Systematic development of coarse-grained polymer models Patrick ...

3.8. Limiting Behavior 35
α 1/2 δˆz
1.00
0.10
0.01
0.1 1.0
ˆf
10.0
Fig. 3.8: Calculation of the longitudinal root-mean-squared fluctuations at different
levels of coarse-graining. The Marko and Siggia potential was used with
λ =1. The curves correspond to ν = 400 ( dashed), ν =20( dotted), and
ν =10( dash-dot). The solid line corresponds to the high-force asymptotic
behavior, 1/(2 ˆ f 3/4 ).
average of the radial coordinate of a single spring, it can be shown that
∂
lim
ˆf→0 ∂ ˆ f 〈ˆztot〉m
= ν
1
0
3
dˆr ˆr4
−ν exp λ
Ueff(ˆr)
−ν
λ . (3.23)
Ueff(ˆr)
1
0 dˆr ˆr2 exp
This expression was used previously in Section 3.6 to calculate the “best-fit” λ at zero force as seen
in Figure 3.5, and it will be used extensively to understand rheological properties in Chapter 5.
We also note here that Ladoux and Doyle [58] derived an expression similar to equation (3.19)
based on scaling arguments and a single spring. Based on the scaling argument, they developed a
model which compared favorably to experimental data and lends support to the results presented
here.
3.8 Limiting Behavior
We have seen thus far that the F-E behavior of bead-spring chains can be written analytically
as integral formulae for arbitrary spring force law. This has allowed for the determination of
the important dimensionless groups that determine the behavior, as well as provide for rapid and
accurate calculation through numerical integration. However, another important advantage to
having analytical formulae is that expansions can be performed. Those expansions can be used
to illustrate limiting and universal behavior as well as obtain approximate algebraic formulae that