Modelling the time evolution of the structure factor in polymer blends

Polymer blends can exhibit a range of phase behaviour. During phase transitions,
the evolution of the microstructure can be monitored using small-angle scattering.
Information about the microstructure can be deduced from measurements of the
structure factor – a quantity directly proportional to the scattered intensity. While
the time evolution of the structure factor can be measured relatively easily, modelling
it has proved to be much more difficult. We believe the latter could be impeding
our ability to control the underlying phase transitions.

In this thesis, we are primarily concerned with thermally-induced polymeric spinodal decomposition and dissolution. The equation of motion for the structure factor
during these phase transitions is known to be unclosed, i.e. an infinite hierarchy
of coupled differential equations. Existing attempts to model the time evolution of
the structure factor during spinodal decomposition and dissolution have focussed on
deriving approximate equations of motion based on truncation schemes.

Arguably, the most advanced approximate equation of motion was derived by
Akcasu et al. We refer to this as the Akcasu equation. There is very little literature
aimed at testing the Akcasu equation. To rectify this, we tested the Akcasu equation
using synthetic structure factor snapshots derived from simulations. In the case of
dissolution, the Akcasu equation performed well at describing the time evolution
of the synthetic structure factor snapshots. In the case of spinodal decomposition,
we determined that improvements are required. We hope these respective findings
motivate further experimental testing and modelling work.

Embracing the duality between the fact the structure factor is hard to model
but relatively easy to measure, we investigated the application of system identification techniques to the problem of modelling the time evolution of the structure
factor during spinodal decomposition. One technique we considered is dynamic
mode decomposition. We demonstrated the ability of dynamic mode decomposition
to make accurate future predictions of synthetic structure factor snapshots based on
the knowledge of previous ones. While further research is required, we believe our
findings could be promising for developing a system to control spinodal decomposition.

Dynamic mode decomposition is a linear and equation-free system identification
technique. This prompted us to investigate system identification techniques that
output parsimonious non-linear governing equations. We had mixed success in this
direction, demonstrating that one such technique could not be applied to the problem
while another showed promising signs. We provide a comprehensive outline of how
one could build on these findings.