Linear polymer molecules in dilute solution are highly flexible
and can be self-entangled. In more concentrated solutions, or
in the melt, there can be important entanglement effects both
within and between polymers, and these entanglements can influence
the rheological properties of the system as well as the crystallization
properties, and hence the properties of the polymeric system
in more ordered states. Although scientists have been aware
of these problems for thirty years or so, it is only recently
that the powerful methods of algebraic topology have been used
systematically to characterise and describe these entanglements.
Starting from the simplest possible system (a ring polymer in
dilute solution) one needs to ask how badly knotted the polymer
will be, as a function of the degree of polymerization, the
stiffness, the solvent quality, etc. To some extent these questions
have been answered by a combination of rigorous mathematical
arguments (combining ideas from combinatorics and from algebraic
topology) and numerical methods (primarily Monte Carlo techniques).
However, as the concentration increases, linking between rings
becomes possible and these links (or catenanes) will influence
the static and dynamic properties of the solution. If the polymer
is linear then, from a strictly topological point of view, it
is unknotted. However it is clear that entanglements occur and
their characterization is an important problem. As we pass from
dilute solutions to melts the characterization of the entanglements
becomes more difficult though some progress could be made using
Monte Carlo methods.
Having characterized the entanglement complexity one then needs to know how it will affect rheological properties. For instance, what is the contribution of entanglements to the elastic properties of a rubbery polymer? How do the dynamics of polymers in solution or in the melt depend on entanglement? These problems also suggest questions about the time scales associated with entanglements in linear chains.
Topological problems also occur in the modeling of polymeric membranes. These are closely related to self-avoiding random surfaces, an area in which rapid progress has recently been made, although many important questions still remain open. Closely related are the properties of vesicles and the response of vesicles in flow fields. In this case the topology of the surface can have an important influence on the behavior of the vesicle.
The conformations of polymers are strongly influenced by any applied geometrical constraints. Polymers behave quite differently in pores or when confined in a slab geometry and their properties in these environments influence their behavior as, for instance, stabilizers of colloidal dispersions. Approximate theories of colloidal stability have been available for many years, but it is only recently that simple models of polymers in confined geometries have been analyzed rigorously. In this area there is considerable room for collaboration and cross-fertilization between combinatorialists and people working in statistical mechanics. There can be interesting interactions between topological properties and these geometrical constraints. E.g., how does the knot probability in a ring polymer change when the polymer is confined to lie in a pore or slab?
The workshop will bring together topologists, combinatorialists and members of the theoretical polymer physics and chemistry communities.
The final two days of the workshop will be hosted by The Geometry Center. In this aspect of the workshop participants will be able to demonstrate the present state of software development and its use in the study of the problems related to the workshop. Discussions of the advancements needed in software development will help give direction to the staff at The Geometry Center in extending its expertise to the wider mathematical and scientific community.
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