Pore coordination number

Figure 3. Pore stability is determined by the dihedral angle and the pore coordination number, (a) The pore with concave surfaces will shrink while (b) the pore with convex surfaces will grow (or become metastable).

The term morphology is used in a general sense to include any parameter that quantifies pore-scale structure pore geometry, pore coordination number, grain shape, porosity, spatial correlations in pore size, etc. Pore morphology varies dramatically from one material to another and has a very strong influence on continuum-scale behavior. 

The Delaunay tessellation is particularly important because it provides a tool to decompose the continuum void space into discrete pores, which is essential for pore-scale modeling. An important drawback to using the Delaunay tessellation for extracting pore structure is that it leads to a fixed pore coordination number of 4, which is a geometric artifact in a real packing, not all voids conform to the tessellation s tetrahedral structure. Experimentally, the average... 

Matrix structure. The size, dimensionality, pore coordinate number, and topology of the matrix influence the phase distributions significantly. 

In two dimensions, a pore can have a dihedral angle y/ = 120°, which is surrounded by N other grains. The number N is called the pore coordination number (CN). Similar to the case of a grain surrounded by other grains, if IV = 6, the pore has straight sides otherwise, it has convex sides for A < 6 and concave sides for IV > 6, as shown in Fig. 8.28. The surface of the pore will move toward its center of curvature, so the pore with < 6 will shrink, whereas the one with N>6 will expand. The pore is metastable for N = 6, so that the number is called critical pore CN, or Nc-Accordingly, if the pores have convex sides, i.e., N < 6, the decrease in the pore... 

Figure 9.40 Conditions for pore stability in three dimensions as a function of pore coordination number. (From Ref. 78.)...

Figure 11.8 Schematic diagram illustrating the reduction in the pore coordination number as a result of normal grain growth.

To further address the effect of particle packing on sintering behaviour and to develop a more general relationship between pore coordination number... 

Closed arrays that contain a single pore better represent conditions within a powder compact. The pore within the array is defined by the number of coordinating particles (pore coordination number, n) and its size (Rp) defined as the radius of the circumscribed circle (or sphere) as shown in Figure 4. 

With the exception of the reasons given in Figure 9.22, porosity significantly depends on particle orientation and packing that can be characterized by a coordination number of packing of particles, np, or pores, Zc, which will be discussed in Section 9.6 and Section 9.7. 

The model of clusters or ensembles of sites and bonds (secondary supramolecular structure), whose size and structure are determined on the scale of a process under consideration. At this level, the local values of coordination numbers of the lattices of pores and particles, that is, number of bonds per one site, morphology of clusters, etc. are important. Examples of the problems at this level are capillary condensation or, in a general case, distribution of the condensed phase, entered into the porous space with limited filling of the pore volume, intermediate stages of sintering, drying, etc. 

Model of a granule of a porous solid as a lattice (labyrinth) of pores and particles, which takes into account the average values of coordination number of bonds and distribution of sites and bonds over the characteristic sizes. 

Unfortunately, the relations (9.61) and (9.62) do not allow establishment of an unequivocal interrelation between the coordination numbers of packings of particles ( ,) and pores (Zc) for corresponding V- and D-lattices, but, for undegenerated D-polyhedra of tetrahedron form Zc = 4. Typical values of nF for random packings and regular packings of monospheres are discussed in Section 9.7.3 and Section 9.7.2, respectively. 

The first model of porous space as a 2D lattice of interconnected pores with a variation of randomness and branchness was offered by Fatt [220], He used a network of resistors as an analog PS. Further, similar approaches were applied in a number of publications (see, e.g., Refs. [221-223]). Later Ksenjheck [224] used a 3D variant of such a model (simple cubic lattice with coordination number 6, formed from crossed cylindrical capillaries of different radii) for modeling MP with randomized psd. The plausible results were obtained in these works, but the quantitative consent with the experiment has not been achieved. 

Moreover, pore and channel size and shape can be tuned by using different alkaline-earth cations (Mg2+, Ca2+, Sr2+, Ba2+) having different sizes and coordination numbers. The heat of water adsorption ranges from —46 to —52 kj mol-1, values similar to those of silica-containing zeolites. As recently reported, the properties of these materials can be further differentiated through the incorporation of transition (i.e. Co2+) or alkaline cations (i.e. K+) into the channels of barium-linked materials composed of metal-assembled cages. 

The overall pore volume of the bed of packed spheres is mainly determined by the. size of the inner cavities, which is in turn dependent on the particle radius, R, and the coordination number, N. The effective Kelvin radius corresponding to this major stage of capillary condensation is approximately given by the radius, r, of the inscribed sphere within the cavity (Karnaukhov 1971). 

In the experimental investigations of Avery and Ramsay (1973), it was found that progressive compaction of very small spheroidal oxide particles led to the sequential change in nitrogen isotherm type from II to IV and finally to I. This was a clear demonstration of a decrease in pore width as a result of increase in particle coordination number. 

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. 

The hydrogel has an open structure (i.e. a low particle coordination number) and is mechanically weak. The removal of the aqueous liquid phase normally leads to a drastic shrinkage of the silica framework and die formation of additional siloxane bonds (Fenelonov et al., 1983). The resulting xerogel is therefore much stronger and more compact, but inevitably has a lower pore volume. 

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