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Adsorption Isoterms and Capillary Condensation in a Nanoslit with Rough Walls: A Density Functional Theory Gersh O. Berim Department of Physics, Canisius College, Buﬀalo, New York 14208, United States

Eli Ruckenstein* Department of Chemical and Biological Engineering, State University of New York at Buﬀalo, Buﬀalo, New York 14260, United States ABSTRACT: Adsorption isoterms and capillary condensation in an open slit with walls decorated with arrays of pillars are examined using the density functional theory. Compared with the main substrate, the pillars can have the same or diﬀerent parameters in the Lennard-Jones interaction potential between them and the ﬂuid in the slit. The roughness of the solid surface, deﬁned as the ratio between the area of the actual surface and the area of the surface free of pillars, is controlled by the height of the pillars. It is shown that the capillary condensation pressure ﬁrst increases with increasing roughness, passes through a maximum, and then decreases. The amount of adsorbed ﬂuid at constant volume of the slit has, in general, a nonmonotonic dependence on roughness. These features of adsorption and capillary condensation are results of increased surface area and changes in the ﬂuid−solid potential energy due to changes in roughness.

1. INTRODUCTION Experimental and theoretical studies have shown that the properties of a ﬂuid are aﬀected by changes in spatial conﬁnements. Such changes aﬀect the phase behavior of ﬂuids resulting in phenomena such as capillary condensation, capillary freezing, and layering transition. The behavior of conﬁned ﬂuids is well understood for one-component ﬂuids as well as binary mixtures and simple models of the pores (long slits, spherical or cylindrical pore shapes, etc.) with smooth uniform walls. However, the surfaces of real pores usually possess either physical roughness (grooves, asperities, etc.) or chemical roughness (nonuniform chemical composition of the substrate), being geometrically or chemically heterogeneous. To estimate the role of such heterogeneities, several recent studies employed the quenched−annealed model of the ﬂuid−solid system and developed a suitable version of the density functional theory (DFT).1−3 One of the main features of that approach was the modeling of roughness as a layer (matrix) of quenched molecules on the solid surface, with nonuniform density along the normal to the solid surface. In lateral directions the layer was considered uniform. The adsorbate molecules were allowed to penetrate into this layer.3 The density distribution of the matrix molecules across the layer was selected in a simple form and used as an input into the theory.1−3 A diﬀerent procedure was used in ref 4 where the density distribution of the surface layer was calculated by minimizing the free energy of the system using the density © 2012 American Chemical Society

functional theory. The approach used in refs 1−4 allowed reduction of the Euler−Lagrange equation for the ﬂuid density distribution to a one-dimensional form, thus providing a major simpliﬁcation of the calculations. Among the results obtained in refs 1−3, one can mention the damping of the oscillations of the adsorbate density which occurs in the absence of the layer, the change in the capillary condensation pressure, and the shift in the hysteresis loops of the adsorption isoterms. The capillary condensation pressure increases (decreases) and the hysteresis loops are shifted to higher (lower) pressures when the adsorbate−matrix interaction is weaker (stronger) than the adsorbate−substrate one. The increase in capillary condensation pressure can become so large that the capillary condensation can be replaced by capillary evaporation.2 The latter phenomenon occurs typically when the ﬂuid is conﬁned between hydrophobic walls.5 One of the main features of the model developed in refs 1−4 is that the surface area of the slit does not depend on the properties of the surface layer and remains equal to that of the smooth surface. Obviously, this is not true for physically rough surfaces for which the surface area increases with increasing roughness, the latter being deﬁned as the ratio between the actual surface area and the area of the smooth surface. Received: April 9, 2012 Revised: June 26, 2012 Published: June 27, 2012 11384

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Nonetheless, the progress in modern nanotechnologies allows one to fabricate physically rough surfaces with prescribed structures, for instance, surfaces decorated with regular arrays of asperities of various shapes, sizes, and compositions.6−11 It is well-known that such modiﬁcations of the surface lead to considerable changes in its wetting properties. In particular, if the ﬂuid penetrates between macroscopic asperities (the socalled Wenzel regime), the contact angle θW, which a macroscopic liquid drop makes with a rough surface, is provided by the Wenzel equation12 cos θ W = r cos θ

(1)

where r is the roughness and θ is the contact angle on a smooth surface. Consequently, a hydrophobic surface (θ > 90°) becomes more hydrophobic and a hydrophilic one (θ < 90°) more hydrophilic with increasing roughness. For nanosize asperities, the hydrophilicity of a hydrophilic surface decreases with increasing roughness and the surface can become even hydrophobic.13,14 Along with changes in wetting, one also expects roughness to aﬀect adsorption. Because the approaches developed in refs 1−4 are not applicable to the systems possessing the above geometrical roughness, in the present paper we examine adsorption and capillary condensation in a planar slit with physically rough surfaces using the traditional DFT, by assuming that the adsorbate does not penetrate into the substrate. The roles of the size of the pillars, which provide the roughness, as well as the roles of the strength of the pillar−ﬂuid interactions are examined in some detail. Because the potential generated by a real heterogeneous solid surface is a complicated function of three spacial coordinates, the numerical solution of the Euler− Lagrange equation providing the ﬂuid density distribution in the system is practically impossible. To simplify the problem, a two-dimensional geometry of the system will be employed. The Euler−Lagrange equation in this case involves two independent spacial variables and is solved by iteration using the procedure developed in ref 14. It is worth noting that the two-dimensional versions of DFT were applyed previously (see refs 15−18) to examine the ﬂuids conﬁned between smooth solid surfaces.

Figure 1. Schematic representation of the considered system. σfs is the hard core diameter of the ﬂuid−solid interactions. All distances between surfaces are measured between the centers of the molecules forming the ﬁrst layers.

constant. The latter volume can be controlled by changing the distance Lh between the walls W1 and W2. For the smooth slit walls (no pillars), Lh was selected 10σff + 2σfs. Hence, when hp is changed, the distance Lh has to be changed according to the formula Lh = 10σff + 2σfs + hp. Note that pillars with height hp > 10σff obstruct the ﬂuid passage in the x-direction because the distance between the pillars located in front of each other on opposite solid walls becomes too small to allow the ﬂuid molecules to move between them. For this reason, only pillars with heights hp < 10σff are considered below. The ﬂuid density distribution (FDD) ρ(r) in this system is considered uniform in the y-direction and nonuniform in the xand h-directions; hence, ρ(r) ≡ ρ(x, h). Because of the regular geometrical location of identical pillars, the rough surface generates a potential which is periodic in the x-direction. If there is no symmetry breaking in the x-direction, the FDD ρ(x, h) has the period Lx = Δp + dp and can be calculated in the ﬁnite range −Lx/2 ≤ x ≤ Lx/2. This reduces considerably the computational time. The ﬂuid is exposed to an external potential due to the ﬂuid−substrate and ﬂuid−pillars interactions. The temperature was ﬁxed at T = 87.3 K. The interaction potentials between the spherical molecules of ﬂuid and between the molecules of ﬂuid and those of the solid are selected in the Lennard-Jones form with hard core repulsion

2. THEORY 2.1. The System. The considered system involves a onecomponent ﬂuid (argon) conﬁned between two semiinﬁnite uniform solid walls W1 and W2 decorated with regular arrays of pillars of height hp, width dp, and distance between pillars Δp (see Figure 1). The pillars are inﬁnite in the y-direction (perpendicular to the plane of the ﬁgure) and are composed of a material which can diﬀer from the material of the substrate. The system is in contact with a reservoir of ﬂuid (argon) at the chemical potential μ < μ0 (at pressure P < P0) where μ0 and P0 are the chemical potential and the pressure at the saturation point of the bulk ﬂuid, respectively. The slit has inﬁnite dimensions in the x and y directions, while the distance Lh between walls is ﬁnite. The origin of the coordinate system in Figure 1 is located at a distance from the solid substrate equal to one hard core diameter σfs of the ﬂuid− solid interaction. All distances between the surfaces of substrate and pillars are measured between the centers of the molecules forming the ﬁrst layers of the corresponding surfaces. All pillars have the same width dp = 2.410σff, where σff is the hard core diameter of the ﬂuid−ﬂuid interactions, and the distance between them Δp = dp. In what follows, the volume of the system accessible to the ﬂuid molecules will be considered

⎧ ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ ⎪ ⎪ 4εα⎢⎜ α ⎟ − ⎜ α ⎟ ⎥ , Δr ≥ σα ⎝ Δr ⎠ ⎦ ϕα(|r − r′|) = ⎨ ⎣⎝ Δr ⎠ ⎪ ⎪ ∞, Δr < σα ⎩

(2)

where the subscript α is “ﬀ”, “fs”, and “fp” for ﬂuid−ﬂuid, ﬂuid− substrate, and ﬂuid−pillars interactions, respectively, εff, εfs, and εfp are energy parameters, σff, σfs, and σfp are hard core diameters, r and r′ provide the locations of the interacting molecules, and Δr = |r − r′| . For argon, εff/kB = 119.76 K, and σff = 3.405 Å, where kB is the Boltzmann constant. For the substrate, the cases εfs = ε0fs and εfs = 2ε0fs will be considered where ε0fs/kB = 153.0 K corresponds to the solid carbon dioxide. For σfs and ρs the values speciﬁc to solid carbon dioxide where employed, namely, σfs = 3.727 Å and ρs = 1.91 × 1028 m−3. In real cases, the parameters σfp, εfp, and ρp (density of pillars) can diﬀer from those of the substrate. However, for simplicity, σfp 11385

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and ρp were selected equal to σfs and ρs, respectively, the diﬀerence between pillars and substrate being provided only by the parameter εfp. The latter parameter was selected ε0fs or 2ε0fs, and three kinds of slits with εfs = εfp = ε0fs (case 1), εfs = εfp = 2ε0fs (case 2), and εfs = ε0fs, εfp = 2ε0fs (case 3) will be considered. Because of the geometry of the system, the external potential Usf generated by the solid (substrate plus pillars) at a point M(x, y, h) depends on the distance h + σfs of this point to the surface of the solid and the position x of point M along this surface, and is independent of y, i.e., Usf ≡ Usf(x, h). The potential Usf can be obtained by integrating the Lennard-Jones potential (eq 2) for the ﬂuid−substrate and ﬂuid−pillars interactions over the entire volume of the solid and can be written as Ufs(r) =

reservoir and F[ρ(r)] is the Helmholtz free energy. The latter can be expressed as the sum of an ideal gas free energy, Fid[ρ(r)], an excess free energy Fhs[ρ(r)] (with respect to the ideal gas) of a reference system of hard spheres, a free energy Fattr[ρ(r)] due to the attractive interactions between the ﬂuid molecules (in the mean ﬁeld approximation), and a free energy Ffs[ρ(r)] due to the interactions between the ﬂuid and solid (substrate plus pillars). Explicit expressions for these contributions, which can be obtained in standard ways,21 are provided in the Appendix. The Euler−Lagrange equation for ρ(x, h) (−Lx/2 ≤ x ≤ Lx/ 2) can be written in the following form22 μ log[Λ3ρ(x , h)] − Q (x , h) = kBT (7)

∫V ρs (r′)ϕfs(|r − r′|) dr′ + ∫V ρp(r′)ϕfp(|r − r′|) s

where the function Q(x, h), which is a functional of ρ(x, h), is given in the Appendix, Λ = hP/(2πmkBT)1/2 is the thermal de Broglie wavelength, hP is the Planck constant, T is the absolute temperature, and m is the mass of a ﬂuid molecule.

p

(3)

dr′

where Vs is the volume of the substrate, Vp is the volume occupied by pillars, ρs(r′) and ρp(r′) are the densities of the substrate and pillars, respectively, which, in general, depend on the coordinates. For a uniform substrate, ρs(r′) ≡ ρs, and the ﬁrst integral in eq 3 can be calculated analytically:

3. RESULTS The aim of the present calculations is to examine the changes in adsorption isoterms and features of capillary condensation due to the physical roughness of the surface, r, and the nature of pillars. The ﬁrst parameter (r) is controlled by the height of the pillars, hp, and for the considered geometry of pillars is given by the equation

∫V ρs (r′)ϕfs(|r − r′|) dr′ s

=

9 ⎡ ⎛ ⎛ σfs ⎞3⎤ σfs ⎞ 2π 3⎢ 2 εfsρσ − ⎜ ⎟ ⎜ ⎟⎥ s fs ⎢ σ σ 3 15 h h + + ⎝ ⎠ ⎝ ⎠ ⎥⎦ fs fs ⎣

r=1+

(4)

The contribution ΔUfp,i(r) of a pillar to the potential Ufs(r) can be represented in the form ΔUfp, i(r) = ρp

∫A

p, i

∫−∞ dy′ϕfp(|r − r′|)

(5)

where Ap,i is the area of the cross section of the pillar in the x−h plane. After integration over y′, eq 5 acquires the form ΔUfp, i(r) =

3π εfpρp σfp3 dx′ dh′ A p, i 2 ⎡ σfp9 ⎢ 31 ⎢⎣ 32 [(x − x′)2 + (h − h′)2 ]11/2

ρav* =

∫

−

⎤ ⎥ [(x − x′)2 + (h − h′)2 ]5/2 ⎥⎦

d p + Δp

(8)

The second parameter (nature of the pillars) is reﬂected in the changes of the energy parameter εfp in the pillar−ﬂuid interactions. First, let us consider the adsorption isoterms obtained for various roughnesses and all considered parameters of ﬂuid− solid interactions. In Figure 2 the dimensionless average density, ρav * ≡ ρavσff3, of the ﬂuid in the slit

∞

dx′ dh′

2hp

Lx /2

∫−L /2 dx ∫0

Lh

dhρ*(x , h)/(LxL h − 2hp d p)

x

(9)

where ρ*(x, h) = ρ(x, is plotted versus the pressure in the reservoir for selected roughnesses (indicated on each curve) and for a smooth surface (r = 1). Note that the denominator in eq 9 represents the total volume of the system ﬁlled with the ﬂuid. The data in Figure 2 are presented only for thermodynamically stable states. The metastable states which exhibit hysteresis loops and correspond to local minima of the free energy are not plotted to avoid too many curves on the ﬁgure. They will be discussed below with regard to Figure 7. The pressure P in the reservoir as a function of chemical potential μ was calculated as in ref 23. Because the volume of the slit accessible to the ﬂuid is constant (does not depend on roughness), the average density provides the amount of ﬂuid adsorbed in the slit as a function of roughness. As expected, the equilibrium capillary condensation pressure Pcond, deﬁned as the pressure at which the vapor and liquid branches of the adsorption isoterm have equal free energies (point of equilibrium phase transition), depends on the roughness of the surface. In Figure 3 its behavior as a function of roughness is presented for all considered surfaces. For case 1, Pcond monotonically increases with increasing roughness and at r ≃ 1.65 becomes larger than P0; that is, the equilibrium h)σff3

σfp3

(6)

The integral on the right-hand side of eq 6 was calculated numerically. To calculate the ﬂuid−solid potential generated by all pillars in the range −Lx/2 ≤ x ≤ Lx/2, the contributions of the upper and lower pillars from that interval as well as the contributions of their six nearest neighbors along the corresponding surface were taken into account. The contributions of all other pillars were negligibly small. Note that more general but more complicated analytical calculations of the potential for parallelepipedal and cylindrical pillars are provided in ref 19 and for stepped surfaces in ref 20. 2.2. Euler−Lagrange Equation for the Fluid Density Distribution. The equation for the ﬂuid density distribution (FDD) ρ(x, h) is obtained through the minimization of the grand canonical potential of the system Ω[ρ(r)] = F[ρ(r)] − μ∫ ρ(r)dr where μ is the chemical potential of the ﬂuid in the 11386

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Figure 3. Capillary condensation pressure Pcond as function of roughness for case 1 (εfs = εfp = ε0fs) (solid line), case 2 (εfs = εfp = 2ε0fs) (dashed line), and case 3 (εfp = 2ε0fs, εfs = ε0fs) (dashed-dotted solid line).

above the pillars decreases rapidly with increasing pillar height, and the contact angle becomes dominated by the attractive interaction between ﬂuid and pillars. The latter attraction increases rapidly with increasing hp, approaching an asymptotic value which remains smaller than the ﬂuid−substrate one. In this case, the equilibrium capillary condensation transition occurs at bulk pressures larger than P0. In cases 2 and 3, the ﬂuid−pillars interaction constant is stronger than that in case 1 (εfp = 2ε0fs) and the increase in pillar height is accompained by two competitive tendencies. One of them is a decrease in hydrophilicity due to the decrease of the ﬂuid−substrate attraction above the pillars, which results in increasing Pcond. The other one is an increase in hydrophilicity due to increasing attraction between ﬂuid and pillars and consequently Pcond decreases. The ﬁrst tendency dominates at small pillar heights and the second dominates at suﬃciently large pillar heights. As a result, the dependence of Pcond on hp passes through a maximum (see Figure 3). In Figure 4, the dependence of the average ﬂuid density, ρav *, in the slit on roughness is presented for several values of the pressure in the reservoir P < Pcond for all considered cases of the ﬂuid−solid interaction parameters. For cases 1 and 2 where the pillars are made of the same material as the substrate, the average density as a function of roughness has a minimum at r ≃ 1.6 for all considered values of P/P0 (see Figure 4a,b). In case 1, the dependence has a maximum at r ≃ 3.5 (Figure 4a). For case 2, such a maximum is most likely exhibited at a higher roughness not considered in the present paper (Figure 4b). In case 3 where the substrate and pillars are made from diﬀerent materials, the average density increases monotonically with increasing roughness (see Figure 4c). For Pcond < P < P0, the average density in the slit for all cases monotonically increases with increasing roughness. In Figure 5, examples of such a dependence for case 2 and case 3 are presented for P/P0 = 0.88. For the same roughness, the diﬀerence between liquid densities in those cases is very small and the corresponding curves almost coincide. To explain qualitatively the behavior of ρav * under diﬀerent conditions, one has to take into account the two factors which aﬀect adsorption in the slit. The ﬁrst one is the area of the surface of the slit which increases with increasing roughness. Because of large ﬂuid densities near the solid, the increase of that area increases the amount of adsorbed ﬂuid. The second

Figure 2. Adsorption isoterms as functions of the pressure in the reservoir for various roughnesses indicated on each curve. (a) Case 1 (εfs = εfp = ε0fs), (b) case 2 (εfs = εfp = 2ε0fs), and (c) case 3 (εfs = ε0fs, εfp = 2ε0fs). Solid lines are for r = 1, long-dashed line is for r = 1.41, dashed lines are for r = 1.83, dot-dashed line is for r = 2.66, and dotted lines are for r = 3.49.

capillary condensation transition becomes a capillary evaporation one. In the two other cases, Pcond slightly increases with increasing roughness, passes through a maximum at r ≃ 1.6 for case 3 and r ≃ 2.25 for case 2, and then monotonically decreases. Qualitatively, the behavior of Pcond can be associated with changes in the wetting properties of the surface with increasing roughness. In all considered cases, the surface without pillars is hydrophilic (θ < 90°).14,22 It was shown in ref 14 that for ﬂuid−substrate and ﬂuid−pillar interaction constants corresponding to case 1 (εfs = ε0fs, εfp = εfs), the hydrophilicity of the surface decreases with increasing pillar height and the surface becomes hydrophobic for suﬃciently large hp. This occurs because the ﬂuid−substrate attraction 11387

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Figure 5. Average density ρav * as function of roughness for P > Pcond for case 2 (εfs = εfp = 2ε0fs, solid line) and case 3 (εfp = 2ε0fs, εfs = ε0fs, dashed line).

Figure 6. Potential of ﬂuid−solid interaction at distance σfs above the middle of a pillar as a function of pillar height for case 1 (εfs = εfp = ε0fs) (solid line) and case 3 (εfp = 2ε0fs, εfs = ε0fs) (dashed line).

monotonic increase of ρav * with increasing hp. For this reason, there is no extremum in the ρav * vs r dependence in Figure 4c. In Figure 7 the hysteresis loops in the dependence of the average density of adsorbed ﬂuid on the pressure in the reservoir is presented at various roughnesses for all considered cases. The dotted vertical lines indicate the location of the bulk saturation pressure (P = P0). In case 1, the adsorption branches of the hysteresis loops are located in the range P > P0 for all considered roughnesses. Because we consider only the range P < P0, those corresponding to P > P0 are not displayed in the ﬁgure. In case 2, the hysteresis loops are located completely in the range P < P0. However, their location depends on roughness (see Figure 7b). In case 3, for small roughnesses (r ≤ 3.0) the adsorption branches of the hysteresis loops are located in the range P > P0. For r > 3.0 the entire hysteresis loops are located in the vapor-phase range of the bulk ﬂuid. This means that the increase in roughness in the last case favors capillary condensation.

Figure 4. Average density ρav * as a function of roughness for various values of the pressure in the reservoir (indicated on each curve) for (a) case 1 (εfs = εfp = ε0fs), (b) case 2 (εfs = εfp = 2ε0fs), and (c) case 3 (εfp = 2ε0fs, εfs = ε0fs). In all cases, P is smaller than Pcond.

factor is the attractive potential near the solid surface, which depends in the considered cases on the height and nature of the pillars. If this potential becomes less attractive, the ﬂuid densities close to the surface decrease, thus decreasing ρav * . Vice versa, if this potential becomes more attractive, ρ*av increases. As an illustration, in Figure 6 the ﬂuid−solid potential is presented at distance σfs above the center of a pillar for cases 1 and 3. In the former case, the potential increases for 0 ≤ hp/σff ≤ 1.2 (becomes less attractive) with increasing pillar height (roughness) and then remains almost constant. However, the area of the solid surface continuously increases with increasing pillar height (roughness). The competition between these two factors is resposible for the behaviors of ρ*av presented in Figure 4a,b. In case 3, the potential decreases for 0 ≤ hp/σff ≤ 1.2 (becomes more attractive) with increasing pillar height (roughness). Along with the increasing area of the surface, this provides a

4. CONCLUSION It was shown that the presence of pillars changes the phase behavior of conﬁned ﬂuids. First, the roughness due to pillars changes the equilibrium capillary condensation pressure, Pcond, in nontrivial ways, resulting in some cases in maxima in the r dependence of Pcond (see Figure 3). Second, the r dependence of ρav * for P < Pcond can exhibit nonmonotonic behavior resulting in the existence of extrema (see Figure 4). The reason for such 11388

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where Λ = hP/(2πmkBT)1/2 is the thermal de Broglie wavelength, hP and kB are the Planck and Boltzmann constants, respectively, T is the absolute temperature, m is the mass of a ﬂuid molecule, ΔΨhs(r) = kBTηρ

4 − 3ηρ ̅ (1

̅

− ηρ )2 ̅

(A.3)

ηρ̅ = is the packing fraction of the ﬂuid molecules, σff is the ﬂuid hard core diameter, and ρ̅(r) is the smoothed density deﬁned as (1/6)πρ̅(r)σff3

ρ ̅ (r) =

∫ dr′ρ(r′) W (|r − r′|)

(A.4)

The weighting function W(|r − r′|) is selected in the form25 ⎧ 3 ⎛ r ⎞ ⎪ 1 − ⎟ , r ≤ σff 3⎜ σff ⎠ W (|r − r′|) = ⎨ πσff ⎝ ⎪ 0, r > σff ⎩

where r = |r − r′|. The contribution to the excess free energy due to the attraction between the ﬂuid−ﬂuid molecules is calculated in the mean-ﬁeld approximation Fattr[ρ(r)] =

1 2

∫ ∫ dr dr′ ρ(r) ρ(r′) ϕff (|r − r′|)

(A.5)

where ϕff(|r − r′|) is the Lennard-Jones potential of the ﬂuid− ﬂuid interactions. The last contribution, Ffs[ρ(r)], is given by the expression Ffs[ρ(r)] =

∫V dr ρ(r) Ufs(r)

(A.6)

where V is the volume occupied by the ﬂuid and Ufs(r) is the net potential generated by the solid (substrate plus pillars). This potential is provided by eq 3. The Euler−Lagrange equation for the ﬂuid density distribution ρ(x, h), obtained by minimizing the grand canonical potential, can be written in the following general form μ log[Λ3ρ(x , h)] − Q (x , h) = kBT (A.7) where μ is the chemical potential in the reservoir and the function Q(x,h) is given by

Figure 7. Hysteresis loops in adsorption isoterm for various roughnesses as a function of the pressure in the reservoir. (a) Case 1 (εfs = εfp = ε0fs), (b) case 2 (εfs = εfp = 2ε0fs), and (c) case 3 (εfp = 2ε0fs, εfs = ε0fs). Solid line is for r = 1, dashed lines are for r = 1.83, dot-dashed line is for r = 2.66, and dotted lines are for r = 3.49.

Q (x , h) = −

1 [ΔΨhs(x , h) + ΔΨ′hs (x , h) + Uff (x , h) kBT

+ Ufs(x , h)]

features is the competition between the increased area of the rough surface and the decrease in the net ﬂuid−solid interaction potential.

where Uff (x , h) =

5. APPENDIX: EULER−LAGRANGE EQUATION To derive the Euler−Lagrange equation, one has to consider the contributions to the free energy mentioned in section 2.2, which can be represented as follows:21,24 Fid[ρ(r)] = kBT Fhs[ρ(r)] =

∫ dr ρ(r){log[Λ3ρ(r)] − 1}

∫ dr ρ(r) ΔΨhs(r)

(A.8)

∫ ∫ dx′ dh′ ρ(x′, h′) ϕff,y(|x − x′|, |h − h′|) (A.9)

ΔΨ′hs (x , h) =

∫∫

dx′ dh′ ρ(x′, h′)Wy(|x − x′| ,

|h − h′|)

(A.1)

∂ ΔΨhs(ρ ̅ )| ρ ̅ = ρ ̅ (x , h ) ′ ′ ∂ρ ̅

(A.10)

ϕff,y(|x − x′|,|h − h′|) and Wy(|x − x′|,|h − h′|) are obtained by integrating the potential ϕff(|r − r′|) and the weighted function W(|r − r′|) with respect to y from −∞ to + ∞.

(A.2) 11389

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(19) Wu, H.; Borhan, A.; Fichthorn, K. A. Interaction of fluids with physically patterned solid surfaces. J. Chem. Phys. 2010, 133 (5), 054704. (20) Bojan, M. J.; Steele, W. A. Computer-Simulation of Physical Adsorption on Stepped Surfaces. Langmuir 1993, 9 (10), 2569−2575. (21) Tarazona, P. Free-energy density functional for hard-spheres. Phys. Rev. A 1985, 31 (4), 2672−2679. (22) Berim, G.; Ruckenstein, E. Nanodrop on a nanorough solid surface: Density functional theory considerations. J. Chem. Phys. 2008, 129 (1), 014708. (23) Ancilotto, F.; Faccin, F.; Toigo, F. Wetting transitions of He-4 on alkali-metal surfaces from density-functional calculations. Phys. Rev. B 2000, 62 (24), 17035−17042. (24) Tarazona, P.; Marconi, U. M. B.; Evans, R. Phase-equilibria of fluid interfaces and confined fluids - nonlocal versus local density functionals. Mol. Phys. 1987, 60 (3), 573−595. (25) Nilson, R. H.; Griffiths, S. K. Condensation pressures in small pores: An analytical model based on density functional theory. J. Chem. Phys. 1999, 111 (9), 4281−4290.

When calculating the term Uff(x,h) of the Euler−Lagrange equation arising due to the long-range ﬂuid−ﬂuid interactions, a cutoﬀ at a distance equal to four molecular diameters σff for the range of Lennard-Jones attraction was employed.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]ﬀalo.edu. Phone: (716) 645-1179. Fax: (716) 645-3822. Notes

The authors declare no competing ﬁnancial interest.

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REFERENCES

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dx.doi.org/10.1021/la3014464 | Langmuir 2012, 28, 11384−11390