Non-Local Density Functional Theory NLDFT

Essential progress has been made recently in the area of molecular level modeling of capillary condensation. The methods of grand canonical Monte Carlo (GCMC) simulations [4], molecular dynamics (MD) [5], and density functional theory (DFT) [6] are capable of generating hysteresis loops for sorption of simple fluids in model pores. In our previous publications (see [7] and references therein), we have shown that the non-local density functional theory (NLDFT) with properly chosen parameters of fluid-fluid and fluid-solid intermolecular interactions quantitatively predicts desorption branches of hysteretic isotherms of nitrogen and argon on reference MCM-41 samples with pore channels narrower than 5 nm. 

The non-local density functional theory (NLDFT) with properly chosen parameters of fluid-fluid and fluid-solid intermolecular interactions quantitatively predicts both adsorption and desorption branches of capillary condensation isotherms on MCM-41 materials with the pore sizes from 5 to 10 nm. Both experimental branches can be used for calculating the pore size distributions in this pore size range. However for the samples with smaller pores, the desorption branch has an advantage of being theoretically accurate. Thus, we recommend to use the desorption isotherms for estimating the pore size distributions in mesoporous materials of MCM-41 type, provided that the pore networking effects are absent. 

The non-local density functional theory (NLDFT) is well established and widely presented in the literature. The distribution of density in a confined pore can be obtained for an open system in which a pore is allowed to exchange mass with the surroundings. From the thermodynamic principle, the density distribution is obtained by minimization of the following grand potential written below for the one-dimensional case [170] ... 

We suggest a model of adsorption in pores with amorphous and microporous solid walls, named the quenched solid non-local density functional theory (QSNLDFT) model. We consider a multicomponent non-local density functional theory (NLDFT), in which the solid is treated as a quenched component with a fixed spatially distributed density. Drawing on several prominent examples, we show that QSNLDFT model produces smooth Isotherms of mono- and polymolecular adsorption, which resemble experimental isotherms on amorphous surfaces. The model reproduces typical behaviors of N2 isotherms on micro- mesoporous materials, such as SBA-15. QSNLDFT model offers a systematic approach to the account for the surface roughness/heterogeneity in pore structure characterization methods. 

Recent progress in the theory of adsorption on porous solids, in general, and in the adsorption methods of pore structure characterization, in particular, has been related, to a large extent, to the application of the density functional theory (DFT) of Inhomogeneous fluids [1]. DFT has helped qualitatively describe and classify the specifics of adsorption and capillary condensation in pores of different geometries [2-4]. Moreover, it has been shown that the non-local density functional theory (NLDFT) with suitably chosen parameters of fluid-fluid and fluid-solid interactions quantitatively predicts the positions of capillary condensation and desorption transitions of argon and nitrogen in cylindrical pores of ordered mesoporous molecular sieves of MCM-41 and SBA-15 types [5,6]. NLDFT methods have been already commercialized by the producers of adsorption equipment for the interpretation of experimental data and the calculation of pore size distributions from adsorption isotherms [7-9]. 

In this paper, we suggest a systematic approach that extends the applicability of NLDFT models to heterogeneous surfaces of amorphous and microporous solids. The main idea is to use a multicomponent NLDFT, in which the solid is treated as one of the components with a fixed spatially distributed density. The model, named quenched solid non-local density functional theory (QSNLDFT), is an extension of the quenehed-annealed DFT model of systems with hard-core interactions recently proposed by Schmidt and coworkers [23,24]. Drawing on several prominent examples, we show that the proposed model produces smooth isotherms in the region of multiplayer adsorption. Moreover, the effects of wall microporosity can be naturally incorporated into the model. Although the parameters of the model have not been yet optimized to describe quantitatively a particular experimental system, the model generates adsorption isotherms which are in qualitative agreement with experimental isotherms of N2 or Ar adsorption on amorphous silica materials. 

The model (2)-(4) is referred to as the quenched solid non-local density functional theory (QSNLDFT). There are several advantages in considering the solid as a quenched component of the system rather than a source of the external field. On the one hand, this approach offers flexibility in the description of the fluid-solid boundary by varying the solid density and the thickness of the diffuse solid surface layer. On the other hand, it retains the main advantage of NLDFT computational efficiency because even a one-dimensional solid density distribution ceui include the effects of surface roughness and heterogeneity. For example, the solid density distribution can be taken from simulations of amorphous silica surfaces [29,30]. 

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