Gyration radii

In dilute polymer solutions, hydrodynamic interactions lead to a concerted motion of tire whole polymer chain and tire surrounding solvent. The folded chains can essentially be considered as impenneable objects whose hydrodynamic radius is / / is tire gyration radius defined as... 

With the total number of monomers and the volume of the system fixed, a number of statistical averages can be sampled in the course of canonical ensemble averaging, like the mean squared end-to-end distance Re), gyration radius R g), bond length (/ ), and mean chain length (L). 

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... 

Eventually, one should also note that even in the case of dead polymers one has also observed a variation of the gyration radius (i g) with D, which goes through a minimum as D —> 0 [48] although the contour length of the chains L does not change. Thus computer simulations, being capable of... 

Consider normal Rouse dynamics of a polymer of length A chosen such that, during its lifetime, the chain MSQ is equal to its gyration radius Rg. Within a time interval Tbreak = cf. Eq. (24), where = o exp(-7) is... 

Let us eonsider briefly the latter for the ease of a tube. If the diameter D of the narrow tube in whieh the polymer moves is smaller than the average size (or gyration radius of the polymer eoil in bulk solution, there is a... 

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). 

FIG. 13 (a) Log-log plot of the longitudinal part of the mean gyration radius, R, ... 

FIG. 14 (a) Plot of the normalized parallel component of the mean gyration radius i gll/i gb (open symbols), and end-to-end distance i /i gb (full symbols), vs the ratio D/i gb [19] (chain lengths N = 128, 256, 512 are distinguished by different symbols). The dashed straight line indicates the asymptotic slope of the scahng function for small D/i gb, namely —2(z/2 — (h) The same for the perpendicular components. 

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... 

FIG. 21 (a) Mean square gyration radius Rg vs total density C of the system for... 

FIG. 25 (a) Variation of the mean-square gyration radius Rg with the intensity of the field (bias) and with host matrix density Cob- (b) The same for the ratio of its longitudinal and transversal components (c) The ratio between the end-to-... 

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 -... 

Fig. 5.17. Number average (Rg)n of the gyration radius of PE with Mn = 20 effective bonds plotted vs. temperature. Parameters of the symbols show the values of d. From [184].

Figure 5.8. Rate of coalescence as a function of the number density of internal droplets in the globules. Globule diameter =11.5 pm droplet diameter = 0.36 pm initial droplet volume fraction in the globules = 25% globule volume fraction = 10% SDS concentration = 2.4 10 mol/1 SMO concentration in the oil phase = 2 wt% 0.1% of silicone oil with a gyration radius of 12 nm was added to dodecane. (Reproduced from [44], with permission.)...

At low water content from vv = 2 to 5.5, a homogeneous reverse micellar solution (the L2 phase) is formed. In this range, the shape of the water droplets changes from spheres (below ir = 4) to cylinders. At tv — 4, the gyration radius has been determined by SAXS and found equal to 4 nm. Syntheses in isolated water-in-oil droplets show formation of a relatively small amount of copper metallic particles. Most of the particles are spherical (87%) with a low percentage (13%) of cylinders. The average size of spherical particles is characterized by a diameter of 12 nm with a size polydispersity of 14%. 

RgH partial mean-square gyration radius of hydrophobic monomers... 

Fig. 11 a Mean square gyration radius and b Jensen-Shannon divergence measure as a function of the attraction energy pp between hydrophilic segments, after the sequence evolution procedure. The characteristic energy of H - H interactions is fixed at hh = 2/cbT, thus stabilizing a dense globular core. Adapted from [70]... 

The formation of a compact globule in copolymers requires them to have specific conformations, which are reached through local conformational fluctuations. A characteristic collapse time may be defined as for instance the time for which the gyration radius reaches its equilibrium value. This time measures the approach to equilibrium for the system and is related to the mean first passage time. 

Note that steady-state situations are investigated, so that the end-to-end distance (R2) and the mean gyration radius... 

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