2,564 research outputs found
Blister patterns and energy minimization in compressed thin films on compliant substrates
This paper is motivated by the complex blister patterns sometimes seen in
thin elastic films on thick, compliant substrates. These patterns are often
induced by an elastic misfit which compresses the film. Blistering permits the
film to expand locally, reducing the elastic energy of the system. It is
natural to ask: what is the minimum elastic energy achievable by blistering on
a fixed area fraction of the substrate? This is a variational problem involving
both the {\it elastic deformation} of the film and substrate and the {\it
geometry} of the blistered region. It involves three small parameters: the {\it
nondimensionalized thickness} of the film, the {\it compliance ratio} of the
film/substrate pair and the {\it mismatch strain}. In formulating the problem,
we use a small-slope (F\"oppl-von K\'arm\'an) approximation for the elastic
energy of the film, and a local approximation for the elastic energy of the
substrate.
For a 1D version of the problem, we obtain "matching" upper and lower bounds
on the minimum energy, in the sense that both bounds have the same scaling
behavior with respect to the small parameters. For a 2D version of the problem,
our results are less complete. Our upper and lower bounds only "match" in their
scaling with respect to the nondimensionalized thickness, not in the dependence
on the compliance ratio and the mismatch strain. The upper bound considers a 2D
lattice of blisters, and uses ideas from the literature on the folding or
"crumpling" of a confined elastic sheet. Our main 2D result is that in a
certain parameter regime, the elastic energy of this lattice is significantly
lower than that of a few large blisters
The coarsening of folds in hanging drapes
We consider the elastic energy of a hanging drape -- a thin elastic sheet,
pulled down by the force of gravity, with fine-scale folding at the top that
achieves approximately uniform confinement. This example of energy-driven
pattern formation in a thin elastic sheet is of particular interest because the
length scale of folding varies with height. We focus on how the minimum elastic
energy depends on the physical parameters. As the sheet thickness vanishes, the
limiting energy is due to the gravitational force and is relatively easy to
understand. Our main accomplishment is to identify the "scaling law" of the
correction due to positive thickness. We do this by (i) proving an upper bound,
by considering the energies of several constructions and taking the best; (ii)
proving an ansatz-free lower bound, which agrees with the upper bound up to a
parameter-independent prefactor. The coarsening of folds in hanging drapes has
also been considered in the recent physics literature, using a self-similar
construction whose basic cell has been called a "wrinklon." Our results
complement and extend that work, by showing that self-similar coarsening
achieves the optimal scaling law in a certain parameter regime, and by showing
that other constructions (involving lateral spreading of the sheet) do better
in other regions of parameter space. Our analysis uses a geometrically linear
F\"{o}ppl-von K\'{a}rm\'{a}n model for the elastic energy, and is restricted to
the case when Poisson's ratio is zero.Comment: 34 page
Failure Processes in Embedded Monolayer Graphene under Axial Compression
Exfoliated monolayer graphene flakes were embedded in a polymer matrix and
loaded under axial compression. By monitoring the shifts of the 2D Raman
phonons of rectangular flakes of various sizes under load, the critical strain
to failure was determined. Prior to loading care was taken for the examined
area of the flake to be free of residual stresses. The critical strain values
for first failure were found to be independent of flake size at a mean value of
-0.60 % corresponding to a yield stress of -6 GPa. By combining Euler mechanics
with a Winkler approach, we show that unlike buckling in air, the presence of
the polymer constraint results in graphene buckling at a fixed value of strain
with an estimated wrinkle wavelength of the order of 1-2 nm. These results were
compared with DFT computations performed on analogue coronene/ PMMA oligomers
and a reasonable agreement was obtained.Comment: 28 pages. Manuscript 20 pages, 8 figures. Supporting information 10
pages, 6 figure
How metal films de-wet substrates - identifying the kinetic pathways and energetic driving forces
We study how single-crystal chromium films of uniform thickness on W(110)
substrates are converted to arrays of three-dimensional (3D) Cr islands during
annealing. We use low-energy electron microscopy (LEEM) to directly observe a
kinetic pathway that produces trenches that expose the wetting layer. Adjacent
film steps move simultaneously uphill and downhill relative to the staircase of
atomic steps on the substrate. This step motion thickens the film regions where
steps advance. Where film steps retract, the film thins, eventually exposing
the stable wetting layer. Since our analysis shows that thick Cr films have a
lattice constant close to bulk Cr, we propose that surface and interface stress
provide a possible driving force for the observed morphological instability.
Atomistic simulations and analytic elastic models show that surface and
interface stress can cause a dependence of film energy on thickness that leads
to an instability to simultaneous thinning and thickening. We observe that
de-wetting is also initiated at bunches of substrate steps in two other
systems, Ag/W(110) and Ag/Ru(0001). We additionally describe how Cr films are
converted into patterns of unidirectional stripes as the trenches that expose
the wetting layer lengthen along the W[001] direction. Finally, we observe how
3D Cr islands form directly during film growth at elevated temperature. The Cr
mesas (wedges) form as Cr film steps advance down the staircase of substrate
steps, another example of the critical role that substrate steps play in 3D
island formation
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