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Keywords
- Anisotropic surface energy; high order partial differential equations; pseudospectral methods; surface diffusion (1)
- Gravity and surface tension driven liquid flows (1)
- Landau-Levich drag-out problem (1)
- Lubrication theory (1)
- capillary meniscus (1)
- coating flows (1)
- convective Cahn-Hilliard (1)
- dynamical systems (1)
- exponential asymptotics (1)
- matching (1)
- quantum dots (1)
- rarefaction fans (1)
- undercompressive waves (1)
Project
- C10 (5)
Application Area
- C (5)
Dynamics of a surface-tension-gradient-driven liquid film rising from a reservoir onto a substrate
(2006)
On a tilted heated substrate, surface tension gradients can draw liquid up
out of a reservoir. The resulting film thickness profile is controlled by the tilt
of the substrate, the imposed temperature gradient, and the thickness of a
postulated thin precursor layer. The evolution of this film in time is studied
using a lubrication model. A number of distinct behaviours are possible as
the substrate tilt angle and other parameters are varied. Recent results for
the multiple stationary profiles possible near the meniscus are used, and the
interaction of these profiles with the advancing front is examined. It is shown
how to systematically determine the evolution of the entire film profile from
the meniscus to the apparent contact line. This allows a categorisation of the
range of behaviours for a transversely uniform profile, in a twodimensional
parameter space. In addition to capillary fronts, and double shock structures,
a new combination of a Type I meniscus with a rarefaction fan, and either an
undercompressive or a classical wave at the advancing front, that arises for
certain ranges of large substrate tilt and precursor thickness is described.
We revisit here the situation of a thin liquid film driven up an
inclined substrate by a thermally induced Marangoni shear stress against
the counter-acting parallel component of gravity. In contrast to previous
studies, we focus here on the meniscus region, in the case where the substrate
is nearly horizontal, so there is a significant contribution from the
normal component of gravity. Our numerical simulations show that the
time-dependent lubrication model for the film profile can reach a steady
state in the meniscus region that is unlike the monotonic solutions found
in [MÄunch, SIAM J. Appl. Math., 62(6):2045-2063, 2002]. A systematic
investigation of the steady states of the lubrication model is carried out by
studying the phase space of the corresponding third order ODE system. We
find a rich structure of the phase space including multiple non-monotonic
solutions with the same far-field film thickness.
We investigate the dewetting rates of thin liquid films using a lubrication
model that describes the dewetting process of polymer melts on hydrophobized
substrates. We study the effect of different boundary conditions at the liquid/solid
interface, in particular, of the no-slip and the Navier slip boundary condition, and
compare our numerical solutions for the no-slip and the slip dominated cases to
available results that originate from scaling arguments, simplified
ow assumptions
and energy balances. We furthermore consider these issues for an extended lubrication
model that includes nonlinear curvature.
We study the instability arising at the moving ridge of holes appearing
upon dewetting of thin polymer films on hydrophobized substrates, giving
special attention to the role of slippage at the liquid/solid for the appear-
ance of the instability. We compare here numerical results for a lubrication
model of the dewetting film assuming either a no-slip or a free slip condition
at the liquid/solid interface. Linear stability analysis reveals that in both
cases, perturbations of the ridge are amplied, but by orders of magnitude
more in the free slip case. Furthermore, the perturbations become much
more asymmetrical in the free slip case, while they develop symmetrical
patterns without slip. Additional computations that solve the lubrication
model for the full three-dimensional
ow confirm that these findings carry
over into the nonlinear regime.
New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the
monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function.
Using phase space analysis for the corresponding fifth-order dynamical
system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input.
Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.
In this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies.
We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature.
Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov
growth that include surface energies also with large anisotropy as well as the effect of wetting energy,
elastic energy and a randomly perturbed atomic deposition flux.
A linear stability analysis for the resulting sixth-order semilinear evolution equation for the thin film surface shows that that the new model allows
for large anisotropy and gives rise to the formation of anisotropic quantum dots. The nonlinear three-dimensional evolution is investigated via numerical solutions.
These suggest that increasing anisotropy stabilizes the faceted surfaces and may lead to a dramatic slow-down of the coarsening of the dots.