Gaussian coil approximation

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,... 

Fig. 2.1. A random walk chain of 50 monomers of fixed length (thin polygon), leaking the position of every fifth monomer as segment coordinates, we find our model chain (fal polygon), which is approximated by a Gaussian coil...

Most synthetic polymers in which the monomer units are connected via single bonds have rather flexible chains. The bond torsion energy is relatively small and the units can rotate around their bonds [14,30,31]. Each molecule can adopt a large number of energetically equivalent conformations and the resulting molecular geometry is that of a statistical coil, approximately described by a Gaussian density distribution. This coil conformation is the characteristic secondary structure of macromolecules in solution and in the melt. It is entropically favoured because of its... 

Recently, in a number of theoretical computational studies the shape of the Gaussian coil has been characterized by the components of radii of gyration Ri, Rj and R3 in the three main directions of the coil (fixed for each conformation). Calculations reveal that the average sh of the coil n be approximated by a three-axial ellipsoid with the ratio of squares of axes... 

The WSL theory developed by Leibler has been shown to be incorrect because of deviations from the fundamental underlying mean-field assumption. Figure 13.14 shows experimental results for a poly(ethylene-propylene/ethylethylene) (PEP-PEE) diblock copolymer that has been fit to the predictions of the Leibler theory without any adjustable parameters, since the ODT and / were calculated from rheological measurements (Bates et al., 1990). This mean-field theory does not qualitatively describe the behavior of this material. Other experiments have indicated that the RPA approximation (Sttihn and Stickel, 1992) and the Gaussian coil assumption (Bates and Hartney, 1985 Holzer et al., 1991) are inaccurate near the ODT. 

Dynamic light scattering from dilute solutions provides the value of the diffusion coefficient, which can be converted to hydrodynamic radius J h,star of the star polymer. The ratio Rh/(Rg) characterizes the compactness of the macromolecule for the uniform hard sphere impenettable for the flow, it is Rb/Rg= (5/3) 1.29, whereas for the Gaussian coil, Rh/(Rg) = 3a- / /8 0.66. For ideal stars (without excluded-volume interactions), the ratio Rh/(Rg) can be derived within the Kirkwood-Riseman approximation, which gives the value of Rh/(Rg) 0.93. Reported experimental values of the Rh/(Rg) ratio for star polymers and starlike block copolymer micelles are usually found close... 

If the stretching ratio A is large, then the increase in conductivity can be substantial. Pearson calculated that for A = 3.6 we obtain an order of magnitude increase in conductivity. The maximum stretch ratio for a Gaussian coil is k,n LlaY, Hence near this point we note that a varies approximately with L. ... 

Polymer molecules in the melt are very coiled up in fact, they are often approximately Gaussian coils. In other words, the radius of gyration for a linear chain is given by Eq. (2.2), which also applies to an isolated coil in solution. In the melt, the density of surrounding molecules prevents an individual molecule from stretching, and it adopts a compact conformation. 

The study of the viscosity and translational diffusion of mesogenic combshaped macromolecules and the estimation of the equilibrium rigidity of their chains thus indicate that with respect to the hydrodynamic properties, the chain molecules of the polymers in Tables 3.2 and 3.3 can be approximated in a wide range of molecular weights by a nonflowable Gaussian coil with elevated rigidity of the main chain. The hydrodynamic diameter of these molecules reflects the... 

At constant molecular weight, the expansion of the star from its Gaussian coil size is larger than the expansion of the linear polymer because increases with /[8]. For linear polybutadiene hf = 0.003359 is found experimentally. With M set at 10 (approximating the high molecular weight limit) values of are cal-... 

Typically in solution, a polymer molecule adopts a conformation in which segments are located away from the centre of the molecule in an approximately Gaussian distribution. It is perfectly possible for any given polymer molecule to adopt a very non-Gaussian conformation, for example an all-trans extended zig-zag. It is, however, not very likely. The Gaussian set of arrangements are known as random coil conformations. 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

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