Diffuse Interface Boundary

The main theoretical results derived from the ideal two-phase model, i.e., Equation (5.70) relating the invariant Q to the phase volumes and the Porod law (5.71), are no longer valid and need be modified when in the two-phase system the phase boundaries are diffuse. To see the modifications necessary to these theoretical results, we represent by p r) the scattering length density distribution in a two-phase material with diffuse boundaries and by p d(r) the density distribution in the (hypothetical) system in which all the diffuse boundaries in the above have been replaced by sharp boundaries. The two are then related to each other25 by a convolution product 

Therefore, with I (q) obeying the Porod law (5.71), a plot of q4I(q) against q2 should give, at large q, a linear relationship with its intercept at q = 0 equal to 27r(Ap)2S and its slope equal to — [2tt(Ap)2S]a2, thus permitting the evaluation of a. 

To see more clearly the physical meaning of a and the three-dimensional smoothing function (5.98), consider a small area of the boundary region and take the direction normal to it as the x direction. If the diffuse boundary is now replaced by a sharp planar boundary, p d(r) in the immediate neighborhood can be represented by 

The volume fraction of phase 1 excluding the transition zone is now equal to 0i — e/2, and the volume fraction of phase 2 is equal to 02 — e/2. We therefore have 

Substituting for p(x) the density profile defined by (5.103), (5.104) and (5.105), and after some regrouping of terms, we find 

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D. Fan, L.-Q. Chen. Diffuse-interface description of grain boundary motion. Phil Mag Lett 75 187, 1997. 

Although this may look very unpleasant, techniques for solving equations of this sort both analytically and numerically are very well established in fact it is usually possible to solve the equation analytically, at least for the simpler geometries often encountered in cell design. Moreover, the structure of the electrode-electrolyte interface allows us to make a substantial simple fication under normal circumstances in the region beyond the diffuse-layer boundary. We can take E as zero in this region, as indicated above, since the potential in the electrolyte bulk must be constant, and so ... 

Diffuse interfaces of certain types can move by means of self-diffusion. One example is the motion of diffuse antiphase boundaries which separate two ordered regions arranged on different sublattices (see Fig. 18.7). Self-diffusion in ordered alloys allows the different types of atoms in the system to jump from one sublattice to the other in order to change the degree of local order as the interface advances. This mechanism is presented in Chapter 18. 

Figure 18.7 Interfaces resulting from two types of continuous transformation, (a) Initial structure consisting of randomly mixed alloy, (b) After spinodal decomposition. Regions of B-rich and B-lean phases separated by diffuse interfaces formed as a result of long-range diffusion, (c) After an ordering transformation. Equivalent ordering variants (domains) separated by two antiphase boundaries (APBs). The APBs result from A and B atomic rearrangement onto different sublattices in each domain.

For quenching experiments, which were necessary to freeze-in the equilibrium states between the phase bands, the samples were reheated in a vacuum furnace under 500 mbar N2 to the original annealing temperature and dropped onto a copper plate kept below 470 K (200°C). With this treatment the Group 5 transition metal nitrides underwent a slight change in surface composition because it was impossible to adjust the appropriate nitrogen pressures. This, however, did not influence the compositions at the interface boundaries inside the diffusion couples. 

Reactions in biphasic systems can take place either at the interface or in the bulk of one of the phases. The reaction at the interface depends on the reactants meeting at the interface boundary. This means, the interface area as well as the diffusion rate across the bulk of the phase plays an important role. On the contrary, in reactions that take place in the bulk phase, the reactants have to be transferred first through the interface before the reactions take place. In this case, the rate of diffusion across the interface is an important factor. Diffusion across the interface is more complicated than the diffusion across a phase, as the mass transfer of the reactant across the interface must be taken into account. Hence, the solubility of the reactants in each phase has to be considered, as this has an effect on diffusion across the interface. In a system where the solubility of a reactant is the same in both phases, the reactant diffuses from the concentrated phase to the less concentrated phase across the interface. This takes into account the mass transfer of the reactant from one phase into the other through the interface. The rate of diffusion J in such systems is described in Equation 4.1, where D is the diffusion coefficient, x is the diffusion distance and l is the interface thickness (Figure 4.9). 

Figure I Interface between water and an agar (a), methylcellulose (b), and pectin (c) gel. The agar (a) was prepared by pouring 1.5% hot sol into a l-in.-diameter plastic die and allowing it to cool. Methylcellulose and pectin (b and c) were prepared similarly. The dies were then immersed in ethanol and placed in Petri dishes in preparation for photographing. The photographs were taken after 6 months (a), 24 h (b), and 4 h (c). Note the sharp boundary in (a), the adhering air bubbles in (b), and the diffuse interface in (c).

A schematic representation of the boundary layers for momentum, heat and mass near the air—water interface. The velocity of the water and the size of eddies in the water decrease as the air—water interface is approached. The larger eddies have greater velocity, which is indicated here by the length of the arrow in the eddy. Because random molecular motions of momentum, heat and mass are characterized by molecular diffusion coefficients of different magnitude (0.01 cm s for momentum, 0.001 cm s for heat and lO cm s for mass), there are three different distances from the wall where molecular motions become as important as eddy motions for transport. The scales are called the viscous (momentum), thermal (heat) and diffusive (molecular) boundary layers near the interface. 

Discrepancies between experimentally obtained and theoretically calculated data for cadmium concentration in the strip phase are 10-150 times at feed or strip flow rate variations. These differences between the experimental and simulated data have the following explanation. According to the model, mass transfer of cadmium from the feed through the carrier to the strip solutions is dependent on the diffusion resistances boundary layer resistances on the feed and strip sides, resistances of the free carrier and cadmium-carrier complex through the carrier solution boundary layers, including those in the pores of the membrane, and resistances due to interfacial reactions at the feed- and strip-side interfaces. In the model equations we took into consideration only mass-transfer relations, motivated by internal driving force (forward... 

In equimolar diffusion, a binary mixture is assumed where the two components diffuse in opposite directions at equal rates. These conditions exist in binary distillation where a mole of component 1 is vaporized for each mole of component 2 that is condensed. In unimolar diffusion, one component diffuses through a second, stagnant one. This is typical of an absorption process where one component diffuses through the gas phase to the interface boundary, is absorbed by the liquid, and then diffuses to the bulk of the liquid. The other gas components are assumed to remain in the gas and the liquid components to remain in the liquid. 

This condition describes the surfactant flux from the bulk to the interface during adsorption. If the surface concentration T is higher than the equilibrium value Fq a desorption process creates a surfactant flux in the opposite direction. Neglecting again any flow and fluxes other than bulk diffusion, for example surface diffusion, the boundary condition becomes very simple. 

A real material never completely fulfills the idealization stipulated in the ideal two-phase model. Therefore certain modifications have to be introduced either to the theoretical expressions derived or to the experimentally observed intensity data before the two can be compared. We discuss separately the effects arising from the presence of heterogeneities within each phase and from the diffuseness of the interface boundaries. 

Figure 5.14 Plots illustrating the relationship among the one-dimensional Gaussian smoothing function g (x), the Heaviside function H(x) representing the sharp boundary, and the convolution product of gi(.x) and H(x). The broken line in the last shows how the effective thickness t of the diffuse interface is defined.

FIGURE 38.16 Schematic of the distribution of ions in the diffused interface between leading and trailing electrolyte in an ITP system. S is the characteristic length scale of this diffused boundary, obtained from the balance of electromigration and diffusion fluxes. 

Fig. 5 Schematic representation of the diffusion and boundary layers at the electrode-solution interface as applied in the diffusion layer model.

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. 

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