Boundary approximations

Fig. 9.4. Dependence of piezoelectric properties of PbZrOj-PbTiOj on composition. The zirconate-rich phase is rhombohedral, whereas the titanate-rich phase is tetrahedral. The piezoelectric coefficients reach a maximum near the morphotropic phase boundary, approximately 45% PbZrOj and 55% PbTiOj. (After Jaffe et al., 1954.)...

For a polycrystalline sample with highly resistive, identical grain boundaries the situation is very similar. The grain boundary resistance between the two microelectrodes on adjacent grains Rgbt me is for identical grain boundaries - approximately given by... 

Case boundary Approximate analytical expressions for the current Eqn... 

Tungsten films have a large density of grain boundaries (approximate grain size is about 10(X) A). Comment on the effect of grain size and orientation on indentation depth and hence the polish rate. 

The diffraaion effects directly attributable to periodic arrays of dislocations that form low-angle boundaries in olivine have been studied by Ricoult and Kohlstedt (1983). Figure 8.35 shows an edge-on tilt boundary approximately parallel to (100) and its associated SAD pattern. The extra... 

Unlike many other continuum approaches, PCM adopts cavities of realistic shape modelled on the solute atoms they are built according to GePol algorithm, [114] in which the cavity is defined as the envelope of spheres centred on solute atoms or atomic groups. Besides the atomic spheres, other spheres are added by GePol to smooth the solute-solvent boundary, approximating the so-called solvent accessible surface proposed by Connolly. 

A large number of boundary approximations exist for use in molecular simulations. All of these can be adapted for use with hybrid potentials. 

Using the linear-elasticity and free boundary approximations used in the early work in the field, which is only valid for small strains (<10%), the change in thickness is given by [141] ... 

Ebert, U., van Saarloos, W. Universal algebraic relaxation of fronts propagating into an unstable state and implications for moving boundary approximations. Phys. Rev. Lett. 80(8), 1650—1653 (1998). http //dx.doi.org/10.1103/PhysRevLett.80.1650... 

The shape of the boundary approximates a Gaussian profile, but the exact description of the shape depends on the starting conditions of the experiment, the mobilities of the various ions, and whether sharpening of the boundary occurs as a result of electrical effects (Kohlrausch regulating... 

The finite integration technique suffers somewhat from a deficiency in being able to model very complicated cavities including curved boundaries with high precision, but the usage of the perfect boundary approximation eliminate this deficiency [123],... 

The probes are assumed to be of contact type but are otherwise quite arbitrary. To model the probe the traction beneath it is prescribed and the resulting boundary value problem is first solved exactly by way of a double Fourier transform. To get managable expressions a far field approximation is then performed using the stationary phase method. As to not be too restrictive the probe is if necessary divided into elements which are each treated separately. Keeping the elements small enough the far field restriction becomes very week so that it is in fact enough if the separation between the probe and defect is one or two wavelengths. As each element can be controlled separately it is possible to have phased arrays and also point or line focussed probes. 

We need to point out that, if the wavelengths of laser radiation are less than the size of typical structures on the optical element, the Fresnel model gives a satisfactory approximation for the diffraction of the wave on a flat optical element If we have to work with super-high resolution e-beam generators when the size of a typical structure on the element is less than the wavelengths, in principle, we need to use the Maxwell equations. Now, the calculation of direct problems of diffraction, using the Maxwell equations, are used only in cases when the element has special symmetry (for example circular symmetry). As a rule, the purpose of this calculation in this case is to define the boundary of the Fresnel model approximation. In common cases, the calculation of the diffraction using the Maxwell equation is an extremely complicated problem, even if we use a super computer. 

For many-electron systems such as atoms and molecules, it is obviously important that approximate wavefiinctions obey the same boundary conditions and symmetry properties as the exact solutions. Therefore, they should be antisynnnetric with respect to interchange of each pair of electrons. Such states can always be constmcted as linear combinations of products such as... 

The boundaries separating these principal types of phase behaviour are shown on X,C, diagram (for equalsized molecules) in figure A2.5.13. For molecules of different size, but with the approximation of equation (A2.5.10). more global phase diagrams were calculated using a third parameter,... 

The above approximation, however, is valid only for dilute solutions and with assemblies of molecules of similar structure. In the event that concentration is high where intemiolecular interactions are very strong, or the system contains a less defined morphology, a different data analysis approach must be taken. One such approach was derived by Debye et al [21]. They have shown tliat for a random two-phase system with sharp boundaries, the correlation fiinction may carry an exponential fomi. 

Hor the periodic boundary conditions described below, the ctitoff distance is fixed by the nearest image approximation to be less than h alf th e sm allest box len gth. W ith a cutoff an y larger, more than nearest images would be included. 

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

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. 

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