Accumulation kinetics

The pH dependence of HIV-1 protease has been assessed by measuring the apparent inhibition constant for a synthetic substrate analog (b). The data are consistent with the catalytic involvement of ionizable groups with pK values of 3.3 and 5.3. Maximal enzymatic activity occurs in the pH range between these two values. On the basis of the accumulated kinetic and structural data on HIV-1 protease, these pK values have been ascribed to the... 

For bi-exponential accumulation kinetics the contribution of the fast alpha phase (Ralpha = 1/(1 - exp(-alpha Tau))) decreases (Ralpha Ca Rbeta Cb) as compared with the contribution of the slow beta-phase (Rbeta = 1/(1 - exp(-beta Tau))). 

Stehly, G. R., Hayton, W. L. (1990) Effect of pH on the accumulation kinetics of pentachlorophenol in goldfish. Arch. Environ. Contam. Toxicol. 19, 464-470. 

Newman, M.C. and D.K. Doubet. 1989. Size-dependence of mercury (II) accumulation kinetics in the mosquitofish, Gambusia affinis (Baird and Girard). Arch. Environ. Contam. Toxicol. 18 819-825. 

Mobile H centres in alkali halides are known to aggregate in a form of complex hole centres [64] this process is stimulated by elastic attraction. It was estimated [65, 66] that for such similar defect attraction the elastic constant A is larger for a factor of 5 than that for dissimilar defects - F, H centres. Therefore, elastic interaction has to play a considerable role in the colloid formation in alkali halides observed at high temperatures [67]. In this Section following [68] we study effects of the elastic interaction in the kinetics of concentration decay whereas in Chapter 7 the concentration accumulation kinetics under permanent particle source will be discussed in detail. 

As in previous Chapters, for practical use this infinite set (7.1.1) has to be decoupled by the Kirkwood - or any other superposition approximation, which permits to reduce a problem to the study of closed set of densities pm,m with indices (m + mr) 2. As earlier, this results in several equations for macroscopic concentrations and three joint correlation functions, for similar, X (r,t),X-s r,t), and dissimilar defects Y(r,t). However, unlike the kinetics of the concentration decay discussed in previous Chapters, for processes with particle sources direct use of Kirkwood s superposition approximation gives good results for small dimesionless concentration parameters Uy t) = nu(t)vo < 1 only (vq is d-dimensional sphere s volume, r0 is its radius). The accumulation kinetics predicted has a very simple form [30, 31]... 

It was demonstrated in [31] that it is namely the black sphere model which introduces large errors into this accumulation kinetics treated in terms of the superposition approximation. The way of avoiding the superposition approximation s shortcomings was developed in [33] and discussed below. 

The direct consequence of this statement for Kirkwood s superposition approximation is as follows. Substitution of equation (2.3.62) into p2,i yields correct order of its magnitude, a 1, provided f] — rfl < ro, r% — r[ > ro (i.e., there is a single A in the recombination sphere around B), since two-point density p p(f, r[ t) oc and pi,i (r 2i t) oc (<7o)° (i.e., is limited as well as another density />2,o> which does not fall into category of virtual configurations). On the other hand, for coordinates satisfying ri - f[ < ro, r 2 — rj < ro (i.e., defect B has in its recombination sphere two defects A) substitution of equation (2.3.62) results in p p oc instead of the correct M Due to this the superposition approximation neglects in the limit (To oo a number of terms in equations which finally leads to a considerable error in the accumulation kinetics. 

Equation (7.1.16) is asymptotically (cto — oo) exact. It shows that the accumulation kinetics is defined by (i) a fraction of AB pairs, 1 — u>, created at relative distances r > r0, (ii) recombination of defects created inside the recombination volume of another-kind defects. The co-factor (1 - <5a - <5b ) in equation (7.1.16) gives just a fraction of free folume available for new defect creation. Two quantities 5a and <5b characterizing, in their turn, the whole volume fraction forbidden for creation of another kind defects are defined entirely by quite specific many-point densities pmfl and po,m > he., by the relative distribution of similar defects only (see equation (7.1.17)). 

If we succeeded in calculating the series in equation (7.1.17), the accumulation kinetics problem under question would be solved. However, an infinite set of coupled equations for pm turns out to be too complicated and thus we restrict ourselves to its cut-off by means of Kirkwood s superposition approximation, in order to get a closed set of non-linear equations for macriscopic densities n (t) and nB(t), as well as for the three joint correlation functions XA(r,t),XB(r,t) and Y(r,t). 

Fig. 7.1. Accumulation kinetics of immobile defects for different spatial dimensions d = 1 (curve 1), d = 2 (curve 2), d = 3 (curve 3). Concentration and irradiation time are in dimensionless units, time scale logarithmic.

It should be stressed once more that the accumulation curve n(t) (or U(t)), especially at high doses, cannot be described by a simple equation (7.1.53) which is often used for interpreting the real experimental data (e.g., [19, 20]). Despite there is the only recombination mechanism, the A + B —> 0 accumulation kinetics at long t due to many-particle effects is no longer exponential function of time (dose). Therefore, successful expansion of the experimental accumulation curve U = U(t) in several exponentials (stages) does not mean that several different mechanisms of defect creation are necessarily involved (as sometimes they suggest, e.g., [39, 40]). 

The accumulation kinetics of correlated (geminate) pairs is less studied. In [30, 41] it was demonstrated that in this case the aggregation effect is weakened and the saturation concentration is reduced essentially. This makes use of Kirkwood s superposition approximation more reliable. Employing the latter, the following relation was derived [30]... 

Integration here is over the recombination sphere. The accumulation kinetics remains to be equation (7.1.53), but with Uq from equation (7.1.56). It follows from (7.1.56) that creation of the Frenkel partners separated by the distinctive... 

The accumulation kinetics under study is defined by three dimensionless parametrs (, rA/r0 and rE/ro all three depend on the temperature. The physical meaning of the former parameter has been discussed earlier, in Section 6.3 and Section 7.1. The joint correlation functions XA, Xr characterize, as before, an aggregation effect of similar particles their random distribution is taken as the initial condition, XA(r,0) — X (r,0) = 1. 

The quantity Uo characterizes the degree of aggregation only on the average. Therefore it is important in each particular case to analyze also the spatial distribution of the defects. Thus, the low-temperature accumulation of the Frenkel defects in the two- and three-dimensional cases was simulated in [36, 114] and the obtained values of Uo for d = 2 considerably exceed the same in [113]. In contrast to the latter, the authors of [36, 114] used a circle as the recombination region. In its turn, the values of u0 obtained in [115] are considerably larger than those found in [114]. We note that it was assumed in [115] that, when an interstitial atom occurs at a site where the recombination spheres of several vacancies overlap, it recombines with the closest vacancy. This demonstrates very well how any details, insignificant at first glance, can affect considerably the accumulation kinetics. 

Lastly, in [118] the accumulation kinetics was simulated for immobile particles for d — 1,2 and 3 in a continuum model. The saturation concentrations obtained, Uq = 4.2,2.07 and 1.04 respectively, agree with results presented above. 

Fig. 7.17. Accumulation kinetics restricted by the tunnelling recombination. Curve 1 shows correlated pairs (r = 10a), p = 7 x 1016 cm 3s l, curve 2 - same with pj = 10pi, curve 3 - uncorrelated particles with pi = pi, curve 4 - uncorrelated particles with P4 = 5 x 1020 cm-3 s-1, curve 5 - as curve 4 with pi = 2.2 x 1016, curve 6 - dependence of the saturation concentration on the dose rate p.

Some analytic expressions are collected in Table 7.6 that have been used in literature to describe the experimental accumulation kinetics, n(t), or dn/dt, the rate of concentration accumulation. Experimentally such kinetics have been studied, both in the alkali-halide crystals [13, 17] and in many metals [43-45] in a wide temperature interval, starting with low (liquid-helium, 4 K) temperatures. Since often a succesful approximation of the accumulation curve is associated with better understanding of a micromechanism of defect formation (see, e.g., [40]) and with other important physical conclu-... 

Equation (3) is the most widely used in analyzing experimental curves, since its form is intuitively clear the rate of the defect accumulation is determined by the fraction of free volume of the crystal not occupied by previously created defects, without taking account of the overlap of the annihilation volumes of similar defects. Evidently it is applicable only in the initial stage of accumulation kinetics at relatively low concentrations of defects, nvo superposition approximation corresponds to the first two terms of expansion (2) in powers of nvo-... 

In our opinion, this book demonstrates clearly that the formalism of many-point particle densities based on the Kirkwood superposition approximation for decoupling the three-particle correlation functions is able to treat adequately all possible cases and reaction regimes studied in the book (including immobile/mobile reactants, correlated/random initial particle distributions, concentration decay/accumulation under permanent source, etc.). Results of most of analytical theories are checked by extensive computer simulations. (It should be reminded that many-particle effects under study were observed for the first time namely in computer simulations [22, 23].) Only few experimental evidences exist now for many-particle effects in bimolecular reactions, the two reliable examples are accumulation kinetics of immobile radiation defects at low temperatures in ionic solids (see [24] for experiments and [25] for their theoretical interpretation) and pseudo-first order reversible diffusion-controlled recombination of protons with excited dye molecules [26]. This is one of main reasons why we did not consider in detail some of very refined theories for the kinetics asymptotics as well as peculiarities of reactions on fractal structures ([27-29] and references therein). 

Minh, N.H., Someya, M., Minh, T.B., Kunisue, T., Watanabe, M., Tanabe, S., Viet, P.H., Tuyen, B.C., 2004. Persistent organochlorine residues in human breast milk from Hanoi and Hochiminh City, Vietnam Contamination, accumulation kinetics and risk assessment for infants. Environ. Pollut. 129, 431 441. 

Malarvannan, G., Kunisue, T., Isobe, T., Takahashi, S., Sudaryanto, A., Prudente, M., Tanabe, S., 2007. Specific Accumulation of Organohalogen Compounds in Human Breast Milk from the Philippines Levels, Distribution, Accumulation Kinetics and Infant Health Risk. Proceedings of the International Symposium on Pioneering Studies of Young Scientists on Chemical Pollution and Environmental Changes, November 17-19, 2006, Ehime University, Matsuyama, lapan, pp. 175-178. 

Sudaryanto, A., Kunisue, T., Kajiwara, N., Iwata, H., Adibroto, T.A., Hartono, P., Tanabe, S., 2006. Specific accumulation of organochlorines in human breast milk from Indonesia Levels, distribution, accumulation kinetics and infant health risk. Environ. Pollut. 139, 107-117. 

Figure 4.6 Kinetic curves of cyclohexene consumption and reaction product accumulation at 580 °C (1 cyclohexene consumption 2 cyclohexadiene accumulation (kinetic curve) and 3 benzene yield).

Neither the maximum nor the descending branches of the upper curves, representing geminate recombination, are reproduced in the Markovian theory. It predicts the monotonous ion accumulation and still further decrease in the ionization quantum yield /. This is because the Markovian theory does not account for either static or subsequent nonstationary electron transfer. When ionization is under diffusional control, both these are faster than the final (Markovian) transfer. EM is a bit better in this respect. As a non-Markovian theory, it accounts at least for static ionization and qualitatively reproduces the maximum in the charge accumulation kinetics. However, the subsequent geminate recombination develops exponentially in EM because the kinematics of ion separation is oversimplified in this model. It roughly contradicts an actual diffusional separation of ions, characterized by numerous recontacts and the power dependence of long-time separation kinetics studied in a number of works [20,21,187],... 

Equations (3.437) and (3.439) together with the original UT equations (3.432) constitute the formal basis of the generalized unified theory (GUT) [195]. The latter can be used to find the system response to the ( pulse, provided the acceptor concentration is sufficiently large. In this way one can obtain the accumulation kinetics of excitations and free ions and their stationary concentrations ... 

---