Diffuse Interface Model

The Allen-Cahn equation applies to the kinetics of a diffuse-interface model for a nonconserved order parameter—for example, the order-disorder parameter r](f,t)... 

Ganesan V, Brenner H (2000) A diffuse interface model of two-phase flow in porous media. Proc R Soc Lond A 456 731-803... 

Diffuse interface model Free-surface flow Front capmring Front tracking Interface tracking Interfacial flow Level set method Phase-field method Volume-of-fluid (VOF) method... 

In this contribution, first a number of fundamental concepts that are central to interface capturing are presented, including definitions of level set functions and unit normal and curvature at an interface. This is followed by consideration of kinematic and dynamic boundary conditions at a sharp interface separating two immiscible fluids and various ways of incorporating those conditions into a continuum, whole-domain formulation of the equations of motion. Next, the volume-of-fluid (VOE) and level set methods are presented, followed by a brief outlook on future directions of research and other interface capturing/tracking methods such as the diffuse interface model and front tracking. 

Continuous models including intermolecular forces, in particular, the diffuse interface model provide a sound theoretical basis for studying equilibrium capillary phenomena in fluids. We have shown that these models can be extended in a natural way to study a thoroughly dynamical spreading process. The lubrication limit, where the contact angle is small, allows us to derive consistently an equation of motion for the liquid-vapor interface interacting with the solid surface. In the static limit, this equation yields back the equilibrium Young-Laplace theory. 

Volume of fluid (VOF) method Level set method Phase fleld method Diffuse interface model Interfacial flow Free surface flow Interface tracking Front tracking Front capturing... 

Kim J (2005) A continuous surface tension force formulation for diffuse-interface models, J Comput Phys 204 784-804... 

Wise, S.M. (2003) Diffuse Interface Model for Microstructural Evolution of Stressed, Binary Thin Films on Patterned Substrates. PhD thesis, University of Vhginia. 

The classic nucleation theory is an excellent qualitative foundation for the understanding of nucleation. It is not, however, appropriate to treat small clusters as bulk materials and to ignore the sometimes significant and diffuse interface region. This was pointed out some years ago by Cahn and Hilliard [16] and is reflected in their model for interfacial tension (see Section III-2B). 

One possibility for increasing the minimum porosity needed to generate disequilibria involves control of element extraction by solid-state diffusion (diffusion control models). If solid diffusion slows the rate that an incompatible element is transported to the melt-mineral interface, then the element will behave as if it has a higher partition coefficient than its equilibrium partition coefficient. This in turn would allow higher melt porosities to achieve the same amount of disequilibria as in pure equilibrium models. Iwamori (1992, 1993) presented a model of this process applicable to all elements that suggested that diffusion control would be important for all elements having diffusivities less than... 

The different behaviour of contact resistance in the two cases can be examined through the two models the just described acoustic mismatch model and the diffuse mismatch model which suppose that all the phonons are scattered at the interface. Hence these two models define two limits in the behaviour of phonons at a discontinuity. 

Some experiments outlined the frequency dependence of phonon scattering on surfaces [74]. Thus, Swartz made the hypothesis that a similar phenomenon could take place at the interface between solids and proposed the diffuse mismatch model [72]. The latter model represents the theoretic limit in which all phonons are heavily scattered at the interface, whereas the basic assumption in the acoustic mismatch model is that no scattering phenomenon takes place at the interface of the two materials. In the reality, phonons may be scattered at the interface with a clear reduction of the contact resistance value as calculated by the acoustic model. 

Fig. 15 Two of the simplest theories for the dissolution of solids (A) the interfacial barrier model, and (B) the diffusion layer model, in the simple form of Nemst [105] and Brunner [106] (dashed trace) and in the more exact form of Levich [104] (solid trace). c is the concentration of the dissolving solid, cs is the solubility, cb is the concentration in the bulk solution, and x is the distance from the solid-liquid interface of thickness h or 8, depending on how it is defined. (Reproduced with permission of the copyright owner, John Wiley and Sons, Inc., from Ref. 1, p. 478.)...

The meehanism of Mo removal in suboxie systems is unelear, and so the fundamental nature of this fraetionation requires further study. However, the effeet may be rmderstood in terms of a two layer diffusion-reaetion model in whieh a reaetion zone in the sediment (where Mo is ehemieally removed) is separated from seawater by a purely diffusive zone in which there is no chemical reaction (Braudes and Devol 1997). The presence of a diffusive zone is likely because Mo removal presumably occurs in suMdic porewaters that lie a finite distance L below the sediment-water interface (Wang and van Cappellen 1996 Zheng et al. 2000a). If HjS is present in the reactive zone such that Mo is removed below this depth, then Mo isotope fractionation in the diffusive zone may be driven by isotope effects in the reactive zone. 

On the basis of these results and predictions, a diffusion-adsorption model has been proposed to explain the results of water-to-droplet MT of FeCp-PrOH [53,97]. Assumptions of the model are as follows. Mass transfer of FeCp-X across the droplet/water interface competes with adsorption on the droplet interface. As illustrated in Figure 20, electrolysis of an FeCp-X/NB droplet renders distribution of the molecule to the water phase as FeCp-X+ at t = 0. At = , although FeCp-X transfers to the droplet interface with a diffusion-limited rate in water, redistribution of the molecule to the droplet interior competes with adsorption on the droplet/water interface. When the droplet surface is occupied by adsorbed FeCp-X to some extent at t = t", distribution of the compound to the droplet interior is assumed to be controlled by the fraction of the interfacial area adsorbed by FeCp-X (r/rx), where V is the amount of FeCp-X adsorbed on the droplet/water interface. 

The two-film theory supposes that the entire resistance to transfer is contained in two fictitious films on either side of the interface, in which transfer occurs by molecular diffusion. This model leads to the conclusion that the mass-transfer coefficient kL is proportional to the diffusivity DAB and inversely proportional to the film thickness Zy as... 

Penetration theory (Higbie, 1935)assumes that turbulent eddies travel from the bulk of the phase to the interface where they remain for a constant exposure time te. The solute is assumed to penetrate into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion. This model predicts that the mass-transfer coefficient is directly proportional to the square root of molecular diffusivity... 

Following the work of Amovilli and Mennucci [21] a model for repulsion interactions at diffuse interfaces has been developed. Since the repulsion energy depends on the solvent density it is then natural to replace the constant density p with a position-dependent density p(z). The first attempt made use of p(z) in the final expression for the repulsion energy [17]. Such a model has subsequently been improved by a derivation of a new repulsion expression [18]. 

The need for a more flexible model was met by developing the formalism for diffuse interfaces [15,17,18]. First, the electrostatic solvation for a diffuse interface has been... 

Because of their crucial role in the comparison between theoretical modelling and experiments, nonelectrostatic interactions have recently been reconsidered for diffuse interfaces. In particular, the introduction of repulsion proved to be the key to obtaining agreement between the CM and experimental findings. In particular, it allowed the surfactant behaviour of halides [17] and [18] to be modelled. 

L. Frediani, R. Cammi, S. Corni, J. Tomasi, A polarizable continuum model for molecules at diffuse interfaces, J. Chem. Phys. 120, 3893 (2004)... 

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