Douglas-Freed scheme

After a complex euialysis Douglas and Freed showed that the solutions to eq 3.26 and 3.27 correct in the Douglas-Freed scheme are given by... 

Substitution of eq 3.29 tind 3.30 into eq 3.21 yields the final result correct in the Douglas-Freed scheme. It reads... 

By taking into account that eq 3.29 is correct in the Douglas-Freed scheme and that u is 0(e) we may rewrite eq 3.27 as... 

As explained in Section 3.8, eq 3.47 is inconsistent with the Douglas-Freed scheme. 

Thus, in the Douglas-Freed scheme, it follows that if eq 3.49 holds, the end distance expansion factor for 3-dimensional chains in poor solvents obeys the two-parameter theory. This consequence may be utilized to determine the function /r(w). With eq 3.36 substituted for this function, the right-hand side of eq 3.50 is expanded in powers of z and the result is compared with the familiar z expansion for in the two-parameter theory, i.e., or = l4-(4/3)z —2.075z - -... (see Section 1.3 of Chapter 2). Then we find... 

These are exact in the Douglas-Freed scheme under the condition specified by eq 3.44. 

This formula indicates that for 3-dimensional chains becomes a universal function of as in the two-parameter theory, in the Douglas-Freed scheme and under the condition (27rL/A) > 1. It predicts 0.269 for at the selfavoiding limit ( = oo). This limiting value is in accordance with 0.268 by Witten and Schafer [12] and 0.269 by des Cloizeaux [13] from different RG calculations. Though slightly laiger than the experimental estimates 0.22 — 0.25, it represents a remarkable success of the RG theory. 

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