Interfacial concentration profile

As a first application for the use of the free energy functional, we will discuss the calculation of interfacial concentration profiles (x) between coexisting unmixed phases and interfacial tension [132-134]. For a symmetric mixture (Na = Nb = N) phase coexistence occurs for p = 0, and since the interfacial profile 4HX) also must be found by minimizing Eq. (47), we look for a solution of (cja = ctb = a)... 

The general problem of polymer-polymer interdiffusion is studied and used to illustrate several unique aspects of the orientation and motion of macromolecules at interfaces. Specific new results are obtained for the short time evolution of the Interfacial concentration profile which is shown, following the reptation... 

Suppose we assume a step function for the interfacial concentration profile, as described in Figure 6.3. The concentration remains Cai from the interface to a distance and then abruptly changes to CAb The quantity Cai is therefore... 

I. Caucheteux, H. Hervet, F. Rondelez, L. Auvray, and J. P. Connon, Polymer adsorption at the solid-liquid interface the interfacial concentration profile, in New Trends in Physics and Physical Chemistry of Polymers (L. H. Lee, ed.). Plenum Press, New York (1989). 

POLYMER ADSORPTION AT THE SOLID-LIQUID INTERFACE THE INTERFACIAL CONCENTRATION PROFILE... 

On the other hand, the H-W analysis was developed based on the NIA, i.e., the interfacial thickness of a block copolymer is the same as that of an immiscible polymer blend. Thus, the H-W analysis is applicable only to the SSL, but near %N 10 the interfacial thickness would be much larger than that for an immiscible polymer blend. Furthermore, in the H-W analysis the mean-fleld potential cok (r) was treated on an ad hoc basis and thus the H-W analysis is applicable to very large values of %N. The phase boundary between the lamellar and cylindrical microdomains predicted by the V-W theory is the same as that predicted by the H-W analysis when %N > 50. Also, the phase boundary between the cylindrical and spherical microdomains predicted by both theories is the same when > 70. However, even when is approximately 80, the phase boundary between the spherical microdomain and homogeneous state predicted by both theories is quite different. Helfand and Wasserman [12] reported that for %N > 40 the interfacial concentration profiles for lamellar microdomains obtained with NIA are essentially the same as those obtained by numerical simulation. However, it is expected that for cylindrical or spherical microdomains, NIA is valid for values of much larger than about 40. The main result of the V-W analysis is that for > 70, the order-order transitions are almost independent of and the transition between lamellar and cylindrical microdomains occurs at / 0.67 and the transition between cylindrical and spherical microdomains occurs at / 0.85. 

To ensure that the detector electrode used in MEMED is a noninvasive probe of the concentration boundary layer that develops adjacent to the droplet, it is usually necessary to employ a small-sized UME (less than 2 /rm diameter). This is essential for amperometric detection protocols, although larger electrodes, up to 50/rm across, can be employed in potentiometric detection mode [73]. A key strength of the technique is that the electrode measures directly the concentration profile of a target species involved in the reaction at the interface, i.e., the spatial distribution of a product or reactant, on the receptor phase side. The shape of this concentration profile is sensitive to the mass transport characteristics for the growing drop, and to the interfacial reaction kinetics. A schematic of the apparatus for MEMED is shown in Fig. 14. 

Typical theoretical concentration profiles, observed at a probe electrode, for the consumption of a receptor phase species in a first-order interfacial reaction are shown in Fig. 16. The simulation involved solving Eq. (30) with appropriate boundary conditions. 

Fig. 5.9 Linear concentration profiles for the extraction of by BH(org) in presence of an interfacial chemical reaction.

As noted previously, most environmental flows are turbulent. The diffusive sublayer, where only diffusion acts to transport mass and the concentration profile is linear, is typically between 10 /xm and 1 mm thick. Measurements within this sublayer are not usually feasible. Thus, the interfacial flux is typically expresses as a bulk transfer... 

In reality, the concentration gradient is constant for only a short distance from the interface and then becomes asymptotic to zero in the bulk. But one can resort to a linearization of the concentration profile, and then one can use the artifice of an imagined (i.e., simplified) diffusion layer in which the concentration is taken as if it changed in a linear fashion from the interfacial value to the bulk value c0. The effective thickness 8 of the diffusion layer, which can be taken as a constant independent of time only under steady-state conditions in which natural convection occurs, proved a useful quantity. With its aid, one can write out the flux-equality condition in the form... 

Figure 15.1 Effect of /3-phase particle size on the concentration, Xeq, of component B in the a phase in equilibrium with a /3-phase particle in a binary system at the temperature T. assuming that, 6 is pure B. (a) Schematic free-energy curves for a phase and three /3-phase particles of different radii, R > R2 > Rs. The free energies (per mole) of the particles increase with decreasing radius due to the contributions of the interfacial energy, which increase as the ratio of interfacial area to volume increases, (b) Corresponding phase diagram. The concentration of B in the a phase in equilibrium with the /0-phase particles, as determined by the common-tangent construction in (a), increases as R decreases, as shown in an exaggerated fashion for clarity, (c) Schematic concentration profiles in the a matrix between the three /3-phase particles.

Let us first consider what would result if the impulse described earlier were created near an impenetrable inert barrier. Such a barrier would behave as a reflector. In Figure 2.6a, an impulse of molecules is suddenly created in solution a short distance away from the interface. The dashed line is the profile that would result in the absence of the barrier. The dotted line results from folding this concentration profile about the interfacial plane. Because diffusion is a linear process, the actual concentration profile in the presence of the barrier is the sum of the dotted and dashed lines, as shown in Figure 2.6b. 

Figure 3.6 Examples of two different concentration profiles leading to the same interfacial excess concentration I )1 ...

Fig. 5.8 Concentration profile for an interfacial reaction with reagent depletion in the diffusion film.

This case of the semi-infinite slab can be solved to yield both a concentration profile and an interfacial flux which ... 

To evaluate C](5 ) and C1(L ), we introduce what has been designated [28, 29] the interfacial zone equilibrium approximation the concentration profiles of all charged defect species within the two interfacial regions (0interior zone (5 local space-charge neutrality can be approximated by... 

CLM method can also be combined with various kinds of spectroscopic methods. Fluorescence lifetime of an interfacially adsorbed zinc-tetra-phenylporphyrin complex was observed by a nanosecond time-resolved laser induced fluorescence method [25]. Microscopic resonance Raman spectrometry was also combined with the CLM. This combination was highly advantageous to measure the concentration profile at the interface and a bulk phase [14]. Furthermore, circular dichroic spectra of the liquid-liquid interface in the CLM could be measured [19]. 

Solution of the coupled mass-transport and reaction problem for arbitrary chemical kinetic rate laws is possible only by numerical methods. The problem is greatly simplified by decoupling the time dependence of mass-transport from that of chemical kinetics the mass-transport solutions rapidly relax to a pseudo steady state in view of the small dimensions of the system (19). The gas-phase diffusion problem may be solved parametrically in terms of the net flux into the drop. In the case of first-order or pseudo-first-order chemical kinetics an analytical solution to the problem of coupled aqueous-phase diffusion and reaction is available (19). These solutions, together with the interfacial boundary condition, specify the concentration profile of the reagent gas. In turn the extent of departure of the reaction rate from that corresponding to saturation may be determined. Finally criteria have been developed (17,19) by which it may be ascertained whether or not there is appreciable (e.g., 10%) limitation to the rate of reaction as a consequence of the finite rate of mass transport. These criteria are listed in Table 1. 

Figure 22. L.h.s. Four basic space charge situations involving ionic conductors (here silver ion conductor) a) contact with an isolator, b) contact with a second ion conductor, c) grain boundary, d) contact with afluid phase. R.h.s. Bending of energy levels and concentration profiles in space charge zones ( = 0 refers to the interfacial edge).

FIGURE 1.8 Schematic representation of the concentration profile (c) as a function of distance (spatial coordinate) normal to the phase boundary full line (bold) in the real system broken line in the reference system chain-dotted line boundaries of the interfacial layer. 

In order to discuss a number of important quantities we consider the interfacial profile for the case of positive adsorption. Such a profile is sketched in fig. 5.6. It represents the polymer concentration dz) as a function of the distance z from the interface. The quantity c(z) is related to the volume fraction concentration profile of segments belonging to free molecules, having no contact with the surface. The excess adsorbed amount r (the amount of... 

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