The hysteresis loop

With many adsorbents a hysteresis loop occurs between the adsorption and desorption branches of the isotfierm (Hgure 3.1). This is due to capillary condensation augmenting multilayer adsorption at the pressures at which hysteresis is present, the radii of curvature being different during adsorption from the radii of curvature during desorption. Since the desorption branch is thermodynamically more stable than the adsorption branch it is usual to use the desorption branch for pore size determinatioa 

Fifteen shape groups of capillaries were analyzed by de Boer [2] from a consideration of five types of hysteresis loop which he designated Type A to Type E (Rgure 3.2). 

Type A. Both adsorption and desorption branches are steep at intermediate relative pressures. This type of isotherm is designated Type HI by x lUPAC [8]. These include tubular capillaries open at both ends tubular capillaries with slightly wider parts tubular capillaries of two main dimensions wide ink bottle jrares provided f/t w 2r tubular capillaries with one narrow part wedge shaped capillaries. lUPAC state that Type HI can te associated with agglomerates or compacts of uniform spheres in fairly regular array and hence to a narrow distribution of pore sizes. 

Type B. The adsorption branch is steep at saturation pressure, the desorption branch at intennediate relative pressure (lUPAC Type H3). These include open slit-shaped capillaries with parallel walls capillaries with very wide bodies and narrow short necks. lUPAC state that this type is observed with aggregates of plate-like particles giving rise to slit-shaped pores 

Type C. The adsorption branch is steep, at intermediate pressures the desorption branch is sloping. These are typical of a heterogeneous distribution of pores of some of the following shapes tapered or double tapered capillaries and wedge formed capillaries with closed sides and open ends. 

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It frequently happens that the micropore effect, the enhancement of interaction potential and the resultant adsorption, ceases to appear when the value of w (and the corresponding relative pressure) is still below the beginning of the hysteresis loop. Within recent years, the micropore range... 

The model proposed by Zsigmondy—which in broad terms is still accepted to-day—assumed that along the initial part of the isotherm (ABC of Fig. 3.1), adsorption is restricted to a thin layer on the walls, until at D (the inception of the hysteresis loop) capillary condensation commences in the finest pores. As the pressure is progressively increased, wider and wider pores are filled until at the saturation pressure the entire system is full of condensate. 

Examples are provided by the work of Carman and Raal with CF2CI2 on silica powder, of Zwietering" with nitrogen on silica spherules and of Kiselev" with hexane on carbon black and more recently of Gregg and Langford with nitrogen on alumina spherules compacted at a series of pressures. In all cases, a well defined Type II isotherm obtained with the loose powder became an equally well defined Type IV isotherm with the compact moreover both branches of the hysteresis loop were situated (drove the isotherm for the uncompacted powder, but the pre-hysteresis region was scarcely affected (cf. Fig. 3.4). The results of all these and similar... 

The hysteresis loops to be found in the literature are of various shapes. The classification originally put forward by de Boer S in 1958 has proved useful, but subsequent experience has shown that his Types C and D hardly ever occur in practice. Moreover in Type B the closure of the loop is never characterized by the vertical branch at saturation pressure, shown in the de Boer diagrams. In the revised classification presented in Fig. 3.5, therefore. Types C and D have been omitted and Type B redrawn at the high-pressure end. The designation E is so well established in the literature that it is retained here, despite the interruption in the sequence of lettering. 

In calculations of pore size from the Type IV isotherm by use of the Kelvin equation, the region of the isotherm involved is the hysteresis loop, since it is here that capillary condensation is occurring. Consequently there are two values of relative pressure for a given uptake, and the question presents itself as to what is the significance of each of the two values of r which would result from insertion of the two different values of relative pressure into Equation (3.20). Any answer to this question calls for a discussion of the origin of hysteresis, and this must be based on actual models of pore shape, since a purely thermodynamic approach cannot account for two positions of apparent equilibrium. 

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. 

Before proceeding to detail, however, it is necessary to consider the question as to which branch of the hysteresis loop—the adsorption or the desorption branch—should be used. Though the mode of calculation is... 

Fig. 3.19 Contrast between the pore size distribution curves based on the adsorption and the desorption branch of the hysteresis loop respectively.

The evidence obtained in compaction experiments is of particular interest in the present context. Figure 3.22 shows the results obtained by Avery and Ramsay for the isotherms of nitrogen on compacts of silica powder. The hysteresis loop moved progressively to the left as the compacting pressure increased, but the lower closure point did not fall below a relative pressure of 0-40. Similar results were obtained in the compaction of zirconia powder both by Avery and Ramsay (cf. Fig. 4.5), and by Gregg and Langford, where the lower closure point moved down to 0-42-0-45p° but not below. With a mesoporous magnesia (prepared by thermal decomposition of the hydrated carbonate) the position of the closure point... 

Thus the hysteresis loop should close at a relative pressure determined by the tensile strength of the liquid adsorptive, no matter whether the pore system extends to finer pores than those characterized by or not. 

Fig. 3.24 Test of the tensile strength hysteresis of hysteresis (Everett and Burgess ). TjT, is plotted against — Tq/Po where is the critical temperature and p.. the critical pressure, of the bulk adsorptive Tq is the tensile strength calculated from the lower closure point of the hysteresis loop. C), benzene O. xenon , 2-2 dimethyl benzene . nitrogen , 2,2,4-trimethylpentane , carbon dioxide 4 n-hexane. The lowest line was calculated from the van der Waals equation, the middle line from the van der Waals equation as modified by Guggenheim, and the upper line from the Berthelot equation. (Courtesy Everett.)...

It was noted earlier (p. 115) that the upward swing in the Type IV isotherm characteristic of capillary condensation not infrequently commences in the region prior to the lower closure point of the hysteresis loop. This feature can be detected by means of an a,-plot or a comparison plot (p. 100). Thus Fig. 3.25(a) shows the nitrogen isotherm and Fig. 3.25(h) the a,-plot for a particular silica gel the isotherm is clearly of Type IV and the closure point is situated around 0 4p° the a,-plot shows an upward swing commencing at a = 0-73, corresponding to relative pressures of 013 and therefore well below the closure point. 

Figure 3.26(a) and (h) gives results for nitrogen on a compact of silica. Curve (a) is a comparison plot in which the adsorption on the compact (ordinates) is plotted against that on the uncompacted powder (abscissae) there is a clear break followed by an increased slope, denoting enhanced adsorption on the compact, at p/p° = 0-15, far below the lower closure point ( 0-42) of the hysteresis loop in the isotherm (curve (b)). A second compact, prepared at 64 ton in" rather than 130 ton in", showed a break, not quite so sharp, at p/p° = 0-35. 

A difficulty in using the method is that of identifying the ptoint F, where capillary condensation commences. This is usually taken as the lower closure point of the loop but as was pointed out in Section 3.5, capillary condensation can occur without hysteresis if the pores are of an appropriate shape—such as wedge-like—before the irreversible condensation responsible for the hysteresis loop sets in. The uncertainty arising from this cause is considerable, since the curve of ln(p°/p) is very steep in this region (cf. Fig. 3.28). 

Fig. 3.28 The Kiselev method for calculation of specific surface from the Type IV isotherm of a compact of alumina powder prepared at 64 ton in". (a) Plot of log, (p7p) against n (showing the upper (n,) and lower (n,) limits of the hysteresis loop) for (i) the desorption branch, and (ii) the adsorption branch of the loop. Values of. 4(des) and /4(ads) are obtained from the area under curves (i) or (ii) respectively, between the limits II, and n,. (6) The relevant part of the isotherm.

Perhaps the most serious limitation, however, arises from the fact neither branch of the hysteresis loop corresponds to thermodynamic reversibility. The value of the integral J Va(p°jp)dn = /, say, differs considerably for the two branches of the loop. In fact... 

Low-pressure hysteresis is not confined to Type I isotherms, however, and is frequently superimposed on the conventional hysteresis loop of the Type IV isotherm. In the region below the shoulder of the hysteresis loop the desorption branch runs parallel to the adsorption curve, as in Fig. 4.26, and in Fig. 4.2S(fi) and (d). It is usually found that the low-pressure hysteresis does not appear unless the desorption run commences from a relative pressure which is above some threshold value. In the study of butane adsorbed on powdered graphite referred to in Fig. 3.23, for example, the isotherm was reversible so long as the relative pressure was confined to the branch below the shoulder F. 

In general, therefore, there are three processes, prior to the kind of capillary condensation associated with the hysteresis loop of a Type IV isotherm, which may occur in a porous body containing micropores along with mesoporesia primary process taking place in very narrow micropores a secondary, cooperative process, taking place in wider micropores, succeeded by a tertiary process governed by a modified Kelvin equation. 

The limits of pore size corresponding to each process will, of course, depend both on the pore geometry and the size of the adsorbate molecule. For slit-shaped pores the primary process will be expected to be limited to widths below la, and the secondary to widths between 2a and 5ff. For more complicated shapes such as interstices between small spheres, the equivalent diameter will be somewhat higher, because of the more effective overlap of adsorption fields from neighbouring parts of the pore walls. The tertiary process—the reversible capillary condensation—will not be able to occur at all in slits if the walls are exactly parallel in other pores, this condensation will take place in the region between 5hysteresis loop and in a pore system containing a variety of pore shapes, reversible capillary condensation occurs in such pores as have a suitable shape alongside the irreversible condensation in the main body of pores. 

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). 

The computation of mesopore size distribution is valid only if the isotherm is of Type IV. In view of the uncertainties inherent in the application of the Kelvin equation and the complexity of most pore systems, little is to be gained by recourse to an elaborate method of computation, and for most practical purposes the Roberts method (or an analogous procedure) is adequate—particularly in comparative studies. The decision as to which branch of the hysteresis loop to use in the calculation remains largely arbitrary. If the desorption branch is adopted (as appears to be favoured by most workers), it needs to be recognized that neither a Type B nor a Type E hysteresis loop is likely to yield a reliable estimate of pore size distribution, even for comparative purposes. 

The hysteresis loop, ia general, suppHes information about the magnetic properties such as Af, Af, preferred direction of the magnetisation or... 

The slope of the hysteresis loop in is also an important parameter. From this slope, the parameter S can be derived (17). In Figure 3 a part of the hysteresis loop (M as a function of the appHed field H) is given. The point at which M is constant as the function of the appHed field is defined as saturation magnetisation (M.). From the slope at can be written tan0 = = 1/ 1 — S ) ot dM/dH = — S ). Thus the S is defined in... 

For perm anen t-m agnet materials where the coercivity is large, the demagaetizatioa curve, which correspoads to the secoad quadrant of the hysteresis loop, sometimes is plotted as the polarization J(= B — vs H(B — H vs H) to show the intrinsic characteristics of the material. The value of... 

Fig. 1. Schematic of the hysteresis loop associated with a shape-memory alloy transformation, where M. and Afp correspond to the martensite start and finish temperatures, respectively, and and correspond to the start and finish of the reverse transformation of martensite, respectively. The physical property can be volume, length, electrical resistance, etc. On cooling the body-centered cubic (bcc) austenite (parent) transforms to an ordered B2 or E)02...

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