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

1.3. Progression of Coarse-grained Models 9
random walk in Brownian motion, but where the number of steps is like the time of the walk.
The Gaussian chain has the advantage that many calculations can be done analytically. Also the
formulas can be examined in the limit of an infinite number of springs. This limit of Nspr →∞can
be done at constant H, which corresponds to chains of progressively larger LAK, or at constant
LAK, which means H must be be made larger as Nspr increases. The first case represents the
behavior of progressively larger chains while the second case represents a finer discretization of the
same polymer. The fact that these two view points are equivalent is because of the scale invariant
nature of the Gaussian distribution. For this reason, the limit of an infinite number of springs
is typically taken and the methods of renormalization group theory are often applied to this and
related models. We will see later in this thesis similar ideas but with nonlinear bead-spring chains.
It is important to note that this scale invariance is due to the linearity of the springs. Recall that
the freely jointed chain only matched the behavior of the polymer if the polymer contained a large
number of Kuhn lengths and was near equilibrium. Similarly the Gaussian chain only matches the
behavior of the freely jointed chain if the freely jointed chain contains a large number of Kuhn
lengths. This is true for the Gaussian chain no matter how small L/(NsprAK) is because of the
structure of the Gaussian distribution.
Nonlinear bead-spring chains (out of equilibrium)
Recall the discussion of flexible polymers out of equilibrium. Although we found that all flexible
polymers looked the same near equilibrium, equivalent to the freely jointed chain, the details of
the chain mattered out of equilibrium. Similarly, the Gaussian chain can be used to model flexible
polymers near equilibrium, but fails to describe the behavior out of equilibrium. The details of
the micromechanical model will determine the type of bead-spring chain needed to describe the
response out of equilibrium.
This is clearly evident from the fact that the Gaussian chain can be infinitely extended, while
real polymers and micromechanical models such as the freely jointed chain and the worm-like chain
can not be extended past the contour length. To represent this finite extensibility, the spring force
law can be changed from a linear spring to one that becomes infinite at a finite distance. The
diverging force prevents the chain from being extended past that point. Extensions near this finite
extent are far from equilibrium and thus the central limit theorem does not hold; the behavior
depends on the details of the individual steps, even for flexible chains which contain many Kuhn
lengths.
The development of the nonlinear force laws to represent coarse-grained models can proceed in
a similar manner as with the Gaussian chain. The spring force law can be chosen to give the correct
probability distribution of the end-to-end vector or the correct force-extension behavior (how much
the molecule extends under application of an external force). These methods are equivalent and
are typically performed in the limit of an infinite number of Kuhn lengths in the chain. This has
been done for the two common micromechanical models, the freely jointed chain and the worm-like
chain. For the freely jointed chain, the result is that the spring force should be taken to be [23]
Fspr = kBT
L −1 (r/L) , (1.17)
AK