Coil limit

In the rod limit, Le = L, and de = d. On the other hand, in the coil limit, Le - V6de, if we assume the chain to be in the unperturbed state. It is to be noted that the axial ratio of the fuzzy cylinder is greater than unity even in the coil limit. At intermediate N, L and the axial ratio Le/de may be calculated, respectively, from the Kratky-Porod equation [102,103] for the (unperturbed)... 

It should be pointed out that in the Zimm, or impermeable coil, limit the specific value of R ff is uninfluential since the friction coefficient C( ) = C/v( ) is independent of R ff, see Eqs. (3.1.7 ) and (3.1.9), v( ) > 1. This is not true in the partial-draining case, in which v(q) is of order unity although larger than the Rouse free-draining limit v(q) = 1, and its full expression must be considered. 

The inhomogeneity of the excitation RF fields from transmit-receive surface coils can be used to advantage for selecting a sensitive volume slab based on approximate distance from the coil. Limited spatial selectivity can be achieved by adjusting the flip angle to best select the depth of interest. 

Values of (5 )/M for polymers not as stiff as PHIC and DNA level off to a coil limit at relatively low M. Hence, scattering measurements have to be extended to low M in order to determine the parameters q and Ml for... 

The best value of d consistent with the [7/] data was found to be 1.6 ( 0.2) nm. The solid line A in Figure 5-10, calculated with q = A2 nm, Ml = 715 nm, and c/ — 1.6 nm, is seen to fit the data points veiy accurately. Its close fit even for Mw above 3x10 appears to indicate that [77] undeigoes less excluded-volume effect than does (5 ) (compare with Figure 5-3). However, this result is an accident due to the defect of the Yamakawa-Fujii-Yoshizaki theory which overestimates [77] near the coil limit. 

Equation 5.13 for provides a bridge between the rod and the coil limits of the polymer. In the rod limit... 

Comparing the coil limit to that for the random coil by Equation 5.9 provides the connection between the statistical segment length and the persistence length... 

The coil limit of this expression is frequently useful... 

A useful, although nonrigorous, approach to assessing the possibility of branching, cross-linking, or collapse of a polymer is to use the experimental value of<5 >, which will inseparably contain any excluded volume effects, and equate this in the coil limit to an expression involving the apparent persistence length, L, which itself embodies both the intrinsic and excluded volume effects in<5">[12, 13] ... 

As may be seen in Figure 8.7, approaches the rodlike limit of unity for L /a<< 1 and the coil limit 2d 3L for LJd >> 1, with transition developing for LJd l. A numerically similar result is obtained for the model with free rotation about N bonds with valance angles n - a ). As seen in Figure 8.7, the results for the model, with a=cos(ct ), are essentially equivalent to the results for the wormlike chain model [88]. 

Thus, the KP chain can take on a variety of conformations, intermediate between straight rods and Gaussian coils, depending on the magnitude of AL. We note that A (equal to the ratio R /L in the coil limit) is often called the Kuhn segment length. 

Figm 15 Reduced third virial coefficients for PS in benzene at 25 °C (unfilled circles) and PIB in cyclohexane at 25 °C (filled circles), compared with theoretical cunres. Solid line, for 26=0.25 and 2 =0.025 dashed line, dotted-dashed line, coil limit of P2 ( a. two-parameter theory). 

The intrinsic viscosity (at zero y) of a cylindrical KP chain was formulated by Yamakawa and Fujii ° on the basis of the Oseen-Burgers method with the preaveraged Oseen hydro-dynamic interaction (HI) tensor. The integral equation derived has the same form as Kirkwood and Riseman s with the mean reciprocal distance (between a point on the cylinder surface and a contour point) as the kernel. Its solution gives in the coil limit... 

The viscosity factor of 2.86 x 10 mol" is 3-18% larger than experimental values for typical flexible polymers " in theta solvents and about 12% larger than the value obtained by Zimm" from Monte Carlo calculations for Gaussian chains without the preaveraging approximation of the HI tensor. Hence, the viscosity theories ofYamakawa and co-workers ° ° ° based on the cylinder and touched-bead models involve appreciable errors near the coil limit and data analysis based on them may lead to tmderestimation of either 2" or ljM.1 or both for flexible chains. 

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