^{*}

Edited by: Daniel Bonamy, Commissariat á l'Energie Atomique et aux Energies Alternatives (CEA), France

Reviewed by: Alberto Rosso, Centre National de la Recherche Scientifique (CNRS), France; Reinaldo Roberto Rosa, Instituto Nacional de Pesquisas Espaciais (INPE), Brazil

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We define the pressure of a porous medium in terms of the grand potential and compute its value in a nano-confined or nano-porous medium, meaning a medium where thermodynamic equations need be adjusted for smallness. On the nano-scale, the pressure depends in a crucial way on the size and shape of the pores. According to Hill [

The description of transport processes in porous media poses many challenges that are well described in the literature (see e.g., [

A central element in the derivation of the equations of transport on the macro-scale is the definition of a representative elementary volume (REV) (see e.g., [

As discussed in Kjelstrup et al. [_{B}_{g}, where _{B} is Boltzmann's constant and _{B}_{g} = Υ = −

where ^{f} and ^{f} are the pressure and the volume of the fluid in the REV, ^{r} and ^{r} are the pressure and the volume in the grains in the REV, and γ^{fr} and Ω^{fr} are the surface tension and the surface area between the fluid and the grain. The assumption behind the expression was the additive nature of the grand potential. This definition of the REV, and the expression for the grand potential, opens up a possibility to define the pressure on the hydrodynamic scale. The aim of this work is to explore this possibility. We shall find that it will work very well for flow of a single fluid in a porous medium. As a non-limiting illustrative example, we use grains positioned in a fcc lattice. The work can be seen as a continuation of our earlier works [

The work so far considered transport processes in micro-porous, not nano-porous media. In micro-porous media, the pressure of any phase (the surface tension of any interface) is independent of the volume of the phase (the area between the phases). This was crucial for the validity of equation 1. For nano-porous systems, we need to step away from Equation (1). Following Hill's procedure for small systems' thermodynamics [

The pressure is not uniquely defined at molecular scale. This lack of uniqueness becomes apparent in molecular dynamics (MD) simulations, for which the computational algorithm has to be carefully designed [

The paper is organized as follows. In section 2 we derive the pressure of a REV for one solid grain surrounded by fluid particles (Case I) and for a three-dimensional face-centered cubic (fcc) lattice of solid grains (Case II). Section 3 describes the molecular dynamics simulation technique when the system is in equilibrium and in a pressure gradient. In section 4 we use the theory to interpret results of equilibrium molecular dynamics simulations for one solid grain and for an array of solid grains in a fluid. Finally we apply the results to describe the system under a pressure gradient. We conclude in the last section that the expressions and the procedure developed provide a viable definition of the pressures and pressure gradients in nano-porous media.

Equation (1) applies to a micro-porous medium, a medium where the pore-size is in the micrometer range or larger [

The symbol

The integral and differential pressures connect to different types of mechanical work on an ensemble of small systems. The differential pressure times the change of the small system volume is the work done on the surroundings by this volume change. The name differential derives from the use of a differential volume. This work is the same, whether the system is large or small. The integral pressure times the volume per replica, however, is the work done by adding one small system of constant volume to the remaining ones, keeping the temperature constant. This work is special for small systems. It derives from an ensemble view, but is equally well measurable. The word integral derives from the addition of a small system.

From statistical mechanics of macro-scale systems, we know that _{B}

where ^{f} and ^{r}, and

We consider here a nano-porous medium, so integral pressures and integral surface tensions apply. The integral pressure and integral surface tension normally depend on the system size. In the porous medium there are two characteristic sizes: the size of a grain and the distance between the surfaces of two grains^{1}

In the following, we consider a single spherical grain confined by a single phase fluid (Case I) and a face-centered cubic (fcc) lattice of spherical grains confined by a single phase fluid (Case II). The size of the REV does not need to be large, and we will show in section 4.2 that the smallest REV is a unit cell in the direction of the pore in the fcc lattice.

Consider the inclusion of a spherical grain ^{f} and phase ^{r}. The total volume is ^{f} + ^{r}. The surface area between phase ^{fr}. The compressional energy of system A has contributions, in principle, from all its small parts

where ^{f}, meaning that ^{fr} [^{r}. This gives

We now introduce a system B in contact with A. System B has volume V, contains pure fluid, and is tuned so that it is in thermodynamic equilibrium with A. The equilibrium condition requires that their grand canonical partition functions are equal, which implies

The fluid pressure ^{f} is the same in phases A and B. We obtain

and by rearranging the terms,

where we have used that ^{r} + ^{f} =

A particle in a confined system

The pressure of the rock particle depends on the volume of the particle. The relation of the two pressures is according to Hill

When this is combined with the equation right above, we find the relation we are after

which is the familiar Young-Laplace's law. By subtracting Equation (10) from Equation (8), we obtain an interesting new relation

The expression relates the integral and differential pressure for a spherical phase

We see from this example how the integral pressure enters the description of small systems. The integral pressure is not equal to our normal bulk pressure, called the differential pressure by Hill,

The above explanation concerned a single spherical grain and was a first step in the development of a procedure to determine the pressure of a nano-porous medium. To create a more realistic model, we introduce now a lattice of spherical grains. The integral pressure of a REV containing

For each grain one may follow the same derivation for the integral and differential pressure as for the single grain. By using Equation (8), we obtain

where the last identity applies to spherical grains only. The differential pressure of the grains is given by a generalization of Equation (10)

where the last identity is only for spherical grains. The differential pressures again satisfy Young-Laplace's law at equilibrium.

When all grains are identical spheres and positioned on a fcc lattice, a properly chosen layer covering half the unit cell can be a proper choice of the REV. We shall see how this can be understood in more detail from the molecular dynamics simulations below. The REV is larger if the material is amorphous.

Cases I and II were simulated at equilibrium, while case II was simulated also away from equilibrium.

The simulation box was three-dimensional with side lengths _{x}, _{y}, _{z}.The box was elongated in the _{x} > _{y} = _{z}. Periodic boundary conditions were used in all directions in the equilibrium simulations. In the non-equilibrium simulation, reflecting particle boundaries [_{y}, _{z}, where Δ_{x}/_{l} = Δ_{y}_{z}. There are two regions A and B in the simulation box. Region A contains fluid (red particles) and grains (blue particles) and region B contains only fluid, see _{1} + B_{2} and A do not have the same size, but the layers have the same thickness, Δ

A slice of the simulation box in case II. The box has side lengths _{x}, _{y}, _{z}, and properties are calculated along the _{1} and _{2} are the two shortest surface-to-surface distances.

The simulation was carried out with LAMMPS [^{*} = 2.0 (in Lennard-Jones units). The critical temperature for the Lennard-Jones/spline potential (LJ/s) is approximately ^{*} = 0.01 to ρ^{*} = 0.7.

In case I the single spherical grain was placed in the center of the box. A periodic image of the spherical grain is a distance _{x}, _{y} and _{z} away in the _{α} − 2_{0} and _{0}, where σ_{0} is the diameter of the fluid particles.

In case II, the spherical grains were placed in a fcc lattice with lattice constant _{2} = _{1} < _{2}. We used _{1} = 4.14σ_{0} and _{1} = 11.21σ_{0}, which is almost the same as the distances considered in case I. The corresponding other distances were _{2} = 10σ_{0} and _{2} = 20σ_{0}. Each grain has 12 nearest neighbors at a distance _{1}.

In all cases we computed the volume of the grains

The particles interact with the Lennard-Jones/spline potential,

Each particle type has a hard-core diameter _{ii} and a soft-core diameter σ_{ii}. There were two types of particles, small particles with σ_{ff} = σ_{0}, _{ff} = 0 and large particles with σ_{rr} = 10σ_{0}, _{rr} = 9σ_{0}. The small particles are the fluid (

We define the radius of the grain particles as _{ff} + σ_{rr})/2 = 5.5σ_{0}, which is the distance from the grain center where the potential energy is zero. Fluid particles can occupy a position closer to the grain than this, this is illustrated in ^{*} = 0.1 and ρ^{*} = 0.7. This shows that the average distance from the grain particle and the closest fluid particle is approximately 5.5σ_{0}, but the fluid particles are able to occupy positions closer to the grain particle.

The radial distribution function of fluid particles around a grain, as shown in ^{*} = 0.1 and ρ^{*} = 0.7.

The interaction strength ϵ_{ij} was set to ϵ_{0} for all particle-particle pairs. The potential and its derivative are continuous in _{c,ij}. The parameters _{ij}, _{ij} and _{s,ij} were determined so that the potential and the derivative of the potential (the force) are continuous at _{s,ij}.

The contribution of the fluid to the grand potential of layer

where _{i} and _{i} are the mass and velocity of fluid particle _{ij} ≡ _{i} − _{j} is the vector connecting particle _{ij} = −∂_{ij}/∂_{ij} is the force between them. The · means an inner product of the vectors. The computation gives

We used the reflecting particle boundary method developed by Li et al. [_{x} with probability (1 − α_{p}) and reflected with probability α_{p}, whereas particles moving from left to right pass freely through the boundary. A large α_{p} gives a high pressure difference and a low α_{p} gives a low pressure difference.

The results of the molecular dynamics simulations are shown in

_{0}. ^{r}, ^{fr} and

We computed the compressional energy, _{l}_{l}, in the bulk liquid (region B) and in the nano-porous medium (region A). In the bulk liquid we computed the pressure directly from the compressional energy, because

The grain pressure ^{fr} were fitted such that the pressure is everywhere the same and are plotted as a function of the fluid pressure ^{f}. The results for case II were next used in

The single sphere case is illustrated in ^{fr}, along the ^{REV}^{REV} we summed _{l}_{l} over all the layers in the REV. At equilibrium, ^{REV} =

where we used that ^{fr}. The values of ^{fr} are fitted such that the pressure, _{l}_{l} of each layer from

The contributions to the compressional energy in this equation for case I are shown in the bottom

_{0}. For our grains with _{0}, this does not happen.

The plots of ^{fr} as functions of _{0} and _{0} are given in the same plots. We see that the plots fall on top of each other. This shows that the integral pressure and the surface tension are independent of the distance ^{fr} were functions of the distance _{0}, deviations may arise, for instance due to contributions from the disjoining pressure. Such a contribution is expected to vary with the surface area, and increase as the distance between interfaces become shorter. In plots like

Fitted grain pressure ^{fr} as a function of pressure _{0} and _{0}).

Consider next the lattice of spherical grains, illustrated in ^{fr}, along the

_{0}. ^{r}, ^{fr} and

When the REV in region A is properly chosen, we know that ^{REV} = _{1} or _{2}, where ^{REV} in region A, we sum _{l}_{l} over all the layers that make up the REV, and obtain

To proceed, we find first the values of all the elements in this equation, except ^{fr}. The values of ^{fr} are fitted such that the pressure is everywhere the same. Using these fitted values, we next calculated

The contributions to the compressional energy in this equation are shown in three stages in

From _{l}_{l} in these REVs is the same and equal to ^{REV}. The layers

The values for ^{fr} are shown as a function of ^{f} for case II in _{1} = 4.14σ_{0} and _{1} = 11.21σ_{0}. We see now a systematic difference between the values of ^{fr} in the two cases. The integral pressure and the surface tension increases as the distance between the grains decreases. The difference in one set can be estimated from the other. Say, for a difference in surface tension Δγ^{fr} we obtain for the same fluid pressure from equation 11, a difference in integral pressure of _{0} while the value in _{0}. The difference may be due to the disjoining pressure. Its distribution is not spherically symmetric, which may explain the difference between 6.5σ_{0} and 5.5σ_{0}.

Fitted grain pressure ^{fr} as a function of pressure _{1} = 4.14σ_{0} and _{1} = 11.21σ_{0}).

The results should be the same as for case I for the larger distance, and indeed that is found, cf. _{0} and _{1} = 11.21σ_{0}, respectively. The curves for the single grain and lattice of grains overlap.

Fitted grain pressure ^{fr} as a function of pressure _{0}) and a lattice of spheres (characteristic length _{1} = 11.21σ_{0}).

The knowledge gained above on the various pressures at equilibrium is needed to construct the REV. The size of the REV includes the complete range of potential interactions available in the system, but not more. To find a REV-property, we need to sample the whole space of possible interactions. The thickness of the REV is larger than the layer thickness used in the simulations.

Our analysis therefore shows that the pressure inside grains in a fcc lattice and the surface tension, depends in particular on the distances between the surfaces of the spheres, including on their periodic replicas. A procedure has been developed to find the pressure of a REV, from information of the (equilibrium) values of ^{fr} as a function of ^{f}. It has been documented in particular for nano-porous medium, but is likely to hold for other lattices, even amorphous materials when the REV can be defined properly.

To show first how a REV-property is determined from the layer-property, consider again the compressional energies of each layer. In the analysis we used the fcc lattice with lattice parameter _{0}. The volume of the grain, ^{r}, and the surface area, Ω^{fr}, varied of course in the exact same way as in _{1} and B_{2} was large, giving a gradient with order of magnitude 10^{12} bar/m. The fluid on the left side is liquid-like, while the fluid on the right side is gas-like. The smallest REV as obtained in the analysis at equilibrium is indicated in the figure.

In order to compute a REV variable away from equilibrium, we therefore follow the procedure described by Kjelstrup et al. [

Compressional energy

We have seen that a nano-porous medium is characterized by pressures in the fluid and the solid phases, as well as the surface tension between the fluid and the solid. When one reduces the size of a thermodynamic system to the nano-meter size, the pressures and the surface tensions become dependent on the size of the system. An important observation is then that there are two relevant pressures rather than one. Hill [

In a macro-scale description, the so-called representative elementary volume (REV) is essential. The REV makes it possible to obtain thermodynamic variables on this scale. We have here discussed how the fact that the macro-scale pressure is constant in equilibrium makes it possible to obtain the integral pressure in the solid, as well as the surface tension, of the liquid-solid contacts in the REV. An observation which confirms the soundness of the procedure is that we recover Young-Laplace's law for the differential pressures. The existence of a REV for systems on the nano-scale supports the idea of a REV that can be defined for pores also of micrometer dimension [

The following conclusions can be drawn from the above studies

We have obtained the first support for a new way to compute the pressure in a nano-porous medium. The integral pressure of the medium is defined by the grand potential. The definition applies to the thermodynamic limit, as well as to systems which are small, according to the definition of Hill [

It follows that nano-porous media need two pressures in their description, the integral and the differential pressure. This is new knowledge in the context of nano-porous media.

For a spherical rock particle of radius

We have constructed two models of a porous medium, case I with a single spherical grain and case II with a fcc lattice of spherical grains. The new method to compute the pressure in these nano-porous mediums is not specific to these two cases, it is general. The method can be used on, e.g., a random distribution of spherical grains, but the REV will need to be larger in order to include all possible microstates. The REV needs in general to be larger as the heterogeneity of the porous medium increases.

To illustrate the concepts, we have constructed a system with a single fluid. The rock pressure and the surface tension are constant throughout the porous medium at equilibrium. The assumptions were confirmed for a porosity change from ϕ = 0.74 to 0.92, for a REV with minimum size of a unit cell.

From the assumption of local equilibrium, we can find the pressure internal to a REV of the porous medium, under non-equilibrium conditions, and a continuous variation in the pressure on a macro-scale.

To obtain these conclusions, we have used molecular dynamics simulations of a single spherical grain in a pore and then for face-centered lattice of spherical grains in a pore. This tool is irreplaceable in its ability to test assumptions made in the theory. The simulations were used here to compute the integral rock pressure and the surface tension, as well as the pressure of the representative volume, and through this to develop a procedure for porous media pressure calculations.

Only one fluid has been studied here. The situation is expected to be more complicated with two-phase flow and an amorphous medium. Nevertheless, we believe that this first step has given useful information for the work to follow. We shall continue to use the grand potential for the more complicated cases, in work toward a non-equilibrium thermodynamic theory for the nano-scale.

All authors contributed equally to the work done. OG carried out the simulations.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The calculation power was granted by The Norwegian Metacenter of Computational Science (NOTUR). We thank the Research Council of Norway through its Centres of Excellence funding scheme, project number 262644, PoreLab.

^{1}Another valid characteristic size is the size of the pores between the grains, but this follows from the two we have chosen.