Macroscopic continuum

The continuum electrostatic approximation is based on the assumption that the solvent polarization density of the solvent at a position r in space is linearly related to the total local electric field at that position. The Poisson equation for macroscopic continuum media... 

In Section III we described an approximation to the nonpolar free energy contribution based on the concept of the solvent-accessible surface area (SASA) [see Eq. (15)]. In the SASA/PB implicit solvent model, the nonpolar free energy contribution is complemented by a macroscopic continuum electrostatic calculation based on the PB equation, thus yielding an approximation to the total free energy, AVP = A different implicit... 

Most theoretical studies of osmosis and reverse osmosis have been carried out using macroscopic continuum hydrodynamics [5,8-13]. The models used include those that treat the wall as either nonporous or porous. In the nonporous models the membrane surface is assumed homogeneous and nonporous. Transport occurs by the molecules dissolving in the membrane phase and then diffusing through the membrane. Mass transfer across the membrane in these models is usually described using the solution-diffusion... 

FIGURE 4.7. A schematic description of the different contributions to the PDLD model. The figure considers the energetics of an ion pair inside a protein interior. The upper part describes the protein permanent dipoles, the middle part describes the induced dipoles of the protein, while the lower part describes the surrounding water molecules and the bulk region, which is represented by a macroscopic continuum model. 

Apart from obvious features such as laminarity, there are speculations that flows in micro channels exhibit a behavior deviating from predictions of macroscopic continuum theory. In the case of gas flows, these deviations, manifesting themselves as, e.g., velocity slip at solid surfaces, are comparatively well understood (for an overview, see [130]). However, for liquid flows on a length scale above 1 pm, there is no clear theoretical foundation for deviations from continuum behavior. Nevertheless, various unexpected phenomena such as friction factors deviating from the continuum prediction [131-133] have been reported. A more detailed discussion of this still unsettled matter is given in Section 2.2. At any rate, one has to be careful here since it may be that measurements in small systems lack precision, essentially because of the incompatibility of analysis in a confined space and with large measuring equipment... 

Simonson, T. Briinger, A.T., Solvation free energies estimated from macroscopic continuum theory an accuracy assessment, J. Phys. Chem. 1994, 98, 4683-4694... 

Archontis, G. Simonson, T. Dielectric relaxation in an enzyme active site molecular dynamics simulations intepreted with a macroscopic continuum model, J. Am. Chem. Soc. 2001,123, 11047-11056. 

The characterization of the flow in existing DPF materials has been assessed by experiments and macroscopic continuum flow in porous media approaches. However, when it comes to material design it is essential to employ flow simulation techniques in geometrically realistic representations of DPF porous media. Some first applications were introduced in Konstandopoulos (2003) and Muntean et al. (2003) and this line of research is especially important for the development of new filter materials, the optimization of catalyst deposition inside the porous wall and for the design of gradient-functional filter microstructures where multiple functionalities in terms of particle separation and catalyst distribution (for combined gas and particle emission control) can be exploited. 

The Theory of Porous Media (TPM) is a macroscopic continuum theory which is based on the theory of mixtures and the concept of volume fractions. For more details see [1] and citations therein. 

Briefly, reconsider the a-b-c logical development of the macroscopic-continuum picture of van der Waals forces ... 

The opportunity to use whole-material dielectric susceptibilities comes at a price. It assumes that the two interacting bodies A and B are so far apart that they do not see molecular or atomic features in their respective structures. This is the "macroscopic-continuum" limit Materials are treated as macroscopic bodies on the laboratory scale all polarizability properties are averaged out much as they average out in a capacitance... 

Other continuous profiles in e produce similarly intriguing behaviors. The nondivergence of free energy and of pressure, qualitatively different from the power-law divergences in Lifshitz theory, occurs here when there is no discontinuity in s itself or its z derivative. Deeper consideration of such behaviors would require going beyond macroscopic-continuum language. 

The material properties of any substance are measured by a deviation of e from unity. E, P, and D as used here are averages over a small volume inside the material, a volume large enough compared with molecular sizes and spacings so as to be able to treat the material content as a macroscopic continuum. 

It is unfortunate that this macroscopic-continuum limitation is sometimes forgotten in overzealous application. The same limitation also holds in the theory of the electrostatic double layers for which we often make believe that the medium is a featureless continuum. Neglect of structure in double layers is equally risky, though, and even more common than in the computation of van der Waals forces. 

Creating spatially varying dielectric susceptibilities and solving the charge-fluctuation equations with these more detailed structures sometimes circumvents the macroscopic-continuum limitation. 

See the seminal paper by B. W. Ninham and V. Yaminsky, "Ion binding and ion specificity The Hofmeister effect and Onsager and Lifshitz theories," Langmuir, 13, 2097-108 (1997), for the connection between solute interaction and van der Waals forces from the perspective of macroscopic continuum theory. 

One of the models for the hydration force, the polarization model,5 assumes that the hydration force is generated by the local correlations between neighboring dipoles present on the surface and in water. The macroscopic continuum theory, in which water is assumed to be a homogeneous dielectric, predicts that there is no electric field above or below a neutral surface carrying a uniform dipolar density. However, at microscopic level the water is hardly homogeneous, and the electric interactions... 

Recently, there has been strong interest in multigrid-type hybrid multiscale simulation. As depicted in Fig. 6, a coarse mesh is employed to advance the macroscopic, continuum variable over macroscopic length and time scales. At each node of the coarse mesh, a microscopic simulation is performed on a finer mesh in a simulation box that is much smaller than the coarse mesh discretization size. The microscopic simulation information is averaged (model reduction or restriction or contraction) to provide information to the coarser mesh by interpolation. On the other hand, the coarse mesh determines the macroscopic variable evolution that can be imposed as a constraint on microscopic simulations. Passing of information between the two meshes enables dynamic coupling. 

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