Population Inversion of Binary Lennard-Jones Mixtures in Nanoslit

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Population Inversion of Binary Lennard-Jones Mixtures in Nanoslit Pores (A Density Functional Theory Study) Ameneh Taghizadeh and Ezat (Tahmineh) Keshavarzi* Department of Chemistry, Isfahan University of Technology, Isfahan, Iran 8415683111 ABSTRACT: The aim of this work is to investigate the population inversion of binary asymmetric Lennard-Jones mixtures inside nanoslit pores due to confinement effects for both vapor and liquid phases. For this purpose we have used mean field fundamental measure theory, and the effect of different parameters such as interaction strength and size ratios of the components, confinement size, and thermodynamic state on the population distribution of molecules have been studied. It has been shown that in the case of bulk liquid mixtures, increasing the role of confinement effects can lead to preferential adsorption of the component with larger size and weaker intermolecular interactions into the nanopore in spite of its minority in the bulk which is referred as population inversion. This population inversion phenomenon is terminated by a sudden condensation which, interestingly, involves a simultaneous adsorption and desorption for more and less bulk concentrated species, respectively. We have demonstrated that this condensation phenomenon shifts to higher bulk densities with increasing the role of confinement effects such that in some cases population inversion is observable for the whole range of densities. In consideration of the conditions in which vapor Lennard-Jones mixtures undergo capillary condensation, the population distribution of components in the vapor- and liquidlike phases was studied. It has been shown that variation of parameters such as interaction strength and size ratios, temperature, and confinement size can lead to conditions in which capillary condensation is accompanying with a population inversion phenomenon. In these cases, whereas the composition of vaporlike phases is the same as bulk fluid, liquidlike phases are richer in the component with less bulk concentration.

I. INTRODUCTION Confined systems have attracted a great deal of attention during the past years because of their theoretical and practical relevance. It is well-known that confinement effects can induce new types of phase transitions1 including wetting, capillary condensation, and layering in fluids, whereas there are not any counterparts for them in the bulk. Understanding the mechanism of such phenomena is of importance in industrial applications in such areas as phase separation, chromatography, and oil recovery. These phase behaviors are more complicated and even richer in the case of confined mixtures. Although there are a number of studies24 on the phase behavior of confined mixtures, theoretical description of this subject still requires further investigation. Selective adsorption from mixtures is a frequently encountered process especially in medicine, biology, natural gas purification, and design of porous polymeric membranes in fuel cells. Studying the factors affecting adsorption process and also searching for the most appropriate conditions under which separation can be done is, therefore, an active area of research.516 A large number of theoretical and simulational works58 have shown that the adsorption of fluid molecules into nanopores can be significantly affected by energy effects. Among recent studies, Thompson and co-workers5 have predicted adsorption of binary mixtures of polar and nonpolar molecules using mean field perturbation theory. Lu et al6 have performed simulation to study the effects of temperature, pressure, and pore r 2011 American Chemical Society

size on the adsorption behavior of equimolar CO2/CH4 mixtures in carbon nanotubes. Preferential adsorption of one component of binary liquid mixtures near liquidliquid phase separa tion at inner surface of mesoporous silica glasses has been investigated by Schoen et al.7 But it has been shown that even in the absence of any long-range wallfluid and fluidfluid interactions confinement can cause a population inversion phenomenon for mixtures.9,10 Jimenez-Angeles et al.9 have indicated that increasing total bulk density can lead to an entropy driven population inversion for nonadditive binary hard sphere mixtures confined inside hard cylindrical nanopores. Kim et al.10 observed this phenomenon for nonadditive hard-sphere mixtures inside nanoslit pores. In this work, we have investigated the population distribution of binary asymmetric Lennard-Jones (LJ) mixtures confined in slitlike pores by considering the effect of different parameters such as interaction strength and size ratios of the components, confinement size, and thermodynamic state. Although adsorption of LJ mixtures has been studied frequently in the literatures,8,11,14 our main aim in this article is to investigate the effect of condensation on the population inversion of this kind of fluids. In fact, population inversion for LJ mixtures can be influenced by condensation Received: December 9, 2010 Revised: January 30, 2011 Published: March 15, 2011 3551

dx.doi.org/10.1021/jp1116883 | J. Phys. Chem. B 2011, 115, 3551–3559

The Journal of Physical Chemistry B

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phenomenon in different ways dependent on the bulk fluid phase which would be examined in this study. Among different theoretical methods for studying inhomogeneous systems we have used density functional theory (DFT) whose ability for predicting a large variety of phase transitions of confined fluids has been proved.17,18 Modified fundamental measure theory19,20 (MFMT) is one of the most successful versions of DFT that provides improved structure for hard sphere fluids near the walls21 and in confined geometries.18,22 The weight functions in this theory are characteristics of each type of molecules making it applicable to mixtures. We have incorporated the excess Helmholtz free energy functional in terms of MFMT for the hard sphere repulsion and mean field approximation23 for long-range attractions to construct the Helmholtz free energy functional of the mixtures. The rest of this paper has been organized as follows: In section II, a brief review of the model and theory is presented. In section III, the results for the population inversion of liquid and vapor LJ mixtures inside nanoslits have been summarized. In each subsection, the effect of different parameters such as interaction strength and size ratios of the components, nanoslit size, and thermodynamic state would be examined. Finally, we end with some conclusions in section IV.

In the present study the modified fundamental measure theory19,20 based on MansooriCarnahanStarligLeland equation of state24 has been used to represent the excess hard sphere free energy of the mixture Z hs Fex ½F1 ðrÞ, F2 ðrÞ ¼ kB T

∑

i¼1

where represents the external field affecting component ith and μi is the chemical potential of this component .F[F1(r),F2(r)] denotes the intrinsic Helmholtz free energy of the mixture consisting of two parts, ideal gas contribution plus an excess term arising from intermolecular interactions, as follows vext i

F½F1 ðrÞ, F2 ðrÞ ¼ Fid ½F1 ðrÞ, F2 ðrÞ þ Fex ½F1 ðrÞ, F2 ðrÞ

ð3Þ

The ideal contribution is given by the exact relation23 2 Z Fi ðrÞfln Fi ðrÞΛ3i 1g dr ð4Þ Fid ½F1 ðrÞ, F2 ðrÞ ¼ kB T

∑

i¼1

where kB is the Boltzmann constant and Λi represents the thermal de Broglie wavelength. The excess Helmholtz free energy can be approximated as a sum of the reference hard sphere contribution and an attractive part hs att ½F1 ðrÞ, F2 ðrÞ þ Fex ½F1 ðrÞ, F2 ðrÞ ð5Þ Fex ½F1 ðrÞ, F2 ðrÞ ¼ Fex

ð6Þ

where Φ(nR(r))is the Helmholtz free energy density of the hard sphere fluid mixture Φhs ðnR ðrÞÞ ¼ n0 lnð1 n3 Þ þ

n1 n2 nV1 nV2 1 n3

" # 1 n3 2 ðn2 3 3n2 nV2 :nV2 Þ n3 lnð1 n3 Þ þ þ 36π n3 3 ð1 n3 Þ2 ð7Þ In above equation, nR(r) and wR(r)are weighted densities and weight functions, respectively, which are given by following equations19,20

II. THE MODEL AND THEORY We consider a two-component fluid mixture consisting of spherical particles of species 1 and 2 in which the molecules interact via the (126) LJ potential " 6 # σij 12 σij LJ ð1Þ uij ðrÞ ¼ 4εij r r where i and j refer to the components of the mixture. In the above equation, σij and εij represent LJ parameters, and r is the distance between centers of the particles. The equilibrium structure of components inside nanopore is calculated using density functional theory. According to this theory, the grand potential of the mixture Ω[F1(r),F2(r)], is a functional of its component density distributions,23 F1(r) and F2(r) Ω½F1 ðrÞ, F2 ðrÞ ¼ F½F1 ðrÞ, F2 ðrÞ 2 Z Fi ðrÞðvext þ ð2Þ i ðrÞ μi Þ dr

Φhs ðnR ðrÞÞdr

nR ðrÞ ¼

2

∑

i¼1

Z

Fi ðr0 ÞwiR ðjr r0 jÞ dr0

wi2 ðrÞ ¼ 2πσii wi1 ðrÞ ¼ πσii 2 w0 ðrÞ ¼ δðσ ii =2 rÞ w3 ðrÞ ¼ θðσ ii =2 rÞ wiV2 ðrÞ ¼ 2πσii wiV1 ðrÞ ¼

r δðσii =2 rÞ jrj

ð8Þ

ð9Þ ð10Þ ð11Þ

where θ(r) is the Heaviside step function and δ(r) denotes the Dirac delta function. The attractive part of the excess free energy has been treated in the mean field approximation23 in which ZZ 1 att 0 0 Fi ðrÞFj ðr0 Þuatt Fex ½FðrÞ ¼ ij ðjr r jÞ dr dr ð12Þ 2 i, j ¼ 1, 2

∑

where uatt ij (r) represents the attractive part of LJ potential between fluid particles. Here we have split the potential at rij,min = 21/6σij in the WeeksChandlerAnderson (WCA) fashion,25 therefore uatt ij (r)is given by ( εij r < rij, min uatt ð13Þ ij ðrÞ ¼ uij ðrÞ r g rij, min Finally, by minimizing the grand potential of the system, the equilibrium density distribution of each component is obtained as 21 Z DΦ i Fi ðrÞ ¼ Fib exp β kB T dr0 wR ðr r0 Þ Dn R R !! Z 0 att 0 0 ext ex þ Fj ðr Þuij ðjr r jÞdr þ vi ðrÞ μi ð14Þ j ¼ i, j

∑

∑

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dx.doi.org/10.1021/jp1116883 |J. Phys. Chem. B 2011, 115, 3551–3559


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Figure 1. Comparison of density profiles resulting from DFT (lines) with MD simulations23 (circles) for an argonkrypton mixture inside a nanoslit with the length of H = 5σ11. The calculations have been carried out at T* = (kBT)/(ε11) = 2 and for Ftσ311 = 0.444, X2=0.738.

In our study the interactions between fluid molecules and nanoslit wlls is expressed by 8 σ ii σii > < ε < z < H wi 2 2 ð15Þ vext i ðzÞ ¼ > :¥ otherwise where εwi is the wallfluid potential parameter for ith component. Equation 15 indicates existence of a uniform field inside the nanoslit. The mixture is characterized by considering the interaction and diameter ratios for LJ parameters of the components denoted by β ((ε22)/(ε11)) and R ((σ22)/(σ11)). All of the length and energy parameters have been reduced to σ11 and ε11, respectively. We have used the LorentzBerthelot combining rules for calculating the cross interaction parameters. The mixture is at a total number density, Ft = F1 þ F2, and molar fraction Xi = (Fi)/(Ft), where Fi specifies the number density of species ith. The molar fraction of species ith inside the nanoslit is defined as Xi,np = (i)/(t) where t = 1 þ 2 and i denotes the average number density of species ith inside the nanoslit calculated as Z 1 H ð16Þ Fi ¼ F ðzÞ dz H 0 i

III. RESULTS AND DISCUSSIONS We consider an open nanoslit pore consisting of two parallel walls with infinite area and the length of H immersed in a bulk binary LJ mixture. The walls, assumed to be structureless, are

located at z = 0 and z = H affecting the components as an external field. The local density distributions of the species, i = 1,2, inside the nanoslit have been calculated using DFT. To ensure the accuracy of our results, in Figure 1 we have compared our DFT results for the structure of a binary LJ mixture with reduced temperature of T* = (kBT)/(ε11) = 2 inside a nanoslit with the length of H = 5σ11 with molecular dynamic simulation data.23 The total bulk number density Ftσ311 and X2 are equal to 0.444 and 0.738, respectively, and the size and interaction strength ratios have chosen to be R = 1.0661 and β = 1.3614 corresponding to an argonkrypton mixture. The external potential due to wallfluid interactions has been modeled by a (9,3) LJ potential with the same parameters as in ref 23. It is clear from Figure 1 that the agreement between DFT results and simulation data is reasonable for both components at the whole range of r. In the next sections we have used DFT to investigate the influence of confinement on the population distribution of LJ mixture components inside nanoslits. Our study consists of two parts in which the bulk fluids are in the liquid and vapor phases with the results to be presented in the following sections, respectively. A. Population Distribution of Liquid LJ Mixtures Inside Nanoslit Pores. The population distribution of LJ mixtures inside nanoslits is controlled by a number of parameters such as interaction strength and size ratios of the components, confinement size, and thermodynamic state. In this part, we have examined the effect of these parameters on the population distribution of liquid mixtures inside nanoslits with hard walls (εw1 = εw2 = 0) whose results have been presented in detail in the next subsections. It should be noted that all of the studied mixtures are in a single liquid phase.26 3553



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Figure 2. Molar fraction of species 1 of LJ mixtures with different values of β inside a hard nanoslit (H = 7σ11, εw1 = εw2 = 0) as a function of Ft for T* = 0.85, R = 1.3, and X2 = 0.008.

A.1. Effect of Intermolecular Interactions on the Population Distribution. To investigate the effect of intermolecular interactions on the population distribution, we have considered mixtures with reduced temperature of T* = (kBT)/(ε11) = 0.85 and bulk composition equal to X2 = 0.008 confined in a nanoslit with the length of H = 7σ11. The size ratio of all the mixtures is R = 1.3, while the interaction strength between 2 species molecules has been changed by variation of β((ε22)/(ε11)) from 0.600 to 0.250. Figure 2 shows the molar fraction of species 1 inside the nanoslit, X1,np, as a function of total bulk number density for different values of β. As it is clear from this figure, different behaviors are observed for the mixture composition with total bulk density which may be divided into three categories. The first category includes the values of β equal to 0.600 and 0.570 in which component 1 with more bulk concentration has been adsorbed more by the nanopore for the whole range of densities. Variation of total bulk density in this category has been lead to a condensation phenomenon at Ftσ311 = Ft*σ311, which is identified by a sudden change in the fluid composition. According to this figure, decreasing β reduces the adsorption of 1 species molecules in such a way that population distribution of molecules inside the nanoslit for the second category, β = 0.320, 0.300, 0.293, is essentially different from the first one. In fact, in these cases, species 2 with less bulk concentration is preferentially adsorbed by the nanopore at lower values of bulk density. We refer to this phenomenon as population inversion in which X2,np > X1,np opposite to the bulk fluid. As in the previous category, increasing total bulk density in this case leads to condensation, a phenomenon which suddenly increases the molar fraction of species 1 inside the nanoslit. This condensation phenomenon terminates the population inversion such that composition of the confined fluid becomes similar to the bulk for Ftσ311 > Ft*σ311 means X1,np > X2,np. It is interesting to note that decreasing β shifts the condensation to higher bulk densities and increases the range of bulk densities in which population inversion occurs. Finally, the third category consists of mixtures with lower values of β, in which population inversion is observable for the whole range of densities, as in the cases of β = 0.280 and 0.250. Parts a and b of Figure 3 show the adsorbed fluid number densities 1σ311 and 2σ311 as a function of total bulk density for different values of β including β = 0.570, 0.293, and 0.250 corresponding to three mentioned categories in Figure 2.

Figure 3. Adsorption curves (iσ311,i = 1,2) as a function of Ftσ311 for LJ mixtures with T* = 0.85, R = 1.3, and X2 = 0.008 for different values of β: β = 0.570 (a) and β = 0.293 and 0.250 (b). The nanoslit length is H = 7σ11.

According to Figure 3a, condensation phenomenon at Ft*σ311 = 0.731 for β = 0.570 is accompanying with a little increment in 2σ311, whereas average density of component 1 is significantly increased. But it is interesting to note that β = 0.293, the condensation phenomenon at Ft*σ311 = 0.7907, is associated with simultaneous adsorption and desorption for components 1 and 2, respectively, as it has been shown in Figure 3b. In fact, in the second category in which component 2 has increased its average density inside the nanopore at lower values of Ft, adsorption of component 1 into the nanoslit requires desorption of 2 species molecules. We observe no condensation phenomenon for β = 0.250 and component 2 has a higher concentration inside the nanoslit at the whole range of densities. These different behaviors may be interpreted by considering the effects of confinement, which in some cases can essentially influence the population distribution of molecules inside the nanoslit. In fact, for confined fluids, introduction of the hard walls can essentially change the system’s energy compared to the bulk through cutting off of some intermolecular interactions.18 This effect of walls on the moleculemolecule interactions for fluids with long-range forces may favor the adsorption of species with weaker attractive intermolecular interactions into the nanopore in comparison to other components in the mixtures. In our studied systems in Figure 2 the population distribution of molecules inside the nanoslit is determined by the competition between two factors, bulk concentration, and the energy effect. In this manner, it seems that for the first category, where the difference between strength of intermolecular interactions of 3554



Figure 4. The number density profiles of species of the LJ mixtures inside a hard nanoslit with the length of H = 7σ11 with the values of (a) β = 0.293, Ft*σ311 = 0.7907, (b) β = 0.293, Ftσ311 = 0.7945, (c) β = 0.570, Ftσ311 = 0.7945, and (d) β = 0.250, Ftσ311 = 0.7945. All of the mixtures are at reduced temperature of T* = 0.85 with the values of R and X2, respectively, equal to 1.3 and 0.008.

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the components is not so important, component 1 with more bulk concentration has been adsorbed more by the nanopore at the whole range of densities. But by weakening the attractive interactions between 2 species molecules, energy effect increases the adsorption of this kind of molecules into the nanoslit, which subsequently can lead to their preferential adsorption. Occurring condensation can end the selective adsorption of 2 species molecules; however it is postponed with increasing the adsorption tendency of this component. In this manner in the third category, where component 2 has much weaker attractive intermolecular interactions, population distribution of molecules is determined by the energy effects for the whole range of densities. As well as the important role of confinement effects in determining the composition of confined fluids, they can significantly influence the structure of fluid molecules inside the nanoslit. To clarify this subject, the number density profiles of species 3 of the LJ mixture with β = 0.293 just before (F*σ t 11 = 0.7907) 3 and after (Ftσ11 = 0.7945) condensation phenomenon (the points A and B in Figures 2 and 3b) have been presented in parts a and b of Figures 4. It should be noted that in both figures, regardless of the composition of the confined fluid, the system adopts structures in which component 2 has more population at the walls. Also we observe a depletion phenomenon for a species 1 molecule at the walls in both figures. In fact these behaviors of molecules can be explained by considering the competition between energy and entropy effects. For confined fluids, the tendency of the molecules to maximize the system’s entropy favors accumulation of them at the walls which significantly increases the accessible volume for other molecules. On the other hand, the long-range attractive interactions hold back the molecules due to cutting off of intermolecular interactions by the walls. In the competition between these entropy and energy effects for component 2 with larger size and weaker intermolecular interactions, the entropy effect dominates, and we observe accumulation of species 2 molecules at the walls, which reduces the excluded volume at the middle space. However for component 1 with more strong attractive interactions, the energy effect prevails and density profiles show depletion. In this manner, comparing parts a and b of Figure 4 indicates that when 3 3 going from F*σ t 11 = 0.7907 to Ftσ11 = 0.7945, species 1 increases its population inside the nanoslit mostly by accumulation in the middle of that. Also we observe a significant decrease in the population of 2 species molecules at the middle space and increased accumulation of the remaining molecules at the walls with increasing total bulk density. Parts c and d of Figure 4 exhibit the number density profiles of species of the LJ mixtures with β = 0.570 and 0.250 at Ftσ311 = 0.7945 inside the nanoslit. As it is clear from Figure 4c, component 1 has a greater population at all regions inside the nanopore for β = 0.570, whereas we observe more concentration for species 2 molecules for the whole space in the case of β = 0.250. In fact, the greater tendency of 1 and 2 species molecules to accumulate respectively at the middle space and the walls can be realized again by doing a comparison between parts bd of Figure 4. As it is clear, populations of species 1 and 2 molecules inside the nanoslit decrease and increase, respectively, with decreasing β. In this way, it is observed that component 1 at first decreases its concentration at the walls and then at the middle space, while population of 2 species molecules at first increases at the walls and then at the middle space. 3555



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Figure 5. Molar fraction of species 1, X1,np, of the LJ mixtures with different values of R inside a nanoslit with the length of H = 6σ11 as a function of Ftσ311. The calculations have been carried out at T* = 0.85 and for β = 0.3, X2 = 0.004.

Figure 6. Molar fraction of species 1 of the LJ mixtures with R = 1.3 and different bulk compositions versus total bulk density. The values of other parameters are the same as Figure 5.

Figure 7. Molar fraction of species 1 of the LJ mixtures with β = 0.3, R = 1.3, and X2 = 0.008 as a function of Ftσ311 for (a) different temperatures and H = 7σ11 and (b) different nanoslit lengths and T* = 0.85.

A.2. Effects of Size Ratio, Fluid Composition, Temperature, and Confinement Size on the Population Distribution. To examine the effect of size ratio of mixture components on the population distribution, in Figure 5 we have presented the molar fraction of species 1, X1,np, of the LJ mixtures with different values of R as a function of total bulk density. The mixtures are at the reduced temperature of T* = 0.85; the values of X2 and β are 0.004 and 0.3, respectively, and the nanoslit length is equal to 6σ11. According to Figure 5, increasing the role of confinement effects by R leads to three pervious mentioned behaviors for X1,np with total bulk density. It is observed that for R = 0.70 component 1 with more bulk concentration is the richer species inside the nanopore for the whole range of densities. But increasing R increases the accumulation of 2 species molecules at the walls as a result of their tendency to maximize the system’s entropy. This behavior of 2 species molecules consequently increases their adsorption into the nanoslit with R such that population inversion phenomenon is observable for higher values of R including R = 1.28, 1.32, 1.34, and 1.36. The figure shows that condensation phenomenon is delayed with R because of the excluded volume effects to the extent that for R = 1.36, population inversion exists for the entire range of densities. The structures of two components inside the nanoslit in this section are similar to those presented in Figure 4 such that while

we observe preferences of 2 species of molecules to the walls due to entropy effects, energy effects may induce depletion phenomenon for species 1. In addition the same as Figure 4, condensation phenomenon is associated with accumulation and depletion of, respectively, 1 and 2 species molecules at the middle space. Figure 6 is the same as Figure 5 but for LJ mixtures with R = 1.3 and different bulk molar fractions. It is clear that at low enough values of X2 , the bulk concentration specifies the main species inside the nanopore at all densities as in the case of X2 = 0.0001. But according to this figure, increasing X 2 may lead to population inversion phenomenon in a limited or entire range of densities. In addition, the figure shows that condensation phenomenon shifts to higher bulk densities as the adsorption of 2 species molecules is increased with X2 . In parts a and b of Figure 7 we have presented the effects of temperature and confinement size on the population distribution. The figures confirm the occurrence of population inversion as the role of confinement effects increases with decreasing T* and H. In fact, the more significant role of cutting off of attractive intermolecular interactions at lower temperatures and also smaller nanopores favors adsorption of the 2 species molecules into the 3556



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B. Population Distribution of Vapor LJ Mixtures Inside Nanoslit Pores. In this section, we have investigated the

Figure 8. (a) Molar fraction of species 1 of LJ mixtures with different values of βinside an attractive nanoslit (H = 11σ11, εw1 = εw2 = 3) as a function of Ftσ311 for T* = 1, R = 1, and X2 = 0.1. (b) Grand potential and molar fraction of the LJ mixture with β = 1.6 vs Ftσ311 for adsorption and desorption branches. A denotes the surface area of the system. (c) Number density profiles of vaporand liquidlike phases of the LJ mixture with β = 1.6 inside the nanoslit.

nanoslit from energy point of view. Also, while condensation is delayed by confinement effects, the inversion may exist for the whole range of densities in some cases.

population distribution of vapor LJ mixtures27 as they are adsorbed into attractive nanoslits by considering the effect of different parameters such as interaction strength and size ratios, confinement size, and thermodynamic state. The interactions between fluid molecules and nanoslit walls are expressed by eq 15 with the parameter values being selected as εw1 = εw2 = 3 throughout this section. Figure 8a depicts the molar fraction of LJ mixtures with different values of β confined in a nanoslit with the length of H = 11σ11 as a function of total bulk density. The mixtures are at the reduced temperature of T* = 1 with the values of X2 and R respectively equal to 0.1 and 1. Increasing total bulk density in our studied systems can lead to capillary condensation which suddenly decreases the molar fraction of component 1 inside the nanopore as can be seen in the figure. The location of these transitions has been determined from the condition of equality of the values of grand potential for the two vapor- and liquidlike phases in equilibrium.17 This subject has been shown in Figure 8b, which presents the grand potential and molar fraction of the confined mixture with β = 1.6 as a function of total bulk density for adsorption and desorption branches. Clearly the cross on the curve of grand potential indicates that the vapor and liquidlike phases are in equilibrium at the specified total bulk density. The number density profiles of the vapor- and liquidlike phases in equilibrium for the mixture have also been plotted in Figure 8c. As it is clear from Figure 8a, component 1 is the more concentrated species in the vaporlike phases for all values of β such that the composition of these phases is almost the same as bulk fluid. According to the figure, confined mixtures are still richer in component 1 upon condensation in the cases of β = 1.3 and 1.4. But, clearly variation of β can significantly alter the composition of liquidlike phases such that these phases become richer in component 2 with stronger intermolecular interactions. Considering that the bulk fluid is in the vapor phase, the longrange interactions between molecules can not play a significant role in the adsorption of different molecules into the nanopore, while for liquefaction of the adsorbed molecules, the component with stronger intermolecular attraction is preferred. In this manner we observe that in the cases of β = 1.5 and 1.6, component 2 is the more concentrated species in the liquidlike phases in spite of its minority in the bulk. In fact, capillary condensation is accompanying with a population inversion phenomenon for these cases such that the liquidlike phases become richer in component 2 . According to Figure 8a, increasing the strength of intermolecular interactions has shifted capillary condensation to lower values of total bulk density as it is expected. The effect of size ratio of mixture components on the composition of vapor- and liquidlike phases has been studied in Figure 9. This figure is the same as Figure 8, but for LJ mixtures with different values of R and the value of β equal to 1.5. Like the pervious systems, capillary condensation in this figure is identified by sudden decrease of X1,np. It should be noted that, however, the composition of the vaporlike phases are almost the same as the bulk fluid for all cases, but variation of R can significantly influence the adsorption of the components into the condensed phases. According to this figure, in the cases of R = 0.70 and 0.76, both the vapor- and liquidlike phases are richer in component 1 with more bulk concentration. But obviously increasing the size of 2 species molecules increases their adsorption into the nanoslit. This adsorption increment of the component with 3557



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In parts a and b of Figure 10, we have examined the effects of temperature and confinement size on the composition of vaporand liquidlike phases. As it is clear, adsorption of 2 species molecules in liquidlike phases increases with decreasing temperature (in the studied range of T*) and also increasing nanoslit length. Therefore, we can obtain conditions in which capillary condensation is simultaneous with population inversion phenomenon for lower temperatures and wider nanopores as it has been shown in the figures. Also it is observed that increasing the size of the nanoslit prompts capillary condensation due to walls potential, which is uniformly attractive inside the nanopore.

Figure 9. Molar fraction of species 1 of LJ mixtures with β = 1.5 and different values of R inside an attractive nanoslit as a function of Ftσ311. The values of other parameters are the same as Figure 8.

Figure 10. Molar fraction of species 1 of LJ mixtures with R = 1 and X2 = 0.1 inside an attractive nanoslit as a function of Ftσ311 for (a) different temperatures with β = 1.5 and H = 11σ11 and (b) different nanoslit lengths with β = 1.4 and T* = 1.

more strong attractive interactions prompt vaporliquid phase transition and also increases the molar fraction of component 2 inside the condensed phase. Therefore, for R = 0.90 and 1, we observe a population inversion for confined fluids upon condensation such that liquidlike phases are richer in component 2. The figure also indicates that the influence of total density on the composition of liquidlike phases increases in the case of mixtures involving components with different size.

IV. CONCLUSION In this work we applied density functional theory to investigate confinement effects on the population inversion of LJ mixtures with vapor and liquid phases in the bulk. For this purpose, the influence of different parameters including β, R, X2, H, and T* on the adsorption behavior of molecules were studied and it was observed that each of them can significantly alter the behavior of mixture composition with total bulk density. Considering liquid LJ mixtures, we observed that energy and entropy effects may lead to a population inversion phenomenon by preferential adsorbing of 2 species molecules with less bulk concentration. In fact, cutting off of some intermolecular interactions by the walls for the case of confined fluids changes the system’s energy in comparison to the bulk fluid. This confinement effect favors adsorption of the component with weaker intermolecular interactions from energy point of view. We showed that this energy effect can lead to selective adsorption of component 2 by decreasing the parameters such as β, T*, and H. Furthermore, we observed such a phenomenon with increasing R as a result of increased tendency of 2 species molecules to adsorb into the nanoslit due to entropy effects. The mentioned population inversion can be terminated by a condensation phenomenon involving simultaneous adsorption and desorption for 1 and 2 species molecules. But our results for vapor LJ mixtures indicated that variation of mentioned parameters can lead to conditions in which capillary condensation is accompanying with a population inversion phenomenon such that liquidlike phases become richer in component 2 with less bulk concentration. In fact, although the composition of the vaporlike phases are governed mainly by bulk concentration, interaction strength and size ratios, confinement size and temperature can significantly alter the composition of the liquidlike phases. We observed that increasing temperature postpones vapor liquid phase transition, but it shifts to lower densities with β, R, and H. But in the case of liquid LJ mixtures, it was shown that the increased tendency of 2 species molecules to adsorb, the higher bulk density at which condensation occurs. In this manner, although increasing T* prompts condensation, it is delayed with decreasing β. In addition, condensation shifts to higher bulk densities with R and X2 increasing and H decreasing. Therefore, by varying the mentioned parameters, we can come to conditions in which population inversion exists for the whole range of densities. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Phone: þ98-311-391-3281. Fax: þ98-311-391-2350. 3558



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’ ACKNOWLEDGMENT The authors acknowledge the financial support by the Isfahan University of Technology Research Council. ’ REFERENCES (1) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkoviak, M. Rep. Prog. Phys. 1999, 62, 1573. (2) Martinez, A.; Pizio, O.; Sokolowski, S. J. Chem. Phys. 2003, 118, 6008. (3) Bucior, K. Colloids Surf. A 2004, 243, 105. (4) Marconi, U. M. B.; Van Swol, F. Mol. Phys. 1991, 72, 1081. (5) Kotdawala, R. R.; Kazantzis, N.; Thompson, R. W. J. Chem. Phys. 2005, 123, 244709. (6) Huang, L.; Zhang, L.; Shao, Q.; Lu, L.; Lu, X. J. Phys. Chem. C 2007, 111, 11912. (7) Rother, G.; Woywod, D.; Schoen, M.; Findenegg, G. H. J. Chem. Phys. 2004, 120, 11864. (8) Shevade, A. V.; Jiang, S.; Gubbins, K. E. J. Chem. Phys. 2000, 113, 6933. (9) Jimenez-Angeles, F.; Duda, Y.; Odriozola, G.; Lozada-Cassou, M. J. Phys. Chem. C 2008, 112, 18028. (10) Kim, S. C.; Suh, S. H.; Seong, B. S. J. Korean Phys. Soc. 2009, 54, 660. (11) Patrykiejew, A.; Sokolowski, S.; Pizio, O. J. Phys. Chem. B 2005, 109, 14227. (12) Sweatman, M. B.; Quirke, N. J. Phys. Chem. B 2005, 109, 10389. (13) Jamnik, A. J. Phys. Chem. B 2007, 111, 3674. (14) Kurniawan, Y.; Bhatia, S. K.; Rudolph, V. AIChE J. 2006, 52, 957. (15) Tovbin, Y. K.; Zhidkova, L. K.; Komarov, V. N. Russ. Chem. Bull. 2001, 50, 786. (16) Cracknell, R. F.; Nicholson, D.; Quirke, N. Mol. Phys. 1993, 80, 885. (17) Yu, Y.; Peng, B. J. Phys. Chem. B 2008, 112, 15407. (18) Keshavarzi, E.; Kamalvand, M. J. Phys. Chem. B 2009, 113, 5493. (19) Yu, Y.; Wu, J. J. Chem. Phys. 2002, 117, 10156. (20) Roth, R.; Evans, R.; Lang, A.; Khal, G. J. Phys.: Condens. Matter 2002, 14, 12063. (21) Keshavarzi, E.; Taghizadeh, A. J. Phys. Chem. B 2010, 114, 10126. (22) Kamalvand, M.; Keshavarzi, E.; Mansoori, G. A. Int. J. Nanosci. 2008, 7, 245. (23) Choudhury, N.; Ghosh, S. K. Phys. Rev. E 2001, 64, 021206. (24) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Laland, T. W. J. Chem. Phys. 1971, 54, 1523. (25) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237. (26) Georgoulaki, A. M.; Ntouros, I. V.; Tassios, D. P.; Panagiotopoulos, A. Z. Fluid Phase Equilib. 1994, 100, 153. (27) Lamm, M. H.; Hall, C. K. AIChE J. 2001, 47, 1664.

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Population Inversion of Binary Lennard-Jones Mixtures in Nanoslit

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