Theoretically Based Model for Competitive Adsorption of Subcritical

Article pubs.acs.org/JPCC

Theoretically Based Model for Competitive Adsorption of Subcritical Mixtures Julien Collell and Guillaume Galliero* Laboratoire des Fluides Complexes et leurs Réservoirs, UMR-5150 with CNRS and TOTAL, Université de Pau et des Pays de l’Adour, BP 1155, 64013 Pau, France ABSTRACT: Numerous works have been achieved on the prediction of subcritical fluids, but most of them are about gas adsorption at low pressures or low concentrations, where adsorption consists in the formation of one adsorption layer. However, at higher pressures or concentrations, these mixtures can form an important number of adsorbed fluid layers and may exhibit a gas/liquid transition when the capillary condensation pressure is reached. In this work, we derive from the statistical mechanics a model predicting subcritical mixtures in such conditions. In the proposed model, the adsorbed molecules are supposed to be distributed in two distinct adsorption layers. Two parameters per adsorbed species characterize the interactions of the species in the adsorption layers. One additional parameter characterizes the adsorption capacity of the porous structures. To check the consistency of the model, Monte Carlo simulations on pure compound and mixtures adsorption are used as reference data. Our model shows improvements compared to the Ideal Adsorbed Solution Theory, whose adsorption isotherm are modeled with the BET model developed by Gritti et al. [J. Chromatogr. A 2002, 978, 81−107]. Especially, the predictions of the adsorbed phases compositions are in good agreement with molecular simulation results, in both gas and liquid states. The adsorbed amount in the gas phase are also correctly predicted. By introducing an additional empirical parameter which characterizes the liquid adsorption capacity, it is also possible to predict the adsorbed amounts for the liquid phase, providing a consistent approach to model the adsorption of liquid and gas mixtures.

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INTRODUCTION The study of adsorption phenomena is a relevant subject of research in both industrial and academic fields, with recent developments in zeolites,1 in metal organic framework,2 and more generally in nanoporous systems.3−5 In most cases of practical interest, not only pure compounds but mixtures are involved. Therefore, adsorption equilibria involving competition between molecules of different types are needed for the understanding of the systems, as well as for the design of porous materials. Several theories have been developed to deal with mixture adsorption phenomena, as the Langmuir theory,6−8 the ideal adsorbed solution theory (IAST)9 and its derivatives,10−12 the multicomponent potential theory of adsorption,13 or the density functional theory.14,15 At low and high pressures, the modeling of subcritical fluid adsorption is often limited to systems where adsorption can be represented by the formation of one dense fluid layer on top of the adsorbent surface, which is relevant: (1) for highly confined systems as zeolites1,16,17 or activated carbons,18−20 where the size of the porous structures does not enable the formation of multiple adsorbed layers and (2) for fluids at low pressures or solutes at low concentrations,21−23 where the formation of multiple adsorption layers is statistically unfavorable. However, there are many fields in which the adsorption process engages the formation of an important number of adsorbed layers, as in food engineering,30,31 in separation process,11,32−34 or in the refining industry.35 More recently, a new field of applications has emerged due to the drop of the gas price, especially in North America.36 The energy industry is © XXXX American Chemical Society

there looking for nonconventional resources which contain liquid-rich fluids such as shale oils. In such resources, the hydrocarbons can form relatively thick adsorbed layers, and under typical reservoir conditions, capillary condensation may lead to a gas−liquid transition. Consequently, the capability of precisely predicting the composition and the amount of the adsorbed phase is a key point that may strongly impact the process design and ultimately the profitability. Statistical mechanics provide a convenient theoretical framework to depict the adsorption phenomena and enable the development of theoretical models based on a Langmuirlike description of the adsorption process. Former works (see for examples Hill24 and Ruthven25) have successfully applied the statistical mechanics to the case of monolayer and multilayer adsorption. More recently, Ally and Braunstein26 or Dutcher et al.27,28 extended these works to systems containing multiple adsorption sites and where different types of adsorption layers can be formed. Ruthven et al.17,25 and Jaroniec et al.29 have extended the monolayer approach to treat multicomponent systems with more than one adsorbate type (competitive adsorption) but are limited to monolayer adsorption. In this work, we propose to extend these former works to multicomponent systems, where several adsorbed layers are likely to be formed (high pressures, high solute concentrations, Received: August 13, 2014 Revised: October 21, 2014

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dx.doi.org/10.1021/jp508194d | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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corresponds to a multilayer, where molecules can sorb on top of each other, with no limitations on the number of heaped up molecules. The model therefore considers the total number of molecules stored in the pore (absolute amount). It is assumed that each adsorbed molecule occupies one adsorption site and that the number of adsorption sites is the same for each adsorbed species. It is an usual assumption present for example in the extensions of the Langmuir theory6−8 or in the IAST.9 We introduce χa,1, the number of molecules of a adsorbed in the monolayer. The adsorbed molecules are represented by one single sphere. The number of distinguishable configurations Ωa1 that can sorb onto Ns adsorption sites is

micro-mesoporous materials). We propose a description of the adsorption process which describes the distribution of the adsorbed species within the adsorbed layers. The proposed competitive model has been applied to the adsorption of subcritical mixtures, in both gas and liquid states. To check the consistency and the limitations of the model, Monte Carlo simulations on pure compounds and synthetic mixture adsorption are used as reference data. This article is structured as follows: we first detail the theoretical development of our competitive adsorption model. Then, we describe the molecular simulations used in this study. Afterward, our results are discussed. Finally, the last section is dedicated to the conclusions.

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Ωa ,1 =

STATISTICAL MECHANICS DEVELOPMENT In this section is described the development of our model of competitive adsorption. The development is made in the frame of the statistical mechanics and is based on the works of Ally and Braunstein26 and Dutcher et al.27,28 This approach consists in determining the number of possible arrangements of the adsorbed molecules to evaluate the Gibbs free energy of the adsorbed phase. To evaluate the Gibbs free energy of an adsorbed phase composed of two species a and b onto a surface containing Ns adsorption sites, we first need to determine the entropy of the adsorbed phase. It is defined by S = k ln Ω

(Ns)! (Ns − χa ,1 ) ! (χa ,1 )!

(2)

Similarly, we introduce χb,1, the number of molecules of b adsorbed in the remaining Ns−χa,1 sites of the monolayer. The number of distinguishable configurations that χb,1 molecules can sorb onto the solid surface is Ωb ,1 =

(Ns − χa ,1 )! (Ns − χa ,1 − χb ,1 ) ! (χb ,1 )!

(3)

The total number of combinations in the monolayer Ω1 is then Ω1 = Ωa,1 × Ωb,1. The number of distinguishable configurations of the molecules adsorbed in the multilayer has to be counted. The molecules of the first layer form χa,1 + χb,1 sorption sites for the multilayer. As stated before, there is no limitation on the number of molecules adsorbed on each site of the multilayer, as shown in Figure 1. It is assumed that each site can indifferently sorb molecules of a and b. We introduce χa,2 and χb,2, the number of molecules of a and b adsorbed in the multilayer. The number of configurations in the multilayer is

(1)

where S is the entropy, k is the Boltzmann constant, and Ω is the number of accessible configurations of the adsorbed phase. Ω is the fundamental quantity that needs to be determined. To do so, the adsorption phenomena is represented with a Langmuir-like description where successive layers of adsorbed fluid are created on top of the solid surface. Two different kinds of adsorbed layers are introduced, depending on the nature of the adsorption sites as illustrated in Figure 1. The molecules sorbed directly onto the solid surface constitute the first adsorption layer, referred thereafter as the monolayer. The adsorbed molecules on the monolayer create sorption sites for the second layer, which lies on top of the monolayer. It

Ω2 = ≈

(χa ,1 + χb ,1 + χa ,2 + χb ,2 − 1)! (χa ,1 + χb ,1 − 1) ! (χa ,2 ) ! (χb ,2 )! (χa ,1

(Na + Nb)! + χb ,1 )! (Na − χa ,1 ) ! (Nb − χb ,1 )!

(4)

where Na = χa,1 + χa,2 and Nb = χb,1 + χb,2 represent the number of adsorbed molecules of a and b. The total number of configurations Ω is Ω= (Ns − χa ,1

(Ns)! (Na + Nb)! − χb ,1 )! (χa ,1 )! (χb ,1 )! (χa ,1 + χb ,1 )! (Na − χa ,1 )! (Nb − χb ,1 )!

(5)

Using the Stirling approximation [ln(N!) = N ln N − N], the entropy is given by Ns N + Nb N + Nb S = Ns ln + Na ln a + Nb ln a k Ns − χa ,1 − χb ,1 Na − χa ,1 Nb − χb ,1

Figure 1. Sketch representing the mixture adsorption onto a flat surface. We suppose that the adsorbed phase split into two adsorbed layers. The first layer, in contact with the solid surface, consists of a monolayer: each site can only sorb one fluid molecule. The second layer, on top of the first layer, consists of a multilayer. The molecules of the first layer act as adsorption sites, with no limitation on the number of molecules adsorbed per site in the multilayer.

+ χa ,1 ln + χb ,1 ln

(Na − χa ,1 )(Ns − χa ,1 − χb ,1 ) χa ,1 (χa ,1 + χb ,1 ) (Nb − χb ,1 )(Ns − χa ,1 − χb ,1 ) χb ,1 (χa ,1 + χb ,1 )

(6) B



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Ns N + Nb + Na ln a Ns − χa ,1 − χb ,1 Na − χa ,1 εa ,1 εb ,1 N + Nb − χa ,1 − χb ,1 + Nb ln a Nb − χb ,1 kT kT

Now, the internal energy of adsorption has to be determined. The interactions between molecules adsorbed in the same layers are assumed to be negligible. It is a common assumption of most adsorption models.6,9−11,34 The internal energy of the adsorbed phase is a function of the interactions between the adsorption sites and the adsorbed molecules. Each adsorbed layer is characterized by one interaction parameter. One which characterizes the interactions in the monolayer between the adsorbed species and the adsorbent and another one which defines the interactions in the multilayer between the adsorbed molecules and the adsorption sites. In this case, the adsorption sites correspond to molecules adsorbed in the first layer. Contrary to some statistical models proposed in the literature,17,25,32 it is assumed in our model that the energy of an adsorbed molecule on the multilayer is independent of the nature of the sorption site, which can be either a or b. This assumption is not harmless and, even if it has not been the case in this study, it might be inadequate when applied to asymmetric mixtures. As a result, the adsorbed layers are considered as ideal mixtures (no excess quantities due to mixing) and the activity coefficients γi are equal to unity. However, this is a common assumption made in the adsorption models and is relatively convenient as the prediction of mixture adsorption does not necessitate the use of additional crossparameters to transcript interaction between unlike species.32 Thus, the knowledge of the pure compound adsorption isotherms is sufficient to predict mixture adsorption in such approximation. In that frame, the internal energy change ΔE can be formulated as ΔE = −εa ,1χa ,1 − εb ,1χb ,1 − Nada − Nbdb

ln Ω* = Ns ln

(10)

The activities of the adsorbed species ai are defined as ⎛ f ⎞ ∂ΔG /kT ln ai = ln⎜⎜ i ⎟⎟ = f ∂Ni ⎝ i , sat ⎠

Nj , T

(11)

where f i,sat is the fugacity of the species i at the gas−liquid saturation. Assuming that the Poynting term (PV) is negligible,37 the Gibbs free energy ΔG reduces to ΔG/kT ≈ ΔE/kT − ln Ω*. Using eq 10 and the definition of the Gibbs free energy, the activity of the species in the mixture reduces to ai =

Ni − χi ,1 1 × Ki Na + Nb

(12)

with Ki = exp(di/kT). Equation 12 is in agreement with the pure compound activity27 derived by Dutcher.27 If Ki = 1, the activity corresponds to the fraction of adsorbed molecules in the second layer, in agreement with Stokes and Robinson.26,38 Solving the system of eqs 9 and 12, the competitive adsorption isotherm for the species a is given by Na = Ns ×

(7)

where εi,1 + di is the energy change of an adsorbed molecule of i in the monolayer from the liquid state, i being either a or b. di is the energy change of an adsorbed molecule of i in the multilayer from the liquid state. The constant εi,1 characterizes the adsorbent/monolayer interaction. The constant di characterizes the interaction in the multilayer, which is a combination two contributions, due to the solid/multilayer interactions and to the monolayer/multilayer interactions. These two contributions are usually a function of the distance of the adsorbed layer to the solid. They are represented here with a single parameter, di, which can be viewed as a mean field value of these two contributions. The Lagrangian multipliers method is used to determine the most probable distribution of configurations, Ω*, which maximize the entropy of the system:

Kaaa[Ca ,1 + (Cb ,1 − Ca ,1)Kbab] (1 − Kaaa − Kbab)[1 + (Ca ,1 − 1)Kaaa + (Cb ,1 − 1)Kbab] (13)

The adsorbed amount of b is obtained by permuting the a and b subscript in eq 13. The constants Ci,1 and Ki characterize the couple adsorption site/adsorbed species. As mentioned previously, they are assumed to be independent of the other species in the mixture. Consequently, they can be determined from the pure compound adsorption isotherms. The pure compound adsorption isotherm is given by27 Ni = Ns

Ci ,1K iai (1 − K iai)[1 + (Ci ,1 − 1)K iai]

(14)

This corresponds to the Guggenheim-Anderson-de Boer model39−41 (GAB), prevalent in the food engineering science to described the adsorption of water.30,31 The extension of our model to a mixture of n species is given by n

∂ ln Ω ∂χi ,1

θ

1 ∂ΔE = kT ∂χi ,1

Ni = Ns ϕ

(8)

χi ,1 (χa ,1 + χb ,1 ) (Ni − χi ,1 )(Ns − χa ,1

n

n

(1 − ∑ j Kjaj)[1 + ∑ j (Cj ,1 − 1)Kjaj]

(15)

This equation provides an analytical expression to predict mixture adsorption using 2n + 1 parameters. In this model, the adsorbed amount of a given species in the mixture is a function of (1) the interactions of the species in the mono- and multilayer and (2) the concurrence between adsorbed species, by means of competing terms [Cj,1 − Ci,1]. If the energy of sorption of the second layer equals the condensation energy, (Ki = 1) then eq 13 reduces to

θ indicates that the differentiation is carried out at a Na,Nb,T,χj≠i,1,χa,2,χb,2 constant. ϕ indicates that the differentiation is carried out at Na,Nb,χj≠i,1 constant. We then obtain a system of two equations given by ⎛ εi ,1 ⎞ = exp⎜ ⎟ = Ci ,1 ⎝ kT ⎠ − χb ,1 )

K iai(Ci ,1 + ∑ j ≠ i (Cj ,1 − Ci ,1)Kjaj)

(9)

i being either a or b. Ci,1 is a constant related to the interaction of the solid with the monolayer. Equation 9 is introduced in eq 6 to get the most probable distribution:

Na = Ns ×

aa(Ca ,1 + (Cb ,1 − Ca ,1)ab) (1 − aa − ab)[1 + (Ca ,1 − 1)aa + (Cb ,1 − 1)ab] (16)

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This corresponds to the ideal adsorbed solution theory9 (IAST), whose pure compound adsorption isotherm are described by a Brunauer−Emmett−Teller42 (BET) model. This equation is referred thereafter as the IAST/BET model, developed by Gritti et al.34 Despite the differences in the theoretical frameworks of the IAST/BET model and our model, the same expressions for the mixture adsorption isotherm are obtained. By the mean of the Ki constants, our model adds one more degree of freedom compared to the IAST/BET model: the adsorption energy in the multilayer can differ from the condensation energy. In the proposed development, the number of adsorbed layers can theoretically go to infinity. Extension of the BET model, which can be recast for the GAB model, have been proposed to introduce limitations on the maximum number of layers that can be created.16,24 However, the imposition of a maximum number of adsorbed layers seems a bit arbitrary and adds an additional parameter to the models. Moreover, in our systems, it does not address the problems raised by the capillary condensation43 which occurs before the total filling of the pore by adsorption.

taken into account in the model but has been, however, introduced, as it is relevant in real systems. The porous structure under study is a slit pore. This system has been chosen to easily identify the adsorption regimes (gas or liquid) and the transition between these regimes, when the capillary pressure is reached. A schematic representation of the porous structure is given in Figure 2. The walls correspond to a FCC lattice. The pore width H* is set to 10 × σ1 and each wall consists of 6 layers, of 8 unit cells in length (Lx) and depth (Lz).

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MOLECULAR SIMULATIONS In this work, Monte Carlo44,45 simulations have been performed to investigate the modeling capabilities of our model when applied to binary synthetic mixtures. The Medea/ Gibbs software (Medea/Gibbs license from IFP-EN and Laboratory of Chemical Physics, CNRS-Université Paris Sud) has been used for that purpose. Gibbs ensemble Monte Carlo simulations in its isothermal−isobaric version are used to obtain fluid phase equilibria of pure compounds and binary mixtures. Grand canonical Monte Carlo simulations are used to compute the adsorption isotherms of pure compound and mixtures. The dispersion/repulsion interactions are modeled with a single site Lennard-Jones 12−6 potential.46 The Lennard-Jones parameters are listed in Table 1. The solid

Figure 2. Sketch of the simulation box of the porous structure used to compute adsorption isotherms.

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RESULTS AND DISCUSSION In this section, the application of our model given by eq 15 is discussed. It is referred as the extended Guggenheim− Anderson−de Boer (E-GAB) model thereafter. To emphasize the benefits of our development, the E-GAB model is confronted to the widespread IAST,51−53 whose pure compound adsorption is described with the BET model, given in eq 16. In the first part of this section, we discuss the fit of pure compound adsorption isotherms. In the second part of this section, we discuss the predictions of mixture adsorption with both E-GAB and IAST/BET models. Pure Compound Adsorption. The fit of pure compound adsorption isotherms is of primary importance, as it provides the parameters used to predict the adsorption of mixtures. The pure compound adsorption isotherms need to be fitted for both pure compound GAB model, given in eq 14 and BET model, given by42 Ci ,1ai Ni = Ns × (1 − ai)[1 + (Ci ,1 − 1)ai] (17)

Table 1. Lennard-Jones Parameters of the Species Involved in the Simulations species

σ (Å)

ε/k (K)

wall 1 2

3.16 3.73 4.38

91.70 149.92 166.58

phase parameters mimics a simplified silicate,47 which represents the most abundant fraction of the inorganic matter in shales.48 The solid is considered rigid during the grand canonical Monte Carlo simulations. The parameters of component 1 are those corresponding to a methane molecule,49 which represents more than 60% of the mole fraction in a typical condensate gas.50 Lorentz−Berthelot mixing rules44 are used for interactions between unlike species. The cutoff radius is set to half the box length. The fluid phase is composed of two compounds, namely 1 and 2. The parameters and thermodynamic conditions have been chosen such that kT/ ε1 = 1 and ε1/ε2 = 0.9. For convenience, σ2 is chosen such as the saturation pressure of 2 is roughly the third of the saturation pressure of 1. This also implies different capillary condensation pressures for the two pure compounds adsorption isotherms as shown in Figure 3. By introducing different sizes for the adsorbed species, the “steric hindrance” of the two species in the adsorbed layers may differ. This is not

The fits of the simulation results are achieved by a least mean square algorithm up to the capillary condensation pressure. The quality of the fits is estimated using the relative root mean square error (rRMS err.) RMS% given by RMS% = 100 × D

1 n

⎛ Nj ,model − Nj ,MS ⎞ ⎟⎟ N ⎝ ⎠ j ,model j=1 n

∑ ⎜⎜

(18)



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weak affinity of the fluid with the adsorbent. The species 2 presents a type IV adsorption isotherm, with more affinity of the adsorbed fluid with the adsorbent surface. Type V adsorption isotherms are more difficult to reproduce with a BET model than the type IV isotherms. The only adjustable parameter Ci,1 of this model enables one to reproduce the filling of the monolayer (shouldering at low activities), while the filling of the multilayer cannot be adjusted because Ki equals unity. This explains a higher relative root-mean-square error for the BET fit of species 1, compared to the other fits. Compared to the BET model, the GAB isotherm with its additional parameter enables a better fit of the adsorption isotherm on the whole range of activities. However, except at very low pressures and close to the capillary condensation, the differences between the two models remain relatively limited for the pure compound. Mixture Adsorption. In this part is an evaluation of the modeling of mixture adsorption. The ability of the E-GAB model to predict the molecular simulation results is compared to the IAST/BET model. The discussion is organized as follows: first is discussed the capability of the models to reproduce the composition of the adsorbed phase, second is discussed the prediction on the adsorbed amounts below and above the pore filling. Adsorbed Phase Composition. In the theoretical development, it is assumed that the parameters Ci,1 and Ki depict only the adsorption sites/adsorbed molecules interactions. Thereby, it should be possible to correctly predict the composition of the adsorbed phase, whatever the fluid state. To verify this point, the adsorption selectivity is introduced:

where n is the number of data points used for the fit, Nj,MS corresponds to the adsorbed amount of the jth data point obtained from molecular simulations, and Nj,model corresponds to the adsorbed amount predicted by the model. Results and errors associated with the fit of pure compound adsorption isotherms of species 1 and 2 are listed in Table 2. Table 2. BET and GAB Parameters Determined on Pure Compound Adsorption Isothermsa model

species

Ns

Ci,1

Ki

rRMS err.

BET

1 2 1 2

61 61 375 375

4.47 16.4 0.860 6.22

1 1 0.496 0.125

4.05 0.60 0.25 0.40

GAB a

The BET model correspond to a GAB model in which Ki = 1.

As mentioned before, one limitation of the two models is that each adsorbed species has to occupy one adsorption site and the monolayer capacity Ns has to be the same for the two species. It is a common limitation shared with other adsorption models such as the extended Langmuir. The monolayer capacity Ns(i) is determined from the fit of one species i is used as initial guess for the fit of the second species Ns(j). This procedure is repeated until two monolayer capacities converge to the same value from one step to another (with a tolerance of ±1 molecule). However, if Ns(i) and Ns(j) do not converge to the same value, the value of Ns chosen is the one that minimizes the relative root-mean-square deviation of the two pure compound adsorption isotherms. As can be seen in Table 2, this only imposes some small deviations on the quality of the fits on the two species. Figure 3 presents the results for pure compound adsorption. It should be emphasized that the adsorbed amounts reported here correspond to the total number of molecules within the slit pore (absolute amounts). The adsorbed quantities are presented as a function of the pure compound activity, defined by eq 12. The uncertainties on the simulations results for the pure compound adsorption isotherms are smaller than the simulations symbols. Due to different Lennard-Jones parameters, the two species exhibit different adsorption behaviors. The species 1 presents a type V adsorption isotherm according to the IUPAC classification.54 The formation of the monolayer is then unfavorable compared to the multilayer formation, due to a

S=

x1 × y2 x 2 × y1

(19)

where xi is the fraction of component i in the adsorbed phase and yi its fraction in the bulk free phase. In accordance with the E-GAB model, the adsorption selectivity is given by SE ‐ GAB =

K1a1[C1,1 + (C2,1 − C1,1)K 2a 2] y2 K 2a 2[C2,1 + (C1,1 − C2,1)K1a1] y1

(20)

The selectivity with the IAST model is obtained by imposing K1 = K2 = 1 in eq 20. The adsorption selectivity of three fluids is studied: in Figure 4a, a mixture with 10% of species 1 and 90% of species 2 on molar basis in the bulk-free phase; in Figure 4b, an equimolar mixture; and in Figure 4c, a mixture

Figure 3. Results and fit of the pure compound adsorption isotherms of the two species. (a) Pure compound adsorption isotherm of species 1. (b) Pure compound adsorption isotherm of species 2. Black ● are results from molecular simulations. Red ■ are for the fit with the BET model. Blue ▲ are for the fit with the GAB model. Dashed lines are guides for the eyes. E



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Figure 4. Adsorption selectivity as a function of pressure. Three bulk-free phase compositions are studied. (a) Adsorption selectivity of a mixture with 10% of species 1 and 90% of species 2 on molar basis in the bulk free phase. (b) Adsorption selectivity of a mixture with 50% of species 1 and 50% of species 2 on molar basis in the bulk free phase. (c) Adsorption selectivity of a mixture with 90% of species 1 and 10% of species 2 on molar basis in the bulk free phase. Black ● are for molecular simulation results, blue ▲ for the E-GAB model, and the red ■ for the IAST/BET model.

IAST at 0.1 MPa. At the same pressure, the average error for the E-GAB model is 12%. We further study the prediction of the adsorbed phase composition as a function of the bulk fluid composition in Figure 5. For a bulk pressure of 0.21 MPa, the selectivity of

with 90% of species 1 and 10% of species 2. The pressure ranges from 0.1 to 1.1 MPa. The uncertainties of the simulation results are not presented here, for the sake of clarity but are not negligible above the capillary condensation for the mixture with 10% of species 1 in the bulk-free phase (Figure 4a). The selectivity curves strongly differ when the composition of the bulk free phase is changed. The capillary condensation from the gas-to-liquid state leads to a transition in these curves, at approximately 0.3 MPa in Figure 4b, 0.4 MPa in Figure 4b, and 0.7 MPa in Figure 4c. Despite these nonmonotonic evolutions, the E-GAB model provides consistent predictions of the selectivity for both gas and liquid states. Some deviations appear for nonequimolar mixtures, where the proportion of the dominant species is overestimated, but the predictions are surprisingly good on the whole range of pressures investigated. We introduce the average absolute deviation (AAD) Di% of species i: D%i

1 = n

n

∑ j=1

Figure 5. Adsorption selectivity at P = 0.21 MPa, as a function of the free phase composition. y1 is the fraction of species 1 in the bulk-free phase, on a molar basis. Black ● are for molecular simulation results, blue ▲ for the E-GAB model, and the red ■ for the IAST/BET model. The right y axis represents the errors in selectivity predictions of the IAST/BET (red line) and E-GAB (blue dashed line) model, compared to simulation results.

|N ji ,MS − N ji ,mod el| N ji ,MS

(21)

where n is the number of data points used for the fit, Nij,MS corresponds to the adsorbed amount of the jth data point of species i obtained from molecular simulations, and Nij,model corresponds to the adsorbed amount of species i predicted by the model. Results and errors associated with the fit of pure compound adsorption isotherms of species 1 and 2 are listed in Table 2. For the three mixtures studied here, the average absolute deviation is 8% for the E-GAB model and 19% for the IAST/ BET model. The IAST/BET and E-GAB predict a similar selectivity at high pressures. However, for the lowest pressures, the capability of the IAST/BET model worsened. The averaged error for the three compositions under study is 43% for the

mixtures with bulk compositions of species 1 (y1) ranging from 10 to 90% is studied. The average error of the E-GAB model compared to the molecular simulations results are below 10%, while it goes from 20 to 45% for the IAST/BET model as y1 increases. Adsorbed Amount. Despite non-negligible deviations for the IAST/BET model at low pressures, the prediction of the adsorbed phase compositions of both models follows the evolution given by the molecular simulation results. Figure 6 F



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same for both species in the mixtures. The asymmetry in size for real systems may be more important, as for example in alkane mixtures. It is out the scope of this study, but such a limit could probably be overcome by treating the species with a chainlike approach, based on the Flory−Huggins theory.55,56 For the adsorption above the capillary condensation, the two models are clearly unable to predict quantitatively the adsorbed amounts. Yet, the E-GAB model remains the most robust since the IAST/BET model diverges to infinity as the sum of the activities approach the unity. This is obvious for the mixture rich in one species: in Figure 6 (panels c and d), where the adsorbed amount is strongly overestimated by the IAST/BET model. However, the selectivity remains well-predicted as seen in Figure 4c. As a result, the crossing of the capillary condensation should not affect the interaction parameters. The remaining parameter which can be reconsidered to improve the modeling above the capillary condensation is then the monolayer capacity Ns. Several authors showed that the adsorption isotherms of liquid mixtures present BET-like shapes.21,26,28,57,58 Most of the theoretical development consist in reformulating the BET model, to overcome the inconsistency when activity is greater or equal to the unity. Another trouble in this kind of approach resides in the meaning of the parameter Ns in the liquid. Thus, using the fact that for the three mixtures studied, the E-GAB model always underestimates the simulation results above the capillary condensation, we introduce a scaling factor α defined by

presents the adsorption isotherms of two mixtures. Adsorbed amounts correspond to the number of molecules in the porous structure.

Figure 6. Adsorption isotherms of binary mixtures. (a) Adsorption of species 1 in a mixture with 50% of species 1 and 50% of species 2 on molar basis in the bulk free phase. (b) Adsorption of species 2 in a mixture with 50% of species 1 and 50% of species 2 on molar basis in the bulk free phase. (c) Adsorption of species 1 in a mixture with 90% of species 1 and 10% of species 2 on molar basis in the bulk free phase. (d) Adsorption of species 1 in a mixture with 90% of species 2 and 10% of species 2 on molar basis in the bulk free phase. Black ● are for molecular simulation results, blue ▲ for the E-GAB model, and the red ■ for the IAST/BET model. Dashed lines are guides for the eyes.

α=

Table 3. Average Deviations of the IAST/BET and E-GAB Model from Molecular Simulation Resultsa AAD species 2 (%)

y1 (%)

IAST/ BET

EGAB

IAST/ BET

EGAB

pressure range (MPa)

10 50 90

24 24 14

10 10 5

5 8 54

3 3 10

0.1 to 0.25 0.1 to 0.35 0.1 to 0.65

(22)

where Ni(MS) is the adsorbed amount of the species i obtained by molecular simulations and Ni(E-GAB) is the adsorbed amount of i predicted by the E-GAB model. Values of the scaling factor above the capillary condensation are reported in Figure 7a. This factor is roughly the same for all mixtures studied here and is nearly constant with pressure. This constant can be viewed as an “engineering” parameter used to extend our model above capillary condensation. However, the relevance of this parameter is supported by the facts that (1) the selectivity independent of the “storage” capacity of the medium is wellpredicted by the E-GAB model in the liquid phase: the parameters Cα,i and Ki remain valid. (2) Liquid adsorption isotherms are generally represented by a BET model.32,34,57 However, these systems are binary systems made of the solvant and the solute species. So the modeling of the adsorption in the liquid phase could reasonably be performed with a rescaled monolayer capacity. The parameter α is used to define a new empirical constant Qs. It corresponds to the value of the parameter Ns, which minimizes the deviation of the E-GAB model with the simulation results above the capillary condensation. It is given by

The average deviations from simulation results in this range of pressures are reported in Table 3. In the gas phase, the

AAD species 1 (%)

Ni(MS) Ni(E‐GAB)

a

They are reported for the three mixtures investigated in this work, as a function of the fraction of species 1 imposed in the bulk free phase. The pressure ranges correspond to the adsorption of gas up to capillary condensation.

average errors of the E-GAB model are at least twice lower than those of the IAST/BET model. This can be explained by the fact that the E-GAB model with an additional adjustable parameter Ki enables one to better reproduce the interaction between adsorbed species in the multilayer. Another cause of discrepancy of the IAST/BET model also comes from the fit of the pure compound, whose root-mean-square error are slightly larger than those of the GAB model. However, both models enable a qualitatively consistent prediction of the adsorbed amounts up to the capillary condensation. Yet, as discussed previously, the current formulation of the model imposes that each adsorbed species occupy one single adsorption site and that the number of adsorption sites is the

Qs =

Ns α

(23)

From molecular simulations, Qs = 644 molecules. Hence, if Ns can be viewed as a two-dimensional surface parameter (along x⃗ and z⃗ in Figure 2), which characterizes the monolayer capacity, then the rescaled empirical parameter Qs can be viewed as a three-dimensional volume parameter, which takes into account the increase of adsorbed species due to the pore G



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Figure 7. Modeling of the adsorbed amount above the capillary pressure. (a) Scaling factor between the mixture above the capillary condensation. Black is for species 1 and red for species 2. + for mixtures with 10% of species 1 in the bulk free phase, × for equimolar mixtures, and * for mixtures with 90% of species 1. (b) Complete adsorption isotherm with adjusted monolayer capacities above capillary condensation. Black is for simulations results, red for E-GAB predictions below capillary condensation (Ns), and blue for predictions above capillary condensation (Qs). By considering two different parameters, Ns and Qs, it has been possible to reproduce the adsorbed amount below and above the capillary condensation.

filling. It characterizes not only the molecules adsorbed at the surface of the porous solid but also those present at the center of the pore due to the capillary condensation, and whose composition is described with an adsorption model. Interestingly, as seen from Figure 7, it is possible to predict the adsorbed phase for gas and liquid mixtures with one set of interaction parameters Ci,1 and Ki, using this empirical correction in the liquid state. The Qs parameter allows reasonable results for the three gas composition presented in this study. This corresponds to a wide range of species 1 mole fraction in the adsorbed phase (x1), ranging from 4.05% to 89.90%. Moreover, such a correction cannot be defined as simply for the IAST/BET model because of its singularity when the sum of the activities approaches the unity. Thereby, the empirical parameter Qs can be used in complement to Ns to reproduce the mixture adsorption at the gaseous and liquid states.

not too different in interaction energy or size.2,18,34,59 The theoretical framework used here seems so well-suited as an alternative way to provide analytic models of mixture adsorption, complementary to other approaches such as the ideal adsorbed solution theory,9 the multicomponent potential theory of adsorption (MPTA),13,33 or approaches based on the Langmuir theory32 to treat the multilayer mixture adsorption. To investigate the capabilities of this model, we study the adsorption of three binary mixtures from gaseous to liquid state using synthetic data generated by molecular simulations. Despite its simplicity, the model enables a very good prediction of adsorption selectivity: the average absolute deviation between molecular simulations results and the model predictions being 8%. For comparison, the average absolute deviation of the IAST/BET34 model is 19%. The model enables also to predict quantitatively the adsorbed amount for pressures up to the capillary condensation with average absolute deviations less than 10%. Furthermore, using one extra empirical parameter to take into account the increase of adsorbed species due to the pore filling above capillary condensation, we have been able to extend the prediction of the adsorbed amount above the capillary pressure.

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CONCLUSIONS On the basis of a statistical description of the adsorption phenomena in mixtures, we derive a simple and theoretically grounded model for competitive adsorption modeling. This model extends existing work26−28 to address the multilayer sorption of systems containing an arbitrary number of adsorbed species. The simplicity of the proposed model, which relies on two parameters per species only, enables fast calculations of the adsorbed amounts. The adsorbed molecules are assumed to be distributed in two adsorbed layers. The first one is a monolayer, where each adsorption site embeds one adsorbed molecule. The second one consists of a multilayer, which lies on top of the monolayer. Moreover, we believe that the description of the species distribution in the adsorption layers promotes the understanding of the competitive adsorption process. Our development relies on the following assumptions. (1) The number of accessible sites has to be the same for each species. (2) Each species occupies one single sorption site. (3) The adsorbed phase is treated as an ideal mixture (no cross interaction parameters). Thus, in this frame, the mixture adsorption can be predicted from the knowledge of pure compound adsorption isotherms, without the need of additional parameters to quantify interactions between unlike species. However, these assumptions may lead to strong limitations, but they are shared with other models from the literature and seem to induce relatively small deviations in numerous systems, as long as the species are

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge TOTAL S.A. for financial support and for a grant for J.C. Computer time for this study was provided by the computing facilities MCIA (Mésocentre de Calcul Intensif Aquitain) of the Université de Bordeaux and of the Université de Pau et des Pays de l’Adour.

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