The Mean-Square Radius of Gyration

The first direct calculations for randomly branched polycondensates were carried out by Zimm and Stockmayer13) by direct evaluation of the double sum 

A list of various mean-square radii of gyration as function of the molecular weight is given in Table 3. In many cases this molecular weight dependence can be described by the approximation of a scaling law, i.e. 

The exponents are also given in Table 3. In most cases one obtains 2 v = 1 (no excluded volume effect is taken into account). Only in the two cases of the ABC polycondensates with the stringent constraint aB + etc = a a, and for the fractions of the random polycondensates, is the much lower exponent of 2v = 1/2 obtained90,104 (see Fig. 31). 

Moreover, the overcrowding effect can be avoided in the cascade theory by introducing a second shell substitution effect. This was done by Gordon and Parker177. In the 

Randomly cross-linked chains (polydisperse (m = 1) primary chains) 

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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... 

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. 

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]... 

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...

Average configuration-dependent physical properties are evaluated for tri- tetra-, and hexafunctional polyethylene stars perturbed by electrostatic repulsion of charges placed at the free chain ends. Configuration-dependent properties evaluated are the probability for a trarts placement, expansion of , the mean-square radius of gyration, asymmetry of the distribution of the chain atoms, and asymmetry of the distribution described by the atoms considered to bear the charges. 

In these eqs. index 2 stands for two branches, i.e. for the linear molecule. is the mean square radius of gyration which is equal to < h2 >/6 for linear chains of sufficient length. As an example, the values for a star molecule of four equal branches are quoted. One obtains for the free-draining case ... 

Knowing these functions, the mean-square radius of gyration (S2)z and the translational diffusion coefficient Dz can easily be derived eventually by application of the Stokes-Einstein relationship an effective hydrodynamic radius may be evaluated. These five... 

In the last chapter, equations were derived for the particle-scattering factor, the mean-square radius of gyration, the diffusion coefficient and the first cumulant of the dynamic structure factor. All these have the common feature that, for homopolymers at least, they can be written in the following form ... 

For the mean-square radius of gyration, one obtains after application of the differential operation (outlined in Eq. C.5)... 

The weight average degree of polymerization is found from Eq. (C.42) by setting q = 0, and the mean-square radius of gyration after differentiation of Pz (q2) with respect to q2... 

This leads to the conclusion that polydisperse Unear chains cannot be distinguished from randomly branched chains using only the shape of their scattering curves. Indeed, when the link probabilities are expressed in terms of the mean-square radius of gyration, the particle-scattering factor is given in both cases by... 

Fig. 32. Theoretically predicted molecular-weight dependence of the mean-square radius of gyration for star-molecules with rays grafted onto a large nucleus. Full line nucleus is a hard sphere chain curve ABC nucleus dotted line A3 nucleus1145...

While the mean-square radius of gyration (S2)z can be obtained from the procedure of differentiating the particle-scattering factor Pz(q2) with respect of q2, the translational diffusion coefficient is obtained by integration of Pz(q2) over the whole q2 region94 ... 

The characterization of branched or cyclic structures by the g- and h-factors has the disadvantage that the properties of the analogous linear chains must be known. Such characterization requires a great deal of work in preparative chemistry as well as accurate physical chemical measurements, and in some cases the linear analogs are not even known. In such cases, a direct combination of the mean-square radius of gyration and the hydrodynamic radius leads to the very useful dimensionless parameter188. ... 

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