Flat Interface

In the mean field considerations above, we have assumed a perfectly flat interface such that the first tenn in the Hamiltonian (B3.6.21) is ineffective. In fact, however, fluctuations of the local interface position are important, and its consequences have been studied extensively [, 58]. 

One can regard the Hamiltonian (B3.6.26) above as a phenomenological expansion in temis of the two invariants Aiand//of the surface. To establish the coimection to the effective interface Hamiltonian (b3.6.16) it is instnictive to consider the limit of an almost flat interface. Then, the local interface position u can be expressed as a single-valued fiinction of the two lateral parameters n(r ). In this Monge representation the interface Hamiltonian can be written as... 

Figure 8.15 Experimental gas-liquid flat interface reactor (Wachi and Jones, 1991b)...

Wachi and Jones (1991b) used a gas-liquid flat interface reactor as a semi-batch precipitation cell for the experimental measurement of calcium carbonate precipitation, as shown in Figure 8.15. 

These concepts were implemented according to the following scheme the liquid element surrounding the bubble and the bulk are considered as two separate dynamic reactors that operate independent of each other and interact at discrete time intervals. In the beginning of the contact time, the interface is being detached from the bulk. When overcome by the bubble, it returns to the bulk and is mixed with it. Hostomsky and Jones (1995) first used such a framework for crystal precipitation in a flat interface stirred cell. To formulate it for a... 

The initial development of a cellular structure from an originally flat interface has been at least partially understood [130]. Let us look only at the large-wavelength A limit (for more details see [122]). In the numerical calculations it was found [123] that for fixed cell-spacing A at increasing velocity a tail instability occurs. A side branch in the groove between two... 

In concentrated NaOH solutions, however, the deviations of the experimental data from the Parsons-Zobel plot are quite noticeable.72 These deviations can be used290 to find the derivative of the chemical potential of a single ion with respect to both the concentration of the given ion and the concentration of the ion of opposite sign. However, in concentrated electrolyte solutions, the deviations of the Parsons-Zobel plot can be caused by other effects,126 279"284 e.g., interferences between the solvent structure and the Debye length. Thus various effects may compensate each other for distances of molecular dimensions, and the Parsons-Zobel plot can appear more straight than it could be for an ideally flat interface. 

The flat interface model employed by Marcus does not seem to be in agreement with the rough picture obtained from molecular dynamics simulations [19,21,64-66]. Benjamin examined the main assumptions of work terms [Eq. (19)] and the reorganization energy [Eq. (18)] by MD simulations of the water-DCE junction [8,19]. It was found that the electric field induced by both liquids underestimates the effect of water molecules and overestimates the effect of DCE molecules in the case of the continuum approach. However, the total field as a function of the charge of the reactants is consistent in both analyses. In conclusion, the continuum model remains as a good approximation despite the crude description of the liquid-liquid boundary. 

Marcus model for flat interfaces [Eqs. (17)-(19)], it was realized that the continuum model for a mixed solvent layer also approaches these experimental values [3-6]. 

If the pressure dependence of the molar volume of the liquid is neglected, integration from a flat interface (r = oo) yields... 

For the special case of straight pores growing orthogonal to the electrode surface forming a flat interface to the bulk, the pore length l becomes equivalent to the layer thickness D. Equation (6.1) then also defines the growth rate of the whole porous layer rPS. The growth rate rPS of a porous layer depends on several... 

When a drop (water) falls to a flat interface (benzene-water) the entire drop does not always join the pool (water). Sometimes a small droplet is left behind and the entire process, called partial coalescence, is repeated. This can happen several times in succession. High-speed motion pictures, taken at about 2000 frames per second, have revealed the details of the action (W3). The film (benzene) ruptures at the critical film thickness and the hole expands rapidly. Surface and gravitational forces then tend to drag the drop into the main pool (water). But the inertia of the high column of incompressible liquid above the drop tends to resist this pull. The result is a horizontal contraction of the drop into a pillar of liquid above the interface. Further pull will cause the column to be pinched through, leaving a small droplet behind. Charles and Mason (C2) have observed that two pinches and two droplets occurred in a few cases. The entire series of events required about 0.20 sec. for aniline drops at an aniline-water interface (C2, W3). 

Locally electro-neutral concentration polarization of a binary electrolyte at an ideally cation-permselective homogeneous interface. Consider a unity thick unstirred layer of a univalent electrolyte adjacent to an ideally cation-permselective homogeneous flat interface. Let us direct the x-axis normally to this interface with the origin x = 0 coinciding with the outer (bulk) edge of the unstirred layer. Let a unity electrolyte concentration be maintained in the bulk. 

Equation C.23 is the form of the Gibbs-Thomson equation introduced in Eq. C.17. It is a conditionfor mechanical equilibrium in a two-phase system with a curved interface. The phase located on the side of the interface toward its center of curvature (e.g. the (3 phase in Fig. C.5), has the higher pressure. Note also that for a flat interface, Eq. C.23 gives Pa = P0, as expected. 

Earlier, the first studies by Cockbain and McRoberts (C7) indicated that the time necessary for coalescing, measured from the point that the drop apparently has come to rest on the flat interface till the moment the first coalescence sets in, is spread statistically around an average value. Both the standard deviation and the mean value depend on the phase system and decrease with increasing temperature and with decreasing drop size, while surface active agents and small impurities that collect at the interface have a strong retarding effect. 

The theory of Groothuis and Zuiderweg is confirmed for drops coalescing on a flat interface by MacKay and Mason (Mia) and for pairs of drops rising in an extraction column by Smith et al. (S3). Dora Thiessen (T3)... 

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