Fracture of materials undergoing solid-solid phase transformation

A large number of technologically important materials undergo solid-solid phase transformations. Examples range from ferroelectrics (transducers and memory devices), zirconia (Thermal Barrier Coatings) to nickel superalloys and (lithium) iron phosphate (Li-ion batteries). These transformations involve a change in the crystal structure either through diffusion of species or local rearrangement of atoms. This change of crystal structure leads to a macroscopic change of shape or volume or both and results in internal stresses during the transformation. In certain situations this stress field gives rise to cracks (tin, iron phosphate etc.) which continue to propagate as the transformation front traverses the material. In other materials the transformation modifies the stress field around cracks and effects crack growth behavior (zirconia, ferroelectrics). These observations serve as our motivation to study cracks in solids undergoing phase transformations. Understanding these effects will help in improving the mechanical reliability of the devices employing these materials.

In this thesis we present work on two problems concerning the interplay between cracks and phase transformations. First, we consider the directional growth of a set of parallel edge cracks due to a solid-solid transformation. We conclude from our analysis that phase transformations can lead to formation of parallel edge cracks when the transformation strain satisfies certain conditions and the resulting cracks grow all the way till their tips cross over the phase boundary. Moreover the cracks continue to grow as the phase boundary traverses into the interior of the body at a uniform spacing without any instabilities. There exists an optimal value for the spacing between the cracks. We ascertain these conclusion by performing numerical simulations using finite elements.

Second, we model the effect of the semiconducting nature and dopants on cracks in ferroelectric perovskite materials, particularly barium titanate. Traditional approaches to model fracture in these materials have treated them as insulators. In reality, they are wide bandgap semiconductors with oxygen vacancies and trace impurities acting as dopants. We incorporate the space charge arising due the semiconducting effect and dopant ionization in a phase field model for the ferroelectric. We derive the governing equations by invoking the dissipation inequality over a ferroelectric domain containing a crack. This approach also yields the driving force acting on the crack. Our phase field simulations of polarization domain evolution around a crack show the accumulation of electronic charge on the crack surface making it more permeable than was previously believed so, as seen in recent experiments. We also discuss the effect the space charge has on domain formation and the crack driving force.

Dendrites Inhibition in Rechargeable Lithium Metal Batteries

The specific high energy and power capacities of rechargeable lithium metal (Li⁰) batteries are ideally suited to portable devices and are valuable as storage units for intermittent renewable energy sources. Lithium, the lightest and most electropositive metal, would be the optimal anode material for rechargeable batteries if it were not for the fact that such devices fail unexpectedly by short-circuiting via the dendrites that grow across electrodes upon recharging. This phenomenon poses a major safety issue because it triggers a series of adverse events that start with overheating, potentially followed by the thermal decomposition and ultimately the ignition of the organic solvents used in such devices.

In this thesis, we developed experimental platform for monitoring and quantifying the dendrite populations grown in a Li battery prototype upon charging under various conditions. We explored the effects of pulse charging in the kHz range and temperature on dendrite growth, and also on loss capacity into detached “dead” lithium particles.

Simultaneously, we developed a computational framework for understanding the dynamics of dendrite propagation. The coarse-grained Monte Carlo model assisted us in the interpretation of pulsing experiments, whereas MD calculations provided insights into the mechanism of dendrites thermal relaxation. We also developed a computational framework for measuring the dead lithium crystals from the experimental images.

Stability of Electrode-Electrolyte Interfaces during Charging in Lithium Batteries

In this thesis we study the growth of a Li electrode-electrolyte interface in the presence of an elastic prestress. In particular, we focus our interest on Li-air batteries with a solid electrolyte, LIPON, which is a new type of secondary or rechargeable battery. Theoretical studies and experimental evidence show that during the process of charging the battery the replated lithium adds unevenly to the electrode surface. This phenomenon eventually leads to dendrite formation as the battery is charged and discharged numerous times. In order to suppress or alleviate this deleterious effect of dendrite growth, we put forth a study based on a linear stability analysis. Taking into account all the mechanisms of mass transport and interfacial kinetics, we model the evolution of the interface. We find that, in the absence of stress, the stability of a planar interface depends on interfacial diffusion properties and interfacial energy. Specifically, if Herring-Mullins capillarity-driven interfacial diffusion is accounted for, interfaces are unstable against all perturbations of wavenumber larger than a critical value. We find that the effect of an elastic prestress is always to stabilize planar interfacial growth by increasing the critical wavenumber for instability. A parametric study results in quantifying the extent of the prestress stabilization in a manner that can potentially be used in the design of Li-air batteries. Moreover, employing the theory of finite differences we numerically solve the equation that describes the evolution of the surface profile and present visualization results of the surface evolution by time. Lastly, numerical simulations performed in a commercial finite element software validate the theoretical formulation of the interfacial elastic energy change with respect to the planar interface.

Topology optimization of silicon anode structures for lithium-ion battery applications

This thesis presents a topology optimization methodology for the systematic design of optimal multifunctional silicon anode structures in lithium-ion batteries. In order to develop next generation high performance lithium-ion batteries, key design challenges relating to the silicon anode structure must be addressed, namely the lithiation-induced mechanical degradation and the low intrinsic electrical conductivity of silicon. As such, this work considers two design objectives of minimum compliance under design dependent volume expansion, and maximum electrical conduction through the structure, both of which are subject to a constraint on material volume. Density-based topology optimization methods are employed in conjunction with regularization techniques, a continuation scheme, and mathematical programming methods. The objectives are first considered individually, during which the iteration history, mesh independence, and influence of prescribed volume fraction and minimum length scale are investigated. The methodology is subsequently extended to a bi-objective formulation to simultaneously address both the compliance and conduction design criteria. A weighting method is used to derive the Pareto fronts, which demonstrate a clear trade-off between the competing design objectives. Furthermore, a systematic parameter study is undertaken to determine the influence of the prescribed volume fraction and minimum length scale on the optimal combined topologies. The developments presented in this work provide a foundation for the informed design and development of silicon anode structures for high performance lithium-ion batteries.

I. Fundamental limitations in microelectronics. II. Power Schottky diode design and comparison with the junction diode. III. Permittivity of strontium titanate

Part I

The physical phenomena which will ultimately limit the packing density of planar bipolar and MOS integrated circuits are examined. The maximum packing density is obtained by minimizing the supply voltage and the size of the devices. The minimum size of a bipolar transistor is determined by junction breakdown, punch-through and doping fluctuations. The minimum size of a MOS transistor is determined by gate oxide breakdown and drain-source punch-through. The packing density of fully active bipolar or static non-complementary MOS circuits becomes limited by power dissipation. The packing density of circuits which are not fully active such as read-only memories, becomes limited by the area occupied by the devices, and the frequency is limited by the circuit time constants and by metal migration. The packing density of fully active dynamic or complementary MOS circuits is limited by the area occupied by the devices, and the frequency is limited by power dissipation and metal migration. It is concluded that read-only memories will reach approximately the same performance and packing density with MOS and bipolar technologies, while fully active circuits will reach the highest levels of integration with dynamic MOS or complementary MOS technologies.

Part II

Because the Schottky diode is a one-carrier device, it has both advantages and disadvantages with respect to the junction diode which is a two-carrier device. The advantage is that there are practically no excess minority carriers which must be swept out before the diode blocks current in the reverse direction, i.e. a much faster recovery time. The disadvantage of the Schottky diode is that for a high voltage device it is not possible to use conductivity modulation as in the p i n diode; since charge carriers are of one sign, no charge cancellation can occur and current becomes space charge limited. The Schottky diode design is developed in Section 2 and the characteristics of an optimally designed silicon Schottky diode are summarized in Fig. 9. Design criteria and quantitative comparison of junction and Schottky diodes is given in Table 1 and Fig. 10. Although somewhat approximate, the treatment allows a systematic quantitative comparison of the devices for any given application.

Part III

We interpret measurements of permittivity of perovskite strontium titanate as a function of orientation, temperature, electric field and frequency performed by Dr. Richard Neville. The free energy of the crystal is calculated as a function of polarization. The Curie-Weiss law and the LST relation are verified. A generalized LST relation is used to calculate the permittivity of strontium titanate from zero to optic frequencies. Two active optic modes are important. The lower frequency mode is attributed mainly to motion of the strontium ions with respect to the rest of the lattice, while the higher frequency active mode is attributed to motion of the titanium ions with respect to the oxygen lattice. An anomalous resonance which multi-domain strontium titanate crystals exhibit below 65°K is described and a plausible mechanism which explains the phenomenon is presented.