Particle coordination number

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. 

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. 

An even larger pore volume can be obtained if the liquid phase is removed under supercritical conditions to give an aerogel . This type of gel has an extremely high surface area and pore volume (see Table 10.7), but it tends to be mechanically weak and unstable when exposed to water vapour because the particle coordination number is low. The upper limiting area of a silica composed of discrete primary particles would be =2000 m2 g 1, but specific surface areas of this magnitude are unlikely to be attained. 

The alcogel featured in Figure 44.3 had a BET area of 641 m /g and a pore volume of 0.93 cm (liquid)/g. An even larger pore volume can be obtained if the fluid phase is removed under supercritical conditions to give an aerogel, that is, a product having a very low particle coordination number. Such materials are macroporous and have a high surface area (Table 44.1), but they are... 

Hierarchical packing of colloidal particles (A), random packing of colloidal particles, coordination number = 3 (B), and TEM micrograph of colloidal xerogel prepared by single-step base-catalyzed hydrolysis of TEOS (r = 2.25) followed by drying at 50°C (C). Bar = 100 nm. 

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. 

B.J.H. methods) (iii) the average diameter (T.E.M.) and/or the dispersion (chemisorption of probe molecule) of the metallic particle. EXAFS will also provide average coordination numbers, which decrease sharply as the particle size decreases. 

From a structural point-of-view the bulk metallic state, that is, fee lattice (with varying densities of defects such as twins and stacking faults) is generally established in gold nanoparticles of about 10 nm diameter and upwards. However, such particles still display many unusual physical properties, primarily as the result of their small size. Shrinking the size of gold particles has an important effect it increases both the relative proportion of surface atoms and of atoms of even lower coordination number, such as edge atoms [49] and these atoms in turn are relatively mobile and reactive. 

Instead, we believe the electronic structure changes are a collective effect of several distinct processes. For example, at surfaces the loss of the bulk symmetry will induce electronic states with different DOS compared to bulk. As the particle sizes are decreased, the contribution of these surface related states becomes more prominent. On the other hand, the decrease of the coordination number is expected to diminish the d-d and s-d hybridization and the crystal field splitting, therefore leading to narrowing of the valence d-band. At the same time, bond length contraction (i.e. a kind of reconstruction ), which was observed in small particles [89-92], should increase the overlap of the d-orbitals of the neighboring atoms, partially restoring the width of the d-band. 

Changing the size of metal particles leads to a change in the atom s mean coordination number, as in smaller particles the number ratio of surface atoms to bulk atoms increases. As a result, the metal s valence band becomes... 

The electrostatic valence rule usually is met rather well by polar compounds, even when considerable covalent bonding is present. For instance, in calcite (CaC03) the Ca2+ ion has coordination number 6 and thus an electrostatic bond strength of s(Ca2+) =. For the C atom, taken as C4+ ion, it is s(C4+) =. We obtain the correct value of z for the oxygen atoms, considering them as O2- ions, if every one of them is surrounded by one C and two Ca particles, z = -[2s(Ca2+) + s(C4+)] = -[2 j + ] = -2. This corresponds to the actual structure. NaN03 and YBOs have the same structure in these cases the rule also is fulfilled when the ions are taken to be Na+, N5+, Y3+, B3+ and 02. For the numerous silicates no or only marginal deviations result when the calculation is performed with metal ions, Si4+ and 02 ions. 

In garnet, Mg3Al2Si3012, an O2- ion is surrounded by 2 Mg2+, 1 Al3+ and 1 Si4+ particle. There are cation sites having coordination numbers of 4, 6 and 8. Use Pauling s second rule to decide which cations go in which sites. 

Mercury(II) arsenate Hg3(As04)2 has three independent Hg particles with coordination numbers five and seven, but each of them has two short Hg O bonds (206 < r(HgO) < 214 pm) in a nearly linear arrangement linking of these polyhedra including the As04 tetrahedra leads to a 3-D network.300... 

Rumpf (R4) has derived an explicit relationship for the tensile strength as a function of porosity, coordination number, particle size, and bonding forces between the individual particles. The model is based on the following assumptions (1) particles are monosize spheres (2) fracture occurs through the particle-particle bonds only and their number in the cross section under stress is high (3) bonds are statistically distributed across the cross section and over all directions in space (4) particles are statistically distributed in the ensemble and hence in the cross section and (5) bond strength between the individual particles is normally distributed and a mean value can be used to represent each one. Rumpf s basic equation for the tensile strength is... 

Fig. 10.4. Frequency of observation of states versus energy, E, and number of particles, N, for a homopolymer of chain length r = 8 and coordination number z = 6onal0xl0xl0 simple cubic lattice. Conditions, following the notation of [48] are T = 11.5, ji = —60.4. In order to reduce clutter, data are plotted only for every third particle. Reprinted by permission from [6], 2000 IOP Publishing Ltd...

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