Statistical rate theory description of beam-dosing adsorption kinetics

The Procedure for Evaluating the Adsorption Energy Distribution from an Analysis of Thermodesorption Spectra Based on the Statistical Rate Theory

We have recently proposed a new method for evaluating the adsorption energy distribution function χ(ε) from the numerical analysis of thermodesorption spectra. This method is based on an application of the Statistical Rate Theory developed by Ward et al. (1982). In this paper, we present some simplifications of this method thereby allowing its ready application. Illustrative calculations for the nitrogen thermodesorption spectra from the surface of molecular sieve carbon fibres (MSCF) are presented.

Vapour adsorption kinetics: statistical rate theory and zeta adsorption isotherm approach

The zeta adsorption isotherm may be combined with statistical rate theory to formulate an expression for vapour adsorption kinetics that is in terms of a rate constant. For heptane adsorbing on silica, the rate constant is experimentally shown to depend only on temperature.

Effects of nonlinear interfacial kinetics and interfacial thermal resistance in planar solidification

Large temperature discontinuities were recently measured at a solid-liquid interface during heat transport processes. These observations suggest that when heat flows between two phases, the interface is not well characterized by assuming thermal equilibrium. This can be of importance in rapid solidification processes. In this paper we consider a planar front model that solidifies from its undercooled melt. We use a generalized interfacial boundary condition that includes nonlinear kinetic effects and allows for a temperature discontinuity. The effects of the new boundary condition on the solidification rates and the temperature profile are reported as a function of time. Our analysis shows that the undercooling regime where constant phase-front velocities are observed at steady states (traveling-wave solutions) are unaffected by the new boundary conditions. These solutions arise when the Stephan number is larger than 1. On the other hand, the solidification rates and the steady-state velocities are greatly affected by the assumed conditions at the interface. Incorporation of an interface thermal resistance, or Kapitza resistance, generates temperature discontinuities at the interface, leads to reduced solidification rates and the Mullins-Sekerka instability arises at longer wavelengths deformation of the planar front.

Statistical rate theory insight into evaporation and condensation in multicomponent systems

Current approaches to mathematically modeling liquid-vapor mass transport (e.g., film theory, penetration theory, boundary layer theory) treat bulk phase transport accurately with diffusion models, but leave the transport across the interface to be described by empirically determined mass transfer coefficients. In multicomponent systems, this requires empirical mixing rules for the single-component mass transfer coefficients. Such approaches can only give estimates of net rates at the interface but cannot examine the movement of individual components. Here we use statistical rate theory to provide new physical insight into evaporation and condensation at interfaces in systems containing multiple volatile components. In contrast to the traditional multicomponent mass transfer approach, we show ranges where one component evaporates while the other condenses even when the net transport is unidirectional.

Comparative vibrational study on alkali coadsorption with CO and O on Ni(111) and Cu(111)

High-resolution electron energy loss spectroscopy was used to investigate alkali (Na, K) coadsorption with CO and O on Cu(111) and Ni(111). Measurements provided new insights in these systems. A CO-induced weakening of the alkali-substrate bond was revealed on both substrates. The effect is more pronounced for the Na+CO/Ni(111) system. Submonolayers of alkalis were found to promote the preferential population of the subsurface site for O/Cu(111) but not for O/Ni(111).

Recent advances on thermocapillary flows and interfacial conditions during the evaporation of liquids

Thermocapillary convection has a very different history for water than for other liquids. For water, several studies have pointed to the lack of evidence supporting the existence of thermocapillary (or Marangoni) convection. Other studies have given clear evidence of its existence and of the role it plays during steady-state water evaporation. We examine both sets of data and suggest a reason for the difference in the interpretation of the experimental data. For organic liquids, the evidence of thermocapillary convection has been clearly documented, but the issues are the type of flow that it generates during steady-state evaporation. We review the measurements and show that the flow field of the evaporating liquid is strongly affected by the presence of the thermocapillary convection. When the results obtained from both water and organic liquids are compared, they give further insight into the nature of thermocapillary convection.

Kinetics of solute adsorption at solid/solution interfaces: a theoretical development of the empirical pseudo-first and pseudo-second order kinetic rate equations, based on applying the statistical rate theory of interfacial transport

For practical applications of solid/solution adsorption processes, the kinetics of these processes is at least as much essential as their features at equilibrium. Meanwhile, the general understanding of this kinetics and its corresponding theoretical description are far behind the understanding and the level of theoretical interpretation of adsorption equilibria in these systems. The Lagergren empirical equation proposed at the end of 19th century to describe the kinetics of solute sorption at the solid/solution interfaces has been the most widely used kinetic equation until now. This equation has also been called the pseudo-first order kinetic equation because it was intuitively associated with the model of one-site occupancy adsorption kinetics governed by the rate of surface reaction. More recently, its generalization for the two-sites-occupancy adsorption was proposed and called the pseudo-second-order kinetic equation. However, the general use and the wide applicability of these empirical equations during more than one century have not resulted in a corresponding fundamental search for their theoretical origin. Here the first theoretical development of these equations is proposed, based on applying the new fundamental approach to kinetics of interfacial transport called the Statistical Rate Theory. It is shown that these empirical equations are simplified forms of a more general equation developed here, for the case when the adsorption kinetics is governed by the rate of surface reactions. The features of that general equation are shown by presenting exhaustive model investigations, and the applicability of that equation is tested by presenting a quantitative analysis of some experimental data reported in the literature.

Sticking coefficient and pressure dependence of desorption rate in the statistical rate theory approach to the kinetics of gas adsorption. Carbon monoxide adsorption/desorption rates on the polycrystalline rhodium surface

The statistical rate theory (SRT) fundamental kinetic equation for the Langmuir model of adsorption reveals some features which are difficult to understand. These are the infinitely high adsorption rate for zero coverage limit, infinitely high desorption rate at full saturation of the surface and dependence of the desorption rate on the adsorbate pressure. These features do not allow for a correct formulation of such important physical parameters as sticking and initial sticking probabilities. In this work these peculiar features of the fundamental SRT kinetic equation are discussed and explained. It is shown that the non-physical behavior of the SRT kinetic equation follows from neglecting possible changes of adsorbate concentration near the adsorbing surface. A simple model accounting for the changes of adsorbate concentration close to the adsorbing surface is discussed. As a result, it was shown that application of even such a simple model leads to fully physical behavior of the SRT kinetic equation for the Langmuir model of adsorption. The model is used to build up equations describing the kinetics of adsorption/desorption of carbon monoxide on the energetically heterogeneous rhodium surface. The values of the parameters in these equations were determined from the analysis of equilibrium adsorption isotherms and from the description of experimental conditions. One additional parameter (having well defined limiting values) had to be adjusted in order to satisfactorily reproduce kinetic data. The experimental data used in this work were taken from the article by Yamada and Tamaru (T. Yamada and K. Tamaru, Surf. Sci., 1984, 138, L155).

Collisions of ideal gas molecules with a rough/fractal surface. A computational study

The frequency of collisions of ideal gas molecules (argon) with a rough surface has been studied. The rough/fractal surface was created using random deposition technique. By applying various depositions, the roughness of the surface was controlled and, as a measure of the irregularity, the fractal dimensions of the surfaces were determined. The surfaces were next immersed in argon (under pressures 2 x 10(3) to 2 x 10(5) Pa) and the numbers of collisions with these surfaces were counted. The calculations were carried out using a simplified molecular dynamics simulation technique (only hard core repulsions were assumed). As a result, it was stated that the frequency of collisions is a linear function of pressure for all fractal dimensions studied (D = 2, ..., 2.5). The frequency per unit pressure is quite complex function of the fractal dimension; however, the changes of that frequency with the fractal dimension are not strong. It was found that the frequency of collisions is controlled by the number of weakly folded sites on the surfaces and there is some mapping between the shape of adsorption energy distribution functions and this number of weakly folded sites. The results for the rough/fractal surfaces were compared with the prediction given by the Langmuir-Hertz equation (valid for smooth surface), generally the departure from the Langmuir-Hertz equation is not higher than 48% for the studied systems (i.e. for the surfaces created using the random deposition technique).

Thermocapillary transport of energy during water evaporation

When evaporation occurs at a spherical water-vapor interface maintained at the circular mouth of a small funnel, studies of the energy transport have indicated that thermal conduction alone does not provide enough energy to evaporate the liquid at the observed rate. If the Gibbs model of the interface is adopted and the "surface-thermal capacity" is assigned a value of 30.6+/-0.8 kJ/(m2 K), then for evaporation experiments with the interfacial temperature in the range -10 degrees C< or =TLV< or =3.5 degrees C and Marangoni number (Ma) in the range 100<Ma<22,000, it was found that if energy transport by both thermocapillary convection and thermal conduction were taken into account, conservation of energy was fully satisfied. The question addressed herein is whether the assigned value of the surface-thermal capacity is an ad hoc empirical parameter or a property of the water-vapor interface that can be used in other circumstances. Accordingly, a series of experiments has been conducted in which water evaporated at cylindrical interfaces that were, on average, 4.4 times larger in area than that of the spherical interfaces used to measure the surface-thermal capacity initially. It is shown that using the value of the surface-thermal capacity determined at a spherical interface, the energy transported by thermocapillary convection and thermal conduction at a cylindrical interface is sufficient to evaporate the liquid at the observed rate. Knowing the value of the surface-thermal capacity also allows the local evaporation flux to be calculated from the measured temperature profiles in the liquid and vapor phases. The calculated local evaporation flux can then be used with statistical rate theory to calculate the vapor-phase pressure along the interface. The predicted mean vapor-phase pressure is in close agreement with that measured, and the predicted pressure gradient is consistent with that expected when thermocapillary convection is present.

Surface-thermal capacity of from measurements made during steady-state evaporation

When D2O(l) evaporates into its vapor under steady-state conditions with the temperature field in the liquid arranged so that there is no buoyancy-driven convection and the Marangoni number is less than approximately 100, it is found that the interface is quiescent and thermal conduction to the interface supplies energy at a sufficient rate to evaporate the liquid. However, if the evaporation rate is raised so that the Marangoni number goes above approximately 100, the interface is transformed: a fluctuating thermocapillary flow occurs, and thermal conduction no longer supplies energy at a sufficient rate to evaporate the liquid. An energy analysis indicates conservation of energy can be satisfied only if thermocapillary convection is taken into account, and the surface-thermal capacity csigma is assigned a value of 32.5+/-0.8 kJ/(m2 K) when the temperature is in the range -10 degrees C< or =TLV< or =3.7 degrees C. This value is consistent with that found previously for H2O, and application of the Gibbs model gives a qualitative explanation for the value. Once the value of the surface-thermal capacity is known, the local heat flux along the interface can be calculated and statistical rate theory can be used to predict the local vapor-phase pressure on the interface. Since this theory introduces no adjustable parameters, the predicted pressure can be compared directly with that measured: this comparison indicates the mean of the pressures predicted to exist on the interface is in close agreement with those measured approximately 20 cm above the interface, and the small pressure gradient along the interface is consistent with the thermocapillary convection predicted from the interfacial temperature gradient.

Kinetics of Isothermal Gas Adsorption on Heterogeneous Solid Surfaces: Equations Based on Generalization of the Statistical Rate Theory of Interfacial Transport

Two generalizations of the statistical rate theory (SRT) approach have been developed for the kinetics of adsorption/desorption on/from heterogeneous surfaces and then compared to each other. The first of them is an improved version of the description based on the condensation approximation (CA) used by us. The other one is an exact generalization for the Langmuir model of adsorption. The latter generalization does not lead to simple analytical expressions for the rate of adsorption/desorption as the CA approach does for typical adsorption energy distributions. The comparison of these two approaches suggests, however, that at small and very high surface coverages the CA approach may not be sufficiently accurate. Very intriguing results are obtained when the developed SRT kinetic equations are applied to describe the kinetics of sorption in carbon molecular sieves. It appears that the fit of experimental data is then equally good as that obtained by assuming that the rate of sorption is controlled by surface diffusion in pores. Also, it is shown that the square-root dependence on time of adsorption at small initial coverages cannot be treated as a definite proof for that sorption proceeds via surface diffusion, as is commonly assumed.

Thermodesorption studies of energetic properties of Ni/MgO-Al2O3 catalysts. Determination of adsorption energy distribution functions

The thermodesorption spectra of hydrogen from coprecipitated catalysts (70-x)NiO-xMgO-30Al(2)O(3) (x = 0-50%(wt)) are reported. The catalysts were calcined at 400 degrees C and reduced with H(2) at 20-800 degrees C and for 3 h at 800 degrees C. NiO reduction degree was between 49.3 and 92.1%. The active surface areas changed from 8.4 to 32.4 m(2)/g whereas mean size of nickel crystallites was between 3.7 and 9.7 nm. The TPD spectra were next analyzed in order to determine the adsorption energy distributions functions. To obtain these functions a theoretical model of adsorption/desorption kinetics based on the statistical rate theory (SRT) was applied. This approach allows for determination of the adsorption energy at nonequilibrium conditions as well as at quasiequilibrium conditions. The resulting distribution functions reveal the presence of two main bands of adsorption energy. Some correlation is found between the determined distributions of adsorption energy and the size of nickel crystallites determined using the XRD method. The presence of MgO favors creation of high energy adsorption sites on Ni crystallites.

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.

Heats of adsorption using temperature programmed adsorption equilibrium: application to the adsorption of CO on Cu/A12O3 and H2 on Pt/Al2O3

A new simple analytical procedure is described that allows the determination of the heats of adsorption (denoted E(theta)) of adsorbed species at several coverages (theta's) using a single experiment. This procedure is an extension of an original method previously developed (denoted AEIR: adsorption equilibrium infrared spectroscopy). A mass spectrometer is used to determine the amounts of gas (in the present study, CO and H2) either desorbed from or adsorbed on a metal supported catalyst (4.7% Cu/Al2O3 and 2.9% Pt/Al2O3) during the perturbation of the adsorption equilibrium due to a controlled change of the adsorption temperature (Ta) at a quasi-constant adsorption pressure (Pa). These amounts allow us to follow the evolution of the adsorption equilibrium coverage (theta(e)) with Ta at the quasi-constant partial pressure (Pa). Then, the curve theta(e) = f(Ta) provides Etheta = f(theta) with the support of an adsorption model. This procedure presents several advantages as compared to the TPD methods, in particular, considering the theoretical supports linked to the exploitation of the experimental data. As compared to AEIR, the TPAE procedure allows one to study the heats of adsorption of adsorbed species that are not detectable by IR. However, it is not adapted if surface reactions occur in parallel to adsorption/desorption processes.

Interfacial conditions during evaporation or condensation of water

Steady-state evaporation and condensation experiments have been conducted with water under conditions where buoyancy-driven convection is not present. The temperature profile in each phase has been measured. At the interface, independently of the direction of the phase change, a temperature discontinuity has been found to exist in which the interfacial vapor temperature is greater than that in the liquid. In a thin layer immediately below the interface the temperature is uniform in a layer ( approximately 0.5 mm) and below that the temperature profile is linear, indicating thermal conduction. The uniform temperature layer indicates a mixing process occurs near the interface that could result from surface-tension driven (Marangoni-Bénard) convection and/or from "energy partitioning" that is necessary to account for the measured temperature discontinuity near the interface. When the measured interfacial properties are used with the expression for the phase change rate that is obtained from statistical rate theory, it is found that the predictions are in close agreement with the measurements.

On the applicability of Arrhenius plot methods to determine surface energetic heterogeneity of adsorbents and catalysts surfaces from experimental TPD spectra

Recovering adsorption energy distribution from experimental data belongs to most difficult problems of adsorption science. In the case when thermodesorption data are used as a source of information, that difficult problem is overcome by the common use of the Arrhenius plot methods. So, we decided to carry out an extensive model investigation to show, how reliable information concerning the surface energetic heterogeneity is obtained by using the Arrhenius plot methods. Like in our previous publications we have used the Statistical Rate Theory of Interfacial Transport to describe the adsorption/desorption kinetics. Our model investigations showed, that the Arrhenius plot methods, cannot provide reliable information about the surface energetic heterogeneity. Moreover, for strongly heterogeneous surfaces a linear relationship exists between the logarithm of the pre-exponential constant and the adsorption energy, for certain adsorption coverages. That kind of compensation effect has, so far, been ascribed to interactions between the adsorbed molecules. The failure of the popular Arrhenius plot method puts, as an urgent agenda, the development of reliable methods for recovering adsorption energy distribution from the thermodesorption data.