320 The Monte Carlo method
represent segments of the polymer; the segments in turn represent groups of atoms.
These atomic groups have a strong short-range interaction, as they are chemically
bonded. Remote segments influence each other through a Van der Waals attraction,
and they repel when they overlap. However, the solvent effect described above
represents another type of repulsion which forces the beads to remain at a minimum
distance of at least a few solvent layers. All these characteristics can be summarised
in the following polymer model:
- The polymer consists ofNbeads, which are represented as point particles.
Neighbouring beads have a fixed mutual distance. This is the only interaction
between them.
- Remote beads feel a repulsion at short distances and a Van der Waals attraction
at longer distances. Their interaction can be modelled by a Lennard–Jones
shape (see Eq. (7.33)).
Note that if we switch off the Lennard–Jones interaction, the polymer represents a
random walk: this is called theideal chaincase. If, on the other hand, the beads repel
each other, we are dealing with a self-avoiding walk (SAW) as the chain cannot cross
itself. Often, polymers are studied on a lattice, which is an even further restriction
of the model, but asymptotic (large length scale) behaviour is not sensitive to this.
Forareviewoflatticealgorithms,seeRef. [40]. Here we shall discuss algorithms
for simulating the off-lattice model. Most of the methods used in this field have a
very similar counterpart in the lattice case.
Note that the behaviour of the polymer is independent of temperatures for high
temperatures. This is because the Lennard–Jones interaction is then dominated by
the repulsive term, which is always noticeable, even when the temperature is very
high. The quantities of interest are the end-to-end distance, which is the distance
from the first to the last bead, and the radius of gyration (see Problem 1.2). We shall
restrict ourselves here to studying the end-to-end distance as a function ofNfor
polymers in two dimensions.
Our aim is to generate (an ensemble of) polymer configurations which are dis-
tributed according to the canonical distribution at a given temperature. Then we can
calculate physical properties for this temperature as an average over the ensemble.
Now let us consider possible steps in a standard Metropolis algorithm. We cannot
move a single bead, as we should keep the distance to its neighbours constant. An
obvious alternative is to choose a bead at random and change the angle between
this bead and its two neighbours by some amount. If the polymer is ‘curled up’,
which turns out to be not unlikely in typical simulations, there are very few such
moves that will be accepted. Algorithms based on this type of moves are called
‘pivot algorithms’: the selected bead acts as a pivot for changing bending or (in
three dimensions) dihedral angles.
