I. Rigid body penetration into brittle materials. II. Phase change effect on shock wave propagation

Part I.

We have developed a technique for measuring the depth time history of rigid body penetration into brittle materials (hard rocks and concretes) under a deceleration of ~ 10⁵ g. The technique includes bar-coded projectile, sabot-projectile separation, detection and recording systems. Because the technique can give very dense data on penetration depth time history, penetration velocity can be deduced. Error analysis shows that the technique has a small intrinsic error of ~ 3-4 % in time during penetration, and 0.3 to 0.7 mm in penetration depth. A series of 4140 steel projectile penetration into G-mixture mortar targets have been conducted using the Caltech 40 mm gas/ powder gun in the velocity range of 100 to 500 m/s.

We report, for the first time, the whole depth-time history of rigid body penetration into brittle materials (the G-mixture mortar) under 10⁵ g deceleration. Based on the experimental results, including penetration depth time history, damage of recovered target and projectile materials and theoretical analysis, we find:

1. Target materials are damaged via compacting in the region in front of a projectile and via brittle radial and lateral crack propagation in the region surrounding the penetration path. The results suggest that expected cracks in front of penetrators may be stopped by a comminuted region that is induced by wave propagation. Aggregate erosion on the projectile lateral surface is < 20% of the final penetration depth. This result suggests that the effect of lateral friction on the penetration process can be ignored.

2. Final penetration depth, P_max, is linearly scaled with initial projectile energy per unit cross-section area, e_s , when targets are intact after impact. Based on the experimental data on the mortar targets, the relation is P_max(mm) 1.15e_s (J/mm² ) + 16.39.

3. Estimation of the energy needed to create an unit penetration volume suggests that the average pressure acting on the target material during penetration is ~ 10 to 20 times higher than the unconfined strength of target materials under quasi-static loading, and 3 to 4 times higher than the possible highest pressure due to friction and material strength and its rate dependence. In addition, the experimental data show that the interaction between cracks and the target free surface significantly affects the penetration process.

4. Based on the fact that the penetration duration, t_max, increases slowly with e_s and does not depend on projectile radius approximately, the dependence of t_max on projectile length is suggested to be described by t_max(μs) = 2.08e_s (J/mm² + 349.0 x m/(πR²), in which m is the projectile mass in grams and R is the projectile radius in mm. The prediction from this relation is in reasonable agreement with the experimental data for different projectile lengths.

5. Deduced penetration velocity time histories suggest that whole penetration history is divided into three stages: (1) An initial stage in which the projectile velocity change is small due to very small contact area between the projectile and target materials; (2) A steady penetration stage in which projectile velocity continues to decrease smoothly; (3) A penetration stop stage in which projectile deceleration jumps up when velocities are close to a critical value of ~ 35 m/s.

6. Deduced averaged deceleration, a, in the steady penetration stage for projectiles with same dimensions is found to be a(g) = 192.4v + 1.89 x 10⁴, where v is initial projectile velocity in m/s. The average pressure acting on target materials during penetration is estimated to be very comparable to shock wave pressure.

7. A similarity of penetration process is found to be described by a relation between normalized penetration depth, P/P_max, and normalized penetration time, t/t_max, as P/P_max = f(t/t_max, where f is a function of t/t_max. After f(t/t_max is determined using experimental data for projectiles with 150 mm length, the penetration depth time history for projectiles with 100 mm length predicted by this relation is in good agreement with experimental data. This similarity also predicts that average deceleration increases with decreasing projectile length, that is verified by the experimental data.

8. Based on the penetration process analysis and the present data, a first principle model for rigid body penetration is suggested. The model incorporates the models for contact area between projectile and target materials, friction coefficient, penetration stop criterion, and normal stress on the projectile surface. The most important assumptions used in the model are: (1) The penetration process can be treated as a series of impact events, therefore, pressure normal to projectile surface is estimated using the Hugoniot relation of target material; (2) The necessary condition for penetration is that the pressure acting on target materials is not lower than the Hugoniot elastic limit; (3) The friction force on projectile lateral surface can be ignored due to cavitation during penetration. All the parameters involved in the model are determined based on independent experimental data. The penetration depth time histories predicted from the model are in good agreement with the experimental data.

9. Based on planar impact and previous quasi-static experimental data, the strain rate dependence of the mortar compressive strength is described by σ_f/σ⁰_f = exp(0.0905(log(έ/έ_0) ^1.14, in the strain rate range of 10^-7/s to 10³/s (σ⁰_f and έ are reference compressive strength and strain rate, respectively). The non-dispersive Hugoniot elastic wave in the G-mixture has an amplitude of ~ 0.14 GPa and a velocity of ~ 4.3 km/s.

Part II.

Stress wave profiles in vitreous GeO₂ were measured using piezoresistance gauges in the pressure range of 5 to 18 GPa under planar plate and spherical projectile impact. Experimental data show that the response of vitreous GeO₂ to planar shock loading can be divided into three stages: (1) A ramp elastic precursor has peak amplitude of 4 GPa and peak particle velocity of 333 m/s. Wave velocity decreases from initial longitudinal elastic wave velocity of 3.5 km/s to 2.9 km/s at 4 GPa; (2) A ramp wave with amplitude of 2.11 GPa follows the precursor when peak loading pressure is 8.4 GPa. Wave velocity drops to the value below bulk wave velocity in this stage; (3) A shock wave achieving final shock state forms when peak pressure is > 6 GPa. The Hugoniot relation is D = 0.917 + 1.711u (km/s) using present data and the data of Jackson and Ahrens [1979] when shock wave pressure is between 6 and 40 GPa for ρ₀ = 3.655 gj cm³ . Based on the present data, the phase change from 4-fold to 6-fold coordination of Ge⁺⁴ with O^-2 in vitreous GeO₂ occurs in the pressure range of 4 to 15 ± 1 GPa under planar shock loading. Comparison of the shock loading data for fused SiO₂ to that on vitreous GeO₂ demonstrates that transformation to the rutile structure in both media are similar. The Hugoniots of vitreous GeO₂ and fused SiO₂ are found to coincide approximately if pressure in fused SiO₂ is scaled by the ratio of fused SiO₂to vitreous GeO₂ density. This result, as well as the same structure, provides the basis for considering vitreous Ge0₂ as an analogous material to fused SiO₂ under shock loading. Experimental results from the spherical projectile impact demonstrate: (1) The supported elastic shock in fused SiO₂ decays less rapidly than a linear elastic wave when elastic wave stress amplitude is higher than 4 GPa. The supported elastic shock in vitreous GeO₂ decays faster than a linear elastic wave; (2) In vitreous GeO₂ , unsupported shock waves decays with peak pressure in the phase transition range (4-15 GPa) with propagation distance, x, as α 1/x^-3.35 , close to the prediction of Chen et al. [1998]. Based on a simple analysis on spherical wave propagation, we find that the different decay rates of a spherical elastic wave in fused SiO₂ and vitreous GeO₂ is predictable on the base of the compressibility variation with stress under one-dimensional strain condition in the two materials.

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.

Enhanced Thermoelectric Performance at the Superionic Phase Transitions of Mixed Ion-Electron Conducting Materials

The quality of a thermoelectric material is judged by the size of its temperature de- pendent thermoeletric-figure-of-merit (zT ). Superionic materials, particularly Zn₄Sb₃ and Cu₂Se, are of current interest for the high zT and low thermal conductivity of their disordered, superionic phase. In this work it is reported that the super-ionic materials Ag₂Se, Cu₂Se and Cu_1.97Ag_0.03Se show enhanced zT in their ordered, normal ion-conducting phases. The zT of Ag₂Se is increased by 30% in its ordered phase as compared to its disordered phase, as measured just below and above its first order phase transition. The zT ’s of Cu₂Se and Cu_1.97Ag_0.03Se both increase by more than 100% over a 30 K temperatures range just below their super-ionic phase transitions. The peak zT of Cu₂Se is 0.7 at 406 K and of Cu_1.97Ag_0.03Se is 1.0 at 400 K. In all three materials these enhancements are due to anomalous increases in their Seebeck coefficients, beyond that predicted by carrier concentration measurements and band structure modeling. As the Seebeck coefficient is the entropy transported per carrier, this suggests that there is an additional quantity of entropy co-transported with charge carriers. Such co-transport has been previously observed via co-transport of vibrational entropy in bipolaron conductors and spin-state entropy in Na_xCo₂O₄. The correlation of the temperature profile of the increases in each material with the nature of their phase transitions indicates that the entropy is associated with the thermodynamcis of ion-ordering. This suggests a new mechanism by which high thermoelectric performance may be understood and engineered.

Phase behavior of complex superprotonic solid acids

Superprotonic phase transitions and thermal behaviors of three complex solid acid systems are presented, namely Rb3H(SO4)2-RbHSO4 system, Rb3H(SeO4)2-Cs3H(SeO4)2 solid solution system, and Cs6(H2SO4)3(H1.5PO4)4. These material systems present a rich set of phase transition characteristics that set them apart from other, simpler solid acids. A.C. impedance spectroscopy, high-temperature X-ray powder diffraction, and thermal analysis, as well as other characterization techniques, were employed to investigate the phase behavior of these systems.

Rb3H(SO4)2 is an atypical member of the M3H(XO4)2 class of compounds (M = alkali metal or NH4+ and X = S or Se) in that a transition to a high-conductivity state involves disproportionation into two phases rather than a simple polymorphic transition [1]. In the present work, investigations of the Rb3H(SO4)2-RbHSO4 system have revealed the disproportionation products to be Rb2SO4 and the previously unknown compound Rb5H3(SO4)4. The new compound becomes stable at a temperature between 25 and 140 °C and is isostructural to a recently reported trigonal phase with space group P3̅m of Cs5H3(SO4)4 [2]. At 185 °C the compound undergoes an apparently polymorphic transformation with a heat of transition of 23.8 kJ/mol and a slight additional increase in conductivity.

The compounds Rb3H(SeO4)2 and Cs3H(SeO4)2, though not isomorphous at ambient temperatures, are quintessential examples of superprotonic materials. Both adopt monoclinic structures at ambient temperatures and ultimately transform to a trigonal (R3̅m) superprotonic structure at slightly elevated temperatures, 178 and 183 °C, respectively. The compounds are completely miscible above the superprotonic transition and show extensive solubility below it. Beyond a careful determination of the phase boundaries, we find a remarkable 40-fold increase in the superprotonic conductivity in intermediate compositions rich in Rb as compared to either end-member.

The compound Cs6(H2SO4)3(H1.5PO4)4 is unusual amongst solid acid compounds in that it has a complex cubic structure at ambient temperature and apparently transforms to a simpler cubic structure of the CsCl-type (isostructural with CsH2PO4) at its transition temperature of 100-120 °C [3]. Here it is found that, depending on the level of humidification, the superprotonic transition of this material is superimposed with a decomposition reaction, which involves both exsolution of (liquid) acid and loss of H2O. This reaction can be suppressed by application of sufficiently high humidity, in which case Cs6(H2SO4)3(H1.5PO4)4 undergoes a true superprotonic transition. It is proposed that, under conditions of low humidity, the decomposition/dehydration reaction transforms the compound to Cs6(H2-0.5xSO4)3(H1.5PO4)4-x, also of the CsCl structure type at the temperatures of interest, but with a smaller unit cell. With increasing temperature, the decomposition/dehydration proceeds to greater and greater extent and unit cell of the solid phase decreases. This is identified to be the source of the apparent negative thermal expansion behavior.

References

[1] L.A. Cowan, R.M. Morcos, N. Hatada, A. Navrotsky, S.M. Haile, Solid State Ionics 179 (2008) (9-10) 305.

[2] M. Sakashita, H. Fujihisa, K.I. Suzuki, S. Hayashi, K. Honda, Solid State Ionics 178 (2007) (21-22) 1262.

[3] C.R.I. Chisholm, Superprotonic Phase Transitions in Solid Acids: Parameters affecting the presence and stability of superprotonic transitions in the MHnXO4 family of compounds (X=S, Se, P, As; M=Li, Na, K, NH4, Rb, Cs), Materials Science, California Institute of Technology, Pasadena, California (2003).

Interplay of martensitic phase transformation and plastic slip in polycrystals

Inspired by key experimental and analytical results regarding Shape Memory Alloys (SMAs), we propose a modelling framework to explore the interplay between martensitic phase transformations and plastic slip in polycrystalline materials, with an eye towards computational efficiency. The resulting framework uses a convexified potential for the internal energy density to capture the stored energy associated with transformation at the meso-scale, and introduces kinetic potentials to govern the evolution of transformation and plastic slip. The framework is novel in the way it treats plasticity on par with transformation.

We implement the framework in the setting of anti-plane shear, using a staggered implicit/explict update: we first use a Fast-Fourier Transform (FFT) solver based on an Augmented Lagrangian formulation to implicitly solve for the full-field displacements of a simulated polycrystal, then explicitly update the volume fraction of martensite and plastic slip using their respective stick-slip type kinetic laws. We observe that, even in this simple setting with an idealized material comprising four martensitic variants and four slip systems, the model recovers a rich variety of SMA type behaviors. We use this model to gain insight into the isothermal behavior of stress-stabilized martensite, looking at the effects of the relative plastic yield strength, the memory of deformation history under non-proportional loading, and several others.

We extend the framework to the generalized 3-D setting, for which the convexified potential is a lower bound on the actual internal energy, and show that the fully implicit discrete time formulation of the framework is governed by a variational principle for mechanical equilibrium. We further propose an extension of the method to finite deformations via an exponential mapping. We implement the generalized framework using an existing Optimal Transport Mesh-free (OTM) solver. We then model the $\alpha$--$\gamma$ and $\alpha$--$\varepsilon$ transformations in pure iron, with an initial attempt in the latter to account for twinning in the parent phase. We demonstrate the scalability of the framework to large scale computing by simulating Taylor impact experiments, observing nearly linear (ideal) speed-up through 256 MPI tasks. Finally, we present preliminary results of a simulated Split-Hopkinson Pressure Bar (SHPB) experiment using the $\alpha$--$\varepsilon$ model.

Synthesis and characterization of high-spin organic materials: prototypes for the polaronic ferromagnet

The design, synthesis and magnetic characterization of thiophene-based models for the polaronic ferromagnet are described. Synthetic strategies employing Wittig and Suzuki coupling were employed to produce polymers with extended π-systems. Oxidative doping using AsF_5 or I_2 produces radical cations (polarons) that are stable at room temperature. Magnetic characterization of the doped polymers, using SQUID-based magnetometry, indicates that in several instances ferromagnetic coupling of polarons occurs along the polymer chain. An investigation of the influence of polaron stability and delocalization on the magnitude of ferromagnetic coupling is pursued. A lower limit for mild, solution phase I_2 doping is established. A comparison of the variable temperature data of various polymers reveals that deleterious antiferromagnetic interactions are relatively insensitive to spin concentration, doping protocols or spin state. Comparison of the various polymers reveals useful design principles and suggests new directions for the development of magnetic organic materials. Novel strategies for solubilizing neutral polymeric materials in polar solvents are investigated.

The incorporation of stable bipyridinium spin-containing units into a polymeric high-spin array is explored. Preliminary results suggest that substituted diquat derivatives may serve as stable spin-containing units for the polaronic ferromagnet and are amenable to electrochemical doping. Synthetic efforts to prepare high-spin polymeric materials using viologens as a spin source have been unsuccessful.

A fundamental approach to phase noise reduction in hybrid Si/III-V lasers

Spontaneous emission into the lasing mode fundamentally limits laser linewidths. Reducing cavity losses provides two benefits to linewidth: (1) fewer excited carriers are needed to reach threshold, resulting in less phase-corrupting spontaneous emission into the laser mode, and (2) more photons are stored in the laser cavity, such that each individual spontaneous emission event disturbs the phase of the field less. Strong optical absorption in III-V materials causes high losses, preventing currently-available semiconductor lasers from achieving ultra-narrow linewidths. This absorption is a natural consequence of the compromise between efficient electrical and efficient optical performance in a semiconductor laser. Some of the III-V layers must be heavily doped in order to funnel excited carriers into the active region, which has the side effect of making the material strongly absorbing.

This thesis presents a new technique, called modal engineering, to remove modal energy from the lossy region and store it in an adjacent low-loss material, thereby reducing overall optical absorption. A quantum mechanical analysis of modal engineering shows that modal gain and spontaneous emission rate into the laser mode are both proportional to the normalized intensity of that mode at the active region. If optical absorption near the active region dominates the total losses of the laser cavity, shifting modal energy from the lossy region to the low-loss region will reduce modal gain, total loss, and the spontaneous emission rate into the mode by the same factor, so that linewidth decreases while the threshold inversion remains constant. The total spontaneous emission rate into all other modes is unchanged.

Modal engineering is demonstrated using the Si/III-V platform, in which light is generated in the III-V material and stored in the low-loss silicon material. The silicon is patterned as a high-Q resonator to minimize all sources of loss. Fabricated lasers employing modal engineering to concentrate light in silicon demonstrate linewidths at least 5 times smaller than lasers without modal engineering at the same pump level above threshold, while maintaining the same thresholds.

Advanced density functional theory methods for materials science

In this work we chiefly deal with two broad classes of problems in computational materials science, determining the doping mechanism in a semiconductor and developing an extreme condition equation of state. While solving certain aspects of these questions is well-trodden ground, both require extending the reach of existing methods to fully answer them. Here we choose to build upon the framework of density functional theory (DFT) which provides an efficient means to investigate a system from a quantum mechanics description.

Zinc Phosphide (Zn₃P₂) could be the basis for cheap and highly efficient solar cells. Its use in this regard is limited by the difficulty in n-type doping the material. In an effort to understand the mechanism behind this, the energetics and electronic structure of intrinsic point defects in zinc phosphide are studied using generalized Kohn-Sham theory and utilizing the Heyd, Scuseria, and Ernzerhof (HSE) hybrid functional for exchange and correlation. Novel 'perturbation extrapolation' is utilized to extend the use of the computationally expensive HSE functional to this large-scale defect system. According to calculations, the formation energy of charged phosphorus interstitial defects are very low in n-type Zn₃P₂ and act as 'electron sinks', nullifying the desired doping and lowering the fermi-level back towards the p-type regime. Going forward, this insight provides clues to fabricating useful zinc phosphide based devices. In addition, the methodology developed for this work can be applied to further doping studies in other systems.

Accurate determination of high pressure and temperature equations of state is fundamental in a variety of fields. However, it is often very difficult to cover a wide range of temperatures and pressures in an laboratory setting. Here we develop methods to determine a multi-phase equation of state for Ta through computation. The typical means of investigating thermodynamic properties is via ’classical’ molecular dynamics where the atomic motion is calculated from Newtonian mechanics with the electronic effects abstracted away into an interatomic potential function. For our purposes, a ’first principles’ approach such as DFT is useful as a classical potential is typically valid for only a portion of the phase diagram (i.e. whatever part it has been fit to). Furthermore, for extremes of temperature and pressure quantum effects become critical to accurately capture an equation of state and are very hard to capture in even complex model potentials. This requires extending the inherently zero temperature DFT to predict the finite temperature response of the system. Statistical modelling and thermodynamic integration is used to extend our results over all phases, as well as phase-coexistence regions which are at the limits of typical DFT validity. We deliver the most comprehensive and accurate equation of state that has been done for Ta. This work also lends insights that can be applied to further equation of state work in many other materials.