Asymmetric particle mobilities

As it is seen in these figures, the higher n(0), the faster the asymptotics is achieved. For the immobile reactant A and d = 1, a(t) systematically exceeds that for the equal mobilities which leads to faster concentration decay in time. The results for d = 2 and 3 are qualitatively similar. Their comparison with the one-dimensional case demonstrates that the concentration decay is now much faster since the critical exponents strive for a = 3/4 and a = 1/2 for the symmetric and asymmetric cases, respectively, which differ greatly from the classical value of a = 1. Respectively, the gap between symmetric and asymmetric decay kinetics grows much faster than in the d = 1 case. Therefore, the conclusion could be drawn that the effect of the relative particle mobility is pronounced better and thus could be observed easier in t ree-dimensional computer simulations rather than in one-dimensional ones, in contrast to what was intuitively expected in [33]. 

In the more difficult case of asymmetry in both mobilities (DA = 0, >b > 0) and initial concentrations, Sn > 0, a kind of master equation was derived and solved numerically, along with the Monte Carlo simulations [32], As it is seen in Fig. 6.26, for equal particle concentrations the decay asymptotics, shown by dashed lines, is reached very fast for both symmetrical and asymmetrical reactant mobilities (curves 1). It is no longer tme, however, for asymmetrical concentrations shown in curves 2. These latter demonstrate clearly an existence of the two different classes of universality for reactions with unequal concentrations one class corresponds to the case when both (A and B) reactants are mobile or those which are in majority whereas... 

A shape of such clusters or domains containing alternatively A or B particles transforms rapidly to the trapezoidal profile, in a complete agreement with both computer simulations and theory presented in [27]. Note also that in the asymmetric case (broken lines and r.h.s. scale in Fig. 6.15(b) immobile particles A form much more dense and compact clusters them mobile B s [26],... 

In calculations presented below we assume first one kind of defects to be immobile (Da = 0, k = 0) and their dimensionless initial concentration n(0) = 0.1 is not too high it is less than 10 per cent of the defect saturation level accumulated after prolonged irradiation [41]. Its increase (decrease) does not affect the results qualitatively but shorten (lengthen, respectively) the distinctive times when the effects under study are observed. To stress the effects of defect mobility, we present in parallel in Sections 6.3.1 and 6.3.2 results obtained for immobile particles A (D = 0, asymmetric case) and equal mobility of particles A and B (Da = Dq, symmetric case). In both cases only pairs of similar particles BB interact via elastic forces, (6.3.5), but not AA or AB. The initial distribution (t = 0) of all defects is assumed to be random, Y(r > 1,0) = -X (r,0) = 1 i/ = A,B. 

The asymptotic (t —> 00) treatment of the set of non-linear integro-differential equations is quite difficult even as a computational problem since solution stability requires the use of small time increments in a mesh and thus we could reach t — 105 only (in the dimensionless units). Besides, in the particular case of the asymmetric mobility, DA — 0, we observe the spatial particle distribution revealing strongly developed singular properties, which requires the additional reduction of the coordinate increment. 

Joint distribution of BB and AB pairs is shown in Fig. 6.44. The distribution of similar mobile particles B at long times in the asymmetric case practically is the same as in the symmetric case (when X = Xb). The behaviour of Xb (r, t) is determined by the Coulomb repulsion of B s for which the non-equilibrium screening effect does not take place. In its turn, some deviation for the joint dissimilar functions Y(r, t) seen in Fig. 6.44 for the symmetric and asymmetric cases is a direct consequence of different screening effects in the latter case the effective recombination radius increases in time which results in an increase of the Y(r,t) gradient at r = ro at long times this correlation function itself strives for the Heaviside step-like form. 

In this Section we continue studies of particle dynamical interactions. For this purpose the formalism of many-particle densities is applied to the study of the cooperative effects in the kinetics of bimolecular A -f B —> 0 reaction between oppositely charged particles (reactants) interacting via the Coulomb forces. We show that unlike the Debye-Hiickel theory in statistical physics, here charge screening has essentially a non-equilibrium character. For the asymmetric mobility of reactants (Da = 0, 0) the joint spatial distri-... 

We can still neglect the vibrational portion of the partition function and the portion for the electronically excited states. In the rotation portion of the partition function a symmetry number enters. This emerges because certain symmetries in transitions are not permitted. The entropy for a symmetrical molecule is thus as smaller, as more symmetrical such a molecule is, with otherwise same characteristics. We experience here a strange contradiction If the elementary particles would be freely mobile in a molecule, then we would expect that they distribute equally. That means an asymmetrical molecule should want itself to convert into a more symmetrical molecule. On the other hand, the law of symmetry in entropy tells to us that an asymmetrical molecule has larger entropy. 

Figure 6 demonstrates a continuous electroki-netic sorting of polyst3rene particles by size and surface charge simultaneously in an asymmetric double-spiral microchaimel [3]. The mixture of nonfluorescent 5 pm, nonfluorescent 10 pm, and fluorescent 10 pm particles is resuspended in 0.1 mM phosphate buffer. As the buffer solution is more conductive, all three t3q>es of particles undergo negative C-iDEP in the spiral. Under the application of a 33 kV/m DC electric field, the initially scattered particles in Fig. 6b are focused by C-iDEP to a tight stream flowing near the outer sidewall of the 100 pm wide first spiral as seen from Fig. 6c. Subsequently in the second spiral whose width increases from 100 pm to 200 pm, the focused particles are all displaced from the inner to the outer wall by C-iDEP at a particle size- and charge-dependent rate or, more accurately, the particle dielectrophoretic to electrokinetic mobility ratio in Eq. 6.

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