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

CHAPTER 3
Force-extension Behavior
As was mentioned in the introduction, one of the most important and widely known properties of
polymers is elasticity, and in particular the presence of an “entropic restoring force.” Furthermore,
with the advent of optical and magnetic tweezer technologies, much more attention is being paid
to the relation between force and extension [55]. In particular, these experiments have been used
to test polymer models which are then used in other contexts. Making quantitative calculations of
the force-extension behavior of bead-spring chains for comparison with the polymers they represent
is the goal of this chapter. This chapter was reproduced in part with permission from Underhill,
P.T. and Doyle, P.S., J. Non-Newtonian Fluid Mech., 122, 3 (2004), copyright 2004 Elsevier B.V.
3.1 System Definition
The typical set-up used to calculate the restoring force using statistical mechanics is shown in
Figure 3.1. One end of the polymer is tethered at the origin, and a constant external force, f, is
applied to the other end of the polymer. The direction of this constant force defines the z-direction
of the coordinate system. The x and y coordinates are therefore in the plane perpendicular to the
applied force. The expectation value of the polymer’s z displacement, 〈z〉, can be calculated as a
function of the applied force. This function, 〈z〉 vs. f, defines the polymer’s force-extension (F-E)
behavior. Note that this is different from the behavior found by performing the analysis shown in
Figure 3.2, in which the ends of the polymer are held fixed at points that are z (or equivalently r)