Topics in Multiscale Modeling of Metals and Metallic Alloys

In a number of areas of application, the behavior of systems depends sensitively on properties that pertain to the atomistic scale, i. e. , the angstrom and femtosecond scales. However, generally the behaviors of interest are macroscopic and are characterized by slow evolution on the scale of meters and years. This broad disparity of length and time scales places extraordinary challenges in computational material science. The overarching objective of this dissertation is to address the problem of multiple space and time scales in atomistic systems undergoing slow macroscopic evolution while retaining full atomistic detail. Our approach may be summarized as follows: (1) The issue of accounting for finite temperature in coarse grained systems has not been solved entirely. For finite temperature systems at equilibrium, constructing an effective free energy in terms of a reduced set of atomic degrees of freedom is still an open area of research. In particular, the thermal vibrations of the missing degrees of freedom need to be accounted for. This is specially important if the aim of the simulation is to determine the dynamic properties of a system, or to allow the transmission of dynamic information between regions of different spatial discretization. To this end, we introduce a framework to simulate (spatially) coarse dynamic systems using the Quasicontinuum method (QC). The equations of motion are strictly derived from dissipative Lagrangian mechanics, which provides a classical Langevin implementation where the characteristic time is governed by the vibrations of the finest length scale in the computational cell. In order to assess the framework's ability to transmit information across scales, we study the phonon impoverish spectra in coarse regions and the resulting underestimation of thermal equilibrium properties. (2) Atomistic simulations have been employed for the past thirty years to determine structural and thermodynamic (equilibrium) properties of solids and their defects over a wide range of temperatures and pressures. The traditional Monte Carlo (MC) and Molecular Dynamics (MD) methods, while ideally suited to these calculations, require appreciable computational resources in order to calculate the long-time averages from which properties are obtained. In order to permit a reasonably quick, but accurate determination of the equilibrium properties of interest, we present an extension of the “maximum entropy” method to build effective alloy potentials while avoiding the treatment of all the system's atomic degrees of freedom. We assess the validity of the model by testing its ability to reproduce experimental measurements. (3) Based upon these effective potentials, we present a numerical framework capable of following the time evolution of atomistic systems over time windows currently beyond the scope of traditional atomistic methods such as Molecular Dynamics (MD) or Monte Carlo (MC). This is accomplished while retaining the underlying atomistic description of the material. We formulate a discrete variational setting in which the simulation of time-dependent phenomena is reduced to a sequence of incremental problems, each characterized by a variational principle. In this fashion we are able to study the interplay between deformation and diffusion using time steps or strain rates that are orders of magnitude larger or smaller than their MD|MC counterparts. (4) We formulate a new class of “Replica Time Integrators” (RTIs) that allows for the two-way transmission of thermal phonons across mesh interfaces. This two-way transmission is accomplished by representing the state of the coarse region by a collection of identical copies or “replicas” of itself. Each replica runs at its own slow time step and is out-of-phase with respect to the others by one fast time step. Then, each replica is capable of absorbing from the fine region the elementary signal that is in phase with the replica. Conversely, each replica is capable of supporting --and transmitting to the fine region-- an elementary signal of a certain phase. Since fine and coarse regions evolve asynchronously in time, RTIs permit both spatial and temporal coarse graining of the system of interest. Using a combination of phase-error analysis and numerical testing we find that RTIs are convergent, and allow step waves and thermal phonons to cross mesh interfaces in both directions losslessly.