##
1A hybrid cylindrical model for characterization of MCM-41 by density functional theory
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A hybrid cylindrical model for characterization of MCM-41 by density functional theory

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A hybrid cylindrical model for characterization of MCM-41 by density functional theory (DFT) is proposed in this work, where the surface heterogeneity of MCM-41 is taken into account by using a hybrid potential model to represent the interactions between a pore wall and molecules inside the pore.

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This model consists of two parts: (1) the potential energies from the oxygen atoms inside the wall, represented by the potential model proposed by our group; (2) the potential energies from the silanol coverage and/or other unknown factors in the surface of the channel of MCM-41, represented by the cylindrical surface potential function of Tjatjopoulos

*et al.*(G. J. Tjatjopoulos, D. L. Feke and J. A. Mann,*J. Phys. Chem.*, 1988,**92**, 4006–4007).
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To test the new model, the DFT method was used to calculate the adsorption isotherm of nitrogen in MCM-41 at 77 K. The isotherm calculated is compared with the experimental data as well as the calculated results of Maddox

*et al*., who divided the surface of MCM-41 into eight sectors and adopted different parameters for each sector to consider the heterogeneity of the surface.
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Compared with the work of Maddox

*et al*.
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(M. W. Maddox, J. P. Olivier and K. E. Gubbins,

*Langmuir*, 1997,**13**, 1737–1745), our model gives a much better fit to the experimental isotherm of nitrogen at 77 K in the pressure range of*P*/*P*_{0}= 0.2–0.5 with much less parameter and computation effort, where phase transition and capillary condensation occur.
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Furthermore, the relationship between the reduced pressure, at which capillary condensation takes place, and the pore diameter by the hybrid model is in good agreement with that obtained by Maddox

*et al*.
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In addition, adsorption and phase behavior of methane and ethane are studied by the model, and the calculated results also coincide well with the experimental isotherms of methane and ethane at 264 K–373 K. Therefore, the hybrid potential model incorporating into the DFT method provides a useful tool for characterization of MCM-41.

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## Introduction

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Porous materials have been applied in many fields, such as gas separation, purification, and reaction processes,

*etc*.
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In the meantime, when a fluid is confined to a region of a molecular scale, its phase behavior can be strongly affected, and a rich variety of new types of phase transitions can occur.

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Therefore, investigations on characterization and the adsorption behavior of porous materials are of great importance from both scientific and practical points of view.

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In 1992, a new family of mesoporous molecular sieves designated as M41S was discovered by Mobil scientists.

^{1,2}
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As is pointed out by Ravikovitch

*et al*.,^{3}the main advantages of MCM-41, a representative of M41S, are (1) of an array of uniform hexagonal channels with very narrow pore size distributions, (2) the pore length is significantly larger than the pore diameter, (3) pore networking effects are negligible.
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Therefore, because of its ideal pore structure, there exists a strong incentive for researchers to study and use it.

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Not only can it be used in adsorption, separation and catalysis, it is also regarded as the most suitable model adsorbent currently available for verification of theoretical predictions for cylindrical pores.

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Many experimental investigations have been performed on the adsorption properties of MCM-41,

^{1,4–8}which provides deep insights into the adsorption behavior of simple fluids in MCM-41, and also serves as an experimental basis for the test and development of theoretical models.
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On the other hand, both the molecular simulation and density functional theory (DFT) methods have been used to study the adsorption behavior of simple fluids in MCM-41 and its pore size distribution (PSD).

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For example, Ravikovitch

*et al*^{3,9,10}. carried out many investigations to study the adsorption isotherms of nitrogen in MCM-41 and PSD of MCM-41 by the DFT method, Maddox*et al*^{11–14}. used the grand canonical Monte Carlo (GCMC) model to study the adsorption behavior of a pure fluid and binary mixtures in MCM-41.
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In both the DFT and MC methods, a potential model for the wall–fluid interactions is required, which plays a key role in the accuracy of the calculated properties.

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As a result, many potential models have been proposed for Lennard-Jones (LJ) cylindrical pores, such as the cylindrical surface potential model of Tjatjopoulos

*et al*.,^{15}the potential model as the analog of 10–4–3 for planar graphite surface,^{11–13}the pseudoatomic model of Nicholson and Gubbins,^{16}the model of Peterson*et al*.,^{17}and a simplified model proposed by Zhang and Wang.^{18}
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Recently, a complete analytical potential model for a cylindrical pore of finite thickness was further proposed by our group.

^{19}
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It gives identical results to the model of Peterson

*et al*.,^{17}while inconvenient numerical integration in the latter is avoided.
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In addition, it reduces to the cylindrical surface model of Tjatjopoulos

*et al*.,^{15}when the wall thickness approaches zero.
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Previous works

^{9,14,18}show that the experimental isotherms can not be reproduced very well by using the existing potential models in the whole pressure range if the surface heterogeneity of MCM-41 is not taken into account.
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Therefore, Maddox

*et al*^{14}. divided the surface of MCM-41 into eight sectors and adopted different parameters for each sectors to consider the heterogeneity of its surface.
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Their work gives a good representation of the adsorption isotherms of nitrogen in MCM-41 by the GCMC method.

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However, considerably greater parameter and computation effort is needed.

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In this work, we focus on the heterogeneity of MCM-41 by using the DFT method, because the DFT method is remarkably efficient in the computation of the meso-porous materials, which are of simple, low-dimensional geometry and their external potentials,

*i.e.*the interactions between the solid surface and fluid molecules, can be described by explicit analytical functions.
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A hybrid potential model is proposed here to consider the surface heterogeneity of MCM-41, which consists of two parts: (1) the potential energies from the oxygen atoms inside the wall, represented by the complete analytical model developed by our group recently.

^{19}
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(2) the potential energies from the silanol coverage and/or other unknown factors in the surface of the channels of MCM-41, represented by the cylindrical surface potential function of Tjatjopoulos

*et al.*^{15}
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To test this new model, first, the adsorption isotherm of nitrogen in MCM-41 at 77 K is calculated.

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Then, calculated and experimental adsorption isotherms of methane and ethane confined in MCM-41 pores at different temperatures are compared.

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Based on our results, the hybrid potential model incorporated into the DFT method is recommended as a useful tool for characterization of MCM-41.

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In addition, some discussion is also addressed.

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## Density functional theory and potential models

### Density functional theory

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In the DFT approach, the grand potential (GP) of the fluid in a pore at a given temperature

*T*, and the chemical potential*μ*is a functional of the local fluid density*ρ*(**):where***r**F*[*ρ*(**)] is the intrinsic Helmholtz free energy functional and***r**v*(*r*) is the potential of the pore walls.
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The criterion for equilibrium is that the grand potential reaches a minimum, namely, the density profile

*ρ*(**) of the adsorbate within the pore satisfies the condition:In this paper the expression of free energy in eqn. (1) is split into attractive and repulsive contributions.***r*
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The attractive part of the fluid–fluid potential is given by the Weeks–Chandler–Anderson perturbation scheme

^{20}for a cut and shifted LJ potential.
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The repulsive part of hard spheres,

*F*_{hs}, is then split into an ideal part,*F*_{hs}^{id}, and an excess part,*F*_{hs}^{ex}:The ideal part is trivial, and the excess part is given by Tarazona’s recipe.^{21,22}
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Detailed information can be referred to in our previous work

^{23}.
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### The potential model

#### The potential model between a pair of fluid molecules

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In this work, we use the LJ potential to represent the interaction between a pair of fluid moleculeswhere

*r*is the interpartical distance,*ε*is the well depth , and*σ*is the molecular size distance between a pair of molecules when the potential is zero.
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The full description can be referred to the ref. .

^{24}
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Parameters of the LJ potential are listed in Tables 1 and 2.

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#### The potential model between the fluid molecules and MCM-41 pore wall

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A hybrid potential model is proposed to represent the interactions between the pore wall and the molecules inside the pore.

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This model consists of two parts: (1) the potential energies from the oxygen atoms inside the wall, represented by the complete analytical potential model proposed by our group

^{19}(see Appendix A); (2) the potential energies from the silanol coverage and/or other unknown factors on the surface of the channels of MCM-41, represented by the cylindrical surface potential model of Tjatjopoulos*et al.*^{15}
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A schematic diagram of this model is shown in Fig. 1(a).

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#### The interactions between the fluid molecules and the oxygen atoms in MCM-41

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Traditionally, we ignore the silicon atoms in the adsorbent structure, and just take into account the interactions between the fluid molecules and the oxygen atoms in MCM-41.

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In this work, a complete analytical potential model proposed

^{19}(see Appendix A) is used to represent this interaction, in which the interaction energy,*U*, of the testing fluid molecule within a cylindrical pore of infinite thickness is given bywhere*F*is the hypergeometric series,*R*is the radius of the cylindrical pore,*l*is the radial distance from the testing particle to the center of the pore, and*ρ*_{solid}is the number density of interaction sites inside the pore wall.
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A skeletal density of 2.0 g cm

^{−3}is used in this work, as adopted in the theoretical model of Feuston and Higgins.^{25}
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Parameters

*ε*_{sf},*σ*_{sf}for solid–fluid interactions are obtained by the Lorentz–Berthelot combining rules:If the wall thickness is finite, the interaction energy of the testing fluid molecule experienced can be calculated by*U*(*R*_{1},*l*) −*U*(*R*_{2},*l*), where*R*_{1}and*R*_{2}are the inside radius and outside radius of the cylindrical pore, respectively.
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#### The interactions between the fluid molecules and the silanol coverage and/or other unknown factors on the MCM-41 surface

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As an approximation, we made the surface heterogeneity of MCM-41 caused by the silanol coverage and/or other unknown factors on the surface of the pores of MCM-41 “smooth”, that is, it is assumed that all the heavy cations and/or functional groups attached to the surface of MCM-41 are uniformly distributed.

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Therefore, the interactions between the attractive sites and a fluid molecule inside MCM-41,

*U*(*r*,*R*), can be represented by the cylindrical surface potential function proposed by Tjatjopoulos*et al*.:where*R*is the radius of the cylindrical pore,*r*is the radial distance from the test particle to the surface of the wall and*F*is the hypergeometric series.
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Because we know little about the composition, density and array mode of these attractive sites present on the surface of MCM-41, we take

*ε*_{surf,f}*ρ*_{surf}as an adjustable parameter.
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It consists of two parts:

*ε*_{surf,f}is the surface heavy cations and/or functional groups–fluid interaction parameter,*ρ*_{surf}is the density of these attractive sites on the wall.
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The values of the parameters obtained are listed in Tables 1 and 2.

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The parameter for the surface molecule,

*σ*_{surf}is set to 0.276 nm,^{24}and parameter*σ*_{surf,f}is obtained by the Lorentz–Berthelot combining rules.
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## Results and discussion

### Adsorption isotherm of N_{2} in MCM-41 at 77 K

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The DFT method combined with the hybrid cylindrical potential model was used to calculate the adsorption isotherm of nitrogen in MCM-41 initially at 77 K. The calculated results and the experimental data

^{14}are shown in Fig. 2.
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Parallel calculations were also carried out for the models of the complete analytical potential model and the model of Tjatjopoulos

*et al*.
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(see Table 1 for the parameters) to see the improvement of the hybrid model.

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From Fig. 2 it is found that the hybrid potential model shows improved results over the two models.

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For a detailed comparison between the three models, results are shown in Fig. 3.

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Obviously, when the reduced pressure

*P*/*P*_{0}varying from 0 to 0.1, the model of Tjatjopoulos*et al*. works better than the complete analytical model, and the hybrid model gives nearly identical results to that of the model of Tjatjopoulos*et al*., as is shown in Fig. 3a.
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The results from these three models are comparable, while the hybrid model being slightly better in the range of

*P*/*P*_{0}= 0.1–0.3.
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When

*P*/*P*_{0}is in the range of 0.3–0.6, the complete analytical model is better than the model of Tjatjopoulos*et al*., and the hybrid model works slightly better than the complete analytical model.
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Consequently, it is concluded that the hybrid model can incorporate the advantages of the two constituent models, leading to a better description of the experimental adsorption isotherm in the pressure ranges.

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Our calculated results are further compared with the work of Maddox

*et al*.,^{14}who divided the surface of MCM-41 into eight sectors and adopted different parameters for each sector to consider the heterogeneity of its surface (see Fig. 1b).
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Their results from GCMC method are shown in Fig. 4, along with our calculated results.

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Obviously, in the very low pressure region, the method of Maddox

*et al*. is better than our model.
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This observation is not surprising, because adsorption at very low pressures is very sensitive to the potential model used, and the cylindrical wall is divided into eight sectors with different parameters in the method of Maddox

*et al*.
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In contrast, in the range of

*P*/*P*_{0}= 0.2–0.4, our model gives better results.
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More impressively, our hybrid model describes phase transition and capillary condensation very well, while the model of Maddox

*et al*. shows large discrepancies in the region.
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Besides, it is noticed that much more complicated potential functions and more parameters are used in their method.

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### Relationship between pore size and capillary condensation pressure for N_{2} at 77 K

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Furthermore, the relationship between the reduced pressure at which capillary condensation takes place and the pore diameter is determined with our model.

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Fig. 5 presents eight isotherms by DFT method for pore size ranging from 2.5 nm to 4.2 nm.

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The reduced pressure at which capillary condensation takes place can then be accurately determined here.

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Fig. 6 shows the relationship between the capillary condensation pressure and the pore diameter.

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The results are in good agreement with those obtained by the Maddox

*et al*. using the Monte Carlo method^{14}.
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### Adsorption isotherms for methane and ethane

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To further test the hybrid model, the experimental adsorption isotherms of methane and ethane in MCM-41 pores with a pore diameter of 4.1 nm and a wall thickness of 1.0 nm measured by Yun

*et al*^{27}. were collected from the literature.
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They measured two isotherms,

*T*= 303.15 and 373.15 K, for methane and four isotherms,*T*= 264.75, 273.55, 303.15 and 373.15 K, for ethane.
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In addition, they used the GCMC approach to calculate the isotherms and obtained good agreement between simulated and experimental values.

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In this work, these experimental data are adopted to further test the new model.

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The parameters for methane and ethane molecules are listed in Table 2.

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The calculated results by using the DFT method with the hybrid model are shown in Figs. 7 and 8.

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Obviously, the calculated isotherms are in good agreement with the experimental data for methane, while the DFT predictions of ethane adsorptions are somewhat less accurate, possibly caused by the simplified treatment of the adsorbate–adsorbate interactions.

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Nevertheless, even under these conditions the DFT predictions show a consistent tendency with experimental observations.

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This again illustrates that the hybrid potential model incorporating into the DFT method provides a useful tool for characterization of MCM-41.

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## Conclusions

87

A hybrid cylindrical model for characterization of MCM-41 is proposed in this work, where the heterogeneity of the pore of MCM-41 is taken into account by dividing the interactions between the fluid molecules in the pore and the wall into two parts: (1) the potential energies from the oxygen atoms inside the wall, represented by the complete analytical potential model proposed by our group recently;

^{19}(2) the potential energies from the silanol coverage and/or other unknown factors in the surface of the channel of MCM-41, represented by the cylindrical surface potential function of Tjatjopoulos*et al.*^{15}
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(see Fig. 1a).

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The hybrid potential model proposed enables the DFT method to represent better the adsorption isotherm of nitrogen in MCM-41 at 77 K than the two constituent potential models, and gives a better fit in the range of

*P*/*P*_{0}= 0.2–0.4, and accurate capillary condensation pressure, compared with the work of Maddox*et al*.,^{14}in which the pore of MCM 41 consists of eight sectors with different potential parameters (see Fig. 1b).
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Impressively, the dependence of the capillary condensation pressure on pore diameter for adsorption of nitrogen at 77 K can be reproduced very well with the hybrid model.

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To test the new model, the adsorption isotherms of methane and ethane ranging from 264–373 K in MCM-41 were predicted with good agreement.

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As a result, the present work demonstrates that the hybrid potential model incorporating into the DFT method provides a useful tool for characterization of MCM-41.

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It should be noted that this work addresses the heterogeneity induced by a smooth potential energy on the surface of MCM-41.

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In fact, there exist some defects and domains of active groups on the surface of MCM-41, which can not be simply represented by a smooth potential function.

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We probably should introduce a more complicated method to take into account the effects of both the energy and geometry heterogeneity.

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This will be discussed in our future work.

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