Isotherms multilayer adsorption

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

The adsorption isotherms are often Langmuirian in type (under conditions such that multilayer formation is not likely), and in the case of zeolites, both n and b vary with the cation present. At higher pressures, capillary condensation typically occurs, as discussed in the next section. Some N2 isotherms for M41S materials are shown in Fig. XVII-27 they are Langmuirian below P/P of about 0.2. In the case of a microporous carbon (prepared by carbonizing olive pits), the isotherms for He at 4.2 K and for N2 at 77 K were similar and Langmuirlike up to P/P near unity, but were fit to a modified Dubninin-Radushkevich (DR) equation (see Eq. XVII-75) to estimate micropore sizes around 40 A [186]. 

Sing (see Ref. 207 and earlier papers) developed a modification of the de Boer r-plot idea. The latter rests on the observation of a characteristic isotherm (Section XVII-9), that is, on the conclusion that the adsorption isotherm is independent of the adsorbent in the multilayer region. Sing recognized that there were differences for different adsorbents, and used an appropriate standard isotherm for each system, the standard isotherm being for a nonporous adsorbent of composition similar to that of the porous one being studied. He then defined a quantity = n/nx)s where nx is the amount adsorbed by the nonporous reference material at the selected P/P. The values are used to correct pore radii for multilayer adsorption in much the same manner as with de Boer. Lecloux and Pirard [208] have discussed further the use of standard isotherms. 

Surface areas are deterrnined routinely and exactiy from measurements of the amount of physically adsorbed, physisorbed, nitrogen. Physical adsorption is a process akin to condensation the adsorbed molecules interact weakly with the surface and multilayers form. The standard interpretation of nitrogen adsorption data is based on the BET model (45), which accounts for multilayer adsorption. From a measured adsorption isotherm and the known area of an adsorbed N2 molecule, taken to be 0.162 nm, the surface area of the soHd is calculated (see Adsorption). 

Adsorption of dispersants at the soHd—Hquid interface from solution is normally measured by changes in the concentration of the dispersant after adsorption has occurred, and plotted as an adsorption isotherm. A classification system of adsorption isotherms has been developed to identify the mechanisms that may be operating, such as monolayer vs multilayer adsorption, and chemisorption vs physical adsorption (8). For moderate to high mol wt polymeric dispersants, the low energy (equiUbrium) configurations of the adsorbed layer are typically about 3—30 nm thick. Normally, the adsorption is monolayer, since the thickness of the first layer significantly reduces attraction for a second layer, unless the polymer is very low mol wt or adsorbs by being nearly immiscible with the solvent. 

Fig. 17 shows the adsorption isotherms of all (undimerized and dimerized) particles. Except for a very fast increase of adsorption connected with filling of the first adlayer, the adsorption isotherm for the system A3 is quite smooth. The step at p/k T 0.28 corresponds to building up of the multilayer structure. The most significant change in the shape of the adsorption isotherm for the system 10, in comparison with the system A3, is the presence of a jump discontinuity at p/k T = 0.0099. Inspection of the density profiles attributes this jump to the prewetting transition in the... 

Adsorption phenomena from solutions onto sohd surfaces have been one of the important subjects in colloid and surface chemistry. Sophisticated application of adsorption has been demonstrated recently in the formation of self-assembhng monolayers and multilayers on various substrates [4,7], However, only a limited number of researchers have been devoted to the study of adsorption in binary hquid systems. The adsorption isotherm and colloidal stabihty measmement have been the main tools for these studies. The molecular level of characterization is needed to elucidate the phenomenon. We have employed the combination of smface forces measmement and Fomier transform infrared spectroscopy in attenuated total reflection (FTIR-ATR) to study the preferential (selective) adsorption of alcohol (methanol, ethanol, and propanol) onto glass surfaces from their binary mixtures with cyclohexane. Om studies have demonstrated the cluster formation of alcohol adsorbed on the surfaces and the long-range attraction associated with such adsorption. We may call these clusters macroclusters, because the thickness of the adsorbed alcohol layer is about 15 mn, which is quite large compared to the size of the alcohol. The following describes the results for the ethanol-cycohexane mixtures [10],... 

The principle underlying surface area measurements is simple physisorb an inert gas such as argon or nitrogen and determine how many molecules are needed to form a complete monolayer. As, for example, the N2 molecule occupies 0.162 nm at 77 K, the total surface area follows directly. Although this sounds straightforward, in practice molecules may adsorb beyond the monolayer to form multilayers. In addition, the molecules may condense in small pores. In fact, the narrower the pores, the easier N2 will condense in them. This phenomenon of capillary pore condensation, as described by the Kelvin equation, can be used to determine the types of pores and their size distribution inside a system. But first we need to know more about adsorption isotherms of physisorbed species. Thus, we will derive the isotherm of Brunauer Emmett and Teller, usually called BET isotherm. 

Figure 5.19 shows an idealized form of the adsorption isotherm for physisorption on a nonporous or macroporous solid. At low pressures the surface is only partially occupied by the gas, until at higher pressures (point B on the curve) the monolayer is filled and the isotherm reaches a plateau. This part of the isotherm, from zero pressures to the point B, is equivalent to the Langmuir isotherm. At higher pressures a second layer starts to form, followed by unrestricted multilayer formation, which is in fact equivalent to condensation, i.e. formation of a liquid layer. In the jargon of physisorption (approved by lUPAC) this is a Type II adsorption isotherm. If a system contains predominantly micropores, i.e. a zeolite or an ultrahigh surface area carbon (>1000 m g ), multilayer formation is limited by the size of the pores. 

In this chapter, we are going to show that using the one- and the two-component multilayer adsorption isotherm models or the models taking into the account lateral interactions among the molecules in the monolayer (discussed in Section 2.1), the overload peak profiles presented in Section 2.4 can be qualitatively modeled. 

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. 

Of special interest in liquid dispersions are the surface-active agents that tend to accumulate at air/ liquid, liquid/liquid, and/or solid/liquid interfaces. Surfactants can arrange themselves to form a coherent film surrounding the dispersed droplets (in emulsions) or suspended particles (in suspensions). This process is an oriented physical adsorption. Adsorption at the interface tends to increase with increasing thermodynamic activity of the surfactant in solution until a complete monolayer is formed at the interface or until the active sites are saturated with surfactant molecules. Also, a multilayer of adsorbed surfactant molecules may occur, resulting in more complex adsorption isotherms. 

PVA and TaM -for the 88%-hydrolyzed PVA. The same dependence was found for the adsorbed layer thickness measured by viscosity and photon correlation spectroscopy. Extension of the adsorption isotherms to higher concentrations gave a second rise in surface concentration, which was attributed to multilayer adsorption and incipient phase separation at the interface. The latex particle size had no effect on the adsorption density however, the thickness of the adsorbed layer increased with increasing particle size, which was attributed to changes in the configuration of the adsorbed polymer molecules. The electrolyte stability of the bare and PVA-covered particles showed that the bare particles coagulated in the primary minimum and the PVA-covered particles flocculated in the secondary minimum and the larger particles were less stable than the smaller particles. 

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. 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

Figure 10.2. Types of adsorption isotherms a) Langmuir type (Z ) physisorption, multilayer.

Adsorption isotherms for n-decylamine on Ni, Fe, Cu, Pb, and Pt at the potential of maximum adsorption are shown in Figure 10.4. It is seen that a limiting coverage is approached in each case except on Pt, where multilayer formation occurs. The coverage 0 in this case is defined as... 

For multilayer adsorption of gases on a solid, the B.E.T. adsorption isotherm can be written in a slightly different notation as... 

It is convenient to divide the extent of adsorption into three categories submonolayer, monolayer, and multilayer. We discuss them in this order. The thermodynamics of adsorption may be developed around experimental isotherms or around calorimetric data. We begin with the definition of adsorption isotherms and how they are determined experimentally (Section 9.2). 

Until now, we have focused our attention on those adsorption isotherms that show a saturation limit, an effect usually associated with monolayer coverage. We have seen two ways of arriving at equations that describe such adsorption from the two-dimensional equation of state via the Gibbs equation or from the partition function via statistical thermodynamics. Before we turn our attention to multilayer adsorption, we introduce a third method for the derivation of isotherms, a kinetic approach, since this is the approach adopted in the derivation of the multilayer, BET adsorption isotherm discussed in Section 9.5. We introduce this approach using the Langmuir isotherm as this would be useful in appreciating the common features of (and the differences between) the Langmuir and BET isotherms. 

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. 

It is easy to see that the BET adsorption isotherm has the correct limits at very high [A] and when multilayer adsorption is negligible. First, consider the case where the pressure of A approaches the value for saturated vapor pressure of A in equilibrium with the liquid. Let the corresponding concentration be designated [A]sa/. The vapor/liquid equilibrium process is written... 

Now consider the form of the BET adsorption isotherm written in Eq. 11.59. If multilayer adsorption were not possible, then Km would be zero. The adsorbed site fraction from Eq. 11.59 becomes... 

Equation 1 can be used to determine the pore diameter of an MCM-41 sample which exhibits capillary condensation at a certain relative pressure, or to determine the capillary condensation pressure for an MCM-41 sample of a certain pore diameter. To construct model adsorption isotherms for MCM-41, one also needs a description of the monolayer-multilayer formation on the pore walls. This description can be based on the experimental finding that the statistical film thickness in MCM-41 pores of different sizes (especially above 3 nm) is relatively constant for pressures sufficiently lower from those of the capillaiy condensation and can be adequately approximated by the t-curve for a suitable reference silica [29-31], for instance that reported in Ref. 35. In these studies [29-31], the statistical film thickness in MCM-41 pores, tMcM-4i, was calculated according to the following equation [29] ... 

Kuge and Yoshikawa (3) related a change in the gas chromatographic peak shape to the beginning of multilayer adsorption on the surface of the solid. For small adsorbate volumes, the peak shape is symmetrical. As the adsorbate volume is increased, a sharp front, diffuse tail, and a defect at the front of the peak top is observed (Figure 11.2). It then acquires a diffuse front and sharp tail. This point corresponds to the B point of the BET Type II adsorption isotherm at which the relative surface area may be calculated. 

Figure 9.4 Left Adsorption isotherm for benzene (CeLL) adsorbing to graphitized thermal blacks at 20° C. The insert shows the adsorption isotherm for low coverages in more detail. Dotted lines indicate mono- or multilayer coverages at multiples of 4.12 //mol/m2. The equilibrium vapor pressure of benzene at 20°C is Po = 10.2 kPa. Right Differential heat of adsorption versus adsorbed amount. The dashed line corresponds to the heat of condensation of bulk benzene. Redrawn after Ref. [369].

In Langmuir model, the maximal adsorption is that of a monolayer. Langmuir adsorption isotherms all saturate at high vapor pressures. This is unrealistic for many cases. In order to consider the adsorption of multilayers, Brunauer, Emmett, and Teller extended the Langmuir theory and derived the so-called BET adsorption isotherm [378], The basic idea in the BET theory was to assume a Langmuir adsorption for each of the layers (Fig. 9.8). 

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. 

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