Capillary Condensation and Capillary Pressure of Methane in Carbon Nanopores: Molecular Dynamics Simulations of Nanoconﬁnement Effects

Fluid Phase Equilibria 459 (2018) 196e207

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Fluid Phase Equilibria

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Capillary condensation and capillary pressure of methane in carbon nanopores: Molecular Dynamics simulations of nanoconﬁnement effects

* Mohammad Sedghi , Mohammad Piri

Department of Petroleum Engineering, University of Wyoming, 1000 E. University Avenue, Laramie, WY 82071, United States article info abstract

Article history: Two main groups of thought have emerged over the past decade to improve thermodynamic modeling of Received 20 August 2017 capillary condensation of nanoconfined fluids. One approach has been developed on the premise of using Received in revised form shifted critical parameters for the condensed phase to account for the wall-fluid interactions, while the 29 October 2017 other one considers a pressure difference across the interface between the condensed and the bulk phases. Accepted 16 December 2017 This pressure difference is the capillary pressure that exists across a curved meniscus formed in a capillary Available online 18 December 2017 pore. For nanoconfinement, a modified Laplace equation has been utilized to calculate the capillary pressure to take into account the confinement effects that become more prominent as the pore size reduces. For small Keywords: fi Capillary condensation pores of a few nanometers in size, however, the impact of structural forces known as nanocon nement fi fi Capillary pressure effects, on the pressure of the con ned phase becomes more signi cant. In this work, we studied the Modified laplace equation capillary pressure of methane at the capillary condensation point in graphite pores smaller than 7 nm, to Nanoconfinement effects verify whether the confined phase can experience a negative pressure at capillary condensation point and whether we can accurately predict this pressure from thermodynamic equations. For this purpose, we used Molecular Dynamics (MD) simulations to investigate the pressure of methane molecules confined in graphite pores of various sizes. Furthermore, normal and tangential pressures of methane in selected pore sizes were obtained during capillary condensation at constant pressure. Our results indicated that for small pores there is a critical size below which capillary condensation did not occur. For larger pores, on the other hand, capillary condensation could be identified with an abrupt drop in the pressure of the confined phase. The capillary pressure attained from the MD simulations were similar to the thermodynamic calculations provided the adsorbed phase was thick enough to screen out wall-fluid interactions. © 2017 Elsevier B.V. All rights reserved.

1. Introduction condensation, a ubiquitous phenomenon occurring in both natural and synthetic nano-dimensional porous media. Capillary conden- Phase behavior of hydrocarbons in nanopores has attracted sation describes condensation of the entrapped fluids at pressures significant attention in recent years, largely due to the development below their bulk vapor saturation points [9e12]. of gas production from porous media with nanometer-sized pores The main challenge in modeling capillary condensation of in ultra-tight formations (e.g., shale and tight sandstones) [1e9]. nanoconfined fluids is to incorporate wall-fluid interactions, Accurate assessment of hydrocarbon storage in and recovery from generally referred to as the confinement effect, into the equations these reservoirs is of direct relevance to the thermodynamic phase of state. Two schools of thought have been developed to include the prediction of nanoconfined fluids [5]. It is known that in nanopores, confinement effect in the phase equilibrium calculations. One the effect of wall-fluid interactions on the phase behavior of approach, which has thus far been exclusive to the cubic equations confined fluids can be significant, causing a divergence from the of state, considers shifted critical parameters for the confined phase behavior of the macroscopic bulk materials [9]. One partic- (condensed) phase [13e15]. In this approach, the same pressure ularly important example of this disparity is called capillary can be used for both vapor and condensed phases. The second method, on the other hand, considers different pressures for the vapor (bulk) and condensed phases [6e8,16]. This pressure differ- * Corresponding author. ence, called capillary pressure P , is induced by highly curved E-mail address: [email protected] (M. Sedghi). cap https://doi.org/10.1016/j.fluid.2017.12.017 0378-3812/© 2017 Elsevier B.V. All rights reserved. M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207 197 menisci formed between the two phases. Implementation of this models. Lu and Jiang derived a theoretical model that can calculate method in the PC-SAFT equation of state has shown promising re- the surface tension of nano-sized droplets based on the readily sults in modeling capillary condensation of associating and non- attainable thermodynamic parameters of a given element [19]. associating fluids in nanopores [6e8]. One of the main objectives of this work is to examine the validity of this technique, as outlined 1 2Sb 1 g ¼ g∞ 1 exp (5) in the following paragraphs, for pore spaces of size < 7 nm. 4rp h 1 3R 4rp h 1 For a cylindrical pore with a radius rp, Pcap can be calculated by using the Laplace equation: where Sb is the entropy of vaporization, R is the ideal gas constant, and h is the effective molecular diameter. This model has been 2g successfully applied to determine the surface tension of mercury Pcap ¼ PNW PW ¼ cosðqÞ (1) rp droplets at various radii [20]. However, since in this work we are studying hemicylindrical surfaces, Lu-Jiang method cannot be used where g is the surface tension and q is the contact angle that the as it was developed for droplets with hemispherical surfaces. liquid-vapor interface creates with the pore wall and is measured Using the confined surface tension (Eq. (3)) in the modified through the condensed phase. Since the pore walls are completely Laplace equation (Eq. (2)), we can determine the pressure of the covered by adsorbed layer(s) of the fluid, the contact angle can be condensed phase. Subsequently, by integrating this capillary pres- reasonably considered zero at the condensation point [6]. W and sure into the phase equilibrium calculations, we can implicitly take NW subscripts denote wetting and non-wetting phases, respec- into account the wall-fluid interactions. The underlying assumption tively. It has been discussed that for the pores smaller than 45 nm, in this method is that at capillary condensation, the pressure of the Eq. (1) becomes inaccurate based on two phenomena emerging in condensed phase is negative due to the large capillary pressure that nanometer pores [6]: (1) strongly adsorbed fluid layers on the pore exists across the interface between the confined and the bulk phase wall considerably reduce the effective size of the pore, and (2) (Eq. (2)). However, before the condensation, the density of the curvature of the liquid surface leads to a reduction in the surface confined phase is greater than that of the bulk phase (due to the tension. Thus, a modified Laplace equation was introduced to ac- tendency of gas molecules to strongly adsorb on the walls), there- count for these effects: fore, the confined phase may have a slightly higher pressure than the bulk pressure. Immediately following the condensation, however, a 2g rp curved interface will form between the confined and the bulk pha- Pcap ¼ (2) rp tp ses, across which the capillary pressure exists (Eq. (2)). Since the confined phase is the wetting phase, its pressure becomes less than where tp is the thickness of the adsorbed fluid and gðrpÞ is the pore- the bulk pressure. Depending on the magnitude of the capillary size-dependent surface tension. It should be noted that at capillary pressure and the bulk pressure, the confined pressure could then condensation point inside slit pores (as considered in this study) become negative. To the best of our knowledge, this abrupt change of the condensed phase forms a hemicylindrical surface. For this ge- pressure during capillary condensation has not been studied in any ometry, the principal radius normal to the walls is equal to the half modeling work. More importantly, the faithfulness of the modified ¼ d of the pore width (r1 2 ), while the other radius parallel to the Laplace equation and the thickness of the adsorbed phase has not walls is infinite (r ¼ ∞). Thus, the curvature (k 1 þ 1 ¼ 1 ) of this 2 r1 r2 r been rigorously studied for extremely small pore sizes of less than surface is half of the curvature of a hemispherical surface formed in 10 nm. For such small pore spaces, physical and chemical properties a cylindrical pore with radius of r1. Therefore, Eq. (2) can be applied of the confined materials are impacted by the high pressures they to slit pores by removing coefficient 2 from the numerator. may experience after condensation (due to nanoconfinement ef- To find the surface tension of the confined phase, Tan et al. [6,7] fects) [21,22]. In a recent study by Long et al. [23e25], Monte Carlo successfully used the well-known Parachor equation [17]to (MC) simulations were employed to investigate the pressure tensor calculate the confined surface tension: components of confined argon in carbon nanoslits with various sizes n o between 0.7 and 2.8 nm. The simulations were run at the argon 4 L L V V saturation point of 87 K and 1 bar so that the bulk phase would be g T; rp ¼ § r T; P r T; P (3) liquid and no interface would exist between the confined and the bulk phases. Their simulation results demonstrated oscillation be- where § is the parachor parameter that can be derived from the tween positive and negative values in the normal pressure profile of surface tension values of the bulk phase. In this equation, the sur- the confined phase as the pore size varied. This oscillation was face tension is related to the difference between the density of the ascribed to the structural (solvent) forces described previously by condensed phase (rLÞ and that of the bulk vapor phase (rV Þ. One Israelachvili [26]. These forces, as discussed further in Section 3,are should note that calculating capillary pressure from Eqs. (2) and (3) originated from layered-structure formation of liquids entrapped in requires an iterative method since the pressure of the condensed small nanoscale spaces. Similar oscillations were also observed in phase (PL) is dependent upon the capillary pressure value as the normal pressure of argon, carbon tetrachloride, and water at or PL ¼ Pv P . cap above their saturation points, confined in carbon nanopores of Other methods have also been proposed in the literature to different geometries including cylindrical and spherical shapes [25]. estimate the surface tension of nanoconfined surfaces. A thermo- Although the existence of the structural forces has been verified in dynamic equation developed by Gibbs-Tolman-Koenig-Buff (GTKB) both experimental and modeling studies, their impact on the phase incorporates the Tolman length (d) to relate the surface tension of a behavior of confined materials has not been examined in the liter- nanoscale surface (g) to its macroscopic value (g ). A simplified ∞ ature. Furthermore, it is not fully understood whether capillary version of the GTKB can be derived by assuming (d=r ) ≪ 1[18]. p pressure exists during capillary condensation of a confined phase under the influence of significant structural forces and if so whether ¼ g∞ g (4) the modified Laplace equation can be used to estimate its value. 1 þ 2d rp In this study, using Molecular Dynamics (MD) simulations, we However, the difficulty in calculating the Tolman length has aimed to investigate validity of applying the modified Laplace hindered the application of this equation in the thermodynamic equation to obtain the capillary pressure of methane in small carbon 198 M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207 nanopores. To this end, we first examined the structural forces of MD Simulations were performed in NVT ensemble, although the methane in graphite pores of various sizes. Thereafter, we probed the pressure was controlled by using piston-like particles, as explained existence of capillary pressure during capillary condensation of later. Two series of simulations were conducted. The first series was methane in different pore sizes with and without the dominant designed to find the normal pressure of condensed methane in presence of structural forces. The capillary pressure values obtained graphite pores of various sizes. The simulations were run at a con- from the MD simulations were then compared with the thermody- stant temperature and pressure condition of 111.5 K and 1 bar namic calculations. Although we considered the slab pores in this (methane's vapor saturation point). At this T and P condition, the study, the choice of pore geometry does not affect the fundamental fluid outside the pore existed as liquid and thus no clear interface, physics behind the capillary condensation and the capillary pressure like the one between gas and liquid, could form between the across liquid-vapor interfaces. The difference between slab and cy- confined and the bulk phases (as shown in Fig.1). This makes it easier lindrical pores is that the curvature of the interface in a cylindrical to evaluate the effects of structural forces on the pressure of the pore is twice of that in a slab pore with the same inscribed radius, as confined phase. For these simulations, initially we carried out noted earlier. Here, we aimed to verify whether the confined phase equilibration simulations for 12 ns to bring the system into equilib- assumed a negative pressure during capillary condensation and rium. Subsequently, the production simulations were conducted for whether we can accurately predict this pressure from thermody- 50 ns where we recorded the simulation trajectories at every 20 ps. In namics equations. Therefore, we studied a simple system of methane the second series, for selected pore sizes, we performed simulations and carbon pores [27,28] to answer these fundamental questions. at various temperatures to track the pressure of confined methane The arrangement of this paper is as follows. In the Method section, during capillary condensation. After establishing the pressure trend, details of our MD simulations as well as the pressure calculation we determined the capillary pressure of methane and compared it methods deployed in this study are introduced. Subsequently, the against the calculation of thermodynamic equations. For these sim- results of MD simulations are presented and discussed in detail in ulations, first, we performed a hybrid of MD and Grand Canonical the Results and Discussion section and the summary of our main Monte Carlo (GCMC) simulations using LAMMPS simulation package findings is provided in the Conclusions section. [32] to establish equilibrium between the confined and the bulk methane phases at the pressure of 1 bar and different temperatures 2. Methods ranging between 108 and 150 K. For the simulations involving adsorption/desorption occurrences, it is easier to use GCMC 2.1. Molecular dynamics simulations approach to reach equilibrium conditions since we can insert or delete particles during the simulation. Each simulation was run for 4 All the MD simulations were performed using the GROMACS ns, during which 500 insertion/deletion attempts took place at every 5.1.1 simulation package [29] on a super-computer cluster with 40 time steps. Following the equilibrium simulation, a production both CPU and GPU processors. Similar to the work of Long et al. simulation was carried out for 60 ns in an NVTensemble to collect the [23], we considered slit geometry for the nanopores. The pore data for pressure analysis. The pressure of the bulk phase was fixed at space was created between two graphite walls each composed of 1 bar using the accelerated piston as explained below. three graphene layers laying parallel to each other with an inter- The time step of 1 fs was used in our MD simulations. To conduct layer spacing of 0.344 nm [30]. The graphene layers placed normal our MD simulations, we employed an efficient leap-frog stochastic to the Y direction with dimensions of 5.16 and 10.15 nm in the X dynamics integrator which uses a friction force to remove the and Z directions, respectively. The farthest layer of graphene in excess heat during the simulation in order to keep the simulation each wall was kept frozen whilst the carbon atoms in other layers temperature constant. For non-equilibrium simulations where were restrained to their lattice positions with a spring constant of there is a translational movement of the whole system, using sto- 18.1 N/m [23,30]. The cutoff distance for the van der Waals in- chastic thermostat is more suitable than non-stochastic thermo- teractions was set to 1.40 nm. Periodic boundary conditions were stats such as Nose-hoover or Berendsen to prevent the flying-ice- applied in all directions, however, since the thickness of the pore cube effects [33,34]. In some of our MD simulations, within the wall was larger than the cutoff distance, the fluid particles inside first few nano seconds, we observed a net momentum transfer of the pore could not feel their periodic boundary images. The di- methane phase before its pressure becomes constant at 1 bar. That mensions of the simulation box in the X and Z directions were set is why we considered the stochastic thermostat in our simulations. at 5.16 and 900 nm, respectively. Fig. 1 displays the end snapshot To ensure accurate temperature maintenance, the temperature of of a MD simulation where methane molecules filled a 2 nm fluids, graphite walls, and the pistons, were controlled separately graphite pore at temperature and pressure of 111.5 K and 1 bar. with relaxation times of 10, 10, and 2 ps, respectively. The pressure Here, the pore width is defined as the distance between the center of the bulk fluid (fluid outside the pore) was modulated by placing of mass of the carbon atoms of the innermost graphene wall two pistons at its both ends as shown in Fig. 1 (orange particles). layers. As discussed in section 3.3.1, we employed a different Using pistons to regulate the pressure of MD simulations is well method called Gibbs dividing surface [31], to define the effective documented in our previous studies [35,36]. Interactions between width of the graphite pores. fluid particles and fluid-wall particles were modeled using Lennard

Fig. 1. The end snapshot of a MD simulation where methane molecules ﬁlled a graphite pore of 2.0 nm width at T and P of 111.5 K and 1 bar. Methane molecules, graphite carbons, and piston particles are colored as yellow, cyan, and orange, respectively. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the Web version of this article.) M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207 199

Jones (LJ) equation. United Atom OPLS force field [37] was used for between two pistons normal to the Z direction. The size of the methane while Steele's Lennard Jones parameters [38] were methane slab was 5 5 10 nm in the X, Y, and Z directions, applied for graphite's carbon: εcc =kB ¼ 28:0 K and scc ¼ 0:34 nm. respectively. We applied pressures of 100 bar on the pistons while the temperature was fixed at 180 K. From Eqs. (7) and (8), the 2.2. Pressure calculations normal and tangential pressures were calculated to be 101 and 104 bar, respectively. These results suggest that the error in our Pressure is a second-order tensor property of materials where pressure calculations is less than 5%. an element Pab indicates a pressure that exists based on forces in b fi direction exerted on a surface normal to the a direction. For a bulk 3.2. Structural forces (nanocon nement effect) fluid under equilibrium conditions, since there is no net momentum transfer and due to its geometrical symmetry, the off- Before we determine the capillary pressure of methane in car- diagonal elements of its pressure tensor are zero. If the fluid is bon nanopores, it was important to study the structural forces that homogenous, the diagonal elements are equal and hence its pres- become predominant inside these pore spaces. It is well- fi sure can be expressed by a scalar quantity. However, for confined documented that the structural forces can signi cantly enhance fi fi fluids in nanopores, the density may not be constant throughout the pressure of the con ned phase as the con nement space be- e the entire fluid. In this case, the diagonal pressures would not comes comparable to that of the molecular layer [23 25]. There- fi fi necessarily be equal and should be defined separately. For nanoslit fore, in the rst step, we calculated the normal pressure of con ned pores at equilibrium conditions, due to mechanical stability, the methane inside nanopores of varying sizes by performing MD simulations at a constant temperature and pressure condition of normal pressure of the confined phase, i.e., Pyy, is constant throughout the pore [23]. The tangential pressures, i.e., average of 111.5 K and 1 bar, which is the bulk saturation point of methane. The results, as presented in Fig. 2, demonstrate that for the pores Pxx and Pzz, however, are equal and only dependent on the distance from the pore wall (i.e., the Y coordinate) [23]. smaller than 2.8 nm, the structural forces become dominant Each pressure component consists of kinetic and configurational causing the normal pressure to oscillate substantially between positive and negative values. As pore size increases, however, the contributions (P ¼ Pkin þ Ppot ). The kinetic contribution (Pkin) ay ay ay ay amplitude of this oscillation decreases until it is disappeared arises from momentum transfer due to the particles' movements completely in larger pores. The structural forces, which hereafter fi pot while the con gurational contribution (Pay ) arises from inter- and will be referred to as nanoconfinement effects, reflect the layering intra-molecular forces. The kinetic pressure can be calculated using structure of the confined fluid; when the (discrete) fluid layers exist the ideal gas approximation [39,40]. securely inside the pore, their normal pressure is close to that of the bulk phase. As the pore size shrinks, the layers are propelled into kinð Þ¼ rð Þ Pay y y kbT (6) each other causing the pressure to increase to a level that one of the layers is forced to leave the pore. At this point, extra space is left where rðyÞ is the local density of fluid at y and kb is the Boltzmann available for the remaining layers causing the normal pressure to constant. The configuration pressure, however, is not as unambig- drop to a negative value as the layer(s) is (are) pulled toward the uous as the kinetic one since its expression can change depending walls. It is evident from Fig. 2 that the nanoconfinement effects on how one defines the forces that are responsible for the pressure. become insignificant for the pore sizes larger than 3.8 nm. Also, it is Therefore, different expressions for this contribution have been notable that for the pores between 2.5 and 3.8 nm, the confinement proposed in the literature. For a system with solely pairwise in- pressure is either close to 1 bar or a large negative value. This trend teractions, using Irving-Kirkwood definition [41,42], the configu- reveals an interesting phenomenon in which the confined methane ration contribution to the normal and tangential pressures can be at the center of the pore forms layering structure only when it leads expressed by the following equations [40]. to a negative pressure. In other words, when confined methane is under stress, discrete layers cannot be formed at the center of the 2 0 1 X y U r y y y y pore. To illustrate this point, cross-sectional view of confined PpotðyÞ¼ ij ij q i q j (7) N 2A r y y methane at three different pores are displayed inside Fig. 2. Inside i sj ij y ij ij ij 2.6 and 3.6 nm pores, methane forms distinctive layers, while for 3.0 nm pore, methane is liquid-like at the center of the pore. We 2 2 0 X x þ z U r y y should note that the simulations of methane in the pores sizes of potð Þ¼ 1 ij ij ij y yi j PT y q q (8) 3.2 and 3.6 nm were repeated twice and similar results (with less 4A rij yij yij i sj yij than 3% variations) were observed. This layering formation of the confined methane phase was also evident by its greater density at where A is the surface area, q is the step function, and xij, yij, and zij the center of 3.6 nm pore compared with 3.0 nm pore. The density r are the components of the intermolecular distance vector ij. The of methane in the region bounded between the strongly adsorbed bracket in a represents the average ensemble of the quantity a.In layers inside 3.0 nm pore was 17.0 nm 3, while for the same region this study, we applied Eqs (6)e(8) at every 0.01 nm along the Y inside 3.6 nm pore, it was 19.5 nm 3. direction. For each simulation, the pressure calculations were per- We should also mention that although examining the shape of formed over 2500 frames taken at every 20 ps. The standard de- the pressure oscillation is outside the scope of this paper, from viation of the normal pressure (Pyy) calculations over the last 2000 Fig. 2 we can estimate that the periodic length of structural forces frames was found to be smaller than 1 bar. for methane is between 3 and 4 angstroms, which is close to the effective diameter of methane molecules (i.e., 3.99 angstrom). 3. Results and Discussion Using an alternative approach to calculate the configurational part of the pressure component, we showed that similar pressure 3.1. Validation of pressure calculations calculations can be performed for larger and more complicated hydrocarbon molecules. The pressure calculation results for pure To evaluate the accuracy of our pressure calculation methods, pentane at T and P of 300 K and 1 bar are presented in the sup- we carried out simulations with methane molecules placed porting information (Figures S1). 200 M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207

Fig. 2. Normal pressure of methane at different pore sizes. Simulations were run at T and P of 111.5 K and 1 bar. The inset pictures are cross-sectional view of conﬁned methane inside three different pores: 2.6, 3.0, and 3.6 nm.

3.3. Capillary condensation 111.5 K (see Fig. 2). The fact that the pressure in different pores does not converge to the bulk pressure can be ascribed to the nano- As stated previously, the main goal of this research was to confinement effects which are still present in these pores, although provide new insights on the capillary pressure of condensed hy- at a much smaller magnitude compared to the smaller pores. The drocarbon phases formed as the capillary condensation occurs in pressure trends presented in Fig. 3a verify the thermodynamics small nanopores. To this end, we selected different pore sizes in the approach that considers capillary pressure to account for the wall- range of 1.9e6.4 nm to study the capillary condensation of fluid interactions, as results showed the pressure of the confined methane. This selection was based on the results presented in the phase became negative following the capillary condensation. previous section (see Fig. 2) where we showed that the nano- To evaluate the accuracy of the modified Laplace equation for confinement effects were negligible for pores larger than 3.8 nm such small pore spaces, we compared the capillary pressure (Pcap) while they became dominant for the smaller pores. For instance, for calculated from Eq. (2) against the MD simulation results. Using Eq. 1.9 and 2.0 nm pores, the normal pressure of confined methane was (2), however, requires a precise definition of the pore size (pore found to be -1247 and 2069 bar, respectively, as shown in Fig. 2. width ¼ 2rp). In the previous section, we considered the pore width This suggests that the structural forces might have different effects is equal to the distance between the centers of carbon atoms in the on methane phase behavior inside these pores. most inner graphene layers (d). In this approach, since we neglect the radius of the carbon atoms, we slightly overestimate the true 3.3.1. Capillary condensation without the strong presence of pore space available to the confined fluids. This error becomes more nanoconfinement effects significant as the pore size becomes comparable to the molecular In this section, we first discuss the capillary pressure of the size. condensed methane in the six pores larger than 3.8 nm. To deter- Different methods have been introduced to define the accessible mine the capillary pressure for each pore, we obtained the normal pore volume to the confined molecules. Kaneko et al. [43] pressure of the confined methane before and after the capillary acknowledged that since carbon atoms are not hard spheres, one condensation. The normal pressure profile of confined methane should not define the pore size based on the distance between the inside the pores with respect to temperature is presented in Fig. 3a. surfaces of carbon atoms on the opposite pore walls. They instead The results show that the normal pressure becomes abruptly proposed that the wall-fluid interface should be defined at a point negative at a certain point. This sharp drop in the pressure coincides where the interaction energy between the adsorbate and the wall with a jump in the density of the condensed phase, which indicates particles becomes zero (i.e., zero-potential distance) [43e45]. that the pressure change happens as the result of capillary Another method is to consider Gibbs dividing surface, as condensation. The pressure and the density profile of the confined commonly used in other molecular studies [46e48], to accurately fl methane in 4.0 nm pore are shown in Fig. 3b. The density of the determine wall- uid interfaces (Ysurf ): confined methane, as explained later, was determined in the region bounded by the strongly adsorbed layers. It is worth noting that ZYsurf Z∞ prior to capillary condensation, the confined pressure was slightly ½r rð Þ ¼ rð Þ higher than the bulk pressure of 1 bar. This difference is expected bulk y dy y dy (9) fi considering that the density of the con ned phase is greater than 0 Ysurf that of the bulk phase due to strong adsorption of methane mole- r rð Þ fi cules on the pore walls. After the capillary condensation, the normal where bulk and y are the bulk and local densities of the con ned pressure starts to rebound (becoming less negative) quickly as phase, respectively. For each pore, we used the trajectory of the temperature decreases, until it reaches its value at temperature of simulation conducted after the condensation point to calculate the M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207 201

), bar -20 yy 4.0 nm -40 4.4 nm -60 4.8 nm -80 5.2 nm Normal pressure (P Normal -100 5.6 nm -120 6.4 nm

-140 100 110 120 130 140 150 Temperature, K

20 18

0 16

14 -20 Pressure 12 ), bar

yy -40 Density -3 10 -60 8

-80 Density, nm 6

Normal pressure (P pressure Normal -100 4

-120 2

-140 0 100 110 120 130 140 150 Temperature, K

Fig. 3. (a) Normal pressure of confined methane vs temperature for different pore sizes between 4 and 6.4 nm. (b) Normal pressure and density profile of confined methane in 4.0 nm pore vs temperature.

fl wall- uid interface. Ysurf value was averaged over the last 100 The most important parameter of Eq. (2) that should be rigor- simulation frames to have a standard error of less than one percent. ously defined is the thickness of the adsorbed phase (tp). To this 0 The actual pore width (d ) was then defined as the distance be- aim, we intuitively considered a surface (Y} ) which splits the fl surf tween the wall- uid interfaces determined for each pore wall. confined methane into two regions: the region in the center of the fi fl Using both methods to de ne the wall- uid interfaces, we ob- pore where methane has similar structure as in the bulk tained similar effective pore sizes, within at most of 2 angstrom and the region near the wall where it has a distinctive structure due difference from each other (1.0 angstrom on average). Therefore, to the wall-fluid interactions. Accordingly, the adsorption normal pressure calculations, particularly for pores larger than thickness is defined as the difference between Y and Y} 4 nm, were not affected considerably by the choice of pore size surf surf ¼ } fi } definition. Here, we showed the results based on the Gibbs dividing (tp Ysurf Ysurf ). To nd Ysurf for each pore, we calculated the surface methodology, since we believe this method is more robust tangential pressure of confined methane at its capillary conden- and flexible to be applied when the overlap between the fluid and sation point. The pressure profile for 4.0 nm pore obtained at the wall particles varies due to enhanced nanoconfinement pres- temperature of 116 K (capillary condensation point) is shown in sure or when the confined phase is composed of particles with Fig. 4. The tangential pressure profile demonstrates two large peaks different sizes (e.g., mixture of methane and ethane). The effective on each side of the pore in the Y axis (i.e., normal direction to the pore width obtained from this method is reported in Table S1 for pore walls). The high tangential pressure is reflective of fluid par- different pores. ticles being propelled into each other due to their strong 202 M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207

Fig. 4. Tangential pressure profile (top) and the end snapshot (bottom) of the simulation of methane in 4.0 nm pore at capillary condensation point of 116 K and 1 bar. interactions with the wall particles. Therefore, one can conclude methane is -113.9 bar (see Fig. 3b), at temperature of 116 K, hence that two strongly adsorbed layers exist on each pore wall. Also, the the absolute value of the capillary pressure can be estimated as simulation snapshot provided in Fig. 4, clearly illustrates this point. 114.9 bar. The capillary pressures obtained from the MD simula- Similar conclusion was reached from the tangential pressure profile tions were then compared with the above-mentioned calculations, and the simulation trajectories obtained from other pores (larger as presented in Fig. 5. than 3.8 nm). For all the pores (larger than 3.8 nm), the calculated Fig. 5 shows that the capillary pressures calculated by using the adsorption thickness varied between 0.8 and 0.81 nm, meaning confined surface tension, obtained from the Parachor method, in that the thickness of each layer is, on average, equal to 0.4 nm, the modified Laplace equation, agree very well with the MD which is equal to the effective diameter of methane. simulation results. It is notable that using the macroscopic (bulk) Finally, the confined surface tension ðgðrpÞÞ should be obtained surface tension values in the modified Laplace equation, the cal- before using Eq. (2). We calculated the confined surface tension culations still match the simulation results reasonably well. from the Parachor method (Eq. (3)). To use this equation, we However, the original Laplace equation (Eq. (1)) significantly un- determine the density of the condensed and the vapor phases from derestimates the capillary pressure, especially for smaller pores. the MD simulations. To estimate the density of the condensed The results imply that the exclusion the adsorption thickness from phase, we considered the central region of the pore that is bounded the pore size is the main improvement in the capillary pressure between the Y} surfaces. For 4.0 nm pore, for instance, the den- estimation. surf fi sity of the condensed phase, as shown in Fig. 3b, was determined as Overall, the consistency between the modi ed Laplace equation fi 14.93 nm 3 at temperature of 116 K (the capillary condensation and the MD results, veri es the validity of this equation for small point) while the density of vapor phase outside the pore was found pore sizes of a few nanometers, provided that the structural forces fi to be 0.02 nm 3 at the same temperature. Using the Parachor co- are not signi cant. Considering the Laplace based equation was fi fi efficient of 72.6 [mN cm 12/m mol4] for methane [49], we found the initially developed for smooth surfaces, it was signi cant to nd surface tension of 11.83 mN/m from Eq. (3). The fact that this value this thermodynamic equation can still be applied for small is lower than the surface tension of bulk methane (12.25 mN/m) nanometer-sized pores where the condensed phase could not form can be ascribed to the curvature of the interface, as pointed out smooth and perfectly curved surfaces as depicted in the snapshot fi previously [50,51]. shown in Fig. 4. The results also con rm that, for these pores, fl The capillary pressure can also be obtained directly from the MD beyond the adsorbed phase, the wall- uid interactions are not fi simulations by subtracting the pressure of the gas phase outside the signi cant, which allows us to reasonably account for these in- pore (here,1 bar) from the normal pressure of the condensed phase. teractions by subtracting the adsorption thickness from the pore For instance, for 4.0 nm pore, the normal pressure of the confined width. M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207 203

140 MD simulation

120 Mod Lap w/ bulk γ r

a Mod Lap w/ Par γ

b 100 , e

r Lap Eq w/ bulk γ u

s 80 s e r p y

r 60 a l l i p

a 40 C

0 3.5 4 4.5 5 5.5 6 6.5 7 Pore size, nm

Fig. 5. Capillary pressures obtained from MD simulations compared with thermodynamic calculations for different pore sizes between 4.0 and 6.4 nm.

3.3.2. Capillary condensation under nanoconfinement effects disappears, has been observed in both experimental and modeling In this section, we discuss the capillary condensation of methane studies [52e58]. For instance, based on a set of experimental in the pore sizes of smaller than 4.0 nm where the nanoconfinement sorption isotherms, Neimark et al. [57] suggested a critical diameter effects can become significant. The normal pressure of the confined of 4 nm for argon adsorption in MCM-41 pores at 87 K, while Las- methane over the range of temperature (between 108 and 170 K) is toskie et al. [58], through Monte Carlo simulations, found a critical shown in Fig. 6 where two distinguishable pressure trends can be pore size of 1.4 nm for nitrogen adsorption in graphite pores at 77 K. observed. For pores equal to or larger than 2.4 nm, at a certain Although the shift in the critical properties of confined materials temperature, pressure abruptly drops to a negative value, which with respect to the size of nanoconfinement has been discussed in indicate capillary condensation as discussed in the previous section. the literature [52e60], the pressure behavior of a confined super- For smaller pores (i.e., 2 and 1.9 nm pores), on the other hand, no critical phase during its adsorption process is less discussed. To clear breaking point can be seen in the pressure trend. This indicates study the continuous filling process occurring in supercritical that instead of capillary condensation, the continuous filling phe- conditions, the pressure and density profiles for 1.9 and 2 nm pores nomenon is occurring in these pores, suggesting that there is a were obtained as shown in Fig. 7a&b. For 1.9 nm pore, the normal critical pore size, between 2 and 2.4 nm, below which capillary pressure becomes continuously more negative as the temperature condensation cannot take place at pressure of 1 bar. In other words, decreases, although the change in the pressure is directly influ- the critical pressure has shifted below 1 bar for this pore. enced by the change in the density of the condensed phase. For this The existence of the critical pore size below which the adsorp- pore, it is not possible to distinguish between the capillary pressure tion isotherms follow a smooth and continuous trend and the and the nanoconfinement effects since they have similar impact on hysteresis between adsorption and desorption isotherms the confinement pressure (i.e., reducing the pressure to a more

200

100

-100 ), bar

yy -200 Capillary 4.0 nm -300 condensation 3.6 nm -400 Continuous filling 3.2 nm -500 2.8 nm -600 Normal pressure (P Normal 2.4 nm -700 2.0 nm -800 1.9 nm -900 100 110 120 130 140 150 160 170 Temperature, K

Fig. 6. Normal pressure of confined methane vs temperature for different pore sizes between 1.9 and 4.0 nm. 204 M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207 negative value). Although, it is worth noting the significant nega- surface tension. Also, the uncertainty in calculating the average tive pressure of the confined methane at high temperatures (above density of the condensed phase increases for smaller pores, 150 K) where there is no condensed phase present at the center of rendering the applicability of this method questionable for these the pore, as indicated by the density profile in Fig. 7a. For 2 nm pore, pores. However, as discussed earlier, the main parameter in Eq. (2) as the pore becomes saturated with methane, the normal pressure is the thickness of the adsorbed phase (tp). To determine tp we of the confined phase becomes negative until it reaches a minimum obtained the tangential pressure profiles at the capillary conden- point at temperature of 134 K. Thereafter, we observe a very gradual sation point, as shown in Fig. 8. It is clear that for the smaller pores, increase in the normal pressure up to temperature of 120 K where the second pressure peak starts to diminish, meaning that the the pressure jumps to a large positive value due to crystallization of second adsorbed layer starts to disappear. This is because for the confined methane, as shown in the supporting information smaller pores, capillary condensation occurs at higher tempera- (Figure S2). The layering formation in the crystalized methane re- tures where increase in entropy (lack of ordering) become more sults in a high (positive) pressure due to the structural forces. The favorable than decrease in enthalpy. It is notable that because of the broad differences in the pressure and density profiles of methane strong graphene-methane interactions, the fist layer remains un- inside these pores with only one angstrom difference in their sizes, affected by the temperature. It is apparent from Fig. 8 that for demonstrate that the phase behavior of confined fluids is greatly 2.4 nm pore there is only one adsorbed layer on each wall, while for impacted by the pore size and the nanoconfinement effects. We 2.8 nm pore, there is a second layer that is weakly adsorbed on the should note that Eq. (2) should only be applied at capillary wall since its pressure is between that of 2.4 and 3.6 nm pores. condensation point and thus is not suitable for these pores. Therefore, we considered tp to be 0.4 nm (equal to one layer) and To evaluate the accuracy of the modified Laplace equation for 0.6 nm (equal to one and a half layers) for 2.4 and 2.8 nm pores, pores larger than 2.0 nm, we performed a similar quantitative respectively. For 3.2 and 3.6 nm pores, we set tp to 0.8 nm similar to analysis as presented for the pores larger than 3.8 nm. We only the larger pores. Using these tp values, capillary pressure was considered bulk surface tension in modified Laplace equation since calculated and compared with the MD simulation results, as dis- only a small improvement was achieved by considering Parachor played in Fig. 9. Except for 2.4 nm pore, calculations of the modified

0 14

-200 12 Pressure

), bar -400 10 yy Density -600 8

-800 6 Density, nm-3 -1000 4 Normal pressure (P pressure Normal -1200 2

-1400 0 100 110 120 130 140 150 160 170 Temperature, K

2500 25

2000 Pressure 20

), bar Density yy 1500 15 Phase transition 1000 from liquid to crystalized solid 10

500 Density, nm-3

Normal pressure (P pressure Normal 5 0

-500 0 100 110 120 130 140 150 160 170 Temperature, K

Fig. 7. Normal pressure and density of conﬁned methane vs temperature in: (a) 1.9 nm pore; and (b) 2.0 nm pore. M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207 205

14000 4.0 nm 12000 3.6 nm 10000 2.8 nm 8000 2.4 nm

6000

4000

2000 Temperature

0 Tangentail pressure ((Pxx+Pzz)/2), bar ((Pxx+Pzz)/2), Tangentail pressure -2000

-4000 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Y-axis, nm

Fig. 8. Tangential pressure proﬁles near the pore wall calculated at the capillary condensation point for different pore sizes between 2.4 and 4.0 nm.

250

MD simulation Strong wall-fluid interactions 200 Mod Lap w/ bulk γ r a b , e r

u 150 s s e r p y r

a 100 l l i p a C 50

0 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 Pore size, nm

Fig. 9. Capillary pressures obtained from MD simulations compared with thermodynamic calculations for different pore sizes between 2.4 and 6.4 nm.

Laplace equation agree well with the MD simulations. For 2.4 nm negative and positive values due to the layering structure of the pore, the modified Laplace equation significantly underestimates confined phase. For pores between 2.5 and 3.8 nm, layering struc- the capillary pressure, which is due to the fact that the methane ture occurs only when it results in a negative pressure. Finally, for molecules beyond the adsorbed layer are still under the influence of pores larger than 3.8 nm, the structural forces become marginalized. the graphene carbons, meaning that they are pulled towards the Subsequently, capillary condensation and capillary pressure of wall. This results in a capillary pressure much greater than the methane in pore sizes between 1.9 and 6.4 nm was studied at a thermodynamic prediction. For larger pores, the adsorbed phase is constant pressure of 1 bar through MD simulations. For small pores thick enough (0.8 nm) to effectively screen out graphene attraction of 1.9 and 2 nm no capillary condensation was observed as they field resulting in a good agreement between the modified Laplace were continuously filled with methane as temperature decreased. equation and the MD simulations. The results also imply that the This suggested that there was a critical pore size between 2.0 and nanoconfinement effects are not significant at the capillary 2.4 nm, below which capillary condensation did not occur. For these condensation point for these pore spaces. pores, the confined pressure and density profiles were greatly affected by the pore size and nonconfinement effects. For large 4. Conclusions pores, capillary condensation was observed as an abrupt phase transition through which the normal pressure of the confined phase We presented the results of MD simulations utilized to investi- drops to a large negative value due to the capillary pressure. The gate the capillary pressure of condensed methane in graphite pores capillary pressures obtained from the MD simulations at the of less than 7 nm. Structural forces (i.e., nanoconfinement effects) condensation point were then compared with the modified Laplace for methane were first studied through normal pressure calcula- equation calculations. To determine the adsorption thickness (tp) tions of the confined phase at different pore sizes. The MD results used in the modified Laplace equation, we obtained the tangential indicate that below 2.8 nm pressure oscillates between large pressure profile of the confined methane at different pores. For all 206 M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207 pores larger than 2.8 nm, two strongly adsorbed layers of methane [19] H.M. Lu, Q. Jiang, Size-dependent surface tension and Tolman's length of e with a total thickness of 0.8 nm was found on each pore wall. Using droplets, Langmuir 21 (2) (2005) 779 781. [20] S. Wang, F. Javadpour, Q. Feng, Confinement correction to mercury intrusion the tp values obtained from MD simulation, a good agreement was capillary pressure of shale nanopores, Sci. Rep. 6 (2016) 20160. found between MD results and the thermodynamic calculations, [21] K. Kaneko, N. Fukuzaki, K. Kakei, T. Suzuki, S. Ozeki, Enhancement of nitric oxide dimerization by micropore fields of activated carbon fibers, Langmuir 5 with exception of the 2.4 nm pore. For this pore, since only one layer e fi (4) (1989) 960 965. was adsorbed on the wall, a signi cantly larger capillary pressure [22] C. Alba-Simionesco, B. Coasne, G. Dosseh, G. Dudziak, K.E. Gubbins, was obtained from MD simulation than the modified Laplace R. Radhakrishnan, M. Sliwinska-Bartkowiak, Effects of confinement on fl freezing and melting, J. Phys. Condens. Matter 18 (6) (2006) R15. equation. The results implied that we can account for the wall- uid [23] Y. Long, J.C. Palmer, B. Coasne, M. Sliwinska-Bartkowiak, K.E. Gubbins, Pres- interactions by subtracting the adsorption thickness from the pore sure enhancement in carbon nanopores: a major confinement effect, Phys. size only when the adsorbed phase is consisting of more than one Chem. Chem. Phys. 13 (38) (2011) 17163e17170. layer of molecules. Also, it was notable that using confined surface [24] Y. Long, J.C. Palmer, B. Coasne, M. Sliwinska-Bartkowiak, K.E. Gubbins, Under pressure: quasi-high pressure effects in nanopores, Microporous Mesoporous tension obtained from the Parachor method, the improvement in Mater. 154 (2012) 19e23. the prediction of the modified Laplace equation was not significant. [25] Y. Long, M. Sliwinska-Bartkowiak, H. Drozdowski, M. Kempinski, K.A. Phillips, J.C. Palmer, K.E. Gubbins, High pressure effect in nanoporous carbon materials: effects of pore geometry, Colloid. Surface. Physicochem. Eng. Aspect. 437 Acknowledgements (2013) 33e41. [26] J. Israelachvili, Solvation forces and liquid structure, as probed by direct force We gratefully acknowledge the financial support of Hess Cor- measurements, Acc. Chem. Res. 20 (11) (1987) 415e421. poration, Saudi Aramco, and the School of Energy Resources at the [27] A. Vishnyakov, E.M. Piotrovskaya, E.N. Brodskaya, Capillary condensation and melting/freezing transitions for methane in slit coal pores, Adsorption 4 (3e4) University of Wyoming. We also would like to thank Dr. Sugata Tan (1998) 207e224. for valuable discussions on the phase behavior of nanoconfined [28] S. Jiang, C.L. Rhykerd, K.E. Gubbins, Layering, freezing transitions, capillary fluids. condensation and diffusion of methane in slit carbon pores, Mol. Phys. 79 (2) (1993) 373e391. [29] M.J. Abraham, T. Murtola, R. Schulz, S. Pall, J.C. Smith, B. Hess, E. Lindahl, Appendix A. Supplementary data GROMACS: high performance molecular simulations through multi-level parallelism from laptops to supercomputers, Software 1e2 (2015) 19e25. [30] S.J. Stuart, A.B. Tutein, J.A. Harrison, A reactive potential for hydrocarbons with Supplementary data related to this article can be found at intermolecular interactions, J. Chem. Phys. 112 (14) (2000) 6472e6486. https://doi.org/10.1016/j.fluid.2017.12.017. [31] J.G. Kirkwood, F.P. Buff, The statistical mechanical theory of surface tension, J. Chem. Phys. 17 (3) (1949) 338e343. [32] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, References J. Comput. Phys. 117 (1) (1995) 1e19. [33] M. Sedghi, M. Piri, L. Goual, Molecular dynamics of wetting layer formation [1] Z. Li, Z. Jin, A. Firoozabadi, Phase behavior and adsorption of pure substances and forced water invasion in angular nanopores with mixed wettability, and mixtures and characterization in nanopore structures by density func- J. Chem. Phys. 141 (19) (2014) 194703. tional theory, SPE J. 19 (06) (2014) 1096e1109. [34] J.E. Basconi, M.R. Shirts, Effects of temperature control algorithms on transport [2] D. Devegowda, K. Sapmanee, F. Civan, R.F. Sigal, Phase Behavior of Gas Con- properties and kinetics in molecular dynamics simulations, J. Chem. Theor. densates in Shales Due to Pore Proximity Effects: Implications for Transport, Comput. 9 (7) (2013) 2887e2899. Reserves and Well Productivity, Society of Petroleum Engineers, 2012. [35] G. Javanbakht, M. Sedghi, W. Welch, L. Goual, Molecular dynamics simulations [3] B. Rahmani Didar, I.Y. Akkutlu, Confinement Effects on Hydrocarbon Mixture of CO2/water/quartz interfacial properties: impact of CO2 dissolution in wa- Phase Behavior in Organic Nanopore, Society of Petroleum Engineers, 2015. ter, Langmuir 31 (21) (2015) 5812e5819. [4] X. Dong, H. Liu, J. Hou, K. Wu, Z. Chen, Phase equilibria of confined fluids in [36] W.R.W. Welch, M. Piri, Molecular dynamics simulations of retrograde nanopores of tight and shale rocks considering the effect of capillary pressure condensation in narrow oil-wet nanopores, J. Phys. Chem. C 119 (18) (2015) and adsorption film, Ind. Eng. Chem. Res. 55 (3) (2016) 798e811. 10040e10047. [5] K. Dhanapal, D. Devegowda, Y. Zhang, A.C. Contreras-Nino, F. Civan, R. Sigal, [37] W. Jorgensen, J. Madura, C. Swenson, Optimized intermolecular potential Phase Behavior and Storage in Organic Shale Nanopores: Modeling of Multi- functions for liquid hydrocarbons, J. Am. Chem. Soc. 106 (22) (1984) 6638e6646. component Hydrocarbons in Connected Pore Systems and Implications for [38] W.A. Steele, The physical interaction of gases with crystalline solids, Surf. Sci. Fluids-in-place Estimates in Shale Oil and Gas Reservoirs, Society of Petroleum 36 (1) (1973) 317e352. Engineers, 2014. [39] M. Rao, B.J. Berne, On the location of surface of tension in the planar interface [6] S.P. Tan, M. Piri, Equation-of-State modeling of associating-fluids phase between liquid and vapour, Mol. Phys. 37 (2) (1979) 455e461. equilibria in nanopores, Fluid Phase Equil. 405 (2015) 157e166. [40] J.P.R.B. Walton, D.J. Tildesley, J.S. Rowlinson, J.R. Henderson, The pressure [7] S.P. Tan, M. Piri, Equation-of-State modeling of confined-fluid phase equilibria tensor at the planar surface of a liquid, Mol. Phys. 48 (6) (1983) 1357e1368. in nanopores, Fluid Phase Equil. 393 (2015) 48e63. [41] J.H. Irving, J.G. Kirkwood, The statistical mechanical theory of transport processes. [8] S.P. Tan, M. Piri, Heat of capillary condensation in nanopores: new insights IV. The equations of hydrodynamics, J. Chem. Phys. 18 (6) (1950) 817e829. from the equation of state, Phys. Chem. Chem. Phys. 19 (7) (2017) 5540e5549. [42] B.D. Todd, D.J. Evans, P.J. Daivis, Pressure tensor for inhomogeneous fluids, [9] E. Barsotti, S.P. Tan, S. Saraji, M. Piri, J.-H. Chen, A review on capillary Phys. Rev. E 52 (2) (1995) 1627e1638. condensation in nanoporous media: implications for hydrocarbon recovery [43] K. Kaneko, R.F. Cracknell, D. Nicholson, Nitrogen adsorption in slit pores at from tight reservoirs, Fuel 184 (2016) 344e361. ambient temperatures: comparison of simulation and experiment, Langmuir [10] B. Coasne, R.J.-M. Pellenq, A Grand canonical Monte Carlo study of capillary 10 (12) (1994) 4606e4609. condensation in mesoporous media: effect of the pore morphology and to- [44] D.D. Do, H.D. Do, Evaluation of 1-site and 5-site models of methane on its pology, J. Chem. Phys. 121 (8) (2004) 3767e3774. adsorption on graphite and in graphitic slit pores, J. Phys. Chem. B 109 (41) [11] M.M. Kohonen, N. Maeda, H.K. Christenson, Kinetics of capillary condensation (2005) 19288e19295. in a nanoscale pore, Phys. Rev. Lett. 82 (23) (1999) 4667e4670. [45] A.V. Neimark, P.I. Ravikovitch, Calibration of pore volume in adsorption ex- [12] M. Tuller, D. Or, L.M. Dudley, Adsorption and capillary condensation in porous periments and theoretical models, Langmuir 13 (19) (1997) 5148e5160. media: liquid retention and interfacial configurations in angular pores, Water [46] S. Wang, F. Javadpour, Q. Feng, Molecular dynamics simulations of oil trans- Resour. Res. 35 (7) (1999) 1949e1964. port through inorganic nanopores in shale, Fuel 171 (2016) 74e86. [13] N.S. Alharthy, T. Nguyen, T. Teklu, H. Kazemi, R. Graves, Multiphase Compo- [47] A. Bot¸ an, B. Rotenberg, V. Marry, P. Turq, B. Noetinger, Hydrodynamics in clay sitional Modeling in Small-scale Pores of Unconventional Shale Reservoirs, nanopores, J. Phys. Chem. C 115 (32) (2011) 16109e16115. Society of Petroleum Engineers, 2013. [48] I.C. Bourg, C.I. Steefel, Molecular dynamics simulations of water structure and [14] G.J. Zarragoicoechea, V.A. Kuz, Critical shift of a confined fluid in a nanopore, diffusion in silica nanopores, J. Phys. Chem. C 116 (21) (2012) 11556e11564. Fluid Phase Equil. 220 (1) (2004) 7e9. [49] W. Porteous, Calculating surface tension of light hydrocarbons and their [15] L. Travalloni, M. Castier, F.W. Tavares, S.I. Sandler, Thermodynamic modeling mixtures, J. Chem. Eng. Data 20 (3) (1975) 339e343. of confined fluids using an extension of the generalized van der Waals theory, [50] B.T. Bui, A.N. Tutuncu, Effect of capillary condensation on geomechanical and Chem. Eng. Sci. 65 (10) (2010) 3088e3099. acoustic properties of shale formations, Journal of Natural Gas Science and [16] B. Nojabaei, R.T. Johns, L. Chu, Effect of Capillary Pressure on Fluid Density and Engineering 26 (2015) 1213e1221. Phase Behavior in Tight Rocks and Shales, Society of Petroleum Engineers, 2012. [51] K. Bui, I.Y. Akkutlu, Nanopore wall effect on surface tension of methane, Mol. [17] O. Exner, Conception and significance of the parachor, Nature 196 (4857) Phys. 113 (22) (2015) 3506e3513. (1962) 890e891. [52] E.A. Müller, K.E. Gubbins, Molecular simulation study of hydrophilic and hy- [18] R.C. Tolman, The effect of droplet size on surface tension, J. Chem. Phys. 17 (3) drophobic behavior of activated carbon surfaces, Carbon 36 (10) (1998) (1949) 333e337. 1433e1438. M. Sedghi, M. Piri / Fluid Phase Equilibria 459 (2018) 196e207 207

[53] A.V. Shevade, S. Jiang, K.E. Gubbins, Molecular simulation study of wateremethanol [57] C. Lastoskie, K.E. Gubbins, N. Quirke, Pore size heterogeneity and the carbon mixtures in activated carbon pores, J. Chem. Phys. 113 (16) (2000) 6933e6942. slit pore: a density functional theory model, Langmuir 9 (10) (1993) [54] B. Li, A. Mehmani, J. Chen, D.T. Georgi, G. Jin, The Condition of Capillary 2693e2702. Condensation and its Effects on Adsorption Isotherms of Unconventional Gas [58] A.V. Neimark, P.I. Ravikovitch, A. Vishnyakov, Bridging scales from molecular Condensate Reservoirs, Society of Petroleum Engineers, 2013. simulations to classical thermodynamics: density functional theory of capil- [55] C.L. McCallum, T.J. Bandosz, S.C. McGrother, E.A. Müller, K.E. Gubbins, lary condensation in nanopores, J. Phys. Condens. Matter 15 (3) (2003) 347. A molecular model for adsorption of water on activated carbon: comparison [59] S.K. Singh, A. Sinha, G. Deo, J.K. Singh, VaporLiquid phase coexistence, critical of simulation and experiment, Langmuir 15 (2) (1999) 533e544. properties, and surface tension of conﬁned alkanes, J. Phys. Chem. C 113 (17) [56] E. Kierlik, P.A. Monson, M.L. Rosinberg, G. Tarjus, Adsorption hysteresis and (2009) 7170e7180. capillary condensation in disordered porous solids: a density functional study, [60] J. Jiang, S.I. Sandler, B. Smit, Capillary phase transitions of N-Alkanes in a J. Phys. Condens. Matter 14 (40) (2002) 9295. carbon nanotube, Nano Lett. 4 (2) (2004) 241e244.