Continuum description

Secondly, the unidirectional compact growth geometry must be considered as shown in Fig. 6.31. The velocity v of the compact/suspension boundary must be defined as 

12 I am grateful to Professor A.P. Philipse for drawing my attention to this approach. 

Darcy s law for the flow through a porous medium reads, in differential form (see for example Ref.[55])  

The superficial velocity q is the same in the support and the compact, hence 

6 — PREPARATION OF ASYMMETRIC CERAMIC MEMBRANE SUPPORTS BY DIP-COATING 

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A layer of explicit solvent molecules surrounded by a continuum description for the highest possible accuracy. 

Sometimes, the system of interest is not the inhnite crystal, but an anomaly in the crystal, such as an extra atom adsorbed in the crystal. In this case, the inhnite symmetry of the crystal is not rigorously correct. The most widely used means for modeling defects is the Mott-Littleton defect method. It is a means for performing an energy minimization in a localized region of the lattice. The method incorporates a continuum description of the polarization for the remainder of the crystal. 

We will attempt to address a number of these phenomena in terms of their micromechanical origins, and to give the essential quantitative ideas that connect the macroscale (continuum description) with the microscale. We also will discuss the importance of direct observations, wherever possible, in establishing uniqueness of scientific interpretation. 

Mullins and Sekerka (17) were the first to construct continuum descriptions like the Solutal Model introduced above and to analyze the stability of a planar interface to small amplitude perturbations of the form... 

Many other, less obvious physical consequences of miniaturization are a result of the scaling behavior of the governing physical laws, which are usually assumed to be the common macroscopic descriptions of flow, heat and mass transfer [3,107]. There are, however, a few cases where the usual continuum descriptions cease to be valid, which are discussed in Chapter 2. When the size of reaction channels or other generic micro-reactor components decreases, the surface-to-volume ratio increases and the mean distance of the specific fluid volume to the reactor walls or to the domain of a second fluid is reduced. As a consequence, the exchange of heat and matter either with the channel walls or with a second fluid is enhanced. 

The principle quantity determining the flow regime of gases and deviations from the standard continuum description is the Knudsen number, defined as... 

The most successful continuum description of membrane elasticity, dynamics, and thermodynamics is based on the smectic bilayer model (for examples of different versions and applications of this approach see Ref. 76-82 and references therein). We introduce this model in conjunction with the question of membrane undulations. 

These results show that hydrodynamic interactions and the spatial dependence of the friction tensor can be investigated in regimes where continuum descriptions are questionable. One of the main advantages of MPC dynamics studies of hydrodynamic interactions is that the spatial dependence of the friction tensor need not be specified a priori as in Langevin dynamics. Instead, these interactions automatically enter the dynamics from the mesoscopic particle-based description of the bath molecules. 

This formulation assumes that the continuum diffusion equation is valid up to a distance a > a, which accounts for the presence of a boundary layer in the vicinity of the catalytic particle where the continuum description no longer applies. The rate constant ky characterizes the reactive process in the boundary layer. If it approximated by binary reactive collisions of A with the catalytic sphere, it is given by kqf = pRGc(8nkBT/m)1 2, where pR is the probability of reaction on collision. 

Precisely owing to the continuum description of the dispersed phase, in Euler-Euler models, particle size is not an issue in relation to selecting grid cell size. Particle size only occurs in the constitutive relations used for modeling the phase interaction force and the dispersed-phase turbulent stresses. 

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

In this contribution, we describe and illustrate the latest generalizations and developments[1]-[3] of a theory of recent formulation[4]-[6] for the study of chemical reactions in solution. This theory combines the powerful interpretive framework of Valence Bond (VB) theory [7] — so well known to chemists — with a dielectric continuum description of the solvent. The latter includes the quantization of the solvent electronic polarization[5, 6] and also accounts for nonequilibrium solvation effects. Compared to earlier, related efforts[4]-[6], [8]-[10], the theory [l]-[3] includes the boundary conditions on the solute cavity in a fashion related to that of Tomasi[ll] for equilibrium problems, and can be applied to reaction systems which require more than two VB states for their description, namely bimolecular Sjy2 reactions ],[8](b),[12],[13] X + RY XR + Y, acid ionizations[8](a),[14] HA +B —> A + HB+, and Menschutkin reactions[7](b), among other reactions. Compared to the various reaction field theories in use[ll],[15]-[21] (some of which are discussed in the present volume), the theory is distinguished by its quantization of the solvent electronic polarization (which in general leads to deviations from a Self-consistent limiting behavior), the inclusion of nonequilibrium solvation — so important for chemical reactions, and the VB perspective. Further historical perspective and discussion of connections to other work may be found in Ref.[l],... 

Similarly to catalysis, the properties of these composite materials are also determined by a hierarchy of structures on very different length/time scales. Therefore, linking mesoscale molecular models and continuum descriptions is relevant for their understanding and optimization. Together with advanced synthesis methods and functional testing, it is thus necessary also to develop new improved computational methods to provide an understanding of materials properties and to assist in the development of new functional materials. 

TvaroSka, KoS r and Hricovini in this book). One way to account for the effect of solvent on conforxnation might be to represent the molecule without environmental influences, and then explicitly include the solvent or other environmental molecules in the calculation. While avoiding built-in influences of environment is a satisfying concept, it is difficult to obtain by experiment parameters that lack those influences. Several methods have been used to study solvation effects, including continuum descriptions (24) and the explicit treatment of solvent molecules in Monte Carlo and molecular dynamics simulation. 

So far, we tacitly assumed that the upper and lower terraces next to the step are below their roughening transition temperature. By fixing the boundary heights of the terraces, away from the step, at, say, level 0 for the lower and level 1 (in units of the lattice spacing) for the upper terrace, one can study the time evolution of the step width w, defined, for instance, as the second moment of the gradient of the step profile also above roughening. Then one obtains s =1/4 for terrace diffusion and 1/2 for evaporation kinetics, as predicted by the continuum description of Mullins and confirmed by our Monte Carlo simulations. 

Assuming the width of the terrace W is much larger than the lattice constant of the crystal, we can use a continuum description of the step) motion. The problem can be simplified by using the relative separation between the tno lines. x(y, t), where y runs along the strip. The equation of motion for x is given by. 

Rigorous scale homogenization procedures lead to continuum models for the entire DPF (Bissett, 1984 Konstandopoulos et al., 2001, 2003) exploiting (as is common in continuum descriptions) a suitable scale disparity, namely the ratio of the channel hydraulic radius to the entire DPF diameter. The smallness of this parameter is invoked to formulate a perturbation expansion of the discrete multichannel equations. The continuum multichannel description of the DPF can accommodate various regeneration methods (thermal, catalytic and N02-assisted) and can provide spatio-temporal information of several quantities of interest (e.g. filter temperature, soot mass distribution, flow distribution, etc.) as illustrated in Fig. 38. 

The problem of linking atomic scale descriptions to continuum descriptions is also a nontrivial one. We will emphasize here that the problem cannot be solved by heroic extensions of the size of molecular dynamics simulations to millions of particles and that this is actually unnecessary. Here we will describe the use of atomic scale calculations for fixing boundary conditions for continuum descriptions in the context of the modeling of static structure (capacitance) and outer shell electron transfer. Though we believe that more can be done with these approaches, several kinds of electrochemical problems—for example, those associated with corrosion phenomena and both inorganic and biological polymers—will require approaches that take into account further intermediate mesoscopic scales. There is less progress to report here, and our discussion will be brief. 

CVD reactors operate at sufficiently high pressures and large characteristic dimensions (e.g., wafer spacing) such that Kn (Knudsen number) << 1, and a continuum description is appropriate. Exceptions are the recent vacuum CVD processes for Si (22, 23) and compound semiconductors (156, 157, 169) that work in the transition to the free molecular flow regime, that is, Kn > 1. Figure 7 gives an example of SiH4 trajectories in nearly free molecular flow (Kn 10) in a very low pressure CVD system for silicon epitaxy that is similar to that described by Meyerson et al. (22, 23 Meyerson and Jensen, manuscript in preparation). Wall collisions dominate, and be-... 

Buehler et al. presented a preliminary study on formation of water from molecular oxygen and hydrogen using a series of atomistic simulations based on ReaxFF MD method.111 They described the dynamics of water formation at a Pt catalyst. By performing this series of studies, we obtain statistically meaningful trajectories that permit to derive the reaction rate constants of water formation. However, the method requires calibrations with either ab initio simulation results in order to correctly evaluate the energetics of OER on Pt. Thus, this method is system specific and less reliable than the ab initio methods and will not replace ab initio methods. Nevertheless, this work demonstrates that atomistic simulation to continuum description can be linked with the ReaxFF MD in a hierarchical multiscale model. 

When the experimentalist set an ambitious objective to evaluate micromechanical properties quantitatively, he will predictably encounter a few fundamental problems. At first, the continuum description which is usually used in contact mechanics might be not applicable for contact areas as small as 1 -10 nm [116,117]. Secondly, since most of the polymers demonstrate a combination of elastic and viscous behaviour, an appropriate model is required to derive the contact area and the stress field upon indentation a viscoelastic and adhesive sample [116,120]. In this case, the duration of the contact and the scanning rate are not unimportant parameters. Moreover, bending of the cantilever results in a complicated motion of the tip including compression, shear and friction effects [131,132]. Third, plastic or inelastic deformation has to be taken into account in data interpretation. Concerning experimental conditions, the most important is to perform a set of calibrations procedures which includes the (x,y,z) calibration of the piezoelectric transducers, the determination of the spring constants of the cantilever, and the evaluation of the tip shape. The experimentalist has to eliminate surface contamination s and be certain about the chemical composition of the tip and the sample. 

If the polarization is mainly due to dipoles, we must recognize that the solvent does not look very homogeneous from the viewpoint of an embedded solute, because the distance between the individual dipoles is similar to the dimensions of the solute (see Fig. 3.1). Thus, a continuum description for the nearest-neighbor solvent atoms in a polar solvent appears to be inadequate. 

In solution things are more complex. The reaction partners are no longer free in their translational motion as they are in the gas phase they have to move in a condensed medium, and their motion is governed by other physical phenomena which for economy of exposition we shall not consider in detail. It is sufficient to recall that the physical models for the most important terms, Brownian motions, diffusion forces, are expressed in their basic form using a continuum description of the medium. 

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