Finite-cubic lattice

Table III.5 gives a comparison of these two representations of the data for (n) for nearest-neighbor random walks on finite, cubic lattices with a centrosymmetric trap and subject to periodic boundary conditions. A similar analysis [14] shows that for d = 3, tetrahedral lattices (v — 4)...

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). 

If we assume a face-centered cubic lattice for H2O, this calculation gives about 4.5 Kcal/mole as the dipole contribution to the energy of vaporization of water. On correcting for finite size of dipole (i.e., assuming d = 1 A), this becomes 6 Kcal. 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

Fig. 40. Scaling for finite-sized lattices. Computer calculations of scaling properties for bond percolation on the three-dimensional simple cubic lattice. When p is the fraction of connected bonds, p = p (p,b) is the fraction of cubic samples of edge length b that contain a continuous path of connected bonds (a spanning cluster) which links opposite faces of the sample. From Kirkpatrick (1979).

The chirality quantification technique proposed by Harary and Mezey [54,55] is motivated by the Resolution Based Similarity Measure (RBSM) approach used in more general molecular similarity analysis [243]. This method does not rely on a single reference object. Instead, it characterizes shape on any desired finite level of resolution by considering various A(J,n) parts of square lattices, called lattice animals or P(G,n) parts of cubic lattices called polycubes which can be inscribed within the two- or three-dimensional objects J or G, respectively. In the above... 

Abstract In a solid with orbital degree of freedom, an orbital configuration does not minimize simultaneously bond energies in equivalent directions. This is a kind of frustration effect which exists intrinsically in orbital degenerate system. We review in this paper the intrinsic orbital frustration effects in Mott insulating systems. We introduce recent our theoretical studies in three orbital models, i.e. the cubic lattice orbital model, the two-dimensional orbital compass model and the honeycomb lattice orbital model. We show numerical results obtained by the Monte-Carlo simulations in finite size systems, and introduce some non-trivial orbital states due to the orbital frustration effect. 

A Bethe-tree is a particular case of more general networks considered in percolation theory. Sahimi and Tsotsis [1985] applied percolation theory and Monte Carlo simulation to deactivation in zeolites, approximated by a simple cubic lattice. Beyne and Froment [1990, 1993] applied percolation theory to reaction, diffusion and deactivation in the real ZSM-5 lattice. The finite rate of growth was described in terms of a polymerization mechanism. Pore blockage was reached in this small pore zeolite. It also affects the path followed by the diffusing molecules that becomes more tortuous, so that the effective diffusivity has to be expressed in terms of the blockage probability. 

Sectional and class methods for the solution of the collisional KE are generally called discrete-velocity methods (DVM). These methods are based on the simple idea of discretizing the velocity space into a grid constituted by a finite number of points. The existing methods are characterized by different grid structures (Aristov, 2001). For example, lattice Boltzmann methods discretize the velocity space into a regular cubic lattice with a constant lattice size (Li-Shi, 2000), whereas other methods employ different discretization schemes (Monaco Preziosi, 1990). By using a similar approach to that used with PBE, it is possible to define A,- as the number density of the particles with velocity and the discretized KE becomes... 

An application of the molecular cubic lattice of systematically constructed distributed basis sets to the linear H%+ system, for which finite element results are available(68), has also been described(69). 

MacDonald, I P. Kaufmann, and F.A.L. Dullien. 1986b. Quantitative image analysis of finite porous media. II. Specific genus of cubic lattice models and Bera sandstone J. Micros. 144 297-316. 

It is in the dependence of (n) on the (average) valency (u) that the results here stand in contrast to the analytic and numerical results obtained for lattices subject to periodic boundary conditions. From studies on periodic lattices, n) should decrease systematically with increase in the uniform valency v. This result pertains as well to random walks on ci = 3 dimensional periodic lattices of unit cells and can also be demonstrated analytically and numerically for walks on higher-dimensional [d < 8) cubic lattices [15,16]. In these problems, v = 2d and hence the higher the dimensionality of the space, the greater the number of pathways to a centrally located deep trap in a periodic array of (cubic) cells the decrease in (n) is found to be quite dramatic with increase in d, and hence v. However, an increase in v will also result in a greater number of pathways that allow the random walker to move away from the trap. For periodic lattices, this latter option positions the random walker closer to the trap in an adjacent cell. For finite lattices, moving away from the trap does not position the walker closer to a trap in an adjacent unit cell it positions the walker closer to the finite boundary of the lattice from whence it must (eventually) work itself back. It is evident, therefore, why the v dependence for periodic lattices is modified when one studies the same class of nearest-neighbor random-walk problems on finite lattices. 

To solve the PB equation for arbitrary geometries requires some type of discretization, to convert the partial differential equation into a set of difference equations. Finite difference methods divide space into a cubic lattice, with the potential, charge density, and ion accessibility defined at the lattice points (or grid points ) and the permittivity defined on the branches (or grid lines ). Equation [1] becomes a system of simultaneous equations referred to as the finite difference Poisson-Boltzmann (FDPB) equation ... 

Fig. 45. Normalized position of the structure factor maximum, x (e,N) = q Rg(e,N), plotted vs eN for N — 20 (squares) and N = 40 (triangles), for the model of block copolymers with only repulsive interactions between nearest neighbor A, B-pairs on the simple cubic lattice with < >v = 0.2. Note that the order-disorder transition is estimated to occur for eN w 7-8 in this Monte Carlo simulation (due to finite size effects and equilibration problems for eN S 6 no more accurate estimation was possible). From Fried and Binder [325].

Mackay [9-73, 9-74] called attention to yet another limitation of the 230-space-group system. It covers only those helices that are compatible with the three-dimensional lattices. All other helices that are finite in one or two dimensions are excluded. Some important virus structures with icosahedral symmetry are among them. Also, there are very small particles of gold that do not have the usual face-centered cubic lattice of gold. They are actually icosahedral shells. The most stable configurations contain 55 or 147 atoms of gold. However, icosahedral symmetry is not treated in the International Tables, and crystals are only defined for infinite repetition. 

Buzza and Gates (102) also addressed the question whether disorder or the increased dimensionality from two to three dimensions is responsible for the observed experimental behavior of the shear modulus. In particular, they explored the lack of the sudden jump in G from zero to a finite value at 0 = 0Q that is predicted by the perfectly ordered 2-D model. We have seen above that disorder appears to remove that abrupt jump in two dimensions (90). For drops on a simple cubic lattice, Buzza and Cates analyzed the drop deformation in uniaxial strain close to 0 = 0q, first using the model of truncated spheres . (For reasons given above, we believe this to be a very poor model.) They showed that this model did not eliminate the discontinuous jump in G. An exact model, based on a theory by Morse and Witten (103) for weakly deformed drops, led to G a 1/ In (0 - 0q), which eliminates the discontinuity, but still shows an unrealistically sharp rise at 0 = 0q and is qualitatively very different from the experimentally observed linear dependence of G on (0 - 0q). Similar conclusions were reached by Lacasse and coworkers (49, 104). A simulation of a disordered 3-D model (104) indicated that the droplet coordination number increased from 6 at to 10 at 0 = 0.84, qualitatively similar to what is seen in disordered 2-D systems (90). Combined with a suitable (anharmonic) interdroplet force potential, the results of the simulation were in close agreement with experimental shear modulus and osmotic pressure data. It therefore appears again that disor-... 

The above approximations make the calculation very simple. At least in the case of a face centered or a body centered lattice, it is a very good approximation to replace the cell by a sphere of equal volume and to assume that the neutron density has zero radial derivative on the surface of the sphere. This is a less good approximation for a simple cubic lattice and the corresponding correction will be given below. This correction constitutes, from the point of view of thermal utilization, the only difference between the simple cubic and face and body centered lattice. For a finite lattice, corrections must be introduced which decrease the thermal utilization. These will be also discussed below. 

This scaling law was compared with the results of self-consistent field theory by van der Linden and Leermakers (1992) they found that the profiles did follow a power law over the central region. In the limit of vanishing bulk volume fraction and infinitely long chains the power law exponent did indeed tend towards 2 as predicted by equation (5.2.38), but the corrections for finite relative molecular mass and bulk volume fractions are considerable. For calculations on a cubic lattice they found that the power law exponent a could be represented by... 

---