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Nonequilibrium free-energy calculation of solids using LAMMPS

https://doi.org/10.1016/j.commatsci.2015.10.050Get rights and content

Highlights

  • State-of-the-art nonequilibrium free-energy calculation methods are presented.

  • Methods are implemented in LAMMPS and can be used as simple script commands.

  • Superior efficiency compared to standard equilibrium methods is demonstrated.

  • Parameter optimization for calculation of free energies of solids and T dependence.

  • Example application to polymorphic transitions for an interatomic potential for iron.

Abstract

This article describes nonequilibrium techniques for the calculation of free energies of solids using molecular dynamics (MD) simulations. These methods provide an alternative to standard equilibrium thermodynamic integration methods and often present superior efficiency. Here we describe the implementation in the LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) code of two specific nonequilibrium processes that allow the calculation of the free-energy difference between two different system Hamiltonians as well as the free-energy temperature dependence of a given Hamiltonian, respectively. The theory behind the methods is summarized, and we describe (including fragments of LAMMPS scripts) how the process parameters should be selected to obtain the best-possible efficiency in the calculations of free energies using nonequilibrium MD simulations. As an example of the application of the methods we present results related to polymorphic transitions for a classical potential model of iron.

Introduction

The calculation of free energies and derivative thermodynamic quantities for condensed-phase systems is a common and widespread application of atomistic simulation techniques. The continuous improvement of methods and the steady increase in computational power have enabled the efficient and reliable computation of free energies for complex systems of interest in materials science, including extended defects and interfaces. Free-energy calculation methods have benefited substantially with the introduction of nonequilibrium approaches such as the adiabatic switching [1] (AS) method, which allow the calculations to be performed along explicitly time-dependent processes that can lead to significant efficiency gains compared to standard equilibrium methods. Moreover, with the derivation of Jarzynski’s equality the connection between equilibrium free-energy differences and nonequilibrium processes has been placed on a firm theoretical basis [2].

In this paper we describe the implementation of state-of-the-art nonequilibrium techniques for calculating free energies of solids in the highly optimized LAMMPS [3] (Large-scale Atomic/Molecular Massively Parallel Simulator) molecular dynamics code. Specifically, we focus on two particular kinds of nonequilibrium routes which, respectively, allow the calculation of free-energy differences between systems described by different Hamiltonians as well as the temperature dependence of the free energy of a given system Hamiltonian from a single nonequilibrium simulation. We demonstrate their application to the calculation of free energies of different crystalline structures for a classical interatomic potential model of iron [4] and discuss extensions of the approaches to the calculations of interfacial free energies, which can be readily undertaken using the methods implemented in LAMMPS [5].

The paper has been organized as follows. In Section 2 we give a short and self-contained presentation of the theoretical framework underlying the nonequilibrium processes used to compute free-energy differences between equilibrium states. In Section 3 we describe the implementation of these methods into LAMMPS, and discuss how to optimize the parameters for efficiency, using bcc iron as an example system. In Section 4 we describe the application of the methods to the study of polymorphism in iron and we end with a summary and a discussion of the extension of the methodology to more complex crystalline systems as well as surfaces and interfaces in Section 5.

Section snippets

Nonequilibrium free-energy estimation

Standard equilibrium free-energy calculations are often based on thermodynamic integration (TI), [6], [7] which is a general class of methods based on the construction of a sequence of equilibrium states on a path between two thermodynamic states of interest. The free-energy difference between two equilibrium states is then determined by computing ensemble averages of the relevant thermodynamic driving force for these states by means of a set of independent equilibrium simulations, followed by

Calculation of bulk free energy

In this section we describe the implementation of the methods described above in the widely used Molecular Dynamics code LAMMPS [3]. We demonstrate the application of the methods to the calculation of the temperature-dependent free energy of a body-centered-cubic solid described by an embedded-atom-method (EAM) [13] many-body interatomic potential model of iron developed by Meyer and Entel [4]. Free energies are calculated at zero pressure for a range of temperatures between 100 and 1600K; the

Iron polymorphism

The formalism described in this article provides an efficient framework for computing the free energies of crystalline solids described by classical interatomic potential models. Such calculations are useful in many contexts in computational materials science. For example, a central issue in the modeling of crystalline solids is polymorphism, i.e., the thermodynamic stability of crystalline phases with different crystal structures as a function of temperature. Many materials used in engineering

Summary and discussion

In this paper a detailed account has been presented of the use of state-of-the-art nonequilibrium simulation methods to compute free energies of solids in LAMMPS using the Frenkel–Ladd [10] and Reversible Scaling [11] paths. The approach was demonstrated in free energy calculations for different polymorphs (fcc, bcc, and hcp) in a classical potential model of iron [4]. It was demonstrated that a precision of tenths of meV/atom or better can be achieved in nonequilibrium simulations for systems

Acknowledgments

This work was supported by the FAPESP grant 2010/13902-4. MK acknowledges support from FAPESP and FAPESP/CEPID 2013/00293-7. The research of RF and MA at UC Berkeley was supported by the US National Science Foundation (Grant No. DMR-1105409).

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