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Ben Murphy's Project SummaryAt the beginning of the summer, there were several topics of research to choose from. My studies in physics introduced me to many topics in applied mathematics. This previous experience biased my decision to topics related to electromagnetic fields. The two most tantalizing related subjects available were the dynamics of electrorheological fluids and electromagnetic scattering models of sea ice.Initially, I chose to study electrorheological fluids. I am fascinated with the mathematics and the many possible industrial applications of such a bizarre composite. A brief overview of this research topic follows. Electrorheological fluids are dual component composites, which have an amazing critical behavior. Dielectric (respectively, conducting) spheres are first immersed in an oil of low viscosity. An alternating electric field is then applied. Resultant electric dipolar (respectively, and magnetic dipolar) interactions cause the spheres to attract, forming small clusters. As the field strength increases, these clusters increase in size, thus slightly increasing the shear viscosity of the composite. It is observed that, at a critical field strength, these clusters form coherent chains that coalesce into columns with crystalline lattice arrangements of the spheres (respectively, fractal nets), increasing the shear viscosity by several orders of magnitude in a matter of milliseconds. Thus, the composite undergoes a discontinuous phase change from a liquid to a solid. The idea seems easy enough. There are many parallels to molecular phase changes, which are predicted very accurately using statistical mechanics. The problem is, we are on length scales much larger than in the statistical mechanics regime, so these powerful tools cannot be directly applied. There are also many subtle interactions at play that are not yet fully understood. Additionally, the application of an alternating field causes many of the parameters, that control the critical behavior, to be inherently complex. These and many other unanswered questions cause this to be a very formidable problem. The mathematics underlying this problem display many parallels to the sea ice problem. Both deal with discontinuous phase transitions of multi-component random media. This implies probabilistic models and statistical analysis are necessary. Both deal with alternating electromagnetic waves and their interaction with the random media in question. Complex analysis of the relevant parameters is therefore also necessary. The solutions of each also depend on knowing the percolation properties of the composites in question. The parallels go on and on. I was extremely surprised, while reading papers on both topics, just how closely related the two problems fundamentally are mathematically! To prepare us for the work to be done, our research project leader, Ken Golden, has been piling up the reading material. This material is essential to the understanding of necessary tools and concepts needed for the problem in question. I have read several extremely well written papers on electrorheological fluids. These papers more or less explain the problems' historical background, laboratory research, and interpretation of the results. The topics of concern are the possible geometric ground state configurations of the columns (respectively, fractal nets), what configurations are observed and, analytically, what these data tell us about the parameters determining the critical behavior. I have also done a considerable amount of reading on the Stieltjes integral representation of the complex effective permittivity developed by Bergman, Milton, Golden, and Papanicolaou. This gives a promising approach to the problem. This integral representation allows bounds on the effective parameters to be obtained. It also suggests many similarities to a statistical mechanics approach to the problem -- methods that would be greeted with open arms if available. We have also had numerous weekly meetings discussing these topics with Ken. These meetings started with students presenting topics introduced in a book we've been reading with him on an independent study basis. Topics include random walks in one and two dimensionsand their relation to the effective conductivity of electrical networks, harmonic functions, and several methods of solving such functions, including the method of Marcov chains. Ken gave informal lectures on related topics in the theory of composite materials, and meetings usually ended with discussions of topics pertinent to the progression of concepts summarized in the papers mentioned above. In order to focus on a specific problem closely related to but not requiring as much background as ER fluids, I shifted my focus to the interaction of electromagnetic waves with sea ice and other composites. There was nothing wasted due to the stunning mathematical similarities between the two problems. I have, since then, been reading papers on the modeling of electromagnetic scattering off of sea ice. This problem has several topics of particular interest. Obtaining remote sensing data in the field allows a better understanding of the problem and verifies correct bounds on relevant parameters. This component in the research is immensely important to the progression of mathematical models. The analytical approach to the problem is based on creating methods to interpret the data obtained in the field. Modeling the interaction of microwaves (and other frequency ranges) with sea ice or other composite materials is the core of what I will be doing in the future. The mathematics alredy created, to approximate the resultant scattering from multi-layer random media with rough interfaces, is very similar to the models previously encountered in my electrorheological fluid reading. The sea ice papers have also introduced me to many fields of mathematical physics that I have not yet encountered. These include the distorted Born approximation and radiative transfer theory. I am now working on finding a Stieltjes integral representation for solutions of the one dimensional Hemholtz equation, similar to the one mentioned above. The fundamental problem is that the above Stieltjes representation only applies to electromagnetic waves in the quasistatic regime where the wavelength is very long compared to the micro structural scale, while for many important phenomena, such as the interaction of microwaves with sea ice or composite photonic crystals, this condition is violated, and the wavelengths are of similar scale to the microstructure. This is a problem that no one has done before, which is an intriguing change of pace. I am really enjoying the work I'm doing and I'm looking forward to continuing next semester. It is my goal to make enough progress on this problem to publish while still an undergraduate. I am confident that, in the future, I will compose a Ph.D. thesis that will make quite a splash in the physics and mathematics community. |
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VIGRE Steering Committee Department of Mathematics University of Utah 155 South 1400 East; Room 233 Salt Lake City, UT 84112 email: viscom@math.utah.edu |
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