Mesoscopic averages

The mesoscopic averages are designated by enclosure in the brackets ([ ]). The mesoscop-ically averaged order parameter S is defined as... 

In the structure with all the surfactant molecules located at monolayers, the volume fraction of surfactant should be proportional to the average surface area times the width of the monolayer divided by the volume, i.e., Ps (X Sa/V. The proportionality constant is called the surfactant parameter [34]. This is true for a single surface with no intersections. In our mesoscopic description the volume is measured in units of the volume occupied by the surfactant molecule, and the area is measured in units of the area occupied by the amphiphile. In other words, in our model the area of the monolayer is the dimensionless quantity equal to the number of amphiphiles residing on the monolayer. Hence, it should be identified with the area rescaled by the surfactant parameter of the corresponding structure. 

Since MPC dynamics yields the hydrodynamic equations on long distance and time scales, it provides a mesoscopic simulation algorithm for investigation of fluid flow that complements other mesoscopic methods. Since it is a particle-based scheme it incorporates fluctuations, which are essential in many applications. For macroscopic fluid flow averaging is required to obtain the deterministic flow fields. In spite of the additional averaging that is required the method has the advantage that it is numerically stable, does not suffer from lattice artifacts in the structure of the Navier-Stokes equations, and boundary conditions are easily implemented. 

The mesoscopic description is introduced by defining functions 4> (q) and 4>B(q) that have the meaning of averaged over some mesoscopic volume values of the microscopic concentration operators. The conditional partition function, Z(4>t) (y =A,B), is the partition function for the system subject to the constraint that the microscopic operators 4>T(q) are fixed at some prescribed... 

Fig. 3 Scanning electron microscope image of a typical mesoscopic Ti02 film employed in DSC. Note the bipyramidal shape of the particles having (101) oriented facets exposed. The average particle size is 20 nm...

Abstract The discussion of relaxation and diffusion of macromolecules in very concentrated solutions and melts of polymers showed that the basic equations of macromolecular dynamics reflect the linear behaviour of a macromolecule among the other macromolecules, so that one can proceed further. Considering the non-linear effects of viscoelasticity, one have to take into account the local anisotropy of mobility of every particle of the chains, introduced in the basic dynamic equations of a macromolecule in Chapter 3, and induced anisotropy of the surrounding, which will be introduced in this chapter. In the spirit of mesoscopic theory we assume that the anisotropy is connected with the averaged orientation of segments of macromolecules, so that the equation of dynamics of the macromolecule retains its form. Eventually, the non-linear relaxation equations for two sets of internal variables are formulated. The first set of variables describes the form of the macromolecular coil - the conformational variables, the second one describes the internal stresses connected mainly with the orientation of segments. 

In Section V we shall show that the syston described by Eq. (4.11) coincides with a nonlinear tension of the popular model of the intinerant oscillator. Note that we are exploring a mesoscopic regime, implying averaging processes which significantly reduce the nonlinear character of the real microscopic interaction (consider, for instance, how nonlinear the L-J potential is). 

Reactions that occur between components in the bulk solution and vesicle-bound components, i.e., reactions occurring across the membrane interface, can be treated mathematically as if they were bimolecular reactions in homogeneous solution. However, kinetic analyses of reactions on the surface of mesoscopic structures are complicated by the finiteness of the reaction space, which may obviate the use of ordinary equations of chemical kinetics that treat the reaction environment as an infinite surface populated with constant average densities of reactant molecules. As was noted above, the kinetics of electron-transfer reactions on the surface of spherical micelles and vesicles is expressed by a sum of exponentials that can be approximated by a single exponential function only at relatively long times [79a, 81], At short times, the kinetics of the oxidative quenching of excited molecules on these surfaces are approximated by the equation [102]... 

Although we expect for dimensional reasons that the average magnitude of h, and hence the magnitude of ([nh]), will be proportional to pv, the tensorial form of ([nh]) is unknown. To obtain ([nh]), without having to revert back to (an almost impossible) microscopic calculation of the director field, Larson and Doi (1991 Kawaguchi 1996) assumed Aat ([nh]) is a function of the mesoscopic order parameter S— that is, that ([nh]) = Ka f(S). Dimensional reasoning then leads to the ansatz that... 

Because it is not possible to reproduce on experiment with exactly the same configuration, we are not only not interested in the precise position of the atoms, we are not even interested in specific configurations, but only in characteristic ones. Although there may be differences on a microscopic scale, the behavior of a system on a macroscopic, and often also on a mesoscopic, scale will be the same. So we do not look at individual trajectories in phase space, but we average over all possible trajectories. This means that we have a phase space density p and a probabihty Pa of finding the system in configuration a. These cire related via... 

An important class of technical electrodes are those based on dispersed catalyst particles. For these, the relation between structure and reactivity is very important, but unraveling the complexities involved is slowed by the problem that reactivity is usually referred to a macroscopic sample while the structural characterization gives local information. Thus, the modification of conductive surfaces by nanometer-sized particles, i.e., to control the size and distribution of the catalyst particles on the substrate, allows one to control the average mesoscopic structure of these electrodes. Recently a technique for the preparation of catalyst particles with a narrow size... 

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