Theory multilayer adsorption

BET method. The most commonly used method for determining the specific surface area is the so-called BET method, which obtained its name from three Nobel prize winners Brunauer, Emmett and Teller (1938). It is a modification of the Langmuir theory, which, besides monolayer adsorption, also considers multilayer adsorption. The equation allows easy calculation of the surface area, commonly referred to as the BET surface area ( bet). From the isotherms also pore-radii and pore-volumes can be calculated (from classical equation for condensation in the pores). 

Conventional bulk measurements of adsorption are performed by determining the amount of gas adsorbed at equilibrium as a function of pressure, at a constant temperature [23-25], These bulk adsorption isotherms are commonly analyzed using a kinetic theory for multilayer adsorption developed in 1938 by Brunauer, Emmett and Teller (the BET Theory) [23]. BET adsorption isotherms are a common material science technique for surface area analysis of porous solids, and also permit calculation of adsorption energy and fractional surface coverage. While more advanced analysis methods, such as Density Functional Theory, have been developed in recent years, BET remains a mainstay of material science, and is the recommended method for the experimental measurement of pore surface area. This is largely due to the clear physical meaning of its principal assumptions, and its ability to handle the primary effects of adsorbate-adsorbate and adsorbate-substrate interactions. 

A major advance in adsorption theory generalized the treatment of monolayer adsorption and incorporated the concept of multilayer adsorption. This is known as the BET theory after Brunauer, Emmett and Teller [32], The adsorption of a gas on a solid surface can be described by... 

The other molecular probe method is the single-probe method (SP method), which is separately proposed by Avnir and Jaroniec,93 and Pfeifer et al.108-112 In the SP method, a single adsorption isotherm is analyzed using a modified FHH theory. The FHH model was developed independently by Frenkel,113 Halsey,114 and Hill,115 and describes the multilayer adsorption coverage. Since the SP method uses only one probe molecule, this method is more convenient than the MP method. However, there are many theoretical limitations in applying the SP method to determination of the surface fractal dimension. Therefore, it is really necessary to discuss whether the SP method is an adequate tool to investigate the surface fractal dimension or not before applying the SP method to certain system. 

Brunauer, Emmett and Teller, in 1938, extended Langmuir s kinetic theory to multilayer adsorption. The BET theory assumes that the uppermost molecules in adsorbed stacks are in dynamic equilibrium with the vapor. This means that, where the surface is covered with only one layer of adsorbate, an equilibrium exists between that layer and the vapor, and where two layers are adsorbed, the upper layer is in equilibrium with the vapor, and so forth. Since the equilibrium is dynamic, the actual location of the surface sites covered by one, two or more layers may vary but the number of molecules in each layer will remain constant. 

As noted above, the range of pressures over which gas adsorption studies are conducted extends from zero to the normal vapor pressure of the adsorbed species p0. An adsorbed layer on a small particle may readily be seen as a potential nucleation center for phase separation at p0. Thus at the upper limit of the pressure range, adsorption and liquefaction appear to converge. At very low pressures it is plausible to restrict the adsorbed molecules to a mono-layer. At the upper limit, however, the imminence of liquefaction suggests that the adsorbed molecules may be more than one layer thick. There is a good deal of evidence supporting the idea that multilayer adsorption is a very common form of physical adsorption on nonporous solids. In this section we are primarily concerned with an adsorption isotherm derived by Brunauer, Emmett, and Teller in 1938 the theory and final equation are invariably known by the initials of the authors BET. 

Usually adsorption is more realistically described by the BET model. BET theory accounts for multilayer adsorption. The adsorption isotherm goes to infinity at relative partial pressures close to one, which corresponds to condensation. 

The theory of Brunauer, Emmett and Teller167 is an extension of the Langmuir treatment to allow for multilayer adsorption on non-porous solid surfaces. The BET equation is derived by balancing the rates of evaporation and condensation for the various adsorbed molecular layers, and is based on the simplifying assumption that a characteristic heat of adsorption A Hi applies to the first monolayer, while the heat of liquefaction, AHL, of the vapour in question applies to adsorption in the second and subsequent molecular layers. The equation is usually written in the form... 

Multilayer adsorption models have been used by Asada [147,148] to account for the zero-order desorption kinetics. The two layers are equilibrated. Desorption goes from the rarefied phase only. This model has been generalized [148] for an arbitrary number of layers. The filling of the upper layer was studied with allowance for the three neighboring molecules being located in the lower one. The desorption frequency factor (CM) was regarded as being independent of the layer number. The theory has been correlated with experiment for the Xe/CO/W system [149]. Analysis of the two-layer model has been continued in Ref. [150], to see how the ratios of the adspecies flows from the rarefied phases of the first and the second layers vary if the frequency factors for the adspecies of the individual layers differ from one another. In the thermodynamic equilibrium conditions these flows were found to be the same at different ratios of the above factors. 

The BET theory, developed by Brunauer, Emmett, and Teller [26], is based on the kinetic model of adsorption proposed by Langmuir [24,25] and was extended to describe the multilayer adsorption by the introduction of some assumptions listed below ... 

The method as a rule used for the determination of the specific surface of a material is the Brunauer-Emmet-Teller (BET) method [2,4,5], The BET theory of multilayer adsorption for the calculation of specific surface area, S, was originally developed by Brunauer, Emmett, and Teller [2,4,5], The adsorption process, within the frame of the BET theory, is considered as a layer-by-layer process. In addition, an energetically homogeneous surface is assumed so that the adsorption field is the same in any site within the surface. Additionally, the adsorption process is considered to be immobile, that is, each molecule is adsorbed in a concrete adsorption site in the surface. Subsequently, the first layer of adsorbed molecules has an energy of interaction with the adsorption field, and a vertical interaction between molecules after the first layer,, is explicitly analogous to the liquefaction heat of the adsorbate. Besides, adsorbed molecules do not interact laterally. 

In recent years a radical change has been taking place in the interpretation of the Type I isotherm for porous adsorbents. According to the classical Langmuir theory, the limiting adsorption w (at the plateau) represents completion of the monolayer and may therefore be used for the calculation of the surface area. The alternative view, which is now widely accepted, is that the initial (steep) part of the Type I isotherm represents micropore filling (rather than surface coverage) and that the low slope of the plateau is due to multilayer adsorption on the small external area. 

It has always seemed appropriate to this writer that the elbow in the adsorption isotherms of nitrogen on finely divided materials at liquid air temperatures should have been called Point B, because it was the outcome of a brilliant series of experimental measurements of such isotherms by the man we honor today. It was correctly identified as the approximate position of the completion of a mono-layer of adsorption on the surface of the material and the beginning of multilayer adsorption. It is now common knowledge to students of even elementary physical chemistry that, in 1938, a theory of adsorption of gases in multimolecular layers was published. Point B was merged into the BET equation. The Washington Section of the American Chemical Society made him the Hillebrand prize winner in 1945 for this work. 

It would be difficult to overestimate the historical importance of the BET theory since for over SO years it has attracted an enormous amount of attention (Davis, 1991). Indeed, the BET method is now accepted as a standard procedure for the determination of the surface area of a wide range of fine powders and porous materials. On the other hand, it is generally recognized that the theory is based on an oversimplified model of multilayer adsorption and that the reliability of the BET method is questionable unless certain conditions are fulfilled. 

By introducing a number of simplifying assumptions, Brunauer, Emmett and Teller (1938) were able to extend the Langmuir mechanism to multilayer adsorption and obtain an isotherm equation (the BET equation), which has Type II character. The original BET treatment involved an extension of the Langmuir kinetic theory of monomolecular adsorption to the formation of an infinite number of adsorbed layers. 

BET surface area the surface area calculated from the Brunauer, Emmett, and Teller theory of multilayer adsorption of a gas on a solid surface. 

Particle Surface Area Determination Methods From the standard definitions of particle surface area, it can be seen that various determination methods are used for surface area measurement, such as adsorption (including Langmuir s equation for monolayer adsorption and the BET equation for multilayer adsorption), particle size distribution, and permeability methods. The different methods are rarely in agreement because the value obtained depends upon the procedures used and also on the assumptions made in the theory relating the surface area to the phenomena measured. The most common methods used for measuring particle surface area are described below. 

Brunauer, Emmett, and Teller were the first to propose a theory for multilayer adsorption (BET theory). Since the behavior of adsorbed molecules is even more difficult to describe in detail than that of molecules in the liquid state, the BET theory contains some rather drastic assumptions. In spite of this, it is still a generally useful theory of physical adsorption. The BET theory gives a correct semiquantitative description of the shape of the isotherm and provides a good means of evaluating (which is then used to estimate the surface area of the soM). 

Water vapor adsorption isotherms have been obtained on cotton from room temperature up to 150°C [303,304]. Theoretical models for explaining the water vapor sorption isotherms of cellulose have been reviewed [303]. Only adsorption theories will be discussed here at ambient temperatures. The shape of the isotherm indicates that multilayer adsorption occurs and thus the Brunauer, Emmett and Teller (BET) or the Guggenheim, Anderson and deBoer (GAB) theory can be applied. In fact, the BET equation can only be applied at relative vapor pressures (RVPs) below 0.5 and after modification up to a RVP of 0.8 [305]. The GAB equation, which was not discussed in the chapter in the book Cellulose Chemistry and Its Applications [303], can be applied up to RVPs above 0.9 [306]. Initially as the RVP... 

Multilayer adsorption on heterogeneous surfaces has been analysed in some detail by Jaronlec et al. ). A statistical multilayer theory by Ash et al. ) also deserves special attention. Here the authors considered mixtures of monomers... 

It is easy to see that adsorption energies are dependent on the curvature of the interface. Consider first adsorption on a planar interface. At low pressures, p, a sub-monolayer, gas-like, and eventually a two-dimensional liquid described by a Langmuir isotherm (or decorations thereof) forms. At higher pressures still (p/ps>0.35, where ps is the saturated vapour pressure) multilayer adsorption isotherms can occur depending on adsorbate, molecular size and adsorbate-substrate interactions. This regime is usually described by the theory of Brxmauer-Emmet-Teller (BET). In this domain, ln(p/pg) = 1/t, where t is the thickness of the film. 

These assumptions are justifiable as the heat of adsorption of the small inert sorbate (e.g., N2 or Ar) is rather low and, hence, differences between sorption sites at the surface will be very small. Similarly, the interaction between the first and the following layers will be close to the heat of condensation, as the effect of polarization by the surface will be small beyond the first layer (screening of the long-range van der Waals forces). From its conception, the BET theory extends the Langmuir model to multilayer adsorption. It postulates that under dynamic equilibrium conditions the rate of adsorption in each layer is equal to the rate of desorption from that layer. Molecules in the first layer are located on sites of constant interaction strength and the molecules in that layer serve as sorption sites for the second layer and so forth. The surface is, therefore, composed of stacks of sorbed molecules. Lateral interactions are assumed to be absent. With these simplifications one arrives at the BET equation... 

Gas-solid equilibria have been studied for over 200 years, since Fontana showed that activated charcoal adsorbs gases and vapors at room temperature [1]. A considerable amoxmt of theoretical and experimental literature is available. The Gibbs isotherm [2] and the multilayer adsorption theory of Brunauer, Emmett and Teller [3], provide serious theoretical guidelines and support in understanding the results of experimental studies. Although, gas-sohd isotherms are difficult to predict quantitatively [4], this branch of adsorption thermod3mamics is much easier than liquid-solid adsorption because of the relative simplicity of the gas-sohd interface as compared to the liquid-solid interface. The Gibbs equation relates the amoimt of a compoimd adsorbed per unit surface area of a hquid-gas or a hquid-hquid interface and the surface or interfacial tensions [2]. This relationship provides a useful theoretical framework. 

Brunauer-Emmett-Teller (BET) [20], Langmuir isotherm theory is applied when a monolayer of molecules is sorbed on a homogeneous surface where each site requires the same energy to adsorb molecules. The Freundlich isotherm also assumes a monolayer, but allows for an exponential distribution of adsorption energies. BET isotherms provide for multilayer adsorption. 

According to a multilayer adsorption theory developed by Champion and Halsey such adsorption on a homogeneous surface leads inevitably to step like isotherms and isotherm continuity should be explained by surface heterogeneity [48]. Differences in the relationship course q°( = f(%PEG) between non-polar hexane, polar chloroform, 5r-complexing benzene, acetone and methanol which are capable of hydrogen bonds formation become comprehensible in view of the fact that specific and non-specific interactions, having different adsorption energies, take part in the adsorption process. Formation of distinct extreme on hexane heat of adsorption curve can be explained by hexane non polarity... 

Shekhovtsova and Fomkin [41] developed a discrete site model to describe the adsorption of methane on microporous adsorbents. The model was tested with adsorption on zeoUte NaX and an activated carbon. A sharp decrease in the heats of adsorption was observed at high adsorbed amounts even in the supercritical temperature range. A multilayer adsorption theory was developed by Wang and Hwang [42] to describe the behavior of several adsorbates on activated carbons. The adsorbates employed included several alkanes, hydrogen sulfide, and carbon monoxide. The isosteric heats of adsorption for all gases were determined using the Clausius-Clapeyron equation. 

Rowley, L.A., Nicholson, D., and Parsonage, N.G. (1976). Grand ensemble Monte Carlo studies of physical adsorption II. The stmcture of the adsorbate. Critique of theories of multilayer adsorption for 12-6 argon on a plane homogeneous solid. Mol. Phys., 31, 389-407. 

The adsorption isotherms provide information about the surface properties of the fibres or paper in the column. For example, the surface area in the column may be deduced from the shape of the isotherm according to the Brunauer-Emmett-Teller theory for multilayer adsorption (13, lJO. This theory gives the number of molecules, Nm, which forms a layer one molecule thick on the surface of the adsorbent. If the area, am, occupied by each vapour molecule adsorbed on the surface is known, then the total surface area, A, is simply given by... 

The total surface area is calculated from the amount of physical adsorption of nitrogen at 77 K. During the thirties Brunauer, Emmett, and Teller [1,2] presented a theory dealing with the multilayer adsorption of gases on solids. They assumed that the first layer of gas molecules is adsorbed more strongly than subsequent layers, and that the heat of adsorption of subsequent layers is constant. They also assumed the absence of lateral interaction between adsorbed molecules. On the basis of these much criticized assumptions they derived an adsorption isotherm, which describes the experimentally determined adsorption isotherms excellently. From the adsorption isotherm a value corresponding to the volume of the adsorbed monolayer is calculated. With physical adsorption the amount of gas adsorbed is usually plotted as a function of the relative pressure, that is the pressure... 

In Secs. II and III we shall summarize the present state of the theory of physical adsorption, starting with the simplest case of essentially independent molecules on a surface (very low equilibrium gas pressure) and then considering monolayer adsorption with interactions, and finally multilayer adsorption. In Sec. IV, the thermodynamics of adsorption will be discussed. 

Now even in the completely idealized case of an adsorbent which is mathematically plane and uniform the theory of multilayer adsorption is... 

---