Possible Research will include the following
Dr. Clay's research is aimed at understanding the effect of electron-electron interactions in materials. Electrons in materials determine many important properties of the material, for example whether the material is conducting, insulating, or semiconducting. Electrons are charged particles and interact with each other through the Coulomb interaction. In many materials, the approximation that electrons do not interact with each other works surprisingly well. However, in some cases this approximation breaks down. Often such "strongly correlated" materials exhibit very unusual electronic properties such as antiferromagnetism or superconductivity.
While the Schroedinger equation describing electrons in a material can be written down, a solution of the full problem is not feasible. However, much simpler model systems can be used to understand the essential effects electron-electron interactions that determine the properties of materials. The most well-known of these models is the Hubbard model, which keeps electron-electron interactions at only very short distances. These models can still not be solved analytically, and much of our knowledge of them comes from numerical solutions. Dr. Clay develops numerical methods to solve these interacting models. Some of the current topics under research in his group are the effect of electron-electron interactions on superconducting pairing, the effect of phonons (crystal vibrations) on electronic properties, and electronic transitions in multi-band systems.
Many applications in materials science, physics, and biology can be described as multi-phase processes whereby two (or more) phases are separated by an interface which may evolve over time. A few examples of such a process is the separation of two components such as oil and water, the modeling of the movement of vesicle membranes, and the modeling of certain formations of crystal growth. Although systems with moving interfaces are extremely interesting, the difficulties in modeling these systems is well known, particularly with regard to tracking the interface. Phase field models have become a popular modeling tool as they eliminate explicitly tracking an interface by introducing an indicator function which tracts the phases instead of tracking the interface. The phase field approach is generally derived from an energy-based variational formulation and has the major advantage of leading to well-posed nonlinear coupled systems that satisfy thermodynamic-consistent energy dissipation laws. Creating and implementing numerical schemes which mimic these energy dissipation laws is the main focus of the proposed project.
In this project, REU students would first be taught the basics of a specific phase field model. The students would then be introduced to a computational program such as the FEniCS project, which utilizes python as the main programming language, or the FELICITY MATLAB/C++ Toolbox. The goal of the REU project would be the completion of a few numerical experiments to be included as part of a paper which supports the accuracy and usefulness of the phase field model.
A key to designing high-performance materials, especially composites, is to understand and control the interface between the various components of the system. Gwaltney and his colleagues have successfully modeled the interfaces between graphitic sheets and vinyl ester resins and between graphitic sheets and polyethylene. Part of this effort involved developing a unique algorithm to include chemical kinetics and selectivity into modeling the curing process. This algorithm, the Relative Reactivity Volume (RRV) model, has now been built into the commercial materials modeling software MAPS.
The next step is to apply these methods to the interface between graphitic sheets and epoxy resins. The REU student, under the guidance of a graduate student with experience in materials modeling, will build models of the graphitic sheets in the presence of uncured epoxy resin. After equilibrating the resin, the RRV algorithm will be used to cure the resin to a variety of conversion percentages typical of epoxies found in real composites. The physical and mechanical properties of these cured resins will then be determined. Parameters of interest include densities, glass transition temperatures, Youngs moduli, and yield stress and strain. It is not currently well understood how these properties vary with conversion percentage and with temperature. Subsequent REU students will then determine how these properties vary with epoxy resin and with pretreatment of the graphitic sheets (for example, oxidized versus pristine sheets).
A large range of denoising and segmentation methods cover various fields of mathematics and engineering. However, conventional methods do not work very well on complex material images since they can involve different types of noise and unclear edges. If a conventional segmentation method is used to detect the edges of pores on the X-ray image of a steel sample, annular pattern artificial noise can be produced due to the artifacts which inevitably appear in the original image during the X-ray process. This could result in inaccurate measurements of pore statistics. Partial differential equation (PDE) based denoising methods can effectively remove noise but there are still some remaining artifacts which cannot be removed using conventional denoising methods.
For the REU projects, we will develop new models and their associated algorithms for denoising and segmentation on material images. One non-conventional denoising/segmentation method that we can consider for material images is by treating the noise as being multiplicative. The programming language for our research project is MATLAB. During the first few weeks of the program, the REU fellows will learn basic tools on MATLAB and test simple codes to get familiar with MATLAB programming. The participants will also learn about the conventional denoising/segmentation methods during these weeks. Once the students get familiar with the use of MATLAB with developing simple codes, they will start writing their own codes for conventional methods and test the numerical examples using materials images. Students will be intensively discussing the development of the new methods/algorithms to overcome the drawbacks on traditional methods for materials images. The newly developed methods will be numerically tested and compared to the conventional methods.
Pattern recognition and machine learning play a prominent role in today's technological world. These activities are important in both basic science and the industry. Increased computational capabilities and advances in algorithms in recent years have taken these tools mainstream. In both academic research and industrial application, one often needs to minimize a cost function by training some parameters on a given data set to make future predictions. The parameters might be describing some subatomic reactions rates or modeling buying patterns in a population. Bayesian analysis is crucial in such situations to estimate the parameters, and also to compare competing models.
In the summer program the students would apply Bayesian analysis for parameter estimation in some simple physical models. Several examples from two-particle elastic and inelastic scattering exits. The students are not expected to derive the model expressions. Prof. Rupak would discuss the expressions, explaining the general physical ideas behind them. In physics, the relevant cost function that one minimizes is the chi-square that produces the likelihood probability distribution of the parameters. Minimizing the chi-square maximizes the likelihood. In the Bayesian formulation, one also incorporates prior xpectation about the sizes of parameters when estimating their values. The students will use Monte Carlo methods that become indispensable when the number of parameters become large (greater than a few). As the program develops, future projects would involve model comparisons. Often problems of interest at low energies cannot be solved directly from the known fundamental forces of nature. The method of effective field theory allows a simpler description at the cost of introducing extra parameters. Competing low energy theories with different number of parameters are possible. While a larger number of parameters can in principle give a better description of the data in terms of a smaller chi-square, in comparing models one should look beyond the goodness-of-fit. There should be some cost associated with introducing extra parameters when a simpler description might suffice. Bayesian model comparison incorporates the Occam's razor principle quantitatively. The Nested Sampling (NS) method will be used for model comparisons. The classical algorithm for NS is straightforward to understand and implement, even for undergraduate students with minimal programming experience. More sophisticated implementations of NS in C++ and Python are publicly available. Prof. Rupak would guide the students to write the interface for the model expressions.
Polycyclic aromatic hydrocarbons (PAHs) have gained significant attention in materials chemistry for their unique photophysical properties useful for organic light emitting diodes (OLEDs), photovoltaics, field-effect transistors, organic electronics, and sensors. Pyrene, a highly aromatic molecule comprised of four fused benzene rings, is a ubiquitous building block for synthesizing exceptionally conjugated compounds, or polymeric materials. While many of these fused-ring systems are planar, nonplanar or twisted conjugated molecules are rapidly emerging due to new applications in nonlinear optics, semiconductors, OLEDs and electronic devices. Notably, twisted molecules have displayed improved stability, enhanced solubility, and provide chiroptical properties deeming them versatile in new materials. A new category of distorted ribbon-like PAHs termed pyrene-fused twistacenes has recently been developed and shows imminent promise toward incorporation in OLED devices. Twisted pyrene-fused molecules are often characterized by the degree of their torsion angle at the carbon atoms in the K-region of pyrene.
In collaboration with Professor Robert Gilliard (University of Virginia) whose groups work on the synthesis of these novel materials, we will study the molecular and electronic structure of these class of molecules with respect to their light emitting properties. In this project undergraduate students will be introduced to main-group organometallic chemistry, data organization and scripting, computational methodologies for electronic structure.
Rapid advances in data collection and processing have provided opportunities to investigate various types of high-dimensional data available in many scientific fields including materials science. The goal of the proposed project is to statistically identify significant features and quantify differences/similarities between materials in high-dimensional data. In many cases, data collection is expensive/time-consuming and therefore the sample size n is less than the data dimension p.
In two-sample tests of high dimensional means, it is well known that the Hotelling's T test fails due to the singularity of the sample covariance matrix S. To tackle this problem, the idea is to project the p-dimensional data onto a subspace of lower dimension d < n using p-by-d random matrices. The special case d = 1 will be carefully investigated in this project. Two resulting questions need to be answered here. Firstly, what is the optimal choice of the projection vector (d = 1). Secondly, we need to study the limiting distribution of the test statistic T to conduct the hypothesis test for the test purpose. An extreme value theorem might be needed for this purpose. The students will learn the answers through numerical experiments by using the statistical language R. First, students will be trained in using R. The students will learn to formulate data projections in terms of hypothesis tests and verify them using simulations.