Colloids are particles with a diameter that lies in the range of tens of nanometers to a few micrometers, and which are suspended in a liquid. As well as having practical applications in areas as diverse as food, paint and cosmetic formulations, they are also useful as experimental model systems. Their dimensions are much larger than those of atoms and molecules, while their dynamics and the structures they form are still governed by thermal fluctuations.

My research uses computer simulations and theoretical
models to study the phase behaviour of various colloidal
particles, focusing on how particle shape can affect the
phases a system will form. In addition, I am interested in
elastic and structural properties of liquid, liquid
crystalline, and crystalline phases.

Colloidal crystals: mimicking atomic behaviour

The earliest computer simulations dealt with the simplest of shapes, such as hard-discs in 2d and hard-spheres in 3d. Even these basic particles can undergo phase transitions, forming a crystalline phase upon compression. Modern computers allow for increasingly more complex shapes to be studied. Even slight particle anisotropy is sufficient for a rich phase behaviour: hard-sphere dumbbells, for example, which consist of two rigidly fused hard spheres of equal size, can form liquid phases, plastic crystalline phases (where the particle centres of mass are aligned but they can rotate [relatively] freely), aperiodic crystals (where the constituent spheres of the dumbbells form a crystal, but the particle centres of mass do not) and, finally, periodic crystals (where both the constituent spheres and the centres of mass lie on lattice sites).

The phase behaviour becomes even richer if one considers
anisotropic dumbbells: two hard spheres of uneven size,
rigidly fused
together. Using
computer simulations, we have shown that such particles,
often named snowmen-shaped particles, can form a wide range
of stable crystalline phases depending on the ratio of the
diameters of the constituent spheres. Here the constituent
spheres form crystals with lattices that resemble such
atomic crystals as NaCl
(salt), CrB (chromium
boride), αIrV
and γCuTi. Another
interesting feature is that the phase behaviour of systems
of snowmen particles is richer than that of the
corresponding binary mixture of hard spheres. Binary hard
sphere mixtures will tend to only form the best packed
crystalline structure, while for snowman particles, there
are multiple ways of arranging the bonds connecting the two
spheres, giving rise to a degeneracy entropy that can
stabilize non-best packed structures.

Snapshots of various colloidal crystals formed ... Snapshots of various colloidal crystals formed by snowman-shaped particles. Larger constituent spheres are shown in blue, smaller constituent spheres in red. Top left: Rotator (or plastic) crystal. Top right: NaCl. Bottom row: Two planes of a CrB crystal.

Isotropic to nematic phase transition. Isotropic to nematic phase transition.

Onsager theory: thermal, elastic and structural properties

The Onsager theory of nematic liquid crystals describes the phase transition from a positionally and orientationally disordered isotropic phase to a positionally disordered but orientationally ordered nematic phase. In this theory, the transition is described by the competition between two different contributions to the free energy: an orientation entropy, where there is a free energy cost for aligning particles, and an interaction entropy, where alignment is favoured in order to reduce the number of interactions between particles. Onsager gave a virial expansion of the Helmholtz free energy which he truncated at second order, making the theory exact for infinitely long rods.

By including higher order virial terms, we have used
Onsager theory to accurately predict the phase behaviour
for various particle shapes, such
as hard
spheroids, cut-spheres, Saturn-shaped
particles
and cuboids. The
theory can also be expanded to study other phases, such as
the
novel cubatic
liquid crystalline phase, where discotic particles form
a distinct phase in which they are positionally disordered but
align along three orthogonal axes, as opposed to the single
axis for nematic liquid crystals. Furthermore, it is
possible to use the theory to calculate the
three Frank
elastic constants for nematic phases, which describe how
a nematic phase resists twist, splay and bend
deformations. These theoretical calculations have shown that
the three coefficients can differ widely from one another
depending on the particle shape. Another application is for
the calculation of
the pair
correlation function for liquid and nematic
phases

Colloidal rods: the effects of shape and flexibility

The fd-virus is a semi-flexible virus particles often
used in experimental studies as a model colloidal rod. One
reason is its dimensions, as its aspect ratio (the ratio of
length to diameter) is very large (L/D
∼ 100), while its persistence length (the length
over which a particle's orientation remains correlated) it a
few times its length (L_{P} ∼
2.5L), meaning that it is still relatively
stiff. Another is the fact that it is highly monodisperse,
whereas synthesized colloidal rods tend to be much more
polydisperse. These properties mean that their phase
behaviour can be accurately predicted by a simple expansion
of
the Onsager
theory to describe semi-flexible rods.

We
have generalized
the Onsager theory still further to describe binary
mixtures of semi-flexible rods. This theory can be used to
accurately predict both qualitatively and quantitatively the
phase behaviour of thick-thin fd-virus mixtures, which have
been shown
in experimental
studies to exhibit a very rich phase behaviour. The
theoretical calculations can go beyond the experiments in
terms of
precision, locating
the cross-overs between different types of phase
behaviour. Another interesting application of this model
is to study the alignment of guest particles immersed in a
host nematic phase. The nematic field causes the guest
particles to align, and experimental studies have shown that
flexible and semi-flexible polymers will undergo a coil-rod
transition, straightening out under the influence of the
nematic field of the host. Our theory can account for and
describe this behaviour for
both long,
flexible guest particles
and short,
rigid ones.

Coil-rod transition for a ... Coil-rod transition for a flexible polymer in a host solution of colloidal rods. When the host solution undergoes an isotropic-nematic phase transition, the nematic field straightens and aligns the polymer.