Kamien
# Kamien Group

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Department of Physics and Astronomy

209 South 33rd Street

University of Pennsylvania

Philadelphia, PA 19104-6396

*Theoretical Condensed Matter Physics*

Our research interests center on problems in condensed matter theory.
We are currently exploring problems in liquid crystals, foams, soft self-assembly, and biological
physics.

## Crystal Structure

Great advances in synthetic chemistry have produced a new class
of monodisperse, yet highly complex, molecules. These molecules
self-assemble into a variety of crystalline lattices with
lattice constants on the order of 10nm.
Unlike colloidal crystals, these materials form
lattices that are not close-packed such as the face-centered-cubic
(FCC) lattice. Instead the body-centered-cubic (BCC) lattice as well as
the more exotic beta-Tungsten lattice form.
While detailed modeling can lead to precise intermolecular
potentials, progress has been made on a generic approach to
crystal formation based on the mathematics of minimal surfaces.
By considering the interface between the Voronoi cells that
contain each molecule, one can argue that the complex molecular
coats favor a minimum area. This leads to the BCC
lattice among others.

## Liquid Crystals

The theory of liquid crystals and liquid crystalline polymers
runs the gamut from the rheology of complex fluids to the esoterica of
homotopy and topological defects. All aspects of theory are
intimately influenced and connected
with experiment. This places liquid crystal physics in an exciting position:
it is driven by both theory and experiment and by both the abstract
and the applied.
Specifically, chiral molecules are ubiquitous in nature and exhibit many liquid crystal
phases.
They are extensively studied by biologists, chemists and biophysicists.
There is still much theoretical work to be done on the wealth of
experimental data.
Smectic liquid crystals as well as columnar liquid crystals
have the analogue of the Meissner phase in superconductors, in which
chirality is excluded from the bulk (just as the magnetic field is
expelled in a superconductor). The constraints imposed by topology
and geometry can screen out chirality as well. We are studying the nonlinear
elasticity of these phases through exact solutions and geometric principles.

## Minimal Surfaces, Smectics, and Foams

Minimal surfaces have a long history of lying in the interface between
Physics and Mathematics. Their use has had somewhat of a
revival due to the ever increasing complexity of lyotropic
phases of matter. Block copolymers and amphiphiles form
structures that resemble doubly-
and triply-periodic minimal surfaces. This
is no surprise: because physics
problems can be posed as extremal problems and because the
mean curvature is the lowest-degree, rotationally-invariant
scalar, minimal surfaces arise as solutions. However, nonlinearities
should play an important role as the lengthscales of the structures
become comparable to molecular sizes.
Generalizing to constant curvature structures, we are working closely with the Durian group to study the properties of
random foams using a mean-field model for the polyhedral cells.

## Packing

Hard sphere systems are pure: their physics is governed solely by entropic depletion interaction with the only adjustable
parameter being the volume fraction. Despite their simplicity, many details of
the freezing transition are not understood. We are interested in the rigorous viability of the
so-called *random-close-packed* state. Through geometric methods we are probing this
state via analytic and numerical approaches. We study the equation of state via modifications
to the virial expansion. We are also interested in the packing of long, stiff and semi-stiff
polymers in the presence of entropic depletors.

*
This work has been or is supported, in part, through the Simons
Foundation, the Charles E. Kaufman Foundation a supporting organization
of The Pittsburgh Foundation, the ACS Petroleum Research Fund, and the
National Science Foundation Through Grants DMR-9732963, INT-9910017,
DMR-0102459, DMR-0129804, DMR-0520020, DMR-0547230, CMMI-0900468,
DMR-1120901, DMR-1262047, and EFRI-1331583.*

*Last modified 31 May 2017*