excess Helmholtz free energy density. It is a functional of six fundamental measure densities nj (r) of the inhomogeneous fluid, Φex,hs [nj (r)] = −n0 ln(1 − n3 ) +

n1 n2 nv1 · nv2 n2 (nv2 · nv2 ) n2 3 + − 2 − 1 − n3 24π(1 − n3 ) 1 − n3 8π(1 − n3 )2

(6)

where nj (r) are calculated by fundamental weights,

nj (r) =

n Z X i=1

ρi (r)ωi (j) (r − r′ )dr′ ;

j = 0, 1, 2, 3, v1, v2

(7)

At each position, the summation runs over all the species in the mixture. The weight

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where the chain free energy functional is formed by complete association of spheres with directional association sites. The free energy functional for mixtures of associating spheres was developed by Chapman 20 and Segura et al. 21 based on Wertheim’s thermodynmaic perturbation theory (TPT1). 16–19 A schematic representation of chain formation is shown in Fig.2 where spheres with directional association sites are forced to bond in a specific order to form chainlike molecules. The resulting chain free energy functional for homonuclear chains is n

∆Ach X = 2(mi − 1) · kT i=1 where

′

δ(|r1 −r2 |−σ αα ) 4π(σ

αα′ 2

)

Z

1 dr1 ρα (r1 ) − ln 2 (i)

′ δ(|r1 − r2 | − σ αα ) αα′ 1 (i) dr2 y (r1 , r2 )ρα (r2 ) + 2 4π(σ αα′ )2 (11)

Z

enforces tangential bonding. The cavity correlation function y αα (r1 , r2 ) ′

is only needed at contact distance of σ αα , and we evaluate it by the bulk reference fluid ′

radial distribution function at some averaged density ρ(i) α (r1 ),i.e., αα′

ycontact (ρ(i) (r1 ), σ α

αα′

HS ) = ycontact (ρ(i) α (r1 ), σ

αα′

)

(12)

Here we use the simplest weight within a sphere, i.e., ρ(i) α (r1 )

3 = 4π(σα )3

Z

|r1 −r2 |σαβ

dr1 dr2 uatt αβ (|r2 − r1 |) · mi ρi (r1 ) · mj ρj (r2 )

(14)

and the pair potential is given by a WCA-type perturbation,

uatt αβ

   0 0 < r < σ αβ    = −εαβ − uLJ σ αβ < r < rmin cut      4εαβ [(σ αβ r)12 − (σ αβ r)6 ] − uLJ rmin < r < rcut cut

(15)

in which uLJ cut is the Lennard-Jones pair potential evaluated at cutoff distance, and here we use rcut = 4σ αβ . For mixtures, the Lorentz-Berthelot mixing rule σ αβ = (σ α + σ β )/2 and √ εαβ = εαα εββ is applied. Grand potential The formalism to calculate the equilibrium density profile can be found directly from the original paper by Tripathi and Chapman. 22 To calculate the grand potential, we use the form in modified iSAFT developed by Jain et al. 31 that is more convenient and does not assume homonuclear molecules. Although modified iSAFT is more rigorous and predicts the exact ideal chain distribution, it requires an integer number of segments whereas original iSAFT (like the bulk SAFT EoS) allows non-integer values. In our tests of both methods using integer numbers of segments (m = 1, 2, 3...), we find there is hardly any difference between the converged density profiles for a wide range of temperatures and pressures, which ensures the reliability to transfer a grand potential functional from modified iSAFT to original iSAFT.

βΩ[{ρ(r)}] =

n Z X i=1

Di (r) =

n X i=1

(1 − mi )

Z

[ρi (r) (mi Di (r) − 1)] dr + β∆Ahs + β∆Adisp

(16)

′

αα [{ρi (r1 )}] δ ln ycontact δ∆Ahs δ∆Adisp ρi (r1 ) dr1 − − δρi (r) δρi (r) δρi (r)

11


(17)

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Note the terms that involve mi are just simplified from Equation (32) in modified iSAFT (i)

grand potential 31 with segment densities ρα (r) identical in each chain. External potential As we see from Eqn. 1, the effect of fluid-solid interaction is only modeled by the external potential. For a graphitic wall, this external field is modeled by a Steele 10-4-3 potential 51 and for slit pores, the potential from both walls can be added to give Vjext (z) = Vj10−4−3 (z) + Vj10−4−3 (H − z) , Vj10−4−3 (z) −3

where ρs = 0.114Å

= 2πρs ∆s σsj

2

2 σsj 10 σsj 4 σsj 4 εsj − − 5 z z 3∆(z + 0.61∆)3

(18)

is the carbon atom density,∆s = 3.35Å is the separation distance

between two graphite layers, σsj and εsj are the segment diameter and interaction energy √ that are estimated from Lorentz-Berthelot mixing rules: σsj = (σs + σj )/2 and εsj = εs εj , where ǫs and ǫj are the interaction energies for solid-solid and fluid-fluid interactions; σs and σj are the carbon atom diameter and segment diameter of component respectively. The commonly used parameters for graphite are σs = 3.345 Å and εs /k = 28 K. 52 However, this external potential was derived for a spherical molecule interacting with a surface assuming the molecule has only one interaction site. For a chain molecule with multiple segments, the interacting force at each position is determined also by the configurational change within the molecule. Previous studies using an external potential in density functional theory usually treat ǫsj as an adjustable parameter to fit the experimental data. For example, in the work by Shen et al., 43 the interaction energy between solid and fluid was adjusted along with pore size to achieve the best fit of the experimental isotherm on porous carbon. However, such an approach can mask the deficiency of the free energy functional and confound the interpretation of other contributions to the functional. Here we determine the interacting force between a single molecule and a surface by

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molecular simulation and apply the simulated potential of mean force to the DFT model. The adaptive biasing force method 53 is selected, in which a force profile is applied to the center of mass of the molecule and eventually the molecule is allowed to move freely in the specified domain as if there is no attractive surface. In this way the force is averaged for all possible configurations of the molecule, which has a different magnitude from a simple summation of the interacting forces on each site with only one configuration. Detailed information on potential of mean force simulation can be found in Supplementary Material A.

Equilibrium density profile Solving Eqn. (2) results in the equilibrium density profiles for n species in the system. Based on the functional contributions described above, the density profiles are obtained from 

µi mi

Viext (r) mi

{0,1,2,3,v1,v2} P

R

(j) ωi (|r

ex,hs r1 |) ∂Φ ∂nj (r1 )

− − − dr1  j   P n R  − dr1 uatt  ij (|r1 − r|)mj ρj (r1 ) ρi (r) = exp  j=1 |r−r1 |>σij h i   + mi ln(y cont (r)λ (r)) − 1 + R dr ρ (r ) δ ln λii (r)  ii 1 i 1 ii 2 δρi (r)  R cont δ ln y (r ) dr1 ρi (r) δρiii (r) 1 + mmi −1 i



      ; i = 1, 2, ...n    

(19)

where λii (r) =

Z

dr1

δ(|r1 − r| − σi ) ρi (r1 ) 4π(σi )2

(20)

It should be noted that Eqn. (19) has been simplified from Tripathi and Chapman. 22 For the ith chain, even though the two end segments would have different chain contribution from the middle segment, we take the average over all the segments and assume that all the segment density distributions are identical. The averaged density inside the slit pore is defined as, ρavg i

1 = H − σs

Z

H−σs /2

ρi (z)dz, σs /2

13


(21)

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which is used to calculate the adsorption isotherms.

Numerical procedures An advantage of the original iSAFT is the simplicity in the chain free energy that requires no recursive integrals. This method shows stable and satisfactory convergence performance with a simple Picard iteration starting from a uniform density profile in the pore. The discretization resolution we use is dz = 0.01σ and a symmetric density profile is assumed to impose a reflective boundary condition. In calculating the adsorption or desorption isotherms, we take the converged density profile from the last pressure (Pi ) as an initial guess for calculation at the next pressure (Pi+1 ).The initial guess plays an important role since the metastable state in hysteresis can only be established with an appropriate initial guess density profile.

Results and discussion Parameters for n-alkanes and external potentials Within the framework of iSAFT, the alkane molecules is described using three parameters, which are the chain length or number of segments (m), the segment size (σ), and the dispersion energy of each segment (ǫk ). As was described above, the dispersion contribution is solely accounted by mean field approximation, which means the radial distribution function is set equal to one outside of hard sphere contact. It is therefore expected that the dispersion energy between molecules will not be the same as parameters from bulk SAFT. To retain both efficiency in inhomogeneous calculation and accuracy in bulk property prediction, we refitted parameters for alkanes based on their saturated liquid densities and vapor pressures. Unlike in modified iSAFT 31 and polymer DFT 25 , the formalism of the chain contribution according to Tripathi and Chapman 22 allows us to have non-integer numbers for the chain length, which has the advantage to more accurately model the phase behavior of a full-range of alkanes. The reference data is from the online library of National Institute of Standards 14


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and Technology (NIST). 54 The absolute average deviation (AAD% =

100 Nref

PNref qical −qiref i=1 q ref i

) from saturated liquid densities and vapor pressures is minimized to find the three model parameters. As mentioned in Dominik A. et al., 32 the chain length and segment diameter were not significantly changed in comparison with the parameters regressed by Gross and Sadowski 42 , so we firstly fix the chain length and segment diameter parameters keeping them the same as PC-SAFT parameters and only adjust the attraction energy. We then adjust the chain length and segment diameter to obtain the smallest error to the reference data. The fitted parameters and fitting errors are presented in Table 1 . As we see the errors are fairly large in comparison with the PC-SAFT EoS, 42 especially for the vapor pressures. This is expected since the dispersion term has only one parameter while the PC-SAFT EoS dispersion term is correlated to simulation results of linear alkanes. Overall we consider this mean field approximated equation of state accurate enough for the current study. For fluid-solid interactions, we find that the potential of mean force (PMF) predicted by molecular dynamics (MD) simulation nearly conforms to the Steele 10-4-3 potential, but with a different well depth. Considering the Steele potential has an analytical formula, we use this equation after calibrating the interaction energy parameters according to the potential well values from simulation. The solid fluid interaction energies are also listed in Table 1.

Pure component adsorption, desorption and phase equilibrium Before predicting the phase behavior of different components in nanopores, we first verify our model in predicting density profiles by comparing with grand canonical Monte Carlo (GCMC) and MD simulations. As previous work presented, this model shows good performance in model molecules like hard chains. 22 In this work, we mainly focus on the modeling of real molecules, in which cases the dispersion contribution plays an important role. Particularly the mean field approximation may bring up some inaccuracy, so it is necessary to compare the density profile to molecular simulation, i.e. grand canonical Monte Carlo (GCMC) simulation or molecular dynamics (MD) simulation. We have performed GCMC simulation for vapor 15


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Table 1: Parameters for DFT modeling of the studied components.

species

m

methane 1.0000 ethane 1.5969 n-propane 2.0049 n-butane 2.3650 n-pentane 2.6636 iso-pentane 2.5321 neo-pentane 2.3459 n-hexane 3.0334 n-heptane 3.4502 n-octane 3.9598

σ(Å) 3.6754 3.4557 3.4915 3.5923 3.6808 3.7448 3.8459 3.7295 3.7357 3.7034

AAD% εj /k(K) vapor liquid pressure density 202.8 2.2 3.1 243.0 5.6 2.2 260.0 5.7 1.7 275.5 5.3 2.1 290.0 6.0 2.1 289.9 5.6 2.1 281.7 5.7 2.1 296.1 7.9 2.3 298.5 7.4 2.1 296.2 8.1 2.1

temperature fluid-solid range interaction (K) εsj /k(K) 100 ∼ 180 63.9 200 ∼ 300 65.7 250 ∼ 350 66.6 250 ∼ 350 67.5 250 ∼ 350 69.1 250 ∼ 350 68.9 260 ∼ 360 64.1 245 ∼ 365 70.7 250 ∼ 350 70.4 250 ∼ 350 69.7

methane adsorption and MD simulation with NAMD 55 for liquid heptane adsorption. The parameters in simulation are from a united-atom model TraPPE force field, 56 which is widely used for alkanes. In MD simulation, the system setup follows the work by Hu et al., 11 where the simulation box includes both confined alkane between two blocks of graphite and a bulk region. We start with methane, a spherical molecule, in which there is no contribution from the chain free energy. FIG. 3 compares the density profiles of adsorbed methane with those from GCMC simulation for a slit pore of H = 5 nm at 155 K and different bulk pressures. Although different molecular models are used in the two methods, the agreement between them is satisfactory. We have normalized the position with respect to the diameter of the molecule in each case. The success is mainly attributed to fundamental measure theory since the structure is primarily determined by repulsion. The theory and GCMC simulation both show growing layers and then capillary condensation of vapor methane as the pressure increases, that is due to the strong attraction from the graphite surface. However, it is much more challenging to obtain excellent agreement between simulation and theory for a chain molecule. As we see the density distribution of heptane in a slit pore of H = 3 nm at 298

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we show the iSAFT predicted critical properties in comparison with literature data for different pore sizes. The predictions are qualitatively consistent with Monte Carlo simulations. For methane, the theory predicts critical temperatures closer to NVT-Gibbs MC and the predicted critical pressure lies between the two simulation studies. It is known that a bulk equation of state overpredicts the critical point, and the difference between theory and simulation critical points appears to be due, in part, to differences in bulk critical points. The theory predicts the trend of critical point with respect to pore size consistent with simulated data. The dataset in FIG. 8 is listed in Table 1 and Table 2 in Supplementary Material B.

Binary systems Prediction of hydrocarbon mixture partitioning between the kerogen matrix and fractures is essential to production management in shale reservoirs. Pitakbunkate et al. 8 performed GCMC study of the pore filling process of a methane/ethane mixture at 250 K in a 5 nm graphite pore, and the phase boundaries and critical point were determined from simulation data. Here we calculate the same properties with iSAFT and compare the results with their simulation data. The adsorption and desorption isotherms with different bulk ethane concentrations are shown in FIG. 9(a). As the bulk ethane concentration decreases, the phase transition pressure increases and the hysteresis disappears as the pore fluid reaches a critical point. The calculated critical point is 50 bar at a bulk ethane fraction of 0.22 as adsorption/desorption hysteresis disappears when the bulk ethane concentration decreases from 0.23 to 0.22. The equilibrium transition was determined from the grand potential of adsorption/desorption isotherms. The equilibrium dew point and bubble point curves of ethane are also shown in the same figure in comparison with the bulk ethane phase boundaries. As the exact bulk concentration corresponding to each isotherm was not reported in Pitakbunkate et al., 8 we only compare the predicted bubble/dew curves of ethane with their simulated ones. We see that our calculated equilibrium phase boundary is in very good agreement with the GCMC adsorption isotherm. We present the bulk phase ethane density 21


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of a Cn /Cn+1 binary mixture and characterize the pore preference of the longer alkane Cn+1 by the ratio of pore fraction and bulk fraction. In FIG. 10(a), gas mixtures of different compositions at 298 K, 1 atm show different selectivity behavior in a 3 nm pore. For C4/C5 mixture, the maximum C5 fraction is 0.5, which is below the mixture saturation composition. The preference for the longer alkane partitioning to the pore is ranked as C1/C2 < C2/C3 < C4/C5 < C3/C4. As we notice, an interesting transition is seen in that C4/C5 mixture lies between C2/C3 and C3/C4. This implies that an entropy loss of pentane plays a more important role than in the C3/C4 mixture.The selectivity has a completely different trend for liquid alkane mixtures at ambient conditions, as shown in FIG. 10(c) with C7/C8 < C6/C7 < C5/C6. For the C5/C6 mixture, C6 still partitions to the pore more than C5. However, for the C6/C7 mixture, the pore fluid has almost the same composition as that in the bulk. What is surprisingly predicted is that for the C7/C8 mixture, the preference to longer alkane no longer holds as the fraction of C8 in the pore is smaller than that in the bulk. The results demonstrate the competition between enthalpy benefit from surface attraction and entropy loss due to geometric confinement. We are not aware of experimental studies that would provide insight in long alkane adsorption in nanopores. For longer alkanes, the advantage in surface affinity becomes less dominant as the long alkanes lose more configurational entropy entering the pore space. In FIG. 10(b) and (d), the pore size effect is studied for a C1/C2 gas mixture and C7/C8 liquid mixture. As the pore size decreases from 8 nm to 2 nm, an increasing selectivity for the larger molecule is seen for C1/C2 while the opposite is observed in C7/C8. This suggests that in short alkane adsorption, the selectivity is mainly determined by surface affinity, however, for long alkanes in the liquid state, adsorption is a competition between attraction for the surface and loss in configurational entropy.

Competitive adsorption of multicomponent systems Competitive adsorption of a multicomponent (n>3) system is of interest both for shale reservoir characterization and other applications such as gas separation and chromatography. 24


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Jiang et al. 57 performed configurational-bias Monte Carlo simulation of linear (C1−nC5) and branched (C5) alkane mixtures on single walled carbon nanotube bundles. They discussed the different competitive adsorption behavior at low and high pressures, where short alkanes partition to the pore more strongly at high pressures. In this section we have performed a similar study and find a similar trend in a slit pore. We firstly study the adsorption of an equimolar mixture of C5 isomers, namely n-/iso/neo-pentane = 1:1:1, at 300 K in a 1.5 nm slit pore. Due to the different arrangements of carbon atoms, the surface attraction force for each molecule is different. This can be verified by the potential of mean force calculation from single molecule simulations shown in FIG. 11. The radius of gyration of each isomer is also presented as we see n-pentane is mostly stretched and thus has the strongest interaction with the surface. In partitioning to the pore space, we see in FIG. 12 that n-pentane is most strongly adsorbed with iso-pentane being the next. The calculation is conducted both for vapor and liquid bulk phase, with vapor phase showing larger selectivity of linear pentane to neo-pentane. Here we show the selectivity as the molar density ratio between one species to neo-pentane. A similar trend is seen in the adsorption in carbon nanotubes. At higher pressure, decreased selectivity is due to the packing limit when approaching adsorption saturation. Although n-pentane is preferentially adsorbed by the pore, as the pressure increases the configurational entropy loss of entering the pore would hinder the continuous increase of stretched molecule in pores. If we take a look at a mixture of more dispersed chains, as in the next example, a mixture of C1/C2/C3/C4/C5 = 5/4/3/2/1 , this pressure effect is more clearly seen. FIG. 13(a) xpore

and (b) show the partition coefficient ( yibulk ) of this mixture in a 1.5 nm pore at 300 K i

with bulk being a vapor in FIG. 13(a) and a liquid in FIG. 13(b). Comparing to the bulk composition, we see in (a) that pentane strongly adsorbs in the pore with a large difference from the bulk concentration. Before capillary condensation in the pore, the npentane fraction continuously increases. However, the n-pentane fraction decreases after condensation while other molecules increase in adsorption. The calculation is stopped before

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fluid properties in shale networks is an active area of research. In this work, we present a model to provide physical understanding of confined alkane properties in shale. As a computationally efficient method, iSAFT has been applied to predict the phase behavior of pure and mixed hydrocarbons in graphite slit pores. In quantifying the fluid-solid interactions, we have introduced the potential of mean force from molecular simulation to evaluate the external potential in a Non-local DFT framework. To summarize, we have examined the density distribution of adsorbed molecules in adsorption of methane and heptane by comparing to molecular simulation. The pore size dependency of pure methane and pure ethane critical properties has been studied and the phase diagrams were established for methane and ethane in a 3 nm pore. The mixture phase behavior was studied taking methane and ethane mixture as a showcase. The predicted properties are comparable with molecular simulation, with the errors mainly arisen from the mean field approximation in treating the dispersion energy. After the initial verification of the theory on pure and binary systems, we predicted the selective adsorption of a series of binary(Cn /Cn+1 ) mixtures up to n-octane. Selectivities have been shown in different sized pores, where we see for short alkanes in gaseous state, Cn+1 always partitions more to the pore while for longer alkane mixtures in liquid state, this is not always true. This observation implies that selective adsorption is not only determined by surface affinities but also by change of configurational entropies, which is also mentioned by Jiang et al.. 57 Further studies on selectivies of multicomponent systems, which are branched pentanes and C1 − C5 mixtures, have shown the similar trends as reported by simulation, 57 and also imply the same mechanism. Selectivities from molecular simulation will be considered for future work.As fluid in shale is a complex mixture confined in nano-scale pores, partitioning of different components between fractures and pore matrix should be considered to obtain a realistic fluid composition in a reservoir. As reported by other works 7,8,40 , an adjustment of fluid thermodynamic properties is inevitable in reservoir simulation.

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Acknowledgement The authors thank The Robert A. Welch Foundation (Grant No. C-1241) for financial support.

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