From molecular sieving to gas effusion through nanoporous 2D graphenes: Comparison between analytical predictions and molecular simulations.

In this paper, we study the permeation of polyatomic gas molecules through 2D graphene membranes. Using equilibrium molecular dynamics simulations, we investigate the permeation of pure gas compounds (CH4, CO2, O2, N2, and H2) through nanoporous graphene membranes with varying pore sizes and geometries. Our simulations consider the recrossing mechanism, often neglected in previous studies, which has a significant effect on permeation for intermediate pore size to molecular diameter ratios. We find that the permeation process can be decoupled into two steps: the crossing process of gas molecules through the pore plane and the escaping process from the pore region to a neighboring adsorption site, which prevents recrossing. To account for these mechanisms, we use a permeance model expressed as the product of the permeance for the crossing process and the probability of molecule escape. This phenomenological model is extended to account for small polyatomic gas molecules and to describe permeation regimes ranging from molecular sieving to effusion. The proposed model captures the temperature dependence and provides insights into the key parameters of the gas/membrane interaction controlling the permeance of the system. This work lays the foundation for predicting gas permeance and exploring membrane separation factors in 2D materials such as graphene.

Modeling gas permeation mechanisms through 2D membranes: Comparison between a phenomenological model and extensive molecular simulations.

Two-dimensional (2D) membranes based on perforated graphene have great potential in the field of separation of chemical species for a variety of applications, including gas treatment. In addition to recent experimental studies, several works simulate the mechanisms of gas permeation through this type of membrane using molecular dynamics, but few combine different techniques to ensure that their method of choice captures all relevant mechanisms. In particular, the re-crossing mechanism leading a gas molecule that has crossed the plane of the membrane to rapidly re-cross it in the opposite direction has never been documented. In this work, we study gas permeation through a simplified 2D membrane model. We combine equilibrium and non-equilibrium molecular dynamics simulations to quantify the impact of these re-crossing mechanisms on the values of the computed transport coefficients. Using non-equilibrium simulations as reference, we show that the equilibrium simulation techniques commonly used can lead to a significant overestimation of the transport properties of the membrane. We propose a simple method to probe the re-crossing dynamics during equilibrium simulations, making it possible to compute correct values of the transport coefficient without the need for non-equilibrium simulations. Furthermore, by analyzing the phenomenology observed in the simulations, we derive an analytical formula for the permeance that takes the form of an Arrhenius law with a non-trivial temperature dependent prefactor. In excellent agreement with our simulation results, this model provides a simple theoretical framework that captures the main mechanisms involved in gas permeation through 2D membranes, including the effect of re-crossing.

Free Volume Model for Transport in Flexible Kerogen of Source Rock's Organic Matter.

We build a model of transport of adsorbed fluid within the microporous network of kerogen's porosity, especially accounting for the adsorption-induced swelling exhibited by flexible kerogen structures. This model, based on Fujita-Kishimoto free volume theory that was historically developed for swellable polymers, is built over extensive results for the self-diffusion coefficients obtained by molecular dynamics calculations for a representative molecular model of kerogen designed to study the importance of flexibility effects on the properties of kerogen. To do so, we first highlight that transport within flexible kerogen incorporating the coupling between the dynamics of the fluid molecules and the kerogen matrix atoms does not introduce any significant collective effects in the usual long time limit. Then, we show that despite the slightly anisotropic diffusion properties, averaging over all the dimensions can still be performed in order to model the behavior of the transport properties with the amount of adsorbed fluid. Lastly, we link the increase of the self-diffusion coefficients and that of the accessible free volume with the fluid loading via the Fujita-Kishimoto model. We conclude by commenting on the evolution and significance of the model parameters over a broad range of thermophysical conditions.

Predicting thermodiffusion in simple binary fluid mixtures.

The predictive capabilities of some existing theoretical models to quantify thermodiffusion have been investigated in this work. To do so, the tests have been performed on two model fluids, the hard-sphere and the Lennard-Jones (including spheres and dimers) ones, exploring different mixtures and thermodynamic conditions thanks to extensive molecular simulations. It has been confirmed that the thermal diffusion factor should be expressed as the sum of one term related to the isotope effect and one term related to the "chemical" effects and that a kinetic term is required to quantify thermodiffusion from the gas state to the liquid state. In addition, regarding the isotope effects, it has been obtained that none of the available theoretical models are able to yield a reasonable prediction relatively to the molecular simulations results and that the moment of inertia contribution is one order of magnitude smaller than the mass contribution in the liquid state. Finally, concerning the chemical effects, it has been shown the Shukla and Firoozabadi model, complemented with a kinetic term, is probably the most reasonable option to estimate the chemical contribution to the thermal diffusion factor, even if it fails in capturing the effect of the asymmetry in size and in shape between the species. Overall, this works confirms that there is still a lack of a generic model able to predict accurately thermal diffusion factors, or equivalently Soret coefficient, in simple binary mixtures from the gas state to the liquid state.

Dynamic Crossover in Fluids: From Hard Spheres to Molecules.

We propose a simple and generic definition of a demarcation reconciling structural and dynamic frameworks when combined with the entropy scaling framework. This crossover line between gas- and liquid-like behaviors is defined as the curve for which an individual property, the contribution to viscosity due to molecules' translation, is exactly equal to a collective property, the contribution to viscosity due to molecular interactions. Such a definition is shown to be consistent with the one based on the minima of the kinematic viscosity. For the hard sphere, this is shown to be an exact solution. For Lennard-Jones spheres and dimers and for some simple real fluids, this relation holds very well. This crossover line passes nearby the critical point, and for all studied fluids, it is well captured by the critical excess entropy curve for atomic fluids, emphasizing the link between transport properties and local structure.

An entropy scaling demarcation of gas- and liquid-like fluid behaviors.

In this work, we propose a generic and simple definition of a line separating gas-like and liquid-like fluid behaviors from the standpoint of shear viscosity. This definition is valid even for fluids such as the hard sphere and the inverse power law that exhibit a unique fluid phase. We argue that this line is defined by the location of the minimum of the macroscopically scaled viscosity when plotted as a function of the excess entropy, which differs from the popular Widom lines. For hard sphere, Lennard-Jones, and inverse-power-law fluids, such a line is located at an excess entropy approximately equal to -2/3 times Boltzmann's constant and corresponds to points in the thermodynamic phase diagram for which the kinetic contribution to viscosity is approximately half of the total viscosity. For flexible Lennard-Jones chains, the excess entropy at the minimum is a linear function of the chain length. This definition opens a straightforward route to classify the dynamical behavior of fluids from a single thermodynamic quantity obtainable from high-accuracy thermodynamic models.

Elemental and isotopic fractionation of noble gases in gas and oil under reservoir conditions: Impact of thermodiffusion⋆.

Noble gases, and the way they fractionate, is a promising approach to better constrain origin, migration and initial state distributions of fluids in gas and oil reservoirs. Thermodiffusion, is one of the phenomena that may lead to isotope and elemental fractionation of noble gases. However, this effect, assumed to be small, has not been quantified, nor measured, in oil and gas under reservoir conditions. Thus, in this work, molecular dynamics simulations have been performed to compute the thermal diffusion factors of noble gases, in a dense gas (methane) and in an oil (n-hexane) under high pressures. Interestingly, it has been found that thermal diffusion factors, associated to both isotopic (36Ar, 40Ar) and elemental fractionations of noble gases (4He, 20Ne, 40Ar, 84Kr and 131Xe) in gas and oil, could be expressed as linear functions of the reduced masses. Regarding the amplitude of the phenomena, it has been found that, in a stationary 1D oil or gas fluid column, thermodiffusion due to a typical geothermal gradient has an impact on noble gas isotopic and elemental fractionation which is of the same order of magnitude than gravity segregation, but opposite in sign. In addition, the relative impact of thermodiffusion on isotopic and elemental fractionations depends on the fluid type which is another interesting feature. Thus, these first numerical results on isotopic and elemental fractionation of noble gases by thermodiffusion in simple pure gas and oil emphasize their interest as natural tracers that could be used to improve the pre-exploitation description of oil and gas reservoirs.

Linking up pressure, chemical potential and thermal gradients.

Petroleum reservoirs are remarkable illustrations of the impact of a thermal gradient on fluid pressure and composition. This topic has been extensively studied during the last decades to build tools that are required by reservoir engineers to populate their models. However, one can get only a very limited number of representative samples from a given reservoir and assessing connectivity between all sampling points is often a key issue. In some extreme cases, the whole reservoir fluid properties must be derived from a single point to define the field development plan. To do so, available models are usually not satisfactory as they need too many parameters and so cannot be considered as predictive tools. We propose in this work a comprehensive approach based on the irreversible thermodynamics principles to derive the relationships between pressure, chemical potentials and thermal gradients in porous media. It appears that there is no need for additional assumptions, it is just a matter of a making the right choices along theoretical developments. One of the most important steps is to express the full pressure gradient. As a final result, we obtain the chemical potential gradients for all components of the mixtures that can be easily translated in term of compositions through Equation of State modelling. The most important features of the final expressions are: i) the species relative separation in a thermal field is sensitive to the relative diffusion coefficients at stationary state. In porous media, the separation is sensitive to the permeability when the overall mobility is similar to diffusive mobility; ii) the magnitude of the separation depends on the residual entropy of the species; iii) the separation is not simply balanced by the average residual entropy. The balance is modified by the relative diffusion mobility of the components; iv) in low permeability porous media, the thermal gradient induces a pressure gradient proportional to the fluid residual entropy. As a validation, the proposed approach has been applied on a reservoir fluid subjected to a geothermal gradient and compared with non-equilibrium molecular dynamics simulation results at the stationary state.

Diffusion of Supercritical Fluids through Single-Layer Nanoporous Solids: Theory and Molecular Simulations.

With the advent of graphene material, membranes based on single-layer nanoporous solids appear as promising devices for fluid separation, be it liquid or gaseous mixtures. The design of such architectured porous materials would greatly benefit from accurate models that can predict their transport and separation properties. More specifically, there is no universal understanding of how parameters such as temperature, fluid loading conditions, or the ratio of the pore size to the fluid molecular diameter influence the permeation process. In this study, we address the problem of pure supercritical fluids diffusing through simplified models of single-layer porous materials. Basically, we investigate a toy model that consists of a single-layer lattice of Lennard-Jones interaction sites with a slit gap of controllable width. We performed extensive equilibrium and biased molecular dynamics simulations to document the physical mechanisms involved at the molecular scale. We propose a general constitutive equation for the diffusional transport coefficient derived from classical statistical mechanics and kinetic theory, which can be further simplified in the ideal gas limit. This transport coefficient relates the molecular flux to the fluid density jump across the single-layer membrane. It is found to be proportional to the accessible surface porosity of the single-layer porous solid and to a thermodynamic factor accounting for the inhomogeneity of the fluid close to the pore entrance. Both quantities directly depend on the potential of mean force that results from molecular interactions between solid and fluid atoms. Comparisons with the simulations data show that the kinetic model captures how narrowing the pore size below the fluid molecular diameter lowers dramatically the value of the transport coefficient. Furthermore, we demonstrate that our general constitutive equation allows for a consistent interpretation of the intricate effects of temperature and fluid loading conditions on the permeation process.

Thermodiffusion in multicomponent n-alkane mixtures.

Compositional grading within a mixture has a strong impact on the evaluation of the pre-exploitation distribution of hydrocarbons in underground layers and sediments. Thermodiffusion, which leads to a partial diffusive separation of species in a mixture due to the geothermal gradient, is thought to play an important role in determining the distribution of species in a reservoir. However, despite recent progress, thermodiffusion is still difficult to measure and model in multicomponent mixtures. In this work, we report on experimental investigations of the thermodiffusion of multicomponent n-alkane mixtures at pressure above 30 MPa. The experiments have been conducted in space onboard the Shi Jian 10 spacecraft so as to isolate the studied phenomena from convection. For the two exploitable cells, containing a ternary liquid mixture and a condensate gas, measurements have shown that the lightest and heaviest species had a tendency to migrate, relatively to the rest of the species, to the hot and cold region, respectively. These trends have been confirmed by molecular dynamics simulations. The measured condensate gas data have been used to quantify the influence of thermodiffusion on the initial fluid distribution of an idealised one dimension reservoir. The results obtained indicate that thermodiffusion tends to noticeably counteract the influence of gravitational segregation on the vertical distribution of species, which could result in an unstable fluid column. This confirms that, in oil and gas reservoirs, the availability of thermodiffusion data for multicomponent mixtures is crucial for a correct evaluation of the initial state fluid distribution.

Communication: A method to compute the transport coefficient of pure fluids diffusing through planar interfaces from equilibrium molecular dynamics simulations.

The computation of diffusion coefficients in molecular systems ranks among the most useful applications of equilibrium molecular dynamics simulations. However, when dealing with the problem of fluid diffusion through vanishingly thin interfaces, classical techniques are not applicable. This is because the volume of space in which molecules diffuse is ill-defined. In such conditions, non-equilibrium techniques allow for the computation of transport coefficients per unit interface width, but their weak point lies in their inability to isolate the contribution of the different physical mechanisms prone to impact the flux of permeating molecules. In this work, we propose a simple and accurate method to compute the diffusional transport coefficient of a pure fluid through a planar interface from equilibrium molecular dynamics simulations, in the form of a diffusion coefficient per unit interface width. In order to demonstrate its validity and accuracy, we apply our method to the case study of a dilute gas diffusing through a smoothly repulsive single-layer porous solid. We believe this complementary technique can benefit to the interpretation of the results obtained on single-layer membranes by means of complex non-equilibrium methods.

Thermodynamic scaling of the shear viscosity of Mie n-6 fluids and their binary mixtures.

In this work, we have evaluated the applicability of the so-called thermodynamic scaling and the isomorph frame to describe the shear viscosity of Mie n-6 fluids of varying repulsive exponents (n = 8, 12, 18, 24, and 36). Furthermore, the effectiveness of the thermodynamic scaling to deal with binary mixtures of Mie n-6 fluids has been explored as well. To generate the viscosity database of these fluids, extensive non-equilibrium molecular dynamics simulations have been performed for various thermodynamic conditions. Then, a systematic approach has been used to determine the gamma exponent value (γ) characteristic of the thermodynamic scaling approach for each system. In addition, the applicability of the isomorph theory with a density dependent gamma has been confirmed in pure fluids. In both pure fluids and mixtures, it has been found that the thermodynamic scaling with a constant gamma is sufficient to correlate the viscosity data on a large range of thermodynamic conditions covering liquid and supercritical states as long as the density is not too high. Interestingly, it has been obtained that, in pure fluids, the value of γ is directly proportional to the repulsive exponent of the Mie potential. Finally, it has been found that the value of γ in mixtures can be deduced from those of the pure component using a simple logarithmic mixing rule.

Couplings between swelling and shear in saturated slit nanopores: a molecular simulation study.

In this article, the coupling between swelling and shear in liquid saturated slit nanopores is studied using molecular dynamics simulations on Lennard-Jones systems. First, the consistency of the simulations using thermodynamics and direct routes is validated when dealing separately with swelling and shear. Then, the coupling between swelling and shear is explored by displacing the solid walls in one direction while letting them move freely on the other. Results indicate that shear can induce swelling and vice versa because of the confined fluid phase structure. This phenomenon, which is neglected in poromechanics modeling, may be non-negligible in highly structured microporous systems, such as clays. It implies that the response to a variation in the external load can be a combination of volumetric and shear deformations, because of the fluid. Finally, we explore the behavior induced by solid walls moving at a constant velocity. Interestingly, when the wall velocity exceeds the swelling velocity, the instantaneous states of the system are no longer at equilibrium and the averaged pore width slightly increases with increasing shear rate.

Determination of the thermodynamic correction factor of fluids confined in nano-metric slit pores from molecular simulation.

The multi-component diffusive mass transport is generally quantified by means of the Maxwell-Stefan diffusion coefficients when using molecular simulations. These coefficients can be related to the Fick diffusion coefficients using the thermodynamic correction factor matrix, which requires to run several simulations to estimate all the elements of the matrix. In a recent work, Schnell et al. ["Thermodynamics of small systems embedded in a reservoir: A detailed analysis of finite size effects," Mol. Phys. 110, 1069-1079 (2012)] developed an approach to determine the full matrix of thermodynamic factors from a single simulation in bulk. This approach relies on finite size effects of small systems on the density fluctuations. We present here an extension of their work for inhomogeneous Lennard Jones fluids confined in slit pores. We first verified this extension by cross validating the results obtained from this approach with the results obtained from the simulated adsorption isotherms, which allows to determine the thermodynamic factor in porous medium. We then studied the effects of the pore width (from 1 to 15 molecular sizes), of the solid-fluid interaction potential (Lennard Jones 9-3, hard wall potential) and of the reduced fluid density (from 0.1 to 0.7 at a reduced temperature T* = 2) on the thermodynamic factor. The deviation of the thermodynamic factor compared to its equivalent bulk value decreases when increasing the pore width and becomes insignificant for reduced pore width above 15. We also found that the thermodynamic factor is sensitive to the magnitude of the fluid-fluid and solid-fluid interactions, which softens or exacerbates the density fluctuations.

Local shear viscosity of strongly inhomogeneous dense fluids: from the hard-sphere to the Lennard-Jones fluids.

This work aims at providing a tractable approach to model the local shear viscosity of strongly inhomogeneous dense fluids composed of spherical molecules, in which the density variations occur on molecular distance. The proposed scheme, which relies on the local density average model, has been applied to the quasi-hard-sphere, the Week-Chandler-Andersen and the Lennard-Jones fluids. A weight function has been developed to deal with the hard-sphere fluid given the specificities of momentum exchange. To extend the approach to the smoothly repulsive potential, we have taken into account that the non-local contributions to the viscosity due to the interactions of particles separated by a given distance are temperature dependent. Then, using a simple perturbation scheme, the approach is extended to the Lennard-Jones fluids. It is shown that the viscosity profiles of inhomogeneous dense fluids deduced from this approach are consistent with those directly computed by non-equilibrium molecular dynamics simulations.

Influence of confinement on thermodiffusion.

This work focuses on a possible influence of a nanoporous medium on the thermodiffusion of a fluid "isotopic" mixture. To do so, we performed molecular dynamics simulations of confined Lennard-Jones binary equimolar mixtures using grand-canonical like and non-equilibrium approaches in sub- and super-critical conditions. The study was conducted in atomistic slit pore of three adsorbent natures for various widths (from 5 to 35 times the size of a molecule). The simulation results indicate that for all thermodynamic conditions and whatever the pore characteristics, the confinement has a negligible effect on the thermal diffusion factor/Soret coefficient. However, when considered separately, the mass diffusion and thermodiffusion coefficients have been found to be largely influenced by the pore characteristics. These two coefficients decrease noticeably when adsorption is stronger and pore width smaller, a behavior that is consistent with a simple hydrodynamic explanation.

Shear behavior of a confined thin film: influence of the molecular dynamics scheme employed.

In this work, we have considered and compared two molecular dynamics schemes widely used when studying a thin fluid film confined between solid surfaces and undergoing boundary shear. In the first approach, the non-equilibrium simulations are performed on a confined fluid explicitly connected to bulk reservoirs. In the second one, non-equilibrium simulations are carried out on the confined fluid only, in which the average density is deduced from a prior simulation in the grand canonical ensemble. We have found that the apparent properties (average density and effective viscosity) of a strongly confined Lennard-Jones liquid are significantly different using one scheme or the other when the solid surfaces induce a strong structure in the whole fluid, i.e., for small separations between the solid surfaces. Furthermore, the shear velocity dependence of the friction force has been found to be as well very sensitive to the approach chosen and can be well understood in terms of the fluid structure, which can even lead to a visco-plastic behavior of the fluid in some cases. Finally, it is shown that the first scheme is the only one usable to explore the history-dependence of the friction force as observed in experiments.

Local viscosity of a fluid confined in a narrow pore.

In this paper, molecular dynamics simulations of a simple Lennard-Jones fluid confined in narrow slit pores and undergoing shear have been performed. The aim is to investigate the effects of density inhomogeneities at the fluid-solid interfaces on the shear viscosity profiles. It has been found that the local viscosity was varying strongly with the distance from the solid walls for both dilute and dense fluid states with oscillations correlated to the density ones. To describe the computed viscosity profiles, we propose a scheme that uses the local average density model, combined with an adequate weight function, for the configurational viscosity and a semiempirical model for the translational viscosity. It is shown that the proposed approach is able to provide viscosity profiles in good agreement with those coming from simulations for different pore widths and for different fluid states (dilute to dense).

Grand canonical-like molecular dynamics simulations: application to anisotropic mass diffusion in a nanoporous medium.

In this work, we describe two grand canonical-like molecular dynamics approaches to investigate mass diffusion phenomenon of a simple Lennard-Jones fluid confined between solid surfaces and in direct contact with reservoirs. In the first method, the density is used as the control variable in the reservoir whereas it is the pressure in the second method. Both methods provide consistent results, however, the constant density approach is the most efficient with respect to the computational time and implementation. Then, employing the constant density approach, we have studied the transient behavior of the diffusion process associated with the migration of one fluid into another one confined between parallel solid walls. Results have shown that the evolution of molar fraction of the invading fluid follows roughly a 1D diffusion model when the solid phase is weakly or moderately adsorbent with a characteristic time increasing when the pore width decreases. However, when the adsorption is high and the pore width small (i.e., below ten molecular sizes), the apparent mass diffusion in the adsorbed layer is reduced compared to that in the center of the slit pore. Hence, this mass diffusion process becomes a two-dimension phenomenon that must take into account an effective mass diffusion coefficient varying locally.

Shear viscosity of inhomogeneous fluids.

Using molecular dynamics simulations on inhomogeneous fluids, we have studied the effects of strong density inhomogeneities of varying wavelengths on the shear viscosity computed locally. For dense fluids, the local average density model combined with an adequate weight function yields a good description of the viscosity profiles obtained by simulations. However, for low density inhomogeneous fluids, the local average density model is unable to describe correctly the viscosity profiles obtained by simulations. It is shown that this weakness can be overcome by taking into account the density inhomogeneity in the local translational contribution to the viscosity using a density gradient like approach.