1.    Introduction and Brief Overview
 

The Random Walk Polymer has three main goals.  First, by watching random walk
polymers being generated one gains an appreciation for the differences that exist between
the sizes and shapes of individual molecules and the average size calculated from a large
collection or ensemble of molecules.  Second, the program illustrates the effects that a
variety of physical obstacles have on the average size and shape of random walk
polymers.  Third, some of the systems appearing in The Random Walk Polymer have been
studied theoretically.  In particular, three of the systems have been studied using the so-
called diffusion equation method.  Some of the theoretically predicted quantities will be
compared to analogous quantities calculated from the configurations generated by the
program.  Hopefully the few systems appearing in The Random Walk Polymer will
indicate the variety of polymer phenomena that can be studied using this simple model of
a polymer molecule.

A random walk polymer is a very simple model of a polymer.  At its most basic level,
this model omits nearly all chemical details (identities of the atoms, bond angles, side
groups, intermolecular interactions)  retaining the following characteristics of
macromolecules: they are made up of a large number of small units, the units are
connected end-to-end, and they can exist in a large number of configurations.  The units
are analogous to monomers and their end-to-end connections give a linear structure.

The configurational aspect of these polymers can be appreciated by looking at the way
The Random Walk Polymer generates "molecules".  One end of the polymer is placed at
the origin of a simple cubic lattice.  The program then generates a random integer from 1
to 6 corresponding to the six possible positions adjacent to the origin (left, right, above,
below, front, back) and draws a line segment from the origin to this randomly selected
site.  This site becomes the new origin, another random number is generated, and the
second segment or unit is drawn.  This process continues until a predetermined number of
segments have been drawn.  Each segment can be thought of as one step and since the
direction of each step is determined at random the entire polymer is the result of a random
walk.  But this particular random walk is only one of many possibilities.  If one went back
to the starting point of the first polymer and took another random walk, its path would
most likely be very different from the first.  Each random walk is one configuration of a
polymer.

The random walk model completely ignores identifying specific atoms.  The model,
as it is implemented in The Random Walk Polymer has unrealistic bond angles -- 180⁰,
90⁰, and 0⁰ -- where this last bond angle occurs whenever a step backtracks on to the
immediately preceding step.  The model also allows two or more segments to occupy the
same site, even when they are far away from each other along the contour of the polymer.
These features of the model may seem to be weaknesses.  However, ignoring the atomic
identities and allowing these bond angles, which are often thought of as local or short
range details, gives a model that captures some of the global or long range properties of
polymers, properties that all flexible linear polymers are expected to possess.  Indeed,
under appropriate solvent and temperature conditions (theta conditions), the size of a real
flexible polymer, as measured by light scattering experiments, is proportional to the
square root of the molecular weight.  The average size of random walk polymers is also
proportional to the square root of the number of segments (i.e., molecular weight).
Random walk polymers have also been used in theories of gel permeation
chromatography, rubber elasticity, and the stabilization/destabilization of colloids.

The Random Walk Polymer uses this simple model to investigate the configurational
characteristics of isolated polymer molecules in a variety of environments.  The first part
of The Random Walk Polymer generates polymer configurations in bulk solution,
determines the probability distribution of the end-to-end distances, and calculates several
average sizes.  The second part considers a random walk polymer near a nonadsorbing
planar boundary such as an air-solution or solid-solution interface.  The polymer
concentration profile near the boundary and the component of the mean square end-to-end
distance perpendicular to the barrier are displayed.  Part three generates configurations
between two planar boundaries.  The concentration profiles of  the polymers are displayed
and the equilibrium constants for the partitioning of polymers between bulk solution and
the confined region are calculated for each molecular weight species.  A simple model of
size exclusion chromatography completes this part.  In parts four and five the random
walks start at the surface of a solid object, either an infinitely long square fiber or a
rectangular particle, and the resulting polymers are classified as either allowed or
forbidden.  The fraction of polymers in each class and their average sizes are calculated
and displayed.  Since the dimensions of the objects can be varied systematically, one can
look for trends in the fractions of allowed configurations and their average sizes.

Some mention must be made of two length scales used throughout The Random Walk
Polymer.  First, the length of a single step or segment, l, is taken to be unity.  A second
length (2 N / 3)^1/2 = (6 N )^1/2 / 3 = l / l  arises repeatedly in the theoretical treatment discussed
later and is used as a scale factor in many of the figures.  This second length is
proportional to the root mean square end-to-end distance,  N ^1/2 l, of a random walk
polymer comprised of N segments each of length l.

The remainder of this booklet describes the features of  The Random Walk Polymer in
more detail.  The diffusion equation model and all of the theoretical calculations that
appear in the program are discussed in the final section.  Some suggestions for using this
program are also provided.  Finally, some references to books and articles that provide
additional information or more complete discussions are listed in the bibliography.
 
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