From zebra stripes to snow flakes to the Devil's Post Pile, regular patterns exist throughout the natural world. Patterns form when a uniform, spatially extended system is driven far enough from equilibrium that the dynamics are nonlinear, but not so far that the dynamics are completely turbulent. Another area of interest in nonlinear dynamics is the study of chaos. Loosely speaking, the term chaos describes deterministic systems that exhibit unpredictable temporal evolution. We are interested in phenomena that combine pattern formation and chaos. Along these lines, we are pursuing a number of experiments that study spatio-temporal chaos: dynamics that are disordered in both space and time.
The simplest example of pattern formation is the transition from a spatially uniform state to a state that is composed of a striped pattern. When such a system is driven farther from equilibrium, the stripe pattern develops defects. When the defects occur randomly in space and time, this is a particular type of spatio-temporal chaos known as defect chaos. Other types of spatio-temporal chaos are referred to as amplitude chaos or phase chaos. Despite these classifications, spatio-temporal chaos remains a poorly understood phenomena.
One of the difficulties in studying spatio-temporal chaos is that it typically occurs when a system is strongly nonlinear. A reduced control parameter e=R/Rc-1 characterizes how nonlinear the system is. Here R is the external driving force and Rc is the value of R at which the initial transition from a uniform state to a pattern occurs. For small values of e, the system is said to be weakly nonlinear. Much of our understanding of regular patterns comes from studies in the weakly nonlinear regime because they afford quantitative comparisons between theory and experiment. This kind of comparison is not possible for large values of e, i.e. strongly nonlinear systems. We will be studying a system, electroconvection in nematic liquid crystals, where there is a direct transition from a spatially uniform state to a state of spatio-temporal chaos. Therefore, the spatio-temporal chaos occurs in the weakly nonlinear regime, and we will be able to use many of the techniques that have been successful in the study of regular patterns. Electroconvection consists of applying an ac voltage across a sample of a nematic liquid crystal. Nematic liquid crystals are familiar from their use in display technology. They flow like fluids, but on average, the molecules are aligned along a particular axis known as the director. By confining a nematic liquid crystal between properly treated glass plates, a uniform alignment of the director is obtained. At a critical value of an applied ac voltage, there is a transition from this uniform state to one in which the director varies periodically and in which convection rolls are formed. The variation of the director has a wavelength that is typically 50 microns, and it can be observed with an optical microscope. Images of three of the possible patterns are shown in the figure.
Another area of research is the flow behavior of two-dimensional fluids. For this, Langmuir monolayers are used as a model system. They are composed of organic molecules that are have a hydrophilic head group and a hydrophobic tail. The molecules are placed on the surface of water and are confined to the air-water interface. These systems are interesting because they form a number of two-dimensional analogs of three-dimensional smectic liquid crystals. A smectic liquid crystal is a fluid in which the molecules are aligned on average along a particular direction and organized into layers. For Langmuir monolayers, the molecules in the liquid crystal phases are tilted. The average alignment of the tilt direction is the analog of the average alignment of the molecules in the smectic liquid crystal.
These liquid crystal phases have a number of interesting properties. First, the molecular order of liquid crystals provides the phase with elastic properties, in addition to the viscous nature of the phase. Therefore, these materials are inherently viscoelastic. Second, the liquid crystal phases are composed of randomly-oriented domains. The tilt direction varies from domain to domain. Many materials in nature are composed of a random-domain structure. Some examples are foams, emulsions, granular materials and metal alloys. Because of the domain structure, these materials exhibit interesting viscoelastic behavior. We have designed a number of experiments to study the interplay between the intrinsic viscoelasticity of the liquid crystal phases and the viscoelastic properties that are due to the domain structure.
We are still at the beginning of the research effort and there are many opportunities for new students, both at the undergraduate and graduate level. In addition, both areas offer the possibility of interdisciplinary work. The study of spatio-temporal chaos is closely related to the study of turbulence and chaos in plasma physics. The work with Langmuir monolayers has possible applications in biology, especially the behavior of membranes. Our lab is located in Physical Science I, room B13, and if you give us a call at (714)824-1714, we would be happy to arrange a tour of our lab.