Modeling of adsorption dynamics at air-liquid interfaces using statistical rate theory (SRT)

A large number of natural and technological processes involve mass transfer at interfaces. Interfacial properties, e.g., adsorption, play a key role in such applications as wetting, foaming, coating, and stabilizing of liquid films. The mechanistic understanding of surface adsorption often assumes molecular diffusion in the bulk liquid and subsequent adsorption at the interface. Diffusion is well described by Fick's law, while adsorption kinetics is less understood and is commonly described using Langmuir-type empirical equations. In this study, a general theoretical model for adsorption kinetics/dynamics at the air-liquid interface is developed; in particular, a new kinetic equation based on the statistical rate theory (SRT) is derived. Similar to many reported kinetic equations, the new kinetic equation also involves a number of parameters, but all these parameters are theoretically obtainable. In the present model, the adsorption dynamics is governed by three dimensionless numbers: psi (ratio of adsorption thickness to diffusion length), lambda (ratio of square of the adsorption thickness to the ratio of adsorption to desorption rate constant), and Nk (ratio of the adsorption rate constant to the product of diffusion coefficient and bulk concentration). Numerical simulations for surface adsorption using the proposed model are carried out and verified. The difference in surface adsorption between the general and the diffusion controlled model is estimated and presented graphically as contours of deviation. Three different regions of adsorption dynamics are identified: diffusion controlled (deviation less than 10%), mixed diffusion and transfer controlled (deviation in the range of 10-90%), and transfer controlled (deviation more than 90%). These three different modes predominantly depend on the value of Nk. The corresponding ranges of Nk for the studied values of psi (10(-2)相似文献   

Kinetics of solute adsorption at solid/aqueous interfaces: searching for the theoretical background of the modified pseudo-first-order kinetic equation

It is shown that the modified pseudo-first-order (MPFO) kinetic equation proposed recently by Yang and Al-Duri simulates well the behavior of the kinetics governed by the rate of surface reaction and described by our general kinetic equation, based on the statistical rate theory. The linear representation with respect to time, offered by the MPFO equation seems to be a convenient tool for distinguishing between the surface reaction and the diffusional kinetics. Also, a method of distinguishing between the surface reaction and the intraparticle diffusion model based on analyzing the initial kinetic isotherms of sorption is proposed. The applicability of these procedures is demonstrated by the analysis of adsorption kinetics of the reactive yellow dye onto an activated carbon.     

Application of a single model to study the adsorption equilibrium of prednisolone on six carbonaceous materials

The knowledge of sorption processes of nonelectrolytes in solution by solid adsorbents implies the study of kinetics, equilibrium, and thermodynamic functions. However, quite frequently the equilibrium isotherms are studied by comparing them with those corresponding to the Giles et al. classification (1); these isotherms are also analyzed by fitting them to equations based on thermodynamic or kinetic criteria, and even to empirical equations. Nevertheless, information obtained is more coherent and satisfactory if the adsorption isotherms are fitted by using an equation describing the equilibrium isotherms according to the kinetic laws. These mentioned laws would determine each one of the unitary processes (one or more) which condition the global process. In this paper, an adsorption process of prednisolone in solution by six carbonaceous materials is explained according to a previously proposed single model, which allows to establish a kinetic law which fits satisfactorily most of C vs t isotherms (2). According to the above-mentioned kinetic law, equations describing sorption equilibrium processes have been deducted, and experimental data points have been fitted to these equations; such a fitting yields to different values of adsorption capacity and kinetic equilibrium constants for the different processes at several temperatures. However, in spite of their practical interest, these constants have no thermodynamic signification. Thus, the thermodynamic equilibrium constant (K) has been calculated by using a modified expression of the Gaines et al. equation (3). Global average values of the thermodynamic functions have also been calculated from the K values. Information related to variations of DeltaH and DeltaS with the surface coverage fraction was obtained by using the corresponding Clausius-Clapeyron equations.     

Adsorption kinetics at the solid/solution interface: statistical rate theory at initial times of adsorption and close to equilibrium

The kinetics of solute adsorption at the solid/solution interface has been studied by statistical rate theory (SRT) at two limiting conditions, one at initial times of adsorption and the other close to equilibrium. A new kinetic equation has been derived for initial times of adsorption on the basis of SRT. For the first time a theoretical interpretation based on SRT has been provided for the modified pseudo-first-order (MPFO) kinetic equation which was proposed empirically by Yang and Al-Duri. It has been shown that the MPFO kinetic equation can be derived from the SRT equation when the system is close to equilibrium. On the basis of numerically generated points ( t, q) by the SRT equation, it has been shown that we can apply the new equation for initial times of adsorption in a larger time range in comparison to the previous q vs radical t linear equation. Also by numerical analysis of the generated kinetic data points, it is shown that application of the MPFO equation for modeling of whole kinetic data causes a large error for the data at initial times of adsorption. The results of numerical analysis are in perfect agreement with our theoretical derivation of the MPFO kinetic equation from the SRT equation. Finally, the results of the present theoretical study were confirmed by analysis of an experimental system.     

Application of a single model to study the adsorption kinetics of prednisolone on six carbonaceous materials

The knowledge of the adsorption processes of nonelectrolytes from liquid solution on solid materials involves the study of their kinetic and equilibrium aspects as well as the understanding of their thermodynamic functions. However, in most published papers adsorption isotherms are analyzed by using the Giles classification and other proposed equations which are either empirical or based on kinetic or thermodynamic criteria. Our opinion is that both the kinetic and the equilibrium studies must be complementary and that, in general, equations describing the adsorption isotherms come from the kinetic laws governing the different partial processes which determine the global process. These kinetic laws may be derived from single models. In this paper a single model is proposed, which makes it possible to establish a kinetic law satisfactorily fitting a great number of C (concentration) vs t (time) isotherms. This model has been applied to study the adsorption process of prednisolone by six carbonaceous materials from ethanol solution, the specific adsorption rate, and the activation thermodynamic functions being calculated. The results obtained have also been used to analyze the influence of the intraparticle diffusion on the kinetics of the process.     

From Langmuir kinetics to first- and second-order rate equations for adsorption

So far, the first- and second-order kinetic equations have been most frequently employed to interpret adsorption data obtained under various conditions, whereas the theoretical origins of these two equations still remain unknown. Using the Langmuir kinetics as a theoretical basis, this study showed that the Langmuir kinetics can be transformed to a polynomial expression of dtheta t /d t = k 1(theta e - theta t ) + k 2(theta e - theta t ) (2), a varying-order rate equation. The sufficient and necessary conditions for simplification of the Langmuir kinetics to the first- and second-order rate equations were put forward, which suggested that the relative magnitude of theta e over k 1/ k 2 governs the simplification of the Langmuir kinetics. In cases where k 1/ k 2 is greater than theta e or k 1/ k 2 is very close to theta e, adsorption kinetics would be reasonably described by the first-order rate equation, whereas the Langmuir kinetics would be reduced to the second-order equation only at k 1/ k 2 < theta e. It was further demonstrated that both theta e and k 1/ k 2 are the function of initial adsorbate concentration ( C 0) at a given dosage of adsorbent, indicating that simplification of the Langmuir kinetics indeed is determined by C 0. Detailed C 0-depedent boundary conditions for simplifying the Langmuir kinetics were also established and were verified by experimental data.     

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.     

The Langmuirian Adsorption Kinetics Revised: A Farewell to the XXth Century Theories?

A brief historical review of the development of the theoretical approaches to the kinetics of gas adsorption/desorption on/from the solid surfaces is presented. The attention is focused on new approaches, challenging the classical theories based on the ideas of Absolute Rate Theory (ART). These new approaches relate the adsorption/desorption kinetics to the chemical potentials of the molecules in the gas and adsorbed states. Among them the so-called Statistical Rate Theory (SRT) has the most rigorous theoretical foundations. That new approach predicts that depending on experimental conditions one can have a variety of kinetic equations corresponding to the Langmuir equilibrium adsorption isotherm.     

On the applicability of the pseudo-second order equation to represent the kinetics of adsorption at solid/solution interfaces: a theoretical analysis based on the statistical rate theory

It is shown that the empirical pseudo-second order kinetic equation is a very efficient formula to correlate the kinetic data generated by applying theoretical expressions developed from the fundamental SRT (Statistical Rate Theory) approach to the interfacial transport. This is especially true when the most popular linear representation is used in which time/adsorbed amount is plotted vs. time. However, the commonly observed goodness of such linear plots does not necessarily speak for the applicability of the pseudo-second order kinetic equation. A reliable estimation, for instance, of the equilibrium adsorbed amount is possible only when a substantial part of a kinetic isotherms corresponds to the conditions close to equilibrium. Energetic surface heterogeneity increases the goodness of these linear regressions. Then, experimental errors have only little effect on the pseudo-second linear plots. This article is dedicated to Professor Mietek Jaroniec on the occasion of his 60^th birthday.     

Dynamic surface tension and adsorption kinetics of beta-casein at the solution/air interface

A diffusion model is proposed to describe the adsorption kinetics of proteins at a liquid interface. The model is based on the simultaneous solution of the Ward-Tordai equation and a set of recently developed equations describing the equilibrium state of the adsorption layer: the adsorption isotherm, the surface layer equation of state, and the function of adsorption distribution over the states with different molar areas. The new kinetics model is compared with dynamic surface tensions of beta-casein solutions measured with the drop/bubble profile and maximum bubble pressure methods. The adsorption process for low concentrations is governed by the diffusion mechanism, while at large protein concentrations this is only the case in the initial stage. The effective diffusion coefficients agree fairly well with literature data. The adsorption values calculated from the dynamic surface tension data agree very well with the used equilibrium adsorption model.     

The load dependence of rate constants

As experimental techniques in biophysics have progressed at the single molecule level, there has been considerable interest in understanding how external mechanical influences (such as load) affect chemical reactions. The majority of biophysical studies investigating load-dependent kinetics use an equation where the rate constant exponentially depends on force, which is sometimes called Bell's equation. This equation requires the determination of two parameters that describe the potential energy-strain function: k(0), which is the reaction rate in the absence of load, and x(c), which is the difference in strain between the reactant and transition states. However, there have been theoretical studies based on Kramers' theory suggesting that the rate constant should have load-dependent pre-exponential terms and nonlinear load-dependent terms in the exponential. Kramers' theory requires an exact knowledge of the potential energy-strain function, which is in general not known for an experimental system. Here, we derive a general approximation of Kramers' theory where the potential energy-strain function is described by five parameters, which can, for small loads, be reduced to four-, three-, and finally to two parameters (Bell's equation). We then use an idealized physical system to validate our approximations to Kramers' theory and show how they can predict parameters of interest (such as k(0) and x(c)) better than Bell's equation. Finally, we show previously published experimental data that are not well fitted by Bell's equation but are adequately fitted by these more exact equations.     

Kinetic models of sorption: a theoretical analysis

The kinetics of sorption from a solution onto an adsorbent has been explored theoretically. The general analytical solution was obtained for two cases. It has been shown that at high initial concentration of solute (sorbate) the general equation converts to a pseudo-first-order model and at lower initial concentration of solute it converts to a pseudo-second-order model. In other words, the sorption process obeys pseudo-first-order kinetics at high initial concentration of solute, while it obeys pseudo-second-order kinetics model at lower initial concentration of solute. The theoretical results (derived equations) show that the observed rate constants of pseudo-first-order and pseudo-second-order models are combinations of adsorption and desorption rate constants and also initial concentration of solute. The obtained theoretical equations are used to correlate experimental data for sorption kinetics of some solutes on various sorbents. The predictions of the theory are in excellent agreement with the experimental data.     

Kinetics and Thermodynamics of the Adsorption of Copper(Ⅱ) onto a New Fe-Si Adsorbent

A new kind of Fe-Si adsorbent was synthesized by iron oxide and diatomite after calcining and hydrothermal process. The influences of the initial Cu2+ concentration, p H and adsorption time on the Cu2+ removal efficiency were discussed. Three adsorption empirical kinetics equations and two thermodynamics equations were used to simulate the adsorption process. The microstructures of newly developed copper removal materials and properties of copper removal are characterized in details by SEM and EDS. Adsorption mechanism of the adsorbent was discussed. The suitable p H value for Cu2+ removal is 5.0 to 6.0 and the adsorption capacity increases with increasing the initial Cu2+ concentration. The adsorption kinetics of the adsorbent could be better described by pseudo second order kinetic model, whereas the adsorption isotherms highly conform to the Freundlich equation. The main crystalline phase of the adsorbent is Fe(Si O3) which can build porous structures conducive to the Cu2+ adsorption.     

Modeling adsorption of colloids and proteins

Recent developments in the modeling of particle and protein adsorption kinetics on solid surfaces are discussed. Emphasis is focused on the coarse-grained methods, where protein molecules are treated as particles having a regular shape (spheres, spheroids) or a system of spherical beads of various sizes. Using such approaches hydrodynamic radii and diffusion coefficients of protein molecules are calculated in an exact way using the linear Stokes equation. Additionally, the surface blocking functions and jamming coverages for such molecule shapes are determined using the random sequential adsorption simulations. Theoretical results obtained in this way for various molecule shapes, including the bead models of fibrinogen are discussed. Knowing the jamming coverage and blocking functions one can formulate boundary conditions for bulk transport equations. Solutions of these equations for the convection and diffusion-controlled transport are presented. These theoretical predictions proved adequate for interpreting experimental data obtained for fibrinogen using AFM, ellipsometry and fluorescence methods. It is, therefore, concluded that these coarse grained approaches combined with solutions of the continuity equation can be efficiently used for quantitatively predicting protein adsorption kinetics for the time scale met under practical situations.     

自然水体生物膜吸附Co,Ni和Cu的特征研究

研究了在自然水体中培养的生物膜吸附Co,Ni和Cu等3种重金属的热力学和动力学特征,并对生物膜吸附各重金属的热力学数据进行了非线性拟合.结果表明,3种重金属的吸附过程均符合Langmuir吸附等温曲线.在溶液中重金属浓度<0.5μmol/L时,生物膜对3种重金属元素的吸附能力顺序是Co>Cu>Ni;在重金属浓度>0.5μmol/L时,顺序是Cu>Co>Ni.对动力学数据进行非线性拟合的结果表明,生物膜对Co,Ni和Cu的吸附均在数小时内达到平衡,吸附过程符合Langmuir等动力学曲线.