Helical Wormlike Chains in Polymer Solutions by Hiromi Yamakawa & Takenao Yoshizaki

Author:Hiromi Yamakawa & Takenao Yoshizaki
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

j

a j

b j

c j

d j

0

1.9379

68.381

⋯

-0.197997609403

1

-17.412

63.638

8.7456

-0.059664102410

2

26.565

30.812

-0.42137

-0.03384594250

3

-46.347

-47.432

3.7180

0.06596504601

4

55.487

2.6680

-4.0179

-0.0154304201201

5

-37.040

⋯

2.5937

⋯

6

10.218

⋯

⋯

⋯

Next we consider the range (2). The most probable configuration for a given Wr ranging from 0 to 1 was determined by Le Bret [4]; it changes from the circle with Wr = 0 to the 8-shaped configuration with Wr = 1. Figure 7.9 shows as an example the most probable configuration with Wr = 0. 37 and Wr(z) = 1 (determined following Le Bret). In this section the most probable configuration with 0 ≪ Wr ≤ 1 is also referred to as the 8-shaped configuration, for convenience. Further, the contour points A1 and A2 corresponding to the crossing in the xy plane are called the “nodes” of the 8-shaped configuration.

Before proceeding to make further developments, we must make two remarks. The first concerns a kind of asymmetry of the shape of the 8-shaped configuration. When the most probable configuration is an 8-shaped configuration, it is necessary to constrain h in addition to u 0 and R M ( = 0) in contrast to the case of the circle. The imposition of the constraint on h is equivalent to specifying the segment number , or the contour distance , of one of the nodes. Thus the integration over in Eq. (7.87) may be carried out first over with h fixed, and then over h, where in the latter integration we may change variables from h to (). The second remark concerns the value of Wr(z). Consider the most probable configuration for Wr = Wr ∗ (0 ≪ Wr ∗ ≤ 1) and Wr(z) = 1, and allow the fluctuations around it under the restriction that the first term on the right-hand side of Eq. (7.86) is equal to Wr ∗− 1. If we consider formally the mathematical fluctuations in , the chain, which is then phantom, is allowed to cross itself in the course of the deformation from the most probable configuration. The configurations that result may then be classified into two types: one with Wr(z) = 1 and the other with (with the configurations with | Wr(z) | ≥ 2 being ignored). From Eq. (7.86), we have Wr = Wr ∗ and for the configurations with Wr(z) = 1 and , respectively. In order to evaluate P(Wr = Wr ∗; L), we must therefore inhibit the fluctuations leading to . We note that even the small fluctuations may actually lead to the latter case for the phantom chain with , and that the inhibited configurations make contribution to (see below). This requirement may be taken into account, although only approximately, by imposing the constraint that only the fluctuations that satisfy e 12 ∗⋅ R 12 > 0 are allowed, where R 12 is the vector distance between the contour points (corresponding to A1) and (corresponding to A2), and e 12 ∗ is the unit vector in the direction of R 12 = R 12 ∗ in the most probable configuration. This constraint can be imposed on the configuration integral by the use of a Fourier representation of a unit step function.

Thus, considering the above two remarks, we may evaluate P(Wr; L) from Eq. (7.87) by introducing in the integrand a factor Δ,

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.