Heating isothermal sintering

Because the effects become less difficult to predict than the more realistic case of constant heating rate sintaing, we shall limit our discussion to isothermal sintering. The linear densification rate of the free matrix (i.e., the glass) can be found from the models for viscous sintaing described in Chapter 8. The Mackenzie and Shuttleworth equation is used because of its simple form, giving... 

Heating schedules can be simple, as in sintering studies involving isothermal sintering or constant heating rate sintering of single-phase powders in laboratory-scale experiments, or have a more complex temperature-time relationship, as in... 

Figure 12.6 Sketch of the temperature-time schedule for isothermal sintering and constant heating rate sintering.

A representation of constant heating rate data should take into account the simultaneous occurrence of densification and coarsening. As a first approximation, we can modify the theoretical equations for isothermal sintering to account for the effect of changing temperature on the densification and coarsening processes. Following Eq. (8.118), the linear densification rate (equal to one third the volumetric densification rate) can be written... 

Truly isothermal operation of a tubular reactor may not be feasible in practice because of large enthalpies of reaction or poor heat transfer characteristics. Nor is it always desirable, as, for example, in the case of a reversible exothermic reaction (see Sect. 3.2.4). In an exothermic catalytic reaction, it may be necessary to provide adequate means for heat transfer to prevent the development of local hot-spots on which coking may occur and reduce the catalyst activity. An excessive temperature rise may also cause the catalyst particles to sinter, thereby reducing their surface area and causing an irreversible decrease in catalytic activity. 

This well accepted method [28] has been used extensively in the characterization of M41S materials [11-12,14]. From the application of this method to MCM-41, it has been concluded that this material contains no significant amounts of microporosity. This is the main evidence presented so far in order to conclude that MCM-41 is exclusively mesoporous. As it happens with any good method its limitations need to be considered in order to avoid misinformation. In the case of the a method the choice of the reference isotherm is crucial. All the reference silicas should be nonporous in order to allow a reliable analysis of MCM-41. Unfortunately, we observed that most of them have a steep rise in their N2 adsorption isotherms at 77 K at low relative pressures and BET surface areas varying from 40 [29] to 400 mVg [30], For this reason, our sample of MCM-41 was heat-treated so as to sinter the silica particles and thus obtain a nonporous silica (BET surface area 1.5 mVg) and as similar as possible to our MCM-41. The N2 adsorption isotherms for a reference silica [29] and our sintered MCM-41 are shown in Figure 7. 

For the ACs the data are representative of the samples after heat-treatment at all three temperatures since during their fabrication these materials have already been treated at temperatures in excess of 850°C. However, for the alumina and clay samples the surface areas and pore volumes are shown after treatment at each temperature as these materials undergo various phase transitions that lead to sintering of the samples and shifts in their relative pore size distributions with heat-treatment. The particle size was determined from the corresponding MIP curve for the powder raw material. The Sbet in the case of microporous ACs should be considered as an apparent surface area due to the micropore filling mechanism associated with these materials [15]. The external area and micropore volumes were calculated from the slope and intercept of the t-plots of the corresponding isotherms. The total pore volume was taken as the amount of gas adsorbed at a relative pressure of 0.96 on the desorption isotherm, equivalent to a pore diameter of 50 nm. The mesopore volume was calculated from the difference in the total pore volume and the micropore volume. 

Figure 13. Sintering parameter (1/K) and viscosity according to Scherer model (85) of viscous sintering for two-step acid-catalyzed xerogel. Samples were heated to indicated temperatures at 2 or 20 °C/min and held isothermally. Corresponding bulk densities are plotted on the abscissa. (Reproduced with permission from reference 79. Copyright 1984.)...

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