Boundary effect

Approaching the crystallites by spheres of radius R, in the case of diffusion-limited exchange Eq. (3.1.14) can be shown to lead to [10, 15] 

Therefore, the difference in the location of coke deposition may be rationalized as a simple consequence of the different sizes of the starting substances for coking. 

For intracrystalline diffusion paths sufficiently small in comparison with the crystallite radii, the effective diffusivity as defined by Eq. (3.1.6) maybe expanded in a power series [9, 63, 64], leading to 

The influence of the confining boundaries on the effective diffusivity, as reflected by Eqs. (3.1.16) and (3.1.17), has been applied repeatedly to determine the pore 

It is worth noting that within a range of 20 %, five different methods of analyzing the crystallite size, viz., (a) microscopic inspection, (b) application of Eq. (3.1.7) for restricted diffusion in the limit of large observation times, (c) application of Eq. (3.1.15) to the results of the PFG NMR tracer desorption technique, and, finally, consideration of the limit of short observation times for (d) reflecting boundaries [Eq. (3.1.16)] and (e) absorbing boundaries [Eq. (3.1.17)], have led to results for the size of the crystallites under study that coincide. 

As stated, the reason for formulating the SSE is to remove the dependence of the entropy metric on the number of histogram bins. However, 

It would be tempting to attempt to define another (and even more bin number independent) metric to be the peak SSE value. However, that peak occurs at different values depending on both the dataset and the design of the chemical descriptor. For example, an information-rich but sufficiently narrow valued descriptor might require the number of bins to be on the order of half of the number of data points before that peak is reached. Therefore, the 

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Teleman, O. An efficient way to conserve the total energy in molecular dynamics simulations boundary effects on energy conservation and dynamic properties. Mol. Simul. 1 (1988) 345-355. 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

Thermal conductivity of foamed plastics has been shown to vary with thickness (197). This has been attributed to the boundary effects of the radiant contribution to heat-transfer. 

Both extreme models of surface heterogeneity presented above can be readily used in computer simulation studies. Application of the patch wise model is amazingly simple, if one recalls that adsorption on each patch occurs independently of adsorption on any other patch and that boundary effects are neglected in this model. For simplicity let us assume here the so-called two-dimensional model of adsorption, which is based on the assumption that the adsorbed layer forms an individual thermodynamic phase, being in thermal equilibrium with the bulk uniform gas. In such a case, adsorption on a uniform surface (a single patch) can be represented as... 

B. J. Brosilow, E. Gulari, R. Ziff. Boundary effects in a surface reaction model for CO oxidation. J Chem Phys 99 1-5, 1993. 

Figure 8.21 (in which site values are suppressed for clarity) shows the continued development of this system. Though boundary effects begin to appear by t = 25, the characteristic manner in which this particular F restructures the initial graph is clear ... 

This flow field is somewhat idealized, and cannot be exactly reproduced in practice. For example, near the planar surfaces, shear flow is inevitable, and, of course, the range of % and y is consequently finite, leading to boundary effects in which the extensional flow field is perturbed. Such uniaxial flow is inevitably transient because the surfaces either meet or separate to laboratory scale distances. 

Certainly two-dimensional techniques have far greater peak capacity than onedimensional techniques. However, the two-dimensional techniques don t utilize the separation space as efficiently as one-dimensional techniques do. These theories and simulations utilized circles as the basis function for a two-dimensional zone. This was later relaxed to an elliptical zone shape for a more realistic zone shape (Davis, 2005) with better understanding of the surrounding boundary effects. In addition, Oros and Davis (1992) showed how to use the two-dimensional statistical theory of spot overlap to estimate the number of component zones in a complex two-dimensional chromatogram. 

Now consider the classical average of the second derivative appearing in (11.38). This average can be integrated by parts if we assume a very large system where boundary effects are negligible [43]... 

There is a large literature discussing the effects of grain boundaries on plastic deformation. The essential effects for clean boundaries have just been discussed, but there are many additional effects when the boundaries are contaminated with impurities and precipitates. All this will not be discussed further here. Books that have differing viewpoints on grain boundary effects are Baker (1983), and Meyers and Chawla (1998). 

We have given some highlights of a theory which combines the familiar multistate VB picture of a molecular system with a dielectric continuum model for the solvent which accounts for the solute s boundary effects — due to the presence of a van der Waals cavity which displays the solute s shape — and includes a quantum model for the electronic solvent polarization. 

In the preceding section, it was demonstrated that the walls of the system can be neglected when the system is in the hydrodynamic regime, i.e., when D is sufficiently large. In this situation, it is often desirable to treat the system as a bulk fluid and apply shear directly without any boundary effects. In this section, we describe two different methods with which to shear bulk systems. 

Choice of lattice linear dimensions Lx, Ly (usually Lx = Ly = L, apart from physically anisotropic situations, such as the ANWII model ). Only finite lattices can be simulated. Usually boundary effects are diminished by the choice of periodic boundary conditions, but occasionally studies with free boundaries are made. Note that Lx, Ly must be chosen such that there is no distortion of the expected orderings in the system e.g. for the model of where due to a third nearest neighbor interaction superstructures with unit cells as large as 4 x 4 did occur, L must be a multiple of 4. 

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