Lattice size effects

Figure 8.4. Graphical separation of lattice size and lattice distortion effects according to Warren-Averbach... 

In many catalytic systems, nanoscopic metallic particles are dispersed on ceramic supports and exhibit different stmctures and properties from bulk due to size effect and metal support interaction etc. For very small metal particles, particle size may influence both geometric and electronic structures. For example, gold particles may undergo a metal-semiconductor transition at the size of about 3.5 nm and become active in CO oxidation [10]. Lattice contractions have been observed in metals such as Pt and Pd, when the particle size is smaller than 2-3 nm [11, 12]. Metal support interaction may have drastic effects on the chemisorptive properties of the metal phase [13-15]. Therefore the stmctural features such as particles size and shape, surface stmcture and configuration of metal-substrate interface are of great importance since these features influence the electronic stmctures and hence the catalytic activities. Particle shapes and size distributions of supported metal catalysts were extensively studied by TEM [16-19]. Surface stmctures such as facets and steps were observed by high-resolution surface profile imaging [20-23]. Metal support interaction and other behaviours under various environments were discussed at atomic scale based on the relevant stmctural information accessible by means of TEM [24-29]. 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

Next, let us consider the fact that a given solid of known crystal structure has at least two additional degrees of freedom which may change its behavior. The presence of lattice defects, such as dislocations, and any alteration of particle size or specific surface will change its Gibbs energy. Since our present knowledge of the influence of lattice defects on solubility is rather limited, we shall restrict ourselves to a discussion of the particle size effect only. 

There have been a number of computer simulations of block copolymers by Binder and co-workers (Fried and Binder 1991a,ft), and this work was reviewed in Binder (1994). Although computer simulations are limited due to the restriction on short chain lengths that can be studied, finite size effects and equilibration problems at low temperatures, the advantages are that the models are perfectly well characterized and ideal (monodisperse, etc.) and microscopic details of the system can be computed (Binder 1994). In the simulations by Binder and co-workers, diblocks were modelled as self- and mutually-avoiding chains on a simple cubic lattice, with chain lengths N = 14 to 60 for/ = 1.A purely repulsive pairwise interaction between A and B segments on adjacent sites was assumed. A finite volume fraction of vacancies was included to speed the thermal equilibration process (Binder 1994). 

The necessity of introducing a combinatorial contribution to the chemical potential is a result of the neglect of size effects in the thermodynamics of pairwise interacting surface models. It also appears in lattice models that do not allow for a realistic representation of molecular sizes and are often simplified to models of equally sized lattice objects. The task of the combinatorial contribution is to represent the chemical potential of virtually homogeneous interacting objects of different size in 1 mol of a liquid mixture of a given composition with respect to the size and shape of the molecules. 

Thus, the interaction between structural entities of molecular and ionic crystals does not lead to appreciable delocalization of electrons strongly bonded to these entities. Therefore, the electronic structure of molecular and ionic crystals is practically not influenced by the size of such crystals. Size effects arise only in M/SC crystals with covalent (or at least partly covalent) bonds and depend on a relationship between the crystal size a and the length of delocalization (or, otherwise, electronic correlation) for valent electrons in a lattice. 

At x > 2 atomic hydrogen begins to incorporate into interstices of the crystal lattice due to quantum-size effects in the gap and that of H2 molecules repulsion. 

At x = 2 and T < 80K this state of H2 in the gap can be treated as quazy-liquid monolayer . At temperatures of liquid helium molecular hydrogen in the van der Waals gap and layer crystal forms a supperlattice consisting from a layered crystal lattice and a lattice of molecular hydrogen cryocrystal built in its van-der-Waals gap. At x>2 atomic hydrogen begins to incorporate into interstices of the crystal lattice due to quantum-size effects arise in the gap and strong repulsion of between H2 molecules. 

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