Effects of the confinement on wall pressure, interfacial tension, and excess adsorption at the nanocylindrical wall

Journal of Molecular Liquids 223 (2016) 182–191

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Effects of the confinement on wall pressure, interfacial tension, and excess adsorption at the nanocylindrical wall Ezat Keshavarzi ⁎, Fatemeh Namdari Department of Chemistry, Isfahan University of Technology, Isfahan 8415683111, Iran

a r t i c l e

i n f o

Article history: Received 10 November 2015 Accepted 30 June 2016 Available online 01 July 2016 Keywords: Bicylindrical pore Interfacial tension Excess adsorption Wall pressure Weighted densities MFMT

a b s t r a c t In this article, a hard sphere fluid confined between two homocentric cylinders which formed a bicylindrical pore has been investigated. Our aim is investigation of the effect of an outer cylinder on the values of wall pressure, interfacial tension, and excess adsorption of fluid at a convex nanocylindrical wall, inner wall. To investigate this effect, the modified fundamental measure theory (MFMT) used. To do so at first we presented a general solution for weighted density integrals in cylindrical coordinate which is applicable for infinite and infinite lengths of cylindrical pore, bicylindrical pore, cylindrical wall, and even truncated cone. In the second step, the wall pressures, interfacial tensions, and excess adsorptions at a convex nanocylindrical wall are obtained for confined fluids in bicylindrical pores and compared with those values for bulk fluids. Our results showed that confinement leads to an oscillatory behavior for wall pressure, interfacial tension, and excess adsorption of the fluid at the wall. The reason for these oscillations lies in structural changes that occur for a fluid as a result of confinement. Variation of the type of interaction of inner wall with fluid from hard to attractive one can reverse the behavior of interfacial tension and adsorption versus size. Also in some cases, it changes their signs while the change in the type of interaction of the outer wall only leads to an increase in amplitudes of the oscillations. © 2016 Published by Elsevier B.V.

1. Introduction Nano-confined fluids and their properties have been the focus of much research because of the significant role they play in both industrial and biological systems [1,2]. Adsorption of fluids in nanopores has important applications in protein extraction, phase separation, and chromatographic processes [3–5]. Moreover, interfacial tension plays a remarkable role in pharmaceutical production and plant nutrition. Certain phenomena such as the capillary rise in plants, ionic transfer through nanochannels in biological cells, and preparation of monodisperse emulsions in industrial processes strongly depend on interfacial tension which may exhibit different values for different nanopores [6–8]. It is expectable that the shapes and curvatures of nanopores and nanowalls should play major roles in determining the values of both fluid interfacial tension and adsorption. A number of studies have been conducted to determine the interfacial tension and excess adsorption of fluids in different geometries [9–12]. Dong Fu [13] investigated the structure, interfacial tension, and excess adsorption of a Lennard-

⁎ Corresponding author. E-mail address: [email protected] (E. Keshavarzi).

http://dx.doi.org/10.1016/j.molliq.2016.06.104 0167-7322/© 2016 Published by Elsevier B.V.

Jones fluid confined between two planar walls for different sizes of slit-like pores. Bryk et al. [14] studied the interfacial tension of a hard sphere fluid in contact with hard spherical and cylindrical walls for a wide range of sizes and densities. Detailed investigations have also been carried out in our previous works on the structure, interfacial tension, and excess adsorption of a bulk fluid at a convex spherical wall as well as the effects of the concave and convex walls of a spherical pore on the structure of fluids [15,16]. Also, Keshavarzi and Taghizadeh studied the structure of the fluid around a nanocylindrical wall including convex and flat walls and the relevant edges [17]. Among the different nanopores, the nanocylindrical pore is the one receiving more attention in theoretical studies due to its geometry which is commonly found in nature. Moreover, it has found wide applications in industrial processes because of the lower friction it causes. In this work, we will focus on a bicylindrical pore to investigate the wall pressure, interfacial tension, and excess adsorption of a hard sphere fluid at a convex cylindrical wall in the presence of another concave cylindrical wall. The presence of the second (concave cylindrical) wall affects the values of the above properties at the convex cylindrical wall. In our case, the fluid is confined between these two concave and convex walls which form a bicylindrical pore. Additionally, the fluid used in this study is taken to be a hard sphere one and the cylindrical wall is considered as being both a hard and an attractive wall. We will obtain

E. Keshavarzi, F. Namdari / Journal of Molecular Liquids 223 (2016) 182–191

and compare the values for wall pressure, interfacial tension, and excess adsorption for the following two different cases: one is the case of a fluid in a nanobicylindrical pore for which the radius of the outer cylinder is infinity (i.e., a bulk fluid at the contact of a convex cylindrical wall), and the second being the case of a fluid in a nanobicylindrical pore for which the radius of the outer cylinder, Rb, is in the order of the molecular diameter (i.e., a confined fluid at the contact of the inner cylindrical wall with a radius identical to that of the first case). It should be noted that the differences observed among the properties thus obtained are directly related to the confinement effects since the inner cylinder will be the same for both cases. Thanks to its higher accuracy, the modified fundamental measure theory (MFMT), which is the most successful version of the DFT approach, is used in this study to determine the structure of the fluid in the bicylindrical pore. Bryk and co-workers [18] studied the adsorption and phase behavior of fluids in pores confined between two uniaxial cylinders; however, the radius of the inner cylinder in their work was larger than the molecular size. To overcome this problem, Tarazona proposed the cavity fundamental measure theory (CFMT) based on his definition of a new free energy density proposed for this kind of nanopore [19,20]. Gonzalez and co-workers used Fourier technique to solve the weighted densities in the original FMT of Rosenfeld (OFMT) for cylindrical and spherical pores [21]. Kong et al. applied a solution for weighted density integrals in a narrow cylinder with infinity length [22]. Malijevsky solved the integration of the weighted densities in an effective one-fold numerical integration that ends up with elliptic functions [23]. Recently, a solution for spherical geometries was presented by Keshavarzi and Helmi [24]. In this study, we will initially present the solution of weighted density integrals in cylindrical coordinate for a cylindrical geometry before we proceed to deal with our main objective of investigating the effects of fluid confining on contact density, interfacial tension, and adsorption of fluid around a cylindrical wall. Although there are some reports for weighted density integrals in MFMT in the literature, our solution has some advantages. We present a general solution which is applicable for cylindrical geometries with finite or infinite lengths including cylindrical pores, bicylindrical pores, cylindrical walls, and even for truncated cone. While some of the reported solution are only applicable for their studied case. It will also be shown that this solution is easier than the two Fourier and Tarazona techniques and also using elliptic functions so far presented elsewhere. The rest of the paper is organized as follows. Section 2 presents a brief review of the MFMT. The solution of weighted density integrals for cylindrical geometries will be presented in Section 3. Section 4 investigates the effects of confinement on the values of wall pressure, interfacial tension, and excess adsorption of a fluid at a cylindrical wall. Finally, conclusions will be provided in Section 5.

183

given by the exact equation as: Z F id ½ρðrÞ ¼ kT

h   i drρðrÞ ln ρðrÞΛ 3 −1

ð3Þ

where, k is the Boltzmann constant, T is absolute temperature, and Λ is the de Broglie wavelength. According to MFMT for a hard sphere fluid, the excess part of the Helmholtz free energy is expressed as follows: Z F ex ½ρðrÞ ¼ kT

dr½Φhs ðnα ðrÞÞ

ð4Þ

In the above equation, Φhs (nα (r)) is the Helmholtz free energy density of a hard sphere fluid. The term Φhs (nα (r)) is divided up into two scalar and vector parts as follows [26,27]: n1 n2 ΦShs ¼ −n0 ln ð1−n3 Þ þ 1−n3 " # 1 1 n32 þ ln ð 1−n Þ þ 3 36πn23 36πn3 ð1−n3 Þ2

ΦVhs ¼ −

" # nv1 nv2 1 1 n2 ðnv2 nv2 Þ − ln ð 1−n Þ þ 3 1−n3 12πn23 12πn3 ð1−n3 Þ2

ð5Þ

ð6Þ

where, nα (r) is the weighted density defined as: Z nα ðrÞ ¼

dr0ρðr0Þwα ðr−r0Þ

ð7Þ

where, wα is the weight function that involves two vector functions, wαv , and four scalar functions, wα, as follows: σ  w3 ðrÞ ¼ θ −r 2 σ  w2 ðrÞ ¼ 2πσw1 ¼ πσ 2 w0 ¼ δ −r  2 r σ −r w2v ðrÞ ¼ 2πσw1v ¼ δ jrj 2

ð8Þ

where, θ(r) is the Heaviside step function, δ(r) represents the Dirac delta function, and r is the distance between two points in the system. By minimizing the grand canonical potential, the equilibrium local density of confined fluids obtains [28]:     ∂Φ 0 0 wα r−r −βV ext ðrÞ ρðrÞ ¼ ρb exp βμ hs ex −∫dr ∑α ∂nα

ð9Þ

2. Modified fundamental measure theory (MFMT) The grand canonical free energy, Ω[ρ(r)], of an inhomogeneous fluid is a functional of the one-body distribution function, ρ(r), which can be defined as [25]: Ω½ρðrÞ ¼ F int ½ρðrÞ þ ∫drρðrÞ½V ext ðrÞ−μ 

ð1Þ

where, Vext (r) is the external potential, μ is the chemical potential of the inhomogeneous fluid, and Fint [ρ(r)] is the intrinsic Helmholtz free energy. The functional Fint [ρ(r)] is expressed as:

In Eq. (9), μhs ex is the hard sphere chemical potential obtained via Mansoori–Carnahan–Starling–Laland (MCSL) equation of state for the pure hard sphere fluid as [29]: βμ hs ex ¼

  η 8−9η þ 3η2 ð1−ηÞ3

ð10Þ

in which, η is the packing fraction whose value is equal to πρbσ3/6. According to the above equations, the grand potential, Ω, is: Z Φhs ðnα ðr ÞÞdr þ V ext ðr Þρðr Þdr  Z Z μb þ kT ρðrÞlnρðr Þdr−kT ρðr Þ 1 þ ln ρb þ ex dr kT Z

Ω ¼ kT F int ½ρðrÞ ¼ F id ½ρðrÞ þ F ex ½ρðrÞ

ð2Þ

where, Fex [ρ(r)] and Fid [ρ(r)] are excess and ideal contributions to the intrinsic Helmholtz free energy, respectively. The ideal contribution is

ð11Þ

where, μbex is the excess chemical potential of the bulk fluid. The external

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potential in our study is: 8 > ∞ > > > 2 3   > > σ  > > r− Rs þ > > 6 7 2 > < −εwRs exp4−λw 5 σ V ext ðrÞ ¼ > 2 3     > > σ > > −r Rb − > > 6 7 > 2 > −ε wRb exp4−λw 5 > > σ :

r ≤Rs þ

σ σ ; r ≥Rb − 2 2 r ≥Rs þ

σ 2

r ≤Rb −

σ 2

ð12Þ where, λw is the wall-fluid potential parameter whose value in our work has been taken to be 1.8 for the case of attractive pore walls. εwRs and εwRb are the wall depths of the wall-fluid potential of the inner and outer cylinders in a bicylindrical pore with radii of Rs and Rb, respectively. The values of these two parameters in our study are εwRs = 1 and εwRb = 1 for attractive walls, but equal to zero for hard walls. 3. Solution of the Rosenfeld weighted density integrals for cylindrical geometries Before moving to our main goal of investigating the confinement effects on the values of wall pressure, interfacial tension, and excess adsorption of a fluid at a convex cylindrical wall, we need to exploit MFMT to obtain the structure of fluids in pores with cylindrical geometries. For this purpose, the weighted density integrals need to be initially calculated for this geometry. In this section, therefore, we initially present our solution for weighted density integrals in a bicylindrical pore. Let us assume a bicylindrical pore with radii Rs and Rb for its inner and outer cylinders, respectively, with H denoting the distance between the two walls, H = Rb − Rs. The bicylindrical pore is selected because it can generate a cylindrical pore when the radius of the inner cylinder becomes equal to zero. Also, it can form a convex cylindrical wall in a bulk fluid when the radius of the outer cylinder becomes infinity. Moreover, in some cases, it can generate a slit-like pore when the radii of the inner and outer cylinders are both far larger than the molecular size. To obtain the weighted density integrals, nα (r), it is appropriate to define our system in the cylindrical coordinate because of the cylindrical symmetry of the nanopores. Clearly, each point in the pore is defined in the cylindrical coordinate by the variables R, z, and φ. We fix the origin of the coordinate on the bicylindrical mandrel and put the vector of r on the x axis as shown in Fig. 1 it should be noted that r is the presentation of each point in the pore. Based on this definition, r is equal to R in our cylindrical coordinate, and φ and z are, therefore, equal to zero. It is well established that n2(R) and n3(R) are the densities of the molecules located on the surface and those in the volume of a sphere with a radius of σ/2 around each point of R, respectively. As we have shown in Fig. 1, r′ is the locus of the points whose distance from R is equal to σ/2. The variable r′ is transformed into R′, φ', and z′ in the cylindrical coordinate as follows: Z n2 ðRÞ ¼

Z ρðR0Þ R0dR0

Z n3 ðRÞ ¼

Z  σ dφ0 δ r− dz0 2

ð13Þ

Z  σ θ r− dz0 2

ð14Þ

Z ρðR0Þ R0dR0

dφ0

Because of the Dirac delta and Heaviside step functions in Eqs. (13) and (14), z′ can only get either of the two values +a or –a for the inte1 = gral n2 (R), where a ¼ ððσ=2Þ2 −R2 −R02 þ 2RR0cosφ0Þ 2 þ z . It can, however, admit all the values from + a to –a for n3(R). After solving the third integral on the z′ variable, we have two dimensional integrals for which the values of R′ and φ' have been chosen in such a way that they generate a circle with a radius equal to σ/2 around each point of R in the xy plane. When R is smaller than σ/2, the integral limits of R′

Fig. 1. A bicylindrical pore with radii RS and Rb for the inner and outer cylinders, respectively; H is Rb − RS.

and φ' will be different from those of cases in which R is larger than σ/ 2. The results for both cases may be summarized as follows: When R ≥ σ/2, we have:

Z n2 ðRÞ ¼ 2σ Z n3 ðRÞ ¼ 4

Rþσ2 R−σ2

Rþσ2 R−σ2

  ρ R0 R0 dR0

  ρ R0 R0 dR0

Z

Z

 arccos

2 2 R2 þR0 −σ

 4

RR0

  σ 2

0

 arccos

2 2 R2 þR0 −σ



RR0

4

2

 2 σ 2

0

1 2

0

−R2 −R0 þ 2RR0 cosφ0

−R2 −R0 þ 2RR0 cosφ0 2

1 =2 dφ

0:5

dφ0

ð15Þ ! The integral of n 2 ðRÞ is: ! n 2 ðRÞ ¼

Z

Z ρðR0Þ R0dR0

Z ! r  σ  δ r− =2 dz0 jrj

ð16Þ

dφ0

This may be solved similar to the way n2 (R) is, but its vectorial character must nevertheless be taken into account. The final result can be summarized below:

! Z 2eR ! n 2 ðRÞ ¼ R

Rþσ2 R−σ2

Z ρðR0ÞR0dR0

 arccos 0

R2 þR02 −σ RR0

2

 4

  σ 2 2

2R2 −2RR0 cosφ0 −R2 −R02 þ 2RR0 cosφ0

1 =2 dφ0

ð17Þ ! where, e R is the unit vector in the direction of R.

E. Keshavarzi, F. Namdari / Journal of Molecular Liquids 223 (2016) 182–191

185

For R b σ/2, we have: n2 ðRÞ ¼ Z σ −R Z 2   2σ ρ R0 R0 dR 0

Z 2σ

Rþσ2 σ −R 2

  ρ R0 R0 dR0

Z

π 0

1

  σ 2

0

2

2 0 0 0 2 −R −R þ 2RR cosφ

  2

arccos

2 R2 þR0 −σ RR0

0

1 =2 dφ þ

4

  σ 2

1 2

02

0

0

cosφ0

1 =2 dφ

2 −R −R þ 2RR n3 ðRÞ ¼ 1=2 Z σ −R Z π  2 2   σ 2 4 ρ R0 R0 dR0 −R2 −R0 þ 2RR0 cosφ0 þ 2 0 0

  2 σ2 2 0 1=2 Z Rþσ Z arccos R þR −0 4  2 2   RR σ 2 4 ρ R0 R0 dR0 −R2 −R0 þ 2RR0 cosφ0 σ 2 −R 0 2 and ! n 2 ðRÞ ¼ Z π ! Z σ 2 e R 2 −R  0  0 2R2 −2RR0 cosφ0 0 ρ R R dR0   1 =2 dφ þ R 2 2 02 0 0 0 σ 0 −R −R þ 2RR cosφ 2

  2 02 σ2 Z arccos R þR −0 4 ! Z Rþσ2   RR 2eR 2R2 −2RR0 cosφ0 0 ρ R0 R0 dR0  1 =2 dφ   σ R 2 02 0 −R 0 σ 2 0 2 2 −R −R þ 2RR cosφ

ð18Þ ! n0(r), n1(r), and n 1 ðrÞ can be calculated via the following equations: n0 ðRÞ ¼ 2n1 ðRÞ=σ ¼ n2 ðRÞ=πσ 2 ! ! n 1 ðRÞ ¼ n 2 ðRÞ=2πσ

ð19Þ

To verify the accuracy of our derivation, we predicted the density profile of a hard sphere fluid confined in a hard cylindrical pore whose results are shown in Fig. 2a and compared with MC simulation data [30]. The results in this figure represent those for a cylindrical pore with a radius equal to 2.2σ while the confined fluid is in equilibrium with its bulk at a density equal to 0.7. Clearly, a very good agreement is observed between our results and the corresponding values obtained from the Monte Carlo (MC) simulation. Also, our DFT results have been compared with the data obtained via CFMT and OFMT methods [21]. In addition to the good agreement found between the two sets of data, our solution is seen to be far easier than the other solutions via the DFT method which use the Fourier technique or approaches based on the definition of a new free energy function. Fig. 2b shows the density profile of a hard sphere fluid confined in a hard bicylindrical pore with inner and outer cylinder radii of 1σ and 4σ respectively and a bulk density equal to 0.7. Comparison of our results with the corresponding values obtained from MC simulation [18] in this figure shows a very good agreement between the two sets of data. Finally, the results for a hard sphere fluid confined in a hard cylindrical pore with Rb = 4σ and its bulk density equal to 0.7 are plotted in the Fig. 2b. 4. Effects of fluid confining on wall pressure, interfacial tension, and excess adsorption of a fluid at a cylindrical wall We consider a hard sphere fluid confined in an open bicylindrical pore with radii equal to Rs and Rb for the inner and outer cylinders, respectively. The height of the bicylindrical pore is infinity and the confined fluid in it is in equilibrium with the bulk fluid. To investigate the confinement effect on wall pressure, interfacial tension, and excess adsorption of the hard sphere fluid at a convex cylindrical wall, we obtained the values for these same properties in two different cases. In the first, the radius of the outer cylinder, Rb, goes to infinity. Therefore, the behavior of the fluid is the same as that of the bulk fluid at contact of a cylindrical wall with a radius of Rs. In the second, the radius of Rb is reduced from infinity to the order of molecular diameter. In this way, the effects of the outer cylinder wall of the bicylindrical pore on the values of wall pressure, interfacial tension, and excess adsorption gradually

Fig. 2. Density distributions of a) a hard-sphere fluid confined in a hard cylindrical pore with radius equal to Rb = 2.2σ and ρbσ3 = 0.7 which is compared with MC simulation data [30], CFMT [21] (×) and OFMT [21] (Δ), and b) a hard-sphere fluid confined in a hard bicylindrical pore with Rs = 1σ and Rb = 4σ compared with MC simulation data [18] as well as a hard sphere fluid confined in a hard cylindrical pore with Rb = 4σ both of which are in equilibrium with the bulk fluid with ρbσ3 =0.7.

appear. In fact, the fluid properties change from the characteristic bulk ones to those of a confined fluid when Rb is changed from infinity to the order of molecular diameter. In this paper, the first case is hereafter named ‘the bulk fluid’ and the second is designated as ‘the confined fluid’. It is well known that the degrees of adsorption and interfacial tension of a fluid at a wall depend on the fluid structure as well as the wall's curvature, temperature (T), and bulk density (ρb). In our case, a fluid is considered with two different structures, one homogeneous bulk fluid (the first case) and the other an inhomogeneous confined one (the second case), at the contact of the same wall. Therefore, the differences between the above-mentioned quantities for these two cases are directly related to differences in the structure of the fluids, that is the confinement effect. In the following subsections, we will concentrate on this effect on wall pressure, interfacial tension, and excess adsorption. 4.1. Effect of confinement on wall pressure The structure of a hard sphere fluid confined in a hard bicylindrical pore is shown in Fig. 3 for different nanopore sizes at a bulk density

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equal to 0.6. In this figure, the radius of the inner cylinder is fixed at 1σ while that of the outer cylinder varies from 3σ to 10σ. It should be noted that for Rb equal to 10σ, the contact density at the wall is the same as that for the bulk fluid. As expected, the density profile of the confined fluid has an inhomogeneous structure with different values for the contact densities at the concavo-convex walls of the cylinders. Clearly, the fluid molecules have a strong tendency to accumulate at the outer cylinder (i.e., the concave wall) rather than at the inner one (i.e. the convex wall). This is due to the effect of curvature which may be explained by entropy effect [15]. One point of great importance at this juncture is the effect of fluid confinement on the properties of the fluid at the wall. It is clear from Fig. 3 that confinement effects appear when the radius of the outer cylinder reduces to the order of molecular diameter while fluid behavior will be the same as that of the bulk fluid for larger radii. In fact, the presence of the second wall changes contact density at the inner wall. The second wall is also capable of varying values of contact density versus H to maximum and minimum levels for integer and half-integer values, respectively, of H (H = Rb − Rs). In fact, the contact density at the inner wall versus H exhibits an oscillatory behavior. To gain a better understanding of the effects of confinement on wall pressure, Pwσ3, relevant diagrams have been plotted in Fig. 4a. The diagrams for contact density, ρcσ3, have been drawn in this figure because of the relationship that holds between them, PwkTσ ¼ ρc σ 3 . This figure compares the contact densities of a hard sphere bulk fluid at a hard cylindrical wall and the hard sphere confined fluid at the hard inner cylinder of a bicylindrical pore for the two different bulk densities of 0.4 and 0.6. The radius of the cylindrical wall is equal to 1σ, while that of the outer cylinder in the bicylindrical pore varies from 2.25σ to 10σ. It should be noted that the confined fluid is in equilibrium with the bulk one. The results in Fig. 4a are reported for both hard and attractive walls of the outer cylinder. Although in the real systems only for fluid with attraction potential and the attractive wall-fluid interaction is expectable, but in modeling we are able to consider the wall-fluid interaction for hard sphere fluids. The later help us to separate the effect of wall fluid attraction from fluid-fluid attraction. Fig. 4b shows the effect of the changes in the inner wall from hard to attractive on its interactions with the fluid. In fact, Fig. 4b is similar to Fig. 4a except that the former illustrates the case for the attractive inner cylindrical wall (i.e., the convex wall) while the latter shows the 3

Fig. 3. Density profiles of the hard sphere fluid confined in a hard bicylindrical pore with the inner cylinder radius equal to 1σ and the outer cylinder radius varying from 3σ to 10σ and a bulk density of 0.6.

Fig. 4. a) Comparison of the contact densities of hard sphere bulk (.….) and confined fluids in contact with a hard convex cylindrical wall. The radius of the cylindrical wall is fixed at 1σ while that of the outer cylinder varies from 2.25σ to 10σ. In this figure, contact densities are presented for which the outer cylinder is hard (– – –) and attractive (––––) with two different bulk densities equal to 0.4 and 0.6. Figure b is similar to figure (a), but it only illustrates the attractive cylindrical wall.

case for the hard inner cylindrical wall. It is clear from these figures that contact density at identical cylindrical walls and for a fixed bulk density will be different for the two bulk and confined fluids, with the difference arising from differences in their structures. As expected, however, the difference will disappear at high values of H. According to Fig. 4a and b, the contact density of the confined fluid has an oscillatory behavior whose amplitude decreases when either the radius of the outer cylinder or the value of H increases. This oscillatory behavior is also observed when the outer cylinder acts as an attractive wall, but only with higher and deeper bumps compared to those for the hard wall. Maximum and minimum values of the oscillations are expectedly due to the integer and half-integer values of H, respectively. This may be explained by the fact that the number of molecular layers in the pore becomes equal to H/sigma for integer values. Fig. 5 can also be exploited to gain a better understanding of the reason underlying the differences in contact densities observed at the identical hard walls for bulk and confined fluids. Clearly, the structure of the hard sphere fluid in a hard cylindrical pore with Rb is equal to 6σ and its bulk density is equal to 0.6. Moreover, local density in the pore is inhomogeneous only near the cylinder wall but it is homogeneous and nearly equal to 0.6 around the cylinder mandrel. If a small cylindrical wall were inserted in the mandrel of this cylindrical pore, the resulting

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187

Fig. 5. The density profile for a hard sphere fluid confined in a hard cylindrical pore with a radius equal to 6σ and a bulk density equal to 0.6.

state would then be the same as that when this cylindrical wall were inserted in a bulk fluid; thus, there is no difference between the wall pressures in the two cases. By increasing the size of the inner cylindrical wall, however, the values of wall pressure become different. This is because the inner cylinder (i.e., the cylindrical wall) is located in the region in which the fluid has an inhomogeneous structure. For example, according to Fig. 5, when the size of the inner cylinder is equal to or larger than 2σ, the effect of the outer cylinder on the value of contact density at the inner wall becomes apparent. In other words, contact density at a wall may be affected by another wall when the distance between the two walls is in the order of 3 or 4 times the molecular diameter. Fig. 6a and b are the same as Fig. 4a and b, except that in these figures, Rb is fixed at 6σ while the radius of the cylindrical wall, Rs, varies from 0.25σ to 4.75σ. It is clear from both figures that in the case of the bulk fluid, contact density increases at a fixed density with the wall size; in the case of the confined fluid, however, it exhibits an increasing and oscillatory behavior. For values of Rs far smaller than Rb or for large values of H, contact density at identical walls remains the same for both bulk and confined fluids. But these values become different when Rs increases so that confinement effects appear. Similar to Fig. 4a and b, maximum and minimum values in Fig. 6a and b belong to integer and halfinteger values of H, respectively. It is evident from these figures that the attraction potential of the outer cylinder can only lead to an increase in the amplitude of oscillations in contact density at a fixed density as compare to those for a hard wall. Another point of interest in Figs. 4 and 6 is the values of contact density and the depths and heights of oscillations which increase with increasing bulk density. The contact density rising with increasing size of cylindrical wall may be explained with recourse to the wall curvature along the following lines. Since the contact density of the fluid at a flat wall is greater than that at a convex wall, its convexity decreases with increasing Rs until it approaches a flat wall for very large values of Rs. Thus, contact density should also increase with Rs as seen in Fig. 6a and b.

4.2. Effect of fluid confinement on interfacial tension Fluid structure is one of the most important parameters affecting interfacial tension. In this subsection, we investigate the effects of confining a hard sphere fluid on the values of interfacial tension at a cylindrical

Fig. 6. Comparison of the contact densities of hard sphere bulk (.….) and confined fluids in contact with a hard convex cylindrical wall. The radius of the outer cylinder is equal to 6σ while that of the inner cylinder, Rs, varies from 0.25σ to 4.75σ. In this figure, contact densities are presented for which the outer cylinder is hard (– – –) and attractive (––––) with two different bulk densities equal to 0.4 and 0.6. Figure b is similar to figure (a), but it only illustrates the attractive cylindrical wall.

wall. The interfacial tension of the bulk fluid near a cylindrical wall, γ, can be obtained via the following equation: γ¼

Ωbicyl −Ωb A

ð20Þ

where, Ωbicyl is the grand potential of the confined fluid in a bicylindrical pore with a very large value of Rb calculated using Eq. (11) and Ωb is the grand potential of the bulk fluid equal to –pV with p designating the pressure of the bulk fluid, V denoting the bulk volume corresponding to the fluid around the wall, and A representing the surface area of the wall which is a convex cylindrical surface in this work. The upper and lower limits of the grand potential in Eq. (11) have been taken to be equal to Rs and Rs + 5σ, respectively. Also, the interfacial tension for the confined fluid in a bicylindrical pore around its inner cylinder, γcon, were obtained from the following equation: γcon ¼

Ωbicyl −Ωcyl A

ð21Þ

where, Ωbicyl is the grand potential for the fluid confined in a bicylindrical pore for which Rb is in the order of molecular diameter

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and Ωcyl is the grand potential for the fluid confined in a cylindrical pore. To calculate the integrals in Eq. (11) for both cases of cylindrical and bicylindrical pores, the lower limits of the integrals were set equal to Rs. Also, the upper limits were set equal to Rs + 5σ for the cases in which Rb ≥ 6σ. However, it was set equal to Rb for Rb b 6σ because Rs is equal to 1σ. To verify the accuracy of our results, the interfacial tensions of a hard sphere fluid around a hard cylindrical wall with a radius equal to Rs = 300σ were obtained for several bulk densities (Fig. 7). This value of Rs was selected because MC simulation data were available for it [31]. Clearly, our results are in good agreement with MC simulation data up to a reduced bulk density equal to 0.7. The interfacial tensions of the hard sphere bulk and confined fluids in the bicylindrical pore at the contact of a hard cylindrical wall have been obtained and compared with each other in Fig. 8a. In this figure, the radius of the cylindrical wall is taken to be equal to Rs = 1σ while that of the outer cylinder in the bicylindrical pore varies from 2.25σ to 10σ. For fluids in equilibrium with that, the relevant bulk densities are 0.4 and 0.6. Also, the potentials of the outer cylinder in the bicylindrical pore have been considered both as a hard and as an attractive wall. Clearly, the interfacial tension exhibits an oscillatory behavior for the confined fluid with maximum and minimum values for the integer and half-integer values of H. Comparison of the values of the interfacial tension for the confined fluid with that for the bulk fluid shows that they converge for larger values of Rb. The effect of wall-fluid attraction of the inner wall on interfacial tension is illustrated in Fig. 8b. This figure is the same as Fig. 8a but only for an attractive cylindrical wall. Comparison of Fig. 8a and b reveals that the change in the potential from hard to attractive can change the sign of the interfacial tension. It should be remembered that the interfacial tension of a fluid in contact with a wall is determined by two different terms. The first is related to the work required to produce an inhomogeneous structure per unit area of the wall, which is always positive for all thermodynamic states [16]. The second is related to the wall-fluid interactions and has a negative contribution only for an attractive wall [16]. This means that its contribution for the hard sphere fluid at the hard cylindrical wall is zero. Thus, for the cases shown in Fig. 8a, the contribution arising from uncontrolled changes in the structure of the fluid near the wall has the dominant role in determining the value of the interfacial tension, indicating that the interfacial tension has a positive sign. For an attractive cylindrical wall (Fig. 8b), however, both terms play their roles in the determination of both the values and the signs

Fig. 7. Comparison of MC simulation data and the values of interfacial tension versus reduced bulk density for a hard sphere fluid at a convex cylindrical wall with a radius equal to 300σ [31].

Fig. 8. a) Comparison of the interfacial tensions of a hard sphere bulk fluid (.….) and a confined fluid at a hard convex cylindrical wall. The radius of the cylindrical wall is fixed at 1σ while that of the outer cylinder varies from 2.25σ to 10σ. This figure presents the contact density results obtained for an outer cylinder taken to be hard (– – –) and an attractive (––––) one for two different bulk densities equal to 0.4 and 0.6. Figure (b) is similar to figure (a) but it only illustrates the case for an attractive cylindrical wall.

of interfacial tension. In fact, if the second term overcomes the first one, interfacial tension takes a negative sign, and vice versa. Evidently, interfacial tension has maximum and minimum values for integer and half-integer values, respectively. This may be explained by the values of contact density for these values of H. We know that when H is an integer, the contact density of the fluid at the cylindrical wall and, thereby, the required work to change the fluid structure per unit area of the wall, increases as compared to the case with half-integer values. But the wallfluid contribution decreases for integer values of H compared to half-integer values. Also seen in Fig. 8b is the confinement effect for a bulk density equal to 0.6 which changes the sign of interfacial tension for the case of H equal to 2σ. In fact, at this density, ρbσ3 = 0.6 for all values of H, expect for pores with H sizes near 2σ for which the second contribution has the dominant role in the interfacial tension. This is while for cases in which H is around 2σ, the work required to change the structure is so large that the first term overcomes the second one. Fig. 9a is similar to Fig. 8a except that in this case, Rb is fixed at 6σ while Rs changes from 0.25σ to 4.75σ. As in Fig. 9a, the interfacial tensions for both bulk and confined fluids are expectedly positive around a hard cylindrical wall. Although the interfacial tensions in both cases decrease with Rs, the one for the confined fluid exhibits an oscillatory

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behavior with large amplitudes for large values of Rs. Clearly, the interfacial tensions of both bulk and confined fluids merge at each density when H becomes greater than 3 or 4 times the molecular diameter. In the absence of the wall-fluid attraction, the work required to produce an inhomogeneous structure per unit area of the cylindrical wall only has a contribution to the sign of the interfacial tension. Since this required work decreases with size, it is expectable that the interfacial tension decreases with the size of the cylindrical wall. Moreover, the work is greater for cases in which H is an integer as compared to those with half-integer values. This is because of the larger values for the contact density of confined fluids in cases where H is an integer value. Fig. 9b is similar to Fig. 9a except that this figure illustrates the fluid at the contact of an attractive cylindrical wall. It should be noted that the interfacial tension is negative in this case (the attractive wall) and increases with size. As expected, the values of interfacial tension for bulk and confined fluids also converge at larger values of H. Similar to Fig. 9a, the maximum and minimum values of interfacial tension are related to integer and half-integer values of H, respectively. In fact, the trend in increasing interfacial tension with size has been reversed by the change of wall potential from a hard to an attractive one (See Fig. 9a and b). This reversed behavior may be explained by reference to the contribution of

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wall–fluid attraction to the interfacial tension. For our present cases, the second contribution of the wall–fluid attraction is the dominant factor in determining interfacial tension; the sign of the interfacial tension is, therefore, negative. Increasing interfacial tension with size for an attractive cylindrical wall may be explained by the fact that more molecules exist per unite area of the wall when size increases. Figs. 8 and 9 also reveal the effect of the outer wall–fluid attraction in a bicylindrical pore on interfacial tension. It increases the amplitude of oscillations with size. This increase may be explained by the higher tendency of fluid molecules to enter the nanopore under the attraction of the outer wall. As it is clear from these figures that the increasing values of bulk density lead to increasing values of interfacial tension and oscillation amplitudes. 4.3. Effect of fluid confining on excess adsorption In this subsection, we investigate the behavior of the excess adsorption of a hard sphere fluid at the contact of both a hard and an attractive cylindrical wall as a result of its confinement in a cylindrical pore. For this purpose, we consider a hard sphere fluid in a bicylindrical pore for which Rb is rather large. We calculate the excess adsorption of the fluid at the inner cylindrical wall, Γ, via the following equation: Γ¼

1 Rs

Z

RS þ5  RS

 ρbicyl ðRÞ−ρbulk RdR

ð22Þ

where, ρbicyl(R) is the local density of the fluid in a bicylindrical pore with a radius of Rs for its inner cylinder and a very large value of Rb. To investigate the effect of fluid confinement on excess adsorption, we reduce the size of the outer cylinder from 10σ to the order of a few times the molecular diameter. The excess adsorption of the confined fluid in a bicylindrical pore around its inner cylinder, Γcon, is obtained as follows: Γ con ¼

Fig. 9. a) Comparison of the interfacial tensions of a hard sphere bulk fluid (.….) and a confined fluid at a hard convex cylindrical wall. The radius of the outer cylinder is fixed at 6σ while that of the inner cylinder varies from 0.25σ to 4.75σ. This figure presents the contact density results obtained for an outer cylinder taken to be hard (– – –) and an attractive (––––) one for two different bulk densities equal to 0.4 and 0.6. figure (b) is similar to figure (a) but it only illustrates the case for an attractive cylindrical wall.

1 Rs

Z 

 ρbicyl ðRÞ−ρcyl ðRÞ RdR

ð23Þ

where, in the above equation ρbicyl(R) is the local density of the fluid confined in a bicylindrical pore with a radius of inner cylinder equal to Rs and outer cylinder, Rb, in order of molecular diameter and ρcyl(R) is the local density of the fluid confined in a cylindrical pore. It should be noted that the lower limit of the integral of Eq. (21) is Rs and its upper limit is Rb for cases for which Rb b 6σ (Rs = 1σ), this is while we have considered the case of Rs + 5σ when Rb ≥6σ. The excess adsorption of hard sphere bulk and confined fluids in contact with a hard cylindrical wall with a radius equal to 1σ (Rs = 1σ) have been obtained and plotted in Fig. 10a. The size of the outer cylinder, Rb, varies from 2.25σ to 10σ and the bulk densities are taken equal to 0.4 and 0.6. The figure also shows the results for a fluid confined in a nanobicylindrical pore for which the outer wall is both hard and attractive. As is seen, the values of excess adsorption for both bulk and confined fluids are negative. Also, the adsorption for the confined fluid oscillates around its bulk value and reaches the bulk values for large values of Rb. According to Fig. 10a, the amplitude of excess adsorption oscillations for the confined fluid becomes smaller for larger values of Rb, or for larger values of H. This is due to the reduced effect of confinement on adsorption. It is expected that excess adsorption of the confined fluid should have a greater value for integer rather than halfinteger values of H, which may be explained by the higher contact densities for integer values of H. Our results also indicate that the change in the potential of the outer cylindrical wall from hard to attractive only leads to increasing height and depth of oscillations. Fig. 10b is similar to Fig. 10a but it is different in that it illustrates an attractive convex cylindrical wall with a radius equal to Rs = 1σ. Comparison of these two figures reveals that wall attraction may give rise to an increase in excess adsorption since it enhances contact density.

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Fig. 10. a) Comparison of the values for excess adsorption of a hard sphere bulk (.….) and a confined fluid at a hard convex cylindrical wall. The radius of the cylindrical wall is fixed at 1σ while that of the outer cylinder varies from 2.25σ to 10σ. The figure presents the results of contact density for both a hard (– – –) and an attractive (––––) outer cylinder for two different bulk densities equal to 0.4 and 0.6. Figure (b) is similar to figure (a) but it only illustrates the results for an attractive cylindrical wall.

Moreover, the excess adsorption of a bulk fluid usually is positive for a density equal to 0.4 (Fig. 10b). In fact, the excess adsorption decreases with bulk density that this may be due to the greater molecular packing of the fluid in the pore as compared to its bulk for the same density at smaller bulk density. To investigate the effect of cylindrical wall curvature on excess adsorption, the results obtained for excess adsorption with different values of Rs are presented in Fig. 11a. In this figure, the radius of the cylindrical wall varies from 0.25σ to 4.75σ while that of the outer cylinder is 6σ for the two different bulk densities of 0.4 and 0.6. Also, the potential of the outer cylindrical wall in this figure is considered as both hard and attractive. It is obvious that the excess adsorption of the bulk fluid at the cylindrical wall is negative and that it increases with Rs for a fixed density. The excess adsorption of the confined fluid initially increases with size (similar to what happens with a bulk fluid), but it exhibits an oscillatory behavior for smaller values of H, or for larger values of Rs. Similar to previous cases, maximum and minimum values of oscillations are related to integer and half-integer values of H, respectively. Fig. 11b is similar to Fig. 11a but it illustrates an attractive cylindrical wall. As seen in Fig. 11b, the excess adsorption may get both negative and positive values. Comparison of these two figures reveals that the trend in the variation of excess adsorption with Rs may be reversed

Fig. 11. a) Comparison of the values for excess adsorption of a hard sphere bulk (.….) and a confined fluid at a hard convex cylindrical wall. The radius of the outer cylinder is fixed at 6σ while that of the inner cylinder varies from 0.25σ to 4.75σ. The figure presents the results of contact density for both a hard (– – –) and an attractive (––––) outer cylinder for two different bulk densities equal to 0.4 and 0.6. Figure (b) is similar to figure (a) but it only illustrates the results for an attractive cylindrical wall.

when wall potential shifts from a hard to an attractive one. The different trends observed for excess adsorption versus size in Fig. 11a and b can be explained by entropy and energy effects. According to the entropy effect, molecules of a larger size have a greater tendency to accumulate at the hard convex cylindrical wall. Thus, excess adsorption increases with size while the energy effect overcomes on entropy effects in the presence of wall-fluid attraction and excess adsorption, therefore, decreases with Rs.

5. Conclusion In this work, the density functional theory was exploited to investigate the confinement effects on wall pressure, interfacial tension, and excess adsorption of a hard sphere fluid in contact with a convex cylindrical wall. Initially we presented a solution for the weighted density integrals that one advantage of using our solution in MFMT is its generality capturing all pores with a cylindrical geometry. This means that a unique formulation can be used for infinite or infinite lengths of cylindrical, bicylindrical, truncated cone and also for slit-like pores. The method proposed in this article was also shown to be far easier

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than the Fourier technique, using the elliptic function, and also there is no need either to define a new free energy function as has been proposed elsewhere. In a subsequent section, the wall pressure, interfacial tension, and excess adsorption of bulk and confined fluids in a bicylindrical pore at a convex cylindrical wall were obtained and compared. Results showed that these properties exhibit an oscillatory behavior in the case of the confined fluid as compared to the bulk fluid. Moreover, it was shown that the same properties exhibit a different behavior in the case of confined fluids from the corresponding ones of the bulk fluids when H is in the order of the molecular diameter. These two different sets of values were found to converge for large values of H. This difference was attributed to the structural differences between the confined fluid and the bulk fluid at the convex cylindrical wall of a bicylindrical pore. Based on our results, the wall pressure of a bulk fluid at the convex cylindrical wall exhibit increasing trends with Rs for both hard and attractive convex cylindrical walls. In contrast, the interfacial tension and excess adsorption of a bulk fluid at a hard convex cylindrical wall showed a decreasing and increasing trend respectively. Changing the potential of the convex cylindrical wall from hard to attractive was found to reverse the trend in interfacial tension and excess adsorption versus Rs or, in certain cases, to changes in the signs of these quantities. The same behaviors were observed for the wall pressure, interfacial tension, and excess adsorption of a confined fluid at the same cylindrical wall but only with some oscillation. Confining a fluid in a bicylindrical pore was also found to lead to an increase or a decrease in the values for wall pressure, interfacial tension, and excess adsorption depending on the integer or half-integer values of H, respectively (the maximum and minimum values in Figs. 3–11). The change in the potential of the outer cylinder from hard to attractive, however, was only observed to increase the amplitude of the oscillations in the case of the confined fluid.

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References [1] A. Waghe, J.C. Rasaiah, G. Hummer, J. Chem. Phys. 137 (2012) 044709. [2] H. Abe, T. Takekiyo, M. Aono, H. Kishimura, Y. Yoshimura, N. Hamaya, J. Mol. Liq. 210 (2015) 200. [3] Z. Ma, J. Bai, Y. Wang, X. Jiang, Appl. Mater. Interfaces 6 (2014) 2431. [4] S. Van der Perre, T. Duerinck, P. Valvekens, D.E. De Vos, G.V. Baron, J.F.M. Denayer, Microporous Mesoporous Mater. 189 (2014) 216. [5] W. Bi, M. Tian, K.H. Row, J. Chromatogr. B 880 (2012) 108. [6] P.E. Boukany, A. Morss, W. Liao, B. Henslee, H. Jung, X. Zhang, B. Yu, X. Wang, Y. Wu, L. Li, K. Gao, X. Hu, X. Zhao, O. Hemminger, W. Lu, G.P. Lafyatis, L.J. Lee, Nat. Nanotechnol. 6 (2011) 747. [7] C. Amine, J. Dreher, T. Helgason, T. Tadros, Food Hydrocoll. 39 (2014) 180. [8] J. Zhu, R.C. Hayward, J. Colloid Interface Sci. 365 (2012) 275. [9] F.M. Rocha, J.A. Miranda, Phys. Rev. E 87 (2013) 013017. [10] K. Bui, I.Y. Akkutlu, Mol. Phys. (2015) 1. [11] P.T.M. Nguyen, D.D. Do, D. Nicholson, J. Colloid Interface Sci. 396 (2013) 242. [12] N. Nijem, P. Canepa, U. Kaipa, K. Tan, K. Roodenko, S. Tekarli, J. Halbert, I.W.H. Oswald, R.K. Arvapally, C. Yang, T. Thonhauser, M.A. Omary, Y.J. Chabal, J. Am. Chem. Soc. 135 (2013) 12615. [13] D. Fu, J. Chem. Phys. 124 (2006) 164701. [14] P. Bryk, R. Roth, K.R. Mecke, S. Dietrich, Phys. Rev. E 68 (2003) 031602. [15] Helmi, E. Keshavarzi, J. Chem. Phys. 433 (2014) 67. [16] E. Keshavarzi, A. Taghizadeh, J. Phys. Soc. Jpn. 80 (2011) 044605. [17] E. Keshavarzi, A. Taghizadeh, J. Phys. Chem. B 114 (2010) 10126. [18] P. Bryk, L. Lajtar, O. Pizio, Z. Sokolowska, S. Sokolowski, J. Colloid Interface Sci. 229 (2000) 526. [19] P. Tarazona, Phys. Rev. Lett. 84 (2000) 694. [20] P. Tarazona, Phys. A 306 (2002) 243. [21] Gonzalez, J.A. White, F.L. Roman, S. Velasco, J. Chem. Phys. 125 (2006) 064703. [22] X. Kong, J. Wu, D. Henderson, J. Colloid Interface Sci. 449 (2015) 130. [23] J. Malijevsky, Chem. Phys. 126 (2007) 134710. [24] E. Keshavarzi, A. Helmi, J. Phys. Chem. B 119 (2015) 3517. [25] Y.X. Yu, F.Q. You, Y. Tang, G.H. Gao, Y.G. Li, J. Phys. Chem. B 110 (2006) 334. [26] X. Yu, J.Z. Wu, J. Chem. Phys. 117 (2002) 10156. [27] R. Roth, R. Evans, A. Lang, G. Kahl, J. Phys. Condens. Matter 14 (2002) 12063. [28] Y. Tang, J.Z. Wu, J. Chem. Phys. 119 (2003) 7388. [29] G.A. Mansoori, N.F. Carnahan, K.E. Starling, T.W. Laland, J. Chem. Phys. 54 (1971) 1523. [30] F.L. Roman, J.A. White, A. Gonzalez, S. Velasco, J. Chem. Phys. 124 (2006) 154708. [31] E. Keshavarzi, A. Helmi, J. Mol. Liq. 198 (2014) 246.