Performance Modeling and Design of Ultra-High Power Microbatteries

Figures 1a and 1b show a schematic and SEMs of the high power battery modeled in this work.⁵ The battery has interdigitated electrodes composed of a highly porous bicontinuous nickel current collector (blue) conformally coated with electrochemically active materials (active materials). The active material is nickel-tin (red) in the anode and lithiated manganese oxide (yellow) in the cathode. The electrolyte fills the volume between the electrodes and inside the electrode pores. Figure 1c shows the lithium and electron transport paths during discharge. Lithium is stored at a high energy state in the anode. An oxidation reaction at the anode-electrolyte surface releases lithium ions and electrons that flow to the lower energy state cathode where they undergo reduction. Electrons do not travel through the electrolyte and instead travel from the anode active material surface through the anode active material, anode current collector, external circuit where they power a load, cathode current collector and cathode active material until they react at the cathode surface (black). Lithium ions flow from the anode to the cathode through the electrolyte (light blue). As lithium ions are released or inserted into the active material, lithium stored in the active material bulk diffuses toward or away from the active material surface according to the concentration gradient (dark blue). The microbattery architecture achieves high power density by simultaneously reducing the ion and electron transport lengths.

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The battery power performance depends upon the battery voltage, which is the difference in anode and cathode electrochemical potentials minus any internal voltage drops. The electrochemical potential difference depends on the lithium concentrations at the active material surfaces. Internal voltage drops are due to the ohmic conduction of electrons through the electrodes, the ohmic conduction of ions across the electrolyte, and the electrochemical kinetics at the anode and cathode surface. Energy density is the product of the battery voltage and amount of charge transferred between electrodes per battery volume. Power density is the product of the battery voltage and charge transfer rate between electrodes per battery volume. As the charge transfer rate increases, the electrochemical potential difference decreases and the internal voltage drops increase, so that the battery voltage falls below its equilibrium value and reaches a cutoff voltage before all of the energy can be fully extracted. The reduced battery voltage at high charge transfer rates decreases power performance, or amount of energy extracted at a given power density.

Here, we model the power performance of high power interdigitated bicontinuous microbatteries using a 1-D model of ion and electron transport.^19,20 Figure 1d shows the model space, which includes the porous anode, porous cathode, and separator. The model space simplifies the interdigitated electrode design to two electrodes bounded by the electrode centerlines with a symmetric boundary condition. The electrodes are approximated as rectangular, with widths (W_neg and W_pos), height (H), and cross sectional areas equal to the measured cross sectional areas of the experimental electrodes. Fig. 1d shows the 1-D transport of ions and electrons. Ion transport through the electrolyte is governed by migration and diffusion using concentrated solution theory,^19,20 and electron transport is ohmic. The total current through each differential length, dx, along the battery is the sum of the current due to local conduction of electrons and ions and is constant across the entire pitch. At each differential length in the electrodes, 1-D diffusion models predict lithium concentration through the active materials. The active material coating is approximated as a thin film with zero lithium flux to the nickel scaffold and flux at the electrode-electrolyte interface determined by Fick's law. The amount of lithium inserted or removed from the active material is determined by Butler-Volmer kinetics which link the electrolyte simulation to the electrochemical potential determined by the active material surface concentration. The model space and boundary conditions are chosen to match conventional battery formats so that the results can be generalized to other electrode architectures.

The nickel current collector volume fraction, ε_Ni, active material volume fraction, ε_act, and electrolyte volume fraction, ε_e, are important model inputs that determine the energy density of the batteries and impact the transport of ions and electrons. ε_Ni, ε_act and ε_e depend on the bicontinuous electrode structure and are calculated from a geometric model of self-assembled polystyrene (PS) spheres organized in a FCC unit cell. The current collector volume is the cubic unit cell volume minus the void volume left by the sintered PS spheres after PS etching, V_void. The active material volume, V_act, is the volume of a thin layer coated on the current collector, calculated from thickness t. Figures 2a and 2b show an inverse opal unit cell and the simplified geometry of two neighboring spheres used to calculate the volume fractions. PS spheres in contact after opal self-assembly are represented by circles with radius R and have 0.74 volume fractions. Sintering the PS increases the radius from R to R_n such that neighboring radii overlap by a length b. V_void is the volume of 4 sintered PS spheres of radii R_n in a FCC unit cell minus the volume of 48 overlapping spherical caps of height h or

where and h is R_n – R. V_act is the volume of 4 spheres with radius R_v subtracted from V_void, where R_v is R_n – t. Additionally, the volume where the active material does not deposit, marked with hatching, is integrated and subtracted out of V_act, but not including the volume of the spherical cap marked by h2.

where h2 is R_v (1 – R / R_n). ε_Ni, ε_act and ε_e are calculated from the V_act and V_void normalized by the unit cell volume, .

and

Equations 3–5 calculate ε_Ni, ε_act and ε_e from fitting parameter t and experimental measurements of R and b. ε_Ni = 0.12 for the experimental batteries where b is 200 nm and R is 500 nm.

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An effective electrode conductivity across the electrode width, k^eff, is approximated by equating the conductance across the width to the conductance down the length, using

as electrical conduction occurs primarily down the length (normal to the model space in Fig. 1d), L, in the interdigitated architecture. The bicontinuous electrode conductivity, σ^eff, is approximated by conduction through the two media in parallel with their volume fractions normalized to 1, such that

σ^eff_pos is 3.3 × 10⁶ S m⁻¹ and σ^eff_neg is 1.1 × 10⁷ S m⁻¹. k^eff is then corrected for the porosity of the electrode with a Bruggeman exponent of 1.5.

The model uses electrolyte material properties from available electrolyte data. The experimental electrolyte is 1:1 ethylene carbonate: dimethyl carbonate (EC:DMC) with initial 1000 mol m⁻³ LiClO₄ concentration, c⁰_e. The electrolyte interdiffusion coefficient is approximated as 2.6 × 10⁻¹⁰ m² s⁻¹ based on 1 M LiClO₄ in propylene carbonate (PC).³⁸ The electrolyte conductivity depends on concentration and is taken from data on LiClO₄ in EC:PC,³⁹ where the 8.5 × 10⁻³ S cm⁻¹ maximum conductivity, σ_e, is close to that of 1 M LiClO₄ in EC:DMC.^40,41 The lithium transference number, t⁺, is 0.363 based on LiPF₆ in EC:DMC because the transference number depends on the solvation radius of the anion, which is similar for PF₆⁻ and ClO₄⁻.^19,40,42 The electrolyte conductivity and diffusivity through the porous electrodes are corrected for the increased path length using a Bruggeman exponent of 1.5.^19,20 Changes to the pore shape or volume fraction in the microbattery electrodes would change the tortuosity and affect the Bruggeman exponent.⁸

The following are material properties for the electrochemically active materials. Figure 3 shows the open circuit voltage (OCV), versus lithium, as a function of the state of charge for the anode and cathode active materials. The OCV is measured from batteries discharged at low rates. The maximum capacity of the cathode, c^max_pos, is 23,000 mol m⁻³ based on a 145 mAh g⁻¹ manganese oxide cathode.⁴³ The initial cathode concentration, c⁰_pos, is approximated as 50 mol m⁻³ for all batteries. The nickel-tin anode is composed of 80% tin. At a full state of charge, the anode OCV corresponds to Li₃Sn with a 55,000 mol m⁻³ volumetric capacity.⁴⁴ c⁰_neg is set so that the low rate voltage plateau matches between predicted and measured batteries.

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Table I shows the active material thickness, t, and diffusivity, D, used to simulate each battery in addition to experimentally measured parameters. The active material thickness and diffusivity are fitting parameters bound by half the interconnect pore size (100 nm) and PITT diffusivity measurements in the anode and cathode materials. Measured anode diffusivities varied between 9.34 × 10⁻¹⁸ and 8.51 × 10⁻¹⁷ m² s⁻¹, with an average of 3.35 × 10⁻¹⁷ m² s⁻¹. Cathode diffusivities varied between 1.57 × 10⁻¹⁹ and 1.14 × 10⁻¹⁶ m² s⁻¹, with an average of 1.83 × 10⁻¹⁷ m² s⁻¹. The anode and cathode diffusivities in the simulation vary by a maximum of 3.35 and 3.66 X the average measured diffusivities for each battery. The model was simulated in COMSOL using a 1-D battery module.

Figure 4 shows predicted and experimental discharge curves of Batteries 1–4, which have the highest combined energy and power densities of our previously tested microbatteries.⁵ At low C rates, predictions and measurements compare very well. At greater than 30 C rate discharges, there is a slight departure between prediction and measurement, up to 3 percent of total capacity. The measured and predicted discharge curves for Batteries 2 and 3 closely overlap at low and high discharge rates, but the higher predicted capacities at 8.6 and 20 C rates are likely due to error in the thin film approximation as the amount of capacity extracted from a thin film is larger than the concave geometries in the curved electrode pores at moderate discharge rates. The lower average voltages of the discharge curves in Battery 3 indicate that the anode was not fully charged before cycling, which caused to cathode to be overcharged when Battery 3 was charged to a 4.0 volt cutoff. The overcharging caused a higher voltage for the first 0.15 Ah m⁻² of experimental discharge. The overall good agreement between predicted and measured discharge curves indicates that the model captures the major physics for high rate discharging and that the 1-D transport model provides an accurate foundation for design studies of batteries discharged at up to 600 C rates.

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We use the calculated lithium concentration throughout the battery to understand how different parameters affect battery performance. Figure 5a shows the electrolyte concentration in Battery 1 at the end of discharge at multiple C rates. The change in electrolyte concentration increases as the C rate increases, except at 457 C, where the battery quickly shuts off before a significant amount of ions can be removed from the electrolyte. This quick shut off is due to limits in the solid active material diffusion and will be discussed later. The maximum concentration change is 250 mol m⁻³ at 146 C, which decreases the electrolyte conductivity from 8.6 × 10⁻³ S cm⁻¹ to 8.2 × 10⁻³ S cm⁻¹ in the cathode region and 8.5 × 10⁻³ S cm⁻¹ in the anode region. The minimal changes in conductivity indicate that diffusion in the electrolyte did not limit the power performance of Battery 1. Figures 5b and 5c show the final lithium concentrations in the solid electrode active materials in Battery 1 at each discharge rate. The differences in surface and center concentrations increase as the discharge rate increases. The surface and center concentrations in the cathode are 22,200 mol m⁻³ and 2,600 mol m⁻³ at 30.5 C, which corresponds to a 1.26 volt difference, or 63% of the battery voltage window. At 457 C, the cathode surface concentration reaches a maximum in 0.17 seconds and limits the capacity to 0.058 Ah m⁻². The large concentration differences across the anode and cathode active materials and the associated overpotential at moderate and high discharge rates show that diffusion in the active material limits the power performance of Battery 1. Diffusive limitations in the active materials limit the power performance of all simulated batteries.

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The design process that follows informs how microbattery geometry influences the trade-offs between energy and power density, and develops design rules that guide these trade-offs. The power density of a battery is proportional to the energy density times the C rate, Power ∼ Crate × Energy, so an improved power performance results from an improved C rate and energy density. Figure 6 shows the microbattery geometries that govern ion and electron transport. The electrode width, W, governs diffusion through the electrolyte because the largest electrolyte concentration gradients occur from depletion and generation of lithium ions across the electrode width. The electrode pitch, P, governs ionic conduction in the electrolyte. The active material thickness, t, governs active material diffusion. The particle radius could function as the thickness in other electrode designs. The bicontinuous electrode architecture and the electrode length, L, have the largest impact on electrode electron conduction in this work. Simulating the energy density of Battery 1 at various C rates for a range of values of a single geometric component or governing transport property isolates the influence of each geometric component on the power performance. Data from Battery 1 provides the primary comparison between experiment and simulation in many of the design plots because the experimental microbatteries cover a small subset of geometries compared to the large variation required to cover the simulation range.

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Electrode width (W)

Figure 7a shows the predicted energy and power density of Battery 1 for various values of the electrode width (5, 12.6 and 50 μm) and electrolyte diffusivity (scaling factor 0.04, 0.2, 1, 5, and 25X from the value in Table I). For batteries with wide electrodes and low electrolyte diffusivities, the Ragone curves show large drops in power density for a given energy density. This results from lithium ions severely depleting in the cathode region such that the concentration is near zero in the electrolyte close to the electrode center. The near zero concentration significantly reduces the local electrolyte conductivity and causes a large voltage to develop across the electrolyte, which brings the battery to the shut off voltage before more energy can be extracted. At higher C rates, the Ragone curves return to normal because diffusion in the active material causes the battery to shut off before severe electrolyte depletion occurs. We conclude that diffusion in the electrolyte has a minimal effect on the battery performance unless the concentration in the cathode region depletes to near zero, at which point the effect is dramatic.

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The point when the concentration in the cathode region depletes to zero is predicted by solving for steady state diffusion in the cathode electrolyte region. The governing equation for 1-D diffusion in the electrolyte with constant and uniform depletion is

The depletion rate in the electrolyte, , is assumed constant when discharged at a constant C rate and uniformly distributed throughout the electrode region. is the amount of lithium ions that enter the cathode from the electrolyte, per electrode volume, or

Equation 9 is valid when the capacities in the battery electrodes are balanced, or when the cathode limits capacity. The discharge time is approximated as the time it takes to fully discharge a battery at a constant current density (3600 / Crate). Solving Equation 8 yields the concentration profile

where x is the distance between the separator, x = 0, and the electrode centerline, x = W_pos. The diffusivity is corrected using the Bruggeman approximation. The initial concentration of the electrolyte, c⁰_e, is the concentration at x = 0, which is valid for most batteries because the concentration change across the separator is small. A symmetric boundary condition exists at x = W_pos. Equation 10 solves for the critical electrode width, W_c, where the steady state electrolyte concentration depletes to zero by setting x = W_c = W_pos when c = 0. W_c is then

Figure 7b shows the predicted energy densities of Battery 1 discharged at multiple C rates with varying electrode width (3–100 μm) and constant W_sep = 19.8 μm. The dashed line shows the energy density of batteries with electrode widths equal to W_c. A maximum energy density occurs for each C rate because of competition between the increases in electrode volume fractions (W/P) that improve energy density and the increase in ion diffusion length through the electrode width. The energy density at 12, 20 and 30.5 C dramatically decreases after the maximum energy density is reached because the concentration of lithium is depleted to near zero in the cathode electrolyte. An electrolyte with increased or decreased diffusion coefficient would increase or decrease the electrode width required to deplete to zero concentration. We conclude that W_c is a good design parameter for the cathode width as it predicts the largest allowable cathode electrode width before severe ion depletion occurs.

Electrode pitch (P)

Figure 7c shows the predicted energy density of Battery 1 discharged at 1–500 C rates with electrolyte conductivities varied between 4 × 10⁻⁵ S cm⁻¹ and 1 × 10⁻¹ S cm⁻¹. The electrolyte conductivity was scaled by 1 × 10⁵ to study the effect of the pitch on power performance without affecting other physics in the system, and also represents the affect of changing the battery electrolyte. The dashed lines represent a constant potential drop across the electrolyte calculated using Ohm's law, where the voltage drop, V_e, and electrolyte conductivity, σ_e, are related to the pitch, P, by

The current density is

i is the electrode with the lowest total capacity. m = 0.5 in the positive and negative electrode region because on average half the current is transported by ionic conduction and the other half by electron conduction in the electrode. m = 1 in the separator region. ε_e is the volume fraction of the electrolyte in each region. Equation 12 is valid when the electrolyte concentration has little variance during discharge, which is true if W_pos < W_c. The energy density at each C rate in Fig. 7c remains constant as the conductivity decreases until the voltage drop in the electrolyte, calculated from Equation 12, is greater than 0.1 volts.

Figure 7d further demonstrates how the electrolyte conductance affects power performance by plotting power density versus energy density for various electrode pitches (30, 45, and 120 μm). The electrode separation is held constant. At each pitch, the peak electrolyte conductivity is scaled by a factor of 0.01, 0.1, 1 or 10X. At low power density, battery performance has minimal dependence upon electrode pitch or conductivity. The point at which the power performance begins to rapidly change (marked by a black diamond for each battery) is predicted by Equation 12 when V_e = 0.1 volts. We conclude that high power performance microbatteries should be designed so that V_e < 0.1 volts.

Electrode electron path

Figure 8a shows how the electrode conductivity affects the energy density of Battery 1 at 1–500 C discharge rates. The conductivity is varied by up to 1 × 10⁻⁸ times the conductivity in Table I. The dashed lines in Fig. 8a correspond to the energy density of batteries where the voltage drop across the electrodes, V_s, is 0.3 or 0.1 V. The voltage drop is estimated using Ohm's law,

where the current density is calculated from Equation 13. Figure 8b shows predicted power density versus energy density for Battery 1 with varying electrode conductivities. The point at which the power performance begins to rapidly change (marked by a black diamond for each battery) is predicted by Equation 14 when V_s = 0.1 volts. We conclude that high power performance microbatteries should be designed so that V_s < 0.1 volts.

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Active material thickness (t)

The active material thickness influences the fraction of lithium extracted or inserted into the active material at a discharge rate. A smaller active material thickness allows more energy to be extracted in less time; there are however practical limits on the thickness. Figure 9a shows how the cathode active material thickness in Battery 1 affects the energy density at 1–500 C rate discharges. A maximum energy density occurs because of competing effects between an increase in the total volume fraction of the active material and an increase in the distance that lithium travels. The maximum can be predicted by comparing the diffusion time, τ, to the diffusion distance, l. The active material thickness corresponding to the maximum power performance, t_o, is predicted from the diffusivity, D_s, and C rate using Dτ/l², or

where n is 1 for a slab and for a sphere. The dashed line in Fig. 9a is the energy density corresponding to t_o, when n = 1, which predicts the maximum with reasonable accuracy. The amount of lithium extracted or inserted into the active material is constant for a constant Dτ/l² when the discharge is stopped at a set surface concentration, which is valid for batteries discharged at a constant C rate.⁴⁵ It follows that the extracted capacity of a microbattery with t_o active material thickness is a constant fraction of the total capacity if the shutoff concentration is set to the maximum concentration in the cathode or minimum in the anode. The fraction of extracted capacity is 2/3 the maximum when n = 1 and the active material thickness is t_o. Figure 9b is a simulated Ragone plot for Battery 1 with cathode active material thicknesses varied between 13 and 100 nm. The power curve for 15.5 C shows the energy and power density of each battery when discharged at a 15.5 C rate. The maximum power performance occurs at the apex of the 15.5 C power curve and corresponds to t_o = 55 nm, which is the optimal active material thickness predicted from Equation 15 at 15.5 C rate. The energy and power density corresponding to t_o for each battery is marked with a diamond. We conclude that power performance is near optimal when the active material thickness is t_o.

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Figures 9c and 9d show how the active material thickness affects the power performance independent of the volume fraction by changing the pore diameters. The active material thickness is varied with the pore size to maintain a constant volume fraction so that the pore diameters in Figs. 9c and 9d are approximately proportional to the active material thickness. The dashed line in Fig. 9c shows battery performances when the active material thickness is t_o. Increasing the active material thickness past t_o dramatically reduces the power performance at all C rates, including 1 C. Decreasing the pore size (PS diameter) improves the power performance because the diffusion distance through the solid active material is reduced (higher power) while the active material volume fraction remains the same (constant energy). A battery fabricated with 109 nm PS diameter particles can be discharged at 300 C and maintain 55% of its total energy density.

In the simulated batteries, the electrode pores are FCC closed packed. If the pore distribution deviated from FCC close packed toward random close packed, the nickel volume fraction would increase and the active material volume fraction would decrease. If the active material thickness was maintained, and thus the C rate performance held constant, the power density would still decrease because the reduced active material volume fraction would decrease the energy density.

The design parameters (t_o, W_c, V_e, and V_s) provide a simple tool to optimize the power performance of a battery using only battery geometry and material properties. To illustrate the design utility, we optimize Battery 1 for a 50 C rate discharge using the design parameters and compare the optimized performance to experimental data. Figure 10a shows a diagram of the optimized microbattery. The microbattery geometry was chosen by setting the active material thickness to t_o and cathode width to W_c, while decreasing the pore size to 250 nm to increase the active material volume fraction. Additionally, the pitch was set so V_e ⩽ 0.1, and the electrode length so that V_s ⩽ 0.1. Figures 10b and 10c show the power performance of the 50 C rate optimized microbattery. The microbattery capacity at 50 C is 56% of the 1 C rate capacity, which agrees well with the 2/3 performance predicted by Equation 15. The 50 C rate optimized design has ∼ 10 X increase in energy density compared to Battery 1 at 1,000 mW cm⁻² μm⁻¹ power density and ∼ 10 X increase in power density at 10 μWh cm⁻² μm⁻¹ energy density. The most significant power performance improvements came from increasing the cathode width and optimizing the active material thickness while reducing the pore size to 250 nm. Figure 10d shows sensitivity analysis results for the 50 C rate optimized design when all combinations of the active material thickness and electrode width are varied by ±10 and 25%. Microbatteries with better power performance than the 50 C optimized design are to the right of the red line. A majority of the microbatteries have lower power performance than the 50 C rate optimized design. We conclude that the design parameters can be used to optimize microbattery geometry for maximum power performance.

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The design optimization presented here is for a single porous anode and cathode separated by electrolyte. Stacking many of these anode, cathode, and electrolyte sections together would increase the total battery energy and power, but maintain the same energy and power density, and thus performance, as long as the electrodes are stacked so that the geometric properties that control transport (represented by t_o, W_c, V_e, and V_s) remain the same.

In conclusion, the performance of microbatteries with up to 7.4 mW cm⁻² μm⁻¹ power densities were predicted using 1-D transport models combined with a geometric model of the bicontinuous electrodes. The 1-D model showed good agreement between predicted and measured discharge curves at up to 600 C rate discharges and indicated that diffusion through the electrochemically active material limited battery power performance. The accuracy of predictions could be improved by accounting for material property changes, including capacitive transport, and accounting for pore wall or surface film mass transport limitations. The microbattery geometry was related to the power performance using four design parameters, t_o, W_c, V_e, and V_s. The microbattery geometry can be optimized for a desired output power by setting the active material thickness to t_o, the cathode width to W_c, the pitch so V_e ⩽ 0.1, and the electrode length so that V_s ⩽ 0.1. A 50 C rate optimized microbattery based on the interdigitated bicontinuous architecture showed a ∼ 10 X improvement in power performance over the experimental batteries, indicating that significant improvements in power performance are possible with advances in microbattery design and fabrication. The 1-D electrochemical model and design parameters provide a new tool for developing novel material architectures, nano composites, and full battery cells with improved power performance, which is critical for the maximum exploitation of energy storage technologies in many applications. In addition, the performance limitations and design parameters developed in this work are based on the same physical models used for macroscopic batteries; therefore, these design principles might be applicable for improving the performance of macroscopic batteries.

List of Symbols

1-D	One dimensional
A	Area
b	Interconnect diameter
c	Concentration
Crate	C rate of discharge
D	Diffusion coefficient
Dc	Polystyrene diameter corresponding to to
DMC	Dimethyl carbonate
EC	Ethylene carbonate
F	Faraday constant [96,485.3365 A s mol−1]
FCC	Face centered cubic
h, h2	Spherical cap height
H	Electrode height
i,I	Current
k	Electrode conductivity
l	Length
L	Electrode length
n	Shape factor
OCV	Open circuit voltage
P	Electrode pitch
PC	Propylene carbonate
PITT	Potentiostatic intermittent titration technique
PS	Polystyrene
	Depletion rate (in electrolyte)
R	Polystyrene radius, or resistance
ρ	Resistivity
SEI	Solid electrolyte interphase
SEM	Scanning electron microscopy
t	Active material thickness
τ	Time
t+	Transference number
V	Voltage or volume
W	Electrode width

Greek

Subscripts

Superscripts

This research was supported by the National Science Foundation Engineering Research Center for Power Optimization of Electro Thermal Systems (POETS) with cooperative agreement EEC-1449548 and in part by the Department of Energy Office of Science Graduate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under contract no. DE-AC05-06OR23100.