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

3.2. Decoupled Springs 23
fixed
solvent(temperature bath)
r
fixed
Fig. 3.2: Illustration of a polymer and bead-spring model in the constant extension
ensemble. Both ends are held fixed at a distance r apart. The external
force required to hold one of the ends fixed is averaged.
depend on the magnitude of extension. It should then be clear that the effective Hamiltonian can
be separated into a sum over each spring
Heff =
Us(rj) − fzj , (3.2)
j
where j denotes each spring, Us(rj) is the potential energy of each spring as a function of the radial
extension of the spring, and zj is the z displacement of spring j. Because the effective Hamiltonian
can be decomposed into a sum over each spring, the partition function for the whole chain, Zw,
splits into a product of the partition functions for single springs, Zs,
where Ns is the number of springs in the chain, and Zs is given by
Zs =
〈f〉
Zw =(Zs) Ns , (3.3)
−Us(r)+fz
exp
d
kBT
3 r. (3.4)
This separation of the partition function has two important consequences. First, the computational
effort needed to calculate the F-E behavior is greatly reduced because the properties of
any size chain can be determined by knowing the properties of a single spring (a single integral).
Second, it illustrates that for this set of conditions the springs are decoupled. In particular, it will
be shown later that the F-E behavior of these bead-spring chain models does not depend explicitly
on the number of beads, which act as free hinges, but only depends on the level of coarse-graining
for each spring. This is counter to other investigators who have argued the importance of the
number of springs in the bead-spring chain model [33].