Energy Effects on the Structure and Thermodynamic Properties of

Mar 30, 2009 - Telephone: +98-311-391-3281., † ... (5) However, in the absence of any long-range attractions between wall−fluid and fluid−fluid ...
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J. Phys. Chem. B 2009, 113, 5493–5499

5493

Energy Effects on the Structure and Thermodynamic Properties of Nanoconfined Fluids (A Density Functional Theory Study) Ezat Keshavarzi*,†,‡ and Mohammad Kamalvand† Department of Chemistry, Isfahan UniVersity of Technology, Isfahan, Iran 8415683111, and Center of Excellence in Nanotechnology in the EnVironment, Isfahan UniVersity of Technology, Isfahan, Iran 8415683111 ReceiVed: September 24, 2008; ReVised Manuscript ReceiVed: January 11, 2009

The structure and properties of fluids confined in nanopores may show a dramatic departure from macroscopic bulk fluids. The main reason for this difference lies in the influence of system walls. In addition to the entropic wall effect, system walls can significantly change the energy of the confined fluid compared to macroscopic bulk fluids. The energy effect of the walls on a nanoconfined fluid appears in two forms. The first effect is the cutting off of the intermolecular interactions by the walls, which appears for example in the integrals for calculation of the thermodynamic properties. The second wall effect involves the wall-molecule interactions. In such confined fluids, the introduction of wall forces and the competition between fluid-wall and fluid-fluid forces could lead to interesting thermodynamic properties, including new kinds of phase transitions not observed in the macroscopic fluid systems. In this article, we use the perturbative fundamental measure density functional theory to study energy effects on the structure and properties of a hard core two-Yukawa fluid confined in a nanoslit. Our results show the changes undergone by the structure and phase transition of the nanoconfined fluids as a result of energy effects. 1. Introduction Nowadays, with salient progress witnessed in nanotechnology and the significant importance of nanoscopic systems, understanding of the behaviors of such systems has found exceptional importance. In nanoscopic systems, fluids confined in nanoscopic pores have been the subject of much research in statistical thermodynamics.1-10 When a fluid is confined inside a nanoscopic pore with molecular dimensions, its behavior becomes dramatically different from that of the macroscopic fluid.4 If we postulate that decreasing the dimension of a system does not change the nature of intermolecular interactions, decreasing the number of confined molecules in a nanopore should be the first reason for the discrepancy observed between the properties of confined and unconfined fluids. In such systems, the small number of molecules cause increased fluctuations in the thermodynamic properties of confined fluids.11 But, for a number of confined fluids in which the confinement happens only in one or two dimensions and whose number of molecules tends toward the macroscopic limit, thermodynamic properties also show a significant difference from the macroscopic fluid.4 From among the various crude models available for studying confined fluids with a macroscopic number of molecules, nanoslit is one of the most well-known. A nanoslit consists of two parallel plates whose areas tend toward the macroscopic limit, with the plate distancing decreasing to a few molecular diameters. Consequently, the confinement of the fluid within the nanoslit occurs only in one dimension.10 Since the number of molecules confined in these systems tends toward the macroscopic limit, fluctuations in their thermodynamic properties are negligible. * To whom correspondence should be addressed. E-mail: keshavrz@ cc.iut.ac.ir. Fax: +98-311-391-2350. Telephone: +98-311-391-3281. † Department of Chemistry, Isfahan University of Technology. ‡ Center of Excellence in Nanotechnology in the Environment, Isfahan University of Technology.

Therefore, the difference between its thermodynamic properties and those of the macroscopic fluid must be due to other factors. To understand the differences between the properties of confined and macroscopic fluids, the study of these properties can be useful when the size of the confined dimension in the nanoscopic system tends toward the macroscopic limit. When the confined dimension of a nanopore increases, its thermodynamic properties tend to rise to macroscopic values.10 While the effects of system walls are ignored in the calculation of thermodynamic properties of macroscopic fluids, these effects can be the main reason for the differences observed between the two systems. System walls may have three effects on the thermodynamic properties of a fluid. The first is due to the wall-molecule interactions.12 Introduction of these interactions, which can be ignored in macroscopic systems compared to intermolecular interactions, may give rise to new properties such as wetting or new kinds of phase transitions. In addition to this direct wall effect, the range of intermolecular interactions will be greater than the confined dimensions of the pore when the distance between nanopore walls tends to a few molecular diameters. Therefore, part of the intermolecular interactions, which are usually long-range attractive forces, will remain unaccounted for in the calculation of thermodynamic properties. The two effects just mentioned have effects on the energy of the system, hence their designation “energy effects”.5 However, in the absence of any long-range attractions between wall-fluid and fluid-fluid molecules in a confined system, its thermodynamic properties show a significant difference with those of the macroscopic fluid. An example would be the differences observed between a hard sphere fluid confined between two parallel hard walls and a macroscopic hard sphere fluid.10 In the absence of any long-range forces and for specific numbers of molecules, the tendency of the system to maximize its entropy leads to these changes in the behavior of the system,13 which can be designated as “entropy effects”.5,14

10.1021/jp808466p CCC: $40.75  2009 American Chemical Society Published on Web 03/30/2009

5494 J. Phys. Chem. B, Vol. 113, No. 16, 2009

Keshavarzi and Kamalvand

We may conclude that the presence of walls in nanopores is the only reason for changing the behavior of a confined fluid with a macroscopic number of molecules. Although these wall effects in confined fluids are universally recognized; the pure energy effects have not been reported in the literature. “Entropy effects” in confined fluids are explained in detail elsewhere,14 but the main goal of the present work is to study the “energy effects” on the structure and thermodynamic properties of a confined fluid. The difficulty, or in cases the impossibility, of the experimental study of confined fluids has made application of theoretical methods a more appropriate and practical solution for studying their behavior.15 In addition to molecular dynamics simulation (MD), which usually yields reliable data, density functional theory (DFT) is also a rather powerful tool for studying confined fluids,16 basically for hard spheres. More accurate investigations of the structure and thermodynamic properties of more realistic fluids have become possible by perturbation theory.1 Fluids whose intermolecular interactions obey the hard core multi-Yukawa potential are important models in this respect. While the direct correlation function is available for hard sphere fluids with the Yukawa tail, the perturbative DFT can be used in their study.1,2 For a hard core two-Yukawa potential, the repulsive part of the potential is a hard sphere fluid. Application of DFT for this fluid is very convenient, yielding accurate results.1,2 Along these lines, the perturbative density functional theory is used in the present paper to study the structure and properties of a hard core two-Yukawa fluid confined in nanoslits with different sizes and wall-fluid potentials. The tail of these two-Yukawa potentials is selected such that it is equivalent to the Lennard-Jones (LJ) attractive tail.17 The rest of this paper is organized as follows: In section 2, the perturbative density functional theory will be briefly described. In section 3, the effect of wall-molecule interactions on the structure and properties of a hard sphere fluid will be studied, and the effect of the unaccounted for portion of the intermolecular interactions in a confined fluid will be investigated in section 4. Section 5 will be dedicated to a discussion of the energy effects on the phase transition of a confined twoYukawa fluid. Finally, conclusions will be presented in section 6. 2. Perturbative Density Functional Theory In the density functional theory, the grand potential, Ω, of an inhomogeneous system relates to its Helmholtz free energy, F, via a Legendre transformation,1

Ω[F(r)] ) F[F(r)] +

∫ dr F(r)[Vext(r) - µ]

(1)

where F(r) is the one body distribution function of the system, Vext(r) is the external field, and µ is the chemical potential. In various versions of the DFT, the Helmholtz free energy of the system splits into two ideal and excess parts:1

F[F(r)] ) Fid[F(r)] + Fex[F(r)]

(2)

∫ dr F(r){ln Λ3F(r) - 1}

Fex[F(r)] ) kBT

∫ dr Φhs[nR(r)] + F attex[F(r)]

(3)

(4)

where Φhs(r) is the excess Helmholtz free energy density of a hard sphere fluid and nR(r) is a weighted density, defined as19

nR(r) )

∫ dr′ F(r′) w(R)(r - r′)

(5)

where R ) 0, 1, 2, 3, V1, and V2, the first four weights of which are scalar and the last two of which are vector weights defined as follows:20

w(2)(r) ) 2πσw(1)(r) ) πσ2w(0)(r) ) δ(σ/2 - r)

(6)

w(3)(r) ) θ(σ/2 - r)

(7)

w(V2)(r) ) 2πσw(V1)(r) )

r δ(σ/2 - r) |r|

(8)

where σ is the hard sphere molecule diameter, δ(r) is the Dirac delta function, θ(r) is the Heaviside step function, and |r| ) r represents the distance between the centers of two molecules. There are a number of reports on the excess Helmholtz free energy density, Φhs, indicating that one of the most successful versions of the theory is the modified fundamental measure density functional theory (M-FMDFT).20,21 This is based on the accurate equation of state of MCSL for pure and mixture hard sphere fluids.24 Similar to the original version,19 Φhs in MFMDFT contains two scalar and vector parts,20,21

Φhs[nR(r)] ) Φhs(S)[nR(r)] + Φhs(V)[nR(r)]

(9)

where the superscripts S and V indicate scalar and vector parts, respectively. According to M-FMDFT, these two parts are given by20,21

n1n2 + 1 - n3 1 1 ln(1 - n3)+ n 3 (10) 2 2 2 36πn3 36πn3(1 - n3)

Φhs(S)[nR(r)] ) -n0 ln(1 - n3) +

where the ideal part is exactly given by12

Fid[F(r)] ) kBT

in which kB is Boltzmann’s constant, T is absolute temperature, and Λ ) h/(2πmkT)1/2 is the thermal de-Broglie wavelength, where h is the universal Plank constant and m is the particle mass. The excess part of the Helmholtz free energy includes both repulsive and attractive contributions.1 According to the perturbation theory, intermolecular interactions in a fluid can be divided into a reference and a perturbation part. Usually, the reference part is assumed to be a hard sphere potential with an effective diameter that depends on the fluid temperature and density. The remaining long-range Van-der-Waals attraction part of the potential energy is taken to be the perturbation contribution. One of the most successful approaches commonly used to obtain the Helmholtz free energy of a fluid that obeys hard core repulsive plus long-range attractive potentials is the Rosenfeld perturbative fundamental measure theory.18 According to this approach,

[

]

Energy Effects on Nanoconfined Fluids

Φhs(V)[nR(r)] ) -

J. Phys. Chem. B, Vol. 113, No. 16, 2009 5495

[

nV1nV2 1 ln(1 - n3) + 1 - n3 12πn32 1 n2(nV2nV2) (11) 12πn3(1 - n3)2

]

According to the perturbative method of Rosenfeld, the contribution of the attraction potential can be expressed as1

ex F att [F(r)]

)

ex δFatt + dr ∆F(r) + δF(r) ex δ2F att dr dr′ ∆F(r) ∆F(r′) + ... (12) δF(r) δF(r′)

∫

ex F att (Fb)

1 2

∫

which is obtained from Tailor expansion of the Helmholtz free energy of the inhomogeneous system about the homogeneous fluid. In this equation, Fb is the homogeneous bulk fluid density and ∆F(r) ) F(r) - Fb. On the other hand, the attractive part of the direct correlation function (DCF) of the homogeneous bulk fluid is given by22

(2),b ∆Catt (|r

- r′|) ) -β

ex δ2Fatt

δF(r) δF(r′)

achieved when r ) σ. According to FMSA results, the attractive part of the DCF for a hard core multi-Yukawa potential is23 (2),b (2),b ∆Catt (|r - r′|) ) CMY (|r - r′|) - C(2),b hs (|r - r′|) (17)

in which the first term represents the DCF of a multi-Yukawa fluid and the second part is the DCF of its reference hard sphere fluid. Therefore, the attractive part of a multi-Yukawa DCF will be23

(2),b ∆Catt (r)

)

{

-λi(r-σ)/σ

M

∑ βεi e i)1 M

r/σ

∑ Y(r, εi, λi)

r>σ ;

r ≡ |r - r′|

reσ

i)1

(18)

where

[

Y(r, εi, λi) ) βεi

(13)

]

e-λi(r-σ)/σ - Q(λi) P(r, λi) r/σ

Q(t) ) [S(t) + 12ηL(t)e-t]-2

and in this way, the attractive part of the excess chemical potential of the system is22

(19)

(20)

S(t) ) ∆2t3 + 6η∆t2 + 18η2t - 12η(1 + 2η) (21) (1),b ∆Catt ) -β

ex δFatt ex ) -βµatt δF(r)

(14)

where β ) 1/kT. If the contributions of derivatives of F with a higher order than 2 are ignored, the attractive part of the excess Helmholtz free energy will be18 ex ex ex F att [F(r)] ) F att (Fb) + µatt

1 kT 2

∫ dr ∆F(r) -

∫ dr dr′ ∆Catt(2),b(|r - r′|) ∆F(r) ∆F(r′) + ... (15)

which is totally independent of any weight functions. According to this equation, if a reliable relation could be developed for the direct correlation function, this perturbation version of the DFT could then be easily employed to study inhomogeneous fluids. An analytical equation for the DCF of a hard core multiYukawa fluid has been recently reported based on the first order mean spherical approximation (FMSA).23 A hard core multiYukawa potential is expressed as1

U(r) )

{

L(t) ) (1 + η/2)t + 1 + 2η

(22)

e-t(r-σ)/σ et(r-σ)/σ + 144η2L2(t) r/σ r/σ 2 2 4 5 3 3 12η [(1 + 2η) t + ∆(1 + 2η)t ]r /σ + 12η[S(t) L(t)t2 ∆2(1 + η/2)t6]r/σ - 24η[(1 + 2η)2t4 - ∆(1 + 2η)t5] + 24ηS(t) L(t)σ/r (23)

P(r, t) ) S2(t)

where, η ) π/6Fbσ3 is the packing fraction of the fluid, ∆ ) 1 - η, and t ) λi. Finally, the one body distribution function of the inhomogeneous fluid can be obtained via minimization of the grand potential.1

[

F(r) ) Fb exp β µex -

]

hs

w(R)(r - r′) ∫ dr′ ∑ R ∂Φ ∂nR

Vext(r) +

∫ dr′ ∆Catt(2),b(|r - r′|)∆F(r′) (24)

where

∞

r σ/2 w w Figure 1 shows the M-FMDFT results for the density profile of a hard sphere fluid with a reduced bulk fluid density of Fσ3 ) 0.6 and near a structureless wall with different wall-molecule attractions, εw. The screening length, λw, for all cases is taken to be 1.8. According to this figure, the tendency of the molecules to accumulate near the wall increases with increasing wallmolecule attractions, εw. In addition, the amplitude of the density profile oscillations increases with increased attraction between the wall and the molecules. This behavior shows that the layering structure of a fluid near the wall will be amplified by an increase in wall-molecule attractions. In addition to εw, the screening length, λw, may affect the thermodynamic properties of a confined fluid. The effect of the screening length on the structure of a fluid near a structureless

wall is presented in Figure 2. This figure shows the density profile of a hard sphere fluid near a structureless wall with different wall-molecule potential screening lengths. The density of the mentioned fluid is Fσ3 ) 0.6 and εw/kT ) 1.0 for all cases. These structures are compared with the density profile of a hard sphere fluid against a structureless hard wall (or a wall-molecule interaction with εw/kT ) 0.0 in eq 26). It is clearly seen that increasing λw causes the value of the density profile to enhance at the contact point. According to eq 26, the density profile will be affected more greatly at higher values of λw in the range very close to the wall, since increased λw values lead to corresponding decreases in the wall-molecule interaction range. The enhanced contact value of the density profile due to increasing values of λw is a result of this fact. Evidently, a change in the first layer causes a corresponding change in the next layers, too. On the other hand, the next layers may also be affected directly as the range of wall-molecule interactions increases. As seen in Figure 2, the height of the second peak of the density profile will increase with decreasing λw or, equivalently, with increasing wall-molecule interaction range. It should be pointed out that, for the case of the nanoslit, variations in λw with a constant pore size are similar to variations in the confined dimension when λw is fixed. Therefore, we will not consider the role of λw variations in nanoslits in the rest of this paper. The wall-molecule interaction not only affects the fine structure of a fluid near the wall but can also change the thermodynamic properties of confined fluids. However, a similar role is expected for the effect of these interactions on the structure and thermodynamic properties of a fluid confined between two parallel plates. One of these thermodynamic properties is excess adsorption per unit area, which is defined as1

Γ)

∫0H [F(z) - Fb] dz

(27)

where H is the distance between two parallel plates. In Figure 3, the excess adsorption of a confined hard sphere fluid with a reduced bulk density of Fσ3 ) 0.6 is shown for various slits with different wall-molecule attractions. A screening length, λw, of 1.8 is assumed for all systems. As shown in this figure, increased attraction between the walls and the confined fluid

Energy Effects on Nanoconfined Fluids

Figure 3. Excess adsorption of a hard sphere fluid confined between two structureless plates at a reduced bulk density of Fσ3 ) 0.6. The wall-molecule interaction is a hard core with an attractive tail with λw ) 1.8 and different values of εw/kT.

J. Phys. Chem. B, Vol. 113, No. 16, 2009 5497

Figure 4. Density profile of a hard core two-Yukawa (equivalent to the LJ attractive tail) fluid against a structureless hard wall at a density of Fσ3 ) 0.6 and with different values of ε. For all potentials, λ1 ) 2.9637 and λ2 ) 14.0167. These structures are compared with a hard sphere fluid density profile with Fσ3 ) 0.6.

molecules leads to enhancement of this quantity in all slits. When the attraction of wall-fluid molecules increases by a considerable degree, the excess adsorption may select a positive value, which means that the average density of the pore is greater than the bulk density. This situation indicates the very high affinity of the nanopore for attracting the molecules. One may see in Figure 3 that all the excess adsorptions tend toward zero when the confined dimension, H, increases. This means that the effect of wall-molecule interactions will diminish at macroscopic limits. In fact, increasing the wall-fluid attractions in a nanopore leads to a decreased local chemical potential of the confined system. Therefore, the tendency of the molecules to enter the pore will increase. Consequently, the average density of the molecules inside the pore will increase correspondingly. 4. Effect of Unaccounted for Interactions Perhaps one of the best ways to evaluate the effect of the unaccounted for portion of the interactions is to compare the structures of the confined and unconfined fluids. In addition to the confined fluids, unaccounted for interactions occur in molecules accumulating near the walls of a macroscopic system. In such systems, the energy and entropy wall effects cause the density profile of the fluid near the wall to vary in comparison with the case of the bulk fluid. However, in the case of a structureless hard wall, while the entropy effect plays a significant role in the inhomogeneity of the structure of the fluid near a wall,14 its role should be distinguished from that of the unaccounted for portion of the interactions. For example, the structure of a hard sphere fluid near a hard wall is determined purely by the entropy effect.13,14 In order to ignore the entropy effect because of the wall for the systems with attractions, the structure of a hard sphere fluid may be compared with that of a hard core two-Yukawa fluid whose hard core diameter is equal to the hard sphere molecule diameter. This is because the entropy effects caused by the walls (which are due to the excluded volume near the walls) must be equal for spheres with identical hard core diameters. Moreover, to avoid conflicts with the effect of wall-molecule interactions, the wall-molecule potential is assumed to be hard without any long-range interactions. Figure 4 shows the perturbative M-FMDFT results for the structure of a hard core two-Yukawa fluid at a reduced bulk density of Fσ3 ) 0.6 near a hard wall. These structures

Figure 5. Excess adsorption of a hard core two-Yukawa fluid (equivalent to the LJ attractive tail) confined between two structureless hard plates at a reduced bulk density of Fσ3 ) 0.6, λ1 ) 2.9637, and λ2 ) 14.0167, with different intermolecular attractions, ε. These are compared with the excess adsorptions of a confined hard sphere fluid with Fσ3 ) 0.6.

have been calculated for fluids with different molecule-molecule attractions. The selected potential parameters are λ1 ) 2.9637, λ2 ) 14.0167, and ε1 ) -ε2 ) ε, which correspond to those of the tail of a Lennard-Jones potential with a diameter of σ and a potential wall depth of ε.17 It is obvious from this figure that the contact point of the density profile for all two-Yukawa fluids, with any molecule-molecule attraction, is lower than the contact point for the hard sphere fluid. In like manner, the tendency of molecules to accumulate near the walls increases with decreasing intermolecular attraction. Also, decreasing intermolecular attraction leads to amplification of the layering structure of molecules against the wall. A similar behavior can be expected for the structure of this fluid confined within a nanoslit with structureless hard walls. The perturbative M-FMDFT results are presented in Figure 5 for excess adsorption of the mentioned fluids confined between two structureless plates at different distances. According to these results, reduced intermolecular attractions cause increased numbers of molecules accumulated inside the nanoslit. To explain the behavior of the mentioned confined fluid, we need

5498 J. Phys. Chem. B, Vol. 113, No. 16, 2009 to note that, in the DFT calculations, the confined fluid is in chemical equilibrium with a macroscopic bulk fluid. An entropy effect always causes the molecules to aggregate near the walls.14 Also, according to our results reported in the previous section, wall-molecule attraction leads to more molecules entering the nanoslit while molecule-molecule attractions essentially play an opposite role. In a bulk fluid, each molecule is surrounded symmetrically by a number of similar molecules. Therefore, this molecule has an equal tendency to move to all directions. In other words, this tendency is totally isotropic. However, the situation is different for a molecule located near the system wall. A molecule placed near the wall cannot be surrounded symmetrically by bulk molecules. In this case, the number of molecules in the bulk side of the original molecule will be greater than that of molecules located on its wall side. Therefore, due to molecule-molecule attraction (which is at all times greater than the hard wall-molecule attraction), the molecule will be attracted to the bulk region located far from the wall. This factor leads to a decreased density profile at the contact point. Another finding is that the layering structure of the fluid will be thinned via this energy effect. As shown in Figure 5, decreasing the tendency of molecules to enter the nanoslit is another result of this energy effect. In this figure, the excess adsorption for a hard core two-Yukawa fluid at a reduced bulk density of Fσ3 ) 0.6 confined in a nanoslit with structureless hard walls is reported for different intermolecular attractions. In all the interatomic potentials investigated, we assumed λ1 ) 2.9637 and λ2 ) 14.0167. According to this figure, increased intermolecular attraction causes the excess adsorption to decrease. In other words, when attraction between fluid molecules decreases, the entropy effect dominates over the effect of the unaccounted for portion of attractions in the absence of wall-molecule attractions. Consequently, in the case of zero attractions (i.e., a hard sphere fluid), excess adsorption will have its maximum value. From a different viewpoint, an increased chemical potential, µ, of the fluid in the reservoir will usually lead to an increase in the number of fluid molecules adsorbed.26 This is while attractive interactions usually cause a decrease in the chemical potential of the fluid. Therefore, the chemical potential difference between the unoccupied pore and the bulk fluid will decrease, which in turn decreases the tendency of the molecules to penetrate the nanoslit. This is the main reason for variations in the structure of the confined fluid with increasing intermolecular attractions. 5. Energy Effects on the Phase Transition in Confined Fluids In addition to the structure of the confined fluids, their phase transitions are significantly affected by energy effects. To examine the influence of the energy effects on the phase transition, we studied the liquid-gas phase transition of a twoYukawa fluid confined between two structureless walls. Similar to what was said earlier, the selected potential parameters included λ1 ) 2.9637, λ2 ) 14.0167, and ε1 ) -ε2 ) ε. The selected bulk fluid has Fσ3 ) 0.6. It should be pointed out that gas or liquid terms may not be appropriate for the description of confined fluids. However, following the conventions and using the confined fluid density, we identified the phase with a higher density as the liquidlike one and the one with a low density phase as the gaslike phase. To investigate the effect of the unaccounted for interactions on the phase transition in confined fluids, we examined the adsorption of the mentioned

Keshavarzi and Kamalvand

Figure 6. Excess adsorption of a hard core two-Yukawa (equivalent to the LJ attractive tail) fluid confined between two structureless hard plates at a reduced bulk density of Fσ3 ) 0.6, λ1 ) 2.9637, and λ2 ) 14.0167, with different values of intermolecular attraction. Clear breakage of the excess adsorption at ε/kT ) 0.91 indicates a first order phase transition. The confined fluid is in the liquidlike state at ε/kT ) 0.5 and in the gaslike state at ε/kT ) 1.0 in the slits with H < 14σ, which is not shown here for the sake of visual clarity.

two-Yukawa fluid confined between two structureless hard walls. In Figure 6, the M-FMDFT results are presented for adsorptions of this two-Yukawa fluid with three different values of molecule-molecule attractions, ε/k. It is seen that for large values of attractions (e.g., ε/kT ) 1.0), a gaslike phase exists inside the pore and desorption of the fluid happens. However, our results for ε/kT ) 1 show that the gaslike phase is the only possible phase for slits with H < 14σ. For small attraction values (ε/kT ) 0.5), on the contrary, the chemical potential of the bulk fluid increases and molecules adsorb inside the pore and a liquidlike phase will be created within all slits with different values of H. At intermediate attraction values (e.g., ε/kT ) 0.91), the size of the pore has a determining role to play in adsorption and desorption of the fluid inside the pore. In such cases, when the confined size of the nanoslit, H, is so small, the unaccounted for portion of attractions becomes considerably large and confined molecules fail to accumulate around each other within the slit, giving rise to a gaslike phase inside the pore. Upon enhancement of H, the contribution of the unaccounted for attractions decreases and confined molecules create a liquidlike phase inside the pore. For this fluid, a liquid-gas phase transition occurs in a nanoslit with H ≈ 5.1σ. In addition, Figure 7 depicts the solvation force for the confined systems mentioned above. The solvation force for a confined fluid is defined as follows:26

f)-

1 ∂Ω A ∂H

( )

µ,T,A

-p)-

∫0H

∂Vext F(z) dz - p (28) ∂H

where f is the solvation force, T is the absolute temperature, A is the area of one of the parallel plates, p is the pressure of the confined fluid at the macroscopic limit, and µ is the chemical potential. Similar to adsorption, solvation force shows a breakage at H ≈ 5.1σ when ε/kT ) 0.91. This behavior clearly indicates a first order phase transition. In this figure, the oscillatory feature of the solvation force for the fluid with ε/kT ) 0.5 is very similar to the hard sphere fluid solvation force, which indicates that, at small attractions, the entropy effect dominates over the energy effect of the unaccounted for interactions.

Energy Effects on Nanoconfined Fluids

J. Phys. Chem. B, Vol. 113, No. 16, 2009 5499 6. Conclusions

Figure 7. Solvation force for a number of systems mentioned in Figure 6. Similar to the excess adsorption, clear breakage of the solvation force at ε/kT ) 0.91 indicates a first order phase transition.

The effects of the wall-fluid and unaccounted for portion of intermolecular interactions on the structure of a fluid in contact with a wall or confined between two structureless walls were examined using the modified fundamental measure density functional theory. Our results confirmed the idea that energy effects may significantly affect the structure and phase behavior of the confined fluids. Our M-FMDFT results indicate that decreasing intermolecular attractions lead to increasing values of the contact point in the density profile of a two-Yukawa fluid near a hard wall. Likewise, increased wall-molecule attractions play a similar role. Both these two factors cause the layering structure of the confined fluid to amplify. In addition, decreased intermolecular attractions lead to the accumulation of more molecules inside the pore. Therefore, at adequately high intermolecular attractions, a gaslike phase is the only phase present inside the pore. Conversely, at lower intermolecular attractions, a liquidlike phase may be created within all pores with any confined dimension, H. However, in addition to these attractions, wall-molecule attractions may give rise to a liquid-gas phase transition inside a pore or cause the phase transitions to shift. According to our results, the normal liquid-gas phase changes in nanoslits will shift to smaller values of H, since the tendency of molecules to accumulate inside the pore increases due to the wall-fluid molecules attractions. Acknowledgment. The authors acknowledge the financial support by the Isfahan University of Technology Research Council and the Iranian Nanotechnology Initiative. References and Notes

Figure 8. Excess adsorption of a hard core two-Yukawa (equivalent to the LJ attractive tail) fluid confined between two structureless hard plates at a reduced bulk density of Fσ3 ) 0.6 and the reduced temperature T* ) 1.0. Increased wall-molecule attractions give rise to the liquid-gas phase transition at lower values of H.

The only issue remaining to be addressed is the effect of wall-molecule interactions on the phase transition of the confined fluids. Since wall-molecule attractions cause accumulation of more molecules inside the pore, these interactions can compensate for part of the unaccounted for interactions. It is, therefore, expected that wall-molecule attractions lead to the gas-liquid phase transition happening at higher values of intermolecular attractions. In fact, there is a competition between the bulk intermolecular attractions (outside the pore) and wall-molecule attractions (inside the pore) to transfer the molecules between these two regions. In Figure 8, the excess adsorption is shown for a two-Yukawa fluid with λ1 ) 2.9637 and λ2 ) 14.0167 at a reduced bulk density of Fσ3 ) 0.6 and a reduced temperature T* ) kT/ε ) 1.0 for different wallmolecule attractions. Clearly, when the walls attract more molecules, the phase change occurs at lower values of H. This means that wall-molecule attractions strongly affect the positions of phase transitions. In conclusion, the competition among the unaccounted for interactions, wall-molecule interactions, and entropy effects may considerably change the phase diagram of a confined fluid.

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