Coil statistical

The disordered state of a statistical coil is what is displayed by polymers in the molten and amorphous states and also in solution. To describe the conformation of a macromolecule consisting of a main chain N +l atoms, the positions of all them have to be determined. Using vectorial 

The objective is to obtain the mean value of the end-to-end distance corresponding to the set of conformations that define the statistical coil state. For this, it is possible to think physically in two ways that might seem different at first sight but are in fact the same (ergodic hypothesis)  

Determine the values that r adopts with time for a particular macromolecule and then calculate a time average. 

Determine the values of r at a particular instant for N macromolecules of the sample, and thus calculate the instantaneous mean value of r for the set of macromolecules. In this way, the mean square value of the end-to-end distance of the chain is defined as 

The result of the two methods is the same for a polymer in the liquid state, because a macromolecule statistically takes all the possible conformations with time, which is equivalent to observing a large number of macromolecules at a particular instant. In the glassy state, each molecule adopts one specific conformation, and therefore the value of r does not change with 

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The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

For a statistical coil, the product of polymer intrinsic viscosity and molecular weight is directly proportional to the cube of the root-mean-square radius of gyration RG 77137... 

Flory and Huggins developed an interaction parameter that may be used as a measure of the solvent power of solvents for amorphous polymers. Flory and Krigbaum introduced the idea of a theta temperature, which is the temperature at which an infinitely long polymer chain exists as a statistical coil in a solvent. 

Aside from chemical composition and chain length the properties of macromo-lecular substances are substantially determined by the conformation and configuration of the individual macromolecules. Isolated macromolecules do not take up a precisely defined three-dimensional shape they rather assume a statistically most probable form which approximates to the state of maximum possible entropy. This is neither a compact sphere nor an extended rigid chain, but rather a more or less loose statistical coil (Fig. 1.7). 

N 090 "Matrix Formulation of the Transition from a Statistical Coil to an Intramolecular Antiparallel p Sheet"... 

As is well-known, this pair of expressions will not be valid for the most general case of a second order fluid, since p22 — tzi must not necessarily vanish for such a fluid. Eq. (2.9) states that the first normal stress difference is equal to twice the free energy stored per unit of volume in steady shear flow. In Section 2.6.2 it will be shown that the simultaneous validity of eqs. (2.9) and (2.10) can probably quite generally be explained as a consequence of the assumption that polymeric liquids consist of statistically coiled chain molecules (Gaussian chains). In this way, the experimental results shown in Figs. 1.7, 1.8 and 1.10, can be understood. 

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). 

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. ... 

Kuhn for statistically coiled molecules. The two dotted lines denoted by F and N stand for the free-draining and the non-draining case of Zimm s theoty for Gaussian coils. The hatched area indicates the area where the experimental points obtained on solutions of anionic polystyrenes are located (See Fig. 3.1). 

We can now ask the question What is the magnitude of the coil density How far is the statistical coil, which we have considered so far, diluted How many monomer units are present in a unit volume It is possible to calculate how the density, p, depends on the distance to the centre of gravity and on the number of links n. It appears that the volume fraction in the centre equals /o = Ihfn, i.e. ... 

An alternative approach to calculate the configurational entropy, involving a scanning simulation method in the absence of solvent, has been carried out by Meirovitch101 and applied to the two-dimensional freely jointed chain of hard disks102 and to decaglycine in the a helix,103-104 hairpin,103 and statistical coil forms.104-105... 

Besides this statistical mechanical approach to the question of helix stability, the problem has also been addressed by conformational energy calculations. First, the helix-breaking tendencies of such residues as serine and aspartic acid can be accounted for by the tendency toward formation of side chain-backbone hydrogen bonds in nonhelical conformations163 (Figures 20 and 21). Second, the free energies of the helical and statistical coil forms in water have... 

H. Meirovitch, M. Vasquez and H. A. Scheraga, Biopolymers, 27, 1189 (1988). Stability of Polypeptide Conformational States. II. Folding of a Polypeptide Chain by the Scanning Simulation Method, and Calculation of the Free Energy of the Statistical Coil. 

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