Population Inversion, Selective Adsorption, and Demixing of Lennard

Apr 10, 2014 - Ezat Keshavarzi* and Abbas Helmi. Physical Chemistry Group, Chemistry Department, Isfahan University of Technology, Isfahan 8415683111,...
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Population Inversion, Selective Adsorption, and Demixing of Lennard-Jones Fluids in Nanospherical Pores Ezat Keshavarzi* and Abbas Helmi Physical Chemistry Group, Chemistry Department, Isfahan University of Technology, Isfahan 8415683111, Iran ABSTRACT: The density functional theory has been employed to investigate the population inversion, selective adsorption, and demixing of confined mixture fluids in a spherical nanocavity. In the case of hard sphere fluids for which only the entropy effect has the dominant role, the selective adsorption process strongly depends on size ratio, population of the adsorbed component, and pore size. The effects of such parameters as interaction strength, size ratio, and thermodynamic state on population inversion and selective adsorption have been investigated for L-J mixture fluids. The results for L-J asymmetric binary mixture fluids indicate that the mole fraction of large species (molecules with bigger radii) inside the cavity becomes greater with increasing size ratio or with decreasing temperature than does that for the other component despite its lower population in the bulk fluid (i.e., the so-called population inversion phenomenon). Our results indicate that the inversion population density decreases with size ratio, and the mole fraction of the component with the bigger radius in the pore increases with temperature. Thus, by selecting a small spherical cavity under special conditions, it will be possible to give rise to the selective separation of component 2 in spite of its lower concentration in the bulk asymmetric L-J mixture. Finally, we have investigated the phase separation, demixing phenomenon, of an asymmetric L-J mixture inside a spherical cavity. Also we investigated the cases for which the layered demixing phenomenon occurs in the asymmetric L-J fluid in a nanospherical pore as a result of the difference between the entropy and energy effects.

I. INTRODUCTION The thermodynamic properties of confined fluids has recently been a topic of great interest among scientists working in both physics and chemistry. 1−5 It is well-known that the thermodynamic properties of a confined fluid in nanoscale systems are drastically different from those of the bulk fluid6,7 and that these properties are completely different for fluids confined in three-dimensional (small) systems with one or two dimensions. It is known that the molecular structures and, therefore, the thermodynamic properties of fluids in small systems strongly depend on pore size, pore geometry, and the number of molecules confined in the pore. A theoretical description of such properties of confined fluids in threedimensional (small) systems is important for industrial applications. Examples of such materials with collective pores of nanometer size in three dimensions include zeolites, clathrates, and fullerenes that are mostly used as the substrate in the selective separating process and as catalyst supports.8,9 In this work, we focus on the population inversion phenomenon, selective adsorption, and phase separation of mixture fluids confined in nanoscale spherical pores. Most studies in the past have concentrated on the population inversion phenomenon and selective adsorption of mixtures in slit pores and cylindrical nanopores.4,10 However, their behavior in spherical cavities still awaits further investigation. The selective separation of mixtures has numerous applications in medicine, industry, bioscience, biotechnology, © 2014 American Chemical Society

and biomedicine. In biological systems and in the food industry, for example, the separation of bioproducts such as proteins and enzymes usually poses serious problems that may be resolved by selective adsorption in small pores.11 Another important phenomenon observed in mixtures is demixing, sometimes called phase separation, which is of interest to the present work. Phase separation has been experimentally studied in such systems as colloid polymer mixtures, latex−polystyrene, silica−polydimethylsiloxane colloidal dispersions, and in biological systems containing proteins or DNA.12 Examples of its applications in food industry and biological systems include separation of milk protein (casein) and amylopectin13 and separation of the tobaccomosaic virus from plants by adding nonadsorbing polymers to the tobaccomosaic virus,14 respectively. Hence, it is interesting to study these properties of mixtures confined in nanoscale pores. This paper aims to investigate the above-mentioned properties, namely population inversion phenomenon, selective adsorption, and phase separation of binary asymmetric mixtures confined in nanoscale spherical cavities. The fluid in a spherical pore is in equilibrium with a bulk fluid via a tiny hole which cannot affect the spherical symmetry of the cavity.15,16 Received: November 24, 2013 Revised: April 5, 2014 Published: April 10, 2014 4582

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and nα(r) is the weighted density. This is a function of fundamental geometric measures of the particles defined by

A number of theoretical methods are available for studying fluid structure and, thereby, the thermodynamic properties of fluids confined in nanoscale spherical cavities. The density functional theory (DFT) is one of the precipitate tools for investigating confined systems. The results obtained from this theory are reportedly in good agreement with those obtained from molecular dynamic simulation.17 The fundamentalmeasure theory has been successfully employed for 3D fluids in bulk and in slit-like pores, but it was not successful in describing solids.18 Therefore, Gonzalez et al. employed both the quasi zero-dimension (a cavity that cannot hold more than one particle) functional approximation and the original fundamental measure theory of Rosenfeld to study the properties of hard sphere fluids confined in spherical pores.19 Soon Chul Kim et al. used DFT to study density profiles of binary hard sphere mixtures confined in spherical pores.20 In 2002, Ioannis A Hadjiagapiou employed the density-functional theory to investigate the equilibrium local densities, structure, and wetting of a one-component fluid in a spherical cavity.9 Recently, modified fundamental measure theory, MFMT, has attracted more attention for its ability to predict certain inhomogeneous phenomena such as adsorption, wetting, and freezing that are not predictable by the integral equation theory.17 MFMT is, therefore, used in the present work. The paper is organized as follows: In section II, a brief review of MFMT is provided. Section III explores the population inversion and selective adsorption of binary hard sphere and binary asymmetric L-J mixtures and investigates the effects of such parameters as reduced temperature, interaction strength, and size ratio. Section IV investigates the phase separation behavior of L-J mixture fluids inside a spherical cavity. Finally, conclusions are presented in Section V.

n

nα(r) =

∑ ∫ d r′ ρi (r′) wi(α)(r − r′)

(5)

i−1

and w(α) i (r − r′) is a weight function that involves two vectorial functions and the following four scalar functions:17,21 wi(3)(r) = θ(σii /2 − r ) wi(2)(r) = δ(σii − r ), wi(1)(r) =

wi(2)(r ) (0) w(2)(r ) , wi (r) = i 2 2πσii πσii (6)

where, θ is the Heaviside step function and δ is the Dirac delta function. The vectorial functions are17,21 wiv(2)(r) =

r δ(σii /2 − r ), |r|

wiv(1)(r) =

wiv(2)(r) 2πσii

In the density functional theory, the grand canonical potential of an inhomogeneous system is defined as2 Ω[ρ1(r), ρ2 (r)] =

1 F[ρ (r), ρ2 (r)] + kT 1

2

∑ ∫ d rρi (r)[Viext(r) − μi ] i=1

(8)

where, μi is the chemical potential of component i and is the external potential for this component. It should be noted that Fex [ρ1(r), ρ2(r)] in eq 1 takes the form Fhs ex[ρ1(r), ρ2(r)] for a hard sphere mixture but is defined as in 9 below for a Lennard-Jones fluid: Vext i

II. THEORY The intrinsic Helmholtz free energy functional of an inhomogeneous binary mixture is defined as

Fex[ρ1(r), ρ2 (r)] = Fexhs[ρ1(r), ρ2 (r)] + Fexatt[ρ1(r), ρ2 (r)] (9)

F[ρ1(r), ρ2 (r)] = Fex[ρ1(r), ρ2 (r)] + Fid[ρ1(r), ρ2 (r)]

In the study of the Lennard-Jones mixture, use was made in this study of the perturbation theory in which Fhs ex[ρ1(r), ρ2(r)] is the reference part and Fatt ex [ρ1(r), ρ2(r)] is the perturbative part defined in the mean-field approximation for a binary mixture as follows:2

(1)

where, ρ1(r) and ρ2(r) are the distribution functions of components 1 and 2,. Fid [ρ1(r), ρ2(r)] is defined by 2

Fid[ρ1(r), ρ2 (r)] = kT ∑

∫ d r ρi (r){ln Λι3ρi (r) − 1}

Fexatt[ρ1(r), ρ2 (r)] 1 ∑ = d r′ d r ρi (r) ρj (r′) uij(att)(r − r′) 2 i , j = 1,2

i=1

(2)

∫∫

In the above equation, Λi = h/(2πmikT)1/2 is the de Broglie wavelength of component i, k is the Boltzmann constant, and T is the absolute temperature. The excess part of the free-energy function of an inhomogeneous mixture of a hard sphere is expressed as Fexhs[ρ1(r), ρ2 (r)] = kT

∫ d r Φhs[{nα(r)}]

(7)

(10)

where, u(att) ij (r − r′) is the attraction between the particles while i and j denote the components of the mixture. This version of perturbation theory was used because of the higher precision of the Weeks−Chandler−Anderson (WCA) method in contrast to other perturbative methods. According to WCA, u(att) is ij given by22

(3)

where, Φhs[{nα(r)}] is the excess free-energy density. The final expression for the excess free energy density includes both scalar and vector contributions:17 n n − nv1nv2 Φ hs[{nα(r)}] = − n0 ln(1 − n3) + 1 2 1 − n3 2 ⎡ ⎤ n3 (n 3 − 3n2nv2 . nv2) 1 ⎢n3 ln(1 − n3) + ⎥ 2 + 2 36π ⎣ (1 − n3) ⎦ n3 3

⎧ r < rij ,min −εij ⎪ uij(att)(r ) = ⎨ (LJ) ⎪ ⎩ uij (r ) r ≥ rij ,min

(11)

where εij is the LJ parameter and r is the distance from the center of one particle to another. We divided the potential at rij,min = 21/2σij in the WCA method. u(LJ) ij (r) is the (12−6) L-J potential between the particles of the fluid:

(4) 4583

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⎡⎛ σij ⎞12 ⎛ σij ⎞6 ⎤ uij(LJ )(r ) = 4εij⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r ⎠⎦ ⎣⎝ r ⎠

In this study, all the results are obtained in the grand canonical ensemble. The Lorentz−Berthelot combining rules are applied here for the Lennard-Jones mixture to calculate the cross interaction parameter (σ12 = (σ22 + σ11)/2, ε12 = (ε11ε22)1/2). The interaction ratio and size ratio for the L-J parameter of the components are β = ε22/ε11 and α = σ22/σ11, respectively. The mole fraction of component i in the bulk mixture is Xi,out = ρi/ρt, where ρi is the number density of the ith species and ρtσ311 = ρ1ρ311 + ρ2ρ311. The mole fraction of the 2 ith component is Xi,in = ρi /ρt in which ρi = 4π/V ∫ Rs 0 ρi(r) r dr is the average number density of the ith species inside the spherical cavity and ρt σ311 = ρ1σ311 + ρ2 σ311. In this work, the Picard iterative method is used according to which the iteration is repeated until the difference between density profiles at each point becomes less than 0.0001.

(12)

where, σij is the LJ molecule diameter. In this work the effective hard sphere diameter has been calculated by d (T ) =

∫0

rij,min

⎛ ⎛ (0) ⎞⎞ ⎜1 − e⎜ −uij (r ) ⎟⎟ dr ⎜ kT ⎟⎟ ⎜ ⎝ ⎠⎠ ⎝

where ⎧ u (LJ )(r ) + ε r < r ⎪ ij ij ij ,min uij(0)(r ) = ⎨ ⎪ 0 r ≥ rij ,min ⎩

It should be noted that this integral is based on the Barker− Henderson perturbation theory, which accounts for the actual repulsive part of the Lennard-Jones intermolecular potential, but the value of hard sphere diameter obtained from this integral is closed to the WCA ones.23,24 Generally, the equilibrium density profile of inhomogeneous fluids can be obtained by minimizing the grand canonical potential as follows:4

III. POPULATION DISTRIBUTION OF A BINARY FLUID MIXTURE IN A NANOSPHERICAL CAVITY A. Hard Sphere Mixtures. In this section, we study the population distribution of hard sphere mixtures in a spherical cavity which is in equilibrium with bulk mixture fluids. It is clear that the density profile and population distribution of hard sphere fluids in spherical cavities depend on various parameters such as the radius of the spherical cavity, the size ratio of the hard sphere mixture, bulk density, and mole fraction. We consider a bulk binary fluid mixture with species 1 and 2 and a size ratio equal to 1.5 which is in equilibrium with a fluid mixture in a hard spherical cavity and a structureless wall at a radius equal to 5σ11 (Rs = 5σ11). The bulk density was chosen such that average numbers of particles (i.e., N1 = 23 and N2 = 66) would enter the pores because the MD results are available for such conditions. In Figure 1, the density profiles of species

⎡ ρi (r) = ρbi exp⎢ −V iext(r)/kT + μi ext /kT ⎢⎣ ∂Φ (α) − ∑ d r′ wi (r − r′) ∂ nα α

∫

⎤

− 1/kT

∑ ∫ ∫ d r′ρj (r′)uij(att)(r − r′)]⎥⎥ ⎦

i , j = 1,2

(13)

In this study, the external potential is given by ⎧ ⎡ ⎤ σ R s − 2ii − r ⎥ ⎪ ⎢ σii − εw , i exp⎢ − λ w ⎪ ⎥ r < Rs − ⎪ σ 2 ii ⎢⎣ ⎥⎦ V iext(r ) = ⎨ ⎪ σ ⎪ ∞ r > R s − ii ⎪ ⎩ 2

((

)

)

(14)

where, εw,i is the value of the potential energy at the contact point for each species and λw is the wall−fluid potential parameter which is equal to λw = 1.8. The equations and integration limits reported in ref 17 are not usable for r < σ/2. In this work integration limits are the same as ref 17 for r > σ/2. It means for r > σ/2 they are from r − σ/2 to r + σ/2 for n2(r) and n3(r) which represent the surface and the volume of sphere with radius σ/2 around r. But theses values cannot be used for ranges less than σ/2. Therefore, in all previous references all the calculations have been reduced to convolutions that can be conveniently handled by means of the Fourier techniques.19 It is very interesting that with the limits of the integrals for r < σ/2 defined in such a way to generate the volume and surface of a sphere with radius σ/2, for n3(r) and n2(r), respectively, we get the correct from of density profile. It should be noted that the results obtained in this way are in a very good agreement with those obtained by the Fourier techniques or MD simulation.19 We will present more results about this subject in our future work.

Figure 1. Density profiles of a hard sphere mixture in a spherical cavity. The bulk densities of species 1 and 2 are chosen in such a way to have N1,in = 23 and N2,in = 66, the size ratio being equal to α = 1.5, and the radius of the spherical cavity to be Rs = 5σ11. The circles are the Monte Carlo simulation.20 The solid and medium dash lines are DFT results.

1and 2 in the spherical cavity have been plotted versus r, and compared with simulation data.20 Clearly, the density profiles of the hard sphere fluids in the pore exhibit an oscillatory behavior. The number of peaks in the density profiles represents the number of molecular layers inside the spherical 4584

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pore. According to this Figure, the DFT results are in good agreement with simulation data. It is known that the structure of fluid mixtures in nanopore systems depends on such parameters as pore size, size ratio, intermolecular interactions of different species, and the wall− fluid interactions in addition to the thermodynamic states of the bulk fluid mixture with which it is in equilibrium. The effects of these parameters may be summarized and explained by two thermodynamic effects, entropy and energy effects. Since the simultaneous study of these two effects is difficult, we consider a hard sphere fluid mixture in a hard spherical cavity, which is only affected by entropy.25 To investigate the role of size ratio and pore size on the population distribution of the hard sphere mixture, we consider a bulk of equimolar hard sphere mixture in equilibrium with a fluid mixture in a spherical cavity. In Figure 2, the mole fraction of species 2 inside the spherical Figure 3. The mole fraction of component 2 of a hard sphere mixture inside a spherical cavity as a function of the radius of the spherical cavity with α = 1.6 and the bulk composition of the hard sphere mixture being X2,out = X1,out = 0.5 for different values of total bulk density including 0.1, 0.15, and 0.2.

Figure 2. Mole fraction of components 1 and 2 inside a spherical cavity as a function of size ratio with Rs = 5σ11. The bulk composition of the hard sphere mixture is X2,out = X1,out = 0.5 for different values of total bulk density including 0.1, 0.15, and 0.2. Figure 4. Mole fraction of species 2 of a hard sphere mixture inside a hard spherical cavity as a function of X2,out with different values of size ratio for Rs = 5σ11 and ρ2σ311 = 0.1.

cavity (X2,in) has been plotted versus size ratio (α) for a spherical nanocavity with a radius equal to 5σ11 for the different bulk densities of 0.1, 0.15, and 0.2. Evidently, the mole fraction of species 2 inside of the spherical cavity increases with both size ratio and bulk density. In fact, by increasing the size of species 2, the excluded volume inside the spherical cavity and, thereby, the entropy of the system increases by the adsorption of species 2. It should be noted that in nanospherical pores, the entropy effect is strongly dependent on pore size because of the 3D confinement. To obtain a better understanding of this effect on the population distribution of hard sphere fluid mixtures, the mole fraction of species 2 is plotted in Figure 3 as a function of the radius of a spherical cavity for an equimolar bulk fluid mixture at a fixed size ratio equal to 1.6 for the different total bulk densities of 0.1, 0.15, and 0.2. According to this Figure, when the spherical cavity is comparable in size with hard sphere molecules, the mole fraction of species 2 in the spherical cavity increases with variations in the radius of the spherical cavity, but for large values of Rs, it increases more slowly with variations in Rs. The mole fraction of the fluid mixture inside the nanospherical pores strongly depends on α and Rs; therefore, it is possible to observe a selective separation by varying these two

parameters. To do so, the mole fraction of component 2 inside the spherical cavity has been plotted in Figure 4 as a function of the mole fraction of species 2 in a bulk fluid with Rs = 5σ11 for different values of size ratio at a total bulk density of ρtσ311 = 0.2. It is clearly seen that the adsorption of species 2 increases with α and, as expected, by the mole fraction of species 2 in the bulk fluid. According to this Figure, for low concentrations of species 2 in the bulk fluid (X2,out < 0.2), component 1 has a higher concentration inside the cavity. This may be explained by the greater entropy generated by component 1 because of its size as compared to the mixing entropy. In fact, the concentration in the pore is determined by the comparison of the mixing entropy and the entropy produced by the inhomogeneous distribution of each component with respect to the wall of nanospherical pore. For our case when x < 0.2, the entropy generated by the inhomogeneous distribution of small components is more than the entropy generated by the mixing of two components. Therefore, component 1 is individually adsorbed into the spherical cavity. In certain cases, say for α = 2, the mole fraction of species 2 rapidly increases inside the 4585

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beyond the total density for which the mole fractions of components 1 and 2 inside the spherical cavity become equal. Therefore, population inversion is observable for ρtσ311 > ρt,invσ311 as shown in Figure 5. Note that the concentration of species 2 in the bulk is very low (0.008). It is also clear from this figure that inversion density decreases with size ratio and that the density range at which population inversion occurs in the nanopores, therefore, increases with size ratio. In fact, the entropy effect causes more adsorption of the molecules of species 2 into the nanospherical pore. This effect becomes more pronounced with increasing size ratio. Population inversion, therefore, occurs at lower values of bulk density. Another point of interest in this figure is the observation that the selective adsorption phenomenon also occurs for high bulk density limits. This means component 2 is selectively adsorbed into the spherical cavity from the mixture, giving rise to selective separation. Clearly, the selective adsorption phenomenon strongly depends on size ratio such that the density at which selective separation occurs decreases with size ratio. To understand the effect of intermolecular interactions on the population distribution of a L-J mixture, the mole fraction of component 2 which is in equilibrium with a bulk fluid with X2,out = 0.008 has been plotted in Figure 6 as a function of total

spherical cavity with increasing mole fraction of species 2 in the bulk fluid. This may give rise to the selective separation phenomenon for species 2 due to the entropy effect. As already shown, the population distribution is controlled by the entropy effect for a hard-sphere mixture, while it is controlled by both energy and entropy effects for a LennardJones mixture. In some cases, therefore, we may observe a population inversion or demixing phenomena. In the next section, we will investigate the population distribution of Lennard-Jones mixture fluids into a spherical cavity. B. Lennard-Jones Mixtures. In this section, we study the effects of interaction strength, size ratio, pore size, and thermodynamic states on the population distribution of a Lennard-Jones fluid in a nanospherical cavity. For a Lennard-Jones mixture fluid which is confined in a nanospherical cavity and is in equilibrium with a bulk fluid mixture, the wall becomes hard and structureless. To determine the effect of size ratio on the population distribution of a L-J fluid mixture, we have plotted in Figure 5 the mole fraction of

Figure 5. Mole fraction of species 2 of a L-J mixture fluid (boron trifluoride and water (α =1.4), diborane and water (α = 1.5)) in a hard spherical cavity versus total bulk density for T* = 1.4, β = 0.3, Rs = 5σ11, and X2,out = 0.008 for different values of size ratio including 1.4, 1.5, 1.6, and 1.7. Figure 6. Mole fraction of species 2 of a L-J mixture fluid (diborane and water) in a hard spherical cavity as a function of total bulk density with α = 1.5, β = 0.3, Rs = 5σ11, and X2,out = 0.008 for different values of reduced temperature including 1.4, 1.5, and 1.6.

species 2, X2,in, inside a spherical pore as a function of the total bulk density for different values of size ratio including 1.4, 1.5, 1.6, and 1.7. In this figure, the radius of the spherical cavity is equal to 5σ11, the reduced temperature, interaction ratio, and mole fraction of component 2 in the bulk fluid being T* = (kT)/ε11 = 1.4, β = 0.3, and 0.008, respectively. The values of β and α have been chosen according to the binary mixture of born trifluoride and water for α = 1.4 and also water and diborane26,27 for α = 1.5. The mole fraction of component 2 in the spherical pore increases with both the total bulk density and the size ratio. As expected, the concentration of species 1 in the cavity is higher for a fixed value of α than that of species 2 at low total bulk densities. But, the mole fraction of species 2 inside the spherical cavity increases with total bulk density until it becomes greater than that of species 1 for high values of total bulk density; this is the population inversion phenomenon. In fact, the inversion happened when the mole fraction of one component in the spherical pore became greater than that of the other in spite of its lower concentration in the bulk. An inversion total density, ρt,out, (dotted line in Figure 5) occurs

bulk density for a nanopore with Rs = 5σ11 and different values of T* = (kBT)/ε11 including 1.4, 1.5, and 1.6. The values for α and β are 1.5 and 0.3, respectively. The values of β and α have been chosen according to binary mixture of water and diborane26,27 for α = 1.5. On the basis of this Figure, the mole fraction of component 2 in the nanopore increases with bulk density until it nears unity, but it decreases with temperature. We know that the composition of a fluid mixture inside a spherical cavity may be determined by the difference between the two entropy and energy effects. In this case, the intermolecular interaction of component 1 is found to be stronger than that of component 2. Clearly, the concentration of species 2 is higher for high density limits than that of species 1; hence, the population inversion occurs. 4586

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however, no study has been dedicated to the effect of energy on the demixing of fluids into spherical pores. For this purpose, we investigated the asymmetric L-J mixture fluid confined inside a spherical cavity which is in equilibrium with the bulk fluid. It is clear that the energy effect, which influences the structure of the fluid inside a spherical pore, may be characterized in either of two ways: by the wall energy effect or by the molecule− molecule interaction. In our case of a L-J fluid in a hard spherical pore, however, the energy effect is determined only by molecule−molecule interactions. To investigate a demixing phase separation in a L-J fluid mixture, we consider a L-J fluid confined in a spherical cavity with a radius of 5σ11 (Rs = 5σ11) in equilibrium with a bulk fluid. The bulk density of species 2 at a bulk fluid is equal to ρbσ311 = 0.1. The mole fraction of component 2 is equal to 0.95, and the reduced temperature and interaction ratio are T* = (kT)/ε22 = 1 and β = 0.2, respectively. In Figure 8, the equilibrium density profiles of the

Figure 6 also indicates that inversion density increases with temperature. It follows then that the bulk density range at which population inversion occurs decreases with temperature. Also, selective separation in the mixture occurs at a lower bulk density when temperature increases. This may be explained by the energy effect; in fact, particularly the component with the weaker intermolecular interaction is adsorbed into the spherical pore at low temperatures. To investigate the effect of the mole fraction of component 2 in the bulk fluid on the population distribution of the L-J mixture, the mole fraction of component 2 in a nanospherical pore has been plotted in Figure 7 versus total bulk density for

Figure 7. Mole fraction of species 2 of a L-J mixture fluid (diborane and water) into a hard spherical cavity as a function of total bulk density with α = 1.5, β = 0.3, Rs = 5σ11, and T* = 1.4 for different values of X2,out including 0.005, 0.008, 0.025, and 0.05. Figure 8. Density profile of L-J mixture fluids with 0.95 for the value of the mole fraction of component 2 at a temperature equal to T* = 1 inside a spherical pore with a radius of Rs = 5σ11 for different values of size ratio including 1.95, 2, and 2.1; the bulk density of species 2 and interaction ratio of the fluid are ρbσ311 = 0.1 and 0.2, respectively.

different values of the mole fraction of component 2 in the bulk including 0.005, 0.008, 0.025, and 0.05 for Rs = 5σ11. The reduced temperature of the mixture fluid is T* = (kT)/ε11 = 1.4 and the size ratio and β are equal to 1.5 and 0.3, respectively. The values of β and α have been chosen according to the binary mixture of water and diborane26,27 for α = 1.5. It is clear from this figure that the mole fraction of component 2 inside the nanospherical pore slowly increases with total bulk density, then it increases faster, before it finally becomes nearly flat. By increasing the mole fraction of component 2 in the bulk fluid, the inversion density is shifted to the lower bulk density. Also, by increasing X2,out, selective adsorption occurs at lower values of total bulk density. Since entropy and energy effects in L-J cases may influence the structure of the fluid mixtures inside a spherical pore, demixing or phase separation may occur in certain cases. We will explore these cases in the next section.

L-J mixture inside a nanopore has been plotted versus r for different values of size ratio including 1.95, 2, and 2.1 for species 1 and 2. It is interesting that the molecules of species 2 exhibit a strong tendency to aggregate around the wall of the spherical pore such that its density in other spaces of the pore is zero. The molecules of species 1, on the other hand, aggregate considerably at the center of the spherical pore. It is clear that the structure of fluid mixtures inside a spherical pore is determined by the difference between the entropy and energy effects. In fact, the distribution of the two components inside the spherical cavity is controlled by size ratio (entropy effect) and the intermolecular interaction strength (energy effect) between the two species. Since species 1 has a stronger intermolecular interaction (energy effect) than species 2, it has a greater tendency to aggregate at the center of the spherical cavity. Also by increasing the size of component 2, it preferentially adsorbs to the cavity wall due to the entropy effect. Thus, the phase separation of the L-J mixture fluid into the spherical pore is seen in this figure. In this way, it is possible to find some thermodynamic states for which the molecules of the L-J mixture fluid show phase separation or demixing. Our study shows that this phase separation happens when α = 1.95,

IV. DEMIXING OF ASYMMETRIC L-J MIXTURE FLUIDS INSIDE A NANOSPHERICAL CAVITY Widom and Rowlinson were the first to study phase separation in nonadditive binary mixtures.28 For many years, it was believed that additive fluid mixtures demixed in no way.29 In 1991, however, Biben and Hansen used the integral equation to predict the demixing of additive fluid mixtures.30 The phase separation process of hard sphere and asymmetric L-J mixtures in bulk fluids has been the subject of much research reported in the literature.30−32 To the best of the authors’ knowledge, 4587

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2, 2.1; T* = (kT)/ε22 = 1; and β = 0.2 for all examined radii of spherical cavity up to 50σ11. Figure 9a, depicts the equilibrium density profiles of the LJ mixture with α = 2.4 inside a nanopore with a radius of 5.5σ11

layer of molecules in the spherical pore has vanished. In fact, the accessible volume in the pore becomes small in comparison with the size of the molecules when the pore size is decreased. This causes the densities of species 1 and 2 to become zero in the center of the pore. To understand the role of wall energy effects on the density profile of a L-J mixture inside a spherical cavity, we consider the L-J mixture fluid confined inside an hard spherical cavity with an attractive tail with a wall potential ratio of ζ = (εw,2)/(εw,1) = 9 (see eq 14) which is in equilibrium with a bulk fluid at a density of 0.11 and 0.9 for the mole fraction of component 2. In Figure 10, the density profile of the L-J mixture with α = 1.1,

Figure 10. The density profile of a L-J mixture with α = 1.1, T*=2, and β = 0.1 inside an hard spherical cavity with an attractive tail with a wall potential ratio of ζ = (εw,2)*/(εw,1) = 9 with εw,1 = 1, which is in equilibrium with a bulk fluid at a density equal to 0.11 with 0.9 for the mole fraction of component 2. The radius of the spherical pore is equal to Rs = 5 σ11.

T* = (kT)/ε22 = 2 and β = 0.1 is plotted versus r. Clearly, the two layers of component 2 are near the wall of the spherical cavity and the molecules of component 1 have aggregated at its center; this is the so-called fluid−fluid demixing phenomenon. In fact, component 2 is especially adsorbed by the wall of the cavity because of the wall energy effect, and the molecule− molecule interaction causes the molecules of component 1 to congregate at the center of the pore. It should be noted that the wall energy effect gives rise to the phase separation process which occurs for a low size ratio (α = 1.1) in Figure 10.

Figure 9. Equilibrium density profiles of an LJ mixture with α = 2.4 inside a nanopore in equilibrium with a bulk fluid at a density of 0.072 with 0.972 for the mole fraction of component 2 at a reduced temperature and an interaction ratio of T* = 1.25 and β = 0.2, respectively: radius of the spherical pore for (a) Rs = 5.5σ11 and for (b) Rs = 5σ11.

(Rs = 5.5σ11). The mixture is in equilibrium with a bulk fluid at a density equal to 0.072 with the mole fraction of component 2 equal to 0.972. The reduced temperature and interaction ratio are T* = (kT)/ε22 = 1.25 and β = 0.2, respectively. It is seen that species 2 of the mixture inside the nanocavity is periodically demixed. In this case, the species with a big diameter, component 2, is specifically adsorbed onto the cavity wall while the other species stands in the second layer with respect to the wall. The same sequence repeats for the other layers. It should be noted that species 1 is expected to be located in the center of the pore due to the entropy effect. In fact, the layered demixing phenomenon occurs in the asymmetric L-J fluid in a nanospherical pore as a result of the difference between the entropy and energy effects. Figure 9b is the same as 9a with the exception that it depicts the case of a L-J mixture inside a nanospherical cavity with a radius equal to 5.5σ11. It is clear from this figure that the central

V. CONCLUSION MFMT was used to investigate the effect of various parameters such as density, interaction strength, size ratio, and pore size on population inversion and selective adsorption of a binary asymmetric Lennard-Jones fluid in a spherical cavity. Also since there is no study on the effect of energy on the demixing of fluids into spherical pores, we investigated it for the asymmetric L-J mixture fluid confined inside a spherical cavity. For further clarification of the role of the entropy effect, the population inversion and selective adsorption of a binary hard sphere fluid in a hard spherical cavity was specifically selected for study. The entropy effect was not able to give rise to the population inversion phenomenon in the case of a hard sphere mixture but it did for the Lennard-Jones fluid possibly due to the difference between the two entropy and energy effects. The 4588

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component with the bigger radius was spatially adsorbed into the spherical pore as a result of increasing β and decreasing T* with a lower bulk concentration. Our results show that by increasing the size ratio, the component with the bigger radius is selectively adsorbed into the spherical pore due to the entropy effect. Therefore, population inversion and selective separation phenomena can be restricted for each of the two species by making changes in one of these parameters. Finally, we investigated the phase separation process of a L-J mixture inside a nanospherical pore for which the Lorentz− Bertolet combining rules were employed to calculate the pair energy parameter. Our results show that, for high values of size ratio, the component with the bigger radius is spatially adsorbed unto the wall of the spherical pore because of the entropy effect. At low temperatures or for low values of β, however, the component with the smaller radius aggregates at the center of the spherical cavity because of the energy effect. Therefore, for high values of size ratio and low interaction ratio, the L-J mixture will be separate over the whole range of Rs due to the confinement effect. Finally, it was found that it would be possible to find some thermodynamic states for which the molecules of a L-J mixture fluid would be demixed.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +98-311-391-3281. Fax: +98-311-391-2350. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to acknowledge the Isfahan University of Technology Research Council for their financial support. We also thank Dr. Ezzatollah Roustazadeh from The English Language Center of Isfahan University of Technology for having edited the final English manuscript.

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REFERENCES

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