David (Qiang) Wang,  Colorado State University

Some Mathematical/Numerical Problems in the Study of Inhomogeneous Polymers

Polymers are large molecules made up of many small chemical units (monomers)
joined together by chemical bonds. Properties of polymers depend not only on
the properties of monomers, but also on how monomers are connected (chain
architecture) and how many monomers are connected (chain length). The unique
properties of polymers make them indispensable in our everyday life.
Predicting their structure-property relations is therefore of paramount
importance, particularly for inhomogeneous systems such as polymers at
surfaces and interfaces.

The self-consistent field (SCF) theory has been widely applied to various
inhomogeneous polymeric systems with great success. In this talk, I will
first introduce the model of Gaussian chain in an external field, which is
the basis of most polymer field theories. I will then discuss some
mathematical/numerical problems encountered in the SCF calculations of two
inhomogeneous systems: block copolymers and charged polymers. In particular,
we are interested in advanced numerical methods for solving the modified
diffusion equation and Poisson equation, and optimization methods for
searching for saddle points. The latter is a generic feature of field
theories where the effective Hamiltonian is complex. I will also give some
examples on the application of SCF calculations in these two systems.