Moving the focal plane approximately 0.2 m away from the mid-plane
of the cell allows us to differentiate between spheres which are
displaced upward and downward from the mid-plane
(Figs. 2(g) and (h)).
The dependence of the spheres' appearance on their vertical
displacement can be calibrated [28,25].
For example, the integrated brightness of a
sphere's image depends monotonically on
its displacement from the focal plane; this is the measure we use.
Not surprisingly, we find that the spheres in the triangular
crystal layer do not lie rigidly in a single plane.
The suspension is too dense to form a monolayer in equilibrium.
Nor, moreover, are the vertical fluctuations of the layer purely
random.
Instead, we observe regions of correlated buckling
whose structure is apparent in Fig. 9.
Larger circles in this figure correspond to spheres which are
closer to the viewing window.
While the vertical displacement information is continuous, we
have selected an arbitrary threshold in computing Fig. 9
to emphasize the buckled structure.
The highlighted region displays a superlattice structure with
ridges of displaced spheres running from the upper right corner
to the lower left.
Superimposed polygons help to demonstrate that the underlying lattice
is still triangular, although distorted.
This qualitative observation is supported by the observed 90 percent
of the total field of view classified as 6-fold coordinated at this
time in Fig. 5.
The same sort of buckling has been observed in equilibrium confined suspensions
at phase boundaries such as that between the single
triangular layer and the two-layer square crystal [17].
Chou and Nelson [19] have analyzed
this system and have identified three
transverse phonon modes lying on the Brillouin zone boundary
which can be responsible for buckling of the confined single layer.
Of those, the one we see corresponds to the
edge phonon.
This also is observed in the equilibrium case [20].
Its appearance in the Chou and Nelson theory is favored when
the sphere-wall interaction is symmetric about the mid-plane
of the system and is much stronger than gravity.
The other
corner phonon
and
edge phonon modes
are asymmetric about the mid-plane and are favored by
symmetry-breaking conditions.
Different regions of the crystal exhibit domains
of differing buckling orientations,
as can be seen in Fig. 9 when comparing the highlighted
region to the region below.
In addition to randomizing defects which suggest small admixtures of
and
phonon modes,
buckling domains of the
edge symmetry meet at four
types of coherent domain walls which are comparable to twin
boundaries.
These appear schematically in Fig. 10.
Each type of domain boundary appears at various times in our system.
Boundaries need not follow crystallographic axes
as drawn but tend to do so in our observations.
Systematic studies of buckling domain wall motion is not possible
in our nonequilibrium system.