Aggregation isotherm

As the theoretical models are comparatively complex, only numerical methods allow to interpret experimental data. A software package is available that allows to make model calculations for any type of the above discussed diffusion-controlled mechanism [223]. In addition to the theory for a Langmuir isotherm, where the collocation solution by Ziller and Miller can serve as analytical solution, the programme gives access also to calculations based on the Frumkin, the reorientation and aggregation isotherms. 

The analysis of the kinetic data was performed on the basis of the diffusion-controlled model, using the Langmuir and the aggregation isotherm, given by Eq. (2.16) and Eqs. (2.107) -(2.111), respectively. As one can see, the agreement with the theory is not satisfactory. The models developed mainly by the Bulgarian school [33] requires extensive numerical calculations so that its application to experimental data will be possible only after the elaboration of effective computer programmes. 

Figure 13. Dynamic surface pressure for an aggregation isotherm with Cq= 510 mol/cm, ...

An interesting example of a large specific surface which is wholly external in nature is provided by a dispersed aerosol composed of fine particles free of cracks and fissures. As soon as the aerosol settles out, of course, its particles come into contact with one another and form aggregates but if the particles are spherical, more particularly if the material is hard, the particle-to-particle contacts will be very small in area the interparticulate junctions will then be so weak that many of them will become broken apart during mechanical handling, or be prized open by the film of adsorbate during an adsorption experiment. In favourable cases the flocculated specimen may have so open a structure that it behaves, as far as its adsorptive properties are concerned, as a completely non-porous material. Solids of this kind are of importance because of their relevance to standard adsorption isotherms (cf. Section 2.12) which play a fundamental role in procedures for the evaluation of specific surface area and pore size distribution by adsorption methods. 

Figure 20 shows the plot of the surface tension vs. the logarithm of the concentration (or-lg c-isotherms) of sodium alkanesulfonates C,0-C15 at 45°C. In accordance with the general behavior of surfactants, the interfacial activity increases with growing chain length. The critical micelle concentration (cM) is shifted to lower concentration values. The typical surface tension at cM is between 38 and 33 mN/m. The ammonium alkanesulfonates show similar behavior, though their solubility is much better. The impact of the counterions is twofold First, a more polarizable counterion lowers the cM value (Fig. 21), while the aggregation number of the micelles rises. Second, polarizable and hydrophobic counterions, such as n-propyl- or isopropylammonium and n-butylammonium ions, enhance the interfacial activity as well (Fig. 22). Hydrophilic counterions such as 2-hydroxyethylammonium have the opposite effect. Table 14 summarizes some data for the dodecane 1-sulfonates.

Nevertheless, the shape of the isotherm in Fignre 3 is qnite similar to those of molecular films in their solid condensed state. This reflects the strong tendency of the particles to aggregate at the air-water interface. Visnal inspection dnring spreading indicates the for-... 

Low-frequency conductivity data [37] obtained along this 45°C isotherm are illustrated in Fig 2. The initial oscillatory variation in the conductivity for a > 0.9 can be assigned to variations in AOT partitioning among dimers and other low aggregates and reverse micelles, as reverse micelles are nucleated by added water (brine). These variations will be discussed in greater detail in another publication. The key behavior for the purposes of this exposition is the onset of the electrical conductivity percolation at a = 0.85. The conductivity increases two orders as a decreases from 0.85 to 0.70, and as shown in the inset, the conductivity increases another two orders as a a decreases from 0.7 to 0.3. 

Figure 15.4(A) shows the effect of the R = Zn2+/Al3+ ratio, which determines the charge density of the LDH layer, on the Freundlich adsorption isotherms. K values are far higher than those measured for smectite or other inorganic matrices. The increase in Kf with the charge density (Kf= 215, 228, 325mg/g, respectively, for R = 4, 3 and 2) is supported by a mechanism of adsorption based on an anion exchange reaction. The desorption isotherms confirm that urease is chemically adsorbed by the LDH surface. The aggregation of the LDH platelets can affect noticeably their adsorption capacity for enzymes and the preparation of LDH adsorbant appears to be a determinant step for the immobilization efficiency. [ZnRAl]-urease hybrid LDH was also prepared by coprecipitation with R = 2, 3 and 4 and Q= urease/ZnRAl from 1 /3 up to 2.5. For Q < 1.0,100 % of the urease is retained by the LDH matrix whatever the R value while for higher Q values an increase in the enzyme/LDH weight ratio leads to a decrease in the percentage of the immobilized amount.

Fig. 4 Idealized surface pressure n versus area A isotherm detailing the inferred molecular orientation and aggregation states during a compression cycle. Reprinted with permission from Arnett et al, 1989. Copyright 1989 American Chemical Society.

The moisture induced isothermal transition are also observed in the complex films. The speed of the isothermal transition are also strongly dependent on the chemical structure of the counterpart polymer (Figure 27). For example, the J-aggregate in the complex film with polymer 2 is more stable than in the original cast film without polymer. 

The phenomena of rapid particle movement and the intimate contact between solids and at least a portion of the gas give rise to a series of characteristics of aggregative fluidization such as good mixing, near isothermal conditions and high rates of heat and mass transfer which are exploited in a wide range of unit operations. 

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