Unperturbed mean-square radius of gyration

The equation for the unperturbed mean-square radius of gyration, 0, of copolymers is obtained for two cases by using the RIS method. For one case it is assumed that the total mass of each structural unit of the chain is situated on the skeletal atom. For the other case the deviation is considered of the center of mass of each structural unit from each skeletal atom. 

SANS allows o, the unperturbed mean-square radius of gyration, to be measured. For a linear unperturbed chain, it is well-known that... 

This shows that (fi )g is proportional to the molecular weight and that R )o /M is thus a constant for a given polymer. Combined with Eq. 2.2 it shows that the unperturbed mean-square radius of gyration is also proportional to the molecular weight ... 

Bonchev etal. [11] used graph theory to develop quantitative relationships between g and several indices of branching complexity. The most important result of their work is that for a mono-disperse polymer with no rings, the unperturbed mean square radius of gyration is related very simply to two of these indices, the Wiener number , W, and the Kirchhoff number , Kf. For acyclic molecules, these two numbers are related by ... 

Baumann [ 18] proposed an empirical equation for calculating the unperturbed mean square radius of gyration, R )q, based on light scattering measurements made in a good solvent rather than a theta solvent ... 

Experimental Assessment, The most direct experimental assessment of the persistence length may be obtained from the root-mean-square radius of gyration measured by light scattering. The analytical expression for the unperturbed Rq of a. wormlike chain has been given by Benoit and Doty (5) ... 

Figure 3.1 Typical dependence of viscosity (rj) on molecular weight (Mw) for linear polymer melts. is defined as X = [s ) Z(p- /M v, where (So) is the mean square radius of gyration of an unperturbed molecule, Z is the number of chain atoms in the polymer, v is the specific volume of polymer, and

The factor a is 1 for the unperturbed coil defined in this way, and is larger (or smaller) when a polymer molecule is expanded (or compressed) due to polymer-solvent interactions. Thus, for an idealized freely jointed chain in theta solvent, we have = nl, and the corresponding mean square radius of gyration = nf-/G. When one accounts for the steric effects that prevent distant chain segments from overlapping (excluded volume effect), the dependence of (r ) is predicted to be (closer to experimental observation), rather than [Eq. (5)]. This polymer coil size is much larger than that based on polymer density, and hence markedly influences the viscosity behavior of polymers. 

But for an unperturbed random coil, ie., in the theta state, the root-mean-square radius of gyration should be proportional to the square root of the molecular weight. Thus, Eq. 2.35 indicates that this system deviates slightly from its theta state. Using an extrapolation procedure based on Baumann s equation. Fetters etal. [19] obtained the following expression for the radius of gyration in the theta state ... 

An important length-scale associated with polymer solutions is the root-mean-square radius of the polymer coil in the solution. For an unperturbed coil, this is known as the unperturbed radius of gyration, / g, and is given by the following (1) ... 

Fig. 4.8. Mean square displacement of the center of mass during the simulation of C3i6H634 [167]. The horizontal arrow denotes the mean square unperturbed radius of gyration...

RIS theory, in the form appropriate for branched molecules, is used to calculate the mean-square unperturbed radius of gyration, < s2>0, for cross-linked polyglycine, poly(L-alanine),poly(L-proline),poly(i-alanyl-D-alanine),poly(i.-prolyl-L-pro lylglycine),poly(L-prolyl-i.-alanylglycine ,poly(glycyl-L-alanyl-L-pro line), andpoly(L-aianyl-L-alanylgIycine).Thecentral amino acid residue in each polypeptide chain is replaced by the L-cysteinyl residue involved in cross-link formation. Each cross-linked molecule is considered to contain two trifunctional branch points, the a-carbon atoms of the two... 

Fig. 23. Molecular weight dependence of the mean-square weight average radius of gyration in the unperturbed state relative to Mw (

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