Dissimilar Particles

When considering dissimilar particles, even taking into account only VW and electric interactions, there are various possible situations that may arise for the energy/distance curve, depending on the nature of interactions, dielectric constant, ionic strength, potential of both surfaces, and so on. Working similarly to the example shown in 

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Polymers are suspended as microparticles in the latex and interactions between these microparticles are prevented by the presence of adsorbed suspending agent and soap molecules. Blending results in a random suspension of dissimilar particles in the mixture of latexes, each unaffected by the other. Rate of flocculation depends entirely on the stabilizer and not on the polymer characteristics as such. Coagulated mass contains an intimate mixture of the polymers. Acrylonitrile butadiene styrene (ABS) polymers [23-25] may be prepared by this method. 

This simple thermodynamic picture is substantially altered if we introduce dissimilar particles into our dispersion. The various interactions now depend on the nature of the two particles, relative to the solvent, and can either favour dispersal or aggregation. Again, this could be the basis for a natural control mechanism as the number and composition of the colloidal building blocks evolve, subtle changes in the interactions could switch a dispersion from stable to unstable. 

Charging by contact electrification is an active mechanism whenever dissimilar particles make and break contact with each other, or whenever they slide over a chute or an electrode. This charging mechanism is most frequently used to charge selectively and obtain an electrostatic separation of two species of dielectric materials as realized in a free fall electrostatic separator. 

The interactions between similar particles, dissimilar particles, and the dispersion medium constitute a complex but essential part of dispersion technology. Such interparticle interactions include both attractive and repulsive forces. These forces depend upon the nature, size, and orientation of the species, as well as on the distance of separation between and among the particles of the dispersed phase and the dispersion medium, respectively. The balance between these forces determines the overall characteristics of the system. 

With the assumed uniformity of d0 values, Equation (84) shows that the coagulation of dissimilar particles becomes energetically unfavorable when the surface tension of the medium is intermediate between the surface tensions of the two kinds of dispersed units. 

Consider now the fluctuations of the order parameter in the system possessing the chemical reaction this problem could be perfectly illustrated by computer simulations on lattices. We start with the bimolecular A + B -y 0 reaction discussed above, and first of all froze particle diffusion. Let the recombination event happen instantly when a pair AB of dissimilar particles occupies the nearest lattice sites (assume lattice to be squared). Immobile particles enter into reaction as a result of their creation with the equal probabilities in empty lattice sites from time to time a newly created particle A(B) finds itself nearby pre-created B(A) and they recombine. (Since this recombination event is instant, the creation rate is of no importance.) This model describes, in particular, Frenkel defect accumulation in solids under... 

The said allows us to understand the importance of the kinetic approach developed for the first time by Waite and Leibfried [21, 22]. In essence, as is seen from Fig. 1.15 and Fig. 1.26, their approach to the simplest A + B —0 reaction does not differ from the Smoluchowski one However, coincidence of the two mathematical formalisms in this particular case does not mean that theories are basically identical. Indeed, the Waite-Leibfried equations are derived as some approximation of the exact kinetic equations due to a simplified treatment of the fluctuational spectrum a complete set of the joint correlation functions x(rJ) for kinds of particles is replaced by the only function xab (a t) describing the correlation of chemically reacting dissimilar particles. Second, the equation defining the correlation function X = Xab(aO is linearized in the function x(rJ)- This is analogous to the... 

Diffusion stimulates the dissimilar particle approach to each other and reaction between them, in this process, as it was said above, diffusion acts to smooth out density inhomogeneities, but on the other hand, the reaction creates them and often the latter trend is predominant. 

Taking into account the statistical independence in coordinates of dissimilar particles, we arrive at... 

Fig. 2.20. Change pmtmi due to particle recombination, (a) Reaction of two dissimilar particles from the (m, m )-group. (b) Reaction of one of the particles from the (m, m )-group with...

In fact, the latter is a functional of the correlation function of dissimilar particles, i.e., to calculate K(t) we need to know either Y(r, t) or p. In its turn, equation (4.1.16) demonstrates that these latter are coupled with three-point densities etc. Therefore, to solve the problem, we have to cut off the infinite equation hierarchy, thus only approximately describing the fluctuation spectrum. Usually it is done by means of the complete Kirkwood superposition approximation, equations (2.3.62) and (2.3.63), or the shortened approximation, equations (2.3.64) and (2.3.65). 

The black sphere approximation permits us to obtain the most simple and physically transparent results for the kinetics of diffusion-controlled reactions. We should remind that this approximation involves a strong negative correlation of dissimilar particles at r ro, where Y(r, t) = 0, described by the Smoluchowski boundary condition... 

Due to the instant recombination all the dissimilar particles with relative distances r ro disappear, which results in the Smoluchowski boundary condition... 

Analysis of the correlation functions demonstrates also impressive general agreement between the superposition approximation and computer simulations. Note, however certain overestimate of the similar particle correlations, X r,t), at small r, especially for d = 1. In its turn the correlation function of dissimilar particles, Y(r,t), demonstrates complete agreement with the statistical simulations. Since the time development of concentrations is defined entirely by Y(r, t), Figs 5.2 and 5.3 serve as an additional evidence for the reliability of the superposition approximation. An estimate of the small distances here at which the function Y (r, t) is no longer zero corresponds quite well to the earlier introduced correlation length o, equation (5.1.47) as one can see in fact that at moment t there are no AB pairs separated by r < o-... 

Quantitative deviations are seen also from the correlation shown in Fig. 5.9. The correlation functions of dissimilar particles Y (r, t) are in good agreement with simulations, which results also in a reliable reproduction of the decay kinetics for nA(t) - unlike behaviour of the correlation functions of the similar particles Xv r,t) which is very well pronounced for XA(r,t). Positive correlations, Xu(r,t) > 1 as r < , argue for the similar particle aggregation, and the superposition approximation tends to overestimate their density. The obtained results permit to conclude that the approximation (2.3.63) of the three-particle correlation function could be in a serious error for the excess of one kind of reactants. 

Figures 1.20 and 1.21 show that at long time t 105 aggregates of similar particles are well pronounced. Their formation is associated with emergence of the non-Poisson fluctuation spectrum in a system. Similar particle aggregation leads to the effective dissimilar particle separation (as compared to their random distribution) and thus - to a reduced reaction rate which is essentially less than expected by the linear approximation.

In simulations [9] Sierpinski gaskets on the 12th stage, containing 177147 or 265722 sites, were used respectively. The number No of randomly distributed A or B particles was 10 percent of the total number of sites. The random mutual annihilation of dissimilar particles was simulated through a minimal process method [10] from all AB pairs at each reaction step one pair was selected randomly, according to its reaction rate (3.1.2) the time... 

Figure 6.5 displays a typical distribution of A and B particles at t = 100 for a realization of the annihilation process on the Sierpinski gasket of the first kind at the 9th stage. The segregation of dissimilar particles, resulting from initial concentration fluctuations, is clearly visible at this reaction stage. 

For the multipole interaction (4.1.44) the dissimilar correlation function could be also presented in a form of product (6.1.4), where Yo(r,t) = exp[—cr(r)f]. Neglecting indirect correlation mechanism, the dissimilar particle function Yo(r, t) — exp[— (r/ o)-m], with o defined by (4.1.45), is stationary in term of variable r)0 = r/ o- Indirect mechanism of the correlation formation, as follows from a solution of equations derived in the superposition approximation, results at long times in Y(r, t) z r) t),... 

The accuracy of the Kirkwood superposition approximation was questioned recently [15] in terms of the new reaction model called NAN (nearest available neighbour reaction) [16-20], Unlike previous reaction models, in the NAN scheme AB pairs recombine in a strict order of separation the closest pair in an initially random distribution is removed first, then the next one and so on. Thus for NAN, the recombination distance R, e.g., the separation of the closest pair of dissimilar particles at any stage of the recombination, replaces real time as the ordering variable time does not enter at all the NAN scheme. R is conveniently measured in units of the initial pair separation. At large R in J-dimensions, NAN scaling arguments [16] lead rapidly to the result that the pair population decreases asymptotically as cR d/2 (c... 

The temporal evolution of spatial correlations of both similar and dissimilar particles for d = 1 is shown in Fig. 6.15 (a) and (b) for both the symmetric, Da = Dft, and asymmetric, Da = 0 cases. What is striking, first of all, is rapid growth of the non-Poisson density fluctuations of similar particles e.g., for Dt/r = 104 the probability density to find a pair of close (r ro) A (or B) particles, XA(ro,t), by a factor of 7 exceeds that for a random distribution. This property could be used as a good aggregation criterion in the study of reactions between actual defects in solids, e.g., in ionic crystals, where concentrations of monomer, dimer and tetramer F centres (1 to 3 electrons trapped by anion vacancies which are 1 to 3nn, respectively) could be easily measured by means of the optical absorption [22], Namely in this manner non-Poissonian clustering of F centres was observed in KC1 crystals X-irradiated for a very long time at 4 K [23],... 

Below we take into account the non-linear terms in the kinetic equations containing functionals J (coupling spatial correlations of similar and dissimilar particles) but neglect the perturbation of the pair potentials assuming that il(r, t) = l3U(r). This is justified in the diluted systems and for the moderate particle interaction which holds for low reactant densities and loose aggregates of similar particles. However, potentials of mean force have to be taken into account for strongly interacting particles (defects) and under particle accumulation when colloid formation often takes place [67],... 

Fig. 6.35. The joint correlation functions for dissimilar particles Y(r,t) (full curves), immobile similar particles Xa r,t) (broken curve) and mobile particles Xb(r,t) (dotted curve). Parameters used are given. The dimensionless time (in units rg/D) is (a) t = 101 (1) 102 (2) 103 (3) 104 (4) (b) Da = De, t = 102 (1) 104 (curve 2).

As it was mentioned above, up to now only the dynamic interaction of dissimilar particles was treated regularly in terms of the standard approach of the chemical kinetics. However, our generalized approach discussed above allow us for the first time to compare effects of dynamic interactions between similar and dissimilar particles. Let us assume that particles A and B attract each other according to the law U v(r) = — Ar-3, which is characterized by the elastic reaction radius re = (/3A)1/3. The attraction potential for BB pairs is the same at r > ro but as earlier it is cut-off, as r ro. Finally, pairs AA do not interact dynamically. Let us consider now again the symmetric and asymmetric cases. In the standard approach the relative diffusion coefficient D /D and the potential 1/bb (r) do not affect the reaction kinetics besides at long times the reaction rate tends to the steady-state value of K(oo) oc re. 

Fig. 6.36. The defect concentration vs time. The dotted line shows neglect of the similar particle correlation in the full line it is incorporated for the case Da = 0 and in the broken line - for the Da = DB case. The elastic interaction constant A of similar and dissimilar particles is the same UAb = UBB = -Ar-3, whereas Uaa = 0.

In conclusion of Section 6.3 we wish to stress that the elastic attraction of similar defects (reactants) leads to their dynamic aggregation which, in turn, reduces considerably the reaction rate. This effect is mostly pronounced for the intermediate times (dependent on the initial defect concentration and spatial distribution), when the effective radius of the interaction re = - JTX exceeds greatly the diffusion length = y/Dt. In this case the reaction kinetics is governed by the elastic interaction of both similar and dissimilar particles. A comparative study shows that for equal elastic constants A the elastic attraction of similar particles has greater impact on the kinetics than interaction of dissimilar particles. 

The equation for the time development of macroscopic concentrations formally coincides with the law of mass action but with dimensionless reaction rate K(t) = K(t)/ AnDr ) which is, generally speaking, time-dependent and defined by the flux of the dissimilar particles via the recombination sphere of the radius tq, equation (5.1.51). Using dimensionless units n(t) = 4nrln(t), r = t/tq, t = Dt/r, and the condition of the reflection of similar particles upon collisions, equation (5.1.40) (zero flux through origin), we obtain for the joint correlation functions the equations (6.3.2), (6.3.3). Note that we use the dimensionless diffusion coefficients, a = 2k, IDb = 2(1 — k), k = Da/ Da + Dq) entering equation (6.3.2). 

Fig. 6.44. The joint correlation function of dissimilar particles, Y(r,t) - full curve, and that of similar particles Xb (r, t) - dotted curve. Parameters are L — 1, n(0) =0.1. Curves 1 -asymmetric case, curves 2 - symmetric case.

The non-equilibrium particle distribution is clearly observed through the joint correlation functions plotted in Fig. 6.47. Note that under the linear approximation [74] the correlation function for the dissimilar defects Y (r, t) increases monotonically with r from zero to the asymptotic value of unity Y(r —y oo,t) = 1. In contrast, curve 1 in Fig. 6.47 (f = 101) demonstrates a maximum which could be interpreted as an enriched concentration of dissimilar pairs, AB, near the boundary of the recombination sphere, r tq. With increasing time this maximum disappears and Y(r, t) assumes the usual smoothed-step form. The calculations show that such a maximum in Y(r, t) takes place within a wide range of the initial defect concentrations and for a random initial distribution of both similar and dissimilar particles used in our calculations X (r, 0) = Y(r > 1,0) = 1. The mutual Coulomb repulsion of similar particles results in a rapid disappearance of close AA (BB) pairs separated by a distance r < L (seen in Fig. 6.47 as a decay of X (r, t) at short r with time). On the other hand, it stimulates strongly the mutual approach (aggregation) of dissimilar particles leading to the maximum for Y(r, t) at intermediate distances observed in Fig. 6.47. 

The role of the non-equilibrium charge screening is emphasized by calculations neglecting such screening, i.e., when equations (5.1.54) are omitted and Uv r) — L/r is postulated. In this case mutual repulsion of similar particles accompanied by the attraction between dissimilar particles are characterized by the infinite interaction radius between particles which leads immediately to the Coulomb catastrophe - an infinite increase in K(t) in time shown in Fig. 6.47. This effect is independent of the choice of the initial defect distributions for both similar and dissimilar particles. On the other hand, incorporation of the Coulomb screening makes equations (6.4.1), (6.4.2) asymptotically valid for any initial distribution of particles. 

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