Geometrical confinement has long been known to influence even qualitative aspects of materials' phase behavior [1]. Freezing point suppression in fluid-filled pores [2] and crystallization of confined fluids under shear [3] are examples with ramifications for practical applications such as catalysis and lubrication. Often, the microscopic mechanisms by which geometrical confinement affects phase behavior are not accessible to direct observation. Our understanding of phase transformations, furthermore, is not yet complete even for unconfined systems. Charge-stabilized colloidal suspensions provide one instance of a system whose structural phases are strongly influenced by confinement and yet whose structure and dynamics can be analyzed with `atomic' resolution using optical microscopy.
The charge-stabilized colloid in this study consists of 0.325
0.005
m
diameter polystyrene spheres suspended in water.
Each sphere has a large number of ionizable groups chemically bonded
to its surface which dissociate when the spheres are dispersed in water.
The resulting surface charge
engenders mutual repulsions among the spheres which are moderated
by the counterions in solution.
This electrostatic interaction stabilizes the suspension
against flocculation driven by van der Waals attractions.
The most widely accepted theory describing
the pair-wise interaction between
colloidal microspheres was formulated almost 50 years ago by
Derjaguin, Landau, Verwey, and Overbeek (DLVO) [4].
In addition to the electrostatic interaction,
the full form of the DLVO potential
includes contributions from both van der Waals and
hydrodynamic forces.
These, however, play a relatively minor role in the behavior of highly
charged colloid in suspensions of low ionic strength such
as the ones we discuss below.
In the limit that the Poisson-Boltzmann equation
for the distribution of counterions can be linearized and under
the assumption that the charge on a sphere is fixed at Z, the
dominant electrostatic
contribution to the DLVO potential has the form [4]
Here r is the center-to-center separation between spheres
of radius a, 
is the Debye-Hückel screening length
which measures the range of electrostatic interactions in an electrolyte
of z-valent counterions at concentration n,
and 
is the dielectric constant of the water.
Eqn. (1) resembles a Yukawa repulsion, with the term in
brackets accounting for exclusion of counterions from the spheres' interiors.
The pairwise interaction between colloidal microspheres has been measured directly only very recently. Measurements on colloidal polystyrene spheres [5,6] and ferrofluid emulsion droplets [7] find excellent agreement with eqn. (1) at least in the strongly interacting limit. The single-sphere charges extracted from these and related neutron-scattering experiments on micellar suspensions [8] are orders of magnitude smaller than their titratable charges [9]. This large discrepancy is consistent with a recently proposed charge renormalization theory [10] which relates a sphere's effective charge to its size as
provided its bare charge is sufficiently large.
The Bjerrum length, 
,
measures the equilibrium separation between counterions,
and C is a constant both
predicted [10] and measured [8,5,6]
to be around 10.
For an electrolyte with z = 1 at 
C, 
= 0.715 nm.
The ensemble of colloidal microspheres in
a charge-stabilized suspension
undergoes phase transitions among fluid and ordered phases as a function of
the spheres' volume fraction, 
, and the
screening length, 
[11,12].
The fluid phase appears in low density suspensions with
short screening lengths while crystals form in more strongly interacting
suspensions.
Of the two ordered phases, crystals of face-centered cubic (fcc) symmetry
appear in dense suspensions while
body-centered cubic (bcc) crystals occur in the strongly-interacting dilute
limit.
The conditions for bcc crystallization approximate
those of the one-component
plasma model [13]
and thus provide a practical realization of the classical Wigner crystal
[14].
There also is experimental evidence that charge-stabilized suspensions
can fall into a glassy state at
particularly high volume fractions [12].
All features of this phase diagram except for the glass
have been reproduced at least
qualitatively in molecular dynamics simulations of Yukawa
particles [15].
This relatively simple phase diagram becomes extremely rich
when charge-stabilized suspensions are sandwiched between clean
glass surfaces [16,17].
The ensemble
of spheres in a single layer acts as an ideal two-dimensional
system whose ordered phase undergoes a two-stage second-order
Kosterlitz-Thouless melting transition [18].
As the separation between glass surfaces is increased and out-of-plane
motion becomes possible, the single-layer triangular crystal 
is replaced
by a two-layer crystal with square symmetry in the plane 
[17].
The transition between 
and 
structures involves
strongly correlated out-of-plane fluctuations [17]
which have
been analyzed by Chou and Nelson [19].
As the separation between confining walls is further increased, the
suspension undergoes a cascade of phase transitions in the following
sequence [17]:
This sequence proceeds to roughly 10 layers before
the bulk structure is recovered.
The structural fluctuations at the transition points
have been studied in detail only for the
transition [20].
In the sections which follow, we study the series of local
structural transformations which occur when a 
crystal
is shear-melted to a supercooled confined fluid and then is allowed
to freeze.
Rather than achieving its equilibrium structure in one step,
the ensemble of spheres
first freezes to a buckled single layer triangular crystal
and then passes through a 
martensitic
[21] phase transition.
The triangular crystal's presence in the reaction path is
an example of the more general phenomenon of one symmetry's
intervention in the evolution of a structure with another symmetry.
The formation of quasicrystals during splat cooling of metal
alloys provides a dramatic example [22],
while metallic glass phases prevent
supercooled metallic fluids from crystallizing at all [23].
