Mean-square-radius of gyration

For free particles, the mean square radius of gyration is essentially the thennal wavelength to within a numerical factor, and for a ID hamionic oscillator in the P ca limit. 

The above radius of gyration is for an isotropic system. If the system is anisotropic, the mean square radius of gyration is equal to... 

The procedure is obviously a very simple one, but suffers from the fact that the success rate diminishes very rapidly as the pore radius becomes smaller. The root-mean-square radius of gyration of... 

Figure 3. Partition coefficient of freely jointed chains between the bulk solution and a cylindrical pore. The chains have different numbers of mass-points (n) and different bond lengths, and are characterized by the root-mean-square radius of gyration measured in units of the pore radius. See text for details.

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... 

Fig. 9. Kinematic viscosity v = r /p (r viscosity, p density) divided by S2/M (S2 = mean square radius of gyration, M molecular mass) as a function of M for polyethylene melts at the same monomeric friction coefficient. (Reprinted with permission from [48]. Copyright 1987 American Chemical Society, Washington)...

Table 5. Scaling predictions for the mean square radius of gyration and the mean square correlation lengths < 2> in the different regimes (see Fig. 38) of polymer solutions [102-104]...

Fig. 10 Equilibrium radius of gyration of a molecule plotted as a function of temperature the molecule is composed of 1000 beads. The radius of gyration shows a steep increase and a large fluctuation above 700 K. The insets show typical chain conformations at indicated temperatures. Note that the ideal random coil state of this fully flexible chain should have the mean-square radius of gyration R2 = 1000 x (1.54/3.92)2/6 = 25.7, the value is around 800 K...

Bueche (16,172) proposed that the viscosity is proportional to the fourth power of the polymer concentration and a complex function of the free volume of the mixture. Kraus and Gruver (170) find that the 3.4 power fits experimental data better than does the fourth power. They used equation (58) with (r2) replaced by the mean-square radius of gyration (s2). The term r2)/(rf) indicates that poor solvents should lower the viscosity more than a good solvent. As the temperature increases, the factor increases as a function of the ratio (T - 7 (tJJ)/(7 - 7 ). The glass transition temperatures of the polymer and diluent are andT o, respectively. 

The fundamental effect of LCB on the polymer is a reduction of the size of the polymer relative to the size of a linear polymer with the same MW as measured, for example by its mean-square radius of gyration ... 

Debye scattering deals with the situation wherein the relative size parameter is large. Since the root mean square radius of gyration of the particle, (s2)1/2, is a... 

An estimation of the local ligand concentration, [N]coii> could be achieved by assuming free movement of the ligands in the interior of a sphere with radius i, the root mean square radius of gyration of the polymer chain ... 

A further remark has to be made when the stars contain polydisperse arms. The radius of gyration is now based on the z-average of the mean square radius of gyration over the molar mass distribution while the degree of polymerization is the weight average Also for this case the molar mass dependence of this radius could be calculated and was found to be [83]... 

Since the mean square radius of gyration requires a z-average but the molar mass a weight average the fractal dimension remains unchanged only if the ratio is independent of the molar mass or close to unity. These conditions are mostly fulfilled with polydisperse linear chains but not for the randomly branched ones. Here this ratio increases strongly with the molar mass. 

The leading parameter that characterizes the distributions of randomly branched samples is an exponent Tthat is defined in the next section. The average procedures for the z-average of the mean square radius of gyration and the weight average molar mass results in the relationship [7]... 

The close connection that exists between polymer morphology and mechanical properties stimulated extensive research in this field. In amorphous polymers, elastic neutron scattering led to important results. Using mixtures of conventional and deuterated macromolecules, the mean square radius of gyration < > of several amorphous polymers in bulk has been determined (237). This... 

Parameter characterizing the effect of long-chain branches on the size of a branched molecule in solution and defined as the ratio of the mean-square radius of gyration of a branched molecule, si), to that of an otherwise identical linear molecule si), with the... 

By extrapolating scattering data for each angle to zero concentration, the mean-square radius of gyration may be measured [9,10,15-18]... 

Chapter C deals with molecular dimensions of interrupted helices. Typical theories for mean-square radius of gyration and mean-square end-to-end distance are reviewed. Important predictions from theory are compared with the results of recent light-scattering measurements. Complications attendant upon the analysis of light-scattering data for polypeptides in the helix-coil transition region are discussed. 

Fig. 11. Dependence of the mean-square radius of gyration on helical fraction for chains with various N, a = 2 x 10-4, a0 = 22.4 A, and a, = 1.5 A...

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