Extracting material response from simple mechanical tests on hardening-softening-hardening viscoplastic solids

Compliant foams are usually characterized by a wide range of desirable mechanical properties. These properties include viscoelasticity at different temperatures, energy absorption, recoverability under cyclic loading, impact resistance, and thermal, electrical, acoustic and radiation-resistance. Some foams contain nano-sized features and are used in small-scale devices. This implies that the characteristic dimensions of foams span multiple length scales, rendering modeling their mechanical properties difficult. Continuum mechanics-based models capture some salient experimental features like the linear elastic regime, followed by non-linear plateau stress regime. However, they lack mesostructural physical details. This makes them incapable of accurately predicting local peaks in stress and strain distributions, which significantly affect the deformation paths. Atomistic methods are capable of capturing the physical origins of deformation at smaller scales, but suffer from impractical computational intensity. Capturing deformation at the so-called meso-scale, which is capable of describing the phenomenon at a continuum level, but with some physical insights, requires developing new theoretical approaches.

A fundamental question that motivates the modeling of foams is ‘how to extract the intrinsic material response from simple mechanical test data, such as stress vs. strain response?’ A 3D model was developed to simulate the mechanical response of foam-type materials. The novelty of this model includes unique features such as the hardening-softening-hardening material response, strain rate-dependence, and plastically compressible solids with plastic non-normality. Suggestive links from atomistic simulations of foams were borrowed to formulate a physically informed hardening material input function. Motivated by a model that qualitatively captured the response of foam-type vertically aligned carbon nanotube (VACNT) pillars under uniaxial compression [2011,“Analysis of Uniaxial Compression of Vertically Aligned Carbon Nanotubes,” J. Mech.Phys. Solids, 59, pp. 2227–2237, Erratum 60, 1753–1756 (2012)], the property space exploration was advanced to three types of simple mechanical tests: 1) uniaxial compression, 2) uniaxial tension, and 3) nanoindentation with a conical and a flat-punch tip. The simulations attempt to explain some of the salient features in experimental data, like
1) The initial linear elastic response.
2) One or more nonlinear instabilities, yielding, and hardening.

The model-inherent relationships between the material properties and the overall stress-strain behavior were validated against the available experimental data. The material properties include the gradient in stiffness along the height, plastic and elastic compressibility, and hardening. Each of these tests was evaluated in terms of their efficiency in extracting material properties. The uniaxial simulation results proved to be a combination of structural and material influences. Out of all deformation paths, flat-punch indentation proved to be superior since it is the most sensitive in capturing the material properties.

Design, Fabrication, and Mechanical Property Analysis of 3D Nanoarchitected Materials

Recent developments in micro- and nanoscale 3D fabrication techniques have enabled the creation of materials with a controllable nanoarchitecture that can have structural features spanning 5 orders of magnitude from tens of nanometers to millimeters. These fabrication methods in conjunction with nanomaterial processing techniques permit a nearly unbounded design space through which new combinations of nanomaterials and architecture can be realized. In the course of this work, we designed, fabricated, and mechanically analyzed a wide range of nanoarchitected materials in the form of nanolattices made from polymer, composite, and hollow ceramic beams. Using a combination of two-photon lithography and atomic layer deposition, we fabricated samples with periodic and hierarchical architectures spanning densities over 4 orders of magnitude from ρ=0.3-300kg/m³ and with features as small as 5nm. Uniaxial compression and cyclic loading tests performed on different nanolattice topologies revealed a range of novel mechanical properties: the constituent nanoceramics used here have size-enhanced strengths that approach the theoretical limit of materials strength; hollow aluminum oxide (Al₂O₃) nanolattices exhibited ductile-like deformation and recovered nearly completely after compression to 50% strain when their wall thicknesses were reduced below 20nm due to the activation of shell buckling; hierarchical nanolattices exhibited enhanced recoverability and a near linear scaling of strength and stiffness with relative density, with E∝ρ^1.04 and σy∝ρ^1.17 for hollow Al₂O₃ samples; periodic rigid and non-rigid nanolattice topologies were tested and showed a nearly uniform scaling of strength and stiffness with relative density, marking a significant deviation from traditional theories on “bending” and “stretching” dominated cellular solids; and the mechanical behavior across all topologies was highly tunable and was observed to strongly correlate with the slenderness λ and the wall thickness-to-radius ratio t/a of the beams. These results demonstrate the potential of nanoarchitected materials to create new highly tunable mechanical metamaterials with previously unattainable properties.

Parylene-C as a new piezoelectric material

The goal of this thesis is to develop a proper microelectromechanical systems (MEMS) process to manufacture piezoelectric Parylene-C (PA-C), which is famous for its chemical inertness, mechanical and thermal properties and electrical insulation. Furthermore, piezoelectric PA-C is used to build miniature, inexpensive, non-biased piezoelectric microphones.

These piezoelectric PA-C MEMS microphones are to be used in any application where a conventional piezoelectric and electret microphone can be used, such as in cell phones and hearing aids. However, they have the advantage of a simplified fabrication process compared with existing technology. In addition, as a piezoelectric polymer, PA-C has varieties of applications due to its low dielectric constant, low elastic stiffness, low density, high voltage sensitivity, high temperature stability and low acoustic and mechanical impedance. Furthermore, PA-C is an FDA approved biocompatible material and is able to maintain operate at a high temperature.

To accomplish piezoelectric PA-C, a MEMS-compatible poling technology has been developed. The PA-C film is poled by applying electrical field during heating. The piezoelectric coefficient, -3.75pC/N, is obtained without film stretching.

The millimeter-scale piezoelectric PA-C microphone is fabricated with an in-plane spiral arrangement of two electrodes. The dynamic range is from less than 30 dB to above 110 dB SPL (referenced 20 µPa) and the open-circuit sensitivities are from 0.001 – 0.11 mV/Pa over a frequency range of 1 - 10 kHz. The total harmonic distortion of the device is less than 20% at 110 dB SPL and 1 kHz.

Dynamics of cell–matrix mechanical interactions in three dimensions

The forces cells apply to their surroundings control biological processes such as growth, adhesion, development, and migration. In the past 20 years, a number of experimental techniques have been developed to measure such cell tractions. These approaches have primarily measured the tractions applied by cells to synthetic two-dimensional substrates, which do not mimic in vivo conditions for most cell types. Many cell types live in a fibrous three-dimensional (3D) matrix environment. While studying cell behavior in such 3D matrices will provide valuable insights for the mechanobiology and tissue engineering communities, no experimental approaches have yet measured cell tractions in a fibrous 3D matrix.

This thesis describes the development and application of an experimental technique for quantifying cellular forces in a natural 3D matrix. Cells and their surrounding matrix are imaged in three dimensions with high speed confocal microscopy. The cell-induced matrix displacements are computed from the 3D image volumes using digital volume correlation. The strain tensor in the 3D matrix is computed by differentiating the displacements, and the stress tensor is computed by applying a constitutive law. Finally, tractions applied by the cells to the matrix are computed directly from the stress tensor.

The 3D traction measurement approach is used to investigate how cells mechanically interact with the matrix in biologically relevant processes such as division and invasion. During division, a single mother cell undergoes a drastic morphological change to split into two daughter cells. In a 3D matrix, dividing cells apply tensile force to the matrix through thin, persistent extensions that in turn direct the orientation and location of the daughter cells. Cell invasion into a 3D matrix is the first step required for cell migration in three dimensions. During invasion, cells initially apply minimal tractions to the matrix as they extend thin protrusions into the matrix fiber network. The invading cells anchor themselves to the matrix using these protrusions, and subsequently pull on the matrix to propel themselves forward.

Lastly, this thesis describes a constitutive model for the 3D fibrous matrix that uses a finite element (FE) approach. The FE model simulates the fibrous microstructure of the matrix and matches the cell-induced matrix displacements observed experimentally using digital volume correlation. The model is applied to predict how cells mechanically sense one another in a 3D matrix. It is found that cell-induced matrix displacements localize along linear paths. These linear paths propagate over a long range through the fibrous matrix, and provide a mechanism for cell-cell signaling and mechanosensing. The FE model developed here has the potential to reveal the effects of matrix density, inhomogeneity, and anisotropy in signaling cell behavior through mechanotransduction.

Crystal Coulomb energies: I. Organic donor-acceptor complexes--a review. II. Classical Ewald calculations of the Coulomb binding energy of four donor-acceptor crystals and Wurster's blue perchlorate. III. A rapid-convergence quantum-mechanical formalism for crystal electronic energies

Chapter I

Theories for organic donor-acceptor (DA) complexes in solution and in the solid state are reviewed, and compared with the available experimental data. As shown by McConnell et al. (Proc. Natl. Acad. Sci. U.S., 53, 46-50 (1965)), the DA crystals fall into two classes, the holoionic class with a fully or almost fully ionic ground state, and the nonionic class with little or no ionic character. If the total lattice binding energy 2ε₁ (per DA pair) gained in ionizing a DA lattice exceeds the cost 2ε_o of ionizing each DA pair, ε₁ + ε_o less than 0, then the lattice is holoionic. The charge-transfer (CT) band in crystals and in solution can be explained, following Mulliken, by a second-order mixing of states, or by any theory that makes the CT transition strongly allowed, and yet due to a small change in the ground state of the non-interacting components D and A (or D⁺ and A^-). The magnetic properties of the DA crystals are discussed.

Chapter II

A computer program, EWALD, was written to calculate by the Ewald fast-convergence method the crystal Coulomb binding energy E_C due to classical monopole-monopole interactions for crystals of any symmetry. The precision of E_C values obtained is high: the uncertainties, estimated by the effect on E_C of changing the Ewald convergence parameter η, ranged from ± 0.00002 eV to ± 0.01 eV in the worst case. The charge distribution for organic ions was idealized as fractional point charges localized at the crystallographic atomic positions: these charges were chosen from available theoretical and experimental estimates. The uncertainty in E_C due to different charge distribution models is typically ± 0.1 eV (± 3%): thus, even the simple Hückel model can give decent results.

E_C for Wurster's Blue Perchl orate is -4.1 eV/molecule: the crystal is stable under the binding provided by direct Coulomb interactions. E_C for N-Methylphenazinium Tetracyanoquino- dimethanide is 0.1 eV: exchange Coulomb interactions, which cannot be estimated classically, must provide the necessary binding.

EWALD was also used to test the McConnell classification of DA crystals. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine: 7,7,8,8-Tetracyanoquinodimethan) E_C = -4.0 eV while 2ε_o = 4.6₅ eV: clearly, exchange forces must provide the balance. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine:para-Chloranil) E_C = -4.4 eV, while 2ε_o = 5.0 eV: again EC falls short of 2ε₁. As a Gedankenexperiment, two nonionic crystals were assumed to be ionized: for (1:1)-(Hexamethyl- benzene:para-Chloranil) E_C = -4.5 eV, 2ε_o = 6.6 eV; for (1:1)- (Napthalene:Tetracyanoethylene) E_C = -4.3 eV, 2ε_o = 6.5 eV. Thus, exchange energies in these nonionic crystals must not exceed 1 eV.

Chapter III

A rapid-convergence quantum-mechanical formalism is derived to calculate the electronic energy of an arbitrary molecular (or molecular-ion) crystal: this provides estimates of crystal binding energies which include the exchange Coulomb inter- actions. Previously obtained LCAO-MO wavefunctions for the isolated molecule(s) ("unit cell spin-orbitals") provide the starting-point. Bloch's theorem is used to construct "crystal spin-orbitals". Overlap between the unit cell orbitals localized in different unit cells is neglected, or is eliminated by Löwdin orthogonalization. Then simple formulas for the total kinetic energy Q^(XT)_λ, nuclear attraction [λ/λ]^XT, direct Coulomb [λλ/λ'λ']^XT and exchange Coulomb [λλ'/λ'λ]^XT integrals are obtained, and direct-space brute-force expansions in atomic wavefunctions are given. Fourier series are obtained for [λ/λ]^XT, [λλ/λ'λ']^XT, and [λλ/λ'λ]^XT with the help of the convolution theorem; the Fourier coefficients require the evaluation of Silverstone's two-center Fourier transform integrals. If the short-range interactions are calculated by brute-force integrations in direct space, and the long-range effects are summed in Fourier space, then rapid convergence is possible for [λ/λ]^XT, [λλ/λ'λ']^XT and [λλ'/λ'λ]^XT. This is achieved, as in the Ewald method, by modifying each atomic wavefunction by a "Gaussian convergence acceleration factor", and evaluating separately in direct and in Fourier space appropriate portions of [λ/λ]^XT, etc., where some of the portions contain the Gaussian factor.

Effect of pressure on mechanical behavior of filled elastomers

The Young's modulus, stress-strain curves, and failure properties of glass bead-filled EPDM vulcanizates were studied under superposed hydrostatic pressure. The glass bead-filled EPDM was employed as a representation of composite systems, and the hydrostatic pressure controls the filler-elastomer separation under deformation. This separation shows up as a volume change of the system, and its infuence is reflected in the mechanical behavior as a reinforcing effect of variable degree.

The strain energy stored in the composite system in simple tension was calculated by introducing a model which is described as a cylindrical block of elastomer with two half spheres of filler on each end with their centers on the axis of the cylinder. In the derivation of the strain energy, assumptions were made to obtain the strain distribution in the model, and strain energy-strain relation for the elastomer was also assumed. The derivation was carried out for the case of no filler-elastomer separation and was modified to include the case of filler-elastomer separation.

The resulting strain energy, as a function of stretch ratio and volume of the system, was used to obtain stress-strain curves and volume change-strain curves of composite systems under superposed hydrostatic pressure.

Changes in the force and the lateral dimension of a ring specimen were measured as it was stretched axially under a superposed hydrostatic pressure in order to calculate the mechanical properties mentioned above. A tensile tester was used which is capable of sealing the whole system to carry out a measurement under pressure. A thickness measuring device, based on the Hall effect, was built for the measurement of changes in the lateral dimension of a specimen.

The theoretical and experimental results of Young's modulus and stress-strain curves were compared and showed fairly good agreement.

The failure data were discussed in terms of failure surfaces, and it was concluded that a failure surface of the glass-bead-filled EPDM consists of two cones.

Nonlinear Effects in Granular Crystals with Broken Periodicity

When studying physical systems, it is common to make approximations: the contact interaction is linear, the crystal is periodic, the variations occurs slowly, the mass of a particle is constant with velocity, or the position of a particle is exactly known are just a few examples. These approximations help us simplify complex systems to make them more comprehensible while still demonstrating interesting physics. But what happens when these assumptions break down? This question becomes particularly interesting in the materials science community in designing new materials structures with exotic properties In this thesis, we study the mechanical response and dynamics in granular crystals, in which the approximation of linearity and infinite size break down. The system is inherently finite, and contact interaction can be tuned to access different nonlinear regimes. When the assumptions of linearity and perfect periodicity are no longer valid, a host of interesting physical phenomena presents itself. The advantage of using a granular crystal is in its experimental feasibility and its similarity to many other materials systems. This allows us to both leverage past experience in the condensed matter physics and materials science communities while also presenting results with implications beyond the narrower granular physics community. In addition, we bring tools from the nonlinear systems community to study the dynamics in finite lattices, where there are inherently more degrees of freedom. This approach leads to the major contributions of this thesis in broken periodic systems. We demonstrate the first defect mode whose spatial profile can be tuned from highly localized to completely delocalized by simply tuning an external parameter. Using the sensitive dynamics near bifurcation points, we present a completely new approach to modifying the incremental stiffness of a lattice to arbitrary values. We show how using nonlinear defect modes, the incremental stiffness can be tuned to anywhere in the force-displacement relation. Other contributions include demonstrating nonlinear breakdown of mechanical filters as a result of finite size, and the presents of frequency attenuation bands in essentially nonlinear materials. We finish by presenting two new energy harvesting systems based on our experience with instabilities in weakly nonlinear systems.

Effect of Microstructural Interfaces on the Mechanical Response of Crystalline Metallic Materials

Advances in nano-scale mechanical testing have brought about progress in the understanding of physical phenomena in materials and a measure of control in the fabrication of novel materials. In contrast to bulk materials that display size-invariant mechanical properties, sub-micron metallic samples show a critical dependence on sample size. The strength of nano-scale single crystalline metals is well-described by a power-law function, σαD^-n, where D is a critical sample size and n is a experimentally-fit positive exponent. This relationship is attributed to source-driven plasticity and demonstrates a strengthening as the decreasing sample size begins to limit the size and number of dislocation sources. A full understanding of this size-dependence is complicated by the presence of microstructural features such as interfaces that can compete with the dominant dislocation-based deformation mechanisms. In this thesis, the effects of microstructural features such as grain boundaries and anisotropic crystallinity on nano-scale metals are investigated through uniaxial compression testing. We find that nano-sized Cu covered by a hard coating displays a Bauschinger effect and the emergence of this behavior can be explained through a simple dislocation-based analytic model. Al nano-pillars containing a single vertically-oriented coincident site lattice grain boundary are found to show similar deformation to single-crystalline nano-pillars with slip traces passing through the grain boundary. With increasing tilt angle of the grain boundary from the pillar axis, we observe a transition from dislocation-dominated deformation to grain boundary sliding. Crystallites are observed to shear along the grain boundary and molecular dynamics simulations reveal a mechanism of atomic migration that accommodates boundary sliding. We conclude with an analysis of the effects of inherent crystal anisotropy and alloying on the mechanical behavior of the Mg alloy, AZ31. Through comparison to pure Mg, we show that the size effect dominates the strength of samples below 10 μm, that differences in the size effect between hexagonal slip systems is due to the inherent crystal anisotropy, suggesting that the fundamental mechanism of the size effect in these slip systems is the same.

The Mechanical Genome in Regulation and Infection

Biological information storage and retrieval is a dynamic process that requires the genome to undergo dramatic structural rearrangements. Recent advances in single-molecule techniques have allowed precise quantification of the nano-mechanical properties of DNA [1, 2], and direct in vivo observation of molecules in action [3]. In this work, we will examine elasticity in protein-mediated DNA looping, whose structural rearrangement is essential for transcriptional regulation in both prokaryotes and eukaryotes. We will look at hydrodynamics in the process of viral DNA ejection, which mediates information transfer and exchange and has prominent implications in evolution. As in the case of Kepler's laws of planetary motion leading to Newton's gravitational theory, and the allometric scaling laws in biology revealing the organizing principles of complex networks [4], experimental data collapse in these biological phenomena has guided much of our studies and urged us to find the underlying physical principles.

Quantum squeezing of motion in a mechanical resonator

Quantum mechanics places limits on the minimum energy of a harmonic oscillator via the ever-present "zero-point" fluctuations of the quantum ground state. Through squeezing, however, it is possible to decrease the noise of a single motional quadrature below the zero-point level as long as noise is added to the orthogonal quadrature. While squeezing below the quantum noise level was achieved decades ago with light, quantum squeezing of the motion of a mechanical resonator is a more difficult prospect due to the large thermal occupations of megahertz-frequency mechanical devices even at typical dilution refrigerator temperatures of ~ 10 mK.

Kronwald, Marquardt, and Clerk (2013) propose a method of squeezing a single quadrature of mechanical motion below the level of its zero-point fluctuations, even when the mechanics starts out with a large thermal occupation. The scheme operates under the framework of cavity optomechanics, where an optical or microwave cavity is coupled to the mechanics in order to control and read out the mechanical state. In the proposal, two pump tones are applied to the cavity, each detuned from the cavity resonance by the mechanical frequency. The pump tones establish and couple the mechanics to a squeezed reservoir, producing arbitrarily-large, steady-state squeezing of the mechanical motion. In this dissertation, I describe two experiments related to the implementation of this proposal in an electromechanical system. I also expand on the theory presented in Kronwald et. al. to include the effects of squeezing in the presence of classical microwave noise, and without assumptions of perfect alignment of the pump frequencies.

In the first experiment, we produce a squeezed thermal state using the method of Kronwald et. al.. We perform back-action evading measurements of the mechanical squeezed state in order to probe the noise in both quadratures of the mechanics. Using this method, we detect single-quadrature fluctuations at the level of 1.09 +/- 0.06 times the quantum zero-point motion.

In the second experiment, we measure the spectral noise of the microwave cavity in the presence of the squeezing tones and fit a full model to the spectrum in order to deduce a quadrature variance of 0.80 +/- 0.03 times the zero-point level. These measurements provide the first evidence of quantum squeezing of motion in a mechanical resonator.

Exploring the Kinetics of Domain Switching in Ferroelectrics for Structural Applications

The complex domain structure in ferroelectrics gives rise to electromechanical coupling, and its evolution (via domain switching) results in a time-dependent (i.e. viscoelastic) response. Although ferroelectrics are used in many technological applications, most do not attempt to exploit the viscoelastic response of ferroelectrics, mainly due to a lack of understanding and accurate models for their description and prediction. Thus, the aim of this thesis research is to gain better understanding of the influence of domain evolution in ferroelectrics on their dynamic mechanical response. There have been few studies on the viscoelastic properties of ferroelectrics, mainly due to a lack of experimental methods. Therefore, an apparatus and method called Broadband Electromechanical Spectroscopy (BES) was designed and built. BES allows for the simultaneous application of dynamic mechanical and electrical loading in a vacuum environment. Using BES, the dynamic stiffness and loss tangent in bending and torsion of a particular ferroelectric, viz. lead zirconate titanate (PZT), was characterized for different combinations of electrical and mechanical loading frequencies throughout the entire electric displacement hysteresis. Experimental results showed significant increases in loss tangent (by nearly an order of magnitude) and compliance during domain switching, which shows promise as a new approach to structural damping. A continuum model of the viscoelasticity of ferroelectrics was developed, which incorporates microstructural evolution via internal variables and associated kinetic relations. For the first time, through a new linearization process, the incremental dynamic stiffness and loss tangent of materials were computed throughout the entire electric displacement hysteresis for different combinations of mechanical and electrical loading frequencies. The model accurately captured experimental results. Using the understanding gained from the characterization and modeling of PZT, two applications of domain switching kinetics were explored by using Micro Fiber Composites (MFCs). Proofs of concept of set-and-hold actuation and structural damping using MFCs were demonstrated.

Fabrication, Characterization, And Deformation of 3D Structural Meta-Materials

Current technological advances in fabrication methods have provided pathways to creating architected structural meta-materials similar to those found in natural organisms that are structurally robust and lightweight, such as diatoms. Structural meta-materials are materials with mechanical properties that are determined by material properties at various length scales, which range from the material microstructure (nm) to the macro-scale architecture (μm – mm). It is now possible to exploit material size effect, which emerge at the nanometer length scale, as well as structural effects to tune the material properties and failure mechanisms of small-scale cellular solids, such as nanolattices. This work demonstrates the fabrication and mechanical properties of 3-dimensional hollow nanolattices in both tension and compression. Hollow gold nanolattices loaded in uniaxial compression demonstrate that strength and stiffness vary as a function of geometry and tube wall thickness. Structural effects were explored by increasing the unit cell angle from 30° to 60° while keeping all other parameters constant; material size effects were probed by varying the tube wall thickness, t, from 200nm to 635nm, at a constant relative density and grain size. In-situ uniaxial compression experiments reveal an order-of-magnitude increase in yield stress and modulus in nanolattices with greater lattice angles, and a 150% increase in the yield strength without a concomitant change in modulus in thicker-walled nanolattices for fixed lattice angles. These results imply that independent control of structural and material size effects enables tunability of mechanical properties of 3-dimensional architected meta-materials and highlight the importance of material, geometric, and microstructural effects in small-scale mechanics. This work also explores the flaw tolerance of 3D hollow-tube alumina kagome nanolattices with and without pre-fabricated notches, both in experiment and simulation. Experiments demonstrate that the hollow kagome nanolattices in uniaxial tension always fail at the same load when the ratio of notch length (a) to sample width (w) is no greater than 1/3, with no correlation between failure occurring at or away from the notch. For notches with (a/w) > 1/3, the samples fail at lower peak loads and this is attributed to the increased compliance as fewer unit cells span the un-notched region. Finite element simulations of the kagome tension samples show that the failure is governed by tensile loading for (a/w) < 1/3 but as (a/w) increases, bending begins to play a significant role in the failure. This work explores the flaw sensitivity of hollow alumina kagome nanolattices in tension, using experiments and simulations, and demonstrates that the discrete-continuum duality of architected structural meta-materials gives rise to their flaw insensitivity even when made entirely of intrinsically brittle materials.

I. Quantum mechanics of one-dimensional two-particle models. II. A quantum mechanical treatment of inelastic collisions

Part I

Solutions of Schrödinger’s equation for system of two particles bound in various stationary one-dimensional potential wells and repelling each other with a Coulomb force are obtained by the method of finite differences. The general properties of such systems are worked out in detail for the case of two electrons in an infinite square well. For small well widths (1-10 a.u.) the energy levels lie above those of the noninteresting particle model by as much as a factor of 4, although excitation energies are only half again as great. The analytical form of the solutions is obtained and it is shown that every eigenstate is doubly degenerate due to the “pathological” nature of the one-dimensional Coulomb potential. This degeneracy is verified numerically by the finite-difference method. The properties of the square-well system are compared with those of the free-electron and hard-sphere models; perturbation and variational treatments are also carried out using the hard-sphere Hamiltonian as a zeroth-order approximation. The lowest several finite-difference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hard-sphere eigenfunctions. The finite-difference solutions in general yield expectation values and matrix elements as accurate as those obtained using the “best” variational function.

The system of two electrons in a parabolic well is also treated by finite differences. In this system it is possible to separate the center-of-mass motion and hence to effect a considerable numerical simplification. It is shown that the pathological one-dimensional Coulomb potential gives rise to doubly degenerate eigenstates for the parabolic well in exactly the same manner as for the infinite square well.

Part II

A general method of treating inelastic collisions quantum mechanically is developed and applied to several one-dimensional models. The formalism is first developed for nonreactive “vibrational” excitations of a bound system by an incident free particle. It is then extended to treat simple exchange reactions of the form A + BC →AB + C. The method consists essentially of finding a set of linearly independent solutions of the Schrödinger equation such that each solution of the set satisfies a distinct, yet arbitrary boundary condition specified in the asymptotic region. These linearly independent solutions are then combined to form a total scattering wavefunction having the correct asymptotic form. The method of finite differences is used to determine the linearly independent functions.

The theory is applied to the impulsive collision of a free particle with a particle bound in (1) an infinite square well and (2) a parabolic well. Calculated transition probabilities agree well with previously obtained values.

Several models for the exchange reaction involving three identical particles are also treated: (1) infinite-square-well potential surface, in which all three particles interact as hard spheres and each two-particle subsystem (i.e. BC and AB) is bound by an attractive infinite-square-well potential; (2) truncated parabolic potential surface, in which the two-particle subsystems are bound by a harmonic oscillator potential which becomes infinite for interparticle separations greater than a certain value; (3) parabolic (untruncated) surface. Although there are no published values with which to compare our reaction probabilities, several independent checks on internal consistency indicate that the results are reliable.