Diffuse Interface Theory

In addition, because of the Gibbs-Duhem relation, c a dp a + cb d/xs = 0, the chemical potential gradients are interdependent  

With the aid of Eqs. 18.7 and 18.8, the flux of B in the volume-fixed frame is then 

The conditions for continuous phase transformations (described in Chapter 17) derive from considerations of the molar free energy of a homogeneous, spatially uniform system with no interfaces. For phase transformations involving nonconserved 

3Spinodals occur at points where second derivatives vanish and for molar free energy and compositions give rise to the sharp cusps in Fig. 17.1 [2], Spinodal derives from the Latin spina, for thorn (the plural, spinae, meant difficulties or perplexities). 

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The Kinetics of Spinodal Decomposition. Cahn s kinetic theory of spinodal decomposition (2) was based on the diffuse interface theory of Cahn and Hilliard (13). By considering the local free energy a function of both composition and composition gradients, Cahn arrived at the following modified linearized diffusion equation (Equation 3) to describe the early stages of phase separation within the unstable region. In this equation, 2 is an Onsager-type... 

The liquid wets the solid completely (0 -> 0) when Agps > Aipi. A similar computation can be carried out in the diffuse interface theory, but is somewhat more problematic, since one has to define a boundary condition for the fluid density on the solid surface. 

The shift of the free energy due to interactions with the solid is given by the liquid-solid interaction term (30) minus the lost part of the fluid-fluid interaction term in Eq. (9). We will compute it here for a liquid layer with a sharp interface at 2 = /i parallel to the solid surface at 2 = /i (the diffuse interface theory is discussed in Section 5.2). Setting p = p/ = const at 0 < 2 < /i and neglecting the gas density, the modified free energy integral (9) can be computed by integrating the van der Waals interactions laterally as in Eq. (11). The correction due to the interaction with the solid is computed as... 

The diffuse interface theory goes back to van der Waals ... 

Section 5. Diffuse interface theory coupled with hydrodynamics is reviewed by... 

Saylor, D.M., Kim, C.S., Patwardhan, D.V., Warren, J.A. Diffuse-interface theory for structure formation and release behavior in controlled dmg release systems. Acta Biomater. 3, 851-864 (2007)... 

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). 

The most possible reason may be in the higher free energy of the protein adsorption on PolyPROPYL A materials. Chemisorbed neutral poly(succinimide) of molecular weight 13000 apparently forms a diffuse interface as predicted by theory (see Sect. 2.2). Controversially, a short polyethyleneimine exists on a surface in a more flat conformation exhibiting almost no excluded volume and producing... 

The problem of morphological instability was solved theoretically by Mullins and Sekerka [20], who proposed a linear theory demonstrating that the morphology of a spherical crystal growing in supercooled melt is destabilized due to thermal diffusion the theory dealt quantitatively with and gave linear analysis of the interface instability in one-directional solidification. 

Now, the cosh function gives inverted parabolas [Fig. 6.65(b)]. Hence, according to the simple diffuse-charge theory, the differential capacity of an electrified interface should not be a constant. Rather, it should show an inverted-parabola dependence on the potential across the interface. This, of course, is a welcome result because the major weakness of the Helmholtz-Perrin model is that it does not predict any variation in capacity with potential, although such a variation is found experimentally [Fig. 6.65(b)],... 

Because experimental study of the structure of crystal/liquid interfaces has been difficult due to the buried nature of the interface and rapid structural fluctuations in the liquid, it has been investigated by computer simulation and theory. Figure B.3 provides several views of crystal/liquid (or amorphous phase) interfaces, which must be classified as diffuse interfaces because the phases adjoining the interface are perturbed significantly over distances of several atomic layers. 

The average thickness E of the diffuse interface layer could be determined (see Table 1) by the above-mentioned two methods. The results obtained with the new method presented in this letter, i.e. by analysis of the negative deviation from Debye s theory (see Fig. 5), were denoted as Eq. The results derived from the negative deviation from Porod s law were denoted as Ep. It is obvious that Ed is close with Ep for various samples, but in each case the former is slightly higher and the reasons are thus not fully understood. With the increase of the average diameter of the colloidal nuclei, E (Ed and Ep) reasonably increased. 

Diffusion path theory. predicts a linear relationship between interface positions and t1 when there is no convection. Based on the correlation coefficients in Table I, this relationship appears to hold for the systems at these conditions. The low coefficients for the upper interfaces resulted from the measurement uncertainty ( 0.05 mm) being the same order of magnitude as the total movement of the interfaces, which is also why these plots are pot included in Figure 7. Division of the best-fit slopes by 2t gives an estimate of the interface velocities at any elapsed time t. 

In contrast, the brine phase tended to form large pockets in the TRS system, causing extensive movement of the liquid crystal to the brine-microemulsion interface. This behavior is schematically illustrated in Figure 9. Not surprisingly, interface movements were inconsistent with diffusion path analysis. Figure 10 shows such a plot for the TRS/C12 system at 1.5 gm/dl salinity. The nonlinearity results from convection, which speeds up equilibration. Experimentally, the inconsistency with diffusion path theory was evident from the time-dependent appearance of the upper microemulsion interface, an indication of variable interface compositions. 

A comparison between experimental and theoretical results shows that diffusion path analysis can qualitatively predict what is observed when an anionic surfactant solution contacts oil. Experimentally, one or two intermediate phases formed at all salinities. The growth of these phases was easily observed through the use of a vertical-orientation microscope. Except when convection occurred due to an intermediate phase being denser than the phase below it, interface positions varied as the square root of time. As a result, diffusion path theory could generally he used to correctly predict the direction of movement and relative speeds of the interfaces. 

Surface/Diffusion Potential Theory. " The transmembrane potential, E, is expressed as a difference between the electrical potential, Ei, and Eo of the two bulk phases in the two aqueous compartments separated by a membrane or as a sum of phase boundary potentials produced at the membrane-electrolyte interfaces and the diffusion potential within the membrane arising from the movement of ionic species through the membrane (Figure 28). 

The diffusion potential theory has its origins in the Nernst theory formulated to describe the liquid junction potential developed at the interface of two electrolytes with different ionic concentrations and ionic mobilities. For two univalent ionic solutions the liquid junction potential is given by... 

Diffusion. This theory proposes that adhesive macromolecules diffuse into the substrate, thereby eliminating the interface, and so can only apply to compatible polymeric substrates. It requires that the chain segments of the polymers possess sufficient mobility and are mutually soluble. The solvent welding of thermoplastics such as PVC (polyvinyl chloride), softened with a chlorinated solvent, is an example of such conditions being met. Diffusion will also take place when two pieces of the same plastic are heat-sealed. The joining of plastic service pipes for carrying gas and water makes use of the diffusion mechanism. 

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