Multilayer isotherms

Because of their prevalence in physical adsorption studies on high-energy, powdered solids, type II isotherms are of considerable practical importance. Bmnauer, Emmett, and Teller (BET) [39] showed how to extent Langmuir s approach to multilayer adsorption, and their equation has come to be known as the BET equation. The derivation that follows is the traditional one, based on a detailed balancing of forward and reverse rates. 

The very considerable success of the BET equation stimulated various investigators to consider modifications of it that would correct certain approximations and give a better fit to type II isotherms. Thus if it is assumed that multilayer formation is limited to n layers, perhaps because of the opposing walls of a capillary being involved, one... 

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 relatively uniform success of these various plots suggests that, except as modified by changes in I, the shape of the isotherm in the multilayer region tends to be characteristic of the adsorbate and independent of the nature of the... 

The existence of this situation (for nonporous solids) explains why the ratio test discussed above and exemplified by the data in Table XVII-3 works so well. Essentially, any isotherm fitting data in the multilayer region must contain a parameter that will be found to be proportional to surface area. In fact, this observation explains the success of Ae point B method (as in Fig. XVII-7) and other single-point methods, since for any P/P value in the characteristic isotherm region, the measured n is related to the surface area of the solid by a proportionality constant that is independent of the nature of the solid. 

The multilayer isotherms illustrated thus far have all been of a continuous appearance—it was such isotherms that the BET, FHH, and other equations treated. About 30 years ago, however, multilayer adsorption on smooth sur-... 

As pointed out in Section XVII-8, agreement of a theoretical isotherm equation with data at one temperature is a necessary but quite insufficient test of the validity of the premises on which it was derived. Quite differently based models may yield equations that are experimentally indistinguishable and even algebraically identical. In the multilayer region, it turns out that in a number of cases the isotherm shape is relatively independent of the nature of the solid and that any equation fitting it can be used to obtain essentially the same relative surface areas for different solids, so that consistency of surface area determination does not provide a sensitive criterion either. 

There is little doubt that, at least with type II isotherms, we can tell the approximate point at which multilayer adsorption sets in. The concept of a two-dimensional phase seems relatively sterile as applied to multilayer adsorption, except insofar as such isotherm equations may be used as empirically convenient, since the thickness of the adsorbed film is not easily allowed to become variable. 

As a general rule, adsorbates above their critical temperatures do not give multilayer type isotherms. In such a situation, a porous absorbent behaves like any other, unless the pores are of molecular size, and at this point the distinction between adsorption and absorption dims. Below the critical temperature, multilayer formation is possible and capillary condensation can occur. These two aspects of the behavior of porous solids are discussed briefly in this section. Some lUPAC (International Union of Pure and Applied Chemistry) recommendations for the characterization of porous solids are given in Ref. 178. 

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. 

In considering isotherm models for chemisorption, it is important to remember the types of systems that are involved. As pointed out, conditions are generally such that physical adsorption is not important, nor is multilayer adsorption, in determining the equilibrium state, although the former especially can play a role in the kinetics of chemisorption. 

In the present study we try to obtain the isotherm equation in the form of a sum of the three terms Langmuir s, Henry s and multilayer adsorption, because it is the most convenient and is easily physically interpreted but, using more a realistic assumption. Namely, we take the partition functions as in the case of the isotherm of d Arcy and Watt [20], but assume that the value of V for the multilayer adsorption appearing in the (5) is equal to the sum of the number of adsorbed water molecules on the Langmuir s and Henry s sites ... 

Langmuir referred to the possibility that the evaporation-condensation mechanism could also apply to second and higher molecular layers, but the equation he derived for the isotherm was complex and has been little used. By adopting the Langmuir mechanism but introducing a number of simplifying assumptions Brunauer, Emmett and Teller in 1938 were able to arrive at their well known equation for multilayer adsorption, which has enjoyed widespread use ever since. 

It is also questionable how far the molecules in all layers after the first should be treated as completely equivalent.From Section 1.2 it follows that the interaction must diminish significantly as distance from the surface increases this falling-off is, indeed, the basis of Halsey s treatment for the multilayer region of the isotherm, which is dealt with in Section 2.11. 

A number of attempts have been made to modify the BET equation so as to obtain better agreement with the experimental isotherm data in the multilayer region. One of the most recent is that of Brunauer and his co-workers ... 

With nitrogen, the departure from spherical symmetry combined with the relatively strong quadrupole moment, leads to a blurring of the step-like character of the isotherm in the multilayer region (cf. Fig. 2.29(b)). 

When the film thickens beyond two or three molecular layers, the effect of surface structure is largely smoothed out. It should therefore be possible, as Hill and Halsey have argued, to analyse the isotherm in the multilayer region by reference to surface forces (Chapter 1), the partial molar entropy of the adsorbed film being taken as equal to that of the liquid adsorptive. By application of the 6-12 relation of Chapter 1 (with omission of the r" term as being negligible except at short distances) Hill was able to arrive at the isotherm equation... 

The validity of Equation (2.30) may be tested by plotting log log (p°/p) against log ( / ) (or, if more convenient, against log n) when a straight line of slope -s should be obtained for the multilayer region of the isotherm. In... 

The intercept on the adsorption axis, and also the value of c, diminishes as the amount of retained nonane increases (Table 4.7). The very high value of c (>10 ) for the starting material could in principle be explained by adsorption either in micropores or on active sites such as exposed Ti cations produced by dehydration but, as shown in earlier work, the latter kind of adsorption would result in isotherms of quite different shape, and can be ruled out. The negative intercept obtained with the 25°C-outgassed sample (Fig. 4.14 curve (D)) is a mathematical consequence of the reduced adsorption at low relative pressure which in expressed in the low c-value (c = 13). It is most probably accounted for by the presence of adsorbed nonane on the external surface which was not removed at 25°C but only at I50°C. (The Frenkel-Halsey-Hill exponent (p. 90) for the multilayer region of the 25°C-outgassed sample was only 1 -9 as compared with 2-61 for the standard rutile, and 2-38 for the 150°C-outgassed sample). 

Both Type III and Type V isotherms are characterized by convexity towards the relative pressure axis, commencing at the origin. In Ty )e III isotherms the convexity persists throughout their course (Fig. 5.1(a), whereas in Type V isotherms there is a point of inflection at fairly high relative pressure, often 0-5 or even higher, so that the isotherm bends over and reaches a plateau DE in the multilayer region of the isotherm (cf. Fig. 5.1 (b)) sometimes there is a final upward sweep near saturation pressure (see DE in Fig. 5.1(b)) attributable to adsorption in coarse mesopores and macropores. 

I (curve D). Thus the micropores had been able to enhance the adsorbent-adsorbate interaction sufficiently to replace monolayer-multilayer formation by micropore filling and thereby change the isotherm from being convex to being concave to the pressure axis. 

The first stage in the interpretation of a physisorption isotherm is to identify the isotherm type and hence the nature of the adsorption process(es) monolayer-multilayer adsorption, capillary condensation or micropore filling. If the isotherm exhibits low-pressure hysteresis (i.e. at p/p° < 0 4, with nitrogen at 77 K) the technique should be checked to establish the degree of accuracy and reproducibility of the measurements. In certain cases it is possible to relate the hysteresis loop to the morphology of the adsorbent (e.g. a Type B loop can be associated with slit-shaped pores or platey particles). 

Henry s law corresponds physically to the situation in which the adsorbed phase is so dilute that there is neither competition for surface sites nor any significant interaction between adsorbed molecules. At higher concentrations both of these effects become important and the form of the isotherm becomes more complex. The isotherms have been classified into five different types (9) (Eig. 4). Isotherms for a microporous adsorbent are generally of type I the more complex forms are associated with multilayer adsorption and capillary condensation. 

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. 

The classical isotherm for multilayer adsorption on a homogeneous, flat surface is the BET isotherm [Brunauer, Emmett, and Teller, J. Am. Chem. Soc., 60, 309 (1938)]... 

Brunauer further developed the Langmuir isotherm expression to include multilayer adsorption ... 

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],... 

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