Square radii

Deutsch and Mark compared the classical expression with a theory developed by Bethe.37 Bethe s calculations showed that the ionization cross section for an atomic electron is approximately proportional to the mean square radius of the appropriate n,l electronic shell. Experiment had also shown a correlation between the maximum in the atomic cross section and the sum of the mean square radii of all outer electrons. This led to the replacement of the Bohr radius with the radius of the corresponding subshell the ionization cross section is now given by,... 

In their seminal work in 1949, Zimm and Stockmayer [69] defined the ratio of the mean square radii of gyration of a branched and a linear polymer of equal molecular weight as the parameter g and is related to the parameter g, which is the ratio of the intrinsic viscosities of a branched and a linear polymer [65-69]. 

Effect of PVA Molecular Weight on Adsorbed Layer Thickness. Figure 4 shows the variation of reduced viscosity with volume fraction for the bare and PVA-covered 190nm-size PS latex particles. For the bare particles, nre(j/ is independent of and the value of the Einstein coefficient is ca. 3.0. For the covered particles, rired/ t increases linearly with tp. Table IV gives the adsorbed layer thicknesses calculated from the differences in the intercepts for the bare and covered particles and determined by photon correlation spectroscopy, as well as the root-mean-square radii of gyration of the free polymer coil in solution. The agreement of the adsorbed layer thicknesses determined by two independent methods is remarkable. The increase in adsorbed layer thickness follows the same dependence on molecular weight as the adsorption density, i.e., for the fully hydrolyzed PVA s and... 

For the contraction factors g of the mean square radii of gyration for fractions of randomly branched materials Zimm and Stockmayer [49] obtained... 

Altenburg also introduced52c) the squared radii of alkanes ... 

Formulas for the mean-square radii of various branched and ringed polymer molecules are developed under the usual assumptions regarding the statistics of chain configuration. For branched molecules, it Is shown that the mean square radii vary less rapidly with molecular weight than for strictly linear molecules, while for systems containing only rings and unbranched chains the variation is more rapid than for the linear case. 

The Eqs. (B.15) and (B.19) are the basic relationships for the calculation of static conformational properties. Having solved the various sums occuring in these relationships, the correct and apparent molecular weight is obtained by setting q = 0, and the mean-square radii of gyration are the corresponding first coefficients in the series expansion of MwPz(q) and MwPapp (q), respectively... 

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. 

Let us compare the kinetics of the selective-solvent-induced collapse of protein-like copolymers with the collapse of random and random-block copolymers [18]. Several kinetic criteria were examined using Langevin molecular dynamics simulations. There are some general results, which seem to be independent of the nature of interactions or the kinetic criteria monitored during the collapse. Here, we restrict our analysis to the evolution of the characteristic ratio f = (Rgp/Rg ) that combines the partial mean-square radii of gyration calculated separately for hydrophobic and hydrophilic beads, k2n and Rg . This ratio takes into account both the properties of compactness and solubility for a heteropolymer globule [70] (compactness is directly related to the mean size of the hydrophobic core, whereas solubility should be dependent on the size of the hydrophilic shell). 

In Fig. 42, we show the ratio R2P/R2 U (i< 2n and R2V are the partial mean-square radii of gyration calculated separately for hydrophobic and hydrophilic beads) as a function of the interaction parameter /. We see that this ratio is an increasing function of /. Qualitatively the same picture is observed for isolated chains. Such behavior is due to the fact that, as the attraction... 

Fig. 3. Mean square radii of diamond lattice chains Monte Carlo calculations of Wall and Erpenbeck (201), plotted according to Eq. (30) (open circles) and Eq. (31) (filled circles)...

In the flow mode, the instruments may be used to collect, simultaneously, the dynamic and classical light scattering data from which the molar mass and root mean square radii are calculated from each slice (See section on fractionation). 

Fig. 15. Change of mean-squared radii of gyration of attractive block copolymers (Nbiock= 12) with different interaction during the phase separation [68]...

Table 2.3 Mean-square radii of gyration of ideal polymers with N Kuhn monomers of length b linear chain, ring,/-arm star with each arm containing A //Kuhn monomers, and -H-polymer with ail linear sections containing W/S Kuhn monomers ...

Table 2.4 Square radii of gyration of rigid objects uniform thin disc of radius R, uniform sphere of radius R, thin rod of length L, and uniform right cylinder of radius R and length L...

Table I. Unperturbed Root-mean-square Radii of Gyration (nm) for IWo Disordered Proteins...

To obtain averaged root-mean-square radii of gyration, partition coefficients (K ) were transformed point by point to values as appropriate integrals were summed. Integrations were by a trapezoidal algorithm. Transformations were obtained from the following calibration curves ... 

In order to interpret observations concerning gyration swelling in terms of (z), we must transform the square radii of gyration / G, into square end-to-end distances R1. We shall perform this change, using a perturbation expansion, and we shall discuss the result using the method of thermic sequences. 

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