"Macroscopic Mechanical Properties of Materials Based on a Computational Hierarchy Starting with Quantum Mechanics."

William A. Goddard, III, Tahir Cagin, Alejandro Strachan, and Richard P. Muller

Materials and Process Simulation Center,

California Institute of Technology,

Pasadena, California 91125, USA.

For first principles modeling of the mechanical properties of materials, we employ a multi-scale strategy, with an overlapping sequence of techniques starting from first principles quantum mechanics (QM), Figure 1. The foundation for predicting all materials properties is QM. This requires no input from experiment and thus is suitable for predicting new materials never yet synthesized. Unfortunately, it is not practical to use QM for systems at the spatial and time scales of interest in materials applications. Instead we find a force field (FF) that accurately reproduces the QM results, but in terms of atomic coordinates rather than electrons. The FF might include charges, force constants, polarizability, dispersion attractions, and Pauli Repulsion. This converts the equations of motion from the Schrodinger equation to Newton's equation, allowing molecular dynamics (MD) simulations to explore both static properties (as a function of temperature, pressure, and stress) along with the time dependent processes, such as the behavior of materials under strain rates, variable load conditions, even the behavior of materials far from equilibrium (hypervelocity impact, shock). With MD, it is practical to model the properties of materials with a million particles, making it possible to predict the fundamentals of crack initiation, dislocation migrations etc. However a cube with 25 nm per side is already ~ million particles, so that the description of microstructure, interaction of grains, and other critical aspects of materials controlling the mechanical properties requires a coarser description referred to as mesoscale modeling. Here the elements are dislocations, cracks, grain surfaces, rather than atoms, but the mesoscale parameters are derived from the MD. With the mesoscale, we can handle the length and time scale required to properly address macroscopic mechanical behavior.

With these techniques we have studied mechanical properties of metals and their alloys, oxides (for both catalysis and structural properties), ceramics, and polymers.
In this paper, we will illustrate the strategies and results from recent applications on describing the phase behavior, phase transformations, equation of state and mechanical properties of oxides such as SiO_{2}, Al_{2}O_{3,} and MgO over a wide range of pressures and temperatures.
In addition, we will describe a first principles based multi-scale modeling of the mechanical properties of metals over a wide range of pressures and temperatures.
This will focus on the atomistic details of core structure of dislocations, their energetics, and mobilities.
The understanding of dislocations in a metal is very important for single crystal plasticity of metals and major factor in their mechanical properties.
These studies present case studies for Ta and Ni, where the FF is derived from QM.
Based on these FF we will examine such failure mechanisms in metals and alloys as spallation and fracture.

We used first principles QM (no empirical parameters) to establish the phase diagram for the B1 (NaCl), B2 (CsCl), and liquid phases of MgO.
The QM was at the level of density functional theory (DFT) with the generalized gradient approximation (GGA) to predict the equation-of-state [volume versus pressure] at 0 K for MgO in the low-density B1 phase and the high-density B2 phase.
We find a pressure induced phase transition at P=400 GPa. These quantum results were used to derive the qMS-Q FF. The accuracy of the fit of the qMS-Q FF to the QM validates the functional form of the qMS-Q FF in which the charges are obtained from charge equilibration (QEq) and the nonelectrostatic forces are described with simple two-body Morse potentials.
This qMS-Q FF, was then used in MD simulations to investigate the phase coexistence curves of the B1-B2 and B1-liquid phases. This leads to a first-principles phase diagram for MgO for pressures up to 500GPa and temperatures up to 8000K.
A similar approach is used to investigate the phase behavior, equation of state, phase transformations and mechanical properties of SiO_{2} and Al_{2}O_{3 } [1,2].

We have performed extensive quantum mechanical calculations on structures, equation of state and the elastic moduli of Ta. Based on these calculations, we developed an embedded atom model (EAM) FF for Ta. The qEAM FF was fitted to the QM zero temperature equation of state for BCC, FCC, and A15 crystal structures over the pressure range, from -10 to 500 GPa. In addition, it was fitted to the QM vacancy formation energy. Using the qEAM FF, we calculated the melting curve for Ta, leading to Tmelt = 3150 ± 100K (zero pressure) in good agreement with experiment. The qEAM FF for Ta was then used in atomistic simulations to determine the core structure, core energy and Peierls energy barrier for ½<111> screw dislocations. Equilibrated core structures were obtained by relaxation of dislocation quadrupoles (in a large periodic cell). The equilibrium dislocation core has three-fold symmetry and spreads in three <112> directions on {110} planes. The Core energy is 1.36 eV per Burger’s vector b. Motions of dislocation were studied in a dislocation dipole cell and the Peierls energy barrier was computed: initial (static) Peierls potential was 0.08 eV per unit dislocation and dynamic Peierls potential was 0.06 eV per unit dislocation. The Peierls stress was estimated to be about 0.03 m , where m is the bulk shear modulus of perfect crystal.

Using the QM-SC many body FF for Ni, we used MD simulations to study the <110>/2 screw dislocation in fcc Ni.
This uses the MPSim/MPI code, which is efficient for systems with 10^{6} atoms per periodic cell.
Here we examine such configurations as an isolated dislocation in a cylinder with free surfaces, as well as dipole and quadrupole systems with 3-D periodic boundary conditions.
The relaxed structures show dissociation into two partials on {111} planes. The equilibrium separation distance between the two partials is 2.1 nm, in good agreement with experiment: 2.6 ±
0.8nm.
We also studied the motion and annihilation process of opposite signed dislocations with a various combinations of dissociation planes.
This allows the analysis of the influence of the presence of impurities on kink formation energy and mobility.

As a final example of first principles multiscale studies of the mechanical properties of materials, we describe the rapid expansion of Ta and Ni following the high compression (~50 GPa) induced by impact of high velocity (2 to 4 km/s) flyer plates. We find that catastrphic failure in this system coincides with a critical behavior characterized by a void distribution of the form N(v) ~ V^{-t}, with t ~ 2.2. This corresponds to a threshold in which percolation of the voids leads to tensile failure. Defining the order parameter as the ratio of the volume of the largest void to the total void volume, we find that it changes rapidly from ~0 to ~1 at the metal point, scaling as a power law with exponent b »
0.4. The similar behavior found for fcc Ni indicates that the critical behavior underlying spall failure is characteristic of failure upon rapid expansion conditions [3].

- E. Demiralp, T. Cagin, W. A. Goddard,
*Phys. Rev. Lett*. 82, 1708 (1999) - A. Strachan, T. Cagin, W. A. Goddard,
*Phys. Rev*.**B**60, 15084 (1999) - A. Strachan, T. Cagin, W. A. Goddard, "Critical behavior of spallation failure in metals
*," Phys. Rev. Lett*. submitted.

Figure 1. The Multiscale Hierarchy of Materials Simulation