Cohan equation

The method devised by Barrett, Joyner, and Halenda (BJH) [35] is one of the earhest methods developed to address the pore size distribution of mesoporous sohds. This method assumes that adsorption in mesoporous solid (cylindrical pore is assumed) follows two sequential processes — building up of adsorbed layer on the surface followed by a capillary condensation process. Karnaukhov and Kiselev [45] accounted for the curvature in the first process, but Bonnetain et al. [46] found that this improvement has httle influence on the determination of pore size distribution. The second process is described by either the Cohan equation (for adsorption branch) or the Kelvin equation (for desorption branch). 

It should be noted that the derivation in Chapter 3 assumed a 3D form for the Kelvin-Cohan equation. In other words, the meniscus that is causing... 

The use of the Kelvin-Cohan equation does not necessarily imply that a liquid film, with a sharp gas-liquid interface exists before commencement of capillary filling. It only implies, that given other alternatives, that the lowest Gibbs free energy situation is for the sudden appearance of a 2D or 3D interface. Thus, a continuous correction for the surface tension before capillary filling may not be justified if the theory does not depend upon an interface before the transition. This is the case for most conventional calculations of capillary filling and x theory is in this respect conventional. 

This then should, according the BdB theory, yield the desorption branch. There is a very close resemblance between Eqs. (206) and (211) with the former containing the 2D form of the Kelvin-Cohan equation and the latter the 3D form. Notice that by L Hospital s rule as Z, right-hand side will approach F(t ) thus yielding the 3D form. 

It would be instructive to show some plots of the isotherm predicted by Eq. (206) to see what this equation means. Fig. 108 shows some plots in terms of moles adsorbed for a 2, 5 and 10 nm pore radius. This calculation uses the cc-s plot for the function F(t). At the points marked with a K the critical thickness is reached and the isotherm follows the dotted lines. The point of capillary filling as predicted by Eq. (208) and the amount of capillary filling are indicated by the dashed fines. Fig. 109 shows the dependance of t and P/P on the pore radius. A comparison of the BdB theory with the Kelvin-Cohan equation, both the 2D and 3D form, is shown in Fig. 110. 

As may be observed in Fig. 5, the simpler theory (the KELVIN equation for desorption, the COHAN equation for adsorption) gives the best, although not perfect, fit. 

Thus, as pointed out by Cohan who first suggested this model, condensation and evaporation occur at difi erent relative pressures and there is hysteresis. The value of r calculated by the standard Kelvin equation (3.20) for a given uptake, will be equal to the core radius r,. if the desorption branch of the hysteresis loop is used, but equal to twice the core radius if the adsorption branch is used. The two values of should, of course, be the same in practice this is rarely found to be so. 

Fig. 104.—Tension r at O = 1.5 for GR-S synthetic rubber containing various proportions of calcium carbonate (particle diameter 3900 mju), but vulcanized under otherwise identical conditions. The solid curve has been calculated according to Ed. (52) the broken curve by neglecting the third term in this equation. (Cohan. s)...

According to the classical treatment of Cohan [8], which is the basis of the conventional BJH method [14], capillary condensation in an infinite cylindrical pore is described by the Kelvin equation using cylindrical meniscus, while desorption is associated with spherical meniscus. In large pores the following asymptotic equation is expected to be valid [8] Pd/Po = (PA/P0)2, where Pd/Po and Pa/Po are the relative pressures of the desorption and adsorption, respectively. An improved treatment [9-11, 13], originated from Deijaguin [9], takes into... 

Figure 2. Capillary hysteresis of nitrogen in cylindrical pores at 77 K. Equilibrium desorption (black squares) and spinodal condensation (open squares) pressures predicted by the NLDFT in comparison with the results of Cohan s equation (the BJH method) for spherical (crosses and line) and cylindrical (line) meniscus.

Although the Kelvin equation (3) has been confirmed i for small water drops by the Millikan method ( 2.VII F), the results with other liquids are very puzzling and have not been satisfactorily explained they seem to indicate a value of or about 100 times that for a flat surface, but. this seems likely to be due to experimental errors. The effect of electric charge on the drops seems improbable, since Shereshefsky found a similar result with capillary tubes. Cohan and Mayer, with capillaries of 2 radius, giving a rise of 255 cm., found, however, that both a and density were normal for water and toluene. 

Attempts at imderstanding adsorption hysteresis have a long history (Everett, 1967 Steele, 1973 Gregg and Sing, 1982). An important e2U ly contribution was made by Cohan (1938) who applied the Kelvin equation to adsorption in pores. Cohan suggested that the occurrence of hysteresis in a single pore is related to differences in the geometry of the Uquid-vapor meniscus in condensation and evaporation. 

Historically, however, considerable attention has been given to corrections to the Kelvin equation arising from the thickness of adsorbed layer and the dependence of surface tension on curvature of interface. The first problem was initially considered as monolayen by Foster [48] and more recently as a function of equHibrium pressure of the system by Cohan [49], Degaguin [50], Foster [51], and Brockhoff and de Boer [52, 53]. The initial approaches of Foster and Cohan... 

In the mid-1940 s Guth (1945) and Smallwood (1944) developed a widely employed (Cohan, 1947 Kraus, 1965c, Chapter 4) equation expressing rubber reinforcement directly in terms of filler concentration. An important form of the equation can be written as... 

The capillary condensation equation for adsorption (eq. 3.9-7) can also be obtained from the free energy argument of Cohan discussed in the following section. 

An alternative view was suggested by Cohan who suggested that capillary condensation occurs along both adsorption and desorption branches of the isotherm, the difference being due to a difference in the shape of the meniscus. During adsorption the pore fills radially and a cylindrical meniscus is formed as sketched in Figure 2.12a. Under these conditions dv/ds = r-(rather than jl as assumed in the Kelvin equation) and with 0 = 0 ... 

The interpretation of the hysteresis loop is a matter of some current discussion. The primary explanation is based upon the Kelvin equation as modified by Cohan [22]. which is ... 

The capillary filling equation theory, that is the Kelvin equation as modified by Cohan [28], can be expressed for cylindrical pores as... 

Nearly all of the analysis of mesoporosity starts with the Kelvin-Cohan [14] formulation. Foster [15] proposed the Kelvin equation for the effect of vapor pressure on capillary rise but did not anticipate its use for very small capillaries where the adsorbate thickness is a significant geometrical perturbation. Cohan formulation subtracts the adsorbate film thickness from the radius of the pore to yield the modified Kelvin equation... 

The enhanced filling associated with mesoporosity is dependent on all cases upon the Kelvin equation in some way. Some of the theories, such as the original Cohan formulation or the BdB theory, assume a fully formed liquid film with a sharp liquid-gas interface. 

The physical adsorption isotherm in the p/po range for multilayer adsorption usually shows a hysteresis on porous solids, as indicated in Figure 1.17. Although several theories have been put forward to explain the hysteresis, it is not yet totally resolved. However, the theory advanced by Cohan (1944) is helpful in understanding why the desorption isotherm is used for the determination of pore size distributions. He argues that, upon adsorption, pores are not filled vertically but rather radially. On desorption, on the other hand, pores are emptied vertically, for which the Kelvin equation of Eq. 1.55 applies. 

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