Diffusion-Limited Capture

In the spherical polar coordinate system, this equation is written as 

The net flux of the chains on the surface of the spherical sink is the surface area (4ttR ) times the steady-state flux into the sink at the distance R, (Ddc r)/dr) /,  

The time evolution of the polymer concentration near the absorbing sink is 

The integral in the above equation is the error function and is tabulated in mathematical handbooks (Abramowitz and Stegun 1965). The net flux of polymer chains arriving on the surface of the sink at time t follows as 

The same procedure can be carried out if the sink is an absorbing circular disk of radius R embedded in a planar wall of infinite dimensions (Pigure 9.2b), although the mathematical details are lengthier (Carslaw and Jaeger 1986). Por this situation, which is directly pertinent to the capture of polymer chains from the donor compartment by the pore, the steady-state flux in the number concentration of polymer molecules is 

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Figure 9.4 Diffusion-limited capture of polymer molecules by (a) an absorbing spherical sink of radius Rand (b) an absorbing circular disk of radius R embedded on a thin infinite membrane.

The key message of the above derivations is that the steady-state net flux of the polymer molecules into an absorbing region, namely the diffusion-limited capture rate, is simply proportional to the product of the linear size of the absorbing region, the diffusion coefiicient of the polymer, and the average polymer concentration co in the donor compartment. 

Carbonic anhydrase II, present in human red blood cells (RBCs), catalyzes the reversible hydration of C02. It is one of the most efficient enzymes and only diffusion-limited in its turnover numbers. The catalytic Zn11 is ligated by three histidine residues and OH this ZnOH+ structure renders the zinc center an efficient nucleophile which is able to attack the C02 molecule and capture it in an adjacent hydrophobic pocket. The catalytic mechanism is shown in Figure 9.5. 

The concept of selectivity is most commonly encountered (and most useful in mechanistic investigations, see Chapter 2) when a reactant or a reactive intermediate has alternative bimolecular routes it is then also very useful in yield optimisation in chemical process development [12]. The reaction in Scheme 4.3 involves an electrophilic intermediate (X) which is captured by nucleophilic reagent C (which could be solvent). If another nucleophilic species (D) is added to the reaction mixture, the additional product D—X is formed in competition with C—X. If Ap is known (e.g. if D is known to react with electrophiles at the diffusion limit), then values for [D], [C] and the product ratio [C—X]/[D—X] allow determination of kc, i.e. quantitative information about the reactivity of X with C, and information about the selectivity of X in reactions with nucleophiles. 

The model and the results presented here illustrate the physicochemical processes involved in char gasification with simultaneous sulfur capture. In particular, they demonstrate that diffusion limitations in the gasification reactions enable the conversion of CaO to CaS within the char even though CaS formation is not feasible at bulk gas conditions. Furthermore, this first version of the model correctly predicts the trends observed experimentally. Future effort in this area will focus on quantitative comparisons of model predictions with results from carefully designed gasification experiments. 

To obtain the overall uncoupling rate constant, ku, we can again make use of the capture probability y [see Eq. (35)], such that kc = yk+. Recall that y quantifies the extent to which receptor/ligand association is rate-limited by the reaction step. As y approaches 0, association is severely reaction-limited, while as y nears 1, binding is almost purely diffusion-limited. ku is thus given by... 

Three thousand separate trajectories were simulated for each receptor density and at each truncation height. In contrast to molecular dynamics, in BDS the solvent is treated as a viscous continuum and, hence, time steps on the order of 0.1 ps are appropriate. Diffusion-limited binding or instantaneous reaction upon ligand-receptor collision was assumed. Computationally, this condition was considered met when the center of the ligand particle and the center of the receptor were less than 20 A apart. Trajectories were truncated when the ligand s height above the membrane surpassed Q, the truncation distance. Sample trajectories of both a successful capture and an escape are shown in Figs. 16a and 16b. 

Usually the defects reduce the lifetime of non-equilibrium carriers and, consequently, their diffusion length in semiconductors. However, because of the presence of the closely spaced AlGaAs barriers, the carrier capture by the QDs in our samples is not diffusion-limited. That is why a difference in the quenching factor of the PL intensity at a given irradiation dose for the above- and below-bandgap excitation for all energies above the m = 2 QD excited state (Fig. lb) is not observed. Thus, the loss of carriers occurs mainly in the dots due to tunneling of caniers from the dots to adjacent non-radiative recombination centers. 

In the model of diffusion limited aggregation, growth occurs by capture of a diffusing particle. Two sets of components, X, (as previously) and diffusers (/),) are postulated. The reaction scheme is... 

In the diffusion-limited regime, the capture rate decreases with chain length and is linear in the polymer concentration. It is, of course, independent of the electric field or convective velocity fields in this regime. 

Thus, in addition to the diffusion-limited result, there emerges a multiplicative factor due to flow, which is proportional to the one-third power of the Peclet number Pe = vR/D. In the flow-dominated regime, the capture rate is proportional to. This is to be contrasted with the Smoluchowski result... 

Overall, the capture rate falls either within the barrier-dominated regime or drift-dominated regime, and the diffusion-limited behavior is not dominant for polyelectrolyte transport under an applied voltage. 

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