MANIPULATION OF IONS, ELECTRONS, AND PHOTONS IN 2D MATERIALS BY ION INTERCALATION

2D materials have attracted tremendous attention due to their unique physical and chemical properties since the discovery of graphene. Despite these intrinsic properties, various modification methods have been applied to 2D materials that yield even more exciting results. Among all modification methods, the intercalation of 2D materials provides the highest possible doping and/or phase change to the pristine 2D materials. This doping effect highly modifies 2D materials, with extraordinary electrical transport as well as optical, thermal, magnetic, and catalytic properties, which are advantageous for optoelectronics, superconductors, thermoelectronics, catalysis and energy storage applications. To study the property changes of 2D materials, we designed and built a planar nanobattery that allows electrochemical ion intercalation in 2D materials. More importantly, this planar nanobattery enables characterization of electrical, optical and structural properties of 2D materials in situ and real time upon ion intercalation. With this device, we successfully intercalated Li-ions into few layer graphene (FLG) and ultrathin graphite, heavily dopes the graphene to 0.6 x 10^15 /cm2, which simultaneously increased its conductivity and transmittance in the visible range. The intercalated LiC6 single crystallite achieved extraordinary optoelectronic properties, in which an eight-layered Li intercalated FLG achieved transmittance of 91.7% (at 550 nm) and sheet resistance of 3 ohm/sq. We extend the research to obtain scalable, printable graphene based transparent conductors with ion intercalation. Surfactant free, printed reduced graphene oxide transparent conductor thin film with Na-ion intercalation is obtained with transmittance of 79% and sheet resistance of 300 ohm/sq (at 550 nm). The figure of merit is calculated as the best pure rGO based transparent conductors. We further improved the tunability of the reduced graphene oxide film by using two layers of CNT films to sandwich it. The tunable range of rGO film is demonstrated from 0.9 um to 10 um in wavelength. Other ions such as K-ion is also studied of its intercalation chemistry and optical properties in graphitic materials. We also used the in situ characterization tools to understand the fundamental properties and improve the performance of battery electrode materials. We investigated the Na-ion interaction with rGO by in situ Transmission electron microscopy (TEM). For the first time, we observed reversible Na metal cluster (with diameter larger than 10 nm) deposition on rGO surface, which we evidenced with atom-resolved HRTEM image of Na metal and electron diffraction pattern. This discovery leads to a porous reduced graphene oxide sodium ion battery anode with record high reversible specific capacity around 450 mAh/g at 25mA/g, a high rate performance of 200 mAh/g at 250 mA/g, and stable cycling performance up to 750 cycles. In addition, direct observation of irreversible formation of Na2O on rGO unveils the origin of commonly observed low 1st Columbic Efficiency of rGO containing electrodes. Another example for in situ characterization for battery electrode is using the planar nanobattery for 2D MoS2 crystallite. Planar nanobattery allows the intrinsic electrical conductivity measurement with single crystalline 2D battery electrode upon ion intercalation and deintercalation process, which is lacking in conventional battery characterization techniques. We discovered that with a “rapid-charging” process at the first cycle, the lithiated MoS2 undergoes a drastic resistance decrease, which in a regular lithiation process, the resistance always increases after lithiation at its final stage. This discovery leads to a 2- fold increase in specific capacity with with rapid first lithiated MoS2 composite electrode material, compare with the regular first lithiated MoS2 composite electrode material, at current density of 250 mA/g.

POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.