Effective binary approach

With each level of increased sophistication, the complexity of treating a diffusion problem increases tremendously, especially for natural silicate melts that have a large number of components. The simplest method (effective binary approach) is the most often used, but it cannot treat the diffusion of many components (such as those that show uphill diffusion). All the other methods have not been applied much to natural silicate melts. 

the effective binary approach and the concentration-based diffusivity matrix are introduced. The modified effective binary approach (Zhang, 1993) has not been followed up. The approach using the activity-based diffusivity matrix, similar to activity-based diffusivity T (Equations 3-61 and 3-62), is probably the best approach, but such diffusivities require systematic effort to obtain. 

In this section, neutral oxide or endmember species are used as components. If ionic species are used as diffusion components, which as often the case for aqueous solutions, then one must also consider electroneutrality. 

Pick s law is an empirical diffusion law for binary systems. For multicomponent systems. Pick s law must be generalized. There are several ways to generalize Pick s law to multicomponent systems. One simple treatment is called the effective binary treatment, in which Pick s law is generalized to a multicomponent system in the simplest way  

Despite the various drawbacks, the effective binary approach is still widely used and will be widely applied to natural systems in the near future because of the difficulties of better approaches. For major components in a silicate melt, it is possible that multicomponent diffusivity matrices will be obtained as a function of temperature and melt composition in the not too distant future. For trace components, the effective binary approach (or the modified effective binary approach in the next section) will likely continue for a long time. The effective binary diffusion approach may be used under the following conditions (but is not limited to these conditions) with consistent and reliable results (Cooper, 1968)  

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Zhang (1993) proposed the modified effective binary approach (also called activity-based effective binary approach). In this approach, the diffusive flux of a component is related to its activity gradient and all other components are treated as one combined component. The diffusive flux for any component i is expressed as (by analogy with Equation 3-61)... 

The effective binary approach is sometimes excellent, and often not good enough. These points are summarized here again ... 

For the dissolution of a crystal into a melt, if one wants to predict the interface melt composition (that is, the composition of the melt that is saturated with the crystal), the dissolution rate, and the diffusion profiles of all major components, thermodynamic understanding coupled with the diffusion matrix approach is necessary (Liang, 1999). If the effective binary approach is used, it would be necessary to determine which is the principal equilibrium-determining component (such as MgO during forsterite dissolution in basaltic melt), estimate the concentration of the component at the interface melt, and then calculate the dissolution rate and diffusion profile. To estimate the interface concentration of the principal component from thermod5mamic equilibrium, because the concentration depends somewhat on the concentrations of other components, only... 

To quantify the diffusion profiles is a difficult multicomponent problem. The activity-based effective binary diffusion approach (i.e. modified effective binary approach) has been adopted to roughly treat the problem. In this approach. 

As long as care is taken so that effective binary diffusivity obtained from experiments under the same set of conditions is applied to a given problem, the approach works well. Although the limitations mean additional work, because of its simplicity and because of the unavailability of the diffusion matrices, the effective binary diffusion approach is the most often used in geological systems. Nonetheless, it is hoped that effort will be made in the future so that multi-component diffusion can be handled more accurately. 

In principle, the diffusion matrix approach can be extended to trace elements. My assessment, however, is that in the near future diffusion matrix involving 50 diffusing components will not be possible. Hence, simple treatment will still have to be used to roughly understand the diffusion behavior of trace elements the effective binary diffusion model to handle monotonic profiles, the modified effective binary diffusion model to handle uphill diffusion, or some combination of the diffusion matrix and effective binary diffusion model. 

Figure 3-24 Calculated diffusion-couple profiles for trace element diffusion and isotopic diffusion in the presence of major element concentration gradients using the approximate approach of activity-based effective binary treatment. The vertical dot-dashed line indicates the interface. The solid curve is the Nd trace element diffusion profile (concentration indicated on the left-hand y-axis), which is nonmonotonic with a pair of maximum and minimum, indicating uphill diffusion. The dashed curve is the Nd isotopic fraction profile. Note that the midisotopic fraction is not at the interface.

A review is presented of techniques for the correlation and prediction of vapor-liquid equilibrium data in systems consisting of two volatile components and a salt dissolved in the liquid phase, and for the testing of such data for thermodynamic consistency. The complex interactions comprising salt effect in systems which in effect consist of a concentrated electrolyte in a mixed solvent composed of two liquid components, one or both of which may be polar, are discussed. The difficulties inherent in their characterization and quantitative treatment are described. Attempts to correlate, predict, and test data for thermodynamic consistency in such systems are reviewed under the following headings correlation at fixed liquid composition, extension to entire liquid composition range, prediction from pure-component properties, use of correlations based on the Gibbs-Duhem equation, and the recent special binary approach. 

The molar flux in a binary mixture is given by Eq. 5.3.4 the corresponding result for a multicomponent mixture, assuming an effective diffusivity approach, therefore, is... 

The form of Eq. 3.2.C-1 is too complex for many engineering calculations, and a common approach is to define a mean effective binary diffiisivity for species j diffusing through the mixture ... 

Stegbyt is the value for the step in Kalle (or Kille) where one switches from binary approach to chord shooting. Since the steps take the values 2 ", any value between 0.50 and 0.25 (e.g., 0.45, 0.4, 0.3) gives the same effect. For equations of high degree nj g, a stegbyt value around 1/ Dd g seems appropriate, but the time does not seem very sensitive to the choice of stegbyt. 

In the case of a multi-component system such as a nanodielectric, an effective medium approach can be used to define the real permittivity of the whole based upon the composition and the properties of the individual components Myroshnychenko and Brosseau (2005) provide a good overview of the topic, as applied to binary systems. The Lichtenecker-Rother equation is an example of such a relationship ... 

The classical computer tomography (CT), including the medical one, has already been demonstrated its efficiency in many practical applications. At the same time, the request of the all-round survey of the object, which is usually unattainable, makes it important to find alternative approaches with less rigid restrictions to the number of projections and accessible views for observation. In the last time, it was understood that one effective way to withstand the extreme lack of data is to introduce a priori knowledge based upon classical inverse theory (including Maximum Entropy Method (MEM)) of the solution of ill-posed problems [1-6]. As shown in [6] for objects with binary structure, the necessary number of projections to get the quality of image restoration compared to that of CT using multistep reconstruction (MSR) method did not exceed seven and eould be reduced even further. 

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

The other class of phenomenological approaches subsumes the random surface theories (Sec. B). These reduce the system to a set of internal surfaces, supposedly filled with amphiphiles, which can be described by an effective interface Hamiltonian. The internal surfaces represent either bilayers or monolayers—bilayers in binary amphiphile—water mixtures, and monolayers in ternary mixtures, where the monolayers are assumed to separate oil domains from water domains. Random surface theories have been formulated on lattices and in the continuum. In the latter case, they are an interesting application of the membrane theories which are studied in many areas of physics, from general statistical field theory to elementary particle physics [26]. Random surface theories for amphiphilic systems have been used to calculate shapes and distributions of vesicles, and phase transitions [27-31]. 

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