Ryan S. Elliott

Stability of thermally-induced martensitic transformations in bi-atomic crystals

Abstract Some of the most interesting, and technologically important solid–solid transformations are the first order diffusionless transformations that occur in certain equiatomic, ordered, bi-atomic crystals. These displacive transformations include thermally-induced, reversible, proper martensitic transformations as seen in shape memory alloys such as NiTi (where group–subgroup relationships exist between the symmetry groups of the crystal phases) and the reconstructive martensitic transformations seen in certain ionic compounds such as CsCl (where no group–subgroup relationship exists between the phases). In contrast to existing continuum mechanics approaches, the present work constructs a continuum energy density function W( F ,θ) (as a function of a uniform deformation gradient and temperature) of a perfect periodic bi-atomic crystal from temperature-dependent atomic pair-potentials. Equilibrium solutions and their stability are examined as a function of temperature for crystals under no external stress. The full problem is solved numerically, and an asymptotic theory is employed to guide the numerical solution near multiple bifurcation points. Using pair-potentials and enforcing constrained kinematics (uniform deformation of a cubic CsCl-type crystal), lower symmetry crystals, such as orthorhombic, monoclinic, and rhombohedral structures are predicted. The first two of these are unstable within the chosen temperature window for our particular case, while the third is stable for higher temperatures. In addition, a hysteretic transformation was discovered in which the CsCl structure is stable at high temperatures and the NaCl structure is stable at low temperatures. These two cubic phases are connected by an unstable rhombohedral path corresponding to the transformation mechanism proposed by Buerger (1951). The CsCl–NaCl transformation suggested by the numerical results is a reconstructive transformation with a group–nonsubgroup relationship between the symmetry groups of the two phases.

A spectral scheme for Kohn-Sham density functional theory of clusters

Starting from the observation that one of the most successful methods for solving the Kohn-Sham equations for periodic systems - the plane-wave method - is a spectral method based on eigenfunction expansion, we formulate a spectral method designed towards solving the Kohn-Sham equations for clusters. This allows for efficient calculation of the electronic structure of clusters (and molecules) with high accuracy and systematic convergence properties without the need for any artificial periodicity. The basis functions in this method form a complete orthonormal set and are expressible in terms of spherical harmonics and spherical Bessel functions. Computation of the occupied eigenstates of the discretized Kohn-Sham Hamiltonian is carried out using a combination of preconditioned block eigensolvers and Chebyshev polynomial filter accelerated subspace iterations. Several algorithmic and computational aspects of the method, including computation of the electrostatics terms and parallelization are discussed. We have implemented these methods and algorithms into an efficient and reliable package called ClusterES (Cluster Electronic Structure). A variety of benchmark calculations employing local and non-local pseudopotentials are carried out using our package and the results are compared to the literature. Convergence properties of the basis set are discussed through numerical examples. Computations involving large systems that contain thousands of electrons are demonstrated to highlight the efficacy of our methodology. The use of our method to study clusters with arbitrary point group symmetries is briefly discussed.

A local quasicontinuum method for 3D multilattice crystalline materials: Application to shape-memory alloys

The quasicontinuum (QC) method, in its local (continuum) limit, is applied to materials with a multilattice crystal structure. Cauchy–Born (CB) kinematics, which accounts for the shifts of the crystal motif, is used to relate atomic motions to continuum deformation gradients. To avoid failures of CB kinematics, QC is augmented with a phonon stability analysis that detects lattice period extensions and identifies the minimum required periodic cell size. This approach is referred to as Cascading Cauchy–Born kinematics (CCB). In this paper, the method is described and developed. It is then used, along with an effective interaction potential (EIP) model for shape-memory alloys, to simulate the shape-memory effect and pseudoelasticity in a finite specimen. The results of these simulations show that (i) the CCB methodology is an essential tool that is required in order for QC-type simulations to correctly capture the first-order phase transitions responsible for these material behaviors, and (ii) that the EIP model adopted in this work coupled with the QC/CCB methodology is capable of predicting the characteristic behavior found in shape-memory alloys.

Stability of pressure-dependent, thermally-induced displacive transformations in bi-atomic crystals

An important property of some metallic alloys, such as NiTi, for technological applications is their coupled thermomechanical shape memory behavior. This is due to temperature-dependent first-order displacive (martensitic) transformations in which their crystal structures transform between a higher symmetry cubic phase and lower symmetry phases (rhombohedral, tetragonal, orthorhombic, or monoclinic). In a recent paper, Elliott et al. (J. Mech. Phys. Solids, in press) proposed a nano-mechanical model based on temperature-dependent atomic potentials to explicitly construct an energy density W ðF; hÞ to find all the different equilibrium paths and their stability of a stress-free bi-atomic perfect crystal as a function of temperature. In this work we investigate the influence of hydrostatic pressure. In general, hydrostatic compression increases the critical temperatures on the principal branches. For the same absolute value, hydrostatic tension is found to have a more pronounced effect on the equilibrium paths than hydrostatic compression. 2002 Elsevier Science Ltd. All rights reserved.

Symmetry-adapted phonon analysis of nanotubes

The characteristics of phonons, i.e. linearized normal modes of vibration, provide important insights into many aspects of crystals, e.g. stability and thermodynamics. In this paper, we use the Objective Structures framework to make concrete analogies between crystalline phonons and normal modes of vibration in non-crystalline but highly symmetric nanostructures. Our strategy is to use an intermediate linear transformation from real-space to an intermediate space in which the Hessian matrix of second derivatives is block-circulant. The block-circulant nature of the Hessian enables us to then follow the procedure to obtain phonons in crystals: namely, we use the Discrete Fourier Transform from this intermediate space to obtain a block-diagonal matrix that is readily diagonalizable. We formulate this for general Objective Structures and then apply it to study carbon nanotubes of various chiralities that are subjected to axial elongation and torsional deformation. We compare the phonon spectra computed in the Objective Framework with spectra computed for armchair and zigzag nanotubes. We also demonstrate the approach by computing the Density of States. In addition to the computational efficiency afforded by Objective Structures in providing the transformations to almost-diagonalize the Hessian, the framework provides an important conceptual simplification to interpret the phonon curves. Our findings include that, first, not all non-optic long-wavelength modes are zero energy and conversely not all zero energy modes are long-wavelength; second, the phonon curves accurately predict both the onset as well as the soft modes for instabilities such as torsional buckling; and third, unlike crystals where phonon stability does not provide information on stability with respect to non-rank-one deformation modes, phonon stability in nanotubes is sufficient to guarantee stability with respect to all perturbations that do not involve structural modes. Our finding of characteristic oscillations in the phonon curves motivates a simple one-dimensional geometric nonlocal model of energy transport in generic Objective Structures. The model shows the interesting interplay between energy transport along axial and helical directions.

Stability of dispersive bi-atomic crystals

Understanding thermoelastic martensitic transformations is a fundamental component in the study of shape memory alloys. These transformations involve a hysteretic change in stability of the crystal lattice between an austenite (high symmetry) phase and a martensite (low symmetry) phase within a small temperature range. In previous work, a continuum energy density W(U;θ) (as a function of the right stretch tensor U and temperature θ) for a perfect bi-atomic crystal was derived based on temperature-dependent atomic pair-potentials. For this model, only high symmetry cubic configurations were found to be stable (local energy minimizers). The present work derives an energy density W(U,P(1),P(2),...;θ) that explicitly accounts for a set of internal atomic shifts P(i). In addition, the model permits the calculation of the crystals dispersion relations which determine the stability of the crystal with respect to bounded perturbations of all wavelengths (Bloch-waves). Using a specific model of a bi-atomic crystal with the temperature serving as the loading parameter, a stress-free bifurcation diagram is generated. Stable equilibrium branches corresponding to the B2 (cubic) and B19 (orthorhombic) crystal structures are found to exist and overlap for certain temperatures. The group-subgroup relationship between these two crystal structures is necessary for the shape memory effect. Thus, our results are consistent with the transformations that occur in shape memory alloys such as AuCd and NiTi.

A force-matching Stillinger-Weber potential for MoS2: Parameterization and Fisher information theory based sensitivity analysis

Two-dimensional molybdenum disulfide (MoS2) is a promising material for the next generation of switchable transistors and photodetectors. In order to perform large-scale molecular simulations of the mechanical and thermal behavior of MoS2-based devices, an accurate interatomic potential is required. To this end, we have developed a Stillinger-Weber potential for monolayer MoS2. The potential parameters are optimized to reproduce the geometry (bond lengths and bond angles) of MoS2 in its equilibrium state and to match as closely as possible the forces acting on the atoms along a dynamical trajectory obtained from ab initio molecular dynamics. Verification calculations indicate that the new potential accurately predicts important material properties including the strain dependence of the cohesive energy, the elastic constants, and the linear thermal expansion coefficient. The uncertainty in the potential parameters is determined using a Fisher information theory analysis. It is found that the parameters are...

Anomalous phonon behavior of carbon nanotubes: First-order influence of external load

External loads typically have a indirect influence on phonon curves, i.e., they influence the phonon curves by changing the state about which linearization is performed. In this paper, we show that in nanotubes, the axial load has a direct first-order influence on the long-wavelength behavior of the transverse acoustic (TA) mode. In particular, when the tube is force-free the TA mode frequencies vary quadratically with wave number and have curvature (second derivative) proportional to the square-root of the nanotubes bending stiffness. When the tube has non-zero external force, the TA mode frequencies vary linearly with wave number and have slope proportional to the square-root of the axial force. Therefore, the TA phonon curves -- and associated transport properties -- are not material properties but rather can be directly tuned by external loads. In addition, we show that the out-of-plane shear deformation does {\em not} contribute to this mode and the unusual properties of the TA mode are exclusively due to bending. Our calculations consist of 3 parts: first, we use a linear chain of atoms as an illustrative example that can be solved in close-form; second, we use our recently-developed symmetry-adapted phonon analysis method to present direct numerical evidence; and finally, we present a simple mechanical model that captures the essential physics of the geometric nonlinearity in slender nanotubes that couples the axial load directly to the phonon curves. We also compute the Density of States and show the significant effect of the external load.

Universal equilibrium solutions

In this chapter we study solutions to the equations of continuum mechanics instead of the equations themselves. In particular, our aim will be to obtain general equilibrium solutions to the field equations of continuum mechanics that are independent, in a specific sense, of the material from which a body is composed. Such solutions are of fundamental importance to the practical application of the theory of continuum mechanics. This is because they provide valuable guidance to the experimentalist who would like to design experiments for the determination of a particular materials constitutive relations. Generally, in an experiment it is only possible to control and measure (to a greater or lesser extent) the tractions and displacements associated with the boundary of the body being studied. From this information one would like to infer the stress and deformation fields within the body and ultimately extract the functional form of the materials constitutive relations and the values of any coefficients belonging to this functional form. However, if the interior stress and deformation fields explicitly depend on the functional form of the constitutive relations, then it is essentially impossible to infer this information from a practical experiment. According to Saccomandi [Sac01], a deformation which satisfies the equilibrium equations with zero body forces and is supported by suitable surface tractions alone is called a controllable solution . A controllable solution that is the same for all materials in a given class is a universal solution .

Continuum Mechanics and Thermodynamics: Scalars, vectors and tensors

Continuum mechanics seeks to provide a fundamental model for material response. It is sensible to require that the predictions of such a theory should not depend on the irrelevant details of a particular coordinate system. The key is to write the theory in terms of variables that are unaffected by such changes; tensors (or tensor fields ) are the measures that have this property. Tensors come in different flavors depending on the number of spatial directions that they couple. The simplest tensor has no directional dependence and is called a scalar invariant to distinguish it from a simple scalar. A vector has one direction. For two directions and higher the general term tensor is used. Tensors are tricky things to define. Many books define tensors in a technical manner in terms of the rules that tensor components must satisfy under coordinate system transformations. While certainly correct, we find such definitions unilluminating when trying to answer the basic question of “what is a tensor?”. In this chapter, we provide an introduction to tensors from the perspective of linear algebra. This approach may appear rather mathematical at first, but in the end it provides a far deeper insight into the nature of tensors. Before we can begin the discussion of the definition of tensors, we must start by defining “space” and “time” and the related concept of a “frame of reference,” which underlie the description of all physical objects.