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

1.3. Progression of Coarse-grained Models 5
made which bring two terminal hydrogen atoms into very close contact. Steric repulsions prevent
this from occurring. Including this effect also will only change the prefactor to the scaling with
N, as long as N is large enough. These type of “local” interactions are accounted for by using the
rotational isomeric state (RIS) model [14].
To this point in our analysis we have examined the configuration space of the polymer. The
solvent has only acted as a temperature bath. No explicit interactions have been included between
the polymer and solvent. Other than the local (along the contour) steric interactions, we have also
not included interactions between segments of polymer. In a real system, segments of polymer can
not occupy the same position in space. This has been neglected in our previous analysis, a so-called
“phantom chain”. The inclusion of these other effects can cause a change in the exponent of the
scaling of the size with length (or number of steps). Under certain conditions, known as “theta
conditions”, these two additional interactions cancel in the sense that the scaling exponent is the
same as if both effects were neglected. In this special circumstance, both effects can be neglected
while obtaining an accurate prediction. This should be done with caution in nonequilibrium situations
because there may be subtle variations that have not been explored yet in the literature
in nonequilibrium situations [15]. This also neglects physics such as knots which can be present in
theta conditions.
If the polymer has a size which is smaller than the theta size, the solvent is considered poor.
If the size is larger than under theta conditions, it is considered a good solvent. It is important to
note that the solvent quality is also a function of temperature. For a given polymer-solvent pair,
there is only a single temperature which corresponds to theta conditions.
The simplest model to show an example of a good solvent is to consider a self avoiding random
walk instead of the random walk already considered. For a self avoiding walk, the scaling behavior
is approximately N 0.6 [9].
1.3 Progression of Coarse-grained Models
1.3.1 Necessity of coarse-graining
In the previous section we discussed the progression of polymer models, in which each new model
correctly included more and more details about the polymer behavior. The result of this is a good
understanding of the static properties of the polymer. However, we are also interested in dynamic
properties of very long polymers over a wide range of time scales. For these purposes, the models
such as RIS are still too intensive to deal with.
Flexible polymers near equilibrium
Consider a polymer under theta conditions. Suppose some detailed model, such as RIS, can calculate
the average end-to-end distance squared. It can be written as
〈r 2 〉 =(N − 1)a 2 CN , (1.3)
which serves as a definition of CN. If N is large enough, CN becomes close to the infinite value
C∞ [15]. For these large N
〈r 2 〉(N − 1)a 2 C∞ , (1.4)