Scaling and universality

FIGURE 5.9 See color insert. Conformation-dependent rate of stretching. Image spaced every 0.13 A at the highest strain rate investigated. (Source T. T. Perkins, D. E. Smith, and S. Chu, Science 276, 2016 (1997). With permission from AAAS. Original in color.) 

These experiments confirm that a long chain moves not only by discrete segments, but also as a whole molecule. The movement is like that of snake. Even in a tangled environment, an individual chain molecule can move about as a whole by diffusion just as reptation predicted. 

The results also indicate that identical polymers beginning from the same state can uncoil in different ways and get hung up in strange knots and bends. 

To summarize what we have discussed so far, the configuration of a polymer chain depends on two basic quantities molecular weight M (which, in turn, is related to 

Although the random motion of a flexible polymer chain is complex, recent theoretical developments predict that the universal behavior should be observed in certain dynamic regimes. Such a behavior is related to the critical exponents as expressed in power laws. 

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Schroder, T.B. and J.C. Dyre, Scaling and universality ofac conduction in disordered solids. Physical Review Letters, 2000. 84 p. 310... 

Schrpder TB, Dyre JC (2000) Scaling and universality of ac conduction in disordered solids. 

Gonventional Scaling and Universality in a Disordered Bilayer Quantum Heisenberg Antiferromagnet. 

In recent years, studies of solutions of polymer blends and of copolymers have aroused a substantial theoretical and experimental interest. This is motivated by both numerous applications and more fundamental issues concerning the usefulness of the scaling and universality concepts to describe the thermodynamic properties and the phase transitions in these systems. In this lecture, chain interactions in dilute and semidilute solutions are reviewed and it is discussed how and when the interactions between chemically different monomers lead to a macroscopic phase separation in the case of ternary polymer A-polymer B- solvent systems and to a mesophase formation in diblock-copolymer solutions. The important conclusion is that due to both the overall monomer concentration fluctuations (excluded volume effects) and the composition fluctuations, the classical Flory theory often fails. This requires the use of the renormalization method and of scaling concepts to give a correct description of the phase diagrams and the critical phenomena observed in these complex systems. We give only here a brief outline, a complete review has been published elsewhere, ... 

As noted earlier in section A2.5.6.2. the assumption of homogeneity and tlie resnlting principle of two-scale-factor universality requires the amplitude coefficients to be related. In particnlar the following relations can be derived ... 

Bruce A D and Wilding N B 1992 Scaling fields and universality of the liquid-gas critical point Phys. Rev.L 68 193-6... 

Note 1 B. Kamp [Z. tech. Fhysik, 12, 30 (1931)] skews tke effect of increased porosity in decreasing tkermal conductivity of boiler scale. Partridge [University of Michigan, Eng. Research Bull., 15, 1930] has published a 170-page treatise on Formation and Properties of Boiler Scale. 

C being another universal constant. It should be emphasized, however, that all these relations neglect corrections to scaling and hence are only asymptotically valid in the limit where both D / and /. The scahng rela-... 

Although there is no universal consensus as to the scale of production and use of chemical substances, it is estimated that the average annual world production of such substances is in excess of 450 million tonnes. Other estimates indicate that there are currently identified over five million distinct chemical compounds, with this number increasing at the rate of over a third of a million per year. Whilst many of these compounds are clearly not in everyday commercial or industrial use, it is estimated that at least 100,000 chemical substances can be considered to be in everyday use on a substantial scale, and that this number is being added to at the rate of at least several hundred per year, in the case of substances which are produced in quantities in excess of one tonne per year. 

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-... 

One of the authors (ML) is grateful to Fukui Institute for Fundamental Chemistry, Kyoto University for a Fukui Institute Fellowship. The present work was in part supported by a CREST (Core Research for Evolutional Science and Technology) grant in the Area of High Performance Computing for Multi-scale and Multi-physics Phenomena from the Japan Science and Technology Agency (JST). 

Viscoelastic and transport properties of polymers in the liquid (solution, melt) or liquid-like (rubber) state determine their processing and application to a large extent and are of basic physical interest [1-3]. An understanding of these dynamic properties at a molecular level, therefore, is of great importance. However, this understanding is complicated by the facts that different motional processes may occur on different length scales and that the dynamics are governed by universal chain properties as well as by the special chemical structure of the monomer units [4, 5],... 

The /1CDM paradigm for structure formation in the Universe, described in many hundreds of published papers, is very effective at reproducing observed large scale structure, based on a boundary condition of a scale-free Gaussian random power spectrum. Yet ACDM contains no information on the physics of whatever makes up CDM, and remains deficient in its description of galaxies and small-scale structures thus it is on galaxy scales and smaller where we can still learn the most, and hopefully attach some (astro-)physics to an ab initio power spectrum. 

P. Meakin, Fractals, Scaling and Growth Far from Equilibrium, Cambridge University Press, Cambridge, 1998. 

Since the publication of the first edition in 1955 there has been a substantial increase in the relevant technical literature but the majority of developments have originated in research work in government and university laboratories rather than in industrial companies. As a result, correlations based on laboratory data have not always been adequately confirmed on the industrial scale. However, the section on absorption towers contains data obtained on industrial equipment and most of the expressions used in the chapters on distillation and evaporation are based on results from industrial practice. 

Last not least, there is a great variety of proprietary lO-design software that has been developed at research facilities and university institutes. E.g., at the TU Delft the S-matrix oriented software Photonic CAD was established several years ago, an innovative 10 design framework based on Hewlett Packard s Microwave Design System. Actual academic work e.g. addresses modes of bent waveguides, BMS-3D" ° is a quite new bend mode solver of the IRE, Prague, or FDTD-schemes with non-uniform grids, a topic of special importance to improve computational efficiency when multi-scale feature sizes are requested, to name a few of recent tasks, only. 

D1V.6.1. Prigogine, Universal constants, time scales and self-interaction. Bull. Cl. Sciences, Acad. Roy. Belg. 48, 1322-1332 (1962). 

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