Reaction of immobile particles

Let us illustrate an idea of the numerical solution of the set (5.1.14) - (5.1.16) (concentration equation (4.1.18) can be integrated trivially) making use of the basic equation 

In the symmetric difference scheme (5.1.18) integrals in (5.1.7), (5.1.8) are calculated within accuracy of O(Ar ), therefore an accuracy of the difference 

Disadvantage of this direct method to solve (5.1.14) to (5.1.16) is that the required computational time is proportional to due to an increase in coordinate parameter m (functions 5 are non-zero at r , where increases in time). This is why another coordinate variable q = r/, is more useful. Its use results in the correlation functions which are practically stationary at long times [8]. The basic equation is 

The accuracy of this difference scheme is 0 Ar -F Ax ), provided the integrals entering (5.1.7) and (5.1.8) are calculated with accuracy of O(Ax ). The difference equations are solved first from the left to the right with the boundary condition g = 0 (no correlations at r ). Similar to the discussed scheme (5.1.18), non-line u effects are taken here into account iteratively. 

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Let us consider accuracy of the superposition approximation for the two quite different classes of problems a long-range reaction of immobile particles and a diffusion-controlled one, where the diffusion length Id arises. 

The analysis of the diffusion-controlled computer simulations confirms once more conclusions drawn above for the static reactions of immobile particles. In particular, the superposition approximation gives the best lower bound estimate of the kinetics reaction, n = n(i). Divergence of computer simulations and analytical theory being negligible for equal concentrations become essential for large depths and when one of reactants is in excess. The obtained results allow us to use the superposition approximation for testing the applicability of simple equations of the linear theory in those cases when computer simulations because of some reasons cannot be performed. Examples will be presented in Chapter 6. 

As it was shown above, in Section 6.1.1 using the reaction of immobile particles as example, the kinetic equations (5.1.2) to (5.1.4), being derived... 

The existence of the (quasi) steady-state in the model of particle accumulation (particle creation corresponds to the reaction reversibility) makes its analogy with dense gases or liquids quite convincing. However, it is also useful to treat the possibility of the pattern formation in the A + B —> 0 reaction without particle source. Indeed, the formation of the domain structure here in the diffusion-controlled regime was also clearly demonstrated [17]. Similar patterns of the spatial distributions were observed for the irreversible reactions between immobile particles - Fig. 1.20 [25] and Fig. 1.21 [26] when the long range (tunnelling) recombination takes place (recombination rate a(r) exponentially depends on the relative distance r and could... 

The kinetics of the diffusion-controlled reaction A + B —> 0 under study is defined by the initial conditions imposed on the kinetic equations. Let us discuss this point using the production of geminate particles (defects) as an example. Neglecting for the sake of simplicity diffusion and recombination (note that even the kinetics of immobile particle accumulation under steady-state source is not a simple problem - see Chapter 7), let us consider several equations from the infinite hierarchy of equations (2.3.43) ... 

Accuracy of the difference scheme is 0(Af + Ar2), which could be reduced to 0(At2 4- Ar2) by means of the symmetrical difference scheme. In practice schemes with monotonously increasing spatial and temporal steps are usually used for these purposes [1, 9-11]. As r 1, Ar is small but increases with r whereas At increment is limited by the condition that the relative change of gm at any step should not exceed a given small value. Unlike the case of immobile particle reaction, the calculation of the functionals J[Z], (5.1.37) and (5.1.38), requires one-dimensional integration only which is not time-consuming. 

Computer simulations of bimolecular reactions for a system of immobile particles (incorporating their production) has a long history see, e.g., [18-22]. For the first time computer simulation as a test of analytical methods in the reaction kinetics was carried out by Zhdanov [23, 24] for d, = 3. Despite the fact that his simulations were performed up to rather small reaction depths, To < 1, it was established that of all empirical equations presented for the tunnelling recombination kinetics (those of linear approximation - (4.1.42) or (4.1.43)) turned out to be mostly correct (note that equations (5.1.14) to (5.1.16) of the complete superposition approximation were not considered.) On the other hand, irrespective of the initial reactant densities and space dimension d for reaction depths T To his theoretical curves deviate from those computer simulated by 10%. Accuracy of the superposition approximation in d = 3 case was first questioned by Kuzovkov [25], it was also... 

The multipole interaction of immobile particles (4.1.44) is an additional way to check up advantages of the superposition approximation [8]. The reason is that the tunnelling recombination (3.1.2) serves better as an example of short-range reaction. Indeed, the distinctive scale tq characterizing distant (non-contact) interaction could be defined as... 

New reaction asymptotic law (2.1.78) emerges due to formation during the reaction course of a new spatial scale - the correlation length = Id- Similar to the case of immobile particles, we can expect here that at long times the coordinate r enters into the correlation function in a scaling form rj = r/Io, so that Y(r,t) —> Y(t, t), X (r,t) -> where the second variable... 

Electrochemical calorimetry — is the application of calorimetry to thermally characterize electrochemical systems. It includes several methods to investigate, for instances, thermal effects in batteries and to determine the -> molar electrochemical Peltier heat. Instrumentation for electrochemical calorimetric studies includes a calorimeter to establish the relationship between the amount of heat released or absorbed with other electrochemical variables, while an electrochemical reaction is taking place. Electrochemical calorimeters are usually tailor-made for a specific electrochemical system and must be well suited for a wide range of operation temperatures and the evaluation of the heat generation rate of the process. Electrochemical calorimeter components include a power supply, a device to control charge and discharge processes, ammeter and voltmeter to measure the current and voltage, as well as a computerized data acquisition system [i]. In situ calorimetry also has been developed for voltammetry of immobilized particles [ii,iii]. 

Potcntiomctric Biosensors Potentiometric electrodes for the analysis of molecules of biochemical importance can be constructed in a fashion similar to that used for gas-sensing electrodes. The most common class of potentiometric biosensors are the so-called enzyme electrodes, in which an enzyme is trapped or immobilized at the surface of an ion-selective electrode. Reaction of the analyte with the enzyme produces a product whose concentration is monitored by the ion-selective electrode. Potentiometric biosensors have also been designed around other biologically active species, including antibodies, bacterial particles, tissue, and hormone receptors. 

The preparation of immobilized CdTe nanoparticles in the 30-60 nm size range on a Te-modified polycrystalline Au surface was reported recently by a method comprising combination of photocathodic stripping and precipitation [100], Visible light irradiation of the Te-modified Au surface generated Te species in situ, followed by interfacial reaction with added Cd " ions in a Na2S04 electrolyte. The resultant CdTe compound deposited as nanosized particles uniformly dispersed on the Au substrate surface. 

When performing catalytic reactions or reactions with immobilized reactants, a bed or support has to be fiUed into a tube or capillary. The fiUing may be a bed of powder, a bed of granules or a three-dimensional material network (e.g. a polymerized foam). By special choice of the filling, e.g. very regularly sized particles, it is attempted to improve the flow characteristics. 

Seki, M., Naito, K. I., and Furusaki, S., Effect of Co-Immobilization of Microporous Particles on the Overall Reaction Rate of Immobilized Cell Biocatalysts, J. Chem. Eng. Jpn., 26 662 (1993)... 

The reactions used for coupling affinity ligands to nanoparticles or microparticles basically are the same as those used for bioconjugation of molecules or for immobilization of ligands onto surfaces or chromatography supports. However, with particles, size can be a major factor in how a reaction is performed and in its resultant reaction kinetics. Since particle types can vary from the low nanometer diameter to the micron size, there are dramatic differences in how such particles behave in solution and how the density of reactive groups or functional groups affects reactions. 

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