Local Isotherm Research Articles

LOCAL DENSITY DISTRIBUTION OF NONUNIFORM POLYMER MELTS AT SOLID-FLUID INTERFACES

We have developed the density functional theory (DFT) and density functional perturbation theory (DFPT) which are based both on the hybrid weighted-density approximation and on the higher-order weighted-density approximation, respectively, to study the structural and thermodynamic properties of polymer melts at interfaces. They were applied to predict the local density distributions, adsorption isotherms, surface excesses, and solvation forces of a freely jointed tangent hard-sphere chain in hard slit pores. The Wertheim's first-order perturbation theory extended by Yu and Wu [J. Chem. Phys.117, 2368 (2002)] was used to calculate the excess free energy and second-order direct correlation function due to the chain connectivity. The weight functions were calculated from the second-order direct correlation functions. The calculated results show that the DFT is in excellent agreement with the computer simulations and slightly better than the DFPT.

Monolayer moisture, free energy change and fractionation of bound water of red chillies

The equilibrium moisture content and water activity data of red chillies within the temperature range 25–45 °C were used to determine the monolayer moisture, free energy change and fractionation of bound water. The BET equation, Caurie equation and local isotherms fixed the safe storage moisture limits of red chillies as 3.84–6.68%, 5.2–6.01% and 6.03–7.69% d.b. in adsorption conditions and 5.85–9.17%, 5.66–6.77% and 6.02–9.76% d.b. in desorption conditions within a temperature range of 45–25 °C, respectively. The free energy was higher in the desorption process than in the adsorption process of chillies at all the temperatures and moisture contents. The maximum value of free energy was 4437.35 kJ/kg mol at 5% d.b. moisture content and the minimum value was 238.37 kJ/kg mol at 50% d.b. moisture content during desorption of chillies in the temperature range of 25–45 °C. It was observed that the primary, secondary and tertiary bound waters end at moisture contents of 1.5%, 14% and 53.6% d.b., respectively. The bound water in red chillies was limited up to a moisture content of 53.6% d.b. and beyond that was free water, which can be removed easily.

Adsorption in Slit Pores and Pore-Size Distribution: A Molecular Layer Structure

Anew approach is developed to analyze the thermodynamic properties of a sub-critical fluid adsorbed in a slit pore of activated carbon. The approach is based on a representation that an adsorbed fluid forms an ordered structure close to a smoothed solid surface. This ordered structure is modelled as a collection of parallel molecular layers. Such a structure allows us to express the Helmholtz free energy of a molecular layer as the sum of the intrinsic Helmholtz free energy specific to that layer and the potential energy of interaction of that layer with all other layers and the solid surface. The intrinsic Helmholtz free energy of a molecular layer is a function (at given temperature) of its two-dimensional density and it can be readily obtained from bulk-phase properties, while the interlayer potential energy interaction is determined by using the 10–4 Lennard-Jones potential. The positions of all layers close to the graphite surface or in a slit pore are considered to correspond to the minimum of the potential energy of the system. This model has led to accurate predictions of nitrogen and argon adsorption on carbon black at their normal boiling points. In the case of adsorption in slit pores, local isotherms are determined from the minimization of the grand potential. The model provides a reasonable description of the 0–1 monolayer transition, phase transition and packing effect. The adsorption of nitrogen at 77.35 K and argon at 87.29 K on activated carbons is analyzed to illustrate the potential of this theory, and the derived pore-size distribution is compared favourably with that obtained by the Density Functional Theory (DFT). The model is less time-consuming than methods such as the DFT and Monte-Carlo simulation, and most importantly it can be readily extended to the adsorption of mixtures and capillary condensation phenomena.

Pore size distribution analysis of activated carbons: Application of density functional theory using nongraphitized carbon black as a reference system

The application of nonlocal density functional theory (NLDFT) to determine pore size distribution (PSD) of activated carbons using a nongraphitized carbon black, instead of graphitized thermal carbon black, as a reference system is explored. We show that in this case nitrogen and argon adsorption isotherms in activated carbons are precisely correlated by the theory, and such an excellent correlation would never be possible if the pore wall surface was assumed to be identical to that of graphitized carbon black. It suggests that pore wall surfaces of activated carbon are closer to that of amorphous solids because of defects of crystalline lattice, finite pore length, and the presence of active centers, etc. Application of the NLDFT adapted to amorphous solids resulted in quantitative description of N 2 and Ar adsorption isotherms on nongraphitized carbon black BP280 at their respective boiling points. In the present paper we determined solid–fluid potentials from experimental adsorption isotherms on nongraphitized carbon black and subsequently used those potentials to model adsorption in slit pores and generate a corresponding set of local isotherms, which we used to determine the PSD functions of different activated carbons.

Effects of surface heterogeneity on the adsorption of nitrogen on graphitized thermal carbon black

In this paper, we study the surface heterogeneity and the surface mediation on the intermolecular potential energy for nitrogen adsorption on graphitized thermal carbon black (GTCB). The surface heterogeneity is modeled as the random distribution of “effective” carbonyl functional groups on the graphite surface. The molecular parameters and the discrete charges of this carbonyl group are taken from Jorgensen, et al. (J. Am. Chem. Soc., (1984) 106, 6638) while those for nitrogen (dispersive parameters and discrete charges) are taken from Murthy et al. (Mol. Phys., (1983) 50, 531) in our Grand Canonical Monte Carlo (GCMC) simulation. The solid surface mediation in the reduction of intermolecular potential energy between two fluid molecules was taken from a recent work by Do et al. (Langmuir, (2004) 20, 7623). Our simulation results accounting for the surface heterogeneity and surface mediation on intermolecular potential energy were compared with the experimental data of nitrogen at 77 and 90 K. The solid-fluid dispersive parameters are determined from the Lorentz-Berthelot (LB) rule. The fraction of the graphite surface covered with carbonyl functional groups was then derived from the consideration of the Henry constant, and for the data of Kruk et al. (Langmuir, (1999) 15, 1435) we have found that 1% of their GTCB surface is covered with “effective” carbonyl functional groups. The damping constant, due to surface mediation, was determined from the consideration of the portion of the adsorption isotherm where the first layer is being completed, and it was found to take a value of 0.0075. With these parameters, we have found that the GCMC simulation results describe the data over the complete range of pressure substantially better than any other MC models in the literature. The implication of this work is demonstrated with local adsorption isotherms of 10 and 20 A slit pores. One was obtained without allowance for surface mediation, while the other correctly accounts for these factors. The two local isotherms differ substantially, and the implication is that if we used incorrect local isotherms (i.e. without the surface mediation) the pore size distribution would be incorrectly derived.

Weighted-density approaches for polymer structure at solid–polymer interfaces

Weighted-density approximations (WDAs), which are based on the weighting function for the second-order direct correlation functions (DCFs) of the uniform polymeric fluids, have been developed to investigate the structural and thermodynamic properties of polymer melts at interfaces. The advantage is the simplicity of calculation of the weighting functions and their accuracies in the applications. They were applied to study the local density distributions and adsorption isotherms of the freely jointed tangent hard-sphere chain, Yukawa chain, and hard-sphere chain mixture in slit pores. The polymer reference interaction model (PRISM) integral equation with the Percus–Yevick (PY) closure has been used to calculate the second-order DCF of the polymeric fluids required as inputs. The mean-field approximation (MFA) has been used to calculate the weighting function for the attractive contribution of a freely jointed tangent Yukawa chain fluid, having attraction among the beads. The calculated results show that (i) for the freely jointed tangent hard-sphere chain, the present theory is in excellent agreement with the computer simulations over a wide range of chain lengths and bulk densities, (ii) the WDA approach for the attraction provides an accurate method for the local density distributions of a freely jointed tangent Yukawa chain fluid, and that (iii) the present theory also yields a reasonably good result for the structural properties of the freely jointed hard-sphere chain mixtures composed of the chain and monomer.

Adsorption of Quadrupolar, Diatomic Nitrogen onto Graphitized Thermal Carbon Black and in Slit-Shaped Carbon Pores. Effects of Surface Mediation

In this paper, we study the effect of solid surface mediation on the intermolecular potential energy of nitrogen, and its impact on the adsorption of nitrogen on a graphitized carbon black surface and in carbon slit-shaped pores. This effect arises from the lower effective interaction potential energy between two particles close to the surface compared to the potential energy of the same two particles when they are far away from the surface. A simple equation is proposed to calculate the reduction factor and this is used in the Grand Canonical Monte Carlo (GCMC) simulation of nitrogen adsorption on graphitized thermal carbon black. With this modification, the GCMC simulation results agree extremely well with the experimental data over a wide range of pressure; the simulation results with the original potential energy (i.e. no surface mediation) give rise to a shoulder in the neighbourhood of monolayer coverage and a significant over-prediction of the second and higher layer coverages. The influence of this surface mediation on the dependence of the pore-filling pressure on the pore width is also studied. It is shown that such surface mediation has a significant effect on the pore-filling pressure. This implies that the use of the local isotherms obtained from the potential model without surface mediation could give rise to a serious error in the determination of the pore-size distribution.

Prediction of multicomponent adsorption equilibrium of gas mixtures including supercritical components

A new model to predict the adsorption equilibrium of mixtures containing supercritical components was presented. The difference in the adsorption mechanism between the supercritical components and the condensable components was accounted for in the new model. Lateral interaction in the adsorbed phase was also considered in the local isotherm. The new model was verified by the experimental data of the mixture CH 4/CO 2/C 2H 6 on activated carbon JX101 collected for 283–313 K and pressures up to 10 × 10 5 Pa . It was also verified with the experimental data published previously in literature. Considerable improvement in the prediction precision was achieved in using the new model to predict the adsorption of mixtures on activated carbon.

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.

Non‐Additivity of Attractive Potentials in Modeling of N2 and Ar Adsorption Isotherms on Graphitized Carbon Black and Porous Carbon by Means of Density Functional Theory

AbstractWe present a new approach accounting for the non‐additivity of attractive parts of solid–fluid and fluid–fluid potentials to improve the quality of the description of nitrogen and argon adsorption isotherms on graphitized carbon black in the framework of non‐local density functional theory. We show that the strong solid–fluid interaction in the first monolayer decreases the fluid–fluid interaction, which prevents the two‐dimensional phase transition to occur. This results in smoother isotherm, which agrees much better with experimental data. In the region of multi‐layer coverage the conventional non‐local density functional theory and grand canonical Monte Carlo simulations are known to over‐predict the amount adsorbed against experimental isotherms. Accounting for the non‐additivity factor decreases the solid–fluid interaction with the increase of intermolecular interactions in the dense adsorbed fluid, preventing the over‐prediction of loading in the region of multi‐layer adsorption. Such an improvement of the non‐local density functional theory allows us to describe experimental nitrogen and argon isotherms on carbon black quite accurately with mean error of 2.5 to 5.8% instead of 17 to 26% in the conventional technique. With this approach, the local isotherms of model pores can be derived, and consequently a more reliable pore size distribution can be obtained. We illustrate this by applying our theory against nitrogen and argon isotherms on a number of activated carbons. The fitting between our model and the data is much better than the conventional NLDFT, suggesting the more reliable PSD obtained with our approach.

Investigation of the surface heterogeneity of solids from reversed-flow inverse gas chromatography

In the present work, the novel methodology of the inverse gas chromatographic technique of reversed-flow gas chromatography (RF-GC) was applied to the well-studied catalytic oxidation of carbon monoxide over silica supported Pt, Rh and Pt–Rh alloy catalysts. Adsorption energies, local isotherms, local monolayer capacities, surface diffusion coefficients, lateral interaction energies and energy distribution functions are simultaneously determined in a single experiment. The variation of the determined physicochemical parameters against the nature of the studied catalysts (Pt content) is consistent with the observed catalytic activity. The energy distribution functions, estimated by means of RF-GC, give useful information about the “ topography” and the nature of the active sites on the catalyst surface, similar to those of experimental techniques, such as Thermal Desorption Spectroscopy studies of the adsorption of CO on group VIII noble metal surfaces. The experimentally found results explain the superior activity of Pt 0.25 + Rh 0.75 alloy, in comparison to that of the pure Pt and Rh catalysts.

Determination of Chemical Kinetic Properties of Heterogeneous Catalysts

AbstractFor Abstract see ChemInform Abstract in Full Text.

Prediction of the adsorption equilibrium of mixtures composed of supercritical gases.

A new model was proposed to predict the adsorption equilibrium of mixtures composed of supercritical gases. The adsorbed phase was visualized as a two-dimensional nonideal compressed gas. Pore size distribution was used to describe the energetic heterogeneity of the surface, and the two-dimensional virial equation was used as the local adsorption isotherm. The new model obtained is thermodynamically rigorous because it reduces to Henry's law as pressure approaches zero. The prediction performance of the new model was verified and compared with other models using the experimental data of a ternary mixture of CH4/N2/H2 and two binary mixtures of CH4/C2H4 and CH4/N2. Better performance was shown for all systems tested.

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.

Physicochemical characterization of interfaces

Reversed-flow inverse gas chromatography (RF-IGC) was used to measure, directly from experimental data, adsorption energies, local adsorption isotherms, the probability density function for the adsorption energies, and lateral interaction parameters, as distributed over experimental time. Local isotherms and the distribution energy function were correlated with adsorption energy. The results obtained are comparable to those calculated on the basis of the well-known integral equation. The RF-IGC method was used with the two most common hydrocarbons, acetylene and 1-butene, as probe gases, in the presence and absence of ozone, and with magnesium oxide and silicon oxide as solid adsorbents.

Effects of soil organic matter on pH-dependent phosphate sorption by soils

Effects of soil organic matter (80M) on P sorption of soils still remain to be clarified because contradictory results have been reported in the literature. In the present study, pH-dependent P sorption on an allophanic Andisol and an alluvial soil was compared with that on hydrogen peroxide (H202)-treated, acid-oxalate (OX)-treated, and dithionite-citrate- bicarbonate (DCB)-treated soils. Removal of 80M increased or decreased P sorption depending on the equilibrium pH values and soil types. In the H2O2 OX-, and DCB-treated soils, P sorption was pH-dependent, but this trend was not conspicuous in the untreated soils. It is likely that 80M affects P sorption of soils through three factors, competitive sorption, inhibition of polymerization and crystallization of metals such as AI and Fe, and flexible structure of metal-80M complexes. As a result, the number of available sites for P sorption would remain relatively constant in the wide range of equilibrium pH values in the presence of 80M. The P sorption characteristics were analyzed at constant equilibrium pH values (4.0 to 7.0) using the Langmuir equation as a local isotherm. The maximum number of available sites for P sorption (Q max) was pH-dependent in the H202-, OX-, and DCBtreated soils, while this trend was not conspicuous in the untreated soils. Affinity constants related to binding strength (K) were less affected by the equilibrium pH values, soil types, and soil treatments, and were almost constant (log K ≈ 4.5). These findings support the hypothesis that 80M plays a role in keeping the number of available sites for P sorption relatively constant but does not affect the P sorption affinity. By estimating the Q max and K values as a function of equilibrium pH values, pH-dependent P sorption was well simulated with four or two adjustable parameters. This empirical model could be useful and convenient for a rough estimation of the pH-dependent P sorption of soils.

Heterogeneous adsorption of 1-indanol on cellulose tribenzoate and adsorption energy distribution of the two enantiomers.

The distributions of the adsorption energies (AED) of two enantiomers, (R)-1- indanol and (S)-1-indanol, on a chiral stationary phase were measured and the results are discussed. The chiral phase used is made of cellulose tribenzoate coated on porous silica. The AEDs were determined using the expectation maximization method, a numerical method that uses directly the raw experimental isotherm data, inverts this set of data into an AED, and introduces no arbitrary information in the calculation. However, it uses the Langmuir equation as the local isotherm. The experimental data fit very well to the bi-Langmuir isotherm model for the more retained enantiomer. Our results show that the AEDs of these two enantiomers have no energy modes that would be identical (same mean energy, mode profile, and mode area), in contrast to numerous cases previously studied, e.g., that of the beta-blockers on a Cel7A column. This indicates a significantly different retention mechanism.

Refined Method of Potential Enhancement in the Equilibria Characterization of Activated Carbon. Comparison with GCMC and DFT

In this paper, we present a technique for equilibria characterization of activated carbon having slit-shaped pores. This method was first developed by Do (Do, D. D. A new method for the characterisation of micro−mesoporous materials. Presented at the International Symposium on New Trends in Colloid and Interface Science, September 24−26, 1998, Chiba, Japan) and applied by his group and other groups for characterization of pore size distribution (PSD) as well as adsorption equilibria determination of a wide range of hydrocarbons. It is refined in this paper and compared with the grand canonical Monte Carlo (GCMC) simulation and density functional theory (DFT). The refined theory results in a good agreement between the pore filling pressure versus pore width and those obtained by GCMC and DFT. Furthermore, our local isotherms are qualitatively in good agreement with those obtained by the GCMC simulations. The main advantage of this method is that it is about 4 orders of magnitude faster than the GCMC simula...

High-Pressure Adsorption of Supercritical Gases on Activated Carbons: An Improved Approach Based on the Density Functional Theory and the Bender Equation of State

Adsorption of nitrogen, argon, methane, and carbon dioxide on activated carbon Norit R1 over a wide range of pressure (up to 50 MPa) at temperatures from 298 to 343 K (supercritical conditions) is analyzed by means of the density functional theory modified by incorporating the Bender equation of state, which describes the bulk phase properties with very high accuracy. It has allowed us to precisely describe the experimental data of carbon dioxide adsorption slightly above and below its critical temperatures. The pore size distribution (PSD) obtained with supercritical gases at ambient temperatures compares reasonably well with the PSD obtained with subcritical nitrogen at 77 K. Our approach does not require the skeletal density of activated carbon from helium adsorption measurements to calculate excess adsorption. Instead, this density is treated as a fitting parameter, and in all cases its values are found to fall into a very narrow range close to 2000 kg/m3. It was shown that in the case of high-pressure adsorption of supercritical gases the PSD could be reliably obtained for the range of pore width between 0.6 and 3 nm. All wider pores can be reliably characterized only in terms of surface area as their corresponding excess local isotherms are the same over a practical range of pressure.

The Thermal Regulation of Gravitational Instabilities in Protoplanetary Disks

We present a series of high-resolution, three-dimensional hydrodynamics simulations of a gravitationally unstable solar nebula model. The influences of both azimuthal grid resolution and the treatment of thermal processes on the origin and evolution of gravitational instabilities are investigated. In the first set of simulations, we vary the azimuthal resolution for a locally isothermal simulation, doubling and quadrupling the resolution used in a previous study; the largest number of grid points is (256, 256, 64) in cylindrical coordinates (r, , z). At this resolution, the disk breaks apart into a dozen short-lived condensations. Although our previous calculations underresolved the number and growth rate of clumps in the disk, the overall qualitative, but fundamental, conclusion remains: fragmentation under the locally isothermal condition in numerical simulations does not in itself lead to the survival of clumps to become gaseous giant protoplanets. Since local isothermality represents an extreme assumption about thermal processes in the disk, we also present several extended simulations in which heating from an artificial viscosity scheme and cooling from a simple volumetric cooling function are applied to two different models of the solar nebula. The models are differentiated primarily by disk temperature: a high-Q model generated directly by our self-consistent field equilibrium code and a low-Q model generated by cooling the high-Q model in a two-dimensional version of our hydrodynamics code. Here, high-Q and low-Q refer to the minimum values of the Toomre stability parameter Q in each disk, Qmin = 1.8 and 0.9, respectively. Previous simulations, by ourselves as well as others, have focused on initial states that are already gravitationally unstable, i.e., models similar to the low-Q model. This paper presents for the first time the numerical evolution of an essentially stable initial equilibrium state (the high-Q model) to a severely unstable one by cooling. The additional heating and cooling are applied to each model over the outer half of the disk or the entire disk. The models are subject to the rapid growth of a four-armed spiral instability; the subsequent evolution of the models depends on the thermal behavior of the disk. The cooling function tends to overwhelm the heating included in our artificial viscosity prescription, and as a result the spiral structure strengthens. The spiral disturbances transport mass at prodigious rates during the early nonlinear stages of development and significantly alter the disk's vertical surface. Although dense condensations of material can appear, their character depends on the extent of the volumetric cooling in the disk. In the simulation of the high-Q model with heating and cooling applied throughout the disk, thin, dense rings form at radii ranging from 1 to 3 AU and steadily increase in mass; later companion formation may occur in these rings as cooling drives them toward instability. When heating and cooling are applied only over the outer radial half of the disk, however, a succession of single condensations appears near 5 AU. Each clump has roughly the mass of Saturn, and some survive a complete orbit. Since the clumps form near the artificial boundary in the treatment of the disk gas physics, the production of a clump in this case is a numerical artifact. Nevertheless, radially abrupt transitions in disk gas characteristics, for example, in opacity, might mimic the artificial boundary effects in our simulations and favor the production of stable companions in actual protostellar and protoplanetary disks. The ultimate survival of condensations as eventual stellar or substellar companions to the central star is still largely an open question.