39 research outputs found
Interpolants of Lattice Functions for the Analysis of Atomistic/Continuum Multiscale Methods
We introduce a general class of (quasi-)interpolants of functions defined on
a Bravais lattice, and establish several technical results for these
interpolants that are crucial ingredients in the analysis of atomistic models
and atomistic/continuum multi-scale methods
Elinvar effect in Ti simulated by on-the-fly trained moment tensor potential
A combination of quantum mechanics calculations with machine learning (ML)
techniques can lead to a paradigm shift in our ability to predict materials
properties from first principles. Here we show that on-the-fly training of an
interatomic potential described through moment tensors provides the same
accuracy as state-of-the-art {\it ab inito} molecular dynamics in predicting
high-temperature elastic properties of materials with two orders of magnitude
less computational effort. Using the technique, we investigate high-temperature
bcc phase of titanium and predict very weak, Elinvar, temperature dependence of
its elastic moduli, similar to the behavior of the so-called GUM Ti-based
alloys [T. Sato {\ it et al.}, Science {\bf 300}, 464 (2003)]. Given the fact
that GUM alloys have complex chemical compositions and operate at room
temperature, Elinvar properties of elemental bcc-Ti observed in the wide
temperature interval 1100--1700 K is unique.Comment: 15 pages, 4 figure
Numerical Methods for Multilattices
Among the efficient numerical methods based on atomistic models, the
quasicontinuum (QC) method has attracted growing interest in recent years. The
QC method was first developed for crystalline materials with Bravais lattice
and was later extended to multilattices (Tadmor et al, 1999). Another existing
numerical approach to modeling multilattices is homogenization. In the present
paper we review the existing numerical methods for multilattices and propose
another concurrent macro-to-micro method in the numerical homogenization
framework. We give a unified mathematical formulation of the new and the
existing methods and show their equivalence. We then consider extensions of the
proposed method to time-dependent problems and to random materials.Comment: 31 page
Constrained Density Functional Theory: A Potential-Based Self-Consistency Approach
Chemical reactions, charge transfer reactions, and
magnetic materials are notoriously difficult to describe within
Kohn−Sham density functional theory, which is strictly a groundstate technique. However, over the last few decades, an
approximate method known as constrained density functional
theory (cDFT) has been developed to model low-lying excitations
linked to charge transfer or spin fluctuations. Nevertheless, despite
becoming very popular due to its versatility, low computational
cost, and availability in numerous software applications, none of the
previous cDFT implementations is strictly similar to the
corresponding ground-state self-consistent density functional
theory: the target value of constraints (e.g., local magnetization)
is not treated equivalently with atomic positions or lattice
parameters. In the present work, by considering a potential-based formulation of the self-consistency problem, the cDFT is
recast in the same framework as Kohn−Sham DFT: a new functional of the potential that includes the constraints is proposed, where
the constraints, the atomic positions, or the lattice parameters are treated all alike, while all other ingredients of the usual potentialbased DFT algorithms are unchanged, thanks to the formulation of the adequate residual. Tests of this approach for the case of spin
constraints (collinear and noncollinear) and charge constraints are performed. Expressions for the derivatives with respect to
constraints (e.g., the spin torque) for the atomic forces and the stress tensor in cDFT are provided. The latter allows one to study
striction effects as a function of the angle between spins. We apply this formalism to body-centered cubic iron and first reproduce the
well-known magnetization amplitude as a function of the angle between local magnetizations. We also study stress as a function of
such an angle. Then, the local collinear magnetization and the local atomic charge are varied together. Since the atomic spin
magnetizations, local atomic charges, atomic positions, and lattice parameters are treated on an equal footing, this formalism is an
ideal starting point for the generation of model Hamiltonians and machine-learning potentials, computation of second or third
derivatives of the energy as delivered from density-functional perturbation theory, or for second-principles approaches
Continuum Surface Energy from a Lattice Model
We investigate connections between the continuum and atomistic descriptions
of deformable crystals, using certain interesting results from number theory.
The energy of a deformed crystal is calculated in the context of a lattice
model with general binary interactions in two dimensions. A new bond counting
approach is used, which reduces the problem to the lattice point problem of
number theory. The main contribution is an explicit formula for the surface
energy density as a function of the deformation gradient and boundary normal.
The result is valid for a large class of domains, including faceted (polygonal)
shapes and regions with piecewise smooth boundaries.Comment: V. 1: 10 pages, no fig's. V 2: 23 pages, no figures. Misprints
corrected. Section 3 added, (new results). Intro expanded, refs added.V 3: 26
pages. Abstract changed. Section 2 split into 2. Section (4) added material.
V 4, 28 pages, Intro rewritten. Changes in Sec.5 (presentation only). Refs
added.V 5,intro changed V.6 address reviewer's comment
Machine-learning Driven Synthesis of TiZrNbHfTaC5 High-Entropy Carbide
Synthesis of high-entropy carbides (HEC) requires high temperatures that can
be provided by electric arc plasma method. However, the formation temperature
of a single-phase sample remains unknown. Moreover, under some temperatures
multi-phase structures can emerge. In this work we developed an approach for a
controllable synthesis of HEC TiZrNbHfTaC5 based on theoretical and
experimental techniques. We used canonical Monte Carlo (CMC) simulations with
the machine learning interatomic potentials to determine the temperature
conditions for the formation of single-phase and multi-phase samples. In full
agreement with the theory, the single-phase sample, produced with electric arc
discharge, was observed at 2000 K. Below 1200 K the sample decomposed into
(Ti-Nb-Ta)C and a mixture of (Zr-Hf-Ta)C, (Zr-Nb-Hf)C, (Zr-Nb)C, and (Zr-Ta)C.
Our results demonstrate the conditions for the formation of HEC and we
anticipate that our approach can pave the way towards targeted synthesis of
multicomponent materials.Comment: 16 pages, 8 figure
Building nonparametric -body force fields using Gaussian process regression
Constructing a classical potential suited to simulate a given atomic system
is a remarkably difficult task. This chapter presents a framework under which
this problem can be tackled, based on the Bayesian construction of
nonparametric force fields of a given order using Gaussian process (GP) priors.
The formalism of GP regression is first reviewed, particularly in relation to
its application in learning local atomic energies and forces. For accurate
regression it is fundamental to incorporate prior knowledge into the GP kernel
function. To this end, this chapter details how properties of smoothness,
invariance and interaction order of a force field can be encoded into
corresponding kernel properties. A range of kernels is then proposed,
possessing all the required properties and an adjustable parameter
governing the interaction order modelled. The order best suited to describe
a given system can be found automatically within the Bayesian framework by
maximisation of the marginal likelihood. The procedure is first tested on a toy
model of known interaction and later applied to two real materials described at
the DFT level of accuracy. The models automatically selected for the two
materials were found to be in agreement with physical intuition. More in
general, it was found that lower order (simpler) models should be chosen when
the data are not sufficient to resolve more complex interactions. Low GPs
can be further sped up by orders of magnitude by constructing the corresponding
tabulated force field, here named "MFF".Comment: 31 pages, 11 figures, book chapte