Binary mixture Phase separation Thin liquid films Free surface Model-H Linear stability analysis Bifurcations of steady state Critical and off-critical mixture Marangoni forces at free surface Finite element method
Model-H is used to describe structures found in the phase separation in films of binary liquid mixture that have a
surface that is free to deform and also may energetically prefer one
of the components. The film rests on a solid smooth substrate that has no preference for any component. On the one hand the study
focuses on static aspects by investigating steady states that are
characterised by their concentration and film height profiles. A large variety of such states are systematically analysed by numerically constructing bifurcation diagrams in dependence of a number of control parameters. The numerical method used is based on minimising the free energy functional at given constraints within a finite element method for a variable domain shape. The structure of the bifurcation diagrams is related to the symmetry properties of the individual solutions on the various branches. On the other hand the full time dependent model-H is linearised about selected steady states, in particular, the laterally invariant, i.e.\ layered states. The resulting dispersion relations are discussed and related to the corresponding bifurcation points of the steady states. In general, the results do well agree and confirm each other.
The described analysis is performed for a number of important cases
whose comparison allows us to gain an advanced understanding of the
system behaviour: We distinguish the critical and off-critical case
that correspond to zero and non-zero mean concentration,
respectively. In the critical case the investigation focuses on (i)
flat films without surface bias, (ii) flat films with surface bias,
(iii) height-modulated films without surface bias, and (iv)
height-modulated films with surface bias. Each case is analysed for several mean film heights and (if applicable) energetic bias at the free surface using the lateral domain size as main control parameter. Linear stability analyses of layered films and symmetry considerations are used to understand the structures of the
determined bifurcation diagrams. For off-critical mixtures our study is more restricted. There we consider height-modulated films without and with surface bias for several mean film heights and (if applicable) energetic bias employing the mean concentration as main control parameter.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.