3. Polymers In Bulk Solution

The bulk solution menu has three options which differ in the amount and format of
information displayed.  Briefly, the first option is useful when one is interested in
comparing the theoretical probability distribution to the distribution for some number of
configurations, large or small, in order to determine an appropriate sample size for a later
simulation.  Options 2 and 3 calculate and display several measures of polymer size and
their averages.  These two options differ mainly in the format in which this information is
displayed: option 2 gives numerical information while option 3 displays the information
graphically.  They also differ in execution time; option 1 is the fastest and option 3 is the
slowest.

In bulk solution an N-step random walk polymer can take on any one of
configurations since there are 6^N possible directions that a step can take on a simple cubic
lattice.  Since even a fairly small number of steps leads to a huge number of
configurations, (N = 10 leads to over sixty million configurations) it is generally
impossible to calculate the true average by generating all of the possible configurations.
Instead, a set of configurations is generated and averages are calculated for the sample.
As a guide to the quality of a particular sample size, the first option allows one to
compare the theoretical gaussian probability distribution to the sample's distribution.  For
example, suppose that one is interested in polymers containing N = 200 segments.  If only
ten random walks are generated, the probability distribution might look something like
the first plot in figure 3-1.  Increasing the number of polymers in the sample improves the
agreement between the actual and theoretical distributions as shown by the other two
plots in figure 3-1.  Figure 3-2 shows that very good agreement can be obtained by taking
a large number of configurations of a large polymer.

The size of a random walk polymer molecule can be defined in several ways.  The
measures of polymer size that are calculated, averaged, and compared to theoretical
predictions are listed below in table 3-1.  The end-to-end distance is the magnitude of the
vector that connects the two ends of the polymer and the components of the end-to-end
distance are the magnitudes of the x, y and z components.  The x, y and z spans can be
thought of as the dimensions of the smallest box that is able to contain the polymer.  Details
of these calculations can be found in the theory section of this document.
 

Table 3-1:  Average Sizes Of An N-Step Random Walk Polymer In Bulk Solution
 

It should be noted at this point that the method of generating random walks does not
check to see if a particular walk is identical to another walk in the sample.  Each step of a
walk is determined by a random number generator that selects an integer from one to six
corresponding to the six possible directions that a step may take.  Thus, if one generates
36 ( = 6² ) configurations of a two-segment polymer there is no guarantee that the
configurations are all different.  In fact, it is very likely that at least two of the
configurations are the same.  As one looks at large polymers, any sample size that can be
run in a reasonable amount of time should have very few, if any, duplicate configurations.
Since this program is a teaching aid rather than a research tool, the rare occurances of
repeated configurations should not detract from its usefulness.  In fact, by observing small
polymers (oligomers) the user can appreciate how the occurance of repeated
configurations affects a sample's averages.
 

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