Flux-difference methods, homogeneous

The explicit, finite difference method (9,10) was used to generate all the simulated results. In this method, the concurrent processes of diffusion and homogeneous kinetics can be separated and determined independently. A wide variety of mechanisms can be considered because the kinetic flux and the diffusional flux in a discrete solution "layer" can be calculated separately and then summed to obtain the total flux. In the simulator, time and distance increments are chosen for convenience in the calculations. Dimensionless parameters are used to relate simulated data to real world data. 

In experimental studies of diffusion, the diffusion-couple technique is often used. A diffusion couple consists of two halves of material each is initially uniform, but the two have different compositions. They are joined together and heated up. Diffusive flux across the interface tries to homogenize the couple. If the duration is not long, the concentrations at both ends would still be the same as the initial concentrations. Under such conditions, the diffusion medium may be treated as infinite and the diffusion problem can be solved using Boltzmann transformation. If the diffusion duration is long (this will be quantified later), the concentrations at the ends would be affected, and the diffusion medium must be treated as finite. Diffusion in such a finite medium cannot be solved by the Boltzmann method, but can be solved using methods such as separation of variables (Section 3.2.7) if the conditions at the two boundaries are known. Below, the concentrations at the two ends are assumed to be unaffected by diffusion. 

At typical flow rates, the concentration in the dialysate, Cout, is less than the actual concentration in the extracellular fluid, Cext (23). The ratio of Cout/Cext is defined as relative recovery, R, and must be considered for probe calibration and sampling optimization. In vitro, R is easily calculated because the dialysate and the extracellular fluid are homogenous therefore, probe calibration is easily obtained. However, in in vivo studies, calculation of R is difficult because of the active removal of neurotransmitters by uptake and tortuosity. Movement of analytes is impeded by tissue that surrounds the probe, and this movement cannot be easily accounted for with in vitro calibrations. Therefore, the most common method to determine concentrations in vivo is the zero-net flux method, in which known analyte concentrations are added to the perfusate (Cin), and then the analyte concentration is measured at the probe outlet (Cout)- The difference between analyte concentration at the inlet and outlet is used to establish the actual analyte concentration in the tissue, and the relative recovery rate can be calculated. This calibration method can be used to estimate basal levels of neurotransmitters. For example, the zero-net flux method has been used to determine that basal concentrations of dopamine are approximately 1-3.5 nM (24, 25). Although basal level concentrations... 

Cross-section sets for Assembly 6A were generated in the manner described in Ref. 3 using ENDF/B Version 1 i.e., MC homogeneous cross sections, corrected tor resonance spatial self-shielding in U and D, and wei ited with the unit-cell fluxes. The keff of the unitcell calculated with 1-D, Sn (n > 16) and the anisotropic option (Pt only) was 1% greater than that calculated In the same manner with the transport approximation. This difference disappeared, however, when the transverse dimension was made Infinite this indicated that the decrease in leakage associated with the use of S total Instead of S transport was responsible for this difference. Similar difference in keff between the two methods was found for the as-built system. 

In addition to the amperometric feedback mode described above, other amperometric operation modes are also possible. For example, in the substrate generation/tip collection (SG/TC) mode, ip is used to monitor the flux of electroactive species from the substrate and vice versa for the tip generation/substrate collection (TG/SC) mode. These operation modes will be described in Section 12.3.1.3 and are useful in studies of homogeneous reactions that occur in the tip-substrate gap (see Section 12.4.2) and also in the evaluation of catalytic activities of different materials for useful reactions, e.g., oxygen reduction and hydrogen oxidation (see Section 12.4.3). In addition to the amperometric methods, other techniques, e.g., potentiometric method is also applicable for SECM and will be discussed in Section 12.3.2. We will also update the techniques suitable for the preparation of SECM amperometric tips in Section 12.3.1.1 and potentiometric probes in Section 12.3.2.2. 

These two different scales are related using the homogenization method in mesoscale computation. For example, effective values of charge flux, elasticity tensor, and conductivities in macroscale are derived using values defined at the microscale and porosity of the region under consideration. 

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