Random environment, polymers

A natural extension of the studies in Sec. VIIA would be the investigation of the drift of a polymer chain in random environment when a constant external field B is applied in one direction [21,22]. 

Ebert U, Baumgartner A, Schafer L (1996) Universal short-time motion of a polymer in a random environment Analytical calculations, a blob picture, and Monte Carlo results. 

Type A includes the classical approach to chiral polymeric catalysts a monomeric ligand is synthesized and attached to a random coil polymer. If a binding position for the metal is incorporated in every constitutional repeating unit, the catalytically active centres are randomly oriented, which is quite a delicate situation. Their micro-environment is dependent on the position on the polymer, and it can be expected that this is true for their asymmetric induction as well. Such a catalyst can hardly be optimized rationally. 

This fractured surface may be modelled as a random surface passing through the cutting bonds, so that the surface has the least total energy. Obviously, the surface can not have any overhang the height z x y) can not be multi-valued at any point (x,y). This random surface problem is therefore analogous to that of a directed polymer in a random environment, in two dimensions, as mentioned before. 

In diluted polymer solution macromolecules are in random environment, influencing on the process of either aggregation or degradation (more rarely on both of them) that results in the stochastic equation instead of the Eq. (6) [13] ... 

The conformational statistics of a polymer chain will change if the chain is placed in a random environment. We can model the effect of the random environment by introducing an interaction energy... 

In this chapter we have demonstrated the rich behavior of polymer chains embedded in a quenched random environment. As a starting point, we considered the problem of a Gaussian chain free to move in a random potential with short-ranged correlations. We derived the equilibrium conformation of the chain using a replica variational ansatz, and highlighted the crucial role of the system s volume. A mapping was established to that of a quantum particle in a random potential, and the phenomenon of localization was explained in terms of the dominance of localized tail states of the Schrodinger equation. 

We shall then survey the important ways in which fractal structures, including polymers, interact with their environments. This leads to a discussion of the thermodynamics and hydrodynamics of a polymer solution. With the fractal properties in mind we discuss the interaction of a random-walk polymer with itself, finding that these interactions change the spatial arrangement of the molecule considerably. Having described the behavior of individual polymer molecules, we can discuss the behavior of solutions, notably those where the polymer chains interpenetrate strongly. The spatial, energetic and dynamic behavior of these solutions can be understood in its major outlines from the fractal properties of isolated polymers treated earlier. 

We close this section with a caveat The above discussion applies to random coil polymers. Polymers in orders environments such as channels or crystals might be expect l to have quite different dynamks [86-91]. In such cases, the RIS description of conformational transitions may be adequate. 

The presence of anomalous diffusion can be explained by the structure of the polymer matrix, which - at the length-scale of a few hopping distances - restricts the penetrant motion to the effect that - on this scale - the penetrant s paths cannot be truly random. The polymer environment, at the same time, causes a separation of time-scales consistent with the hopping mechanism (short-time in-cavity motion vs long-time diffusive motion). This, in turn, is another cause for anomalous diffusion. 

Various authors have developed methods based on Raman spectroscopy for determining crystallinity. In some polymers, specific vibrational bands have been attributed to the crystalline phase. In others, the ratio of the intensities of trans conformation vibrations to gauche conformation vibrations has been used. Still others have used the width of a particular vibrational band to indicate crystallinity. The last method is based on the effect of the local environment on the frequency of the band. In a random environment, many different local environments exist and the band is broadened by these variations. In a crystal, the local environment is the same for all of the chain segments. Then all of these segments have (nearly) the same frequency and the band is narrow. For a band that is sensitive to the local environment, a narrow band indicates high crystallinity and a broad band indicates low crystallinity. 

Comets, F., Shiga, T. and Yoshida, N. (2004). Probabihstic Analysis of Directed Polymers in Random Environment a Review, Advanced Studies in Pure Mathematics 39, pp. 115-142. 

In addition to an array of experimental methods, we also consider a more diverse assortment of polymeric systems than has been true in other chapters. Besides synthetic polymer solutions, we also consider aqueous protein solutions. The former polymers are well represented by the random coil model the latter are approximated by rigid ellipsoids or spheres. For random coils changes in the goodness of the solvent affects coil dimensions. For aqueous proteins the solvent-solute interaction results in various degrees of hydration, which also changes the size of the molecules. Hence the methods we discuss are all potential sources of information about these interactions between polymers and their solvent environments. 

DEPTH PROFILE. The secondary electrons produced by ionization processes from an incident beam of high-energy electrons are randomly directed in space. Spatial "equilibrium" is achieved only after a minimum distance from the surface of a polymer in contact with a vacuum or gaseous environment (of much lower density). Consequently, the absorbed radiation dose increases to a maximum at a distance from the surface (2 mm for 1 MeV electrons) which depends on the energy of the electrons. The energy deposition then decreases towards zero at a limiting penetration depth. 

In this section, we describe the role of fhe specific membrane environment on proton transport. As we have already seen in previous sections, it is insufficient to consider the membrane as an inert container for water pathways. The membrane conductivity depends on the distribution of water and the coupled dynamics of wafer molecules and protons af multiple scales. In order to rationalize structural effects on proton conductivity, one needs to take into account explicit polymer-water interactions at molecular scale and phenomena at polymer-water interfaces and in wafer-filled pores at mesoscopic scale, as well as the statistical geometry and percolation effects of the phase-segregated random domains of polymer and wafer at the macroscopic scale. 

The complications for fhe fheoretical description of proton fransporf in the interfacial region befween polymer and water are caused by the flexibility of fhe side chains, fheir random distributions at polymeric aggregates, and their partial penetration into the bulk of water-filled pores. The importance of an appropriate flexibilify of hydrated side chains has been explored recently in extensive molecular modeling studies. Continuum dielectric approaches and molecular dynamics simulations have been utilized to explore the effects of sfafic inferfacial charge distributions on proton mobility in single-pore environments of Molecular level simulations were employed... 

For a semi-flexible tube in a dilute environment, local repulsive potentials among parts of the fiber induce a self-avoiding random walk configuration (swollen coil [151]). In a crowded environment, the depletive action may dominate and the fiber will tend to collapse on itself, forming a globular phase. We know from standard statistical physics of polymers that this latter phase... 

CA of Polysaccharides. Polysaccharides adopt a wide variety of shapes that depend on their composition and their environment. In solution, polymers are almost always random coils that have local regions that might be similar to conformations that are found in the solid state. The chapter by Brant and Christ discusses conformations of polysaccharides in solutions both in terms of these local regions and by the overall shape of the random coil in terms of end-to-end distance, etc. The following discussion concerns only linear (unbranched) molecules, and refers only to regular polymers, i.e., those that have repeated sequences of monomeric residues located by screw-axis (helical) symmetry. 

In crystalline oxides and hydroxides of iron (III) octahedral coordination is much more common than tetrahedral 43). Only in y-FegOs is a substantial fraction of the iron (1/3) in tetrahedral sites. The polymer isolated from nitrate solution is the first example of a ferric oxyhydroxide in which apparently all of the irons are tetrahedrally coordinated. From the oxyhydroxide core of ferritin, Harrison et al. 44) have interpreted X-ray and electron diffraction results in terms of a crystalline model involving close packed oxygen layers with iron randomly distributed among the eight tetrahedral and four octahedral sites in the unit cell. In view of the close similarity in Mdssbauer parameters between ferritin and the synthetic poljmier it would appear unlikely that the local environment of the iron could be very different in the two materials, whatever the degree of crystallinity. Further study of this question is needed. 

For random coils, is directly proportional to the contour length. If n is the number of main chain atoms in the chain, = an. The parameter a is relatively insensitive to environment (21), and has been calculated for a number of polymers from strictly intramolecular considerations using the rotational isomeric model (22). The root-mean-square distance of segments from the center of gravity of the coil is called the radius of gyration S. The quantity S3 is an approximate measure of the pervaded volume of the coil. For Gaussian coils,... 

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