Local Isotherm Research Articles

Toward Solving the Unstable Linear Fredholm Equation of the First Kind: A New Procedure Called the Adsorption Stochastic Algorithm (ASA) and Its Properties

The paper consists of three main parts. In the first, a new “adsorption stochastic algorithm” (called ASA) for solving the unstable linear Fredholm integral equation of the first kind is proposed. In this program, some procedures of estimating the relative minimum in one dimension are tested. The newly developed algorithm is applied in the second part for reconstructing some pore size distribution functions (monomodal and multimodal). Moreover, the influence of a random noise on the stability of the solution of the inverse problem is studied. In the third part, the experimental verification of the above-mentioned method is presented. The results calculated by ASA are compared with those obtained by applying advanced regularization CONTIN and INTEG algorithms. It is shown that the developed ASA method always provides stable and very similar results to Tikhonov's regularization method. Moreover, the ASA computations obtained for the Nguyen and Do local isotherms as the kernel are very similar to the results calculated by the most sophisticated regularization density functional theory software. Summing up, the method can be very useful for evaluating the pore size distribution from experimental data.

Morphology and surface heterogeneities in synthetic goethites

In the framework on a study of the acido-basic and sorption properties of iron oxides, a thorough characterization of two types of goethite powders was performed in several laboratories joined in a common project. Chemical analysis by ICPAES; high-resolution SEM, TEM, and AFM observations; XRD with line width analysis; and argon and nitrogen sorption isotherms were used for that purpose. The main crystallographic faces of goethite particles could be identified as {001}, {101}, and {121}, and their abundance correlated with the distribution of low-pressure argon adsorption local isotherms. These results will be very useful for further studies on the relationship between surface reactivity in aqueous solution and orientation of solid surfaces.

An Inverse Gas Chromatographic Instrumentation for the Study of Carbon Monoxide's Adsorption on Rh/SiO2 Catalyst, Under Hydrogen‐Rich Conditions

A new and simple methodology for the study of the adsorption of carbon monoxide over a silica supported rhodium catalyst, referring to a wide range of hydrogen‐rich atmosphere (25%–75% H2), in experimental conditions compatible with the operation of fuel‐cells is presented. Experimentally, it is based on an inverse gas chromatographic technique known as reversed flow gas chromatography (RFGC), which is combined with the appropriate mathematical analysis which permits the measurement of adsorption, desorption, and surface reaction rate constants, as well as local adsorption energies, local isotherms, and surface diffusion coefficients. The values of the parameters of the present work vary with the hydrogen amount into the H2‐rich gas atmosphere and they are comparable to those obtained by other experimental, as well as theoretical techniques.

Phenol adsorption on porous and non-porous carbons

Phenol physisorption on a series of porous and non-porous amorphous carbons was studied at 298 K. Grand Canonical Monte Carlo computer simulations were performed to simulate phenol adsorption from the gas phase. Phenol is adsorbed in a solid-like state within the pores and there is no well-developed multilayer regime. The ‘ t’ plot method was adapted to phenol adsorption and the results obtained are in agreement with the model solids employed. The simulated adsorption isotherms were compared with experimental results obtained for adsorption from aqueous solutions of phenol. BET surface areas were calculated. Other characteristics of the adsorption system analyzed were: adsorption energy distribution functions, density profiles, distribution of molecules according to gas–solid energy, and local isotherms.

Adsorption Energy Distribution Model for VOCs onto Activated Carbons

An adsorption isotherm model was proposed for two types of volatile organic compounds (VOCs) on a heterogeneous carbon surface. The Langmuir isotherm was used as a local isotherm for describing heterogeneous surfaces to obtain the adsorption energy distribution. The adsorption temperature studied ranged from 30 to 50°C, and the pressure of VOCs varied from 0 to 0.35 atm. The present model differed from pervious studies in assuming that the pre-exponential factor was not to be a constant. The pre-exponential factors were determined directly from the experimental data, and the result empirically showed that the pre-exponential factor was correlated with the adsorption energy by a simple exponential function. We found that both adsorption energy distributions of two VOCs were essentially step functions over the restricted pressure range, indicating adsorbates on the carbon surface with its own uniform distribution. By incorporation of the energy distribution and the relationship between the pre-exponential factor and the adsorption energy, the adsorption isotherms for the two VOCs on the carbons can be well predicted.

Nitrogen Physisorption on Defective C60

Nitrogen physical adsorption, from 50 up to 120 K, on a defective C 6 0 fullerene solid is studied with Grand Canonical Monte Carlo computer simulations. New adsorption sites are produced due to the creation of a vacancy in the fcc crystal structure of C 6 0 . The distributions of molecules with respect to their gas-solid and gas-gas interaction energies are analyzed. The contribution of the internal cavity of the solid to the total surface area is estimated from the local isotherms. The adsorbate cross-sectional area is calculated at all temperatures. The isosteric heat of adsorption is calculated from the Henry's Law constant. The adsorption energy distribution functions are discussed comparing the simulation results with experimental isotherms.

Adsorption in Slit-Like Pores of Activated Carbons: Improvement of the Horvath and Kawazoe Method

Anew thermodynamic approach has been developed in this paper to analyze adsorption in slitlike pores. The equilibrium is described by two thermodynamic conditions: the Helmholtz free energy must be minimal, and the grand potential functional at that minimum must be negative. This approach has led to local isotherms that describe adsorption in the form of a single layer or two layers near the pore walls. In narrow pores local isotherms have one step that could be either very sharp but continuous or discontinuous benchlike for a definite range of pore width. The latter reflects a so-called 0 --> 1 monolayer transition. In relatively wide pores, local isotherms have two steps, of which the first step corresponds to the appearance of two layers near the pore walls, while the second step corresponds to the filling of the space between these layers. All features of local isotherms are in agreement with the results obtained from the density functional theory and Monte Carlo simulations. The approach is used for determining pore size distributions of carbon materials. We illustrate this with the benzene adsorption data on activated carbon at 20, 50, and 80 degreesC, argon adsorption on activated carbon Norit ROX at 87.3 K, and nitrogen adsorption on activated carbon Norit R1 at 77.3 K.

Pore size distribution model derived from a modified DR equation and simulated pore filling for nitrogen adsorption at 77 K

The Dubinin–Radushkevich (DR) equation has been used extensively to characterize microporous materials. However, it assumes a continuous pore filling mechanism and does not reduce to Henry’s law at low adsorptive pressures. In this study, we modify the DR equation by introducing adsorption density and the correlation between pore filling pressure and critical pore size for N 2 adsorption at 77 K. This modified DR equation is used as the local isotherm in the generalized adsorption isotherm (GAI) to interpret N 2 adsorption isotherm at 77 K. To solve the GAI, a modified normal distribution is developed and used as a distribution function of pore size on the domain [0, ∞). Pore size distribution (PSD) analyses of several adsorbent samples are carried out using this model. The results are found comparable with other popular PSD methods (e.g. MP, JC, HK and DFT).

Adsorption of Dissociating Aromatic Compounds by Activated Carbon: Effects of Ionization on the Adsorption Capacity

The adsorption of three dissociating aromatic compounds on an untreated commercially available activated carbon was investigated systematically. All adsorption experiments were carried out in pH-controlled aqueous solutions. The experimental isotherms were analyzed using the homogeneous Langmuir model. The effects of surface heterogeneity were investigated via the theory of localized physical adsorption, using the Langmuir equation as the local isotherm equation and assuming a Gaussian adsorption energy distribution. A new method of utilizing the extracted parameters from the homogeneous and heterogeneous models is presented. This new approach provided a better understanding of the change in monolayer capacity and other adsorption characteristics of a carbon/solute system with pH.

Adsorption of Spherical Molecules in Probing the Surface Topography. 1. Patchwise Heterogeneity Model

The Fowler−Guggenheim local isotherm corresponding to the lattice model of localized adsorption on a patchwise surface was used to evaluate the statistical distribution of adsorption sites with respect to their adsorption energies from an experimental adsorption isotherm. An accurate low-pressure argon adsorption isotherm at 77 K on muscovite was used as initial information. The calculations were carried out in two ways that imply different surface constructions. The first one is based on the classical hypothesis of Langmuir, who assumed a finite manifold of adsorption sites and wrote an overall isotherm as a series summing the additive contributions of the different sites. The second is derived on the assumption of an infinite manifold of sites, when the sum turns into an integral. Both representations are equivalent only with the exact adsorption model. However, as shown, in a number of cases they can give similar results even with the approximate model. The influence of the ω parameter expressing the reduced energy of lateral interactions was tested for different ω values ranging between 0 and 4 in units of RT. Muscovite possesses crystallinity. Therefore, one may expect that its surface consists of recurring adsorption sites. Peaks on the energy distribution could mean at least a short range ordering (SRO) on this surface. A derivative of a local isotherm for sites of each kind is known to be a bell-shaped curve. The greater the energy of lateral attraction is, the narrower is the bell; that is, the Langmuir local isotherm (the case of ω = 0) corresponds to the widest bell. Hence, if the hypothesis concerning SRO holds true, an energy distribution obtained with ω = 0 must be discrete, inasmuch as a broadening of the calculated partial distributions can lead only to a deteriorating of the approximation accuracy of local derivatives and, correspondingly, of the overall derivative and the overall isotherm. Indeed, the distributions calculated with a small ω represent sums of δ-like peaks. With a small ω, both algorithms show the same results; therefore, any numerical artifacts are excluded. However, SRO on the surface is not proven yet, because with a large ω the algorithms lead to continuous distributions rather than to discrete ones. The relative approximation error of the overall isotherm with ω = 0 is not too large (≅2−3%). However, the large approximation error of the overall derivative (≅15%) unambiguously indicates the unacceptability of the assumption ω = 0 (local Langmuir isotherm). Thus, to make final conclusions about the structure of the surface and validate such an analysis, it is necessary to improve the patchwise model, to test the method by using an independent molecular simulation, and to determine realistic values of the energies of lateral interactions being different and not equal for adsorption sites of each kind.

Effective states approximation (ESA) for adsorption on homogeneous and heterogeneous surfaces

A simple approximation for the local and overall adsorption isotherms of monoatomic species on heterogeneous surfaces with different topographies is presented. The model relies on the lattice-gas approach where the site adsorption energies generated by the gas–solid potential, split into a set of independent energy substates because of the ads–ads interactions. The distinctive feature of the approximation is that it preserves in a simple manner the energy states for single particles by making more crude assumptions in counting the total number of configurations compared to standard approaches (mean-field, quasi-chemical) in which the energetics of single particles are oversimplified while the counting of lattice configurations is carried out as accurately as possible. At a comparable degree of approximation this approach, which is denoted as ESA leads to appreciably better results than mean-field. ESA can also be applied to adsorption on heterogeneous surfaces with distinct energy topographies. The effect of lateral interactions and surface topography on the local isotherm at fixed overall coverage and temperature (the so-called energetic structure of the adsorbate) is analyzed and the results are compared with Monte Carlo simulations, the Bragg–Williams and the quasi-chemical approximations. The assumption of an intermediate energy topography characterized by nearest-neighbor site energy correlation, rather than random or patch-wise distributions, is shown to have a significant effect on the distribution of molecules over adsorption sites.

AN INVERSE GAS CHROMATOGRAPHIC TOOL FOR THE MEASUREMENT OF LOCAL ISOTHERMS ON HETEROGENEOUS SURFACES

ABSTRACT A new application of an inverse gas chromatographic technique, known as Reversed Flow Gas Chromatography (RFGC), which permits the experimental measurement of local isotherms, θ( p, T, ϵ) is presented. The method does not use analytical or numerical solutions of the classical integral equation comprising the energy distribution function, f(ϵ), as unknown, but it is based on a time function of gas chromatographic (GC) peaks obtained by short flow reversals of the carrier gas. It can be used for the measurement of local adsorption isotherms θ, as well as for the estimation of the related parameters such as local energies of adsorption ϵ, local adsorbed concentrations c s *, and local maximum monolayer capacities c max *. As a model system, we selected oxygen's adsorption onto three-way silica supported 25% Pt+75% Rh alloy catalyst.

The Use of Liquid Phase Adsorption Isotherms for Characterization of Activated Carbons

The characterization of three commercial activated carbons was carried out using the adsorption of various compounds in the aqueous phase. For this purpose the generalized adsorption isotherm was employed, and a modification of the Dubinin–Radushkevich pore filling model, incorporating repulsive contributions to the pore potential as well as bulk liquid phase nonideality, was used as the local isotherm. Eight different flavor compounds were used as adsorbates, and the isotherms were jointly fitted to yield a common pore size distribution for each carbon. The bulk liquid phase nonideality was incorporated through the UNIFAC activity coefficient model, and the repulsive contribution to the pore potential was incorporated through the Steele 10-4-3 potential model. The mean micropore network coordination number for each carbon was also determined from the fitted saturation capacity based on percolation theory. Good agreement between the model and the experimental data was observed. In addition, excellent agreement between the bimodal gamma pore size distribution and density functional theory-cum-regularization-based pore size distribution obtained by argon adsorption was also observed, supporting the validity of the model. The results show that liquid phase adsorption, using adsorptive molecules of different sizes, can be an effective means of characterizing the pore size distribution as well as connectivity. Alternately, if the carbon pore size distribution is independently known, the method can be used to “measure” critical molecular sizes.

Application of heterogeneous vacancy solution theory to characterization of microporous solids

The vacancy solution theory of adsorption is re-formulated here through the mass-action law, and placed in a convenient framework permitting the development of thermodynamically consistent isotherms. It is shown that both the multisite Langmuir model and the classical vacancy solution theory expression are special cases of the more general approach when the Flory–Huggins activity coefficient model is used, with the former being the thermodynamically consistent result. The improved vacancy solution theory approach is further extended here to heterogeneous adsorbents by considering the pore-width dependent potential along with a pore size distribution. However, application of the model to numerous hydrocarbons as well as other adsorptives on microporous activated carbons shows that the multisite model has difficulty in the presence of a pore size distribution, because pores of different sizes can have different numbers of adsorbed layers and therefore different site occupancies. On the other hand, use of the classical vacancy solution theory expression for the local isotherm leads to good simultaneous fit of the data, while yielding a site diameter of about 0.257 nm, consistent with that expected for the potential well in aromatic rings on carbon pore surfaces. It is argued that the classical approach is successful because the Flory–Huggins term effectively represents adsorbate interactions in disguise. When used together with the ideal adsorbed solution theory the heterogeneous vacancy solution theory successfully predicts binary adsorption equilibria, and is found to perform better than the multisite Langmuir as well as the heterogeneous Langmuir model.

Adsorption integral equation via complex approximation with constraints: kernel of general form

For the monolayer adsorption on a homogeneous surface, including arbitrary range lateral interactions, the isotherm can be written as a power series of the Langmuir isotherm. If this isotherm is used as the kernel in the adsorption integral equation, this integral equation can be solved in an analytical form. Because the global isotherm is usually known as a set of experimental values, the use of a numerical method is inevitable. A new numerical method for solving the adsorption integral equation with a kernel of general form is developed. It is based on recent results concerning the structure of the local isotherm and on the ideas of complex approximation with constraints, and allows reduction of the problem under consideration to a linear-quadratic programming problem. Results of numerical experiments are presented. The method can be useful for the evaluation of the adsorption energy distribution from experimental data. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1058–1066, 2001

Interrelations between Adsorption Energies and Local Isotherms, Local Monolayer Capacities, and Energy Distribution Functions, as Determined for Heterogeneous Surfaces by Inverse Gas Chromatography

Physicochemical parameters for adsorption of gases at the submonolayer regions of heterogeneous solid surfaces are measured experimentally as a function of time, and then interrelated as local isotherms θ against adsorption energy ε, fractional changes of adsorption sites f(ε)/c*max against ε, θ against f(ε)/c*max, and distribution functions θ f(ε)/c*max over adsorption energy values ε, without using at all the well-known integral equation Θ(p, T)=∫∞0θ(p, T, ε)f(ε)dε and assumptions concerning the pair f(ε) and θ(p, T, ε). The method uses only chromatographic experimental data obtained by the inverse gas chromatography technique known as reversed-flow gas chromatography. It has been applied to the adsorption of cis-2-butene and trans-2-butene onto particles of Penteli marble at temperatures of 302, 314, 323, and 333 K. The results obtained are comparable with those calculated on the basis of the well-known integral equation.

Adsorption-Energy Distribution of Heterogeneous Surfaces Predicted by Projections onto Convex Sets

The method of projections onto convex sets (POCS) is used to calculate the adsorption-energy distribution function from the adsorption integral (using a modified Langmuir local isotherm) for energetically heterogeneous surfaces. The POCS method, originally developed in the 1960s, has been successfully applied for many years to estimation problems, mainly in the fields of image processing, signal recovery, and optics. It allows one to incorporate into an iteration scheme available information about the experimental data and the measurement error as well as a priori constraints (such as nonnegativity) based on physical reasoning. It is important to note that the POCS method does not lead to a unique “optimum” solution. Rather, a feasible solution that is consistent with all imposed contraints is found within a “solution space”. The “size” of this solution space depends on how large the measurement errors are; it also depends on the accuracy of the error statistics and the number and significance of a priori constraints used. In several examples, the POCS method is used to recover energy distributions from simulated adsorption data containing normally distributed errors. The excellent recoveries obtained demonstrate the value of the POCS method as a robust and reliable tool for adsorption-integral inversions.

Adsorption integral equation via complex approximation with constraints: kernel of general form

AbstractFor the monolayer adsorption on a homogeneous surface, including arbitrary range lateral interactions, the isotherm can be written as a power series of the Langmuir isotherm. If this isotherm is used as the kernel in the adsorption integral equation, this integral equation can be solved in an analytical form. Because the global isotherm is usually known as a set of experimental values, the use of a numerical method is inevitable. A new numerical method for solving the adsorption integral equation with a kernel of general form is developed. It is based on recent results concerning the structure of the local isotherm and on the ideas of complex approximation with constraints, and allows reduction of the problem under consideration to a linear‐quadratic programming problem. Results of numerical experiments are presented. The method can be useful for the evaluation of the adsorption energy distribution from experimental data. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1058–1066, 2001

Protostellar Disk Instabilities and the Formation of Substellar Companions

Recent numerical simulations of self-gravitating protostellar disks have suggested that gravitational instabilities can lead to the production of substellar companions. In these simulations, the disk is typically assumed to be locally isothermal; i.e., the initial, axisymmetric temperature in the disk remains everywhere unchanged. Such an idealized condition implies extremely efficient cooling for outwardly moving parcels of gas. While we have seen disk disruption in our own locally isothermal simulations of a small, massive protostellar disk, no long-lived companions formed as a result of the instabilities. Instead, thermal and tidal effects and the complex interactions of the disk material prevented permanent condensations from forming, despite the vigorous growth of spiral instabilities. In order to compare our results more directly with those of other authors, we here present three-dimensional evolutions of an older, larger, but less massive protostellar disk. We show that potentially long-lived condensations form only for the extreme of local isothermality, and then only when severe restrictions are placed on the natural tendency of the protostellar disk to expand in response to gravitational instabilities. A more realistic adiabatic evolution leads to vertical and radial expansion of the disk but no clump formation. We conclude that isothermal disk calculations cannot demonstrate companion formation by disk fragmentation but only suggest it at best. It will be necessary in future numerical work on this problem to treat the disk thermodynamics more realistically.

Statistical fractal adsorption isotherms, linear energy relations, and power-law trapping-time distributions in porous media

Drazer and Zanette [Phys. Rev. E 60, 5858 (1999)] have reported on interesting experiments which show that trapping-time distributions in porous media obey a scaling law of the negative power-law type. Unfortunately, their theoretical interpretation of the experimental data has physical and mathematical inconsistencies and errors. Drazer and Zanette assume the existence of a distribution of local adsorption isotherms for which the random parameter is not a thermodynamic function, but a kinetic parameter, the trapping time. Moreover, they mistakenly identify the reciprocal value of a rate coefficient with the instantaneous (fluctuating) value of the trapping time. Their approach leads to mathematically inconsistent probability densities for the trapping times, which they find to be non-normalizable. We suggest a different theory, which is physically and mathematically consistent. We start with the classical patch approximation, which assumes the existence of a distribution of adsorption heats, and introduce two linear energy relationships between the activation energies of the adsorption and desorption processes and the adsorption heat. If the distribution of the adsorption heat obeys the exponential law of Zeldovich and Roghinsky, then both the adsorption isotherm and the probability density of trapping times can be evaluated analytically in terms of the incomplete beta and gamma functions, respectively. Our probability density of the trapping times is mathematically consistent; that is, it is non-negative and normalized to unity. For large times it has a long tail which obeys a scaling law of the negative power-law type, which is consistent with the experimental data of Drazer and Zanette. By using their data we can evaluate the numerical values of the proportionality coefficients in the linear energy relations. The theory suggests that experimental study of the temperature dependence of the fractal exponents helps to elucidate the mechanism of the adsorption-desorption process.