Concentration profile spherical interface

The concentration profile for a reactant A which must migrate from a drop or bubble into the continuous phase to react might be as shown in Figure 12-10. There is a concentration drop around the spherical drop or bubble because it is migrating outward, but, as with a planar gas-liquid interface in the falling film reactor, there should be a discontinuity in at the interface due to the solubility of species A and a consequent equilibrium distribution between phases. 

Figure 2.3 compares the profiles of the electrostatic potential and the concentration of redox molecules at interfaces of spherical electrodes with radii of 1 and 100 nm, respectively, obtained by solving Equation 2.2 and 2.3 for one-electron reduction of a -1 valence reactant at an electrode potential of ca. -0.5 V with respect to the potential of zero charge (PZC). Unless specifically stated, we will assume that the formal potential of the considered redox reaction (E°) is equal to the PZC for the sake of simplicity. The horizontal ordinate represents the distance from the OHP normalized by 5, the distance at which the concentration of reactant reaches 95% of its bulk value. We may consider 5 the thickness of the entire interfacial region. One can see that the potential profile (EDL) is nearly sqneezed to a vertical line at OHP at an electrode of 100 nm in radius as compared with the concentration profile (CDL), whereas significant penetration of the potential profile into the concentration profile occurs at electrode of 1 nm in radius, which indicates the transition of the interface from a CDL dominated natnre to EDL nature as electrode size approaches nanometer scales. 

Figure 4-21 The concept of boundary layer and boundary layer thickness 5. (a) Compositional boundary layer surrounding a falling and dissolving spherical crystal. The arrow represents the direction of crystal motion. The shaded circle represents the spherical particle. The region between the solid circle and the dashed oval represents the boundary layer. For clarity, the thickness of the boundary layer is exaggerated, (b) Definition of boundary layer thickness 5. The compositional profile shown is "averaged" over all directions. From the average profile, the "effective" boundary layer thickness is obtained by drawing a tangent at x = 0 (r=a) to the concentration curve. The 5 is the distance between the interface (x = 0) and the point where the tangent line intercepts the bulk concentration.

Although the shape of the profile of a "spherical diffusion couple" is similar to that of a one-dimensional diffusion couple, one difference is that, whereas the midconcentration position stays mathematically at the initial interface for the normal diffusion couple, the midconcentration position moves with time in the "spherical diffusion couple." Initially, the concentration at the initial interface (r = a) is the mid-concentration Cmid = (Ci + C2)/2. However, as diffusion progresses, the concentration at r = a is no longer the mid-concentration. Rather, the location of the mid-concentration moves to a smaller r. Define the mid-concentration location as Tq. Then Tq x a(l — z /2) for small times. If layer 1 is the solid core (meaning r extends to 0), the concentration at the center begins... 

The authors review the theoretical analysis of the hydrodynamic stability of fluid interfaces under nonequilibrium conditions performed by themselves and their coworkers during the last ten years. They give the basic equations they use as well as the associate boundary conditions and the constraints considered. For a single interface (planar or spherical) these constraints are a Fickean diffusion of a surface-active solute on either side of the interface with a linear or an erfian profile of concentration, sorption processes at the interface, surface chemical reactions and electrical or electrochemical constraints for charged interfaces. General stability criteria are given for each case considered and the predictions obtained are compared with experimental data. The last section is devoted to the stability of thin liquid films (aqueous or lipidic films). 

The exponential dependence of -1/3 is consistent with geometric scaling arguments for close-packed spheres of diameter De [18,19]. Thus, models used to describe these forces are based on the successive removal of spherical micelles as they are progressively confined between approaching interfaces [17,20,21]. Later, the TIRM technique was utilized to study similar oscillating force profiles at lower surfactant concentrations but, of course, still above the cmc [22]. 

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