4. Polymers Near A Single Nonadsorbing Planar Interface
A random walk polymer near a nonadsorbing planar interface is the simplest
model
for a polymer solution in contact with either a gas phase or a nonadsorbing
solid. Near
this barrier an N-step random walk polymer no longer has 6N
allowed configurations. In
fact, as the polymer origin approaches the barrier, the number of allowed
configurations
steadily decreases. This portion of The Random Walk Polymer
investigates the
consequences of an impenetrable barrier on the fraction of allowed
configurations that
start at various distances from the barrier, a quantity that is related
to the polymer
concentration or the concentration of the first segment of the
polymer. The component
of the mean square end-to-end distance perpendicular to the barrier
is also calculated and
plotted.
Figure 4-1 shows the output screen after one
configuration of a 100-segment polymer
has been generated. The barrier is oriented perpendicular to
the screen and forms the left
boundary of the configuration grid. Because of this, only a two
dimensional projection of
the configuration is shown. For display purposes the configuration
starts at the center of
the grid and after the last segment is placed the two ends are connected,
in this case, by
the diagonal line. In The Random Walk Polymer the segments
and the end-to-end line are
different colors. The dark bar along the bottom of the grid indicates
the set of grid lines
at which the configuration could have started and never crossed the
barrier. The
numerical value for this minimum distance between the barrier and the
polymer origin is
given in the upper left portion of the output screen.
Figure 4-2 shows the output screen after the
second configuration has been generated.
Only this second configuration can exist near the barrier while far
from the boundary both
configurations are allowed. The graphs display a running summary
of the configurations
being generated allowing one to watch the simulation sample approach
the theoretical
curves.
Figure 4-3 displays the cumulative results
of 200 configurations of a 100-segment
polymer and shows that they agree, at least qualitatively, with the
theoretical predictions.
As expected, the fraction of allowed configurations decreases from
unity far from the
barrier to near zero close to the barrier. In other words, the
concentration of polymer, or
polymer end segments, drops from its bulk value far from the barrier
to near zero near the
barrier. At the same time, the component of the square end-to-end
distance starts at its
bulk value far from the barrier, drops to a minimum at an intermediate
distance, and then
increases as the polymer end approaches the barrier.