Mesoscopic approach

A similar approach, in spirit, has been proposed [212] for the study of two-component classical systems, for example poly electrolytes, which consist of mesoscopic, highly-charged, poly ions, and microscopic. 

Due to the complexity of macromolecular materials computer simulations become increasingly important in polymer science or, better, in what is now called soft matter physics. There are several reviews available which deal with a great variety of problems and techniques [1-7]. It is the purpose of the present introduction to give a very brief overview of the different approaches, mainly for dense systems, and a few apphcations. To do so we will confine ourselves to techniques describing polymers on a molecular level. By molecular level we mean both the microscopic and the mesoscopic level of description. In the case of the microscopic description (all)... 

In ternary mixtures of oil, water, and surfactant the ordering properties of the system follow from the vectorial character of the interactions of the surfactant molecules with both the oil and the water molecules. The typical size of the ordered domains, much larger than the molecular size, justifies the application of the mesoscopic Landau-Ginzburg approach to the ordering. In the simplest approach of Gompper and Schick [3,12], which we call here the basic Landau-Ginzburg model, the orientational degrees of free-... 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

Another important area of future development concerns copying the supramolecular principle of cell envelopes of archaea, which have evolved in the most extreme and hostile ecosystems. This biomimetic approach is expected to lead to new technologies for stabilizing fnnctional lipid membranes and their nse at the mesoscopic and macroscopic scales [200]. Along the same line, liposomes coated with S-layer lattices resemble archaeal cell envelopes or virns envelopes. Since liposomes have a broad application potential, particu-... 

Landauer proposed in 1957 the first mesoscopic theoretical approach to charge transport [176]. Transport is treated as a scattering problem, ignoring initially all inelastic interactions. Phase coherence is assumed to be preserved within the entire conductor. Transport properties, such as the electrical conductance, are intimately related to the transmission probability for an electron to cross the system. Landauer considered the current as a consequence of the injection of electrons at one end of a sample, and the probability of the electrons reaching the other end. The total conductance is determined by the sum of all current-carrying eigenmodes and their transmission probability, which leads to the Landauer formula of a ID system ... 

While Heisig et al. solved the diffusion equation numerically using a finite volume method and thus from a macroscopic point of view, Frasch took a mesoscopic approach the diffusion of single molecules was simulated using a random walk [69], A limited number of molecules were allowed moving in a two-dimensional biphasic representation of the stratum corneum. The positions of the molecules were changed with each time step by adding a random number to each of the molecule s coordinates. The displacement was related... 

Of course, the macroscopic equations cannot actually be derived from the microscopic ones. In practice they are pieced together from general principles and experience. The stochastic mesoscopic description must be obtained in the same way. This semi-phenomenological approach is remarkably successful in the range where the macroscopic equations are linear, see chapter VIII. In the nonlinear case, however, difficulties appear, which can only be resolved by the improved, but still mesoscopic, method of chapter X. 

We understand very well that any book inavoidably reflects authors interests and scientific taste this fact is, first of all, usually seen in the selection of material which in our case is very plentiful and diverse. For instance, Chapter 2 gives examples of different general approaches used in chemical kinetics (macroscopic, mesoscopic and microscopic) and numerous methods for solving particular problems. We focus here on the microscopic approach based on the concept of active particles (structure elements, reactants, defects) whose spatial redistribution arises due to their diffusion affected by... 

Therefore, we tried to develop the adequate mathematical formalism of the fluctuation-controlled chemical kinetics based on a concept of active particles. Simultaneously, the mesoscopic theory of concentration field fluctuations was developed by a number of investigators (see Chapter 2) having more qualitative character. Undoubtedly, these two approaches - microscopic and mesoscopic - overlap, since a lot of fundamental results like asymptotic... 

Applying these methods to systems in the vicinity of the non-equilibrium critical points, the conclusion was drawn [72] that the mesoscopic approach contains excess information about spatial particle distribution the details of how the whole system s volume is divided into cells become unimportant as — oo. The possibility to employ expansion in inverse powers of vo -similarly to a complete mixing case - was also discussed. Asymptotically it leads to the Focker-Planck equation equivalent to the Langevin-like equation. 

Summing this Section up, we would like to note that in the approach discussed here the introduction of stochasticity on a mesoscopic level restricts the applicability of a method by such statements of a problem where subtle details of particle interaction become unimportant. First of all, we mean that kinetic processes with non-equilibrium critical points, when at long reaction time the correlation length exceeds all other spatial dimensions. This limitation makes us consider in the next Section 2.3 the microscopic level of the kinetic description. 

Note here that the relation between mesoscopic and microscopic approaches is not trivial. In fact, the former is closer to the macroscopic treatment (Section 2.1.1) which neglects the structural characteristics of a system. Passing from the micro- to meso- and, finally, to macroscopic level we loose also the initial statement of a stochastic model of the Markov process. Indeed, the disadvantages of deterministic equations used for rather simplified treatment of bimolecular kinetics (Section 2.1) lead to the macro- and mesoscopic models (Section 2.2) where the stochasticity is kept either by adding the stochastic external forces (Section 2.2.1) or by postulating the master equation itself for the relevant Markov process (Section 2.2.2). In the former case the fluctuation source is assumed to be external, whereas in the latter kinetics of bimolecular reaction and fluctuations are coupled and mutually related. Section 2.3.1.2 is aimed to consider the relation between these three levels as well as to discuss problem of how determinicity and stochasticity can coexist. 

Following the approach discussed in Section 2.2.2, let us divide the whole reaction volume V of the spatially extended system into N equivalent cells (domains) [81]. However, there is an essential difference with the mesoscopic level of treatment in Section 2.2.2 a number of particles in cells were expected to be much greater than unity. Note that this restriction is not imposed on the microscopic level of system s treatment. Their volumes are chosen to be so small that each cell can be occupied by a single particle only. (There is an analogy with the lattice gas model in the theory of phase transitions [76].) Despite the finiteness of vq coming from atomistic reasons or lattice discreteness, at the very end we make the limiting transition vo - 0, iV - oo, v0N = V, to the continuous pattern of point dimensionless particles. 

The critical dose rate pc necessary for initiating the aggregation process is the smaller, the lower the temperature, the stronger elastic attraction of similar particles and the slower the diffusion (greater the activation energy for hopping). This conclusion is in a complete qualitive agreement with the results obtained recently in terms of a quite different mesoscopic approach [63-65],... 

Both equations (7.2.16) and (7.2.18) have the same dependence on the relative diffusion coefficient, D — D + D, but different dependence on the elastic interaction between defects. However, in both cases the stronger similar defect attraction, the lower is the critical dose rate. In the mesoscopic approach this effect is less pronounced (logarithmic vs. linear dependence) and here pc is considerably higher. It seems that this approach is able to detect only those mesoscopic-size aggregates which are already well-developed - unlike the microscopic formalism able to detect even the marginal aggregation effects. 

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