Radius expansion factor

Recalling that i/ = (R )/N ) is the expansion factor for the mean-square end-to-end distance and the radius of gyration Rg is - /(R2)76 within the uniform expansion approximation, we have... 

Ratio of a dimensional characteristic of a macromolecule in a given solvent at a given temperature to the same dimensional characteristic in the theta state at the same temperature. The most frequently used expansion factors are expansion factor of the mean-square end-to-end distance, Ur = (/o) expansion factor of the radius oj gyration, as = (/0) relative viscosity, = ([ /]/[ /]o), where [ ] and [ /]o are the intrinsic viscosity in a given solvent and in the theta state at the same temperature, respectively. 

The radius of gyration is expected to be different under theta and nontheta conditions since the extent of coil swelling due to imbibed solvent changes with solvent goodness. We define a coil expansion factor a as follows ... 

Empirically, the expansion factor a, for the radius of gyration of a polyion, due to electrostatic volume exclusion, is given by120-123)... 

Yamakawa-Stockmayer-Shimada (YSS) theory [75-77] predicts that the radius expansion factor as (=(/0)1/2) is a universal function of the scaled excluded-volume parameter z defined by... 

Utilizing the intermolecular interaction potential described by Eq. (2) and assuming a uniform expansion for the chain the expansion factor a for the radius of gyration... 

Figure 5. (a) Expansion factor of mean-square end-to-end distance aj and fbj radius of gyration as function of z = xBJN for open (Op.) and periodic (Per.) chain. Other results shown here are Domb and Barrett (DB) [67], Douglas and Freed (DF) [66], Muthukumar and Nickel (MN) [64], and Flory modified (Fm) [17,69]. (From ref. 68, by permission of the publishers, Butterworth Co. (Publishers) Ltd. .)... 

O is the so-called Flory s constant, a is the expansion factor of the polymer molecule, which depends from the thermodynamic quality of the solvent (a = 1 in ideal solvent), ( o> is the mean-square radius of gyration, is Avogadro s number, and is the volume of the equivalent hydrodynamic sphere. 

The parameter a is close to but not equal to the linear expansion factor of the excluded volume effect with respect to the end-to-end distance aM or that with respect to the radius of gyration as (46). 

Yamakawa, ff., and Yoshizaki, T., Effects of fluctuating hydrodynamic interaction on the hydrodynamic radius expansion factor of polymer chains, Macromolecides, 28, 3604—3608 (1995). 

If eq 1.18 is used with eq 1.28, it is possible to calculate as as a function of 2 over the entire range of positive 2. We call the combined equation (not written explicitly here) the Domb-Barrett interpolation formula for the radius expansion factor. 

The continuous chain version of eq 1.23 gives for the radius expansion factor... 

Exactly the same analysis as that described above can be made regarding the radius expansion factor as. Corresponding to eq 3.21, we can derive... 

With eq 1.4 the end distance expansion factor and the radius expansion factor as can be calculated analytically (in the continuous chain limit). The results are as follows ... 

The radius expansion factor as(r) for perturbed spring-bead ring chains was calculated to first order in the excluded-volume variable 2 by Casassa [77], who obtained... 

In the early 1960s, Yamakawa [23], Eizner [24], and Grimley [25] showed by perturbation calculations valid to first order in c that the radius expansion factor Os decreases with increasing c. Subsequently, Fixm u tmd Peterson [26] predicted by an approximate theory that the shrinkage of chain dimensions occurs fiist rapidly and then gradually as the solution is concentrated. However,... 

The unperturbed dimensions refers to molecular size exclusive of solvent effects. It arises from intramolecular polar and steric interactions and free rotation. The expansion factor is the result of solvent and polymer molecule interaction. For linear polymers, the square of the radius of gyration is related to the mean-square end-to-end distance by the following relationship ... 

This follows from the expansion factor, a is greater than unity in a good solvent where the actual perturbed dimensions exceed the unperturbed ones. The greater the value of the unperturbed dimensions the better is the solvent. The above relationship is an average derived at experimentally from numerous computations. Because branched chains have multiple ends it is simpler to describe them in terms of the radius of gyration. The volume that a branched polymer molecule occupies in solution is smaller than a linear one, which equals it in molecular weight and in number of segments. 

Both repulsive and attractive forces exist in real random coils. The effect on the radius of gyration of the coil expansion caused by the interaction of these two opposing forces can be formally described by the use of an expansion factor ... 

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