Soft Condensed Matter

Dr Karl Saunders

• Disordered Liquid Crystals
• Vortex Lattices in Ferromagnetic Type II Superconductors
• The Dynamics of Driven Disordered Extended Media
• Viscoelasticity

In my doctoral work, completed at the University of Oregon under the guidance of Professor John Toner (http://materialscience.uoregon.edu/toner.html) and in collaboration with Professor Leo Radzihovsky (http://spot.colorado.edu/~radzihov/) at the University of Colorado, I studied problems in soft condensed matter physics, particularly in the area of liquid crystals. There is a vast array of different types of liquid crystals, each sharing the feature of being neither completely liquid-like nor crystalline in nature. A smectic-A liquid crystal (see Figure 1) for instance consists of a periodic stack of liquid (translationally disordered) layers; thus, they are liquid-like in two directions (within the layers) but solid-like in one (normal to the layers). This combination leads to large fluctuations, which have been the subject of a great deal of theoretical and experimental study over the past two decades. A columnar liquid crystal (see Figure 2) on the other hand consists of a periodic (usually triangular) array of liquid-like columns and is thus solid-like in two directions and liquid-like in one. A nematic phase (see Figure 3) is basically made up of rod-like molecules that have their long axes aligned along a common direction but in which the arrangement of their centres is random. The existence of the special direction in the nematic phase means it is optically birefringent, a property that is exploited in liquid crystal displays.

Figure 1: Schematic of a Smectic-A Liquid Crystal

Figure 2: Schematic of a Columnar Liquid Crystal

Figure 3: Schematic of a Nematic Liquid Crystal

Exotic New Bragg Glass Phases
My work focused on the effects of quenched disorder on the equilibrium properties of liquid crystals. In this work I showed, using renormaliztion group methods, that confinement of a smectic-A or columnar liquid crystal to a random medium such as

aerogel (a very low density frozen network of silica strands) can result in exotic new ``Bragg Glass'' phases. This is a phase in which long-ranged translational order of the arrangement of layers/columns is destroyed by the quenched disorder. Nonetheless, the liquid crystal remains free of unbound defects so that the topological order (and hence elasticity) of the layers/columns is retained. The resulting elasticity is, however, strongly anomalous and nonlinear. The existence of such a phase is a dramatic illustration of the often unknown fact that a solid does not melt when long range order (and hence sharp x-ray scattering peaks) is lost, but rather when topological order is lost.

Different types of quenched disorder (i.e. isotropic or anisotropic) can, it turns out, lead to very different glassy phases that lie in different universality classes. It was also possible to predict the existence of a new phase resulting from the confinement of a columnar liquid crystal phase to anisotropic aerogel. For certain types of anisotropy, one direction of order in the columnar phase can be destroyed, resulting in a phase which has the translational order of a smectic-A liquid crystal but the elasticity (albeit anomalous) of a columnar liquid crystal. From x-ray scattering, which probes the

translational order, one may see a columnar phase disguised as a smectic phase. Experimental research on these predicted phases is being carried out by Professor Robert Leheny at Johns Hopkins University and related experiments on smectics in isotropic aerogel have been carried out by Professor Noel Clark at the University of Colorado.

In an interesting off-shoot of our work we found that the discoveries which we made for disordered columnar liquid crystals are also relevant to ferromagnetic type IIsuperconductors. These ferromagnetic superconductors exhibit a spontaneous vortex state. This is essentially a state in which the magnetic field can penetrate the superconductor but only in single flux lines. These flux lines arrange themselves in a triangular lattice. It turns out that the elastic properties of this flux lattice are just like

those of a columnar liquid crystal phase, more precisely it lies in the same universality class. If the superconductor is dirty, due to the presence of magnetic impurities, then a stable Bragg glass phase of the flux lattice could result. Its properties would include those of the columnar Bragg glass that we have already studied and could be probed using external magnetic fields.

As a postdoctoral fellow I conducted theoretical research on a range of problems within the fields of soft condensed matter and statistical physics, in particular,

the dynamics of driven disordered extended media. Extended condensed matter systems driven over quenched disorder exhibit a rich dynamics, including non-equilibrium phase transitions and history dependent behaviour. Such systems include vortex arrays in type-II superconductors, charge density waves in anisotropic conductors and cracks in heterogeneous solids, to cite a few. Closely related behaviour also arises in friction and lubrication. Most of the theoretical work to date has focused on the dissipative dynamics of driven elastic media that are distorted by disorder, but cannot tear. At zero temperature such systems exhibit a continuous depinning transition from a pinned to a unique sliding state.

Experiments and simulations show, however, that many physical systems with strong disorder depin plastically, with fluid-like regions flowing around pinned solid regions. In this regime the response is strongly history dependent, with long-term memory and switching as the system explores a variety of non-equilibrium sliding states. While at Syracuse I, along with my collaborators Professor Cristina Marchetti (http://www.phy.syr.edu/%7Emcm/), Professor Alan Middleton (http://www.phy.syr.edu/%7Eaam/) and Dr. Jennifer Schwarz (now at the University of California Los Angeles), worked on developing and understanding new theoretical models that attempt to capture these features. An underlying principle of these models is the inclusion of interactions within the medium that reflect the presence of topological defects due to strong disorder, thus allowing for plastic response.

One such model uses the well-known phenomenology of viscoelasticity in dense fluids, in that the elastic coupling between the driven degrees of freedom is replaced by a coupling that is non-local in time and allows for elastic restoring forces to turn into dissipative fluid flow on short time scales. This allows for spatially inhomogeneous response, with the coexistence of moving and pinned degrees of freedom.

A Phase Slip Model
A second model that allows for plasticity in a driven disordered extended medium uses an interaction that allows for ``phase slip'' between degrees of freedom. This is intended to incorporate the introduction of a topological defect, or tear, in response to an excessive strain caused by a region of very strong disorder. Upon the introduction of the defect one domain will jump or slip relative to a neighbouring domain. An elastic restoring force is linear in the relative displacement (or phase difference) between two neighbouring domains, and therefore never allows a slip or jump of one domain relative to another. However non-montonic, e.g. sinusoidal, coupling between domains will, for sufficiently large relative displacement, change sign and allow a slip. The changing of sign is equivalent to the introduction of a defect. This captures nicely the features seen in experiments and simulations, and even within mean field theory the resulting phase diagram is very rich.

Relationship to Coupled Oscillators
It is also worthwhile pointing out that models of driven disordered systems with non-monotonic interactions are also relevant for arrays of nonlinearly coupled oscillators. An example is the Kuramoto model used to describe the onset of synchronization in many biological and chemical systems. The model consists of a large number of oscillators with random natural frequencies and a sinusoidal coupling in their local phase differences.

Viscoelasticity
In related research I, along with Cristina Marchetti, provided some justification for the use of viscoelastic interactions to reflect the presence of defects. We did this by showing that the dynamics of a two-dimensional crystal with a finite concentration of free dislocations, as well as vacancy and interstitial defects, is governed by the hydrodynamic equations of a viscoelastic medium