Gyrate scaling

This immediately gives v = 0.60 implying water was a good solvent for HEC. The measured hydrodynamic and radii gyration scaled with molecular weight as (Fig. 26)... 

Fig. XI-6. Polymer segment volume fraction profiles for N = 10, = 0-5, and Xi = 1, on a semilogarithinic plot against distance from the surface scaled on the polymer radius of gyration showing contributions from loops and tails. The inset shows the overall profile on a linear scale, from Ref. 65.

Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

In order to get a better notion of the scaling relationship between gyration radius and average chain length for different density regimes, it is convenient... 

Single chains confined between two parallel purely repulsive walls with = 0 show in the simulations the crossover from three- to two-dimensional behavior more clearly than in the case of adsorption (Sec. Ill), where we saw that the scaling exponents for the diffusion constant and the relaxation time slightly exceeded their theoretical values of 1 and 2.5, respectively. In sufficiently narrow slits, D density profile in the perpendicular direction (z) across the film that the monomers are localized in the mid-plane z = Djl so that a two-dimensional SAW, cf. Eq. (24), is easily established [15] i.e., the scaling of the longitudinal component of the mean gyration radius and also the relaxation times exhibit nicely the 2 /-exponent = 3/4 (Fig. 13). 

Due to difficulties and uncertainties in the experimental separation of the porous media [93], and the inevitability of approximations in the analytical treatment [87,89], the nature of the chain movement in a random environment is still far from being well understood, and theoretical predictions are controversial [87,89]. Thus, on the ground of replica calculations within a variational approach, one predicts three regimes [87] in which the chain gyration radius Rg scales with the number of repeatable units N as rI (X for low, R x N for medium, and R x for high... 

Besides crystalline order and structure, the chain conformation and segment orientation of polymer molecules in the vicinity of the surface are also expected to be modified due to the specific interaction and boundary condition at the surface between polymers and air (Fig. 1 a). According to detailed computer simulations [127, 128], the chain conformation at the free polymer surface is disturbed over a distance corresponding approximately to the radius of gyration of one chain. The chain segments in the outermost layers are expected to be oriented parallel to the surface and chain ends will be enriched at the surface. Experiments on the chain conformation in this region are not available, but might be feasible with evanescent wave techniques described previously. Surface structure on a micrometer scale is observed with IR-ATR techniques [129],... 

Note that the accuracy with which the MD calculation can estinate the gyration radius is only about 10%, and thus it is not clear whether the slight disagreement between the MC and MD results for (Rg) in Fig. 5.2 is significant. We empharize a comparison of dynamic properties here because the usefulness of MC to estinate any dynamic properties of polymers is doubled often in the literature. Comparisons her static properties on smaller length scales -... 

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]...

Elastic and quasi-elastic (NSE) neutron scattering experiments were performed on dilute solutions of linear poly(isoprene) (PIP) polymers and of PIP stars (f = 4,12,18) [150]. In all cases the protonated polymers were dissolved in d-benzene and measured at T = 323 K, where benzene is a good solvent. Figure 50 shows the results of the static scattering profile in a scaled Kratky representation. In this plot the radii of gyration, obtained from a fit of the... 

Fig. 33 Scaling of the molar mass of PNIPAM mesoglobules (M 8) vs. their radius of gyration (i g) with fractal dimensionality 2.7 (filled symbols) and the shape factor J g/J h (open symbols). The conditions at which mesoglobules were formed correspond to those in Table 2 Mw = 27300gmol 1, non-equilibrium heated (circles) Mw = 160000gmol 1, nonequilibrium heated (triangles) Mw = 160000gmoL1, equilibrium heated (squares). (Reprinted with permission from Ref. [ 147] copyright 2005 Elsevier)...

The sizes of the dendrimers have been determined by calculating the molecular volumes, as defined by the van der Waals radii of the atoms, and by calculating the radii of gyration for several configurations of the dendrimers, as obtained from a molecular dynamics simulation at room temperature. The solvent influence on the calculated radii was estimated by scaling the nonbonded interactions between the atoms. Molecular volumes and average radii for ensembles of 500 conformations of the BAB-dendr-(NH2)D dendrimers have been collected in Table 26.2. 

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