A variational model for diblock copolymer-homopolymer blends
Y. van Gennip
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Polymers are long chainlike molecules common in nature and
technological applications. When two different types of polymers, say
type A and B, are chemically bonded together making an even longer
chain, a diblock copolymer is formed. Often the A and B polymers repel
each other and this repulsion grows stronger when the temperature is
lowered. However, the polymers cannot completely separate because of
the chemical bond between them. This interplay between two different
forces gives rise to interesting phenomena. While repulsion between A
and B polymers favours separation on a macroscale, the bond between A
and B wants to keep both polymers together on the microscale. As a
result regions with mainly A-polymer and regions with mainly B-polymer
will form patterns on an intermediate scale.
Because of their pattern forming behaviour models for diblock
copolymers are often studied by mathematicians. Also more complex
systems have attracted attention. One type of such systems is a blend
of a diblock copolymer and a homopolymer, i.e. a blend of A-B
diblock-copolymers and a third polymer, C, which is not bonded to the
other two. We study a variational model proposed for such blends, given
by the energy
and are non-negative constants. and represent the volume fractions of the A- and B-polymers. The remaining space is filled by C-polymers, so that is their volume fraction.
Inspired by work on a related model for lipid bilayers, the
membranes found in living cells, we hope to prove some interesting
properties of the patterns that minimise the energy .
In this related model it is found that the resulting patterns of the
model have solid-like properties, like bending stiffness and resistance
to fracturing and stretching, just like the membranes in our cells.
Moreover, the mass of these patterns congregates along lower
dimensional structures, something which corresponds well to the
experimentally observed fact that lipid bilayers form membranes,
two-dimensional objects in three-dimensional space. This is called
partial localisation. We hope to uncover similar properties for
minimising structures in our model.