Analytical theory particles

Thus, dendrimers exhibit a unique combination of (a) high molecular weights, typical for classical macromolecular substances, (b) molecular shapes, similar to idealized spherical particles and (c) nanoscopic sizes that are larger than those of low molecular weight compounds but smaller than those of typical macromolecules. As such, they provide unique rheological systems that are between typical chain-type polymers and suspensions of spherical particles. Notably, such systems have not been available for rheological study before, nor are there yet analytical theories of dense fluids of spherical particles that are successful in predicting useful numerical results. 

An increase of the standard deviation at r 3 due to small number of survived particles, demonstrates a limited possibility of the direct statistical simulations for a system with a variable number of particles. However, certain conclusions could be drawn even for such limited statistical information. Say, if for equal concentrations the analytical theory based on the superposition approximation seems to be quite adequate, for unequal concentrations its deviation from the computer simulations greatly increases in time. The superposition approximation gives the lower bound estimate of the actual kinetic curves tia( ) but if for d = 2 shown in Fig. 5.8 the deviation is considerable, for d, = 1 (Fig. 5.7) it is not observed, at least for the reaction depths considered. 

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. 

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). 

Evident progress in studies of liquids has been achieved up to now with the use of computer simulations and of the models based on analytical theory. These methods provide different information and are mutually complementary. The first method employs rather rigorous potential functions and yields usually a chaotic picture of the multiple-particle trajectories but has not been able to give, as far as we know, a satisfactory description of the wideband spectra. The analytical theory is based on a phenomenological consideration (which possibly gives more regular trajectories of the particles than arise in reality ) in terms of a potential well. It can be tractable only if the profile of such a well is rather... 

In this contribution we have reviewed the recent results concerning the collective dynamics of charged liquids. In order to establish the role of long-range Coulombic interactions we have concentrated our attention on the comparison of the results obtained for binary mixtures of neutral and charged particles. Such a comparison has been performed on two levels of consideration - on the level of analytical theories and numerical simulations. The main conclusions from our studies are as follows. 

Substantial interest has been raised in the problem of the structure and dynamics of suspensions in shear hydrodynamic fields. ° ° The experiments showed that both shear-induced melting and shear-induced ordering can be observed at different particle volume fractions and shear rates. The nonequilibrium microstructure of the suspension under shear can be investigated in these experiments and compared with the predictions from analytical theories and computer simulations. 

Below, the analytical one-particle theory will be used. It is simple and yields good agreement between the calculated and experimental spectra. 

The main purpose of an analytic approach to the theory of chemical reactions is to furnish a detailed imderstanding of the elements involved, rather than to provide a quick comparison with experimentally determined macroscopic rates. It is not essential that the particular divisions made in this review be followed in constructing an analytic theory, but some such organization of the problem as presented here will be required. Within the current framework of physical theory it seems natural to consider in sequence the following four steps identification and characterization of species interaction of species collision dynamics and many-particle dynamics. 

Our understanding of diffusion and reaction in single-file systems is impaired by the lack of a comprehensive analytical theory. The traditional way of analytically treating the evolution of particle distributions by differential equations is prevented by the correlation of the movement of distant particles. One may respond to this restriction by considering joint probabilities covering the occupancy and further suitable quantities with respect to each individual site. These joint probabilities may be shown to be subject to master equations. 

It was Lewis who, already back in 1933, contrasted the different features of theories in chemistry and physics. He presented structural organic chemistry as the paradigm of a chemical theory, as an analytical theory in the sense it was groimded on a large body of experimental material from which the chemist attempted to deduce a body of simple laws that were consistent with the known phenomena. He called the paradigm of a physical theory a synthetic theory to stress that the mathematical physicist starts by postulating laws governing the mutual behavior of particles and then "attempts to synthesize an atom or a molecule" (Lewis 1933, 17). He maintained that... 

The deep-channel surface viscometer is a frequently used experimental method for measuring interfacial shear viscosity owing to its sensitivity (irish S 10 surface poises) and relatively simple analytical theory. The main drawback of this technique is the necessity of placing a small tracer particle within the interfacial flow field for tracking the central surface velocity. This may be particularly cumbersome with heavy-oil systems, for which the particle may require several hours or more to execute a complete revolution, as well as with liquid -liquid systems, for which the placement of the particle at the interface may be difficult. For more details, see Refs. 58 and 151-156. 

These examples also show that simulation is an important tool for many nanoscale materials problems. Although brute force simulation is not usually effective because the time scales are too long and the number of particles is too large, a combination of simulation in conjunction with analytical theory and simple models can be quite effective. 

Figure 1.5 The probability for N particles to exit a wedge hopper has a broad power-law tail with P N) N (dashed line), seen in experiments, Monte Carlo simulation, and analytic theory. This is in distinct contrast to the distribution function found in conical hoppers, which decays exponentially. Shown are distribution functions for experimental hoppers of lengths L = 16.2-22.2 cm. Simulation and theory assume = 3 adjacent, statistically independent cells. (From Saraf, S. and Franklin, S.V., Physical Review E, 83(3), 030301, March 2011.)...

For completeness, note that several researchers have exploited the well-developed analytical theories of the stmcture of fluids to model percolation in mixtures of interacting particles. By proposing various extensions of the multicomponent Omstein-Zernike equation, coupled with connectivity definitions from continuum percolation theories, simplified analytical expressions are derived for the percolation threshold of a composite system subjected to interparticle and medium-induced interactions. However, to date, simulations dominate the study of dynamic percolation. 

---