L. F. Herrera

A new method to determine pore size and its volume distribution of porous solids having known atomistic configuration

A simple method, based on Monte Carlo integration, is presented to derive pore size and its volume distribution for porous solids having known configuration of solid atoms. Because pores do not have any particular shape, it is important that we define the pore size in an unambiguous manner and the volume associated with each pore size. The void volume that we adopt is the one that is accessible to the center of mass of the probe particle. We test this new method with porous solids having well defined pores such as graphitic slit pores and carbon nanotubes, and then apply it to obtain the pore volume distribution of complex solids such as disordered solids, rectangular pores, defected graphitic pores, metal organic framework and zeolite.

A Monte Carlo integration method to determine accessible volume, accessible surface area and its fractal dimension

We present a self-consistent Monte Carlo integration scheme to determine the accessible volume and the accessible surface area of a porous solid with known atomistic configuration. The new feature of this method is the determination of the variation of volume not only with respect to the distance from the surface (geometrical factor) but also with respect to the energy of the closest solid atom type. The variation with respect to distance gives us information about the area of the solid-fluid boundary (which is defined as one on which a spherical particle has zero solid-fluid potential energy) while the variation of the interfacial area of a contour at any distance from the surface, yields the surface curvature, for both convex and concave surfaces. On the other hand, the variation with respect to the type of solid atom yields information about the distribution of the area in terms of the heterogeneity of the surface. We illustrate our new method with a number of examples, ranging from a simple channel pore to complex solids, such as metal organic frameworks (MOF) and bundles of carbon nanotubes.

A Monte Carlo scheme based on mid-density in a hysteresis loop to determine equilibrium phase transition

A new and simple method to determine equilibrium phase transition in adsorption systems exhibiting a hysteresis loop is presented as an alternative to methods such as multiple histogram reweighting, gauge cell method and thermodynamic integration. This method is based on the NVT-grand canonical Monte Carlo mid-density scheme to determine the coexistence chemical potential and coexistence densities of an adsorption system. We illustrate this new scheme with argon and methane adsorption in a number of model solids having slit and cylindrical pores. This method does not have a strong basis on thermodynamic ground, but it does provide a simple heuristic approach that is simpler to understand physically.

Microscopic analysis of adsorption in slit-like pores: layer fluctuations of particle number, layer isosteric heat and histogram of particle number

A number of measures are proposed as a microscopic means to analyse adsorption of gas on a surface and in slit pores under subcritical and supercritical conditions. Layer fluctuation of particle number provides us with information on where most of the mass interchange occurs, which can then be used as an indicator of the position of the interface separating the adsorbed phase and gas phase. The layer compressibility can be used to compare the adsorbed phase density with that of the bulk liquid. The layer isosteric heat provides an indication of the relative contribution of each layer to the overall isosteric heat. Finally, a histogram of particle number as a function of fluid–fluid particle energy is utilised to yield valuable information about the energetic structure of the adsorbed phase, for example (1) the number of neighbouring particles and (2) the evolution of the arrangement of particles.

Comparative simulation study of nitrogen and ammonia adsorption on graphitized and nongraphitized carbon blacks.

Grand canonical Monte Carlo simulation is used to study the adsorption of nitrogen at 77 K and ammonia at 240 K to represent weakly polar and polar molecules, respectively, on infinite and finite graphite surfaces. These graphite surfaces were modeled with different percentages of carbons removed (defects) from the top graphite layer. Increasing the number of defects increases the adsorption and the isosteric heat of nitrogen at low pressure. At moderate pressures the amount adsorbed is less due to the disruption in the packing of the nitrogen in the first layer. In contrast, the adsorption of ammonia at all pressures is reduced as the percentage of defects is increased. This is due to the disruption in ammonia bonding caused by the defects. The condensation-like step change in the ammonia isotherm on the perfect graphite surface is not observed for any of these surfaces with defects even for the case of only 10% defects. At high percentage of defects the adsorption isotherm is close to Henry law behavior for much of the pressure range. The adsorption on finite surfaces shows that the amount adsorbed for both molecules decreases compared with that of the infinite surfaces, resulting from interaction potentials with the surface and other fluid molecules at the edge. The decrease is much greater for the ammonia adsorption because the bonding between ammonia molecules is disrupted, meaning that the adsorption cannot follow the mechanism of condensation seen for the infinite surface.

Histogram of number of particles as an indicator for 2D phase transition in adsorption of gases on graphite

Grand canonical Monte Carlo simulation is used to study the adsorption of gases with strong and weak molecular interaction on graphite. We choose nitrogen adsorption at 77 K, ethylene at 104 K, methanol at 240 K and ammonia at 300 K as model examples. The adsorption mechanism of these species can be studied by analysing the radial distribution and the ‘number of particles histogram’ as a function of loading. At low pressures, at which the surface is barely covered with molecules, nitrogen and ethylene adsorb in a similar manner, while ammonia and methanol show a distinct difference because of the formation of clusters, resulted from the hydrogen bonding. Small clusters are observed for methanol and larger ones for ammonia, which is in agreement with the fact that hydrogen bonding is more significant in ammonia than in methanol. Analysis of the number of particles distribution can identify 2D phase transition as a sudden shift of the peak in the number histogram as exemplified with the adsorption of ethylene at 104 K and ammonia at 300 K.

Monte Carlo optimization scheme to determine the physical properties of porous and nonporous solids.

A new method, based on a Monte Carlo scheme, is developed to determine physical properties of nonporous and porous solids. In the case of nonporous solids, we calculate the surface area. This surface area is found as the sum of areas of patches of different surface energy on the solid, which is assumed to take a patchwise topology (i.e., adsorption sites of the same energy are grouped together in one patch). As a result of this assumption, we derive not only the surface area, but also the accessible volume and the surface energy distribution. In the case of porous solids, the optimization method is used to derive the surface area and the pore size distribution simultaneously. The derivation of these physical properties is based on adsorption data from a volumetric apparatus. We test this novel idea with the inversion problem of deriving surface areas of patches of different energies for a number of nonporous solids. The method is also tested with the derivation of the pore size distribution of some porous solid models. The results are very encouraging and demonstrate the great potential of this method as an alternative to the usual deterministic optimization algorithms which are known to be sensitive to the choice of the initial guess of the parameters. Since the geometrical parameters are physical quantities (i.e., only positive values are accepted), we also propose a scheme to enforce the positivity constraint of the solution.

Characterization of Virtual Nano-Structures through the Use of Monte Carlo Integration

The determination of the properties of porous solids remains an integral element to the understanding of adsorption, transport and reaction processes in new and novel materials. The advent of molecular simulation has led to an improved understanding and prediction of adsorption processes using molecular models. These molecular models have removed the constraints of traditional adsorption theories, which require rigid assumptions about the structure of a material. However, even if we possess a full molecular model of a solid, it is still desirable to define the properties of this solid in a standard manner with quantities such as the accessible volume, surface area and pore size distribution. This talk will present Monte Carlo integration methods for calculating these quantities in a physically meaningful and unambiguous way. The proposed methods for calculating the surface area and pore size distribution were tested on an array of idealised solid configurations including cylindrical and cubic pores. The method presented is adequate for all configurations tested giving confidence to its applicability to disordered solids. The method is further tested by using several different noble gas probe molecules. Finally, the results of this technique are compared against those obtained by applying the BET equation for a range of novel materials.