Length scales macroscopic/microscopic

All the macroscopic properties of polymers depend on a number of different factors prominent among them are the chemical structures as well as the arrangement of the macromolecules in a dense packing [1-6]. The relationships between the microscopic details and the macroscopic properties are the topics of interest here. In principle, computer simulation is a universal tool for deriving the macroscopic properties of materials from the microscopic input [7-14]. Starting from the chemical structure, quantum mechanical methods and spectroscopic information yield effective potentials that are used in Monte Carlo (MC) and molecular dynamics (MD) simulations in order to study the structure and dynamics of these materials on the relevant length scales and time scales, and to characterize the resulting thermal and mechanical proper-... 

This strategy permits one to test the validity of macroscopic theories on microscopic length scales, the reliability of experimental techniques and, vice versa, the appropriateness of the CFD treatment. Furthermore, having put the simulations on a safe basis also enables one to predict transport features outside the experimentally accessible parameter range with some confidence of reliability [8]. 

A traditional explanation of solid friction, which is mainly employed in engineering sciences, is based on plastic deformation.12 Typical surfaces are rough on microscopic length scales, as indicated in Figure 3. As a result, intimate mechanical contact between macroscopic solids occurs only at isolated points, typically at a small fraction of the apparent area of contact. 

Conventional dimensional analysis uses single length and time scales to obtain dimensionless groups. In the first section, a new kind of dimensional analysis is developed which employs two kinds of such scales, the microscopic (molecular) scale and the macroscopic scale. This provides some physical significance to the exponent of the Reynolds number in the expression of the Sherwood number, as well as some bounds of this exponent for both laminar and turbulent motion. 

An important objective in materials science is the establishment of relationships between the microscopic structure or molecular dynamics and the resulting macroscopic properties. Once established, this knowledge then allows the design of improved materials. Thus, the availability of powerful analytical tools such as nuclear magnetic resonance (NMR) spectroscopy [1-6] is one of the key issues in polymer science. Its unique chemical selectivity and high flexibility allows one to study structure, chain conformation and molecular dynamics in much detail and depth. NMR in its different variants provides information from the molecular to the macroscopic length scale and on molecular motions from the 1 Hz to 1010 Hz. It can be applied to crystalline as well as to amorphous samples which is of particular importance for the study of polymers. Moreover, NMR can be conveniently applied to polymers since they contain predominantly nuclei that are NMR sensitive such as H and 13C. 

Just as there is a correspondence between the symmetry of crystals and that of their physical properties, there is also a connection between the symmetry exhibited by a crystal at the macroscopic and microscopic length scales, in other words, between the external crystal morphology and tme internal crystal structure. Under favorable circumstances, the point group (but not the space group) to which a crystal belongs can be determined solely by examination of... 

The computational advantages of such multigrid methods arise from two key factors. First, microscopic simulations are carried out over microscopic length scales instead of the entire domain. For example, if the size of fine grid is 1% of the coarse grid in each dimension, the computational cost of the hybrid scheme is reduced by 10 2rf, compared with a microscopic simulation over the entire domain, where d is the dimensionality of the problem. Second, since relaxation of the microscopic model is very fast, QSS can be applied at the microscopic grid while the entire system evolves over macroscopic time scales. In other words, one needs to perform a microscopic simulation at each macroscopic node for a much shorter time than the macroscopic time increment, as was the case for the onion-type hybrid models as well. 

Percolation theory provides a well-defined model applicable to a wide variety of spatially random phenomena, both macroscopic and microscopic (Table X). The characteristic length scales for these phenomena... 

An important common feature of macroion solutions is that they are characterized by at least two distinct length scales determined by the size of macroions (an order up to lOnm in the case of ionic micellar solutions) and size of the species of primary solvent (water molecules and salt ions, i.e. few Angstroms). Considering practical colloidal macro-dispersions, like foams, gels, emulsions, etc., usually we are dealing with as many as four distinct length scales molecular scale (up to lnm) that characterizes the species of the primary solvent (water or simple electrolytes) submicroscopic or nano scale (up to lOOnm) that characterizes nanoparticles or surfactant aggregates called micelles microscopic or mesoscopic scale (up to lOO m) that encompasses liquid droplets or bubbles in emulsion and foam systems as well as other colloidal suspensions, and macroscopic scale (the walls of container etc). 

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