Joël Puibasset
email: puibasset@cnrs-orleans.fr

Thermodynamic pressure of simple fluids confined in cylindrical nanopores by isothermal-isobaric Monte Carlo: influence of fluid/substrate interactions

The thermodynamic pressure or grand potential density is calculated by isobaric-isothermal Monte Carlo algorithm for simple Lennard-Jones fluid confined in cylindrical pores presenting chemical heterogeneities along their axis. Heuristic arguments and simulation results show that the thermodynamic pressure of the confined fluid contains two contributions. The first term is the usual pressure of the bulk fluid for a density equal to the confined fluid density defined as the total number of confined particles divided by the accessible volume due to thermal agitation. A second term has to be added, which is empirically shown to be proportional to the fluid/wall interface area, and almost constant

along the adsorption and desorption branches. This interfacial contribution, calculated for various pore models, has small variations reminiscent of the fluid adsorption/desorption properties calculated in the various pores. In particular, it is shown that this interfacial quantity is maximum for fluid/substrate interaction intensity of the same order as fluid/fluid one, while the thermodynamic pressure at which rapid desorption occurs presents a minimum.

Stronger or weaker fluid/wall affinity favors gas state nucleation on desorption of confined fluids.

INTRODUCTION

In many natural or industrial situations, a fluid is confined in a porous material. In such situations, the thermodynamic properties of the fluid are expected to be influenced.

Depending on the cases, it may be desirable to enhance or minimize these effects. They are also used to characterize the porous substrate. As a consequence, in order to optimize industrial processes and characterization methods, many efforts are devoted to understand adsorption/desorption phenomena. For a recent review see Ref [1]. Molecular simulation approaches are well suited to study fluid confinement in nanometric pores, allowing a description at the molecular level taking into account all intermolecular interactions. Various simulation methods allow the evaluation of all thermodynamic quantities. However, the calculations are sometimes cumbersome, and some simplifications in the pore model, chosen to keep the main pore features, may greatly simplify the calculations.

The properties of fluids confined in nanopores are influenced by the fluid/wall interactions.

For large pores, where the fluid/wall contributions are expected to be small, it is natural to write the confined fluid free energy as the bulk contribution plus an interfacial correction. It is actually always possible to consider any thermodynamic property of a confined fluid in terms of bulk plus interfacial contributions. However, for strong confinement, the interfacial contributions are expected to be large and strongly dependent on the nature of the substrate and the external parameters like the temperature and the chemical potential in the grand canonical ensemble. It is emphasized that this dependence on many parameters makes a systematic study of fluid/substrate interfacial properties extremely difficult. In comparison, the systematic study of liquid/gas interfacial properties is easier because the substrate is absent and the external parameters generally reduce to only one due to the coexistence condition. References [2-10] give some examples of such studies using molecular simulation.

In a recent paper, 11 a new empirical approach in terms of bulk plus interfacial contributions was proposed. It was shown, on few examples, to greatly simplify the problem since this approach reduces the dependence of the interfacial contributions to only one external parameter instead of two. In this paper, it is attempted to give a theoretical justification of the approach, which allow giving the conditions for its validity. Another point is addressed: the influence of the fluid/wall interaction. It is shown that the thermodynamic pressure for which abrupt desorption occurs is not monotonous with the fluid/wall interaction intensity. The interfacial contribution to the thermodynamic pressure is shown to reproduce this behavior.

THEORY

For fluid adsorbed in large pores, the thermodynamic properties are expected to be close to the bulk ones with a correction arising from the fluid/wall interactions. For nanometric pores, the interfacial terms dominate. However, within the context of the capillary approximation, it is always possible to write any thermodynamic quantity in terms of bulk plus interfacial contributions. For the grand potential ( )

N TS E T µ µ - - = Ω ,
, where Τ is the temperature, µ the chemical potential, E the internal energy, S the entropy, and N the number of molecules, one has:

( ) ( ) ( ) A T V T p T µ γ µ µ , , , bulk + - = Ω (1)
where bulk p is the pressure of the bulk fluid, V is the volume occupied by the fluid, γ is the fluid/substrate interfacial tension and A in the fluid/substrate interfacial area. The grand potential and the pressure are calculated for the same temperature T and chemical potential µ.

As a consequence, the interfacial tension γ is expected to depend on T and µ, and is therefore not a constant along the isotherm. In a recent paper 11 , another interfacial quantity ζ was introduced such that:

( ) ( ) ( ) ( ) A T V T T p T µ ζ µ µ , , , , bulk can + Γ - = Ω (2)
where ( )

V N T p / , bulk can
is the bulk pressure as a function of the canonical variables NVT, and Γ is the confined fluid density, obtained as the average number of particles in the system divided by the pore volume.

( ) ( ) ( ) ( ) ( ) ( ) A V T T p T p T T / , , , , bulk can bulk µ µ ζ µ γ Γ - + = (3)
Let us consider simple examples to illustrate this approach.

The first case to be considered is an ideal fluid (fluid-fluid interactions are neglected)

confined in a given external potential ( )

r ext 1
. This approximation is available at low chemical potential, where the fluid coverage is low enough so that fluid particles are almost isolated and fluid/fluid interactions can be neglected compared to fluid/wall contributions. Within this approximation, the integration of the "barometric law", [START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF] gives the amount of ideal fluid adsorbed in the substrate:

( )       Λ = = Γ ∫ Ψ - kT kT e d V e V N / 3 / id id ext 1 r r µ ( 4
)
where µ is the chemical potential, Λ is de Broglie's wave length, and the integral of the exponential of the external potential ext 1 is performed over the accessible volume V of the pore. Considering the Gibbs equation along the adsorption/desorption isotherms µ Nd d -= Ω at constant volume V and temperature T, and performing the integration using Eq. 4 gives:

kTV d N id id id Γ - = - = Ω ∫ ∞ - µ µ (5)
Taking into account the ideal fluid equation of state

( ) ( ) ( ) ( ) ( ) [ ] ∫ Ψ - - Λ = Ω + = kT kT e d V A e kT A T V T p T / 3 / id bulk id ext / , , , r r µ µ µ µ γ (6)
and ( ) 0 , = µ ζ T

. In this example, the interfacial free energy γ depends on T and µ, while the quantity ζ is constant (equal to zero) along the isotherms. This result is expected to be valid along the gaslike branches of adsorption as long as the fluid density is not too high.

The second case concerns a real fluid (fluid-fluid interactions are now taken into account) in a uniform external potential

( ) 0 ext ext 1 1 = r
. In this case, the energetic contribution

0 ext 1 N
has to be added to the fluid free energy. In the canonical ensemble, the chemical potential is shifted by 0 ext 1 , while in the grand canonical ensemble the average density reads:

( ) ( )

0 ext bulk 1 , , - = Γ µ ρ µ T T (7)
Integration of the Gibbs equation gives:

( ) ( ) ( ) ∫ ∫ ′ - ′ - = ′ ′ Γ - = Ω µ µ ρ µ µ µ d T V d V T T 0 ext bulk 1 , , , ( ) ( )
( )V T T p V T p µ µ , , 1 , bulk can 0 ext bulk Γ - = - - = (8)
As a consequence,

( ) ( ) ( ) ( ) A V T p T p T / 1 , , , 0 ext bulk bulk - - = µ µ µ γ (9)
and ( ) 0 , = µ ζ T . In this second example, the interfacial free energy γ depends on T and µ, while the quantity ζ is constant (equal to zero) along the isotherms.

Finally, let us now consider the general case of a fluid confined in an external potential ( )

r ext 1
. The low density part of the isotherm was considered in the first case; let us now focus on the liquidlike portion of the isotherm where the fluid is dense. It is assumed that the external potential varies slowly compared to the fluid characteristic length. The local density is then given by:

( ) ( ) r r ext bulk local 1 - , ) , , ( µ ρ µ T T = Γ (10)
A grand potential density may be defined using the Gibbs equation and performing integration along the chemical potential path at a given position r :

( ) ( ) ( ) ( ) r r r ext bulk local local 1 , , , , , 2 - - = ′ ′ Γ - = ∫ µ µ µ µ T p d T T ( 11
)
A second integration over space gives the grand potential:

( ) ( ) ∫ ′ - = Ω r r d T T , , 2 , local µ µ (12)
Let us now write the external potential as a constant plus a fluctuating contribution:

( ) ( ) r r ext 0 ext 1 1 1 δ + = (13)
For instance, the external potential produced by a cylindrical pore exhibits a plateau close to the pore center and a fluctuating contribution which changes sign close to the pore wall. Note that 0 1 is not specified by Eq. 13. Additional specifications are given later. It is however assumed that 0 1 is chosen so that the fluctuating contribution is minimized and close to the adsorbent surface. Using Eqs. 11 and 13, the grand potential density may be written as:

( )

) ( 1 ) , , ( 1 - , ) , , ( 2
ext local 0 bulk local r r r δ µ µ µ T T p T Γ + - ≈ (14)
Equation 14 then shows that the grand potential decomposes in two contributions:

( ) ( ) ( ) ( ) r r r d T V T p T ext local 0 bulk 1 , , 1 - , , δ µ µ µ ∫ Γ + - ≈ Ω (15)
Equation 15 may be rewritten as:

(

Γ + Γ - ≈ Ω r r r d T V T T p T ext local 0 bulk can 1 , , , , , δ µ µ µ ) ( ) ( ) ( ) ( ) ∫
where ( ) ( )

0 bulk 0 1 - , , µ ρ µ T T = Γ corresponds
( ) ( ) ( ) ∫ Γ ≈ r r r d T A T ext local 1 , , 1 , δ µ µ ζ (17)
This proportionality was empirically confirmed based on simulations results. 11 Let us now focus on the variations of ζ with the chemical potential along the liquidlike portion of the adsorption isotherm. According to Eqs. 17 and 10, these variations are given by the variations of the local density with µ, i.e. the slope of the grand canonical bulk adsorption isotherm. This quantity is a priori small in the dense fluid state. In conclusion, along the high density portion of the adsorption isotherm, the interfacial quantity ζ is expected to have small variations with µ. One may then write:

( ) ( ) ( ) ( ) ( ) V A T T T p V T T / , , / , , 0 bulk can ζ µ µ µ - Γ ≈ Ω - = Π (18)
It is possible to make these variations even smaller: As previously mentioned, the choice of 0 1 is actually not unique, and then ext 1 δ may be tuned to minimize the variations of ζ with µ.

This will be empirically shown to be possible in next section. The main consequence of such a choice is that the value of ( )

µ , 0 T Γ
entering into Eq. 18 may not be the local fluid density in the pore center. The new value will be empirically shown to remain however very close to that value, and compatible with the confined fluid global density N/V, where N is the total number of adsorbed molecules and V the accessible pore volume where the external potential is less than roughly 10 3 in reduced units.

PORE MODELS AND SIMULATION METHODS

As mentioned in introduction, the molecular simulation methods allow in principle the determination of all thermodynamic quantities of confined fluid. For instance, the grand potential is obtained by acquiring adsorption/desorption isotherms at various temperatures (sub-and super-critical temperatures) in the grand canonical ensemble, and then performing integration along thermodynamic paths. [START_REF] Peterson | [END_REF][START_REF] Frenkel | Understanding Molecular Simulation[END_REF] This method is however quite lengthy, and simplifications in the pore model allow the use of much less time consuming direct methods.

The thermodynamic pressure (minus the grand potential density) is directly accessible in the isobaric-isothermal statistical ensemble [START_REF] Frenkel | Understanding Molecular Simulation[END_REF][START_REF] Nicholson | Computer simulation and the statistical mechanics of adsorption[END_REF][START_REF] Allen | Computer simulation of liquids[END_REF] (or its generalization [START_REF] Puibasset | [END_REF][18][19] ) which requires a translational (respectively rotational) invariance of the external potential. More details are given below. The atomic roughness has then to be smoothed. However, nanometric scale modulation of the external potential is possible if the required invariance is satisfied. The pores are then chosen to be tubular capillaries presenting fluid/wall interaction intensity modulated along the pore axis while satisfying the rotational invariance around the pore axis.

All intermolecular interactions are supposed to be Lennard-Jones like. The interaction parameters entering the potential are chosen to be those of argon for the fluid, and argon/solid carbon dioxide to model the fluid/substrate interactions. This model is frequently used in literature for fundamental issues concerning fluids at an interface. [START_REF] Peterson | [END_REF][20][21][22][23][24] For more details see also Ref [25]. The Lennard-Jones parameters are given in Table 1. All thermodynamic quantities are normalized to the Ar-Ar parameters (reduced quantities, denoted by an asterisk). The interaction sites of the solid matrix are supposed to be uniformly distributed (smooth wall approximation), and the fluid/adsorbent interactions give rise to an external potential, denoted ( )

r cyl ψ
for an infinite cylindrical pore. In this study, the reduced pore diameter is chosen to be 5 for all pore models. The obtained external potential fields was given elsewhere. 11 In order to evaluate the effect of the fluid/substrate interaction intensity, the external potential may be multiplied by a factor denoted a:

) ( ) ( cyl cyl a r a r ψ ψ = (19)
The factor a varies between 0.5 and 1.2 in this work. The chemical heterogeneity is directly introduced by modulating the amplitude of the external potential along the axial direction.

This introduces an explicit dependence of the potential on r and z, the radial and axial coordinates in a cylindrical representation. The characteristic spatial extension of the chemical heterogeneity is taken equal to λ . The chemically heterogeneous external potential is given by:

( ) { } ) ( cos 1 ) , ( cyl 2 heter r b z r z ψ ψ λ π + = ( 20
)
where b measures the amplitude of the heterogeneity (less than 0.2 in this study). This potential mimics the existence of more or less attractive sites distributed alternatively along the pore. The simulation box size along the axial coordinate is 12 in reduced units, large enough to avoid finite system size effects. The values of λ considered in this study are 2.4, 4.0 and 12.0 in reduced units. The calculations are performed at the reduced temperature T* = kT/ε Ar-Ar = 0.77.

As previously mentioned, the smooth wall approximation allows to produce models presenting a rotational invariance around the pore axis. In such situations, a recently developed generalization of the isobaric-isothermal ensemble may be used, [START_REF] Puibasset | [END_REF][18][19] where the

thermodynamic pressure ( ) ( ) V V T V T ∂ Ω -∂ = Π / , , , , µ µ
is imposed and the system is allowed to equilibrate in the corresponding statistical ensemble. Due to the invariance, the thermodynamic pressure is actually independent of V and the grand potential is directly given by ( )

( )V T V T µ µ , , , Π - = Ω
It is emphasized that the thermodynamic pressure Π (scalar quantity) differs from the mechanical pressure, which is an anisotropic tensor for confined fluids. However, for a homogeneous fluid, Π reduces to the bulk scalar mechanical pressure denoted p bulk , and, for a fluid at an interface, the thermodynamic pressure also contains fluid/substrate contributions, or interfacial tension contributions in a classical capillarity theory point of view (see Eq. 1).

For each simulation run, a minimum of 5. volume is defined as the region of the pore where the potential does not reach infinity, which is equivalent to taking the pore diameter equal to 5 in our case. It will be emphasized later that another more physical convention will emerge, taking into account that the whole volume is

actually not accessible at usual temperatures due to the effect of the repulsive part of the external potential.

SIMULATION RESULTS

The adsorption/desorption isotherms obtained in the various pores are given in Figs. 123. In Fig. 1 are reported the homogeneous cylindrical pore results for various fluid/wall interaction intensity. Figures 2 and 3 concern heterogeneous pores, and show the effect of the heterogeneity amplitude (Fig. 2) and wavelength (Fig. 3). As can be seen, the isotherms for homogeneous cylindrical pores (Fig. 1) exhibit large hysteresis for all fluid/wall interaction intensities considered in this study, which is expected at low temperature. Starting from low thermodynamic pressure Π, the amount adsorbed increases continuously with Π (gaslike branch), until the so-called capillary condensation point is reached: at this point, the amount adsorbed is discontinuous (vertical adsorption branch). The Monte Carlo procedure jumps irreversibly to the liquidlike state. On desorption, this dense state persist below the capillary condensation point, down to the so-called desorption pressure where the fluid rapidly desorbs via a mechanism which is expected to be cavitation (meniscus recession from the boundaries is impossible due to the periodic boundary conditions used in the simulations). As can be seen, the capillary condensation and desorption pressures are modulated by the fluid/wall interaction intensity. These quantities are shown in Fig. 4 and 5 as a function of a (see Eq.

19)

. The thermodynamic pressure, at which capillary condensation occurs, increases with fluid/wall interaction intensity. This shows that the gaslike state is stabilized by higher fluid/wall interactions. On the other hand, the cavitation thermodynamic pressure presents a minimum (see Fig. 5) for a around 0.8-0.9. Note that fluid/wall interactions are equal to the fluid/fluid interactions for a = 0.78 (see Table 1). This suggests that fluid/wall interactions stabilize the liquidlike state when they are similar to fluid/fluid interactions. Weaker or stronger interactions tend to destabilize the fluid. This may be specific to strong confinement in nanopores, and is not necessarily valid close to a plane interface. 26 The chemical heterogeneities along tubular pores are expected to strongly affect adsorbed fluid properties. Three pore models are then compared, where the amplitude of the chemical heterogeneity is varied: b = 0 (perfectly homogeneous pore), b = 0.1, and b = 0.2 (see Eq. 20).

In these cases, the heterogeneity wavelength is constant and equal to 12σ Ar-Ar . The adsorption isotherms are given in Fig. 2. As can be seen, the high density parts of the isotherms are quite similar. However, below a reduced thermodynamic pressure Π* = 0.2 the chemical heterogeneity strongly distorts the isotherm. The hysteresis width slightly reduces when b increases, and the gaslike part of the isotherm exhibits a reversible step reminiscent of a third state of intermediate global density. This state was clearly observed in larger pores where multiple hystereses were possible. 25,27 It corresponds to a liquidlike meniscus trapped in the most attractive part of the tubular pore, while the less attractive part contains few molecules (gaslike fluid). The global density is then intermediate between gaslike and liquidlike. In the present case, the pore size is small and the transition between the purely gaslike state and this third state of intermediate density is continuous. The last point to be mentioned is that the hysteresis shifts towards larger thermodynamic pressure for stronger chemical heterogeneity. This is due to the presence of weakly attractive regions in the pore which disfavors the then introduce the pore size as a free parameter which links ( )

µ , 0 T Γ in Eq. 18 with ( ) µ , T Γ
given by the simulation results. Note that the area A and volume V are also affected by this redefinition of the pore radius. For a given temperature, the quality of the fit is measured by the quantity:

( ) ( ) ∑             - Γ - Γ Π = n i i i V A p n 1 2 0 bulk can 0 1 ζ χ (21)
where the sum is performed over the n simulation points, labeled by i, for a given isotherm.

This quantity evaluates the difference between the calculated thermodynamic pressure and its phenomenological expression in terms of bulk pressure (taken from Ref [28]) plus interfacial contribution. Minimization of the quantity χ allows the determination of both the effective pore size and interfacial term ζ with uncertainties of the order of few percent and 5 to 10 percent respectively.

The result of the minimizations along the adsorption branches always exhibit very small values for ζ, compatible with zero within error bars. This is expected since, at low coverage, the fluid follows the ideal gas approximation (see section 2). The results obtained along the liquidlike branches are given in Tables 234. As can be seen, the effective pore size is equal to 2.06 in all situations, except for very weak fluid/wall interactions where the effective pore radius reaches 2.08. For comparison, the external potential is equal respectively to zero, 10 3

and reaches infinity for a reduced radial coordinate equal to respectively 1.73, 2.04 and 2.5.

The effective pore size deduced from the minimizations is then very close to the accessible pore size where the external potential reaches 10 3 in reduced units. When decreasing the external potential amplitude a, the effective pore size slightly increases because the repulsive part of the potential also slightly decreases. The value obtained for the interfacial term ζ * varies between -1.20 and -0.99. The Figs. 6-8 show 0 Γ (see Eq. 18) as a function of the

quantity V A / ζ + Π
(symbols) and the bulk fluid equation of state (solid line, T* = 0.77). 28 As can be seen, a good agreement is obtained over the whole liquidlike branch of the isotherm, which validates the analysis. Systematic small deviations are however visible at low thermodynamic pressure, showing that ζ is actually slightly dependent on the chemical potential µ. It is however emphasized that this dependence is quite small compared to the variations of the true interfacial tension γ with µ (see Eq. 1). 11 As can be seen, the modulation of the fluid/wall external potential does not drastically changes the values of ζ, which are almost all identical within the uncertainty of few percent.

The most important effect is observed when a 10 percent amplitude modulation of the external potential of wavelength λ* = 12 is produced (b = 0.1). In this case, ζ is 14% larger in magnitude (Table 3). However, further increase of the modulation amplitude does not increase ζ by much. The reduction of the heterogeneity wavelength has the opposite effect (see Table 4). Reducing this wavelength seems to produce values of ζ closer to the perfectly cylindrical case. This is reminiscent of the discussion on the adsorption/desorption isotherms where it was shown that the chemical heterogeneity effect was the strongest for λ* = 12 and that short wavelength heterogeneities reduce to simple surface roughness. The evolution of ζ with the fluid/wall interaction intensity is shown in Fig. 9 with error bars. Despite the fact that the variations are within error bars, a maximum is observable. This is reminiscent of the minimum previously observed in desorption pressure of the dense confined fluid (see Fig. 6).

  to the plateau value, i.e. a typical value for the density of the fluid confined in the pore. Considering the fact that ext 1 δ is the fluctuating contribution of the external potential, which is a priori localized close to the substrate wall, the last term in Eq. 16 is expected to be proportional to the surface area A. It is then possible to define an interfacial contribution to the grand potential, which corresponds to the definition of ζ (Eq. 2):

10 6

 10 Monte Carlo trials were performed to acquire statistical data. The thermodynamic pressure (imposed parameter) is varied between -0.5 and 10. Negative values are possible in the liquidlike states at low temperature due to the existence of cohesive forces. The highest values of the thermodynamic pressure are reached at high density due to the repulsive part of the intermolecular potential. During these simulations, the fluctuating variable is the volume (isobaric-isothermal ensemble). After averaging, one introduces the global reduced density Γ of the confined fluid, defined as the total number of particles N divided by the average reduced volume V. By convention, this

The parameter ζ 5 CONCLUSIONFIG. 1 .FIG. 2 .FIG. 3 .FIG. 4 .FIG. 5 .FIG. 6 .FIG. 7 .FIG. 8 .FIG. 9 .

 5123456789 FIGURE CAPTIONS

Figure 2 reduced

 2 Figure 2

TABLE 1 :

 1 Lennard-Jones argon/argon and argon/solid CO 2 interaction parameters.

		ε/k (K)	σ (Å)
	argon-argon	119.8	3.405
	argon-CO 2	153.0	3.725

TABLE 2 :

 2 Reduced interfacial contribution liq ζ * and effective reduced pore size R* defined by Eq. 18 along the liquidlike portion of the isotherms for a Lennard-Jones fluid confined in perfectly homogeneous cylindrical pores, with various values for the fluid/wall interaction intensity a (see Eq. 19).

	a	0.5	0.7	0.8	0.9	1.0	1.1	1.2
	R*	2.08	2.07	2.06	2.06	2.06	2.06	2.06
	ζ*	-1.06	-1.06	-1.02	-0.99	-1.04	-1.03	-1.07

TABLE 3 :

 3 Reduced interfacial contribution liq ζ * and effective reduced pore size R* defined by Eq. 18 along the liquidlike portion of the isotherms for a Lennard-Jones fluid confined in heterogeneous pores (see Eq. 20) with λ * = 12 and various values for b.

	b	0.0	0.1	0.2
	R*	2.06	2.06	2.06
	ζ*	-1.03	-1.17	-1.20

TABLE 4 :

 4 Reduced interfacial contribution liq ζ * and effective reduced pore size R* defined by Eq. 18 along the liquidlike portion of the isotherms for a Lennard-Jones fluid confined in heterogeneous pores (see Eq. 20) with b = 0.1 and various values for λ * .

	λ*	12	4.0	2.4
	R*	2.06	2.06	2.06
	ζ*	-1.17	-1.12	-1.10

/ε ε ε ε Ar-Ar 0.00 0.05 0.10 reduced density Γ Γ

σ Ar-Ar 3 /ε ε ε ε Ar-Ar 0.00 0.05 0.10 0.15 0.20 reduced density Γ Γ

ACKNOWLEDGEMENTS

Dr. A. Delville, Dr. E. Kierlik, Dr. P. Porion and Dr. G. Tarjus are gratefully acknowledged for fruitful discussions on adsorption in porous materials. The simulations were performed on workstations purchased thanks to grants from Région Centre (France), and on IBM SP4 supercomputers at the Centre de Ressources Informatique de Haute Normandie (CRIHAN, St Etienne du Rouvray, France) and the Institut de Développement des Ressources en Informatique Scientifique (IDRIS-CNRS, Orsay, France).

capillary condensation at low thermodynamic pressure (region of the pore where the cosine in Eq. 20 is less than zero).

The last point concerns the influence of the heterogeneity wavelength. Figure 3 shows the adsorption/desorption isotherms for three reduced wavelength λ* = 2.4, 4.0, and 12 and the same amplitude b = 0.1 (see Eq. 20). In the case of the largest wavelength, the system exhibits one single hysteresis. However, as previously mentioned, the gaslike branch presents a step reminiscent of an intermediate state stabilized by the chemical heterogeneity. For shorter wavelengths λ* = 2.4 and 4.0, the system does not exhibit this state: the hysteresis is wider, and close to the perfectly homogeneous case b = 0. According to these results, it seems that the heterogeneity wavelength λ* = 4.0 is too short to allow the stabilization of the intermediate state: in this case, the heterogeneity has the same effect as a simple atomic roughness λ* around 1.

The previous analysis on confined fluid thermodynamic pressure in terms of bulk plus interfacial contributions (see Eq. 18) is now applied to the adsorption/desorption isotherms obtained by simulation. As mentioned in section 2, the effective confined fluid density ( ) µ , 0 T Γ depends on the choice of 0 1 . It is however expected to be close to the global density of the confined fluid ( )

. This last quantity is also ill defined, since it relies on the exact definition of the pore volume. For instance, ( )

was calculated with a reduced pore radius of 2.5 where the external potential is infinite. However, for a reasonable temperature around 1 in reduced units, the accessible pore volume, due to thermal fluctuations, is expected to be delimited to the pore region where the external potential is less than few hundreds in reduced units (the external potential varies sharply close to the pore wall). The quantities


Joël Puibasset
email: puibasset@cnrs-orleans.fr

Surface excess free energy of simple fluids confined in cylindrical pores by isothermal-isobaric Monte Carlo: influence of pore size

Confined fluid properties are mainly determined by interfacial phenomena characterized by surface quantities. Based on a simple model of Lennard-Jones particles confined in a cylindrical pore, this study introduces a grand potential surface quantity to quantify the difference in thermodynamic pressure between the bulk and the confined fluids. The usual surface tension γ defined as this grand potential difference for the same chemical potential in both confined and bulk states is generally strongly dependent on both the chemical potential and temperature. It is proposed to introduce another surface quantity ζ which measures the thermodynamic pressure difference between confined and bulk states for identical densities. It is shown that this quantity is much less dependent on confined fluid density or chemical

potential. It is actually constant along the gaslike and liquidlike adsorption / desorption branches for an irreversible isotherm (hysteresis), with a different value for each branch. For reversible supercritical isotherms, ζ is shown to remain constant in the low and high density parts of the isotherm. This independence on chemical potential (or equivalently fluid density) is believed to be of great interest for practical applications when one desires to calculate thermodynamic quantities like the usual surface tension γ or the thermodynamic pressure of a confined fluid for any given chemical potential and temperature. Such calculations are required to determine fundamental properties like metastability or coexistence. The effects of temperature, fluid / substrate interaction strength and pore size are studied.

INTRODUCTION

The problem of fluid confinement in porous materials is the subject of numerous theoretical and experimental investigations. The fields of application are catalysis, gas separation, or porous materials characterization for example. In between experiment and theory, simulation tools are able to produce significant results in the field of fluids adsorption. The main advantage is that an accurate model can be used to describe molecular interactions, and realistic numerical porous materials can be achieved. However, in order to reach some statistical significance in terms of porous material structural and topological properties, it is generally desired to consider large systems. In this case, the atomistic simulations may become time consuming, and some phenomenological approaches may become useful in order to extrapolate simulation results to larger systems or to guess the fluid behavior for a set of parameters (like temperature and chemical potential) which have not been explicitly considered in the simulations. Essentially, such phenomenological approaches consist in separating the thermodynamic functions in bulk and surface excess contributions.

It is well established that for a fluid confined in mesoscopic pores (larger than few nanometers), its thermodynamic properties in the center of the pore (far from the matrix wall) are close to its bulk properties. If one is mainly interested in global (in opposition to local) properties, it may be useful to introduce (surface) quantities to quantitatively characterize the difference between the adsorbed and bulk fluids. Within this context, the thin wall approximation consists in rewriting the adsorbed fluid properties as a bulk plus a surface contribution. One of the most important quantities is the excess free energy or surface tension which quantifies the fluid / substrate interactions contribution to free energy. The importance of surface tension cannot be underestimated. The knowledge of the amount adsorbed and the corresponding surface tension allows calculating all thermodynamic properties of the confined fluid.

Many papers have focused on the determination of liquid / gas interfacial free energies in a molecular simulation approach. Refs [1][2][3][4][5][6][7][8] give some examples published in the last decade.

However, few are concerned with a systematic study of fluid / substrate surface tension at the molecular level. There are many reasons to this. The first point is that for a given fluid, many substrates may be envisaged. It is emphasized that both the chemical as well as the structural properties of the adsorbent are relevant to surface free energy. The second point, and certainly the most important, is that, even for a given fluid / substrate system, the free energy still depends on both the temperature T and the chemical potential µ, which makes its full determination cumbersome. This is not the case for liquid / gas interfacial properties which are only defined at coexistence, which reduces the number of independent parameters. Despite these difficulties, this paper focuses on a molecular approach to the calculation of fluid / wall interfacial free energy for argon confined in cylindrical pores of solid carbon dioxide as a function of temperature and chemical potential. After the presentation of the molecular model and techniques, a new quantity ("effective surface tension") is introduced to characterize the effect of the fluid / wall interface, which allows the determination of the surface tension. This quantity is calculated along the adsorption / desorption isotherms for the confined fluid. It is shown that this quantity is actually constant along the adsorption and desorption branches, which greatly simplifies the calculations, since the double parameter dependence of the surface tension (on T and µ) has been reduced to a single dependence. The effects of pore size and fluid / wall interaction strength are then studied.

SIMULATION RESULTS

PORE MODELS

The choice of the simulation model is guided by simplicity, because the aim is to give a comprehensive phenomenological interpretation of the calculated fluid properties in terms of bulk pressure and fluid / substrate surface tension. The application to more realistic substrates is straightforward, and will be investigated in a forthcoming work. The model capillaries are chosen to be cylindrical pores of radius R ranging from mesoscopic (few nanometers) down to microscopic scale (<1nm). For the sake of simplicity, all intermolecular interactions are supposed to be Lennard-Jones like. The interaction parameters entering the potential are chosen to be those of argon for the fluid, and argon / solid carbon dioxide to model the fluid / substrate interactions. This model is frequently used in literature by many authors for fundamental issues concerning fluids at an interface. [9][10][11][12][13][14]. For more details see also Ref [15]. The Lennard-Jones parameters are given in Table 1. All thermodynamic quantities are normalized to the Ar-Ar parameters (reduced quantities, denoted by an asterisk). The interaction sites of the solid matrix are supposed to be uniformly distributed (smooth wall approximation), and the fluid / adsorbent interactions give rise to an external potential, denoted

( ) r R cyl *
ψ for an infinite cylindrical pore of reduced radius R*. Figure 1 shows the different external potential fields obtained for various pore radii. As expected, the smaller the pore, the deeper the resulting fluid / wall interaction.

Due to computer time limitations, the case of very large pores cannot be considered.

However, the limit of infinite radius is modeled by plane surfaces. The potential created by such a plane surface (not shown) is very close to that created by the largest R* = 8 pore shown in Fig 1 (when both are given as a function of the distance to the wall). In order to enhance fluid / wall contribution and avoid spurious effects due to the presence of a free interface, two opposite parallel plane substrates are considered, which can be regarded as a slit pore. The pore width is taken large enough (equal to 8 and 16 σ Ar-Ar ) so that the two interfaces can be considered as independent. As a consequence, the external potential is negligible in the median plane of the slit pore, and the potential in the surrounding of one of the solid surfaces is not influenced by the opposite face.

SIMULATION TECHNIQUES

The aim of this paper is to calculate thermodynamic quantities giving information on the adsorbed fluid (confinement effect), especially the contribution due to fluid / substrate interactions. For instance, determination of the excess Helmholtz free energy would give the interfacial surface tension. Similarly, the calculation of the grand potential (associated to the grand canonical ensemble) also allows the determination of the fluid / substrate surface tension. These thermodynamic quantities are in principle accessible through general integration procedures. [11,[START_REF] Frenkel | Understanding Molecular Simulation[END_REF] However, in order to avoid lengthy calculation of complete sets of adsorption / desorption isotherms, direct algorithms are required. Perturbative methods (performed in canonical ensemble) have been proposed [START_REF] Eppenga | [END_REF][18][19], which have proven to be very efficient as simulation algorithms. The isobaric-isothermal ensemble is another direct method where the grand potential density (thermodynamic pressure) is imposed and the system is allowed to equilibrate in the corresponding statistical ensemble [START_REF] Frenkel | Understanding Molecular Simulation[END_REF][START_REF] Nicholson | Computer simulation and the statistical mechanics of adsorption[END_REF][START_REF] Allen | Computer simulation of liquids[END_REF]. These powerful techniques are generally limited to systems presenting translational invariance, which is the case for homogeneous fluids, or fluids confined in very simple geometries (slit or cylinders).

This paper focuses on such simples geometries. However, this preliminary work will be extended to more realistic pores, taking for instance into account possible physico-chemical heterogeneities. In this case, the translational invariance is broken, especially for tubular pores. However, in the case where the physico-chemical heterogeneities are along the pore axis, the axial symmetry may be preserved. In this case, an original approach was proposed to calculate the thermodynamic pressure as a function of the density of a confined fluid using a generalized isothermal-isobaric algorithm. [START_REF] Puibasset | [END_REF][23][24] These algorithms are shortly presented now.

Let us consider a fluid characterized by its (imposed) volume V, temperature T, and chemical potential µ (grand canonical ensemble). The differential of the corresponding grand canonical potential ( )

N TS E V T µ µ - - = Ω , ,
where E is the internal energy, S the entropy, and N the number of molecules, is given by

µ Nd SdT dV d - - Π - = Ω (1)
where the quantity dV Π is the mechanical work associated to a volume variation dV. If the system presents translational invariance along at least one direction (for instance parallel to the substrate surface for a fluid confined in a slit pore), then the partial derivatives (like the thermodynamic pressure Π) of an extensive thermodynamic quantity (like the grand potential Ω) with respect to the volume V of the system is independent of its size (homogeneous function of degree zero). In this case, the extensive quantity Ω is a homogeneous function of the first degree of its volume (the only extensive parameter), and Euler's theorem gives:

V Π - = Ω (2)
The grand potential in the slit pores is then calculable by Monte Carlo simulations in the usual isobaric-isothermal ensemble (constant N, Π, and T) [START_REF] Frenkel | Understanding Molecular Simulation[END_REF][START_REF] Nicholson | Computer simulation and the statistical mechanics of adsorption[END_REF][START_REF] Allen | Computer simulation of liquids[END_REF]. The corresponding Monte-Carlo algorithm is made of translation trials in order to thermalize the system, and volume variations trials in order to equilibrate the system thermodynamic pressure with the external reservoir. It is emphasized that the thermodynamic pressure Π (scalar quantity) defined by Eq.

1 should not be confused with the mechanical pressure, which is an anisotropic tensor for confined fluids. However, for a homogeneous fluid, Π reduces to the bulk scalar mechanical pressure denoted p bulk . Most generally, for a fluid at an interface, the thermodynamic pressure also contains the fluid / substrate contributions, or surface tension contributions in a classical capillarity theory point of view (see later).

For pores presenting an axial symmetry, the previous remarks have been shown to remain valid. [START_REF] Puibasset | [END_REF][23][24] In this case, if one considers, as the system, the portion of the pore contained in a dihedron of angle Θ radians, the volume is

π π 2 / 2 Θ = V V where π 2 V is the volume of the complete ( π 2 = Θ
) pore of the same length. Due to rotational invariance, all extensive quantities (like volume itself) are proportional to Θ. For instance the energy of the system is given by

π π 2 / 2 Θ = E E
where π 2 E is the energy of the complete

π 2 = Θ
pore. Considering the fluid contained in this dihedron, its thermodynamic potential (associated to the grand

canonical Θ T µ ensemble) is the grand potential N TS E µ - - = Ω
, where N is the number of particles in the dihedron, µ the chemical potential, S the entropy, T the temperature. Its differential is given by:

µ π π Nd SdT d V d - - Θ Π - = Ω 2 2 ( 3
)
where Π (thermodynamic pressure for axisymmetric system) is such that -( V is a constant), Euler's theorem gives:

) Θ Π d V π π 2 / 2 corresponds
V V Π - = Θ Π - = Ω π π 2 2 (4)
According to this result, a Monte Carlo simulation in the (N, Π, T)-ensemble allows to calculate directly the grand potential Ω of the confined fluid. The Monte Carlo algorithm to sample this generalized isothermal-isobaric ensemble is very similar to the classical isothermal-isobaric one. [23,24] It decomposes in two parts, corresponding to thermalization, and volume variations. Thermalization is performed through particle displacements trials.

Since

                  Θ ∆Θ + + + ∆Θ Π + ∆ - = ) / 1 ln( ) 1 ( / ) 2 ( exp , 1 min 2 vol acc N kT LR E P (5)
For further details concerning the algorithm implementation, see Ref [24]. In the case of perfectly cylindrical pores, both usual and generalized algorithms give exactly the same results. However, since we are also interested in the effect of physico-chemical heterogeneities in tubular pores, the generalized algorithm is necessary.

The simulations were performed in cylindrical pores of reduced length equal to 12. For slit pores, the typical reduced size has been chosen around 12 in each direction parallel to the pore walls. These lengths have been shown to be large enough to avoid finite system size effects.

For each run, a minimum of 5. width of 8 and 16 in reduced units. For comparison, the bulk Lennard-Jones equation of state [25] is also given for the same temperature. As can be seen, the bulk isotherm shows a van der Waals loop (subcritical temperature), while confinement produces reversible isotherm in the cylindrical pores, and irreversible adsorption / desorption for the large slit pore (H* = 16).

ISOTHERMS

The small slit pore (H* = 8) presents a vertical rise in adsorption similar to the one in the large slit pore, with a very small hysteresis.

A general trend is an increase of the amount of fluid adsorbed with increasing thermodynamic pressure. For a given thermodynamic pressure, the larger the pore, the larger the density Γ = N/V of adsorbed fluid. For small pores, the fluid density is much smaller than the bulk, due to the definition of the accessible pore volume V. It will be shown later that a more physical definition allows reconciling the densities with the bulk ones. As can be seen, the low density curves are perfectly superimposed. This suggests a universal behavior in the low density limit.

In the case of low coverage, the probability to have two interacting particles on the surface is negligible since particles are far from each other. The ideal gas approximation may then be applied. The integration of the "barometric law" [START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF], gives the amount of adsorbed ideal fluid:

[ ] ∫ Ψ - Λ = kT kT e d e N / 3 / id ext r µ (6)
where µ is the chemical potential, Λ is de Broglie's wave length, and the integral of the exponential of the external potential ext 1 is performed over the accessible volume of the pore.

Considering the differential given by Eq 1 along the Gibbs adsorption isotherm (constant volume V and temperature T), and performing the integration using Eq. 6 one gets:

kT N d N id id id - = - = Ω ∫ µ (7)
As a consequence, the ideal thermodynamic pressure is given by:

kT V id id id Γ = Ω - = Π (8)
This law explains why all isotherms (bulk and confined) are superimposed in the low density limit when plotted as a function of the thermodynamic pressure.

Figure 2 also shows that the large fluid uptake which takes place in the largest pores occurs for thermodynamic pressure slightly above the bulk spinodal. This suggests that the pore filling may occur within the pore by a nucleation mechanism in the gas phase. It is also noticeable that the confined fluid isotherms depart from the bulk one with lower adsorption values for a given thermodynamic pressure. This is easily interpreted by the attractive fluid wall interactions which confine the fluid close to the wall. For a given density, the more attractive the substrate, the larger the density profile at the wall, and the larger the mechanical work associated to volume variations, and hence the larger the thermodynamic pressure.

ANALYSIS AND DISCUSSION

FLUID / SUBSTRATE EFFECTIVE SURFACE TENSION

In the classical capillarity theory, the interfacial phenomena (liquid / gas interface or fluid / substrate interactions) are generally treated in the framework of the thin wall approximation.

In this context, the changes in the free energy induced by the presence of this interface are entirely contained in the definition of the surface tension associated to this interface. The previously introduced grand potential of a fluid confined in a capillary then reduces to:

A pV γ + - = Ω ( 9
)
where p is the pressure of the fluid, V is the volume occupied by the fluid, γ is the fluid / substrate surface tension and A is the fluid / substrate interfacial area. The thermodynamic pressure then reads:

V A p γ - = Π (10)
For a cylindrical pore of radius R, the surface to volume ratio is equal to 2/R. For a slit pore, this ratio is equal to 2/H with H the pore width. The thermodynamic pressure Π is given by the simulation results. If one identifies p with the bulk fluid pressure bulk p , the surface term γ can be determined. The usual definition of the surface tension requires that the bulk pressure bulk p be calculated for the same chemical potential as that for which Π has been calculated. In other words, [25]). As can be seen, the obtained surface tension is negative, as expected for an attractive surface. It is also emphasized that γ varies between -1.02 and -0.16, depending on the chemical potential, i.e.

( ) ( ) µ µ γ Π - = bulk p V A ( 11
)
the amount of adsorbed fluid. Such a strong dependence of surface tension with surface coverage is also expected in the ideal gas approximation (where only fluid substrate interactions are taken into account). However, it was previously shown that within this ideal approximation, the confined fluid thermodynamic pressure is given by the same formula as the bulk fluid ideal pressure. As a consequence, let us introduce another quantity ζ such that

( ) ( ) Γ Π - Γ = bulk p V A ζ (12)
with Γ the confined fluid density. This quantity is always equal to zero for any fluid in a uniform external potential, and for an ideal fluid confined in any given external potential. This quantity then measures the combined effect of both the external potential variations (close to the substrate) and the fluid particles interactions, on the difference between the bulk and confined thermodynamic pressures. In this definition, the surface to volume ratio has been introduced like for surface tension, because it is expected that the difference between the bulk and confined thermodynamic pressures is merely localized at the fluid / substrate interface where the external potential varies rapidly. The simulation results confirm this expected behavior (see later). ζ is referred to as an "effective surface tension", but it should not be confused with the true surface tension γ which can easily be deduced from ζ using Eqs. 11 and 12. The evaluation of ζ using formula 12 for each of the previously calculated simulation points produces values which are much less dependent on the fluid state (determined in this case by its density Γ) than for the true surface tension ( )

µ γ
. This is actually the main interest for the introduction of this quantity. However, it is reminded that the density of a confined fluid depends on the exact definition of the accessible pore volume.

Changing this definition (in other words changing the radius of a cylindrical pore by α)

changes the density by 1/α 2 and, as a consequence, changes the value of ζ . Considering the fact that the pore volume definition is quite arbitrary, it has been decided in this work to relax this definition and look for a value of α which minimizes the (already small) variations of ζ with the density. For a given pore, especially those presenting a hysteresis, one can easily get convinced that a unique value for α and ζ is not possible to verify Eq. 12 (a priori, a twovalue hysteresis is also expected for α and ζ). However, it is shown below that the two effective surface tension values and only one effective pore radius are enough to perform the fits. The two effective surface tensions correspond to the adsorption and desorption branches.

The advantage of this approach is twofold: firstly, a unique effective pore radius is introduced, and secondly, an effective surface tension is introduced which takes only two values corresponding to the gas and liquid branches, whatever the chemical potential.

Let us introduce the first free parameter α such that αR (resp. αH) is equal to the effective pore radius (resp. slit pore size), while R (resp. H) is defined as previously as the radius (resp.

pore width) where the external potential diverges. The new effective pore volume is then multiplied by α 2 for cylindrical pores (resp. α for slit pores). The new effective fluid density in the pore is then ρ = Γ/α 2 in cylindrical pores (resp. ρ = Γ/α in slit pores), where Γ = N/V is the initial fluid density as previously introduced. The physical isotherm Π(Γ) is not affected by this redefinition of volume. However, the mathematical expression of the function is affected. To avoid any confusion, let us introduce this new function such that ( ) ( )

ρ Π′ = Γ Π .
The density of the corresponding bulk fluid is taken equal to ρ. The second free parameter is

ζ . The equation 12 now reads:

( ) ( ) ( ) ( ) ρ ρ ρ α ζ Π′ - = Γ Π - = bulk bulk p p V A ( 13
)
Note that the surface to volume ratio is also affected by this redefinition of the pore radius, by the same factor 1/α for cylindrical and slit pores. The quality of the fit is measured by the quantity:

( ) ( ) ∑             - - Γ Π = n i i i V A p n 1 2 bulk 1 α ζ ρ χ ( 14
)
where the sum is performed over the n simulation points, labeled by i, for a given isotherm;

Π i is the thermodynamic pressure of point i, Γ i the corresponding density given in Figs 234, and ρ i the rescaled density. This quantity evaluates the difference between the calculated isotherm Π(Γ) and its phenomenological expression in terms of bulk pressure and surface tension.

The first case to be considered is the low temperature, liquidlike branch of argon confined in the slit pore of reduced width 8. The Fig. 5 shows the quantity χ around its minimum in the two directions defined by αH and ζ . As can be seen, the minimum is well pronounced, especially in the αH direction which allows defining the effective width of the slit pore to be around 7.3 in reduced units. The relative uncertainty is few percent. It is remarkable that this effective pore width corresponds to the pore portion where the potential is lower than a value of the order of 10 3 kT. In other words, it corresponds to the pore volume really accessible to the fluid particles at temperature T with a non negligible probability. Note that the external potential varies very rapidly close to the pore surface. As a consequence, the pore volume definition only requires an order of magnitude for the external potential limit. As can be seen, the gaslike branches are compatible with the ideal gas behavior since the surface tensions are equal to zero within error bars. The effective pore width obtained in the small slit pore is also compatible with the previous value obtained at low temperature, which validates the approach. Another point to be emphasized is that both surface tensions obtained in the two slit pores of different sizes are identical within errors bars, as expected for an interfacial quantity. Note that if the surface to volume ratio is not introduced in Eqs. 12 and 13, then this equality is lost. This gives a strong argument for the introduction of this ratio which was initially introduced based on qualitative arguments. As previously, the quantity

V A α ζ / + Π
is shown for the liquidlike branches as a function of ρ in Fig. 6 (up and down triangles, T* = 1.20), as well as the bulk isotherm. The good agreement validates the approach.

INFLUENCE OF PORE DIAMETER

The previous analysis has been performed for all isotherm branches obtained at T* = 0.77 in the cylindrical pores of various sizes shown in Fig. 3. The results are given in Table 3 and Fig. 7. For pores presenting a hysteresis, the parameter optimization is performed over the whole liquidlike and the whole gaslike branches. For pores presenting reversible curves, the optimization of the liquidlike parameters has been performed for a reduced thermodynamic pressure Π larger than 0.1, and the optimization of the gaslike parameters has been performed for Π below 0.02. As for slit pores, the optimizations along the gaslike branches give surface tensions ζ gaz compatible with zero. As can be seen on Fig. 7 all liquidlike branches can be superimposed to the bulk behavior after the effective pore size and surface contributions have been taken into account. It should also be noted that the effective pore radii obtained along the gaslike branches are equal to those obtained along the liquidlike branches within error bars. These common values are different from the initial pore radius R. They are smaller, but are not equal to R-σ Ar-CO2 . For comparison, the pore radii defined by an external potential equal to 10 3 kT are also given in Table 4. The good agreement with the effective radii αR validates the previous analysis and shows that this effective pore size has physical grounds and corresponds essentially to the region of the pore where the fluid has a non negligible probability to be found. The same analysis has been performed for the high temperature isotherms (T* = 1.20).

As previously for reversible isotherms, it is impossible to perform a good quality fit with one single surface tension parameter. Two sets of parameters are then calculated, corresponding to the low and high density parts of the isotherms. The arbitrary cuts are 0.02 for the thermodynamic pressure upper limit of the gaslike branch, and 0.1 for the lower limit of the liquidlike branches. In the intermediate region, the steep increase of the adsorbed amount versus thermodynamic pressure does not allow the determination of a constant value of the surface tension term. As previously, the effective surface tensions obtained along the low density part of the isotherm are compatible with zero. The results for the liquidlike branches are given in Table 5 and Fig. 8 for a comparison of

V A α ζ / + Π
as a function of ρ with the bulk isotherm. As can be seen, a good agreement is obtained, and the effective pore radii are again very close to the value of the radius determined as the region of the pore where the external potential reaches value of the order of 10 3 kT.

The values obtained for the liquidlike branches are significantly larger in magnitude than those obtained for the gaslike states (compatible with zero). This is expected since fluid / substrate contributions are larger for a dense fluid adsorbed at the surface of a solid. It is also emphasized that the obtained effective surface tensions are negative. This is due to attractive fluid / wall interactions which tend to spread the fluid at the surface.

INFLUENCE OF FLUID SUBSTRATE INTERACTIONS

When considering several pore diameters as previously, both pore size and fluid / wall interactions vary. Figure 1 shows that the external potential is deeper for smaller pores. In order to disentangle these effects, the effective surface tension ζ has been calculated for various values of the fluid / substrate interaction parameter. The results obtained with fluid / wall interactions multiplied by 0.8, 1.0 (initial model) and 1.2 are given in Table 6 and the comparison of

V A α ζ / + Π
as a function of ρ with the bulk isotherm is shown in Fig 9. As can be seen, the geometric parameter αR is not influenced by these small modifications of the external potential amplitude, and the effective surface tension ζ is also constant within error bars. However, it is noted that a trend is observed: ζ slightly increases in magnitude (more negative) with the external potential (deeper). This is expected since ζ is directly related to fluid interfacial interactions.

DISCUSSION

The Figs. 10 and 11 show the effective surface tension of liquidlike branches as a function of the inverse pore radius 1/αR at T* = 0.77 and 1.20. As can be seen, the effective surface tension contributions vary significantly with the pore size. The smaller the pore size, the smaller the magnitude of ζ . It is emphasized that this dependence is not due to deeper fluid / wall external potential in narrower pores, since Table 6 shows that if this was the case the surface tension ζ would be more negative for smaller pores. It is suggested that the observed behavior might be similar to the curvature dependence of liquid gas surface tension (Tolman's law). [START_REF] Tolman | [END_REF] As can be seen, a linear dependence of the surface tension ζ with the inverse pore radius is compatible with the data, except for very small pores. This behavior is to be interpreted in terms of the Gibbs-Tolman-Koenig-Buff equation [START_REF] Tolman | [END_REF][28][29] applied to the surface tension ζ:

( ) r r δ ζ ζ - = ∞ 1 ( 15
)
where ∞ ζ denotes the surface tension for the plane fluid / substrate interface, and δ is the displacement of the surface of zero mass density relative to the surface of tension. Inspection of Figs. 10 and 11 shows that a value of δ = σ Ar-Ar is compatible with the data (δ = σ Ar-Ar gives lines which pass through the upper right corner of the figure). This formula is not expected to be correct for extremely small pores, which explains the observed discrepancy for αR < 2 σ Ar-Ar (i.e. pore diameter less than 1.4 nm)

CONCLUSION

In order to understand confined fluid properties, the thermodynamic pressure or grand potential density is calculated for various adsorbed amounts (corresponding to various chemical potentials) at two temperatures. The difference with the bulk properties for a given chemical potential can be interpreted as an excess interfacial free energy, or surface tension. It pore large enough to present a hysteresis in adsorption. Obviously, the thermodynamic pressure at which desorption occurs is smaller than that for which adsorption occurs.

However, if the pore size is reduced, then the excess thermodynamic pressure increases for the liquidlike part of the isotherm, while it remains essentially unchanged (close to zero) along the gaslike branch. As a consequence, the smaller the pore size, the smaller the difference between the thermodynamic pressures at which adsorption and desorption occur.

For a pore small enough, this difference reaches zero, and both gaslike and liquidlike branches merge to give a single reversible isotherm. This occurs for the so-called critical pore at the considered temperature. This study proposes to interpret this critical pore size as a direct consequence of the curvature effect in the effective fluid interfacial tension ζ for corresponding densities.

  the size Θ of the simulation box is the conjugate parameter associated to the thermodynamic pressure Π, the second kind of trials consists in changing the volume of the simulation box by small angular increments denoted ∆Θ. A new molecular configuration occupying the new volume is deduced from the initial configuration by simple homothetic transformation, as standard procedure in isobaric Monte Carlo simulation. The change in energy is denoted ∆Ε. The acceptance probability for this volume change trial is given by the standard formula where the volume is replaced by ΘR 2 L/2 :[23,24]

10 6

 10 Monte Carlo trials were performed to acquire statistical data. During these simulations, the fluctuating variable is the volume. After averaging, one can introduce the global reduced density Γ of the confined fluid, defined as the total number of particles N divided by the averaged reduced volume V. By convention, this volume is defined as the region of the pore where the potential is not infinite. Note that this definition could be generalized to more complex molecules, taking into account the global molecule / wall interaction energy. It will be emphasized later that another more physical definition could be considered. It is emphasized that this volume definition does not affect the Monte-Carlo acceptance probability for volume changes (Eq 5). The simulations were performed at two reduced temperatures: T* = kT/ε Ar-Ar = 0.77 and 1.20. In both cases, the bulk fluid is subcritical (the Lennard-Jones bulk critical reduced temperature is around 1.3)[25]. The thermodynamic pressure is varied between -0.5 and 10. Negative values are possible in the liquid state for low temperatures due to the existence of cohesive forces. The highest values of the thermodynamic pressure are reached at high density due to the repulsive part of the intermolecular potential.

Figure 2

 2 Figure 2 gives the total amount of fluid adsorbed in the pores as a function of the

Figure 3

 3 Figure 3 gives the adsorption / desorption isotherms for a reduced temperature T* = 0.77 in

  For the effective surface tension, the obtained value is ζ * = -2.3 in reduced units with an uncertainty around 5 to 10 percent. The pore width being large, this value can be considered to give an estimate of the effective surface tension for a flat fluid / substrate interface. The Fig. 6 shows the quantity V A α ζ / + Π as a function of ρ (up triangles, T* = 0.77) and the bulk fluid equation of state (solid line, T* = 0.77). As can be seen, an excellent agreement is obtained over the whole liquidlike branch of the isotherm, which validates the analysis. The optimization of Eq. 13 over the gaslike branch of the same isotherm gives an effective pore radius of 7.4 and an effective surface tension ζ = 0.0 within errors bars. The fact that this effective pore radius is comparable to the previous one (along the liquidlike branch) shows the robustness of the method. The value obtained for the effective surface tension along the gaslike branch was expected to be compatible with zero since, at low coverage, the fluid follows the ideal gas approximation. The next optimization has been performed for the two slit pores of reduced width 8 and 16 at high temperature (T* = 1.20). This situation is very similar to the previous one with two branches (gaslike and liquidlike) giving two different values for ζ and a common value for the effective pore radius for each pore. For the large slit pore (H* = 16), liq ζ = -1.5 , gas ζ = 0.0 and αH* = 15.2; for the small slit pore (H* = 8), liq ζ = -1.4 , gas ζ = 0.0 and αH* = 7.2.

  is shown that, as expected, this quantity depends significantly on the chemical potential and the temperature, which reduces considerably its practical use. However, another quantity ζ (see Eqs 12-13) can be introduced which measures the difference between the confined and bulk thermodynamic pressure for identical densities (different chemical potentials). This quantity relies on the definition of the accessible pore volume for a confined fluid which has implications in the calculation of its density. It is shown that a particular choice of pore radius, corresponding to the region of the pore where the external potential is less than approximately 10 3 kT, allows ζ to be independent of the chemical potential or fluid density, within errors, along each gaslike and liquidlike branch (only the temperature dependence remains). The obtained constants are however not identical for the gaslike and liquidlike branches. For reversible isotherms (supercritical fluid) ζ is constant along the low and high density parts of the isotherms, but varies in the intermediate region where rapid adsorption occurs. This quantity is shown to be close to zero along the gaslike branches, as expected in the ideal gas approximation. For dense confined fluid, ζ is negative for both considered temperatures. Its absolute value is shown to increase with the fluid / substrate interaction strength. It is also shown that the curvature effect in cylindrical pores dominates the effect of external potential depth varying with pore size, since, for decreasing pore size, ζ decreases in absolute value while the external potential depth increases. The curvature effect is shown to follow a linear behavior for mesoscopic pores (> 1nm) analogous to the Gibbs-Tolman-Koenig-Buff equation for liquid / gas interface. This simple dependence is responsible for the appearance of a critical radius for a given temperature. Let us consider a fluid confined in a

FIG. 1 . 5 (FIG. 4 .FIG. 5 .FIG. 6 .FIG. 7 .

 154567 FIG.1. External potential in the cylindrical solid CO 2 pores (see text) of various reduced radii

FIG. 8 .

 8 FIG.8. Same as Fig. 7 for T* = 1.20. For symbols: see Fig 2.

Figure 1 reduced radial distance r * = r /σ σ

 1 Figure 1

  Figure 6

Table 2

 2

	gives the values obtained for different chemical potentials in the cylindrical pore of
	reduced radius 4. The thermodynamic pressure ( ) µ Π	is obtained by combination of the grand
	canonical adsorption isotherms (amount adsorbed versus chemical potential, see Ref [15]) and
	the generalized isothermal-isobaric isotherm (amount adsorbed versus thermodynamic
	pressure, obtained in this work). The relation	p	bulk	( ) µ	is taken from Ref

TABLE 1 :

 1 Lennard-Jones argon / argon and argon / solid CO 2 interaction parameters.

		ε/k (K)	σ (Å)
	argon-argon	119.8	3.405
	argon-CO 2	153.0	3.725

TABLE 2 :

 2 Reduced chemical potential µ*, density Γ*, thermodynamic pressure Π*, and surface tension γ * (defined by Eq. 11) for a Lennard-Jones fluid confined in a cylindrical pore of radius R* = 4, at T* = 0.77, and for three different states. The bulk pressure used inEq. 11 is also given (Ref[25]).

	µ *	Γ*(µ * )	Π*(µ * )	p bulk * (µ * )	γ *(µ * )
	-10.8	0.54	-0.10	-0.61	-1.02
	-10.4	0.57	+0.10	-0.31	-0.82
	-8.9	0.64	+1.0	0.92	-0.16

TABLE 3 :

 3 Reduced effective surface tension liq ζ * and effective pore size αR* = αV*/A* defined by Eq. 13 along the liquidlike portion of the isotherm (T* = 0.77) for argon confined in the cylindrical pores.

	pore radius R *	liq ζ *	α R *
	1.5	-0.90	1.09
	2.0	-0.91	1.59
	2.5	-1.0	2.06
	3.0	-1.3	2.57
	4.0	-1.7	3.58
	8.0	-1.9	7.64

TABLE 4 :

 4 Reduced radii obtained for the six cylindrical pores. Three definitions are proposed, corresponding to the distance at which the external potential is equal to zero, 10 3 kT,

	or tends to +∞.
	Ψ = 0.0	Ψ = 10 3 kT	Ψ = +∞.
	0.73	1.04	1.5
	1.23	1.54	2.0
	1.73	2.04	2.5
	2.23	2.54	3.0
	3.22	3.54	4.0
	7.22	7.54	8.0

TABLE 5 :

 5 Reduced Effective surface tension liq ζ * and effective pore size αR* = αV*/A* defined by Eq. 13 along the liquidlike portion of the isotherm (T* =1.20) for argon confined in the cylindrical pores.

	pore radius R *	liq ζ *	α R *
	1.5	-0.89	1.12
	2.0	-0.98	1.61
	3.0	-1.0	2.60
	4.0	-1.1	3.60
	8.0	-1.3	7.47

TABLE 6 :

 6 Reduced effective surface tension liq ζ * and effective pore size αR* = αV*/A* defined by Eq. 13 along the liquidlike portion of the isotherm (T* = 0.77) for argon confined in the cylindrical pore of radius R * = 2.5, and for various magnification of the fluid / substrate interaction (external potential).

	magnification	liq ζ *	α R *
	0.8	-1.02	2.06
	1.0	-1.04	2.06
	1.2	-1.07	2.06

/ε ε ε ε Ar-Ar 0.01 0.1 1 10 reduced density Γ Γ

10 -0.5 0.0 reduced density Γ Γ

/ε ε ε ε Ar-Ar -0.02 0.00 0.02 0.04 0.06 0.08 reduced density Γ Γ

ACKNOWLEDGEMENTS

Drs. A. Delville, P. Porion, E. Kierlik and G. Tarjus are gratefully acknowledged for fruitful discussions. The simulations were performed on workstations purchased thanks to grants from Région Centre (France), and on IBM SP4 supercomputers at the Centre de Ressources Informatique de Haute Normandie (CRIHAN, St Etienne du Rouvray, France) and the Institut de Développement des Ressources en Informatique Scientifique (IDRIS-CNRS, Orsay, France), which are gratefully acknowledged for the CPU grant n°2005002 and 051153