Elsevier

Applied Surface Science

Volume 253, Issue 13, 30 April 2007, Pages 5596-5600
Applied Surface Science

Adsorption in cylindrical pores: Mixed lattice-site/off-site Monte Carlo simulations in pores with heterogeneous wall structure

https://doi.org/10.1016/j.apsusc.2006.12.096Get rights and content

Abstract

We present results of grand canonical Monte Carlo simulations of adsorption in cylindrical pores with rough surface modeled by lattice-site approach. Each site is characterized by two parameters: structural and energetic, which locally modify the structure and energy properties of the surface. There are three types of sites, randomly distributed over the wall: attractive, neutral and repulsive with respect to the smooth pore model. The results presented here show how this model affects the mechanism of adsorption and how it changes the forms of adsorption isotherm. We compare our numerical results with the experimental data of adsorption of a simple fluid (CH4, T = 77 K) in cylindrical silica pore of diameter d = 4 nm (MCM-41 material).

Introduction

Modeling isotherms of adsorption is an important and very difficult task. Important, because the observed isotherms can have apparently similar forms but, at the same time, they may result from different particular microscopic mechanisms. The problem is difficult because most of the adsorbing surfaces are strongly heterogeneous, in addition to possible pore size distributions and/or interconnectivity in real materials. In such situations, the models must take into account the disordered structures of the adsorbing system. An extensive review of the theoretical models of adsorption of gases on flat heterogeneous surfaces is given in the book by Rudzinski and Everett [1]. The increased complexity of the adsorption mechanism makes computer simulations methods the most effective tools in studying the problem theoretically. Among them, the Monte Carlo simulations in grand canonical ensemble are very well adapted to the study of adsorption and the most frequently used.

The first numerical studies of adsorption on heterogeneous plane surfaces used simple lattice site models [2], [3]. They proposed realistic models of the surface, based on more or less disordered pattern of the initially crystalline structure [4]. In all these models, heterogeneity of the adsorbing surface was introduced in one of the following ways: (1) an ordered surface was created by cutting of a known silica polymorph; (2) an unrelaxed amorphous surface was obtained by cutting bulk amorphous silica; (3) a relaxed amorphous surface was created by relaxing the amorphous surface; (4) a random surface was created by Monte Carlo simulations. These approaches are general and may be also used to model porous materials.

However, curved surfaces typical for porous materials introduce a new level of difficulty. In such situation not only the heterogeneity of the surface plays a role. The form of the pore is also an important factor that may modify the adsorption mechanism as compared to a flat surface. The real adsorbents are structured in a regular way (like graphite or carbon nano-tubes) or have amorphous, disordered and heterogeneous wall structures (like MCM-41 silica pores). As discussed in [5], the most realistic models have been based either on explicit atomic structures with different level of randomness of their positions [2], [3], [6], [7], [8], [9], [10], [11], [12], [13], [14] or on a distribution of the adsorption sites in the cylindrical pore, without any direct references to the underlying atomic positions [14], [15], [16], [17], [18], [19]. The first approach has a merit of representing more accurately the structure of pore walls. The second approach was often much faster in numerical applications and allowed one to simulate the systems with larger number of adsorbed particles. The models were also easily parameterized and the influence of a particular choice of parameter value on the mechanism of adsorption was easy to analyze [16], [19].

In this paper, we propose a simple model of surface heterogeneity that combines together these two approaches. We also discuss an influence of heterogeneity of surfaces on the mechanism of adsorption and, consequently, on the form of isotherms, in cylindrical pores. The simplicity of the model consists in a small number of parameters, which is needed to control the adsorbing sites distribution and to define the structural and energetic parameters that modify the adsorption mechanism. There are two features of adsorption mechanism that are primary addressed in our analysis: (i) the low pressure adsorption which seems to depend mostly on the energetic distribution of the adsorption sites and determines the first layer formation mechanism [16], [19]; (ii) the step-wise character of the multilayer adsorption which is sensitive to the structural heterogeneity of the surface. The temperature effect is superimposed on the structural and energetic heterogeneity [16], [20], [21], [22]. Its entropic effect makes the structure more disordered and seems to match the heterogeneity, that is, the increasing temperature facilitates low-pressure adsorption and rounds-off the step-wise form of isotherms. However, the influence of the temperature is purely kinetic. Although it leads to macroscopic effects similar to the influence of the heterogeneity (for example, similar variations of the isotherm forms), the microscopic mechanisms are totally different in these two situations. Generally, the lower the temperature (and/or the size of the pore is smaller) the more difficult is to reproduce adsorption isotherms in heterogeneous pores. It is the goal of this paper to study these aspects of adsorption using a mixed lattice-site/off-site model defined in the following chapter. Methane, with its bulk melting temperature of 90 K, is an interesting model of adsorbent to study the mechanism of adsorption in heterogeneous pores at temperatures around 77 K.

Section snippets

Heterogeneity model

To define surface heterogeneity in the pore we have used a lattice-site like model. The surface of the smooth cylinder with the radius R (R = 2 nm) and the length Lz (Lz = 5 nm) has been divided into small areas (2D cells) si,j of rectangular shape. The size of each rectangle is defined by two parameters: an angle Δϕ giving the length of the curved side of rectangle (=R Δϕ) and a length Δz, along the pore axis (see Fig. 1). Consequently, we have a total number of lattice site elements si,j (i = 1, (2πϕ

Numerical model

The conventional grand canonical MC ensemble was applied. The simulation box contained one pore of diameter d = 4 nm. The periodic boundary conditions along the tube axis were applied. Most of the simulations were performed in the pore of length Lz = 5 nm; some calculations were done in the boxes with Lz = 10 nm to verify an importance of the size effect. The adsorbed system was assumed to be in equilibrium with the bulk gas, which obeyed the ideal gas law. This allowed us to use the external gas

Results

Our reference situation is a cylindrical pore with smooth wall. In such system, at 77 K, the simulated isotherm has a step-wise form: the first layer formation and the capillary condensation are observed at well-defined pressures. As a consequence, the adsorption is negligible below the pressure of the first layer formation. Similar form of the isotherm was observed experimentally for adsorption on smooth or ordered surfaces such as graphite or carbon nano-tube walls. On the contrary, step-wise

Conclusions

The advantage of the computer simulation methods is its capability to give a microscopic insight into experimental data, which, by definition, are macroscopic and of statistical nature. In this paper, we have shown how the heterogeneity of the pore surface affects the microscopic mechanism and stability of adsorbed system. If the system may form regular layers, the step-wise isotherm is observed. It is the case of ideal smooth surface. The continuous isotherms are observed if the heterogeneity

References (24)

  • D. Nicholson et al.

    J. Colloid Inter. Sci.

    (1977)
  • J.M. Stallons et al.

    Chem. Eng. Science

    (2001)
  • B. Kuchta et al.

    Colloid. Surf. A

    (2004)
  • M.J. Bojan et al.

    Surf. Sci. Lett.

    (1988)
  • B. Kuchta et al.

    Stud. Surf. Sci. Catal.

    (2005)
  • W. Rudzinski et al.

    Adsorption of Gases on Heterogeneous Surfaces

    (1992)
  • J.A. O’Brien et al.

    J. Chem. Soc. Far. Trans.

    (1985)
  • C.G. Sonwane et al.

    J. Phys. Chem.

    (2005)
  • Y. He et al.

    Langmuir

    (2003)
  • J.-H. Yun et al.

    Langmuir

    (2002)
  • B.P. Feutson et al.

    J. Phys. Chem.

    (1994)
  • B. Coasne et al.

    J. Chem. Phys.

    (2004)
  • Cited by (12)

    • Molecular simulation of methane adsorption in nanoscale rough slits

      2022, Journal of Natural Gas Science and Engineering
      Citation Excerpt :

      The fractal dimension of nanopores ranges from 2–3 with those in kerogen larger than others (Li et al., 2019; Liu et al., 2019; Toghraie et al., 2019). Kuchta et al. (2007) stated that regular adsorption layers were formed on smooth surfaces with stepped isotherms, while continuous isotherms were shown for rough surfaces as the subsequent ordered adsorption layers were inhibited. Gao et al. (2021) built the rough pore models by rotating the graphene face (0001) to different inclination angles.

    • Simplified local density model for gas adsorption in cylindrical carbon pores

      2019, Applied Surface Science
      Citation Excerpt :

      Furthermore, if the pore size is small enough to contain only one fluid molecule or the distance of fluid molecule to the surface of cylinder wall is smaller than σff, it is more complicated to calculate attractive potential directly. The curved surface of cylindrical pore increases the calculation difficulty to a new level [12]. Thence, a novel method is introduced below.

    • Study of heat of adsorption across the capillary condensation in cylindrical pores

      2011, Colloids and Surfaces A: Physicochemical and Engineering Aspects
      Citation Excerpt :

      This is typical of the isosteric heat behavior on a heterogeneous surface [15,17], in contrast to the isosteric heat obtained from simulations on a homogenous surface [29]. In order to simulate the heterogeneity of the MCM-41 surface, we have developed a new model which is simpler than other models proposed in the literature [17,22,30–32]. By considering the heterogeneity of MCM-41, we allow for structural and energetic variation along the pore (Fig. 15).

    View all citing articles on Scopus
    View full text