Geoscience Reference
In-Depth Information
Bubbles and minimal surfaces
In
plate tectonic animations
one sees
that the plates are constantly changing shape
and size much like a froth on a beer. Foams
are collections of surfaces that minimize area
under volume constraints. Confined bubbles have
12 pentagonal faces and consist of a basic pen-
tagonal dodecahedron with two, three or four
extra hexagonal faces. These structures have only
pentagonal and hexagonal faces, with no adja-
cent hexagons. Clathrates and zeolites also adopt
these structures. Carbon cages involve the basic
pentagonal dodecahedron unit supplemented by
hexagonal faces with an isolated pentagon rule
for the bigger molecules. These structures are
constrained minimal energy surfaces and may
serve as models for the
tessellation of the
Earth
. All vertices of these structures are triple-
junctions, meeting at 120
◦
; all faces are pen-
tagons or hexagons. Sheared bubbles in foams
can adopt complex shapes, such as boomerangs,
but they are still minimal surfaces. In close-
packed arrays of bubbles the dominant hexago-
nal coordination is interrupted by linear defects
characterized by five-coordination. The midpoints
between objects packed on a sphere often define
a pentagonal network.
geometry initially as a tiling, packing or isoperi-
metric exercise. The problems of
tessella-
tions of spheres and global tectonic
patterns
are venerable ones. No current theory
addresses these issues. The planform of a freely
convecting spherical shell may have little to do
with the sizes, shapes and number of plates. The
plates may self-organize and serve as the tem-
plate that organizes mantle flow. This would def-
initely be true if the outer shell contains most
of the buoyancy of the system and most of the
dissipation.
In the Earth sciences, perceived polyhedra
forms on the surface have historically been used
to support global contraction, expansion and
drift theories [e.g.
mantleplumes
]. But, identi-
cal geometric forms can often be generated by
very different forces; it is impossible to deduce
from observed patterns alone which forces are
acting. The corollary is that pattern formation
may be understood, or predicted, at some level
without a complete understanding of the physi-
cal details.
The classic problem of convection goes back
to
experiments by Benard in 1900
. The
hexagonal pattern he observed was attributed to
thermal convection for many decades but is now
known to be due to variable surface tension at
the top of the fluid [
Marangoni convection
].
Plan views of foams also exhibit this pattern. In
all of these cases space must be filled and princi-
ples of economy are at work.
The optimal arrangement of tiles on a
sphere has a correspondence with many dis-
tinct physical problems (
carbon clusters,
clathrates, boron hydrides, quasi-
crystals, distribution of atoms about
a central atom
and bubbles in foam). It is
often found that straight lines (great circles)
and equant cells (squares, pentagons or hexagons
rather than rectangles) of identical size serve
to minimize such quantities as perimeter, sur-
face area, energy and so on. There is an energy
cost for creating boundaries and larger entities
often grow at the expense of smaller ones. In
phenomena controlled by surface tension, sur-
face energy, stress, elasticity and convection one
often finds tripartite boundaries (called triple
junctions or valence-3 vertices) and hexagonal
patterns.
Jamming
Bubble rafts, or 2D foams, are classic minimum
energy systems and show many similarities to
plate tectonics. They are examples of
soft matter
.
They readily deform and
recrystallize
(coarsen).
Foams are equilibrium structures held together
by surface
tension
. A variety of systems, includ-
ing granular media and colloidal suspensions are
fragile
; they exhibit non-equilibrium transitions
from a fluid-like to a solid-like state character-
ized by
jamming of the constituent par-
ticles
. A jammed solid can be refluidized by
heat, vibration, or by an applied stress. Although
plates are often treated as rigid objects, in
the long term they act more like
fragile
or soft matter
, with ephemeral shapes and
boundaries.
Granular material and colloids tend to self-
organize so as to be compatible with the load
on them. They are held together by
compression
.
They are rigid or elastic along compressional
stress chains
but they collapse and reorganize in
