A Monte Carlo study of proton adsorption at the heterogeneous oxide/electrolyte interface

Adsorption of protons on a heterogeneous solid surface is modeled using the Monte Carlo (MC) simulation method. The surface of an oxide is assumed to consist of adsorption sites with pK assigned according to a quasi-Gaussian distribution. The influence of the electrostatic interactions combined with the energetic heterogeneity of the surface is examined, and the MC results are compared with the predictions of the mean field theory (MFT). It is demonstrated that the heterogeneity affects strongly the shape of the isotherms while it does not change the location of the common intersection point of the isotherms. On the other hand, introduction of repulsive interactions into the system is found to shift the CIP toward lower values of pH. It is also shown that the MFT, in general, describes correctly the behavior of the system. On the contrary the condensation approximation, used to derive relatively simple expressions for the adsorption isotherms, introduces serious errors unless the surface is strongly heterogeneous. Some practical remarks how to eliminate the errors associated both with the MC simulations and with the theory are also presented.     

Application of density functional theory to analysis of energetic heterogeneity and pore size distribution of activated carbons

A new approach based on the nonlocal density functional theory to determine pore size distribution (PSD) of activated carbons and energetic heterogeneity of the pore wall is proposed. The energetic heterogeneity is modeled with an energy distribution function (EDF), describing the distribution of solid-fluid potential well depth (this distribution is a Dirac delta function for an energetic homogeneous surface). The approach allows simultaneous determining of the PSD (assuming slit shape) and EDF from nitrogen or argon isotherms at their respective boiling points by using a set of local isotherms calculated for a range of pore widths and solid-fluid potential well depths. It is found that the structure of the pore wall surface significantly differs from that ofgraphitized carbon black. This could be attributed to defects in the crystalline structure of the surface, active oxide centers, finite size of the pore walls (in either wall thickness or pore length), and so forth. Those factors depend on the precursor and the process of carbonization and activation and hence provide a fingerprint for each adsorbent. The approach allows very accurate correlation of the experimental adsorption isotherm and leads to PSDs that are simpler and more realistic than those obtained with the original nonlocal density functional theory.     

Heterogeneity of multiwalled carbon nanotubes based on adsorption of simple aromatic compounds from aqueous solutions

The surface heterogeneity of multiwalled carbon nanotubes (MWCNTs) is studied on the basis of adsorption isotherms from dilute aqueous phenol and dopamine solutions at various pH values. The generalized Langmuir–Freundlich isotherm equation was applied to investigate the cooperative effect of the surface heterogeneity and the lateral interactions between the adsorbates. The theoretical isosteric heats of adsorption were obtained assuming that the heat of adsorption profile reveals both the energetic heterogeneity of the adsorption system and the strength of the interactions between the neighboring molecules. The adsorption energy distribution functions were calculated by using algorithm based on a regularization method. The great advantage of this method is that the regularization makes no assumption about the shape of the obtained energy distribution functions. Analysis of the isosteric heats of adsorption for MWCNTs showed that the influence of the surface heterogeneity is much stronger than the role of the lateral interactions. The most typical adsorption heat is 20–22 kJ/mol for both phenol and dopamine. After purification of nanotubes, heat value for phenol dropped to 16–17 kJ/mol. The range of the energy distribution is only slightly influenced by the surface chemistry of the nanotubes in the aqueous conditions.     

Some remarks on the calculation of the pore size distribution function of activated carbons

Different authors investigated the effects of geometric and energetic heterogeneities on adsorption and on carbon characterization methods. In most theoretical studies carbon structure is modeled as parallel infinite graphite walls that form ideal slit-shaped pores of the fixed widths. In the literature there is the lack of systematic studies showing the influence of pore structural and Lennard-Jones (LJ) potential parameters on the pore-size distribution functions. Moreover, the parameters characterizing the properties of the adsorbed phase and the heterogeneity of the adsorbent surface should be taken into account. The Nguyen and Do method with proposed by us ASA algorithm, were utilized for the assessment of the porosity from the series of almost few thousands numerically generated local adsorption isotherms. The values of the mentioned-above parameters are varied over the wide range (ca. +/-20%) of the reference ones. Different types of the theoretical and experimental adsorption isotherms (nitrogen at 77 K) were taken into account as the global ones. They were related to the mechanism of the primary, secondary or mixed micropore filling. The variations in some above-mentioned parameters have significant effects only for PSDs (and for average pore widths) corresponding to the primary micropore filling mechanism. On the other hand, for the process of the secondary micropore filling, the influence of these parameters (without the BET coefficient for adsorption on a "flat" surface, c(s,B)) is rather insignificant. Nevertheless the differences between local and global adsorption isotherms (in the whole range of relative pressures) the absence of micropores having pore half width equal to ca. 1 nm on PSDs was observed for studied adsorbate-adsorbent systems with exceptions of the strictly microporous adsorbents and/or the low values of c(s,B). Comparison of the experimental data with the generated theoretical isosteric enthalpy of adsorption indicates that the phenomenal uptake observed from experiment can be explained in terms of the reasonable solid-fluid interaction parameters. Therefore, we varied the heterogeneity of the adsorbent surface via the strength and the range of the solid-fluid potential and the parameter c(s,B) in order to reproduce the experimental data of enthalpy of adsorption. Note that similar procedure was applied by Wang and Johnson to reproduce some hydrogen adsorption data measured for carbon nanofibres. The analysis of the obtained results shows that the selection of the values of the parameters of the intermolecular interactions and the quantities characterizing the properties of the adsorbed phase and the heterogeneity of the adsorbent walls for molecular simulations should be made with care and the influence of possible errors should be considered.     

Determination of isotherms by gas-solid chromatography. Applications

This literature review of the fundamental developments in gas-solid adsorption isotherms includes articles published from 1933 until now. Analytical and numerical methods used for calculating the adsorption energy distribution function, as a quantitative measure of surface heterogeneity, are included. Special attention is paid to inverse gas chromatography (IGC) and more precisely to a new version of IGC known as reversed-flow gas chromatography (RF-IGC or RF-GC). RF-GC is presented as a quick, precise and effective method to investigate physicochemical properties of different kinds of adsorbents, through adsorption isotherms and related energetic parameter determinations. Advantages of the RF-GC method over traditional chromatographic methods are discussed.     

Characterization of surface energetic heterogeneity of pure and poly (acrylic acid)‐adsorbed carbon nanotubes by deconvoluting the nitrogen adsorption isotherm

The surface energetic heterogeneity of pure and poly (acrylic acid) (PAA)‐adsorbed carbon nanotubes (CNTs) were studied by a nitrogen probe adsorption technique in a wide range of pressures. The adsorption energy distributions (AEDs) were calculated from the low‐pressure data of isotherms by deconvoluting the low‐pressure experimental nitrogen adsorption isotherms. The surface of pure CNTs is heterogeneous as its AED presents four peaks at 42, 52, 57 and 78 K. It is observed that the AED of CNTs can be evidently modified by PAA adsorption. While the PAA adsorption amount increases, the high‐energy peaks at 52, 57 and 78 K gradually weaken and diminish at last, whereas the low ones such as at 42 K strengthen and new peaks arise at 27 and 32 K. It is proposed that PAA molecules prefer interacting with and screening the higher energetic sites to the lower ones. It will facilitate the understanding of the polymer adsorption on energetic heterogeneity surfaces. Copyright © 2006 John Wiley & Sons, Ltd.     

Investigation of activated carbon surface heterogeneity by argon and nitrogen low-pressure quasi-equilibrium volumetry

Surface heterogeneity can be assessed by adsorption of different gaseous probes on solid materials. In the present study, four types of activated carbons were analyzed by classical N2 Brunauer-Emmett-Teller (BET) measurements and by low-pressure quasi-equilibrium volumetry (LPQEV) (Villieras, F.; Michot, L. J.; Bardot, F.; Cases, J. M.; Francois, M.; Rudzinski, W. Langmuir 1997, 13, 1104). Three methods of data evaluation were applied: (a) the Frenkel-Halsey-Hill method for estimation of fractal dimensions from BET data, (b) the Horwath-Kawazoe method to calculate the pore size distribution from LPQEV Ar and N2 adsorption isotherms, and (c) the derivative isotherm summation (DIS) method to describe the solid's surface heterogeneity by a concept of local derivative isotherms. Similar Ar and N2 adsorption energy distributions were obtained on all carbons, which indicates the presence of mainly nonpolar surfaces. When adsorption was described by the van der Waals equation, the ratio between the interaction energy of different energetic sites with argon and nitrogen was 0.88. This value corresponded very well with a slope obtained when Ar and N2 positions of local isotherms by the DIS method were compared. This relationship has an important impact because it enables one to constrain the modeling of local isotherms. This study, besides the surface information, showed large possibilities of the DIS method for the surface analysis not only in terms of solid heterogeneity characterization but also in terms of polarity assessment.     

Adsorption mechanism of carbon dioxide in faujasites: grand canonical monte carlo simulations and microcalorimetry measurements

Molecular simulations have been coupled with adsorption microcalorimetry measurements in order to understand more deeply the interactions between carbon dioxide and various types of faujasite surfaces. The modeling studies, based on newly derived interatomic potentials for describing the interactions within the whole system, provide isotherms and evolutions of the differential enthalpy of adsorption as a function of coverage for DAY, NaY, and NaLSX which are in very good accordance with those obtained experimentally. The microscopic mechanism of CO2 adsorption was carefully analyzed, with different behaviors proposed, depending on the energetic characteristics of each faujasite surface, which are consistent with the trends observed for the differential enthalpies of adsorption.     

Isotherm analysis of phenol adsorption on polymeric adsorbents from nonaqueous solution

Macroporous poly(methyl methacrylate-co-divinylbenzene) (PMMA), interpenetrating polymer adsorbent based on poly(styrene-co-divinylbenzene) (PS) and poly(methyl methacrylate-co-divinylbenzene) (PMMA/PS), and macroporous cross-linked poly(N-p-vinylbenzyl acetylamide) (PVBA) were prepared for the adsorption of phenol from cyclohexane. The sorption isotherms of phenol on the three polymeric adsorbents were measured and fitted to Langmuir and Freundlich isotherms. It is shown that the Langmuir isotherm, which is based on a homogeneous surface model, is unsuitable to describe the sorption of phenol on the adsorbents from nonaqueous solution and the Freundlich equation fits the tested three adsorption systems well. The isosteric enthalpy was quantitatively correlated with the fractional loading for the sorption of phenol onto the three polymeric adsorbents. The surface energetic heterogeneity patterns of the adsorbents were described with functions of isosteric enthalpy. The results showed that the tested three polymeric adsorbents exhibited different surface energetic heterogeneity patterns. The initial isosteric enthalpy of phenol sorption on polymeric adsorbent has to do with the surface chemical composition and is free from the pore structure of the polymeric adsorbent matrix. Forming hydrogen bonds between phenol molecules and adsorbent is the main driving force of phenol sorption onto PVBA and PMMA adsorbent from nonaqueous solution. When phenol is adsorbed on PMMA/PS, pi-pi interaction resulting from the stacking of the benzene rings of the adsorbed phenol molecules and the pendant benzene ring of adsorbent is involved.     

Monte Carlo simulations of hydrogen adsorption in alkali-doped single-walled carbon nanotubes

Monte Carlo simulations and Widom's test particle insertion method have been used to calculate the solubility coefficients (S) and the adsorption equilibrium constants (K) in single-walled (10,10) armchair carbon nanotubes including single nanotubes, and nanotube bundles with various configurations with and without alkali dopants. The hydrogen adsorption isotherms at room temperature were predicted by following the Langmuir adsorption model using the calculated constants S and K. The simulation results were in good agreement with experimental data as well as the grand canonical Monte Carlo simulation results reported in the literature. The simulations of nanotube bundle configurations suggest that the gravimetric hydrogen adsorption increases with internanotube gap size. It may be attributed to favorable hydrogen-nanotube interactions outside the nanotubes. The effect of alkali doping on hydrogen adsorption was studied by incorporating K+ or Li+ ions into nanotube arrays using a Monte Carlo simulation. The results on hydrogen adsorption isotherms indicate hydrogen adsorption of 3.95 wt% for K-doping, and 4.21 wt% for Li-doping, in reasonable agreement with the experimental results obtained at 100 atm and room temperature.     

Microscopic mechanism of adsorption in cylindrical nanopores with heterogenous wall structure

We study the microscopic mechanism of adsorption in nanometric cylindrical pores with strongly heterogeneous walls using grand canonical Monte Carlo simulations. The pore surface structure is modeled by a new lattice-site approach. Each site is characterized by two amplitudes--structural and energetic--that locally modify the structural and energetic properties of the surface. The amplitudes are randomly distributed over the pore wall. We have shown that different structural and energetic distribution functions lead to different mechanism of adsorption. The energetic site distribution plays the most crucial role in the submonolayer region. The structural site distribution modifies the multilayer adsorption. A method to analyze the stability of the adsorbed system using static susceptibility is proposed. Potential applications in multiscale modeling are discussed.     

Study of proton adsorption at heterogeneous oxide/electrolyte interface. Prediction of the surface potential using Monte Carlo simulations and 1-pK approach

Adsorption of protons on a heterogeneous solid surface is modeled using the Monte Carlo (MC) simulation method. The surface of an oxide is assumed to consist of adsorption sites with pK assigned according to a quasi-Gaussian distribution. The influence of the electrostatic interactions combined with the energetic heterogeneity of the surface is examined and the MC results are compared with the predictions of the analytical 1-pK approach. The surface potential behavior is examined using both "experimental" MC results and "theoretical" results obtained from the application of 1-pK model. The results are compared qualitatively with experimental determination of the surface potential of metal oxide surfaces. They confirm that the relation between the surface potential and the pH of bulk solution should not be described by the Nernst equation but by the equation with the parameter linearly reducing Nerstian potential. The values of this parameter are examined with respect to degree of surface energetic heterogeneity and site density of the surface.     

Characterization of micro-mesoporous materials from nitrogen and toluene adsorption: experiment and modeling

Universal mechanisms of adsorption and capillary condensation of toluene and nitrogen on ordered MCM-41 and PHTS materials are studied by means of high-resolution experiments and Monte Carlo molecular simulations. A molecular simulation model of toluene adsorption in silica nanopores, which accounts for surface heterogeneity, and a hybrid molecular-macsroscopic method for pore size distribution (PSD) calculations have been developed. For a range of reference materials, the PSD results obtained from toluene isotherms are consistent with the results of nitrogen adsorption using the nonlocal density functional theory method.     

Modeling of binary adsorption on heterogeneous surfaces characterized by a quasi-gaussian adsorption energy distribution

The integral equation (IE) approach coupled with a quasi-Gaussian adsorption energy distribution is used to model the adsorption of single gases and their binary mixture on a heterogeneous solid surface. The adsorbing surface is assumed to be characterized by two, generally different in width, quasi-Gaussian distribution functions, each of them related to a single component of the mixture. The influence of correlations between the distribution functions associated with different components on the corresponding adsorption isotherms and phase diagrams is discussed. In particular, it is demonstrated that a lack of microscopic correlations between the adsorption energies of the components may lead to the formation of an azeotropic mixture. The predictions of the theory are also compared with the results of the grand canonical Monte Carlo (GCMC) simulations carried out for the system studied.     

Adsorbed solution model for prediction of normal-phase chromatography process with varying composition of the mobile phase

The adsorbed solution model has been used to predict competitive adsorption equilibria of the solute and the active component of mobile phase in a normal-phase liquid chromatography system. The inputs to the calculations were the single adsorption isotherms accounting for energetic heterogeneity of the adsorbent surface and non-ideality of the mobile phase solution. The competitive adsorption model has been coupled with a model of the column dynamics and used for simulating of chromatography process at different mobile phase composition. The predictions have been verified by comparing the simulated and experimental chromatograms. The model allowed quantitative prediction of chromatography process on the basis of the pure-species adsorption isotherms.     

Thermodynamics of CO2 adsorption on functionalized SBA-15 silica. NLDFT analysis of surface energetic heterogeneity

Siliceous SBA-15 mesoporous molecular sieves were functionalized with different amounts of 3-aminopropyl-trimethoxysilane. To obtain a more detailed insight into the material properties of the prepared samples, their textural parameters were combined with results of thermal analysis. Adsorption isotherms of carbon dioxide on parent and functionalized SBA-15 were measured in the temperature range from 273 to 333 K. From the temperature dependence of CO(2) isotherms the isosteric adsorption heats of CO(2) were determined and discussed. Information about the surface energetic heterogeneity caused by tethered 3-aminopropyl groups were obtained from CO(2) adsorption energy distributions calculated using the theoretical CO(2) adsorption isotherms derived from the non-local density functional theory. The values of isosteric heats and the energy distributions of CO(2) adsorption detect highly energetic sites and enabled quantification of their concentrations.     

Modelling adsorbent heterogeneity for adsorption from binary liquid mixtures

It is demonstrated that the detailed structure of the surface energy or selectivity distribution function is not critical to obtaining adequate analytical expressions for surface excess isotherms for adsorption from binary liquid mixtures on heterogeneous adsorbents. The gamma and the uniform selectivity distribution functions, which are very different in form, were successfully used in conjunction with the monolayer-pore filling model for adsorption on a homogeneous site to describe adsorption of various binary liquid mixtures on silica gel. Both models described the salient features of the surface characteristics of the silica gel.     

Effects of Energetic Surface Heterogeneity on Ion Adsorption at the Electrolyte/Oxide Interfaces: Non-symmetrical Binding-to-Surface Energy Dispersions

The equations developed by us for the surface complexation models, taking into account energetic heterogeneity of surface oxygens, are applied here to study the effects of the shape of adsorption energy distribution on the ion adsorption at the oxide/electrolyte interface. The paper presents comparison of two models: one assuming that the energetical heterogeneity of oxide surface is described by a symmetrical and next a non-symmetrical adsorption energy dispersion for the formation of various surface complexes. The comparison of these two models was based on the obvious assumption that the variances of both the symmetrical and the non-symmetrical distributions are equal. The potentiometric titration data are not sensitive enough to choose the right adsorption model. So, in addition the individual isotherms of adsorption of cation and anion, of the inert electrolyte measured radiometrically, have been taken into consideration. The symmetrical adsorption energy distribution seems to represent the features of these adsorption systems better.     

Kinetics of metal ions adsorption at heterogeneous solid/solution interfaces: A theoretical treatment based on statistical rate theory

The statistical rate theory combined with a two-component competitive adsorption model is applied to describe the effect of pH on the kinetics of metal ions adsorption at energetically heterogeneous solid/solution interfaces. The surface heterogeneity has been represented by both Gaussian-like and rectangular functions of the adsorption energy distribution. A concept of effective heterogeneity parameters is found to represent very well the combined effects of surface energetic heterogeneity and of the electrostatic lateral interactions in the adsorbed phase, described by using the mean field approximation. The applicability of our approach is demonstrated by a quantitative analysis of two sets of experimental data reported in literature. Our theoretical expressions have been able to successfully correlate kinetic and equilibrium data in both these cases.     

Hydrogen adsorption on nickel (100) single-crystal face. A Monte Carlo study of the equilibrium and kinetics

We propose a model of the dissociative adsorption of hydrogen on nickel single-crystal face. In this model, we treat the Ni(100) surface as a strongly correlated energetically heterogeneous surface, because the density functional theory (DFT) studies indicate that hydrogen atoms may adsorb either on hollow sites (energetically more favorable, binding energy 48 kJ/mol H) or bridge sites with the binding energy less by 11 kJ/mol H. The essential assumption of the proposed model is that the dissociation of the hydrogen molecule is possible only over the topmost Ni atom, and the resulting H atoms may adsorb either on two free hollow sites (but the adjacent bridge sites must be free) or two bridge sites (the adjacent hollow sites must be free). If the above condition is not fulfilled, then the dissociation and adsorption are impossible. The second assumption is that the rate (probability) of the associative desorption is limited by the rate of diffusion of H atoms on the surface. This is because the two H atoms desorb, giving an H2 molecule, only when they meet on two adjacent hollow-bridge sites. Our model recovers very well the behavior of the experimental equilibrium adsorption isotherms as well as kinetic isotherms. As a result, we stated that hydrogen atoms are not completely free on the surface, but they cannot also be considered localized at room and elevated temperatures. Additionally, while analyzing the kinetic adsorption isotherms, we stated that the rate-limiting step during the dissociative adsorption of H2 is the disintegration of the activated complex and the subsequent adsorption of hydrogen atoms.