Pore filling theory

According to the pore filling theory, the shrinkage of a compact occurs by a successive accommodation and recovery of grain shape. Explain the process of sample shrinkage in detail and its related driving force. 

Lee, S.-M. and Kang, S.-J. L., Theoretical analysis of liquid phase sintering pore filling theory, Acta Mater., 46, 3191-202, 1998. 

Various mechanisms of coke poisoning active site coverage, pore filling as well as pore blockage have been observed in FCC [18, 19, 43] and Percolation theory concepts have been proposed for the modelling here of [45, 46, 47, 48]. This approach provides a framework for describing diffusion and accessibility properties of randomly disordered structures. 

A systematic study of the adsorption of nitrogen by packed assemblages of spheroidal particles was undertaken by Adkins and Davis (1986, 1987). After the consideration of various pore filling models, it was concluded that the desorption process can be adequately described by the instability of a Kelvin, hemispherical meniscus in the neck (i.e. the window) of the structure and the adsorption process can be viewed as a delayed Kelvin condensation in the largest dimension of the void structure. This reasoning is consistent with the network-percolation theory of hysteresis, which is discussed in Section 7.5. 

Computer modelling of physisorption hysteresis is simplified if it is assumed that pore filling occurs reversibly (i.e. in accordance with the Kelvin equation) along the adsorption branch of the loop. Percolation theory has been applied by Mason (1988), Seaton (1991), Liu et al., (1993, 1994), Lopez-Ramon et al., (1997) and others (Zhdanov et al.,1987 Neimark 1991). One approach is to picture the pore space as a three-dimensional network (or lattice) of cavities and necks. If the total neck volume is relatively small, the location of the adsorption branch should be mainly determined by the cavity size distribution. On the other hand, if the evaporation process is controlled by percolation, the location of the desorption branch is determined by the network coordination number and neck size distribution. 

A novel approach is reported for the accurate evaluation of pore size distributions for mesoporous and microporous silicas from nitrogen adsorption data. The model used is a hybrid combination of statistical mechanical calculations and experimental observations for macroporous silicas and for MCM-41 ordered mesoporous silicas, which are regarded as the best model mesoporous solids currently available. Thus, an accurate reference isotherm has been developed from extensive experimental observations and surface heterogeneity analysis by density functional theory the critical pore filling pressures have been determined as a function of the pore size from adsorption isotherms on MCM-41 materials well characterized by independent X-ray techniques and finally, the important variation of the pore fluid density with pressure and pore size has been accounted for by density functional theory calculations. The pore size distribution for an unknown sample is extracted from its experimental nitrogen isotherm by inversion of the integral equation of adsorption using the hybrid models as the kernel matrix. The approach reported in the current study opens new opportunities in characterization of mesoporous and microporous-mesoporous materials. 

Figure 3. (a) Isotherm calculated by density functional theory for a 4.1 run wide cylindrical pore with uniform surface potential (solid points). The solid line is the reconstructed isotherm for a flat surface having the adsorptive potential distribution of MCM-41. (b) Normalized isotherm for a 4.1 run MCM-41 (solid points) compared to the composite model for the same pore size. Note that the height of the pore-filling step is accurately accounted for. 

It is by now clear that several mechanisms and phases are present in the adsorption process in micropores, due to the interplay between gas-gas and gas-solid interactions, depending on their geometry and size. For this reason all those methods assuming a particular pore-filling mechanism, or adsorption model, should show shortcomings in some regions of the relevant parameters and their predictions should be compared with those based on more fundamental formulations of the adsorption process, like Density Functional Theory (DFT) [13] or Monte Carlo simulation [13,15]. Then, one question that arises is how the adsorption model affects the determination of the MSD ... 

I ic. 11. Relationship between the pore filling pressure and the pore width predicted by the modified Kelvin equation (MK). the Horvath-Kawazoe method (HK), density functional theory (DFT). and Gibbs ensemble Monte Carlo simulation (points) for nitrogen adsorption in carbon slit pores at 77 K [11]. 

The mutual attraction through the slit gap affects liquid film stability, and at a certain critical vapor pressure (or film thickness) the two films form a liquid bridge (Fig. 1-1 c) followed by a spontaneous filling up of the slit (assuming the film is in contact with the bulk liquid phase). The liquid-vapor interface moves to the plate boundaries. This phase transition from dilute vapor to a dense liquid is known as capillary condensation and was observed experimentally with the surface force apparatus by Christenson (1994) and Curry and Christenson (1996). Extensive theories for this phenomenon and its critical points are provided by Derjaguin and Chu-raev (1976), Evans et al. (1986), Forcada (1993), and Iwamatsu and Horii (1996). In general, slit-shaped pores fill up at a film thickness of about HI3, or when <) l(H,h)/dh = 0, such that... 

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