Statistical coil model

From the point of view that the statistically coiled model chain is built up of rigid rods (random links), it seems that eq. (5.10) must be truncated, as eq. (5.11a). 

Jha AK et al (2005) Statistical coil model of the unfolded state resolving the reconciliation problem. Proc Natl Acad Sci USA 102(37) 13099-13104... 

In any case, it can be demonstrated with the aid of the dumb-bell model that eq. (5.10) is a much better approximation for statistical coil molecules than eq. (5.11 a) for rigid rods. Two cases are considered for the purpose A rigid dumb-bell of fixed length hr and an elastic dumb-bell, according to the usual definition, possessing a root mean square length (Kyi. ... 

Figure 2 (a) Sketch of the variables determining the rotation isomeric state model of a statistical coil 9 = bond angle, = bond torsion angle [3], (b) Representation of one of the possible conformations of a two-dimensional statistical coil (33 monomers) with bond-angle restriction ( — 90° <0 <90°) [3],... 

The main quantitative developments of the random coil model of flexible polymers began in 1934 with the work of E. Guth and H. E Mark [12] and W. Kuhn [13]. Using the concept of free rotation of the carbon-carbon bond, Guth and Mark developed the idea of the random walk or random flight of the polymer chain, which led to the familiar Gaussian statistics of today, and eventually to the famous relationship between the end-to-end distance of the main chain and the square root of the molecular weight, described below. 

Completely statistical (random) arrangements of the macromolecules without a regular order or orientation, i.e., without constant distances, are known as amorphous states. There is no long-range order whatsoever. The valid model for such states is the statistical coil. This is the dominating secondary structure in synthetic polymers and polymeric solutions. Its determinant parameter is coil density. 

Rubbers and gels are three-dimensional networks composed of mutually cross-linked polymers. They behave like solids, but they still have high internal degrees of freedom that are free from constraints of external force the random coils connecting the cross-links are free in thermal Brownian motion. The characteristic elasticity of polymeric materials appears from the conformational entropy of these random coils. In this chapter, we study the structures and mechanical properties of rubbers on the basis of the statistical-mechanical models of polymer networks. 

In the Ising model, each atomic magnet occupies one lattice site with a spin that is either up or down. This model is named for physicist E Ising, who solved it in his Ph.D. thesis work with W Lenz in 1925. If you model each spin as an f or I on a one-dimensional lattice, the partition function can be computed in the same way as the partition fimction for a sequence of W or C units in the helix-coil model. The difference between the one-dimensional Ising model and the helix-coil model is just the statistical weights. For the magnet model, the weights are cj(tt) = and (tl) = dUt) = where... 

In 1953 Rouse published a paper to describe theoretically the flow of polymers in dilute solutions. The polymer molecule is assumed to exist as a statistical coil and is subdivided into N submolecules. Each submolecule is thought of a solid bead. The beads behave as Gaussian chains and their entropy-elastic recovery can be described by a spring with a spring constant hkT/cP-, where a is the average end-to-end distance of a submolecule and k is the Boltzmann constant. The model is shown in Figure 8.9. 

Unperturbed Statistical Coils, in the Kirkewood and Riseman model, the polymer is represented as a collection of beads interconnected by bonds of length L... 

Statistical Coils in Perturbed Mode. This model is also called the Flory-Fox model. In the previous model, the effect of excluded volume was not taken into consideration. According to the Flory and Fox analysis, the latter model applies only to the case of unperturbed chains under 8 conditions the case of chains in a good solvent requires a separate freatment. The Flory-Fox model is based on the assumption that long-range interactions and the perturbations that they cause do not modify the flow of a solution ... 

The statistical model of the random coil discussed in Chap. 1 illustrates many of these items. 

Use of random flight statistics to derive rg for the coil assumes the individual segments exclude no volume from one another. While physically unrealistic, this assumption makes the derivation mathematically manageable. Neglecting this volume exclusion means that coil dimensions are underestimated by the random fight model, but this effect can be offset by applying the result to a solvent in which polymer-polymer contacts are somewhat favored over polymer-solvent contacts. 

Before discussing details of their model and others, it is useful to review the two main techniques used to infer the characteristics of chain conformation in unordered polypeptides. One line of evidence came from hydrodynamic experiments—viscosity and sedimentation—from which a statistical end-to-end distance could be estimated and compared with values derived from calculations on polymer chain models (Flory, 1969). The second is based on spectroscopic experiments, in particular CD spectroscopy, from which information is obtained about the local chain conformation rather than global properties such as those derived from hydrodynamics. It is entirely possible for a polypeptide chain to adopt some particular local structure while retaining characteristics of random coils derived from hydrodynamic measurements this was pointed out by Krimm and Tiffany (1974). In support of their proposal, Tiffany and Krimm noted the following points ... 

Models of the polymer coil are based on the end-to-end distance, which is generally not directly available as a quantitative feature. Coils in dilute solution can be characterized in terms of the radius of gyration, Rg, which is a statistical measure of the distribution of mass about the center of gravity or in terms of the hydrodynamic radius, Rh, that is usually determined through the use of Stokes law and a measurement of a drag coefficient or friction factor, /drag/ for the coil,... 

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