Mesoscopic length scales

Coarse-grained models have a longstanding history in polymer science. Long-chain molecules share many common mesoscopic characteristics which are independent of the atomistic stmcture of the chemical repeat units [4, 5 and 6]. The self-similar stmcture [7, 8, 9 and 10] on large length scales is only characterized by a single length scale, the chain extension R. 

The surface dividing the components of the mixture formed by a layer of surfactant characterizes the structure of the mixture on a mesoscopic length scale. This interface is described by its global properties such as the surface area, the Euler characteristic or genus, distribution of normal vectors, or in more detail by its local properties such as the mean and Gaussian curvatures. 

Finally, we assume that the fields 4>, p, and u vary slowly on the length scale of the lattice constant (the size of the molecules) and introduce continuous approximation for the thermodynamical-potential density. In the lattice model the only interactions between the amphiphiles are the steric repulsions provided by the lattice structure. The lattice structure does not allow for changes of the orientation of surfactant for distances smaller than the lattice constant. To assure similar property within the mesoscopic description, we add to the grand-thermodynamical potential a term propor-tional to (V u) - -(V x u) [15], so that the correlation length for the orientational order is equal to the size of the molecules. 

FIG. 1 Sketch of a colloidal suspension. Mesoscopic particles float in an atomic liquid. Water molecules are drawn schematically. Note the difference in length scales between solvent and solute. 

Classical surface and colloid chemistry generally treats systems experimentally in a statistical fashion, with phenomenological theories that are applicable only to building simplified microstructural models. In recent years scientists have learned not only to observe individual atoms or molecules but also to manipulate them with subangstrom precision. The characterization of surfaces and interfaces on nanoscopic and mesoscopic length scales is important both for a basic understanding of colloidal phenomena and for the creation and mastery of a multitude of industrial applications. 

The self-organization or assembly of nnits at the nanoscale to form supramolecnlar ensembles on mesoscopic length scales comprises the range of colloidal systems. There is a need to understand the connection between structure and properties, the evolution and dynamics of these structures at the different levels—supramolecnlar, molecular, and sub-molecular— by learning from below. ... 

The characteristic times on which catalytic events occur vary more or less in parallel with the different length scales discussed above. The activation and breaking of a chemical bond inside a molecule occurs in the picosecond regime, completion of an entire reaction cycle from complexation between catalyst and reactants through separation from the product may take anywhere between microseconds for the fastest enzymatic reactions to minutes for complicated reactions on surfaces. On the mesoscopic level, diffusion in and outside pores, and through shaped catalyst particles may take between seconds and minutes, and the residence times of molecules inside entire reactors may be from seconds to, effectively, infinity if the reactants end up in unwanted byproducts such as coke, which stay on the catalyst. 

As has already been emphasized in Fig. 1.1, there is the further problem of connecting the mesoscopic scale, where one considers length scales from the size of effective monomers to the scale of the whole coils, to still much larger scales, to describe structures formed by multichain heterophase systems. Examples of such problems are polymer blends, where droplets of the minority phase exist on the background of the majority matrix, etc. The treatment of... 

The occurrence of kinetic instabilities as well as oscillatory and even chaotic temporal behavior of a catalytic reaction under steady-state flow conditions can be traced back to the nonlinear character of the differential equations describing the kinetics coupled to transport processes (diffusion and heat conductance). Studies with single crystal surfaces revealed the formation of a large wealth of concentration patterns of the adsorbates on mesoscopic (say pm) length scales which can be studied experimentally by suitable tools and theoretically within the framework of nonlinear dynamics. [31]... 

Such long times usually appear when the observed objects are large compared to single atoms, but are still mesoscopic, e.g. polymer molecules or aggregates of smaller molecules, or when dynamic processes at all length scales slow down in the vicinity of phase transitions or due to a glass transition. 

Fig. 23 Schematic illustration of the development of LSBR. White background, shaded regions and hatched regions denote the paraelectric regions, non-ferroelectric LSBR and ferroelectric regions, respectively. The area simroimded hy solid line in g represents the ferroelectric region with mesoscopic length scale...

In fact, there are various scenarios for a real or incipient tricritical point in ionic fluids. In contrast to the case of polymer tricriticality, which is well-established from theory and experiment, for ionic fluids these scenarios are still speculative and call for more detailed theoretical and experimental investigations. In particular, it remains to identify the physical origin of the mesoscopic, second length scales. 

TDFRS allows for experiments on a micro- to mesoscopic length scale with short subsecond diffusion time constants, which eliminate almost all convection problems. There is no permanent bleaching of the dye as in related forced Rayleigh scattering experiments with photochromic markers [29, 30] and no chemical modification of the polymer. Furthermore, the perturbations are extremely weak, and the solution stays close to thermal equilibrium. 

Catalyst quality on a mesoscopic length scale (diffusion length, loading, profiles)... 

The formal homogenization process is accomplished by considering every property depending on both global and meso-length scales in the form / = f(x,y), where x and y are the macroscopic and mesoscopic coordinates respectively. The relation between the length scales is y = e 1x. This shows that quantities vary e-1 faster on the meso level than those the macro one. We then postulate two-scale asymptotic expansions for the unknowns in terms of the perturbation parameter e... 

Reductionist or projective approaches to reduce large datasets (or systems) at the atomic length scale to important components at mesoscopic or macroscopic length scales... 

Yet another promising line of research lies in creating mesoscopic representations whose fundamental scale is somewhere within the 1010 range of distance scales and for which one defines closed (or fully consistent) equations of motion. At the macroscopic limit, hydrodynamic models are a very successful and standard example. More recent approaches include the Cahn-Hillard coarse-grained models and phase-field models. In some cases, one aims to ascertain the degree to which the systems exhibit self-similarity at various length scales hence the lack of a specific parameterization—which would be necessary using reduceddimensional models—is not of much importance. 

The structure of polymer surfaces and thin polymeric films at the mesoscopic scale is of interest, both for application and basic research [1] As the size of many technological devices decreases, the natural length scales of many typical polymers such as the radius of gyration, the persistence length, or the domain size in block copolymers, match the feature size and thus the materials are expected to display a new behaviour [2-5], On the other hand, the tendency towards spontaneous structure forma-... 

Carbon Black Networking on Mesoscopic Length Scales... 

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